Stokes Parameters - Properties

Properties

For purely monochromatic coherent radiation, one can show that

\begin{matrix} Q^2+U^2+V^2 = I^2, \end{matrix}

whereas for the whole (non-coherent) beam radiation, the Stokes parameters are defined as averaged quantities, and the previous equation becomes an inequality:

\begin{matrix} Q^2+U^2+V^2 \le I^2. \end{matrix}

However, we can define a total polarization intensity, so that

\begin{matrix} Q^{2} + U^2 +V^2 = I_p^2, \end{matrix}

where is the total polarization fraction.

Let us define the complex intensity of linear polarization to be

\begin{matrix} L & \equiv & |L|e^{i2\theta} \\ & \equiv & Q +iU. \\ \end{matrix}

Under a rotation of the polarization ellipse, it can be shown that and are invariant, but

\begin{matrix} L & \rightarrow & e^{i2\theta'}L, \\ Q & \rightarrow & \mbox{Re}\left(e^{i2\theta'}L\right), \\ U & \rightarrow & \mbox{Im}\left(e^{i2\theta'}L\right).\\ \end{matrix}

With these properties, the Stokes parameters may be thought of as constituting three generalized intensities:

\begin{matrix} I & \ge & 0, \\ V & \in & \mathbb{R}, \\ L & \in & \mathbb{C}, \\ \end{matrix}

where is the total intensity, is the intensity of circular polarization, and is the intensity of linear polarization. The total intensity of polarization is, and the orientation and sense of rotation are given by

\begin{matrix} \theta &=& \frac{1}{2}\arg(L), \\ h &=& \sgn(V). \\ \end{matrix}

Since and, we have

\begin{matrix} |L| &=& \sqrt{Q^2+U^2}, \\ \theta &=& \frac{1}{2}\tan^{-1}(U/Q). \\ \end{matrix}

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