# Broadband Spectral Domain Interferometry for Optical Characterization of Nematic Liquid Crystals
## 3 Chromatic Index Variation of \(Mlc2132\)
### Extended Cauchy Coefficients  
According to [31; 32], the off-resonance chromatic dispersion of a LC mixture can be fitted for both indices by the extended Cauchy equation, written as:  
\[n_{o,e}(\lambda)=A_{o,e}+\frac{B_{o,e}}{\lambda^{2}}+\frac{C_{o,e}}{\lambda^{4}} \tag{8}\]  
where \(A_{o,e},B_{o,e},C_{o,e}\) are constants depending on the LC mixture and the polarization state. Fitting the measured spectral phase according to Equation (7) enables determining the coefficients \(B_{o,e}\) and \(C_{o,e}\). \(A_{o,e}\) is defined with a refractive index value for a given wavelength and lab temperature \(n(\lambda_{ref},T_{lab})\), known in another way. From the extensive \(E7\) study of Li [31] are taken the values \(n_{e}=1.7390\), \(n_{o}=1.5222\) for \(\lambda_{ref}=0.55893\) um and \(T_{lab}=20^{\circ}\). The resulting fits are displayed in Figure 2 and the fitting coefficient \(A_{o,e},B_{o,e},C_{o,e}\) are gathered in Table 2. The determined values are consistent with the characterizations reported previously. \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\lambda_{0}\) = 750 nm & \(\varphi^{2}\) & \(\varphi^{3}\) & \(\sigma_{\varphi^{2}}\) & \(\sigma_{\varphi^{3}}\) & \(\varphi^{2}\)[31] & \(\varphi^{3}\)[31] \\ \hline \(E7_{e}\) & 337 & 158 & 1 & 6 & 338 & 318 \\ \(E7_{o}\) & 126 & 20 & 2 & 7 & 125 & 69 \\ \(MLC2132_{e}\) & 387 & 231 & 2 & 9 & & \\ \(MLC2132_{o}\) & 124 & 69 & 1 & 3 & & \\ \hline \end{tabular}
\end{table} 
Table 1: GVD (\(\varphi^{2}\)) and TOD (\(\varphi^{3}\)) coefficients, for \(\lambda_{0}\) = 750 nm, for \(E7\) and \(MLC2132\) on both axis. \(\sigma\) is the standard deviation of the fitted parameters. GVD (\(\phi^{2}\)) and TOD (\(\varphi^{3}\)) values calculated from [31] are shown for comparison. GVD and TOD are respectively expressed in (\(fs^{2}/mm\)) and (\(fs^{3}/mm\)). The _MLC_2132 mixture is characterized following the same procedure (Figure 2c,d), with the following refractive index values, provided by Merck : \(n_{e}=1.7657,n_{o}=1.5094\), measured for \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **A** & \(\sigma_{A}\) & **B** & \(\sigma_{B}\) & **C** & \(\sigma_{C}\) \\ \hline \(E7_{e}\) & 1.687 & 0.005 & 0.0079 & 0.0001 & 0.0027 & 0.0001 \\ \(E7_{o}\) & 1.495 & 0.005 & 0.0073 & 0.0001 & 0.0004 & 0.0001 \\ \(E7_{e}\)[31] & 1.6933 & & 0.0078 & & 0.0028 & \\ \(E7_{o}\)[31] & 1.499 & & 0.0072 & & 0.0003 & \\ \(MLC2132\,_{e}\) & 1.712 & 0.001 & 0.0095 & 0.0001 & 0.0032 & 0.0001 \\ \(MLC2132\,_{o}\) & 1.487 & 0.001 & 0.0072 & 0.0001 & 0.0003 & 0.0001 \\ \hline \end{tabular}
\end{table} 
Table 2: \(E7\) and \(MLC2132\) extended Cauchy coefficients, after fitting the experimental data, for both LC axes. \(\sigma\) is the standard deviation of the fitted parameters. The data are acquired at a lab temperature of 20 \({}^{\circ}\)C. Parameters values from [31] are shown for comparison. Figure 2: **(a,b)** E7 measured and fitted spectral phase (\(\varphi_{o,e}^{FTSI}(\lambda)\), with the linear group delay term subtracted) for two different wavelength ranges. On both sub-plots are displayed the measurement for the extraordinary (black) and ordinary (grey) axes and the fit (following Equations (7) and (8), see text) for the extraordinary (orange) and ordinary (green) axes. The top insets display the overall phase \(\varphi_{o,e}(\lambda)\) (extraordinary (black) and ordinary (grey)). **(c,d)** \(MLC2132\) measured and fitted spectral phase (\(\varphi_{o,e}^{FTSI}(\lambda)\), with the linear group delay term subtracted) for two different wavelength ranges. On both sub-plots are displayed the measurement for the extraordinary (black) and ordinary (grey) axes and the fit (see text) for the extraordinary (orange) and ordinary (green) axes. The top insets display the overall phase \(\varphi_{o,e}(\lambda)\) (extraordinary (black) and ordinary (grey)). \(\lambda_{ref}=0.5893\)\(\upmu\)m and \(T_{lab}=20^{\circ}\).