Polarized Radiation Transport Equation in Anisotropic Media

Abstract

In particular, in this paper polarized radiation transport equation in media composed of randomly spatially distributed discrete non-spherical scatterers, the scatterer sizes being comparable with the electromagnetic radiation wavelength, has been derived from the equations of classical electrodynamics.

The paper is dedicated to the memory of Evgraph Sergeevich Kuznetsov.

Supplement

Eigenvalues and eigenvectors of the operator ${\hat{\mathbf{\mathcal{A}}}} ({\mathbf{k}} \to {\mathbf{k}})$

Consider first the case of Hermitian matrix ${\hat{\mathbf{\mathcal{A}}}} ({\mathbf{k}} \to {\mathbf{k}})$, representing it as

$$ {\hat{\mathbf{\mathcal{A}}}} ({\mathbf{k}} \to {\mathbf{k}}) = \left[ {\begin{array}{*{20}c} {{\mathbf{\mathcal{A}}}_{{{\mathbf{11}}}}^{\prime } } & {{\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime } } \\ {{\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime } } & {{\mathbf{\mathcal{A}}}_{22}^{\prime } } \\ \end{array} } \right] + i\left[ {\begin{array}{*{20}c} 0 & {{\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } } \\ { - {\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } } & 0 \\ \end{array} } \right], $$

(S.1)

where

$$ {\mathbf{\mathcal{A}}}_{\alpha \beta }^{\prime } = {\text{Re}}\,{\mathbf{\mathcal{A}}}_{\alpha \beta } ,\quad {\mathbf{\mathcal{A}}}_{\alpha \beta }^{\prime \prime } ={\text{Im}}\,{\mathbf{\mathcal{A}}}_{\alpha \beta } . $$

The eigenvalues ${\mathbf{\mathcal{A}}}^{{{\mathbf{(1)}}}}$ and ${\mathbf{\mathcal{A}}}^{{{\mathbf{(}}2{\mathbf{)}}}}$ of a Hermitian matrix are real values and are the roots of the characteristic polynomial

$$ \lambda^{2} - t\lambda + d, $$

where

$$ t = {\text{Tr}}\,{\hat{\mathbf{\mathcal{A}}}},\quad d ={\text{det}}\,\,{\hat{\mathbf{\mathcal{A}.}}}$$

Put at first ${\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } = 0$ (what means from physical viewpoint that non-absorbing non-magnetic anisotropic medium is considered). Then we have the following eigenvalues and and orthonormal eigenvectors of the operator ${\hat{\mathbf{\mathcal{A}}}} ({\mathbf{k}} \to {\mathbf{k}})$

$$ {\mathbf{\mathcal{A}}}^{(1,2)} = \frac{1}{2}\left\{ {\left( {{\mathbf{\mathcal{A}}}_{11}^{{\prime }} + {\mathbf{\mathcal{A}}}_{22}^{{\prime }} } \right) \pm \sqrt {\left( {{\mathbf{\mathcal{A}}}_{11}^{{\prime }} - {\mathbf{\mathcal{A}}}_{22}^{{\prime }} } \right)^{2} + 4{\mathbf{\mathcal{A}}}_{11}^{{{\prime }\,{2}}} } } \right\}, $$

(S.2)

$$ {\mathbf{e}}^{(0)\left( 1 \right)} = \left( {C_{o}^{2} + {\mathbf{\mathcal{A}}}_{12}^{{{\prime }\,\text{2}}} } \right)^{ - 1/2} \left[ {\begin{array}{*{20}l} {{\mathbf{\mathcal{A}}}_{12}^{{\prime }} } \hfill \\ {C_{o} } \hfill \\ \end{array} } \right], $$

(S.3)

$$ {\mathbf{e}}^{(0)\left( 2 \right)} = \left( {C_{o}^{2} + {\mathbf{\mathcal{A}}}_{12}^{{{\prime }\,{2}}} } \right)^{ - 1/2} \left[ {\begin{array}{*{20}l} {C_{o} } \hfill \\ { - {\mathbf{\mathcal{A}}}_{12}^{{\prime }} } \hfill \\ \end{array} } \right]. $$

(S.4)

where

$$ C_{0} = \frac{1}{2}\left\{ { - \left( {{\mathbf{\mathcal{A}}}_{11}^{{\prime }} - {\mathbf{\mathcal{A}}}_{22}^{{\prime }} } \right) + \sqrt {\left( {{\mathbf{\mathcal{A}}}_{11}^{{\prime }} - {\mathbf{\mathcal{A}}}_{22}^{{\prime }} } \right)^{2} + 4{\mathbf{\mathcal{A}}}_{12}^{{{\prime }\,2}} } } \right\}. $$

(S.5)

So, in the anisotropic medium along any direction k two linearly polarized waves in mutually orthogonal polarization states, defined by vectors ${\mathbf{e}}^{(0)\left( 1 \right)}$ and ${\mathbf{e}}^{(0)\left( 2 \right)}$, can propagate without absorption with phase velocities $v_{ph}^{{(\alpha {\mathbf{)}}}} ({\mathbf{k}}){\mathbf{ = }}c/{\mathbf{\mathcal{A}}}^{{{\mathbf{(}}\alpha {\mathbf{)}}}} ({\mathbf{k}}),\,\,\alpha = 1,2.$ Let now ${\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } \ne 0.$ Then in the basis $\{ {\mathbf{e}}^{(0)\left( 1 \right)} ,{\mathbf{e}}^{(0)\left( 2 \right)} \}$ the operator can be written as

$$ {\hat{\mathbf{\mathcal{A}}}} ({\mathbf{k}} \to {\mathbf{k}}) = \left[ {\begin{array}{*{20}l} {{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}1}} } & {i{\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } } \\ { - i{\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } } & {{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}2}} } \\ \end{array} } \right]. $$

(S.6)

The eigenvalues and eigenvectors of ${\hat{\mathbf{\mathcal{A}}}} ({\mathbf{k}} \to {\mathbf{k}})$ are now defined by the expressions:

$$ {\mathbf{\mathcal{A}}}^{{{\mathbf{(}}1,2{\mathbf{)}}}} = \frac{1}{2}\{ ({\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}1}} + {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}2}} ) \pm \sqrt {\left( {{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}1}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}2}} } \right)^{2} + 4{\mathbf{\mathcal{A}}}_{12}^{{{\prime \prime }\,\text{2}}} } \} , $$

(S.7)

$$ {\mathbf{e}}^{\left( 1 \right)} = \left( {C_{1}^{2} + {\mathbf{\mathcal{A}}}_{12}^{{{\prime \prime }\,2}} } \right)^{ - 1/2} \left[ \begin{aligned} {\mathbf{\mathcal{A}}}_{12}^{{\prime \prime }} \hfill \\ - i\,C_{1} \hfill \\ \end{aligned} \right],\quad {\mathbf{e}}^{\left( 2 \right)} = \left( {C_{1}^{2} + {\mathbf{\mathcal{A}}}_{12}^{{{\prime \prime }\,2}} } \right)^{ - 1/2} \left[ \begin{aligned} C_{1} \hfill \\ - i\,{\mathbf{\mathcal{A}}}_{12}^{{\prime \prime }} \hfill \\ \end{aligned} \right], $$

(S.8)

$$ C_{1}^{{}} = \frac{1}{2}\{ - ({\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}1}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}2}} ) + \sqrt {({\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}1}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}2}} )^{2} + 4{\mathbf{\mathcal{A}}}_{12}^{{{\prime \prime }\,2}} } \} . $$

(S.9)

Therefore in the case of Hermittean and complex-valued ${\hat{\mathbf{\mathcal{A}}}} (\,{\mathbf{k}} \to {\mathbf{k}})$ two elliptically polarized waves in mutually opposite polarization states can propagate in anisotropic medium along any direction k without absorption. For these waves the orientations of large axes of polarization ellipse coincide with the directions ${\mathbf{e}}^{(0)\left( 1 \right)}$ and ${\mathbf{e}}^{(0)\left( 2 \right)}$. In particular, if ${\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime } \ll {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}1}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}2}}$ we have

$$ {\mathbf{\mathcal{A}}}^{(1,2)} \simeq {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1,2)}} + \frac{{{\mathbf{\mathcal{A}}}_{12}^{\prime \prime } }}{{{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1)}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(2)}} }}, $$

(S.10)

$$ \begin{aligned} {\mathbf{e}}^{\left( 1 \right)} & = \left[ {1 + \left( {\frac{{{\mathbf{\mathcal{A}}}_{12}^{\prime \prime } }}{{{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1)}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(2)}} }}} \right)^{2} } \right]^{ - 1/2} \left[ {\begin{array}{*{20}c} 1 \\ { - i\frac{{{\mathbf{\mathcal{A}}}_{12}^{\prime \prime } }}{{{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1)}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(2)}} }}} \\ \end{array} } \right], \\ {\mathbf{e}}^{\left( 1 \right)} & = \left[ {1 + \left( {\frac{{{\mathbf{\mathcal{A}}}_{12}^{\prime \prime } }}{{{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1)}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(2)}} }}} \right)^{2} } \right]^{ - 1/2} \left[ {\begin{array}{*{20}c} {\frac{{{\mathbf{\mathcal{A}}}_{12}^{\prime \prime } }}{{{\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1)}} - {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(2)}} }}} \\ i \\ \end{array} } \right]. \\ \end{aligned} $$

(S.11)

Therefore, at small value ${\mathbf{\mathcal{A}}}_{{{\mathbf{12}}}}^{\prime \prime }$ the normal waves are weakly elliptically polarized.

The case ${\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}(1)}} {\mathbf{ = \mathcal{A}}}^{{{\mathbf{(0)}}(2)}}$ corresponds to magneto-optical effect in isotropic medium. Indeed, we have

$$ {\mathbf{\mathcal{A}}}^{(1,2)} \simeq {\mathbf{\mathcal{A}}}^{{{\mathbf{(0)}}}} \pm {\mathbf{\mathcal{A}}}_{12}^{\prime \prime } , $$

(S.12)

$$ {\mathbf{e}}^{{\left( {1,2} \right)}} = 2^{ - 1/2} \left[ \begin{aligned} 1 \hfill \\ \pm i \hfill \\ \end{aligned} \right], $$

(S.13)

and so the circular birefringence takes place in the medium.

In general case the eigenvalues of ${\hat{\mathbf{\mathcal{A}}}} (\,{\mathbf{k}} \to {\mathbf{k}})$ are complex-valued, and the eigenvectors are not orthogonal. The normal matrices, satisfying the condition ${\hat{\mathbf{\mathcal{A}}}}\,{\hat{\mathbf{\mathcal{A}}}}^{{\mathbf{ + }}} { = }{\hat{\mathbf{\mathcal{A}}}}^{{\mathbf{ + }}} {\hat{\mathbf{\mathcal{A}},}}$ possess orthogonal eigenvectors.