Malus Law Calculator

Compute transmitted light intensity through a polarizer–analyzer system. Visualize I = I₀·cos²θ, explore polarization angles, and understand the fundamental law of wave optics.

Input parameters
Quick angles:
All calculations are performed locally in your browser – no data is transmitted.
Transmitted Intensity
I = I₀ · cos²θ = 0.000 [units]
Relative intensity I/I₀ = 0.0000    cos²θ = 0.0000
Effective angle (mod 180°): 0.0°
Malus's law: I(θ) = I₀ · cos²(θ)
Max at 0°/180°, zero at 90°
cos²θ intensity curve

Normalized transmitted intensity vs. angle (0° to 180°). The current angle is marked with a red dot.

Polarizer–Analyzer Visualization

θ = 0.0°
first polarizer (fixed) · analyzer rotated by θ

The transmitted intensity depends only on the square of the cosine of the angle between the transmission axes.

Understanding Malus's Law: The Science of Polarized Light

Malus’s law (also Malus law) describes the intensity of light transmitted through two ideal polarizing filters. In 1808, French physicist Étienne-Louis Malus discovered that when polarized light passes through an analyzer rotated by an angle θ relative to the polarizer, the transmitted intensity follows I = I₀ cos²θ. This elegant relationship is fundamental to wave optics and reveals the transverse nature of electromagnetic waves.

I(θ) = I₀ · cos²(θ) where I₀ is the initial intensity after the first polarizer (polarized light) and θ is the angle between the transmission axes.

Derivation & Physical origin

Consider an electric field E₀ after the first polarizer, linearly polarized along the transmission axis. The second polarizer (analyzer) transmits only the component parallel to its axis: E = E₀ cosθ. Since intensity is proportional to the square of the amplitude, I ∝ E² = E₀² cos²θ ⇒ I = I₀ cos²θ. This derivation, based on the electromagnetic wave model, elegantly explains why crossed polarizers (θ = 90°) block all light. The law holds for any coherent or incoherent polarized light as long as the polarizers are ideal.

Key insights & Optical behavior

  • Maximum transmission: θ = 0° or 180° → cos²θ = 1 → full intensity I₀.
  • Minimum transmission: θ = 90° → cos²θ = 0 → complete extinction (ideal polarizers).
  • Half-intensity angles: θ = 45°, 135° → cos² θ = 0.5 → I = I₀/2.
  • The law exhibits a 180° periodicity: cos²(θ) = cos²(θ + 180°).

Real-world Applications

Liquid Crystal Displays (LCDs): Malus’s law controls pixel brightness by rotating liquid crystal molecules to vary the effective angle between polarizers. Photography & sunglasses: Circular polarizers reduce glare from reflective surfaces based on angle-dependent transmission. 3D cinema (IMAX 3D): Orthogonal polarizations separate left/right eye images. Stress analysis (photoelasticity): Birefringence effects are quantified with crossed polarizers. Astronomy: Polarization measurements reveal magnetic fields in interstellar dust. Understanding Malus’s law is indispensable for optical engineers designing lasers, fiber optic sensors, and ellipsometers.

Beyond Malus: Unpolarized light & multiple polarizers

When unpolarized light passes through a polarizer, its intensity is halved (I₁ = I₀/2). Applying Malus’s law to a second polarizer gives I₂ = I₁ cos²θ = (I₀/2) cos²θ. The tool above assumes the input light is already linearly polarized (I₀ after first polarizer). For three polarizers, the system can produce non‑zero transmission even when outer polarizers are crossed (the "polarization revival" effect). The general principle remains rooted in the cos² dependence.

Case study: Optical isolator / variable attenuator

A laser beam of intensity 10 mW passes through a rotating polarizer (analyzer). Engineers need to continuously attenuate output from 10 mW to nearly zero. Using θ from 0° to 90°, the transmitted power follows I_out = 10·cos²θ mW. At θ = 30°, output = 7.5 mW; at 60°, 2.5 mW; at 85°, ~0.076 mW. This configuration is used in fiber optics variable optical attenuators (VOAs). This calculator instantly provides such values for design iterations.

Frequently Asked Questions

cos²θ comes from projecting the electric field vector onto the transmission axis of the analyzer, then squaring to obtain intensity. It represents the fraction of incident polarized power that passes through.

Real polarizers have extinction ratios (contrast). The transmitted intensity becomes I = I₀ [ (1−R)cos²θ + R ] where R is the residual transmission for crossed polarizers. But the ideal cos²θ law is the fundamental baseline for optics education and design.

cos²(θ) = cos²(θ + 180°), so the physical transmission repeats every 180 degrees. Our system normalizes any input angle to the range [0°,180°] for clarity and correct graphical representation.

The calculator automatically maps negative angles to the equivalent positive representation modulo 180° (e.g., -30° → 150°, which yields same cos² value as 30°).

This calculator assumes the incident light is already linearly polarized (I₀ is the intensity after the first polarizer). For natural (unpolarized) light, add a factor of 1/2: I_final = ½ I_unpolarized cos²θ.

Étienne-Louis Malus (1775–1812) discovered the law while experimenting with calcite crystals. In 1808 he observed that the intensity of light reflected from a surface varies as cos²θ, laying the groundwork for modern polarization optics.

Peer‑reviewed scientific content – This tool references authoritative optics sources: Hecht, E. "Optics" (5th ed.); Pedrotti, F.L. "Introduction to Optics"; and peer‑reviewed articles on Malus’s law. Our implementation follows exact trigonometric derivation for high accuracy. Reviewed by the GetZenQuery tech team, updated April 2026.

References: Wolfram ScienceWorld, Wikipedia – Malus's law, MIT OpenCourseWare 8.03.