Malus's Law

The statement and where the cos² comes from

A polarizer has a transmission axis. When linearly polarized light hits it, the polarizer transmits only the component of the electric field along that axis, and absorbs (or reflects) the orthogonal component.

If the incident polarization makes angle θ with the transmission axis, the transmitted field amplitude is E·cos θ. Intensity is proportional to |E|², so:

E_out  = E_in · cos(θ)        (amplitude projection)
I_out  = I_in · cos²(θ)       (Malus's law)

That's it. No empirical fit. The squared cosine follows from amplitude projection times the I ∝ |E|² rule of electromagnetism. Malus discovered it in 1809 by observing sunlight reflected off a Paris window — an early hint that light is transverse.

Reference table — angle vs transmission

θ (angle)	cos θ	cos² θ	Transmission %
0° (aligned)	1.000	1.000	100.0%
15°	0.966	0.933	93.3%
30°	0.866	0.750	75.0%
45°	0.707	0.500	50.0%
60°	0.500	0.250	25.0%
75°	0.259	0.067	6.7%
90° (crossed)	0.000	0.000	0% ideal

The three-polarizer "paradox"

Cross two polarizers — say, first vertical, second horizontal. Transmission is zero. Now insert a third polarizer at 45° between them. Transmission jumps to 12.5%.

It looks paradoxical: adding a filter increased the light. But Malus tells you exactly why:

Unpolarized input I₀
After 1st polarizer (vertical):       I₀ / 2
After 2nd polarizer (45° to vertical): (I₀/2) · cos²(45°) = I₀/4
After 3rd polarizer (horizontal, 45° to second): (I₀/4) · cos²(45°) = I₀/8

Final transmission = 12.5%

The middle polarizer repolarizes the field along a new axis, so the last polarizer sees a 45° angle instead of 90°. This is also the textbook demonstration of why quantum measurements don't commute: it's the same projection algebra that runs Stern-Gerlach cascades and Bell experiments.

Worked example — costed transmission claims

Setup	Calculation	Result
Crossed polarizers, ideal	cos²(90°)	Transmission = 0 (perfect block)
Crossed Polaroid film (real)	extinction ratio ~10⁻⁴	~0.01% leak — visibly dark
Crossed Glan-Thompson prism	extinction ratio ~10⁻⁶	1 ppm leak — laser-grade
Polaroid sunglasses on LCD phone (rotated 90°)	cos²(90°) at LCD output polarizer	Screen looks black
Unpolarized light through one Polaroid	½ ideal, ~38% real	Half intensity, fully polarized output
LCD pixel mid-gray (60°)	cos²(60°)	25% — produces midtone grayscale

Unpolarized input — why it's always 50%

Unpolarized light is a mix of every polarization angle, randomly distributed. The fraction transmitted is the angular average of cos² θ:

⟨cos² θ⟩ = (1 / 2π) · ∫₀^{2π} cos²(θ) dθ = 1/2

So an ideal polarizer transmits exactly 50% of any unpolarized beam regardless of its rotation. That's why rotating polarizer sunglasses doesn't change overall brightness when looking at unpolarized scenery — but does when you tilt them against a partially polarized source (water reflection, blue sky 90° from the sun).

JavaScript — Malus's law calculations

// Malus: transmitted intensity through one polarizer
function malus(I0, thetaDeg) {
  const t = thetaDeg * Math.PI / 180;
  return I0 * Math.cos(t) ** 2;
}

console.log(`0°: ${malus(1, 0).toFixed(3)}`);    // 1.000
console.log(`30°: ${malus(1, 30).toFixed(3)}`);  // 0.750
console.log(`45°: ${malus(1, 45).toFixed(3)}`);  // 0.500
console.log(`60°: ${malus(1, 60).toFixed(3)}`);  // 0.250
console.log(`90°: ${malus(1, 90).toFixed(3)}`);  // 0.000

// Cascade — array of polarizer axis angles in degrees
function cascade(I0, axisAnglesDeg, inputAngleDeg = null) {
  let I = I0;
  let polAxis = inputAngleDeg;  // null = unpolarized
  for (const ax of axisAnglesDeg) {
    if (polAxis === null) {
      I = I / 2;          // unpolarized input → 50%
    } else {
      I = malus(I, ax - polAxis);
    }
    polAxis = ax;
  }
  return I;
}

// Three-polarizer paradox: 0° → 45° → 90° from unpolarized input
console.log(`Three-polarizer 0/45/90: ${cascade(1, [0, 45, 90]).toFixed(4)}`);
// 0.1250 — 12.5%

// Two crossed (0 → 90) from unpolarized
console.log(`Crossed 0/90: ${cascade(1, [0, 90]).toFixed(4)}`);
// 0.0000 — 0%

// Same crossed but 4 polarizers, each rotated 30°: 0 → 30 → 60 → 90
console.log(`4 polarizers (0/30/60/90): ${cascade(1, [0, 30, 60, 90]).toFixed(4)}`);
// 0.2109 — 21% (more steps → less attenuation per step → more transmission)

// In the limit of N polarizers each rotated by 90/N: lim → cos²(0) · cos²(0) · ... → 1
function nStep(N) {
  return cascade(1, Array.from({length: N+1}, (_, i) => (90 * i) / N));
}
console.log(`N=2: ${nStep(2).toFixed(3)}`);    // 0.125
console.log(`N=10: ${nStep(10).toFixed(3)}`);  // ~0.450
console.log(`N=100: ${nStep(100).toFixed(3)}`); // ~0.925
// This is the Quantum Zeno effect for polarization — measure often, change polarization gradually.

// LCD grayscale: voltage → twist angle → cos²
function lcdGray(voltage, V_max = 5) {
  // Simple model: twist angle goes from 90° (off) to 0° (on)
  const twist = 90 - 90 * (voltage / V_max);
  return malus(1, twist);
}
for (let v = 0; v <= 5; v++) {
  console.log(`V=${v}V → ${(lcdGray(v) * 100).toFixed(1)}% transmission`);
}

Where Malus's law matters

• LCD displays. Every pixel's grayscale is a Malus's-law computation between two polarizers and a liquid-crystal cell.
• Photographic polarizing filters. Rotate the filter to find the angle that minimizes glare; transmission of unwanted polarization follows Malus.
• Adjustable neutral-density filters. Two stacked polarizers rotated relative to each other — continuous attenuation from 100% to ~0% via cos²θ.
• Optical isolation. Laser systems use polarizer + Faraday rotator + polarizer to block back-reflections (one-way valve).
• Quantum-mechanics demos. Single-photon polarizer cascades show cos² as a probability, leading to Bell tests.
• Stress-photoelasticity. Birefringent stress patterns rotate polarization differently along different paths; Malus-filtered output reveals the stress map.

Common mistakes

• Using cos instead of cos². Amplitude projects as cos θ, but intensity (what your eye/photodiode measures) scales as cos². Forgetting the square halves your prediction at 45° and undershoots everywhere.
• Applying Malus to unpolarized input. Malus requires a known input polarization. For unpolarized light, the first polarizer always transmits 50%, regardless of axis.
• Forgetting absorption loss in real polarizers. Ideal: 50% from unpolarized. Real Polaroid: ~38%. Use the polarizer's spec sheet for accurate predictions.
• Assuming θ = 90° is perfect zero in real life. Real extinction ratios are 10⁻⁴ to 10⁻⁶. For high-precision experiments, use prism polarizers, not film.
• Ignoring wavelength. Polarizer extinction is best near design wavelength; deep blue or near-IR can be 100× worse. Specify your λ.
• Confusing rotation of polarizer with rotation of polarization. Rotating the polarizer is equivalent to rotating the input polarization in the opposite sense — same cos² formula, but be consistent about which one is moving.