50
W/m²
25
W/m²
0
W/m²
0.25
25.0%
45
°
50
W/m²
25
W/m²
0
W/m²
0.25
25.0%
45
°
The Polarization Calculator computes the transmitted intensity through a system of two or three linear polarizers using Malus's law: I = I₀cos²θ. It handles both unpolarized and pre-polarized incident light, calculates the intensity after each polarizer, and shows the total transmission ratio — essential for understanding polarimetry, LCD displays, photography, and optical instrumentation.
Polarization is a fundamental property of transverse waves describing the orientation of the electric field oscillation. Unpolarized light (from the Sun, incandescent bulbs, LEDs) contains random polarization directions that average to zero net polarization. A linear polarizer selects one component, reducing intensity by half but producing a well-defined polarization state that subsequent polarizers can further manipulate.
The core relationship governing polarizer transmission is Malus's law:
$$I = I_0\cos^2\theta$$
where I₀ is the intensity of polarized light incident on an analyzer and θ is the angle between the light's polarization direction and the analyzer's transmission axis.
First polarizer:
Second polarizer (analyzer):
$$I_2 = I_1\cos^2(\theta_2 - \theta_1)$$
Third polarizer (if used):
$$I_3 = I_2\cos^2(\theta_3 - \theta_2)$$
A remarkable consequence: two crossed polarizers (θ = 90°) transmit zero light. But inserting a third polarizer between them at 45° allows some light through — the intermediate polarizer rotates the polarization direction, giving the second pair a non-zero projection. This three-polarizer arrangement transmits I₀ × (1/2) × cos²(45°) × cos²(45°) = I₀/8 for unpolarized input.
The degree of polarization after passing through any linear polarizer is 100% — the transmitted light is fully linearly polarized regardless of the input state.
The intensity after each polarizer decreases by cos²(angle between consecutive polarizers). If two polarizers are crossed (90° apart), the transmitted intensity is zero — complete extinction. The transmission ratio I_out/I_in gives the overall fraction of light passing through the system. For photography, this ratio determines the exposure adjustment needed when using a polarizing filter (typically 1-2 stops).
Inputs
Results
Unpolarized light (100 W/m²) passes through three polarizers at 0°, 45°, 90°. After P1: 50 (half of unpolarized). After P2: 50×cos²(45°) = 25. After P3: 25×cos²(45°) = 12.5. Total transmission: 12.5%, demonstrating how an intermediate polarizer allows light through otherwise crossed polarizers.
Inputs
Results
Unpolarized light at 200 W/m² passes through a polarizer-analyzer pair at 30° apart. After P1: 100 W/m². After P2: 100×cos²(30°) = 75 W/m². Total transmission: 37.5%.
Malus's law states that when polarized light of intensity I₀ passes through a linear polarizer (analyzer), the transmitted intensity is I = I₀cos²θ, where θ is the angle between the polarization direction and the polarizer's transmission axis. It was discovered by Étienne-Louis Malus in 1809.
Unpolarized light contains all polarization directions equally. For each component at angle θ to the polarizer axis, the transmitted fraction is cos²θ. Averaging cos²θ over all angles from 0° to 360° gives exactly 1/2. So a single polarizer always halves the intensity of unpolarized light.
Two crossed polarizers (90° apart) transmit zero light. Inserting a third polarizer at 45° between them breaks the problem into two 45° steps. Each step transmits cos²(45°) = 50% of the polarized light. The intermediate polarizer rotates the polarization direction, giving the final analyzer a non-zero component to transmit.
Polarization is used in LCD displays (crossed polarizers with liquid crystal between them), photography (reducing glare and reflections), 3D cinema (different polarizations for each eye), optical stress analysis (photoelasticity), sunglasses (blocking horizontally polarized glare), and fiber-optic communications.
Brewster's angle is the angle of incidence at which reflected light is perfectly polarized. It occurs when the reflected and refracted rays are perpendicular: tan(θ_B) = n₂/n₁. For glass (n ≈ 1.5) in air, Brewster's angle is about 56.3°. This is why polarized sunglasses reduce glare from flat surfaces.
Pure linear polarization does not change the wavelength or color. However, birefringent materials (like calcite, stressed plastics, or cellophane) have different refractive indices for different polarization directions, which can create wavelength-dependent phase shifts and produce vivid colors when viewed between crossed polarizers.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
How helpful was this calculator?
Be the first to rate!
