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1401 | Amplitude modulation Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In amplitude modulation, the amplitude (signal strength) of the carrier wave is varied in proportion to that of the message signal being transmitted. The message signal is, for example, a function of the sound to be reproduced by a loudspeaker, or the light intensity of pixels of a television screen. This technique contrasts with frequency modulation, in which the frequency of the carrier signal is varied, and phase modulation, in which its phase is varied. AM was | "Amplitude modulation" | [
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1402 | the earliest modulation method used to transmit voice by radio. It was developed during the first quarter of the 20th century beginning with Landell de Moura and Reginald Fessenden's radiotelephone experiments in 1900. It remains in use today in many forms of communication; for example it is used in portable two-way radios, VHF aircraft radio, citizens band radio, and in computer modems in the form of QAM. "AM" is often used to refer to mediumwave AM radio broadcasting. In electronics and telecommunications, modulation means varying some aspect of a continuous wave carrier signal with an information-bearing modulation waveform, such as | "Amplitude modulation" | [
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1403 | an audio signal which represents sound, or a video signal which represents images. In this sense, the carrier wave, which has a much higher frequency than the message signal, "carries" the information. At the receiving station, the message signal is extracted from the modulated carrier by demodulation. In amplitude modulation, the amplitude or "strength" of the carrier oscillations is varied. For example, in AM radio communication, a continuous wave radio-frequency signal (a sinusoidal carrier wave) has its amplitude modulated by an audio waveform before transmission. The audio waveform modifies the amplitude of the carrier wave and determines the "envelope" of | "Amplitude modulation" | [
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1404 | the waveform. In the frequency domain, amplitude modulation produces a signal with power concentrated at the carrier frequency and two adjacent sidebands. Each sideband is equal in bandwidth to that of the modulating signal, and is a mirror image of the other. Standard AM is thus sometimes called "double-sideband amplitude modulation" (DSB-AM) to distinguish it from more sophisticated modulation methods also based on AM. One disadvantage of all amplitude modulation techniques (not only standard AM) is that the receiver amplifies and detects noise and electromagnetic interference in equal proportion to the signal. Increasing the received signal-to-noise ratio, say, by a | "Amplitude modulation" | [
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1405 | factor of 10 (a 10 decibel improvement), thus would require increasing the transmitter power by a factor of 10. This is in contrast to frequency modulation (FM) and digital radio where the effect of such noise following demodulation is strongly reduced so long as the received signal is well above the threshold for reception. For this reason AM broadcast is not favored for music and high fidelity broadcasting, but rather for voice communications and broadcasts (sports, news, talk radio etc.). Another disadvantage of AM is that it is inefficient in power usage; at least two-thirds of the power is concentrated | "Amplitude modulation" | [
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1406 | in the carrier signal. The carrier signal contains none of the original information being transmitted (voice, video, data, etc.). However its presence provides a simple means of demodulation using envelope detection, providing a frequency and phase reference to extract the modulation from the sidebands. In some modulation systems based on AM, a lower transmitter power is required through partial or total elimination of the carrier component, however receivers for these signals are more complex and costly. The receiver may regenerate a copy of the carrier frequency (usually as shifted to the intermediate frequency) from a greatly reduced "pilot" carrier (in | "Amplitude modulation" | [
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1407 | reduced-carrier transmission or DSB-RC) to use in the demodulation process. Even with the carrier totally eliminated in double-sideband suppressed-carrier transmission, carrier regeneration is possible using a Costas phase-locked loop. This doesn't work however for single-sideband suppressed-carrier transmission (SSB-SC), leading to the characteristic "Donald Duck" sound from such receivers when slightly detuned. Single sideband is nevertheless used widely in amateur radio and other voice communications both due to its power efficiency and bandwidth efficiency (cutting the RF bandwidth in half compared to standard AM). On the other hand, in medium wave and short wave broadcasting, standard AM with the full carrier | "Amplitude modulation" | [
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1408 | allows for reception using inexpensive receivers. The broadcaster absorbs the extra power cost to greatly increase potential audience. An additional function provided by the carrier in standard AM, but which is lost in either single or double-sideband suppressed-carrier transmission, is that it provides an amplitude reference. In the receiver, the automatic gain control (AGC) responds to the carrier so that the reproduced audio level stays in a fixed proportion to the original modulation. On the other hand, with suppressed-carrier transmissions there is "no" transmitted power during pauses in the modulation, so the AGC must respond to peaks of the transmitted | "Amplitude modulation" | [
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1409 | power during peaks in the modulation. This typically involves a so-called "fast attack, slow decay" circuit which holds the AGC level for a second or more following such peaks, in between syllables or short pauses in the program. This is very acceptable for communications radios, where compression of the audio aids intelligibility. However it is absolutely undesired for music or normal broadcast programming, where a faithful reproduction of the original program, including its varying modulation levels, is expected. A trivial form of AM which can be used for transmitting binary data is on-off keying, the simplest form of "amplitude-shift keying", | "Amplitude modulation" | [
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1410 | in which ones and zeros are represented by the presence or absence of a carrier. On-off keying is likewise used by radio amateurs to transmit Morse code where it is known as continuous wave (CW) operation, even though the transmission is not strictly "continuous." A more complex form of AM, quadrature amplitude modulation is now more commonly used with digital data, while making more efficient use of the available bandwidth. In 1982, the International Telecommunication Union (ITU) designated the types of amplitude modulation: Although AM was used in a few crude experiments in multiplex telegraph and telephone transmission in the | "Amplitude modulation" | [
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1411 | late 1800s, the practical development of amplitude modulation is synonymous with the development between 1900 and 1920 of "radiotelephone" transmission, that is, the effort to send sound (audio) by radio waves. The first radio transmitters, called spark gap transmitters, transmitted information by wireless telegraphy, using different length pulses of carrier wave to spell out text messages in Morse code. They couldn't transmit audio because the carrier consisted of strings of damped waves, pulses of radio waves that declined to zero, that sounded like a buzz in receivers. In effect they were already amplitude modulated. The first AM transmission was made | "Amplitude modulation" | [
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1412 | by Canadian researcher Reginald Fessenden on 23 December 1900 using a spark gap transmitter with a specially designed high frequency 10 kHz interrupter, over a distance of 1 mile (1.6 km) at Cobb Island, Maryland, USA. His first transmitted words were, "Hello. One, two, three, four. Is it snowing where you are, Mr. Thiessen?". The words were barely intelligible above the background buzz of the spark. Fessenden was a significant figure in the development of AM radio. He was one of the first researchers to realize, from experiments like the above, that the existing technology for producing radio waves, the | "Amplitude modulation" | [
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1413 | spark transmitter, was not usable for amplitude modulation, and that a new kind of transmitter, one that produced sinusoidal "continuous waves", was needed. This was a radical idea at the time, because experts believed the impulsive spark was necessary to produce radio frequency waves, and Fessenden was ridiculed. He invented and helped develop one of the first continuous wave transmitters - the Alexanderson alternator, with which he made what is considered the first AM public entertainment broadcast on Christmas Eve, 1906. He also discovered the principle on which AM is based, heterodyning, and invented one of the first detectors able | "Amplitude modulation" | [
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1414 | to rectify and receive AM, the electrolytic detector or "liquid baretter", in 1902. Other radio detectors invented for wireless telegraphy, such as the Fleming valve (1904) and the crystal detector (1906) also proved able to rectify AM signals, so the technological hurdle was generating AM waves; receiving them was not a problem. Early experiments in AM radio transmission, conducted by Fessenden, Valdemar Poulsen, Ernst Ruhmer, Quirino Majorana, Charles Harrold, and Lee De Forest, were hampered by the lack of a technology for amplification. The first practical continuous wave AM transmitters were based on either the huge, expensive Alexanderson alternator, developed | "Amplitude modulation" | [
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1415 | 1906-1910, or versions of the Poulsen arc transmitter (arc converter), invented in 1903. The modifications necessary to transmit AM were clumsy and resulted in very low quality audio. Modulation was usually accomplished by a carbon microphone inserted directly in the antenna or ground wire; its varying resistance varied the current to the antenna. The limited power handling ability of the microphone severely limited the power of the first radiotelephones; many of the microphones were water-cooled. The discovery in 1912 of the amplifying ability of the Audion vacuum tube, invented in 1906 by Lee De Forest, solved these problems. The vacuum | "Amplitude modulation" | [
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1416 | tube feedback oscillator, invented in 1912 by Edwin Armstrong and Alexander Meissner, was a cheap source of continuous waves and could be easily modulated to make an AM transmitter. Modulation did not have to be done at the output but could be applied to the signal before the final amplifier tube, so the microphone or other audio source didn't have to handle high power. Wartime research greatly advanced the art of AM modulation, and after the war the availability of cheap tubes sparked a great increase in the number of radio stations experimenting with AM transmission of news or music. | "Amplitude modulation" | [
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1417 | The vacuum tube was responsible for the rise of AM radio broadcasting around 1920, the first electronic mass entertainment medium. Amplitude modulation was virtually the only type used for radio broadcasting until FM broadcasting began after World War 2. At the same time as AM radio began, telephone companies such as AT&T were developing the other large application for AM: sending multiple telephone calls through a single wire by modulating them on separate carrier frequencies, called "frequency division multiplexing". John Renshaw Carson in 1915 did the first mathematical analysis of amplitude modulation, showing that a signal and carrier frequency combined | "Amplitude modulation" | [
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1418 | in a nonlinear device would create two sidebands on either side of the carrier frequency, and passing the modulated signal through another nonlinear device would extract the original baseband signal. His analysis also showed only one sideband was necessary to transmit the audio signal, and Carson patented single-sideband modulation (SSB) on 1 December 1915. This more advanced variant of amplitude modulation was adopted by AT&T for longwave transatlantic telephone service beginning 7 January 1927. After WW2 it was developed by the military for aircraft communication. Consider a carrier wave (sine wave) of frequency "f" and amplitude "A" given by: Let | "Amplitude modulation" | [
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1419 | "m"("t") represent the modulation waveform. For this example we shall take the modulation to be simply a sine wave of a frequency "f", a much lower frequency (such as an audio frequency) than "f": where "m" is the amplitude sensitivity, "M" is the amplitude of modulation. If "m" < 1, "(1 + m(t)/A)" is always positive for undermodulation. If "m" > 1 then overmodulation occurs and reconstruction of message signal from the transmitted signal would lead in loss of original signal. Amplitude modulation results when the carrier "c(t)" is multiplied by the positive quantity "(1 + m(t)/A)": In this simple | "Amplitude modulation" | [
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1420 | case "m" is identical to the modulation index, discussed below. With "m" = 0.5 the amplitude modulated signal "y"("t") thus corresponds to the top graph (labelled "50% Modulation") in figure 4. Using prosthaphaeresis identities, "y"("t") can be shown to be the sum of three sine waves: Therefore, the modulated signal has three components: the carrier wave "c(t)" which is unchanged, and two pure sine waves (known as sidebands) with frequencies slightly above and below the carrier frequency "f". Of course a useful modulation signal "m(t)" will generally not consist of a single sine wave, as treated above. However, by the | "Amplitude modulation" | [
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] |
1421 | principle of Fourier decomposition, "m(t)" can be expressed as the sum of a number of sine waves of various frequencies, amplitudes, and phases. Carrying out the multiplication of "1 + m(t)" with "c(t)" as above then yields a result consisting of a sum of sine waves. Again the carrier "c(t)" is present unchanged, but for each frequency component of "m" at "f" there are two sidebands at frequencies "f + f" and "f - f". The collection of the former frequencies above the carrier frequency is known as the upper sideband, and those below constitute the lower sideband. In a | "Amplitude modulation" | [
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] |
1422 | slightly different way of looking at it, we can consider the modulation "m(t)" to consist of an equal mix of positive and negative frequency components (as results from a formal Fourier transform of a real valued quantity) as shown in the top of Fig. 2. Then one can view the sidebands as that modulation "m(t)" having simply been shifted in frequency by "f" as depicted at the bottom right of Fig. 2 (formally, the modulated signal also contains identical components at negative frequencies, shown at the bottom left of Fig. 2 for completeness). If we just look at the short-term | "Amplitude modulation" | [
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1423 | spectrum of modulation, changing as it would for a human voice for instance, then we can plot the frequency content (horizontal axis) as a function of time (vertical axis) as in Fig. 3. It can again be seen that as the modulation frequency content varies, at any point in time there is an upper sideband generated according to those frequencies shifted "above" the carrier frequency, and the same content mirror-imaged in the lower sideband below the carrier frequency. At all times, the carrier itself remains constant, and of greater power than the total sideband power. The RF bandwidth of an | "Amplitude modulation" | [
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] |
1424 | AM transmission (refer to Figure 2, but only considering positive frequencies) is twice the bandwidth of the modulating (or "baseband") signal, since the upper and lower sidebands around the carrier frequency each have a bandwidth as wide as the highest modulating frequency. Although the bandwidth of an AM signal is narrower than one using frequency modulation (FM), it is twice as wide as single-sideband techniques; it thus may be viewed as spectrally inefficient. Within a frequency band, only half as many transmissions (or "channels") can thus be accommodated. For this reason analog television employs a variant of single-sideband (known as | "Amplitude modulation" | [
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] |
1425 | vestigial sideband, somewhat of a compromise in terms of bandwidth) in order to reduce the required channel spacing. Another improvement over standard AM is obtained through reduction or suppression of the carrier component of the modulated spectrum. In Figure 2 this is the spike in between the sidebands; even with full (100%) sine wave modulation, the power in the carrier component is twice that in the sidebands, yet it carries no unique information. Thus there is a great advantage in efficiency in reducing or totally suppressing the carrier, either in conjunction with elimination of one sideband (single-sideband suppressed-carrier transmission) or | "Amplitude modulation" | [
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1426 | with both sidebands remaining (double sideband suppressed carrier). While these suppressed carrier transmissions are efficient in terms of transmitter power, they require more sophisticated receivers employing synchronous detection and regeneration of the carrier frequency. For that reason, standard AM continues to be widely used, especially in broadcast transmission, to allow for the use of inexpensive receivers using envelope detection. Even (analog) television, with a (largely) suppressed lower sideband, includes sufficient carrier power for use of envelope detection. But for communications systems where both transmitters and receivers can be optimized, suppression of both one sideband and the carrier represent a net | "Amplitude modulation" | [
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1427 | advantage and are frequently employed. A technique used widely in broadcast AM transmitters is an application of the Hapburg carrier, first proposed in the 1930s but impractical with the technology then available. During periods of low modulation the carrier power would be reduced and would return to full power during periods of high modulation levels. This has the effect of reducing the overall power demand of the transmitter and is most effective on speech type programmes. Various trade names are used for its implementation by the transmitter manufacturers from the late 80's onwards. The AM modulation index is a measure | "Amplitude modulation" | [
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1428 | based on the ratio of the modulation excursions of the RF signal to the level of the unmodulated carrier. It is thus defined as: where formula_6 and formula_7 are the modulation amplitude and carrier amplitude, respectively; the modulation amplitude is the peak (positive or negative) change in the RF amplitude from its unmodulated value. Modulation index is normally expressed as a percentage, and may be displayed on a meter connected to an AM transmitter. So if formula_8, carrier amplitude varies by 50% above (and below) its unmodulated level, as is shown in the first waveform, below. For formula_9, it varies | "Amplitude modulation" | [
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1429 | by 100% as shown in the illustration below it. With 100% modulation the wave amplitude sometimes reaches zero, and this represents full modulation using standard AM and is often a target (in order to obtain the highest possible signal-to-noise ratio) but mustn't be exceeded. Increasing the modulating signal beyond that point, known as overmodulation, causes a standard AM modulator (see below) to fail, as the negative excursions of the wave envelope cannot become less than zero, resulting in distortion ("clipping") of the received modulation. Transmitters typically incorporate a limiter circuit to avoid overmodulation, and/or a compressor circuit (especially for voice | "Amplitude modulation" | [
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] |
1430 | communications) in order to still approach 100% modulation for maximum intelligibility above the noise. Such circuits are sometimes referred to as a vogad. However it is possible to talk about a modulation index exceeding 100%, without introducing distortion, in the case of double-sideband reduced-carrier transmission. In that case, negative excursions beyond zero entail a reversal of the carrier phase, as shown in the third waveform below. This cannot be produced using the efficient high-level (output stage) modulation techniques (see below) which are widely used especially in high power broadcast transmitters. Rather, a special modulator produces such a waveform at a | "Amplitude modulation" | [
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] |
1431 | low level followed by a linear amplifier. What's more, a standard AM receiver using an envelope detector is incapable of properly demodulating such a signal. Rather, synchronous detection is required. Thus double-sideband transmission is generally "not" referred to as "AM" even though it generates an identical RF waveform as standard AM as long as the modulation index is below 100%. Such systems more often attempt a radical reduction of the carrier level compared to the sidebands (where the useful information is present) to the point of double-sideband suppressed-carrier transmission where the carrier is (ideally) reduced to zero. In all such | "Amplitude modulation" | [
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1432 | cases the term "modulation index" loses its value as it refers to the ratio of the modulation amplitude to a rather small (or zero) remaining carrier amplitude. Modulation circuit designs may be classified as low- or high-level (depending on whether they modulate in a low-power domain—followed by amplification for transmission—or in the high-power domain of the transmitted signal). In modern radio systems, modulated signals are generated via digital signal processing (DSP). With DSP many types of AM are possible with software control (including DSB with carrier, SSB suppressed-carrier and independent sideband, or ISB). Calculated digital samples are converted to voltages | "Amplitude modulation" | [
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1433 | with a digital-to-analog converter, typically at a frequency less than the desired RF-output frequency. The analog signal must then be shifted in frequency and linearly amplified to the desired frequency and power level (linear amplification must be used to prevent modulation distortion). This low-level method for AM is used in many Amateur Radio transceivers. AM may also be generated at a low level, using analog methods described in the next section. High-power AM transmitters (such as those used for AM broadcasting) are based on high-efficiency class-D and class-E power amplifier stages, modulated by varying the supply voltage. Older designs (for | "Amplitude modulation" | [
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1434 | broadcast and amateur radio) also generate AM by controlling the gain of the transmitter's final amplifier (generally class-C, for efficiency). The following types are for vacuum tube transmitters (but similar options are available with transistors): The simplest form of AM demodulator consists of a diode which is configured to act as envelope detector. Another type of demodulator, the product detector, can provide better-quality demodulation with additional circuit complexity. Amplitude modulation Amplitude modulation (AM) is a modulation technique used in electronic communication, most commonly for transmitting information via a radio carrier wave. In amplitude modulation, the amplitude (signal strength) of the | "Amplitude modulation" | [
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] |
1435 | Augustin-Jean Fresnel Augustin-Jean Fresnel ( , ; ; 10 May 178814 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton's corpuscular theory, from the late 1830s until the end of the 19th century. But he is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric (purely refractive) stepped lens, first proposed by Count Buffon and independently | "Augustin-Jean Fresnel" | [
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] |
1436 | reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors. By expressing Huygens' principle of secondary waves and Young's principle of interference in quantitative terms, and supposing that simple colors consist of sinusoidal waves, Fresnel gave the first satisfactory explanation of diffraction by straight edges, including the first satisfactory wave-based explanation of rectilinear propagation. Part of his argument was a proof that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions. By further supposing that light waves are purely transverse, Fresnel explained | "Augustin-Jean Fresnel" | [
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1437 | the nature of polarization and lack thereof, the mechanism of chromatic polarization (the colors produced when polarized light is passed through a slice of doubly-refractive crystal followed by a second polarizer), and the transmission and reflection coefficients at the interface between two transparent isotropic media (including Brewster's angle). Then, by generalizing the direction-speed-polarization relation for calcite, he accounted for the directions and polarizations of the refracted rays in doubly-refractive crystals of the "biaxial" class (those for which Huygens' secondary wavefronts are not axisymmetric). The period between the first publication of his pure-transverse-wave hypothesis and the submission of his first correct | "Augustin-Jean Fresnel" | [
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1438 | solution to the biaxial problem was less than a year. Later, he coined the terms "linear polarization", "circular polarization", and "elliptical polarization", explained how optical rotation could be understood as a difference in propagation speeds for the two directions of circular polarization, and (by allowing the reflection coefficient to be complex) accounted for the change in polarization due to total internal reflection, as exploited in the Fresnel rhomb. Defenders of the established corpuscular theory could not match his quantitative explanations of so many phenomena on so few assumptions. Fresnel's legacy is the more remarkable in view of his lifelong battle | "Augustin-Jean Fresnel" | [
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] |
1439 | with tuberculosis, to which he succumbed at the age of 39. Although he did not become a public celebrity in his short lifetime, he lived just long enough to receive due recognition from his peers, including (on his deathbed) the Rumford Medal of the Royal Society of London, and his name is ubiquitous in the modern terminology of optics and waves. Inevitably, after the wave theory of light was subsumed by Maxwell's electromagnetic theory in the 1860s, some attention was diverted from the magnitude of Fresnel's contribution. In the period between Fresnel's unification of physical optics and Maxwell's wider unification, | "Augustin-Jean Fresnel" | [
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1440 | a contemporary authority, Professor Humphrey Lloyd, described Fresnel's transverse-wave theory as "the noblest fabric which has ever adorned the domain of physical science, Newton's system of the universe alone excepted." Augustin-Jean Fresnel (also called Augustin Jean or simply Augustin), born in Broglie, Normandy, on 10 May 1788, was the second of four sons of the architect Jacques Fresnel (1755–1805) and his wife Augustine, "née" Mérimée (1755–1833). In 1790, following the Revolution, Broglie became part of the département of Eure. The family moved at least twice — in 1790 to Cherbourg, and in 1794 to Jacques' home town of Mathieu, where | "Augustin-Jean Fresnel" | [
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1441 | Madame Fresnel would spend 25 years as a widow, outliving two of her sons. The first son, Louis (1786–1809), was admitted to the École Polytechnique, became a lieutenant in the artillery, and was killed in action at Jaca, Spain, the day before his 23rd birthday. The third, Léonor (1790–1869), followed Augustin into civil engineering, succeeded him as Secretary of the Lighthouse Commission, and helped to edit his collected works. The fourth, Fulgence Fresnel (1795–1855), became a noted linguist, diplomat, and orientalist, and occasionally assisted Augustin with negotiations. Léonor apparently was the only one of the four who married. Their mother's | "Augustin-Jean Fresnel" | [
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1442 | younger brother, Jean François "Léonor" Mérimée (1757–1836), father of the writer Prosper Mérimée (1803–1870), was a paint artist who turned his attention to the chemistry of painting. He became the Permanent Secretary of the École des Beaux-Arts and (until 1814) a professor at the École Polytechnique, and was the initial point of contact between Augustin and the leading optical physicists of the day . The Fresnel brothers were initially home-schooled by their mother. The sickly Augustin was considered the slow one, hardly beginning to read until the age of eight. At nine and ten he was undistinguished except for his | "Augustin-Jean Fresnel" | [
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1443 | ability to turn tree-branches into toy bows and guns that worked far too well, earning himself the title "l'homme de génie" (the man of genius) from his accomplices, and a united crackdown from their elders. In 1801, Augustin was sent to the "École Centrale" at Caen, as company for Louis. But Augustin lifted his performance: in late 1804 he was accepted into the École Polytechnique, being placed 17th in the entrance examination, in which his solutions to geometry problems impressed the examiner, Adrien-Marie Legendre. As the surviving records of the École Polytechnique begin in 1808, we know little of Augustin's | "Augustin-Jean Fresnel" | [
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1444 | time there, except that he apparently excelled in geometry and drawing — in spite of continuing poor health — and made few if any friends. Graduating in 1806, he then enrolled at the École Nationale des Ponts et Chaussées (National School of Bridges and Roads, also known as "ENPC" or "École des Ponts"), from which he graduated in 1809, entering the service of the Corps des Ponts et Chaussées as an "ingénieur ordinaire aspirant" (ordinary engineer in training). Directly or indirectly, he was to remain in the employment of the "Corps des Ponts" for the rest of his life. Augustin | "Augustin-Jean Fresnel" | [
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1445 | Fresnel's parents were Roman Catholics of the Jansenist sect, characterized by an extreme Augustinian view of original sin. Religion took first place in the boys' home-schooling. In 1802, Mme Fresnel wrote to Louis concerning Augustin: Augustin remained a Jansenist. He indeed regarded his intellectual talents as gifts from God, and considered it his duty to use them for the benefit of others. Plagued by poor health, and determined to do his duty before death thwarted him, he shunned pleasures and worked to the point of exhaustion. According to his fellow engineer Alphonse Duleau, who helped to nurse him through his | "Augustin-Jean Fresnel" | [
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] |
1446 | final illness, Fresnel saw the study of nature as part of the study of the power and goodness of God. He placed virtue above science and genius. Yet in his last days he needed "strength of soul," not against death alone, but against "the interruption of discoveries… of which he hoped to derive useful applications." Jansenism is considered heretical by the Roman Catholic Church , and may be part of the explanation why Fresnel, in spite of his scientific achievements and his royalist credentials, never gained a permanent academic teaching post; his only teaching appointment was at the Athénée in | "Augustin-Jean Fresnel" | [
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1447 | the winter of 1819–20. Be that as it may, the brief article on Fresnel in the old "Catholic Encyclopedia" does not mention his Jansenism, but describes him as "a deeply religious man and remarkable for his keen sense of duty." Fresnel was initially posted to the western département of Vendée. There, in 1811, he anticipated what became known as the Solvay process for producing soda ash, except that recycling of the ammonia was not considered. That difference may explain why leading chemists, who learned of his discovery through his uncle Léonor, eventually thought it uneconomic. About 1812, Fresnel was sent | "Augustin-Jean Fresnel" | [
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1448 | to Nyons, in the southern département of Drôme, to assist with the imperial highway that was to connect Spain and Italy. It is from Nyons that we have the first evidence of his interest in optics. On 15 May 1814, while work was slack due to Napoleon's defeat, Fresnel wrote a ""P.S."" to his brother Léonor, saying in part: At the end of 1814 he still had no information on the subject. (Concerning the name "Institute", note that the French "Académie des Sciences" was merged with other "académies" to form the "Institut de France" in 1795. In 1816 the "Académie | "Augustin-Jean Fresnel" | [
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1449 | des Sciences" regained its name and autonomy, but remained part of the Institute.) In March 1815, perceiving Napoleon's return from Elba as "an attack on civilization", Fresnel departed without leave, hastened to Toulouse and offered his services to the royalist resistance, but soon found himself on the sick list. Returning to Nyons in defeat, he was threatened and had his windows broken. During the Hundred Days he was placed on suspension, which he was eventually allowed to spend at his mother's house in Mathieu. There he used his enforced leisure to begin his optical experiments. The appreciation of Fresnel's reconstruction | "Augustin-Jean Fresnel" | [
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1450 | of physical optics might be assisted by an overview of the fragmented state in which he found the subject. In this subsection, optical phenomena that were unexplained or whose explanations were disputed are named in bold type. The corpuscular theory of light, favored by Isaac Newton and accepted by nearly all of Fresnel's seniors, easily explained rectilinear propagation: the corpuscles obviously moved very fast, so that their paths were very nearly straight. The wave theory, as developed by Christiaan Huygens in his "Treatise on Light" (1690), explained rectilinear propagation on the assumption that each point crossed by a traveling wavefront | "Augustin-Jean Fresnel" | [
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1451 | becomes the source of a secondary wavefront. Given the initial position of a traveling wavefront, any later position (according to Huygens) was the common tangent surface (envelope) of the secondary wavefronts emitted from the earlier position. As the extent of the common tangent was limited by the extent of the initial wavefront, the repeated application of Huygens' construction to a plane wavefront of limited extent (in a uniform medium) gave a straight, parallel beam. While this construction indeed predicted rectilinear propagation, it was difficult to reconcile with the common observation that wavefronts on the surface of water can bend around | "Augustin-Jean Fresnel" | [
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1452 | obstructions, and with the similar behavior of sound waves — causing Newton to maintain, to the end of his life, that if light consisted of waves it would "bend and spread every way" into the shadows. Huygens' theory neatly explained the law of ordinary reflection and the law of ordinary refraction ("Snell's law"), provided that the secondary waves traveled slower in denser media (those of higher refractive index). The corpuscular theory, with the hypothesis that the corpuscles were subject to forces acting perpendicular to surfaces, explained the same laws equally well, albeit with the implication that light traveled "faster" in | "Augustin-Jean Fresnel" | [
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1453 | denser media; that implication was wrong, but could not be directly disproven with the technology of Newton's time or even Fresnel's time . Similarly inconclusive was stellar aberration — that is, the apparent change in the position of a star due to the velocity of the earth across the line of sight (not to be confused with stellar parallax, which is due to the "displacement" of the earth across the line of sight). Identified by James Bradley in 1728, stellar aberration was widely taken as confirmation of the corpuscular theory. But it was equally compatible with the wave theory, as | "Augustin-Jean Fresnel" | [
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1454 | Euler noted in 1746 — tacitly assuming that the aether (the supposed wave-bearing medium) near the earth was not disturbed by the motion of the earth. The outstanding strength of Huygens' theory was his explanation of the birefringence (double refraction) of "Iceland crystal" (transparent calcite), on the assumption that the secondary waves are spherical for the ordinary refraction (which satisfies Snell's law) and spheroidal for the "extraordinary" refraction (which does not). In general, Huygens' common-tangent construction implies that rays are "paths of least time" between successive positions of the wavefront, in accordance with Fermat's principle. In the special case of | "Augustin-Jean Fresnel" | [
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1455 | isotropic media, the secondary wavefronts must be spherical, and Huygens' construction then implies that the rays are perpendicular to the wavefront; indeed, the law of "ordinary" refraction can be separately derived from that premise, as Ignace-Gaston Pardies did before Huygens. Although Newton rejected the wave theory, he noticed its potential to explain colors, including the colors of "thin plates" (e.g., "Newton's rings", and the colors of skylight reflected in soap bubbles), on the assumption that light consists of "periodic" waves, with the lowest frequencies (longest wavelengths) at the red end of the spectrum, and the highest frequencies (shortest wavelengths) at | "Augustin-Jean Fresnel" | [
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1456 | the violet end. In 1672 he published a heavy hint to that effect, but contemporary supporters of the wave theory failed to act on it: Robert Hooke treated light as a periodic sequence of pulses but did not use frequency as the criterion of color, while Huygens treated the waves as individual pulses without any periodicity; and Pardies died young in 1673. Newton himself tried to explain colors of thin plates using the corpuscular theory, by supposing that his corpuscles had the wavelike property of alternating between "fits of easy transmission" and "fits of easy reflection", the distance between like | "Augustin-Jean Fresnel" | [
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1457 | "fits" depending on the color and the medium and, awkwardly, on the angle of refraction or reflection into that medium. More awkwardly still, this theory required thin plates to reflect only at the back surface, although "thick" plates manifestly reflected also at the front surface. It was not until 1801 that Thomas Young, in the Bakerian Lecture for that year, cited Newton's hint, and accounted for the colors of a thin plate as the combined effect of the front and back reflections, which reinforce or cancel each other according to the "wavelength" and the thickness. Young similarly explained the colors | "Augustin-Jean Fresnel" | [
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1458 | of "striated surfaces" (e.g., gratings) as the wavelength-dependent reinforcement or cancellation of reflections from adjacent lines. He described this reinforcement or cancellation as interference. Neither Newton nor Huygens satisfactorily explained diffraction — the blurring and fringing of shadows where, according to rectilinear propagation, they ought to be sharp. Newton, who called diffraction "inflexion", supposed that rays of light passing close to obstacles were bent ("inflected"); but his explanation was only qualitative. Huygens' common-tangent construction, without modifications, could not accommodate diffraction at all. Two such modifications were proposed by Young in the same 1801 Bakerian Lecture: first, that the secondary waves | "Augustin-Jean Fresnel" | [
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1459 | near the edge of an obstacle could diverge into the shadow, but only weakly, due to limited reinforcement from other secondary waves; and second, that diffraction by an edge was caused by interference between two rays: one reflected off the edge, and the other inflected while passing near the edge. The latter ray would be undeviated if sufficiently far from the edge, but Young did not elaborate on that case. These were the earliest suggestions that the degree of diffraction depends on wavelength. Later, in the 1803 Bakerian Lecture, Young ceased to regard inflection as a separate phenomenon, and produced | "Augustin-Jean Fresnel" | [
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1460 | evidence that diffraction fringes "inside" the shadow of a narrow obstacle were due to interference: when the light from one side was blocked, the internal fringes disappeared. But Young was alone in such efforts until Fresnel entered the field. Huygens, in his investigation of double refraction, noticed something that he could not explain: when light passes through two similarly oriented calcite crystals at normal incidence, the ordinary ray emerging from the first crystal suffers only the ordinary refraction in the second, while the extraordinary ray emerging from the first suffers only the extraordinary refraction in the second; but when the | "Augustin-Jean Fresnel" | [
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1461 | second crystal is rotated 90° about the incident rays, the roles are interchanged, so that the ordinary ray emerging from the first crystal suffers only the extraordinary refraction in the second, and vice versa. This discovery gave Newton another reason to reject the wave theory: rays of light evidently had "sides". Corpuscles could have sides (or "poles", as they would later be called); but waves of light could not, because (so it seemed) any such waves would need to be longitudinal (with vibrations in the direction of propagation). Newton offered an alternative "Rule" for the extraordinary refraction, which rode on | "Augustin-Jean Fresnel" | [
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1462 | his authority through the 18th century, although he made "no known attempt to deduce it from any principles of optics, corpuscular or otherwise." In 1808 the extraordinary refraction of calcite was investigated experimentally, with unprecedented accuracy, by Étienne-Louis Malus, and found to be consistent with Huygens' spheroid construction, not Newton's "Rule". Malus, encouraged by Pierre-Simon Laplace, then sought to explain this law in corpuscular terms: from the known relation between the incident and refracted ray directions, Malus derived the corpuscular velocity (as a function of direction) that would satisfy Maupertuis's "least action" principle. But, as Young pointed out, the existence | "Augustin-Jean Fresnel" | [
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1463 | of such a velocity law was guaranteed by Huygens' spheroid, because Huygens' construction leads to Fermat's principle, which becomes Maupertuis's principle if the ray speed is replaced by the reciprocal of the particle speed! The corpuscularists had not found a "force" law that would yield the alleged velocity law. Worse, it was doubtful that any such force law would satisfy the conditions of Maupertuis's principle. In contrast, Young proceeded to show that "a medium more easily compressible in one direction than in any direction perpendicular to it, as if it consisted of an infinite number of parallel plates connected by | "Augustin-Jean Fresnel" | [
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] |
1464 | a substance somewhat less elastic" admits spheroidal longitudinal wavefronts, as Huygens supposed. But Malus, in the course of his experiments on double refraction, noticed something else: when a ray of light is reflected off a non-metallic surface at the appropriate angle, it behaves like "one" of the two rays emerging from a calcite crystal. It was Malus who coined the term polarization to describe this behavior, although the polarizing angle became known as Brewster's angle after its dependence on the refractive index was determined experimentally by David Brewster in 1815. Malus also introduced the term "plane of polarization". In the | "Augustin-Jean Fresnel" | [
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1465 | case of polarization by reflection, his "plane of polarization" was the plane of the incident and reflected rays; in modern terms, this is the plane "normal" to the "electric" vibration. In 1809, Malus further discovered that the intensity of light passing through "two" polarizers is proportional to the squared cosine of the angle between their planes of polarization (Malus's law), whether the polarizers work by reflection or double refraction, and that "all" birefringent crystals produce both extraordinary refraction and polarization. As the corpuscularists started trying to explain these things in terms of polar "molecules" of light, the wave-theorists had "no | "Augustin-Jean Fresnel" | [
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1466 | working hypothesis" on the nature of polarization, prompting Young to remark that Malus's observations "present greater difficulties to the advocates of the undulatory theory than any other facts with which we are acquainted." Malus died in February 1812, at the age of 36, shortly after receiving the Rumford Medal for his work on polarization. In August 1811, François Arago reported that if a thin plate of mica was viewed against a white polarized backlight through a calcite crystal, the two images of the mica were of complementary colors (the overlap having the same color as the background). The light emerging | "Augustin-Jean Fresnel" | [
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1467 | from the mica was ""de"polarized" in the sense that there was no orientation of the calcite that made one image disappear; yet it was not ordinary (""un"polarized") light, for which the two images would be of the same color. Rotating the calcite around the line of sight changed the colors, though they remained complementary. Rotating the mica changed the "saturation" (not the hue) of the colors. This phenomenon became known as chromatic polarization. Replacing the mica with a much thicker plate of quartz, with its faces perpendicular to the optic axis (the axis of Huygens' spheroid or Malus's velocity function), | "Augustin-Jean Fresnel" | [
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1468 | produced a similar effect, except that rotating the quartz made no difference. Arago tried to explain his observations in "corpuscular" terms. In 1812, as Arago pursued further qualitative experiments and other commitments, Jean-Baptiste Biot reworked the same ground using a gypsum lamina in place of the mica, and found empirical formulae for the intensities of the ordinary and extraordinary images. The formulae contained two coefficients, supposedly representing colors of rays "affected" and "unaffected" by the plate — the "affected" rays being of the same color mix as those reflected by amorphous thin plates of proportional, but lesser, thickness. Arago protested, | "Augustin-Jean Fresnel" | [
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1469 | declaring that he had made some of the same discoveries but had not had time to write them up. In fact the overlap between Arago's work and Biot's was minimal, because Arago's was only qualitative and sought to include other topics. But the dispute triggered a notorious falling-out between the two men. Later that year, Biot tried to explain the observations as an oscillation of the alignment of the "affected" corpuscles at a frequency proportional to that of Newton's "fits", due to forces depending on the alignment. This theory became known as "mobile polarization". To reconcile his results with a | "Augustin-Jean Fresnel" | [
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] |
1470 | sinusoidal oscillation, Biot had to suppose that the corpuscles emerged with one of two permitted orientations, namely the extremes of the oscillation, with probabilities depending on the phase of the oscillation. Corpuscular optics was becoming expensive on assumptions. But in 1813, Biot reported that the case of quartz was simpler: the observable phenomenon (now called optical rotation or "optical activity" or sometimes "rotary polarization") was a gradual rotation of the polarization direction with distance, and could be explained by a corresponding rotation ("not" oscillation) of the corpuscles. Early in 1814, reviewing Biot's work on chromatic polarization, Young noted that the | "Augustin-Jean Fresnel" | [
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1471 | periodicity of the color as a function of the plate thickness — including the factor by which the period exceeded that for a reflective thin plate, and even the effect of obliquity of the plate (but not the role of polarization) — could be explained by the wave theory in terms of the different propagation times of the ordinary and extraordinary waves through the plate. But Young was then the only public defender of the wave theory. In summary, in the spring of 1814, as Fresnel tried in vain to guess what polarization was, the corpuscularists thought that they knew, | "Augustin-Jean Fresnel" | [
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1472 | while the wave-theorists (if we may use the plural) literally had no idea. Both theories claimed to explain rectilinear propagation, but the wave explanation was overwhelmingly regarded as unconvincing. The corpuscular theory could not link double refraction to specific surface forces; the wave theory could not yet link it to polarization. The corpuscular theory was weak on thin plates and silent on gratings; the wave theory was strong on both, but under-appreciated. Concerning diffraction, the corpuscular theory did not yield quantitative predictions, while the wave theory had begun to do so by considering diffraction as a manifestation of interference, but | "Augustin-Jean Fresnel" | [
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] |
1473 | had only considered two rays at a time. Only the corpuscular theory gave even a vague insight into Brewster's angle, Malus's law, or optical rotation. Concerning chromatic polarization, the wave theory explained the periodicity far better than the corpuscular theory, but had nothing to say about the role of polarization; and its explanation of the periodicity was largely ignored. And Arago had founded the study of chromatic polarization, only to lose the lead, controversially, to Biot. Such were the circumstances in which Arago first heard of Fresnel's interest in optics. Fresnel's letters from later in 1814 reveal his interest in | "Augustin-Jean Fresnel" | [
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1474 | the wave theory, including his awareness that it explained the constancy of the speed of light and was at least compatible with stellar aberration. Eventually he compiled what he called his "rêveries" (musings) into an essay and submitted it via Léonor Mérimée to André-Marie Ampère, who did not respond directly. But on 19 December, Mérimée dined with Ampère and Arago, with whom he was acquainted through the École Polytechnique; and Arago promised to look at Fresnel's essay. In mid 1815, on his way home to Mathieu to serve his suspension, Fresnel met Arago in Paris and spoke of the wave | "Augustin-Jean Fresnel" | [
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1475 | theory and stellar aberration. He was informed that he was trying to break down open doors (""il enfonçait des portes ouvertes""), and directed to classical works on optics. On 12 July 1815, as Fresnel was about to leave Paris, Arago left him a note on a new topic: Fresnel would not have ready access to these works outside Paris, and could not read English. But, in Mathieu — with a point-source of light made by focusing sunlight with a drop of honey, a crude micrometer of his own construction, and supporting apparatus made by a local locksmith — he began | "Augustin-Jean Fresnel" | [
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1476 | his own experiments. His technique was novel: whereas earlier investigators had projected the fringes onto a screen, Fresnel soon abandoned the screen and observed the fringes in space, through a lens with the micrometer at its focus, allowing more accurate measurements while requiring less light. Later in July, after Napoleon's final defeat, Fresnel was reinstated with the advantage of having backed the winning side. He requested a two-month leave of absence, which was readily granted because roadworks were in abeyance. On 23 September he wrote to Arago, beginning "I think I have found the explanation and the law of colored | "Augustin-Jean Fresnel" | [
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1477 | fringes which one notices in the shadows of bodies illuminated by a luminous point." In the same paragraph, however, Fresnel implicitly acknowledged doubt about the novelty of his work: noting that he would need to incur some expense in order to improve his measurements, he wanted to know "whether this is not useless, and whether the law of diffraction has not already been established by sufficiently exact experiments." He explained that he had not yet had a chance to acquire the items on his reading lists, with the apparent exception of "Young's book", which he could not understand without his | "Augustin-Jean Fresnel" | [
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1478 | brother's help. Not surprisingly, he had retraced many of Young's steps. In a memoir sent to the Institute on 15 October 1815, Fresnel mapped the external and internal fringes in the shadow of a wire. He noticed, like Young before him, that the internal fringes disappeared when the light from one side was blocked, and concluded that "the vibrations of two rays that cross each other under a very small angle can contradict each other…" But, whereas Young took the disappearance of the internal fringes as "confirmation" of the principle of interference, Fresnel reported that it was the internal fringes | "Augustin-Jean Fresnel" | [
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1479 | that first drew his attention to the principle. To explain the diffraction pattern, Fresnel constructed the internal fringes by considering the intersections of circular wavefronts emitted from the two edges of the obstruction, and the external fringes by considering the intersections between direct waves and waves reflected off the nearer edge. For the external fringes, to obtain tolerable agreement with observation, he had to suppose that the reflected wave was inverted; and he noted that the predicted paths of the fringes were hyperbolic. In the part of the memoir that most clearly surpassed Young, Fresnel explained the ordinary laws of | "Augustin-Jean Fresnel" | [
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1480 | reflection and refraction in terms of interference, noting that if two parallel rays were reflected or refracted at other than the prescribed angle, they would no longer have the same phase in a common perpendicular plane, and every vibration would be cancelled by a nearby vibration. He noted that his explanation was valid provided that the surface irregularities were much smaller than the wavelength. On 10 November, Fresnel sent a supplementary note dealing with Newton's rings and with gratings, including, for the first time, "transmission" gratings — although in that case the interfering rays were still assumed to be "inflected", | "Augustin-Jean Fresnel" | [
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1481 | and the experimental verification was inadequate because it used only two threads. As Fresnel was not a member of the Institute, the fate of his memoir depended heavily on the report of a single member. The reporter for Fresnel's memoir turned out to be Arago (with Poinsot as the other reviewer). On 8 November, Arago wrote to Fresnel: Fresnel was troubled, wanting to know more precisely where he had collided with Young. Concerning the curved paths of the "colored bands", Young had noted the hyperbolic paths of the fringes in the two-source interference pattern, corresponding roughly to Fresnel's "internal" fringes, | "Augustin-Jean Fresnel" | [
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1482 | and had described the hyperbolic fringes that appear "on the screen" within rectangular shadows. But Arago erred in his belief that the curved paths of the fringes were fundamentally incompatible with the corpuscular theory. Arago's letter went on to request more data on the external fringes. Fresnel complied, until he exhausted his leave and was assigned to Rennes in the département of Ille-et-Vilaine. At this point Arago interceded with Gaspard de Prony, head of the École des Ponts, who wrote to Louis-Mathieu Molé, head of the Corps des Ponts, suggesting that the progress of science and the prestige of the | "Augustin-Jean Fresnel" | [
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1483 | Corps would be enhanced if Fresnel could come to Paris for a time. He arrived in March 1816, and his leave was subsequently extended through the middle of the year. Meanwhile, in an experiment reported on 26 February 1816, Arago verified Fresnel's prediction that the internal fringes were shifted if the rays on one side of the obstacle passed through a thin glass lamina. Fresnel correctly attributed this phenomenon to the lower wave velocity in the glass. Arago later used a similar argument to explain the colors in the scintillation of stars. Fresnel's updated memoir was eventually published in the | "Augustin-Jean Fresnel" | [
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1484 | March 1816 issue of "Annales de Chimie et de Physique", of which Arago had recently become co-editor. That issue did not actually appear until May. In March, Fresnel already had competition: Biot read a memoir on diffraction by himself and his student Claude Pouillet, containing copious data and arguing that the regularity of diffraction fringes, like the regularity of Newton's rings, must be linked to Newton's "fits". But the new link was not rigorous, and Pouillet himself would become a distinguished early adopter of the wave theory. On 24 May 1816, Fresnel wrote to Young (in French), acknowledging how little | "Augustin-Jean Fresnel" | [
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1485 | of his own memoir was new. But in a "supplement" signed on 14 July and read the next day, Fresnel noted that the internal fringes were more accurately predicted by supposing that the two interfering rays came from some distance "outside" the edges of the obstacle. To explain this, he divided the incident wavefront at the obstacle into what we now call "Fresnel zones", such that the secondary waves from each zone were spread over half a cycle when they arrived at the observation point. The zones on each side of the obstacle largely canceled out in pairs, except the | "Augustin-Jean Fresnel" | [
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1486 | first, which was represented by an "efficacious ray". This approach worked for the internal fringes, but the superposition of the efficacious ray and the direct ray did "not" work for the "external" fringes. The contribution from the "efficacious ray" was thought to be only "partly" canceled, for reasons involving the dynamics of the medium: where the wavefront was continuous, symmetry forbade oblique vibrations; but near the obstacle that truncated the wavefront, the asymmetry allowed some sideways vibration towards the geometric shadow. This argument showed that Fresnel had not (yet) fully accepted Huygens' principle, which would have permitted oblique radiation from | "Augustin-Jean Fresnel" | [
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1487 | all portions of the front. In the same supplement, Fresnel described his well-known double mirror, comprising two flat mirrors joined at an angle of slightly less than 180°, with which he produced a two-slit interference pattern from two virtual images of the same slit. A conventional double-slit experiment required a preliminary "single" slit to ensure that the light falling on the double slit was "coherent" (synchronized). In Fresnel's version, the preliminary single slit was retained, and the double slit was replaced by the double mirror — which bore no physical resemblance to the double slit and yet performed the same | "Augustin-Jean Fresnel" | [
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1488 | function. This result (which had been announced by Arago in the March issue of the "Annales") made it hard to believe that the two-slit pattern had anything to do with corpuscles being deflected as they passed near the edges of the slits. But 1816 was the "Year Without a Summer": crops failed; hungry farming families lined the streets of Rennes; the central government organized "charity workhouses" for the needy; and in October, Fresnel was sent back to Ille-et-Vilaine to supervise charity workers in addition to his regular road crew. According to Arago, Fresnel's letters from December 1816 reveal his consequent | "Augustin-Jean Fresnel" | [
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1489 | anxiety. To Arago he complained of being "tormented by the worries of surveillance, and the need to reprimand…" And to Mérimée he wrote: "I find nothing more tiresome than having to manage other men, and I admit that I have no idea what I'm doing." On 17 March 1817, the Académie des Sciences announced that diffraction would be the topic for the biannual physics "Grand Prix" to be awarded in 1819. The deadline for entries was set at 1 August 1818 to allow time for replication of experiments. Although the wording of the problem referred to rays and inflection and | "Augustin-Jean Fresnel" | [
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] |
1490 | did not invite wave-based solutions, Arago and Ampère encouraged Fresnel to enter. In the fall of 1817, Fresnel, supported by de Prony, obtained a leave of absence from the new head of the Corp des Ponts, Louis Becquey, and returned to Paris. He resumed his engineering duties in the spring of 1818; but from then on he was based in Paris, first on the Canal de l'Ourcq, and then (from May 1819) with the cadastre of the pavements. On 15 January 1818, in a different context (revisited below), Fresnel showed that the addition of sinusoidal functions of the same frequency | "Augustin-Jean Fresnel" | [
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1491 | but different phases is analogous to the addition of forces with different directions. His method was similar to the phasor representation, except that the "forces" were plane vectors rather than complex numbers; they could be added, and multiplied by scalars, but not (yet) multiplied and divided by each other. The explanation was algebraic rather than geometric. Knowledge of this method was assumed in a preliminary note on diffraction, dated 19 April 1818 and deposited on 20 April, in which Fresnel outlined the elementary theory of diffraction as found in modern textbooks. He restated Huygens' principle in combination with the superposition | "Augustin-Jean Fresnel" | [
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1492 | principle, saying that the vibration at each point on a wavefront is the sum of the vibrations that would be sent to it at that moment by all the elements of the wavefront in any of its previous positions, all elements acting separately . For a wavefront partly obstructed in a previous position, the summation was to be carried out over the unobstructed portion. In directions other than the normal to the primary wavefront, the secondary waves were weakened due to obliquity, but weakened much more by destructive interference, so that the effect of obliquity alone could be ignored. For | "Augustin-Jean Fresnel" | [
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] |
1493 | diffraction by a straight edge, the intensity as a function of distance from the geometric shadow could then be expressed with sufficient accuracy in terms of what are now called the normalized Fresnel integrals: The same note included a table of the integrals, for an upper limit ranging from 0 to 5.1 in steps of 0.1, computed with a mean error of 0.0003, plus a smaller table of maxima and minima of the resulting intensity. In his final "Memoir on the diffraction of light", deposited on 29 July and bearing the Latin epigraph ""Natura simplex et fecunda"" ("Nature simple and | "Augustin-Jean Fresnel" | [
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1494 | fertile"), Fresnel slightly expanded the two tables without changing the existing figures, except for a correction to the first minimum of intensity. For completeness, he repeated his solution to "the problem of interference", whereby sinusoidal functions are added like vectors. He acknowledged the directionality of the secondary sources and the variation in their distances from the observation point, chiefly to explain why these things make negligible difference in the context, provided of course that the secondary sources do not radiate in the retrograde direction. Then, applying his theory of interference to the secondary waves, he expressed the intensity of light | "Augustin-Jean Fresnel" | [
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1495 | diffracted by a single straight edge (half-plane) in terms of integrals which involved the dimensions of the problem, but which could be converted to the normalized forms above. With reference to the integrals, he explained the calculation of the maxima and minima of the intensity (external fringes), and noted that the calculated intensity falls very rapidly as one moves into the geometric shadow. The last result, as Olivier Darrigol says, "amounts to a proof of the rectilinear propagation of light in the wave theory, indeed the first proof that a modern physicist would still accept." For the experimental testing of | "Augustin-Jean Fresnel" | [
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1496 | his calculations, Fresnel used red light with a wavelength of 638nm, which he deduced from the diffraction pattern in the simple case in which light incident on a single slit was focused by a cylindrical lens. For a variety of distances from the source to the obstacle and from the obstacle to the field point, he compared the calculated and observed positions of the fringes for diffraction by a half-plane, a slit, and a narrow strip — concentrating on the minima, which were visually sharper than the maxima. For the slit and the strip, he could not use the previously | "Augustin-Jean Fresnel" | [
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1497 | computed table of maxima and minima; for each combination of dimensions, the intensity had to be expressed in terms of sums or differences of Fresnel integrals and calculated from the table of integrals, and the extrema had to be calculated anew. The agreement between calculation and measurement was better than 1.5% in almost every case. Near the end of the memoir, Fresnel summed up the difference between Huygens' use of secondary waves and his own: whereas Huygens says there is light only where the secondary waves exactly agree, Fresnel says there is complete darkness only where the secondary waves exactly | "Augustin-Jean Fresnel" | [
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1498 | cancel out. The judging committee comprised Laplace, Biot, and Poisson (all corpuscularists), Gay-Lussac (uncommitted), and Arago, who eventually wrote the committee's report. Although entries in the competition were supposed to be anonymous to the judges, Fresnel's must have been recognizable by the content. There was only one other entry, of which neither the manuscript nor any record of the author has survived. That entry (identified as "no.1") was mentioned only in the last paragraph of the judges' report, noting that the author had shown ignorance of the relevant earlier works of Young and Fresnel, used insufficiently precise methods of observation, | "Augustin-Jean Fresnel" | [
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1499 | overlooked known phenomena, and made obvious errors. In the words of John Worrall, "The competition facing Fresnel could hardly have been less stiff." We may infer that the committee had only two options: award the prize to Fresnel ("no.2"), or withhold it. The committee deliberated into the new year. Then Poisson, exploiting a case in which Fresnel's theory gave easy integrals, predicted that if a circular obstacle were illuminated by a point-source, there should be (according to the theory) a bright spot in the center of the shadow, illuminated as brightly as the exterior. This seems to have been intended | "Augustin-Jean Fresnel" | [
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1500 | as a "reductio ad absurdum". Arago, undeterred, assembled an experiment with an obstacle 2mm in diameter — and there, in the center of the shadow, was Poisson's spot. The unanimous report of the committee, read at the meeting of the Académie on 15 March 1819, awarded the prize to "the memoir marked no.2, and bearing as epigraph: "Natura simplex et fecunda"." At the same meeting, after the judgment was delivered, the president of the Académie opened a sealed note accompanying the memoir, revealing the author as Fresnel. The award was announced at the public meeting of the Académie a week | "Augustin-Jean Fresnel" | [
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