Étienne-Louis Malus

How to use this course

Étienne-Louis Malus did nearly all his science between 1807 and 1811, in the gaps of a career as a military engineer, and died at 36. In that time he founded the geometrical theory of ray systems, gave the first precise test of Christiaan Huygens’s law of double refraction, discovered that reflection polarizes light, coined the word polarization, and found the law \(I=I_0\cos^2\theta\) that every physics student learns. He interpreted all of it with the particle theory of light. The wave theory that replaced it within a decade kept every one of his laws.

Each technical lesson opens with a paragraph for the general reader, then gives precise statements, derivations and exercises with folded solutions. The order follows his work: the analytical optics of 1807, the double-refraction experiments of 1807–1808, the discovery of polarization by reflection in late 1808, and the prize memoir of 1810. A companion course on Jean-Baptiste Biot continues the story of polarization after Malus’s death.

Timeline

Paris, Mézières, the army, 1775–1798

Étienne-Louis Malus was born in Paris on 23 July 1775. His father, Anne-Louis Malus du Mitry, held the office of treasurer of France; his mother was Louise-Charlotte Desboves. He was educated at home in Greek, Latin and mathematics.

In 1793 he passed the entrance examination of the military engineering school at Mézières with distinction, and Gaspard Monge noticed his talent. The Revolution then caught up with him. His father’s royal office made the family suspect, and Malus was dismissed. He enlisted as a private in a Paris volunteer battalion and was sent to dig fortifications at Dunkirk.

An engineer on the works recognized his ability and sent him to the new École centrale des travaux publics, soon renamed the École Polytechnique. He entered in 1794 with its first students. Monge became his friend, and Joseph Fourier was among his teachers. While still a student Malus began research on the passage of light through media of different refractive index.

He was commissioned sub-lieutenant of engineers on 20 February 1796 and captain on 19 June 1796, taught mathematics at the engineering school in Metz, and served with the army on the Rhine. Stationed at Giessen, he became engaged to Wilhelmine-Louise Koch, eldest daughter of the university’s chancellor. Before they could marry, he was ordered to join an expedition whose destination was secret.

Egypt and Syria, 1798–1801

The destination was Egypt. Malus sailed from Toulon in May 1798 with Napoleon Bonaparte’s army and its party of scholars. He fought at the battle of the Pyramids in July. After the British destroyed the French fleet at Aboukir Bay on 1 August, the army was cut off from France, and Malus wrote that the men began to lose hope of seeing their country again.

On 22 August 1798 Bonaparte founded the Institut d’Égypte in Cairo. Malus was named to its mathematics section, alongside Monge, Fourier and Bonaparte himself. He kept a diary of his work, published long after his death as L’agenda de Malus, and wrote a scientific memoir in Cairo.

Most of his time went to soldiering. He took part in suppressing the Cairo revolt of October 1798 and in the siege of El-Arish during the Syrian campaign of early 1799. At Jaffa, plague broke out among the troops, and Malus caught it. He survived, recovering in quarantine at Lesbeh and then at the post of Qatia. The army in Egypt held out until 1801. Malus landed at Marseilles on 14 October 1801, his health permanently weakened.

Engineer and examiner, 1801–1808

Early in 1802 Malus returned to Giessen and married Wilhelmine-Louise Koch. His military career then took him from fortress to fortress. At Lille (1802–03) he helped found a local society of sciences and arts and became its president. At Antwerp (1804–06) he supervised harbour works. From 1806 to 1808 he was sub-director of fortifications at Strasbourg.

From 1805 he was also an examiner at the École Polytechnique, first in geometry and analysis, then in physics. The post brought him regularly to Paris and into the circle of Pierre-Simon Laplace and Claude-Louis Berthollet, who sought to explain all of physics by short-range forces acting on particles, light included.

On 20 April 1807 he presented his Traité d’optique to the Institut. It treated families of light rays as geometric objects, found the equations of caustic surfaces, and stated the theorem that bears his name (Lessons 04 and 05). In 1807 he began measuring double refraction in Iceland spar. On 4 January 1808 the Institut announced a prize for an experimental and theoretical account of double refraction, and in 1808 Malus was recalled to Paris. He now had the subject, the time and the competition he needed.

Systems of rays and caustics

For the general reader. On a sunny day a coffee cup shows a bright cusped curve on the surface of the coffee. It is a caustic: the curve to which the reflected rays are all tangent, where light piles up. Malus treated a whole family of rays as one mathematical object and found general equations for such surfaces. His Traité d’optique of 1807 began the geometric theory that William Rowan Hamilton later turned into a complete optics.

Ray systems

In space, a system of rays (later called a congruence of lines) is a two-parameter family

where \(\mathbf a\) is a base point, \(\mathbf d\) the ray direction and \(t\) the distance along the ray. Malus asked where neighbouring rays meet. Two rays infinitely close in the direction \((du,dv)\) intersect when \(\mathbf x_u\,du+\mathbf x_v\,dv\) is parallel to \(\mathbf d\).

In optics the focal surfaces are the caustics. For rays in a plane the picture is simpler: a one-parameter family of lines has one focus per line, and the foci trace the envelope of the family.

The circular mirror

Let rays parallel to the \(x\)-axis travel in the \(+x\) direction and reflect from the inside of the unit circle at \(P=(\cos t,\sin t)\), with \(|t|<\pi/2\). The reflected rays envelop the curve

an arc of a nephroid. Its cusp at \(t=0\) is \((\tfrac12,0)\), the paraxial focus halfway between centre and mirror. Rays far from the axis cross it nearer the mirror: this is spherical aberration, visible as the bright curve in the coffee cup.

Malus’s theorem

For the general reader. Light from a point spreads in rays that are all perpendicular to spheres around the point. Malus proved that after one reflection or refraction the rays are still all perpendicular to some surface. He believed that a second refraction could destroy this property. He was wrong: Charles Dupin showed that it survives any number of reflections and refractions. In wave optics the surface is simply the wavefront, so the theorem is the bridge between rays and waves.

Proof sketch. Define \(S(\mathbf x)\) as the optical path length from the initial surface to \(\mathbf x\) along the ray through \(\mathbf x\). Consider a neighbouring ray. By Fermat’s principle the optical path is stationary under small changes of the path, so to first order the difference \(S(\mathbf x+\delta\mathbf x)-S(\mathbf x)\) comes only from the endpoints. At the start both rays leave the surface perpendicularly, so the contribution there is zero. At the end it is \(n\,\mathbf d\cdot\delta\mathbf x\). Hence \(\nabla S=n\,\mathbf d\): the rays are the gradient lines of \(S\) and are normal to its level sets. Each reflection or refraction obeys Fermat’s principle, so the argument applies after any number of them. William Rowan Hamilton made \(S\) the centre of his optics in the 1820s; in modern terms it is the eikonal.

A test for normality

A field of unit ray directions \(\mathbf d\) is normal to a family of surfaces exactly when \(\mathbf d=\nabla S/|\nabla S|\) for some \(S\). For smooth fields this holds if and only if

a special case of the theorem of Frobenius. Not every ray system passes the test, so the theorem has real content.

Double refraction and Huygens’s construction

For the general reader. Look at print through a crystal of Iceland spar and you see it twice. In 1690 Huygens explained the second image by supposing that one of the light waves inside the crystal spreads as a squashed ellipsoid instead of a sphere. A century later René Just Haüy doubted the rule and William Hyde Wollaston’s tests were incomplete. Malus measured the refraction of the extraordinary ray in many directions in 1807 and 1808 and showed that Huygens was right. This won him the prize of 1810.

For calcite, with refractive indices \(n_o=1.658\) and \(n_e=1.486\) for yellow light, \(v_e>v_o\): the spheroid is oblate. Malus measured the deviations of the extraordinary image for many cuts and incidences and found agreement with Huygens’s construction within his errors. For the theory he followed Laplace and used the principle of least action for light particles (Lesson 10).

Walk-off

Let the wave normal make angle \(\theta\) with the axis. The ray makes angle \(\theta+\rho\) with it, where, with \(n=c/v\),

Derivation. Write the spheroid as \(F=(x^2+y^2)n_e^2+z^2n_o^2=c^2\). The wave normal at the tangent point is parallel to \(\nabla F\propto(xn_e^2,\,0,\,zn_o^2)\) in the \(xz\)-plane. If the ray is at angle \(\varphi\) to \(z\), then \(\tan\theta=(n_e^2/n_o^2)\tan\varphi\). Then \(\tan\rho=\tan(\varphi-\theta)\) simplifies to the formula above.

Polarization by reflection

For the general reader. Light passing through Iceland spar emerges in two beams with “sides”: each one, sent into a second crystal, splits differently depending on how the crystal is turned. In 1808 Malus found that ordinary reflection does the same thing. A story told since the nineteenth century says he noticed it while looking through a crystal at the evening sun reflected in the windows of the Luxembourg Palace; the source of the story is uncertain. He announced the discovery on 12 December 1808 and called the property polarization.

In wave language, light “polarized in the plane of incidence” has its electric field perpendicular to that plane: it is s-polarized. The other polarization, with \(\mathbf E\) in the plane of incidence, is p-polarized. Malus’s angle is where \(p\)-light is not reflected.

Malus measured the angle for water. His glass was too uneven to give a clean law, and David Brewster found the general rule in 1815. Augustin-Jean Fresnel derived the amplitude ratios in 1821–23 and explained both.

The cosine-squared law

For the general reader. Turn one pair of polarizing sunglasses in front of another and the view darkens smoothly to black. Malus found the rule for how much light gets through: the intensity goes as the square of the cosine of the angle between the two polarizing directions. He found it first for two crystals of Iceland spar and for reflection, in the years 1808–1810. It is the oldest quantitative law of polarization.

Derivation. In the transverse-wave picture of Young and Fresnel, the light is a vibration vector \(\mathbf E\) perpendicular to the ray. An analyser transmits the component along its axis \(\hat{\mathbf a}\), of amplitude \(E\cos\theta\), and blocks or deflects the rest. Intensity is proportional to amplitude squared. The two-crystal form follows by projecting each beam onto the two perpendicular axes of the second crystal. Malus himself explained the law with particles whose “poles” are realigned (Lesson 10).

Partial polarization by refraction

For the general reader. If reflection at glass favours one polarization, the transmitted light must be left with more of the other. Malus found that refracted light contains light polarized at right angles to the reflected light. By stacking many plates, one can make the transmitted beam almost completely polarized. Such piles of plates were the standard polarizers of the nineteenth century.

Proof. Half the light is \(s\) and half \(p\). Energy conservation gives \(T_s=1-R_s\) and \(T_p=1-R_p\) for each polarization separately. The reflected excess of \(s\) over \(p\) is \(\tfrac12(R_s-R_p)\); the transmitted excess of \(p\) over \(s\) is \(\tfrac12\big((1-R_p)-(1-R_s)\big)=\tfrac12(R_s-R_p)\).

The pile of plates

At the polarizing angle, \(R_p=0\) at every surface, and the \(s\)-component loses a fraction \(R_s\) at each. Ignoring multiple reflections, after \(m\) surfaces the transmitted intensities are \(T_p=1\) and \(T_s=(1-R_s)^m\), so the degree of polarization is

Polarized corpuscles and least action

For the general reader. Malus, like Laplace and Biot, held Isaac Newton’s view that light is a stream of particles. To explain polarization he gave the particles sides, or poles, which reflection and crystals line up. To explain refraction, he and Laplace used the principle of least action for particles instead of the principle of least time for waves. Both principles give the same paths, for a reason that is pleasant to see. The particle theory failed in 1850, when Léon Foucault measured light travelling slower in water than in air.

The second equality is the crux. In the wave theory, light is slower in water; in the particle theory, faster. Newton explained refraction by an attraction that speeds up particles entering the denser medium, and that attraction was the model for Laplace’s short-range forces. Every ray experiment of Malus’s time was blind to the difference, which is why Laplace could derive Huygens’s law for corpuscles in January 1809, to Malus’s irritation, since he regarded it as taking the priority his experiments had earned.

Laplace’s velocity law

For the extraordinary ray Laplace assumed that the corpuscle speed depends on the angle \(\varphi\) between the ray and the optic axis as

and showed by least action that this reproduces Huygens’s construction. Exercise 10.1 shows why: \(u\) is the reciprocal of the ray speed on Huygens’s spheroid.

Prize and Académie, 1810–1811

Malus finished his double-refraction experiments in December 1808, the same month he announced polarization by reflection. In 1809 he joined the Society of Arcueil, the private research circle of Laplace and Berthollet, and published the discovery. His prize memoir, Théorie de la double réfraction de la lumière dans les substances cristallisées, won the Institut’s prize in 1810 and appeared in 1811. The judges valued above all his measurements, his law of relative intensities and his treatment of the forces producing double refraction.

On 18 August 1810 he was elected to the First Class of the Institut. In the same year the Royal Society of London awarded him the Rumford Medal, although Britain and France were at war. Thomas Young, the Society’s foreign secretary and the leading defender of the wave theory, sent him the news in March 1811. Malus was promoted major of engineers in December 1810, and in 1811 he became director of studies at the École Polytechnique.

In 1811 he served with Joseph-Louis Lagrange, Adrien-Marie Legendre, Laplace and Haüy on the committee that judged the prize on heat conduction, which Fourier won. His last memoir was read to the Institut on 19 August 1811. Meanwhile François Arago, Biot and, in Scotland, Brewster had taken up polarization, and Arago’s discovery of chromatic polarization in 1811 opened the line of work that would lead to the wave theory.

Death and legacy

Malus had been ill for several years, his health broken by disease contracted in Egypt and Syria. He died in Paris on 24 February 1812, aged 36, probably of tuberculosis. (Some sources give 23 February.) He was buried in the Père-Lachaise cemetery. His wife Wilhelmine-Louise died on 18 August 1813.

Within a decade, Young and Fresnel had shown that light is a transverse wave. Polarization, which Malus had seen as proof of polarized particles, became the decisive evidence for transversality. Every law he found was kept.

Legacy

• Polarization: the word, the discovery of polarization by reflection, and the cosine-squared law, now used in every polarizer, camera filter and liquid-crystal display.
• Geometrical optics: the theory of ray systems and caustics; the Malus–Dupin theorem, which Hamilton made the foundation of his optics and which in modern geometry says that wavefronts form Lagrangian families.
• Crystal optics: the first precise confirmation of Huygens’s law of double refraction.
• Honours: his name is one of the 72 engraved on the Eiffel Tower; the rue Malus in Paris was named after him in 1879.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Étienne Louis Malus, MacTutor History of Mathematics.
2. E. Frankel, Malus, Étienne Louis, Dictionary of Scientific Biography; online at Encyclopedia.com.
3. Étienne-Louis Malus, Wikipedia; French Wikipedia for the Egyptian campaign.
4. É.-L. Malus, L’agenda de Malus. Souvenirs de l’expédition d’Égypte 1798–1801, ed. General Thoumas, Paris, 1892.
5. M. Crosland, The Society of Arcueil: A View of French Science at the Time of Napoleon I, Heinemann, 1967.

Physics

1. É.-L. Malus, Traité d’optique, Mémoires présentés à l’Institut par divers savants 2 (1811), presented 1807.
2. É.-L. Malus, Sur une propriété de la lumière réfléchie, Mémoires de physique et de chimie de la Société d’Arcueil 2 (1809).
3. É.-L. Malus, Théorie de la double réfraction de la lumière dans les substances cristallisées, Paris, 1810.
4. Malus–Dupin theorem, Wikipedia.
5. W. R. Hamilton, Theory of systems of rays, Trans. Royal Irish Academy 15 (1828).
6. J. Z. Buchwald, The Rise of the Wave Theory of Light, University of Chicago Press, 1989.
7. J. Bain, Arago and Biot and Fresnel’s unification, lecture notes on the history of light.
8. M. Born and E. Wolf, Principles of Optics, Cambridge University Press: chapters on geometrical optics and crystal optics.
9. V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer, on caustics and Lagrangian submanifolds.