Müller matrix

The Müller matrix (after Hans Müller , who introduced it in 1943) is a transformation matrix for the Stokes vector , which describes the polarization state of an electromagnetic wave (including visible light ). It characterizes the optical element with regard to the interaction with the wave, for example the polarization state is influenced during reflection at an interface or during transmission through a body. ${\displaystyle \mathrm {M} }$

Analogous to the Jones formalism from the Jones vector and Jones matrix for completely polarized waves, the Stokes vector and Müller matrix form the Müller formalism .

description

The Müller matrix is ​​a 4 × 4 matrix . It describes the change in the intensity and the state of polarization of partially and fully polarized and unpolarized light (described by the Stokes vector) during reflection, refraction or transmission through a material.

To describe the changes in the polarization state, the Stokes vector of the incident light is multiplied by the Müller matrix . The result is again a Stokes vector that describes the reflected or transmitted light, depending on the choice of the Müller matrix: ${\displaystyle {\vec {S}}_{\mathrm {A} }}$ ${\displaystyle \mathrm {M} }$ ${\displaystyle {\vec {S}}_{\mathrm {B} }}$

${\displaystyle {\vec {S}}_{\mathrm {B} }=\mathrm {M} \cdot {\vec {S}}_{\mathrm {A} }}$

A typical area of ​​application is the description of optical components in optical measurement technology , for example ellipsometry . As a rule, several optical components such as polarizers , delay elements or compensators and a sample are used. This optical system can be calculated by multiplying the Stokes vector of the incident light step by step with the Müller matrices of the respective components:

${\displaystyle {\vec {S}}_{\mathrm {B} }={\bigg (}\mathrm {M} _{3}{\Big (}\mathrm {M} _{2}{\big (}\mathrm {M} _{1}\cdot {\vec {S}}_{\mathrm {A} }{\big )}{\Big )}{\bigg )}}$

Since the matrix multiplication is associative , the system can also be described as follows:

${\displaystyle \Leftrightarrow {\vec {S}}_{\mathrm {B} }=\mathrm {M} _{3}\mathrm {M} _{2}\mathrm {M} _{1}{\vec {S}}_{\mathrm {A} }}$

Examples of ideal optical components

Linear polarizer
For horizontal transmission	For vertical transmission	+ 45 °; transmission	−45 °; transmission
${\displaystyle {1 \over 2}{\begin{pmatrix}1&1&0&0\\1&1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}\quad }$	${\displaystyle {1 \over 2}{\begin{pmatrix}1&-1&0&0\\-1&1&0&0\\0&0&0&0\\0&0&0&0\end{pmatrix}}\quad }$	${\displaystyle {1 \over 2}{\begin{pmatrix}1&0&1&0\\0&0&0&0\\1&0&1&0\\0&0&0&0\end{pmatrix}}\quad }$	${\displaystyle {1 \over 2}{\begin{pmatrix}1&0&-1&0\\0&0&0&0\\-1&0&1&0\\0&0&0&0\end{pmatrix}}\quad }$
Retardation plate
λ / 4 ( fast-axis ; vertical)	λ / 4 ( fast-axis ; horizontal)	λ / 2 ( fast-axis ; vertical)
${\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&-1\\0&0&1&0\end{pmatrix}}\quad }$	${\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&0&1\\0&0&-1&0\end{pmatrix}}\quad }$	${\displaystyle {\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}}\quad }$
Other
Attenuation filter (25% transmission)
${\displaystyle {1 \over 4}{\begin{pmatrix}1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix}}\quad }$

literature

• Bass Michael, Decusatis Casimer, Enoch Jay: Handbook of Optics, Volume I: Geometrical and Physical Optics, Polarized Light, Components and Instruments . 3. Edition. Mcgraw-Hill Professional, 2009, ISBN 978-0-07-149889-0 .
• Edward Collett: Field Guide to Polarization. In: SPIE Field Guides. FG05, SPIE, 2005, ISBN 0-8194-5868-6 .
• Eugene Hecht: Optics. 2nd ed. Addison-Wesley, 1987, ISBN 0-201-11609-X .

Individual evidence

1. ↑ Hans Müller: Memorandum on the polarization optics of the photo-elastic shutter . In: Report Number 2 of the OSRD Project OEMsr-576 . 1943.