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+{"text":"\\section{Introduction}\n\nThe stellarator concept \\citep{spitzerStellaratorConcept1958} offers a path to a steady-state and disruption-free fusion reactor with low recirculating power, but its complex three-dimensional geometry must be carefully designed to guarantee good plasma properties. In particular, the stellarator does not generally guarantee confinement of particles on collisionless trajectories due to its lack of continuous symmetry, leading to large neoclassical transport \\citep{helanderTheoryPlasmaConfinement2014}. However, the use of numerical optimisation techniques has led to advanced stellarator designs with good confinement properties, culminating in the design and construction of the HSX \\citep{andersonHelicallySymmetricExperiment1995} and W7-X stellarators \\citep{beidlerPhysicsEngineeringDesign1990}.\n\nAlthough gradient-based optimisation algorithms are generally more efficient than gradient-free algorithms, because of the large number of parameters (e.g. to represent the plasma boundary) they can be prohibitively expensive computationally if the gradients are evaluated via finite-differences. A more efficient way of obtaining gradient information is provided by adjoint methods, which were recently introduced in the stellarator optimisation field and have already found widespread application \\citep{landremanComputingLocalSensitivity2018, paulAdjointMethodGradientbased2018, antonsenAdjointApproachCalculating2019, paulAdjointMethodNeoclassical2019, paulAdjointApproachCalculating2020, paulAdjointMethodsStellarator2020, paulGradientbasedOptimization3D2021, geraldiniAdjointMethodDetermining2021, giulianiSinglestageGradientbasedStellarator2020}.\n\nPrevious work \\citep{antonsenAdjointApproachCalculating2019, paulAdjointApproachCalculating2020, paulGradientbasedOptimization3D2021} applied adjoint methods to ideal magnetohydrostatic (MHS) equilibria, building in the assumption of integrability, i.e. the existence of a set of nested flux surfaces. However, three-dimensional magnetic fields are generally not integrable due to the lack of continuous symmetry. Moreover, singularities arise at rational surfaces for linearised ideal MHS equilibria, making the computation of derivatives challenging \\citep{paulGradientbasedOptimization3D2021}. To overcome these challenges, different equilibrium models can be considered, such as vacuum or force-free fields. We herein apply adjoint methods to vacuum magnetic fields, relinquishing the assumption of global integrability, and avoiding the singular behaviour of MHS equilibria. Modeling the plasma magnetic field as a vacuum field is justified in the limit of vanishing plasma current and $\\beta$, the ratio of thermal pressure to magnetic pressure. Vacuum fields are thus broadly relevant for stellarators configurations, which tend to operate at low $\\beta$ and low plasma current, as non-axisymmetric shaping of the coils is used to generate rotational transform. Moreover, optimised vacuum solutions can serve as useful starting points for the optimisation of finite-pressure equilibria \\citep{boozerCurlfreeMagneticFields2019}.\n\nWe shall consider two objective functions, one targeting a rotational transform value on the boundary, and another targeting quasisymmetry on the boundary. As a subset of the larger class of omnigenous fields \\citep{hallThreedimensionalEquilibriumAnisotropic1975}, for which particles are confined on collisionless trajectories, quasisymmetric fields \\citep{nuhrenbergQuasihelicallySymmetricToroidal1988} have attracted strong interest, notably leading to the designs of the HSX \\citep{andersonHelicallySymmetricExperiment1995} and NCSX \\citep{zarnstorffPhysicsCompactAdvanced2001} stellarators. Multiple formulations of quasisymmetry exist \\citep{helanderTheoryPlasmaConfinement2014, rodriguezNecessarySufficientConditions2020, burbyMathematicsQuasisymmetry2020}, all of which employ flux coordinates and therefore require the existence of nested flux surfaces. We propose a method of constructing approximate flux coordinates on an isolated flux surface, on which quasisymmetry can then be defined and optimised for. The existence of at least one isolated flux surface will be guaranteed, by imposing the boundary condition that the magnetic field be tangential on a prescribed boundary. Note that we will not consider whether this boundary condition can actually be realised with a set of coils, a task pursued by codes like FOCUS \\citep{zhuNewMethodDesign2018}.\n\nWith the exception of \\cite{landremanMagneticFieldsPrecise2021}, previous optimisation studies \\citep{nuhrenbergQuasihelicallySymmetricToroidal1988, kuNewClassesQuasihelically2011, drevlakESTELLQuasiToroidallySymmetric2013, baderStellaratorEquilibriaReactor2019, hennebergPropertiesNewQuasiaxisymmetric2019, hennebergImprovingFastparticleConfinement2020, landremanStellaratorOptimizationGood2021} targeted quasisymmetry by minimising the symmetry-breaking components of the magnetic field strength in Boozer coordinates, often for vacuum magnetic fields. The most widely-used solver, whether for vacuum fields or plasmas with finite pressure, is the VMEC code \\citep{hirshmanThreedimensionalFreeBoundary1986}, which notably assumes the existence of nested flux surfaces. \nWe will employ the SPEC code \\citep{hudsonComputationMultiregionRelaxed2012}, which does not build in this assumption. Furthermore, in contrast to most previous studies, we use a formulation of quasisymmetry that does not rely on a Boozer coordinate transformation, although it still enables the specification of a desired helicity of the magnetic field strength. \n\nPrevious studies have sought to optimise for quasisymmetry either on a single flux surface \\citep[e.g.][]{hennebergImprovingFastparticleConfinement2020}, or on multiple flux surfaces \\citep[e.g.][]{landremanMagneticFieldsPrecise2021} with the aim of approximating quasisymmetry in a finite volume. We will herein consider quasisymmetry on a single flux surface only. Away from a surface with exact quasisymmetry, the deviation from quasisymmetry will generally increase linearly in the flux difference \\citep{senguptaVacuumMagneticFields2021}. It will thus be of interest to extend the present work on vacuum fields to multi-region relaxed magnetohydrodynamic (MRxMHD) equilibria. In this model, the interfaces between force-free regions are flux surfaces, such that quasisymmetry can be optimised for on multiple flux surfaces.\n\nThis paper is structured as follows. We begin with a brief introduction to adjoint methods in \\S\\ref{sec:basics_adjoint_methods}. A method of constructing approximate flux coordinates on a single flux surface is introduced in \\S\\ref{sec:approximate_flux_coordinates}. The derived adjoint equations for vacuum fields are presented in \\S\\ref{sec:adjoint_formalism_vacuum_fields}, first for a simpler objective function targeting a given rotational transform value on the boundary in \\S\\ref{sec:iota_fom_shape_derivative}, then for one targeting quasisymmetry on the boundary with a given helicity in \\S\\ref{sec:QS_fom_shape_derivative}. The resulting shape gradients are evaluated numerically and benchmarked against finite-difference calculations.\n\n\\section{Basics of adjoint methods}\n\\label{sec:basics_adjoint_methods}\n\nWe are interested in obtaining derivative information for a functional $f(\\mathcal{S}, u(\\mathcal{S}))$, called hereafter the objective function. This functional depends on the surface $\\mathcal{S}$ explicitly and also implicitly through the solution $u(\\mathcal{S})$ to a partial differential equation (PDE) $\\mathcal{P}(\\mathcal{S},u) = 0$. Here, $\\mathcal{P}$ is a general operator and $u$ is member of a Hilbert space with associated inner product $\\langle \\cdot , \\cdot \\rangle$, taken in our case to be the surface integral $\\int_\\mathcal{S} \\diff S \\;(\\cdot)(\\cdot)$. \n\nConsider a displacement of the surface $\\mathcal{S}$ in the direction $\\delta\\mathbf{x}$ with magnitude $\\epsilon$, resulting in a perturbed surface $\\mathcal{S}_\\epsilon = \\{ \\mathbf{x}_0 + \\epsilon \\delta\\mathbf{x}(\\mathbf{x}_0) : \\mathbf{x}_0 \\in \\mathcal{S}\\}$. The shape derivative of an arbitrary function $g(\\mathcal{S})$ in the direction $\\delta\\mathbf{x}$ is now defined as\n\\begin{equation}\n    \\delta g(\\mathcal{S})[\\delta\\mathbf{x}] = \\lim_{\\epsilon\\rightarrow 0} \\frac{g(\\mathcal{S_\\epsilon}) - g(\\mathcal{S})}{\\epsilon}.\n\\end{equation}\nIf $g$ depends only on the geometrical shape of the surface, the shape derivative $\\delta g[\\delta\\mathbf{x}]$ will be a function of only the normal component $\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec}$ of the displacement, as any tangential component of $\\delta\\mathbf{x}$ leaves the shape of $\\mathcal{S}$ unchanged to first order. Here, $\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$ is a normal vector on $\\mathcal{S}$.\n\nTo compute derivatives of the objective function while enforcing the PDE constraint $\\mathcal{P}(\\mathcal{S},u) = 0$, the method of Lagrange multipliers is used. Consider the Lagrangian\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S},u,q) = f(\\mathcal{S},u) + \\int_\\mathcal{S} \\diff S \\; q \\; \\mathcal{P}(\\mathcal{S},u),\n\\end{equation}\nwith the Lagrange multiplier $q$. Its shape derivative $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ contains explicit contributions in the perturbation $\\delta\\mathbf{x}$, as well as implicit contributions through $\\delta q[\\delta\\mathbf{x}]$ and $\\delta u[\\delta\\mathbf{x}]$. The implicit dependencies are removed by making $\\mathcal{L}$ stationary with respect to $\\delta q[\\delta\\mathbf{x}]$, which is equivalent to enforcing the original PDE $\\mathcal{P}(\\mathcal{S},u) = 0$, and $\\delta u[\\delta\\mathbf{x}]$, which leads to an adjoint PDE for $q$. \n\nIf $\\mathcal{L}$ is stationary with respect to both $\\delta u[\\delta\\mathbf{x}]$ and $\\delta q[\\delta\\mathbf{x}]$, the remaining explicit dependence of its shape derivative $\\delta\\mathcal{L}(\\mathcal{S},u,q)[\\delta\\mathbf{x}]$ is equal to the shape derivative of the figure of merit $\\delta f(\\mathcal{S},u(\\mathcal{S}))[\\delta\\mathbf{x}]$ with $u=u(\\mathcal{S})$ satisfying the PDE constraint, as shown in e.g. \\citet{paulAdjointMethodsStellarator2020}. The Hadamard-Zol\\'esio structure theorem \\citep{delfourShapesGeometries2011} further states that the remaining contribution to the Lagrangian's shape derivative, provided $\\mathcal{L}$ is sufficiently smooth, can be expressed as\n\\begin{equation}\n    \\delta\\mathcal{L}(\\mathcal{S},u,q)[\\delta\\mathbf{x}] = \\int_\\mathcal{S} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\; \\mathcal{G},\n\\end{equation}\nwhere $\\mathcal{G}$ is called the shape gradient, and can be interpreted as the local sensitivity of the objective function to perturbations of $\\mathcal{S}$.\n\nIn practice, the surface $\\mathcal{S}$ is typically represented by a finite set of parameters $\\Omega = \\{\\Omega_i,\\; i = 1, 2, \\dots N\\}$, e.g. Fourier coefficients $\\{R_{m,n}, Z_{m,n}\\}$, and the functional $f(\\mathcal{S},u(\\mathcal{S}))$ is approximated by a function $f(\\Omega,u(\\Omega))$. The derivative of $f(\\Omega,u(\\Omega))$ with respect to a parameter $\\Omega_i$ can be approximated as\n\\begin{equation}\n    \\frac{\\partial f(\\Omega, u(\\Omega))}{\\partial \\Omega_i} = \\int_\\mathcal{S} \\diff S\\; \\frac{\\partial \\mathbf{x}}{\\partial \\Omega_i}\\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\; \\mathcal{G}. \\label{eq:parameter_derivatives_shape_grad}\n\\end{equation}\nThe adjoint method of evaluating the parameter derivatives required for optimisation or sensitivity analysis \\citep{paulAdjointMethodsStellarator2020} thus consists in computing $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ to obtain the adjoint PDE for $q$ and the shape gradient $\\mathcal{G}$, which is then used to evaluate the right-hand-side of \\eqref{eq:parameter_derivatives_shape_grad}.\n\nWhen evaluating the parameter derivatives numerically through \\eqref{eq:parameter_derivatives_shape_grad}, errors are introduced from the inexact solutions to the original and adjoint PDEs. Indeed, these PDEs are assumed to be exactly satisfied in the preceding derivation to remove the implicit dependencies of $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ on $\\delta u[\\delta\\mathbf{x}]$ and $\\delta q[\\delta\\mathbf{x}]$, and also typically when deriving an expression for the shape gradient $\\mathcal{G}$.\n\nThe formalism presented above can easily be generalised to multiple PDE constraints, and, for a closed $\\mathcal{S}$, to PDEs satisfied not on $\\mathcal{S}$ but in the volume enclosed by it. This will be done in \\S\\ref{sec:adjoint_formalism_vacuum_fields}, where both the Laplace equation for the vacuum field and the straight field line equation, respectively valid in the volume and on the boundary, will be enforced as constraints, with two corresponding adjoint variables.\n\n\\section{Evaluating approximate flux coordinates on an isolated flux surface}\n\\label{sec:approximate_flux_coordinates}\n\nThe existence of nested flux surfaces is commonly assumed in theoretical studies of magnetically confined plasmas, e.g. to formulate quasisymmetry. In particular, many formulas involve $\\nabla\\psi$, where the toroidal flux $\\psi$ is a global flux surface label. However, three-dimensional magnetic fields lacking a continuous symmetry are not generally integrable. It is desirable to generalise $\\nabla\\psi$ to the case of an isolated flux surface, i.e. a flux surface in whose neighbourhood the field is generally non-integrable.\n\nOn a flux surface $\\mathcal{S}$, the magnetic field's normal component vanishes by definition, i.e.  $\\mathbf{B}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0$ with $\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$ the unit normal vector on $\\mathcal{S}$. The field line label $\\alpha$ on $\\mathcal{S}$ is defined through the straight field line equation $\\mathbf{B}\\cdot\\nabla_\\Upgamma\\alpha = 0$. Here, the tangential gradient $\\nabla_\\Upgamma$, defined in App.~\\ref{app:diff_operators_surfaces}, is the component of the 3D gradient tangential to the surface \\eqref{eq:def_tangential_gradient}. Note that in the integrable case, $\\nabla\\psi$ is normal to the flux surfaces, and the magnetic field satisfies $\\mathbf{B} = \\nabla\\psi\\times\\nabla\\alpha$.\n\nWe now define the generalisation $\\overline{\\nabla\\psi}$ on $\\mathcal{S}$ of the toroidal flux gradient $\\nabla\\psi$, by setting $\\overline{\\nabla\\psi}$ normal to $\\mathcal{S}$, and by requiring $\\mathbf{B} = \\overline{\\nabla\\psi}\\times\\nabla\\alpha$ to be satisfied on $\\mathcal{S}$. Squaring the latter equality and using $\\overline{\\nabla\\psi} = \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} |\\overline{\\nabla\\psi}|$ yields\n\\begin{equation}\n    \\overline{\\nabla\\psi} = \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\; \\frac{B}{\\abs{\\nabla_\\Upgamma \\alpha}}, \\label{eq:def_nabla_tilde_psi}\n\\end{equation}\nwhere $B = \\abs{\\mathbf{B}}$ is the magnetic field strength. Note that $\\overline{\\nabla\\psi}$ is defined through \\eqref{eq:def_nabla_tilde_psi}, and should not be misinterpreted as the gradient of a scalar function.\n\nThe defining expression for $\\overline{\\nabla\\psi}$ \\eqref{eq:def_nabla_tilde_psi} can be evaluated on any flux surface without requiring nested flux surfaces in its neighbourhood, and will revert to $\\overline{\\nabla\\psi} = \\nabla\\psi$ when the field is integrable in the neighbourhood of that flux surface. In practice, one might couple objective functions relying on \\eqref{eq:def_nabla_tilde_psi} with a figure of merit targeting integrability, aiming for a final plasma shape for which the field is integrable, such that $\\overline{\\nabla\\psi} = \\nabla\\psi$ and the minimised objective function represents the physical quantity of interest.\n\n\\begin{figure}\n\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/abs_nabla_psi_VMEC.pdf}\n  \\caption{}\n  \\label{fig:abs_nabla_psi_VMEC}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/abs_nabla_psi_SPEC.pdf}\n  \\caption{}\n  \\label{fig:abs_nabla_psi_SPEC}\n\\end{subfigure}\\\\\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/relative_diff_abs_nabla_psi.pdf}\n  \\caption{}\n  \\label{fig:relative_diff_abs_nabla_psi}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/poincare_surfaces_VMEC_SPEC.pdf}\n  \\caption{}\n  \\label{fig:poincare_surfaces_VMEC_SPEC}\n\\end{subfigure}%\n\\caption{Comparison of (a) the toroidal flux gradient $\\abs{\\nabla\\psi}$ evaluated with VMEC with (b) the generalised toroidal flux gradient $|\\overline{\\nabla\\psi}|$ \\eqref{eq:def_nabla_tilde_psi} obtained in SPEC, for a $5$-period rotating ellipse case with half a rotation per field period, major radius at the ellipse centre $R_0 = 5$\\;m, and ellipse major and minor axes values of $2$\\;m and $1$\\;m, respectively. The relative difference between the two quantities is below $1\\%$, as shown in (c). A small difference is to be expected in this case, where integrability is well satisfied, as attested in (d) by the Poincar\\'e plot at toroidal angle $\\phi=0$ from the SPEC calculation, which agrees well with the flux surfaces computed by VMEC. All data generated in this paper can be obtained from \\cite{niesDataCodesPaper2021}.}\n\\label{fig:abs_nabla_psi}\n\\end{figure}\n\nThe generalised toroidal flux gradient \\eqref{eq:def_nabla_tilde_psi} is evaluated in Fig.~\\ref{fig:abs_nabla_psi_SPEC} for a rotating ellipse configuration computed with the SPEC code. It agrees excellently with the toroidal flux gradient evaluated by VMEC, which can be calculated directly due to the imposed nestedness of flux surfaces, shown in Fig.~\\ref{fig:abs_nabla_psi_VMEC}. The relative difference between the two is below a percentage point in this case, as shown in Fig.~\\ref{fig:relative_diff_abs_nabla_psi}. The difference is expected to be small when integrability is satisfied, which indeed seems to hold here, as attested by the absence of islands and chaotic regions in the SPEC Poincar\\'e plot shown in Fig.~\\ref{fig:poincare_surfaces_VMEC_SPEC}. Note that SPEC solves for a vacuum magnetic field, while VMEC computes an ideal MHS equilibrium with vanishing thermal pressure, and with a plasma current that is small but finite due to the constraint of integrability.\n\nThe generalised toroidal flux gradient $\\overline{\\nabla\\psi}$ can be applied generally in any situation where local flux coordinates need to be evaluated, e.g. in calculations of perpendicular transport or magnetohydrodynamic stability. Isolated flux surfaces occur e.g. in fixed-boundary equilibrium calculations, where the plasma outer boundary is constrained to be a flux surface as a boundary condition on the magnetic field, or at the interfaces of MRxMHD equilibria computed by e.g. SPEC \\citep{hudsonComputationMultiregionRelaxed2012} or BIEST \\citep{malhotraTaylorStatesStellarators2019}. In the following, we will employ \\eqref{eq:def_nabla_tilde_psi} specifically for a fixed-boundary vacuum field to formulate quasisymmetry on the boundary.\n\n\\section{Application of adjoint formalism to vacuum fields}\n\\label{sec:adjoint_formalism_vacuum_fields}\n\nConsider a vacuum magnetic field $\\mathbf{B}$ in a toroidal domain $\\mathcal{V}$ bounded by the surface $\\mathcal{S} = \\partial \\mathcal{V}$. As the vacuum magnetic field is curl-free, it can be expressed as $\\mathbf{B}=\\nabla\\Phi$, with the scalar potential $\\Phi$. Because we consider a simple torus $\\mathcal{V}$, the most general form for the scalar potential is $\\Phi=G (\\omega+\\phi)$, where $G$ is a constant, $\\omega$ is a single-valued function on $\\mathcal{S}$, and $\\phi$ is an arbitrary toroidal angle. By integrating the magnetic field along a toroidal loop around the torus, the constant $G$ is found to be proportional to the net external current through the `hole' of the torus.\n\nAs the magnetic field is divergence-less, the magnetic scalar potential satisfies the Laplace equation. The field's normal component is constrained to vanish on $\\mathcal{S}$ by imposing a Neumann boundary condition on the magnetic scalar potential. Further prescribing e.g. $G$, or the toroidal flux, guarantees a unique solution to Laplace's equation. We herein opt to hold the toroidal flux fixed, although the shape derivative $\\delta G[\\delta\\mathbf{x}]$ will not appear in this study due to our normalisation of the figure of merit for quasisymmetry \\eqref{eq:definition_fQS}. A different choice of normalisation would lead to an additional contribution proportional to $\\delta G[\\delta\\mathbf{x}]$ in the shape derivative of the Lagrangian.\n\nFor convenience, we define the normalised magnetic field $\\mathbf{\\breve{B}}$ as\n\\begin{equation}\n    \\mathbf{\\breve{B}} \\equiv \\mathbf{B}\/G = \\nabla\\big(\\omega + \\phi\\big). \\label{eq:magnetic_field_scalar_pot}\n\\end{equation}\nLet us further assume the toroidal angle $\\phi$ to be the azimuthal angle in cylindrical coordinates, satisfying $\\Updelta\\phi=0$ in the domain of interest. We can thus write\n\\begin{subequations}\n\\begin{align}\n    \\nabla\\cdot\\mathbf{\\breve{B}} = \\Updelta\\omega = 0   \\qquad\\qquad\\qquad\\mathrm{in}\\;&\\mathcal{V}, \\label{eq:vacuum_field_Laplace}\\\\\n    \\mathbf{\\breve{B}}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = \\nabla(\\omega+\\phi)\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0 \\qquad\\mathrm{on}\\; & \\mathcal{S} \\label{eq:vacuum_field_normal_BC},\n\\end{align}\n\\end{subequations}\nwith $\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$ the normal unit vector on $\\mathcal{S}$. Furthermore, the shape derivative $\\delta\\omega[\\delta\\mathbf{x}]$ satisfies\n\\begin{subequations}\n\\begin{align}\n    \\Updelta(\\delta\\omega[\\delta\\mathbf{x}]) = 0   \\qquad\\qquad\\mathrm{in}\\;&\\mathcal{V}, \\label{eq:vacuum_field_pert_Laplace}\\\\\n    \\mathbf{\\breve{B}}\\cdot\\delta\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}[\\delta\\mathbf{x}] + \\nabla(\\delta\\omega[\\delta\\mathbf{x}])\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} + (\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla \\mathbf{\\breve{B}}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0 \\quad\\mathrm{on}\\; & \\mathcal{S} \\label{eq:vacuum_field_pert_normal_BC},\n\\end{align}\n\\end{subequations}\nwhere the Laplace equation is obtained from noting the commutative property of shape and spatial derivatives, and the normal boundary condition on $\\delta\\omega$ was derived in e.g. \\citet[\\S 3.2]{sokolowskiIntroductionShapeOptimization1992}. The shape derivative of the normal vector $\\delta\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}[\\delta\\mathbf{x}]=-\\nabla_\\Upgamma(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})$ is derived in App.~\\ref{app:normal_extension_and_normal_vector_shape_derivative}.\n\nEvaluating the rotational transform and quasisymmetry figures of merit further requires the solution to the straight field line equation\n\\begin{equation}\n    0 = \\mathbf{B}\\cdot\\nabla_\\Upgamma\\alpha \\qquad\\mathrm{on}\\; \\mathcal{S}, \\label{eq:straight_field_line_eq}\n\\end{equation}\nwith the field line label\n\\begin{equation}\n   \\alpha \\equiv \\theta-\\iota\\phi+\\lambda(\\theta,\\phi), \\label{eq:def_field_line_label_alpha}\n\\end{equation}\nwhere $\\theta$ is a general poloidal angle, $\\lambda$ is a single-valued function of $\\theta$ and $\\phi$, and $\\iota$ is a scalar. Note that both $\\lambda$ and $\\iota$ are defined on the boundary $\\mathcal{S}$ only, through \\eqref{eq:def_field_line_label_alpha}.\n\nLet us define the Lagrangian corresponding to an arbitrary objective function $f(\\mathcal{S},\\omega,\\iota,\\lambda)$,\n\\begin{equation}\n    \\mathcal{L}(\\mathcal{S}, \\omega, q_\\omega, \\iota, \\lambda, q_\\alpha ) = f(\\mathcal{S},\\omega,\\iota,\\lambda) + \\mathcal{M}(\\mathcal{S}, \\omega, q_\\omega) + \\mathcal{N}(\\mathcal{S}, \\omega, \\iota, \\lambda, q_\\alpha), \\label{eq:lagrangian_general}\n\\end{equation}\nwith the weak form of the Laplace equation \\eqref{eq:vacuum_field_Laplace}\n\\begin{equation}\n    \\mathcal{M}(\\mathcal{S}, \\omega, q_\\omega) = \\int_{\\mathcal{V}} \\diff V \\; q_\\omega \\Updelta \\omega, \\label{eq:weak_form_Laplace_vac}\n\\end{equation}\nand the weak form of the straight field line equation \\eqref{eq:straight_field_line_eq} normalised by $G$ \n\\begin{equation}\n    \\mathcal{N}(\\mathcal{S}, \\omega, \\iota, \\lambda, q_\\alpha) = \\int_{\\mathcal{S}} \\diff S \\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla_\\Upgamma \\alpha. \\label{eq:weak_form_straight_field_line_eq}\n\\end{equation}\nAs explained in \\S\\ref{sec:basics_adjoint_methods}, $q_\\omega$ and $q_\\alpha$ act as Lagrange multipliers: making the Lagrangian \\eqref{eq:lagrangian_general} stationary with respect to $\\delta q_\\omega[\\delta\\mathbf{x}]$ and $\\delta q_\\alpha[\\delta\\mathbf{x}]$ ensures that the Laplace \\eqref{eq:vacuum_field_Laplace} and straight field line \\eqref{eq:straight_field_line_eq} equations are satisfied, respectively. These trivial variations are omitted in the following under the assumption that \\eqref{eq:vacuum_field_Laplace} and \\eqref{eq:straight_field_line_eq} are satisfied, thus considering only the implicit dependencies of $\\delta\\mathcal{L}[\\delta\\mathbf{x}]$ on $\\delta\\omega[\\delta\\mathbf{x}]$, $\\delta\\iota[\\delta\\mathbf{x}]$, and $\\delta\\lambda[\\delta\\mathbf{x}]$ to obtain the adjoint equations for $q_\\omega$ and $q_\\alpha$.\n\nFirst, the shape derivative of $\\mathcal{M}$ \\eqref{eq:weak_form_Laplace_vac}, derived in App.~\\ref{app:derivation_laplace_eq_shape_derivative}, is\n\\begin{equation}\n    \\delta\\mathcal{M}[\\delta\\mathbf{x}] = \\int_{\\mathcal{V}} \\diff V \\; \\delta\\omega[\\delta\\mathbf{x}] \\Updelta q_\\omega - \\int_{\\mathcal{S}} \\diff S \\; \\bigg[ \\delta \\omega[\\delta\\mathbf{x}]  \\nabla q_\\omega \\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} - (\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;  \\mathbf{\\breve{B}}\\cdot\\nabla q_\\omega  \\bigg]. \\label{eq:variation_M_tot}\n\\end{equation}\n\nSecond, the shape derivative of $\\mathcal{N}$ \\eqref{eq:weak_form_straight_field_line_eq}, derived in App.~\\ref{app:derivation_straight_field_line_eq_shape_derivative}, is\n\\begin{align}\n    \\delta\\mathcal{N}[\\delta\\mathbf{x}]  = \\int_{\\mathcal{S}} & \\diff S\\;  \\bigg[ - \\delta\\omega[\\delta\\mathbf{x}] \\;\\nabla_\\Upgamma \\cdot \\left(q_\\alpha \\nabla_\\Upgamma \\alpha\\right) - \\delta\\iota[\\delta\\mathbf{x}]\\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi   \\label{eq:variation_N_tot}\\\\\n    \\nonumber & - \\delta\\lambda[\\delta\\mathbf{x}]\\; \\nabla_\\Upgamma \\cdot \\left( q_\\alpha \\mathbf{\\breve{B}} \\right) + (\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec}) q_\\alpha \\left( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}}\\cdot\\nabla_\\Upgamma\\alpha - \\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla_\\Upgamma\\alpha \\right) \\bigg].\n\\end{align}\nThe tangential gradient $\\nabla_\\Upgamma(\\cdot)$ and tangential divergence $\\nabla_\\Upgamma\\cdot(\\cdot)$ operators are defined in App.~\\ref{app:diff_operators_surfaces}.\n\nWe now proceed by computing the shape derivatives of two objective functions, first targeting a given rotational transform value on $\\mathcal{S}$ (\\S\\ref{sec:iota_fom_shape_derivative}), and second targeting quasi-symmetry on $\\mathcal{S}$ with a given helicity value (\\S\\ref{sec:QS_fom_shape_derivative}). We will then be able to evaluate the shape derivative of the Lagrangian \\eqref{eq:lagrangian_general}, yielding the adjoint equations and shape gradient formulas. Numerical verification and example shape gradients are shown for each figure of merit.\n\n\\subsection{Rotational transform objective function}\n\\label{sec:iota_fom_shape_derivative}\n\nBefore evaluating the more complicated shape gradient for the quasisymmetry figure of merit in \\S\\ref{sec:QS_fom_shape_derivative}, we consider a simple figure of merit targeting a given target rotational transform $\\iota_T$ on the surface $\\mathcal{S}$. We thus define\n\\begin{equation}\n    f_\\iota(\\iota) = \\frac{1}{2} ( \\iota - \\iota_T )^2  \\label{eq:f_iota},\n\\end{equation}\nwhere $\\iota$ is the rotational transform on $\\mathcal{S}$, obtained by solving the straight field line equation \\eqref{eq:straight_field_line_eq}. The shape derivative of $f_\\iota$ is simply\n\\begin{equation}\n    \\delta f_\\iota [\\delta\\mathbf{x}] = \\delta \\iota[\\delta\\mathbf{x}] \\; ( \\iota - \\iota_T ). \\label{eq:variation_f_iota_tot}\n\\end{equation}\nBy combining \\eqref{eq:variation_M_tot}, \\eqref{eq:variation_N_tot}, and \\eqref{eq:variation_f_iota_tot}, we obtain the shape derivative of the Lagrangian $\\mathcal{L}_\\iota$ [\\eqref{eq:lagrangian_general} with $f=f_\\iota$],\n\\begin{align}\n    \\delta\\mathcal{L}&_\\iota[\\delta\\mathbf{x}]  = \\int_{\\mathcal{V}} \\diff V \\; \\delta\\omega[\\delta\\mathbf{x}] \\; \\Updelta q_\\omega - \\int_{\\mathcal{S}} \\diff S \\;  \\delta \\omega [\\delta\\mathbf{x}] \\bigg[ \\nabla q_\\omega \\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} + \\nabla_\\Upgamma \\cdot \\Big(q_\\alpha \\nabla_\\Upgamma \\alpha \\Big) \\bigg] \\label{eq:variation_L_iota_tot}\\\\\n    \\nonumber & - \\delta\\iota[\\delta\\mathbf{x}]\\; \\bigg[ \\int_{\\mathcal{S}} \\diff S \\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi + ( \\iota - \\iota_T ) \\bigg] - \\int_{\\mathcal{S}} \\diff S \\;\\delta\\lambda[\\delta\\mathbf{x}]\\; \\nabla_\\Upgamma \\cdot \\Big( q_\\alpha \\mathbf{\\breve{B}} \\Big)  \\\\\n    \\nonumber & + \\int_{\\mathcal{S}} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec}) \\bigg[\\nabla q_\\omega \\cdot \\mathbf{\\breve{B}} + q_\\alpha \\left( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}}\\cdot\\nabla_\\Upgamma\\alpha - \\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla_\\Upgamma\\alpha \\right) \\bigg] \\Bigg].\n\\end{align}\n\nFirst, we obtain the adjoint equation for $q_\\alpha$ by requiring the second line of \\eqref{eq:variation_L_iota_tot} to vanish,\n\\begin{subequations}\n\\begin{align}\n   \\nabla_\\Upgamma \\cdot \\Big( q_\\alpha \\mathbf{\\breve{B}} \\Big) = 0 \\label{eq:iota_fom_adjoint_diff_eq_qsfl}, \\\\\n   \\int_{\\mathcal{S}} \\diff S \\; q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi + ( \\iota - \\iota_T ) = 0 \\label{eq:iota_fom_adjoint_integral_eq_qsfl},\n\\end{align}\n\\end{subequations}\nwith both equations defined on $\\mathcal{S}$. Using \\eqref{eq:surface_divergence_theorem}, the surface integral of \\eqref{eq:iota_fom_adjoint_diff_eq_qsfl} yields $0 = \\mathbf{\\breve{B}}\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}$, which is consistent with the boundary condition on the magnetic field \\eqref{eq:vacuum_field_normal_BC}. The first equation \\eqref{eq:iota_fom_adjoint_diff_eq_qsfl} can be recast in the form of a magnetic differential equation $\\mathbf{B}\\cdot\\nabla q_\\alpha = - q_\\alpha\\nabla_\\Upgamma\\cdot\\mathbf{B}$, while the second equation \\eqref{eq:iota_fom_adjoint_integral_eq_qsfl} is an integral condition on $q_\\alpha$ that ensures uniqueness of the solution.\n\nSecond, the adjoint equation for $q_\\omega$ is obtained by requiring the first line of \\eqref{eq:variation_L_iota_tot} to vanish,\n\\begin{subequations}\n\\begin{align}\n    \\Updelta q_\\omega = 0   \\qquad\\qquad\\qquad\\mathrm{in}\\;&\\mathcal{V}, \\label{eq:iota_fom_adjoint_eq_qomega}\\\\\n    \\nabla q_\\omega\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = -\\nabla_\\Upgamma \\cdot \\Big(q_\\alpha \\nabla_\\Upgamma \\alpha \\Big) \\qquad\\mathrm{on}\\; & \\mathcal{S} \\label{eq:iota_fom_adjoint_eq_qomega_normal_BC}.\n\\end{align}\n\\end{subequations}\nLike the magnetic potential $\\omega$, the adjoint variable $q_\\omega$ satisfies the Laplace equation in $\\mathcal{V}$ \\eqref{eq:iota_fom_adjoint_eq_qomega}. However, contrary to $\\omega$, $q_\\omega$ has a non-zero normal boundary condition on $\\mathcal{S}$ \\eqref{eq:iota_fom_adjoint_eq_qomega_normal_BC}, which notably depends on the straight field line adjoint variable $q_\\alpha$. Equations \\eqref{eq:iota_fom_adjoint_eq_qomega_normal_BC} and \\eqref{eq:iota_fom_adjoint_eq_qomega} are consistent, as $\\int_\\mathcal{V} \\diff V\\; \\Updelta q_\\omega = \\int_\\mathcal{S} \\diff S\\; \\nabla q_\\omega\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0$, by \\eqref{eq:surface_divergence_theorem}.\n\nFinally, the remaining contribution from the last line of \\eqref{eq:variation_L_iota_tot} yields the shape gradient\n\\begin{equation}\n   \\mathcal{G_\\iota} = \\frac{1}{G} \\Big[ \\mathbf{B}\\cdot \\nabla q_\\omega + q_\\alpha \\big( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{B} - \\mathbf{B}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\big)\\cdot\\nabla_\\Upgamma \\alpha \\Big], \\label{eq:shape_gradient_iota_fom}\n\\end{equation}\nwith $\\delta\\mathcal{L}_\\iota[\\delta\\mathbf{x}]  = \\int_{\\mathcal{S}} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;\\mathcal{G_\\iota}$.\n\nWe now calculate the shape gradient numerically and verify it against a finite-difference evaluation. The solutions to Laplace's equation for the vacuum magnetic field \\eqref{eq:vacuum_field_Laplace} and adjoint equation for $q_\\omega$ \\eqref{eq:iota_fom_adjoint_eq_qomega} are calculated with the SPEC code \\citep{hudsonComputationMultiregionRelaxed2012}, employing the new Zernike polynomial implementation \\citep{quCoordinateParameterisationSpectral2020}. In all results shown, the radial resolution $L_\\mathrm{rad}$ in SPEC is tied to the poloidal Fourier resolution $M_\\mathrm{pol}$ through $L_\\mathrm{rad} = M_\\mathrm{pol} + 4$. The solutions to the straight field line and $q_\\alpha$ adjoint equations are obtained with a Fourier-Galerkin spectral solver.\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_analytic_iota_target.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_iota_fom_adjoint}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_findiff_iota_target.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_iota_fom_findiff}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_relative_error_iota_target.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_iota_fom_relative_error}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/dfdomega_convergence_epsilon_rel_error_iota_fom.pdf}\n  \\caption{}\n  \\label{fig:convergence_iota_fom}\n\\end{subfigure}\n\\caption{Shape gradient for the rotational transform objective function with $\\iota_T = 1$, evaluated through (a) adjoint methods and (b) a forward finite-difference scheme with step size $\\epsilon_\\mathrm{FD}=10^{-7}$, for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}, with Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol}) = (16,16)$. The relative error, defined as the absolute error normalised by the maximal absolute value of the adjoint shape gradient, is shown in (c). The convergence of the relative error in the parameter derivative \\eqref{eq:parameter_derivatives_shape_grad} for a random direction in $\\Omega$ is shown in (d) as a function of the step-size $\\epsilon_\\mathrm{FD}$ and Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol})$. The black dashed line indicates the linear scaling in $\\epsilon_\\mathrm{FD}$ expected from the employed forward finite-difference scheme.}\n\\label{fig:iota_fom_numerical_eval_and_convergence}\n\\end{figure}\n\nThe shape gradient \\eqref{eq:shape_gradient_iota_fom} is shown in Fig.~\\ref{fig:shape_grad_iota_fom_adjoint} for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}. The localisation at the ellipse tips is unsurprising, as near-axis expansions show that ellipticity of the flux surfaces generates rotational transform \\citep{mercierEquilibriumStabilityToroidal1964}. This shape gradient $\\mathcal{G}_\\mathrm{adjoint}$ can be verified against the direct finite-difference evaluation $\\mathcal{G}_\\mathrm{FD}$ shown in Fig.~\\ref{fig:shape_grad_iota_fom_findiff}, obtained by evaluating the parameter derivatives $\\partial f\/\\partial \\Omega_i$ through finite-differences and inverting \\eqref{eq:parameter_derivatives_shape_grad}, see \\citet{landremanComputingLocalSensitivity2018}. On the scale of the figure, the two shape gradients seem identical. The relative error is shown in Fig.~\\ref{fig:shape_grad_iota_fom_relative_error} to be small, limited to $\\sim 2\\%$ at the ellipse tips, and exhibits oscillations typical of a truncated Fourier resolution. The relative error is here defined as the absolute error normalised by the $L^\\infty$-norm of $\\mathcal{G}_\\mathrm{adjoint}$, i.e. its maximum absolute value. This choice is preferable to e.g. the $L^2$-norm, as the shape gradient and the error thereof have small average values on the boundary compared to their large values at the ellipse tips, such that unreasonably high relative errors would result at these locations if using the $L^2$-norm as normalisation. \n\nFurthermore, we test convergence of the shape gradient by evaluating a parameter derivative \\eqref{eq:parameter_derivatives_shape_grad} for a random direction in $\\Omega$. The parameter derivative is evaluated both through the adjoint shape gradient and by a forward finite-difference scheme. The relative error is shown in Fig.~\\ref{fig:convergence_iota_fom} as a function of the finite-difference step size $\\epsilon_\\mathrm{FD}$ and the Fourier resolution, which is used both in SPEC and the Fourier-Galerkin spectral solver. As $\\epsilon_\\mathrm{FD}$ is reduced, the error initially decreases linearly with $\\epsilon_\\mathrm{FD}$, as expected from the employed forward finite-difference scheme, until it plateaus at a value governed by the finite Fourier or radial resolution. As mentioned in \\S\\ref{sec:basics_adjoint_methods}, errors in the adjoint shape gradient are introduced by the assumption that the constraint and adjoint PDEs are exactly satisfied. In practice, these PDEs are solved only approximately, limited by the finite Fourier and radial resolution, such that a reduction of the error with increasing resolution is to be expected.\n\n\\subsection{Quasisymmetry objective function}\n\\label{sec:QS_fom_shape_derivative}\n\nFor a general (non-vacuum) magnetic field with nested flux surfaces, quasisymmetry can be expressed as\n\\begin{equation}\n    \\frac{\\mathbf{B}\\cdot\\nabla\\psi\\times\\nabla B}{\\mathbf{B}\\cdot\\nabla B} = -\\frac{MG + NI}{N-\\iota M}, \\label{eq:quasisymmetry_magnetic_field_condition}\n\\end{equation}\nwhere $I$ is the net toroidal plasma current and $N\/M$ is the helicity of the field strength in Boozer coordinates, see e.g. \\citet{helanderTheoryPlasmaConfinement2014}. For the vacuum field considered here, $I=0$. In the following, we will not consider quasi-poloidal symmetry, i.e. we will assume $M\\ne 0$. If desired, it would be straight-forward to extend the derived results to include the case $M=0$.\n\nFor magnetic fields with globally nested flux surfaces labelled by $\\psi$, \\eqref{eq:quasisymmetry_magnetic_field_condition} is defined globally. However, we are considering a generally non-integrable field, assuming only that the boundary $\\mathcal{S}$ is a flux surface. Using the generalised toroidal flux gradient defined in \\eqref{eq:def_nabla_tilde_psi}, we are able to define quasisymmetry on the isolated flux surface $\\mathcal{S}$, leading to the quasisymmetry (QS) objective function\n\\begin{equation}\n    f_\\mathrm{QS}(\\mathcal{S}, \\omega, \\iota, \\lambda) =  \\frac{1}{2} \\int_\\mathcal{S} \\diff S\\; v_\\mathrm{QS}^2(\\omega, \\iota, \\lambda), \\label{eq:definition_fQS}\n\\end{equation}\nwith\n\\begin{equation}\n    v_\\mathrm{QS} =  \\mathbf{\\breve{B}}\\cdot\\nabla \\breve{B}  -  \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\cdot\\nabla \\breve{B}  \\left( \\iota - N\/M\\right). \\label{eq:definition_vQS}\n\\end{equation}\nIf $f_\\mathrm{QS}=0$ and the field is integrable in the neighbourhood of $\\mathcal{S}$, \\eqref{eq:quasisymmetry_magnetic_field_condition} will be satisfied on $\\mathcal{S}$, i.e. the field is quasisymmetric on the boundary.\n\nThe shape derivative of $f_\\mathrm{QS}$ is derived in App.~\\ref{app:derivation_QS_fom_shape_derivative}, with the final expression given in \\eqref{eq:delta_fQS_tot}. Combined with the shape derivatives of $\\mathcal{M}$ \\eqref{eq:variation_M_tot} and $\\mathcal{N}$ \\eqref{eq:variation_N_tot}, the shape derivative of the Lagrangian \\eqref{eq:lagrangian_general} with the quasisymmetric figure of merit follows \\eqref{eq:variation_L_QS_tot}.\n\nRequiring the Lagrangian to be stationary with respect to variations in $\\iota$ and $\\lambda$, the first two lines of \\eqref{eq:variation_L_QS_tot} yield the adjoint equations for $q_\\alpha$,\n\\begin{subequations}\n\\begin{align}\n   \\nabla_\\Upgamma \\cdot \\Big( q_\\alpha \\mathbf{\\breve{B}} \\Big) = -\\nabla_\\Upgamma\\cdot\\left[ \\nabla_\\Upgamma\\alpha \\left( v_\\mathrm{QS}\\; \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\cdot\\nabla \\breve{B}\\; \\frac{\\iota - N\/M}{\\abs{\\nabla_\\Upgamma \\alpha}^2}\\right) \\right] \\label{eq:QS_fom_adjoint_diff_eq_qsfl}, \\\\\n   0=\\int_{\\mathcal{S}} \\diff S \\left\\{ q_\\alpha \\mathbf{\\breve{B}} \\cdot \\nabla\\phi + v_\\mathrm{QS}\\; \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\cdot\\nabla \\breve{B} \\left[ \\frac{\\nabla_\\Upgamma\\alpha\\cdot\\nabla_\\Upgamma \\phi}{\\abs{\\nabla_\\Upgamma \\alpha}^2} (\\iota-N\/M) + 1 \\right] \\right\\} \\label{eq:QS_fom_adjoint_integral_eq_qsfl}.\n\\end{align}\n\\end{subequations}\nSimilarly to the rotational transform objective function case, $q_\\alpha$ satisfies a magnetic differential equation \\eqref{eq:QS_fom_adjoint_diff_eq_qsfl} on $\\mathcal{S}$, with integral condition \\eqref{eq:QS_fom_adjoint_integral_eq_qsfl}. By \\eqref{eq:surface_divergence_theorem}, the surface integral of \\eqref{eq:QS_fom_adjoint_diff_eq_qsfl} is consistent with the magnetic field's normal component vanishing on the boundary \\eqref{eq:vacuum_field_normal_BC}.\n\nFurthermore, requiring the Lagrangian to be stationary with respect to variations in $\\omega$, we obtain the adjoint equations for $q_\\omega$ from the third and fourth lines of \\eqref{eq:variation_L_QS_tot}\n\\begin{subequations}\n\\begin{align}\n    \\Updelta q_\\omega & = 0   \\qquad\\qquad\\qquad\\mathrm{in}\\;\\mathcal{V}, \\label{eq:QS_fom_adjoint_eq_qomega}\\\\\n    \\nabla q_\\omega \\cdot \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} &= -\\nabla_\\Upgamma \\cdot \\Bigg\\{ q_\\alpha \\nabla_\\Upgamma \\alpha + v_\\mathrm{QS} \\;\\nabla_\\Upgamma \\breve{B} -  \\frac{\\mathbf{\\breve{B}}}{\\breve{B}} \\nabla_\\Upgamma \\cdot (v_\\mathrm{QS} \\; \\mathbf{\\breve{B}})\\label{eq:QS_fom_adjoint_eq_qomega_normal_BC} \\\\\n    \\nonumber & + \\left(\\iota-N\/M\\right) \\left[ v_\\mathrm{QS} \\; \\frac{\\overline{\\nabla\\psi}}{G}\\times\\nabla_\\Upgamma\\breve{B} - \\mathbf{\\breve{B}}\\; \\nabla_\\Upgamma\\cdot\\left(\\frac{1}{\\breve{B}}v_\\mathrm{QS} \\mathbf{\\breve{B}}\\times\\frac{\\overline{\\nabla\\psi}}{G}\\right)  \\right] \\Bigg\\} \\quad \\text{ on } \\mathcal{S}.\n\\end{align}\n\\end{subequations}\nAgain, $q_\\omega$ satisfies the Laplace equation in $\\mathcal{V}$ \\eqref{eq:QS_fom_adjoint_eq_qomega}, with a normal boundary condition on $\\mathcal{S}$ that is the tangential divergence of a vector tangential to the surface \\eqref{eq:QS_fom_adjoint_eq_qomega_normal_BC}. The boundary condition \\eqref{eq:QS_fom_adjoint_eq_qomega_normal_BC} is consistent with the Laplace equation, as $\\int_\\mathcal{V} \\diff V\\; \\Updelta q_\\omega = \\int_\\mathcal{S} \\diff S\\; \\nabla q_\\omega\\cdot\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} = 0$, by \\eqref{eq:surface_divergence_theorem}. \n\nFinally, we obtain the shape gradient from the last three lines of \\eqref{eq:variation_L_QS_tot},\n\\begin{align}\n    \\mathcal{G}&_\\mathrm{QS}  =  - (\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\breve{B}) \\nabla_\\Upgamma\\cdot\\left( v_\\mathrm{QS} \\mathbf{\\breve{B}} \\right) - v_\\mathrm{QS}\\;\\left(\\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} - \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}} \\right)\\cdot\\nabla_\\Upgamma\\breve{B} \\label{eq:QS_fom_shape_gradient} \\\\\n    \\nonumber & + \\left(\\iota-N\/M\\right)\\; \\frac{|\\overline{\\nabla\\psi}|}{G} \\; \\mathbf{\\breve{B}}\\times\\nabla\\breve{B}\\cdot  \\left[ \\abs{\\nabla_\\Upgamma\\alpha} \\nabla_\\Upgamma\\left( \\frac{v_\\mathrm{QS}}{\\abs{\\nabla_\\Upgamma\\alpha}} \\right) + \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\; v_\\mathrm{QS}\\; \\left( \\frac{\\nabla_\\Upgamma\\alpha\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla_\\Upgamma\\alpha}{\\abs{\\nabla_\\Upgamma \\alpha}^2} -h \\right) \\right]  \\\\\n    \\nonumber & + \\mathbf{\\breve{B}}\\cdot\\nabla q_\\omega + q_\\alpha \\left( \\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}}\\cdot\\nabla\\mathbf{\\breve{B}} - \\mathbf{\\breve{B}}\\cdot\\nabla\\ensuremath{\\boldsymbol{\\hat{\\mathrm{n}}}} \\right)\\cdot\\nabla_\\Upgamma\\alpha + \\frac{h}{2} v_\\mathrm{QS}^2,\n\\end{align}\nwith $\\delta\\mathcal{L}_\\mathrm{QS}[\\delta\\mathbf{x}]  = \\int_{\\mathcal{S}} \\diff S \\;(\\ensuremath{\\delta\\boldsymbol{\\mathrm{x}}\\cdot \\normvec})\\;\\mathcal{G_\\mathrm{QS}}$, and $h$ the summed curvature.\n\n\\begin{figure}\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_analytic_quasisymm.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_QS_fom_adjoint}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_findiff_quasisymm.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_QS_fom_findiff}\n\\end{subfigure}\n\\\\\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/shape_grad_3d_relative_error_quasisymm.pdf}\n  \\caption{}\n  \\label{fig:shape_grad_QS_fom_relative error}\n\\end{subfigure}%\n\\begin{subfigure}{.5\\textwidth}\n  \\centering\n  \\includegraphics[width=1\\linewidth]{plots_pdf\/dfdomega_convergence_epsilon_rel_error_QS_fom.pdf}\n  \\caption{}\n  \\label{fig:convergence_QS_fom}\n\\end{subfigure}\n\\caption{Shape gradient for the quasisymmetry objective function with helicity $N\/M=5$, evaluated through (a) adjoint methods and (b) a forward finite-difference scheme with step size $\\epsilon_\\mathrm{FD}=10^{-9}$, for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}, with Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol}) = (16,16)$. The relative error, defined as the absolute error normalised by the maximal absolute value of the adjoint shape gradient, is shown in (c). The convergence of the relative error in the parameter derivative \\eqref{eq:parameter_derivatives_shape_grad} for a random direction in $\\Omega$ is shown in (d) as a function of the step-size $\\epsilon_\\mathrm{FD}$ and Fourier resolution $(N_\\mathrm{tor}, M_\\mathrm{pol})$. The black dashed line indicates the linear scaling in $\\epsilon_\\mathrm{FD}$ expected from the employed forward finite-difference scheme.}\n\\label{fig:QS_fom_numerical_eval_and_convergence}\n\\end{figure}\n\nThe shape gradient \\eqref{eq:QS_fom_shape_gradient} for targeted quasi-helical symmetry with helicity $N\/M = 5$ is shown in Fig.~\\ref{fig:shape_grad_QS_fom_adjoint} for the example rotating ellipse case introduced in Fig.~\\ref{fig:abs_nabla_psi}. The shape gradient obtained through adjoint methods is verified against a finite-difference evaluation in Fig.~\\ref{fig:shape_grad_QS_fom_findiff}. The error is visibly small, as is attested by the small relative error of the shape gradient shown in Fig.~\\ref{fig:shape_grad_QS_fom_relative error}. Convergence of the relative error for a parameter derivative in a random direction in $\\Omega$, evaluated with the adjoint method and with a centered finite-difference scheme, is shown in Fig.~\\ref{fig:convergence_QS_fom}. Akin to the rotational transform figure of merit convergence study in Fig.~\\ref{fig:convergence_iota_fom}, the error decreases linearly with $\\epsilon_\\mathrm{FD}$ until it plateaus due to finite Fourier or radial resolution. While the lowest resolution of $(N_\\mathrm{tor}, M_\\mathrm{pol}) = (8,8)$ seemed reasonable for the rotational transform figure of merit in Fig.~\\ref{fig:convergence_iota_fom}, a higher resolution is clearly required for the quasisymmetry figure of merit. This could be due to the fact that higher derivatives of the magnetic field are involved in the shape gradient for quasisymmetry \\eqref{eq:QS_fom_shape_gradient} than in the one for rotational transform \\eqref{eq:shape_gradient_iota_fom}, through derivatives of $v_\\mathrm{QS}$. The resulting fine-scale structure of $\\mathcal{G}$ is harder to resolve with a truncated Fourier series. However, the relative errors in Figs.~\\ref{fig:convergence_iota_fom}~and~\\ref{fig:convergence_QS_fom} are similarly small at the highest Fourier resolutions employed.\n\n\\section{Conclusions}\n\nIn this work, we derived the adjoint equations and shape gradient for the rotational transform and quasisymmetry of a vacuum field on a surface. The shape gradients allow fast computation of derivatives with respect to the parameters that describe the geometry of the surface, which are used in optimisation and sensitivity analyses. For a boundary represented by $N$ parameters, the speed-up from the adjoint method is $O(N)$ compared to a finite-difference evaluation. \n\nThis should enable future use of codes such as SPEC \\citep{hudsonComputationMultiregionRelaxed2012} in optimisation calculations, which was hitherto neglected in favour of the more widely-used VMEC code \\citep{hirshmanThreedimensionalFreeBoundary1986}. Contrary to VMEC, SPEC does not rely on the assumption of nested flux surfaces and can therefore model stochastic and island regions. In practice, employing adjoint methods and computing derivatives of quantities arising from ideal MHS equilibria is challenging, as the linearised MHS equilibrium equations possess regular singular points at every rational surface that resonates with the perturbation. These challenges can be avoided by the use of alternative equilibrium models, such as force-free magnetic fields, or the vacuum fields considered in this work. The generality of the results presented herein would also allow for their implementation in other solvers such as BIEST \\citep{malhotraTaylorStatesStellarators2019}. It is left for future work to extend the vacuum field results presented herein to the more general force-free fields modeled by SPEC.  Furthermore, the adjoint methods for vacuum fields introduced in this work could be fruitfully applied to other optimisation problems, e.g. in neoclassical transport calculations.\n\nIt is generally believed that exact quasisymmetry cannot be obtained exactly in a finite volume as near-axis expansions lead to an an overdetermined system of equations \\citep{garrenExistenceQuasihelicallySymmetric1991}, although that can be resolved by allowing for an anisotropic plasma pressure \\citep{rodriguezSolvingProblemOverdetermination2021a, rodriguezSolvingProblemOverdetermination2021}. Exact quasisymmetry on a surface is thought generally possible \\citep{garrenExistenceQuasihelicallySymmetric1991, plunkQuasiaxisymmetricMagneticFields2018}; and indeed, a vacuum solution near axisymmetry was recently found \\citep{senguptaVacuumMagneticFields2021}. The shape gradient for quasisymmetry derived in this work could be used to numerically probe the existence of quasisymmetric solutions on a surface that are not close to axisymmetry. For this purpose, the shape gradient for the rotational transform objective function \\eqref{eq:shape_gradient_iota_fom} could be used to avoid the axisymmetric solution at $\\iota = 0$, or also to avoid low order rationals. Furthermore, the shape gradients derived herein could be used to investigate if and how optimisation for quasisymmetry and for the rotational transform compete with each other. Finally, combining the derivatives of quasisymmetry and rotational transform with previously obtained derivatives of coil shapes \\citep{hudsonDifferentiatingShapeStellarator2018} and island size \\citep{geraldiniAdjointMethodDetermining2021} should, in principle, allow for the efficient search of a stellarator configuration with significant rotational transform, good integrability and neoclassical confinement at the boundary, realised by simple coils.\n\n\\begin{acknowledgments}\nThis work was supported by U.S. DOE DE-AC02-09CH11466, DE-SC0016072 and DE-AC02\u201376CH03073. A.B. acknowledges the generous support of the Simons Foundation.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nKontsevich has associated certain characteristic classes to\nfinite-dimensional $L_\\infty$- or $A_\\infty$-algebras equipped with an\ninvariant inner product, \\cite{Kncsg, KFd}. These are expressed in\nterms of the homology of certain complexes spanned by graphs with some\nadditional structures. This construction is by now well-understood\nboth from the point of view of Lie algebra homology and topological\nconformal field theory; see, for example, \\cite{HaLacc}.\n\nIn this note, we explore a natural generalization of this construction\nto the case of \\emph{curved} algebras, introduced by Positselski in\n\\cite{Pnqdc}. It turns out that a complete description of these\nclasses, and of the homology of the associated graph complexes, is\npossible. We show that these are all obtainable from\none-dimensional algebras, and that these classes are zero for algebras\nwith zero curvature. This contrasts with the corresponding problem for\nconventional graph complexes, which is still widely open.\n\nAs explained by Kontsevich, these graph complexes can be viewed as\ncomputing the stable homology of Lie algebras of symplectic vector\nfields on a vector space $W$ (in the $A_\\infty$ case, one should\ntake \\emph{noncommutative} symplectic vector fields).\nThis motivates us to consider the\nstability maps.  In this direction, we prove that the map from the Lie\nalgebra of symplectic vector fields on $W$ vanishing at the origin to\nthe homology of the Lie algebra of \\emph{all} vector fields on $W\n\\oplus \\CC \\cdot w$, where $w$ is an odd vector, is zero.  Similarly,\nwe show the same for the Lie algebra of noncommutative\nsymplectic vector fields.\n\nThe precise relation to the previous result is as follows. Any cyclic\n$L_\\infty$-algebra structure on $V$ defines an unstable characteristic\nclass in the homology of the Lie algebra of symplectic vector fields\non the shifted vector space $W = \\Pi V$. As $\\dim V \\rightarrow\n\\infty$, the homology of this Lie algebra converges to the graph\nhomology (at least if $V$ is only growing in the purely even or purely\nodd direction: see Theorem \\ref{GFgraph} below), and the image of the\nunstable characteristic class under the stability maps gives, in the\nlimit, the aforementioned (stable) characteristic class.  Hence, our\nresult above says that the unstable curved characteristic class of an\nalgebra with zero curvature already maps to zero under the first\nstability map $W:=\\Pi V \\into W \\oplus \\CC\\cdot w$.\n\n\nA related observation is the following: if $A$ is a curved\n(associative or Lie, or more generally $A_\\infty$- or $L_\\infty$-)\nalgebra with nonzero curvature, then $A$ is gauge equivalent (i.e.,\nhomotopy isomorphic) to the algebra with the same curvature and zero\nmultiplication, in a sense we will recall below.  In the case of\n\\emph{cyclic} curved algebras, we also compute the gauge equivalence\nclasses, which are less trivial: nontrivially curved algebras are\ngauge equivalent to the direct sum of a curved algebra of dimension at\nmost two of a certain form (but having nontrivial multiplications in\ngeneral), with a zero algebra.\n\nThis observation hints at a triviality of\ncurved infinity-algebras from a homological point of view, at least when the cyclic structure is not considered. A similar\nresult on the triviality of the corresponding derived categories was\nobtained recently in \\cite{KLNnv}; another manifestation of this triviality principle is briefly discussed in the last section of this paper.\n\nFinally, we generalize these results to the operadic setting, i.e., to\ntypes of algebras other than associative and Lie algebras. In\nparticular, we can apply it to Poisson, Gerstenhaber, BV, permutation,\nand pre-Lie algebras. For the most part, the generalization is\nstraightforward, and we restrict ourselves with giving only an outline\nof arguments in this section.  There is, however, one important aspect\nwhich is less visible in the special cases of commutative and ribbon\ngraphs: a curved graph complex associated with a cyclic (or even\nmodular) operad $\\mathcal O$ is quasi-isomorphic to a variant of the\n\\emph{deformation complex of a curved $\\mathcal O$-algebra on a\n  one-dimensional space}.  Therefore, this graph complex supports the\nstructure of a differential graded Lie algebra. This differential Lie\nalgebra, and its Chevalley-Eilenberg complex, appeared in various\nguises in the works of Zwiebach-Sen, Costello and\nHarrelson-Voronov-Zuniga on quantum master equation, \\cite{SZbias,\n  Cospftft, HVZocms}.\n\\subsection{Notation and conventions}\\label{notsec}\nIn this paper we work in the category of $\\mathbb Z\/2$-graded vector\nspaces (also known as (super)vector spaces) over $\\CC$ although all\nresults continue to hold in the $\\mathbb Z$-graded context and when\n$\\CC$ is replaced by any field of characteristic zero. We will usually\nrefer to these graded vector spaces simply as ``spaces.'' The parity of a homogeneous vector $v$ in a space will be denoted by $|v|$. The adjective\n`differential graded' will mean `differential $\\mathbb Z\/2$-graded'\nand will be abbreviated as `dg'. A (commutative) differential graded\n(Lie) algebra will be abbreviated as\n(c)dg(l)a.\nAll of\nour unmarked tensors are understood to be taken over $\\CC$. For a $\\mathbb Z\/2$-graded vector space $V=V_0\\oplus V_1$ the symbol $\\Pi\n V$ will denote the \\emph{parity reversion} of $V$; thus $(\\Pi\n V)_0=V_1$ while $(\\Pi V)_1=V_0$.\n\n\n We will make use of the language of \\emph{formal}\\footnote{Here the\n   word ``formal'' is understood in the sense of a formal\n   neighborhood, which differs from the notion of ``formality'' in\n   rational homotopy theory.}  spaces and algebras (which exist since\n \\cite{Lefat} under the name \"linearly compact\"; see, e.g.,\n \\cite{HaLactha} for a recent treatment relevant to the present work,\n under the present name).  A formal space is an inverse limit of\n finite-dimensional spaces. \n\n\n\n\n The functor of taking the linear dual establishes an\n anti-equivalence between the category of (discrete) vector spaces and that of\n formal vector spaces.\n\n It will always be clear from the context whether\n we work with formal or discrete vector spaces, and we will typically\n not mention this specifically later on; the tensor product of two formal spaces is understood to be their \\emph{completed} tensor product.  Furthermore, the symbol $V$\n will be reserved for a discrete space, with its dual $V^*$ therefore\n a formal space.\n\n  In particular, we will work with formal (c)dg(l)as.  The main\n examples will be completed tensor and symmetric algebras on formal\n spaces $W$; these will be denoted by $\\hat{T}W$ and\n $\\hat{S}W$ respectively. Note that we will \\emph{never} consider the\n uncompleted $TW$ and $SW$ when $W$ is formal, and similarly never\n consider the completed $\\hat{T}V$ or $\\hat{S}V$ when $V$ is discrete,\n so as to stay in either the category of formal spaces or that of\n discrete spaces.\n\n\n The Lie algebras of continuous derivations of $\\hat{T}W$ and\n of $\\hat{S}W$ will be denoted by $\\Der(\\hat{T}W)$ and\n $\\Der(\\hat{S}W)$ respectively; we will also consider their Lie\n subalgebras $\\Der^0(\\hat{T}W)$ and $\\Der^0(\\hat{S}W)$ consisting of\n derivations having no constant terms.\n\nWe mention here two potential pitfalls present in this framework. Firstly,\nthe categories of discrete and formal vector spaces are not disjoint: the\nspaces which are both discrete and formal are precisely finite-dimensional\nspaces. And secondly, not every space is either discrete or formal; moreover\nsuch spaces arise as a result of some natural operations with discrete or\nformal spaces. For example if $U$ and $W$ are infinite-dimensional formal or\ndiscrete spaces then the vector space $\\Hom(U,W)$ is neither formal nor\ndiscrete. Similarly, the Lie algebras\n$\\Der(\\hat{T}V^*)$ and $\\Der({T}V)$ will be neither\nformal nor discrete if $V$ is infinite-dimensional.\n\nTherefore, to avoid possible confusion, we make the blanket assumption\nthat the $\\ZZ\/2$-graded vector space $V$ which appears throughout the\npaper, in addition to being discrete as above, is in fact\n\\emph{finite-dimensional}.  This way all the objects we consider will\nlive in either the category of formal spaces or the category of\ndiscrete spaces (but not both: so each finite-dimensional space we\nconsider will be viewed in only one way). The price we pay is that\nsome of our results are not formulated in maximal generality; namely\nTheorem \\ref{curvtrivthm} and Claim \\ref{opclaim} do not need the\nspace $V$ to be finite-dimensional (although essentially none of the\nexposition needs to be modified to obtain this generalization).\n\n\n For a \\emph{formal} dgla $\\g$ its Chevalley-Eilenberg cohomological\n complex will be denoted by $\\CE^\\bullet(\\g)$.  This is defined as\n follows (note that the definition \\emph{differs} from the usual one\n in where completions are taken, since $\\g$ is formal rather than\n discrete):\nthe underlying graded vector space of $\\CE^\\bullet(\\g)$ is $S\\Pi{\\mathfrak g}^*$ and the differential is given as a sum of two maps $d_{\\operatorname I}$ and $d_{\\CE}$. Here $d_{\\operatorname I}$ and $d_{\\CE}$ are both specified by their restriction onto $\\Pi{\\mathfrak g}^*$ and extended to the whole $S\\Pi{\\mathfrak g}^*$ by the Leibniz rule; further $d_{\\operatorname I}:\\Pi{\\mathfrak g}^*\\to \\Pi{\\mathfrak g}^*$ is the shift of the dual of the internal differential on ${\\mathfrak g}$ whereas $d_{\\CE}:\\Pi{\\mathfrak g}^*\\to S^2(\\Pi{\\mathfrak g}^*)$ is induced by the commutator map $[,]:{\\mathfrak g}\\otimes {\\mathfrak g} \\to {\\mathfrak g}$.\n\nThis is in general a $\\ZZ\/2$-graded complex.  In the case that the\ndifferential $d$ is zero, it has an additional grading by\n\\emph{cohomological degree}, i.e., $S^i \\Pi\\mathfrak{ g}^*$ is in degree\n$i$.  The corresponding homological complex is the linear dual:\n$\\CE_\\bullet(\\g)=(\\CE^\\bullet(\\g))^*$, which has the underlying formal space $\\hat S \\Pi \\mathfrak{g}$.\nNote that in some papers (e.g. \\cite{HaLacc}) the Chevalley-Eilenberg complex of a graded Lie algebra ${\\mathfrak g}$ is defined using the (in our\ncase completed) \\emph{exterior} algebra $\\hat \\Lambda {\\mathfrak g}$; this definition is equivalent to ours under a canonical isomorphism $\\hat S(\\Pi{\\mathfrak g})\\cong \\hat \\Lambda {\\mathfrak g}$ where\n\\[\\Pi g_1\\ldots\\Pi g_n\\mapsto (-1)^{|g_{n-1}|+2|g_{n-1}|+\\ldots+(n-1)|g_1|}g_1\\wedge\\ldots\\wedge g_n.\\]\n\n\n\n \\section{Gauge equivalence classes of curved (cyclic) $A_\\infty$- and\n   $L_\\infty$-algebras}\n\\subsection{Curved $A_\\infty$- and $L_\\infty$- algebras}\nWe recall the definition of $A_\\infty$- and $L_\\infty$-algebras\nfollowing \\cite{HaLacc}, as well as their curved analogues; cf.~e.g.,\n\\cite{Nicbd} and references therein.\n\n\nA curved $A_\\infty$-algebra structure on a\n(finite-dimensional\\footnote{As in \\S \\ref{notsec}, $V$ is considered as\na discrete space throughout; in the present subsection\none could allow it to be infinite-dimensional and discrete.\nWe will not make further mention of this.}) space $V$ is\na continuous odd derivation $m$ of the formal dga $\\hat{T}\\Pi V^*$ and\na curved $L_\\infty$-algebra structure on $V$ is a continuous odd\nderivation $m$ of the formal cdga $\\hat{S}\\Pi V^*$; additionally $m$\nis required to square to zero in both cases. An ordinary (i.e. uncurved) $A_\\infty$-\nor $L_\\infty$-structure is specified by the requirement that $m$ have\nno constant term. The components $m_i:{T}^i\\Pi V\\to \\Pi V$ or\n${S}^i\\Pi V\\to \\Pi V$ of the dual of the restriction of $m$ to the\ntensor or symmetric powers $\\Pi V$ are the structure maps of the\ncorresponding $A_\\infty$- or $L_\\infty$-structure.\n\nWe note that sometimes it is convenient to use the more traditional\nway of writing the structure maps of an $A_\\infty$- or\n$L_\\infty$-algebra $V$ as ${T}^iV\\to V$ or ${\\Lambda}^iV\\to V$; these\nmaps will then be even or odd depending on whether $i$ is even or\nodd. To alleviate the notation we will still write $m_i$ for these\nmaps when the meaning is clear from the context.\n\nWe are interested in \\emph{gauge equivalence} classes of $A_\\infty$- or $L_\\infty$- structures on a fixed  space $V$. In particular, this equivalence relation implies other relations found in the literature under the names of homotopy or quasi-isomorphism.\n\nNamely, a gauge equivalence between $(V,m)$ and $(V,m')$ is defined as\na derivation $\\xi \\in \\Der^0(\\hat{T}\\Pi V^*)$ or $\\xi \\in\n\\Der^0(\\hat{S} \\Pi V^*)$ which is even and satisfies $m' = e^{\\ad \\xi}\nm$.\\footnote{In the case that our ground field is not $\\CC$, $e^\\xi$\n  still makes sense if we require additionally that $\\xi_1 = 0$ as\n  well, i.e., $\\xi$ has no linear term.  We can then modify the above\n  by saying that a gauge equivalence is a composition of such an\n  equivalence (for $\\xi_1=0$) with a linear isomorphism of $V$.} In\nparticular, such a gauge equivalence yields an isomorphism of dgas\n$e^\\xi: (\\hat{T} \\Pi V^*,m) \\iso (\\hat{T} \\Pi V^*, m')$ or cdgas\n$e^\\xi: (\\hat{S} \\Pi V^*,m) \\iso (\\hat{S} \\Pi V^*, m')$, and we usually\ndenote the gauge equivalence by $e^\\xi$.\n\\begin{theorem}\\label{curvtrivthm}\n  If $(V, m)$ is a curved $A_\\infty$- or $L_\\infty$- algebra for which\n  the curvature $m_0 = c \\in V_0$ is nonzero, then $m$ is gauge\n  equivalent to the structure $m' = c$ with all higher multiplications\n  $m'_i = 0$ for $i > 0$.\n\\end{theorem}\nRoughly, the above is saying that, when an (odd, noncommutative)\nformal vector field is nonzero evaluated at zero, then it is\nequivalent to a constant vector field up to (generally nonlinear) change of coordinates.\n\nAs a corollary of the theorem, it follows that any two\nnontrivially curved algebras with the same underlying graded vector space $V$\nare gauge equivalent.\n\\begin{proof}[Proof of Theorem \\ref{curvtrivthm}]\n  We consider the $A_\\infty$ case; the $L_\\infty$ case is similar.\n\n  Any $A_\\infty$-algebra structure $(V,m')$ with $m'_0=m_0=c$ can be\n  viewed as a deformation of $(V,m_0)$. Indeed, let us introduce a\n  formal parameter $\\hbar$; then $(V,m')$ is equivalent to the\n  deformed structure $(V,m'_\\hbar)$, where $m'_\\hbar = m'_0 + \\sum_{i\n    \\geq 1} \\hbar^i m'_i$.  This yields an equivalence with deformed\n  structures whose $i$-ary operations are homogeneous of degree $i$ in\n  $\\hbar$.\n\n  The structures of the form $(V,m')$ are governed by the dgla\n  $\\Der(\\hat T \\Pi V^*, [,c])$, where $c \\in V$ is viewed as an odd\n  constant derivation of $\\hat T \\Pi V^*$.  Formal deformations (such\n  as $(V, m'_\\hbar)$) are governed by the dgla $\\Der(\\hat T \\Pi V^*[[\\hbar]],\n  [,c])$.  Gauge equivalences $e^{\\xi}$ of Maurer-Cartan elements of\n  $\\Der(\\hat T \\Pi V^*, [,c])$, which we can assume satisfy $\\xi_0 = 0\n  = \\xi_1$ (so as to not change the curvature) are identified with\n  gauge equivalences $e^{\\frac{1}{\\hbar}\\xi_\\hbar}$ (for $\\xi_\\hbar :=\n  \\sum_{i} \\hbar^i \\xi_i$) of the corresponding Maurer-Cartan elements\n  of $\\Der(\\hat T \\Pi V^*[[\\hbar]], [,c])$, which preserve the grading\n  $|\\hbar|=1=|V^*|$.\n\n Since gauge equivalence classes of deformations of Maurer-Cartan\n elements are preserved under quasi-isomorphisms of dglas, it suffices\n to show that $\\Der(\\hat T \\Pi V^*, [,c])$ is acyclic. Let us write\n down this complex $C^\\bullet$ explicitly; note that $C^\\bullet$ is a version of\n the Hochschild complex in the curved setting.\n\n Set $C^i:=\\Hom((\\Pi V)^{\\otimes i}, \\Pi V)$ with the differential $d:C^i\\to\n C^{i-1}$  given by the formula, for $f\\in\n C^i$, and $x_1, \\ldots, x_{i-1} \\in \\Pi V$:\n\\[\ndf(x_1,\\ldots, x_{i-1})=\\sum_k(-1)^{|x_1|+\\cdots+|x_k|}f(x_1,\\ldots, x_{k},c,x_{k+1},\\ldots, x_i).\n\\]\nWe construct an explicit contracting homotopy.  Choose an odd linear\nmap $\\epsilon:\\Pi V\\to\\CC$ such that $\\epsilon(c)=1$ (here, odd means\nthat $\\epsilon|_{\\Pi (V_1) = (\\Pi V)_0} = 0$). Define maps $s_i:C^i\\to\nC^{i+1}$ by the formula, for $f\\in C^i$:\n\\begin{equation} \\label{siref}\ns_if(x_1,\\ldots,x_{i+1})=\\epsilon(x_1)f(x_2,\\ldots, x_{i+1}).\n\\end{equation}\nThen,\n\\begin{equation}\nd s_i f(x_1,\\ldots,x_{i}) + s_{i-1} d f(x_1, \\ldots, x_{i}) = \\epsilon(c) f(x_1, \\ldots, x_{i}) = f(x_1, \\ldots, x_{i}). \\qedhere\n\\end{equation}\n\\end{proof}\n\\subsection{Cyclic algebras}\nWe now extend the results of the previous subsection to the case of\nalgebras with a cyclic inner product.  By an \\emph{inner product} on a\n$\\ZZ\/2$-graded vector space $V$ we mean a nondegenerate symmetric\nbilinear form $( -, - ): V\\otimes V \\rightarrow \\CC$, where $V$ is\nrequired to be finite-dimensional. An \\emph{inner product space} is a\nspace equipped with an inner product.\n\\begin{definition} Let $(V,m)$ be a finite-dimensional $A_\\infty$- or\n  $L_\\infty$-algebra.  A cyclic inner product on $V$ is an inner\n  product $( -, - ): V\\otimes V \\rightarrow \\CC$ for which the tensors\n  $( m_i(v_1,\\ldots,v_i), v_{i+1})$ are invariant with respect to the\n  signed cyclic permutations of arguments:\n\\[\n(\n  m_i(v_1,\\ldots,v_i), v_{i+1})=\n (-1)^{i+|v_1|(|v_2|+\\ldots+|v_{i+1}|)}( m_i(v_{i+1},v_1\\ldots,v_{i-1}), v_{i})\n \\] We call an algebra $(V,m)$ a \\emph{cyclic} $A_\\infty$- or\n $L_\\infty$-algebra when it is equipped with such a cyclic inner\n product.\n\\end{definition}\nWe need to define a subspace of $\\Der(\\hat{T}\\Pi V^*)$ of\n\\emph{cyclic} derivations.  To do so, first note that $\\Der(\\hat{T}\n\\Pi V^*) \\cong \\Hom(\\Pi V^*, \\hat{T}\\Pi V^*)$ via the restriction map.\nUsing the inner product, we can identify the latter with\n$(\\Pi V^* \\otimes\n\\hat{T}\\Pi V^*)$.  Next, on any tensor power $(\\Pi V^*)^{\\otimes n}$, we\ncan define the (graded) cyclic permutation operator $\\sigma:\n(\\Pi V^*)^{\\otimes n} \\rightarrow (\\Pi V^*)^{\\otimes n}$. This extends to a\ncontinuous linear automorphism of $(\\Pi V^* \\otimes \\hat{T}\\Pi V^*)$, and\nyields a continuous linear automorphism of $\\Der(\\hat{T} \\Pi\nV^*)$.\n\\begin{definition}\\label{cycderdefn}\n  Let $\\CDer(\\hat{T} \\Pi V^*)$ denote the space of derivations\n  which are invariant under cyclic permutation: we call these\n  \\emph{cyclic} derivations.  Similarly, define $\\CDer^0(\\hat{T} \\Pi V^*)$ as those cyclic derivations with zero constant term (i.e., preserving the\naugmentation $\\hat{T}^{\\geq 1} \\Pi V^*$). Finally, define the spaces $\\CDer(\\hat{S} \\Pi V^*)$ and $\\CDer^0(\\hat{S} \\Pi V^*)$ analogously.\n\\end{definition}\nNote that $\\CDer(\\hat{S} \\Pi V^*)$ can be\nviewed as the subspace of $\\CDer(\\hat{T} \\Pi V^*)$ landing in symmetric\ntensors, and similarly for the version with zero constant term.\n\nGiven $V$ together with an inner product, cyclic curved\n$A_\\infty$-structures are the same as odd derivations $\\xi \\in\n\\CDer(\\hat{T} \\Pi V^*)$ which square to zero.  Similarly,\nuncurved structures correspond to odd square-zero $\\xi \\in\n\\CDer(\\hat{T} \\Pi V^*)$, and the $L_\\infty$-versions are\nobtained by replacing $\\hat{T}$ with $\\hat{S}$.  See, e.g.,\n\\cite{HaLactha} for details.\n\\begin{definition} A gauge equivalence of cyclic $A_\\infty$- or\n  $L_\\infty$- structures $(V,m)$ and $(V,m')$ on a fixed inner product\n  space $V$ is a map $e^\\xi$ where $\\xi \\in \\CDer^0(\\hat{T}\\Pi V^*)$\n  or $\\xi \\in \\CDer^0(\\hat{S} \\Pi V^*)$ satisfies $m' = e^{\\ad \\xi}\n  m$.\n\\end{definition}\n\\begin{remark}\n  In the literature, the term \\emph{symplectic} is sometimes used\n  instead of cyclic, the idea being that an inner product\n  on $V$ is equivalent to a (constant) symplectic structure on $\\Pi\n  V$, so that the cyclic derivations on $\\hat S \\Pi V^*$ or $\\hat T \\Pi V^*$ are the same\n  as formal (possibly noncommutative) symplectic vector fields on $\\Pi V$; a cyclic gauge equivalence could then be interpreted as a (formal, noncommutative) symplectomorphism (preserving the $\\ZZ\/2$-grading).\nNote that it follows from this interpretation that\nthe cyclic derivations of $V$ form a Lie superalgebra, and the cyclic\ngauge equivalences a Lie group.\n\\end{remark}\n\\begin{theorem}\\label{cyccurvtrivthm}\\\n \\begin{enumerate}\n\\item[(a)] If $(V, m)$ is a curved $A_\\infty$-algebra with a\n  cyclic inner product for which the curvature $m_0 = c \\in V_0$ is\n  nonzero, and $c' \\in V$ any even element for which $(c, c') = 1$, then $(V,m)$ is gauge equivalent to the\n  structure $m'$ with $m'_0 =c$, and higher multiplications\n\\begin{gather}\n  m_{2i-1}' = 0, i \\geq 1, \\\\\n  m_{2i}'(x_1, \\ldots, x_{2i}) = \\bigl(\\prod_{j=1}^{2i} (c', x_j) \\bigr) \\cdot (m_{2i}(c,c,\\ldots,c), c) \\cdot c'.\n\\end{gather}\n\\item[(b)] If $(V,m)$ is a curved $L_\\infty$-algebra with a cyclic inner product for which the curvature $m_0 = c \\in V_0$ is nonzero, then $(V,m)$ is gauge equivalent to the\n  structure $m'$ with $m'_0 =c$, and higher multiplications $m'_i = 0$ zero for all $i \\geq 1$.\n\\end{enumerate}\n\\end{theorem}\nOne immediately deduces\n\\begin{corollary}\\\n\\begin{enumerate}\n\\item[(a)]\nTwo curved cyclic $A_\\infty$-algebra structures $(V,m)$ and $(V,m')$ on the same underlying inner product space $V$ with nonzero curvature $m_0, m_0'$\nare gauge equivalent if and only if\n\\begin{equation}\n(m_0, m_0) = (m_0', m_0'), \\quad (m_{2i}(m_0, \\ldots, m_0), m_0) = (m_{2i}'(m_0', \\ldots, m_0'), m_0'), \\forall i \\geq 1.\n\\end{equation}\n\\item[(b)] Two curved cyclic $L_\\infty$-algebra structures $(V,m)$ and\n  $(V,m')$ on the same underlying inner product space $V$ with\n  nonzero curvature $m_0, m_0'$ are gauge equivalent if and only if\n  $(m_0, m_0) = (m_0', m_0')$.\n\\end{enumerate}\n\\end{corollary}\n\\begin{remark}\nAnother, perhaps intuitively more clear, way to understand this result is as follows. Consider the curved cyclic $A_\\infty$-algebra $V$ with curvature $c$ whose underlying inner product space is spanned by a single even vector $c$ with $(c,c)=1$ and higher products $m_{2i}(c,\\ldots,c)=t_ic$; $m_{2i+1}=0$ for $i=0,1,\\ldots$. Here $t_i$ are arbitrary numbers.\n\nConsider also  the curved cyclic $A_\\infty$-algebra $V^\\prime$ whose underlying space is spanned by two even vectors $c$ and $c^\\prime$ with $(c,c^\\prime)=(c^\\prime,c)=1, (c,c)=(c^\\prime,c^\\prime)=0$. The $A_\\infty$-structure is given as\n$m_0 = c$ and, for $i > 0$, $m_{2i}(c,\\ldots, c)=t_ic^\\prime$, where $t_i$ are arbitrary, and all other higher products are zero.\n\nThen any cyclic curved $A_\\infty$-algebra with nonzero curvature is gauge equivalent to the direct sum of either $V$ or $V^\\prime$ with an $A_\\infty$-algebra having zero $A_\\infty$-structure.\n\\end{remark}\n\\begin{proof}[Proof of Theorem \\ref{cyccurvtrivthm}]\n (a) We modify the previous\n  obstruction theory argument. Deformations of the algebra with $m_0=c$ and all\nhigher operations zero are governed by the subcomplex\n\\begin{equation}\\label{bbeq}\nB^\\bullet := \\CDer(\\hat{T}\\Pi V^*, [,c]) \\subset C^\\bullet\n\\end{equation}\nof the one\nconsidered in Theorem \\ref{curvtrivthm}.\n\nIn Lemma \\ref{qil} below, we show that $B^{\\bullet}$ is quasi-isomorphic to the subcomplex\n$B^{\\bullet}_0$ spanned by\n the cocycles\n\\begin{equation}\\label{ceq}\n\\epsilon^{i+1}(x_1, \\ldots, x_i) := \\epsilon(x_1) \\cdots \\epsilon(x_i) c', \\quad \\epsilon(v) := (c', v).\n\\end{equation}\nNote that, when $i$ is odd, $\\epsilon^{i+1} = 0$. Moreover, $B^{\\bullet}_0$\nis an abelian sub-dgla with zero differential.\n\nUsing the lemma, the result follows as in the proof of Theorem\n\\ref{curvtrivthm}. In more detail, the gauge equivalence classes of\nMaurer-Cartan elements of $B^\\bullet[[\\hbar]]$ and\n$B^\\bullet_0[[\\hbar]]$ which are zero modulo $\\hbar$ are\nidentified. Moreover, the formal Maurer-Cartan elements which are\nhomogeneous of the form $\\sum_{i \\geq 1} \\hbar^i m_i$ for $m_i \\in B^i$\nor $B^i_0$ are identified with actual Maurer-Cartan elements $\\sum_{i\n  \\geq 1} m_i$ with zero constant term.  Hence, the gauge equivalence\nclasses of cyclic curved $A_\\infty$- structures with $m_0 = c$ are\nidentified with gauge equivalence classes of Maurer-Cartan elements of\n$B^\\bullet_0$ with zero constant term. Since the latter is abelian\nwith zero differential, all elements are Maurer-Cartan, and define\ndistinct gauge equivalence classes.\n\nWe deduce that all curved $A_\\infty$-structures with $m_0 = c$ are gauge\nequivalent to one with all odd operations equal to zero, and even operations given by some multiple of the operation\n\\begin{equation}\n(x_1, \\ldots, x_{2i}) \\mapsto\n\\bigl(\\prod_{j=1}^{2i} (c', x_j) \\bigr) \\cdot c'.\n\\end{equation}\nIt remains to show that this multiple is $(m_{2i}(c,c,\\ldots,c), c)$.  This follows by taking the component of the original operation in the direct summand $B^i_0 \\subseteq B^i = (B^i_0 \\oplus B^i_+)$, using the definition of $B^i_+$ in\n\\eqref{bipdfn} below.\n\n(b) We can apply the same argument as above, except now with\n$B:=\\CDer(\\hat{S}\\Pi V^*, [,c])$.  The same argument as above applies\nand we deduce the same result, except that this time\n$m_{2i}(c,c,\\ldots,c) = 0$ for all $i \\geq 1$ by skew-symmetry of $m_{2i}$.\n\\end{proof}\n\n\\begin{lemma}\\label{qil} Keeping the assumptions and notation\n  of the theorem, the complex $B^\\bullet$ decomposes as $B^\\bullet =\nB^\\bullet_0 \\oplus B^\\bullet_+$, where $B^\\bullet_+$ is acyclic.  Moreover,\nthe inclusion $B^\\bullet_0 \\into B^\\bullet$ is a quasi-isomorphism of dglas.\n\\end{lemma}\n\\begin{proof}\nThe second statement follows from the first, since $B^\\bullet_0 \\into B^\\bullet$ is a dgla map.\n\nTo prove the first statement, we modify the contracting\n  homotopy $s_i$ from Theorem \\ref{curvtrivthm} to act on $B^\\bullet$. The result will \\emph{not}  be a contracting homotopy, but will instead accomplish the desired goal.\n\nDefine maps $s_i':B^i\\to B^{i+1}$ by\n\\begin{equation}\\label{sipfla1}\ns_i' f = \\sum_{j=0}^{i+1} \\sigma^j (s_i f),\n\\end{equation}\nwhere $\\sigma^j$ is the $j$-th power of the cyclic permutation defined above Definition \\ref{cycderdefn}.\nWe now compute $s_{i-1}' d + d s_i'$.  Assume that $f \\in B^i$.  In this\ncase, we may use the formula\n\\begin{multline}\\label{sipfla2}\ns_i' f(x_1,\\ldots,x_{i+1}) = \\sum_{j=1}^{i+1} (-1)^{|x_1|+\\ldots+|x_{j-1}|} \\epsilon(x_j) f(x_1, \\ldots, x_{j-1}, x_{j+1}, \\ldots, x_{i+1})\n\\\\\n+ (-1)^{|x_1| + \\ldots + |x_{i+1}|} (f(x_1, \\ldots, x_i), x_{i+1}) c'.\n\\end{multline}\nThen, we compute\n\\begin{multline}\\label{sipfla3}\n(s_{i-1}' d + d s_i')f(x_1, \\ldots, x_i) =  (i+1) \\epsilon(c) f(x_1, \\ldots, x_i) \\\\ - \\bigl( \\sum_{j=1}^{i+1} \\epsilon(x_j) f(x_1, \\ldots, x_{j-1}, c, x_{j+1}, \\ldots, x_{i}) + (f(x_1, \\ldots, x_{i}), c)c' \\bigr).\n\\end{multline}\nThus, the operator on the RHS is contractible.  Next, for $k \\leq i+1$, define subspaces\n\\begin{gather}\n  B^i_k \\subset B^i, \\quad B^i_k = B^i \\cap \\bigl(\\CC[S_{i+1}] \\cdot \\bigl( (\\CC \\cdot \\epsilon)^{\\otimes (i+1-k)} \\otimes ((\\CC \\cdot c)^\\perp)^{\\otimes k} \\bigr) \\bigr), \\label{bipdfn0}\\\\\n  B^i_+ := \\sum_{k =1}^{i+1} B^i_k. \\label{bipdfn}\n\\end{gather}\nNote that $B^\\bullet_+$ is a subcomplex, and $B^\\bullet = B^\\bullet_0\n\\oplus B^\\bullet_+$.\n\nWe claim that the RHS of \\eqref{sipfla3} acts as $k \\cdot \\Id$ on\n$B^i_k$ for all $k$. This follows directly.  As a result, $s_{i-1}' d + d\ns_i'$ restricts to zero on $B^\\bullet_0$ and to an automorphism on\n$B^\\bullet_+$.  This proves the lemma. \\qedhere\n\n\\end{proof}\n\\begin{remark}\n  Lemma \\ref{qil} is equivalent to the statement that the cyclic\n  (co)homology of any curved $A_\\infty$-algebra $V$ whose structure\n  maps are all zero, except for $m_0$ (which is not zero), is\n  isomorphic to the cyclic (co)homology of a one-dimensional\n  $A_\\infty$-algebra $V_1$ having the same property. See\n  \\cite{GJainfa, HaLactha} for the notion of cyclic (co)homology of\n  $A_\\infty$-algebras. These (co)homologies could be computed in\n  other ways from the above, for instance, with the help of Connes'\n  exact sequence connecting Hochschild (co)homology with cyclic\n  (co)homology (which appears in \\cite{GJainfa} in the curved\n  setting), since Theorem \\ref{curvtrivthm} shows that the Hochschild\n  (co)homologies of $V$ and $V_1$ are trivial.\n\\end{remark}\n\\section{Characteristic classes of curved algebras}\n\\subsection{The uncurved case}\\label{ccs}\nWe briefly recall Kontsevich's construction of characteristic classes\nof finite-dimensional $A_\\infty$- or $L_\\infty$-algebras with cyclic\ninner products.\nFirst, we recall the definition of certain graph complexes, for which\ncyclic $A_\\infty$- or $L_\\infty$-algebras will produce cycles.\n\\begin{definition}\n  A graph is a tuple $(H,V,E, \\varphi_V, \\varphi_E)$ of sets $H, V, E$\n  of \\emph{half-edges}, \\emph{vertices}, and \\emph{edges}, and\n  surjective maps $\\varphi_V: H \\rightarrow V, \\varphi_E: H\n  \\rightarrow E$ such that the fibers of $\\varphi_E$ all have\n  cardinality two.\n\\end{definition}\nGiven $\\Gamma = (H, V, E, \\varphi_V, \\varphi_E)$, we will also write $H_\\Gamma = H, V_\\Gamma = V, E_\\Gamma = E, \\varphi_V^\\Gamma =\n\\varphi_V$, and $\\varphi_E^\\Gamma = \\varphi_E$.\n\\begin{definition}\n  A ribbon graph is a graph together with a cyclic ordering on each\n  fiber $\\varphi_V^{-1}(t)$.\n\\end{definition}\nIntuitively, one may think of the edges of ribbon graphs as slightly\nfattened, which explains the cyclic ordering at vertices.\n\nKontsevich's graph complexes have a basis of graphs of a certain type,\nwith differential taking a graph to the sum over all edges of the\ncontracted graph obtained by shrinking that edge to a point, together\nwith a sign.  To make this precise requires the notion of\n\\emph{orientation}:\n\\begin{definition}\n  An orientation on a graph is a choice of ordering of all the\n  half-edges, ordering of all the vertices, and a sign $\\pm 1$, modulo\n  the relation that applying a transposition to the ordering of either\n  the half-edges or the vertices is the same as changing the sign.\n\\end{definition}\n\\begin{definition}\n  An oriented graph is a graph equipped with an orientation. An\n  isomorphism of oriented graphs is an isomorphism of graphs which\n  preserves orientation.  Similarly, the same definition applies replacing\n  ``graph'' with ``ribbon graph.''\n\\end{definition}\nNext, given a graph $\\Gamma = (H,V,E, \\varphi_V, \\varphi_E)$ and an\nedge $e \\in E$ with endpoints $v_1, v_2 \\in V$ meeting halves $h_1,\nh_2 \\in H$, one defines the contracted graph $d_e(\\Gamma) = (H, V \/\n\\{v_1 = v_2\\}, E, \\varphi_V', \\varphi_E)$ by identifying the endpoints\n$v_1$ and $v_2$. If $\\Gamma$ is moreover a ribbon graph, with the\ncyclically ordered sets $\\varphi_V^{-1}(v_1) = (a_1, a_2, \\ldots, a_i\n= h_1)$ and $\\varphi_V^{-1}(v_2) = (b_1, b_2, \\ldots, b_j = h_2)$,\nthen the cyclic ordering of the half edges at the new vertex $v = v_1\n= v_2$ is defined as $(a_1, a_2, \\ldots, a_{i-1}, b_1, b_2,\n\\ldots, b_{j-1})$. Finally, if $\\Gamma$ is equipped with an\norientation, where the half-edges are ordered as $h_1, h_2, p_1,\n\\ldots, p_m$ and with vertices ordered by $v_1, v_2, w_1, \\ldots,\nw_{k}$, the new orientation is given by the ordering $p_1, \\ldots,\np_m$ and $v, w_1, \\ldots, w_{k}$ of vertices, without changing the\nsign.\n\nConsider the graded vector space with basis the isomorphism classes of\noriented graphs \\emph{whose vertices have valence $\\geq 2$}, modulo\nthe relation that a graph is negative its opposite orientation. The\ngrading is given by the number of vertices.  Let $\\mathcal{G}$ be the\ncompletion of this graded vector space with respect to the number of\nedges (so \\emph{not} with respect to the defining grading on the\nvector space, which is by number of vertices).  Similarly, define\n$\\mathcal{G}_r$ using ribbon graphs rather than graphs.  In other\nwords, these are the spaces of possibly infinite linear combinations\nof isomorphism classes of oriented graphs which are not isomorphic to\nthe graph obtained by reversing the orientation.\n\nThen, it is a result of \\cite{Kncsg} that\n\\begin{equation}\nd(\\Gamma) := \\sum_{e \\in E_\\Gamma} d_e(\\Gamma)\n\\end{equation}\ndefines a differential on $\\mathcal{G}$ and $\\mathcal{G}_r$.\n\\begin{definition}\n  Kontsevich's graph complex is defined as $(\\mathcal{G}, d)$, and his\n  ribbon graph complex is defined as $(\\mathcal{G}_r, d)$.\n\\end{definition}\n\\begin{remark}\n  We could alternatively have used the uncompleted graph complex\n  above; however, the completed version is the one which naturally\n  contains characteristic classes of $L_\\infty$- or\n  $A_\\infty$-algebras.  In particular, taking homology commutes with\n  taking completion, for the following well-known reason: We can write the\n  uncompleted graph complex as a direct sum of the subcomplexes of\n  graphs of a fixed genus (i.e., first Betti number of the graph as a\n  topological space).  For each fixed genus, the completion with\n  respect to number of edges is the same as the completion with\n  respect to the grading, i.e., the number of vertices, since there\n  are only finitely many graphs with a fixed genus and number of\n  vertices.  Hence, the completed graph complex is the same as the\n  direct product of the completions of these subcomplexes with respect\n  to their usual grading.\n\\end{remark}\nFinally, given a cyclic $A_\\infty$-algebra $V$, one constructs an\nelement of $\\mathcal{G}_r$, given by a sum\n\\begin{equation}\n[V]:=\\sum_{\\Gamma} \\frac{1}{|\\Aut(\\Gamma)|} c_\\Gamma(V) \\cdot \\Gamma,\n\\end{equation}\nwhere we sum over isomorphism classes of ribbon graphs $\\Gamma$ (with\ngroup of automorphisms $\\Aut(\\Gamma)$), and $c_\\Gamma$ is given as\nfollows.  Equip $\\Gamma$ with an orientation. We will define\n$c_\\Gamma$ so that the opposite orientation would produce $-c_\\Gamma$.\nNamely, $c_\\Gamma$ is given by contracting the multiplications $m_i$\nof $V$ with the pairings $( -, -)$ according to the graph.\nIn more detail, consider\n\\begin{equation} \\label{multmaps} \\prod_{i=1}^n ( m(-), -\n  ): \\bigotimes_{i=1}^n V^{\\otimes |\\phi_V^{-1}(v_i)|}\n  \\rightarrow \\CC.\n\\end{equation}\nLet $h_1, \\ldots, h_{|H|}$ be the ordering of the half-edges and $v_1,\n\\ldots, v_{|V|}$ the ordering of the vertices defined by the\norientation, and assume that the sign is $1$. Let us pick\n\\emph{ciliations} of each of the vertices $v_1, \\ldots, v_{|V|}$,\nwhich means a linear ordering of the half-edges $\\varphi^{-1}_V(v_i)$\nmeeting each vertex $v_i$, compatible with the cyclic ordering given\nby the ribbon structure.  Up to changing the sign, let us assume that\nthe ordering of the half-edges is $\\varphi^{-1}_V(v_1),\n\\varphi^{-1}_V(v_2), \\ldots, \\varphi^{-1}_V(v_{|V|})$.  Let $f \\in V\n\\otimes V$ be the inverse to the pairing $(-,-): V \\otimes V\n\\rightarrow \\CC$. Finally, pick an arbitrary ordering of the edges\n$e_1, \\ldots, e_{|E|}$. Then, one applies \\eqref{multmaps} to the\nelement obtained by applying the signed permutation of components of\n$f^{\\otimes |E|} \\in V^{\\otimes |H|}$ which rearranges the half-edges\n$\\varphi_E^{-1}(e_1), \\ldots, \\varphi_E^{-1}(e_{|E|})$ into the\nordering $h_1, \\ldots, h_{|H|}$. One can check that the result does\nnot depend on the choices of orderings (but only depends on the\norientation of $\\Gamma$ by a sign, as mentioned above), and we let\n$c_\\Gamma(V)$ to be the result of this computation.\n\nIn a similar manner, one constructs from any cyclic $L_\\infty$-algebra $V$\nan element $[V]$ of $\\mathcal{G}$.  Then, the following result\nis due to Kontsevich:\n\\begin{proposition}\\cite{KFd} If $V$ is a cyclic $A_\\infty$- or $L_\\infty$-algebra then $[V]$ is a cycle on\n  $(\\mathcal{G}, d)$ or $(\\mathcal{G}_r, d)$ respectively.\n\\end{proposition}\n  A direct proof of the proposition in the $A_\\infty$-case\n  is contained, e.g., in \\cite{Igu}. More conceptually, one can view a\n  cyclic $A_\\infty$-algebra $V$ as an algebra over ${\\mathsf\n    F}\\underline{\\mathscr{A}\\textit{ss}}^{1}$, the Feynman transform of the $\\Det$-twisted naive\n  modular closure of the cyclic operad $\\mathscr{A}\\textit{ss}$; see \\cite{GKmo} and\n  \\cite{ChLadft} concerning these notions. Therefore, we obtain a map of\n  modular operads ${\\mathsf F}\\underline{\\mathscr{A}\\textit{ss}}^1\\to\n  \\mathscr{E}(V)$, where $\\mathscr{E}(V)$ is the modular endomorphism\n  operad of $V$. The map between the vacuum parts of the corresponding\n  operads\n\\[{\\mathsf F}\\underline{\\mathscr{A}\\textit{ss}}^1((0))\\to\n\\mathscr{E}(V)((0))\\cong \\CC\n\\]\nis precisely the characteristic class described above, and it follows that it does indeed give a cycle. One can prove the proposition similarly\nin the $L_\\infty$ case.\n\n\n\\subsection{Curved characteristic classes}\nThe preceding results have a natural generalization to the case of\ncurved algebras.  We need to remove the valence $\\geq 2$ condition,\nhowever, and study the graph complexes $(\\widetilde{\\mathcal{G}}, d),\n(\\widetilde{\\mathcal{G}_r}, d)$ of formal linear combinations of\ngraphs and ribbon graphs where vertices are allowed to have valence\n$1$ (we will not allow valence-zero vertices, since they don't add\nanything of value). We can consider this to be the complex of ``graphs\nwith stubs,'' where a stub is an edge incident to a valence-one\nvertex. Note that the conventional graph complex $(\\mathcal{G}, d)$ is\na subcomplex of $(\\widetilde{\\mathcal{G}}, d)$ and similarly in the\nribbon case. Then, everything else goes through exactly as above, and\nwe obtain the following result.\n\\begin{proposition} Any curved cyclic $A_\\infty$- or\n  $L_\\infty$-algebra $V$ induces a cycle $[V]$ on\n  $(\\widetilde{\\mathcal{G}_r}, d)$ or on $(\\widetilde{\\mathcal{G}},\n  d)$ respectively.\n\\end{proposition}\nThe curved graph homology could be expressed in terms of certain\nGelfand-Fuks type homology. Namely, let $W$ be a graded symplectic\nvector space and consider the graded formal Lie algebra $\\g(W)$ of formal\nsymplectic vector fields on $W$. Similarly consider the graded formal Lie\nalgebra $\\g_r(W)$ consisting of formal \\emph{noncommutative}\nsymplectic vector fields on $W$, i.e. the Lie algebra\n$\\CDer(\\Pi W)$. Taking the stable limit as the dimension\nof the even or odd part of $W$ goes to infinity, we arrive at\nfollowing result (which\nis a straightforward adaptation of \\cite[Theorem 1.1]{Kncsg}, up to technical\nproblems stemming from the lack of complete reducibility of finite-dimensional representations of simple Lie superalgebras).  Let\n$\\CC^{m}$ denote the even space of dimension $m$ and $\\Pi \\CC^m$ the odd\nspace of dimension $m$.  We equip $\\CC^{2m}$ with the standard symplectic\nform, and $\\Pi \\CC^m$ with the standard odd symplectic (i.e., orthogonal) form.\n\\begin{theorem}\\label{GFgraph}\n  Let $W$ be a fixed inner product space.\n  There are isomorphisms\n\\begin{equation}\n\\operatorname{H}_\\bullet\n  (\\widetilde{\\mathcal{G}_r}) \\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g_r(W \\oplus \\CC^{2m})), \\quad \\operatorname{H}_\\bullet\n  (\\widetilde{\\mathcal{G}_r}) \\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g_r(W \\oplus \\Pi \\CC^m))\n\\end{equation}\n between the stable Chevalley-Eilenberg\n  homology of the Lie algebra $\\g_r$ and of the\n  corresponding version of the curved graph complex.\nSimilarly, we have isomorphisms\n\\begin{equation}\nH_\\bullet(\\widetilde{\\mathcal{G}})\\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g(W \\oplus \\CC^{2m})), \\quad H_\\bullet(\\widetilde{\\mathcal{G}})\\cong \\lim_{m \\rightarrow \\infty}\n   \\HCE_\\bullet(\\g(W \\oplus \\Pi \\CC^m)).\n\\end{equation}\n\\end{theorem}\n\\begin{remark}\nIt might be possible to further generalize this result\nto (certain) cases where both the even and the odd part have dimension going to\ninfinity, but that creates additional technical difficulties that we\nprefer to avoid (as we do not need such generality). In any case, they\nare the same difficulties that arise in the original uncurved setting\nof \\cite{Kncsg} (note that in \\emph{op.~cit.} only the even case is\nconsidered).\n\\end{remark}\n\\begin{proof}\n  The corresponding result for the uncurved graph complex and vector\n  fields vanishing at the origin was established by Kontsevich\n  \\cite{Kncsg} and his proof carries over to the present context, up\n  to some technical difficulties created by the fact that we are in\n  the super context, where $W_0$ and $W_1$ can both be nonzero. Define\n  a nonnegative grading on $\\CE^\\bullet(\\g_r(W))$ called\n  \\emph{weight}, which is the sum of the homological grading and the\n  degree of polynomial coefficients of the vector fields (this will\n  correspond, on graphs, to the number of half-edges).\n\n  The main tool is that the inclusion of subcomplexes\n\\begin{equation} \\label{invincleq}\n\\CE^\\bullet(\\g_r(W \\oplus \\CC^{2m}))^{\\mathfrak{osp}(W \\oplus\n    \\CC^{2m})} \\into \\CE^\\bullet(\\g_r(W \\oplus \\CC^{2m}))\n\\end{equation}\nis asymptotically a quasi-isomorphism, and similarly replacing\n$\\CC^{2m}$ with $\\Pi \\CC^m$.  By this, we mean that in weights $\\leq\nN$, there exists $M$ such that, if $m \\geq M$, then the inclusion is\nan isomorphism on homology in weights $\\leq N$.\n\nWe carry out the argument with $\\CC^{2m}$; replacing this by $\\Pi\n\\CC^m$ will not affect anything. Let $U_m := W \\oplus \\CC^{2m}$.  First note that $\\mathfrak{osp}(U_m)\n\\subseteq \\g_r(U_m)$ is the subspace of\nlinear vector fields and hence acts trivially on the cohomology of\n$\\CE^\\bullet(\\g_r(U_m))$.  Hence, the statement would\nfollow if it were true that $\\mathfrak{osp}(U_m)$ acted\ncompletely reducibly on $\\CE^\\bullet(\\g_r(U_m))$.  This\nis not, in general, true (when $W$ is not purely even); however, it\nfollows from Lemma\n\\ref{superredlem} below that,\nfor each $N \\geq 0$, there exists $M$ so that $m \\geq M$\nimplies that the weight $\\leq N$ part of $\\CE^\\bullet(\\g_r(U_m))$ is completely reducible as an $\\mathfrak{osp}(U_m)$-representation. This is sufficient\nto deduce that \\eqref{invincleq} is a quasi-isomorphism in weights $\\leq N$.\n\n\nTo complete the proof, it remains to relate the invariant subcomplex\nto graphs.  This part of the argument is nearly identical to\n\\emph{op.~cit.}, so we will be brief.  Associated to each graph is an\n$\\mathfrak{osp}(W)$-invariant element of $\\CE_\\bullet(\\g_r(W))$, as\ndescribed in the previous subsection. However, the resulting map\n$\\mathcal{G}_r \\to \\CE_\\bullet(\\g_r(W))$ does not linearly extend to a\nmap of complexes. Instead, if we attach to the dual of a graph in\n$\\mathcal{G}_r^*$ a canonical element of\n$\\CE^\\bullet(\\g_r(W))^{\\mathfrak{osp}(W)}$, as explained in\n\\emph{op.~cit.}, one obtains a canonical map of complexes\n$(\\mathcal{G}_r^*, d^*) \\to \\CE^\\bullet(\\g_r(W))^{\\mathfrak{osp}(W)}$.\nIn more detail, under the identification $\\mathcal{G}^*_r \\cong\n\\mathcal{G}_r$ using the basis of ribbon graphs (with fixed\norientations), this map sends each oriented ribbon graph $\\Gamma$ to\nthe corresponding functional which contracts elements of\n$\\CE_{|V_\\Gamma|}(\\g_r(W))$ using the symplectic form $W \\otimes\nW\\rightarrow \\CC$, similarly to the construction of \\S \\ref{ccs}.  That is,\nwe view $\\CE_{|V_\\Gamma|}(\\g_r(W)) \\subset \\Hom(W, \\hat{T}(W))$ as a subspace\nof $\\hat{T}(W)$ using the symplectic form, and contract with a\npermutation of $\\omega^{\\otimes |E_\\Gamma|}$, where $\\omega \\in W\n\\otimes W$ is the inverse to the symplectic form, so that the copy of\n$\\omega$ corresponding to each edge contracts the corresponding pair\nof half-edges.  For details on how to prove this indeed yields a map\nof complexes, see, e.g., \\cite[Theorem 4.10]{HaLacc}, and also\n\\cite{HamsaKt}.\n\nBy the fundamental theorems of invariant theory \\cite{Wftit} (see\nalso, e.g., \\cite{Howeftit}) and the super generalization found in\n\\cite{Seracit}, following the reasoning in the proof of Lemma\n\\ref{superredlem} below, it follows that this map of complexes is\nasymptotically an isomorphism: for every fixed $N \\geq 0$, there\nexists $M \\geq 0$ so that, when $\\dim W \\geq M$, this map is an\nisomorphism if we restrict to graphs with at most $N$ edges, and hence\nelements of $\\CE^\\bullet(\\g_r(W))^{\\mathfrak{osp}(W)}$ of weight $\\leq N$.\n\nThis construction goes through completely analogously for commutative\ngraphs and the Lie algebra $\\g$.\n\\end{proof}\nAs in the proof above, let $U_m := W \\oplus \\CC^{2m}$.  Also define $U_m' := W \\oplus \\Pi \\CC^m$.\n\\begin{lemma}\\label{superredlem}\nFor all $N \\geq 0$, there exists $M \\geq 0$ such that, when $m \\geq M$, $U_m^{\\otimes N}$ and $(U_m')^{\\otimes N}$ are completely reducible $\\mathfrak{osp}(U_m)$- and $\\mathfrak{osp}(U_m')$-modules, respectively.\n\\end{lemma}\n\\begin{proof}\n  We carry out the argument for $U_m$; the same applies for $U_m'$.\n  Suppose that one has subrepresentations $0 \\neq \\tau_1 \\subsetneq\n  \\tau_2 \\subseteq U_m^{\\otimes N}$.  We wish to show that the\n  inclusion $\\tau_1 \\into \\tau_2$ splits.  By adjunction, we\n  equivalently need to show that the composition $\\CC \\into \\tau_1^*\n  \\otimes \\tau_1 \\into \\tau_1^* \\otimes \\tau_2$ splits.  Since\n  $U_m^{\\otimes N}$ is self-dual using the pairing, $\\tau_1^* \\otimes\n  \\tau_2 \\subseteq U_m^{\\otimes 2N}$. This means that it suffices to\n  show that the inclusion of the invariant part, $(U_m^{\\otimes\n    2N})^{\\mathfrak{osp}(U_m)} \\into U_m^{\\otimes 2N}$, splits. Let us\n  replace $2N$ by $N$ for convenience.\n\n  This last fact follows using the fundamental theorems of invariant\n  theory.  Namely, by \\cite{Seracit}, all of the $\\mathfrak{osp}(U_m)$-invariants in the tensor algebra $T(U_m)$ are tensor products of the\n  pairing on $U_m$ with an invariant related to the determinant, whose\n  tensor weight (i.e., number of tensor components) is a function of\n  $m$ that goes to infinity when $m$ does.  Hence, for large enough\n  $m$, all of the invariants in weight $U_m^{\\otimes \\leq N}$ are\n  spanned by tensor products of the pairing on $W$. By the classical\n  second fundamental theorem of invariant theory \\cite{Wftit} (see\n  also \\cite{Howeftit}), for large enough $m$, there are no nontrivial\n  relations between these invariants.  The same argument\n  applied to $U_m^*$ shows that the coinvariants have the same\n  description. Moreover, for large enough $m$, the invariants of\n  $U_m^{\\otimes \\leq N}$ and invariants of $(U_m^*)^{\\otimes \\leq N}$\n  have a perfect pairing.  This implies that the composition\n  $(U_m^{\\otimes \\leq N})^{\\mathfrak{osp}(U_m)} \\into U_m^{\\otimes N}\n  \\onto (U_m^{\\otimes N})_{\\mathfrak{osp}(U_m)}$ is an isomorphism for\n  large enough $m$.\n\\end{proof}\n\nUnlike the case for the usual graph homology, it is possible to give a\ncomplete calculation of the homology of the curved graph complex and\ncurved ribbon graph complex:\n\\begin{theorem}\n\\label{acythm}\\\n\\begin{enumerate}\n\\item[(a)]\nThe homology of the complex  $(\\widetilde{\\mathcal{G}_r}, d)$ is identified with the space of\n  formal linear combinations of graphs all of whose connected\n  components are graphs whose vertices have valence one with the exception of at most a\n  single vertex, which has odd valence.\n\\item[(b)]\nThe homology of the complex $(\\widetilde{\\mathcal{G}}, d)$ is identified with the space of\n  formal linear combinations of graphs each of whose connected\n  components is the connected graph with one edge and two vertices.\n\\end{enumerate}\n\\end{theorem}\nWe can reinterpret this theorem as follows. First, it is enough to\ncompute the homology of the subcomplex of connected nonempty graphs,\n$\\widetilde{\\mathcal{G}_{r,c}} \\subset \\widetilde{\\mathcal{G}_r}$ and\n$\\widetilde{\\mathcal{G}_c} \\subset \\widetilde{\\mathcal{G}}$.  We call\nthese complexes the \\emph{connected} ribbon graph complex and the\nconnected graph complex.  Note that $\\widetilde{\\mathcal{G}_r} \\cong\n\\Sym \\widetilde{\\mathcal{G}_{r,c}}$ and $\\widetilde{\\mathcal{G}} \\cong\n\\Sym \\widetilde{\\mathcal{G}_c}$.  Then, the desired result is that the\nhomology of the former is identified with linear combinations of\nstar-shaped graphs with an odd number of edges and at most a single\nvertex of valence $\\geq 2$, and the latter is one-dimensional and\nspanned by the connected graph with two vertices and a single edge.\n\nIn other words, the theorem states that\n$\\widetilde{\\mathcal{G}_{r,c}}$ and $\\widetilde{\\mathcal{G}_c}$ are\nquasi-isomorphic to the subcomplexes spanned by connected graphs with\nat most a single vertex of valence $\\geq 2$.  These complexes are\nidentical with the deformation complexes of Theorem \\ref{cyccurvtrivthm}\nfor the one-dimensional curved $A_\\infty$ and $L_\\infty$ algebras with\ncurvature $c$ satisfying $(c,c) = 1$, and all higher operations zero.\nThat is, they are the graded vector spaces of noncommutative\nsymplectic vector fields and ordinary symplectic vector fields on the\nodd one-dimensional symplectic vector space $\\CC \\cdot c$, equipped\nwith the differential $\\ad(\\frac{\\partial}{\\partial c})$ (which turns out to be zero).\n\\begin{remark}\n  The stable homology of the Lie algebra of symplectic vector fields\n  has been computed by Guillemin and Shnider in \\cite{GSssrc}, and thus,\n  part (b) of the above theorem could be deduced from their\n  calculation, taking into account Theorem \\ref{GFgraph}. However we\n  include this result for completeness, and because the argument we\n  use in part (a) essentially extends without change to this case.\n\\end{remark}\n\n\\begin{proof}[Proof of Theorem \\ref{acythm}]\n    Call a graph a \\emph{line segment} if it is topologically a line\n  segment, i.e., it is connected, and either it is a single vertex\nwith no edges, or all of its vertices have valence\n  two except for two vertices, which have valence one.  Given a\n  connected graph $\\Gamma$, let us call a vertex $v \\in V_\\Gamma$\n  \\emph{exterior} if it either has valence at most one, or one of the\n  connected components of $\\Gamma \\setminus v$ is a line segment. Call\n  all other vertices \\emph{interior}.\n\n  We will make use of a filtration on the connected (ribbon) graph\n  complex given by the number of interior vertices in the graph.  We\n  call this the \\emph{interior vertex filtration}.  The associated\n  graded complex is identified, as a vector space, with the (ribbon)\n  graph complex, and with the differential which is almost the same,\n  but only contracts edges incident to at least one exterior\n  vertex. Moreover, the associated graded complexes $\\gr\n  \\widetilde{\\mathcal{G}_{r,c}}$ and $\\gr \\widetilde{\\mathcal{G}_c}$\n  are graded not only by number of interior vertices, but by the\n  subgraph $\\Gamma_0 \\subseteq \\Gamma$ obtained by restricting to\n  interior vertices and edges which are incident only to interior\n  vertices (allowing here also the empty graph and the graph with a\n  single vertex and no edges).  It is clear that $\\Gamma_0$ is\n  connected (possibly empty).  Let $\\gr_{\\Gamma_0}\n  \\widetilde{\\mathcal{G}_{r,c}}$ or $\\gr_{\\Gamma_0}\n  \\widetilde{\\mathcal{G}_c}$ denote the resulting subcomplex graded by\n  $\\Gamma_0$.\n\n  (a) We claim that, when $\\Gamma_0$ contains at least one edge, then\n  $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$ is acyclic.\n\n  Assume that $\\Gamma_0$ contains an edge.  Pick a half-edge $h$ of\n  $\\Gamma_0$, and let $v$ be the incident vertex.  We construct from\n  $v$ and $h$ a contracting homotopy $s$ on $\\gr_{\\Gamma_0}\n  \\widetilde{\\mathcal{G}_{r,c}}$.  Namely, for every oriented graph\n  $\\Gamma$ whose subgraph on internal vertices is $\\Gamma_0$, let\n  $s \\Gamma$ be the graph obtained from $\\Gamma$ by adding a new\n  univalent vertex together with an edge connecting it to $v$. The\n  resulting new half-edge incident to $v$ is, in the cyclic ordering\n  at $v$, one half-edge counterclockwise away from $h$.  Pick the\n  orientation on the new graph $s \\Gamma$ so that $\\Gamma$ is one of\n  the summands of $d(s \\Gamma)$.\n\n  We claim that $sd + ds = \\Id$.  It suffices to show that, for every\n  oriented graph $\\Gamma$ as above, $sd(\\Gamma) + ds(\\Gamma)=\\Gamma$.\n  In turn, it suffices to show that, for every edge $e \\in E_\\Gamma$,\n  $s d_e \\Gamma = - d_e (s \\Gamma)$.  This follows immediately from\n  our definition of $s$.\n\n\n\nHence, we deduce that $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$\nis acyclic, as claimed.\n\nNext, we compute $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$ where\n$\\Gamma_0$ is either empty or is the graph with a single vertex and no edges.\nCall the first graph ``$\\emptyset$'' and the second one ``$\\pt$.''\n\nFirst, $\\gr_{\\pt} \\widetilde{\\mathcal{G}_{r,c}}$ consists of star-shaped\ngraphs with a single vertex of valence $\\geq 3$, with the usual graph\ndifferential except that we do not allow to contract an edge that\nwould result in a graph without a vertex of valence $\\geq 3$, i.e., an\nedge which is incident to a vertex of valence $1$ and a vertex of\nvalence $3$. Consider the filtration by the valence of\nthe interior vertex. The associated graded complexes have homology\nwhich is one-dimensional, concentrated in the part where all vertices\nbut one have valence $1$.  Moreover, such graphs are actually zero\nwhen the node has even valence (because a cyclic symmetry can reverse\nthe orientation).  Hence, the associated spectral sequence computing\n$H_*(\\gr_{\\pt} \\widetilde{\\mathcal{G}_{r,c}})$ collapses at the first\npage, and the resulting homology is spanned by the star-shaped graphs\nwith a single vertex of odd valence $\\geq 3$ and with all other\nvertices of valence $1$.\n\nNext, $\\gr_\\emptyset \\widetilde{\\mathcal{G}_{r,c}}$ is the subcomplex of line\nsegments, whose homology is one-dimensional and spanned by the line segment\nwith two vertices.\n\nWe deduce that the first page of the spectral sequence of the interior\nvertex filtration on $\\widetilde{\\mathcal{G}_{r,c}}$ is concentrated\nin the part with $\\leq 1$ interior vertices.  The part in degree $1$\nis the span of star-shaped graphs with one vertex of odd valence $\\geq\n3$ and the other vertices of valence $1$, and the part in degree $0$\nis the span of the line segment with two vertices.  Since all of these\ngraphs have only odd numbers of edges, it is clear that the spectral\nsequence collapses at the first page, and the graphs above span the\nhomology of $\\widetilde{\\mathcal{G}_{r,c}}$, as desired.\n\n(b) The same argument as above applies in this case, except that,\nsince our graphs no longer have cyclic orderings of half edges at\nvertices, we modify the construction of the contracting homotopy $s$\n(used to show that $\\gr_{\\Gamma_0} \\widetilde{\\mathcal{G}_{r,c}}$ is\nacyclic when $\\Gamma_0$ contains an edge) accordingly. Namely, we\nremove the condition that the new edge be next to $h$ in the\ncounterclockwise cyclic ordering of half edges at the vertex $v$.\nEverything else goes through without change, except that now the\nstar-shaped graphs with a single vertex of valence $\\geq 1$ are all\nzero except for the one with only two vertices.  This implies the\ndesired result. \\qedhere\n\n\\end{proof}\nWe see that the complex of simply-connected graphs (which splits off\nas a direct summand) carries all of the homology of the complexes\n$(\\widetilde{\\mathcal{G}_r}, d)$ and $(\\widetilde{\\mathcal{G}}, d)$,\nand we obtain the following result.\n\\begin{corollary}\\label{curveduncurved}\n  The inclusions of complexes $(\\mathcal{G}_r,\n  d)\\subset(\\widetilde{\\mathcal{G}_r}, d)$ and $(\\mathcal{G},\n  d)\\subset(\\widetilde{\\mathcal{G}}, d)$ induce the zero maps on\n  homology.\n\\end{corollary}\n\\begin{remark}\n  It is natural to ask whether the nontrivial homology classes in the\n  curved graph complexes $\\widetilde{\\mathcal{G}_r}$ and\n  $\\widetilde{\\mathcal{G}}$ are detected by curved $A_\\infty$- and\n  $L_\\infty$-algebras. The answer is yes; in fact, they are detected by\n  one-dimensional algebras.  For the $A_\\infty$ case, let $\\Gamma(i)$\n  be the star-shaped graph whose central vertex has valence $2i+1$ and\n  $V(i)$ be the one-dimensional cyclic curved $A_\\infty$-algebra\n  spanned by an even vector $c$ such that $( c,c)=1$, $m_0=c$,\n  $m_{2i}(c,\\ldots,c)=c$, and all other $A_\\infty$-products are\n  zero. Then it is easy to see that the cycle\n  $[V(i)]\\in\\widetilde{\\mathcal{G}_r}$ is homologous to\n  $\\pm\\frac{1}{2i+1}\\Gamma(i)$ (the sign depends on the choice of an\n  orientation on $\\Gamma(i)$).  Next, let $\\Gamma(0)$ be the connected\n  graph with one edge and two vertices.  Then, we may let $V(0)$ be\n  the one-dimensional cyclic curved $A_\\infty$-algebra spanned by an\n  even vector $c$ such that $m_0=c$, $( c, c ) = 1$, and all the\n  higher $A_\\infty$-products are zero.  Then,\n  $[V(0)]\\in\\widetilde{\\mathcal{G}_r}$ is again homologous to\n  $\\pm\\frac{1}{2}\\Gamma(0)$.  The same statement holds for the\n  $L_\\infty$-setting, if we now let $\\Gamma(0)$ be an ordinary (not\n  ribbon) connected graph also with two vertices and one edge, and\n  consider the one-dimensional curved cyclic $L_\\infty$-algebra,\n  $V(0)$, with curvature $c$ satisfying $( c, c ) = 1$ and all higher\n  operations equal to zero: $[V(0)]\\in\\widetilde{\\mathcal{G}_r}$ is\n  homologous to $\\pm\\frac{1}{2}\\Gamma(0)$.\n\\end{remark}\n\\section{Homology of Lie algebras of vector fields and stability maps}\\label{unstabsec}\nIn order to refine Theorem \\ref{acythm}, we recall first a more\ngeneral way to view the construction of characteristic classes.\nLet $\\mathfrak{g}$ be a formal dgla.\nConsider a Maurer-Cartan element of $\\mathfrak{g}$, i.e., an element\n$x \\in \\Pi\\mathfrak{g}$ satisfying $dx + \\frac{1}{2} [x,x] = 0$.\nThen, $e^x = \\sum_{i \\geq 0}x^{i}\/i!$ defines a cycle in\n $\\CE_\\bullet(\\mathfrak{g})$, and hence a homology class of even\ntotal degree.  In the situation where the differential $d$ on\n$\\mathfrak{g}$ is zero, using the homological degree $|S^{i}\n\\Pi\\mathfrak{g}| = i$, each element $x^{ i}$ itself is a cycle, and we\nobtain \\emph{unstable} characteristic classes $[x^{ i}] \\in\n\\CE_i(\\mathfrak{g})$.\n\nMoreover, if $\\mathfrak{h} \\subseteq \\mathfrak g_0$ is a pronilpotent\nLie subalgebra of the even part of $\\mathfrak{g}$, then there is\ndefined a notion of \\emph{gauge equivalence} of Maurer-Cartan elements\ncorresponding to the adjoint action of the Lie group of the Lie\nalgebra $\\mathfrak{h}$; then it follows that if two Maurer-Cartan\nelements are gauge equivalent by a gauge in $\\mathfrak{h}$, then their\ncharacteristic classes are homologous. (One can more generally take\n$\\mathfrak{h} \\subseteq \\mathfrak{g}_0$ to be a Lie subalgebra which\nis the Lie algebra of a pro-Lie group).  This statement, as well as a\nmore detailed treatment of characteristic classes, can be found in,\ne.g., \\cite{Hamccms}.\n\nReturning to the situation of a (cyclic or curved) $L_\\infty$- or\n$A_\\infty$-algebra $V$, the corresponding element of the Lie algebra\n$\\Der^0(\\hat{S} (\\Pi V^*)), \\Der(\\hat{T}(\\Pi V^*))$, etc., defines a\ncanonical homology class.\n\nThe relationship to the aforementioned characteristic classes is\nKontsevich's result that the limit as the dimension of $V$ goes to\n$\\infty$ of the Lie homology of $\\CDer^0(\\hat{S} (\\Pi V^*))$ or\n$\\CDer^0(\\hat{T}(\\Pi V^*))$ is the completion (by number of edges) of\nthe homology of the graph complexes $(\\mathcal{G}, d)$. In the curved\nsituation the relevant result is Theorem \\ref{GFgraph}.\n\nFurthermore, given $V$, and any inner product space $W$,\n we can form the trivial extension $V \\oplus\nW$ where all multiplications with the second factor are zero. This\ninduces maps\n\\begin{gather} \\label{vplieeqn}\n\\varphi_{V,W}: \\CE_\\bullet(\\CDer(\\hat{S}(\\Pi V^*)))\n\\rightarrow \\CE_\\bullet(\\CDer(\\hat{S}(\\Pi (V \\oplus W)^*))), \\\\\n\\label{vpasseqn}\n\\varphi_{V,W}: \\CE_\\bullet(\\CDer(\\hat{T}(\\Pi V^*)))\n\\rightarrow \\CE_\\bullet(\\CDer(\\hat{T}(\\Pi (V \\oplus W)^*))),\n\\end{gather}\nand similarly the restrictions $\\varphi_{V,W}^0 :=\n\\varphi_{V,W}|_{\\CE_\\bullet(\\CDer^0(\\hat{S}(\\Pi V^*)))}$ or\n$\\varphi_{V,W}^0 := \\varphi_{V,W}|_{\\CE_\\bullet(\\CDer^0(\\hat{T}(\\Pi\n  V^*)))}$.  It is well known (and easy to check) that Kontsevich's\nconstruction (\\S \\ref{ccs}) is obtained from the above construction in\nthe limit: the image of the unstable characteristic cycle $\\sum_i\n\\xi^{i}\/i! \\in \\CE_\\bullet(\\CDer(\\hat{S}(\\Pi V^*)))$ under\n$\\varphi_{V,W}^0$ as $\\dim W_0 \\rightarrow \\infty$ (for fixed $W_1$)\nidentifies with the characteristic cycle on $(\\mathcal{G}, d)$ given\nin $\\S \\ref{ccs}$; similarly if we fix $W_0$ and let $\\dim W_1 \\to\n\\infty$.  In the curved setting, by Theorem \\ref{acythm}, for each\nfixed degree $i$, if we fix $W_1$, then for large enough $\\dim W_0$,\n$\\varphi_{V,W}$ induces a projection on homology,\n$$\\HCE_i(\\CDer(\\hat{S}(\\Pi V^*)))\n\\onto \\begin{cases} \\CC, & \\text{if $i$ is even}, \\\\ 0, & \\text{if $i$\n    is odd}.\\end{cases}$$\nThe same is true if we fix $W_0$ and consider\nlarge enough $\\dim W_1$.  A similar result is true in the associative\nversion, where now we project onto the span of graphs whose connected\ncomponents are stars with odd valence as in Theorem \\ref{acythm}.\n\n Using Theorem \\ref{cyccurvtrivthm}, we can prove an unstable\n analogue of Corollary \\ref{curveduncurved}, which gives information\n about the maps $\\varphi_{V,W}$ for all $W$ with $W_0 \\neq 0$:\n\\begin{theorem}\\label{unscurveduncurved}\n  If $V$ and $W$ are inner product spaces and $W_0 \\neq\n  0$, then the compositions\n\\begin{gather} \\label{lstabmaps}\n\\CE_i(\\CDer^0(\\hat S(\\Pi V^*))) \\to \\CE_i(\\CDer(\\hat S(\\Pi V^*))) \\mathop{\\to}^{\\varphi_{V,W}} \\CE_i(\\CDer(\\hat S(\\Pi (V \\oplus W)^*))), \\\\ \\label{astabmaps}\n\\CE_i(\\CDer^0(\\hat T(\\Pi V^*))) \\to \\CE_i(\\CDer(\\hat T(\\Pi V^*))) \\mathop{\\to}^{\\varphi_{V,W}} \\CE_i(\\CDer(\\hat T(\\Pi (V \\oplus W)^*)))\n\\end{gather}\nare zero on homology.\n\\end{theorem}\nBefore we prove the theorem, we first prove a lemma which may be\ninteresting in itself.  Given an element $c \\in V$ and a $L_\\infty$- or\n$A_\\infty$-structure given by a Maurer-Cartan element $\\xi \\in\n\\CDer^0(\\hat S(\\Pi V^*))$ or $\\xi \\in \\CDer^0(\\hat T(\\Pi V^*))$, we\nsay that $c$ is \\emph{central} if $[c, \\xi] = 0$. In particular, for\nan ordinary Lie algebra, $c$ is an element satisfying $\\{c, v\\}=0$ for\nall $v \\in V$, and for an ordinary associative algebra, $c$ satisfies\n$c \\cdot v = v \\cdot c$ for all $v \\in V$.\n\\begin{lemma}\nSuppose that $V$ is an (uncurved) $L_\\infty$- (or $A_\\infty$-) algebra with a nonzero even central element $c \\in V_0$.\nThen, the image of the resulting (unstable) characteristic class of $V$ under the appropriate map,\n\\begin{gather}\n\\CE_\\bullet(\\CDer^0(\\hat S(\\Pi V^*))) \\to \\CE_\\bullet(\\CDer(\\hat S(\\Pi V^*))) \\text{ or }\\\\\n\\CE_\\bullet(\\CDer^0(\\hat T(\\Pi V^*))) \\to \\CE_\\bullet(\\CDer(\\hat T(\\Pi V^*))),\n\\end{gather}\nis zero on homology.\n\\end{lemma}\n\\begin{proof}\nWe first consider the $L_\\infty$ case.  Let $\\mathfrak{g} :=\n  \\CDer(\\hat{S}(\\Pi V^*))$.  Let $\\xi\n  \\in \\CDer^0(\\hat{S}(\\Pi V^*)) \\subseteq\n  \\mathfrak{g}$ correspond to the $L_\\infty$-structure on $V$,\n  i.e., $\\xi$ satisfies $[\\xi, \\xi] = 0$, and viewed as an element\nof $\\mathfrak{g}$, $\\xi(v)$ has no\n  constant term for all $v \\in \\Pi V^*$.\n  Let $\\ell$ denote the structure maps for the algebra\n  $V$, with $\\ell_i: V^{\\otimes i} \\rightarrow V$ the $i$-th component.\n\n  Now, consider the $L_\\infty$-structure $\\{\\ell^\\lambda_i\\}$ on $V$\n  which is obtained by $\\ell^\\lambda_i := \\ell_i$ if $i \\geq 1$, and\n  $\\ell_0^\\lambda = \\lambda c$ for $\\lambda \\in \\CC$. Since $c$ is\n  central and even, these indeed define $L_\\infty$-structures.\n\n  Then, by Theorem \\ref{cyccurvtrivthm}, $V$ equipped with\n  $\\ell^\\lambda$ is gauge equivalent to an algebra with all higher\n  multiplications equal to zero, and curvature equal to $\\lambda\n  c$. This gauge equivalence is by vector fields with zero constant\n  and linear term, which form a pronilpotent dgla.  Hence, in the\n  limit as $\\lambda \\rightarrow 0$, we see that the characteristic\n  class of $(V, \\ell^\\lambda)$ becomes a boundary, i.e., the original\n  characteristic class of $(V, \\ell)$ is a boundary in\n  $\\CE_\\bullet(\\CDer(\\hat S(\\Pi V^*)))$, as desired.\n\n  In the $A_\\infty$ case, the same argument applies: as before, one\n  deforms the uncurved $A_\\infty$-structure $\\{m_i\\}$ to the curved\n  structure $\\{m^\\lambda_i\\}$ with $m^\\lambda_i = m_i$ for $i \\geq 1$\n  and $m^\\lambda_0 = \\lambda c$.  The difference is, by Theorem\n  \\ref{cyccurvtrivthm}, the resulting structure $(V, m^\\lambda)$ is\n  gauge equivalent to the algebra described there, which does not have\n  all higher operations zero.  Call this algebra structure\n  $\\{(m')^\\lambda_i\\}$.  Even though these are nonzero for infinitely\n  many $i$, it is still true that, as $\\lambda \\rightarrow 0$,\n  $(m')^\\lambda_i \\rightarrow 0$, and so we still deduce that the\n  original characteristic class was a boundary.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{unscurveduncurved}]\n  We treat only the $L_\\infty$ case, since the $A_\\infty$ case is\n  identical.  If $V$ is an uncurved $L_\\infty$-algebra, then $V \\oplus\n  W$ is an $L_\\infty$-algebra with a nonzero central element, namely,\n  any nonzero element of $W_0$.  Hence, the image of the resulting\n  characteristic class under \\eqref{lstabmaps} is zero.\n\nHowever, in general, not all homology classes of\n$\\CE_\\bullet(\\CDer^0(\\hat{S}(\\Pi V^*)))$ are obtainable in the above manner.\nTo fix this problem, we can consider nontrivial coefficients.  Given\nany cdga $\\mathfrak{a}$, we may consider\n$\\mathfrak{a}$-linear $L_\\infty$-algebra structures on $U := V\n\\otimes_\\CC \\mathfrak{a}$, for a fixed finite-dimensional\nspace $V$.  Denote $U^* := \\Hom_{\\mathfrak{a}}(U,\n\\mathfrak{a})$.  Then, these algebra structures on $U$ are, by\ndefinition, Maurer-Cartan elements of\n$\\Der^0(\\hat{S}_{\\mathfrak{a}}(\\Pi U^*))$, i.e., odd elements $f$\nsatisfying $df + \\frac{1}{2} [f, f] = 0$, where $d$ is the\ndifferential induced by the differential on $\\mathfrak{a}$. If we fix\nan inner product on $V$, we obtain an induced $\\mathfrak{a}$-linear\ninner product on $U$ (of the form $U \\otimes_{\\mathfrak{a}} U\n\\rightarrow \\mathfrak{a}$), and cyclic $L_\\infty$-algebras of this\nform are given by elements of $\\CDer^0(\\hat{S}_{\\mathfrak{a}}(\\Pi\nU^*))$.\n\nAs above, we obtain an unstable characteristic class of\n$\\CE_\\bullet(\\CDer^0(\\hat{S}_{\\mathfrak{a}}(\\Pi U^*)))$.  The above argument\nshows that the images of such classes under the tensor product of \\eqref{lstabmaps} with $\\Id_{\\mathfrak{a}}$ are boundaries.\n\nTo conclude the proof, we will use a universal example.\nQuite generally, if $\\mathfrak{h}$ is a formal dgla and\n$\\mathfrak{a}$ a cdga, then $\\CE^\\bullet(\\mathfrak{h}) =\nS\\Pi \\mathfrak{h}^*$ is naturally a cdga, and\nMaurer-Cartan elements of $\\mathfrak{h} \\otimes \\mathfrak{a}$\nidentify with cdga morphisms $\\CE^\\bullet(\\mathfrak{h})\n\\rightarrow \\mathfrak{a}$.\nIf we set $\\mathfrak{a}\n:= \\CE^\\bullet(\\mathfrak{h})$, then the identity map\n$\\Id_{\\CE^\\bullet(\\mathfrak{h})}$ yields a Maurer-Cartan element of\n$\\mathfrak{h} \\otimes \\mathfrak{a}$, and the resulting cycle $\\sum_i\n\\xi^{ i} \/ i! \\in \\widehat{\\CE_\\bullet(\\mathfrak{h}) \\otimes \\mathfrak{a}}\n\\cong \\End(\\CE^\\bullet(\\mathfrak{h}))$ is nothing but the identity map.\n\nApplying this construction to the case $\\mathfrak{h} =\n\\CDer^0(\\hat{S}(\\Pi V^*))$ and $\\mathfrak{a} =\n\\CE^\\bullet(\\CDer^0(\\hat{S}(\\Pi V^*)))$, we deduce from the above that\nthe $\\mathfrak{a}$-linear maps $\\widehat{\\varphi_{V,W} \\otimes\n  \\Id_{\\mathfrak{a}}}$ send the cycle $\\xi$ corresponding to the\nidentity element of $\\End(\\CE^\\bullet(\\mathfrak{h}))$ to a\nboundary. This implies that \\eqref{lstabmaps} itself is zero on\nhomology, as desired.\n\\end{proof}\n\n\n\n\\section{Operadic generalization}\nIn this section we sketch an operadic generalization of our main\nresults, from the associative and Lie cases to more general\nsettings. As we will show in the following section, these include\nPoisson, Gerstenhaber, BV, permutation, and pre-Lie algebras: see\nExamples \\ref{pgexam}, \\ref{bvexam}, and \\ref{permexam} in the\nfollowing section.\n\nWe may think of curved (cyclic) $A_\\infty$- and $L_\\infty$-algebras as\narising from the following construction, which we think of\nheuristically as a type of ``Koszul duality'' between operads\ngoverning curved algebras and those governing unital algebras (we do\nnot attempt to make this description precise).\n\\begin{remark}\n  For a somewhat related result, see \\cite{HiMickdt}, where resolutions\n  for operads of unital algebras are constructed by defining a Koszul\n  dual curved cooperad and performing a version of the cobar\n  construction. Here, we will not make use of the notion of curved\n  (co)operads defined in \\cite{HiMickdt}, and will only use ordinary (dg)\n  operads.\n\\end{remark}\nLet $\\mathcal{O}$ be a (cyclic) dg operad \\cite{GKcoch}, which we\nassume to be unital with unit $I \\in \\mathcal{O}(1)$. Moreover, we\nwill assume throughout that each $\\mathcal{O}(i)$ is a\nfinite-dimensional $\\ZZ\/2$-graded vector space (this isn't really\nessential, but it makes dualization less technical, and includes all\noperads we have in mind. Properly speaking, we view $\\mathcal{O}(i)$\nas a formal space, and everything generalizes to the\ninfinite-dimensional formal setting.)  Let $m_0 \\in \\mathcal{O}(0)$ be\nan element (corresponding to a ``0-ary'' operation).  Recall that\nevery (cyclic) operad is an $\\SS$ ($\\SS_+$)-module, where an\n$\\SS$-module is defined as a collection $\\{V_m\\}_{m \\geq 0}$ of\n$S_m$-modules for all $m \\geq 0$, and an $\\SS_+$-module is an\n$\\SS$-module where each $V_m$ is actually a module over $S_{m+1}$ (the\nunderlying $\\SS$-module is obtained using the inclusion $S_m \\subseteq\nS_{m+1}$ of permutations fixing $m+1$).\n\nWe will now consider $\\mathcal{O}$ as a nonunital operad and\nperform the cobar construction \\cite{GiKa}. Namely, let $\\mathcal{O}^* := \\{ \\mathcal{O}(m)^* \\}$ be the dual\n$\\SS$ (or $\\SS_+$)-module. Let $C(\\mathcal{O})$ be the free operad\ngenerated by $\\Pi \\mathcal{O}^*$, equipped with a differential\n$d_{C(\\mathcal{O})}$ obtained as follows.  For every $j \\geq 0$ and\nall $1 \\leq i \\leq k$, there is an operadic composition map\n\\begin{equation}\n\\circ^{k,j}_i: \\mathcal{O}(k) \\otimes \\mathcal{O}(j) \\rightarrow \\mathcal{O}(j+k-1),\n\\end{equation}\nwhich corresponds to plugging the element of $\\mathcal{O}(k)$ into the\n$i$-th input of the element of $\\mathcal{O}(j)$.  Let\n$(\\circ^{k,j}_i)^*$ be the linear dual to the above map.\nThen, we\ndefine the differential $d_{C(\\mathcal{O})}$ to be the\nunique extension to a derivation of the operation\n\\begin{equation}\nd_{C(\\mathcal{O})}|_{\\mathcal{O}(i)^*} = d_{\\mathcal{O}}^* +\n\\bigoplus_{j,k} (\\circ^{k,j}_i)^*.\n\\end{equation}\nNow, in the case that $\\mathcal{O}$ is the ordinary (non-dg) operad\ngoverning \\emph{unital} associative or commutative algebras, then\n$C(\\mathcal{O})$ is a dg operad with the property that\ngraded (dg) algebras $V$ over $C(\\mathcal{O})$, equipped\nwith zero differential, are the same as curved $A_\\infty$- or\n$L_\\infty$-algebras. In this case, $m_0 \\in \\mathcal{O}(0)$ is the unit of the\nmultiplication, and the curvature of a graded algebra $V$ over $C(\\mathcal{O})$ is the image of $m_0^* \\in \\mathcal{O}(0)^*$ in $V$.\n\n\nNext, suppose that\n$\\mathcal{O}'$ is the suboperad of $\\mathcal{O}$ in positive arity,\ni.e., $\\mathcal{O}'(0) = 0$ and $\\mathcal{O}'(i) = \\mathcal{O}(i)$ for $i \\geq 1$.\n Further\nsuppose that $\\mathcal{O}'$ is a Koszul operad with augmentation\n$\\overline{\\mathcal{O}'}$ (i.e., $\\overline{\\mathcal{O}'}$ is\na suboperad such that $\\mathcal{O}'(1) = \\overline{\\mathcal{O}'}(1) \\oplus \\CC \\cdot \\Id$, and\n$\\overline{\\mathcal{O}'}(i) = \\mathcal{O}'(i)$ for $i \\neq 1$).\nThen, $C(\\overline{\\mathcal{O}'})$ yields a resolution of the\nKoszul dual operad $(\\mathcal{O}')^!$ (this last fact is equivalent to\nKoszulity; see \\cite{GiKa}).  \\emph{Caution:} This operad $C(\\overline{\\mathcal{O}'})$ is sometimes called the cobar construction on the unital operad\n$\\mathcal{O}'$ itself, but\nhere we consider cobar constructions only on nonunital operads.\n\nWe think of an algebra over $C(\\mathcal{O})$ (with zero differential)\nas a certain type of \\emph{curved} version of an\n$((\\mathcal{O}')^!)_\\infty$-algebra, as opposed to an algebra over\n$C(\\overline{\\mathcal{O}'})$, which, in the case $\\mathcal{O}'$ is\nKoszul, is the same as an ordinary\n$((\\mathcal{O}')^!)_\\infty$-algebra.  (More generally, given a dg\nspace $V$ with possibly nontrivial differential, one could view a\n$C(\\mathcal{O})$-algebra structure on it as a dg generalization of a\ncurved $((\\mathcal{O}')^!)_\\infty$-algebra; thus one could speak, for\ninstance, about a curved dg $A_\\infty$- or $L_\\infty$-algebra. This\nwould be a Maurer-Cartan element in the appropriate differential\ngraded Lie algebras of (formal, noncommutative) vector fields rather\nthan simply square-zero odd derivations.)\n\n\nLet us return to the general setting of an arbitrary (cyclic) operad\n$\\mathcal{O}$. The analogues of Theorems \\ref{curvtrivthm} and\n\\ref{cyccurvtrivthm} are then the following.\n\\begin{definition}\n  A \\emph{weakly unital multiplication} $m_2 \\in \\mathcal{O}(2)$\n  is an element such that the image of the maps\n\\begin{equation}\\label{wtm2map}\nm_2(- \\otimes \\Id), m_2(\\Id \\otimes -): \\mathcal{O}(0) \\rightarrow \\mathcal{O}(1)\n\\end{equation}\nlie in the one-dimensional space $\\CC \\cdot \\Id \\subseteq \\mathcal{O}(1)$, and such that the maps are not both zero.\n\\end{definition}\nIf $m_2$ is weakly unital, then\n\\begin{equation} \\label{m2ideq}\nm_2(- \\otimes \\Id) = m_2(\\Id \\otimes -).\n\\end{equation}\nIndeed, if $m_0 \\in \\mathcal{O}(0)$ is any element such that\n$m_2(m_0 \\otimes \\Id) = \\lambda \\cdot \\Id$ for some nonzero $\\lambda\n\\in \\CC$, then it follows that $m_2(m_0 \\otimes m_0) = \\lambda\\cdot m_0$,\nand hence that $m_2(\\Id \\otimes m_0) = \\lambda \\cdot \\Id$.\n\nTherefore, when $m_2$ is weakly unital, we define $\\widetilde{m_2}\n\\in \\mathcal{O}(0)^*$ to be the resulting map \\eqref{m2ideq}.\n\n\nNext, given a $C(\\mathcal{O})$ algebra $V$, the $0$-ary operations form a map $\\gamma: \\mathcal{O}(0)^* \\rightarrow V$, which we call the \\emph{$0$-ary structure\nmap}.\n\\begin{definition}\nGiven an operad $\\mathcal{O}$ with weakly unital multiplication $m_2$,\na $C(\\mathcal{O})$ algebra $V$ with $0$-ary structure map $\\gamma$\nis said to be\n\\emph{nontrivially curved with respect to $m_2$}\nif the element $\\gamma(\\widetilde{m_2})$ is nonzero.\n\\end{definition}\nWe call the element $\\gamma(\\widetilde{m_2})$ the\n\\emph{curvature} of $V$ (for the element $m_2$).\n\nThen, Theorems \\ref{curvtrivthm} and \\ref{cyccurvtrivthm}\ngeneralize as follows.  Fix a graded vector space $V$ (with zero\ndifferential).  If $\\mathcal{O}$ is any dg operad, we denote a\n$\\mathcal{O}$-algebra structure on $V$ as a pair $(V, \\phi)$ for the\nalgebra structure, where given any $o \\in \\mathcal{O}(k)$, $\\phi(o) \\in\n\\Hom_\\CC(V^{\\otimes k}, V)$.\n\\begin{claim} Fix a graded vector space $V$. \\label{opclaim}\n\\begin{enumerate}\n\\item[(i)] Let $\\mathcal{O}$ be an operad with a weakly unital\n  multiplication $m_2$. Let $(V,\\phi)$ be a\n  $C(\\mathcal{O})$-algebra structure with nonzero curvature $c\n  \\in V$.\n  Then, $(V,\\phi)$ is gauge equivalent to the algebra $(V,\\phi')$ with\n  the same $0$-ary structure map $\\gamma$,\nbut with all\n  higher multiplications equal to zero.\n\\item[(ii)] Let $\\mathcal{O}$ be a cyclic operad with a weakly unital\n  multiplication $m_2$. Let $(V,\\phi)$ be a cyclic\n  $C(\\mathcal{O})$-algebra structure, with nonzero curvature $c\n  \\in V$.  Let $c' \\in V$ be an element such that $(c', c) = 1$.\n  Then, $(V,\\phi)$ is gauge equivalent to the algebra $(V,\\phi')$ with\n  the same $0$-ary structure map $\\gamma$, but with higher\n  multiplications of the form\n\\begin{equation}\\label{opclaim12eq}\n\\phi'(m)(v_1, \\ldots, v_i) = \\bigl(\\prod_{j=1}^i (c',v_j) \\bigr) \\cdot (\\phi(m)(c, c, \\ldots, c), c) \\cdot c'.\n\\end{equation}\n\\end{enumerate}\n\\end{claim}\nHere, gauge equivalence is defined as follows.  For any\n$C(\\mathcal{O})$-algebra structure on $V$, one can form the formal dg\n$\\mathcal{O}$-algebra $\\hat{C}_{\\mathcal{O}}(V)$, which is defined as the completed\nfree $\\mathcal{O}$-algebra generated by $\\Pi V^*$:\n\\[\n\\hat{C}_{\\mathcal{O}}(V)=\\prod_{n=0}^\\infty {\\mathcal\n  O}(n)\\otimes_{S_n}(\\Pi V^*)^{\\otimes n} ,\\] equipped with the\ndifferential $d_{\\hat{C}_{\\mathcal{O}}(V)}$ on\n$\\hat{C}_{\\mathcal{O}}(V)$ defined by the canonical linear map $\\Pi V^*\n\\rightarrow \\hat{C}_{\\mathcal{O}}(V)$, which is shifted dual to a\nrestriction of the structure maps $C(\\mathcal{O}) \\rightarrow\n\\Hom(V^{\\otimes n}, V)$.\n\nNext, assume for the moment that $V$ is equipped with the zero\n$C(\\mathcal{O})$-algebra structure, i.e.,\n$d_{\\hat{C}_{\\mathcal{O}}(V)} = 0$. We may then form the formal dgla\n$\\Der(\\hat{C}_{\\mathcal{O}}(V))$, where the differential is induced by\nthe differential on $V$ (and is zero in the case where $V$ was merely\na graded vector space, as in the setting of (non-dg) curved\n$A_\\infty$- or $L_\\infty$-algebras considered in earlier sections of\nthis paper).  It is then a standard fact (see, e.g., \\cite[Proposition\n2.15]{GJohaii}, Proposition 2.15 where, however, this result is\nformulated in the language of coalgebras) that the above yields a\nbijection between square-zero odd derivations of\n$\\hat{C}_{\\mathcal{O}}(V)$ and $C(\\mathcal{O})$-structures on $V$.\nFinally, we define two $C(\\mathcal{O})$-structures on $V$ to be gauge\nequivalent if the corresponding differentials\n$d_{\\hat{C}_{\\mathcal{O}}(V)}$ and $d_{\\hat{C}_{\\mathcal{O}}(V)}'$ are\ngauge equivalent, i.e., that there exists an even derivation $\\xi \\in\n\\Der^0(\\hat{C}_{\\mathcal{O}}(V))$ with zero constant term such that\n$d_{\\hat{C}_{\\mathcal{O}}(V)}' = e^{\\ad \\xi}\nd_{\\hat{C}_{\\mathcal{O}}(V)}$.\n\nSimilarly, if $\\mathcal{O}$ is a cyclic operad, and $V$ is a cyclic\nalgebra over $\\mathcal{O}$ (i.e., an algebra equipped with a\nnondegenerate inner product compatible with the cyclic structure on\n$\\mathcal{O}$), then there is a natural $\\ZZ\/(m+1)$ action on the\nsubspace $\\Pi V^* \\otimes \\mathcal{O}(m) \\otimes_{S_m} (\\Pi V^*)^{\\otimes m}\n\\cong \\Hom(\\Pi V^*, \\hat{C}_{\\mathcal{O}}(V)) = \\Der(\\hat{C}_{\\mathcal{O}}(V))$\ncoming from the $S_{m+1}$-structure on $\\mathcal{O}(m)$ and the\ncompatible inner product on $V$.  Then, we can define the \\emph{cyclic\n  derivations}, $\\CDer(\\hat{C}_{\\mathcal{O}}(V))$, to be the subspace of\n$\\ZZ\/(m+1)$-invariants in each degree $m$.  Then, a gauge equivalence\n$e^{\\ad\\xi}: d_{\\hat{C}_{\\mathcal{O}}(V)} \\iso d_{\\hat{C}_{\\mathcal{O}}(V)}'$ is\n\\emph{cyclic} if $\\xi$ is cyclic.\n\nFinally, we explain the main idea of the proof of the claim (without\ngoing into full detail).  We explain only the second part, since it is\nmore involved. We first form the cyclic deformation complex for the\n$C(\\mathcal{O})$-structure on $V$\nstructure with all higher operations $C(\\mathcal{O})(i), i > 0$ acting as\nzero, and with $0$-ary structure map $\\gamma: C(\\mathcal{O})(0) = \\Pi\\mathcal{O}(0)^* \\to V$.\nThis complex is $\\CDer(\\hat{C}_{\\mathcal{O}}(V))$, equipped with the\ndifferential $\\ad \\xi$, where $\\xi \\in\n\\CDer(\\hat{C}_{\\mathcal{O}}(V))$ is the derivation obtained from $\\Pi\\gamma^*: \\Pi V^* \\to \\mathcal{O}(0)$, by the formula (for $o \\in \\mathcal{O}(n)$ and $f_1, \\ldots, f_n \\in \\Pi V^*$):\n\\begin{equation}\n\\xi(o \\otimes_{S_n} (f_1 \\otimes \\cdots \\otimes f_n)) = \\sum_{i=1}^n (o \\circ_i \\Pi\\gamma^*(f_i)) \\otimes_{S_{n-1}} (f_1 \\otimes \\cdots \\otimes \\hat f_i \\otimes \\cdots \\otimes f_n),\n\\end{equation}\nwhere $\\hat f_i$ denotes omitting $f_i$ from the tensor product.\n\nThen, we again show that this complex is quasi-isomorphic\nto the subcomplex spanned by cochains such that $\\mathcal{O}(i)$ acts by\nmultiples of the element\n\\begin{equation}\\label{ceq2}\n\\epsilon^{i+1}(x_1, \\ldots, x_i) := \\epsilon(x_1) \\cdots \\epsilon(x_i) c', \\quad \\epsilon(v) := (c', v),\n\\end{equation}\nand hence that the original algebra structure is gauge equivalent to\none where all $\\geq 1$-ary operations are multiples of the operation\n\\begin{equation}\n\\phi'(m)(v_1, \\ldots, v_i) = \\bigl(\\prod_{j=1}^i (c',v_j) \\bigr) \\cdot c'.\n\\end{equation}\nFinally, as before, we can see that this multiple is $(\\phi(m)(c, c, \\ldots, c), c)$.\n\nThe main technical step in the above is showing that the complex is\nquasi-isomorphic to the subcomplex spanned by cochains as in\n\\eqref{ceq2}. This amounts to a generalization of Lemma \\ref{qil}.\nThe main point is that the map $s_i'$ defined in \\eqref{sipfla1} can\nbe generalized to this setting, as follows.  If $f = o\n\\otimes_{S_{n+1}} (f_1 \\otimes \\cdots \\otimes f_{n+1}) \\in\n\\mathcal{O}(n) \\otimes_{S_{n+1}} (\\Pi V^*)^{\\otimes (n+1)}$, viewed as\nan element of $\\CDer(\\hat{C}_{\\mathcal{O}}(V))$, then $s_n f \\in\n\\mathcal{O}(n+1) \\otimes_{S_{n+2}} (V^*)^{\\otimes(n+2)}$ is obtained\nas\n\\[\ns_n f = \\sum_{j=1}^{n+1} (o \\circ_j m_2) \\otimes_{S_{n+2}} (f_1 \\otimes \\cdots \\otimes f_{j-1} \\otimes \\epsilon \\otimes f_j \\otimes \\cdots \\otimes f_{n+1}).\n\\]\nThen, one obtains a similar result to \\eqref{sipfla3}, which gives the desired conclusion.\n\nNext, Theorem \\ref{acythm} generalizes as follows.  Let $\\cO$ be a\ncyclic operad, and let $\\mathcal{G}_{\\mathcal{O}}$ be the graph\ncomplex constructed from the operad $\\mathcal{O}$.  This complex is a\ngeneralization of Kontsevich's graph homology which has many\nequivalent definitions in the literature; one definition is via the\nFeynman transform construction of \\cite{GKmo}.  Namely, consider the\nnaive $\\Det$-twisted modular closure $\\underline{\\mathcal O}^1$ of\n$\\mathcal O$ by considering all contraction operations to be zero and\nall parts of $\\mathcal{O}$ of genus $\\geq 1$ to be zero; see, e.g.,\n\\cite[\\S 2]{ChLadft} for this notion. Then form the Feynman transform\noperad $\\mathsf{F}\\underline{\\mathcal O}^1$. The $0$-ary part\n$\\mathsf{F}\\underline{\\mathcal O}^1((0))$ is the desired graph\ncomplex.  As before, let $\\mathcal{G}_{\\mathcal{O},c} \\subseteq\n\\mathcal{G}_{\\mathcal{O}}$ be the subcomplex spanned by connected\nnonempty graphs; one has $\\mathcal{G}_{\\mathcal{O}} \\cong \\Sym\n\\mathcal{G}_{\\mathcal{O},c}$.\n\\begin{claim}\\label{opclaim2}\n  The complex $\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to\n  the quotient of the subcomplex spanned by graphs with at most one\n  vertex of valence $\\geq 2$ by the span of line segments with three\n  vertices whose central vertex is labeled by $\\Id$.\\footnote{Note\n    that, if $\\mathcal{O}(0)$ is even one-dimensional, as in the\n    preceding cases of $\\mathscr{C}\\!\\mathit{omm}$ or $\\mathscr{A}\\!\\mathit{ss}$, then these\n    line segments with central vertex labeled by $\\Id$ are already\n    zero. More generally, the span of these line segments is\n    isomorphic to $\\wedge^2 \\cO(0)$, by considering the labels at the\n    univalent vertices.}\n\\end{claim}\nIn other words, $\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to a\nquotient of the deformation complex of a certain canonical\ncyclic $C(\\mathcal{O})$-algebra (which we think of as a type of curved\n$((\\mathcal{O}')^!)_\\infty$-algebra), as follows.\\footnote{This\n  interpretation requires that $\\mathcal{O}(0)$ be finite-dimensional,\n  as we are assuming.  For infinite-dimensional formal\n  $\\mathcal{O}(0)$, while the claim above still holds, this\n  interpretation is technically not available.} Let $V :=\n\\mathcal{O}(0)^*$. View $V$ as a $\\mathcal{O}$-algebra with all $\\geq\n1$-ary operations trivial, and with $0$-ary structure $\\Id:\n\\mathcal{O}(0)^* \\to \\mathcal{O}(0)^*$. Since all higher operations\nare trivial, any inner product on $V$ is cyclic; we can fix one but it\nwill not really affect anything.  Then, the deformation complex of $V$\nas a $C(\\mathcal{O})$-algebra is\n$$\\CDer(\\hat{C}_{\\mathcal{O}}(V)) \\cong \\bigoplus_{m \\geq 0} \\mathcal{O}(m) \\otimes_{S_{m+1}} \\mathcal{O}(0),$$\nequipped with the differential $\\ad \\xi$ where $\\xi \\in\n\\CDer(\\hat{C}_{\\mathcal{O}}(V))$ is the element corresponding to $\\Id\n\\in V \\otimes \\mathcal{O}(0) \\cong \\Hom(V, V) \\subseteq\n\\Der(\\hat{C}_{\\mathcal{O}}(V))$.  By the above,\n$\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to the quotient\nof this deformation complex by the subcomplex in arity $2$, $\\Id\n\\otimes_{S_2} \\mathcal{O}(0) \\subseteq \\bigoplus_{m \\geq 0}\n\\mathcal{O}(m) \\otimes_{S_{m+1}} \\mathcal{O}(0)$.\n\nThe proof of Claim \\ref{opclaim2} is a direct generalization of the\nproof of Theorem \\ref{acythm}.  First, we generalize the notion of\ninterior vertex.  Note that $\\mathcal{G}_{\\mathcal{O},c}$ is\nspanned by oriented graphs of the following form: the vertices are\nordered, and each $m$-valent vertex is labeled by an element of\n$\\cO(m-1)$.  Next, the half-edges are also ordered.  Many of these\ngraphs are equivalent: applying a permutation of the half-edges\nincident to a given vertex is set equal to applying the corresponding\npermutation to the element of $\\cO$ labeling that vertex; also,\napplying any permutation of the half-edges is set equal to multiplying\nby the sign of that permutation. Finally, applying a permutation to\nthe vertices is the same as multiplying by the sign of that\npermutation.\n\nThen, an interior vertex of a graph as above is a vertex which has\nvalence $\\geq 2$ and whose removal does not result in a graph one of\nwhose connected components consists only of univalent vertices and\nbivalent vertices labeled by elements of $\\CC \\cdot \\Id \\subseteq\n\\cO(1)$.  As before, the number of interior vertices defines an\nincreasing filtration. Moreover, if we choose an arbitrary basis\n$\\{\\Gamma_i\\}$ of $\\mathcal{G}_{\\mathcal{O},c}$, then the\nassociated graded complex is graded by this basis, where\n$\\gr_{\\Gamma_i} \\mathcal{G}_{\\mathcal{O},c}$ is spanned by graphs\nwhose restriction to interior vertices yields $\\Gamma_i$ (note that\n$\\Gamma_i$ need not be a graph).\n\n\nThe main step is to generalize the construction of the contracting\nhomotopy which shows that $\\gr_{\\Gamma_0}\n\\mathcal{G}_{\\mathcal{O},c}$ is acyclic when $\\Gamma_0$ is a graph\ncontaining an edge. To do so, first fix an element $m_0 \\in \\cO(2)$\nsuch that $m_2 \\circ_1 m_0 = m_2 \\circ_2 m_0 = \\Id \\in \\cO(1)$.  We\nchoose the basis $\\{\\Gamma_i\\}$ to consist of graphs.  Let $\\Gamma_0$\nbe one such graph which contains an edge.  Fix a half edge $h$\nof $\\Gamma_0$\nbased at a vertex $v$.  Let us choose $v$ to be the last vertex in the\nordering of vertices, and $h$ to be the last half-edge in the ordering\nof half-edges based at $v$.  Suppose $v$ is $m$-valent, and let the\nlabel of $v$ be $o_v \\in \\cO(m-1)$. Label the half-edges of $v$, in\norder, $h_1, \\ldots, h_m$, with $h = h_m$.  The contracting homotopy\n$s$ then acts on any graph $\\Gamma \\in \\gr_{\\Gamma_0}\n\\mathcal{G}_{\\mathcal{O},c}$ by first adding a new half-edge\n$h_{m+1}$ to $v$ (last in the ordering at $v$).  Then, the label $o_v$\nis replaced by $o_v \\circ_m m_2$, where $m_2 \\in \\cO(2)$ is the weakly\nunital multiplication. Finally, one adds a new univalent vertex $y$ to\n$\\Gamma$ (which becomes the last vertex), labels it by $m_0$, and\nattaches it to $h_{m+1}$.  It is then straightforward to verify that\n$(sd + ds) \\Gamma = \\Gamma$, and hence $\\gr_{\\Gamma_0}\n\\mathcal{G}_{\\mathcal{O},c}$, is acyclic.\n\nFinally, we explain the appearance of the quotient by line segments\nwith three vertices whose central vertex is labeled by $\\Id$.  Namely,\nthese are exactly the graphs with a single vertex of valence $\\geq 2$\nwhich, nonetheless, have no interior vertices.  By a generalization of\nthe arguments in the proof of Theorem \\ref{acythm}, $\\gr_{\\pt}\n\\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic to the quotient of\nthe subcomplex spanned by star-shaped graphs with a single vertex of\nvalence $\\geq 2$ by the span of these graphs.  On the other hand,\n$\\gr_{\\emptyset} \\mathcal{G}_{\\mathcal{O},c}$ is quasi-isomorphic\nto the subcomplex spanned by graphs with two vertices, each univalent.\nThus, the second page of the spectral sequence for the interior vertex\nfiltration yields, in degree one, the claimed quotient\ncomplex modulo the graphs with only two vertices, and in degree zero,\nthe span of graphs with only two vertices and one edge.  The spectral\nsequence collapses at the third page to the homology of the whole\nquotient subcomplex stated in Claim \\ref{opclaim2}, which proves the\nresult. We omit further details.\n\nFinally, one can deduce an unstable version of Claim \\ref{opclaim2},\nanalogous to Theorem \\ref{unscurveduncurved}:\n\\begin{claim}\\label{opclaim3}\nThe composition\n\\begin{equation}\n\\CE_\\bullet(\\CDer^0(\\hat{C}_{\\mathcal{O}}(V))) \\to \\CE_\\bullet(\\CDer(\\hat{C}_{\\mathcal{O}}(V)))\n\\rightarrow \\CE_\\bullet(\\CDer(\\hat{C}_{\\mathcal{O}}(V \\oplus W)))\n\\end{equation}\nis zero on homology for any uncurved cyclic $C(\\mathcal{O})$-algebra $V$,\nwhere $W$ is an inner product space with $W_0 \\neq 0$.\n\\end{claim}\nWe omit the proof, which is obtained by combining the preceding\nmaterial with \\S \\ref{unstabsec}.\n\\section{Examples and further comments}\nIn this section we provide some remarks and examples regarding the\nmaterial of the previous section.\n\n\\subsection{Generalization to the modular case}\nHere we briefly sketch how the results of the previous section\ncan be extended from the\nsetting of cyclic operads to that of modular operads.\n\nAn analogous result to Claim \\ref{opclaim2} holds for the\n\\emph{twisted} version of the $\\mathcal O$-graph complex which\ncorresponds to the Feynman transform ${\\mathsf F}\\underline{\\mathcal\n  O}((0))$ (as opposed to ${\\mathsf F}\\underline{\\mathcal O}^1((0))$)\nwhich is the usual version of the graph complex.\\footnote{We note that\n  the operads ${\\mathsf F}\\underline{\\mathcal O}^d((n))$ are not,\n  strictly speaking, modular operads in the sense of \\cite{GKmo},\n  since the stability condition is not satisfied; however the\n  construction of the Feynman transform still makes sense and this\n  causes us no trouble. We will ignore this point henceforth.}  For\nexample, in the commutative and ribbon graph case the difference\nbetween two types of graph complex lies in a different notion of\norientation: a twisted orientation corresponds to ordering edges of a\ngraph, as opposed to ordering the half edges and vertices.\n\nMore generally, one can replace cyclic operads $\\cO$ by arbitrary\n(twisted) modular operads. Then, if $\\cO$ admits a weakly unital\nmultiplication, one still obtains in the same manner the result that\nthe graph complex for $\\cO$ is quasi-isomorphic to the subcomplex\ndefined analogously to the above.\n\nSimilarly, we can generalize all the constructions of the preceding\nsection from the cyclic to the modular setting. Let $\\cO$ be an\narbitrary (twisted) modular operad $\\cO$.  In this case, one replaces\n$C(\\cO)$ (used in the cyclic case above) by the Feynman transform\n(twisted) modular operad, ${\\mathsf F}\\mathcal O$, and modular\nalgebras over this operad are then thought of as curved algebras. The\nconstructions, results (Claims \\ref{opclaim}.(ii) and \\ref{opclaim3}),\nand proofs carry over to this setting.  In the special case where one\nhas a cyclic operad and considers it $\\Det$-twisted modular using the\nnaive $\\Det$-twisted modular closure, one recovers the above results.\nAnother example concerns so-called quantum $A_\\infty$-algebras: Let\n$\\mathcal{O}'$ be the $\\Det$-twisted modular closure of\n$\\mathscr{A}\\!ss$ such that ${\\mathsf F}\\mathcal O'$-algebras are\nso-called quantum $A_\\infty$-algebras (see for instance \\cite[Example\n5.2]{ChLafdmm} for an explanation of this notion). Then, if we let\n$\\mathcal{O} = \\mathcal{O}' \\oplus \\CC[0]$ be the unital version\n(adding a $0$-ary operation providing a unit for the associative\nmultiplication), then ${\\mathsf F}\\mathcal O$ defines a notion of\ncurved quantum $A_\\infty$-algebras.  As in the cyclic $A_\\infty$ case,\none sees that nontrivially curved quantum $A_\\infty$-algebra\nstructures on a fixed vector space are gauge equivalent to those where\nall the operations of positive arity are of the form\n\\eqref{opclaim12eq}, and in particular all land in a fixed\none-dimensional vector space. We refrain from making precise\nstatements.\n\n\\subsection{Examples}\nHere we provide examples of the preceding constructions for Poisson, Gerstenhaber, BV, and permutation (or pre-Lie) algebras.\n\\begin{example}\\label{pgexam} Consider the case of Poisson algebras.\n  It is natural to consider unital Poisson algebras, where here a unit\n  $f$ is an element satisfying $\\{1, f\\} = \\{f, 1\\} = 0$ for all $f$,\n  whereas $1 \\cdot f = f \\cdot 1 = f$. Let $\\mathcal{O}$ be the operad\n  governing unital Poisson algebras, i.e., $\\mathcal{O} =\n  u\\mathscr{P}\\!\\textit{oiss} = \\mathscr{P}\\!\\textit{oiss} \\oplus\n  \\CC[0]$ where $\\mathscr{P}\\!\\textit{oiss}$ is the Poisson operad and\n  $\\CC[0]$ is the one-dimensional vector space concentrated in degree\n  zero, which has zero compositions with the bracket of $\\mathcal{O}$,\n  and composition $m_2(m_0 \\otimes \\Id) = \\Id = m_2(\\Id \\otimes m_0)$\n  with the commutative multiplication, i.e.,\n  $\\mathscr{P}\\!\\textit{oiss} \\supset u\\mathscr{C}\\!\\textit{omm}$.\n  Then, by the above, one can think of $C(\\mathcal{O})$-algebras as\n  curved Poisson-infinity algebras (since Poisson is Koszul self-dual,\n  as in the associative case), and it follows from Claim\n  \\ref{opclaim}.(i) that nontrivially curved Poisson-infinity structures\n  on a vector space are all gauge equivalent.\n\n  Similarly, the Gerstenhaber operad $\\mathscr{G}\\!\\textit{erst}$ is\n  Koszul and dual to its suspension $\\mathfrak{s}\n  \\mathscr{G}\\!\\textit{erst}$; here the suspension is defined by\n  tensoring by the endomorphism operad of the one-dimensional odd\n  vector space (as a $\\ZZ\/2$-graded $\\SS$-module, this means that one\n  applies $\\Pi$ to the even-ary part).  As before, one can consider\n  unital $\\mathscr{G}\\!\\textit{erst}$ algebras,\n  $u\\mathscr{G}\\!\\textit{erst}$, and similarly its suspension $\\mathfrak{s}(u\n  \\mathscr{G} \\! \\textit{erst})$. Let $\\cO$ be this latter operad,\n  which we can also think of as $u\n  \\mathscr{G}\\!\\textit{erst}^!$. Here, the unit is odd.  We define\n  curved $\\mathscr{G}\\!\\textit{erst}_\\infty$ algebras as algebras over\n  $C(\\cO)$, where now the curvature is odd, rather than even.  From\n  Claim \\ref{opclaim}.(i), we deduce that any two nontrivially curved\n  Gerstenhaber-infinity algebra structures on the same graded vector\n  space are gauge equivalent.\n\n  In the case of the Poisson operad (but not for the Gerstenhaber operad),\n  in fact $\\mathscr{P}\\!\\textit{oiss}$ is cyclic, and hence one obtains\n  the notion of cyclic curved Poisson-infinity algebras. Claim\n  \\ref{opclaim}.(ii) then implies that all nontrivially curved cyclic\n  Poisson-infinity structures on a vector space are gauge equivalent\n  to those for which all operations of positive arity are of the form\n  \\eqref{opclaim12eq}, and in particular all land in a fixed\n  one-dimensional vector space.\n\n  Moreover, in fact one can compute the associated (unital Poisson)\n  graph homology. By Claim \\ref{opclaim2}, the associated graph\n  complex is quasi-isomorphic to the subcomplex spanned by graphs\n  whose vertices are all univalent except for at most one.  Since the\n  Poisson operad is the associated graded operad of the associative\n  operad, and in particular has the same $\\SS$-module structure, one\n  sees that this subcomplex has zero differential, and is spanned by\n  the star-shaped graphs with central vertex of odd valence.  That is,\n  the graph homology for the unital Poisson operad is isomorphic to\n  the homology of the graph complex $\\widetilde{\\mathcal{G}_r}$ for the unital\n  associative operad.\n\\end{example}\n\\begin{example}\\label{bvexam} The BV operad $\\BV$ \\cite{Getbva} is the homology operad of the operad of framed little discs; it is known to be cyclic. Its algebras, called $\\BV$-algebras, are Gerstenhaber algebras together with an odd operator $\\Delta$ which is a differential operator of second order with respect to the commutative multiplication and a derivation of the odd Lie bracket.  Let $\\overline{\\BV}$, as before, denote the augmentation ideal of the operad $\\BV$.\n  The $C(\\overline{\\BV})$-graph complex computes, according to\n  \\cite{Giafl2d}, the homology of the classifying space of diffeomorphism\n  groups of 3-dimensional oriented handlebodies.  Although the operad\n  $\\BV$ is not defined by (homogeneous) quadratic relations, in\n  \\cite{GTVhbva} it was shown to be Koszul in a more general sense,\n  and its Koszul dual, $\\BV^!$ was described, and shown to be\n  quasi-isomorphic to $C(\\overline{\\BV})$.\n\n  A unital $\\BV$-algebra is a $\\BV$-algebra with a unit with respect\n  to the commutative multiplication and such that the value of\n  $\\Delta$ on the unit is zero.\n\n  Consider the operad ${\\mathcal O} =u\\BV$ governing unital\n  $\\BV$-algebras; then we can view $C(\\mathcal O)$ as the operad\n  governing curved $\\BV^!_\\infty$-algebras. It follows that all\n  nontrivially curved $\\BV^!_\\infty$-algebras are gauge equivalent.\nFurthermore, according to Claim\n  \\ref{opclaim}.(ii) all nontrivially curved cyclic $\\BV^!_\\infty$-algebras are gauge equivalent\n  to those for which all operations of positive arity are of the form\n  \\eqref{opclaim12eq}.\n\n  Finally, by Claim \\ref{opclaim2}, the graph complex for $\\mathcal O$\n  is quasi-isomorphic to the subcomplex with at most one vertex of\n  valence $\\geq 2$.\n\\end{example}\n\\begin{example}\\label{permexam} Consider the case of (right) pre-Lie\n  algebras, i.e., those with a single operation $\\star$ satisfying the\n  relation\n\\begin{equation}\nx \\star (y \\star z) - (x \\star y) \\star z = x \\star (z \\star y) - (x \\star z) \\star y.\n\\end{equation}\nFor such algebras, it makes perfect sense to define the notion of a unit, $1$, such that\n\\begin{equation} \\label{prelieunit}\n1 \\star x = x = x \\star 1.\n\\end{equation}\nWe thus obtain an operad $\\mathcal{O} = u \\mathscr{P}\\!\\textit{re-Lie}\n= \\mathscr{P}\\!\\textit{re-Lie} \\oplus \\CC[0]$ governing \\emph{unital}\npre-Lie algebras, where now the compositions with $\\CC[0]$ are given\nby \\eqref{prelieunit}, or more precisely, $m_2(m_0 \\otimes \\Id) = \\Id\n= m_2(\\Id \\otimes m_0)$ where $m_0 = 1 \\in \\CC[0]$ and $m_2 \\in\n\\mathscr{P}\\!\\textit{re-Lie}[2]$ is the multiplication operation\n$\\star$.  By the above procedure, we then obtain a type of curved\nalgebra, namely algebras over $C(\\mathcal{O})$.  We will call these\n \\emph{curved (right)\n  $\\mathscr{P}\\!\\textit{erm}_{\\infty}$-algebras}, since the operad\n(right) $\\mathscr{P}\\!\\textit{erm}$ is Koszul dual to (right)\n$\\mathscr{P}\\!\\textit{re-Lie}$. In particular, curved\n  $\\mathscr{P}\\!\\textit{erm}_\\infty$ algebras with zero curvature\n  ($m_0=0$, i.e., the corresponding derivation of $C_{\\mathcal{O}}(V)$\n  has zero constant term) are the same as ordinary\n  $\\mathscr{P}\\!\\textit{erm}_\\infty$ algebras.  Let us recall here\n  that ordinary permutation algebras are algebras with an operation\n  $\\circ$ satisfying the relation\n\\begin{equation}\nx \\circ (z \\circ y) = x \\circ (y \\circ z) = (x \\circ y) \\circ z,\n\\end{equation}\ni.e., associative algebras additionally satisfying the first equality\nabove.  The operad $\\mathscr{P}\\!\\textit{erm}$ is the one whose algebras are\npermutation algebras.\n\nFrom the above results, we deduce that any two curved\n$\\mathscr{P}\\!\\textit{erm}_\\infty$-algebra structures on $V$ with nonzero curvature\nare gauge-equivalent.\n\nNote that, strictly speaking the operad $\\mathscr{P}\\!\\textit{re-Lie}$ is not cyclic, it is\nanticyclic \\cite[p. 9]{GKcoch}; however we can ignore this difference; any anticyclic operad gives rise to a cyclic one via the operadic suspension. Therefore, we obtain the corresponding result on the classification of nontrivially curved (anti)cyclic $\\mathscr{P}\\!\\textit{erm}_\\infty$-algebras, and on\n$u\\mathscr{P}\\!\\textit{re-Lie}$-graph homology.\n\\end{example}\n\\subsection{On the cobar construction of unital operads}\nFinally, we remark that there is a certain subtlety associated with\ntaking cobar-constructions of unital operads (such as an operad with a\nweakly unital multiplication), as we do. Let $\\mathcal O$ be\nsuch an operad. Then it is easy to see that for any $n\\geq 0$ the\ncomplex $C{\\mathcal O}(n)$ is contractible. However it does not follow\nthat any $C\\mathcal O$-algebra is gauge equivalent to a trivial\none. Indeed, one can take $\\mathcal O$ to be the operad $u{\\mathscr\n  A}\\!\\textit{ss}$ or $u{\\mathscr C}\\!\\textit{omm}$ governing unital\nassociative or unital commutative algebras; then $C\\mathcal\nO$-algebras (on a graded vector space $V$ with zero differential) will\nbe curved $A_\\infty$- or $L_\\infty$-algebras, respectively, and there is\nno reason for an arbitrary curved $A_\\infty$- or $L_\\infty$-algebra to\nbe gauge equivalent to a trivial one.  In fact, even if we let\n$\\mathcal O$ be the operad ${\\mathscr A}\\!\\textit{ss}$ or ${\\mathscr\n  C}\\!\\textit{omm}$, and take the bar construction of it (i.e., of the\nwhole unital operad, rather than just the augmentation ideal as one\nusually does in these cases), we again obtain that $C \\mathcal\nO$-algebras will be ordinary $A_\\infty$- or $L_\\infty$-algebras, with\nthe usual (nontrivial) gauge equivalence relation, even though $C\n\\mathcal{O}$ remains acyclic.\n\nThe explanation of this apparent paradox is that the operad $C\\mathcal\nO$ is \\emph{not cofibrant}; see \\cite{BMahto} for this notion. An algebra\nover $C\\mathcal O$ is a map from $C\\mathcal O$ to an endomorphism\noperad of a dg vector space and this map is not necessarily homotopic\nto zero even though $C\\mathcal O$ is acyclic. A similar phenomenon\noccurs when considering a cobar-construction $(T\\Pi V^*,d)$ for a\nunital associative algebra $V$; dg maps from $T\\Pi V^*$ to the field\n$\\CC$ are the Maurer-Cartan elements in $V$, i.e. the odd elements\n$v\\in V$ for which $dv+v^2=0$. Such elements need not be gauge\nequivalent to zero despite $(T\\Pi V^*,d)$ being acyclic; again,\nprecisely because $(T\\Pi V^*,d)$ is not a cofibrant dga.\n\nFurthermore, given a cyclic operad $\\mathcal O$ as above we can form its $\\Det^d$-modular closure $\\overline{\\mathcal O}^d$ and its naive $\\Det^d$-modular closure $\\underline{\\mathcal O}^d$. Then the same reasoning\nshows that the complexes ${\\mathsf F}\\overline{\\mathcal O}^d((n))$  and ${\\mathsf F}\\underline{\\mathcal O}^d((n))$ are acyclic\nfor $n>0$; here ${\\mathsf F}{\\mathcal O}$ is the Feynman transform of\n$\\mathcal O$.\n\nNext, the complexes  ${\\mathsf F}\\underline{u{\\mathscr C}\\!\\textit{omm}}^1((0))$ and ${\\mathsf\n  F}\\underline{u{\\mathscr A}\\!\\textit{ss}}^1((0))$ are\nnothing but our graph complexes $\\widetilde{\\mathcal G}$ and\n$\\widetilde{\\mathcal G}_r$. For $n > 0$, ${\\mathsf F}\\underline{u{\\mathscr C}\\!\\textit{omm}}^1((n))$ and ${\\mathsf\n  F}\\underline{u{\\mathscr A}\\!\\textit{ss}}^1((n))$ are similar except they are\nspanned by graphs which are additionally equipped with $n$ external\nlabeled edges (legs) which are not allowed to be contracted. When legs\nare present, these graph complexes are therefore acyclic.  However,\nthe vacuum (legless) part  ${\\mathsf F}\\overline{\\mathcal O}^d((n))$  and ${\\mathsf F}\\underline{\\mathcal O}^d((n))$ of the\nFeynman transform need not be acyclic, as our results demonstrate.\n\n\n\n\\vskip 10 pt\n\\noindent\n\\bibliographystyle{amsalpha}\n\\def\\cprime{$'$}\n\\providecommand{\\bysame}{\\leavevmode\\hbox to3em{\\hrulefill}\\thinspace}\n\\providecommand{\\MR}{\\relax\\ifhmode\\unskip\\space\\fi MR }\n\\providecommand{\\MRhref}[2]\n  \\href{http:\/\/www.ams.org\/mathscinet-getitem?mr=#1}{#2}\n}\n\\providecommand{\\href}[2]{#2}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\tPerovskite photovoltaic solar cells (PPSC) have reached efficiency comparable to those of the best crystalline silicon (c-Si) devices within only a few years \\cite{nrel.gov}, thanks to the tremendous electrical and optical properties of the metal-halide hybrid perovskites materials \\cite{Snaith2013,jeon2015,Tress2017}. For single junction PPSC, the best performance yet demonstrated reach or even overpass a short-circuit current density $J_{sc}$ of 25.8 $mA.cm^{-2}$, an open circuit voltage $V_{oc}$ of 1.179 $V$, and a fill factor $FF$ of 0.846, leading to power conversion efficiency $PCE$ of 25.7\\% \\cite{nrel.gov}. These achievements are already quite close to the theoretically expected maximum values, especially for the $J_{sc}$, which is typically of 27.83 $mA.cm^{-2}$ under AM1.5G illumination, given that the band gap energy is estimated in their studies to 1.53 $eV$. Furthermore, tandem PPSC made of at least two different perovskite materials appears promising bust also challenging and less mature. In the case of the most common 2 terminal (2T) configuration, a net performance increase could have been recently demonstrated compared to single junction PPSC \\cite{lin_all-perovskite_2022}. \n\t\n\tAs for every optoelectronic device, light management (LM) in PPSC is of primary importance to reach the best performance. As schematized in Figure \\ref{fig:Interplays}, the LM is expected to impact three key performance parameters. First, thanks to the LM, impinging sunlight can be efficiently collected and then absorbed leading to a high $J_{sc}$. Then the LM can help to enhance the $V_{oc}$ thanks to the so-called photon recycling (PR)\\cite{miller2012strong}, provided a strong external luminescence \\cite{Ross1967} and considering the semiconductor exhibits weak non radiative recombinations. Finally, as recently proposed, the energy yield (EY) tends to substitute to the two previous criteria usually considered under standard test conditions (STC). For the latter, requested studies, however, remain complex since they intend to couple illumination models and angular-dependant optical, thermal and electrical models of the cell for one year \\cite{Peters2018}. However, simply considering the incidence angle in the evaluation of absorption and thus $J_{sc}$ is a first step towards EY improvement.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.6]{Fig\/InterPlays.png}\n\t\t\\caption{The three main correlated aspects that governed Light Management.}\n\t\t\\label{fig:Interplays}\n\t\\end{figure}\n\t\n\t\n\tAs illustrated in Figure \\ref{fig:Interplays}, the LM in PPSC results from three interdependent considerations: the photonic engineering (PE) at the wavelength scale, the material choices, and the nanofabrication processes. Practically, the last two directly define the PPSC architecture.\n\t\n\tIn brief, still regardless of the envisaged semiconductor, the LM can invoke several kinds of PE. Most frequently, it results in anti-reflection (AR) over the whole absorbed spectral range, leading to a larger $J_{sc}$ as well as possibly larger EY if this the AR effect also occurs over a wide angular domain. Moreover, beyond the AR effects, light trapping (LT) mechanisms can be involved at the band edge of the perovskite material, also leading to a larger $J_{sc}$ \\cite{Bermel2007,Zhou2008}, whereas enhanced PR thanks to absorption suppression just above the band gap could enhance $V_{oc}$ \\cite{Sandhu2013}, still provided previously mentioned conditions. This well established PE is reminded in more details in section \\ref{light_management}. the LM could even be extended to the infrared domain to possibly limit the parasitic sub band gap absorption and enhance the thermal radiation, so as to decrease the operating temperature \\cite{dumoulin_radiative_2021}. Finally, for the specific case of 2T tandem PPSC, the LM is required to ensure at the same time absorption optimization and current matching between the two cells. \n\t\n\tIn the specific case of metal-halide perovskites, intrinsic properties make these materials particularly relevant for LT, compared to the standard case of c-Si solar cells. First, metal-halide perovskites are direct band gap semiconductors, the light is then more easily absorbed in a submicron thick layer, where LT and possibly PR can occur at their band edge, as detailed later. Second, their refractive indices are lower than that of c-Si, thus, AR effect is easier to obtain over the spectral range of interest. Lastly, the ability to recycle photons has been observed\/shown in the specific case of lead iodide perovskite material \\cite{Pazos-Outon2016}.\n\t\n\tOther key properties of metal-halide perovskites, mainly electrical and with regards to aging, have direct consequences on the typical architecture of the PPSC. The diffusion lengths of the carriers in the perovskites imply both contacts should be in the vicinity of the photocarrier generation, thereby covering both sides of the cell {\\cite{yang2019enhancing}}. Additionally, one of these contacts should obviously be transparent to the sunlight. The double heterojunction architecture is widespread, leading to the use of additional intermediate layers, the electron\/hole transport layers (ETL\/HTL). Then, the back metallic contact, ideally made of noble metal, can thus act as a back mirror. In any case, a careful optical design is required to keep low parasitic absorption in these surrounding layers. Moreover, an effective encapsulation should be added to prevent from the degradation of the perovskite. It should be carefully chosen considering the impact of its optical properties. \n\t\n\tLet us finally envisage the wide panel of low-cost elaboration techniques and strategies enabling the structuration of the perovskite thin film at various scales, from the sub-micron to the over-micron one. First PPSC mainly relied on a mesoporous architecture, in which the perovskite infiltrated a scaffold. Then, progress in the elaboration techniques of the perovskite in liquid or vapor phases basically results in a better microcrystallization, and thus larger diffusion lengths. Simple planar architectures have therefore replaced mesoporous ones \\cite{Liu2013,Liu2014b,Zhao2019}; thanks to these far larger diffusion lengths, current planar PPSC overpass the performances of the best mesoporous PPSC. \n\t\n\tFollowing these huge modifications, the LM has evolved: if the mesoporous architecture could have been optimized for efficient LM thanks to the scaffold (typically made of TiO$_2$) to enhance the $J_{sc}$, most of the planar cells demonstrated in the last few years (see e.g. supporting information of ref \\cite{Xu2020}) exhibit performances approaching the limits, up to about 90\\% of the maximum achievable $J_{sc}$, mostly by simply taking care of AR.\n\t\n\tHowever, refined the LM still enables additional improvements of the performances towards the limits. For example, reduced optical losses thanks to LT especially at the band edge helps to gain the last missing $mA\/cm^2$, as detailed in the following. In addition, the $V_{oc}$ could be improved thanks to an accurate treatment of the photocarrier recombination mechanisms and the associated rates.\n\t\n\tA few review papers discussed on various demonstrations of LM for PPSC at the time of their writing \\cite{Zhang2018,Zhang2019}. In this paper, we focus on PE mainly in corrugated dielectric, without any additional metallic material inside or at the vicinity of the perovskite material. Indeed, if using metal can induce LT due to plasmonics effect, other phenomena such parasitic absorption can frequently balance this effect \\cite{SiavashMoakhar2020}.\n\t\n\tAfter a brief presentation of the main PPSC architectures and material choices that impact LM, and summarizing the PE, we propose to further analyze various published structures relying on a wave optics approach. Thus, by unraveling the various mechanisms involved, we are able to compare the various proposed strategies to deduce guidelines for further optimization. These guidelines are finally illustrated thanks to simulations of the optical properties of a few case studies.\n\t\n\t\\section{Key PPSC materials and architectures}\n\t\n\tThe LM in the PPSC directly derive from their architecture, i.e. a stack of layers of various materials and thus optical indices, each layer having a thickness of the order of a few tenth of wavelengths, and with possible structurations from small to large scales compared to the wavelength. The main properties of the perovskite materials impacting LM have already been discussed. However, comparisons will appear difficult and need to be treated carefully, due to discrepancies between the structures investigated, as detailed hereafter. \n\t\n\t\\subsection{Patterning of the PPSC}\n\t\\label{possible_architecture}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/Architecture.png}\n\t\t\\caption{Various typical architectures of complete PPSC obtained using nanofabrication. a. Almost flat (possible nanoscale roughness) b. patterned perovskite using \"bottom-up\" approach, i.e. structuration of the ETL\/HTL, then overcoating with perovskite; c. patterned perovskite using \"top down\" approach, i.e. perovskite coating then structuration and overcoating. d. Flat perovskite but patterned surrounding layers and e. Patterned substrate coated with conformal stack. All envisaged patterned can be either periodic or aperiodic.}\n\t\t\\label{fig:Architecture}\n\t\\end{figure}\n\t\n\tLet us present an overview of the main patterning processes and deriving architectures that have been proposed in the literature.\n\tDue to the low diffusion lengths in the first fabricated metal-halide hybrid perovskites, first PPSC used mesoporous architectures to limit recombination. Indeed, TiO$_2$ scaffolds were infiltrated with the perovskite material, enabling a more efficient collection of the carriers, and, in some cases, LM \\cite{Lee2015,Lin2016,Ramos2016}. Nevertheless, as already mentioned in the introduction, most promising approaches relies nowadays in planar architectures. \n\t\n\tIn the following, layers will be considered as almost flat provided their roughness is at a sub-wavelength, nanometre scale, especially thanks to a good crystallinity in the case of the perovskite. \n\t\n\tHowever, a 2D structuration can be introduced in the PPSC stack, notably for single junction PPSC. In this frame, the perovskite layer itself can be one of the patterned layers, or only surrounding layers are structured, and the perovskite can be considered as flat. \n\tThe various envisaged technical solutions within each of the two strategies are sketched in Figure \\ref{fig:Architecture} b-e.\n\n\tPatterning of the perovskite layer can be achieved by various approaches. Such a structuration can first be generated by depositing this active layer on top of patterned layers, for example the ETL\/HTL standing underneath (\"bottom-up\" like approach, Figure \\ref{fig:Architecture} b.); the pattern can be either random \\cite{Xu2020,Huang2016,Pascoe2016}, or periodic \\cite{Paetzold2015,Abdelraouf2016,Wang2018,Kim2019}. Alternatively, the perovskite layer itself can be directly patterned \\textit{after its deposition} (\"top-down\" like approach, Figure \\ref{fig:Architecture} c.), again either in a random \\cite{Wang2018a,Wang2018b} or periodic way \\cite{Schmager2019,Schmager2019b}. \n\t\n\tAnother approach is to realize flat perovskite layers and to corrugate other adjacent layers (Figure \\ref{fig:Architecture} d.), such as the encapsulation layer \\cite{Nguyen2016,Li2019,kim_light_2021}, transport layers \\cite{kang2016b,Qarony2018,wang_coordinating_2021}, possibly associated with the metal back contact \\cite{Wei2017,Zhang2018b}, or only the front contacts \\cite{Hossain2020,Tockhorn2020}, and finally a glass substrate covering layer \\cite{Tavakoli2015,Dudem2016,Jost2017,Peer2017,doi:10.1002\/adom.201900018,Thangavel2020}.\n\t\n\tFinally, the front substrate or the back contact layer can be patterned within a broad range of sizes, from the hundreds of nanometers to almost the millimeter; then the other layers of the stack are conformally deposited, leading to a fully structured cell \\cite{Qarony2018,Wang2016b,Du2016,Shi2017,Soldera2018,Tooghi2020a,Wang2019,Soldera2020,Haque2020} (Figure \\ref{fig:Architecture} e.). It is noticeable that among these numerous papers, only two \\cite{Tockhorn2020,Soldera2020} are dealing with experimental results. Indeed, spin coating of perovskite on a patterned substrate remains challenging. The former mesoporous architecture has recently been revisited under the form of opal like structuring \\cite{Lobet2020}.\n\t\n\t\\subsection{Influence of the choice of Materials}\n\t\n\t\\subsubsection{Perovskite medium}\n\tThe choice of the absorbing material for PPSC results from multiple constraints. It should first exhibit an optimal band gap energy, in the range 1.1 - 1.4 $eV$\\cite{shockley1961detailed}, as well as optimal electrical properties and stability. Then, various deposition processes can be envisaged for a given metal halide perovskite. It may mainly lead to various grain sizes of the material during the crystallization process. For sizes of the order of the wavelength, this can affect LM. \n\t\n\tMethylammonium lead iodide perovskite material (CH$_3$NH$_3$PbI$_3$, MAPI) has been widely considered in the early stages of PPSC development, thanks to its rather simple chemical composition, and despite $E_g$ slightly above the optimal range. \n\tIt is even noticeable that different $E_g$ are reported for MAPI typically from 1.55 to 1.6 $eV$ (see e.g. \\cite{jiang2015}). This implies a more than 7\\% uncertainty on the maximum achievable $J_{sc}$, that is already of the order of the achievable gain using LM as reported in the following. Thus, the quantitative comparison between different studies, even implying the same material within the meaning of the chemical composition, is rather delicate. Differences of aging of the perovskite could also impact the comparison. Finally, various dispersion models can be used to fit the dielectric function \\cite{Li2020b}, leading to differences that also impact the optical indices that are used for simulations. \n\t\n\tFor all these reasons, it has to be emphasized that, in the following, only the relative impact of LM could be put into perspective, but the various given performances cannot be accurately compared. Finally, if MAPI can still be used as a typical case study, other materials have been developed in the past few years. They are more stable and exhibit a slightly lower $E_g$, mainly thanks to the substitution of MA by FA (Formamidinium, (NH$_2$)$_2$CH), possibly with Cs, as can be seen later.\n\t\n\t\\subsubsection{Transport and contact layer materials}\n\tFor the transport layers, mainly polymer materials are used, as well as thin layers of inorganic materials such as TiO$_2$. It is noticeable than materials such as those based on C$_{60}$ fullerene, acting as ETL, exhibit significant absorption at the shortest wavelengths of the solar spectrum, whereas polymers used as HTL mainly absorb at larger wavelengths, close to and above the band gap of the perovskite. Then, there is often a compromise to be found between suitable band levels, high electrical conductivity, and barrier to possible migrations, and low parasitic absorption, even if the thicknesses of these transport layers remain limited.\n\t\n\t\\section{Photonic engineering}\n\t\\label{Photonic concepts description}\n\t\n\t\\begin{figure}[b]\n\t\t\\begin{center}\n\t\t\t\\includegraphics[width=\\linewidth]{Fig\/LightManagement.png}\n\t\t\t\\caption{Main Light Management strategies a. Without peculiar LM, reflectance can be important. b. Using a flat structure, or a patterned that can be considered as homogeneous (e.g. given its subwavelength dimensions), LM can lead to a rather broadband AR effect as well as to Fabry Perot modes that can enhance the absorption. c. Using a random texturation, LM can results from scattering, possibly Lambertian, that can also enhance the absorption. d. Using a periodic or strongly correlated patterned, in addition to AR and Fabry Perot effects, some Light Trapping can occur, drastically enhancing the absorption especially at band edge of perovskite.}\n\t\t\t\\label{fig:LightManagement}\n\t\t\\end{center}\n\t\\end{figure}\n\t\n\tLet us now summarize the various kinds of PE, mostly at the wavelength scale, that can be envisaged to improve LM in both single-junction and 2T multi-junction PPSC, and their impact on geometrical requirements in the PPSC. They are sketched in Figure \\ref{fig:LightManagement}. Most of the strategies derive from those already developed for other kinds of direct or even indirect band gap material used in thin film \\cite{massiot2020progress}.\n\t\n\tIn this frame, LM results from various light-structure interaction mechanisms. These mechanisms occur mainly either in the vertical direction (i.e. orthogonal to the stack) or in the directions of the patterns. Depending on the scale of the structure with respect to the wavelength, three main models can be used: first, at sub-wavelength scales, an effective medium; then, at wavelength scale, wave optics, or, at larger scales, geometrical optics.\n\t\n\t\\subsection{Photonic engineering in multilayer unpatterned stack}\n\t\\label{photonic concepts multilayer}\n\tIt is well known that geometrical optics are mainly suitable when the dimensions are far larger than the wavelength. Thus, it cannot accurately describe the interference effects that occur within the thickness of the numerous thin films that compose the cell. In contrast, the optical indices of medium textured at subwavelength scales can be homogenized in a approximated way using Effective Medium Theories such as the Bruggeman model. Then, descriptions involving wave optics are the most rigorous, and wave optics mechanisms might also be the most promising ones.\n\t\n\tLet us first briefly recall the two basic mechanisms able to enhance light harvesting in a flat stack, i.e. without the assistance of any patterning. In a rather high index, highly absorbing material, a properly designed stack enables to drastically reduce the impedance mismatch with the surrounding environment, whereas in a finite thickness absorbing layer, especially with a low absorption, it is rather a Fabry Perot like approach that can be used, leading to highly enhanced but narrow band absorption. \n\t\n\tWithin a typical PPSC stack, having a metallic contact that can act has a back mirror, these two approaches can be invoked. Schematically, at short wavelength, the large extinction coefficient of the perovskite enables the single pass absorption in the perovskite film, but AR effect is required. At larger wavelengths close to band gap, where the absorption drops, multiple passes are required, the second approach is well suited (Figure \\ref{fig:LightManagement} b.). \n\t\n\tHowever, the exact nature of the layers, and thus their optical indices and thicknesses are restricted by other constraints such as energy band levels, electrical resistance, risk of shunting, or even diffusion barriers for some species. Consequently, parasitic optical absorption has to be carefully studied. Moreover, broadband enhancement of the absorption under normal incidence is unlikely to be obtained using such simple architectures. \n\t\n\t\\subsection{Photonic engineering in multilayer, patterned stack}\n\t\\label{photonic_concepts_multilayer_patterned}\n\tIn addition to the continuum of propagative waves, light can be confined into the discrete set of in-plane, transverse guided modes existing in the stack. It is noticeable that the in-plane wave vector of any of these guided modes is larger than those of the free space modes. Thus these modes cannot be simply coupled from the free space in a perfectly flat stack, without any periodic or aperiodic corrugation. Then, the various kinds of in-plane structurations - having dimensions from the wavelength scale up to two orders larger - all induce diffraction. Whereas the impinging light lies around the normal incidence and has thus negligible in-plane wave vector $k_{in\\parallel}$, the light inside the device is diffracted. Indeed, the electromagnetic field can be expanded over a set of plane waves thanks to the in-plane spatial frequencies induced in the medium by the structuration. The reflected waves also experience diffraction accordingly. Considering the various kinds of structurations and thus of the resulting spatial frequencies, three main cases can be envisaged:\n\t\n\t\\begin{enumerate}[label=\\roman*)]\n\t\t\\item A random structuration tends to induce isotropic diffraction, so Lambertian scattering. Moreover, this scattering does not depend on the wavelength within the absorbed spectral range. It can typically lead to broadband AR effect (Figure \\ref{fig:LightManagement} c.). However the absorption enhancement remains limited. Indeed, given the already large absorption (except at the band edge) of the perovskite, and the back reflection induced by the metallic contact, it will be at most of the order of $4n^2$ with an index lower than other materials, in particular inorganic ones \\cite{Yablonovitch1982}.\n\t\t\\item A correlated disordered structuration that enhances a specific set of spatial frequencies, leads to a spectral dependent absorption enhancement. A careful choice of the sizes of the patterns is required to enhance the absorption in the desired spectral range \\cite{Vynck2012nmat}.\n\t\t\\item As a particular case from the previous case, periodic, so-called photonic crystal (PC) structurations can be envisaged. Most of them consists in a square or triangular lattice a simple pattern such as pillars or holes with a vertical profile, or smoother pyramidal, or even parabolic profiles. They mainly lead to discrete modal properties. A few complex patterns arranged in a periodic way have been explored \\cite{martins2013deterministic,Oskooi2014,Bozzola2014,ding2016design,Li2020}, enabling an absorption enhancement in a targeted spectral range thanks to a larger local density of modes. \n\t\\end{enumerate}\n\t\n\t\n\tIn any case, the diffraction efficiency, i.e. the amount of light that is effectively diffracted, and that is thus intended to be absorbed, strongly depends on the pattern of the structuration, including the optical index contrast, as well as on the diffraction order $p$. Whereas, as already mentioned, for scattering, the diffraction efficiency hardly depends on the direction, for diffraction by periodic structures, it is generally larger for the first diffraction order.\n\t\n\tIn this frame, it is noticeable that a LT phenomenon rigorously occurs when the impinging light is coupled thanks to, at least, strongly correlated structure, ideally periodic patterns, having respectively a correlation length or a period $\\Lambda$, within at least one guided mode of the stack (Figure \\ref{fig:LightManagement} d.). According to a perturbation approach \\cite{CamarilloAbad2020}, in a 1D case for sake of illustration without losing generality, a phase matching condition can be written under the form:\n\t\n\t\\begin{equation}\n\t\t\\beta_m=k_{in\\parallel}+p \\frac{2\\pi}{\\Lambda}\n\t\t\\label{QGM}\n\t\\end{equation}\n\tWhere $\\beta_m$ is the in-plane wave number of the $m^{th}$ guided mode of the stack. Due to the time reversal symmetry, the light coupled into the guided mode, if not fully absorbed, can be decoupled after a certain length. These modes are thus so-called \"quasi-guided mode\" (QGM).\n\t\n\tSuch modes can drastically enhance the light path inside the waveguide layer. Thus, if the in-coupled QGM is mainly confined in the perovskite layer rather than in the surrounding layers, the useful absorption is enhanced. Moreover, it also depends on the absorption; more precisely, the absorption is optimized at critical coupling, i.e. when an equilibrium is reached between diffraction efficiency and absorption. Consequently, the spectral bandwidth of the LT is set since the quality factor of the mode at the optimal absorption is of the order of $n\/2\\kappa$, $n,\\kappa$ being respectively the real and imaginary parts of the complex optical index \\cite{Park2009}.\n\t\n\tSuch a resonant LT appears promising to enhance the low absorption close to the band gap of the perovskite, provided it is not reduced elsewhere at shorter wavelengths. More precisely, within the width of the resonance, the absorption can be significantly larger than the previously mentioned broadband limit \\cite{Yu2010}. It could also be used in tandem PPSC mainly to optimize the $J_{sc}$, especially provided LT occurs in a guided mode specifically confined in one of the perovskite layers. As already discussed, for the other spatial frequencies that do not lead to LT, patterning can still result in a rather broadband AR effect, in addition to the one resulting from the design of the stack. Further analysis of all these effects can be found elsewhere \\cite{Zanotto2010,Brongersma2014}. \n\t\n\tIn addition to these PE approaches that contribute at first order to enhance the absorption and so the $J_{sc}$, other strategies could be used to enhance the $V_{oc}$, using a PR mechanism. In a simplified way, PR is supposed to induce at the $V_{oc}$ operating point a high density of photons into the absorbing materials, and so to give a chance to electrons-holes pairs to recombine radiatively rather than in a non radiative way. Typically, this requires at first order to inhibit the radiation, and consequently the absorption, in the luminescence spectral domain of the semiconductor. Thus, enhancing simultaneously the $J_{sc}$ using LT and the $V_{oc}$ using PR is a compromise \\cite{Sandhu2013}.\n\t\n\tIn any case, it has to be emphasized that all these effects should ideally target the perovskite layer only and not enhance the parasitic absorption in the other layers.\n\t\n\t\\subsection{Main geometrical requirements on patterns for Light Trapping}\n\t\n\tAs mentioned previously, 2D in-plane structurations are able to diffract the impinging sunlight under a quasi-normal incidence into the large number of modes of the PPSC, possibly including the guided modes.\n\tIndeed, the architectures of both single and multi-junction PPSC reviewed in the section \\ref{possible_architecture} can be schematized as one or several sub-micron thick high index perovskite layers lying between two lower indices ETL and HTL and possibly also lower index layers in the multi-junction case. The whole is lying on medium index TCO on a low index glass substrate and coated on top with metal.\n\tSuch structures exhibit several guided modes in both TE and TM polarization states.\n\tLet us first make the assumption of weak corrugation that doesn't change significantly the effective indices of the stack of active layers lying on a substrate. The largest effective index modes - typically 2.2 - are mainly confined in the perovskite, whereas the other modes, whose effective indices are below 1.9, can be confined in the whole stack of layers.\n\tIn this frame, for normal incident light ($k_{in\/\/}=0$) and for the highest efficient coupling at order $p=1$, equation (\\ref{QGM}) can be rewritten under the form:\n\t\\begin{equation}\n\t\t\\beta_m=\\frac{2\\pi N_{eff,m}}{\\lambda}=\\frac{2\\pi}{\\Lambda}\n\t\t\\label{phase_matching_equation}\n\t\\end{equation}\n\twhere $\\Lambda$ is the period, and could even be a characteristic length in correlated patterns. This 1D case can easily be extended to 2D patterns. \n\t\n\tThus, to be able to efficiently couple into the fundamental guided mode, which is mainly confined into the perovskite, at wavelengths corresponding to its band edge, e.g. 750 $nm$, $\\Lambda$ should be around 340 $nm$, whereas it should be larger than 400 $nm$ for higher order guided modes. This results in a pseudo-guided mode, as discussed at the end of section \\ref{photonic_concepts_multilayer_patterned}. At shorter wavelengths, such structurations can also couple light into the various low effective index modes (including Fabry Perot like modes) of the stack, that may also result in a broadband AR effect. \n\tThis typically implies that all the micron scale or even larger patterns in any of the layers of the stack are rather unlikely to couple the impinging light into guided modes in the low absorption domain, where such a coupling is of main interest. Such patterns only induce coupling into the Fabry Perot like modes of the stack and\/or diffraction at high orders with a limited efficiency.\n\t\n\t\\section{Examples of reported light management strategies in PPSC}\n\t\n\tWe report here on some studies to illustrate concrete LM strategies. For any case, the chosen perovskite material and its thickness are of primary importance. First of all, planar PPSC as envisaged in the section \\ref{possible_architecture} for their architecture and in section \\ref{photonic concepts multilayer} for the corresponding PE are discussed, putting mainly into evidence the importance of the thicknesses on LM. Then, results on in-plane patterned PPSC are reviewed, with focuses on the involved PE.\n\t\n\t\\subsection{Optimization of the thicknesses in planar PPSC}\n\t\n\tMost studies related to thickness optimization only focus on that of the perovskite layer, and report mainly on the derived useful absorption, as detailed just below. However, the whole stack has been considered in a few papers, with a view to investigate the parasitic absorption, as discussed later.\n\t\n\t\\subsubsection{Choice of the perovskite thickness in single junction PPSC}\n\t\n\tAs already mentioned, the optimal thickness of the sole perovskite layer is chosen as a compromise between maximizing optical absorption and keeping low bulk recombination of the carriers. In addition to the analysis of a few seminal cases of single junction PPSC, we present a synthesis of several studies, mainly experimental, that report on the effect of the perovskite layer thickness of planar single junction PPSC, first and mainly made of MAPI, then of other lead-halide perovskites. The detailed performances of the representative cells are summarized in Table \\ref{tableau 1} and, when available, the EQE are plotted in Figure \\ref{fig:EQE_tableau_1} in a spectral range limited from 400 to 800 $nm$ for sake of simple comparison.\\\\\n\t\n\t\\paragraph{MAPI}\n\t\n\tOne of the first high efficiency planar PPSC \\cite{Liu2013} which is made of an about 300 $nm$ thick MAPICl layer exhibited a $J_{sc}$ of 21.5 $mA.cm^{-2}$. In this seminal paper, M. Liu \\textit{et al.} already mentioned the possibility of an optimum thickness due to the balance between absorption and recombination. It was shown later by Y. Da \\textit{et al.} \\cite{Da2018} that this cell suffered from optical losses that represented 11.3\\% of the energy losses, and especially more than 20\\% of the achievable $J_{sc}$. This was mainly related to reflectance and parasitic absorption in the FTO. This thus emphasizes the importance of minimizing these effects by a careful design of the stack.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/T1_final.png}\n\t\t\\caption{EQE of various PPSC also reported in Table \\ref{tableau 1} \\cite{Liu2014b,Momblona2014,Liu2014,Liu2019b,Correa-Baena2016,jeong_pseudo-halide_2021,Ball2015,Lin2015}.}\n\t\t\\label{fig:EQE_tableau_1}\n\t\\end{figure}\n\t\n\tThen, the first high-efficiency PPSC, solution processed at room temperature, proposed by D. Liu \\textit{et al.} \\cite{Liu2014b} was made of an about 300 $nm$ thick MAPI layer. A $J_{sc}$ of 20.4 $mA.cm^{-2}$ was obtained, so about 5 $mA.cm^{-2}$ below the maximum achievable $J_{sc}$ given a typical $E_g$ of MAPI. The EQE (see Figure \\ref{fig:EQE_tableau_1}) was rather flat and remained lower than 0.8. The deficit could be due to both optical losses (too low absorption in the perovskite) and electrical losses (too high recombination). \n\t\n\tC. Momblona \\textit{et al.} \\cite{Momblona2014} conducted an experimental study on PPSC made of MAPI deposited by thermal evaporation, especially by increasing its thickness from 160 to 900 $nm$. If the $J_{sc}$ increased rather monotonously with the perovskite layer thickness, it never exceeded 20.4 $mA.cm^{-2}$. The main parameter that decreased with increasing perovskite layer thickness was the fill factor. As a result, the cell owning optimal performance was obtained for a perovskite layer thickness around 300 $nm$. More precisely, whereas the $J_{sc}$ slightly increases, both the $V_{oc}$ and the $FF$ significantly decreased at thicknesses larger than 300 $nm$. At these early ages of PPSC, the corresponding optimal PCE was 12.7\\%, with a $J_{sc}$ of 18.8 $mA.cm^{-2}$. Then, leading further investigations on the $J_{sc}$ deficit, it could be derived from the EQE (see Figure \\ref{fig:EQE_tableau_1}) that optical losses in almost 300 $nm$ thick perovskite were important at both short wavelengths and from 600 to 700 $nm$, compared to the 900 $nm$ thick perovskite layer that fully absorbs. The enhanced absorption in the range 700 to 750 $nm$ is likely due to a cavity effect. Thus, even if such a thickness of about 300 $nm$ appears to be optimal under the PCE point of view, it remains too thin to avoid optical losses. In addition, even with such thin absorbing layers, such cells still suffers from electrical losses. \n\t\n\tIn a similar way, D. Liu \\textit{et al.} \\cite{Liu2014} also then studied PPSC consisting in a ITO\/ZnO\/MAPI\/P3HT\/Ag stack, in which MAPI was obtained using thermal evaporation or solution processing. Again, in spite of a larger absorption and limited short-circuit for thicker films, a limited diffusion length led to an optimum thickness of the MAPI layer of about 330 $nm$ for the solution processed PPSC. It is noticeable that the corresponding EQE (see Figure \\ref{fig:EQE_tableau_1}) was rather low and flat. This could be related to scattering due to sub-micron crystallinity that mitigates cavity effects at larger wavelengths.\n\t\n\tMore recently, Y. Liu \\textit{et al.} \\cite{Liu2019b}, still using MAPI, similarly concluded that there were more power loss in PPSC when the thickness of the MAPI layer was either less or more than about 300 $nm$. They noticed that the grain size increased with the thickness, in favor of a larger thickness. However, it was noticeable that, at wavelengths larger than 650 $nm$, the EQE of the 303 $nm$ thick MAPI PPSC (see Figure \\ref{fig:EQE_tableau_1}) was lower than the one of the 564 $nm$ thick MAPI PPSC. This is related to too low absorption in this spectral range. \\\\\n\t\n\t\n\t\\paragraph{Other lead-halide perovskites}\n\t\n\tAs an alternative, J.P. Correa-Baena \\textit{et al.} \\cite{Correa-Baena2016} used multi-cations perovskites materials; these were recently under the spotlight, especially considering their better stability \\cite{jeon2015compositional}. Using FA$_{0.83}$MA$_{0.17}$Pb(I$_{0.83}$Br$_{0.17}$)$_3$ together with mesoporous TiO$_2$, the thickness that maximises the PCE was found to be at least 480 $nm$. The corresponding $J_{sc}$ was almost 24.4 $mA.cm^{-2}$. Considering the band gap of this material, reported to be about 1.63 eV \\cite{jacobsson2016exploration}, this is already the maximum achievable value. The EQE (see Figure \\ref{fig:EQE_tableau_1}) is noticeably high (also thanks to the an IQE of almost 100\\%) and flat at large wavelength. This could be related to the mesoporous TiO$_2$ that induces some scattering effect. \n\t\n\tUnlike the previous papers, M. Rai \\textit{et al.} \\cite{Rai2020} focused on the $V_{oc}$ deficit related to non-radiative recombination as a function of the thickness, expressed in terms of molar concentration of the precursor solution. This time, the studied perovskite was Cs$_{0.2}$FA$_{0.8}$Pb(I$_{0.85}$Br$_{0.15}$)$_3$, whose band gap has been found to be 1.62 $eV$, so close to the one of MAPI. They noticed as well that the $V_{oc}$ decreases with the thickness, whereas the $J_{sc}$ increases. As for the previously mentioned paper, the grain size increases with the thickness; the molar concentration of the precursor solution that maximizes PCE correspond to thickness of the perovskite layer of about 400 $nm$, so 30\\% larger than the one usually obtained using MAPI. This might be due to a larger diffusion length of the photocarriers for this perovskite.\n\t\n\tK.B. Nine \\textit{et al.} \\cite{nine_performance_2020} simulated optically and electrically the effect of the thickness of several FACsPI layers, leading to a larger optimal thickness, of almost 600 $nm$, so twice the thickness usually found using MAPI. Indeed, better electrical properties of this perovskite material, especially carrier mobility, allow such a thicker layer. \\\\\n\t\n\t\\paragraph{Synthesis}\n\t\n\t\n\t\n\t\\begin{table}[!h]\n\t\t\\begin{tabular}{@{}lllllll@{}}\n\t\t\t\\hline\n\t\t\tMaterial & Thickness ($nm$) & $J_{sc} (mA.cm^{-2})$ & $V_{oc} (V)$ & $FF$ & PCE (\\%) &  Ref \\\\\n\t\t\t\\hline\n\t\t\tMAPICl & 330 & 21.5 & 1.07 & 0.67 & 15.4 & \\cite{Liu2013} \\\\  \n\t\t\tMAPI & 300  & 20.4 & 1.03 & 0.749 & 15.7 & \\cite{Liu2014b} \\\\    \n\t\t\tMAPI & 285  & 18.8 & 1.07 & 0.63 & 12.7 & \\cite{Momblona2014} \\\\\n\t\t\tMAPI (solution process) & 330  & 17 & 0.94 & 0.62 & 11.8 & \\cite{Liu2014} \\\\\n\t\t\tMAPI & 303 & 21.3 & 1.07 & 0.715 & 18.4 & \\cite{Liu2019b} \\\\\n\t\t\tFAMAPBrI & 480 & 24 & 1.14 & 0.75 & 20.8 & \\cite{Correa-Baena2016} \\\\\n\t\t\tFAPI & 600 & 26.35 & 1.189 & 0.817 & 25.59 & \\cite{jeong_pseudo-halide_2021}\\\\\n\t\t\tMAPI & 492 & 21.56 & nr & nr & nr & \\cite{Ball2015}\\\\\n\t\t\tMAPI & 350 & 21.9 & 1.05 & 0.72 & 16.5 & \\cite{Lin2015} \\\\  \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\t\\caption{Experimental performances of planar PPSC with optimized perovskite thickness (nr: not reported)}\n\t\t\\label{tableau 1}\n\t\\end{table}\n\t\n\tIt is noticeable that these studies tend to an optimal thickness of about 300 $nm$ when using MAPI. This thickness could lead to state-of-the-art PCE of J. Li \\textit{et al.} \\cite{LI2018331}, with noticeably high $J_{sc}$ of 24.1 $mA.cm^{-2}$, mainly thanks to an ETL including graphdiyne, which improves electrical properties. Then, FAPI is the perovskite used for the best up-to-date single junction PPSC \\cite{jeong_pseudo-halide_2021} which performance were mentioned in the introduction, with a thickness of about 600 $nm$, even if, among numerous refinements of the architecture, the rough, about 50 $nm$ thick TiO$_2$ layer already may help to trap the light. \n\t\n\tIt appears that choosing a thickness smaller than the one maximizing the $J_{sc}$ could help to keep low resistance, so high FF and PCE. Then, with such a $J_{sc}$ around 90\\% of the maximum achievable $J_{sc}$, LM can optimize the absorption in the perovskite, especially at photon energies close to the band gap, where the absorption starts to decrease.\n\t\n\tIn any case, it is of high interest to understand the reasons why large variations of the internal perovskite absorption spectrum have been obtained for similar thicknesses of the same perovskite material (MAPI), as can be noticed through the cases 2 to 5 of Table \\ref{tableau 1} and the corresponding EQE plotted in Figure \\ref{fig:EQE_tableau_1}. They might be due to differences in the surroundings layers (thicknesses, indices) or in the perovskite itself, like microcrystallinity that can induce scattering. These phenomena will be described in the following, starting with the analysis of the effect of the surroundings layers in a unpatterned stack. \n\t\n\t\\subsubsection{Single junction unpatterned PPSC stack analysis}\n\tIf there can be an optimal thickness for the perovskite layer, the effect of the other layers, especially on the absorption and reflectance, should also be analyzed. Here, we review a selected set of publications focusing on this issue.\n\t\n\tJ.M. Ball \\textit{et al.} \\cite{Ball2015} studied planar PPSC using an optical model based on the transfer-matrix formalism with experimentally determined complex refractive index data. They focused on a typical stack made of FTO\/TiO$_2$\/MAPI\/SPIRO-OMeTAD\/Au. Under the assumption of $E_g = 1.56 eV$, the detailed analysis revealed that for a calculated $J_{sc}$ of 21.56 $mA.cm^{-2}$, parasitic absorption induces a lack of about 1.72 $mA.cm^{-2}$, and reflection losses corresponding to a $J_{sc}$ decrease of 1.36 $mA.cm^{-2}$, so a total of more than 10\\% of the collected current. On the other hand, the IQE losses can be estimated to another 10\\%. Moreover, most of the optical losses take place in the 420 $nm$ thick FTO. This underlines the importance of taking care of all the layers of the stack.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/lin_cavite.png}\n\t\t\t\\caption{Optical-field distribution for four wavelengths: for $\\lambda < 500 nm$ the optical-field distribution follows the Beer\u2013Lambert law and no optical field reaches to the back electrode as a result of the high absorption coefficient. In such cases the absorption is saturated and no optical interference occurs. For $\\lambda > 500 nm$ the optical field is governed by low-finesse cavity interference.}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]\n\t\t\t{Fig\/lin_eqe.png}\n\t\t\t\\caption{EQE spectra of devices with different MAPI-layer thicknesses. For $\\lambda < 500 nm$ there is minimal influence of the film thickness, but for $\\lambda < 500 nm$ the EQE is strongly thickness dependent because of the optical interference (Fabry Perot like modes)}\n\t\t\\end{subfigure}\n\t\t\\caption{PPSC properties as a function of the MAPI layer thickness \\cite{Lin2015}. Reproduced with permission}\n\t\t\\label{fig:lin eqe}\n\t\\end{figure}\n\t\n\tQ. Lin \\textit{et al.} \\cite{Lin2015} deeply investigated the properties of PPSC made of MAPI. Thanks to their measurement of the optical indices of the MAPI and of the other materials, they simulated the properties of PPSC, assuming an IQE of 100\\%. They identified the two absorption regimes of a flat PPSC, previously discussed in section \\ref{photonic concepts multilayer}, namely the single pass and the cavity regime. Thus, thanks to optical cavity effects at wavelengths larger than 500 $nm$, the noticeably high EQE of their cell (see Figure \\ref{fig:EQE_tableau_1}) appeared to be optimized for a thickness of about 370 $nm$ (see Figure \\ref{fig:lin eqe}). The experimental data noticeably confirmed the simulation study, since a cell having a 350 $nm$ thick MAPI layer led to the best properties, $J_{sc}=21.9 mA.cm^{-2}$, $V_{oc}=1.05 V$, $FF=0.72$ and $PCE=16.5\\%$. Given the resulting limited thickness of the MAPI, the obtained $J_{sc}$ was 87\\% of the maximum achievable value. These remarkable results were also obtained thanks to the optimization of the p and n type interlayers that were used to optimize the heterojunctions. It was especially shown that the PCDTBT p type interlayer has to be as thin as possible ( $\\sim 5 nm$), because of its absorption, even if it enabled a nice crystallinity of the MAPI. Otherwise, given the almost 100\\% IQE, most of the 4 to 5 $mA.cm^{-2}$ decrease of the photocurrent (depending on the exact value of the band gap energy of the considered MAPI) corresponds to optical losses, which could be either parasitic absorption or reflectance.\n\t\n\tM. Van Eerden \\textit{et al.} \\cite{VanEerden2017} analyzed the optical losses of 350 $nm$ thick MAPI layers PPSC. Within the considered cell, most of the losses are related to reflectance (equivalent to 4.2 $mA.cm^{-2}$), whereas the parasitic absorption remains limited to the half, and despite a subwavelength roughness (typically from 10 to 50 nm RMS) at both ITO\/TiO$_2$ and MAPI\/SPIRO-OMeTAD interfaces. \n\t\n\t\\subsubsection{Analysis of unpatterned, up-to-date record 2T tandem PPSC}\n\t\\label{tandem_plan}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/xiao_tandem_schema.png}\n\t\t\t\\caption{Stack of materials }\n\t\t\t\\label{fig:xiao tandem schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/xiao_tandem_jsc.png}\n\t\t\t\\caption{Simulated $J_{sc}$ as function of wide-band gap and narrow-band gap perovskite layer thicknesses.}\n\t\t\t\\label{fig:xiao tandem jsc }\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.3\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/xiao_tandem_eqe.png}\n\t\t\t\\caption{EQE of both subcells.}\n\t\t\t\\label{fig:xiao tandem eqe }\n\t\t\\end{subfigure}\n\t\t\\caption{Best up-to-date 2T tandem, all perovskite PPSC \\cite{xiao_all-perovskite_2020}. Reproduced with permission.}    \n\t\\end{figure}\n\t\n\tWhen compared to other tandem architectures, the 2T case leads to possibly limited optical losses, but such cells require an equilibrium of the current densities delivered by the two subcells. Among recent results \\cite{wang_prospects_2021}, K. Xiao \\textit{et al.} \\cite{xiao_all-perovskite_2020} have deeply investigated 2T tandem PPSC (see Figure \\ref{fig:xiao tandem schema}) both numerically and experimentally. They have shown using optical simulations that the reachable $J_{sc}$, which is indeed the smallest $J_{sc}$ delivered by each of the subcell, strongly depends on the thicknesses of each perovskite layer (see Figure \\ref{fig:xiao tandem jsc }), provided that thicknesses of the other layers have been set taking into account both electrical and optical properties (low parasitic absorption). Indeed, it appears that the top perovskite has to be thick enough to absorb enough light, but that a too thick layer also absorbs too much light, and finally that the overall $J_{sc}$ is limited by the bottom sub cell. The thickness of the perovskite bottom sub cell has to be large enough, more than 1.1 $\\mu m$.\n\tWhen fabricated on a small surface, such a cell exhibits even a slightly larger $J_{sc}$ than simulated, likely due to the too pessimistic predicted absorption. Even if this result remains a record at the time of its publication, its EQE (see Figure \\ref{fig:xiao tandem eqe }) reveals some non-idealities. Indeed, in addition to the relatively low values of the EQE plateaus, there is an overlap of the absorption domains of the two perovskite materials between 500 and 700 $nm$. This means that some undesirable thermalization still occurs below 700 $nm$ in the bottom sub cell. \n\t\n\t\\subsection{In-plane structuration for light management in single junction PPSC}\n\t\\label{light_management}\n\t\n\tAt this stage, it appears that even the best reported cells exhibit an EQE that can still be improved using PE at the wavelength scale, even if most of them already benefit from the roughness of the microcrystalline perovskite.\n\t\n\tIn this section, we will highlight important criteria that should be met to ensure a fair demonstration of LM. The report on selected studies is organized as a function of the various architectures as envisaged in section \\ref{possible_architecture}.\n\t\n\t\\subsubsection{General criteria for fair LM demonstration}\n\tThe impact of LM on the performance of a solar cell can be demonstrated by comparing a patterned device with an unpatterned reference. In case of superficial structurations or with the addition of specific LM layers, the absorption improvement appears generally obvious compared to the same unpatterned stack, or, even more favorable, to the unpatterned stack without the additional flat LM layer. Moreover, integrating LM layer could lead to a better charge collection, by decreasing the carrier path; the specific LM effect would then be difficult to distinguish. Therefore, great care should be taken in the definition of a reference structure or device. In particular, the volume of perovskite materials should be as close as possible in both structures. Additionally, the net enhancement of EQE, efficiency or even yield value can be accurately estimated only if the unpatterned reference has been optimized first. These conditions should be met in order to discuss LM for PPSC performance optimization. \n\t\n\t\\subsubsection{Periodic patterning for resonant LT}\n\t\\paragraph{Superficial structuration and light management layers}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/peer_bismwas_MLA_schema.png}\n\t\t\t\\caption{Schematic of the PPSC.}\n\t\t\t\\label{fig:peer bismwas_schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/peer_bismwas_absorption.png}\n\t\t\t\\caption{Simulated absorption spectrum for microlens array of period $~700 nm$ and height $800 nm$. The absorption of flat solar cell is overlaid for comparison.}\n\t\t\t\\label{fig:peer bismwas_absorption}\n\t\t\\end{subfigure}\n\t\t\\caption{PPSC with microlens array on air-glass side \\cite{Peer2017}. Reproduced with permission.}\n\t\\end{figure}\n\t\n\tA. Peer \\textit{et al.} \\cite{Peer2017} simulated planar PPSC with a stack including a 400 $nm$ thick MAPI layer. The cell lay on top of a 700 $\\mu m$ thick glass substrate. A micro lens array was then added on the top face of the glass substrate, by imprinting a polymer such as polystyrene (see Figure \\ref{fig:peer bismwas_schema}). Each micro lens of the triangular lattice had a smooth profile close to truncated pyramids. For an aspect ratio period\/height of the micro lenses close to 1, and a period of 700 $nm$, optimized gain of 6.3\\% of the $J_{sc}$ was reported compared to the flat reference, whose EQE noticeably shown dips likely due to cavity effects in the cell (see Figure \\ref{fig:peer bismwas_absorption}). Then, according to the EQE of the cell coupled to the micro lens array, the overall resulting broadband enhancement of the absorption is due at lower wavelength to AR effect, and to LT at the band edge. The optimal period of the pattern is rather large, in agreement with a low effective index of the coupled guided mode. It means that LT occurs in the mode that is hardly guided in the perovskite, but that has a nonnegligible overlap with the pattern. This overlap remains anyway limited, as revealed by the high quality factor of the resonance. Therefore, the absorption reaches almost 1, so is close to critical coupling conditions.\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/wei_2017_adv_Energy_mat_afm.png}\n\t\t\t\\caption{SEM (left)  and  AFM (right) images of  the spin-coated layer (top) of grating patterned layer (middle) and moth-eye  patterned layer (bottom).}\n\t\t\t\\label{fig:wei 2017 adv Energy mat afm}    \n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/wei_2017_adv_Energy_mat_EQE2.png}\n\t\t\t\\caption{External quantum efficiency  (EQE) spectra of PPSC, and relative enhancement obtained by dividing the spectra of grating and moth-eye patterned devices by that of flat one.}\n\t\t\t\\label{fig:wei 2017 adv Energy mat perf}\n\t\t\\end{subfigure}\n\t\t\\caption{PPSC with bio inspired PCBM HTL \\cite{Wei2017}. Reproduced with permission.}\n\t\\end{figure}\n\t\n\tJ. Wei \\textit{et al.} \\cite{Wei2017} fabricated bioinspired back electrodes by imprinting the HTL made of PCBM before the conformal deposition of the Bphen\/Ag back contact. Patterns were either a periodic sinusoidal 1D grating or a uniform 2D moth-eye structure (see Figure \\ref{fig:wei 2017 adv Energy mat afm}), with a typical pitch of about 600 $nm$. It has then been experimentally shown (see Figure \\ref{fig:wei 2017 adv Energy mat perf}) that the EQE of the 240 $nm$ thick MAPICl PPSC in increased by 10\\% up to 40\\% at the band edge, mainly thanks to the 2D moth eye pattern. The overall $J_{sc}$ increased by 14.3\\%, mainly due to absorption increase, but a lower series resistance was also measured for patterned cells, which led to a slight increase of the IQE. As shown by there FDTD simulations of periodic structures, this is due light diffraction into the guided modes, rather than plasmonic effects. It can be confirmed by Fourier Analysis of the top view of the moth eye pattern that a significant part of the spatial frequencies lies in the optimal range for LT (see Figure SI-\\ref{fig:Supplemental Fourier Analysis Wei}).\n\t\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/T2_final.png}\n\t\t\\caption{EQE of the patterned PPSC (solid line) and of their planar references (dashed line) (left) and EQE enhancement ((right) of reported simulated $J_{sc}$ enhancements of Table \\ref{tableau pattern mapi}, mainly resulting from Light Management, for MAPI single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified photonic concept \\cite{Peer2017,Wei2017,kim_light_2021,Schmager2019,Du2016,Hossain2020,Qarony2018}}\n\t\t\\label{fig:EQE_table_2}\n\t\\end{figure}\n\t\n\t\n\tH. Kim \\textit{et al.} \\cite{kim_light_2021} simulated nanosphere arrays of TiO$_2$ conformally coated with silica on a standard MAPI PPSC stack, with a MAPI thickness limited to 100 $nm$. Provided suitable spacing and nanosphere diameter, the $J_{sc}$ could jump from 14 to 18.7 $mA.cm^{-2}$. Authors invoked the Mie scattering effect of the array to explain the enhancement, but it appears that the FEM simulation has likely been done using periodic boundary conditions, and thus that a LT effect occurs at several wavelengths, such as 705 or 790 $nm$ for those at the band edge of MAPI. This is in line with the larger EQE enhancement close to the band edge, as can be seen in Figure \\ref{fig:EQE_table_2}. It remains that the overall $J_{sc}$ is anyway limited due to the unusually low thickness. \\\\\n\t\n\t\\paragraph{Structuration of the perovskite} \n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/schmager_solmat_schema.png}\n\t\t\t\\caption{Layer stack of the simulated patterned PPSC, along with associated layer thicknesses. Three configurations of the active layer are simulated (1) a planar reference, (2) a cylindrical indention into the perovskite layer of variable depth; and (3) a hole geometry which corresponds to the maximum cylindrical indention.}\n\t\t\t\\label{fig:schmager solmat schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/schmager_solmat_simul.png}\n\t\t\t\\caption{Absorptance in the perovskite layer of the hole pattern and the 120 $nm$ deep indentation compared to the corresponding planar reference and the theoretical Yablonovitch limit for two different initial perovskite layer thicknesses of (left) 200 $nm$ and (middle) 300 $nm$. $J_{sc}$ for all data are compared (right). The displayed nanophotonic patterns employ a geometrical $ff=0.4$ and a period of 380 $nm$.}\n\t\t\t\\label{fig:schmager solmat simul}\n\t\t\\end{subfigure}\n\t\t\\caption{Various 2D in plane patterns of the MAPI layer in a PPSC \\cite{Schmager2019}. Reproduced with permission}\n\t\\end{figure}\n\t\n\tR. Schmager \\textit{et al.} \\cite{Schmager2019} simulated a classical MAPI PPSC (see Figure \\ref{fig:schmager solmat schema}). Compared to the planar configuration, etching of the MAPI layer was envisaged, thanks to a square lattice of holes, filled with Spiro-OMeTAD, with various etching depths. To avoid any short circuit, an optimized partial etching of 120 $nm$ led to a 5.6\\% increase of the $J_{sc}$, compared to the initially flat 300 $nm$ thick MAPI layer, provided an equivalent volume of perovskite. It results from a LT at the band edge, but without any sharp resonance, as can be observed on the spectral response of other patterns (see Figure \\ref{fig:schmager solmat simul}).\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/schmager_solmat_carac.png}\n\t\t\\caption{External quantum efficiency and reflectance spectra of the planar and nanoimprinted perovskite solar cells. The nanoimprinted perovskite layer has a period of 480 $nm$. The nanoimprinted perovskite solar cell shows enhanced absorption and current generation close to the band gap. For energies below the band gap (wavelength larger than 790 $nm$) discrete peaks are visible in the reflectance measurement. \\cite{Schmager2019b}. Reproduced with permission.}\n\t\t\\label{fig:schmager solmat carac}\n\t\\end{figure}\n\t\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/T3_final.png}\n\t\t\\caption{EQE of the patterned PPSC (solid line) and of their planar references (dashed line) (left) and  EQE enhancement ((right) of Table \\ref{tableau pattern exp}, mainly resulting from Light Management, for  single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified photonic concept \\cite{Schmager2019b,Tockhorn2020,Paetzold2015,Dudem2016,Jost2017,Zhang2018b,Pascoe2016}}\n\t\t\\label{fig:EQE_table_3}\n\t\\end{figure}\n\t\n\tThe same group \\cite{Schmager2019b} also realized nanoimprinted PPSC made of Cs$_{0.1}$(FA$_{0.83}$MA$_{0.17}$)$_{0.9}$Pb(I$_{0.83}$Br$_{0.17}$)$_3$. Their experimental results show a relative improvement of 2\\% of the PCE compared to the planar reference for the complete, gold coated solar cell. This was obtained thanks to an increase of the $J_{sc}$ from 19.1 to 19.4 $mA.cm^{-2}$ and noticeably identical $V_{oc}$ and $FF$.The EQE of the patterned cell was improved at wavelengths larger than 680 $nm$. A coupling to a quasi-guided mode is observed in the perovskite band edge (see Figure \\ref{fig:schmager solmat carac}), leading to a significant EQE enhancement (see Figure \\ref{fig:EQE_table_3}). \\\\\n\t\n\t\\paragraph{Textured substrate}\n\t\n\t\\begin{figure}[!h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/du_john_schema.png}\n\t\t\t\\caption{Three-dimensional perspective  and two-dimensional side view of the inverted vertical cone perovskite unit cell.}\n\t\t\t\\label{fig:du john schema}\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.45\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/du_john_simul.png}\n\t\t\t\\caption{Simulated absorption and reflection spectra of the optimized (period $a = 600 nm$, radius $R = 380 nm$) inverted cone PC and planar PPSC. The MAPI thickness of the planar cell is 180 $nm$, corresponding to the \"equivalent bulk thickness\" of MAPI in the inverted-cone PC device.}\n\t\t\t\\label{fig:du john simul}\n\t\t\\end{subfigure}\n\t\t\\caption{MAPI PPSC patterned with inverted cones\\cite{Du2016}. Reproduced with permission.}\n\t\\end{figure}\n\t\n\tQ. G. Du \\textit{et al.} \\cite{Du2016} simulated a strongly corrugated MAPI layer. It was obtained thanks to cones realized in ITO coated glass substrate with an intermediate conformal PEDOT layer lying on ITO (see Figure \\ref{fig:du john schema}). Thanks to optimized period and radius, a 15-17\\% improvement was expected in the PCE compared to the best flat cells of equivalent volume (see Figure \\ref{fig:du john simul}). The optimal period of about 400 $nm$ fits well the required period for LT into the guided modes, given the fact that the effective index of the modes is lower than our examples, due to the strong corrugation of the MAPI filled with low index material. AR and resonant light trapping are thus simultaneously obtained.\n\t\n\t\\subsubsection{Periodic structures for non resonant LM: anti-reflection and scattering}\n\t\n\t\\paragraph{Superficial structuration and light management layers}\n\t\n\tM. I. Hossain \\textit{et al.} \\cite{Hossain2020} proposed to coat the MAPBI like perovskite, having a thickness from 100 to 400 $nm$, with zinc oxide. More precisely, a first thick, lightly doped ZnO ensured the top contact, then it could be covered by texturations patterned either as pyramids or as non-resonant metasurfaces. The period was set to 800 $nm$. Accordingly to the mechanism typically occurring using such a period, no sharp resonance enhances the absorption at the band-edge, only an AR like effect occurs. Both kinds of patterns lead to simulated equivalent $J_{sc}$ enhancements of about 3 $mA.cm^{-2}$ whatever the thickness of the perovskite between 100 and 400 $nm$, so a relative enhancement roughly from 25\\% to 12\\% within the same range, but only 10\\% for the thickness 300 $nm$.\n\t\n\tP. Tockhorn \\textit{et al.} \\cite{Tockhorn2020} simulated and fabricated a 550 $nm$ thick PPSC made of a mixed cation, mixed halide Cs$_{0.05}$(FA$_{0.83}$MA$_{0.17}$)$_{0.95}$Pb(I$_{0.83}$Br$_{0.17}$) perovskite material on various periodically patterned glass substrates, in p-i-n configuration. The top side of the glass substrates was coated with low index, NaF thin film. Compared to the planar reference, patterned structures could exhibit a $J_{sc}$ up to 1 $mA\/cm^2$ larger. This results from a broadband enhancement of the EQE, including at the band edge, mainly attributed to AR effect, as verified by simulations where the volume of perovskite has been kept constant. This is in line with the flat EQE enhancement, as can be seen in Figure \\ref{fig:EQE_table_3} The resulting PCE reaches 19.7\\%, 1\\% absolute above the one of the planar reference.\\\\\n\t\n\t\\paragraph{Structuration of the perovskite} \n\t\n\tU. W. Paetzold \\textit{et al.} \\cite{Paetzold2015} proposed to pattern the front ITO electrode with a square lattice of pillars. The ETL was then corrugated, as well as the about 320 $nm$ thick MAPICl layer that planarized the stack, before the HTL and Al back contact deposition. They noticed increased absorption and EQE as the lattice period decreased, up to the smallest value envisaged, i.e. 500 $nm$. This led to an increase ofthe $J_{sc}$ by 5\\%. Broadband enhancements were observed, at wavelengths shorter than the band edge of the material, as well as a limited LT at the band edge, according to the EQE enhancement plotted in Figure \\ref{fig:EQE_table_3}. This is likely due to the lack of large enough spatial frequencies to couple the quasi-guided modes of the structure, given a period slightly above the optimal range. In addition, there might be a slight change in the volume of perovskite between the patterned cells compared to the flat reference.\\\\\n\t\n\t\\paragraph{Textured substrate}\n\t\n\tW. Qarony \\textit{et al.} \\cite{Qarony2018} calculated the EQE for three various configurations of PPSC using the same volume of perovskite and including moth eye periodical patterns with a period of about 150 $nm$, typically leading to scattering. The fist one had only a pattern at the top air \/ ZnO interface (thus is more comparable to former structures), but the two others considered a conformally patterned stack, either on a patterned Al substrate, or on a patterned NiO\/ITO layer. The patterned Al substrate clearly led to a lower EQE due to additional parasitic absorption. Moreover, a slightly lower $J_{sc}$ was obtained with the conformal stack on the flat Al substrate compared to the top patterned stack. Even this last case exhibits a low EQE enhancement (see Figure \\ref{fig:EQE_table_3}).\n\t\n\t\\subsubsection{Aperiodic patterning for LM}\n\t\n\t\\paragraph{Superficial structuration and light management layers}\n\tB. Dudem \\textit{et al.} \\cite{Dudem2016} proposed a multifunctional inverted micro-structured pyramidal Polydimethylsiloxane (PDMS) AR layer for enhancing the device efficiency, through AR and self cleaning. The MAPI layer was 340 $nm$ thick. The sizes of the pyramid were in the range from 1 to 10 $\\mu m$, therefore too large for efficient diffraction. Compared to flat PDMS, the $J_{sc}$ increases by 0.38 $mA.cm^{-2}$ up to 21.25 $mA.cm^{-2}$, corresponding to a limited AR effect, as confirmed by the flat EQE enhancement (see Figure \\ref{fig:EQE_table_3}).\n\t\n\tRather similarly, M. Jo\\v{s}t \\textit{et al.} \\cite{Jost2017} fabricated a so-called light management foil on the glass substrate of a planar 270 $nm$ thick MAPI cell, which led to a limited EQE enhancement (see Figure \\ref{fig:EQE_table_3}). At first, the thick glass substrate prevents from a strong overlap between any guided mode into the perovskite and the foil. Moreover, whatever the glass thickness, the lack of LT is also related to the far too low spatial frequencies resulting from the texturation. Indeed, thanks to available top view of the foil, a Fourier transform analysis reveals (see Figure SI-\\ref{fig:Supplemental Fourier Analysis Jost}) that most of Fourier's components lie below 7 $\\mu m^{-1}$, whereas the optimum $\\beta_m$ should be in the range 18 - 34 $\\mu m^{-1}$ at first order of diffraction, and even larger for other orders.\n\t\n\tOn the other side of stack, H. Zhang \\textit{et al.} \\cite{Zhang2018b} measured and modelled the impact of the roughness of the back mirror on the back scattering of MAPICl PPSC. The reference was made of a perovskite layer with a significant roughness after crystallization, coated with a thick Spiro layer that planarizes, then a flat gold mirror. With a thinner HTL, the gold mirror replicated the roughness of the perovskite layer leading to light back scattering. The PCE increased from 19.3\\% to 19.8\\%, mainly thanks to the $J_{sc}$ increase from 22.7 to 23.6 $mA.cm^{2}$. It is noticeable that the obtained EQE enhancement is the largest at the band edge. This could be related to the grain size that appears to be in the range 200 - 500 $nm$, so including the optimal range for efficient LT.\\\\\n\t\n\t\\paragraph{Structuration of the perovskite}\n\t\n\tA. R. Pascoe \\textit{et al.} \\cite{Pascoe2016} proposed textured MAPI at the scale of several hundreds of nanometers thanks to gas crystallization. It enhanced the EQE at wavelengths larger than 550 $nm$ (see Figure \\ref{fig:EQE_table_3}), thanks to an induced so-called scattering. Accordingly, the averaged $J_{sc}$ increased from 21.3 for planar references having an about 300 $nm$ thick MAPI layer to 22.1 $mA.cm^{2}$ for patterned samples of comparable volume. As previously, the noticeably large EQE enhancement close to the band edge can be related to the typical grain size of the order of 500 $nm$.\n\t\n\t\\subsubsection{Synthesis}\n\tIn the various previously described studies reporting on $J_{sc}$ enhancements thanks to LM, it can be noticed that most of the possible architectures envisaged in the section \\ref{possible_architecture} have been considered, using all the PE described in section \\ref{Photonic concepts description}. Tables \\ref{tableau pattern mapi} and \\ref{tableau pattern exp} summarize the $J_{sc}$ enhancements reported in the various previously described studies.\n\t\n\tFrom Table \\ref{tableau pattern mapi}, focusing on simulations of periodic patterning of MAPI based single junction PPSC, it comes out that LT can indeed significantly increase the $J_{sc}$ for very thin perovskite layers. The enhancement is more limited at a thickness of about 300 $nm$. It remains that an accurate comparison of the $J_{sc}$ enhancements is not possible, since performance enhancement strongly depends on the choice of the unpatterned reference, and especially its optimization in terms of LM as discussed previously. However, the EQE enhancements (Figure \\ref{fig:EQE_table_2}) can clearly reach larger values thanks to LT than for AR.\n\t\n\tFrom Table \\ref{tableau pattern exp}, focusing on experimental results, using various perovskite materials, the $J_{sc}$ enhancements are of the same order as the simulated ones, again with the same precautions as above. As for the simulated PPSC, the EQE enhancements (Figure \\ref{fig:EQE_table_3}) also reach larger values using patterns at the scale of the wavelength in the material. Moreover, the periodic case lead to the largest enhancement.\n\t\n\t\\begin{table}[h]\n\t\t\\caption{Reported simulated $J_{sc}$ enhancements mainly resulting from Light Management, for MAPI single junction PPSC, with any kind of in-plane pattern (additional layers, structuration of the perovskite or conformal perovskite) and whatever the main identified PE}\n\t\t\\label{tableau pattern mapi}\n\t\t\\begin{tabular}{@{}lllll@{}}\n\t\t\t\\hline\n\t\t\tPerov.thick. ($nm$ ) & $J_{sc}$ enhanc. (\\%) & Pattern. type & Photonic concept &  Ref \\\\\n\t\t\t\\hline\n\t\t\t400 & 6.3 & periodic add. patterned layer & Broadband AR and LT & \\cite{Peer2017} \\\\  \n\t\t\t120 & 31.7 & periodic add. patterned layer & Broadband AR and LT & \\cite{kim_light_2021} \\\\  \n\t\t\t300 & 5.6 & periodic struct. of the perovskite & Broadband AR and LT & \\cite{Schmager2019}  \\\\\n\t\t\t180 & 17 & periodic conformal perovskite & Broadband AR and LT & \\cite{Du2016} \\\\\n\t\t\t300 & 10 & periodic add. patterned layer & Broadband AR & \\cite{Hossain2020}  \\\\\n\t\t\t300 & 10 & periodic conformal perovskite & Broadband AR & \\cite{Qarony2018}\\\\     \n\t\t\t\\hline\n\t\t\\end{tabular}\n\t\\end{table}\n\t\n\t\\begin{table}[!h]\n\t\t\\caption{Reported experimental $J_{sc}$ enhancements mainly resulting from Light Management, for single junction PPSC, with any kind of in-plane pattern, and whatever the main identified PE}\n\t\t\\begin{tabular}{@{}lllllll@{}}\n\t\t\n\t\t\t\\hline\n\t\t\tMaterial & Perov.thick. ($nm$) & $J_{sc}$ enhanc. (\\%) & Pattern. type & Photonic concept &  Ref \\\\\n\t\t\t\\hline\n\t\t\tMAPI & 240 & 14.3 & periodic add. patterned layer & Broadband AR and LT & \\cite{Wei2017} \\\\\n\t\t\tCsFAMAPBI & 370 & 2 & periodic struct. of the perovskite & Broadband AR and LT & \\cite{Schmager2019b}  \\\\    \n\t\t\tCsFAMAPBI & 550 & 6.3 & periodic, struct. of the ETL & Broadband AR & \\cite{Tockhorn2020} \\\\  \n\t\t\tMAPICl & 320 & 5 & periodic struct. of the perovskite & Broadband AR & \\cite{Paetzold2015}  \\\\\n\t\t\tMAPI & 340 & 1.8 & aperiodic add. patterned layer & Broadband AR & \\cite{Dudem2016} \\\\\n\t\t\tMAPI & 270 & 4.8 & aperiodic add. patterned layer & Broadband AR & \\cite{Jost2017} \\\\\n\t\t\tMAPICl & 300 & 4 & aperiodic add. patterned layer & Broadband AR & \\cite{Zhang2018b} \\\\\n\t\t\tMAPI & 300 & 3 & aperiodic substrate corrugation & Broadband AR & \\cite{Pascoe2016}\\\\\n\t\t\t\\hline\n\t\t\\end{tabular} \\label{tableau pattern exp}\n\t\\end{table}\n\t\n\t\n\t\\subsection{LM for PR}\n\t\n\tLM for $V_{oc}$ enhancement thanks to PR in PPSC is discussed in a limited number of publications.\n\t\n\tS. Nanz \\textit{et al.} \\cite{Nanz2019} investigated mainly theoretically the effect of various kinds of LM strategies on the PR, for multilayer stacks that are part of PPSC, mainly without the HTL and metallic contact. They were thus able to derive an upper limit $\\Delta V_{oc}$ for each case, under the assumption of pure radiative recombination. According to the summarized principle reminded previously, the $\\Delta V_{oc}$ resulting from PR in a patterned multilayer was in between the values obtained with the Lambertian multilayer and the rigorously planar multilayer. Indeed, the Lambertian multilayer, as it consists in a better absorber, also radiates the luminescence, whereas, for targeted thicknesses, luminescence can be partly guided, leading to recycling. Then, the quasi-guided mode of patterned multilayer led to enhanced PR compared to the Lambertian multilayer, and also an enhanced $J_{sc}$ compared to the flat multilayer.\n\tHowever, A. Bowman \\textit{et al.} \\cite{Bowman2020} showed that using a more realistic model including recombination, PR was rather unlikely to occur at maximum peak power even if the cell only interacts with a limited solid angle. In this context, it thus appears more promising to increase the absorption and thus the extraction, to the detriment of recycling.\n\t\n\t\\section{Simulations and Perspective}\n\t\n\tAs shown in the previous section, a limited number of studies, mainly focused on $J_{sc}$ enhancements, evidenced a LM effect in PPSC. This synthesis also illustrates that due to a lack of common references, different kinds of PE can hardly be compared, so as the most promising architectures for AR and light trapping. Therefore, we propose to simulate some of the promising patterns, to be able to compare their performances using typical materials and architecture of a PPSC.\n\t\n\t\\subsection{Methodology}\n\t\n\tThe Rigorous Coupled Wave Analysis \\cite{moharam1981rigorous} suits well for the simulation of the stacks periodically patterned under plane wave illumination. We have used the $S^4$ code \\cite{LIU20122233} available in the Solcore package \\cite{Alonso-Alvarez2018}. The derived $J_{sc}$ are obtained using AM1.5G spectrum \\cite{Gueymard1995} provided an IQE of 1.\n\t\n\tOptical indices used are from S. Manzoor \\textit{et al.} \\cite{Manzoor2018} for MAPI, from K.R. McIntosh \\textit{et al.} \\cite{McIntosh2009} for PMMA, and from J.M. Ball \\textit{et al.} \\cite{Ball2015} for all other materials: Sodalime Glass as a substrate, ITO, TiO$_2$, doped Spiro-OMeTAD (as HTL material), and gold. Thicknesses are set to 100 $nm$ for top contact, 20 $nm$ for ETL, and 300 $nm$ for Au.\n\t\n\tThe ETL is supposed to be a thin TiO$_2$ layer, dense and flat, to avoid scattering, and to limit parasitic absorption compared to other envisaged organic materials.\n\t\\subsection{Optimization of the key layers thicknesses in various planar single-junction PPSC}\n\t\n\tLet us first consider a planar PPSC on an infinitely thick substrate, illuminated through this substrate under normal incidence (see Figure \\ref{fig:perspective_single_junction}). The thicknesses of the MAPI layer $th_{MAPI}$ and of the HTL $th_{HTL}$ are supposed to vary in the ranges 300 - 700 $nm$ and 200 - 400 $nm$ respectively. As can be seen in Figure SI-\\ref{fig:jsc_thicknesses}, the $J_{sc}$ of such a cell increases with $th_{MAPI}$. Moreover, for a given $th_{MAPI}$, the $th_{HTL}$ has a non-negligible influence in the $J_{sc}$; e.g. for the smallest MAPI thickness of the considered range, the $J_{sc}$ can be increased by more than 0.5 $mA.cm^{-2}$, up to about 21.74 $mA.cm^{-2}$.\n\t\n\tIt remains that the low $J_{sc}$ for the thinnest considered MAPI is due to lower absorption at long wavelengths, as can be seen in Figure SI- \\ref{fig:absorption_spectra}, for a given thickness of HTL of 240 $nm$.\n\t\n\tGiven the possible identified advantages of using a thinner MAPI layer, mainly for electrical properties, its thickness will be set to 300 $nm$ in the following.\n\tAn AR PMMA layer is then coated on the glass substrate, which thickness is set to 1 $mm$ (see Figure \\ref{fig:perspective_single_junction}), with a negligible roughness. For the chosen $th_{MAPI}$, the coupled influence of PMMA thickness, $th_{PMMA}$, and $th_{HTL}$ is studied. According to Figure SI- \\ref{fig:jsc_thicknesses_pmma}, a significantly increased $J_{sc}$, up to about 21.92 $mA.cm^{-2}$, can be obtained for $th_{PMMA} = 360\\ nm$ together with $th_{HTL} = 250\\ nm$. This last structure will be then used as a planar but optimized reference for fair estimation of the impact of PE in the following. The corresponding spectrum is drawn in Figure \\ref{fig:A_tous}.\n\t\n\t\\subsection{Introduction of various 2D PC to enhance the current density}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\linewidth]{Fig\/perspective_single_junction.png}\n\t\t\\caption{Various configurations of single junction PPSC simulated in the perspective. (a) planer PPSC on an infinite glass substrate, to study the effect of MAPI and Spiro-OmeTAD thicknesses (see results in Figure SI-\\ref{fig:jsc_thicknesses}), (b) Planar reference having a 300 $nm$ thick MAPI layer, and various PMMA and Spiro-Ometad thicknesses to maximise absorption (see results in Figure SI-\\ref{fig:jsc_thicknesses_pmma}), (c) Cross section of a 2D square lattice of cylindrical holes in MAPI, to study the effect of the corresponding PC parameters (see results in Figures SI-\\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_perovskite} and \\ref{fig:A_tous}), (d) Cross section of a 2D square lattice of cylindrical holes in PMMA, to study the effect of the corresponding PC parameters (see results in Figures SI-\\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_pmma} and \\ref{fig:A_tous}), (e) Cross section of two 2D distinct square lattices of cylindrical holes in PMMA and MAPI, to study the effect of PC parameters (see results in Figures SI-\\ref{fig:scan_N_Nprime} and \\ref{fig:A_tous})  }\n\t\t\\label{fig:perspective_single_junction}\n\t\\end{figure}\n\t\n\t\n\tAs already discussed, to further increase the absorption and thus the $J_{sc}$, the most efficient strategies should be to couple the impinging light into the guided modes thanks to properly designed patterns. In the following, we envisage 2D square lattices of cylindrical patterns. Each resulting 2D PC owns typically three parameters that can be optimized: i) its period $P$, ii) its filling fraction ($ff$) which is the ratio between the hole surface and the period surface, and iii) its thickness $t$. \n\tThese 2D PC can be located either (see Figure \\ref{fig:perspective_single_junction}):\n\t\\begin{itemize}\n\t\t\\item in the MAPI layer, made of holes in the MAPI layer filled with HTL material; for a fair comparison, the volume of MAPI material is the same as the planar PPSC, so its total thickness changes. Moreover, $t < th_{MAPI}$ to prevent from short circuits between HTL and ETL. A $t_{HTL}=250 nm$ thick slab of HTL is kept for planarization;    \n\t\t\\item in the top PMMA layer, patterned in a PC of air holes, with $t = th_{HTL}$, in favor of the diffraction efficiency, given the low index of the PMMA;\n\t\t\\item simultaneously at the two previous locations, but each PC has its own set of parameters.\n\t\\end{itemize}\n\t\n\tIn the following studies, all the PC parameters are scanned over realistic ranges. The step for $P$ and $t$ is 5 $nm$, whereas only 3 $ff$ have been envisaged: 0.3, 0.4 and 0.5; these appear to be the most realistic values compatible with a large area patterning at a reasonable cost. \n\t\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[width=\\textwidth]{Fig\/A_tous_cascade.png}\n\t\t\\caption{Simulated absorption spectra for various 1 mm thick glass substrate PPSC with maximized short circuit current density: the planar PPSC (Figure \\ref{fig:perspective_single_junction} (b)), the PPSC with a 2D PC into the MAPI (Figure \\ref{fig:perspective_single_junction} (c)), the PPSC with a 2D PC in the PMMA (Figure \\ref{fig:perspective_single_junction} (d)), and the PPSC with both 2D PC (Figure \\ref{fig:perspective_single_junction} (e)). Small amplitude, high spectral resolution Fabry Perot resonances take place in the $1 mm$ thick glass substrate.}\n\t\t\\label{fig:A_tous}\n\t\\end{figure}\n\t\n\t\n\t\\subsubsection{Single junction: 2D PC in the perovskite layer}\n\tWith respect to the best planar PPSC coated with PMMA, an increase of almost 1 $mA.cm^{-2}$, leading to a $J_{sc}$ about 22.88 $mA.cm^{-2}$ is obtained thanks to a PC into the MAPI (see Figure SI- \\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_perovskite}), with $P = 400\\ nm$, $ff= 0.4$ and $t= 110\\ nm$. It has been checked that $ff$ of 0.3 and 0.5 lead to lower current densities than the optimal one, but still larger than the reference. The corresponding spectra in Figure \\ref{fig:A_tous} reveal that the improvement is mainly due to a larger band edge absorption, so LT, as confirmed by the absorption enhancement in the same Figure.\n\t\n\tIt is noticeable that other periods, smaller than 550 $nm$, can lead to more limited $J_{sc}$ enhancements. However, it has been checked that using periods around 5 and 10 $\\mu m \\pm 0.5$ lead to a $J_{sc}$ lower than 21.85 $mA.cm^{-2}$. It confirms that such periods, far larger than the sub-micron optimal one, do not enable an efficient diffraction and thus do not lead to any $J_{sc}$ enhancement, when compared to an optimized planar reference.  \n\t\n\t\\subsubsection{Single junction: 2D PC in the PMMA covering layer}\n\t\n\tWith respect to the best planar PPSC coated with PMMA, an increase of almost 0.9 $mA.cm^{-2}$, leading to a $J_{sc}$ about 22.75 $mA.cm^{-2}$, is obtained thanks to a PC in the PMMA layer. The 2D PC parameters are $P = 665\\ nm$, $ff = 0.4$ and $t= 615\\ nm$ (see Figure SI-\\ref{fig:TOTAL_absorption_Jsc_Jo_fn_period_etch_depth40_PC_pmma}). It has been checked that $ff$ of 0.3 and 0.5 lead to lower current densities. It can be noticed on the corresponding spectra in Figure \\ref{fig:A_tous} that the improvement is due to both a AR effect and limited LT at band edge absorption, since it implies low effective index guided modes that are the only able to interact, weakly, with the pattern on top of the thick substrate. The enhancement remains lower than the one induced by the 2D PC in the perovskite layer. \n\tAgain, it has been checked that periods far larger than the optimal one, typically around around 5 and 10 $\\mu m \\pm 0.5$ lead to a $J_{sc}$ lower than 22 $mA.cm^{-2}$, so to a limited enhancement, because of a reduced diffraction efficiency.\n\t\n\t\\subsubsection{Single junction: Combination of the two 2D PC}\n\t\n\tGiven the previous enhancements, a structure that combines the two 2D PC, one in the PMMA and the second at the MAPI\/HTL interface, can be envisaged. It is noticeable that RCWA method implies that the period of such a combined architecture is a integer $N_{PMMA}$ times the period of the 2D PC in the PMMA ($P_{PMMA}$), and another integer $N_{MAPI}$ times the pitch of the 2D PC in the MAPI ($P_{MAPI}$); other PC parameters ($ff_{PMMA}$, $t_{PMMA}$ on the one hand, $ff_{MAPI}$, $t_{MAPI}$ on the other hand) can differ (see Figure \\ref{fig:perspective_single_junction}).\n\tFor the sake of illustration of a possible further $J_{sc}$ enhancement, it has been chosen to set the parameters of the 2D PC at the MAPI\/HTL as for the optimized cell with a flat PMMA layer, i.e. $P_{MAPI} = 400\\ nm$, $ff_{MAPI} = 0.4$, $t_{MAPI} = 110\\ nm$, as well as $ff_{PMMA} = 0.5$ and $t_{PMMA} = 405\\ nm$, among the thinnest most favorable values of the 2D PC in PMMA associated with planar MAPI. Then, $N_{PMMA}$ is scanned from 5 to 8 and $N_{MAPI}$ from 9 to 15, given the fact that $P_{PMMA}$ is typically larger $P_{MAPI}$ according to the previous studies. The simulated $J_{sc}$ displayed in Figure SI-\\ref{fig:scan_N_Nprime} shows that a limited increase, of about 0.26 $mA.cm^{-2}$, up to 23.14 $mA.cm^{-2}$, is possible provided $P_{PMMA} = 530\\ nm$. It can be seen in Figure \\ref{fig:A_tous} that such $J_{sc}$ enhancement results from both LT and AR effects compared to the planar reference. If the full space of various possible PC parameters has not been scanned, and thus the previous parameters not fully optimized, the interest of such a combination is yet demonstrated.\n\t\n\t\\subsection{2T tandem}\n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\includegraphics[scale=0.7]{Fig\/figures_perspective_tandem.png}\n\t\t\\caption{Various configurations of 2T tandem PPSC simulated in the perspective. (i) Planar ref on an infinite glass substrate with varying perovskite materials thicknesses, to reach and equilibrium of $J_{sc}$. (ii) Cross section of a 2D square lattice of ITO pillars in 1.77-eV perovskite, to enhance $J_{sc, 1.77eV-perovskite}$ (iii) Cross section of a 2D square lattice of ITO pillars in the 1.77-eV perovskite, and of dots of 1.22-eV perovskite in HTL. Inspired by the configuration proposed by K. Xiao \\textit{et al.} \\cite{xiao_all-perovskite_2020}}\n\t\t\\label{fig:figures perspective tandem}\n\t\\end{figure}\n\t\n\tA 2T tandem PPSC generally exhibits thermalization, as the one previously described in section \\ref{tandem_plan}. Thus, we will consider this example as a case study for possible improvement. In the following simulations, the optical indices they provided for the considered materials are used. The planar stack (see Figure \\ref{fig:figures perspective tandem}), considered as a reference, is close to the one shown in Figure \\ref{fig:xiao tandem schema}. Oppositely to our previous studies, the glass substrate is again supposed to be infinitely thick to avoid the additional interferences in the substrate. Within the stack, the layer thicknesses (except perovskites) have been set to realistic values such as $t_{ITO}=100\\ nm$, $t_{NiO}=t_{NVPB}= 10\\  nm$ (simplified version of a mixed material ETL), $t_{C60} = 10\\ nm$ for both HTL, as well as $t_{PEDOT-PSS} = 10\\ nm$ and $t_{Cu}=100\\ nm$. As justified later, the SnO$_2$ layer, with $t_{SnO_2}=100\\ nm$, acts as an optical spacer (the 1 $nm$ thick Au layer has thus been neglect).\n\t\n\tSetting the thicknesses of both perovskite layers to $t_{1.77\\ eV\\ PK} = 400\\ nm$ (also the maximum in the considered range) and $t_{1.22\\ eV\\ PK} = 880\\ nm$ (in the range 800 - 1200 $nm$) leads to the highest $J_{sc}$ of 16.25 $mA.cm^{-2}$, that appears to be limited by the 1.77 $eV$ perovskite subcell.\n\tThis value is slightly larger than the one obtained by K. Xiao \\textit{et al.}. Moreover, it is obtained in our case for different perovskite thicknesses, due to a mismatch between the thicknesses of the charge transport layers we have chosen and author's choices. However, our derived absorbance spectra for both sub cells (see Figure \\ref{fig:A_tous_tandem}) still exhibits a thermalization effect. Simply increasing $t_{1.77\\ eV\\ PK}$ could be at the expense of the charges collection. \n\t\n\tIn this frame, the possible enhancement of the $J_{sc,1.77\\ eV\\ PK}$ thanks to a 2D PC at the ITO \/ 1.77 $eV$ perovskite layer interface is studied. The 2D PC consists of a square lattice of ITO pillars, coated with conformal ETL and then with a 1.77 $eV$ perovskite layer that planarizes the corresponding subcell, while keeping an equivalent volume of perovskite as in the planar tandem cell. \n\t\n\t\\begin{figure}[h]\n\t\t\\centering\n\t\t\\begin{subfigure}[b]{0.65\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/tandem.png}\n\t\t\t\\caption{Simulated absorption spectra of the various envisaged 2T tandem PPSC.}\n\t\t\t\n\t\t\\end{subfigure}\n\t\t\\hfill\n\t\t\\begin{subfigure}[b]{0.25\\textwidth}\n\t\t\t\\centering\n\t\t\t\\includegraphics[width=\\textwidth]{Fig\/tandem_zoom.png}\n\t\t\t\\caption{Zoom on LT in 1.77-eV perovskite material}\n\t\t\t\n\t\t\\end{subfigure}\n\t\t\\caption{2T tandem PPSC simulated in the perspective, Inspired by the configuration proposed by K. Xiao \\textit{et al.} \\cite{xiao_all-perovskite_2020}.}\n\t\t\\label{fig:A_tous_tandem}\n\t\\end{figure}\n\t\n\tA significant enhancement of $J_{sc,1.77\\ eV PK}$, up to 17.92 $mA.cm^{-2}$, has been found for $P=325\\ nm$, $ff=0.5$ (here defined as the ITO filling fraction) and $t=160\\ nm$.\n\tThe reason is twofold (see spectra in Figure \\ref{fig:A_tous_tandem}): a broadband AR as well as a limited LT at the band edge of the 1.77 $eV$ perovskite material that slightly reduces the thermalization. LT specifically occurs in one of the absorbing materials, to the detriment of the other one. It results from a guided mode mainly confined in the perovskite of interest, with a limited overlapping in the other perovskite, especially thanks to the rather large, 100 $nm$ SnO$_2$ optical spacer. Indeed, this layer can prevent evanescent coupling between quasi guided modes of the perovskites. Additionally, it could protect the 1.77 $eV$ perovskite material during the fabrication, but the junction between the two subcells might then also be less efficient. \n\t\n\tMoreover, this first step leads to a strong $J_{sc}$ disequilibrium between the two sub cells, since $J_{sc,1.22\\ eV\\ PK}$ even slightly decreases to 15.7 $mA.cm^{-2}$, compared to the planar reference. To increase in a second step $J_{sc,1.22\\ eV\\ PK}$, this low-Eg perovskite layer can also be patterned rather than simply increasing its already large thickness.\n\t\n\tStarting from the last structure, a second 2D PC is introduced at the 1.22 $eV$ perovskite material \/ HTL interface. To target a LT at larger wavelength, close to the band edge of this perovskite, its period has to be increased. However, given the constraint induced by the boundary conditions of the simulation, we simply chose a supercell having a period twice the one of the single PC device, with one pattern in the 1.22 $eV$ perovskite material, and two patterns in the 1.77 $eV$ perovskite material (see Figure \\ref{fig:figures perspective tandem}), while keeping $ff$ and thicknesses constant. Moreover, again, the volume of this perovskite is kept constant, HTL acting as a planariser, with a minimum thickness of 10 $nm$ (with a larger volume as in the planar alternative). \n\t\n\tIt appears that $J_{sc,1.77\\ eV PK}$ remains unchanged, whereas $J_{sc,1.22\\ eV\\ PK}$ is increased up to 16.2 $mA.cm^{-2}$, reducing, but not cancelling the disequilibrium. As expected, this results from LT at the band edge in the 1.22 $eV$ perovskite material (see Figure \\ref{fig:A_tous_tandem}). \n\tThis last result appears as a proof of concept to integrate two PC in order to induce LT, as well as possibly AR, in two different absorbing layers of a given stack, provided each one exhibits guided modes without overlapping. If not yet optimized, it is noticeable that, providing the first subcell is planarized, the two PC could practically have independent parameters, offering additional degrees of freedom to reduce disequilibrium. Given the required spacer, such a concept could be also applied to other kind of three or four terminal multijunctions cells. \n\t\n\t\\section{Conclusion - Perspectives}\n\t\n\tLight management for PV solar cells was mainly intended to increase the $J_{sc}$, simply thanks to a larger absorption. When using metal-halide perovskite materials, LM is shown to result from an interaction between the various material choices, the patterning processes and the PE. Moreover, demonstrating sole LM needs to meet some rigorous criteria to get rid of mixed electrical, material and photonic effects.\n\t\n\tThe investigations presented in this review have demonstrated that the easiest $J_{sc}$ enhancement is indeed the most frequently envisaged effect, especially for single junction cells, compared to the still highly challenging $V_{oc}$ enhancement, and the immature studies of the EY due to the lack of studies at the module scale. As demonstrated by several authors, even thicknesses optimization within flat PPSC can enhance absorption thanks to Fabry Perot effects while taking care of the electrical constraints. To further increase the absorption at the perovskite band edge, an in-plane pattern, at various scales, can be introduced in one of the layers. However, patterns at the wavelength scale appeared as the most efficient. In any case, the $J_{sc}$ enhancement compared to already optimized flat reference remained limited to a few percent. As regards the photonic regimes, the broadband AR effect was the most frequently observed, but only LT was able to significantly enhance the absorption at the band edge, as confirmed by our own simulations. In addition, we confirmed that direct patterning of the perovskite leads to more efficient LT than an additional layer on top of the substrate. Finally, in 2T tandem cells, we showed that one PC in each subcell can enhance each of the $J_{sc}$ separately.\n\t\n\tBeyond, some of the strategies developed for these opaque, single junction PPSC can be adjusted for other applications of hybrid perovskites for sun light harvesting. Indeed, accurately tuned spectral absorption is required in semitransparent single junction. Moreover, the same concepts can be tweaked to tailor reflectivity spectrum for perovskite-based color printing devices \\cite{Gholipour2017,Gao2018,Fan2019,Yoo2021} and engineering absorption management in full-color perovskite detectors \\cite{Hossain2020b,Qarony2020}.\n\t\n\tThis work is even part of far larger context. Indeed, other studies are ongoing concerning materials, which need to be more stable, and processes. In this frame, it can be noticed that record cells might not share the same encapsulation strategies as more realistic, large surface cells and modules \\cite{Wang2021b}, both for aging and safety reasons \\cite{wu_evolution_2021, wu_main_2021}. All perovskite tandem cells are even more challenging on the fabrication point of view \\cite{Zheng2020}, but are also very promising since they combine all the potential of perovskite based cells and modules with high yields.\n\t\n\tFinally, according to the reciprocity relation between absorption and emission \\cite{Rau2007}, light extraction strategies in perovskite LED \\cite{Gholipour2017,Wang:17,C7NR01631J}, as well as resonances of high quality factor in perovskite-based laser \\cite{Chen2016b,Pourdavoud2017,Qin2020} can use similar concepts to those derived in this work. Even LM in 2T tandem cells could be mimicked in white LEDs obtained by stacking several emitting materials. For LEDs, a high light extraction efficiency is a even more the key of good efficiency.\n\t\n\t\\textit{Acknowledgement:}\n\t\n\tRCWA simulations were performed on the Newton computer cluster facilities operated by PMCS2I at Ecole Centrale de Lyon and on the CNRS\/IN2P3 Computing Center in Lyon. R. M. L. acknowledges project EMIPERO (ANR-18-CE24-0016).\n\t\n\t\n\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\nCubeSat space telescopes are a growing asset in astronomical research\\cite{Shkolnik_2018,douglas2019cubesats}. Their standardized format and compact volume enables the rapid development of high-impact mission concepts above the earth's atmosphere. Flight-tested commercial microelectromechanical systems (MEMS)  have further enhanced CubeSat telescopes by providing dynamic pointing and aberration control on-orbit (DeMi, NODE) \\cite{Yenchesky2019OptomechanicalDA,Clements2016}. The expanding capacity of CubeSats for astronomical research invites the development of novel technologies for their optical systems. However, the resolution in CubeSat optics is fundamentally limited by the clear aperture permitted by the CubeSat volume. The rayleigh criterion tells us that resolution ($\\Delta \\theta$) is inversely related to to the entrance pupil diameter ($D$) of the optical system.\n\n\\begin{equation}\n    \\Delta \\theta = 1.22 \\frac{\\lambda}{D}\n\\end{equation}\n\nReflective CubeSat objectives traditionally require mounting hardware that limits the available entrance pupil diameter. Single-point diamond turned (SPDT) mirrors can mitigate this limitation by enabling the manufacturer to machine the mounting hardware directly into the rear of the mirror substrate, eliminating the need for mounting hardware around the edge of the mirror and enhancing the nominal throughput and resolution. This also grants the objective a considerable degree of athermalization by eliminating the difference in the coefficient of thermal expansion between the mirror and mounting hardware\\cite{Zhang:17}. SPDT surfaces are typically used in longer wavelengths (MWIR, LWIR) due to the midspatial frequency errors left by tooling marks. However, recent developments in optical polishing\\cite{Jeon:17} have introduced Magnetorheological Finishing (MRF) to the fabrication of SPDT surfaces, reducing the dominant midspatial frequencies considerably and therefore extending their use into the optical. Athermal CubeSat objectives composed of low surface roughness mirrors with a large entrance pupil invites the design of high-performance optical CubeSat payloads to make the next generation of research and technology development in space more accessible. The Versatile CubeSat Telescope (VCT) concept is a prototypical fore-optic that boasts compatibility with a variety of research payloads. The goal of the VCT is to develop a flexible, large aperture telescope that can be replicated at low cost for easy adaptability to future research payloads and technology demonstrations. \n\n\\section{OPTICAL DESIGN - ON-AXIS V.S. OFF-AXIS IMPLEMENTATIONS}\nThe design of the VCT began with two realizations of a CubeSat telescope that could take advantage of a large primary mirror. The first, an on-axis telescope with a more classical Ritchey-Chretien objective outfitted with a plano-convex aspheric collimator. The second, an all-reflective off-axis Ritchey-Chretien solution with a freeform collimator. We baselined a 95mm entrance pupil diameter with a 20$\\%$ obscuration for the on-axis telescope, and scaled the entrance pupil diameter of the off-axis telescope to be equivalent in collecting area. A high pupil magnification is required for both designs to image the primary mirror onto the small MEMS FSM clear aperture (5mm). The specifications and system layouts are shown in table \\ref{tab:specs} and figure \\ref{fig:layout} respectively.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{95 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Field of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{20$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\qquad\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{88 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Fiel of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{0$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\caption{Optical design specifications for the (left) On-Axis and (right) Off-Axis telescope designs. All specifications are similar except for those in bold, which are set such that the apertures of each design are equal in throughput.}\n    \\label{tab:specs}\n\\end{table}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{vct_drawings.png}\n    \\caption{System layouts of the (left) On-Axis Telescope and the (right) Off-Axis Telescope designs. Each has three optical elements, two fast-steering mirrors (FSM) for fine pointing control, and a volume dedicated to a research payload.}\n    \\label{fig:layout}\n\\end{figure}\n\n Aspheric surfaces were necessary for both telescopes to achieve a high pupil magnification (19x and 17.5x for the on- and off-axis designs respectively) while maintaining a well-corrected field of view. While this slightly complicates the metrology of the mirrors it did not add any additional cost because the mirrors were already going to be diamond-turned. Both telescopes utilized a Ritchey-Chretien objective that forms an image near the primary mirror, which is collimated by an aspheric element to produce an external exit pupil. The size of the exit pupil was chosen to match the clear aperture of a MEMS fast-steering mirror (FSM)\\cite{memsfsm,Serra21} to correct for spacecraft pointing and jitter errors while in orbit. The external pupil also serves as a convenient interface for any instrument suite that would be included on the VCT (e.g. spectrographs, cameras, lasers).  The concept for the on-axis design was to make a well-established telescope format work given the pupil magnification and field of view specifications. The aspheric surfaces granted the design a well-corrected field of view in a compact format suitable for a CubeSat (2U). However, the design has a secondary obscuration which limits the throughput of the system and adds diffraction features to the image. The primary goal for the off-axis design was to offer a viable alternative to the classical on-axis design that would not suffer from undesirable diffraction effects brought upon by the secondary obscuration. Unfortuantely, shifting the primary parabolic mirror off-axis comes at the cost of more field-dependent aberration that limits the system's field of view. To accommodate this we selected the tertiary mirror of the off-axis design to be freeform. Before optimization of the freeform surface, the rotational symmetry of the system was broken by tilting the secondary and tertiary mirrors about their foci to make the system free of linear astigmatism. A linear-astigmatism free three mirror system (LAF-TMS) has been shown by Park et al\\cite{park_development_2020} to be an excellent starting point for freeform designs by using the mirror tilt angles and inter-mirror distances (Eq \\ref{eq:laf}) to mitigate a dominating aberration early in the design process.\n\n\\begin{equation}\n    \\frac{l_2^\\prime}{l_2}\\frac{l_3^\\prime}{l_3}tan{i_1}+\\left(1+\\frac{l_2^\\prime}{l_2}\\right)\\frac{l_3^\\prime}{l_3}tan{i_2+}\\left(1+\\frac{l_3^\\prime}{l_3}\\right)tan{i_3=0}\n    \\label{eq:laf}\n\\end{equation}\n\nIn this equation, parameters i$_{1,2,3}$ are the angles of incidence of the optical axis ray on surfaces 1,2, and 3, while $l_{2,3}$ and ${l^{\\prime}}_{2,3}$ are the front and rear focal lengths the secondary and tertiary mirror. Once the baseline LAF-TMS design had been implemented, the third mirror was changed to a freeform XY-Polynomial surface where coefficients were optimized that maintained the bilateral symmetry of the optical system. The mirrors were shifted off-axis in the $\\hat{y}$ direction, as was the field bias (+0.2$^{o}$). Consequently the freeform surface was constrained to be plane-symmetric about the $\\hat{y}-\\hat{z}$ plane by solely optimizing coefficients of even order in $\\hat{x}$. The resultant design is nearly diffraction-limited across the biased field of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{strhvsfov.png}\n    \\caption{Maps of strehl ratio v.s. field of view for the (left) On-Axis and (right) Off-Axis VCT. The symmetry of the optical system mirrors the performance quite readily, with the on-axis design having a rotationally symmetric field performance whereas the off-axis design exhibits plane-symmetric performance.}\n    \\label{fig:strvfov}\n\\end{figure}\n\n\\section{SENSITIVITY ANALYSIS}\n \nThe misalignment sensitivities of the two designs were explored in Zemax OpticStudio (ZOS) by iteratively perturbing each optic in six degrees of freedom using the Python ZOS-API. Our as-built performance goal for both systems was an average Strehl ratio $>$ 0.7 across the field of view to maintain reasonably diffraction-limited performance. The $0\\%$, $70\\%$, and $100\\%$ field of view in $\\pm \\hat{x}$ and $\\pm \\hat{y}$ were considered in the average calculation, considerably biasing the sensitivity analysis toward the edge of the field of view where performance is expected to degrade faster. This results in a more pessimistic analysis of misalignment sensitivity. The sensitivity analysis fixes the primary mirror while M2 and L3\/M3 were perturbed to better understand how misalignments with respect to the primary would impact the optical performance over all fields of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{qkdsens.png}\n    \\caption{Sensitivity to misalignment curves of the On-Axis (Solid Red) and Off-Axis (Dashed Blue) designs for their respective secondary (top) and tertiary (bottom) optic. The on-axis system is less sensitive for X\/Y decenter of both elements, X\/Y\/Z tilt of M2, and X\/Z tilt of M3.}\n    \\label{fig:my_label}\n\\end{figure}\n\nFigure 2 shows that in most cases the on-axis design was less sensitive to misalignment in six degrees of freedom. This result is consistent with the complexity of each system. The off-axis design employs off-axis conics and a freeform surface, both of which will be sensitive to misalignment due to the higher mirror slopes than the on-axis design. In applications where the secondary obscuration is tolerable, the on-axis design would be a better choice for applications where cost and misalignment risk are limiting factors.\n\n\\section{POLARIZATION ANALYSIS}\nMany spaceborne optical payloads have a degree of sensitivity to polarization effects. From laser experiments in support of quantum communications\\cite{Serra21} to polarimetry of atmospheric ice\\cite{Hart20}, there is a clear need to characterize the response of cubesat payloads to a vector electric field. To offer support for polarization-sensitive research payloads a comprehensive polarization ray trace (PRT) analysis was conducted for each realization of the VCT.  PRT is a ray-based approach to vector field calculations enabled by determining the local fresnel reflection coefficients that alter the transmission and phase of the electric field at each surface in the optical system. These effects are traced from the entrance pupil to the exit pupil of the optical system to determine the cumulative polarization behavior of the instrument. The proposed VCT designs use fast primary mirrors to achieve the high pupil magnification, so some polarization aberration is expected. However mirror substrates and coatings can be selected to mitigate the impact of polarization aberration for a given application. \n\nA PRT tool was written using the ZOS-API to create maps of diattenuation and retardance at the exit pupil of the system. The ZOS-API tool creates a batch raytrace and propagates it from the entrance pupil of the optical system to the surface under investigation. The output of the raytrace gives surface normal vectors in addition to exit ray vectors which are used to calculate the incident ray vectors and angle of incidence. Diattenuation ($D$) and retardance ($\\delta$) data is produced from the Zemax coating files and ray angles of incidence to generate the surface and pupil maps shown in figure \\ref{fig:polmaps_al}.\n\n\\begin{equation}\n    D = \\frac{|r_p|^{2} - |r_p|^{2}}{|r_s|^{2} + |r_p|^{2}};\\phantom{flapjackfacts}\\delta = |\\phi_{p} - \\phi_{s}|\n    \\label{eq:diat}\n\\end{equation}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{polplots.png}\n    \\caption{Polarization pupil maps assuming bare aluminum mirrors and uncoated N-BK7 lens surfaces. (Top) Diattenuation and (Bottom) retardance maps of the (left) On-Axis and (right) Off-axis versatile cubesat telescopes. These data were computed at $\\lambda$ = 780nm. At this wavelength the Off-Axis Aluminum VCT sees the greatest diattenuation, and the Off-Axis Gold sees the greatest retardance. This indicates that the off-axis design would introduce more challenges to polarization-sensitive payloads than the on-axis design would.}\n    \\label{fig:polmaps_al}\n\\end{figure}\n\nThe polarization pupil map permits analysis of the optimal material and configuration for a given polarization-dependent application. Figure \\ref{fig:polmaps_al} demonstrates that aluminum is more diattenuating than gold at $\\lambda=780nm$, and the off-axis configuration performs worse than the on-axis configuration due to the higher angles of incidence on the mirror surfaces. \n\n\\section{MECHANICAL DESIGN}\n\nThe VCT must be designed to deliver a high optical wavefront when operational on-orbit, yet also robust enough to survive the rigors of launch and orbital injection separation.  Achieving both goals in a compact package that affords straight-forward assembly and alignment is quite challenging.  Another important aspect of the design must also consider the mass contribution of the structural elements as this can be costly when considering launch requirements and on-orbit maneuvering.  With the prevalence of small satellite design ( $<$ 12U), development of a telescope that fits within a 2U volume with a mass less than 2 kg would present a low-cost solution that enables rapid prototyping of research payloads. For a prototype realization of the VCT, the optical system design called for SPDT Aluminum primary and secondary mirrors. All mechanical structural elements were also chosen to be aluminum to match the thermal behavior of the mounting structure to the optical elements. The collimating lens, which locates the instrument pupil, could also be easily mounted to the OTA using a common aluminum barrel structure with simple shims for precise alignment. \n\nThe first goal in the mechanical design of the telescope Optical Tube Assembly (OTA) is to support the optical elements with minimal effect due to expected thermal perturbations that would be encountered on-orbit.  These effects would include both rigid body misalignment and wavefront aberrations caused by subtle flexing of the optical elements.  In order to bound the problem, the first design decisions centered on overall sizing to fit within the small-sat form factor. The primary-secondary despace tolerance is very tight, so a truss structure was designed to properly space the two mirrors. The truss structure presents a good strength to weight ratio and is commonly used in ground-based telescope designs. For this space-based application, the truss structure would not allow for simple and easy shrouding of the optical path and thus would likely require additional structural pieces to support stray light baffling elements and thermal mitigation elements (i.e., mylar blankets).  Therefore, an early design decision was made to investigate a tube structure that would serve as both the metering structure that could maintain the mirror separation tolerance, transverse misalignment tolerance, and provide an in-place baffle that could be customized for the particular optical application.  A significant benefit of the tube design is the mass savings realized by the ability to select a very thin tube wall thickness that still retains the desired structural performance for all loading cases.  This tube would also allow for easy installation of thermal mitigation measures and targeted design of thermal conduction paths that further ensure optical performance stability while on-orbit.\n\nThe next major goal in the mechanical design considers the manufacturability of the system.  Considering its small size, the OTA manufacturability includes the fabrication processes for all structural components, as well as thoughtfully planned assembly and alignment processes that ensure the optical performance requirements can be met.  The aluminum mirrors provide a unique opportunity to meet the need for an assembly that is easy to integrate and align.  The primary mirror optical surface is thus applied directly to the structural component (primary mirror) which has appropriate design elements that consider thermal performance and mounting points for both the metering tube and the interface to the downstream optical system via a \\emph{hex plate} interface.  Clearly defining the rear of the primary mirror blank as the OTA support location allows for simple design of flexural components that will not propagate errors from other systems into the OTA, or vice-versa.  Similarly, the secondary mirror surface is applied directly to a head-ring structure that also considers thermal performance and mounting points to the metering tube, but also minimizes the size of support strut spiders that are common in the on-axis  design.  Considering the small size of the OTA, the one-piece secondary mirror\/head-ring simplifies its alignment features by moving them out to the tube diameter, rather than trying to squeeze them into the shadow of the secondary mirror on the entrance aperture.\n\nThe final goal of the mechanical design considers the interface of the OTA to the instrument and the spacecraft.  Most terrestrial optical system applications utilize a breadboard style support that allows for many mounting possibilities for both the OTA and any downstream instrument opto-mechanical elements, and can be purposefully designed to incorporate spacecraft mounting options that will isolate the payload from the spacecraft both thermally and dynamically as necessary.  The hex plate provides all of these interfaces in a single machined aluminum component.  The connection of the OTA to the hex plate is of particular importance for this discussion.  As mentioned earlier, the OTA must remain dynamically and thermally isolated from the other spacecraft or instrument components to preserve the telescope optical performance. A Finite Element Model (FEM) was created using the aluminum tube structure OTA, with both structural and optical components represented, and a Finite Element Analysis (FEA) was performed to determine the model behavior with both structurally dynamic boundary conditions and also varying thermal environmental conditions.  The FEM is shown in figure \\ref{fig:vct_fea}.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{vct_fea.png}\n    \\caption{The FEM of the VCT OTA highlighting the hex plate, flexure, and tube design.}\n    \\label{fig:vct_fea}\n\\end{figure}\n\nA simple three flexure design was selected to mount the OTA to the hex plate.  The flexures were positioned tangentially to the primary mirror blank to better accommodate thermal deformations that could arise at the hex plate. The flexure material properties were varied to determine the best trade of optical path stability, launch survivability, manufacturability, and cost. The materials considered were Aluminum (Al 6061-T6) and Titanium (Ti-6Al-4v).  For the structural analysis the hex plate was fixed at three points on its outer edge simulating spacecraft support, and MAC loads (100g accelerations in three orthogonal directions) were applied to the model.  Various flexure throat cross sectional areas were considered, and the best design was chosen based on the resulting maximum stresses revealed in the FEA.  The results for each material selection under this loading are shown in table \\ref{tab:macload}. \n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n     Load Case Direction & Max Flexure Stress [Al] & Max Flexure Stress [Ti] \\\\\n    \\hline\n        100g $\\hat{x}$ & 103ksi & 112ksi \\\\\n        100g $\\hat{y}$ & 98ksi & 103ksi \\\\ \n        100g $\\hat{z}$ & 42ksi & 46ksi \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{MAC loading in three orthogonal directions and the corresponding maximum flexure stress for Aluminum and Titanium flexures. The Ti flexures were chosen due to their higher maximum stress.}\n    \\label{tab:macload}\n\\end{table}\n\nAn additional analytical consideration for the structure is the modal behavior of the OTA.  The modal FEA reveals the lowest expected resonant frequency of the OTA under the prescribed mounting conditions for given material selections and flexure design choices.  The table below shows the first three modal FEA results for the same conditions discussed above.  These results indicate there would be little chance of exciting resonance in the OTA structure when considering expected launch loads.  And similarly, the telescope itself would not likely impart damaging resonance to any other spacecraft systems or partner satellites if launched as a secondary payload ride-share.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n        Al Flexures & Ti Flexures  \\\\\n    \\hline\n        130Hz & 156Hz \\\\\n        130Hz & 156Hz \\\\\n        363Hz & 434Hz \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Modal behavior of the OTA for Al and Ti flexures for the cases described in \\ref{tab:macload}. Based on the results from the FEA for both materials, the Titanium flexure design was selected based on the extra margin provided on mass and stress.}\n    \\label{tab:my_label}\n\\end{table}\n\n Verification  of the selected flexure design needed to also consider the thermal behavior of the system.  For ease of creating the thermal FEM, the boundary conditions for the structure were preserved from the structural FEA and a 27.8\u00b0C differential in temperature was applied to the hex plate.  This set of conditions would bring into focus the thermal isolation of the OTA from the hex plate.  The results from this thermal analysis were normalized to show an expected mirror sag of 10nm P-V for each 1.0\u00b0C of temperature differential.  The results (shown below) also indicate an axisymmetric deformation of the OTA as would be expected.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{fea_results.PNG}\n    \\caption{Finite Element results of the hexplate-primary mirror temperature differential. The flexure design results in $\\approx$ 10nm of trefoil surface sag per degree Celsius.}\n    \\label{fig:fea_results}\n\\end{figure}\n\nOverall, an iterative analysis process was implemented to determine the final telescope structure geometry.  The results from dynamic survivability, modal, and thermal analyses were considered for each element.  Part of the FEM definition involves application of material properties that also allow for the consideration of mass as a secondary factor in selection of the final geometry.  Resulting optical performance effects are then analyzed in the Zemax OpticStudio STAR module for each iteration by exporting the three-dimensional nodal parameters from the FEA results. Other geometry selections made during the design process were more straightforward than the flexural interface between the OTA and the hex plate.  For instance, given the axisymmetric behavior for all analyses, the tube geometry selections were made without direct optical analyses as the major performance effect would be in focus change due to varying thermal conditions.  For this, common engineering knowledge can be applied to bias the focus alignment for all optical elements prior to integration with an instrument for the given expected on-orbit operational conditions (for this project +\/-1\u00b0C).  Similarly, the hex plate design only needed to preserve the axisymmetric geometry approximation when considering its overall shape and interface definition.  \nWhile not discussed in detail earlier, mass of each components was considered as an important mechanical design decision.  An overall maximum allowable mass budget of 2 kg was allotted to the design and did not present a problem during the iterative analyses.  The final telescope design mass is expected to be less than 1 kg once all hardware is integrated.  The only major design decision based on mass reduction involved light-weighting the primary mirror blank by cutting pockets into the structure and analyzing these pockets for their structural effect on mirror surface figure.  \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{3dprintlw.PNG}\n    \\caption{3D Prints of the primary mirror (left) and hex plate (right) demonstrating the lightweighting scheme used to achieve the mass requirement of $< 2kg$}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\section{STOP EVALUATION - MERGING FEA WITH SEQUENTIAL RAYTRACING}\n\nNumerous loading conditions were studied through FEA software to ensure the overall survival of the VCT payload, however, FEA analysis by itself could not determine how the deformed elements would impact the system's optical performance. Further studies of the wavefront deformation were necessary to confirm that the system would be able to function properly in its environment of operation. Typically, this kind of analysis would require the deformed surface to be decomposed into hundreds of polynomial terms to accurately model the shape of the surface, with no guarantee of the raytracing software being able to support the number of terms needed. However, through use of OpticStudio's STOP (Structural, Thermal, and Optical Performance) analysis feature \\emph{STAR} (Structural, Thermal, Analysis and Results) evaluation of the system's performance in various loading scenarios was made quick and efficient. STAR allows for the data from FEA software to be loaded directly into OpticStudio and onto select optical elements. By using the position and displacement of each node in a 6 column (x, y, z, dx, dy, dz) format STAR accurately deforms the surface shape, allowing for the deformed wavefront to be observed under various loading conditions. For the purposes of our design study we primarily looked at the surface deformation on the primary mirror caused by rotation about the optical axis.\n\nIn figure \\ref{fig:star} the change of the optical system's wavefront and PSF are shown after applying a -237 arcsecond rotation about the optical axis. This causes clear degradation in the system's overall optical performance. This optical analysis is critical for an iterative design process between optical and mechanical design teams. Quick optical performance analysis through STAR allows for the mechanical team to adjust the structures supporting the optical surfaces where necessary, ensuring performance specifications are met in realistic structural and thermal scenarios. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{starinv.png}\n    \\caption{Influence of STAR on the wavefront (left) and the PSF (right). A force tangent to the primary mirror was applied in FEA and the resultant surface deformation was loaded into the OpticStudio raytrace model. This permits evaluation of performance degradation as a function of structural and thermal deformation.}\n    \\label{fig:star}\n\\end{figure}\n\n\n\\section{Laboratory Prototype Status}\n\nThe next step for the development of the VCT is the construction of a laboratory prototype of the on-axis aluminum VCT. Mirrors for the on-axis VCT have been fabricated by Hanbat national university and are shown in figure \\ref{fig:fab_mirrors}. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{mirror_img.PNG}\n    \\caption{The fabricated (left) primary and (right) secondary of the on-axis VCT design manufactured by Hanbat National University in South Korea.}\n    \\label{fig:fab_mirrors}\n\\end{figure}\n\nThe primary and secondary mirror have arrived at the University of Arizona and are awaiting measurement and assembly so that they can be mounted into the OTA described in section 5 for system-level tests. The UArizona Space Astrophysics lab test facilities include a TVAC chamber and Hexapod that can be used for thermal vaccum and jitter tests respectively. Details of the comprehensive characterization and assembly of the Versatile Cubesat Telescope prototype will be published in the future.\n\n\\section{Conclusion}\n\nWe present diffraction-limited on- and off-axis designs for a Versatile CubeSat Telescope that fits within a 2U volume at low cost. Comprehensive analysis of the sensitivity to misalignment and polarization were considered, and a stable mechanical housing was created for the on-axis VCT. The VCT is a well-characterized high-performance design that can be adapted to a variety of space-borne research payloads. Future iterations of the VCT will expand upon the development of MEMS devices in space (e.g. DeMi\\cite{Morgan21}) by replacing the FSM with a Deformable Mirror for a higher degree of active wavefront control. Closed loop thermal simulations will be conducted to demonstrate the behavior of the VCT in response to a dynamic thermal environment.\n\n\\section{Acknowledgements}\nWe thank Zemax for the early access to their STAR analysis feature. This research made use of community-developed core Python packages, including: Numpy\\cite{numpy}, Matplotlib \\cite{matplotlib}, SciPy \\cite{jones_scipy_2001}, Astropy \\cite{the_astropy_collaboration_astropy_2013}, and Jupyter, IPython Interactive Computing architecture \\cite{perez_ipython_2007,kluyver_jupyter_2016}. Portions of this work were supported by the Arizona Board of Regents Technology Research Initiative Fund (TRIF).\n\n\n\\section{INTRODUCTION}\n\nCubeSat space telescopes are a growing asset in astronomical research\\cite{Shkolnik_2018,douglas2019cubesats}. Their standardized format and compact volume enables the rapid development of high-impact mission concepts above the earth's atmosphere. Flight-tested commercial microelectromechanical systems (MEMS)  have further enhanced CubeSat telescopes by providing dynamic pointing and aberration control on-orbit (DeMi, NODE) \\cite{Yenchesky2019OptomechanicalDA,Clements2016}. The expanding capacity of CubeSats for astronomical research invites the development of novel technologies for their optical systems. However, the resolution in CubeSat optics is fundamentally limited by the clear aperture permitted by the CubeSat volume. The rayleigh criterion tells us that resolution ($\\Delta \\theta$) is inversely related to to the entrance pupil diameter ($D$) of the optical system.\n\n\\begin{equation}\n    \\Delta \\theta = 1.22 \\frac{\\lambda}{D}\n\\end{equation}\n\nReflective CubeSat objectives traditionally require mounting hardware that limits the available entrance pupil diameter. Single-point diamond turned (SPDT) mirrors can mitigate this limitation by enabling the manufacturer to machine the mounting hardware directly into the rear of the mirror substrate, eliminating the need for mounting hardware around the edge of the mirror and enhancing the nominal throughput and resolution. This also grants the objective a considerable degree of athermalization by eliminating the difference in the coefficient of thermal expansion between the mirror and mounting hardware\\cite{Zhang:17}. SPDT surfaces are typically used in longer wavelengths (MWIR, LWIR) due to the midspatial frequency errors left by tooling marks. However, recent developments in optical polishing\\cite{Jeon:17} have introduced Magnetorheological Finishing (MRF) to the fabrication of SPDT surfaces, reducing the dominant midspatial frequencies considerably and therefore extending their use into the optical. Athermal CubeSat objectives composed of low surface roughness mirrors with a large entrance pupil invites the design of high-performance optical CubeSat payloads to make the next generation of research and technology development in space more accessible. The Versatile CubeSat Telescope (VCT) concept is a prototypical fore-optic that boasts compatibility with a variety of research payloads. The goal of the VCT is to develop a flexible, large aperture telescope that can be replicated at low cost for easy adaptability to future research payloads and technology demonstrations. \n\n\\section{OPTICAL DESIGN - ON-AXIS V.S. OFF-AXIS IMPLEMENTATIONS}\nThe design of the VCT began with two realizations of a CubeSat telescope that could take advantage of a large primary mirror. The first, an on-axis telescope with a more classical Ritchey-Chretien objective outfitted with a plano-convex aspheric collimator. The second, an all-reflective off-axis Ritchey-Chretien solution with a freeform collimator. We baselined a 95mm entrance pupil diameter with a 20$\\%$ obscuration for the on-axis telescope, and scaled the entrance pupil diameter of the off-axis telescope to be equivalent in collecting area. A high pupil magnification is required for both designs to image the primary mirror onto the small MEMS FSM clear aperture (5mm). The specifications and system layouts are shown in table \\ref{tab:specs} and figure \\ref{fig:layout} respectively.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{95 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Field of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{20$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\qquad\n    \\begin{tabular}{c c}\n        \\hline\n        Specification & Value  \\\\\n        \\hline\n        Entrance Pupil Diameter & \\textbf{88 mm} \\\\\n        Exit Pupil Diameter & 5 mm \\\\\n        Half Fiel of View & 0.2$^{\\circ}$ \\\\ \n        Secondary Obscuration & \\textbf{0$\\%$} \\\\\n        Central Field Strehl & $>$0.99 \\\\\n        Average Field Strehl & $>$ 0.80 \\\\\n        OTA Packaging Volume & $<$ 2U \\\\\n        \\hline\n        \\\\\n    \\end{tabular}\n    \\caption{Optical design specifications for the (left) On-Axis and (right) Off-Axis telescope designs. All specifications are similar except for those in bold, which are set such that the apertures of each design are equal in throughput.}\n    \\label{tab:specs}\n\\end{table}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.9\\textwidth]{vct_drawings.png}\n    \\caption{System layouts of the (left) On-Axis Telescope and the (right) Off-Axis Telescope designs. Each has three optical elements, two fast-steering mirrors (FSM) for fine pointing control, and a volume dedicated to a research payload.}\n    \\label{fig:layout}\n\\end{figure}\n\n Aspheric surfaces were necessary for both telescopes to achieve a high pupil magnification (19x and 17.5x for the on- and off-axis designs respectively) while maintaining a well-corrected field of view. While this slightly complicates the metrology of the mirrors it did not add any additional cost because the mirrors were already going to be diamond-turned. Both telescopes utilized a Ritchey-Chretien objective that forms an image near the primary mirror, which is collimated by an aspheric element to produce an external exit pupil. The size of the exit pupil was chosen to match the clear aperture of a MEMS fast-steering mirror (FSM)\\cite{memsfsm,Serra21} to correct for spacecraft pointing and jitter errors while in orbit. The external pupil also serves as a convenient interface for any instrument suite that would be included on the VCT (e.g. spectrographs, cameras, lasers).  The concept for the on-axis design was to make a well-established telescope format work given the pupil magnification and field of view specifications. The aspheric surfaces granted the design a well-corrected field of view in a compact format suitable for a CubeSat (2U). However, the design has a secondary obscuration which limits the throughput of the system and adds diffraction features to the image. The primary goal for the off-axis design was to offer a viable alternative to the classical on-axis design that would not suffer from undesirable diffraction effects brought upon by the secondary obscuration. Unfortuantely, shifting the primary parabolic mirror off-axis comes at the cost of more field-dependent aberration that limits the system's field of view. To accommodate this we selected the tertiary mirror of the off-axis design to be freeform. Before optimization of the freeform surface, the rotational symmetry of the system was broken by tilting the secondary and tertiary mirrors about their foci to make the system free of linear astigmatism. A linear-astigmatism free three mirror system (LAF-TMS) has been shown by Park et al\\cite{park_development_2020} to be an excellent starting point for freeform designs by using the mirror tilt angles and inter-mirror distances (Eq \\ref{eq:laf}) to mitigate a dominating aberration early in the design process.\n\n\\begin{equation}\n    \\frac{l_2^\\prime}{l_2}\\frac{l_3^\\prime}{l_3}tan{i_1}+\\left(1+\\frac{l_2^\\prime}{l_2}\\right)\\frac{l_3^\\prime}{l_3}tan{i_2+}\\left(1+\\frac{l_3^\\prime}{l_3}\\right)tan{i_3=0}\n    \\label{eq:laf}\n\\end{equation}\n\nIn this equation, parameters i$_{1,2,3}$ are the angles of incidence of the optical axis ray on surfaces 1,2, and 3, while $l_{2,3}$ and ${l^{\\prime}}_{2,3}$ are the front and rear focal lengths the secondary and tertiary mirror. Once the baseline LAF-TMS design had been implemented, the third mirror was changed to a freeform XY-Polynomial surface where coefficients were optimized that maintained the bilateral symmetry of the optical system. The mirrors were shifted off-axis in the $\\hat{y}$ direction, as was the field bias (+0.2$^{o}$). Consequently the freeform surface was constrained to be plane-symmetric about the $\\hat{y}-\\hat{z}$ plane by solely optimizing coefficients of even order in $\\hat{x}$. The resultant design is nearly diffraction-limited across the biased field of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.8\\textwidth]{strhvsfov.png}\n    \\caption{Maps of strehl ratio v.s. field of view for the (left) On-Axis and (right) Off-Axis VCT. The symmetry of the optical system mirrors the performance quite readily, with the on-axis design having a rotationally symmetric field performance whereas the off-axis design exhibits plane-symmetric performance.}\n    \\label{fig:strvfov}\n\\end{figure}\n\n\\section{SENSITIVITY ANALYSIS}\n \nThe misalignment sensitivities of the two designs were explored in Zemax OpticStudio (ZOS) by iteratively perturbing each optic in six degrees of freedom using the Python ZOS-API. Our as-built performance goal for both systems was an average Strehl ratio $>$ 0.7 across the field of view to maintain reasonably diffraction-limited performance. The $0\\%$, $70\\%$, and $100\\%$ field of view in $\\pm \\hat{x}$ and $\\pm \\hat{y}$ were considered in the average calculation, considerably biasing the sensitivity analysis toward the edge of the field of view where performance is expected to degrade faster. This results in a more pessimistic analysis of misalignment sensitivity. The sensitivity analysis fixes the primary mirror while M2 and L3\/M3 were perturbed to better understand how misalignments with respect to the primary would impact the optical performance over all fields of view.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{qkdsens.png}\n    \\caption{Sensitivity to misalignment curves of the On-Axis (Solid Red) and Off-Axis (Dashed Blue) designs for their respective secondary (top) and tertiary (bottom) optic. The on-axis system is less sensitive for X\/Y decenter of both elements, X\/Y\/Z tilt of M2, and X\/Z tilt of M3.}\n    \\label{fig:my_label}\n\\end{figure}\n\nFigure 2 shows that in most cases the on-axis design was less sensitive to misalignment in six degrees of freedom. This result is consistent with the complexity of each system. The off-axis design employs off-axis conics and a freeform surface, both of which will be sensitive to misalignment due to the higher mirror slopes than the on-axis design. In applications where the secondary obscuration is tolerable, the on-axis design would be a better choice for applications where cost and misalignment risk are limiting factors.\n\n\\section{POLARIZATION ANALYSIS}\nMany spaceborne optical payloads have a degree of sensitivity to polarization effects. From laser experiments in support of quantum communications\\cite{Serra21} to polarimetry of atmospheric ice\\cite{Hart20}, there is a clear need to characterize the response of cubesat payloads to a vector electric field. To offer support for polarization-sensitive research payloads a comprehensive polarization ray trace (PRT) analysis was conducted for each realization of the VCT.  PRT is a ray-based approach to vector field calculations enabled by determining the local fresnel reflection coefficients that alter the transmission and phase of the electric field at each surface in the optical system. These effects are traced from the entrance pupil to the exit pupil of the optical system to determine the cumulative polarization behavior of the instrument. The proposed VCT designs use fast primary mirrors to achieve the high pupil magnification, so some polarization aberration is expected. However mirror substrates and coatings can be selected to mitigate the impact of polarization aberration for a given application. \n\nA PRT tool was written using the ZOS-API to create maps of diattenuation and retardance at the exit pupil of the system. The ZOS-API tool creates a batch raytrace and propagates it from the entrance pupil of the optical system to the surface under investigation. The output of the raytrace gives surface normal vectors in addition to exit ray vectors which are used to calculate the incident ray vectors and angle of incidence. Diattenuation ($D$) and retardance ($\\delta$) data is produced from the Zemax coating files and ray angles of incidence to generate the surface and pupil maps shown in figure \\ref{fig:polmaps_al}.\n\n\\begin{equation}\n    D = \\frac{|r_p|^{2} - |r_p|^{2}}{|r_s|^{2} + |r_p|^{2}};\\phantom{flapjackfacts}\\delta = |\\phi_{p} - \\phi_{s}|\n    \\label{eq:diat}\n\\end{equation}\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{polplots.png}\n    \\caption{Polarization pupil maps assuming bare aluminum mirrors and uncoated N-BK7 lens surfaces. (Top) Diattenuation and (Bottom) retardance maps of the (left) On-Axis and (right) Off-axis versatile cubesat telescopes. These data were computed at $\\lambda$ = 780nm. At this wavelength the Off-Axis Aluminum VCT sees the greatest diattenuation, and the Off-Axis Gold sees the greatest retardance. This indicates that the off-axis design would introduce more challenges to polarization-sensitive payloads than the on-axis design would.}\n    \\label{fig:polmaps_al}\n\\end{figure}\n\nThe polarization pupil map permits analysis of the optimal material and configuration for a given polarization-dependent application. Figure \\ref{fig:polmaps_al} demonstrates that aluminum is more diattenuating than gold at $\\lambda=780nm$, and the off-axis configuration performs worse than the on-axis configuration due to the higher angles of incidence on the mirror surfaces. \n\n\\section{MECHANICAL DESIGN}\n\nThe VCT must be designed to deliver a high optical wavefront when operational on-orbit, yet also robust enough to survive the rigors of launch and orbital injection separation.  Achieving both goals in a compact package that affords straight-forward assembly and alignment is quite challenging.  Another important aspect of the design must also consider the mass contribution of the structural elements as this can be costly when considering launch requirements and on-orbit maneuvering.  With the prevalence of small satellite design ( $<$ 12U), development of a telescope that fits within a 2U volume with a mass less than 2 kg would present a low-cost solution that enables rapid prototyping of research payloads. For a prototype realization of the VCT, the optical system design called for SPDT Aluminum primary and secondary mirrors. All mechanical structural elements were also chosen to be aluminum to match the thermal behavior of the mounting structure to the optical elements. The collimating lens, which locates the instrument pupil, could also be easily mounted to the OTA using a common aluminum barrel structure with simple shims for precise alignment. \n\nThe first goal in the mechanical design of the telescope Optical Tube Assembly (OTA) is to support the optical elements with minimal effect due to expected thermal perturbations that would be encountered on-orbit.  These effects would include both rigid body misalignment and wavefront aberrations caused by subtle flexing of the optical elements.  In order to bound the problem, the first design decisions centered on overall sizing to fit within the small-sat form factor. The primary-secondary despace tolerance is very tight, so a truss structure was designed to properly space the two mirrors. The truss structure presents a good strength to weight ratio and is commonly used in ground-based telescope designs. For this space-based application, the truss structure would not allow for simple and easy shrouding of the optical path and thus would likely require additional structural pieces to support stray light baffling elements and thermal mitigation elements (i.e., mylar blankets).  Therefore, an early design decision was made to investigate a tube structure that would serve as both the metering structure that could maintain the mirror separation tolerance, transverse misalignment tolerance, and provide an in-place baffle that could be customized for the particular optical application.  A significant benefit of the tube design is the mass savings realized by the ability to select a very thin tube wall thickness that still retains the desired structural performance for all loading cases.  This tube would also allow for easy installation of thermal mitigation measures and targeted design of thermal conduction paths that further ensure optical performance stability while on-orbit.\n\nThe next major goal in the mechanical design considers the manufacturability of the system.  Considering its small size, the OTA manufacturability includes the fabrication processes for all structural components, as well as thoughtfully planned assembly and alignment processes that ensure the optical performance requirements can be met.  The aluminum mirrors provide a unique opportunity to meet the need for an assembly that is easy to integrate and align.  The primary mirror optical surface is thus applied directly to the structural component (primary mirror) which has appropriate design elements that consider thermal performance and mounting points for both the metering tube and the interface to the downstream optical system via a \\emph{hex plate} interface.  Clearly defining the rear of the primary mirror blank as the OTA support location allows for simple design of flexural components that will not propagate errors from other systems into the OTA, or vice-versa.  Similarly, the secondary mirror surface is applied directly to a head-ring structure that also considers thermal performance and mounting points to the metering tube, but also minimizes the size of support strut spiders that are common in the on-axis  design.  Considering the small size of the OTA, the one-piece secondary mirror\/head-ring simplifies its alignment features by moving them out to the tube diameter, rather than trying to squeeze them into the shadow of the secondary mirror on the entrance aperture.\n\nThe final goal of the mechanical design considers the interface of the OTA to the instrument and the spacecraft.  Most terrestrial optical system applications utilize a breadboard style support that allows for many mounting possibilities for both the OTA and any downstream instrument opto-mechanical elements, and can be purposefully designed to incorporate spacecraft mounting options that will isolate the payload from the spacecraft both thermally and dynamically as necessary.  The hex plate provides all of these interfaces in a single machined aluminum component.  The connection of the OTA to the hex plate is of particular importance for this discussion.  As mentioned earlier, the OTA must remain dynamically and thermally isolated from the other spacecraft or instrument components to preserve the telescope optical performance. A Finite Element Model (FEM) was created using the aluminum tube structure OTA, with both structural and optical components represented, and a Finite Element Analysis (FEA) was performed to determine the model behavior with both structurally dynamic boundary conditions and also varying thermal environmental conditions.  The FEM is shown in figure \\ref{fig:vct_fea}.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{vct_fea.png}\n    \\caption{The FEM of the VCT OTA highlighting the hex plate, flexure, and tube design.}\n    \\label{fig:vct_fea}\n\\end{figure}\n\nA simple three flexure design was selected to mount the OTA to the hex plate.  The flexures were positioned tangentially to the primary mirror blank to better accommodate thermal deformations that could arise at the hex plate. The flexure material properties were varied to determine the best trade of optical path stability, launch survivability, manufacturability, and cost. The materials considered were Aluminum (Al 6061-T6) and Titanium (Ti-6Al-4v).  For the structural analysis the hex plate was fixed at three points on its outer edge simulating spacecraft support, and MAC loads (100g accelerations in three orthogonal directions) were applied to the model.  Various flexure throat cross sectional areas were considered, and the best design was chosen based on the resulting maximum stresses revealed in the FEA.  The results for each material selection under this loading are shown in table \\ref{tab:macload}. \n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n     Load Case Direction & Max Flexure Stress [Al] & Max Flexure Stress [Ti] \\\\\n    \\hline\n        100g $\\hat{x}$ & 103ksi & 112ksi \\\\\n        100g $\\hat{y}$ & 98ksi & 103ksi \\\\ \n        100g $\\hat{z}$ & 42ksi & 46ksi \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{MAC loading in three orthogonal directions and the corresponding maximum flexure stress for Aluminum and Titanium flexures. The Ti flexures were chosen due to their higher maximum stress.}\n    \\label{tab:macload}\n\\end{table}\n\nAn additional analytical consideration for the structure is the modal behavior of the OTA.  The modal FEA reveals the lowest expected resonant frequency of the OTA under the prescribed mounting conditions for given material selections and flexure design choices.  The table below shows the first three modal FEA results for the same conditions discussed above.  These results indicate there would be little chance of exciting resonance in the OTA structure when considering expected launch loads.  And similarly, the telescope itself would not likely impart damaging resonance to any other spacecraft systems or partner satellites if launched as a secondary payload ride-share.\n\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{c c c}\n    \\hline\n        Al Flexures & Ti Flexures  \\\\\n    \\hline\n        130Hz & 156Hz \\\\\n        130Hz & 156Hz \\\\\n        363Hz & 434Hz \\\\\n    \\hline\n    \\end{tabular}\n    \\caption{Modal behavior of the OTA for Al and Ti flexures for the cases described in \\ref{tab:macload}. Based on the results from the FEA for both materials, the Titanium flexure design was selected based on the extra margin provided on mass and stress.}\n    \\label{tab:my_label}\n\\end{table}\n\n Verification  of the selected flexure design needed to also consider the thermal behavior of the system.  For ease of creating the thermal FEM, the boundary conditions for the structure were preserved from the structural FEA and a 27.8\u00b0C differential in temperature was applied to the hex plate.  This set of conditions would bring into focus the thermal isolation of the OTA from the hex plate.  The results from this thermal analysis were normalized to show an expected mirror sag of 10nm P-V for each 1.0\u00b0C of temperature differential.  The results (shown below) also indicate an axisymmetric deformation of the OTA as would be expected.\n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.55\\textwidth]{fea_results.PNG}\n    \\caption{Finite Element results of the hexplate-primary mirror temperature differential. The flexure design results in $\\approx$ 10nm of trefoil surface sag per degree Celsius.}\n    \\label{fig:fea_results}\n\\end{figure}\n\nOverall, an iterative analysis process was implemented to determine the final telescope structure geometry.  The results from dynamic survivability, modal, and thermal analyses were considered for each element.  Part of the FEM definition involves application of material properties that also allow for the consideration of mass as a secondary factor in selection of the final geometry.  Resulting optical performance effects are then analyzed in the Zemax OpticStudio STAR module for each iteration by exporting the three-dimensional nodal parameters from the FEA results. Other geometry selections made during the design process were more straightforward than the flexural interface between the OTA and the hex plate.  For instance, given the axisymmetric behavior for all analyses, the tube geometry selections were made without direct optical analyses as the major performance effect would be in focus change due to varying thermal conditions.  For this, common engineering knowledge can be applied to bias the focus alignment for all optical elements prior to integration with an instrument for the given expected on-orbit operational conditions (for this project +\/-1\u00b0C).  Similarly, the hex plate design only needed to preserve the axisymmetric geometry approximation when considering its overall shape and interface definition.  \nWhile not discussed in detail earlier, mass of each components was considered as an important mechanical design decision.  An overall maximum allowable mass budget of 2 kg was allotted to the design and did not present a problem during the iterative analyses.  The final telescope design mass is expected to be less than 1 kg once all hardware is integrated.  The only major design decision based on mass reduction involved light-weighting the primary mirror blank by cutting pockets into the structure and analyzing these pockets for their structural effect on mirror surface figure.  \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.6\\textwidth]{3dprintlw.PNG}\n    \\caption{3D Prints of the primary mirror (left) and hex plate (right) demonstrating the lightweighting scheme used to achieve the mass requirement of $< 2kg$}\n    \\label{fig:my_label}\n\\end{figure}\n\n\\section{STOP EVALUATION - MERGING FEA WITH SEQUENTIAL RAYTRACING}\n\nNumerous loading conditions were studied through FEA software to ensure the overall survival of the VCT payload, however, FEA analysis by itself could not determine how the deformed elements would impact the system's optical performance. Further studies of the wavefront deformation were necessary to confirm that the system would be able to function properly in its environment of operation. Typically, this kind of analysis would require the deformed surface to be decomposed into hundreds of polynomial terms to accurately model the shape of the surface, with no guarantee of the raytracing software being able to support the number of terms needed. However, through use of OpticStudio's STOP (Structural, Thermal, and Optical Performance) analysis feature \\emph{STAR} (Structural, Thermal, Analysis and Results) evaluation of the system's performance in various loading scenarios was made quick and efficient. STAR allows for the data from FEA software to be loaded directly into OpticStudio and onto select optical elements. By using the position and displacement of each node in a 6 column (x, y, z, dx, dy, dz) format STAR accurately deforms the surface shape, allowing for the deformed wavefront to be observed under various loading conditions. For the purposes of our design study we primarily looked at the surface deformation on the primary mirror caused by rotation about the optical axis.\n\nIn figure \\ref{fig:star} the change of the optical system's wavefront and PSF are shown after applying a -237 arcsecond rotation about the optical axis. This causes clear degradation in the system's overall optical performance. This optical analysis is critical for an iterative design process between optical and mechanical design teams. Quick optical performance analysis through STAR allows for the mechanical team to adjust the structures supporting the optical surfaces where necessary, ensuring performance specifications are met in realistic structural and thermal scenarios. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=\\textwidth]{starinv.png}\n    \\caption{Influence of STAR on the wavefront (left) and the PSF (right). A force tangent to the primary mirror was applied in FEA and the resultant surface deformation was loaded into the OpticStudio raytrace model. This permits evaluation of performance degradation as a function of structural and thermal deformation.}\n    \\label{fig:star}\n\\end{figure}\n\n\n\\section{Laboratory Prototype Status}\n\nThe next step for the development of the VCT is the construction of a laboratory prototype of the on-axis aluminum VCT. Mirrors for the on-axis VCT have been fabricated by Hanbat national university and are shown in figure \\ref{fig:fab_mirrors}. \n\n\\begin{figure}[H]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{mirror_img.PNG}\n    \\caption{The fabricated (left) primary and (right) secondary of the on-axis VCT design manufactured by Hanbat National University in South Korea.}\n    \\label{fig:fab_mirrors}\n\\end{figure}\n\nThe primary and secondary mirror have arrived at the University of Arizona and are awaiting measurement and assembly so that they can be mounted into the OTA described in section 5 for system-level tests. The UArizona Space Astrophysics lab test facilities include a TVAC chamber and Hexapod that can be used for thermal vaccum and jitter tests respectively. Details of the comprehensive characterization and assembly of the Versatile Cubesat Telescope prototype will be published in the future.\n\n\\section{Conclusion}\n\nWe present diffraction-limited on- and off-axis designs for a Versatile CubeSat Telescope that fits within a 2U volume at low cost. Comprehensive analysis of the sensitivity to misalignment and polarization were considered, and a stable mechanical housing was created for the on-axis VCT. The VCT is a well-characterized high-performance design that can be adapted to a variety of space-borne research payloads. Future iterations of the VCT will expand upon the development of MEMS devices in space (e.g. DeMi\\cite{Morgan21}) by replacing the FSM with a Deformable Mirror for a higher degree of active wavefront control. Closed loop thermal simulations will be conducted to demonstrate the behavior of the VCT in response to a dynamic thermal environment.\n\n\\section{Acknowledgements}\nWe thank Zemax for the early access to their STAR analysis feature. This research made use of community-developed core Python packages, including: Numpy\\cite{numpy}, Matplotlib \\cite{matplotlib}, SciPy \\cite{jones_scipy_2001}, Astropy \\cite{the_astropy_collaboration_astropy_2013}, and Jupyter, IPython Interactive Computing architecture \\cite{perez_ipython_2007,kluyver_jupyter_2016}. Portions of this work were supported by the Arizona Board of Regents Technology Research Initiative Fund (TRIF).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:intro}\nCardiac magnetic resonance imaging is a valuable tool for myocardial structure, function, and tissue assessment, providing essential information for clinical diagnosis and treatment decisions in cardiovascular disease. Using standard segmented sequences in which data acquisition is segmented over multiple heart beats, good image quality can be obtained in patients with regular cardiac rhythm and good breath-holding ability; however, image quality can be degraded by motion artifacts when scanning patients with arrhythmia or poor breath-hold compliance. In comparison to segmented acquisitions, single-shot techniques can be applied for rapid image acquisition of a whole slice within a single shot, greatly reducing the scan time. Due to the short acquisition duration of single-shot techniques (typically < 200 ms), artifacts from intra-shot motion are negligible, therefore such methods tend to be robust against cardiac and breathing motion. However, this motion robustness comes at the expense of lower spatial resolution and signal to noise ratio (SNR). An example of the benefit of single-shot over segmented Late Gadolinium Enhanced (LGE) imaging in a patient who could not breath-hold is shown in \\ref{fig:fig1}. Recent techniques proposed to enhance the SNR of single-shot methods by motion correcting and then averaging multiple single-shot images acquired in free-breathing \\cite{Kellman2005}. While this technique shows good results with low acceleration factors, it may not provide optimal image quality for higher undersampling, introducing blurring and undersampling artifacts, mainly due to higher weight given to the regularization.  Batchelor et al. \\cite{Batchelor2005} proposed a first generalized reconstruction framework for motion compensation. The method allows arbitrary motion to be compensated by solving a general matrix inversion problem. This technique, however, requires an adequate knowledge of the displacement fields. The recent GRICS method \\cite{Odille2008a} extended this work by jointly estimating the motion and the recovered image, however, it relied on a motion model provided by external sensors (e.g. ECG, respiratory belt).\n\nIn this work, we sought to develop an efficient motion correction implementation suitable for reconstructing a high-resolution, high-SNR image from multiple accelerated single-shot images. The proposed method combines the benefits of using a hybrid self-navigated sampling scheme (see Fig. \\ref{fig:fig1}) with a joint reconstruction framework. In the image reconstruction step, a highly efficient feature-preserving regularization scheme (Beltrami) is proposed for recovering sharp details. We show that the proposed method is robust to high acceleration factors and yields results with efficient noise reduction and better overall image quality at a low computational cost.\n\n\n\\section{Theory}\n\n\\subsection{General Motion Compensation Framework}\n\nMotion compensation techniques aim to solve the following inverse problem: find an underlying image $\\rho$ free of motion artifacts, given derived measurements $s$ through the system $E$, affected by noise $\\nu: s = E\\rho + \\nu$. Where $E$ is the encoding matrix, generally composed of a sampling operator $\\xi$, a Fourier transform $F$, coil sensitivity maps $\\sigma$ (in case of parallel imaging reconstruction), and a motion warping operator $W$ describing a non-rigid deformation for each shot. Here $\\rho \\in \\mathbb{C}^{n_x \\times n_y}$ and $s \\in \\mathbb{C}^{n_x \\times n_y\/acc \\times n_r \\times n_c}$  are complex data with $n_c$ the number of receiver coils, $n_r$ the number of repetitions and $acc$ the acceleration factor. In this work, the acquired data s represents the k-space data from multiple single-shot images and is generally corrupted by noise. A typical approach to solve this problem is to minimize the squared difference as assessed by the Euclidean norm. However, this problem is generally ill-posed (e.g. due to undersampling and noise), leading to non-uniqueness of the solution, if it exists. Thus, regularity constraints on the unknown solution $\\rho$ have to be considered. Furthermore, motion should also be considered as unknown. The general joint optimization framework is then defined as\n\n\\begin{equation}\n\\label{eq:eq1}\n\\rho = \\argmin\\limits_{\\left( \\rho, \\vec{u}\\right)} \\lbrace \\Vert s - E\\left(\\vec{u}\\right) \\rho \\Vert_2^2 + \\lambda \\phi \\left( \\rho \\right) \\rbrace \\text{ where } E\\left( \\vec{u}\\right) = \\xi F \\sigma W \\left( \\vec{u}\\right)\n\\end{equation}\n\nHere $\\vec{u}$ represents the displacement fields, $\\phi$ is the chosen regularization function and $\\lambda > 0$ is the corresponding regularization parameter. The optimization problem in Equation \\ref{eq:eq1} is solved in four steps: (i) we first use the k-space center of the single-shot images to extract a self-navigation signal and to cluster the raw data into a reduced number of respiratory bins \\cite{Usman2013}; (ii) we reconstruct the images from each bin independently using a Beltrami-regularized SENSE (B-SENSE) reconstruction; (iii) then an estimate of the motion is obtained using a non-rigid registration (minimization of Equation \\ref{eq:eq1} with regards to $\\vec{u}$ \\cite{Odille2014}) and (iv) a high resolution\/SNR image is generated using the proposed motion-compensated reconstruction process (minimization with regards to $\\rho$). A general description of the method is shown in Fig. \\ref{fig:fig2}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.91\\textwidth]{Fig1.png}\n\\caption{(Left) Comparison between 2D segmented LGE (top) and 2D single-shot LGE (bottom) on a 77-year-old patient with breath-holding difficulties. The segmented LGE has higher resolution than the single-shot LGE but shows severe motion artefacts. (Right) Proposed hybrid k-space acquisition scheme including motion-calibration data (fully sampled center) and undersampled periphery, aimed to combine the resolution of segmented LGE with the motion-robustness of single-shot LGE.}\n\\label{fig:fig1}\n\\end{figure}\n\n\n\\subsection{Beltrami-Regularized SENSE}\n\nIn our framework, a respiratory signal is extracted from the motion calibration data itself. This pre-processing step is achieved by stacking the low resolution images along the time dimension. Singular value decomposition is then applied to the stack and the first left-singular vector is used as a good approximation of the true respiratory signal. A specific respiratory phase is then assigned to each acquired shot, as explained in \\cite{Usman2013}. This binning strategy would split the data into fewer motion states $n_b \\left( < n_r\\right)$ with negligible respiration motion and lower undersampling in each of them. Images from each respiratory bin $\\left( \\rho_i \\right)_{i = 1,\\dots,n_b}$ are then individually reconstructed by solving\n\n\\begin{equation}\n\\label{eq:eq2}\n\\rho_i = \\argmin\\limits_{\\rho} \\lbrace \\Vert s_i - E_i \\rho \\Vert_2^2 + \\lambda \\sqrt{1+\\beta^2 \\vert \\nabla \\rho \\vert^2} \\rbrace\n\\end{equation}\n\nThe first term in Equation \\ref{eq:eq2} is a data fidelity term that aims to minimize the difference between the reconstructed image and the acquired data. The Beltrami regularization $\\sqrt{1+\\beta^2 \\vert \\nabla \\rho \\vert^2}$ has been introduced in the field of string theory for physics and has shown high potential in several imaging problems, including image denoising and enhancement \\cite{Polyakov1981} and super-resolution reconstruction \\cite{Odille2015}. In particular, the metric can be chosen such that the Beltrami energy corresponds to an arbitrary interpolation between Gaussian diffusion $\\beta \\to 0$ and total variation (TV) \\cite{Rudin1992a} regularization $\\beta \\gg 1$. In \\cite{Zosso2014}, the authors showed that Beltrami regularization is able to maintain the advantage of TV (edges preserving, noise reduction) as well as reducing the effect of staircasing. B-SENSE is very similar to compressed sensing SENSE (CS-SENSE) methods presented by other authors \\cite{Liang2008}, where here Beltrami is making the image sparse in the gradient domain. Even though this suggests that B-SENSE has a close relationship with the compressed sensing (CS) theory, it is, however, not CS as defined by Cand\\`es et al. \\cite{Candes2006}, especially due to the pseudo-random undersampling pattern used here (i.e. a uniform random pattern is used in \\cite{Candes2006}). We propose to solve Equation \\ref{eq:eq2} by adopting a primal-dual projected gradient approach \\cite{Chan1999} with the potential to converge faster than the classic primal gradient-descent \\cite{Zosso2014}. Respiratory motion estimation is then accomplished using independent non-rigid registration of the images reconstructed from each respiratory bin. Here we use an iterative framework validated in a large patient database for myocardial $T_2$ mapping \\cite{Odille2014}, which is based on minimizing the sum-of-squared differences of the pixel intensities within a multi-resolution Gauss-Newton scheme.\n\n\n\\subsection{Motion Compensated Reconstruction with Preserved-Features}\n\nThis section presents the final step for solving the motion compensated problem in Equation \\ref{eq:eq1}. The aim of the method is to reconstruct the high resolution, high SNR image $\\rho$ from the acquired raw data $s = \\left( s_i \\right)_{i = 1,\\dots,n_b}$. For the motion compensated reconstruction, we solve the following optimization problem, with the acquisition model now including the estimated motion fields:\n\n\\begin{equation}\n\\label{eq:eq3}\n\\rho = \\argmin\\limits_{\\rho} \\lbrace \\Vert s - E\\left(\\vec{u}\\right) \\rho \\Vert_2^2 + \\lambda \\sqrt{1+\\beta^2 \\vert \\nabla \\rho \\vert^2} \\rbrace\n\\end{equation}\n\nAs in the previous section, we use a primal-dual projected gradient approach, employing the Beltrami energy as regularity prior \\cite{Zosso2014}. Note that regularization is always preferred in motion compensated reconstruction due to the ill-conditioning induced by the motion operators, as shown in \\cite{Atkinson2003}.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig2.png}\n\\caption{Schematic illustration of the proposed reconstruction, including the non-rigid motion extraction. Acquisition is performed using complementary trajectories, leading to uniform samplings in the phase encoding direction, which allows for an optimal combination of the k-spaces according to their positions in the breathing signal. The motion model, initialized by registering the images from each respiratory bin, is incorporated into the reconstruction process.}\n\\label{fig:fig2}\n\\end{figure}\n\n\\section{Material and Methods}\n\nThe proposed reconstruction algorithm was applied and validated with different experiments using Matlab (The MathWorks, Natick, MA) on a PC with Intel Xeon 3.3 GHz CPU and 64GB ram. The experiments were performed on 3T MR750w and 1.5T MR450w systems (GE Healthcare, WI, USA).\n\n\n\\begin{table}\n  \\centering\n  \\caption{Parameters used for the different experiments. The acquisition matrix size was 192 x 256.}\n  \\label{tab:table1}\n \\begin{tabular}{cccccccc}\n    \\toprule\n     & \\#repetition & \\#calib & \\#periphery & acc & acc shot & acc shot & NEX\\\\\n     & $n_r$ & lines & lines & peri & pre-bin & post-bin & sequence\\\\\n    \\midrule\n    Simulation 1 & 4 & 32 & 48 & 3.3 & 2.4 & - & 1.67\\\\\n    Simulation 2 & 4 & 32 & 32 & 5 & 3 & - & 1.33\\\\\n    Phantom      & 6 & 32 & 48 & 3.3 & 2.4 & - & 2.5\\\\\n    In vivo      & 15 & 17 & 43 & 4.1 & 3.2 & 1.1 (5 bins) & 4.7\\\\\n    \\bottomrule\n  \\end{tabular}\n\\end{table}\n\n\n\\subsection{Offline Simulation on Synthetic Data}\n\nIn order to perform a realistic simulation, we first created a synthetic dataset based on actual LGE patient images. In one patient with suspected cardiovascular disease, four repetitions of a cardiac-gated, inversion recovery prepared, single-shot LGE scan were acquired in free-breathing 10 minutes after Gadolinium injection. Cardiac images were obtained with a spoiled fast gradient echo sequence and the following parameters: matrix size 192 x 256, in-plane spatial resolution 1.52 mm x 1.52 mm in short axis with slice thickness = 8 mm, readout flip angle = 20 degrees, echo time (TE) = 2.02 ms, mid-diastolic trigger delay, pulse repetition time (TR) = 4.43 ms and SENSE factor = 2 with partial Fourier. Synthetic k-space data were created by the application of synthetic coil sensitivity maps (with Gaussian profiles) to the LGE images, Fourier transformation and undersampling in the phase encoding direction. A full sampling of the central k-space area (17 lines) was used and the peripheral area was undersampled with a Golden Step Cartesian trajectory \\cite{Derbyshire2011} with an acceleration factor $R = 3.3$. Spacing between samples was proportional to the Golden ratio ($p = 0.618$). This trajectory enables an irregular but almost uniform distribution of the acquired data for any arbitrary number of repetitions, leading to incoherent aliasing (Fig. \\ref{fig:fig1}, right). The motion-free image was reconstructed using our reconstruction and compared to a motion correction method similar to that proposed by Kellman et al. \\cite{Kellman2005}, where the motion-free image is recovered by averaging the registered images obtained after the B-SENSE reconstruction step. We call this prior method reconstruction-registration-average (RRA).\n\n\\subsection{Phantom Imaging}\n\nSingle-shot pulse sequences were used to acquire phantom images with a 26-channel cardiac coil. The sequence was modified to take into account the same Golden ratio sampling as in our offline simulation experiment. The protocol was applied to acquire phantom images with a resolution of 1.48 mm x 1.48 mm. A translational motion was imposed to the table to mimic respiratory motion.\n\n\n\\subsection{In Vivo Validation Experiment with Self-Navigation}\n\nIn vivo cardiac datasets from two healthy adult volunteers were acquired on a 1.5T scanner using a 32-channel cardiac coil. A multi-shot slice (15 shots) of a free-breathing, cardiac-gated, spoiled fast gradient echo sequence (without inversion-recovery preparation) was collected with the following parameters: TE = 2.10 ms, TR = 4.52 ms, 8 mm slice thickness, FOV = 253 mm x 338 mm, matrix size 192 x 256 and 1.32 mm x 1.32 mm in-plane resolution, diastolic trigger delay. A fully sampled reference was acquired additionally in breath-hold for visual comparison. Each shot consists of 60 k-space lines: the central k-space was fully sampled with 17 lines and the periphery (43 lines) was undersampled, leading to a global acceleration factor of 3.2. An estimate of the respiratory signal was extracted using the proposed self-navigated technique, and was subsequently used to separate the acquired data into five respiratory bins. An overview of the parameters used in this study is given in Table \\ref{tab:table1}.\n\n\\section{Results}\n\nThe time needed to run the motion-compensated reconstruction for 15 shots of matrix size equal to 192 x 256 with 32-channel cardiac coils was about 1 min 35 s, including the time to compute the motion between shots.\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Fig3.png}\n\\caption{Cardiac short-axis reconstruction of a synthetic dataset generated from 4 single-shot LGE acquisitions in free-breathing on a 37-year-old patient with acceleration factors $r = 3.3$ and $r = 5$. a) One reconstruction using a classic SENSE (192 x 256), b) Sum-of-Squares (all repetitions), c) Reconstruction-Registration-Average, d) proposed reconstruction.}\n\\label{fig:fig3}\n\\end{figure}\n\n\\subsection{Offline Simulation on Synthetic Data}\n\nExample reconstruction results on the simulated data generated from a patient with nonischemic cardiomyopathy are shown in Fig. \\ref{fig:fig3}. One can see a spatially blurred result with a standard reconstruction-registration-average (RRA) method. The proposed reconstruction exhibits significant quality improvements over each method with an acceleration factor $r = 3.3$ while reconstructing sharper edges (arrows) and small structures. For higher acceleration factors the performance of our method is much better compared to RRA, both in terms of reconstruction accuracy and image quality.\n\n\n\\subsection{Phantom Imaging}\n\nSimilar results can be observed in phantom experiments (Fig. \\ref{fig:fig4}). Comparisons with a classic Tikhonov reconstruction are shown in Fig. \\ref{fig:fig4}. The results present the reconstructed phantom motion experiments where here the motion has been applied with the table. The sum-of-squares reconstruction (Fig. \\ref{fig:fig4}, left) clearly exhibits the effect of motion. As in the previous experiment, the RRA method exhibits blurry results (due to the undersampling), although providing a motion-corrected denoised image. A visual improvement can be noticed when applying a motion compensated reconstruction with Tikhonov regularization. The latter method performs well but is, however, unable to recover sharp edges and some residual artifacts can still be seen on the recovered image. The use of a fast primal-dual algorithm combined with Beltrami regularization makes the proposed reconstruction robust with better performance in terms of image quality, reduced artifacts and sharpness (Fig. \\ref{fig:fig4}, Bel).\n\n\n\\begin{figure}[h!]\n\\includegraphics[width=\\textwidth]{Fig4.png}\n\\caption{Reconstructions on a phantom using two different regularization methods with acceleration factor 3.3. Six single-shot repetitions have been acquired. From left to right: Sum-of-Squares (SoS), Reconstruction-Registration-Average (RRA), Tikhonov, Beltrami, Reference.}\n\\label{fig:fig4}\n\\end{figure}\n\n\\subsection{In Vivo Validation Experiment with Self-Navigation}\n\nFigure \\ref{fig:fig5} shows an example of the proposed fast and automatic self-navigated binning method on 150 consecutive slices of liver SPGR acquisition. The temporal rate was 400 ms, corresponding to a total acquisition duration of 1 min. The extracted respiratory signal (red) shows good agreement with the respiratory belt placed on the subject's thorax (blue), with a coefficient of determination $R^2 = 0.76$. Raw data acquired in similar motion states can be clustered into a reduced number of motion states, thereby improving the quality of images from which to extract motion in free-breathing without the need for navigators or external sensors.\nShort-axis images of the myocardium of a healthy subject without Gadolinium injection and without inversion recovery preparation are shown in Fig. \\ref{fig:fig6}. Both cardiac structures (myocardium wall, papillary muscles) and non-cardiac structures (blood vessels) are very well recovered with our reconstruction. The method yields significant sharpening of the myocardium wall and papillary muscles. However, due to the relatively high-undersampling, the RRA method is unable to recover a good quality image, exhibiting blurry structures and losing some of the details in the image such as blood vessels (arrows). This particularity is also seen in Fig. \\ref{fig:fig6} d where a specific intensity profile is plotted. The sharpness of the edges on our motion-corrected reconstruction is confirmed as well as the fidelity to the breath-hold acquisition.\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Fig5.png}\n\\caption{Data binning step of the proposed self-navigated signal obtained from 150 consecutive 2D fast spoiled gradient echo acquisitions in free-breathing liver imaging.}\n\\label{fig:fig5}\n\\end{figure}\n\n\n\\begin{figure}[h!]\n\\centering\n\\includegraphics[width=\\textwidth]{Fig6.png}\n\\caption{Data binning step of the proposed self-navigated signal obtained from 150 consecutive 2D fast spoiled gradient echo acquisitions in free-breathing liver imaging.}\n\\label{fig:fig6}\n\\end{figure}\n\n\\section{Discussion and Conclusion}\n\nWe introduced a new free-breathing single-shot LGE pipeline, including an optimized sampling and the associated joint reconstruction and motion correction algorithm designed for fast and robust cardiac imaging. By incorporating the estimated motion into the reconstruction process, we increased the robustness of the model and exhibited good quality images.\n\nIn this study, we used a fast and automatic self-navigated binning strategy that aims to cluster the acquired raw data into similar motion states. While preliminary results have shown improved image quality and better motion estimation, additional optimization of number of bins and number of repetitions is still required to maintain an optimal tradeoff between reconstruction quality, reconstruction time and accuracy of motion estimates. The motion corrected images show better visual quality than classic reconstructions but appear less sharp than corresponding breath-held acquisitions, especially for high accelerations. Possible explanations are the inaccuracies in motion estimates or other effects related to MR physics, such as spin history or changes in $B_0$ and $B_1$ inhomogeneities induced by breathing.\n\nOne interesting application of the proposed motion correction model is for high-resolution 3D isotropic LGE single-shot imaging of the heart, such as the one proposed recently in \\cite{Dzyubachyk2015} for myocardial scar assessment. This will allow for the reconstruction of 3D isotropic motion corrected volumes by keeping the advantages of a 2D acquisition (high tissue and vessel contrast, short acquisition time), e.g. using super-resolution techniques \\cite{Odille2015}. Other applications, such as abdominal imaging \\cite{Buerger2013} and coronary vessel imaging \\cite{Cruz2016}, are being investigated.\n\nA limitation to the method is that potential through-plane motion cannot be corrected, although it remains small compared to the slice thickness. To overcome this problem, one could consider weighting the images according to the motion amplitude compared to the target image or acquiring 3D slab instead of 2D slice data and applying motion compensation. The preliminary results presented in this work should be confirmed with further patient studies.\n\nThe feasibility of the proposed reconstruction has been evaluated in simulation, phantom and volunteer experiments. The method has been shown to allow non-rigid motion correction while efficiently recovering features, thanks to the Beltrami regularization scheme. The conventional segmented LGE acquisition is limited by the maximum breath-hold time, which limits the signal-to-noise ratio and\/or spatial resolution. This limitation is overcome by the presented free breathing approach. Ultimately, this method could enable accurate motion corrected reconstruction of single-shot images with higher spatial resolution and a higher signal-to-noise ratio compared to conventional segmented methods, with the potential to offer high-quality LGE imaging in challenging patients.\n\n\n\\subsection*{Acknowledgments}\nThe authors thank Mayo Clinic (Rochester, MN), Advanced Cardiovascular Imaging (New York, NY) and Morriston Hospital (Swansea, UK) for providing some of the imaging data. This publication was supported by the European Commission, through Grant Number 605162. The content is solely the responsibility of the authors and does not necessarily represent the official views of the EU.\n\n\n\\bibliographystyle{splncs}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}