The paper by Germer (see above) shows that the Mueller matrix BRDF elements in the xyxy basis can be expressed as an expansion based upon the Zernike polynomials.
The unpolarized 11 element is given by
\begin{equation}
f_{{\rm r},11}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir})=r(\lambda) \sum_{nmkp} a_{11,nm}^{kp}\lambda^p I_{nm}^{k}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir}),
\end{equation}
where $\phi_{\rm ir}$ is the azimuthal scattering angle $\phi_{\rm s}$, $\lambda$ is the wavelength,
\begin{equation}
I_{nm}^{k}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir})= \frac{1}{2\pi}\left[\frac{(n+1)(m+1)}{A_{nm}^k}\right]^{1/2}
\left[R_n^k\left(\sqrt{2}\sin\frac{\theta_{\rm i}}{2}\right)R_m^k\left(\sqrt{2}\sin\frac{\theta_{\rm r}}{2}\right)+\right.
\left.R_m^k\left(\sqrt{2}\sin\frac{\theta_{\rm i}}{2}\right)R_n^k\left(\sqrt{2}\sin\frac{\theta_{\rm r}}{2}\right)\right]
\cos k\Delta\phi_{\rm ir},
\end{equation}
\begin{equation}
A_{nm}^k=
\left\{\begin{array}{ll}
4 & \mbox{if $(n=0)$ or $((n=m)$ and $(k=0))$}\\
2 & \mbox{if $((n=m)$ or $(k=0))$}\\
1 & \mbox{otherwise,}
\end{array}\right.
\end{equation}
and the radial Zernike polynomials are given by
\begin{equation}
R_n^m(\rho) = \sum_{s=0}^{(n-\vert m \vert)/2}(-1)^s\frac{(n-s)!}{s!\left(\frac{n+m}{2}-s\right)!\left(\frac{n-m}{2}-s\right)!}\rho^{n-2s}.
\end{equation}
The parameter $r(\lambda)$ is a scaling factor that accounts for the wavelength dependence of the directional-hemispherical reflectance. The model parameter scale
is used to store $r(\lambda)$.
The other diagonal elements are given by
\begin{equation}
f_{{\rm r},ii}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir})= f_{{\rm r},11}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir}) \sum_{nmklp} a_{ii,nm}^{klp}\lambda^p H_{nm}^{kl}(\theta_{\rm i},\phi_{\rm i},\theta_{\rm r},\phi_{\rm r}),
\end{equation}
where
\begin{equation}
H_{nm}^{kl}(\theta_{\rm i},\phi_{\rm i},\theta_{\rm r},\phi_{\rm r}) = \frac{
K_n^k(\theta_{\rm i},\phi_{\rm i})K_m^l(\theta_{\rm r},\phi_{\rm r})+K_n^k(\theta_{\rm r},\phi_{\rm r})K_m^l(\theta_{\rm i},\phi_{\rm i})
}{(2+2\delta_{nm}\delta_{kl})^{1/2}}.
\end{equation}
The upper-diagonal elements are given by
\begin{equation}
f_{{\rm r},ij}(\theta_{\rm i},\phi_{\rm i},\theta_{\rm r},\phi_{\rm r}) = f_{{\rm r},11}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir}) \sum_{nkmlp} a_{ij,nm}^{klp}\lambda^p K_n^k(\theta_{\rm i},\phi_{\rm i})K_m^l(\theta_{\rm r},\phi_{\rm r}),
\end{equation}
and the lower-diagonal elements are given by
\begin{equation}
f_{{\rm r},ji}(\theta_{\rm i},\phi_{\rm i},\theta_{\rm r},\phi_{\rm r}) = \pm f_{{\rm r},11}(\theta_{\rm i},\theta_{\rm r},\Delta\phi_{\rm ir}) \sum_{nkmlp} a_{ij,nm}^{klp}\lambda^p K_m^l(\theta_{\rm i},\phi_{\rm i})K_n^k(\theta_{\rm r},\phi_{\rm r}),
\end{equation}
where
\begin{equation}
K_n^k(\theta,\phi)=\sqrt{\frac{n+1}{\pi}}R_n^k\left(\sqrt{2}\sin\frac{\theta}{2}\right){\rm az}_k(\phi),
\label{eq:zp}
\end{equation}
and where
\begin{equation}
{\rm az}_k(\phi)=\left\{
\begin{array}{ll}
-\sin k\phi, & k<0\\
1/\sqrt{2}, & k=0\\
\cos k\phi, & k>0.
\end{array}
\right.
\end{equation}
The file containing the coefficients is a column-separated-variable (CSV) file with eight columns:
- 1. Mueller matrix index (1-based), $i$, where the Mueller matrix is $M_{ij}$.
- 2. Mueller matrix index (1-based), $j$, where the Mueller matrix is $M_{ij}$.
- 3. Zernike radial order $m$
- 4. Zernike radial order $n$
- 5. Zernike azimuthal order $k$
- 6. Zernike azimuthal order $l$
- 7. Power of $\lambda$, $p$ (where $\lambda$ is assumed to be in the common length unit, usually micrometers)
- 8. Coefficient, $a_{ij,mn}^{klp}$
Coefficient files appropriate for sintered PTFE are included in the supplementary data for the paper.