# Broadband Spectral Domain Interferometry for Optical Characterization of Nematic Liquid Crystals
## 2 Materials and Methods
### Experimental Set-Up and Analysis  
A versatile, novel instrument to measure the LC refractive indices dependencies is proposed. Our method relies on SDI with a broad spectrum with noticeable advantages compared to the common methods. Indeed, the analysis is resistant to energy fluctuations, provides a good resolution, and is particularly suited to the characterization of dispersive media, as detailed below. The experimental setup is schematized in Figure 1a. The light source is a broadband fiber super-continuum from Leukos (Electro-Visir-250-VIS-IR, 650 mW, 450-2500 nm), with a selected spectral bandwidth extending between 450 nm and 950 nm. A polarizer defines the linear polarization state. The super-continuum seeds a [PERSON] interferometer, with a delay line to detune the respective optical path between the two arms. The spectrally resolved interferogram is finally acquired by two different spectrometers in order to catch the full spectral bandwidth (Ocean Optics 400-800 nm and Avantes for 550-880 nm). The [PERSON] interferometer is first balanced in dispersion. After that, the sample to characterize is inserted in one arm, while an empty LC cell (e.g., two BK7 plates with ITO) is inserted in the reference arm. The only phase difference then originates from the LC mixture. The sample can be rotated in order to scan both refractive indices. A typical fringed spectrum measured for a 200 \(\upmu\)m optical path detuning between the two arms, corresponding to a group delay difference of 600 fs in this spectral range, is shown in Figure 1b. The principle of the analysis is then based on Fourier-Transform Spectral Interferometry (FTSI, [33; 34]) to retrieve the spectral phase from the interferometer output, as briefly summarized in the following. Figure 1: (**a**) Experimental set-up: a super-continuum laser source (Leukos) seeds a [PERSON] interferometer for Spectral Domain Interferometry. A polarizer (P) sets the polarization direction. The LC cell is placed in one arm, and eventually inside an oven (HC photonics corp.) for thermotropy analysis. (F: long-pass filter with \(\lambda_{cutoff}=\) 500 nm, D: short-pass dichroic mirror \(\lambda_{cutoff}=\) 950 nm, M: mirror, BS: beam splitter, C: compensation plates). (**b**) Typical one-arm (black) and interference (red) spectra. (**c**) Fourier Transform of the acquired spectrum (black) and applied temporal filter (blue). (**d**) Measured spectral phase from FTSI analysis (see text). The measured interference spectrum can be written as :  
\[I(\omega,\tau)=I_{0}(\omega)[1+cos(\varphi_{R}(\omega)-\varphi_{S}(\omega)+\omega \tau)] \tag{1}\]  
where \(I_{0}(\omega)\) is the initial light intensity, \(\varphi_{S}(\omega)\) (resp. \(\varphi_{R}(\omega)\)) is the spectral phase introduced by the sample (resp. the reference phase) and \(\tau\) is the group delay difference between the two optical paths. For a proper FTSI analysis, \(\tau\) is chosen so that the two components (t = 0 and t = \(\tau\)) do not overlap in the temporal domain, and is below the upper delay value \(\tau_{max}\) set by the spectrometer resolution (\(\delta\lambda\)), in our case \(\delta\lambda=0.3\) nm and \(\tau_{max}\pm 3\) ps. A Fourier Transform is applied to \(I(\omega,\tau)\) and enables determining \(\tau\), as shown in Figure 1c. This is the first output of the calculation. Indeed, monitoring \(\tau\) enables monitoring the changes of the optical path, relatively to the initial detuning. To go further, temporal filtering of the term centered in \(\tau\) in the Fourier domain and reverse Fourier Transform are applied. The resulting argument \(\phi\) is:  
\[\phi=\varphi_{R}(\omega)-\varphi_{S}(\omega)+\omega\tau \tag{2}\]  
The \(\omega\tau\) term can be subtracted. Knowing \(\varphi_{R}(\omega)\) then gives \(\varphi_{S}(\omega)\), the unknown sample spectral phase ( Figure 1d). This is the second output of the calculation. Let's now consider our sample, a LC layer with thickness \(L\).