# Optical Chirality Determined from Mueller Matrices
## 3 Theory
### Method I: Evaluation of Optical Activity Directly from Mueller-Matrix Elements  
This method works if a sample only exhibits CB and/or CD and is assumed to have no linear anisotropy or depolarization. The parameter \(C_{\text{B}}\) then corresponds to circular \begin{table}
\begin{tabular}{l l l} \hline \hline
**Property** & **L-Matrix** & **m-Matrix** \\ \hline Circular birefringence & \(C_{\text{B}}=\mu d\) & \(\mu=2\pi(n_{\text{I}}-n_{\text{r}})/\lambda\) \\ Circular dichroism & \(C_{\text{D}}=\delta d\) & \(\delta=2\pi(k_{\text{I}}-k_{\text{r}})/\lambda\) \\ \(xy\) birefringence & \(L_{\text{B}}=\eta d\) & \(\eta=2\pi(n_{x}-n_{y})/\lambda\) \\ \(xy\) dichroism & \(L_{\text{D}}=\beta d\) & \(\beta=2\pi(k_{x}-k_{y})/\lambda\) \\ \(\pm 45^{\circ}\) birefringence & \(L^{\prime}_{\text{B}}=\
u d\) & \(\
u=2\pi(n_{45}-n_{-45})/\lambda\) \\ \(\pm 45^{\circ}\) dichroism & \(L^{\prime}_{\text{D}}=\gamma d\) & \(\gamma=2\pi(k_{45}-k_{-45})/\lambda\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Definitions of the fundamental optical properties in the \(\mathbf{L}\)- and \(\mathbf{m}\)-matrices. retardation. In a pure CB sample, like a sugar solution, the Mueller matrix becomes that of a rotator, which rotates polarization by an angle \(\alpha=C_{\text{B}}/2\)[12]  
\[\mathbf{M}_{\text{R}}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&\cos C_{\text{B}}&\mp\sin C_{\text{B}}&0\\ 0&\pm\sin C_{\text{B}}&\cos C_{\text{B}}&0\\ 0&0&0&1\end{array} \right]. \tag{2}\]  
Different authors use different conventions, and, in Equation (2), the upper signs of the \(\sin C_{\text{B}}\) terms correspond to the use of the physics convention with time dependence according to \(e^{-i\omega t}\) whereas the lower signs correspond to the engineering convention using \(e^{i\omega t}\). The signs also depend on the definition of \(C_{\text{B}}\). We will use the definition in Table 1 based on the difference \(n_{1}-n_{\text{r}}\) and not on \(n_{\text{r}}-n_{\text{l}}\), which also is in frequent use. The parameter \(C_{\text{D}}\) is related to circular diattenuation, and the corresponding Mueller matrix becomes that of a circular diattenuator (see Appendix A.1 for the derivation). \[\mathbf{M}_{\text{D}}=\frac{T_{\text{l}}+T_{\text{r}}}{2}\left[ \begin{array}[] {cccc}1&0&0&\tanh C_{\text{D}}\\ 0&\operatorname{sech}C_{\text{D}}&0&0\\ 0&0&\operatorname{sech}C_{\text{D}}&0\\ \tanh C_{\text{D}}&0&0&1\end{array} \right], \tag{3}\]  
where \(T_{\text{l}}\) and \(T_{\text{r}}\) are the transmittances for left-handed and right-handed polarized light, respectively. We also note that \(\tanh C_{\text{D}}=-(T_{\text{l}}-T_{\text{r}})/(T_{\text{l}}+T_{\text{r}})\). In general, both CD and CB are present, and the full Mueller matrix becomes  
\[\mathbf{M}_{\text{C}}=\mathbf{M}_{\text{R}}\mathbf{M}_{\text{D}}=\left[ \begin{array}{cccc}1&0&0&\tanh C_{\text{D}}\\ 0&\operatorname{sech}C_{\text{D}}\cos C_{\text{B}}&\mp\operatorname{sech}C_{ \text{D}}\sin C_{\text{B}}&0\\ 0&\pm\operatorname{sech}C_{\text{D}}\sin C_{\text{B}}&\operatorname{sech}C_{ \text{D}}\cos C_{\text{B}}&0\\ \tanh C_{\text{D}}&0&0&1\end{array} \right], \tag{4}\]  
where the prefactor \((T_{\text{l}}+T_{\text{r}})/2\) in Equation (4) has been omitted for the normalization of \(\mathbf{M}_{\text{C}}\). The matrices \(\mathbf{M}_{\text{R}}\) and \(\mathbf{M}_{\text{D}}\) commute, and thus the order of multiplication in Equation (4) does not matter. For small values of \(C_{\text{B}}\) and \(C_{\text{D}}\), as is often the case in CD spectroscopy and ORD, we can utilize series expansions to approximate \(\operatorname{sech}C_{\text{D}}\approx 1\), \(\cos C_{\text{B}}\approx 1\), \(\sin C_{\text{B}}\approx C_{\text{B}}\) and \(\tanh C_{\text{D}}\approx C_{\text{D}}\). Furthermore, \(C_{\text{B}}=2\alpha\) where \(\alpha\) is the rotation of polarization used in ORD. In addition, we have, from Equation (A12), that \(C_{\text{D}}\approx 2\epsilon\). \(\epsilon\) is named \"ellipticity\" in CD spectroscopy but should more correctly be called the ellipticity angle.