Malus Law Calculator

Malus’s law describes how the intensity of linearly polarized light changes when it passes through an ideal linear polarizer called an analyzer. The transmitted intensity I equals the incident polarized intensity I₀ multiplied by the square of the cosine of angle θ between the light’s polarization direction and the analyzer’s transmission axis. When the axes align at zero degrees, the cosine is one and the ideal analyzer transmits the full polarized intensity. At ninety degrees, the axes are crossed and the ideal transmitted intensity is zero. The calculator can solve for any one of the three variables. To find final intensity, it evaluates I₀ cos²(θ). To recover initial intensity, it divides the measured final intensity by cos²(θ). That inverse becomes undefined at ninety degrees because an ideal crossed analyzer transmits no information about the original magnitude. To determine the principal analyzer angle, it evaluates the inverse cosine of the square root of I/I₀ and reports an angle from zero to ninety degrees. A physical input ratio must lie between zero and one. Angles are entered in degrees and converted internally for the trigonometric calculation. Intensity may use watts per square meter or another consistent intensity unit because the ratio is dimensionless, but the displayed output uses W/m². The initial value in Malus’s law refers to light that is already linearly polarized immediately before the analyzer. If unpolarized light first passes through an ideal polarizer, its average intensity is reduced by half before Malus’s law is applied to a second polarizer. Real optical systems depart from the ideal model. Polarizers have finite transmission, imperfect extinction, spectral dependence, and alignment uncertainty. Scattering, reflections, partially polarized light, detector offsets, and background illumination can leave nonzero readings with crossed axes. For laboratory work, subtract background, measure actual component transmission, and include uncertainty. Malus’s law is nevertheless a foundational and useful model for polarimetry, optical filters, liquid-crystal displays, stress analysis, glare reduction, and physics education. This solver makes forward and inverse calculations transparent while enforcing the intensity ratio needed for a real principal angle.