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Polarisation is usually taught with Polaroid sheets and the rule I = I0 cos²θ. But it was discovered with a lump of Iceland spar and a reflection, more than a century before Polaroid existed. This post covers who Etienne-Louis Malus was, what he saw, how David Brewster turned it into a law, and how to solve the reflection questions that JEE Main asks.
Who was Etienne-Louis Malus?
The MacTutor biography of Malus gives his dates as 23 July 1775 to 24 February 1812, both in Paris. He studied at the military engineering school at Mézières, where Gaspard Monge spotted his talent, then at the École Polytechnique, where he was taught by Joseph Fourier. As an army engineer he went with Napoleon's expedition to Egypt in 1798 and returned in 1801.
His mathematical work was almost all about light. MacTutor records that he published his discovery of the polarisation of light by reflection in 1809 and his theory of double refraction in 1810. He won the Académie des Sciences prize for the double refraction memoir, and the Royal Society of London gave him its Rumford Medal despite the war between England and France. He died at 36.
What was already known about double images?
A crystal of Iceland spar (calcite) splits a ray into two, so you see a double image. Malus's memoir, Sur une propriété de la lumière réfléchie, printed in the 1809 volume of the Mémoires of the Société d'Arcueil, starts there. Pass the two beams from one crystal through a second crystal and the result depends on how the second crystal is turned. At some angles each beam splits again into two; after a quarter turn each beam goes through as a single beam. The light leaving the first crystal has gained a property that ordinary light does not have: it behaves differently depending on how the second crystal is turned.
What did Malus discover?
Malus found that a crystal is not needed. His memoir reports that light reflected from the surface of water at an angle of 52° 45' from the vertical has all the characteristics of one of the two beams produced by a calcite crystal. Looked at through a crystal, it does not split. Turn the crystal and one image fades and vanishes once every quarter turn. He found the same effect for other transparent bodies, at an angle that is generally larger for substances that bend light more.
He also noticed what happens to the light that goes into the surface. When he sent the two polarised beams from a crystal onto water at that angle, one beam was partly reflected as usual, but the other went entirely into the water, with none reflected.
How did Brewster turn it into a law?
In 1815 David Brewster wrote to the Royal Society, in a paper on the laws which regulate the polarisation of light by reflexion, that Malus had "the sole merit of creating" this branch of science. Brewster measured many more substances and found one rule that fitted them all:
"The index of refraction is the tangent of the angle of polarisation."
His table lists water with a calculated polarising angle of 53° 11' and the observed 52° 45'. Today this is Brewster's law: tan θB = n, for light going from air into a medium of refractive index n.
Worked example 1: the Brewster angle for glass
Light in air hits glass with n = 1.50.
Brewster angle. tan θB = 1.50, so θB = 56.3°.
Refracted angle. By Snell's law, sin r = sin 56.3° / 1.50 = 0.832 / 1.50 = 0.555, so r = 33.7°.
Check. 56.3° + 33.7° = 90°. At the Brewster angle the reflected and refracted rays are at right angles. This gives a second route to the answer.
For water, n = 1.33 gives θB = tan⁻¹(1.33) = 53.1°, close to the 52° 45' (52.75°) Malus measured with the tools of his day.
Worked example 2: reflection plus Malus's law
Unpolarised light of intensity I0 passes through a polariser and then an analyser whose axis is at 60° to the first.
First sheet. An unpolarised beam loses half: I0/2.
Second sheet. I = (I0/2) cos² 60° = (I0/2)(0.25) = I0/8.
The NCERT Class 12 chapter Wave Optics derives I = I0 cos²θ for two Polaroids and calls it Malus' law. Our post on single-slit diffraction and polarisation has more Polaroid problems; here the focus is reflection.
Which mistakes should you avoid?
Using sin or cos instead of tan. Brewster's law is tan θB = n. Write it before substituting.
Measuring the angle from the surface. Malus gave his angle from the vertical, which is the normal for a level water surface. JEE questions use the normal too.
Thinking all reflected light is polarised. Only at the Brewster angle is the reflected light fully polarised. At other angles it is partly polarised.
Forgetting the first factor of one half. Unpolarised light through one ideal polariser always drops to I0/2, whatever the sheet's angle.
The JEE (Main) 2026 syllabus lists plane-polarised light, Brewster's law and Polaroids. Check the current year's syllabus, since NCERT's Class 12 chapter covers Polaroids and Malus' law but not reflection in detail.
What can a student take from Malus's way of working?
Look again at something everyday. Reflections off water were ordinary. Viewing one through a crystal made it a discovery.
Test the general case. He did not stop at water; he tried every transparent body he could.
Leave room for the next person. Malus measured angles; Brewster found the law. Good data outlives its first explanation.
If the electric field directions in polarisation are hard to picture, the Fermi Simulation Lab has interactive simulations for ideas that are easier to see than read.
Try it today: find the Brewster angle for diamond (n = 2.42) and check that the reflected and refracted rays are at 90°.
