## Abstract

Biological and chemical materials can be detected, identified, and characterized through their optical activity: optical rotation and circular dichroism. The optical activity of chiral materials leaves their footprints on the elements of the two off-diagonal quadrants of the Mueller matrices, which are proportional to the cross-linearly polarized reflection and transmission coefficients. At the free-space–chiral interface, lateral waves that propagate just below the interface are excited by waves that enter and emerge from the chiral material at the critical angle for total internal reflection. As the waves enter and emerge from the chiral material, the transmitted waves also undergo cross polarization. In the neighborhood of the critical angle, the cross-polarized transmission coefficients are relatively large, and they also exhibit the impact of the optical activity of the material. Therefore, the lateral waves can also be used to identify optically active materials.

© 2011 Optical Society of America

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### Equations (28)

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(1)
$${E}_{r}^{c}={P}_{1}^{c}{T}_{1}^{c}{P}_{L}^{c}{T}_{0}^{c}{P}_{0}^{c}{E}_{i}^{c}\mathrm{.}$$
(2)
$${E}^{c}=\left(\begin{array}{c}{E}^{R}\\ {E}^{L}\end{array}\right)=A{E}^{L}=\left(\begin{array}{cc}1& -j\\ 1& j\end{array}\right)\left(\begin{array}{c}{E}^{V}\\ {E}^{H}\end{array}\right)\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}{E}^{L}={A}^{-1}{E}^{c}\mathrm{.}$$
(3)
$${A}^{-1}=\frac{1}{2}\left(\begin{array}{cc}1& 1\\ j& -j\end{array}\right)\mathrm{.}$$
(4)
$${E}_{r}^{L}={P}_{1}^{c\prime}{T}_{1}^{L}{P}_{L}^{L}{T}_{0}^{L}{P}_{0}^{c\prime}{E}_{i}^{L}\mathrm{.}$$
(5)
$${T}_{j}^{L}={A}^{-1}{T}_{j}^{c}A,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}j=0,1,$$
(6)
$${P}_{0}^{L}={A}^{-1}{P}_{0}^{c}A={P}_{0}^{c},$$
(7)
$${P}_{L}^{L}={A}^{-1}{P}_{L}^{c}A,$$
(8)
$${P}_{1}^{L}={A}^{-1}{P}_{1}^{c}A={P}_{1}^{c},$$
(9)
$${T}^{c}=\left(\begin{array}{cc}{T}^{RR}& {T}^{RL}\\ {T}^{LR}& {T}^{LL}\end{array}\right)\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}{T}^{L}=\left(\begin{array}{cc}{T}^{VV}& {T}^{VH}\\ {T}^{HV}& {T}^{HH}\end{array}\right),$$
(10)
$${P}_{k}^{c}=\left(\begin{array}{cc}{e}^{-i{k}_{0}{L}_{k}}& 0\\ 0& {e}^{-i{k}_{0}{L}_{k}}\end{array}\right),\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}k=0,1.$$
(11)
$${P}_{k}^{L}={P}_{k}^{c}={e}^{-j{k}_{0}{L}_{k}}I,\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}k=0,1.$$
(12)
$${P}_{L}^{c}=\left(\begin{array}{cc}{e}^{-j{\gamma}_{11}L}& 0\\ 0& {e}^{-j{\gamma}_{21}L}\end{array}\right)\mathrm{.}$$
(13)
$${u}_{11}=({\gamma}_{11}^{2}-{q}^{2}{)}^{\frac{1}{2}}={\gamma}_{11}\mathrm{cos}{\theta}_{11},$$
(14)
$${u}_{21}=({\gamma}_{21}^{2}-{q}^{2}{)}^{\frac{1}{2}}={\gamma}_{21}\mathrm{cos}{\theta}_{21}\mathrm{.}$$
(15)
$${\gamma}_{11}=\frac{{k}_{1}}{1-{\beta}_{1}{k}_{1}}={k}_{0}({n}_{1}^{\prime}-j{n}_{1}^{\prime \prime})\phantom{\rule[-0.0ex]{2.0em}{0.0ex}}\phantom{\rule{0ex}{0ex}}{\gamma}_{21}=\frac{{k}_{1}}{1+{\beta}_{1}{k}_{1}}={k}_{0}({n}_{2}^{\prime}-j{n}_{2}^{\prime \prime})\mathrm{.}$$
(16)
$$q={k}_{0}\mathrm{sin}{\theta}_{0}={\gamma}_{11}\mathrm{sin}{\theta}_{11}={\gamma}_{21}\mathrm{sin}{\theta}_{21}\mathrm{.}$$
(17)
$$\mathrm{sin}{\theta}_{11}^{c}=1\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\mathrm{sin}{\theta}_{21}^{c}=1.$$
(18)
$$\mathrm{sin}{\theta}_{10}^{c}=\frac{{\gamma}_{11}}{{k}_{0}}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\text{and}\phantom{\rule[-0.0ex]{1.0em}{0.0ex}}\mathrm{sin}{\theta}_{20}^{c}=\frac{{\gamma}_{21}}{{k}_{0}}\mathrm{.}$$
(19)
$${T}^{c}={T}^{c0}+{k}_{1}{\beta}_{1}{T}^{c0\prime}\mathrm{.}$$
(20)
$${T}^{L}={A}^{-1}{T}^{c}A=\left(\begin{array}{cc}{T}^{VV}& {T}^{VH}\\ {T}^{HV}& {T}^{HH}\end{array}\right)\mathrm{.}$$
(21)
$${R}^{L}=\left(\begin{array}{cc}{R}^{VV}& {R}^{VH}\\ {R}^{HV}& {R}^{HH}\end{array}\right)\mathrm{.}$$
(22)
$${T}^{VH}=-{T}^{HV}\left(\frac{{Z}_{0}}{{Z}_{1}}\right)={R}^{VH}=-{R}^{HV}=\frac{1}{2}j{\beta}_{1}{k}_{1}T{}_{01}^{HH}{T}_{10}^{VV}{\mathrm{tan}}^{2}{\theta}_{1}\equiv \frac{1}{2}j{\beta}_{1}{k}_{1}F=jfF/2.$$
(23)
$${Z}_{0}=(\frac{{\mu}_{0}}{{\u03f5}_{0}}{)}^{\frac{1}{2}}$$
(24)
$${Z}_{1}=(\frac{{\mu}_{1}}{{\u03f5}_{1}}{)}^{\frac{1}{2}}$$
(25)
$${k}_{0}\mathrm{sin}{\theta}_{0}={k}_{1}\mathrm{sin}{\theta}_{1}\mathrm{.}$$
(26)
$$OR+jCD=({\gamma}_{1}-{\gamma}_{2})\frac{l}{2}=\mathrm{Re}({\gamma}_{1}-{\gamma}_{2})\frac{l}{2}+j\mathrm{Im}({\gamma}_{1}-{\gamma}_{2})\frac{l}{2}={k}_{0}({n}_{1}^{\prime}-{n}_{2}^{\prime})\frac{l}{2}+j{k}_{0}({n}_{1}^{\prime \prime}-{n}_{2}^{\prime \prime})\frac{l}{2}\mathrm{.}$$
(27)
$$O{R}^{\prime}+jC{D}^{\prime}={k}_{1}^{2}{\beta}_{1}={k}_{0}^{2}({n}^{\prime}-j{n}^{\prime \prime}{)}^{2}({\beta}_{1}^{\prime}+j{\beta}_{1}^{\prime \prime})\mathrm{.}$$
(28)
$$O{R}^{\prime}+jC{D}^{\prime}={k}_{1}^{2}{\beta}_{1}\mathrm{.}$$