# Broadband Spectral Domain Interferometry for Optical Characterization of Nematic Liquid Crystals
## 4 Thermotropic Characterization of \(MLC2132\)  
### Principle  
Liquid crystal molecules are sensitive to temperature and present a large thermal non-linearity of their different optical properties. Some analysis of the behavior of the molecules when changing the inner temperature have been published so far [21; 31; 32]. A higher temperature contributes to a lower order parameter, leading to a decrease of the overall birefringence up to the clearing point. The isotropic phase temperature, \(T_{c}\) or clearing point, sets the critical temperature for the nematic-to-isotropic phase transition. A few physical models describe the temperature effect on LC refractive indices in the literature. Hallers equation and Vuks model [35; 36] disclose the relationship between the birefringence and temperature, but not for the individual refractive indices. More recently, a _4 parameter parabolic_ model has been proposed [31]. According to this model, both refractive indices evolution with the inner temperature are written as :  
\[n_{e}(T)=D-ET+\frac{2\Delta n_{0}}{3}\left(1-\frac{T}{T_{c}}\right)^{\beta} \tag{9}\]  
\[n_{o}(T)=D-ET-\frac{\Delta n_{0}}{3}\left(1-\frac{T}{T_{c}}\right)^{\beta} \tag{10}\]  
Figure 3: Chromatic variation of the refractive index and \(\Delta n\) in (**a**) and chromatic variation of the group refractive index and \(\Delta n^{8}\) in (**b**). The legend stands for (**a**) and (**b**) : \(E7\) (dotted lines) and \(MLC2132\) (solid lines). Purple dots in (**a**) are the data provided by the furnisher (20 \({}^{\circ}\)C).  
where coefficients \(D,E,\Delta n_{0},\beta\) are constant parameters for a specific mixture. This model has been experimentally validated with several LC mixtures and \(E7\) coefficients are then reported: \(D=1.7546\), \(E=5.361^{-4},Tc=331\), \(\beta=0.2391,\Delta n_{0}=0.3768\)[31].  
The described broadband SDI setup, capable of tracking large dynamic phase changes, is suitable to register the group index evolution with the temperature, for the two axes of the birefringent medium. For this purpose, the LC cell is placed in an oven, whose temperature is constantly monitored. The procedure is the following. First, the cell is heated up until the cell transmission and the fringe contrast are abruptly disrupted because of the contraction of the order parameter with the temperature. When the fringes almost disappear and the acquired signal starts to fluctuate the nematic-to-isotropic transition is reached [37]. From this condition, the cell is cooled down with a \(5^{\circ}\) decreasing step and 20 min of rest time to reach the thermal equilibrium. Ten interference spectra are acquired for temperature. The group index variation \(\Delta n^{\delta}(T)\) Equation (3) between successive acquisition step is then measured:  
\[\Delta n^{\delta}=c\frac{\Delta\tau}{L} \tag{11}\]  
where \(\Delta\tau\) is group delay difference obtained by the FTSI technique and L is the LC thickness. The measurement precision comes from the spectrometer resolution and the LC thickness. Given the first one, an error of \(\pm 0.5\) fs is assumed which gives an overall uncertainty of 0.005 on \(\Delta n^{\delta}\).