How to calculate transmitted intensity with Malus's law lets you estimate the fraction of polarized light passing through a polarizer rotated by angle θ relative to the incident polarization. The formula was derived experimentally in 1809 by Étienne-Louis Malus (École Polytechnique Paris) during studies of light reflected off glass, and remains the cornerstone of polarimetry. The standard form is I = I₀ · cos²(θ), where I₀ is the incident intensity (assuming linearly polarized light) and θ the angle between polarization and polarizer axis. For unpolarized light incident on a first polarizer, intensity halves instantaneously: I_after₁ = I₀/2; a second polarizer rotated by θ transmits I = (I₀/2)·cos²(θ). It is the basis of photographic polarizing filters (Hoya, B+W, Marumi, Tiffen), every smartphone and TV LCD/LED screen, RealD/IMAX 3D cinema glasses, anti-glare Ray-Ban sunglasses, and stress-optics and polarimetry instruments at NIST, MIT, Caltech, Lockheed Martin, Northrop Grumman.

1. Malus's law and light polarization

Polarization of an electromagnetic wave describes the orientation of the oscillating electric field perpendicular to the propagation direction. Linear, circular (right/left), and elliptical polarizations exist. A linear polarizer (Polaroid, Glan-Thompson calcite, Wollaston, stretched polymer) selects only the component along its axis. For light already polarized with intensity I₀ at angle θ, transmitted intensity is I = I₀·cos²(θ). For unpolarized light (sunlight, incandescent), intensity after the first polarizer is I = I₀/2, independent of orientation. Two-polarizer Malus rotation enables sub-percent intensity modulation, basis of many precision optical devices.

2. Typical configurations

• Parallel polarizers (θ=0°) — T = 100%: maximum transmission.
• 30° — T = 75%.
• 45° — T = 50%: reference for many applications.
• 60° — T = 25%.
• Crossed polarizers (θ=90°) — T = 0%: blocks all light, basis of LCD off-pixels.
• Three polarizers at 0°/45°/90° — T = cos²(45°)·cos²(45°) = 25% (counterintuitive: without the middle polarizer T=0).
• Hoya HD photographic polarizer at 45° for water reflection elimination — T = 0.5 nominal plus directional sky selectivity.

3. Validity range and corrections

Malus's law assumes ideal polarizers with infinite extinction ratio (crossed T=0%). Real polarizers have finite ratios: HN-22 Polaroid ER ≈ 10⁴, Glan-Thompson calcite ER ≈ 10⁵-10⁶, dielectric polarizers for lasers ER ≈ 10⁶. They also have transmission losses at θ=0°: Polaroid 35-45%, Glan-Thompson 80-90%, dielectric 95-98%. For polychromatic light, polarizer dispersion introduces wavelength dependence. For partially polarized light with degree p<1, combine linear-Malus with half the unpolarized component. For circular or elliptical polarization, Jones-Mueller formalism with 2×2 (Jones) or 4×4 (Mueller-Stokes) matrices is needed. Strain birefringence in plastics and stressed glass has complex chromatic-angular dependence.

4. Worked numerical examples

Example 1 — Hoya HD polarizing filter on Sony A7IV. Unpolarized sunlight I₀ = 100 units. After first polarizer: I₁ = 50. Rotating the filter by 45° relative to a water surface (p-polarized reflection): the reflection drops from I_refl·cos²(0)=I_refl to I_refl·cos²(90°)=0 — completely eliminated, sky blue saturates. Used in mountain and wildlife photography.

Example 2 — Pixel-level LCD screen. Unpolarized LED backlight, first lower polarizer (T=50%, I=50), liquid crystals rotate polarization 0-90° pixel by pixel based on applied voltage. Second upper polarizer crossed at 90°: "off" pixel T=cos²(90°)=0%, "on" pixel T=cos²(0°)=100%. 0-100% per-pixel modulation produces the image. Samsung, LG, Sharp, BOE technology.

Example 3 — Three polarizers at 0°, 45°, 90° (classical quantum paradox). I₀=100 → after 1st (axis 0°): I=50 → after 2nd (axis 45°): I=50·cos²(45°)=25 → after 3rd (axis 90°): I=25·cos²(45°)=12.5%. Without the middle polarizer, 1st and 3rd crossed give T=0%. Classic demonstration in MIT, Caltech and Stanford physics labs of quantum interference of light.

5. Practical applications

• Photographic polarizing filters: Hoya HD, B+W (Schneider), Marumi DHG, Tiffen, Heliopan, Sigma. Eliminate water/glass reflections, saturate blue skies, reduce haze. Standard for landscape, wildlife, and sports photography
• LCD/LED screens: every smartphone, TV, monitor, laptop uses crossed polarizers with pixel-pixel modulated liquid crystals — Samsung, LG, Sharp, BOE China, AU Optronics Taiwan
• 3D cinema glasses: RealD uses circular polarization (Malus modified), IMAX 3D uses ±45° linear
• Anti-glare sunglasses: Ray-Ban Luxottica, Maui Jim, Costa, Oakley use polarizing filters to reduce road and water glare
• Stress optics and photoelasticity: used at NIST, MIT, Caltech for non-destructive analysis of mechanical components, bridges, dams
• Chemical polarimetry: optical rotation of sugar solutions, AOAC analytical chemistry standard

6. Comparison with other formalisms

The Jones formalism (1941) uses 2×1 vectors and 2×2 matrices to describe polarized states and propagation through optical elements. More powerful than Malus for handling circular/elliptical light. The Mueller-Stokes formalism (1943) uses 4×1 vectors (Stokes I,Q,U,V) and 4×4 matrices, handling partially polarized light and depolarizers. For Brewster polarization (Brewster angle, where R_p=0), Malus is a special case at θ=90° for the polarizer. For circular beams, Malus must be replaced by 50% transmission through linear polarizer. For optically active materials (sugars, peptides, blood glucose), optical rotation is measured via modified Malus. Codes Zemax OpticStudio, COMSOL Multiphysics, Lumerical FDTD, MEEP implement all formalisms.

7. Special cases and advanced applications

For Polaroid film (originals from Edwin Land 1932), the principle is oriented molecular dichroism — absorbing molecules aligned by mechanical stretching. For thin-film dielectric polarizers used in HeNe, Nd:YAG, ArF excimer lasers, ER reaches 10⁶. For Glan-Thompson polarizers (calcite + balsam), 8-10° angular separation between ordinary and extraordinary rays. For electro-optics (Pockels or Kerr cells), polarization is modulated by applied voltage, basis of 40-100 Gb/s telecom optical modulators by Cisco, Nokia, Coriant. For polarizing metasurfaces (developed at MIT, Caltech, Stanford), sub-λ structures control polarization. For quantum optics (MIT, Caltech, IBM Almaden), Malus underpins Bell and GHZ tests with entangled photons.

Updated July 2026