The Stokes parameters are a set of values that describe the polarization state of electromagnetic radiation (including visible light). They were introduced by George Gabriel Stokes in 1852, as a mathematically convenient alternative to the more common description of incoherent or partially polarized radiation in terms of its total intensity (I), (fractional) degree of polarization (p), and the shape parameters of the polarization ellipse. The relationship of the Stokes parameters to intensity and polarization ellipse parameters is shown in the equations and figure below.

$\backslash begin\{matrix\}$

S_0 &=& I \\ S_1 &=& I p \cos 2\psi \cos 2\chi\\ S_2 &=& I p \sin 2\psi \cos 2\chi\\ S_3 &=& I p \sin 2\chi \end{matrix}

Here $I\; p$, $2\backslash psi$ and $2\backslash chi$ are the spherical coordinates of the polarization state in the three-dimensional space of the last three Stokes parameters. Note the factor of two before $\backslash psi$ represents the fact that any polarization ellipse is indistinguishable from one rotated by 180°, while the factor of two before $\backslash chi$ indicates that an ellipse is indistinguishable from one with the semi-axis lengths swapped accompanied by a 90° rotation. The four Stokes parameters are sometimes denoted I, Q, U and V, respectively.

Stokes vectors

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The Stokes parameters are often combined into a vector, known as the Stokes vector:

$$

\vec S \ = \begin{pmatrix} S_0 \\ S_1 \\ S_2 \\ S_3\end{pmatrix} = \begin{pmatrix} I \\ Q \\ U \\ V\end{pmatrix}

The Stokes vector spans the space of unpolarized, partially polarized, and fully polarized light. For comparison, the Jones vector only spans the space of fully polarized light, but is more useful for problems involving coherent light. The four Stokes parameters do not form a preferred basis of the space, but rather were chosen becasue they can be easily measured or calculated.

The effect of an optical system on the polarization of light can be determined by constructing the Stokes vector for the input light and applying Mueller calculus, to obtain the Stokes vector of the light leaving the system.

Examples

Below are shown some Stokes vectors for common states of polarization of light.

$$

\begin{pmatrix} 1 \\ 1 \\ 0 \\ 0\end{pmatrix} \quad \textrm{Linearly\ polarized\ (horizontal)}

$$

\begin{pmatrix} 1 \\ -1 \\ 0 \\ 0\end{pmatrix} \quad \textrm{Linearly \ polarized \ (vertical)}

$$

\begin{pmatrix} 1 \\ 0 \\ 1 \\ 0\end{pmatrix} \quad \textrm{Linearly \ polarized\ } (+45^\circ)

$$

\begin{pmatrix} 1 \\ 0 \\ -1 \\ 0\end{pmatrix} \quad \textrm{Linearly \ polarized\ } (-45^\circ)

$$

\begin{pmatrix} 1 \\ 0 \\ 0 \\ 1\end{pmatrix} \quad \textrm{Right-hand \ circularly \ polarized}

$$

\begin{pmatrix} 1 \\ 0 \\ 0 \\ -1\end{pmatrix} \quad \textrm{Left-hand \ circularly \ polarized}

$$

\begin{pmatrix} 1 \\ 0 \\ 0 \\ 0\end{pmatrix} \quad \textrm{Unpolarized}

See also

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• Mueller calculus
• Jones calculus

Polarization | Radiometry

Stokes-Parameter