## 3 Imperfections of Scalar Approximation
### Reconstruction Errors  
In the above analysis, the scalar approximation has obvious imperfections in the diffraction efficiencies and phase sensitivity when the grating period is less than 5\(\lambda\). These imperfections lead to errors in the geometric reconstruction of the etching depth and duty cycle of the DOC. Hence, these errors for 2, 4, and 6 \(\upmu\)m gratings were analyzed. For geometric parameter reconstruction by the DOC, several data libraries of diffraction efficiencies were established based on rigorous calculations. Using the rigorous simulation method FDTD, the diffraction efficiencies of different orders \(\eta_{\text{rig,i}}\) were obtained for various duty cycles and etching depths. With the use of Equation (4), the scalar approximation data \(\eta_{\text{scalar,}j}\) were calculated, and the diffraction orders from 0 to 7 were selected  
Figure 5: Reconstructed phase based on scalar approximation and rigorous simulation for the grating model with (**a**) a variable duty cycle or (**b**) a variable sidewall slope ratio, for different periods subject to TE polarization. for geometric parameter reconstruction. The data libraries were searched for the minimum difference between \(\eta_{rigj}\) and \(\eta_{scalar}\), which corresponds to the duty cycle \(D\) and etching depth \(h\), according to the following principle:  
\[\min\biggl{\{}\sum_{i=1}^{7}\lvert\eta_{rigj,i}(D,h)-\eta_{scalar,i}(D_{0},h_{0}) \rvert\biggr{\}} \tag{6}\]  
where  
\[\Delta D=D-D_{0}\text{ and }\Delta h=h-h_{0} \tag{7}\]  
As the period decreased, the deviations in the geometric parameters reconstructions obtained via scalar approximation and rigorous calculation increased. The reconstruction errors were 0.105 for duty cycle and 180 nm for etching depth with 2 um period gratings. These deviations in the CGH measurement are unacceptable. For the duty cycle error in Figure 6a-c, the obvious reconstruction errors are concentrated in regions where the depth is large. The areas with obvious depth error distributions are concentrated at the two extremes of the small duty cycle in Figure 6d-f. Even though the reconstruction errors decrease as the period increases, the maximum depth error reaches 53 nm, even for a period of 6 um. In DOCs, geometric errors in the surface relief reconstruction lead directly to the accuracy evaluation for the reference wavefront of a CGH. Thus, it affects the testing and machining accuracy of the aspheric mirrors. Based on the above analysis, the deviation of the wavefront caused by reconstruction geometric errors due to the imperfections of scalar approximation in DOC was analyzed. The deviation is given by the absolute phase difference \(\Delta\psi\) between the ideal grating and  
Figure 6: Reconstruction geometric errors for duty cycle \(D\) (0.1–0.5) and etching depth \(h\) (200–600 nm) with scalar approximation and rigorous calculations. (**a–c**) and (**d–f**) indicate the duty cycle errors \(\Delta\,D\) and etching depth errors \(\Delta\,h\) in reconstruction for gratings with periods of 2, 4, or 6 μm, respectively. The step was set to 0.2 for the duty cycle \(D_{0}\) and 20 nm for the depth \(h_{0}\). The sampling of the search step was 0.001 for \(D\) and 1 nm for \(h\) in the rigorous simulation. The values of the color bars indicate the deviation in \(D\) and \(h\) (nm). the grating with variations in geometric parameters such as the duty cycle and etching depth. The wavefront deviation can be expressed as follows:  
\[\Delta W_{error}=\frac{\Delta\psi}{2\pi} \tag{8}\]  
Duty cycle errors within 10% and depth errors within 100 nm were introduced to the 2, 4, and 6 \(\upmu\)m gratings. As shown in Figure 7a-c, the wavefront deviation caused by duty cycle errors was more sensitive to small-period gratings. A 10% duty cycle error in the 2 \(\upmu\)m-period produced a wavefront deviation of 0.009\(\uplambda\). When the period was greater than 3 \(\upmu\)m, the deviation caused by the duty cycle error was less than 0.005\(\uplambda\).