## 3 Results
### Experimental Methods and Results  
This method was applied to characterize a transmissive SLM (GCS-SLM-T2, Beijing Daheng Corp., China) in the experiments. The size of the pixel of the SLM was 18 \(\upmu\)m \(\times\) 18 \(\upmu\)m. The experimental setup can also be seen in Figure 1. A laser with a wavelength of 532 nm was used here. Polarizers P1 and P2 were adjusted to make sure that the SLM worked in the pure phase mode. Two-part gray-scale images were displayed on the SLM during the experiment, just like in the simulation. Figure 3a is a typical image, and the gray scale in its variable part is 100. The corresponding hologram is shown in Figure 3b. There is a shift in the fringes between the reference and the variable parts. Areas with the same size were selected in both parts for FFT, as mentioned in Section 2. Figure 3c shows the spectra of the selected area in the variable part after FFT. The point with the highest intensity in the +1 st order of the spectra was chosen to get phase \(\varphi_{v}\). Its position continued to be the same for all the gray scales. A similar process was also implemented on the reference part to get \(\varphi_{r}\). Following the same steps as those described in Section 2, the relationships between the gray scale and the added phase could be obtained, as shown in Figure 3d. Figure 2: (**a**) The simulated interferogram with \(\theta_{\text{set}}\) of \(\pi\). Areas around the red line were selected for Fourier transform; the inset is the enlarged patterns surrounded by yellow lines. (**b**) the frequency spectrum of the selected variable areas; the inset is the enlarged image of the +1 st order; (**c**) the relationship of \(\theta_{\text{set}}\) and \(\theta_{\text{cal}}\); (**d**) the relationship of the relative error and the gray scales. From Figure 3d, the phases could be adjusted nonlinearly by the SLM. The adjustable range was between 0 and about 0.8\(\pi\). If the gray scale is below 20, the phase shift increases slowly with the increase in the gray scale value, while it is maintained almost constant with a value of about 0.8\(\pi\) if the gray scale value is beyond 180. For gray scale values between 80 and 130, the phase shift increases nearly linearly with the increasing gray scale value. The traditional method like that in Reference [18] was also used to verify the new method. Figure 4 presents the main process. At first, the interference patterns were rotated parallel to one edge of the image, as closely as possible. The \(x\) axis was set to be parallel to this edge. Taking Figure 3b as an example, by properly cutting and slightly rotating it, Figure 4a could be obtained. It seems that the two figures had the same line orientation, because the angle of the rotation was quite small (about 0.05 rad). A dashed line parallel to the \(y\) axis was chosen in each part of the image (see in Figure 4a). Here, the \(y\) axis is perpendicular to the \(x\) axis. The relationships between the intensities along the lines and the \(y\) coordinates are shown in Figure 4b. Both curves varied periodically and there was a relative shift between them. By measuring the distance of the adjacent peaks or troughs, the period could be obtained. Then it was used to compare the relative shift between the two curves to get the phase shift. Here, as shown in Figure 4b, the period was about 45 pixels, and the relative shift was about 8.1 pixels. The phase shift here could then be calculated as \(\frac{8.1}{45}\times 2\pi\). This means that the relative added phase was about 0.36\(\pi\) when the gray scale was 100. Repeating the same process, the relationships between the gray scale and the added phase could also be obtained. The results are also referred to in Figure 3d. They matched the results obtained by the new method very well, indicating that the new method was effective. The processing time of the two methods were different.