# Progress in Phase Calibration for Liquid Crystal Spatial Light Modulators
## 4 Diffractive Phase Calibration Method
### Diffraction Pattern Analyzed Method  
The diffraction pattern analyzed method is a kind of method using diffraction theory to connect the grayscale-related phase with the directly diffracted pattern changes in the far field. The uploaded grayscale pattern could be a coded image or refreshed simple moving units on a uniform background. The global grayscale-phase relation could be acquired from the diffraction pattern changes, and the spatial nonuniformity could be measured.  
In an original diffraction analyzed method [40], an LC television panel removed from Epson projector (VPJ-700) is calibrated. As shown in Figure 6b, a Ronchi grating with two grayscales \(L_{1}\) and \(L_{2}\) at 50% duty ratio is designed and the grayscale-related phases of the grating are \(\phi_{1}(L_{1})\) and \(\phi_{2}(L_{2})\) respectively, so that the relative grayscale and phase are \(L=L_{1}-L_{2}\), \(\phi(L)=\phi_{1}(L_{1})-\phi_{2}(L_{2})\). The grating period is \(P\), the width of the grating along the \(y\)-direction is \(H\), and \(m\) is an integer. In addition, the phase transmittance equation along the \(x\)-direction can be described as:  
\[t(x,y)=\left\{rect\bigg{(}\frac{x-P/4}{P/2}\bigg{)}+\exp[i\phi(L)]rect\bigg{(} \frac{x-3P/4}{P/2}\bigg{)}\right\}\otimes comb(x/P)\cdot rect\bigg{(}\frac{x}{ mP}\bigg{)}\cdot rect\bigg{(}\frac{y}{H}\bigg{)} \tag{4}\]  
The _rect_ in Equation (4) is the rectangular function, and the \(comb\) is the comb function. With uniform monochromatic plane wave illumination, spectrum intensity distribution can be described as Fourier transform: \(T(f_{x},f_{y})=F.T.[t(x,y)]\cdot 1\). In addition, the spectrum pattern in the Fourier plane along the \(f_{x}\)-axis (\(f_{y}=0\)) is:  
\[\big{|}T(f_{x},0)\big{|}_{2}\propto I(f_{x})\cdot\cos[\phi(L)+f_{x}P/2] \tag{5}\]  
The relation between the spectrum intensity distribution and the phase difference \(\phi(L)\) can be acquired by holding the uploaded \(L_{1}\) with changing \(L_{2}\) from 0 to 255 (8-bit) and measuring the intensity changes of the spectrum. A global grayscale-phase relation can be acquired, and this method can also be modified for spatially nonuniformity measurement.  
The calibration of SLM suffering from spatially-varying phase response could be realized [11]. The calibrated SLM is from BNS (HSPDM512 1064-PCIe, Boulder Nonlinear Systems). As shown in Figure 7, the pixels on the SLM are composed of \(8\times 8\)-pixel subsections. Because of the small grating size compared to the diffraction distance, the captured image could be analyzed as Fraunhofer diffraction pattern [41]. Neglecting the coefficient, the Fraunhofer diffraction pattern could be treated as the spectrum of the grating. The grayscale-phase relation of the \(8\times 8\)-pixel subsection can be measured according to Equation (5). Hence, the nonuniformity could be measured by moving the small grating inside the subsection.  
Figure 6: (a) Schematic of a diffraction pattern analyzed method. MO: microscope objective; PH: pinhole filter; L: collimating lens; P: polarizer; BS: beam splitter; BB: beam blocker; M: mirror; SLM: spatial light modulator; CAM: camera. (b) Uploaded pattern of Ronchi grating.  
The checkerboard pattern can also be diffracted for calibration based on the diffractive method [42]. A checkerboard phase pattern with binary phase distribution is uploaded and the far field Fraunhofer diffraction pattern is captured. The related phase could be measured by analyzing the intensity of first orders in the spectrum domain.  
By using this kind of diffractive method based on spectrum or Fraunhofer diffraction analysis, the global grayscale-phase LUT and nonuniformity measurement could be carried out. However, a desired camera with a high sensitivity to intensity changes and high signal-to-noise ratio is needed, for the reason that the derived phase is related to the measured intensity directly. Some attempts using the phase retrieval algorithm to reconstruct the phase from the diffraction pattern have been presented [43; 44].