Stokes parameters

The Stokes parameters are a set of four values, usually as or and denotes the 1852 by George Gabriel Stokes to describe the polarization state of electromagnetic waves were (usually light) introduced. The special thing about these values ​​is that they can be calculated by simply measuring the radiant power after passing through different polarizers , and so the polarization state can be determined quite easily. ${\displaystyle S_{0}\ldots S_{3}}$ ${\displaystyle I,Q,U}$ ${\displaystyle V}$

The Stokes parameters can be combined to form the Stokes vector . Analogous to the Jones vector and the Jones matrix - also called Jones formalism - the effect of optical systems on Stokes vectors can be treated in the Müller formalism by using appropriate matrices ( Müller matrix ). In contrast to the Jones formalism, the irradiance can be described, but only for incoherent light. This means that no phase information is contained and therefore does not allow the calculation of interference effects .

definition

Examples

polarization	Polarization state	Stokes vector
linear, horizontal		${\displaystyle {\begin{pmatrix}1\\1\\0\\0\end{pmatrix}}}$
linear, vertical		${\displaystyle {\begin{pmatrix}1\\-1\\0\\0\end{pmatrix}}}$
linear, + 45 °		${\displaystyle {\begin{pmatrix}1\\0\\1\\0\end{pmatrix}}}$
left-circular		${\displaystyle {\begin{pmatrix}1\\0\\0\\-1\end{pmatrix}}}$
right-circular		${\displaystyle {\begin{pmatrix}1\\0\\0\\1\end{pmatrix}}}$
unpolarized		${\displaystyle {\begin{pmatrix}1\\0\\0\\0\end{pmatrix}}}$

${\displaystyle S_{0}=I=P_{0^{\circ }}+P_{90^{\circ }}=\langle E_{x}^{2}+E_{y}^{2}\rangle }$

${\displaystyle S_{1}=Q=P_{0^{\circ }}-P_{90^{\circ }}=\langle E_{x}^{2}-E_{y}^{2}\rangle }$

${\displaystyle S_{2}=U=P_{45^{\circ }}-P_{135^{\circ }}=\langle 2E_{x}E_{y}\cos \delta \rangle }$

${\displaystyle S_{3}=V=P_{\rm {RZ}}-P_{\rm {LZ}}\,=\langle 2E_{x}E_{y}\sin \delta \rangle }$

The powers are the measured power after passing through a horizontally (0 °), vertical (90 °), 45 ° and 135 ° oriented, ideal polarizer as well as the right and left circularly polarized portion of the light.

Alternatively, they can be defined using the time-averaged amplitudes , the electrical wave vectors in an orthogonal coordinate system, and their relative phase . ${\displaystyle E_{\mathrm {x} }}$ ${\displaystyle E_{\mathrm {y} }}$ ${\displaystyle \delta }$

Usually, the Stokes parameters are normalized to the incident power by dividing all four values ​​by S ₀ ; in this context one speaks of the normalized Stokes vector .

${\displaystyle {\vec {S}}_{\mathrm {N} }={\frac {1}{S_{0}}}{\begin{pmatrix}S_{0}\\S_{1}\\S_{2}\\S_{3}\end{pmatrix}}}$

Definition of the Stokes parameters in other reference systems

Reference value	${\displaystyle S_{0}}$	${\displaystyle S_{1}}$	${\displaystyle S_{2}}$	${\displaystyle S_{3}}$
Light intensity	${\displaystyle I_{\mathrm {x} }+I_{\mathrm {y} }}$	${\displaystyle I_{\mathrm {x} }-I_{\mathrm {y} }}$	${\displaystyle I_{\mathrm {+45^{\circ }} }-I_{\mathrm {-45^{\circ }} }}$	${\displaystyle I_{\mathrm {R} }-I_{\mathrm {L} }}$
ε-θ system	${\displaystyle 1}$	${\displaystyle \cos {2\epsilon }\cdot \cos {2\theta }}$	${\displaystyle \cos {2\epsilon }\cdot \sin {2\theta }}$	${\displaystyle \sin {2\epsilon }}$
ψ-Δ system	${\displaystyle 1}$	${\displaystyle -\cos {2\psi }}$	${\displaystyle \sin {2\psi }\cdot \cos {\Delta }}$	${\displaystyle -\sin {2\psi }\cdot \sin {\Delta }}$

Definition in spherical coordinates

Another formulation can be found in spherical coordinates:

${\displaystyle {\begin{matrix}S_{0}&=&I\\S_{1}&=&I_{\mathrm {p} }\cos 2\psi \cos 2\chi \\S_{2}&=&I_{\mathrm {p} }\sin 2\psi \cos 2\chi \\S_{3}&=&I_{\mathrm {p} }\sin 2\chi \end{matrix}}}$

Where is the total intensity, the polarized portion of the intensity, the tilt of the polarization ellipse and the ratio of the two main axes of the polarization ellipse. ${\displaystyle I}$ ${\displaystyle I_{\mathrm {p} }}$ ${\displaystyle \psi }$ ${\displaystyle \tan \chi }$

Degree of polarization

The degree of polarization Π indicates how large the ordered part of the wave is. It is defined by:

${\displaystyle \Pi ={\frac {\sqrt {S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}}{S_{0}}}}$

Or for only linearly polarized light:

${\displaystyle \Pi ={\frac {\sqrt {S_{1}^{2}+S_{2}^{2}}}{S_{0}}}}$

The following applies to fully polarized light:

${\displaystyle S_{0}^{2}=S_{1}^{2}+S_{2}^{2}+S_{3}^{2}}$

For unpolarized light, however, the following applies:

${\displaystyle S_{1}=S_{2}=S_{3}=0}$.

Maximum polarization angle

The angle of maximum polarization is defined by:

${\displaystyle \Theta ={\frac {1}{2}}\arctan {\frac {S_{2}}{S_{1}}}+n{\frac {\pi }{2}}}$

where is for , otherwise . In other words, this means that you have to add 90 ° to the angle if is less than . ${\displaystyle n=1}$ ${\displaystyle S_{1}<0}$ ${\displaystyle n=0}$ ${\displaystyle \Theta }$ ${\displaystyle S_{1}}$ ${\displaystyle 0}$

literature

• William A. Shurcliff: Polarized Light: Production and Use . Harvard University Press, Cambridge, Mass. 1962, ISBN 0-674-68250-5 .
• Craig F. Bohren, Donald R. Huffman: Absorption and scattering of light by small particles . Wiley, New York 1998, ISBN 0-471-29340-7 .

Web links

• Jürgen Weiprecht: Polarization and Stokes parameters. In: Compendium for the astronomical internship. Hans-Georg Reimann, Olaf Fischer, Christian Friedemann, Reinhard E. Schielicke, October 29, 2002, accessed on February 2, 2010 . 

Individual evidence

1. ↑ Hiroyuki Fujiwara: Spectroscopic Ellipsometry: Principles and Applications . Wiley-Interscience, 2007, ISBN 978-0-470-01608-4 , pp. 75 .