Polarization Calculator
Calculate the intensity of light transmitted through a polarizer and analyzer using Malus's Law. Perfect for optics students and engineers designing polarized optical systems.
About this calculator
Malus's Law describes how the intensity of polarized light changes as it passes through a second polarizer (the analyzer). For initially polarized light, the transmitted intensity is: I = I₀ · cos²(θ), where I₀ is the incident intensity and θ is the angle between the polarizer's transmission axis and the analyzer's axis. When the input light is unpolarized, the first polarizer transmits exactly half the intensity regardless of orientation, so the formula becomes: I = (I₀ / 2) · cos²(θ). When θ = 0° the polarizer and analyzer are aligned and maximum light passes through; at θ = 90° they are crossed and ideally no light is transmitted. This principle underlies LCD screens, sunglasses, photographic filters, and stress-analysis instruments. Brewster's angle — where reflected light becomes completely polarized — is arctan(n₂/n₁) and can also be calculated for material pairs.
How to use
Suppose unpolarized light with intensity I₀ = 200 W/m² passes through a polarizer set at 0°, then hits an analyzer set at 40°. The angle difference θ = 40° − 0° = 40°. Since the light starts unpolarized, apply: I = (200 / 2) · cos²(40°) = 100 × (0.7660)² = 100 × 0.5868 ≈ 58.7 W/m². Now rotate the analyzer to 90°: I = 100 × cos²(90°) = 100 × 0 = 0 W/m². The two configurations illustrate partial transmission at 40° and complete extinction (in ideal conditions) at 90°.
Frequently asked questions
What does Malus's Law predict when the polarizer and analyzer are perpendicular?
When the transmission axes of the polarizer and analyzer are exactly 90° apart — called the 'crossed polarizer' configuration — Malus's Law predicts zero transmitted intensity, because cos²(90°) = 0. In practice, real polarizers are not perfect and a small residual leakage occurs, but good quality polarizers achieve extinction ratios of 1000:1 or better. Crossed polarizers are used in LCD displays: liquid crystal cells rotate the polarization angle of each pixel, selectively allowing light to pass through the second polarizer to create images. They are also used in photoelastic stress analysis, where stressed transparent materials rotate polarized light, revealing internal stress patterns as colorful fringe patterns.
How does Malus's Law apply to unpolarized light passing through a single polarizer?
When completely unpolarized light strikes a linear polarizer, the transmitted intensity is exactly I₀/2 regardless of the polarizer's orientation. This is because unpolarized light can be modeled as a superposition of equal intensities in all polarization directions; integrating cos²(θ) over all angles from 0° to 360° averages to one-half. This halving of intensity is unavoidable with a single ideal linear polarizer and sets a fundamental limit on how much light can be extracted from an unpolarized source. After passing through the first polarizer, the light is now linearly polarized and subsequent transmission through a second polarizer follows the full Malus cos² relationship.
What is Brewster's angle and how is it related to polarization?
Brewster's angle (θ_B) is the specific angle of incidence at which light reflecting off a dielectric surface (such as glass or water) becomes completely linearly polarized in the plane parallel to the surface. It is given by θ_B = arctan(n₂/n₁), where n₁ and n₂ are the refractive indices of the two media. At this angle, the reflected and refracted rays are perpendicular to each other and the reflected beam contains only the s-polarized (perpendicular) component. Polarized sunglasses exploit this effect: their vertically-oriented polarizers block the horizontally-polarized glare reflected from roads and water surfaces at near-Brewster's angle. For glass (n ≈ 1.5) in air, Brewster's angle is about 56°.