Augustin-Jean Fresnel

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 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 "fits" depending on color and, awkwardly, on the angle of incidence. 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. He similarly explained the colors of "striated surfaces" (e.g., gratings) as the wavelength-dependent reinforcement or cancellation of reflections from adjacent lines. Young described this reinforcement or cancellation as interference.

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 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 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.

Nor was Fresnel the first to suggest replacing a convex lens with a series of concentric annular prisms, to reduce weight and absorption. In 1748, Count Buffon proposed grinding such prisms as steps in a single piece of glass. In 1790 (although secondary sources give the date as 1773 or 1788), the Marquis de Condorcet suggested that it would be easier to make the annular sections separately and assemble them on a frame; but even that was impractical at the time. These designs were intended not for lighthouses, but for burning glasses. Brewster, however, proposed a system similar to Condorcet's in 1811, and by 1820 was advocating its use in British lighthouses.

Their mother's 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 (see below).

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.

Fresnel was not the first person to focus a lighthouse beam using a lens. That distinction apparently belongs to the London glass-cutter Thomas Rogers, who proposed the idea to Trinity House in 1788. The first Rogers lenses, 53 cm in diameter and 14 cm thick at the center, were installed at the Old Lower Lighthouse at Portland Bill in 1789. Further samples followed at Howth Baily, North Foreland, and at least four other locations. But much of the light was wasted by absorption in the glass.

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 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 ordinair 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 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:

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 then sought to explain this law in corpuscular terms: from the known relation between the incident and refracted ray directions, he derived the corpuscular velocity (as a function of direction) that that would satisfy Maupertuis's "least action" principle. But, as Young pointed out, the existence 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 a substance somewhat less elastic" admits spheroidal longitudinal wavefronts, as Huygens supposed.

In 1810, Ara Go found experimentally that the degree of refraction of starlight does not depend on the direction of the earth's motion relative to the line of sight. In 1818, Fresnel showed that this result could be explained by the wave theory, on the hypothesis that if an object with refractive index n {\displaystyle n} moved at velocity v {\displaystyle v} relative to the external aether (taken as stationary), then the velocity of light inside the object gained the additional component v ( 1 − 1 / n 2 ) {\displaystyle \,v(1-1/n^{2})} . He supported that hypothesis by supposing that if the density of the external aether was taken as unity, the density of the internal aether was n 2 {\displaystyle n^{2}} , of which the excess, namely n 2 − 1 {\displaystyle \,n^{2}{-}1\,} , was dragged along at velocity v {\displaystyle v} , whence the average velocity of the internal aether was v ( 1 − 1 / n 2 ) {\displaystyle \,v(1-1/n^{2})} . The factor in parentheses, which Fresnel originally expressed in terms of wavelengths, became known as the Fresnel drag coefficient. (See Aether drag hypothesis.)

In August 1811, François Ara Go 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 from the mica was "depolarized" in the sense that there was no orientation of the calcite that made one image disappear; yet it was not ordinary ("unpolarized") 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), produced a similar effect, except that rotating the quartz made no difference. Ara Go tried to explain his observations in corpuscular terms.

which are none other than Biot's empirical formulae of 1812, except that Biot interpreted U {\displaystyle U} and A {\displaystyle A} as the "unaffected" and "affected" selections of the rays incident on the lamina. If Biot's substitutions were accurate, they would imply that his experimental results were more fully explained by Fresnel's theory than by his own.

In 1813, Brewster observed the simple concentric pattern in "beryl, emerald, ruby &c." The same pattern was later observed in calcite by Wollaston, Biot, and Seebeck.  Biot, assuming that the concentric pattern was the general case, tried to calculate the colors with his theory of chromatic polarization, and succeeded better for some minerals than for others. In 1818, Brewster belatedly explained why: seven of the twelve minerals employed by Biot had the lemniscate pattern, which Brewster had observed as early as 1812; and the minerals with the more complicated rings also had a more complicated law of refraction.

Fresnel's essay Rêveries of 1814 has not survived. While its content would have been interesting to historians, its quality may perhaps be gauged from the fact that Fresnel himself never referred to it in his maturity.

In 1815, Brewster reported that colors appear when a slice of isotropic material, placed between crossed polarizers, is mechanically stressed. Brewster himself immediately and correctly attributed this phenomenon to stress-induced birefringence  — now known as photoelasticity.

In the draft memoir of 30 August 1816, Fresnel mentioned two hypotheses — one of which he attributed to Ampère — by which the non-interference of orthogonally-polarized beams could be explained if polarized light waves were partly transverse. But Fresnel could not develop either of these ideas into a comprehensive theory. According to his later account, both he and Ampère realized as early as September 1816 that the non-interference of orthogonally-polarized beams, together with the phase-inversion rule in chromatic polarization, would be most easily explained if the waves were purely transverse. But that would raise a new difficulty: as natural light seemed to be unpolarized and its waves were therefore presumed to be longitudinal, one would need to explain how the longitudinal component of vibration disappeared on polarization, and why it did not reappear when polarized light was reflected or refracted obliquely by a glass plate. 

By 1817 it had been discovered by Brewster, but not adequately reported, that plane-polarized light was partly depolarized by total internal reflection if initially polarized at an acute angle to the plane of incidence. Fresnel rediscovered this effect and investigated it by including total internal reflection in a chromatic-polarization experiment. With the aid of his first theory of chromatic polarization, he found that the apparently depolarized light was a mixture of components polarized parallel and perpendicular to the plane of incidence, and that the total reflection introduced a phase difference between them. Choosing an appropriate angle of incidence (not yet exactly specified) gave a phase difference of 1/8 of a cycle (45°). Two such reflections from the "parallel faces" of "two coupled prisms" gave a phase difference of 1/4 of a cycle (90°). These findings were contained in a memoir submitted to the Académie on 10 November 1817 and read a fortnight later. An undated marginal note indicates that the two coupled prisms were later replaced by a single "parallelepiped in glass" — now known as a Fresnel rhomb.

This was the memoir whose "supplement", dated January 1818, contained the method of superposing sinusoidal functions and the restatement of Malus's law in terms of amplitudes. In the same supplement, Fresnel reported his discovery that optical rotation could be imitated by passing the polarized light through a Fresnel rhomb (still in the form of "coupled prisms"), followed by an ordinary birefringent lamina sliced parallel to its axis, with the axis at 45° to the plane of reflection of the Fresnel rhomb, followed by a second Fresnel rhomb at 90° to the first. In a further memoir read on 30 March, Fresnel reported that if polarized light was fully "depolarized" by a Fresnel rhomb — now described as a parallelepiped — its properties were not further modified by a subsequent passage through an optically rotating medium or device.

Fresnel was elected to the Société Philomathique de Paris in April 1819, and in 1822 became one of the editors of the Société's  Bulletin des Sciences. As early as May 1817, at Arago's suggestion, Fresnel applied for membership of the Académie des Sciences, but received only one vote. The successful candidate on that occasion was Joseph Fourier. In November 1822, Fourier's elevation to Permanent Secretary of the Académie created a vacancy in the physics section, which was filled in February 1823 by Pierre Louis Dulong, with 36 votes to Fresnel's 20. But in May 1823, after another vacancy was left by the death of Jacques Charles,  Fresnel's election was unanimous. In 1824, Fresnel was made a chevalier de la Légion d'honneur (Knight of the Legion of Honour).

Not included in the Oeuvres  are two short notes by Fresnel on magnetism, which were discovered among Ampère's manuscripts. In response to Ørsted's discovery of electromagnetism in 1820, Ampère initially supposed that the field of a permanent magnet was due to a macroscopic circulating current. Fresnel suggested instead that there was a microscopic current circulating around each particle of the magnet. In his first note, he argued that microscopic currents, unlike macroscopic currents, would explain why a hollow cylindrical magnet does not lose its magnetism when cut longitudinally. In his second note, dated 5 July 1821, he further argued that a macroscopic current had the counterfactual implication that a permanent magnet should be hot, whereas microscopic currents circulating around the molecules might avoid the heating mechanism. He was not to know that the fundamental units of permanent magnetism are even smaller than molecules (see Electron magnetic moment). The two notes, together with Ampère's acknowledgment, were eventually published in 1885.

For the supplement to Riffault's translation of Thomson's System of Chemistry, Fresnel was chosen to contribute the article on light. The resulting 137-page essay, titled De la Lumière (On Light), was apparently finished in June 1821 and published by February 1822. With sections covering the nature of light, diffraction, thin-film interference, reflection and refraction, double refraction and polarization, chromatic polarization, and modification of polarization by reflection, it made a comprehensive case for the wave theory to a readership that was not restricted to physicists.

Fresnel's health, which had always been poor, deteriorated in the winter of 1822-3, increasing the urgency of his original research, and causing him to turn down an invitation from Young to write an article on double refraction for the Encyclopædia Britannica. The memoirs on circular and elliptical polarization and optical rotation, and on the detailed derivation of the Fresnel equations and their application to total internal reflection, date from this period. In the spring he recovered enough, in his own view, to supervise the lens installation at Cordouan. Soon afterwards, it became clear that his condition was tuberculosis.

Fresnel gave details of the "mechanical solution" in a memoir read to the Académie des Sciences on 7 January 1823. Conservation of Energy was combined with continuity of the tangential vibration at the interface. The resulting formulae for the reflection coefficients and reflectivities became known as the Fresnel equations. The reflection coefficients for the s and p polarizations are most succinctly expressed as

In 1824 he was advised that if he wanted to live longer, he needed to scale back his activities. Perceiving his lighthouse work to be his most important duty, he resigned as an examiner at the École Polytechnique, and closed his scientific notebooks. His last note to the Académie, read on 13 June 1825, described the first radiometer and attributed the observed repulsive force to a temperature difference. Although his fundamental research ceased, his advocacy did not; as late as August or September 1826, he found the time to answer Herschel's queries on the wave theory. It was Herschel who recommended Fresnel for the Royal Society's Rumford Medal.

More disturbing is the fate of a long article that Fresnel wrote in 1825, to be translated for the new English journal European Review. The journal failed before Fresnel's contribution could be published. Fresnel unsuccessfully tried to recover the manuscript, and the editors of his collected works were unable to find it. According to Grattan-Guinness, the article was "apparently similar" to the essay De la Lumière of 1821/22. But Fresnel's theories on polarization, total internal reflection, and double refraction had developed since then. Together, the circumstances suggest that the most comprehensive defense of the wave theory of light, by its most influential proponent, has been lost.

Fresnel's cough worsened in the winter of 1826-7, leaving him too ill to return to Mathieu in the spring. In early June he was carried to Ville-d'Avray, 12 km west of Paris. There his mother joined him. On 6 July, Ara Go arrived to deliver the Rumford Medal. Sensing Arago's distress, Fresnel whispered that "the most beautiful crown means little, when it is laid on the grave of a friend." Fresnel did not have the strength to reply to the Royal Society. He died eight days later, on Bastille Day.

Fresnel's "second memoir" on double refraction was not printed until 1827, the year of his death. Until then, the best published source on his work on double refraction was an extract of that memoir, printed in 1822. His final treatment of partial reflection and total internal reflection, read to the Académie in January 1823, was thought to be lost until it was rediscovered among the papers of the deceased Joseph Fourier (1768–1830), and was printed in 1831. Until then, it was known chiefly through an extract printed in 1823 and 1825. The memoir introducing the parallelepiped form of the Fresnel rhomb, read in March 1818, was mislaid until 1846. Most of Fresnel's writings on polarized light before 1821 — including his first theory of chromatic polarization (submitted 7 October 1816) and the crucial "supplement" of January 1818  — were not published in full until his Oeuvres complètes ("complete works") began to appear in 1866. The "supplement" of July 1816, proposing the "efficacious ray" and reporting the famous double-mirror experiment, met the same fate, as did the "first memoir" on double refraction.

The analytical complexity of Fresnel's derivation of the ray-velocity surface was an implicit challenge to find a shorter path to the result. This was answered by James MacCullagh in 1830, and by william Rowan Hamilton in 1832.

Publication of Fresnel's collected works was itself delayed by the deaths of successive editors. The task was initially entrusted to Félix Savary, who died in 1841. It was restarted twenty years later by the Ministry of Public Instruction. Of the three editors eventually named in the Oeuvres, Sénarmont died in 1862, Verdet in 1866, and Léonor Fresnel in 1869, by which time only two of the three volumes had appeared. At the beginning of vol. 3 (1870), the completion of the project is described in a long footnote by " J. Lissajous."

The first large catadioptric lenses were made in 1842 for the lighthouses at Gravelines and Île Vierge; these were fixed third-order lenses whose catadioptric rings (made in segments) were one metre in diameter. The first-order Skerryvore lens, installed in 1844, was only partly catadioptric; it was similar to the Cordouan lens except that the lower slats were replaced by French-made catadioptric prisms, while mirrors were retained at the top. The first fully catadioptric first-order lens, installed at Ailly in 1852, also gave eight rotating beams plus a fixed light at the bottom; but its top section had eight catadioptric panels focusing the light about 4 degrees ahead of the main beams, in order to lengthen the flashes. The first fully catadioptric lens with purely revolving beams — also of first order — was installed at Saint-Clément-des-Baleines in 1854, and marked the completion of Fresnel's original Carte des Phares.

In 1846, George Gabriel Stokes pointed out that there was no need to divide the aether inside a moving object into two portions; all of it could be considered as moving at a Common velocity. Then, if the aether was conserved while its density changed in proportion to n 2 {\displaystyle n^{2}} , the resulting velocity of the aether inside the object was equal to Fresnel's additional velocity component.

Hence, in 1850, when Foucault and Fizeau found by experiment that light travels more slowly in water than in air, in accordance with the wave explanation of refraction and contrary to the corpuscular explanation, the result came as no surprise.

Although he did not become a public Celebrity in his short lifetime, Fresnel 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, 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." 

Production of one-piece stepped lenses (roughly as envisaged by Buffon) eventually became profitable. By the 1870s, in the United States, such lenses were made of pressed glass and used with small Lights on ships and piers. Similar lenses are used in Fresnel lanterns for stage lighting. Lenses with finer steps serve as condensers in overhead projectors. Still finer steps can be found in low-cost plastic "sheet" magnifiers.

The monument to Fresnel at his birthplace (see above) was dedicated on 14 September 1884 with a speech by Jules Jamin, Permanent Secretary of the Académie des Sciences.  "FRESNEL" is among the 72 names embossed on the Eiffel Tower (on the south-east side, fourth from the left). In the 19th century, as every lighthouse in France acquired a Fresnel lens, every one acquired a bust of Fresnel, seemingly watching over the coastline that he had made safer. The lunar features Promontorium Fresnel and Rimae Fresnel were later named after him.

The "ellipsoid of elasticity" indeed gave the correct ray velocities, although the initial experimental verification was only approximate. But it did not give the correct directions of vibration, for the biaxial case or even for the uniaxial case, because the vibrations in Fresnel's model were tangential to the wavefront, which is not generally normal to the ray (for an extraordinary ray). This mistake was corrected in an "extract" that Fresnel read to the Académie a week later, on 26 November. Starting with Huygens' spheroid, Fresnel obtained the 4th-degree "surface of elasticity" which, when sectioned by a plane as above, would yield the wave-normal velocities for a wavefront in that plane, together with their vibration directions. For the biaxial case, he generalized the surface to allow three unequal principal dimensions. But he retained the former "ellipsoid of elasticity" as an approximation, from which he deduced Biot's dihedral law.

Huygens, in his investigation of double refraction, noticed something that he could not explain: when a ray 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 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 his authority through the 18th century, although he made "no known attempt to deduce it from any principles of optics, corpuscular or otherwise." 

where r 2 = x 2 + y 2 + z 2 , {\displaystyle \,r^{2}=x^{2\!}+y^{2\!}+z^{2},\,} and a , b , c {\displaystyle \,a,b,c\,} are the propagation speeds in directions normal to the coordinate axes for vibrations along the axes (the ray and wave-normal speeds being the same in those special cases). Later commentators put the equation in the more compact and memorable form

What Whewell called the "true theory" has since undergone two major revisions. The first, by Maxwell, specified the physical fields whose variations constitute the waves of light. The second, initiated by Einstein's explanation of the photoelectric effect, supposed that the Energy of light waves was divided into quanta, which were eventually identified with particles called photons. But photons did not exactly correspond to Newton's corpuscles; for Example, Newton's explanation of ordinary refraction required the corpuscles to travel faster in media of higher refractive index, which photons do not. Neither did photons displace waves; rather, they led to the paradox of wave–particle duality.

The analogy between light waves and transverse waves in elastic solids does not predict dispersion — that is, the frequency-dependence of the speed of propagation, which enables prisms to produce spectra and causes lenses to suffer from chromatic aberration. Fresnel, in De la Lumière and in the second supplement to his first memoir on double refraction, suggested that dispersion could be accounted for if the particles of the medium exerted forces on each other over distances that were significant fractions of a wavelength. Later, more than once, Fresnel referred to the demonstration of this result as being contained in a note appended to his second memoir on double refraction. But no such note appeared in print, and the relevant manuscripts found after his death showed only that, around 1824, he was comparing refractive indices measured by Fraunhofer with a theoretical formula, the meaning of which was not fully explained. One obvious possibility is that the explanation was given in the missing note.