Polarization of Light: Malus's Law, Circular States & Birefringence

A transverse wave has an orientation

Light is a transverse electromagnetic wave: its oscillating electric field points perpendicular to the direction of travel. Because that perpendicular plane is two-dimensional, the field's oscillation direction within it is a real, physical degree of freedom — polarization. Unpolarized light (sunlight, an incandescent bulb) is a rapid, random jumble of every orientation; a polarizing filter passes only the component of the field aligned with its transmission axis and blocks the rest.

Malus's Law

Once light is already linearly polarized (say, by a first filter) and hits a second filter oriented at angle θ to the first, only the field component along the second filter's axis survives. Since intensity goes as the square of the field amplitude, the transmitted intensity follows the cosine of the angle, squared — Malus's Law (Étienne-Louis Malus, 1808):

I = I₀ cos²θ

At θ = 0 the filters are aligned and all the light that reached the second filter passes; at θ = 90° ("crossed polarizers") cos²θ = 0 and no light passes at all, however bright the source. This is the mechanism behind polarized sunglasses reducing glare, LCD screens controlling pixel brightness, and the classic demo of two crossed filters going dark with a third filter at 45° between them paradoxically letting light back through.

Linear, circular and elliptical: it's about the phase

Polarization state is fully described by two perpendicular field components (conventionally Ex and Ey) and the phase difference between them. If the two components oscillate exactly in phase (or exactly out of phase by 180°), their sum traces a straight line — linear polarization. If they have equal amplitude and are exactly 90° out of phase, the resultant vector traces a circle as time passes — circular polarization (left- or right-handed depending on the sign of the phase offset). Any other combination of amplitude ratio and phase difference traces an ellipse — elliptical polarization, the general case, of which linear and circular are the two special limits.

Ex = E0x cos(ωt)
Ey = E0y cos(ωt + δ)
δ = 0 or π            & E0x=E0y → linear (±45°) / any ratio → linear at angle
δ = ±π/2, E0x = E0y   → circular
otherwise             → elliptical

Birefringence: how you actually make circular light

You can't create the 90° phase offset needed for circular polarization with an ordinary filter — you need a birefringent material, one whose refractive index depends on the light's polarization direction, so light polarized along one crystal axis (the "fast axis") travels faster through the material than light polarized along the perpendicular axis (the "slow axis"). Send linearly polarized light in at 45° to those axes, choose the crystal's thickness so the two components end up exactly a quarter-wavelength out of step, and you have a quarter-wave plate: linear polarization in, circular polarization out. A thicker plate introducing a half-wavelength delay is a half-wave plate, which instead flips the orientation of linear polarization, reflecting it about the fast axis.

Where polarization physics shows up

Reflected glare off water or glass is partially polarized (parallel to the reflecting surface) because of the angle-dependent Fresnel reflection coefficients, which is exactly what polarized sunglasses are cut to block. Liquid-crystal displays sandwich a layer of birefringent liquid crystal between two crossed polarizers and use an applied voltage to twist the crystal's orientation, continuously varying how much light gets through pixel by pixel — the whole display is Malus's Law and birefringence used as an electronically-controlled valve. Stress in transparent plastics becomes visible as coloured fringes between crossed polarizers (photoelasticity) because mechanical stress itself induces birefringence, letting engineers visualise where a physical model concentrates load.

Frequently asked questions

Try it live

Everything above runs in your browser — open Polarization of Light Simulator and change the parameters while it is running. Nothing is installed, nothing is uploaded, the whole model lives in one tab.