Augustin-Jean Fresnel

How to use this course

In 1814 most leading French physicists held that light was a stream of particles. By 1827 the wave theory had won, and the main reason was the work of one man who did optics in the spare time of an engineering career. Augustin-Jean Fresnel turned the qualitative wave ideas of Christiaan Huygens and Thomas Young into a quantitative theory. He computed diffraction patterns that matched measurement, showed that light waves vibrate transversely, derived the laws of reflection and refraction for polarized light, and explained double refraction in crystals. He also designed the lens that still bears his name.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. We use modern notation: complex amplitudes, wavenumber \(k=2\pi/\lambda\) and, where it shortens an argument, Maxwell’s theory. Fresnel reasoned with an elastic aether, and the lessons point out where his mechanical model and the modern electromagnetic one give the same equations.

Timeline

Broglie, Mathieu and Caen, 1788–1804

Augustin-Jean Fresnel was born on 10 May 1788 in Broglie, in Normandy. His father, Jacques Fresnel, was an architect; his mother, Augustine Mérimée, was the daughter of the steward of the Broglie estate. The family were Jansenists, members of a strict Catholic movement that the Church had condemned. Fresnel kept that faith all his life and regarded his talents as gifts he owed to others.

Around 1789 the family moved to Cherbourg, where Jacques worked on the harbour. In 1794, with the works halted by the Revolution, they withdrew to Mathieu, a village north of Caen. Augustine would live there as a widow for twenty-five years. Augustin was the second of four sons. Louis, the eldest, went through the École polytechnique and was killed in action in Spain. Léonor became a civil engineer and later edited his brother’s collected works. Fulgence became an orientalist and diplomat. Their mother’s brother, the painter Léonor Mérimée, taught at the École polytechnique, and his son was the writer Prosper Mérimée, Augustin’s cousin.

The boys were taught at home. Augustin was sickly and was thought slow; the story that he could barely read at eight is disputed. His playmates nevertheless called him “the man of genius” for the toy bows and guns he made and improved. In 1801 he was sent to the École centrale at Caen, mainly to keep Louis company, and there his interest in mathematics took hold.

The engineer, 1804–1814

In late 1804 Fresnel entered the École polytechnique in Paris, placed 17th in the entrance examination. He was strong in geometry and drawing, and in his first year he won a prize for a solution to a geometry problem set by Adrien-Marie Legendre, which was published in 1805. He left in 1806 for the École des ponts et chaussées and graduated in 1809. He then joined the state engineering corps, the Corps des ponts et chaussées, and remained in it for the rest of his life.

His first posting was in the Vendée. There, in 1811, he worked out a process for making soda ash from salt that anticipated the later Solvay process, though without its recycling of ammonia. Around 1812 he was sent to Nyons in the Drôme to build part of the imperial highway linking Spain and Italy. The work was dull and the region remote.

His letters from Nyons are the first evidence of his interest in light. On 15 May 1814, with work slack after Napoleon’s first defeat, he asked his brother Léonor to send him the papers of the French physicists on the polarization of light. He had seen a notice of Jean-Baptiste Biot’s memoir in the official newspaper, the Moniteur. He did not receive it until February 1815. At this point he knew almost nothing of the subject. He had not read Thomas Young, and he could not read English.

The Hundred Days and the prize, 1815–1819

In March 1815 Napoleon escaped from Elba. Fresnel, a royalist, regarded his return as an attack on civilization. He left his post without permission and went to Toulouse to offer his services to the royalist resistance, but fell ill. Back in Nyons he was threatened and his windows were broken. During the Hundred Days he was suspended from duty and went to his mother’s house at Mathieu. The enforced leisure gave him time for experiments.

He studied the fringes at the edges of shadows cast in light from a narrow source. He found that the outer fringes did not move in straight lines as the screen was moved back, which ruled out a simple deflection of particles at the edge. In July 1815, after Waterloo, he passed through Paris and met François Arago, who became his scientific patron. Through Arago he learned, by 1816, that Thomas Young had already explained some of these fringes by interference, and he acknowledged Young’s priority. In October 1815 Fresnel submitted his first memoir on diffraction to the Académie. In a supplement of July 1816 he described the double-mirror experiment, which produced interference with no edges involved at all (Lesson 04).

The prize competition

On 17 March 1817 the Académie announced diffraction as the subject of its Grand Prix. The commission was dominated by supporters of the particle theory, among them Pierre-Simon Laplace, Biot and Siméon Denis Poisson, with Arago and Joseph Louis Gay-Lussac more sympathetic to waves. In 1818 Fresnel was transferred to Paris. He deposited his memoir, under the motto Natura simplex et fecunda, on 29 July 1818, just before the deadline.

The memoir combined Huygens’s construction with Young’s principle of interference and computed fringe positions that matched Fresnel’s own careful measurements. Poisson, reading it, deduced that the centre of the shadow of a small disc should be bright, a consequence he took to be absurd. Arago made the experiment with a disc about 2 mm across and saw the spot. Fresnel won the prize in 1819. Arago later noted that Joseph-Nicolas Delisle and Giacomo Filippo Maraldi had seen the spot a century earlier without understanding it.

Interference and the double mirror

For the general reader. Two waves that meet can add or cancel. Light added to light can give darkness, a fact that no theory of particles explained naturally. Young had shown this with light passing through two openings. Fresnel’s double mirror produced the same effect with nothing but reflection, so that no one could blame the fringes on some action of an edge.

Two-source interference

Represent a monochromatic scalar wave by a complex amplitude \(U\), with intensity \(I=|U|^2\) averaged over a period. Two coherent waves \(U_1=A_1e^{i\phi_1}\), \(U_2=A_2e^{i\phi_2}\) give

Derivation. The distances from \((0,\pm d/2)\) to \((D,y)\) are \(r_\pm=\sqrt{D^2+(y\mp d/2)^2}\approx D+\frac{(y\mp d/2)^2}{2D}\). Then \(r_+-r_-\approx -\,yd/D\), so the phase difference is \(k\,yd/D\) in magnitude. Bright fringes sit where this is a multiple of \(2\pi\).

The double mirror

Light from a narrow slit \(S\) falls on two plane mirrors meeting along a line \(O\) at a small angle \(\alpha\) from being coplanar. Each mirror forms a virtual image of \(S\). Both images lie on a circle about \(O\) of radius \(r=|OS|\), and since rotating a mirror by \(\alpha\) rotates the image by \(2\alpha\) about \(O\), the images are separated by

The reflected beams overlap and give fringes of spacing \(i=\lambda(r+L)/(2r\alpha)\) on a screen at distance \(L\) from \(O\). Since the light never grazes an edge, the fringes come purely from the superposition of two reflected beams. Fresnel used the measured spacing to determine \(\lambda\). He later achieved the same splitting with a thin double prism, the Fresnel biprism.

The Huygens–Fresnel principle and the bright spot

For the general reader. Huygens had imagined every point of a wavefront as the source of a small new wave. He used the idea only to find where the next wavefront lies. Fresnel let the small waves interfere with each other. This single change turned a geometric picture into a calculating machine, and its first striking prediction was a bright spot where common sense expects the darkest shadow.

Fresnel introduced the obliquity factor and fixed the overall constant by requiring that an unobstructed wave reproduce itself. The factor \(1/i\) expresses that the secondary waves must lead the primary wave by a quarter period, a point Fresnel recognised and could not justify mechanically. Kirchhoff supplied the justification in the 1880s.

Fresnel approximation

For an aperture in the plane \(z=0\) and observation at \((x,y,z)\) with small angles, expand \(r\approx z+\frac{(x-x')^2+(y-y')^2}{2z}\) in the phase and set \(r\approx z\), \(K\approx1\) elsewhere:

Half-period zones

Seen from an axial point \(P\) at distance \(z\), divide a plane wavefront into rings whose edges are \(z+\lambda/2,\ z+\lambda,\ z+3\lambda/2,\dots\) from \(P\). The \(n\)-th ring has outer radius \(\rho_n\approx\sqrt{n\lambda z}\). Adjacent zones have nearly equal area and contribute with opposite signs, so the contribution of a free wave is about half that of the first zone alone.

Derivation. On the axis, polar coordinates give for the hole

using \(\lambda k=2\pi\). Then \(|1-e^{i\pi N}|^2=4\sin^2(\pi N/2)\). For the disc, integrate from \(R\) to \(\infty\) (with a convergence factor) or use Babinet’s principle: hole plus disc equals the free wave \(U_0e^{ikz}\), so the disc gives \(U_0e^{ikz}e^{i\pi N}\), of modulus \(|U_0|\).

This is Poisson’s spot. In the Fresnel approximation, the centre of the shadow is as bright as if the disc were absent. In practice the edge must be smooth to a small fraction of a zone width, which is why the spot is easy to miss.

The straight edge and the Fresnel integrals

For the general reader. Near the edge of any shadow there is a set of bright and dark bands, and light creeps a little way into the shadow. Fresnel computed these bands exactly. The calculation needed integrals that cannot be done in closed form, so he tabulated them by hand. His predicted positions matched his measurements to a few thousandths of a millimetre, the decisive evidence in his prize memoir.

Derivation. The \(y'\) integral in the Fresnel formula is a full Gaussian-type integral, \(\int e^{i\pi u^2/2}du=1+i\), after scaling. For the \(x'\) integral, set \(t=(x'-x)\sqrt{2/(\lambda z)}\). The open region \(x'>0\) becomes \(t>-w\), giving \(\int_{-w}^\infty e^{i\pi t^2/2}dt=\big(\tfrac12+C(w)\big)+i\big(\tfrac12+S(w)\big)\). The prefactor works out to \(1/(2i)\) times the \(1+i\) from the other integral, of modulus \(1/\sqrt2\), which gives the formula.

Reading the spiral

The intensity is half the squared length of the chord from the lower eye \((-\tfrac12,-\tfrac12)\) to the point at parameter \(w\). Deep in the shadow (\(w\to-\infty\)) the chord shrinks to zero. At the geometric edge (\(w=0\)) the chord reaches the origin, so \(I=I_0/4\). In the lit region the endpoint winds round the upper eye, and the chord oscillates about \(\sqrt2\), so the intensity oscillates about \(I_0\). The first maximum, at \(w\approx1.22\), is about \(1.37\,I_0\).

Transverse waves and polarization

For the general reader. Sound in air is a longitudinal wave: the air moves back and forth along the direction of travel. Light seemed to be the same kind of wave, but polarization shows that a beam of light has sidedness, as if it had a preferred direction across its path. Fresnel concluded that the vibration of light is entirely across the direction of travel. The idea was strange for a wave in a fluid, and he hesitated before publishing it.

The Fresnel–Arago laws

In 1816 Fresnel and Arago tried to make light beams polarized in perpendicular planes interfere, and failed. Their joint memoir appeared in 1819.

The first law is immediate if the vibration is a vector transverse to the ray: perpendicular vectors add with no cross term, \(|\mathbf E_1+\mathbf E_2|^2=|\mathbf E_1|^2+|\mathbf E_2|^2\). A longitudinal component would always interfere. Arago showed Fresnel a letter in which Young compared light to the waves on a shaken cord. Fresnel then drew the full conclusion, that the vibration has no longitudinal component, and published it in 1821. He acknowledged that Young, bolder in conjecture, had stated the idea first.

Describing polarization

For light travelling along \(z\), write the transverse field as \(\mathbf E=\mathrm{Re}\big[(E_x\hat x+E_y\hat y)e^{i(kz-\omega t)}\big]\) with complex \(E_x,E_y\). If they are in phase the light is linearly polarized. If \(E_y=\pm iE_x\) it is circularly polarized. In general the tip of \(\mathbf E\) traces an ellipse. Fresnel recognised all three cases, and that linear light is the sum of two opposite circular components:

Derivation. Resolve the incident amplitude \(E_0\) along the plate axes: \(E_0\cos\varphi\), \(E_0\sin\varphi\). The plate delays them by \(\delta=2\pi(n_1-n_2)d/\lambda\) relative to each other. Project on the analyser direction, perpendicular to the original: amplitude \(E_0\cos\varphi\sin\varphi\,(1-e^{i\delta})\), intensity \(I_0\tfrac14\sin^2 2\varphi\cdot4\sin^2(\delta/2)\). Since \(\delta\) depends on \(\lambda\), white light emerges coloured, and the colours of the two analyser settings are complementary. This was the phenomenon Arago and Biot had studied, and Fresnel’s account rested on interference after the analyser brings the two components to a common plane: law 4 above.

Optical rotation

Quartz cut perpendicular to its axis rotates the plane of linearly polarized light. Fresnel explained this in 1822 as circular double refraction: the two circular components travel with slightly different indices \(n_L,n_R\). He confirmed it by separating the two circular beams with a compound prism of right- and left-handed quartz.

The Fresnel equations and total reflection

For the general reader. A window reflects a little light and passes most of it. How much it reflects depends on the angle and on the polarization, and at one angle a polarization is not reflected at all. Fresnel found the exact formulas in 1823. He also discovered that light totally reflected inside glass has its polarization changed in a predictable way, and he built a block of glass that turns linear light into circular light.

Light passes from a medium of index \(n_1\) into one of index \(n_2\), with angle of incidence \(\theta_i\) and angle of refraction \(\theta_t\), \(n_1\sin\theta_i=n_2\sin\theta_t\). Call the component with field perpendicular to the plane of incidence s, and the component in that plane p.

Derivation (electromagnetic). For non-magnetic media the tangential components of \(\mathbf E\) and \(\mathbf H\) are continuous across the interface, and \(|\mathbf H|\propto n|\mathbf E|\). For s polarization, \(\mathbf E\) is tangential: \(E_i+E_r=E_t\), and the tangential \(\mathbf H\) gives \(n_1\cos\theta_i(E_i-E_r)=n_2\cos\theta_t E_t\). Solving gives \(r_s\) and \(t_s\). The p case is the same with the roles of \(\mathbf E\) and \(\mathbf H\) exchanged. Fresnel reached the same formulas half a century before Maxwell, from an elastic aether with energy conservation and continuity of the tangential vibration, choosing the density of the aether to differ between media.

Phase under total reflection

Write \(n=n_2/n_1<1\). Beyond the critical angle \(\cos\theta_t=i\sqrt{\sin^2\theta_i-n^2}/n\) is imaginary, and each \(r\) has the form \(\bar z/z\), of modulus 1. One finds

where \(\delta=\delta_p-\delta_s\) is the relative phase between the two components (signs depend on convention; the magnitudes do not). Fresnel saw that two total reflections, each giving \(\delta=45^\circ\), turn linear light at \(45^\circ\) to the plane of incidence into circular light. The glass block that does this is the Fresnel rhomb. Unlike a crystal plate, it works almost equally well for all colours.

Double refraction and the wave surface

For the general reader. A crystal of Iceland spar shows two images of anything seen through it. Huygens explained this in 1690 by supposing that one of the two waves spreads as a sphere and the other as a squashed ellipsoid. For crystals such as topaz, both images misbehave, and Huygens’s picture fails. In 1821 and 1822 Fresnel found a single surface, of fourth degree, that accounts for both waves in every crystal.

Setting up

In modern language, an anisotropic crystal has a symmetric dielectric tensor with principal values \(\varepsilon_x,\varepsilon_y,\varepsilon_z\). Define principal speeds \(a=c/\sqrt{\varepsilon_x}\), \(b=c/\sqrt{\varepsilon_y}\), \(c_3=c/\sqrt{\varepsilon_z}\), and order them \(a>b>c_3\). Fresnel had the same three constants as elasticities of the aether along three axes.

Derivation. For a plane wave \(e^{i(k\mathbf s\cdot\mathbf r-\omega t)}\), Maxwell’s equations give \(\mathbf D=-\frac{1}{\mu_0v^2}\,\mathbf s\times(\mathbf s\times\mathbf E)=\frac{1}{\mu_0v^2}\big(\mathbf E-\mathbf s(\mathbf s\cdot\mathbf E)\big)\). With \(D_j=\varepsilon_0\varepsilon_jE_j\) and \(c^2=1/(\varepsilon_0\mu_0)\), the \(j\)-th component reads \(E_j\big(v^2-v_j^2\big)=-v_j^2\,s_j\,(\mathbf s\cdot\mathbf E)\) with \(v_j\in\{a,b,c_3\}\). Hence \(D_j\propto s_j/(v^2-v_j^2)\). Transversality \(\mathbf s\cdot\mathbf D=0\) is then exactly the stated equation.

Optic axes

In the plane \(s_y=0\), clearing denominators gives \((v^2-b^2)\big[s_x^2(v^2-c_3^2)+s_z^2(v^2-a^2)\big]=0\). The roots are \(v=b\) and \(v^2=s_x^2c_3^2+s_z^2a^2\). They coincide when

So there are two optic axes, symmetric about the \(z\)-axis in the \(xz\)-plane: the crystal is biaxial. If two of the principal speeds are equal, the axes merge into one and the crystal is uniaxial.

Fresnel also found the corresponding ray surface, the shape of a wave spreading from a point:

It is a two-sheeted surface of degree four with four conical points. In 1832 William Rowan Hamilton studied these singular points and predicted conical refraction, which Humphrey Lloyd then observed in aragonite.

Moving media: the drag coefficient

For the general reader. The Earth moves through space at about 30 km/s. If light is a wave in an aether, that motion should show up in optical experiments, yet Arago found in 1810 that the bending of starlight by a prism was unaffected. In 1818 Fresnel explained the null result by supposing that moving glass drags the light inside it only partly. His formula was confirmed by Hippolyte Fizeau in 1851, and the correct explanation arrived only with Einstein’s relativity.

Fresnel obtained \(f\) by supposing that the aether density inside a body is proportional to \(n^2\) and that only the excess over the outside density is carried along. With this coefficient, first-order effects of the Earth’s motion cancel in refraction experiments, which explained Arago’s result.

Max von Laue made this observation in 1907. What Fresnel described as partial dragging of an aether is a kinematic consequence of special relativity, with no aether involved.

Lighthouses and the Académie, 1819–1825

On 21 June 1819 Fresnel was seconded to the Commission des phares, the body responsible for France’s lighthouses. Existing lights used metal reflectors behind lamps and lost about half the light. By the end of August 1819 he proposed replacing them with “lentilles à échelons”, stepped lenses built from concentric prismatic rings. Georges-Louis Leclerc, comte de Buffon, had earlier suggested a lens cut in steps from a single block; Fresnel arrived at the idea independently and made it practical by assembling it from separate pieces. A large lens would otherwise be too thick to cast, too heavy, and would absorb too much light.

A public demonstration on 13 April 1821 showed that the new lens far outperformed reflectors. In 1822 an eight-panel rotating lens was tested on the unfinished Arc de Triomphe. Its first installation was at the Cordouan lighthouse in the Gironde estuary, lit on 25 July 1823. In May 1824 Fresnel became secretary of the commission. He then designed fixed lenses giving a steady beam all round the horizon, drew up a plan for a national network of lights, and in late 1825 proposed catadioptric prisms that use total internal reflection (Lesson 08) to capture rays a lens would miss. These elements remain the basis of lighthouse optics.

Recognition

Fresnel was elected to the Société philomathique in April 1819. His first candidacy for the Académie des sciences, in May 1817, received one vote. In February 1823 he lost an election to Pierre Louis Dulong, 36 votes to 20. In May 1823 he was elected unanimously. He became a chevalier of the Légion d’honneur in 1824 and a Foreign Member of the Royal Society of London on 9 June 1825. He also served as an examiner at the École polytechnique. A note of 1821, found later among André-Marie Ampère’s papers, shows him discussing the molecular currents by which Ampère explained magnetism.

Last years, death and legacy, 1824–1827

Fresnel’s health declined in the winter of 1822–23, and tuberculosis was diagnosed. In 1824 he was advised to reduce his work. He resigned as examiner at the École polytechnique and closed his scientific notebooks, while continuing the lighthouse programme. His last note to the Académie, read on 13 June 1825, concerned the radiometer effect, which he attributed to temperature differences. In 1826 he answered at length a set of questions on the wave theory from John Herschel.

He attended the Académie for the last time on 30 April 1827. In early June he was taken to Ville-d’Avray, west of Paris, where his mother joined him. On 6 July Arago brought him the Rumford Medal of the Royal Society, awarded for 1824. Fresnel died on 14 July 1827, aged 39. He is buried in the Père Lachaise cemetery. A second memoir on double refraction was printed a few months later. His collected works, edited by Henri de Sénarmont, Émile Verdet and his brother Léonor, appeared in three volumes from 1866 to 1870.

Legacy

• The wave theory. By the 1830s the wave theory was accepted across Europe. William Whewell later wrote that Young and Fresnel together made up “the Newton of optical science”.
• Mathematical optics. The Huygens–Fresnel principle, Fresnel diffraction, Fresnel integrals and zones, the Fresnel number, and Fresnel’s wave surface are standard tools. Kirchhoff’s scalar diffraction theory and modern Fourier optics descend from them.
• Polarization and electromagnetism. The Fresnel equations survived intact when James Clerk Maxwell showed that light is an electromagnetic wave; transversality became a property of the field.
• Moving media. The drag coefficient, confirmed by Fizeau, became one of the experimental facts that special relativity had to explain.
• Engineering. Within a century of his proposal, more than ten thousand lights used Fresnel lenses, and the stepped design is used today in headlamps, projectors and solar concentrators.

Connections

People linked to Augustin-Jean Fresnel across this collection. A name without a link has no page here yet. See Augustin-Jean Fresnel in the network of all 337 lives →

Influenced by

• Adrien-Marie Legendre
• Christiaan Huygens
• Thomas Young
• Étienne-Louis Malus

Influenced

• Augustin-Louis Cauchy
• George Biddell Airy
• George Gabriel Stokes
• Hippolyte Fizeau
• James Clerk Maxwell
• William Rowan Hamilton

Colleagues and collaborators

• André-Marie Ampère
• François Arago 1815-1827
• John Herschel
• Joseph Louis Gay-Lussac
• Joseph von Fraunhofer
• Pierre-Simon Laplace

Rivals

• David Brewster
• Jean-Baptiste Biot
• Siméon Denis Poisson 1818-1819

Friends

• André-Marie Ampère 1822-1827

Also mentioned in this life

• Albert Einstein
• Leonhard Euler
• Joseph Fourier
• Gustav Kirchhoff
• Isaac Newton
• Count Rumford
• Max von Laue

Fields

• engineering 6 lives

On CoursesHub

• Wave Optics · Optics & Photonics

Sources and further reading

Biography

1. Augustin Jean Fresnel, MacTutor History of Mathematics, University of St Andrews.
2. Augustin-Jean Fresnel, Wikipedia (detailed, with primary references).
3. Augustin Fresnel, Photoniques, special issue on Fresnel, 2019.
4. F. Arago, Fresnel (éloge), in Œuvres complètes de François Arago, vol. 1, 1854.
5. J. Z. Buchwald, The Rise of the Wave Theory of Light, University of Chicago Press, 1989.
6. T. Levitt, A Short Bright Flash: Augustin Fresnel and the Birth of the Modern Lighthouse, Norton, 2013.

Physics

1. A. Fresnel, Œuvres complètes d’Augustin Fresnel, ed. H. de Sénarmont, É. Verdet and L. Fresnel, 3 vols, Paris, 1866–1870.
2. A. Fresnel, Mémoire sur la diffraction de la lumière (prize memoir, 1818), Mémoires de l’Académie des sciences 5 (1826).
3. F. Arago and A. Fresnel, Mémoire sur l’action que les rayons de lumière polarisés exercent les uns sur les autres, Annales de chimie et de physique 10 (1819).
4. Arago spot, Wikipedia.
5. M. Born and E. Wolf, Principles of Optics, Cambridge University Press (chapters on diffraction and crystal optics).
6. E. Hecht, Optics, Pearson (Fresnel equations, Fresnel diffraction, the Cornu spiral).
7. E. T. Whittaker, A History of the Theories of Aether and Electricity, vol. 1, 1910; revised 1951.
8. W. Whewell, History of the Inductive Sciences, 3rd ed., 1857, vol. II, book IX.