Stokes parameters

The Stokes parameters $(S_0, S_1, S_2, S_3)$ are a four-component vector representation of the polarization state of light. Unlike Jones vectors — which describe only fully-polarized light — Stokes parameters can describe any polarization state, including partially polarized and fully unpolarized light. They are the standard polarimetric description used in optical measurement and remote sensing.

Definitions. The Stokes parameters are measurable quantities:

$$S_0 \;=\; I_{0°} + I_{90°} \quad \text{(total intensity)}$$ $$S_1 \;=\; I_{0°} - I_{90°} \quad \text{(horizontal vs vertical linear)}$$ $$S_2 \;=\; I_{+45°} - I_{-45°} \quad \text{(diagonal linear)}$$ $$S_3 \;=\; I_\text{RCP} - I_\text{LCP} \quad \text{(circular polarization)}$$

where $I_{\theta}$ is the intensity transmitted through a linear polarizer at angle $\theta$ from the horizontal, and $I_\text{RCP}, I_\text{LCP}$ are the intensities transmitted through right- and left-circular polarizers.

All four Stokes parameters have units of intensity (typically W/m²). They are derived from measurable transmission through specific polarizing optics — no electric-field amplitude or phase need be measured.

Standard polarization states.

State	$S_0$	$S_1$	$S_2$	$S_3$
Horizontal linear	1	1	0	0
Vertical linear	1	$-1$	0	0
+45° linear	1	0	1	0
$-45°$ linear	1	0	$-1$	0
Right-circular	1	0	0	1
Left-circular	1	0	0	$-1$
Unpolarized	1	0	0	0

Degree of polarization. The total polarization fraction:

$$P \;=\; \frac{\sqrt{S_1^2 + S_2^2 + S_3^2}}{S_0}.$$

$P = 1$ for fully polarized light; $P = 0$ for fully unpolarized light; intermediate values correspond to partial polarization. The unpolarized fraction is $(1 - P)$.

Poincaré sphere. Normalize the Stokes parameters by $S_0$: $(s_1, s_2, s_3) = (S_1, S_2, S_3)/S_0$. For fully polarized light, these obey $s_1^2 + s_2^2 + s_3^2 = 1$ — lying on a unit sphere called the Poincaré sphere. Each point on the sphere uniquely identifies a fully-polarized state:

• Equator ($s_3 = 0$): linear polarization states, with azimuth angle parameterized by longitude
• North pole ($s_3 = 1$): right-circular polarization
• South pole ($s_3 = -1$): left-circular polarization
• Mid-latitudes: elliptically polarized states

Partially-polarized light corresponds to a point inside the sphere; fully-unpolarized light is at the origin. Polarization-changing optical elements (waveplates, polarizers, optical rotators) trace out trajectories on the sphere as parameters change.

Mueller matrices. A linear optical element acting on Stokes vectors is represented by a 4×4 Mueller matrix $\mathbf{M}$:

$$\mathbf{S}_\text{out} \;=\; \mathbf{M} \mathbf{S}_\text{in}.$$

Common Mueller matrices:

Element	Mueller matrix description
Linear polarizer at angle $\theta$	$4\times4$ with elements depending on $\theta$
Quarter-wave plate at angle $\theta$	Rotates Stokes vector by 90° about appropriate axis
Half-wave plate at angle $\theta$	Rotates Stokes vector by 180° about appropriate axis
Optical rotator (Faraday)	Rotates azimuth on Poincaré sphere
Depolarizer	Reduces magnitude of $(S_1, S_2, S_3)$

A complete polarimetric measurement applies the Mueller matrix in the forward sense (predicting output from input polarization); inverse problems (recovering medium properties from polarimetric measurements) form the basis of polarimetric imaging and remote sensing.

Standard applications.

• Polarimetric remote sensing: identify materials by their polarimetric signatures (Stokes vector imaging)
• Stress/strain measurement: photoelastic effect changes birefringence; Stokes parameters map stress
• Liquid crystal characterization: Stokes-vector analysis of transmission through LC samples
• Spectropolarimetry: combine wavelength-resolved and polarization-resolved measurements
• Fiber-optic polarization monitoring: real-time Stokes parameter measurement at fiber output
• Polarimetric scattering: characterize biological and other complex tissues
• Quantum optics: photon-pair polarization correlations expressed in Stokes basis

Measurement equipment. A complete Stokes polarimeter requires 4 measurements:

Measurement	Optical configuration
$I_{0°}$, $I_{90°}$	Linear polarizer at 0° and 90°
$I_{+45°}$, $I_{-45°}$	Linear polarizer at 45° and $-45°$
$I_\text{RCP}$	Quarter-wave plate at 45° + linear polarizer at 0°
$I_\text{LCP}$	Quarter-wave plate at $-45°$ + linear polarizer at 0°

Modern Stokes polarimeters (Thorlabs PAX series, Schäfter+Kirchhoff PolaScope) use rotating-element or photoelastic-modulator designs to acquire all four parameters simultaneously or in rapid sequence.

Stokes vs Jones representations.

Aspect	Stokes vector	Jones vector
Describes fully-polarized light	Yes	Yes
Describes partially-polarized light	Yes	No
Real or complex	Real (4 components)	Complex (2 components)
Physical units	Intensity (W/m²)	Field amplitude (V/m)
Direct measurement	Yes	No (phase usually unknowable)
Coherent optical operations	Mueller matrix (4×4)	Jones matrix (2×2)
Convention	Includes intensity scale factor	Pure direction

Stokes parameters are the natural choice for measurement and for any application involving incoherent or partially coherent light. Jones vectors are simpler for theoretical calculations of fully-polarized fields.

References: Saleh & Teich, Fundamentals of Photonics (3rd ed., 2019), Ch. 6 (polarization optics); Born & Wolf, Principles of Optics (7th ed., 1999), Ch. 10 (statistical optics) for the comprehensive treatment; Goldstein, Polarized Light (3rd ed., 2010) for the engineering reference.