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'''Augustin-Jean Fresnel''' ({{IPAc-en|f|r|eɪ|ˈ|n|ɛ|l}} {{respell|fray|NEL|'}}; {{IPA-fr|ɔ.ɡy.stɛ̃ ʒɑ̃ fʁɛ.nɛl|lang}}; 10 May 1788 – 14 July 1827) was a [[France|French]] civil [[engineer]] and [[physicist]] whose research in [[optics]] led to the almost universal acceptance of the wave theory of light, and the rejection of any remnant of [[Isaac Newton|Newton]]'s [[corpuscular theory of light|corpuscular theory]], from the 1830s<ref>Darrigol, 2012, pp.
But 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 [[lighthouse|lighthouses]], saving unknown numbers of lives at sea. The simpler ''dioptric'' (purely refractive) stepped lens, first proposed by [[Georges-Louis Leclerc, Comte de Buffon|Count Buffon]]{{r|chisholm-1911-lighthouse}} and independently reinvented by Fresnel, is used in screen [[magnifying glass|magnifiers]] and in condenser lenses for [[overhead projector|overhead projectors]].
By expressing [[Christiaan Huygens|Huygens]]' principle of secondary waves and [[Thomas Young (scientist)|Young]]'s principle of [[interference (wave propagation)|interference]] in quantitative terms, and supposing that simple colors consist of ''[[sine wave|sinusoidal]]'' waves, Fresnel gave the first satisfactory explanation of [[diffraction]] by straight edges, including the first explanation of rectilinear propagation that would satisfy a modern physicist.<ref>Darrigol, 2012, p.{{hsp}}205</ref> By further supposing that light waves are purely ''[[transverse wave|transverse]]'', he explained the nature of [[polarization]] and lack thereof, the mechanism of ''chromatic polarization'' (the colors produced when polarized light is passed through a slice of doubly-refractive crystal followed by a second polarizer), and the [[transmission coefficient|transmission]] and [[reflection coefficient]]s at a boundary between transparent [[isotropy|isotropic]] media (including [[Brewster's angle]]). Then, by generalizing the relationship between wave speed and polarization for [[calcite]], he accounted for the directions and polarizations of the refracted rays in [[birefringence|doubly-refractive]] crystals of the ''biaxial'' class (those for which Huygens' secondary wavefronts are not axisymmetric). The period between the first publication of his pure-transverse-wave hypothesis and the presentation of his solution to the biaxial problem was less than a year. Later, by allowing the reflection coefficient to be ''[[complex number|complex]]'', he 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.
Fresnel's legacy is the more remarkable in view of his lifelong battle with [[tuberculosis]], to which he succumbed at the age of 39. Although he did not become a public celebrity in his short 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 recurs frequently in the modern terminology of optics and waves.
Inevitably, after the wave theory of light was subsumed by [[James Clerk Maxwell|Maxwell]]'s [[electromagnetism|electromagnetic]] theory in the 1860s and '70s, the magnitude of Fresnel's contribution was somewhat obscured. In the period between Fresnel's unification of physical optics and Maxwell's wider unification, a contemporary authority, Professor [[Humphrey Lloyd (physicist)|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."{{r|lloyd-1841}}
== Early life ==
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=== Family ===
Augustin-Jean Fresnel (also called Augustin Jean or simply Augustin), born in [[Broglie, Eure|Broglie]], [[Normandy]], on 10 May 1788, was the second of four sons of the architect Jacques Fresnel (1755–1805){{r|favre}} and his wife Augustine, ''née'' Mérimée (1755?–1833).{{r|jeanelie}} In 1790, following the [[French Revolution|Revolution]], Broglie became part of the [[Departments of France|département]] of [[Eure]]. The family moved twice — in 1790 to [[Cherbourg-Octeville|Cherbourg]],<ref>Levitt, 2013, p.{{hsp}}23.</ref> and in 1794{{r|silliman-2008|p=166}} to Jacques' home town of [[Mathieu, Calvados|Mathieu]], where Madame Fresnel remained as a widow,{{r|boutry|p=590}} outliving two of her sons.
The first son, Louis (1786–1809), was admitted to the [[École Polytechnique]], became a lieutenant in the artillery, and was killed at [[Jaca]], [[Spain]], the day before his 23rd birthday.{{r|jeanelie}} The third, Léonor (1790–1869),{{r|favre}} followed Augustin into civil [[engineer]]ing, succeeded him as Secretary of the Lighthouse Commission,<ref>Levitt, 2013, p.{{hsp}}99.</ref> and helped to edit his collected works
Their mother's brother Léonor Mérimée (1757–1836),{{r|jeanelie}} father of the writer [[Prosper Mérimée]] (1803–1870), was a painter who studied the chemistry of painting. He became the Permanent Secretary of the [[École des Beaux-Arts]] (School of Fine Arts) and a professor at the École polytechnique, and was the initial point of contact between Augustin and the
=== Education ===
Augustin and his brothers were initially home-schooled by their mother. Augustin, a sickly child, was considered the slow one, just beginning to read at the age of eight. At ten he was undistinguished except for his ability to turn tree-branches into toy bows and cannon that were too dangerous to play with, provoking a crackdown from his elders.<ref>Levitt, 2013, pp.{{nnbsp}}24–5.</ref>{{r|boutry|p=590–91}}
In 1801, Augustin was sent to the ''École centrale'' at [[Caen]], as company for Louis. But Augustin lifted his performance: in 1804 he was accepted into the École Polytechnique, being placed 17th in the entrance examination, in which his solutions to geometry problems impressed the examiner, [[Adrien-Marie Legendre]]. As the surviving records of the École Polytechnique begin in 1808, we know little of Augustin Fresnel's time there, except that he apparently excelled in geometry and drawing — in spite of continuing poor health — and made few if any friends. Graduating in 1806, he enrolled at the [[École des ponts ParisTech|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 of Bridges, Waters and Forests|Corps des Ponts et Chaussées]] as an ''ingénieur ordinair aspirant'' (ordinary engineer in training). Directly or indirectly, the "Corps des Ponts" would be his sole or main employer for the rest of his life.{{r|chisholm-1911-fresnel}}<ref>Levitt, 2013, pp.{{nnbsp}}25–7.</ref>{{r|boutry|p=591–2,601}}
=== Influence of religion ===
Fresnel's parents were [[Catholic Church|Roman Catholics]] of the [[Jansenism|Jansenist]] sect, characterized by an extreme [[Augustine of Hippo|Augustinian]] view of [[original sin]]. In the
Augustin Fresnel remained a Jansenist.<ref>Levitt, 2013, p.{{hsp}}24.</ref> He indeed regarded his intellectual talents as a gift from God, and considered it his duty to use them for the benefit of others.{{r|kneller-1911|p=148}} Plagued by poor health, and determined to do his duty before death thwarted him, he shunned pleasures and worked himself to exhaustion.{{r|silliman-2008|p=166}} 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. Yet in his last days he needed "strength of soul," not against death alone, but against "the interruption of discoveries… of which he hoped to derive useful applications."{{r|kneller-1911|p=148–9n}} Although Jansenism is considered [[heresy|heretical]] by the Roman Catholic Church, the brief article on Fresnel in the ''[[Catholic Encyclopedia]]'' (1909) does not mention his Jansenism, but describes him as "a deeply religious man and remarkable for his keen sense of duty."{{r|brock-1909}}▼
His early progress in learning was slow, and when eight years old he was still unable to read. At the age of thirteen he entered the École Centrale in Caen, and at sixteen and a half the École Polytechnique, where he acquitted himself with distinction. Thence he went to the École des Ponts et Chaussées. He served as an engineer successively in the departments of Vendée, Drôme and Ille-et-Vilaine; but his espousal of the cause of the Bourbons in 1814 occasioned, on Napoleon’s reaccession to power, the loss of his appointment. On the second restoration he obtained a post as engineer in Paris.{{r|chisholm-1911-fresnel}}▼
▲Fresnel's parents were [[Catholic Church|Roman Catholics]] of the [[Jansenism|Jansenist]] sect, characterized by an extreme [[Augustine of Hippo|Augustinian]] view of [[original sin]]. In the early home-schooling that the four boys received from their mother, religion took first place. In 1802, Mme Fresnel wrote to Louis concerning Augustin:
▲Augustin Fresnel indeed regarded his intellectual talents as a gift from God, and considered it his duty to use them for the benefit of others. Plagued by poor health, and determined to do his duty before death thwarted him, he shunned pleasures and worked himself to exhaustion.{{r|silliman-2008|p=166}} According to his fellow engineer Alphonse Duleau, who helped to nurse him through his final illness, Fresnel saw the study of nature as the study of the power and goodness of God. He placed virtue above science and genius. Yet in his last days he needed "strength of soul," not against death alone, but against "the interruption of discoveries… of which he hoped to derive useful applications."{{r|kneller-1911|p=148–9n}} Although Jansenism is considered [[heresy|heretical]] by the Roman Catholic Church, the brief article on Fresnel in the ''[[Catholic Encyclopedia]]'' (1909) does not mention his Jansenism, but describes him as "a deeply religious man and remarkable for his keen sense of duty."{{r|brock-1909}}
== Engineering assignments ==
▲
During the [[Hundred Days]] he was ''persona non grata'', but after [[Battle of Waterloo|Waterloo]] he returned to Paris to his former occupation.{{r|rines-1919}}
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In 1818 he read a memoir on diffraction for which in the ensuing year he received the prize of the Académie des Sciences at Paris.{{r|fresnel-1819b}}
The [[Fresnel diffraction]] equation is an approximation of [[Kirchhoff's diffraction formula|Kirchhoff-Fresnel diffraction]] that can be applied to the propagation of waves in the [[near and far field|near field]].<ref>[[Max Born|M. Born]] & E. Wolf, Principles of Optics, 1999, Cambridge University Press, Cambridge</ref> It is used to calculate the [[diffraction pattern]] created by waves passing through an aperture or around an object, when viewed from relatively close to the object. In contrast the diffraction pattern in the [[near and far field|far field]] region is given by the [[Fraunhofer diffraction]] equation.
=== Polarization ===
=== Partial reflection ===
[[|reflectance|reflectivity]], [[reflection coefficient]], [[Fresnel equations]], [[computer graphics]], rendering of water.
Circularly polarized light he obtained by means of a rhomb of glass, known as "Fresnel’s rhomb", having obtuse angles of 126°, and acute angles of 54°.{{r|chisholm-1911-fresnel}}
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demonstration of the phenomenon of the stellar
aberration.{{r|ripley-dana-1879}}
[[aberration of light]], not published; [[aether drag hypothesis]]
=== Reception ===
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=== Prior art ===
Fresnel was not the first person to focus a lighthouse beam using a lens. That distinction apparently belongs to the London glasscutter Thomas Rogers, who proposed the idea to [[Trinity House]] in 1788.{{r|tag-prior}} The first Rogers lenses, 53cm in diameter and 14cm thick at the center, were installed at the [[Old Lower Lighthouse]] at [[Portland Bill]] in 1789.<ref>Levitt, 2013, p.{{hsp}}57.</ref> Further samples followed at [[Baily Lighthouse|Howth Baily]], [[North Foreland]], and at least four other locations.{{r|tag-prior}} But much of the light was wasted by absorption in the glass.
[[File:Fresnel lens.svg|thumb|upright|1: Cross-section of Buffon/Fresnel lens. 2: Cross-section of conventional [[Lens (optics)#Types of simple lenses|plano-convex lens]] of equivalent power. (Buffon's version was [[Lens (optics)#Types of simple lenses|biconvex]].<ref>Levitt, 2013, p.{{hsp}}59.</ref>)]]
Nor was Fresnel the first to suggest replacing a convex lens with a series of concentric annular prisms, to reduce weight and absorption. In 1748, [[Georges-Louis Leclerc, Comte de Buffon|Count Buffon]] proposed grinding such prisms as steps in a single piece of glass.{{r|chisholm-1911-lighthouse}} In 1790{{r|condorcet-1790}} (although secondary sources give the date as 1773{{r|appleton-1861|p=609}} or 1788{{r|tag-2017}}), the [[Marquis de Condorcet]] suggested that it would be easier to make the annular sections separately and assemble them on a frame; but even that was impractical at the time.{{r|tag-fres}}<ref>Levitt, 2013, p.{{hsp}}71.</ref> These designs were intended not for lighthouses,{{r|chisholm-1911-lighthouse}} but for [[burning glass|burning glasses]].{{r|appleton-1861|p=609}} Brewster, however, proposed a system similar to Condorcet's in 1811,{{r|chisholm-1911-lighthouse|ripley-dana-1879|tag-2017}} and by 1820 was advocating its use in British lighthouses.{{r|chisholm-1911-brewster}}
=== Prototypes ===
Meanwhile, in June 1819, Fresnel was engaged by the ''Commission des phares'' (Commission of Lighthouses) on the recommendation of Arago (a member of the Commission since 1813), to review possible improvements in lighthouse illumination.{{r|tag-fres}} The Commission had been established by Napoleon in 1811, and placed under the ''Corps des ponts'' — Fresnel's employer.<ref>Levitt, 2013, pp.
On 29 August 1819, unaware of the Buffon-Condorcet-Brewster proposal,{{r|ripley-dana-1879|tag-fres}} Fresnel presented his first report, in which he recommended what he called ''lentilles à échelons'' (lenses by steps) to replace the reflectors then in use, which reflected only about half of the incident light.<ref>Levitt, 2013, pp.
(Fresnel acknowledged the British lenses and Buffon's invention in a memoir published in 1822.{{r|fresnel-1822-phares|p=2–4}}. The date of that memoir may be the source of the claim that Fresnel's lighthouse advocacy began two years later than Brewster's;{{r|chisholm-1911-brewster}} but the text makes it clear that Fresnel's involvement began no later than 1819.{{r|fresnel-1822-phares|p=1}})
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[[File:Fresnel lighthouse lens diagram.png|thumb|246px|Cross-section of a first-generation Fresnel lighthouse lens, with sloping mirrors ''m,n'' above and below the refractive panel ''RC'' (with central segment ''A''). If the cross-section in every vertical plane through the lamp ''L'' is the same, the light is spread evenly around the horizon.]]
Fresnel's next lens was a rotating apparatus with eight "bull's-eye" panels made in annular arcs by [[Saint-Gobain]],<ref>Levitt, 2013, p.{{hsp}}71.</ref>, giving eight rotating beams, to be seen by mariners as a periodic flash. Above and behind each main panel was a smaller, sloping bull's-eye panel of trapezoidal outline with trapezoidal elements.{{r|gombert-2017}} This refracted the light to a sloping plane mirror, which then reflected it horizontally, 7 degrees ahead of the main beam, increasing the duration of the flash.{{r|fresnel-1822-phares|p=13,25}} Below the main panels were 128 small mirrors arranged in four rings, stacked like the slats of a [[louver]] or [[Venetian blind]]. Each ring, shaped like a [[frustum]] of a [[cone]], reflected the light to the horizon, giving a fainter steady light between the flashes. The official test, conducted on the ''[[Arc de Triomphe]]'' on 20 August 1822, was witnessed by the Commission — and by [[Louis XVIII of France|Louis XVIII]] and his entourage — from 32km away. The apparatus was stored at [[Bordeaux]] for the winter, and then reassembled at [[Cordouan Lighthouse]] under Fresnel's supervision. On 25 July 1823, the world's first lighthouse Fresnel lens was lit.<ref>Levitt, 2013, pp.
In 1824, Fresnel designed the first ''fixed'' lens — for spreading light evenly around the horizon{{r|tag-fres}} while minimizing waste above or below. This had the familiar reflecting (''catoptric'') rings above and below the refracting (''dioptric'') panels. But the curved refracting surfaces were segments of toroids about a common vertical axis, so that the dioptric panel looked like a cylindrical drum and the entire apparatus looked like a beehive.
In May of the same year,{{r|ripley-dana-1879}} Fresnel was promoted to Secretary of the ''Commission des phares'', becoming the first member of that body to draw a salary.<ref>Levitt, 2013, p.{{hsp}}82.</ref> He was also an examiner at the École Polytechnique (since 1821),{{r|brock-1909}} but poor health soon induced him to resign that post and save his energy for his lighthouse work.<ref>Levitt, 2013, p.{{hsp}}97.</ref> The following year he unveiled the ''Carte des phares'' (Lighthouse Map), calling for a system of 51 lighthouses plus smaller harbor lights, in a hierarchy of lens sizes (called ''orders'', the first order being the largest), with different characteristics to facilitate recognition: a constant light (from a fixed lens), one flash per minute (from a rotating lens with eight panels), and two per minute (sixteen panels). On 1 February 1825, the second lighthouse Fresnel lens entered service: a third-order fixed lens at Dunkirk.<ref>Levitt, 2013, pp.
Also in 1825, Fresnel extended his fixed design by adding a rotating array outside the fixed array.{{r|tag-fres}} Each panel of the rotating array refracted part of the fixed light from a horizontal fan into a narrow beam.
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[[File:MuseeMarine-phareFresnel-p1000466.jpg|thumb|left|First-order rotating catadioptric Fresnel lens, dated 1870, on display at the ''[[Musée national de la Marine]]'', Paris. In this case the dioptric prisms (inside the bronze circles) and catadioptric prisms (outside) are arranged to give a purely flashing light with four flashes per rotation. The assembly stands 2.54 metres tall and weighs about 1.5 tonnes.]]
To reduce the loss of light in the reflecting elements, Fresnel proposed to replace the mirrors with ''catadioptric'' prisms, through which the light would pass by two refractions and one [[total internal reflection]].<ref>Levitt, 2013, pp.
The first large catadioptric lenses were made in 1842 for the lighthouses at Gravelines and [[Île Vierge]]; these were fixed third-order lenses whose catadoptric rings (made in segments) were one metre in diameter. The first-order [[Skerryvore]] lens, installed in 1844, was only partly catadoptric; it was similar to the Cordouan lens except that the lower slats were replaced by French-made catadioptric prisms, while mirrors were retained at the top. The first ''fully'' catadioptric first-order lens, installed at Ailly in 1852, also gave eight rotating beams plus a fixed light at the bottom; but its top section had eight catadioptric panels focusing the light about 4 degrees ahead of the main beams, in order to lengthen the flashes. The first fully catadioptric lens with ''purely revolving'' beams — also of first order — was installed at [[Saint-Clément-des-Baleines]] in 1854, and marked the completion of Fresnel's original ''Carte des phares''.<ref>Levitt, 2013, pp.
[[File:Flat flexible plastic sheet lens.JPG|thumb|Close-up view of a thin plastic Fresnel lens.]]
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In 1827 he received the [[Rumford Medal]].{{r|rines-1919|chisholm-1911-fresnel}}
In 1824,<ref>Levitt, 2013, p.{{hsp}}77.</ref> he was made a ''chevalier de la Légion d'honneur'' (Knight of the [[Legion of Honour]]).{{r|academie}}
[[List of the 72 names on the Eiffel Tower|72 names on the Eiffel Tower]] (south-east side, 4th
from the left).
== Decline and death ==
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[[File:Cordouan6.jpg|thumb|The lantern room of the [[Cordouan Lighthouse]], in which the first Fresnel lens entered service in 1823. The current fixed catadioptric "beehive" lens replaced Fresnel's original rotating lens in 1854.{{r|pharedeC}}]]
With a century after Fresnel's initial proposal, more than 10,000 lights with Fresnel lenses marked coastlines around the world.<ref>Levitt, 2013, p.{{hsp}}19.</ref> The numbers of lives saved can only be guessed at. Concerning the other benefits, science historian Theresa H. Levitt has remarked:{{quote|Everywhere I looked, the story repeated itself. The moment a Fresnel lens appeared at a ___location was the moment that region becamed linked into the world economy.<ref>Levitt, 2013, p.{{hsp}}8.</ref>}}
In the history of physical optics, Fresnel's successful revival of the wave theory seems to identify him as the pivotal figure between Newton, who held that light consisted of corpuscles, and [[James Clerk Maxwell|Maxwell]], who established that light waves are electromagnetic. Whereas [[Albert Einstein|Einstein]] described Maxwell's work as "the most profound and the most fruitful that physics has experienced since the time of Newton,"{{r|jamesCMF}} commentators of the era between Fresnel and Maxwell made similarly strong statements about Fresnel:
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* MacCullagh, as early as 1830, wrote that Fresnel's mechanical theory of double refraction "would do honour to the sagacity of Newton".{{r|macCullagh-1830|p=78}}.
* Lloyd, after his experimental confirmation of conical refraction, lived for another 48 years. In 1834, in his ''Report on the progress and present state of physical optics'' for the [[British Association for the Advancement of Science|British Science Association]], he wrote:
* [[William Whewell]], in all three editions of his ''History of the Inductive Sciences'' (1837, 1847, and 1857), at the end of Book IX, compared the histories of physical astronomy and physical optics and concluded:{{quote|It would, perhaps, be too fanciful to attempt to establish a parallelism between the prominent persons who figure in these two histories. If we were to do this, we must consider Huyghens and Hooke as standing in the place of Copernicus, since, like him, they announced the true theory, but left it to a future age to give it development and mechanical confirmation; Malus and Brewster, grouping them together, correspond to [[Tycho Brahe]] and [[Johannes Kepler|Kepler]], laborious in accumulating observations, inventive and happy in discovering laws of phenomena; and Young and Fresnel combined, make up the Newton of optical science.{{r|whewell-1857|p=370-71}} }}
{{clear}}
[[File:David d'Angers - Fresnel.jpg|thumb|upright
What Whewell called the "true theory" has since undergone two major revisions. The first, by Maxwell, specified the physical fields whose variations constitute the waves of light. The second, initiated by Einstein's explanation of the [[photoelectric effect]], supposed that the energy of light waves was divided into [[quantum|quanta]], which were eventually identified with particles called [[photon|photons]]. But photons did not exactly correspond to Newton's corpuscles; for example, Newton's explanation of ordinary refraction required the corpuscles to travel faster in media of higher refractive index, which photons do not. Nor did photons displace waves; rather, they led to the paradox of [[wave–particle duality]].
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<ref name=favre>J.H. Favre, "Augustin Fresnel", gw.geneanet.org, accessed 30 August 2017.</ref>
<ref name=fresnel-1819b>A. Fresnel, "Mémoire sur la diffraction de la lumière" (deposited 1818, "crowned" 1819), in ''Oeuvres complètes'', [https://books.google.com/books?id=1l0_AAAAcAAJ v.1], pp. 247–364
<ref name=fresnel-1822-phares>A. Fresnel, "Mémoire sur un nouveau système d'éclairage des phares", read at the Académie des Sciences
<ref name=gombert-2017>D. Gombert, photograph of the [https://www.flickr.com/photos/gebete29/32970312394/in/photostream/
<ref name=jamesCMF>James Clerk Maxwell Foundation, [http://www.clerkmaxwellfoundation.org/html/about_maxwell.html "Who was James Clerk Maxwell?"], accessed 6 August 2017; [https://web.archive.org/web/20170630003106/http://www.clerkmaxwellfoundation.org/html/about_maxwell.html archived] 30 June 2017.</ref>
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<ref name=pharedeC>Phare de Cordouan, [http://www.phare-de-cordouan.fr/lighting-systems.html "The lighting systems of the Cordouan Lighthouse"], accessed 26 August 2017; [https://web.archive.org/web/20160922153001/http://www.phare-de-cordouan.fr/lighting-systems.html archived] 22 September 2016.</ref>
<ref name=rines-1919>G.E. Rines (ed.), "Fresnel, Augustin Jean", ''Encyclopedia Americana'', 1918–20, v.12 (1919), [https://babel.hathitrust.org/cgi/pt?id=wu.89094370657;view=1up;seq=111 p.93]. (
<ref name=ripley-dana-1879>G. Ripley & C.A. Dana (ed.), "Fresnel, Augustin Jean", ''American Cyclopedia'', [https://archive.org/details/americancyclopae07ripluoft v.7], pp.486–9.</ref> (Contrary to this entry [p.486], calcite and quartz were ''not'' the only doubly refractive crystals known before Fresnel; see text.)
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