Müller Matrix Polarimetry

The 4×4 Müller matrix acts on the Stokes vector [S0,S1,S2,S3] to transform polarization state. Visualize light on the Poincaré sphere and explore wave plates, polarizers, and rotators.

Poincaré Sphere

Polarization Ellipse

Stokes History

Input State

Ellipse angle χ 0°

Ellipticity ψ 0°

Intensity S0 1.00

Optical Element

Element type

Retardance δ 90°

Orientation θ 0°

Stokes Parameters

Input [S0,S1,S2,S3]

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→ M →

Output [S0',S1',S2',S3']

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Stokes properties

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Müller Matrix M:

Müller-Stokes Formalism

Stokes vector: S = [S0, S1, S2, S3] = [I, Q, U, V]

S0=intensity, S1=H/V, S2=±45°, S3=circular polarization

Wave plate M: M_WP(δ,θ) = rotation to θ · diag[1,1,cosδ,−sinδ; ...] · rotation back

Polarizer M: M_LP(θ) = ½·[1,cos2θ,sin2θ,0; cos2θ,...]

Degree of polarization: DOP = √(S1²+S2²+S3²)/S0 ≤ 1