arnosimons/wikipedia-physics-corpus · Datasets at Hugging Face

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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. As early as September 1816, according to his later account, he realized 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, and Ampère "had the same thought" on the phase-inversion rule. But that would raise a new difficulty: as natural light seemed to be un polarized 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.	990	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	100	82	2,024	8	10	30	58
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 published in a supplement to the Encyclopædia Britannica in February 1818, in which he added that Malus's law would be explained if polarization consisted in a transverse motion.	492	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	101	83	2,024	8	10	30	58
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. But these incomplete theories had not reconciled the nature of polarization with the apparent existence of unpolarized light; that achievement was to be Fresnel's alone.	488	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	102	84	2,024	8	10	30	58
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. Two pages later he noted, apparently for the first time in writing, that his phase-inversion rule and the non-interference of orthogonally-polarized beams would be easily explained if the vibrations of fully polarized waves were "perpendicular to the normal to the wave"—that is, purely transverse.	619	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	103	85	2,024	8	10	30	58
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 "unpolarized" state of natural or "direct" light is due to rapid and random variations in that orientation, in which case two coherent portions of "unpolarized" light will still interfere because their orientations will be synchronized.	546	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	104	86	2,024	8	10	30	58
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 in Arago's Annales for May, June, and July 1821. In the first installment, Fresnel described "direct" (unpolarized) light as "the rapid succession of systems of waves polarized in all directions", and gave what is essentially the modern explanation of chromatic polarization, albeit in terms of the analogy between polarization and the resolution of forces in a plane, mentioning transverse waves only in a footnote. The introduction of transverse waves into the main argument was delayed to the second installment, in which he revealed the suspicion that he and Ampère had harbored since 1816, and the difficulty it raised. He continued:	959	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	105	87	2,024	8	10	30	58
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.	276	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	106	88	2,024	8	10	30	58
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".	218	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	107	89	2,024	8	10	30	58
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 distribution was presumed to be static. For Fresnel, the state of polarization of a beam concerned the variation of a displacement over time. That displacement might be constrained but was not static, and rays were geometric constructions, not countable objects. The conceptual gap between the wave theory and selectionism had become unbridgeable.	663	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	108	90	2,024	8	10	30	58
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. If that assumption was to be widely entertained, its explanatory power would need to be impressive.	421	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	109	91	2,024	8	10	30	58
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 refraction at the surface between two isotropic media of different indices, Fresnel decomposed the transverse vibrations into two perpendicular components, now known as the s and p components, which are parallel to the surface and the plane of incidence, respectively; in other words, the s and p components are respectively square and parallel to the plane of incidence. For the s component, Fresnel supposed that the interaction between the two media was analogous to an elastic collision, and obtained a formula for what we now call the reflectivity : the ratio of the reflected intensity to the incident intensity. The predicted reflectivity was non-zero at all angles.	994	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	110	92	2,024	8	10	30	58
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 proviso that the direction of vibration in Fresnel's model was perpendicular to the plane of polarization as defined by Malus. (On the ensuing controversy, see Plane of polarization.) The technology of the time did not allow the s and p reflectivities to be measured accurately enough to test Fresnel's formulae at arbitrary angles of incidence. But the formulae could be rewritten in terms of what we now call the reflection coefficient : the signed ratio of the reflected amplitude to the incident amplitude. Then, if the plane of polarization of the incident ray was at 45° to the plane of incidence, the tangent of the corresponding angle for the reflected ray was obtainable from the ratio of the two reflection coefficients, and this angle could be measured. Fresnel had measured it for a range of angles of incidence, for glass and water, and the agreement between the calculated and measured angles was better than 1.5° in all cases.	1,261	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	111	93	2,024	8	10	30	58
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	416	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	112	94	2,024	8	10	30	58
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.	303	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	113	95	2,024	8	10	30	58
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 appearance of a wide variety of objects under illumination—for example, in computer graphics (see Physically based rendering).	438	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	114	96	2,024	8	10	30	58
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°) out of phase, and elliptical polarization for other cases in which the two components have a fixed amplitude ratio and a fixed phase difference. He then explained how optical rotation could be understood as a species of birefringence. Linearly-polarized light could be resolved into two circularly-polarized components rotating in opposite directions. If these components propagated at slightly different speeds, the phase difference between them—and therefore the direction of their linearly-polarized resultant—would vary continuously with distance.	870	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	115	97	2,024	8	10	30	58
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 polarization), but not that it could vary in kind. Hence it was thought that the degree of polarization was the degree to which the light could be suppressed by an analyzer with the appropriate orientation. Light that had been converted from linear to elliptical or circular polarization (e.g., by passage through a crystal lamina, or by total internal reflection) was described as partly or fully "depolarized" because of its behavior in an analyzer. After Fresnel, the defining feature of polarized light was that the perpendicular components of vibration had a fixed ratio of amplitudes and a fixed difference in phase. By that definition, elliptically or circularly polarized light is fully polarized although it cannot be fully suppressed by an analyzer alone. The conceptual gap between the wave theory and selectionism had widened again.	1,161	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	116	98	2,024	8	10	30	58
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.	1,132	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	117	99	2,024	8	10	30	58
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 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.	834	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	118	100	2,024	8	10	30	58
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 precise angle at which a Fresnel rhomb should be cut in order to convert linear polarization to circular polarization. For a refractive index of 1.51, there were two solutions: about 48.6° and 54.6°.	513	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	119	101	2,024	8	10	30	58
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 significant, so that a chromatic-polarization experiment will show a pattern of concentric rings. But most minerals, when observed in this manner, show a more complicated pattern of rings involving two foci and a lemniscate curve, as if they had two optic axes. The two classes of minerals naturally become known as uniaxal and biaxal —or, in later literature, uniaxial and biaxial.	693	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	120	102	2,024	8	10	30	58
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.	620	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	121	103	2,024	8	10	30	58
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 sphere (for the ordinary wave) and an oblate spheroid (for the extraordinary wave) touching each other at opposite points of a common axis—touching at the north and south poles, if we may use a geographic analogy. But according to Malus's corpuscular theory of double refraction, the ray velocity was proportional to the reciprocal of that given by Huygens's theory, in which case the velocity law was of the form	728	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	122	104	2,024	8	10	30	58
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 (containing the ray) if the ray was extraordinary. In Fresnel's model, the direction of vibration was normal to the plane of polarization. Hence, for the sphere (the ordinary wave), the vibration was along the lines of latitude (continuing the geographic analogy); and for the spheroid (the extraordinary wave), the vibration was along the lines of longitude.	677	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	123	105	2,024	8	10	30	58
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 the two axes (Biot's sine law). And for the polarization of the ordinary ray, the plane of the ray and the axis was replaced by the plane bisecting the dihedral angle between the two planes each of which contained the ray and one axis (Biot's dihedral law). Biot's laws meant that a biaxial crystal with axes at a small angle, cleaved in the plane of those axes, behaved nearly like a uniaxial crystal at near-normal incidence; this was fortunate because gypsum, which had been used in chromatic-polarization experiments, is biaxial.	852	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	124	106	2,024	8	10	30	58
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 law; that is, neither ray was the product of spherical secondary waves.	389	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	125	107	2,024	8	10	30	58
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 prolate spheroid in an ellipse whose major and minor semi-axes give the magnitudes of the extraordinary and ordinary ray velocities in the direction normal to the plane, and (said Fresnel) the directions of their respective vibrations. The direction of the optic axis is the normal to the plane for which the ellipse of intersection reduces to a circle. So, for the biaxial case, Fresnel simply replaced the prolate spheroid with a triaxial ellipsoid, which was to be sectioned by a plane in the same way. In general there would be two planes passing through the center of the ellipsoid and cutting it in a circle, and the normals to these planes would give two optic axes. From the geometry, Fresnel deduced Biot's sine law (with the ray velocities replaced by their reciprocals).	1,098	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	126	108	2,024	8	10	30	58
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 an extraordinary ray, is not generally normal to the ray. This error (which is small if, as in most cases, the birefringence is weak) was corrected in an "extract" that Fresnel read to the Académie a week later, on 26 November. Starting with Huygens's spheroid, Fresnel obtained a 4th-degree surface 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 equation to obtain a surface with three unequal principal dimensions; this he subsequently called the "surface of elasticity". But he retained the earlier ellipsoid as an approximation, from which he deduced Biot's dihedral law.	1,043	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	127	109	2,024	8	10	30	58
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 reaction in the same direction, (ii) the reaction was otherwise a linear function of the displacement, and (iii) the radius of the surface in any direction was the square root of the component, in that direction, of the reaction to a unit displacement in that direction. The last assumption recognized the requirement that if a wave was to maintain a fixed direction of propagation and a fixed direction of vibration, the reaction must not be outside the plane of those two directions.	800	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	128	110	2,024	8	10	30	58
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 possible plane wavefronts that could have crossed the origin one unit of time ago, and he listed the mathematical conditions that it must satisfy. But he doubted the feasibility of deriving the surface from those conditions.	538	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	129	111	2,024	8	10	30	58
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 ellipsoid in two curves: a circle and an ellipse. Thus he found that the "wave surface" is described by the 4th-degree equation	443	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	130	112	2,024	8	10	30	58
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	291	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	131	113	2,024	8	10	30	58
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 noted that in a biaxial crystal, unlike a uniaxial crystal, the directions in which there is only one wave-normal velocity are not the same as those in which there is only one ray velocity. Nowadays we refer to the former directions as the optic axes or binormal axes, and the latter as the ray axes or biradial axes (see Birefringence).	653	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	132	114	2,024	8	10	30	58
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).	271	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	133	115	2,024	8	10	30	58
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 Recueils of the Académie des Sciences for 1824; this was not actually printed until late 1827, a few months after his death. In this work, having established the three perpendicular axes on which a displacement produces a parallel reaction, and thence constructed the surface of elasticity, he showed that Biot's dihedral law is exact provided that the binormals are taken as the optic axes, and the wave-normal direction as the direction of propagation.	775	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	134	116	2,024	8	10	30	58
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 ensuing difficulties drove a long competitive effort to find an accurate mechanical model of the aether. Fresnel's own model was not dynamically rigorous; for example, it deduced the reaction to a shear strain by considering the displacement of one particle while all others were fixed, and it assumed that the stiffness determined the wave velocity as in a stretched string, whatever the direction of the wave-normal. But it was enough to enable the wave theory to do what selectionist theory could not: generate testable formulae covering a comprehensive range of optical phenomena, from mechanical assumptions.	933	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	135	117	2,024	8	10	30	58
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.	275	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	136	118	2,024	8	10	30	58
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 their 90° angles pointing in alternating directions. Two half-prisms were added at the ends to make the whole assembly rectangular. The prisms were separated by thin films of turpentine (térébenthine) to suppress internal reflections, allowing a clear line of sight along the row. When the four prisms with similar orientations were compressed in a vise across the line of sight, an object viewed through the assembly produced two images with perpendicular polarizations, with an apparent spacing of 1.5 mm at one metre.	836	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	137	119	2,024	8	10	30	58
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 seem unpolarized when viewed through an analyzer but, when viewed through a Fresnel rhomb, would be polarized at ±45° to the plane of reflection of the rhomb (indicating that they were initially circularly polarized in opposite directions). This would show directly that optical rotation is a form of birefringence. In the memoir of December 1822, in which he introduced the term circular polarization, he reported that he had confirmed this prediction using only one 14°–152°–14° prism and two glass half-prisms. But he obtained a wider separation of the images by replacing the glass half-prism with quartz half-prisms whose rotation was opposite to that of the 14°–152°–14° prism. He added in passing that one could further increase the separation by increasing the number of prisms.	1,105	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	138	120	2,024	8	10	30	58
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.	586	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	139	121	2,024	8	10	30	58
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-Guinness :	327	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	140	122	2,024	8	10	30	58
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 judgement of posterity, declared that he placed these researches above everything that had been communicated to the Académie for a long time.	461	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	141	123	2,024	8	10	30	58
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's theory as a successful generalization of Malus's ray-velocity law, embracing Biot's laws.	411	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	142	124	2,024	8	10	30	58
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 rather than his own—and that the predicted waves, even if they were initially transverse, became more longitudinal as they propagated. In reply Fresnel noted, inter alia, that the equations in which Poisson put so much faith did not even predict viscosity. The implication was clear: given that the behavior of light had not been satisfactorily explained except by transverse waves, it was not the responsibility of the wave-theorists to abandon transverse waves in deference to pre-conceived notions about the aether; rather, it was the responsibility of the aether modelers to produce a model that accommodated transverse waves. According to Robert H. Silliman, Poisson eventually accepted the wave theory shortly before his death in 1840.	1,058	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	143	125	2,024	8	10	30	58
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 have adopted the theory immediately." Fresnel's other prominent French opponent, Biot, appeared to take a neutral position in 1830, and eventually accepted the wave theory—possibly by 1846 and certainly by 1858.	530	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	144	126	2,024	8	10	30	58
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 sympathetic to the wave theory, although not entirely free of selectionist language. It was circulating privately by 1828 and was published in 1830. Meanwhile, Young's translation of Fresnel's De la Lumière was published in installments from 1827 to 1829. George Biddell Airy, the former Lucasian Professor at Cambridge and future Astronomer Royal, unreservedly accepted the wave theory by 1831. In 1834, he famously calculated the diffraction pattern of a circular aperture from the wave theory, thereby explaining the limited angular resolution of a perfect telescope (see Airy disk). By the end of the 1830s, the only prominent British physicist who held out against the wave theory was Brewster, whose objections included the difficulty of explaining photochemical effects and (in his opinion) dispersion.	1,122	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	145	127	2,024	8	10	30	58
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.	237	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	146	128	2,024	8	10	30	58
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 then found to explain or predict others. In the corpuscular system there is "no unexpected success, no happy coincidence, no convergence of principles from remote quarters"; but in the wave system, "all tends to unity and simplicity."	553	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	147	129	2,024	8	10	30	58
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.	244	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	148	130	2,024	8	10	30	58
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 in about half a dozen other locations by 1804. But much of the light was wasted by absorption in the glass.	428	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	149	131	2,024	8	10	30	58
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 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.	609	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	150	132	2,024	8	10	30	58
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 under the Corps des Ponts—Fresnel's employer.	364	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	151	133	2,024	8	10	30	58
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 commissioners, Jacques Charles, recalled Buffon's suggestion, leaving Fresnel embarrassed for having again "broken through an open door". But, whereas Buffon's version was biconvex and in one piece, Fresnel's was plano-convex and made of multiple prisms for easier construction. With an official budget of 500 francs, Fresnel approached three manufacturers. The third, François Soleil, produced the prototype. Finished in March 1820, it had a square lens panel 55 cm on a side, containing 97 polygonal (not annular) prisms—and so impressed the Commission that Fresnel was asked for a full eight-panel version. This model, completed a year later in spite of insufficient funding, had panels 76 cm square. In a public spectacle on the evening of 13 April 1821, it was demonstrated by comparison with the most recent reflectors, which it suddenly rendered obsolete.	1,177	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	152	134	2,024	8	10	30	58
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 refracted the light to a sloping plane mirror, which then reflected it horizontally, 7 degrees ahead of the main beam, increasing the duration of the flash. Below the main panels were 128 small mirrors arranged in four rings, stacked like the slats of a louver or Venetian blind. Each ring, shaped as a frustum of a cone, reflected the light to the horizon, giving a fainter steady light between the flashes. The official test, conducted on the unfinished Arc de Triomphe on 20 August 1822, was witnessed by the commission—and by Louis XVIII and his entourage—from 32 km away. The apparatus was stored at Bordeaux for the winter, and then reassembled at Cordouan Lighthouse under Fresnel's supervision. On 25 July 1823, the world's first lighthouse Fresnel lens was lit. Soon afterwards, Fresnel started coughing up blood.	1,137	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	153	135	2,024	8	10	30	58
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 season, and anxiety about judging others induced him to resign that post in late 1824, to save his energy for his lighthouse work.	446	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	154	136	2,024	8	10	30	58
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 by reflecting (catoptric) rings above and below the refracting (dioptric) parts, the entire apparatus would look like a beehive. The second Fresnel lens to enter service was indeed a fixed lens, of third order, installed at Dunkirk by 1 February 1825. However, due to the difficulty of fabricating large toroidal prisms, this apparatus had a 16-sided polygonal plan.	687	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	155	137	2,024	8	10	30	58
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.	213	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	156	138	2,024	8	10	30	58
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 flash per minute (from a rotating lens with eight panels), and two per minute (sixteen panels).	413	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	157	139	2,024	8	10	30	58
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 result was the lighthouse lens as we now know it. In 1826 he assembled a small model for use on the Canal Saint-Martin, but he did not live to see a full-sized version.	485	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	158	140	2,024	8	10	30	58
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 lenses made in 1842 for the lighthouses at Gravelines and Île Vierge. The first fully catadioptric first-order lens, installed at Ailly in 1852, gave eight rotating beams assisted by eight catadioptric panels at the top (to lengthen the flashes), plus a fixed light from below. 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 Augustin Fresnel's original Carte des Phares.	807	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	159	141	2,024	8	10	30	58
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 Fresnel lenses as condensers in overhead projectors. Still finer steps can be found in low-cost plastic "sheet" magnifiers.	441	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	160	142	2,024	8	10	30	58
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).	764	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	161	143	2,024	8	10	30	58
Meanwhile, in Britain, the wave theory was yet to take hold; Fresnel wrote to Thomas Young in November 1824, saying in part:	124	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	162	144	2,024	8	10	30	58
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 inner approbation which has always been the sweetest reward of my efforts. Doubtless I have often needed the sting of vanity to excite me to pursue my researches in moments of disgust or discouragement; but all the compliments I received from MM. Arago, Laplace, and Biot never gave me as much pleasure as the discovery of a theoretical truth and the confirmation of my calculations by experiment.	718	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	163	145	2,024	8	10	30	58
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 important discoveries in Physical Optics."	360	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	164	146	2,024	8	10	30	58
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 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.	539	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	165	147	2,024	8	10	30	58
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 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.	639	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	166	148	2,024	8	10	30	58
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.	655	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	167	149	2,024	8	10	30	58
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 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.	586	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	168	150	2,024	8	10	30	58
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 ".	220	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	169	151	2,024	8	10	30	58
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 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, and then attracted such interest that it was soon republished in English. 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.	1,203	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	170	152	2,024	8	10	30	58
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."	536	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	171	153	2,024	8	10	30	58
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.	1,067	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	172	154	2,024	8	10	30	58
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 September 1824. The journal failed before Fresnel's contribution could be published. Fresnel tried unsuccessfully to recover the manuscript. The editors of his collected works were unable to find it, and concluded that it was probably lost.	557	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	173	155	2,024	8	10	30	58
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 velocity 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 ) . 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 , of which the excess, namely n 2 − − 1 , was dragged along at velocity v , whence the average velocity of the internal aether was 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.)	942	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	174	156	2,024	8	10	30	58
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 supposed that the different refractive indices of different media were due to different aether densities, not different elasticities.	449	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	175	157	2,024	8	10	30	58
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. 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.	935	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	176	158	2,024	8	10	30	58
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 that the wave theory was at least compatible with dispersion; if the model of dispersion was to be accurate over a wider range of frequencies, it needed to be modified so as to take account of resonances within the medium (see Sellmeier equation).	568	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	177	159	2,024	8	10	30	58
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.	221	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	178	160	2,024	8	10	30	58
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:	242	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	179	161	2,024	8	10	30	58
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.	161	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	180	162	2,024	8	10	30	58
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 profound and the most fruitful that physics has experienced since the time of Newton," commentators of the era between Fresnel and Maxwell made similarly strong statements about Fresnel:	507	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	181	163	2,024	8	10	30	58
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 discovery of universal gravitation.	350	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	182	164	2,024	8	10	30	58
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 give it development and mechanical confirmation; Malus and Brewster, grouping them together, correspond to Tycho Brahe and Kepler, laborious in accumulating observations, inventive and happy in discovering laws of phenomena; and Young and Fresnel combined, make up the Newton of optical science.	612	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	183	165	2,024	8	10	30	58
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 his methods are applicable to multiple types of waves. The second revision, 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. Moreover, the phenomena studied by Fresnel, which included nearly all the optical phenomena known at his time, are still most easily explained in terms of the wave nature of light. So it was that, as late as 1927, the astronomer Eugène Michel Antoniadi declared Fresnel to be "the dominant figure in optics."	1,211	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	184	166	2,024	8	10	30	58
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 speaking, the symmetry group of the object.	363	Automorphism	https://en.wikipedia.org/wiki/Automorphism	185	0	2,024	8	10	30	58
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.)	301	Automorphism	https://en.wikipedia.org/wiki/Automorphism	186	1	2,024	8	10	30	58
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 structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the trivial automorphism	472	Automorphism	https://en.wikipedia.org/wiki/Automorphism	187	2	2,024	8	10	30	58
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.	188	Automorphism	https://en.wikipedia.org/wiki/Automorphism	188	3	2,024	8	10	30	58
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.	137	Automorphism	https://en.wikipedia.org/wiki/Automorphism	189	4	2,024	8	10	30	58
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:	262	Automorphism	https://en.wikipedia.org/wiki/Automorphism	190	5	2,024	8	10	30	58
so that μ μ is a new fifth root of unity, connected with the former fifth root λ λ by relations of perfect reciprocity.	119	Automorphism	https://en.wikipedia.org/wiki/Automorphism	191	6	2,024	8	10	30	58
In some categories—notably groups, rings, and Lie algebras —it is possible to separate automorphisms into two types, called "inner" and "outer" automorphisms.	158	Automorphism	https://en.wikipedia.org/wiki/Automorphism	192	7	2,024	8	10	30	58
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 automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.	409	Automorphism	https://en.wikipedia.org/wiki/Automorphism	193	8	2,024	8	10	30	58
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.	198	Automorphism	https://en.wikipedia.org/wiki/Automorphism	194	9	2,024	8	10	30	58
The same definition holds in any unital ring or algebra where a is any invertible element. For Lie algebras the definition is slightly different.	145	Automorphism	https://en.wikipedia.org/wiki/Automorphism	195	10	2,024	8	10	30	58
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 processes by which these arrangements change. This comprises ions, neutral atoms and, unless otherwise stated, it can be assumed that the term atom includes ions.	476	Atomic_physics	https://en.wikipedia.org/wiki/Atomic_physics	196	0	2,024	8	10	30	58
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 reactions and special properties of atomic nuclei.	366	Atomic_physics	https://en.wikipedia.org/wiki/Atomic_physics	197	1	2,024	8	10	30	58
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.	228	Atomic_physics	https://en.wikipedia.org/wiki/Atomic_physics	198	2	2,024	8	10	30	58
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 is concerned with processes such as ionization and excitation by photons or collisions with atomic particles.	430	Atomic_physics	https://en.wikipedia.org/wiki/Atomic_physics	199	3	2,024	8	10	30	58

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. As early as September 1816, according to his later account, he realized 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, and Ampère "had the same thought" on the phase-inversion rule. But that would raise a new difficulty: as natural light seemed to be un polarized 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.

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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 published in a supplement to the Encyclopædia Britannica in February 1818, in which he added that Malus's law would be explained if polarization consisted in a transverse motion.

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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. But these incomplete theories had not reconciled the nature of polarization with the apparent existence of unpolarized light; that achievement was to be Fresnel's alone.

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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. Two pages later he noted, apparently for the first time in writing, that his phase-inversion rule and the non-interference of orthogonally-polarized beams would be easily explained if the vibrations of fully polarized waves were "perpendicular to the normal to the wave"—that is, purely transverse.

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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 "unpolarized" state of natural or "direct" light is due to rapid and random variations in that orientation, in which case two coherent portions of "unpolarized" light will still interfere because their orientations will be synchronized.

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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 in Arago's Annales for May, June, and July 1821. In the first installment, Fresnel described "direct" (unpolarized) light as "the rapid succession of systems of waves polarized in all directions", and gave what is essentially the modern explanation of chromatic polarization, albeit in terms of the analogy between polarization and the resolution of forces in a plane, mentioning transverse waves only in a footnote. The introduction of transverse waves into the main argument was delayed to the second installment, in which he revealed the suspicion that he and Ampère had harbored since 1816, and the difficulty it raised. He continued:

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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 distribution was presumed to be static. For Fresnel, the state of polarization of a beam concerned the variation of a displacement over time. That displacement might be constrained but was not static, and rays were geometric constructions, not countable objects. The conceptual gap between the wave theory and selectionism had become unbridgeable.

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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. If that assumption was to be widely entertained, its explanatory power would need to be impressive.

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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 refraction at the surface between two isotropic media of different indices, Fresnel decomposed the transverse vibrations into two perpendicular components, now known as the s and p components, which are parallel to the surface and the plane of incidence, respectively; in other words, the s and p components are respectively square and parallel to the plane of incidence. For the s component, Fresnel supposed that the interaction between the two media was analogous to an elastic collision, and obtained a formula for what we now call the reflectivity : the ratio of the reflected intensity to the incident intensity. The predicted reflectivity was non-zero at all angles.

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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 proviso that the direction of vibration in Fresnel's model was perpendicular to the plane of polarization as defined by Malus. (On the ensuing controversy, see Plane of polarization.) The technology of the time did not allow the s and p reflectivities to be measured accurately enough to test Fresnel's formulae at arbitrary angles of incidence. But the formulae could be rewritten in terms of what we now call the reflection coefficient : the signed ratio of the reflected amplitude to the incident amplitude. Then, if the plane of polarization of the incident ray was at 45° to the plane of incidence, the tangent of the corresponding angle for the reflected ray was obtainable from the ratio of the two reflection coefficients, and this angle could be measured. Fresnel had measured it for a range of angles of incidence, for glass and water, and the agreement between the calculated and measured angles was better than 1.5° in all cases.

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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 equations. The reflection coefficients for the s and p polarizations are most succinctly expressed as

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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 appearance of a wide variety of objects under illumination—for example, in computer graphics (see Physically based rendering).

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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°) out of phase, and elliptical polarization for other cases in which the two components have a fixed amplitude ratio and a fixed phase difference. He then explained how optical rotation could be understood as a species of birefringence. Linearly-polarized light could be resolved into two circularly-polarized components rotating in opposite directions. If these components propagated at slightly different speeds, the phase difference between them—and therefore the direction of their linearly-polarized resultant—would vary continuously with distance.

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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 polarization), but not that it could vary in kind. Hence it was thought that the degree of polarization was the degree to which the light could be suppressed by an analyzer with the appropriate orientation. Light that had been converted from linear to elliptical or circular polarization (e.g., by passage through a crystal lamina, or by total internal reflection) was described as partly or fully "depolarized" because of its behavior in an analyzer. After Fresnel, the defining feature of polarized light was that the perpendicular components of vibration had a fixed ratio of amplitudes and a fixed difference in phase. By that definition, elliptically or circularly polarized light is fully polarized although it cannot be fully suppressed by an analyzer alone. The conceptual gap between the wave theory and selectionism had widened again.

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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 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.

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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 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.

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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 precise angle at which a Fresnel rhomb should be cut in order to convert linear polarization to circular polarization. For a refractive index of 1.51, there were two solutions: about 48.6° and 54.6°.

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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 significant, so that a chromatic-polarization experiment will show a pattern of concentric rings. But most minerals, when observed in this manner, show a more complicated pattern of rings involving two foci and a lemniscate curve, as if they had two optic axes. The two classes of minerals naturally become known as uniaxal and biaxal —or, in later literature, uniaxial and biaxial.

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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 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.

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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 sphere (for the ordinary wave) and an oblate spheroid (for the extraordinary wave) touching each other at opposite points of a common axis—touching at the north and south poles, if we may use a geographic analogy. But according to Malus's corpuscular theory of double refraction, the ray velocity was proportional to the reciprocal of that given by Huygens's theory, in which case the velocity law was of the form

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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 (containing the ray) if the ray was extraordinary. In Fresnel's model, the direction of vibration was normal to the plane of polarization. Hence, for the sphere (the ordinary wave), the vibration was along the lines of latitude (continuing the geographic analogy); and for the spheroid (the extraordinary wave), the vibration was along the lines of longitude.

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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 the two axes (Biot's sine law). And for the polarization of the ordinary ray, the plane of the ray and the axis was replaced by the plane bisecting the dihedral angle between the two planes each of which contained the ray and one axis (Biot's dihedral law). Biot's laws meant that a biaxial crystal with axes at a small angle, cleaved in the plane of those axes, behaved nearly like a uniaxial crystal at near-normal incidence; this was fortunate because gypsum, which had been used in chromatic-polarization experiments, is biaxial.

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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 law; that is, neither ray was the product of spherical secondary waves.

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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 prolate spheroid in an ellipse whose major and minor semi-axes give the magnitudes of the extraordinary and ordinary ray velocities in the direction normal to the plane, and (said Fresnel) the directions of their respective vibrations. The direction of the optic axis is the normal to the plane for which the ellipse of intersection reduces to a circle. So, for the biaxial case, Fresnel simply replaced the prolate spheroid with a triaxial ellipsoid, which was to be sectioned by a plane in the same way. In general there would be two planes passing through the center of the ellipsoid and cutting it in a circle, and the normals to these planes would give two optic axes. From the geometry, Fresnel deduced Biot's sine law (with the ray velocities replaced by their reciprocals).

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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 an extraordinary ray, is not generally normal to the ray. This error (which is small if, as in most cases, the birefringence is weak) was corrected in an "extract" that Fresnel read to the Académie a week later, on 26 November. Starting with Huygens's spheroid, Fresnel obtained a 4th-degree surface 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 equation to obtain a surface with three unequal principal dimensions; this he subsequently called the "surface of elasticity". But he retained the earlier ellipsoid as an approximation, from which he deduced Biot's dihedral law.

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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 reaction in the same direction, (ii) the reaction was otherwise a linear function of the displacement, and (iii) the radius of the surface in any direction was the square root of the component, in that direction, of the reaction to a unit displacement in that direction. The last assumption recognized the requirement that if a wave was to maintain a fixed direction of propagation and a fixed direction of vibration, the reaction must not be outside the plane of those two directions.

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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 possible plane wavefronts that could have crossed the origin one unit of time ago, and he listed the mathematical conditions that it must satisfy. But he doubted the feasibility of deriving the surface from those conditions.

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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 ellipsoid in two curves: a circle and an ellipse. Thus he found that the "wave surface" is described by the 4th-degree equation

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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 noted that in a biaxial crystal, unlike a uniaxial crystal, the directions in which there is only one wave-normal velocity are not the same as those in which there is only one ray velocity. Nowadays we refer to the former directions as the optic axes or binormal axes, and the latter as the ray axes or biradial axes (see Birefringence).

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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 Recueils of the Académie des Sciences for 1824; this was not actually printed until late 1827, a few months after his death. In this work, having established the three perpendicular axes on which a displacement produces a parallel reaction, and thence constructed the surface of elasticity, he showed that Biot's dihedral law is exact provided that the binormals are taken as the optic axes, and the wave-normal direction as the direction of propagation.

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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 ensuing difficulties drove a long competitive effort to find an accurate mechanical model of the aether. Fresnel's own model was not dynamically rigorous; for example, it deduced the reaction to a shear strain by considering the displacement of one particle while all others were fixed, and it assumed that the stiffness determined the wave velocity as in a stretched string, whatever the direction of the wave-normal. But it was enough to enable the wave theory to do what selectionist theory could not: generate testable formulae covering a comprehensive range of optical phenomena, from mechanical assumptions.

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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 their 90° angles pointing in alternating directions. Two half-prisms were added at the ends to make the whole assembly rectangular. The prisms were separated by thin films of turpentine (térébenthine) to suppress internal reflections, allowing a clear line of sight along the row. When the four prisms with similar orientations were compressed in a vise across the line of sight, an object viewed through the assembly produced two images with perpendicular polarizations, with an apparent spacing of 1.5 mm at one metre.

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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 seem unpolarized when viewed through an analyzer but, when viewed through a Fresnel rhomb, would be polarized at ±45° to the plane of reflection of the rhomb (indicating that they were initially circularly polarized in opposite directions). This would show directly that optical rotation is a form of birefringence. In the memoir of December 1822, in which he introduced the term circular polarization, he reported that he had confirmed this prediction using only one 14°–152°–14° prism and two glass half-prisms. But he obtained a wider separation of the images by replacing the glass half-prism with quartz half-prisms whose rotation was opposite to that of the 14°–152°–14° prism. He added in passing that one could further increase the separation by increasing the number of prisms.

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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, 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.

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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-Guinness :

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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 judgement of posterity, declared that he placed these researches above everything that had been communicated to the Académie for a long time.

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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's theory as a successful generalization of Malus's ray-velocity law, embracing Biot's laws.

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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 rather than his own—and that the predicted waves, even if they were initially transverse, became more longitudinal as they propagated. In reply Fresnel noted, inter alia, that the equations in which Poisson put so much faith did not even predict viscosity. The implication was clear: given that the behavior of light had not been satisfactorily explained except by transverse waves, it was not the responsibility of the wave-theorists to abandon transverse waves in deference to pre-conceived notions about the aether; rather, it was the responsibility of the aether modelers to produce a model that accommodated transverse waves. According to Robert H. Silliman, Poisson eventually accepted the wave theory shortly before his death in 1840.

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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 have adopted the theory immediately." Fresnel's other prominent French opponent, Biot, appeared to take a neutral position in 1830, and eventually accepted the wave theory—possibly by 1846 and certainly by 1858.

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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 sympathetic to the wave theory, although not entirely free of selectionist language. It was circulating privately by 1828 and was published in 1830. Meanwhile, Young's translation of Fresnel's De la Lumière was published in installments from 1827 to 1829. George Biddell Airy, the former Lucasian Professor at Cambridge and future Astronomer Royal, unreservedly accepted the wave theory by 1831. In 1834, he famously calculated the diffraction pattern of a circular aperture from the wave theory, thereby explaining the limited angular resolution of a perfect telescope (see Airy disk). By the end of the 1830s, the only prominent British physicist who held out against the wave theory was Brewster, whose objections included the difficulty of explaining photochemical effects and (in his opinion) dispersion.

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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 then found to explain or predict others. In the corpuscular system there is "no unexpected success, no happy coincidence, no convergence of principles from remote quarters"; but in the wave system, "all tends to unity and simplicity."

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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 in about half a dozen other locations by 1804. But much of the light was wasted by absorption in the glass.

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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 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.

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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 under the Corps des Ponts—Fresnel's employer.

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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 commissioners, Jacques Charles, recalled Buffon's suggestion, leaving Fresnel embarrassed for having again "broken through an open door". But, whereas Buffon's version was biconvex and in one piece, Fresnel's was plano-convex and made of multiple prisms for easier construction. With an official budget of 500 francs, Fresnel approached three manufacturers. The third, François Soleil, produced the prototype. Finished in March 1820, it had a square lens panel 55 cm on a side, containing 97 polygonal (not annular) prisms—and so impressed the Commission that Fresnel was asked for a full eight-panel version. This model, completed a year later in spite of insufficient funding, had panels 76 cm square. In a public spectacle on the evening of 13 April 1821, it was demonstrated by comparison with the most recent reflectors, which it suddenly rendered obsolete.

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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 refracted the light to a sloping plane mirror, which then reflected it horizontally, 7 degrees ahead of the main beam, increasing the duration of the flash. Below the main panels were 128 small mirrors arranged in four rings, stacked like the slats of a louver or Venetian blind. Each ring, shaped as a frustum of a cone, reflected the light to the horizon, giving a fainter steady light between the flashes. The official test, conducted on the unfinished Arc de Triomphe on 20 August 1822, was witnessed by the commission—and by Louis XVIII and his entourage—from 32 km away. The apparatus was stored at Bordeaux for the winter, and then reassembled at Cordouan Lighthouse under Fresnel's supervision. On 25 July 1823, the world's first lighthouse Fresnel lens was lit. Soon afterwards, Fresnel started coughing up blood.

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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 season, and anxiety about judging others induced him to resign that post in late 1824, to save his energy for his lighthouse work.

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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 by reflecting (catoptric) rings above and below the refracting (dioptric) parts, the entire apparatus would look like a beehive. The second Fresnel lens to enter service was indeed a fixed lens, of third order, installed at Dunkirk by 1 February 1825. However, due to the difficulty of fabricating large toroidal prisms, this apparatus had a 16-sided polygonal plan.

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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 flash per minute (from a rotating lens with eight panels), and two per minute (sixteen panels).

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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 result was the lighthouse lens as we now know it. In 1826 he assembled a small model for use on the Canal Saint-Martin, but he did not live to see a full-sized version.

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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 lenses made in 1842 for the lighthouses at Gravelines and Île Vierge. The first fully catadioptric first-order lens, installed at Ailly in 1852, gave eight rotating beams assisted by eight catadioptric panels at the top (to lengthen the flashes), plus a fixed light from below. 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 Augustin Fresnel's original Carte des Phares.

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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 Fresnel lenses as condensers in overhead projectors. Still finer steps can be found in low-cost plastic "sheet" magnifiers.

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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 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).

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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 inner approbation which has always been the sweetest reward of my efforts. Doubtless I have often needed the sting of vanity to excite me to pursue my researches in moments of disgust or discouragement; but all the compliments I received from MM. Arago, Laplace, and Biot never gave me as much pleasure as the discovery of a theoretical truth and the confirmation of my calculations by experiment.

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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 important discoveries in Physical Optics."

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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 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.

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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 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.

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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 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.

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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 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.

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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 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, and then attracted such interest that it was soon republished in English. 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.

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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, 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."

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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 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.

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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 September 1824. The journal failed before Fresnel's contribution could be published. Fresnel tried unsuccessfully to recover the manuscript. The editors of his collected works were unable to find it, and concluded that it was probably lost.

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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 velocity 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 ) . 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 , of which the excess, namely n 2 − − 1 , was dragged along at velocity v , whence the average velocity of the internal aether was 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.)

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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 supposed that the different refractive indices of different media were due to different aether densities, not different elasticities.

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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 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. 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.

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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 that the wave theory was at least compatible with dispersion; if the model of dispersion was to be accurate over a wider range of frequencies, it needed to be modified so as to take account of resonances within the medium (see Sellmeier equation).

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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 profound and the most fruitful that physics has experienced since the time of Newton," commentators of the era between Fresnel and Maxwell made similarly strong statements about Fresnel:

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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 discovery of universal gravitation.

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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 give it development and mechanical confirmation; Malus and Brewster, grouping them together, correspond to Tycho Brahe and Kepler, laborious in accumulating observations, inventive and happy in discovering laws of phenomena; and Young and Fresnel combined, make up the Newton of optical science.

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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 his methods are applicable to multiple types of waves. The second revision, 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. Moreover, the phenomena studied by Fresnel, which included nearly all the optical phenomena known at his time, are still most easily explained in terms of the wave nature of light. So it was that, as late as 1927, the astronomer Eugène Michel Antoniadi declared Fresnel to be "the dominant figure in optics."

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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 speaking, the symmetry group of the object.

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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 structures, the two definitions are equivalent; in this case, the identity morphism is simply the identity function, and is often called the trivial automorphism

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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 automorphisms form a normal subgroup of Aut(G), denoted by Inn(G); this is called Goursat's lemma.

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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 processes by which these arrangements change. This comprises ions, neutral atoms and, unless otherwise stated, it can be assumed that the term atom includes ions.

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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 reactions and special properties of atomic nuclei.

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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 is concerned with processes such as ionization and excitation by photons or collisions with atomic particles.

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