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# Equation Summary

General equations

Malus’ Law and Brewster’s Law:

$I\left(\text{θ}\right)={I}_{0}{\mathrm{cos}}^{2}\text{θ}\phantom{\rule{3em}{0ex}}\mathrm{tan}i={n}_{2}/{n}_{1}=n$

Field components propagating in the z direction:

$\begin{array}{l}{E}_{x}\left(z,t\right)={E}_{0x}\mathrm{cos}\left(\text{ω}t-kz+{\text{δ}}_{x}\right)\\ {E}_{y}\left(z,t\right)={E}_{0y}\mathrm{cos}\left(\text{ω}t-kz+{\text{δ}}_{y}\right)\end{array}$

Equation of the polarization ellipse:

$\frac{{E}_{x}{\left(z,t\right)}^{2}}{{E}_{0x}^{2}}+\frac{{E}_{y}{\left(z,t\right)}^{2}}{{E}_{0y}^{2}}-\frac{2{E}_{x}\left(z,t\right){E}_{y}\left(z,t\right)}{{E}_{0x}{E}_{0y}}\mathrm{cos}\text{δ}={\mathrm{sin}}^{2}\text{δ}$

Parameters of the polarization ellipse:

$\begin{array}{cc}\mathrm{tan}2\text{ψ}=\frac{2{E}_{0x}{E}_{0y}}{{E}_{0x}^{2}-{E}_{0y}^{2}}\mathrm{cos}\text{δ}& 0\le \psi \le \pi \\ \mathrm{sin}2\text{χ}=\frac{2{E}_{0x}{E}_{0y}\mathrm{sin}\text{δ}}{{E}_{0x}^{2}+{E}_{0y}^{2}}& -\text{π}/4<\text{χ}\le \text{π}/4\end{array}$

Auxilliary angle definition and polarization parameters:

Stokes polarization parameters:

$\begin{array}{l}{S}_{0}={E}_{0x}^{2}+{E}_{0y}^{2}\\ {S}_{1}={E}_{0x}^{2}-{E}_{0y}^{2}\\ {S}_{2}=2{E}_{0x}{E}_{0y}\mathrm{cos}\text{δ}\end{array}$

Stokes vector:

$S=\left(\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right)=\left(\begin{array}{c}{E}_{0x}^{2}+{E}_{0y}^{2}\\ {E}_{0x}^{2}-{E}_{0y}^{2}\\ 2{E}_{0x}{E}_{0y}\mathrm{cos}\text{δ}\\ 2{E}_{0x}{E}_{0y}\mathrm{sin}\text{δ}\end{array}\right)$

Stokes parameters in complex notation:

$\begin{array}{l}{S}_{0}={E}_{x}{E}_{x}^{*}+{E}_{y}{E}_{y}^{*}\\ {S}_{1}={E}_{x}{E}_{x}^{*}-{E}_{y}{E}_{y}^{*}\\ {S}_{2}={E}_{x}{E}_{y}^{*}+{E}_{y}{E}_{x}^{*}\\ {S}_{3}=i\left({E}_{x}{E}_{y}^{*}+{E}_{y}{E}_{x}^{*}\right)\end{array}$

Stokes vector on the Poincaré sphere:

Stokes vector on the observable polarization sphere:

$S=\left(\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right)=\left(\begin{array}{c}1\\ \mathrm{cos}2\text{α}\\ \mathrm{sin}2\text{α}\mathrm{cos}\text{δ}\\ \mathrm{sin}2\text{α}\mathrm{sin}\text{δ}\end{array}\right)$

Angle relation of 2a and δ to the Stokes parameters:

$\begin{array}{c}2\text{α}={\mathrm{cos}}^{-1}\left(\frac{{S}_{1}}{{S}_{0}}\right)\phantom{\rule{1em}{0ex}}0\le 2\text{α}<\text{π}\\ \text{δ}={\mathrm{tan}}^{-1}\left(\frac{{S}_{3}}{{S}_{2}}\right)\phantom{\rule{1em}{0ex}}0\le \text{δ}<2\text{π}\end{array}$

Ellipticity and orientation angles relations:

$\begin{array}{ll}\mathrm{tan}\left(2\text{ψ}\right)=\mathrm{tan}\left(2\text{α}\right)\mathrm{cos}\text{δ}& \phantom{\rule{1em}{0ex}}\mathrm{sin}\left(2\text{χ}\right)=\mathrm{sin}\left(2\text{α}\right)\mathrm{sin}\text{δ}\\ \mathrm{cos}\left(2\text{α}\right)=\mathrm{cos}\left(2\text{χ}\right)\mathrm{cos}\left(2\text{ψ}\right)& \mathrm{cot}\text{δ}=\mathrm{cot}\left(2\text{χ}\right)\mathrm{sin}\left(2\text{ψ}\right)\end{array}$

Stokes vectors for unpolarized and partially polarized light:

$\begin{array}{c}{S}_{\text{UNP}}={S}_{0}\left(\begin{array}{c}1\\ 0\\ 0\\ 0\end{array}\right)\\ S=\left(\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right)=\left(1-\mathcal{P}\right)\left(\begin{array}{c}{S}_{0}\\ 0\\ 0\\ 0\end{array}\right)+\mathcal{P}\left(\begin{array}{c}{S}_{0}\\ {S}_{1}\\ {S}_{2}\\ {S}_{3}\end{array}\right)\phantom{\rule{1em}{0ex}}0\le \mathcal{P}\le 1\end{array}$

Definition of the degree of polarization P:

$\begin{array}{c}\mathcal{P}=\frac{{I}_{\text{pol}}}{{I}_{\text{tot}}}=\frac{\sqrt{{S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}}}{{S}_{0}}\phantom{\rule{1.5em}{0ex}}0\le \mathcal{P}\le 1\\ {S}_{0}^{2}\ge {S}_{1}^{2}+{S}_{2}^{2}+{S}_{3}^{2}\end{array}$

Mueller matrices

Mathematical form of the Mueller matrix:

$M=\left(\begin{array}{llll}{m}_{00}& {m}_{01}& {m}_{02}& {m}_{03}\\ {m}_{10}& {m}_{11}& {m}_{12}& {m}_{13}\\ {m}_{20}& {m}_{21}& {m}_{22}& {m}_{23}\\ {m}_{30}& {m}_{31}& {m}_{32}& {m}_{33}\end{array}\right)$

Mueller matrix for a polarizer:

$\begin{array}{lll}{M}_{POL}\left({p}_{x},{p}_{y}\right)& =& \frac{1}{2}\left(\begin{array}{cccc}{p}_{x}^{2}+{p}_{y}^{2}& {p}_{x}^{2}-{p}_{y}^{2}& 0& 0\\ {p}_{x}^{2}-{p}_{y}^{2}& {p}_{x}^{2}+{p}_{y}^{2}& 0& 0\\ 0& 0& 2{p}_{x}{p}_{y}& 0\\ 0& 0& 0& 2{p}_{x}{p}_{y}\end{array}\right)\\ & & 0\le {p}_{x}\le 1\phantom{\rule{2em}{0ex}}0\le {p}_{y}\le 1\end{array}$

Mueller matrices for a linear horizontal and vertical polarizer:

$\begin{array}{ll}{M}_{\text{POL}}=\frac{1}{2}\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)& \phantom{\rule{1em}{0ex}}{M}_{\text{POL}}=\frac{1}{2}\left(\begin{array}{cccc}1& -1& 0& 0\\ -1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\end{array}$

Mueller matrix for a linear polarizer with its transmission axis at +45°:

${M}_{\text{POL}}=\frac{1}{2}\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& 0& 0\\ 1& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right)$

Mueller matrix for a wave plate:

${M}_{\text{WP}}\left(\text{ϕ}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{cos}\text{ϕ}& -\mathrm{sin}\text{ϕ}\\ 0& 0& \mathrm{sin}\text{ϕ}& \mathrm{cos}\text{ϕ}\end{array}\right)\text{\hspace{0.17em}}\text{ϕ}=\frac{2\text{π}}{\text{λ}}\left({n}_{o}-{n}_{e}\right)d$

Mueller matrices for a quarter- and half-wave plate:

$\begin{array}{ll}{M}_{\text{QWP}}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 0& -1\\ 0& 0& 1& 0\end{array}\right)& \phantom{\rule{1em}{0ex}}{M}_{\text{HWP}}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)\end{array}$

Mueller matrix for a rotator:

${M}_{\text{ROT}}\left(\text{θ}\right)=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& \mathrm{cos}2\text{θ}& \mathrm{sin}2\text{θ}& 0\\ 0& -\mathrm{sin}2\text{θ}& \mathrm{cos}2\text{θ}& 0\\ 0& 0& 0& 1\end{array}\right)$

Mueller matrix for a rotated polarizing element:

$M\left(\text{θ}\right)={M}_{\text{ROT}}\left(-2\text{θ}\right)\cdot M\cdot {M}_{\text{ROT}}\left(2\text{θ}\right)$

Mueller matrix for a rotated linear polarizer:

${M}_{POL}\left(\text{θ}\right)=\frac{1}{2}\left(\begin{array}{cccc}1& \mathrm{cos}2\text{θ}& \mathrm{sin}2\text{θ}& 0\\ \mathrm{cos}2\text{θ}& {\mathrm{cos}}^{2}2\text{θ}& \mathrm{sin}2\text{θ}\mathrm{cos}2\text{θ}& 0\\ \mathrm{sin}2\text{θ}& \mathrm{sin}2\text{θ}\mathrm{cos}2\text{θ}& {\mathrm{sin}}^{2}2\text{θ}& 0\\ 0& 0& 0& 0\end{array}\right)$

Mueller matrix of a circular polarizer:

${M}_{\text{CP}}=\left(\begin{array}{cccc}1& 0& 1& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\\ 1& 0& 1& 0\end{array}\right)$

Reflection and transmission

Fresnel’s Equations for reflection and refraction:

$\begin{array}{cc}{R}_{p}=\frac{\mathrm{tan}\left(i-r\right)}{\mathrm{tan}\left(i+r\right)}{E}_{p}& {R}_{s}=-\frac{\mathrm{sin}\left(i-r\right)}{\mathrm{sin}\left(i+r\right)}{E}_{s}\\ {T}_{p}=\frac{2\mathrm{sin}r\mathrm{cos}i}{\mathrm{sin}\left(i+r\right)\mathrm{cos}\left(i-r\right)}{E}_{p}& {T}_{S}=\frac{2\mathrm{sin}r\mathrm{cos}i}{\mathrm{sin}\left(i+r\right)}{E}_{S}\end{array}$

Stokes parameters for reflection:

$\begin{array}{c}{S}_{0R}={f}_{R}\left[\left({\mathrm{cos}}^{2}{\text{α}}_{-}+{\mathrm{cos}}^{2}{\text{α}}_{+}\right){S}_{0}+\left({\mathrm{cos}}^{2}{\text{α}}_{-}-{\mathrm{cos}}^{2}{\text{α}}_{+}\right){S}_{1}\right]\\ {S}_{1R}={f}_{R}\left[\left({\mathrm{cos}}^{2}{\text{α}}_{-}-{\mathrm{cos}}^{2}{\text{α}}_{+}\right){S}_{0}+\left({\mathrm{cos}}^{2}{\text{α}}_{-}+{\mathrm{cos}}^{2}{\text{α}}_{+}\right){S}_{1}\right]\\ {S}_{2R}=-{f}_{R}\left(2\mathrm{cos}{\text{α}}_{-}\mathrm{cos}{\text{α}}_{+}\right){S}_{2}\\ {S}_{3R}=-{f}_{R}\left(2\mathrm{cos}{\text{α}}_{-}\mathrm{cos}{\text{α}}_{+}\right){S}_{3}\phantom{\rule{1em}{0ex}}{\text{f}}_{\text{R}}=\frac{1}{2}{\left(\frac{\mathrm{tan}{\text{α}}_{-}}{\mathrm{tan}{\text{α}}_{+}}\right)}^{2}\end{array}$

Stokes parameters for transmission (refraction):

$\begin{array}{c}{S}_{0T}={f}_{T}\left[\left({\mathrm{cos}}^{2}{\text{α}}_{-}+1\right){S}_{0}+\left({\mathrm{cos}}^{2}{\text{α}}_{-}-1\right){S}_{1}\right]\\ {S}_{1T}={f}_{T}\left[\left({\mathrm{cos}}^{2}{\text{α}}_{-}-1\right){S}_{0}+\left({\mathrm{cos}}^{2}{\text{α}}_{-}+1\right){S}_{1}\right]\\ {S}_{2T}=-{f}_{T}\left(2\mathrm{cos}{\text{α}}_{-}\right){S}_{2}\\ {S}_{3T}=-{f}_{T}\left(2\mathrm{cos}{\text{α}}_{-}\right){S}_{3}\phantom{\rule{1em}{0ex}}{f}_{T}=\frac{1}{2}\frac{\mathrm{sin}2i\mathrm{sin}2r}{{\left(\mathrm{sin}{\text{α}}_{+}\mathrm{cos}{\text{α}}_{-}\right)}^{2}}\end{array}$

Mueller matrices for reflection and transmission:

$\begin{array}{ll}{M}_{\text{R}}={\left(\frac{n-1}{n+1}\right)}^{2}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& -1& 0\\ 0& 0& 0& -1\end{array}\right)& {M}_{\text{T}}=\frac{4n}{{\left(n+1\right)}^{2}}\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right)\end{array}$

Mueller matrices for Brewster angle reflection and transmission:

$\begin{array}{c}{M}_{R,B}=\frac{1}{2}{\mathrm{cos}}^{2}2{i}_{B}\left(\begin{array}{cccc}1& 1& 0& 0\\ 1& 1& 0& 0\\ 0& 0& 0& 0\\ 0& 0& 0& 0\end{array}\right)\\ {M}_{T,B}=\frac{1}{2}\left(\begin{array}{cccc}{\mathrm{sin}}^{2}2{i}_{B}+1& {\mathrm{sin}}^{2}2{i}_{B}-1& 0& 0\\ {\mathrm{sin}}^{2}2{i}_{B}-1& {\mathrm{sin}}^{2}2{i}_{B}+1& 0& 0\\ 0& 0& 2\mathrm{sin}2{i}_{B}& 0\\ 0& 0& 0& 2\mathrm{sin}2{i}_{B}\end{array}\right)\end{array}$

Mueller matrix for total internal reflection:

$\begin{array}{ll}{M}_{R}=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{cos}\text{δ}& -\mathrm{sin}\text{δ}\\ 0& 0& \mathrm{sin}\text{δ}& \mathrm{cos}\text{δ}\end{array}\right)& \mathrm{tan}\frac{\text{δ}}{2}=\frac{\mathrm{cos}i\sqrt{{n}^{2}{\mathrm{sin}}^{2}i-1}}{n{\mathrm{sin}}^{2}i}\end{array}$

Mueller matrix of the Fresnel rhomb:

$M=\left(\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& \mathrm{cos}\left({\text{δ}}_{U}+{\text{δ}}_{L}\right)& -\mathrm{sin}\left({\text{δ}}_{U}+{\text{δ}}_{L}\right)\\ 0& 0& \mathrm{sin}\left({\text{δ}}_{U}+{\text{δ}}_{L}\right)& \mathrm{cos}\left({\text{δ}}_{U}+{\text{δ}}_{L}\right)\end{array}\right)$

Jones matrix calculus

Jones vector:

$\mathbf{\text{E}}=\left(\begin{array}{l}{E}_{x}\\ {E}_{y}\end{array}\right)=\left(\begin{array}{l}{E}_{0x}{e}^{i{\text{δ}}_{x}}\\ {E}_{0y}{e}^{i{\text{δ}}_{y}}\end{array}\right)$

Jones matrix formulation of optical intensity:

$I={\mathbf{\text{E}}}^{†}\cdot \mathbf{\text{E}}\phantom{\rule{1.5em}{0ex}}I=\left({E}_{x}{\text{}}^{*}\phantom{\rule{1em}{0ex}}{E}_{y}{\text{}}^{*}\right)\left(\begin{array}{l}{E}_{x}\\ {E}_{y}\end{array}\right)={E}_{x}{E}_{x}{\text{}}^{*}+{E}_{y}{E}_{y}{\text{}}^{*}$

Jones matrix definition:

$\mathbf{\text{J}}=\left(\begin{array}{ll}{j}_{xx}& {j}_{xy}\\ {j}_{yx}& {j}_{yy}\end{array}\right)$

Jones matrices for linear polarizers:

$\begin{array}{ll}{\mathbf{\text{J}}}_{\text{POL}}=\left(\begin{array}{cc}{p}_{x}& 0\\ 0& {p}_{y}\end{array}\right)& 0\le {p}_{x},{p}_{y}\le 1\\ {\mathbf{\text{J}}}_{\text{LHP}}=\left(\begin{array}{cc}1& 0\\ 0& 0\end{array}\right)& {\mathbf{\text{J}}}_{\text{LVP}}=\left(\begin{array}{cc}0& 0\\ 0& 1\end{array}\right)\end{array}$

Jones matrices for a wave plate, quarter-wave plate, and a half-wave plate:

$\begin{array}{l}{\mathbf{\text{J}}}_{\text{WP}}=\left(\begin{array}{cc}{e}^{i\left(\text{ϕ}/2\right)}& 0\\ 0& {e}^{-i\left(\text{ϕ}/2\right)}\end{array}\right)\to \left(\begin{array}{cc}1& 0\\ 0& {e}^{-i\text{ϕ}}\end{array}\right)\\ {\mathbf{\text{J}}}_{\text{QWP}}=\left(\begin{array}{cc}1& 0\\ 0& -i\end{array}\right)\phantom{\rule{2em}{0ex}}{\mathbf{\text{J}}}_{\text{HWP}}=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)\end{array}$

Jones matrix for a rotator:

${\mathbf{\text{J}}}_{\text{ROT}}\left(\text{θ}\right)=\left(\begin{array}{cc}\mathrm{cos}\text{θ}& \mathrm{sin}\text{θ}\\ -\mathrm{sin}\text{θ}& \mathrm{cos}\text{θ}\end{array}\right)$

Jones matrix for a rotated polarizing element:

$\mathbf{\text{J}}\left(\text{θ}\right)={\mathbf{\text{J}}}_{\text{ROT}}\left(-\text{θ}\right)\cdot \mathbf{\text{J}}\cdot {\mathbf{\text{J}}}_{\text{ROT}}\left(\text{θ}\right)$

Polarizer characterization

Transmittance of a single linear polarizer:

${T}_{\text{UNP}}=\frac{{k}_{1}+{k}_{2}}{2}\phantom{\rule{1.5em}{0ex}}{T}_{\text{LHP}}=\frac{{k}_{1}}{2}\phantom{\rule{1.5em}{0ex}}{T}_{\text{LVP}}=\frac{{k}_{2}}{2}$

Transmittance of a pair of linear polarizers:

${T}_{\text{UNP}}={H}_{0}=\frac{1}{2}\left({k}_{1}^{2}+{k}_{2}^{2}\right)\phantom{\rule{1.5em}{0ex}}{T}_{\text{LHP}}=\frac{{k}_{1}^{2}}{2}\phantom{\rule{1.5em}{0ex}}{T}_{\text{LVP}}=\frac{{k}_{2}^{2}}{2}$

Transmittance of a pair of crossed polarizers:

${T}_{\text{UNP}}={H}_{90}={k}_{1}{k}_{2}$

Transmittance for a single rotated polarizer at an angle θ:

$T\left(\text{θ}\right)={k}_{1}{\mathrm{sin}}^{2}\text{θ}+{k}_{2}{\mathrm{cos}}^{2}\text{θ}$

Transmittance for a fixed and rotated polarizer at an angle θ:

$\begin{array}{c}T\left(\text{θ}\right)={k}_{1}{k}_{2}{\mathrm{sin}}^{2}\text{θ}+\frac{1}{2}\left({k}_{1}^{2}+{k}_{2}^{2}\right){\mathrm{cos}}^{2}\text{θ}\\ ={H}_{90}+\left({H}_{0}-{H}_{90}\right){\mathrm{cos}}^{2}\text{θ}\end{array}$

Logarithmic definition of transmittance (density):

${D}_{0}=-{\mathrm{log}}_{10}{H}_{0}\phantom{\rule{2em}{0ex}}{D}_{90}=-{\mathrm{log}}_{10}{H}_{90}$

Contrast ratio of a polarizer:

$C=\frac{{T}_{\mathrm{max}}-{T}_{\mathrm{min}}}{{T}_{\mathrm{max}}+{T}_{\mathrm{min}}}=\frac{{k}_{1}-{k}_{2}}{{k}_{1}+{k}_{2}}$

Kerr and Pockel cell characterization

Birefringence of the Kerr effect:

${n}_{p}-{n}_{s}=\text{λ}B{\text{ε}}^{2}$

Pockel cell phase shift:

$\text{ϕ}={\text{ϕ}}_{y}-{\text{ϕ}}_{x}=\frac{\text{ω}{n}_{0}^{3}{r}_{63}}{c}V$

Stokes vector for a linearly oscillating charge:

$S={\left(\frac{e{z}_{0}}{{c}^{2}R}\right)}^{2}{\mathrm{sin}}^{2}\text{θ}{\text{ω}}_{0}^{4}\left(\begin{array}{l}1\\ 1\\ 0\\ 0\end{array}\right)$

Stokes vector for a randomly oriented oscillating charge:

$S=\frac{8\text{π}}{3}{\left(\frac{eA}{{c}^{2}R}\right)}^{2}{\text{ω}}_{0}^{4}\left(\begin{array}{l}1\\ 0\\ 0\\ 0\end{array}\right)$

Stokes vector for a charge moving in a circle:

$S={\left(\frac{eA}{{c}^{2}R}\right)}^{2}{\text{ω}}_{0}^{4}\left(\begin{array}{c}1+{\mathrm{cos}}^{2}\text{θ}\\ {\mathrm{sin}}^{2}\text{θ}\\ 0\\ 2\mathrm{cos}\text{θ}\end{array}\right)$

Stokes vector for the Zeeman effect:

$S=\frac{2}{3}{\left(\frac{eA}{2{c}^{2}R}\right)}^{2}\left[{\text{ω}}_{-}^{4}\left(\begin{array}{c}1+{\mathrm{cos}}^{2}\text{θ}\\ -{\mathrm{sin}}^{2}\text{θ}\\ 0\\ -2\mathrm{cos}\text{θ}\end{array}\right)+2{\text{ω}}_{0}^{4}\left(\begin{array}{c}{\mathrm{sin}}^{2}\text{θ}\\ {\mathrm{sin}}^{2}\text{θ}\\ 0\\ 0\end{array}\right)+{\text{ω}}_{+}^{4}\left(\begin{array}{c}1+{\mathrm{cos}}^{2}\text{θ}\\ -{\mathrm{sin}}^{2}\text{θ}\\ 0\\ 2\mathrm{cos}\text{θ}\end{array}\right)\right]$

Mueller matrix for Thomson scattering:

$M=\frac{1}{2}{\left(\frac{{e}^{2}}{m{c}^{2}R}\right)}^{2}\left(\begin{array}{cccc}1+{\mathrm{cos}}^{2}\text{θ}& -{\mathrm{sin}}^{2}\text{θ}& 0& 0\\ -{\mathrm{sin}}^{2}\text{θ}& 1+{\mathrm{cos}}^{2}\text{θ}& 0& 0\\ 0& 0& 2\mathrm{cos}\text{θ}& 0\\ 0& 0& 0& 2\mathrm{cos}\text{θ}\end{array}\right)$

Stokes vector for Rayleigh scattering:

${S}^{\prime }=\frac{1}{2}{\left[\frac{{e}^{2}}{m{c}^{2}\left({\text{ω}}^{2}-{\text{ω}}_{0}^{2}\right)}\right]}^{2}{\text{ω}}^{4}\left(\begin{array}{c}{S}_{0}\left(1+{\mathrm{cos}}^{2}\text{θ}\right)-{S}_{1}{\mathrm{sin}}^{2}\text{θ}\\ -{S}_{0}{\mathrm{sin}}^{2}\text{θ}+{S}_{1}\left(1+{\mathrm{cos}}^{2}\text{θ}\right)\\ 2{S}_{2}\mathrm{cos}\text{θ}\\ 2{S}_{3}\mathrm{cos}\text{θ}\end{array}\right)$

Absorbing media—semiconductors and metals

Complex dielectric constant and refractive index for metals:

$\mathbf{\text{ε}}=\text{ε}-i\left(\frac{4\text{π}\text{σ}}{\text{ω}}\right)\phantom{\rule{3em}{0ex}}\mathbf{\text{n}}=n\left(1-i\text{κ}\right)$

Reflectivity at normal incidence for absorbing media:

${\mathcal{R}}_{s}={\mathcal{R}}_{p}=\left[\frac{{\left(n-1\right)}^{2}+{\left(n\text{κ}\right)}^{2}}{{\left(n+1\right)}^{2}+{\left(n\text{κ}\right)}^{2}}\right]$

s- and p- reflectivity at non-normal incidence:

$\begin{array}{l}{\mathcal{R}}_{s}={|\frac{{R}_{s}}{{E}_{s}}|}^{2}=\left[\frac{{\left(n-\mathrm{cos}i\right)}^{2}+{\left(n\text{κ}\right)}^{2}}{{\left(n+\mathrm{cos}i\right)}^{2}+{\left(n\text{κ}\right)}^{2}}\right]\\ {\mathcal{R}}_{p}={|\frac{{R}_{p}}{{E}_{p}}|}^{2}=\left[\frac{{\left(n-1/\mathrm{cos}i\right)}^{2}+{\left(n\text{κ}\right)}^{2}}{{\left(n+1/\mathrm{cos}i\right)}^{2}+{\left(n\text{κ}\right)}^{2}}\right]\end{array}$
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