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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. ...
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Independently, on 12 January 1817, Young wrote to Arago (in English) noting that a transverse vibration would constitute a polarization, and that if two longitudinal waves crossed at a significant angle, they could not cancel without leaving a residual transverse vibration. Young repeated this idea in an article publis...
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Thus Fresnel, by his own testimony, may not have been the first person to suspect that light waves could have a transverse component, or that polarized waves were exclusively transverse. And it was Young, not Fresnel, who first published the idea that polarization depends on the orientation of a transverse vibration. B...
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In a note that Buchwald dates in the summer of 1818, Fresnel entertained the idea that unpolarized waves could have vibrations of the same energy and obliquity, with their orientations distributed uniformly about the wave-normal, and that the degree of polarization was the degree of non -uniformity in the distribution....
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But if he could account for lack of polarization by averaging out the transverse component, he did not also need to assume a longitudinal component. It was enough to suppose that light waves are purely transverse, hence always polarized in the sense of having a particular transverse orientation, and that the "unpolariz...
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It is not known exactly when Fresnel made this last step, because there is no relevant documentation from 1820 or early 1821 (perhaps because he was too busy working on lighthouse-lens prototypes; see below). But he first published the idea in a paper on " Calcul des teintes… " ("calculation of the tints…"), serialized...
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It has only been for a few months that in meditating more attentively on this subject, I have realized that it was very probable that the oscillatory movements of light waves were executed solely along the plane of these waves, for direct light as well as for polarized light.
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According to this new view, he wrote, "the act of polarization consists not in creating these transverse movements, but in decomposing them into two fixed perpendicular directions and in separating the two components".
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While selectionists could insist on interpreting Fresnel's diffraction integrals in terms of discrete, countable rays, they could not do the same with his theory of polarization. For a selectionist, the state of polarization of a beam concerned the distribution of orientations over the population of rays, and that dist...
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The other difficulty posed by pure transverse waves, of course, was the apparent implication that the aether was an elastic solid, except that, unlike other elastic solids, it was incapable of transmitting longitudinal waves. The wave theory was cheap on assumptions, but its latest assumption was expensive on credulity...
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In the second installment of "Calcul des teintes" (June 1821), Fresnel supposed, by analogy with sound waves, that the density of the aether in a refractive medium was inversely proportional to the square of the wave velocity, and therefore directly proportional to the square of the refractive index. For reflection and...
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The third installment (July 1821) was a short "postscript" in which Fresnel announced that he had found, by a "mechanical solution", a formula for the reflectivity of the p component, which predicted that the reflectivity was zero at the Brewster angle. So polarization by reflection had been accounted for—but with the ...
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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 equat...
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where i and r are the angles of incidence and refraction; these equations are known respectively as Fresnel's sine law and Fresnel's tangent law. By allowing the coefficients to be complex, Fresnel even accounted for the different phase shifts of the s and p components due to total internal reflection.
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This success inspired James MacCullagh and Augustin-Louis Cauchy, beginning in 1836, to analyze reflection from metals by using the Fresnel equations with a complex refractive index. The same technique is applicable to non-metallic opaque media. With these generalizations, the Fresnel equations can predict the appearan...
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In a memoir dated 9 December 1822, Fresnel coined the terms linear polarization (French: polarisation rectiligne) for the simple case in which the perpendicular components of vibration are in phase or 180° out of phase, circular polarization for the case in which they are of equal magnitude and a quarter-cycle (±90°) o...
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These concepts called for a redefinition of the distinction between polarized and unpolarized light. Before Fresnel, it was thought that polarization could vary in direction, and in degree (e.g., due to variation in the angle of reflection off a transparent body), and that it could be a function of color (chromatic pol...
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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 chr...
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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 emulated by passing the polarized light through a Fresnel ...
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The connection between optical rotation and birefringence was further explained in 1822, in the memoir on elliptical and circular polarization. This was followed by the memoir on reflection, read in January 1823, in which Fresnel quantified the phase shifts in total internal reflection, and thence calculated the precis...
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When light passes through a slice of calcite cut perpendicular to its optic axis, the difference between the propagation times of the ordinary and extraordinary waves has a second-order dependence on the angle of incidence. If the slice is observed in a highly convergent cone of light, that dependence becomes significa...
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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 ...
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In a uniform crystal, according to Huygens's theory, the secondary wavefront that expands from the origin in unit time is the ray-velocity surface —that is, the surface whose "distance" from the origin in any direction is the ray velocity in that direction. In calcite, this surface is two-sheeted, consisting of a spher...
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where v o and v e were the ordinary and extraordinary ray velocities according to the corpuscular theory, and θ θ was the angle between the ray and the optic axis. By Malus's definition, the plane of polarization of a ray was the plane of the ray and the optic axis if the ray was ordinary, or the perpendicular plane (c...
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On 29 March 1819, Biot presented a memoir in which he proposed simple generalizations of Malus's rules for a crystal with two axes, and reported that both generalizations seemed to be confirmed by experiment. For the velocity law, the squared sine was replaced by the product of the sines of the angles from the ray to t...
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Until Fresnel turned his attention to biaxial birefringence, it was assumed that one of the two refractions was ordinary, even in biaxial crystals. But, in a memoir submitted on 19 November 1821, Fresnel reported two experiments on topaz showing that neither refraction was ordinary in the sense of satisfying Snell's la...
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The same memoir contained Fresnel's first attempt at the biaxial velocity law. For calcite, if we interchange the equatorial and polar radii of Huygens's oblate spheroid while preserving the polar direction, we obtain a prolate spheroid touching the sphere at the equator. A plane through the center/origin cuts this pro...
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The ellipsoid 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, for a...
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Fresnel's initial derivation of the surface of elasticity had been purely geometric, and not deductively rigorous. His first attempt at a mechanical derivation, contained in a "supplement" dated 13 January 1822, assumed that (i) there were three mutually perpendicular directions in which a displacement produced a react...
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In the same supplement, Fresnel considered how he might find, for the biaxial case, the secondary wavefront that expands from the origin in unit time—that is, the surface that reduces to Huygens's sphere and spheroid in the uniaxial case. He noted that this "wave surface" (surface de l'onde) is tangential to all possib...
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In a "second supplement", Fresnel eventually exploited two related facts: (i) the "wave surface" was also the ray-velocity surface, which could be obtained by sectioning the ellipsoid that he had initially mistaken for the surface of elasticity, and (ii) the "wave surface" intersected each plane of symmetry of the elli...
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where r 2 = x 2 + y 2 + z 2 , and 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
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Earlier in the "second supplement", Fresnel modeled the medium as an array of point-masses and found that the force-displacement relation was described by a symmetric matrix, confirming the existence of three mutually perpendicular axes on which the displacement produced a parallel force. Later in the document, he note...
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Fresnel's "second supplement" was signed on 31 March 1822 and submitted the next day—less than a year after the publication of his pure-transverse-wave hypothesis, and just less than a year after the demonstration of his prototype eight-panel lighthouse lens (see below).
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Having presented the pieces of his theory in roughly the order of discovery, Fresnel needed to rearrange the material so as to emphasize the mechanical foundations; and he still needed a rigorous treatment of Biot's dihedral law. He attended to these matters in his "second memoir" on double refraction, published in the...
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As early as 1822, Fresnel discussed his perpendicular axes with Cauchy. Acknowledging Fresnel's influence, Cauchy went on to develop the first rigorous theory of elasticity of non-isotropic solids (1827), hence the first rigorous theory of transverse waves therein (1830)—which he promptly tried to apply to optics. The ...
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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.
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In a memoir read in September 1822, Fresnel announced that he had verified Brewster's diagnosis more directly, by compressing a combination of glass prisms so severely that one could actually see a double image through it. In his experiment, Fresnel lined up seven 45°–90°–45° prisms, short side to short side, with thei...
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At the end of that memoir, Fresnel predicted that if the compressed prisms were replaced by (unstressed) monocrystalline quartz prisms with matching directions of optical rotation, and with their optic axes aligned along the row, an object seen by looking along the common optic axis would give two images, which would s...
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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, diffracti...
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To examine Fresnel's first memoir and supplements on double refraction, the Académie des Sciences appointed Ampère, Arago, Fourier, and Poisson. Their report, of which Arago was clearly the main author, was delivered at the meeting of 19 August 1822. Then, in the words of Émile Verdet, as translated by Ivor Grattan-Gui...
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Immediately after the reading of the report, Laplace took the floor, and… proclaimed the exceptional importance of the work which had just been reported: he congratulated the author on his steadfastness and his sagacity which had led him to discover a law which had escaped the cleverest, and, anticipating somewhat the ...
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Whether Laplace was announcing his conversion to the wave theory—at the age of 73—is uncertain. Grattan-Guinness entertained the idea. Buchwald, noting that Arago failed to explain that the "ellipsoid of elasticity" did not give the correct planes of polarization, suggests that Laplace may have merely regarded Fresnel'...
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In the following year, Poisson, who did not sign Arago's report, disputed the possibility of transverse waves in the aether. Starting from assumed equations of motion of a fluid medium, he noted that they did not give the correct results for partial reflection and double refraction—as if that were Fresnel's problem rat...
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Among the French, Poisson's reluctance was an exception. According to Eugene Frankel, "in Paris no debate on the issue seems to have taken place after 1825. Indeed, almost the entire generation of physicists and mathematicians who came to maturity in the 1820s—Pouillet, Savart, Lamé, Navier, Liouville, Cauchy—seem to h...
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In 1826, the British astronomer John Herschel, who was working on a book-length article on light for the Encyclopædia Metropolitana, addressed three questions to Fresnel concerning double refraction, partial reflection, and their relation to polarization. The resulting article, titled simply "Light", was highly sympath...
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A German translation of De la Lumière was published in installments in 1825 and 1828. The wave theory was adopted by Fraunhofer in the early 1820s and by Franz Ernst Neumann in the 1830s, and then began to find favor in German textbooks.
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The economy of assumptions under the wave theory was emphasized by William Whewell in his History of the Inductive Sciences, first published in 1837. In the corpuscular system, "every new class of facts requires a new supposition," whereas in the wave system, a hypothesis devised in order to explain one phenomenon is t...
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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.
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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, whose first 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 were installed...
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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, the Marquis de Condorcet suggested that it would be easier to make the annular secti...
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Meanwhile, on 21 June 1819, Fresnel was "temporarily" seconded by the Commission des Phares (Commission of Lighthouses) on the recommendation of Arago (a member of the Commission since 1813), to review possible improvements in lighthouse illumination. The commission had been established by Napoleon in 1811 and placed u...
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By the end of August 1819, unaware of the Buffon-Condorcet-Brewster proposal, Fresnel made his first presentation to the commission, recommending what he called lentilles à échelons (lenses by steps) to replace the reflectors then in use, which reflected only about half of the incident light. One of the assembled commi...
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Fresnel's next lens was a rotating apparatus with eight "bull's-eye" panels, made in annular arcs by Saint-Gobain, giving eight rotating beams—to be seen by mariners as a periodic flash. Above and behind each main panel was a smaller, sloping bull's-eye panel of trapezoidal outline with trapezoidal elements. This refra...
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In May 1824, Fresnel was promoted to secretary of the Commission des Phares, becoming the first member of that body to draw a salary, albeit in the concurrent role of Engineer-in-Chief. He was also an examiner (not a teacher) at the École Polytechnique since 1821; but poor health, long hours during the examination seas...
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In the same year he designed the first fixed lens—for spreading light evenly around the horizon while minimizing waste above or below. Ideally the curved refracting surfaces would be segments of toroids about a common vertical axis, so that the dioptric panel would look like a cylindrical drum. If this was supplemented...
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In 1825, Fresnel extended his fixed-lens design by adding a rotating array outside the fixed array. Each panel of the rotating array was to refract part of the fixed light from a horizontal fan into a narrow beam.
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Also in 1825, Fresnel unveiled the Carte des Phares (Lighthouse Map), calling for a system of 51 lighthouses plus smaller harbor lights, in a hierarchy of lens sizes (called orders, the first order being the largest), with different characteristics to facilitate recognition: a constant light (from a fixed lens), one fl...
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In late 1825, to reduce the loss of light in the reflecting elements, Fresnel proposed to replace each mirror with a catadioptric prism, through which the light would travel by refraction through the first surface, then total internal reflection off the second surface, then refraction through the third surface. The res...
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The first fixed lens with toroidal prisms was a first-order apparatus designed by the Scottish engineer Alan Stevenson under the guidance of Léonor Fresnel, and fabricated by Isaac Cookson & Co. from French glass; it entered service at the Isle of May in 1836. The first large catadioptric lenses were fixed third-order ...
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Production of one-piece stepped dioptric lenses—roughly as envisaged by Buffon—became practical in 1852, when John L. Gilliland of the Brooklyn Flint-Glass Company patented a method of making such lenses from press-molded glass. By the 1950s, the substitution of plastic for glass made it economic to use fine-stepped Fr...
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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 oc...
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Meanwhile, in Britain, the wave theory was yet to take hold; Fresnel wrote to Thomas Young in November 1824, saying in part:
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I am far from denying the value that I attach to the praise of English scholars, or pretending that they would not have flattered me agreeably. But for a long time this sensibility, or vanity, which is called the love of glory, has been much blunted in me: I work far less to capture the public's votes than to obtain an...
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But "the praise of English scholars" soon followed. On 9 June 1825, Fresnel was made a Foreign Member of the Royal Society of London. In 1827 he was awarded the society's Rumford Medal for the year 1824, "For his Development of the Undulatory Theory as applied to the Phenomena of Polarized Light, and for his various im...
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A 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 i...
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Fresnel's health, which had always been poor, deteriorated in the winter of 1822–1823, increasing the urgency of his original research, and (in part) preventing him from contributing an article on polarization and double refraction for the Encyclopædia Britannica. The memoirs on circular and elliptical polarization and...
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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 f...
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Fresnel's cough worsened in the winter of 1826–1827, leaving him too ill to return to Mathieu in the spring. The Académie meeting of 30 April 1827 was the last that he attended. In early June he was carried to Ville-d'Avray, 12 kilometres (7.5 mi) west of Paris. There his mother joined him. On 6 July, Arago arrived to ...
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He is buried at Père Lachaise Cemetery, Paris. The inscription on his headstone is partly eroded away; the legible part says, when translated, "To the memory of Augustin Jean Fresnel, member of the Institute of France ".
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Fresnel's "second memoir" on double refraction was not printed until late 1827, a few months after 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 Académ...
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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, Ve...
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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 instea...
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Fresnel's essay Rêveries of 1814 has not survived. The article "Sur les Différents Systèmes relatifs à la Théorie de la Lumière" ("On the Different Systems relating to the Theory of Light"), which Fresnel wrote for the newly launched English journal European Review, was received by the publisher's agent in Paris in Sep...
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In 1810, Arago 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 moved at veloci...
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In his analysis of double refraction, Fresnel supposed that the different refractive indices in different directions within the same medium were due to a directional variation in elasticity, not density (because the concept of mass per unit volume is not directional). But in his treatment of partial reflection, he supp...
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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 ...
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In the 1830s, Fresnel's suggestion was taken up by Cauchy, Baden Powell, and Philip Kelland, and it was found to be tolerably consistent with the variation of refractive indices with wavelength over the visible spectrum for a variety of transparent media (see Cauchy's equation). These investigations were enough to show...
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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 MacCullagh in 1830, and by William Rowan Hamilton in 1832.
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Within a century of Fresnel's initial stepped-lens proposal, more than 10,000 lights with Fresnel lenses were protecting lives and property around the world. Concerning the other benefits, the science historian Theresa H. Levitt has remarked:
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Everywhere I looked, the story repeated itself. The moment a Fresnel lens appeared at a location was the moment that region became linked into the world economy.
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In the history of physical optics, Fresnel's successful revival of the wave theory nominates him as the pivotal figure between Newton, who held that light consisted of corpuscles, and James Clerk Maxwell, who established that light waves are electromagnetic. Whereas Albert Einstein described Maxwell's work as "the most...
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The theory of Fresnel to which I now proceed,—and which not only embraces all the known phenomena, but has even outstripped observation, and predicted consequences which were afterwards fully verified,—will, I am persuaded, be regarded as the finest generalization in physical science which has been made since the disco...
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It would, perhaps, be too fanciful to attempt to establish a parallelism between the prominent persons who figure in these two histories. If we were to do this, we must consider Huyghens and Hooke as standing in the place of Copernicus, since, like him, they announced the true theory, but left it to a future age to giv...
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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. Without the benefit of this knowledge, Fresnel managed to construct the world's first coherent theory of light, showing in retrospect that h...
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In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism group. It is, loosely ...
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In an algebraic structure such as a group, a ring, or vector space, an automorphism is simply a bijective homomorphism of an object into itself. (The definition of a homomorphism depends on the type of algebraic structure; see, for example, group homomorphism, ring homomorphism, and linear operator.)
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More generally, for an object in some category, an automorphism is a morphism of the object to itself that has an inverse morphism; that is, a morphism f : X → → X is an automorphism if there is a morphism g : X → → X such that g ∘ ∘ f = f ∘ ∘ g = id X , where id X is the identity morphism of X. For algebraic structure...
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The automorphisms of an object X form a group under composition of morphisms, which is called the automorphism group of X. This results straightforwardly from the definition of a category.
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The automorphism group of an object X in a category C is often denoted Aut C (X), or simply Aut(X) if the category is clear from context.
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One of the earliest group automorphisms (automorphism of a group, not simply a group of automorphisms of points) was given by the Irish mathematician William Rowan Hamilton in 1856, in his icosian calculus, where he discovered an order two automorphism, writing:
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so that μ μ is a new fifth root of unity, connected with the former fifth root λ λ by relations of perfect reciprocity.
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In some categories—notably groups, rings, and Lie algebras —it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.
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In the case of groups, the inner automorphisms are the conjugations by the elements of the group itself. For each element a of a group G, conjugation by a is the operation φ a : G → G given by φ a (g) = aga (or a ga ; usage varies). One can easily check that conjugation by a is a group automorphism. The inner automorph...
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The other automorphisms are called outer automorphisms. The quotient group Aut(G) / Inn(G) is usually denoted by Out(G); the non-trivial elements are the cosets that contain the outer automorphisms.
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The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.
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Atomic physics is the field of physics that studies atoms as an isolated system of electrons and an atomic nucleus. Atomic physics typically refers to the study of atomic structure and the interaction between atoms. It is primarily concerned with the way in which electrons are arranged around the nucleus and the proces...
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The term atomic physics can be associated with nuclear power and nuclear weapons, due to the synonymous use of atomic and nuclear in standard English. Physicists distinguish between atomic physics—which deals with the atom as a system consisting of a nucleus and electrons—and nuclear physics, which studies nuclear reac...
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As with many scientific fields, strict delineation can be highly contrived and atomic physics is often considered in the wider context of atomic, molecular, and optical physics. Physics research groups are usually so classified.
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Atomic physics primarily considers atoms in isolation. Atomic models will consist of a single nucleus that may be surrounded by one or more bound electrons. It is not concerned with the formation of molecules (although much of the physics is identical), nor does it examine atoms in a solid state as condensed matter. It...
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