# Optical Chirality Determined from Mueller Matrices
## Appendix A Conventions
### Effects of Choice of Time Dependence in the \(\mathbf{L}\)-Matrix  
Various signs due to the choice of conventions are used by different authors in the \(\mathbf{L}\)-matrix in Equation (9). We used the \(e^{-i\omega t}\) convention and definitions of the anisotropic parameters as in Table 1. Derivations of a few of the elements of \(\mathbf{L}\) are given here as examples. Further details are given by [PERSON] [3]. The starting point is the differential propagation matrix defining the matrix \(\mathbf{m}\)  
\[\mathbf{m}=\lim_{\Delta\simeq\to 0}\frac{\mathbf{M}(z,\Delta z)-\mathbf{I}}{ \Delta z}\] (A13)  
where \(\mathbf{M}(z,\Delta z)\) is the Mueller matrix for propagation from \(z\) to \(z+\Delta z\) in a homogeneous material. A linear \(xy\) retarder has the Mueller matrix  
\[\mathbf{M}=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&\cos\delta_{xy}&\sin\delta_{xy}\\ 0&0&-\sin\delta_{xy}&\cos\delta_{xy}\end{array}\right],\] (A14)  
where \(\delta_{xy}=2\pi d_{\text{R}}(n_{x}-n_{y})/\lambda\) and \(d_{\text{R}}\) is the retarder thickness. For a thin slab (\(z\to z+\Delta z\)) with thickness \(\Delta z\), we can expand cosine and sine as \(\cos\delta_{xy}\approx 1\) and \(\sin\delta_{xy}\approx\delta_{xy}\). Equation (A14) then results in the Mueller matrix  
\[\mathbf{M}(z,\Delta z)\approx\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&\delta\\ 0&0&-\delta&1\end{array}\right]=\left[\begin{array}{cccc}1&0&0&0\\ 0&1&0&0\\ 0&0&1&\eta\Delta z\\ 0&0&-\eta\Delta z&1\end{array}\right]\] (A15)  
where we introduced \(\eta=2\pi(n_{x}-n_{y})/\lambda\). The matrix \(\mathbf{m}\) for a linear \(xy\) retarder now follows from its definition in Equation (A13)  
\[\mathbf{m}=\left[\begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&\eta\\ 0&0&-\eta&0\end{array}\right].\] (A16)  
Finally, we obtain the matrix \(\mathbf{L}\) by multiplying with the sample thickness \(d\), and, by also noting that \(\eta d=L_{\text{B}}\), we obtain  
\[\mathbf{L}=d\mathbf{m}=\left[\begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&\eta d\\ 0&0&-\eta d&0\end{array}\right]=\left[\begin{array}{cccc}0&0&0&0\\ 0&0&0&0\\ 0&0&0&L_{\text{B}}\\ 0&0&-L_{\text{B}}&0\end{array}\right].\] (A17)The retarder Mueller matrix for the \(\pm 45^{\circ}\) direction is  
\[\mathbf{M}=\left[\begin{array}{cccc}1&0&0&0\\ 0&\cos\delta_{\pm 45^{\circ}}&0&-\sin\delta_{\pm 45^{\circ}}\\ 0&0&1&0\\ 0&\sin\delta_{\pm 45^{\circ}}&0&\cos\delta_{\pm 45^{\circ}}\end{array}\right], \tag{103}\]  
and with the same procedure as above, we find  
\[\mathbf{L}=\left[\begin{array}{cccc}0&0&0&0\\ 0&0&0&-L_{\text{B}}^{\prime}\\ 0&0&0&0\\ 0&L_{\text{B}}^{\prime}&0&0\end{array}\right]. \tag{104}\]  
The diattenuator Mueller matrix for the \(xy\) direction is  
\[\mathbf{M}=\left[\begin{array}{cccc}1&-\tanh L_{\text{D}}&0&0\\ -\tanh L_{\text{D}}&1&0&0\\ 0&0&\sinh L_{\text{D}}&0\\ 0&0&0&\sinh L_{\text{D}}\end{array}\right], \tag{105}\]  
and with the same procedure as above, we obtain  
\[\mathbf{L}=\left[\begin{array}{cccc}0&-L_{\text{D}}&0&0\\ L_{\text{D}}&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{array}\right]. \tag{106}\]  
If the engineering convention is used, the signs of \(L_{\text{D}}\), \(L_{\text{D}}^{\prime}\) and \(C_{\text{D}}\) will not change in Equation (9), whereas the signs of \(L_{\text{B}}\), \(L_{\text{B}}^{\prime}\), and \(C_{\text{B}}\) will change to the opposite.