# Progress in Phase Calibration for Liquid Crystal Spatial Light Modulators
## 2 Basic Principle of Phase-Only SLM  
The liquid crystal refers to a state of matter between the liquid and solid crystal. There are many phases of liquid crystal. The nematic and smectic C * ferroelectric phases of LCs are the most popular for SLMs. Currently, the ferroelectric LC-based SLM could reach binary phase modulation with a very rapid response, and the SLM based on it could lead to obvious zeroth order and conjugated images at the same time. For the multi-grayscale phase modulation, the nematic phases of LC are suitable with reasonable viscidity and response. The SLM are like sandwiches with different layers. Taking the reflective LCoS for instance, the basic structure of reflective SLM can be shown in Figure 1a. As it is called Liquid Crystal on Silicon, the LC layer is arranged on the silicon substrate. Based on the semiconductor technology, the active matrix circuit is formed on the silicon substrate to control each pixel electrode by the applied electrical potential individually. The pixel electrode could be made by reflective aluminum with pixel circuity, which contains gate lines and buried transistors for a maximum fill factor [14; 15]. Two alignment layers with rubbing direction could be used to initialize the orientation of the modulated LC molecules. The transparent electrode is deployed to work with the pixel electrode for generating the electric field across the LC cell. The birefringence could be modulated by the electric field, which is an electrically controlled birefringence (ECB) mode. Figure 1: (**a**) Basic structure of LCoS. (**b**) Schematic of zero-twisted ECB mode-based LC cell in ON and OFF states. LC: liquid crystal; E\({}_{th}\): threshold of the electric field amplitude. For optical applications, the director of the LC molecules plays an essential role in the birefringence. Before phase modulation, the incident light should be polarized for matching the director of LC. The zero-twisted nematic ECB mode is one of the most widely used phase modulation modes for LCoS. As shown in Figure 1b, the electric field could be generated by the electrodes. The alignment layers provide a pre-orientation of the small angle of \(2^{\circ}\) to prevent reverse tilt switching [16]. The LC molecules are homogeneously quasi-parallel (with \(2^{\circ}\)) aligned to the electrode panel when no electric field is applied. When the electric field is applied and the amplitude is larger than the threshold of the electric field amplitude [17], the LC molecules will tilt. The effective refractive index of the cell varies with the molecules polar angle \(\theta(\mathrm{E})\) related to the applied electric field. Thus, the phase retardation could be modulated by changing the applied electric field. Assuming complete polarization and no absorption, the Jones matrix of the LC cell could be:  
\[J_{\mathrm{ECB}}=\left( \begin{array}{cc}1&0\\ 0&\exp[-i\Gamma(\mathrm{E})]\end{array} \right) \tag{1}\]  
The phase retardation \(\Gamma(\mathrm{E})\) could be:  
\[\Gamma(\mathrm{E})=\frac{2\pi}{\lambda}d\big{(}n_{eff}(\mathrm{E})-n_{o}\big{)} \tag{2}\]  
where \(\lambda\) is the wavelength, \(d\) is the thickness of the LC layer, \(n_{eff}(\mathrm{E})\) is the effective refractive index varying with the applied electric field, and \(n_{o}\) is the ordinary refractive index of the LC [18]. The \(n_{eff}(\mathrm{E})\) ranges from \(n_{o}\) to \(n_{e}\) (extraordinary refractive index), which depends on the applied electric field. The birefringence \(\Delta n=n_{e}-n_{o}\) of positive LC could be from 0.05 to 0.45 [19; 20]. The pattern with the grayscale values uploaded on SLM could be transferred to the applied electric field on the pixel electrodes to control the phase modulation of each pixel. Ideally, the grayscale-phase relation set to the commercial phase-only SLM should be linear, and the gamma curve of the ideal SLM is shown as the red line in Figure 2a. In fact, with the aforementioned imperfectness, the shifted phase modulated by SLM without calibration is represented by the red lines in Figure 2b.