Polarization & Malus Law Calculator
Calculates transmitted light intensity after passing through one or more polarizers using Malus's Law. Use it when designing optical systems, analyzing polarized filters, or studying light attenuation in physics labs.
About this calculator
Malus's Law states that when polarized light passes through a linear polarizer, the transmitted intensity depends on the angle between the light's polarization direction and the polarizer axis. The formula is I = I₀ · cos²(θ), where I₀ is the initial intensity and θ is the polarizer angle. For multiple polarizers each rotated by the same angle θ relative to the previous one, the law applies successively: I = I₀ · cos²ⁿ(θ), where n is the number of polarizers. The calculator also relates to Brewster's angle, θ_B = arctan(n₂/n₁), which gives the angle at which reflected light becomes completely polarized. Together these tools help optical engineers, photographers using polarizing filters, and physicists model how much light survives a multi-stage polarization system.
How to use
Suppose unpolarized light enters with I₀ = 100 W/m², passes through 2 polarizers each set at θ = 30°. Step 1: Enter 100 in Initial Light Intensity. Step 2: Enter 30 in Polarizer Angle. Step 3: Enter 2 in Number of Polarizers. The formula gives: I = 100 × cos²(30°) × cos²(30°)^(2−1) = 100 × (0.866)² × (0.866)² = 100 × 0.75 × 0.75 = 56.25 W/m². So about 56% of the original intensity exits the second polarizer.
Frequently asked questions
What does Malus's Law calculate and how does the angle affect light intensity?
Malus's Law predicts how much light intensity passes through a polarizer based on the angle between the incoming polarization direction and the polarizer's transmission axis. The relationship is proportional to the square of the cosine of that angle, so at 0° all light passes through and at 90° no light passes. Intermediate angles produce partial transmission, making it easy to continuously dim polarized light by rotating a filter. This principle underlies LCD screens, sunglasses, and many optical instruments.
How does adding more polarizers reduce light intensity in a Malus's Law system?
Each successive polarizer applies cos²(θ) attenuation to whatever intensity remains from the previous stage. With n identical polarizers each at angle θ, the total transmitted intensity is I₀ · cos²ⁿ(θ). Even a modest angle like 30° reduces intensity to 75% per stage, so two stages yield ~56% and three stages ~42% of the original. This cascading effect is important when designing optical isolators, variable attenuators, or multi-stage polarimetry setups.
What is Brewster's angle and why does it matter for polarization?
Brewster's angle θ_B is the specific angle of incidence at which light reflected from a dielectric surface (like glass or water) is completely horizontally polarized. It is calculated as 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. Photographers exploit this with polarizing filters to eliminate glare from water or glass, and anti-reflection coatings in optics are also designed around Brewster's angle principles.