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Polarization of Light

Rotate polarizing filters and watch Malus's Law — I = I₀ cos²θ — in real time

Optics Electromagnetism Malus's Law Wave Physics
Mode:
θ = 45° cos²θ = 0.500 I / I₀ = 0.500 Transmitted: 50% Mode: Linear

🔮 Polarization of Light

Natural light is unpolarized — its electric field oscillates in all perpendicular planes simultaneously. A polarizer transmits only the component aligned with its transmission axis, producing linearly polarized light.

When polarized light passes through a second polarizer (analyzer) rotated by angle θ:
I = I₀ · cos²(θ)   — Malus's Law (Étienne-Louis Malus, 1808)

Circular polarization arises when two equal-amplitude, orthogonal oscillations are 90° out of phase (±¼ wavelength retardation). Elliptical polarization is the general case. A quarter-wave plate converts linear to circular; a half-wave plate rotates the polarization direction.

Birefringent materials (calcite, quartz) have two different refractive indices for orthogonal polarizations, splitting beams and creating retardation effects used in LCD screens and optical instruments.

About this simulation

This simulation visualizes Malus's Law — I = I₀ cos²θ — by letting you rotate two polarizing filters and watch the transmitted intensity change in real time. Beyond simple linear polarization, it also models circular and elliptical polarization (orthogonal E-field components with a phase offset) and a birefringence mode showing how a calcite crystal splits an unpolarized beam into ordinary and extraordinary rays.

🔬 What it shows

An animated electric-field vector travels from an unpolarized source through polarizer P₁, then (optionally) through a second polarizer/analyzer P₂ set at angle θ relative to P₁. A live polar diagram traces the cos²θ curve while a readout shows θ, cos²θ, and the percentage of light transmitted.

🎮 How to use

Drag the Polarizer 1 and Polarizer 2 sliders to change their angles, toggle the second polarizer on/off, switch between Linear, Circular, Elliptical, and Birefringence modes, or jump to a preset such as "Crossed (0)" or "45° (½ power)" to see Malus's Law at key angles instantly.

💡 Did you know?

At θ = 90° (crossed polarizers) transmission drops to exactly zero — no light gets through — while at θ = 0° it passes at full intensity. Polaroid sunglasses, LCD screens, and photography filters all rely on this same cos²θ relationship discovered by Étienne-Louis Malus in 1808.

Frequently asked questions

What is Malus's Law?

Malus's Law states that when polarized light of intensity I₀ passes through an analyzer whose transmission axis is at angle θ to the light's polarization direction, the transmitted intensity is I = I₀ cos²θ. It was discovered by French physicist Étienne-Louis Malus in 1808.

Why does no light pass through crossed polarizers?

When two polarizers are crossed at 90°, cos²(90°) = 0, so the analyzer blocks the entire component of the light aligned with the first polarizer's axis. This is the "Crossed (0)" preset in the simulation.

What is the difference between linear, circular, and elliptical polarization?

Linear polarization has the E-field oscillating along a single fixed direction. Circular polarization arises from two equal-amplitude, orthogonal oscillations 90° out of phase, tracing a circle. Elliptical polarization is the general case with unequal amplitudes and/or a different phase offset, tracing an ellipse.

What does a quarter-wave plate or half-wave plate do?

A quarter-wave plate introduces a quarter-wavelength retardation between two orthogonal polarization components, converting linear polarization into circular polarization (or vice versa). A half-wave plate introduces a half-wavelength retardation, effectively rotating the plane of linear polarization.

What is birefringence and where is it used?

Birefringent materials such as calcite and quartz have two different refractive indices for orthogonal polarization directions, splitting an incoming beam into an "ordinary" ray and an "extraordinary" ray that travel at different speeds. This effect underlies wave plates, LCD screens, and many optical instruments.