Light Polarization | mysimulator.uk

Light as a transverse wave, and what "polarized" means

Light is a transverse electromagnetic wave: its electric field oscillates perpendicular to the direction of travel. Ordinary sunlight or a bulb's output is unpolarized — the field's oscillation direction changes randomly, many times per second, with no preferred orientation. Polarizing light means filtering, reflecting or scattering it so that the electric field ends up oscillating along a single, well-defined axis (linear polarization), or tracing out a circle or ellipse as it propagates (circular and elliptical polarization).

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Malus's law: cosine-squared, not cosine

A linear polarizer only transmits the component of the electric field aligned with its transmission axis. If linearly polarized light of intensity I₀ hits a second polarizer (an analyzer) at angle θ to the first, only the field component E₀cos θ gets through — and since intensity is proportional to the square of the field, the transmitted intensity follows a cosine-squared law, not a plain cosine:

I(θ) = I₀ · cos²(θ)         (Malus's law, 1808)

θ = 0°   → I = I₀     (fully aligned, all light passes)
θ = 45°  → I = I₀ / 2
θ = 90°  → I = 0      (crossed polarizers, fully blocked)

Two crossed polarizers (θ = 90°) block all light in the ideal case — the basis of LCD screens, polarized sunglasses tested against each other, and countless optics demonstrations. Étienne-Louis Malus discovered the effect in 1808 while looking at sunlight reflected off windows through a calcite crystal, which is also how he stumbled onto the next phenomenon.

Brewster's angle: polarizing by reflection alone

Unpolarized light hitting a dielectric surface (glass, water) reflects with its two polarization components — parallel and perpendicular to the plane of incidence — at different efficiencies, described by the Fresnel equations. At one specific angle, the Brewster angle, the parallel-polarized component's reflectance drops to exactly zero, so the reflected beam is purely perpendicular-polarized:

θ_B = arctan(n₂ / n₁)      (Brewster's angle, 1815)

for air → glass (n₁=1, n₂≈1.5):  θ_B ≈ 56.3°
at θ_B: reflected ray ⊥ refracted ray exactly (90° between them)

This is precisely why polarized sunglasses cut reflected glare off water and roads — that glare is heavily polarized by near-Brewster-angle reflection, and a polarizer oriented to block that axis removes most of it while barely dimming everything else.

Circular, elliptical, and birefringence

Combine two perpendicular linear waves with a 90° phase offset and equal amplitude, and the resulting field vector traces a circle instead of a line as it propagates — circular polarization; unequal amplitudes or a different phase offset trace an ellipse instead. Wave plates (quarter-wave and half-wave) create this by exploiting birefringence: certain crystals (calcite, quartz) have a refractive index that depends on the light's polarization direction, so the two perpendicular components travel at different speeds through the same physical thickness, building up exactly the phase offset needed to convert linear light into circular or vice versa.

Frequently asked questions

Why does Malus's law use cosine squared instead of just cosine?

The polarizer only lets through the field component along its axis, which scales as cos θ, but the intensity you actually measure is proportional to the square of the field amplitude. Squaring cos θ gives cos²θ, which is why crossed polarizers at 90° block all light rather than half of it.

What makes the Brewster angle special?

At that specific angle of incidence, the Fresnel equations predict zero reflectance for light polarized parallel to the plane of incidence, so the reflected beam contains only the perpendicular component — reflection alone has fully polarized it, with no filter needed.

How does a wave plate turn linear light into circular light?

A birefringent crystal has two different refractive indices for two perpendicular polarization directions, so those two components of the light travel at different speeds through it. A quarter-wave plate is cut to exactly the thickness that builds up a 90° phase lag between them, which converts linear polarization into circular.

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Light Polarization: Malus's Law and Brewster's Angle