# Broadband Spectral Domain Interferometry for Optical Characterization of Nematic Liquid Crystals
## 1 Introduction  
The remarkable optical properties of nematic liquid crystals (LC), e.g., tunable birefringence and transparency, induce a growing development of novel photonic applications based on light phase modulation. The mechanism relies on a tunable change of the LC optical index, either due to electrically-induced molecular orientation, or thermal-control of the nematic order [1; 2]. Increasing the mixture birefringence together with the liquid crystal layer thickness (e.g., above 50 \(\upmu\)m) enables enhancing the induced phase-shift and thus develop innovative applications [3; 4]. Recently, electrically-addressed large cell gap nematics have attracted interests for terahertz phase shifters [5; 6], high-resolution hyperspectral imaging [7; 8; 9; 10], phase and group delay control of ultrashort pulses trains [11], tunable angular shearing in wedge-shaped cells [12] and terahertz vortex beam generators [13], just to cite a few. In such progress, chiral nematics are not in rest with the recent demonstration of sub-picosecond pulse compression [14] and femtosecond pulses dispersion management [15]. Furthermore, self-induced thermo-optical change of the LC birefringence can be exploited to shape infrared femtosecond pulses, through spatial self-phase modulation [16]. Here again, the combination of the LC layer thickness (180 \(\upmu\)m) and its high birefringence enables achieving large phase-shift of several tens of radians. Very recently, this thermo-optical mechanism resulted in an innovative phase shaper for multi-octave light sources [17]. The continuous quest for more performing mixtures is still very active and new ones are regularly coming out on the market. Among these, \(MLC2132\) (Merck) present some interesting features: a large birefringence (\(\Delta n=n_{e}-n_{o}=0.26\), with \(n_{e}\), resp. \(n_{o}\), the refractive index for extraordinary, resp. ordinary, axis) and a high nematic-to-isotropic phase transition temperature (\(T_{c}=387\) K). However, although the mixture is referenced in some publications, little information is available and it originates from the manufacturer [18; 19]. To determine the advantage of \(MLC2132\) for the above-mentioned applications, an in-depth characterization of its anisotropic optical properties is needed. The measurement of the optical indices of LC mixtures is usually adapted from methods commonly employed for bulk medium characterization. Therefore, a widespread method to characterize a LC mixture relies on an Abbe Refractometer: precise measurements can be performed in a short time and for both polarization states [20; 21], but the monochromatic measurements and the narrow working temperature range still limits the system. A wedged-cell refractometer [22] has been recently reported for precise mid-IR and far-IR measurements (up to 10 \(\upmu\)m), but the method requires the fabrication of _'ad hoc'_ LC wedges. Talbot-Rayleigh refractometer [23] is used for polychromatic refractive indices measurement in less than 1 second, but the resulting low precision and resolution still limit its efficiency. Eventually, some LC drops are also characterized through a Jelley-Leitz refractometer, that requires shear flow to orientate the molecules [24]. On the other side, the conventional birefringence measurements rely on a monochromatic illumination of a LC-cell placed between crossed-polarizers [25]. In that case, the transmitted intensity is written as \(I_{t}=I/2(1-cos(\Delta\phi))\) with \(\Delta\phi=\frac{2\uppi}{\lambda}\Delta nL\). The birefringence \(\Delta n\) is written as : \(\Delta n=n_{e}-n_{o}\), with \(n_{e}\) (resp. \(n_{o}\)) the refractive index for extraordinary (resp. ordinary) axis. \(L\) is the nematic layer thickness. This method suffers from two drawbacks: the cosine dependence of the retrieved phase decreases the overall sensitivity and the measurement is affected by potential fluctuations of the light intensity.