quality:correction, structural:expansion
Historical context: From Newton to Biot

The appreciation of Fresnel's reconstruction of physical optics might be assisted by an overview of the fragmented state in which he found the subject. In this subsection, optical phenomena that were unexplained or whose explanations were disputed are named in bold type.

The corpuscular theory of light, favored by Isaac Newton and accepted by nearly all of Fresnel's seniors, easily explained rectilinear propagation: the corpuscles obviously moved very fast, so that their paths were very nearly straight. The wave theory, as developed by Christiaan Huygens in his Treatise on Light (1690), explained rectilinear propagation on the assumption that each point crossed by a traveling wavefront becomes the source of a secondary wavefront. Given the initial position of a traveling wavefront, any later position (according to Huygens) was the common tangent surface (envelope) of the secondary wavefronts emitted from the earlier position. As the extent of the common tangent was limited by the extent of the initial wavefront, the repeated application of Huygens's construction to a plane wavefront of limited extent (in a uniform medium) gave a straight, parallel beam. While this construction indeed predicted rectilinear propagation, it was difficult to reconcile with the common observation that wavefronts on the surface of water can bend around obstructions, and with the similar behavior of sound waves—causing Newton to maintain, to the end of his life, that if light consisted of waves it would "bend and spread every way" into the shadows.

Huygens's theory neatly explained the law of ordinary reflection and the law of ordinary refraction ("Snell's law"), provided that the secondary waves traveled slower in denser media (those of higher refractive index). The corpuscular theory, with the hypothesis that the corpuscles were subject to forces acting perpendicular to surfaces, explained the same laws equally well, albeit with the implication that light traveled faster in denser media; that implication was wrong, but could not be directly disproven with the technology of Newton's time or even Fresnel's time.

Similarly inconclusive was stellar aberration —that is, the apparent change in the position of a star due to the velocity of the earth across the line of sight (not to be confused with stellar parallax, which is due to the displacement of the earth across the line of sight). Identified by James Bradley in 1728, stellar aberration was widely taken as confirmation of the corpuscular theory. But it was equally compatible with the wave theory, as Euler noted in 1746—tacitly assuming that the aether (the supposed wave-bearing medium) near the earth was not disturbed by the motion of the earth.

The outstanding strength of Huygens's theory was his explanation of the birefringence (double refraction) of " Iceland crystal " (transparent calcite), on the assumption that the secondary waves are spherical for the ordinary refraction (which satisfies Snell's law) and spheroidal for the extraordinary refraction (which does not). In general, Huygens's common-tangent construction implies that rays are paths of least time between successive positions of the wavefront, in accordance with Fermat's principle. In the special case of isotropic media, the secondary wavefronts must be spherical, and Huygens's construction then implies that the rays are perpendicular to the wavefront; indeed, the law of ordinary refraction can be separately derived from that premise, as Ignace-Gaston Pardies did before Huygens.

Although Newton rejected the wave theory, he noticed its potential to explain colors, including the colors of " thin plates " (e.g., " Newton's rings ", and the colors of skylight reflected in soap bubbles), on the assumption that light consists of periodic waves, with the lowest frequencies (longest wavelengths) at the red end of the spectrum, and the highest frequencies (shortest wavelengths) at the violet end. In 1672 he published a heavy hint to that effect, but contemporary supporters of the wave theory failed to act on it: Robert Hooke treated light as a periodic sequence of pulses but did not use frequency as the criterion of color, while Huygens treated the waves as individual pulses without any periodicity; and Pardies died young in 1673. Newton himself tried to explain colors of thin plates using the corpuscular theory, by supposing that his corpuscles had the wavelike property of alternating between "fits of easy transmission" and "fits of easy reflection", the distance between like "fits" depending on the color and the medium and, awkwardly, on the angle of refraction or reflection into that medium. More awkwardly still, this theory required thin plates to reflect only at the back surface, although thick plates manifestly reflected also at the front surface. It was not until 1801 that Thomas Young, in the Bakerian Lecture for that year, cited Newton's hint, and accounted for the colors of a thin plate as the combined effect of the front and back reflections, which reinforce or cancel each other according to the wavelength and the thickness. Young similarly explained the colors of "striated surfaces" (e.g., gratings) as the wavelength-dependent reinforcement or cancellation of reflections from adjacent lines. He described this reinforcement or cancellation as interference.

Neither Newton nor Huygens satisfactorily explained diffraction —the blurring and fringing of shadows where, according to rectilinear propagation, they ought to be sharp. Newton, who called diffraction "inflexion", supposed that rays of light passing close to obstacles were bent ("inflected"); but his explanation was only qualitative. Huygens's common-tangent construction, without modifications, could not accommodate diffraction at all. Two such modifications were proposed by Young in the same 1801 Bakerian Lecture: first, that the secondary waves near the edge of an obstacle could diverge into the shadow, but only weakly, due to limited reinforcement from other secondary waves; and second, that diffraction by an edge was caused by interference between two rays: one reflected off the edge, and the other inflected while passing near the edge. The latter ray would be undeviated if sufficiently far from the edge, but Young did not elaborate on that case. These were the earliest suggestions that the degree of diffraction depends on wavelength. Later, in the 1803 Bakerian Lecture, Young ceased to regard inflection as a separate phenomenon, and produced evidence that diffraction fringes inside the shadow of a narrow obstacle were due to interference: when the light from one side was blocked, the internal fringes disappeared. But Young was alone in such efforts until Fresnel entered the field.

Huygens, in his investigation of double refraction, noticed something that he could not explain: when light passes through two similarly oriented calcite crystals at normal incidence, the ordinary ray emerging from the first crystal suffers only the ordinary refraction in the second, while the extraordinary ray emerging from the first suffers only the extraordinary refraction in the second; but when the second crystal is rotated 90° about the incident rays, the roles are interchanged, so that the ordinary ray emerging from the first crystal suffers only the extraordinary refraction in the second, and vice versa. This discovery gave Newton another reason to reject the wave theory: rays of light evidently had "sides". Corpuscles could have sides (or poles, as they would later be called); but waves of light could not, because (so it seemed) any such waves would need to be longitudinal (with vibrations in the direction of propagation). Newton offered an alternative "Rule" for the extraordinary refraction, which rode on his authority through the 18th century, although he made "no known attempt to deduce it from any principles of optics, corpuscular or otherwise."

In 1808, the extraordinary refraction of calcite was investigated experimentally, with unprecedented accuracy, by Étienne-Louis Malus, and found to be consistent with Huygens's spheroid construction, not Newton's "Rule". Malus, encouraged by Pierre-Simon Laplace, then sought to explain this law in corpuscular terms: from the known relation between the incident and refracted ray directions, Malus derived the corpuscular velocity (as a function of direction) that would satisfy Maupertuis 's "least action" principle. But, as Young pointed out, the existence of such a velocity law was guaranteed by Huygens's spheroid, because Huygens's construction leads to Fermat's principle, which becomes Maupertuis's principle if the ray speed is replaced by the reciprocal of the particle speed! The corpuscularists had not found a force law that would yield the alleged velocity law, except by a circular argument in which a force acting at the surface of the crystal inexplicably depended on the direction of the (possibly subsequent) velocity within the crystal. Worse, it was doubtful that any such force would satisfy the conditions of Maupertuis's principle. In contrast, Young proceeded to show that "a medium more easily compressible in one direction than in any direction perpendicular to it, as if it consisted of an infinite number of parallel plates connected by a substance somewhat less elastic" admits spheroidal longitudinal wavefronts, as Huygens supposed.

But Malus, in the midst of his experiments on double refraction, noticed something else: when a ray of light is reflected off a non-metallic surface at the appropriate angle, it behaves like one of the two rays emerging from a calcite crystal. It was Malus who coined the term polarization to describe this behavior, although the polarizing angle became known as Brewster's angle after its dependence on the refractive index was determined experimentally by David Brewster in 1815. Malus also introduced the term plane of polarization. In the case of polarization by reflection, his "plane of polarization" was the plane of the incident and reflected rays; in modern terms, this is the plane normal to the electric vibration. In 1809, Malus further discovered that the intensity of light passing through two polarizers is proportional to the squared cosine of the angle between their planes of polarization (Malus's law), whether the polarizers work by reflection or double refraction, and that all birefringent crystals produce both extraordinary refraction and polarization. As the corpuscularists started trying to explain these things in terms of polar "molecules" of light, the wave-theorists had no working hypothesis on the nature of polarization, prompting Young to remark that Malus's observations "present greater difficulties to the advocates of the undulatory theory than any other facts with which we are acquainted."

Malus died in February 1812, at the age of 36, shortly after receiving the Rumford Medal for his work on polarization.

In August 1811, François Arago reported that if a thin plate of mica was viewed against a white polarized backlight through a calcite crystal, the two images of the mica were of complementary colors (the overlap having the same color as the background). The light emerging from the mica was " de polarized" in the sense that there was no orientation of the calcite that made one image disappear; yet it was not ordinary (" un polarized") light, for which the two images would be of the same color. Rotating the calcite around the line of sight changed the colors, though they remained complementary. Rotating the mica changed the saturation (not the hue) of the colors. This phenomenon became known as chromatic polarization. Replacing the mica with a much thicker plate of quartz, with its faces perpendicular to the optic axis (the axis of Huygens's spheroid or Malus's velocity function), produced a similar effect, except that rotating the quartz made no difference. Arago tried to explain his observations in corpuscular terms.

In 1812, as Arago pursued further qualitative experiments and other commitments, Jean-Baptiste Biot reworked the same ground using a gypsum lamina in place of the mica, and found empirical formulae for the intensities of the ordinary and extraordinary images. The formulae contained two coefficients, supposedly representing colors of rays "affected" and "unaffected" by the plate—the "affected" rays being of the same color mix as those reflected by amorphous thin plates of proportional, but lesser, thickness.

Arago protested, declaring that he had made some of the same discoveries but had not had time to write them up. In fact the overlap between Arago's work and Biot's was minimal, Arago's being only qualitative and wider in scope (attempting to include polarization by reflection). But the dispute triggered a notorious falling-out between the two men.

Later that year, Biot tried to explain the observations as an oscillation of the alignment of the "affected" corpuscles at a frequency proportional to that of Newton's "fits", due to forces depending on the alignment. This theory became known as mobile polarization. To reconcile his results with a sinusoidal oscillation, Biot had to suppose that the corpuscles emerged with one of two permitted orientations, namely the extremes of the oscillation, with probabilities depending on the phase of the oscillation. Corpuscular optics was becoming expensive on assumptions. But in 1813, Biot reported that the case of quartz was simpler: the observable phenomenon (now called optical rotation or optical activity or sometimes rotary polarization) was a gradual rotation of the polarization direction with distance, and could be explained by a corresponding rotation (not oscillation) of the corpuscles.

Early in 1814, reviewing Biot's work on chromatic polarization, Young noted that the periodicity of the color as a function of the plate thickness—including the factor by which the period exceeded that for a reflective thin plate, and even the effect of obliquity of the plate (but not the role of polarization)—could be explained by the wave theory in terms of the different propagation times of the ordinary and extraordinary waves through the plate. But Young was then the only public defender of the wave theory.

In summary, in the spring of 1814, as Fresnel tried in vain to guess what polarization was, the corpuscularists thought that they knew, while the wave-theorists (if we may use the plural) literally had no idea. Both theories claimed to explain rectilinear propagation, but the wave explanation was overwhelmingly regarded as unconvincing. The corpuscular theory could not rigorously link double refraction to surface forces; the wave theory could not yet link it to polarization. The corpuscular theory was weak on thin plates and silent on gratings; the wave theory was strong on both, but under-appreciated. Concerning diffraction, the corpuscular theory did not yield quantitative predictions, while the wave theory had begun to do so by considering diffraction as a manifestation of interference, but had only considered two rays at a time. Only the corpuscular theory gave even a vague insight into Brewster's angle, Malus's law, or optical rotation. Concerning chromatic polarization, the wave theory explained the periodicity far better than the corpuscular theory, but had nothing to say about the role of polarization; and its explanation of the periodicity was largely ignored. And Arago had founded the study of chromatic polarization, only to lose the lead, controversially, to Biot. Such were the circumstances in which Arago first heard of Fresnel's interest in optics.