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

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The atomic number or nuclear charge number (symbol Z) of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (n p) or the number of protons found in the nucleus of every atom of that element. The atomic number can be used to uniquely identify ordinary chemical elements. In an ordinary uncharged atom, the atomic number is also equal to the number of electrons.	460	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	0	0	2,024	8	10	30	57
For an ordinary atom which contains protons, neutrons and electrons, the sum of the atomic number Z and the neutron number N gives the atom's atomic mass number A. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes) and the mass defect of the nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in daltons (making a quantity called the " relative isotopic mass "), is within 1% of the whole number A.	526	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	1	1	2,024	8	10	30	57
Atoms with the same atomic number but different neutron numbers, and hence different mass numbers, are known as isotopes. A little more than three-quarters of naturally occurring elements exist as a mixture of isotopes (see monoisotopic elements), and the average isotopic mass of an isotopic mixture for an element (called the relative atomic mass) in a defined environment on Earth determines the element's standard atomic weight. Historically, it was these atomic weights of elements (in comparison to hydrogen) that were the quantities measurable by chemists in the 19th century.	583	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	2	2	2,024	8	10	30	57
The conventional symbol Z comes from the German word Z ahl 'number', which, before the modern synthesis of ideas from chemistry and physics, merely denoted an element's numerical place in the periodic table, whose order was then approximately, but not completely, consistent with the order of the elements by atomic weights. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge and a physical characteristic of atoms, did the word Atom z ahl (and its English equivalent atomic number) come into common use in this context.	566	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	3	3	2,024	8	10	30	57
The rules above do not always apply to exotic atoms which contain short-lived elementary particles other than protons, neutrons and electrons.	142	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	4	4	2,024	8	10	30	57
Loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order.	154	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	5	5	2,024	8	10	30	57
Dmitri Mendeleev said that he arranged his first periodic tables (first published on March 6, 1869) in order of atomic weight ("Atomgewicht"). However, in consideration of the elements' observed chemical properties, he changed the order slightly and placed tellurium (atomic weight 127.6) ahead of iodine (atomic weight 126.9). This placement is consistent with the modern practice of ordering the elements by proton number, Z, but that number was not known or suspected at the time.	483	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	6	6	2,024	8	10	30	57
A simple numbering based on periodic table position was never entirely satisfactory. In addition to the case of iodine and tellurium, several other pairs of elements (such as argon and potassium, cobalt and nickel) were later shown to have nearly identical or reversed atomic weights, thus requiring their placement in the periodic table to be determined by their chemical properties. However the gradual identification of more and more chemically similar lanthanide elements, whose atomic number was not obvious, led to inconsistency and uncertainty in the periodic numbering of elements at least from lutetium (element 71) onward (hafnium was not known at this time).	669	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	7	7	2,024	8	10	30	57
In 1911, Ernest Rutherford gave a model of the atom in which a central nucleus held most of the atom's mass and a positive charge which, in units of the electron's charge, was to be approximately equal to half of the atom's atomic weight, expressed in numbers of hydrogen atoms. This central charge would thus be approximately half the atomic weight (though it was almost 25% different from the atomic number of gold (Z = 79, A = 197), the single element from which Rutherford made his guess). Nevertheless, in spite of Rutherford's estimation that gold had a central charge of about 100 (but was element Z = 79 on the periodic table), a month after Rutherford's paper appeared, Antonius van den Broek first formally suggested that the central charge and number of electrons in an atom were exactly equal to its place in the periodic table (also known as element number, atomic number, and symbolized Z). This eventually proved to be the case.	943	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	8	8	2,024	8	10	30	57
The experimental position improved dramatically after research by Henry Moseley in 1913. Moseley, after discussions with Bohr who was at the same lab (and who had used Van den Broek's hypothesis in his Bohr model of the atom), decided to test Van den Broek's and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fitted the Bohr theory's postulation that the frequency of the spectral lines be proportional to the square of Z.	459	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	9	9	2,024	8	10	30	57
To do this, Moseley measured the wavelengths of the innermost photon transitions (K and L lines) produced by the elements from aluminium (Z = 13) to gold (Z = 79) used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons (x-rays) increased from one target to the next in an arithmetic progression. This led to the conclusion (Moseley's law) that the atomic number does closely correspond (with an offset of one unit for K-lines, in Moseley's work) to the calculated electric charge of the nucleus, i.e. the element number Z. Among other things, Moseley demonstrated that the lanthanide series (from lanthanum to lutetium inclusive) must have 15 members—no fewer and no more—which was far from obvious from known chemistry at that time.	792	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	10	10	2,024	8	10	30	57
After Moseley's death in 1915, the atomic numbers of all known elements from hydrogen to uranium (Z = 92) were examined by his method. There were seven elements (with Z < 92) which were not found and therefore identified as still undiscovered, corresponding to atomic numbers 43, 61, 72, 75, 85, 87 and 91. From 1918 to 1947, all seven of these missing elements were discovered. By this time, the first four transuranium elements had also been discovered, so that the periodic table was complete with no gaps as far as curium (Z = 96).	535	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	11	11	2,024	8	10	30	57
In 1915, the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood. An old idea called Prout's hypothesis had postulated that the elements were all made of residues (or "protyles") of the lightest element hydrogen, which in the Bohr-Rutherford model had a single electron and a nuclear charge of one. However, as early as 1907, Rutherford and Thomas Royds had shown that alpha particles, which had a charge of +2, were the nuclei of helium atoms, which had a mass four times that of hydrogen, not two times. If Prout's hypothesis were true, something had to be neutralizing some of the charge of the hydrogen nuclei present in the nuclei of heavier atoms.	741	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	12	12	2,024	8	10	30	57
In 1917, Rutherford succeeded in generating hydrogen nuclei from a nuclear reaction between alpha particles and nitrogen gas, and believed he had proven Prout's law. He called the new heavy nuclear particles protons in 1920 (alternate names being proutons and protyles). It had been immediately apparent from the work of Moseley that the nuclei of heavy atoms have more than twice as much mass as would be expected from their being made of hydrogen nuclei, and thus there was required a hypothesis for the neutralization of the extra protons presumed present in all heavy nuclei. A helium nucleus was presumed to be composed of four protons plus two "nuclear electrons" (electrons bound inside the nucleus) to cancel two of the charges. At the other end of the periodic table, a nucleus of gold with a mass 197 times that of hydrogen was thought to contain 118 nuclear electrons in the nucleus to give it a residual charge of +79, consistent with its atomic number.	965	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	13	13	2,024	8	10	30	57
All consideration of nuclear electrons ended with James Chadwick 's discovery of the neutron in 1932. An atom of gold now was seen as containing 118 neutrons rather than 118 nuclear electrons, and its positive nuclear charge now was realized to come entirely from a content of 79 protons. Since Moseley had previously shown that the atomic number Z of an element equals this positive charge, it was now clear that Z is identical to the number of protons of its nuclei.	468	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	14	14	2,024	8	10	30	57
Each element has a specific set of chemical properties as a consequence of the number of electrons present in the neutral atom, which is Z (the atomic number). The configuration of these electrons follows from the principles of quantum mechanics. The number of electrons in each element's electron shells, particularly the outermost valence shell, is the primary factor in determining its chemical bonding behavior. Hence, it is the atomic number alone that determines the chemical properties of an element; and it is for this reason that an element can be defined as consisting of any mixture of atoms with a given atomic number.	630	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	15	15	2,024	8	10	30	57
The quest for new elements is usually described using atomic numbers. As of 2024, all elements with atomic numbers 1 to 118 have been observed. Synthesis of new elements is accomplished by bombarding target atoms of heavy elements with ions, such that the sum of the atomic numbers of the target and ion elements equals the atomic number of the element being created. In general, the half-life of a nuclide becomes shorter as atomic number increases, though undiscovered nuclides with certain " magic " numbers of protons and neutrons may have relatively longer half-lives and comprise an island of stability.	609	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	16	16	2,024	8	10	30	57
A hypothetical element composed only of neutrons has also been proposed and would have atomic number 0, but has never been observed.	132	Atomic_number	https://en.wikipedia.org/wiki/Atomic_number	17	17	2,024	8	10	30	57
Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton 's corpuscular theory, from the late 1830s until the end of the 19th century. He is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric (purely refractive) stepped lens, first proposed by Count Buffon and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors.	714	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	18	0	2,024	8	10	30	58
By expressing Huygens 's principle of secondary waves and Young 's principle of interference in quantitative terms, and supposing that simple colors consist of sinusoidal waves, Fresnel gave the first satisfactory explanation of diffraction by straight edges, including the first satisfactory wave-based explanation of rectilinear propagation. Part of his argument was a proof that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions. By further supposing that light waves are purely transverse, Fresnel explained the nature of polarization, the mechanism of chromatic polarization, and the transmission and reflection coefficients at the interface between two transparent isotropic media. Then, by generalizing the direction-speed-polarization relation for calcite, he accounted for the directions and polarizations of the refracted rays in doubly-refractive crystals of the biaxial class (those for which Huygens's secondary wavefronts are not axisymmetric). The period between the first publication of his pure-transverse-wave hypothesis, and the submission of his first correct solution to the biaxial problem, was less than a year.	1,229	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	19	1	2,024	8	10	30	58
Later, he coined the terms linear polarization, circular polarization, and elliptical polarization, explained how optical rotation could be understood as a difference in propagation speeds for the two directions of circular polarization, and (by allowing the reflection coefficient to be complex) accounted for the change in polarization due to total internal reflection, as exploited in the Fresnel rhomb. Defenders of the established corpuscular theory could not match his quantitative explanations of so many phenomena on so few assumptions.	544	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	20	2	2,024	8	10	30	58
Fresnel had a lifelong battle with tuberculosis, to which he succumbed at the age of 39. Although he did not become a public celebrity in his lifetime, he lived just long enough to receive due recognition from his peers, including (on his deathbed) the Rumford Medal of the Royal Society of London, and his name is ubiquitous in the modern terminology of optics and waves. After the wave theory of light was subsumed by Maxwell 's electromagnetic theory in the 1860s, some attention was diverted from the magnitude of Fresnel's contribution. In the period between Fresnel's unification of physical optics and Maxwell's wider unification, a contemporary authority, Humphrey Lloyd, described Fresnel's transverse-wave theory as "the noblest fabric which has ever adorned the domain of physical science, Newton's system of the universe alone excepted."	849	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	21	3	2,024	8	10	30	58
Augustin-Jean Fresnel (also called Augustin Jean or simply Augustin), born in Broglie, Normandy, on 10 May 1788, was the second of four sons of the architect Jacques Fresnel and his wife Augustine, née Mérimée. The family moved twice—in 1789/90 to Cherbourg, and in 1794 to Jacques's home town of Mathieu, where Augustine would spend 25 years as a widow, outliving two of her sons.	381	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	22	4	2,024	8	10	30	58
The first son, Louis, was admitted to the École Polytechnique, became a lieutenant in the artillery, and was killed in action at Jaca, Spain. The third, Léonor, followed Augustin into civil engineering, succeeded him as secretary of the Lighthouse Commission, and helped to edit his collected works. The fourth, Fulgence Fresnel, became a linguist, diplomat, and orientalist, and occasionally assisted Augustin with negotiations. Fulgence died in Bagdad in 1855 having led a mission to explore Babylon.	502	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	23	5	2,024	8	10	30	58
Madame Fresnel's younger brother, Jean François "Léonor" Mérimée, father of the writer Prosper Mérimée, was a painter who turned his attention to the chemistry of painting. He became the Permanent Secretary of the École des Beaux-Arts and (until 1814) a professor at the École Polytechnique, and was the initial point of contact between Augustin and the leading optical physicists of the day (see below).	404	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	24	6	2,024	8	10	30	58
The Fresnel brothers were initially home-schooled by their mother. The sickly Augustin was considered the slow one, not inclined to memorization; but the popular story that he hardly began to read until the age of eight is disputed. At the age of nine or ten he was undistinguished except for his ability to turn tree-branches into toy bows and guns that worked far too well, earning himself the title l'homme de génie (the man of genius) from his accomplices, and a united crackdown from their elders.	502	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	25	7	2,024	8	10	30	58
In 1801, Augustin was sent to the École Centrale at Caen, as company for Louis. But Augustin lifted his performance: in late 1804 he was accepted into the École Polytechnique, being placed 17th in the entrance examination. As the detailed records of the École Polytechnique begin in 1808, we know little of Augustin's time there, except that he made few if any friends and—in spite of continuing poor health—excelled in drawing and geometry: in his first year he took a prize for his solution to a geometry problem posed by Adrien-Marie Legendre. Graduating in 1806, he then enrolled at the École Nationale des Ponts et Chaussées (National School of Bridges and Roads, also known as "ENPC" or "École des Ponts"), from which he graduated in 1809, entering the service of the Corps des Ponts et Chaussées as an ingénieur ordinaire aspirant (ordinary engineer in training). Directly or indirectly, he was to remain in the employment of the "Corps des Ponts" for the rest of his life.	980	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	26	8	2,024	8	10	30	58
Fresnel's parents were Roman Catholics of the Jansenist sect, characterized by an extreme Augustinian view of original sin. Religion took first place in the boys' home-schooling. In 1802, his mother said:	204	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	27	9	2,024	8	10	30	58
I pray God to give my son the grace to employ the great talents, which he has received, for his own benefit, and for the God of all. Much will be asked from him to whom much has been given, and most will be required of him who has received most.	245	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	28	10	2,024	8	10	30	58
Augustin remained a Jansenist. He regarded his intellectual talents as gifts from God, and considered it his duty to use them for the benefit of others. According to his fellow engineer Alphonse Duleau, who helped to nurse him through his final illness, Fresnel saw the study of nature as part of the study of the power and goodness of God. He placed virtue above science and genius. In his last days he prayed for "strength of soul," not against death alone, but against "the interruption of discoveries… of which he hoped to derive useful applications."	555	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	29	11	2,024	8	10	30	58
Jansenism is considered heretical by the Roman Catholic Church, and Grattan-Guinness suggests this is why Fresnel never gained a permanent academic teaching post; his only teaching appointment was at the Athénée in the winter of 1819–20. The article on Fresnel in the Catholic Encyclopedia does not mention his Jansenism, but describes him as "a deeply religious man and remarkable for his keen sense of duty."	410	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	30	12	2,024	8	10	30	58
Fresnel was initially posted to the western département of Vendée. There, in 1811, he anticipated what became known as the Solvay process for producing soda ash, except that recycling of the ammonia was not considered. That difference may explain why leading chemists, who learned of his discovery through his uncle Léonor, eventually thought it uneconomic.	357	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	31	13	2,024	8	10	30	58
About 1812, Fresnel was sent to Nyons, in the southern département of Drôme, to assist with the imperial highway that was to connect Spain and Italy. It is from Nyons that we have the first evidence of his interest in optics. On 15 May 1814, while work was slack due to Napoleon 's defeat, Fresnel wrote a " P.S. " to his brother Léonor, saying in part:	353	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	32	14	2,024	8	10	30	58
I would also like to have papers that might tell me about the discoveries of French physicists on the polarization of light. I saw in the Moniteur of a few months ago that Biot had read to the Institute a very interesting memoir on the polarization of light. Though I break my head, I cannot guess what that is.	311	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	33	15	2,024	8	10	30	58
As late as 28 December he was still waiting for information, but by 10 February 1815 he had received Biot's memoir. (The Institut de France had taken over the functions of the French Académie des Sciences and other académies in 1795. In 1816 the Académie des Sciences regained its name and autonomy, but remained part of the institute.)	336	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	34	16	2,024	8	10	30	58
In March 1815, perceiving Napoleon's return from Elba as "an attack on civilization", Fresnel departed without leave, hastened to Toulouse and offered his services to the royalist resistance, but soon found himself on the sick list. Returning to Nyons in defeat, he was threatened and had his windows broken. During the Hundred Days he was placed on suspension, which he was eventually allowed to spend at his mother's house in Mathieu. There he used his enforced leisure to begin his optical experiments.	505	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	35	17	2,024	8	10	30	58
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.	271	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	36	18	2,024	8	10	30	58
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.	1,279	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	37	19	2,024	8	10	30	58
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 (see Foucault's measurements of the speed of light).	626	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	38	20	2,024	8	10	30	58
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.	595	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	39	21	2,024	8	10	30	58
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.	813	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	40	22	2,024	8	10	30	58
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.	1,814	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	41	23	2,024	8	10	30	58
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.	1,409	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	42	24	2,024	8	10	30	58
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."	1,176	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	43	25	2,024	8	10	30	58
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.	1,466	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	44	26	2,024	8	10	30	58
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."	1,430	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	45	27	2,024	8	10	30	58
Malus died in February 1812, at the age of 36, shortly after receiving the Rumford Medal for his work on polarization.	118	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	46	28	2,024	8	10	30	58
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.	1,032	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	47	29	2,024	8	10	30	58
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.	512	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	48	30	2,024	8	10	30	58
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.	349	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	49	31	2,024	8	10	30	58
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.	899	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	50	32	2,024	8	10	30	58
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.	516	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	51	33	2,024	8	10	30	58
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.	1,383	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	52	34	2,024	8	10	30	58
Fresnel's letters from later in 1814 reveal his interest in the wave theory, including his awareness that it explained the constancy of the speed of light and was at least compatible with stellar aberration. Eventually he compiled what he called his rêveries (musings) into an essay and submitted it via Léonor Mérimée to André-Marie Ampère, who did not respond directly. But on 19 December, Mérimée dined with Ampère and Arago, with whom he was acquainted through the École Polytechnique; and Arago promised to look at Fresnel's essay.	536	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	53	35	2,024	8	10	30	58
In mid 1815, on his way home to Mathieu to serve his suspension, Fresnel met Arago in Paris and spoke of the wave theory and stellar aberration. He was informed that he was trying to break down open doors (" il enfonçait des portes ouvertes "), and directed to classical works on optics.	287	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	54	36	2,024	8	10	30	58
On 12 July 1815, as Fresnel was about to leave Paris, Arago left him a note on a new topic:	91	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	55	37	2,024	8	10	30	58
I do not know of any book that contains all the experiments that physicists are doing on the diffraction of light. M'sieur Fresnel will only be able to get to know this part of the optics by reading the work by Grimaldi, the one by Newton, the English treatise by Jordan, and the memoirs of Brougham and Young, which are part of the collection of the Philosophical Transactions.	378	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	56	38	2,024	8	10	30	58
Fresnel would not have ready access to these works outside Paris, and could not read English. But, in Mathieu—with a point-source of light made by focusing sunlight with a drop of honey, a crude micrometer of his own construction, and supporting apparatus made by a local locksmith—he began his own experiments. His technique was novel: whereas earlier investigators had projected the fringes onto a screen, Fresnel soon abandoned the screen and observed the fringes in space, through a lens with the micrometer at its focus, allowing more accurate measurements while requiring less light.	589	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	57	39	2,024	8	10	30	58
Later in July, after Napoleon's final defeat, Fresnel was reinstated with the advantage of having backed the winning side. He requested a two-month leave of absence, which was readily granted because roadworks were in abeyance.	227	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	58	40	2,024	8	10	30	58
On 23 September he wrote to Arago, beginning "I think I have found the explanation and the law of colored fringes which one notices in the shadows of bodies illuminated by a luminous point." In the same paragraph, however, Fresnel implicitly acknowledged doubt about the novelty of his work: noting that he would need to incur some expense in order to improve his measurements, he wanted to know "whether this is not useless, and whether the law of diffraction has not already been established by sufficiently exact experiments." He explained that he had not yet had a chance to acquire the items on his reading lists, with the apparent exception of "Young's book", which he could not understand without his brother's help. Not surprisingly, he had retraced many of Young's steps.	780	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	59	41	2,024	8	10	30	58
In a memoir sent to the institute on 15 October 1815, Fresnel mapped the external and internal fringes in the shadow of a wire. He noticed, like Young before him, that the internal fringes disappeared when the light from one side was blocked, and concluded that "the vibrations of two rays that cross each other under a very small angle can contradict each other…" But, whereas Young took the disappearance of the internal fringes as confirmation of the principle of interference, Fresnel reported that it was the internal fringes that first drew his attention to the principle. To explain the diffraction pattern, Fresnel constructed the internal fringes by considering the intersections of circular wavefronts emitted from the two edges of the obstruction, and the external fringes by considering the intersections between direct waves and waves reflected off the nearer edge. For the external fringes, to obtain tolerable agreement with observation, he had to suppose that the reflected wave was inverted ; and he noted that the predicted paths of the fringes were hyperbolic. In the part of the memoir that most clearly surpassed Young, Fresnel explained the ordinary laws of reflection and refraction in terms of interference, noting that if two parallel rays were reflected or refracted at other than the prescribed angle, they would no longer have the same phase in a common perpendicular plane, and every vibration would be cancelled by a nearby vibration. He noted that his explanation was valid provided that the surface irregularities were much smaller than the wavelength.	1,584	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	60	42	2,024	8	10	30	58
On 10 November, Fresnel sent a supplementary note dealing with Newton's rings and with gratings, including, for the first time, transmission gratings—although in that case the interfering rays were still assumed to be "inflected", and the experimental verification was inadequate because it used only two threads.	313	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	61	43	2,024	8	10	30	58
As Fresnel was not a member of the institute, the fate of his memoir depended heavily on the report of a single member. The reporter for Fresnel's memoir turned out to be Arago (with Poinsot as the other reviewer). On 8 November, Arago wrote to Fresnel:	253	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	62	44	2,024	8	10	30	58
I have been instructed by the Institute to examine your memoir on the diffraction of light; I have studied it carefully, and found many interesting experiments, some of which had already been done by Dr. Thomas Young, who in general regards this phenomenon in a manner rather analogous to the one you have adopted. But what neither he nor anyone had seen before you is that the external colored bands do not travel in a straight line as one moves away from the opaque body. The results you have achieved in this regard seem to me very important; perhaps they can serve to prove the truth of the undulatory system, so often and so feebly combated by physicists who have not bothered to understand it.	699	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	63	45	2,024	8	10	30	58
Fresnel was troubled, wanting to know more precisely where he had collided with Young. Concerning the curved paths of the "colored bands", Young had noted the hyperbolic paths of the fringes in the two-source interference pattern, corresponding roughly to Fresnel's internal fringes, and had described the hyperbolic fringes that appear on the screen within rectangular shadows. He had not mentioned the curved paths of the external fringes of a shadow; but, as he later explained, that was because Newton had already done so. Newton evidently thought the fringes were caustics. Thus Arago erred in his belief that the curved paths of the fringes were fundamentally incompatible with the corpuscular theory.	707	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	64	46	2,024	8	10	30	58
Arago's letter went on to request more data on the external fringes. Fresnel complied, until he exhausted his leave and was assigned to Rennes in the département of Ille-et-Vilaine. At this point Arago interceded with Gaspard de Prony, head of the École des Ponts, who wrote to Louis-Mathieu Molé, head of the Corps des Ponts, suggesting that the progress of science and the prestige of the Corps would be enhanced if Fresnel could come to Paris for a time. He arrived in March 1816, and his leave was subsequently extended through the middle of the year.	555	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	65	47	2,024	8	10	30	58
Meanwhile, in an experiment reported on 26 February 1816, Arago verified Fresnel's prediction that the internal fringes were shifted if the rays on one side of the obstacle passed through a thin glass lamina. Fresnel correctly attributed this phenomenon to the lower wave velocity in the glass. Arago later used a similar argument to explain the colors in the scintillation of stars.	383	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	66	48	2,024	8	10	30	58
Fresnel's updated memoir was eventually published in the March 1816 issue of Annales de Chimie et de Physique, of which Arago had recently become co-editor. That issue did not actually appear until May. In March, Fresnel already had competition: Biot read a memoir on diffraction by himself and his student Claude Pouillet, containing copious data and arguing that the regularity of diffraction fringes, like the regularity of Newton's rings, must be linked to Newton's "fits". But the new link was not rigorous, and Pouillet himself would become a distinguished early adopter of the wave theory.	596	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	67	49	2,024	8	10	30	58
On 24 May 1816, Fresnel wrote to Young (in French), acknowledging how little of his own memoir was new. But in a "supplement" signed on 14 July and read the next day, Fresnel noted that the internal fringes were more accurately predicted by supposing that the two interfering rays came from some distance outside the edges of the obstacle. To explain this, he divided the incident wavefront at the obstacle into what we now call Fresnel zones, such that the secondary waves from each zone were spread over half a cycle when they arrived at the observation point. The zones on one side of the obstacle largely canceled out in pairs, except the first zone, which was represented by an "efficacious ray". This approach worked for the internal fringes, but the superposition of the efficacious ray and the direct ray did not work for the external fringes.	851	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	68	50	2,024	8	10	30	58
The contribution from the "efficacious ray" was thought to be only partly canceled, for reasons involving the dynamics of the medium: where the wavefront was continuous, symmetry forbade oblique vibrations; but near the obstacle that truncated the wavefront, the asymmetry allowed some sideways vibration towards the geometric shadow. This argument showed that Fresnel had not (yet) fully accepted Huygens's principle, which would have permitted oblique radiation from all portions of the front.	495	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	69	51	2,024	8	10	30	58
In the same supplement, Fresnel described his well-known double mirror, comprising two flat mirrors joined at an angle of slightly less than 180°, with which he produced a two-slit interference pattern from two virtual images of the same slit. A conventional double-slit experiment required a preliminary single slit to ensure that the light falling on the double slit was coherent (synchronized). In Fresnel's version, the preliminary single slit was retained, and the double slit was replaced by the double mirror—which bore no physical resemblance to the double slit and yet performed the same function. This result (which had been announced by Arago in the March issue of the Annales) made it hard to believe that the two-slit pattern had anything to do with corpuscles being deflected as they passed near the edges of the slits.	833	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	70	52	2,024	8	10	30	58
But 1816 was the " Year Without a Summer ": crops failed; hungry farming families lined the streets of Rennes; the central government organized "charity workhouses" for the needy; and in October, Fresnel was sent back to Ille-et-Vilaine to supervise charity workers in addition to his regular road crew. According to Arago,	323	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	71	53	2,024	8	10	30	58
with Fresnel conscientiousness was always the foremost part of his character, and he constantly performed his duties as an engineer with the most rigorous scrupulousness. The mission to defend the revenues of the state, to obtain for them the best employment possible, appeared to his eyes in the light of a question of honour. The functionary, whatever might be his rank, who submitted to him an ambiguous account, became at once the object of his profound contempt. … Under such circumstances the habitual gentleness of his manners disappeared…	546	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	72	54	2,024	8	10	30	58
Fresnel's letters from December 1816 reveal his consequent anxiety. To Arago he complained of being "tormented by the worries of surveillance, and the need to reprimand…" And to Mérimée he wrote: "I find nothing more tiresome than having to manage other men, and I admit that I have no idea what I'm doing."	307	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	73	55	2,024	8	10	30	58
On 17 March 1817, the Académie des Sciences announced that diffraction would be the topic for the biannual physics Grand Prix to be awarded in 1819. The deadline for entries was set at 1 August 1818 to allow time for replication of experiments. Although the wording of the problem referred to rays and inflection and did not invite wave-based solutions, Arago and Ampère encouraged Fresnel to enter.	399	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	74	56	2,024	8	10	30	58
In the fall of 1817, Fresnel, supported by de Prony, obtained a leave of absence from the new head of the Corp des Ponts, Louis Becquey, and returned to Paris. He resumed his engineering duties in the spring of 1818; but from then on he was based in Paris, first on the Canal de l'Ourcq, and then (from May 1819) with the cadastre of the pavements.	348	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	75	57	2,024	8	10	30	58
On 15 January 1818, in a different context (revisited below), Fresnel showed that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions. His method was similar to the phasor representation, except that the "forces" were plane vectors rather than complex numbers ; they could be added, and multiplied by scalars, but not (yet) multiplied and divided by each other. The explanation was algebraic rather than geometric.	507	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	76	58	2,024	8	10	30	58
Knowledge of this method was assumed in a preliminary note on diffraction, dated 19 April 1818 and deposited on 20 April, in which Fresnel outlined the elementary theory of diffraction as found in modern textbooks. He restated Huygens's principle in combination with the superposition principle, saying that the vibration at each point on a wavefront is the sum of the vibrations that would be sent to it at that moment by all the elements of the wavefront in any of its previous positions, all elements acting separately (see Huygens–Fresnel principle). For a wavefront partly obstructed in a previous position, the summation was to be carried out over the unobstructed portion. In directions other than the normal to the primary wavefront, the secondary waves were weakened due to obliquity, but weakened much more by destructive interference, so that the effect of obliquity alone could be ignored. For diffraction by a straight edge, the intensity as a function of distance from the geometric shadow could then be expressed with sufficient accuracy in terms of what are now called the normalized Fresnel integrals :	1,119	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	77	59	2,024	8	10	30	58
The same note included a table of the integrals, for an upper limit ranging from 0 to 5.1 in steps of 0.1, computed with a mean error of 0.0003, plus a smaller table of maxima and minima of the resulting intensity.	214	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	78	60	2,024	8	10	30	58
In his final "Memoir on the diffraction of light", deposited on 29 July and bearing the Latin epigraph " Natura simplex et fecunda " ("Nature simple and fertile"), Fresnel slightly expanded the two tables without changing the existing figures, except for a correction to the first minimum of intensity. For completeness, he repeated his solution to "the problem of interference", whereby sinusoidal functions are added like vectors. He acknowledged the directionality of the secondary sources and the variation in their distances from the observation point, chiefly to explain why these things make negligible difference in the context, provided of course that the secondary sources do not radiate in the retrograde direction. Then, applying his theory of interference to the secondary waves, he expressed the intensity of light diffracted by a single straight edge (half-plane) in terms of integrals which involved the dimensions of the problem, but which could be converted to the normalized forms above. With reference to the integrals, he explained the calculation of the maxima and minima of the intensity (external fringes), and noted that the calculated intensity falls very rapidly as one moves into the geometric shadow. The last result, as Olivier Darrigol says, "amounts to a proof of the rectilinear propagation of light in the wave theory, indeed the first proof that a modern physicist would still accept."	1,420	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	79	61	2,024	8	10	30	58
For the experimental testing of his calculations, Fresnel used red light with a wavelength of 638 nm, which he deduced from the diffraction pattern in the simple case in which light incident on a single slit was focused by a cylindrical lens. For a variety of distances from the source to the obstacle and from the obstacle to the field point, he compared the calculated and observed positions of the fringes for diffraction by a half-plane, a slit, and a narrow strip—concentrating on the minima, which were visually sharper than the maxima. For the slit and the strip, he could not use the previously computed table of maxima and minima; for each combination of dimensions, the intensity had to be expressed in terms of sums or differences of Fresnel integrals and calculated from the table of integrals, and the extrema had to be calculated anew. The agreement between calculation and measurement was better than 1.5% in almost every case.	942	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	80	62	2,024	8	10	30	58
Near the end of the memoir, Fresnel summed up the difference between Huygens's use of secondary waves and his own: whereas Huygens says there is light only where the secondary waves exactly agree, Fresnel says there is complete darkness only where the secondary waves exactly cancel out.	287	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	81	63	2,024	8	10	30	58
The judging committee comprised Laplace, Biot, and Poisson (all corpuscularists), Gay-Lussac (uncommitted), and Arago, who eventually wrote the committee's report. Although entries in the competition were supposed to be anonymous to the judges, Fresnel's must have been recognizable by the content. There was only one other entry, of which neither the manuscript nor any record of the author has survived. That entry (identified as "no. 1") was mentioned only in the last paragraph of the judges' report, noting that the author had shown ignorance of the relevant earlier works of Young and Fresnel, used insufficiently precise methods of observation, overlooked known phenomena, and made obvious errors. In the words of John Worrall, "The competition facing Fresnel could hardly have been less stiff." We may infer that the committee had only two options: award the prize to Fresnel ("no. 2"), or withhold it.	910	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	82	64	2,024	8	10	30	58
The committee deliberated into the new year. Then Poisson, exploiting a case in which Fresnel's theory gave easy integrals, predicted that if a circular obstacle were illuminated by a point-source, there should be (according to the theory) a bright spot in the center of the shadow, illuminated as brightly as the exterior. This seems to have been intended as a reductio ad absurdum. Arago, undeterred, assembled an experiment with an obstacle 2 mm in diameter—and there, in the center of the shadow, was Poisson's spot.	520	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	83	65	2,024	8	10	30	58
The unanimous report of the committee, read at the meeting of the Académie on 15 March 1819, awarded the prize to "the memoir marked no. 2, and bearing as epigraph: Natura simplex et fecunda." At the same meeting, after the judgment was delivered, the president of the Académie opened a sealed note accompanying the memoir, revealing the author as Fresnel. The award was announced at the public meeting of the Académie a week later, on 22 March.	445	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	84	66	2,024	8	10	30	58
Arago's verification of Poisson's counter-intuitive prediction passed into folklore as if it had decided the prize. That view, however, is not supported by the judges' report, which gave the matter only two sentences in the penultimate paragraph. Neither did Fresnel's triumph immediately convert Laplace, Biot, and Poisson to the wave theory, for at least four reasons. First, although the professionalization of science in France had established common standards, it was one thing to acknowledge a piece of research as meeting those standards, and another thing to regard it as conclusive. Second, it was possible to interpret Fresnel's integrals as rules for combining rays. Arago even encouraged that interpretation, presumably in order to minimize resistance to Fresnel's ideas. Even Biot began teaching the Huygens-Fresnel principle without committing himself to a wave basis. Third, Fresnel's theory did not adequately explain the mechanism of generation of secondary waves or why they had any significant angular spread; this issue particularly bothered Poisson. Fourth, the question that most exercised optical physicists at that time was not diffraction, but polarization—on which Fresnel had been working, but was yet to make his critical breakthrough.	1,263	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	85	67	2,024	8	10	30	58
An emission theory of light was one that regarded the propagation of light as the transport of some kind of matter. While the corpuscular theory was obviously an emission theory, the converse did not follow: in principle, one could be an emissionist without being a corpuscularist. This was convenient because, beyond the ordinary laws of reflection and refraction, emissionists never managed to make testable quantitative predictions from a theory of forces acting on corpuscles of light. But they did make quantitative predictions from the premises that rays were countable objects, which were conserved in their interactions with matter (except absorbent media), and which had particular orientations with respect to their directions of propagation. According to this framework, polarization and the related phenomena of double refraction and partial reflection involved altering the orientations of the rays and/or selecting them according to orientation, and the state of polarization of a beam (a bundle of rays) was a question of how many rays were in what orientations: in a fully polarized beam, the orientations were all the same. This approach, which Jed Buchwald has called selectionism, was pioneered by Malus and diligently pursued by Biot.	1,254	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	86	68	2,024	8	10	30	58
Fresnel, in contrast, decided to introduce polarization into interference experiments.	86	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	87	69	2,024	8	10	30	58
In July or August 1816, Fresnel discovered that when a birefringent crystal produced two images of a single slit, he could not obtain the usual two-slit interference pattern, even if he compensated for the different propagation times. A more general experiment, suggested by Arago, found that if the two beams of a double-slit device were separately polarized, the interference pattern appeared and disappeared as the polarization of one beam was rotated, giving full interference for parallel polarizations, but no interference for perpendicular polarizations (see Fresnel–Arago laws). These experiments, among others, were eventually reported in a brief memoir published in 1819 and later translated into English.	715	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	88	70	2,024	8	10	30	58
In a memoir drafted on 30 August 1816 and revised on 6 October, Fresnel reported an experiment in which he placed two matching thin laminae in a double-slit apparatus—one over each slit, with their optic axes perpendicular—and obtained two interference patterns offset in opposite directions, with perpendicular polarizations. This, in combination with the previous findings, meant that each lamina split the incident light into perpendicularly polarized components with different velocities—just like a normal (thick) birefringent crystal, and contrary to Biot's "mobile polarization" hypothesis.	597	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	89	71	2,024	8	10	30	58
Accordingly, in the same memoir, Fresnel offered his first attempt at a wave theory of chromatic polarization. When polarized light passed through a crystal lamina, it was split into ordinary and extraordinary waves (with intensities described by Malus's law), and these were perpendicularly polarized and therefore did not interfere, so that no colors were produced (yet). But if they then passed through an analyzer (second polarizer), their polarizations were brought into alignment (with intensities again modified according to Malus's law), and they would interfere. This explanation, by itself, predicts that if the analyzer is rotated 90°, the ordinary and extraordinary waves simply switch roles, so that if the analyzer takes the form of a calcite crystal, the two images of the lamina should be of the same hue (this issue is revisited below). But in fact, as Arago and Biot had found, they are of complementary colors. To correct the prediction, Fresnel proposed a phase-inversion rule whereby one of the constituent waves of one of the two images suffered an additional 180° phase shift on its way through the lamina. This inversion was a weakness in the theory relative to Biot's, as Fresnel acknowledged, although the rule specified which of the two images had the inverted wave. Moreover, Fresnel could deal only with special cases, because he had not yet solved the problem of superposing sinusoidal functions with arbitrary phase differences due to propagation at different velocities through the lamina.	1,521	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	90	72	2,024	8	10	30	58
He solved that problem in a "supplement" signed on 15 January 1818 (mentioned above). In the same document, he accommodated Malus's law by proposing an underlying law: that if polarized light is incident on a birefringent crystal with its optic axis at an angle θ to the "plane of polarization", the ordinary and extraordinary vibrations (as functions of time) are scaled by the factors cos θ and sin θ, respectively. Although modern readers easily interpret these factors in terms of perpendicular components of a transverse oscillation, Fresnel did not (yet) explain them that way. Hence he still needed the phase-inversion rule. He applied all these principles to a case of chromatic polarization not covered by Biot's formulae, involving two successive laminae with axes separated by 45°, and obtained predictions that disagreed with Biot's experiments (except in special cases) but agreed with his own.	907	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	91	73	2,024	8	10	30	58
Fresnel applied the same principles to the standard case of chromatic polarization, in which one birefringent lamina was sliced parallel to its axis and placed between a polarizer and an analyzer. If the analyzer took the form of a thick calcite crystal with its axis in the plane of polarization, Fresnel predicted that the intensities of the ordinary and extraordinary images of the lamina were respectively proportional to	425	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	92	74	2,024	8	10	30	58
where i is the angle from the initial plane of polarization to the optic axis of the lamina, s is the angle from the initial plane of polarization to the plane of polarization of the final ordinary image, and ϕ ϕ is the phase lag of the extraordinary wave relative to the ordinary wave due to the difference in propagation times through the lamina. The terms in ϕ ϕ are the frequency-dependent terms and explain why the lamina must be thin in order to produce discernible colors: if the lamina is too thick, cos ⁡ ⁡ ϕ ϕ will pass through too many cycles as the frequency varies through the visible range, and the eye (which divides the visible spectrum into only three bands) will not be able to resolve the cycles.	715	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	93	75	2,024	8	10	30	58
From these equations it is easily verified that I o + I e = 1 for all ϕ ϕ , so that the colors are complementary. Without the phase-inversion rule, there would be a plus sign in front of the last term in the second equation, so that the ϕ ϕ -dependent term would be the same in both equations, implying (incorrectly) that the colors were of the same hue.	354	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	94	76	2,024	8	10	30	58
These equations were included in an undated note that Fresnel gave to Biot, to which Biot added a few lines of his own. If we substitute	136	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	95	77	2,024	8	10	30	58
then Fresnel's formulae can be rewritten as	43	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	96	78	2,024	8	10	30	58
which are none other than Biot's empirical formulae of 1812, except that Biot interpreted U and A as the "unaffected" and "affected" selections of the rays incident on the lamina. If Biot's substitutions were accurate, they would imply that his experimental results were more fully explained by Fresnel's theory than by his own.	328	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	97	79	2,024	8	10	30	58
Arago delayed reporting on Fresnel's works on chromatic polarization until June 1821, when he used them in a broad attack on Biot's theory. In his written response, Biot protested that Arago's attack went beyond the proper scope of a report on the nominated works of Fresnel. But Biot also claimed that the substitutions for U and A , and therefore Fresnel's expressions for I o and I e , were empirically wrong because when Fresnel's intensities of spectral colors were mixed according to Newton's rules, the squared cosine and sine functions varied too smoothly to account for the observed sequence of colors. That claim drew a written reply from Fresnel, who disputed whether the colors changed as abruptly as Biot claimed, and whether the human eye could judge color with sufficient objectivity for the purpose. On the latter question, Fresnel pointed out that different observers may give different names to the same color. Furthermore, he said, a single observer can only compare colors side by side; and even if they are judged to be the same, the identity is of sensation, not necessarily of composition. Fresnel's oldest and strongest point—that thin crystals were subject to the same laws as thick ones and did not need or allow a separate theory—Biot left unanswered. Arago and Fresnel were seen to have won the debate.	1,330	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	98	80	2,024	8	10	30	58
Moreover, by this time Fresnel had a new, simpler explanation of his equations on chromatic polarization.	105	Augustin-Jean_Fresnel	https://en.wikipedia.org/wiki/Augustin-Jean_Fresnel	99	81	2,024	8	10	30	58

The atomic number or nuclear charge number (symbol Z) of a chemical element is the charge number of an atomic nucleus. For ordinary nuclei composed of protons and neutrons, this is equal to the proton number (n p) or the number of protons found in the nucleus of every atom of that element. The atomic number can be used to uniquely identify ordinary chemical elements. In an ordinary uncharged atom, the atomic number is also equal to the number of electrons.

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For an ordinary atom which contains protons, neutrons and electrons, the sum of the atomic number Z and the neutron number N gives the atom's atomic mass number A. Since protons and neutrons have approximately the same mass (and the mass of the electrons is negligible for many purposes) and the mass defect of the nucleon binding is always small compared to the nucleon mass, the atomic mass of any atom, when expressed in daltons (making a quantity called the " relative isotopic mass "), is within 1% of the whole number A.

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Atoms with the same atomic number but different neutron numbers, and hence different mass numbers, are known as isotopes. A little more than three-quarters of naturally occurring elements exist as a mixture of isotopes (see monoisotopic elements), and the average isotopic mass of an isotopic mixture for an element (called the relative atomic mass) in a defined environment on Earth determines the element's standard atomic weight. Historically, it was these atomic weights of elements (in comparison to hydrogen) that were the quantities measurable by chemists in the 19th century.

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The conventional symbol Z comes from the German word Z ahl 'number', which, before the modern synthesis of ideas from chemistry and physics, merely denoted an element's numerical place in the periodic table, whose order was then approximately, but not completely, consistent with the order of the elements by atomic weights. Only after 1915, with the suggestion and evidence that this Z number was also the nuclear charge and a physical characteristic of atoms, did the word Atom z ahl (and its English equivalent atomic number) come into common use in this context.

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The rules above do not always apply to exotic atoms which contain short-lived elementary particles other than protons, neutrons and electrons.

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Loosely speaking, the existence or construction of a periodic table of elements creates an ordering of the elements, and so they can be numbered in order.

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Dmitri Mendeleev said that he arranged his first periodic tables (first published on March 6, 1869) in order of atomic weight ("Atomgewicht"). However, in consideration of the elements' observed chemical properties, he changed the order slightly and placed tellurium (atomic weight 127.6) ahead of iodine (atomic weight 126.9). This placement is consistent with the modern practice of ordering the elements by proton number, Z, but that number was not known or suspected at the time.

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A simple numbering based on periodic table position was never entirely satisfactory. In addition to the case of iodine and tellurium, several other pairs of elements (such as argon and potassium, cobalt and nickel) were later shown to have nearly identical or reversed atomic weights, thus requiring their placement in the periodic table to be determined by their chemical properties. However the gradual identification of more and more chemically similar lanthanide elements, whose atomic number was not obvious, led to inconsistency and uncertainty in the periodic numbering of elements at least from lutetium (element 71) onward (hafnium was not known at this time).

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In 1911, Ernest Rutherford gave a model of the atom in which a central nucleus held most of the atom's mass and a positive charge which, in units of the electron's charge, was to be approximately equal to half of the atom's atomic weight, expressed in numbers of hydrogen atoms. This central charge would thus be approximately half the atomic weight (though it was almost 25% different from the atomic number of gold (Z = 79, A = 197), the single element from which Rutherford made his guess). Nevertheless, in spite of Rutherford's estimation that gold had a central charge of about 100 (but was element Z = 79 on the periodic table), a month after Rutherford's paper appeared, Antonius van den Broek first formally suggested that the central charge and number of electrons in an atom were exactly equal to its place in the periodic table (also known as element number, atomic number, and symbolized Z). This eventually proved to be the case.

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The experimental position improved dramatically after research by Henry Moseley in 1913. Moseley, after discussions with Bohr who was at the same lab (and who had used Van den Broek's hypothesis in his Bohr model of the atom), decided to test Van den Broek's and Bohr's hypothesis directly, by seeing if spectral lines emitted from excited atoms fitted the Bohr theory's postulation that the frequency of the spectral lines be proportional to the square of Z.

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To do this, Moseley measured the wavelengths of the innermost photon transitions (K and L lines) produced by the elements from aluminium (Z = 13) to gold (Z = 79) used as a series of movable anodic targets inside an x-ray tube. The square root of the frequency of these photons (x-rays) increased from one target to the next in an arithmetic progression. This led to the conclusion (Moseley's law) that the atomic number does closely correspond (with an offset of one unit for K-lines, in Moseley's work) to the calculated electric charge of the nucleus, i.e. the element number Z. Among other things, Moseley demonstrated that the lanthanide series (from lanthanum to lutetium inclusive) must have 15 members—no fewer and no more—which was far from obvious from known chemistry at that time.

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After Moseley's death in 1915, the atomic numbers of all known elements from hydrogen to uranium (Z = 92) were examined by his method. There were seven elements (with Z < 92) which were not found and therefore identified as still undiscovered, corresponding to atomic numbers 43, 61, 72, 75, 85, 87 and 91. From 1918 to 1947, all seven of these missing elements were discovered. By this time, the first four transuranium elements had also been discovered, so that the periodic table was complete with no gaps as far as curium (Z = 96).

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In 1915, the reason for nuclear charge being quantized in units of Z, which were now recognized to be the same as the element number, was not understood. An old idea called Prout's hypothesis had postulated that the elements were all made of residues (or "protyles") of the lightest element hydrogen, which in the Bohr-Rutherford model had a single electron and a nuclear charge of one. However, as early as 1907, Rutherford and Thomas Royds had shown that alpha particles, which had a charge of +2, were the nuclei of helium atoms, which had a mass four times that of hydrogen, not two times. If Prout's hypothesis were true, something had to be neutralizing some of the charge of the hydrogen nuclei present in the nuclei of heavier atoms.

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In 1917, Rutherford succeeded in generating hydrogen nuclei from a nuclear reaction between alpha particles and nitrogen gas, and believed he had proven Prout's law. He called the new heavy nuclear particles protons in 1920 (alternate names being proutons and protyles). It had been immediately apparent from the work of Moseley that the nuclei of heavy atoms have more than twice as much mass as would be expected from their being made of hydrogen nuclei, and thus there was required a hypothesis for the neutralization of the extra protons presumed present in all heavy nuclei. A helium nucleus was presumed to be composed of four protons plus two "nuclear electrons" (electrons bound inside the nucleus) to cancel two of the charges. At the other end of the periodic table, a nucleus of gold with a mass 197 times that of hydrogen was thought to contain 118 nuclear electrons in the nucleus to give it a residual charge of +79, consistent with its atomic number.

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All consideration of nuclear electrons ended with James Chadwick 's discovery of the neutron in 1932. An atom of gold now was seen as containing 118 neutrons rather than 118 nuclear electrons, and its positive nuclear charge now was realized to come entirely from a content of 79 protons. Since Moseley had previously shown that the atomic number Z of an element equals this positive charge, it was now clear that Z is identical to the number of protons of its nuclei.

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Each element has a specific set of chemical properties as a consequence of the number of electrons present in the neutral atom, which is Z (the atomic number). The configuration of these electrons follows from the principles of quantum mechanics. The number of electrons in each element's electron shells, particularly the outermost valence shell, is the primary factor in determining its chemical bonding behavior. Hence, it is the atomic number alone that determines the chemical properties of an element; and it is for this reason that an element can be defined as consisting of any mixture of atoms with a given atomic number.

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The quest for new elements is usually described using atomic numbers. As of 2024, all elements with atomic numbers 1 to 118 have been observed. Synthesis of new elements is accomplished by bombarding target atoms of heavy elements with ions, such that the sum of the atomic numbers of the target and ion elements equals the atomic number of the element being created. In general, the half-life of a nuclide becomes shorter as atomic number increases, though undiscovered nuclides with certain " magic " numbers of protons and neutrons may have relatively longer half-lives and comprise an island of stability.

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A hypothetical element composed only of neutrons has also been proposed and would have atomic number 0, but has never been observed.

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Augustin-Jean Fresnel (10 May 1788 – 14 July 1827) was a French civil engineer and physicist whose research in optics led to the almost unanimous acceptance of the wave theory of light, excluding any remnant of Newton 's corpuscular theory, from the late 1830s until the end of the 19th century. He is perhaps better known for inventing the catadioptric (reflective/refractive) Fresnel lens and for pioneering the use of "stepped" lenses to extend the visibility of lighthouses, saving countless lives at sea. The simpler dioptric (purely refractive) stepped lens, first proposed by Count Buffon and independently reinvented by Fresnel, is used in screen magnifiers and in condenser lenses for overhead projectors.

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By expressing Huygens 's principle of secondary waves and Young 's principle of interference in quantitative terms, and supposing that simple colors consist of sinusoidal waves, Fresnel gave the first satisfactory explanation of diffraction by straight edges, including the first satisfactory wave-based explanation of rectilinear propagation. Part of his argument was a proof that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions. By further supposing that light waves are purely transverse, Fresnel explained the nature of polarization, the mechanism of chromatic polarization, and the transmission and reflection coefficients at the interface between two transparent isotropic media. Then, by generalizing the direction-speed-polarization relation for calcite, he accounted for the directions and polarizations of the refracted rays in doubly-refractive crystals of the biaxial class (those for which Huygens's secondary wavefronts are not axisymmetric). The period between the first publication of his pure-transverse-wave hypothesis, and the submission of his first correct solution to the biaxial problem, was less than a year.

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Later, he coined the terms linear polarization, circular polarization, and elliptical polarization, explained how optical rotation could be understood as a difference in propagation speeds for the two directions of circular polarization, and (by allowing the reflection coefficient to be complex) accounted for the change in polarization due to total internal reflection, as exploited in the Fresnel rhomb. Defenders of the established corpuscular theory could not match his quantitative explanations of so many phenomena on so few assumptions.

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Fresnel had a lifelong battle with tuberculosis, to which he succumbed at the age of 39. Although he did not become a public celebrity in his lifetime, he lived just long enough to receive due recognition from his peers, including (on his deathbed) the Rumford Medal of the Royal Society of London, and his name is ubiquitous in the modern terminology of optics and waves. After the wave theory of light was subsumed by Maxwell 's electromagnetic theory in the 1860s, some attention was diverted from the magnitude of Fresnel's contribution. In the period between Fresnel's unification of physical optics and Maxwell's wider unification, a contemporary authority, Humphrey Lloyd, described Fresnel's transverse-wave theory as "the noblest fabric which has ever adorned the domain of physical science, Newton's system of the universe alone excepted."

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Augustin-Jean Fresnel (also called Augustin Jean or simply Augustin), born in Broglie, Normandy, on 10 May 1788, was the second of four sons of the architect Jacques Fresnel and his wife Augustine, née Mérimée. The family moved twice—in 1789/90 to Cherbourg, and in 1794 to Jacques's home town of Mathieu, where Augustine would spend 25 years as a widow, outliving two of her sons.

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The first son, Louis, was admitted to the École Polytechnique, became a lieutenant in the artillery, and was killed in action at Jaca, Spain. The third, Léonor, followed Augustin into civil engineering, succeeded him as secretary of the Lighthouse Commission, and helped to edit his collected works. The fourth, Fulgence Fresnel, became a linguist, diplomat, and orientalist, and occasionally assisted Augustin with negotiations. Fulgence died in Bagdad in 1855 having led a mission to explore Babylon.

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Madame Fresnel's younger brother, Jean François "Léonor" Mérimée, father of the writer Prosper Mérimée, was a painter who turned his attention to the chemistry of painting. He became the Permanent Secretary of the École des Beaux-Arts and (until 1814) a professor at the École Polytechnique, and was the initial point of contact between Augustin and the leading optical physicists of the day (see below).

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The Fresnel brothers were initially home-schooled by their mother. The sickly Augustin was considered the slow one, not inclined to memorization; but the popular story that he hardly began to read until the age of eight is disputed. At the age of nine or ten he was undistinguished except for his ability to turn tree-branches into toy bows and guns that worked far too well, earning himself the title l'homme de génie (the man of genius) from his accomplices, and a united crackdown from their elders.

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In 1801, Augustin was sent to the École Centrale at Caen, as company for Louis. But Augustin lifted his performance: in late 1804 he was accepted into the École Polytechnique, being placed 17th in the entrance examination. As the detailed records of the École Polytechnique begin in 1808, we know little of Augustin's time there, except that he made few if any friends and—in spite of continuing poor health—excelled in drawing and geometry: in his first year he took a prize for his solution to a geometry problem posed by Adrien-Marie Legendre. Graduating in 1806, he then enrolled at the École Nationale des Ponts et Chaussées (National School of Bridges and Roads, also known as "ENPC" or "École des Ponts"), from which he graduated in 1809, entering the service of the Corps des Ponts et Chaussées as an ingénieur ordinaire aspirant (ordinary engineer in training). Directly or indirectly, he was to remain in the employment of the "Corps des Ponts" for the rest of his life.

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Fresnel's parents were Roman Catholics of the Jansenist sect, characterized by an extreme Augustinian view of original sin. Religion took first place in the boys' home-schooling. In 1802, his mother said:

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I pray God to give my son the grace to employ the great talents, which he has received, for his own benefit, and for the God of all. Much will be asked from him to whom much has been given, and most will be required of him who has received most.

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Augustin remained a Jansenist. He regarded his intellectual talents as gifts from God, and considered it his duty to use them for the benefit of others. According to his fellow engineer Alphonse Duleau, who helped to nurse him through his final illness, Fresnel saw the study of nature as part of the study of the power and goodness of God. He placed virtue above science and genius. In his last days he prayed for "strength of soul," not against death alone, but against "the interruption of discoveries… of which he hoped to derive useful applications."

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Jansenism is considered heretical by the Roman Catholic Church, and Grattan-Guinness suggests this is why Fresnel never gained a permanent academic teaching post; his only teaching appointment was at the Athénée in the winter of 1819–20. The article on Fresnel in the Catholic Encyclopedia does not mention his Jansenism, but describes him as "a deeply religious man and remarkable for his keen sense of duty."

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Fresnel was initially posted to the western département of Vendée. There, in 1811, he anticipated what became known as the Solvay process for producing soda ash, except that recycling of the ammonia was not considered. That difference may explain why leading chemists, who learned of his discovery through his uncle Léonor, eventually thought it uneconomic.

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About 1812, Fresnel was sent to Nyons, in the southern département of Drôme, to assist with the imperial highway that was to connect Spain and Italy. It is from Nyons that we have the first evidence of his interest in optics. On 15 May 1814, while work was slack due to Napoleon 's defeat, Fresnel wrote a " P.S. " to his brother Léonor, saying in part:

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I would also like to have papers that might tell me about the discoveries of French physicists on the polarization of light. I saw in the Moniteur of a few months ago that Biot had read to the Institute a very interesting memoir on the polarization of light. Though I break my head, I cannot guess what that is.

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As late as 28 December he was still waiting for information, but by 10 February 1815 he had received Biot's memoir. (The Institut de France had taken over the functions of the French Académie des Sciences and other académies in 1795. In 1816 the Académie des Sciences regained its name and autonomy, but remained part of the institute.)

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In March 1815, perceiving Napoleon's return from Elba as "an attack on civilization", Fresnel departed without leave, hastened to Toulouse and offered his services to the royalist resistance, but soon found himself on the sick list. Returning to Nyons in defeat, he was threatened and had his windows broken. During the Hundred Days he was placed on suspension, which he was eventually allowed to spend at his mother's house in Mathieu. There he used his enforced leisure to begin his optical experiments.

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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.

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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.

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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 (see Foucault's measurements of the speed of light).

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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.

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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.

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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.

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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.

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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."

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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.

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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."

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

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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.

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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.

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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.

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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.

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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.

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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.

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Fresnel's letters from later in 1814 reveal his interest in the wave theory, including his awareness that it explained the constancy of the speed of light and was at least compatible with stellar aberration. Eventually he compiled what he called his rêveries (musings) into an essay and submitted it via Léonor Mérimée to André-Marie Ampère, who did not respond directly. But on 19 December, Mérimée dined with Ampère and Arago, with whom he was acquainted through the École Polytechnique; and Arago promised to look at Fresnel's essay.

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In mid 1815, on his way home to Mathieu to serve his suspension, Fresnel met Arago in Paris and spoke of the wave theory and stellar aberration. He was informed that he was trying to break down open doors (" il enfonçait des portes ouvertes "), and directed to classical works on optics.

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On 12 July 1815, as Fresnel was about to leave Paris, Arago left him a note on a new topic:

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I do not know of any book that contains all the experiments that physicists are doing on the diffraction of light. M'sieur Fresnel will only be able to get to know this part of the optics by reading the work by Grimaldi, the one by Newton, the English treatise by Jordan, and the memoirs of Brougham and Young, which are part of the collection of the Philosophical Transactions.

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Fresnel would not have ready access to these works outside Paris, and could not read English. But, in Mathieu—with a point-source of light made by focusing sunlight with a drop of honey, a crude micrometer of his own construction, and supporting apparatus made by a local locksmith—he began his own experiments. His technique was novel: whereas earlier investigators had projected the fringes onto a screen, Fresnel soon abandoned the screen and observed the fringes in space, through a lens with the micrometer at its focus, allowing more accurate measurements while requiring less light.

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Later in July, after Napoleon's final defeat, Fresnel was reinstated with the advantage of having backed the winning side. He requested a two-month leave of absence, which was readily granted because roadworks were in abeyance.

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On 23 September he wrote to Arago, beginning "I think I have found the explanation and the law of colored fringes which one notices in the shadows of bodies illuminated by a luminous point." In the same paragraph, however, Fresnel implicitly acknowledged doubt about the novelty of his work: noting that he would need to incur some expense in order to improve his measurements, he wanted to know "whether this is not useless, and whether the law of diffraction has not already been established by sufficiently exact experiments." He explained that he had not yet had a chance to acquire the items on his reading lists, with the apparent exception of "Young's book", which he could not understand without his brother's help. Not surprisingly, he had retraced many of Young's steps.

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In a memoir sent to the institute on 15 October 1815, Fresnel mapped the external and internal fringes in the shadow of a wire. He noticed, like Young before him, that the internal fringes disappeared when the light from one side was blocked, and concluded that "the vibrations of two rays that cross each other under a very small angle can contradict each other…" But, whereas Young took the disappearance of the internal fringes as confirmation of the principle of interference, Fresnel reported that it was the internal fringes that first drew his attention to the principle. To explain the diffraction pattern, Fresnel constructed the internal fringes by considering the intersections of circular wavefronts emitted from the two edges of the obstruction, and the external fringes by considering the intersections between direct waves and waves reflected off the nearer edge. For the external fringes, to obtain tolerable agreement with observation, he had to suppose that the reflected wave was inverted ; and he noted that the predicted paths of the fringes were hyperbolic. In the part of the memoir that most clearly surpassed Young, Fresnel explained the ordinary laws of reflection and refraction in terms of interference, noting that if two parallel rays were reflected or refracted at other than the prescribed angle, they would no longer have the same phase in a common perpendicular plane, and every vibration would be cancelled by a nearby vibration. He noted that his explanation was valid provided that the surface irregularities were much smaller than the wavelength.

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On 10 November, Fresnel sent a supplementary note dealing with Newton's rings and with gratings, including, for the first time, transmission gratings—although in that case the interfering rays were still assumed to be "inflected", and the experimental verification was inadequate because it used only two threads.

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As Fresnel was not a member of the institute, the fate of his memoir depended heavily on the report of a single member. The reporter for Fresnel's memoir turned out to be Arago (with Poinsot as the other reviewer). On 8 November, Arago wrote to Fresnel:

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I have been instructed by the Institute to examine your memoir on the diffraction of light; I have studied it carefully, and found many interesting experiments, some of which had already been done by Dr. Thomas Young, who in general regards this phenomenon in a manner rather analogous to the one you have adopted. But what neither he nor anyone had seen before you is that the external colored bands do not travel in a straight line as one moves away from the opaque body. The results you have achieved in this regard seem to me very important; perhaps they can serve to prove the truth of the undulatory system, so often and so feebly combated by physicists who have not bothered to understand it.

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Fresnel was troubled, wanting to know more precisely where he had collided with Young. Concerning the curved paths of the "colored bands", Young had noted the hyperbolic paths of the fringes in the two-source interference pattern, corresponding roughly to Fresnel's internal fringes, and had described the hyperbolic fringes that appear on the screen within rectangular shadows. He had not mentioned the curved paths of the external fringes of a shadow; but, as he later explained, that was because Newton had already done so. Newton evidently thought the fringes were caustics. Thus Arago erred in his belief that the curved paths of the fringes were fundamentally incompatible with the corpuscular theory.

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Arago's letter went on to request more data on the external fringes. Fresnel complied, until he exhausted his leave and was assigned to Rennes in the département of Ille-et-Vilaine. At this point Arago interceded with Gaspard de Prony, head of the École des Ponts, who wrote to Louis-Mathieu Molé, head of the Corps des Ponts, suggesting that the progress of science and the prestige of the Corps would be enhanced if Fresnel could come to Paris for a time. He arrived in March 1816, and his leave was subsequently extended through the middle of the year.

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Meanwhile, in an experiment reported on 26 February 1816, Arago verified Fresnel's prediction that the internal fringes were shifted if the rays on one side of the obstacle passed through a thin glass lamina. Fresnel correctly attributed this phenomenon to the lower wave velocity in the glass. Arago later used a similar argument to explain the colors in the scintillation of stars.

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Fresnel's updated memoir was eventually published in the March 1816 issue of Annales de Chimie et de Physique, of which Arago had recently become co-editor. That issue did not actually appear until May. In March, Fresnel already had competition: Biot read a memoir on diffraction by himself and his student Claude Pouillet, containing copious data and arguing that the regularity of diffraction fringes, like the regularity of Newton's rings, must be linked to Newton's "fits". But the new link was not rigorous, and Pouillet himself would become a distinguished early adopter of the wave theory.

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On 24 May 1816, Fresnel wrote to Young (in French), acknowledging how little of his own memoir was new. But in a "supplement" signed on 14 July and read the next day, Fresnel noted that the internal fringes were more accurately predicted by supposing that the two interfering rays came from some distance outside the edges of the obstacle. To explain this, he divided the incident wavefront at the obstacle into what we now call Fresnel zones, such that the secondary waves from each zone were spread over half a cycle when they arrived at the observation point. The zones on one side of the obstacle largely canceled out in pairs, except the first zone, which was represented by an "efficacious ray". This approach worked for the internal fringes, but the superposition of the efficacious ray and the direct ray did not work for the external fringes.

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The contribution from the "efficacious ray" was thought to be only partly canceled, for reasons involving the dynamics of the medium: where the wavefront was continuous, symmetry forbade oblique vibrations; but near the obstacle that truncated the wavefront, the asymmetry allowed some sideways vibration towards the geometric shadow. This argument showed that Fresnel had not (yet) fully accepted Huygens's principle, which would have permitted oblique radiation from all portions of the front.

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In the same supplement, Fresnel described his well-known double mirror, comprising two flat mirrors joined at an angle of slightly less than 180°, with which he produced a two-slit interference pattern from two virtual images of the same slit. A conventional double-slit experiment required a preliminary single slit to ensure that the light falling on the double slit was coherent (synchronized). In Fresnel's version, the preliminary single slit was retained, and the double slit was replaced by the double mirror—which bore no physical resemblance to the double slit and yet performed the same function. This result (which had been announced by Arago in the March issue of the Annales) made it hard to believe that the two-slit pattern had anything to do with corpuscles being deflected as they passed near the edges of the slits.

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But 1816 was the " Year Without a Summer ": crops failed; hungry farming families lined the streets of Rennes; the central government organized "charity workhouses" for the needy; and in October, Fresnel was sent back to Ille-et-Vilaine to supervise charity workers in addition to his regular road crew. According to Arago,

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with Fresnel conscientiousness was always the foremost part of his character, and he constantly performed his duties as an engineer with the most rigorous scrupulousness. The mission to defend the revenues of the state, to obtain for them the best employment possible, appeared to his eyes in the light of a question of honour. The functionary, whatever might be his rank, who submitted to him an ambiguous account, became at once the object of his profound contempt. … Under such circumstances the habitual gentleness of his manners disappeared…

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Fresnel's letters from December 1816 reveal his consequent anxiety. To Arago he complained of being "tormented by the worries of surveillance, and the need to reprimand…" And to Mérimée he wrote: "I find nothing more tiresome than having to manage other men, and I admit that I have no idea what I'm doing."

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On 17 March 1817, the Académie des Sciences announced that diffraction would be the topic for the biannual physics Grand Prix to be awarded in 1819. The deadline for entries was set at 1 August 1818 to allow time for replication of experiments. Although the wording of the problem referred to rays and inflection and did not invite wave-based solutions, Arago and Ampère encouraged Fresnel to enter.

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In the fall of 1817, Fresnel, supported by de Prony, obtained a leave of absence from the new head of the Corp des Ponts, Louis Becquey, and returned to Paris. He resumed his engineering duties in the spring of 1818; but from then on he was based in Paris, first on the Canal de l'Ourcq, and then (from May 1819) with the cadastre of the pavements.

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On 15 January 1818, in a different context (revisited below), Fresnel showed that the addition of sinusoidal functions of the same frequency but different phases is analogous to the addition of forces with different directions. His method was similar to the phasor representation, except that the "forces" were plane vectors rather than complex numbers ; they could be added, and multiplied by scalars, but not (yet) multiplied and divided by each other. The explanation was algebraic rather than geometric.

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Knowledge of this method was assumed in a preliminary note on diffraction, dated 19 April 1818 and deposited on 20 April, in which Fresnel outlined the elementary theory of diffraction as found in modern textbooks. He restated Huygens's principle in combination with the superposition principle, saying that the vibration at each point on a wavefront is the sum of the vibrations that would be sent to it at that moment by all the elements of the wavefront in any of its previous positions, all elements acting separately (see Huygens–Fresnel principle). For a wavefront partly obstructed in a previous position, the summation was to be carried out over the unobstructed portion. In directions other than the normal to the primary wavefront, the secondary waves were weakened due to obliquity, but weakened much more by destructive interference, so that the effect of obliquity alone could be ignored. For diffraction by a straight edge, the intensity as a function of distance from the geometric shadow could then be expressed with sufficient accuracy in terms of what are now called the normalized Fresnel integrals :

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The same note included a table of the integrals, for an upper limit ranging from 0 to 5.1 in steps of 0.1, computed with a mean error of 0.0003, plus a smaller table of maxima and minima of the resulting intensity.

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In his final "Memoir on the diffraction of light", deposited on 29 July and bearing the Latin epigraph " Natura simplex et fecunda " ("Nature simple and fertile"), Fresnel slightly expanded the two tables without changing the existing figures, except for a correction to the first minimum of intensity. For completeness, he repeated his solution to "the problem of interference", whereby sinusoidal functions are added like vectors. He acknowledged the directionality of the secondary sources and the variation in their distances from the observation point, chiefly to explain why these things make negligible difference in the context, provided of course that the secondary sources do not radiate in the retrograde direction. Then, applying his theory of interference to the secondary waves, he expressed the intensity of light diffracted by a single straight edge (half-plane) in terms of integrals which involved the dimensions of the problem, but which could be converted to the normalized forms above. With reference to the integrals, he explained the calculation of the maxima and minima of the intensity (external fringes), and noted that the calculated intensity falls very rapidly as one moves into the geometric shadow. The last result, as Olivier Darrigol says, "amounts to a proof of the rectilinear propagation of light in the wave theory, indeed the first proof that a modern physicist would still accept."

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For the experimental testing of his calculations, Fresnel used red light with a wavelength of 638 nm, which he deduced from the diffraction pattern in the simple case in which light incident on a single slit was focused by a cylindrical lens. For a variety of distances from the source to the obstacle and from the obstacle to the field point, he compared the calculated and observed positions of the fringes for diffraction by a half-plane, a slit, and a narrow strip—concentrating on the minima, which were visually sharper than the maxima. For the slit and the strip, he could not use the previously computed table of maxima and minima; for each combination of dimensions, the intensity had to be expressed in terms of sums or differences of Fresnel integrals and calculated from the table of integrals, and the extrema had to be calculated anew. The agreement between calculation and measurement was better than 1.5% in almost every case.

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Near the end of the memoir, Fresnel summed up the difference between Huygens's use of secondary waves and his own: whereas Huygens says there is light only where the secondary waves exactly agree, Fresnel says there is complete darkness only where the secondary waves exactly cancel out.

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The judging committee comprised Laplace, Biot, and Poisson (all corpuscularists), Gay-Lussac (uncommitted), and Arago, who eventually wrote the committee's report. Although entries in the competition were supposed to be anonymous to the judges, Fresnel's must have been recognizable by the content. There was only one other entry, of which neither the manuscript nor any record of the author has survived. That entry (identified as "no. 1") was mentioned only in the last paragraph of the judges' report, noting that the author had shown ignorance of the relevant earlier works of Young and Fresnel, used insufficiently precise methods of observation, overlooked known phenomena, and made obvious errors. In the words of John Worrall, "The competition facing Fresnel could hardly have been less stiff." We may infer that the committee had only two options: award the prize to Fresnel ("no. 2"), or withhold it.

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The committee deliberated into the new year. Then Poisson, exploiting a case in which Fresnel's theory gave easy integrals, predicted that if a circular obstacle were illuminated by a point-source, there should be (according to the theory) a bright spot in the center of the shadow, illuminated as brightly as the exterior. This seems to have been intended as a reductio ad absurdum. Arago, undeterred, assembled an experiment with an obstacle 2 mm in diameter—and there, in the center of the shadow, was Poisson's spot.

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The unanimous report of the committee, read at the meeting of the Académie on 15 March 1819, awarded the prize to "the memoir marked no. 2, and bearing as epigraph: Natura simplex et fecunda." At the same meeting, after the judgment was delivered, the president of the Académie opened a sealed note accompanying the memoir, revealing the author as Fresnel. The award was announced at the public meeting of the Académie a week later, on 22 March.

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Arago's verification of Poisson's counter-intuitive prediction passed into folklore as if it had decided the prize. That view, however, is not supported by the judges' report, which gave the matter only two sentences in the penultimate paragraph. Neither did Fresnel's triumph immediately convert Laplace, Biot, and Poisson to the wave theory, for at least four reasons. First, although the professionalization of science in France had established common standards, it was one thing to acknowledge a piece of research as meeting those standards, and another thing to regard it as conclusive. Second, it was possible to interpret Fresnel's integrals as rules for combining rays. Arago even encouraged that interpretation, presumably in order to minimize resistance to Fresnel's ideas. Even Biot began teaching the Huygens-Fresnel principle without committing himself to a wave basis. Third, Fresnel's theory did not adequately explain the mechanism of generation of secondary waves or why they had any significant angular spread; this issue particularly bothered Poisson. Fourth, the question that most exercised optical physicists at that time was not diffraction, but polarization—on which Fresnel had been working, but was yet to make his critical breakthrough.

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An emission theory of light was one that regarded the propagation of light as the transport of some kind of matter. While the corpuscular theory was obviously an emission theory, the converse did not follow: in principle, one could be an emissionist without being a corpuscularist. This was convenient because, beyond the ordinary laws of reflection and refraction, emissionists never managed to make testable quantitative predictions from a theory of forces acting on corpuscles of light. But they did make quantitative predictions from the premises that rays were countable objects, which were conserved in their interactions with matter (except absorbent media), and which had particular orientations with respect to their directions of propagation. According to this framework, polarization and the related phenomena of double refraction and partial reflection involved altering the orientations of the rays and/or selecting them according to orientation, and the state of polarization of a beam (a bundle of rays) was a question of how many rays were in what orientations: in a fully polarized beam, the orientations were all the same. This approach, which Jed Buchwald has called selectionism, was pioneered by Malus and diligently pursued by Biot.

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Fresnel, in contrast, decided to introduce polarization into interference experiments.

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In July or August 1816, Fresnel discovered that when a birefringent crystal produced two images of a single slit, he could not obtain the usual two-slit interference pattern, even if he compensated for the different propagation times. A more general experiment, suggested by Arago, found that if the two beams of a double-slit device were separately polarized, the interference pattern appeared and disappeared as the polarization of one beam was rotated, giving full interference for parallel polarizations, but no interference for perpendicular polarizations (see Fresnel–Arago laws). These experiments, among others, were eventually reported in a brief memoir published in 1819 and later translated into English.

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In a memoir drafted on 30 August 1816 and revised on 6 October, Fresnel reported an experiment in which he placed two matching thin laminae in a double-slit apparatus—one over each slit, with their optic axes perpendicular—and obtained two interference patterns offset in opposite directions, with perpendicular polarizations. This, in combination with the previous findings, meant that each lamina split the incident light into perpendicularly polarized components with different velocities—just like a normal (thick) birefringent crystal, and contrary to Biot's "mobile polarization" hypothesis.

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Accordingly, in the same memoir, Fresnel offered his first attempt at a wave theory of chromatic polarization. When polarized light passed through a crystal lamina, it was split into ordinary and extraordinary waves (with intensities described by Malus's law), and these were perpendicularly polarized and therefore did not interfere, so that no colors were produced (yet). But if they then passed through an analyzer (second polarizer), their polarizations were brought into alignment (with intensities again modified according to Malus's law), and they would interfere. This explanation, by itself, predicts that if the analyzer is rotated 90°, the ordinary and extraordinary waves simply switch roles, so that if the analyzer takes the form of a calcite crystal, the two images of the lamina should be of the same hue (this issue is revisited below). But in fact, as Arago and Biot had found, they are of complementary colors. To correct the prediction, Fresnel proposed a phase-inversion rule whereby one of the constituent waves of one of the two images suffered an additional 180° phase shift on its way through the lamina. This inversion was a weakness in the theory relative to Biot's, as Fresnel acknowledged, although the rule specified which of the two images had the inverted wave. Moreover, Fresnel could deal only with special cases, because he had not yet solved the problem of superposing sinusoidal functions with arbitrary phase differences due to propagation at different velocities through the lamina.

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He solved that problem in a "supplement" signed on 15 January 1818 (mentioned above). In the same document, he accommodated Malus's law by proposing an underlying law: that if polarized light is incident on a birefringent crystal with its optic axis at an angle θ to the "plane of polarization", the ordinary and extraordinary vibrations (as functions of time) are scaled by the factors cos θ and sin θ, respectively. Although modern readers easily interpret these factors in terms of perpendicular components of a transverse oscillation, Fresnel did not (yet) explain them that way. Hence he still needed the phase-inversion rule. He applied all these principles to a case of chromatic polarization not covered by Biot's formulae, involving two successive laminae with axes separated by 45°, and obtained predictions that disagreed with Biot's experiments (except in special cases) but agreed with his own.

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Fresnel applied the same principles to the standard case of chromatic polarization, in which one birefringent lamina was sliced parallel to its axis and placed between a polarizer and an analyzer. If the analyzer took the form of a thick calcite crystal with its axis in the plane of polarization, Fresnel predicted that the intensities of the ordinary and extraordinary images of the lamina were respectively proportional to

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where i is the angle from the initial plane of polarization to the optic axis of the lamina, s is the angle from the initial plane of polarization to the plane of polarization of the final ordinary image, and ϕ ϕ is the phase lag of the extraordinary wave relative to the ordinary wave due to the difference in propagation times through the lamina. The terms in ϕ ϕ are the frequency-dependent terms and explain why the lamina must be thin in order to produce discernible colors: if the lamina is too thick, cos ⁡ ⁡ ϕ ϕ will pass through too many cycles as the frequency varies through the visible range, and the eye (which divides the visible spectrum into only three bands) will not be able to resolve the cycles.

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From these equations it is easily verified that I o + I e = 1 for all ϕ ϕ , so that the colors are complementary. Without the phase-inversion rule, there would be a plus sign in front of the last term in the second equation, so that the ϕ ϕ -dependent term would be the same in both equations, implying (incorrectly) that the colors were of the same hue.

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These equations were included in an undated note that Fresnel gave to Biot, to which Biot added a few lines of his own. If we substitute

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then Fresnel's formulae can be rewritten as

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

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Arago delayed reporting on Fresnel's works on chromatic polarization until June 1821, when he used them in a broad attack on Biot's theory. In his written response, Biot protested that Arago's attack went beyond the proper scope of a report on the nominated works of Fresnel. But Biot also claimed that the substitutions for U and A , and therefore Fresnel's expressions for I o and I e , were empirically wrong because when Fresnel's intensities of spectral colors were mixed according to Newton's rules, the squared cosine and sine functions varied too smoothly to account for the observed sequence of colors. That claim drew a written reply from Fresnel, who disputed whether the colors changed as abruptly as Biot claimed, and whether the human eye could judge color with sufficient objectivity for the purpose. On the latter question, Fresnel pointed out that different observers may give different names to the same color. Furthermore, he said, a single observer can only compare colors side by side; and even if they are judged to be the same, the identity is of sensation, not necessarily of composition. Fresnel's oldest and strongest point—that thin crystals were subject to the same laws as thick ones and did not need or allow a separate theory—Biot left unanswered. Arago and Fresnel were seen to have won the debate.

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Moreover, by this time Fresnel had a new, simpler explanation of his equations on chromatic polarization.

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