## Abstract

Measurements of a reduced Mueller matrix in backscattering from highly diffusive, dielectric samples are reported as a function of the angle of incidence. It was found that the off-diagonal terms depend greatly on the angle of incidence, increasing to a maximum near grazing incidence. We show that, despite a significant scattering originating in the bulk of such diffusive media, the nontrivial behavior of the off-diagonal Muller matrix is primarily due to surface scattering phenomena. The experimental data can be simply explained by assuming a random orientation of small particles and considering only double scattering in the plane of the surface.

© 2002 Optical Society of America

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### Equations (28)

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(1)
$$\mathbf{J}=\left(\begin{array}{c}I\\ Q\\ U\\ V\end{array}\right)=\left(\begin{array}{c}\u3008{E}_{x}{E}_{x}^{*}+{E}_{y}{E}_{y}^{*}\u3009\\ \u3008{E}_{x}{E}_{x}^{*}-{E}_{y}{E}_{y}^{*}\u3009\\ \u3008{E}_{x}{E}_{y}^{*}+{E}_{y}{E}_{x}^{*}\u3009\\ \u3008i({E}_{x}{E}_{y}^{*}-{E}_{y}{E}_{x}^{*})\u3009\end{array}\right),$$
(2)
$${\mathbf{J}}^{\prime}=[\mathbf{M}]\mathbf{J},$$
(3)
$$\left[\begin{array}{cccc}{M}_{11}& {M}_{12}& {M}_{13}& {M}_{14}\\ {M}_{21}& {M}_{22}& {M}_{23}& {M}_{24}\\ {M}_{31}& {M}_{32}& {M}_{33}& {M}_{34}\\ {M}_{41}& {M}_{42}& {M}_{43}& {M}_{44}\end{array}\right]\left(\begin{array}{c}I\\ Q\\ U\\ V\end{array}\right)=\left(\begin{array}{c}{I}^{\prime}\\ {Q}^{\prime}\\ {U}^{\prime}\\ {V}^{\prime}\end{array}\right)$$
(4)
$$[\mathbf{M}]=\prod _{k=n}^{1}[{\mathbf{M}}_{k}],$$
(5)
$$[{\mathbf{A}}_{\pm}]=\left[\begin{array}{cc}1& \pm 1\\ \pm 1& 1\end{array}\right],$$
(6)
$${\mathbf{J}}^{\prime}=[{\mathbf{A}}_{\pm}][\mathbf{M}]\mathbf{J}$$
(7)
$${I}_{\mathrm{HH}}={M}_{11}+{M}_{12}+{M}_{21}+{M}_{22},$$
(8)
$${I}_{\mathrm{HV}}={M}_{11}+{M}_{12}-{M}_{21}-{M}_{22},$$
(9)
$${I}_{\mathrm{VH}}={M}_{11}-{M}_{12}+{M}_{21}-{M}_{22},$$
(10)
$${I}_{\mathrm{VV}}={M}_{11}-{M}_{12}-{M}_{21}+{M}_{22},$$
(11)
$${M}_{11}={I}_{\mathrm{HH}}+{I}_{\mathrm{HV}}+{I}_{\mathrm{VH}}+{I}_{\mathrm{VV}},$$
(12)
$${M}_{12}={I}_{\mathrm{HH}}+{I}_{\mathrm{HV}}-{I}_{\mathrm{VH}}-{I}_{\mathrm{VV}},$$
(13)
$${M}_{21}={I}_{\mathrm{HH}}-{I}_{\mathrm{HV}}+{I}_{\mathrm{VH}}-{I}_{\mathrm{VV}},$$
(14)
$${M}_{22}={I}_{\mathrm{HH}}-{I}_{\mathrm{HV}}-{I}_{\mathrm{VH}}+{I}_{\mathrm{VV}}.$$
(15)
$$I=\sum _{S}{I}^{(S)}.$$
(16)
$${\mathbf{E}}^{(S)}={\mathbf{E}}_{1}^{(S)}\cdot \mathbf{x}+{\mathbf{E}}_{2}^{(S)}\cdot \mathbf{y}+{\mathbf{E}}_{3}^{(S)}\cdot \mathbf{z},$$
(17)
$${\mathbf{M}}^{(S)}(\psi ,\gamma )=R(\gamma )M(\psi )R(-\gamma )R(\gamma )M(\pi -\psi )R(-\gamma ),$$
(18)
$$\mathbf{R}(\gamma )=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& cos(2\gamma )& -sin(2\gamma )& 0\\ 0& sin(2\gamma )& cos(2\gamma )& 0\\ 0& 0& 0& 1\end{array}\right],$$
(19)
$${\mathbf{J}}^{(S)}=R(\gamma )M(\psi )M(\pi -\psi )R(-\gamma ){\mathbf{J}}_{0}.$$
(20)
$${\mathbf{M}}_{\mathrm{tot}}=C{\int}_{0}^{2\pi}{M}^{(S)}(\psi ,\gamma )W(\gamma )\mathrm{d}\varphi $$
(21)
$$\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}cos(\psi )=cos(\varphi )sin(\theta ),$$
(22)
$${sin}^{2}(\psi )=1-{cos}^{2}(\varphi ){sin}^{2}(\theta ),$$
(23)
$$tan(\varphi )=tan(\gamma )cos(\theta ),$$
(24)
$$cos(2\gamma )=\frac{[1+{cos}^{2}(\theta )]{cos}^{2}(\varphi )-1}{1-{cos}^{2}(\varphi ){sin}^{2}(\theta )}.$$
(25)
$$\mathbf{M}(\psi )=\left[\begin{array}{cccc}8+2N-(2+3N){sin}^{2}(\psi )& -(2+3N){sin}^{2}(\psi )& 0& 0\\ -(2+3N){sin}^{2}(\psi )& (2+3N)(2-{sin}^{2}(\psi ))& 0& 0\\ 0& 0& -(2+3N)cos(\psi )& 0\\ 0& 0& 0& 10Ncos(\psi )\end{array}\right],$$
(26)
$$N=\frac{({\alpha}_{1}{\alpha}_{2}^{*}+{\alpha}_{2}{\alpha}_{3}^{*}+{\alpha}_{3}{\alpha}_{1}^{*}+{\alpha}_{2}{\alpha}_{1}^{*}+{\alpha}_{3}{\alpha}_{2}^{*}+{\alpha}_{1}{\alpha}_{3}^{*})}{({\alpha}_{1}{\alpha}_{1}^{*}+{\alpha}_{2}{\alpha}_{2}^{*}+{\alpha}_{3}{\alpha}_{3}^{*})}.$$
(27)
$${M}_{12}=C{\int}_{0}^{2\pi}-2(2+3N)[(6+4N){sin}^{2}\psi -(2+3N){sin}^{4}\psi ]cos2\gamma \mathrm{d}\varphi .$$
(28)
$${M}_{12}={M}_{21}={C}^{\prime}(2+3N){sin}^{2}(\theta )\times \left\{4+N+\frac{(2+3N)[-3{cos}^{2}(\theta )]}{4}\right\}.$$