Polarization

Polarization basics

For a transverse wave (like light), the electric field oscillates perpendicular to propagation direction. Polarization specifies which DIRECTION (or pattern of directions) this oscillation takes.

Type	Description
Unpolarized	Random or rapidly-changing oscillation directions
Linearly polarized	Oscillation in a single fixed direction
Circularly polarized	Direction rotates uniformly around propagation
Elliptically polarized	General case; rotation with non-equal amplitudes
Partially polarized	Mix of polarized and unpolarized

Malus's law

When linearly polarized light passes through a polarizer at angle θ from its transmission axis:

I = I₀ · cos²(θ)

θ (angle)	I/I₀
0° (aligned)	1.00 (100%)
30°	0.75
45°	0.50
60°	0.25
90° (crossed)	0.00

Two perpendicular polarizers transmit nothing. But ADD a third polarizer at 45° between them — transmission becomes nonzero (½ × ½ = 25%).

Brewster's angle

At incidence angle θ_B = arctan(n₂/n₁), the reflected light is 100% s-polarized (perpendicular to plane of incidence).

tan θ_B = n₂ / n₁

Boundary	θ_B
Air to water	53.1°
Air to glass (n=1.5)	56.3°
Air to diamond	67.5°

Sunglasses, polarized photography filters, laser optics all use Brewster's angle.

Real-world polarization

System	How polarization is used
Polarized sunglasses	Block horizontal polarization (reflections off water/roads)
LCD screens	Liquid crystals rotate polarization between two polarizers
3D movies	Two oppositely circularly polarized images; glasses separate them
Camera polarizing filters	Reduce reflections, saturate skies
Microscopy	Polarized light reveals stress in materials, identifies minerals
Astronomy	Polarized light reveals magnetic fields, dust orientation, scattering processes
Photography (glass through windows)	Polarizing filters cut reflections off glass and water
Bee navigation	Bees see polarization patterns in sky to navigate (sky polarized perpendicular to sun)

JavaScript — polarization calculations

// Malus's law
function malusLaw(I_0, angle_deg) {
  const angle = angle_deg * Math.PI / 180;
  return I_0 * Math.cos(angle) ** 2;
}

console.log(`30° polarizer: ${(malusLaw(1, 30) * 100).toFixed(1)}%`);  // 75%
console.log(`45°: ${(malusLaw(1, 45) * 100).toFixed(1)}%`);            // 50%
console.log(`60°: ${(malusLaw(1, 60) * 100).toFixed(1)}%`);            // 25%
console.log(`90°: ${(malusLaw(1, 90) * 100).toFixed(1)}%`);            // 0%

// Three-polarizer paradox: cross-polarizers + middle 45° filter
function threePolarizers(I_0, angle_middle_deg) {
  // First polarizer (vertical) transmits I_0/2 (unpolarized → polarized loses half)
  const after_first = I_0 / 2;
  // Through middle at angle θ from first
  const after_middle = malusLaw(after_first, angle_middle_deg);
  // Through final (90° from first), so angle from middle is 90 - middle_angle
  const after_third = malusLaw(after_middle, 90 - angle_middle_deg);
  return after_third;
}

console.log(`Cross polarizers + 45° between: ${(threePolarizers(1, 45) * 100).toFixed(1)}%`);
// 12.5% transmits (paradox: adding a polarizer increases transmission!)

// Brewster's angle
function brewsterAngle(n_dense, n_rare = 1.0) {
  return Math.atan(n_dense / n_rare) * 180 / Math.PI;
}

console.log(`Air-water Brewster: ${brewsterAngle(1.33).toFixed(2)}°`);
console.log(`Air-glass Brewster: ${brewsterAngle(1.5).toFixed(2)}°`);

// Reduce light by 4x with two cross-polarizers in cascade
function cascade2Polarizers(I_0) {
  // Cross polarizers transmit 0% — actually need offset
  // But for ND-filter style: I = I_0 · cos²(θ)
  return I_0 * Math.cos(60 * Math.PI / 180) ** 2;
}

// Adjustable polarizer ND filter — can reduce 0 to ~99.5%

// Circular polarizer transmission (left or right circularly polarized light)
function circularPolarizerTransmission(input) {
  // Half is transmitted; other half blocked
  if (input.handedness === 'matching') return input.intensity;
  return 0;
}

Where polarization matters

• Sunglasses and photography. Reduce reflections, glare, saturate colors.
• Display technology. All LCDs use polarization. OLEDs typically don't (different physics).
• Microscopy. Polarized light microscopy reveals birefringent materials (crystals, minerals, stressed glass).
• Astronomy. Polarimetry maps magnetic fields, scattering by interstellar dust, exoplanet atmospheres.
• Quantum physics. Photon polarization is a quantum bit (qubit candidate). Bell's inequality experiments measure polarized photons.
• Telecommunications. Polarization-multiplexing in fiber optics doubles data capacity.
• 3D cinema. Circular polarization for 3D images (glasses-based).

Common mistakes

• Trying to polarize sound. Sound is longitudinal — no polarization possible.
• Forgetting that two crossed polarizers = 0%. Crossed (90°) polarizers block all light. Adding a third at 45° makes them transmit 12.5% — a quantum-mechanical-feeling result.
• Confusing absorbing and reflecting polarizers. Most consumer polarizers absorb the wrong polarization. Some optical setups use reflecting polarizers (lasers, lab).
• Using sin² instead of cos² in Malus. When θ is between transmission axis and polarization, use cos². Watch for definitions.
• Treating sunlight as fully polarized. Direct sunlight is essentially unpolarized. Polarized sunglasses cut PARTIAL polarization from reflections.
• Forgetting Brewster only works for one polarization. At θ_B, only s-polarization reflects. P-polarization passes through entirely. Important for laser cavity Brewster windows.