Mathematical Model for the Separation of an Elliptically Polarized Wave in Anisotropic Photonic Crystals into TE- and TM-Waves

Abstract

A one-dimensional anisotropic photonic crystal is considered. A mathematical model is constructed that separates an elliptical polarization wave into TE- and TM-type waves. The transformation matrix is found for a periodic structure with an arbitrary number of layers in a period in the form of a block diagonal matrix. The transformation matrix is found for an arbitrary number of periods of the structure with anisotropic layers. Dispersion relations are found that determine the boundaries of the allowed zones.

APPENDIX

The transformation matrix of a TE-wave [7] has the form

$${\mathbf{M}}\left( z \right) = \left| {\begin{array}{*{20}{c}} { - \frac{{\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} \exp \left( {j{{k}_{1}}z} \right) - \sqrt {{{\chi }_{1}}{{\gamma }_{2}}} \exp \left( {j{{k}_{2}}z} \right)}}{{\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - \sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}}&{\sqrt {{{\gamma }_{1}}{{\gamma }_{2}}} \frac{{\exp \left( {j{{k}_{1}}z} \right) - \exp \left( {j{{k}_{2}}z} \right)}}{{\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - \sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}} \\ { - \sqrt {{{\chi }_{1}}{{\chi }_{2}}} \frac{{\exp \left( {j{{k}_{1}}z} \right) - \exp \left( {j{{k}_{2}}z} \right)}}{{\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - \sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}}&{\frac{{\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} \exp \left( {j{{k}_{1}}z} \right) - \sqrt {{{\chi }_{2}}{{\gamma }_{1}}} \exp \left( {j{{k}_{2}}z} \right)}}{{\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - \sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}} \end{array}} \right|,$$

(A1)

where

$$\begin{gathered} {{\gamma }_{1}} = \frac{{{{a}_{{11}}}{{\xi }_{2}} + {{a}_{{12}}}{{\zeta }_{1}}{{\xi }_{2}} - {{a}_{{21}}} - {{a}_{{22}}}{{\zeta }_{1}}}}{{{{\xi }_{2}} - {{\xi }_{1}}}}, \\ {{\gamma }_{2}} = \frac{{{{a}_{{11}}}{{\xi }_{1}} + {{a}_{{12}}}{{\zeta }_{2}}{{\xi }_{1}} - {{a}_{{21}}} - {{a}_{{22}}}{{\zeta }_{2}}}}{{{{\xi }_{2}} - {{\xi }_{1}}}}, \\ {{\chi }_{1}} = \frac{{{{b}_{{21}}}{{\zeta }_{2}} + {{b}_{{22}}}{{\xi }_{1}}{{\zeta }_{2}} - {{b}_{{11}}} - {{b}_{{12}}}{{\xi }_{1}}}}{{{{\zeta }_{2}} - {{\zeta }_{1}}}},~ \\ {{\chi }_{2}} = \frac{{{{b}_{{21}}}{{\zeta }_{1}} + {{b}_{{22}}}{{\xi }_{2}}{{\zeta }_{1}} - {{b}_{{11}}} - {{b}_{{12}}}{{\xi }_{2}}}}{{{{\zeta }_{2}} - {{\zeta }_{1}}}}, \\ \end{gathered} $$

(A2)

and the factors of influence ${{\xi }_{{1,2}}}$ and ${{\zeta }_{{1,2}}}$ are defined by the expressions

$$\begin{gathered} {{\xi }_{{1,2}}} = \frac{{{{a}_{{22}}}{{b}_{{22}}} - {{a}_{{11}}}{{b}_{{11}}} + {{a}_{{21}}}{{b}_{{12}}} - {{a}_{{12}}}{{b}_{{21}}}}}{{2\left( {{{a}_{{11}}}{{b}_{{12}}} + {{a}_{{12}}}{{b}_{{22}}}} \right)}} \\ \,\, \times \left[ {1 \pm \sqrt {1 + \frac{{4\left( {{{a}_{{22}}}{{b}_{{21}}} + {{a}_{{21}}}{{b}_{{11}}}} \right)\left( {{{a}_{{11}}}{{b}_{{12}}} + {{a}_{{12}}}{{b}_{{22}}}} \right)}}{{{{{\left( {{{a}_{{11}}}{{b}_{{11}}} - {{a}_{{22}}}{{b}_{{22}}} + {{a}_{{12}}}{{b}_{{21}}} - {{a}_{{21}}}{{b}_{{12}}}} \right)}}^{2}}}}} } \right]~, \\ \end{gathered} $$

(A3)

$${{\zeta }_{{1,2}}} = - \frac{{{{b}_{{12}}}{{\xi }_{{1,2}}} + {{b}_{{11}}}}}{{{{b}_{{22}}}{{\xi }_{{1,2}}} + {{b}_{{21}}}}}~.$$

(A4)

The transformation matrix of a TM-wave [7] has the form

$${\mathbf{N}}\left( z \right) = \left| {\begin{array}{*{20}{c}} {\frac{{ - {{\varsigma }_{2}}{{\xi }_{1}}\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} \exp \left( { - j{{k}_{1}}z} \right) + {{\varsigma }_{1}}{{\xi }_{2}}\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} \exp \left( { - j{{k}_{1}}z} \right)}}{{{{\varsigma }_{1}}{{\xi }_{2}}\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - {{\varsigma }_{2}}{{\xi }_{1}}\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}}&{\frac{{{{\xi }_{1}}{{\xi }_{2}}\sqrt {{{\gamma }_{1}}{{\gamma }_{2}}} }}{{{{\varsigma }_{1}}{{\xi }_{2}}\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - {{\varsigma }_{2}}{{\xi }_{1}}\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}\left[ {\exp \left( { - j{{k}_{1}}z} \right) - \exp \left( { - j{{k}_{1}}z} \right)} \right]} \\ {\frac{{ - {{\varsigma }_{1}}{{\varsigma }_{2}}\sqrt {{{\chi }_{1}}{{\chi }_{2}}} \left[ {\exp \left( { - j{{k}_{1}}z} \right) - \exp \left( { - j{{k}_{1}}z} \right)} \right]}}{{{{\varsigma }_{1}}{{\xi }_{2}}\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - {{\varsigma }_{2}}{{\xi }_{1}}\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}}&{\frac{{{{\varsigma }_{1}}{{\xi }_{2}}\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} \exp \left( { - j{{k}_{1}}z} \right) - {{\varsigma }_{2}}{{\xi }_{1}}\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} \exp \left( { - j{{k}_{1}}z} \right)}}{{{{\varsigma }_{1}}{{\xi }_{2}}\sqrt {{{\chi }_{1}}{{\gamma }_{2}}} - {{\varsigma }_{2}}{{\xi }_{1}}\sqrt {{{\chi }_{2}}{{\gamma }_{1}}} }}} \end{array}} \right|.$$

(A5)

Thus, in [7], the transformation matrix is represented as a diagonal block matrix.