# Pursuing High Quality Phase-Only Liquid Crystal on Silicon (LCoS) Devices
## 4 Phase Linearity  
Phase linearity correction is an essential step for phase modulation panels. This is similar to the gamma correction for linear amplitude modulation in LC displays [42], but is a more complex process, covering the full 2 \(\uppi\) radians phase. The linear phase response is a key criterion for satisfying computer-generated hologram (CGH) patterns calculated on the basis of the iterative Fourier transform algorithm (IFTA) or Fresnel (near-field)/Fraunhofer (far-field) diffraction equations, derived from scalar diffraction theory. The linear material for modulation is the main assumption of the theory. However, the intrinsic electro-optical response of liquid crystal is nonlinear. The measured intensity curve can be converted to phase based on following equation:  
\[\frac{I}{I_{0}}=\cos^{2}\chi-\sin 2\phi\sin 2(\phi-\chi)\sin^{2}(\frac{\delta}{ 2}) \tag{1}\]  
where \(I/I_{0}\) is the normalized intensity, \(\chi\) is the angle between the polarizer and the analyzer, \(\phi\) is the angle between the LC director and the polarizer, and \(\delta\) is the phase retardation. A nonlinear phase curve containing the sine or cosine function is tuned to a linear phase response using the look-up table (LUT) method [43]. The CGH simulation cannot correctly project its result without linear correction of LCoS-SLM. To solve this issue, \"Linear Phase Calibration\" is the first step for LCoS-SLM. The ease of calibration is closely related to the driving scheme in the control circuit. The digital driving scheme uses pulse width modulation (PWM) to generate the gray scale that can be divided into finer 0- and 1-bit planes. This makes programming of linear calibration much easier than the backplane usinganalog driving scheme. The average phase accuracy error (APAE%) and the root mean square (RMS) methods can be applied to evaluate the phase linearity of all gray levels:  
\[\text{APAE%}:\ \frac{\sum\limits_{GL=0}^{GL=255}\left(\frac{|\delta_{\text{m}}( LUT(GL))-\delta_{i}(GL)|}{\delta_{\text{max}}-\delta_{\text{min}}}\right)}{256} \tag{2}\]  
\[\text{RMS}:\sqrt{\frac{\sum\limits_{GL=0}^{GL=255}\left(\delta_{\text{m}}(LUT (GL))-\delta_{i}(GL)\right)^{2}}{256}} \tag{3}\]  
where GL is gray level, LUT is look-up table, and \(\delta\) is the measured phase value. Let us take our PCM-2-01-633 LCoS-SLM panels with 1920 \(\times\) 1080 resolution and 60-144 Hz picture frame rate as an example. A design with high-programmed pulse switching at \(\Delta\)V = 5 V of the IC backplane was applied to reach a liquid crystal response within 4 ms. The phase linearity can be adjusted down to 1.08% (APAE%) and 0.024 \(\pi\) radians (RMS). In general, achieving ideal phase linearity in an analog driving LCoS-SLM is more difficult than in a digital driving scheme without additional optical compensation [44]. The Meadowlark Optics LCoS-SLM can achieve high-speed response (2-5 ms) while maintaining a certain degree of linearity in a low-resolution panel (256 \(\times\) 256) using analog driving [45], but not in their higher resolution (HSP512 and HSP1920) LCoS panels. The LCoS panel with an analog drive from Hamamatsu Photonics can achieve ultra-high linearity (0.03 \(\pi\) radians (RMS)) at a lower resolution (792 \(\times\) 600) and 60 Hz picture frame rate. However, to maintain ultra-high phase linearity, a thicker LC cell gap and lower operation voltage are required. The drawback of this setup is a slower response time (>30 ms).