Light Wavefront Polarization & Malus's Law

Because light is a transverse wave, a crystalline analyzer transmits only the projection of E parallel to its optical axis.

Advanced Level · AP / University

Mechanism

Because light is a transverse electromagnetic wave (E ⊥ k), the electric field vector E oscillates in the plane perpendicular to propagation. Unpolarized natural light consists of rapidly varying random polarization orientations, whereas a linear polarizer transmits only the component of E parallel to its transmission axis.

Equations and conservation laws

When linearly polarized light of amplitude E₀ and intensity I₀ ∝ E₀² encounters a second polarizer (analyzer) rotated by angle θ, only the projected amplitude E = E₀ cos θ passes through. Since optical intensity scales as the square of the electric field amplitude, the transmitted intensity obeys Malus’s Law: I(θ) = I₀ cos ²θ, vanishing completely when the axes are crossed (θ = 90°).

Measurements and applications

Polarization also occurs by reflection off a dielectric of refractive index n. At Brewster’s angle tan θ_B = n (where reflected and refracted rays are orthogonal, θ_B + γ = 90°), the p-polarized component cannot radiate along the reflection direction, rendering the reflected beam 100% linearly s-polarized, which is the principle behind anti-glare polarized sunglasses.

Experimental parameters and SI units

Analyzer axis angle θ: 0 … 90 °, standard value 35 °
Polarized intensity I₀: 40 … 160 W/m², standard value 100 W/m²
Light wavelength λ: 420 … 680 nm, standard value 530 nm
Glass refractive index n (for θ_B): 1.3 … 1.9, standard value 1.52

Physical model and application

Light reflected from a horizontal dielectric surface (such as water or glass) at Brewster’s angle becomes completely linearly polarized parallel to the interface. A polarizing analyzer oriented with a vertical transmission axis extinguishes this glare in accordance with Malus’s law I = I₀ cos ²θ.

Eₒᵤₜ = E₀ cos θ, I(θ) = I₀ cos ²θ (Malus's Law) tan θ_B = n₂₁, θ_B + θ_refr = 90° (Brewster's Law)

Question

A linearly polarized light beam of intensity I₀ strikes a second polarizer (analyzer) whose transmission axis is rotated by θ = 60° relative to the beam's polarization plane. What intensity I emerges from the analyzer?

Selected answer Option B

The analyzer passes only the electric field component E = E₀ cos θ along its axis. Since optical intensity is proportional to the square of field amplitude (I ∝ E²), we obtain Malus's Law: I(θ) = I₀ cos ²θ. At θ = 60°, cos 60° = 0.5 so I = 0.25 I₀, and at θ = 90° (crossed polarizers) the wavefront is completely blocked.

Formula

I(θ) = I₀ cos ²θ, E(θ) = E₀ | cos θ|, θ_B = arctan (n)

θ = 35°, I = 67.1 W/m², θ_B = 56.7°

Key indicators 67.1 W/m²

Transmitted I₀cos²θ

67.1 W/m²

Input after P₁ (I₀)

100.0 W/m²

Blocked I₀sin²θ

32.9 W/m²

Fig. 21.1 . Transmitted intensity I = I₀ cos²θ vs. fraction blocked by the analyzer

Intensity I(θ) 67.1 W/m²

Factor cos²θ 67.1 %

Brewster angle θ_B 56.7°

State after analyzer Linear (θ = 35°)