# Alternative Design of Binary Phase Diffractive Optical Element with Non-\(\pi\) Phase Difference
## 3 Simulation  
To further verify the effectiveness of the proposed method, a variety of DOEs were designed to generate different patterns. Figure 3 shows the design results of the DOEs for four patterns with the proposed non 0-\(\pi\) design method and the conventional 0-\(\pi\) design method. Due to the sensitivity of the iterative Fourier transform algorithm to the initial phase distribution, we performed 20 designs for each pattern and selected the best results. It should be pointed out that the uniformity of the reconstructed pattern obtained by both design methods were all above 98%and showed no significant difference, so it is not given in the figure. The achieved phase differences of the non 0-\(\pi\) DOEs were 0.755\(\pi\), 0.755\(\pi\), 0.787\(\pi\), and 0.905\(\pi\), respectively. The corresponding numerically reconstructed light fields indicated that the designed non 0-\(\pi\) DOEs were correct and could produce high-precision diffraction patterns. The calculated diffraction efficiencies were 70.5%, 70.5%, 84.5%, and 77.9%, respectively. For comparison, the obtained efficiencies of the traditional 0-\(\pi\) DOEs were 67.5%, 67.5%, 78.6%, and 77.9%, respectively. We can see that the diffraction efficiency of the non 0-\(\pi\) DOE was obviously higher than that of the 0-\(\pi\) DOE for the first three diffraction patterns. For the fourth 7 \(\times\) 5 beam splitter design, however, the results were similar. This indicates that in some cases, the non-\(\pi\) DOE can indeed improve the diffraction efficiency, and in another cases, it cannot. Although non 0-\(\pi\) DOE does not always get a higher diffraction efficiency, it has a special property [29]. That is, when the step height is less than the theoretical value, the zero-order intensity is greater than the other order. Conversely, when the step height is higher than the theoretical value, the zero-order intensity is less than that of other orders. Figure 4 shows the numerical reconstructed intensity distribution of the designed non 0-\(\pi\) DOE for 1 \(\times\) 7 beam splitting at different etching depth. The optimal phase difference is 0.787\(\pi\). Here we assumed that wavelength was \(\lambda\) = 650 nm, fused silica was used as the substrate with refractive index of n = 1.4565 at this wavelength. Thus, theoretical depth of the designed DOE was d = \(\lambda\varphi/(2(\text{n }-1))\) = 560 nm. It can be seen that when the etching depth is lower than the theoretical depth, the zero-order light intensity will be greater than other orders. Conversely, the zero-order intensity will be smaller. In fact, this phenomenon was already confirmed by the result in Figure 2. The reason mainly lies in that when the phase difference moves close to \(\pi\), the zero-order light intensity will always reduce for the binary-phase DOE. Therefore, when the phase difference is larger than the optimal value, meaning that the phase difference is closer to \(\pi\), the zero order will be weaker. In contrast, when the phase difference is less than the optimal value, it means that the phase difference is farther from \(\pi\), so the zero-order will be enhanced. This unique zero-order adjustable character is very useful in the fabrication process because we can judge whether the etching depth is smaller or larger than the target by comparing the intensity of the zero order with other orders, especially for DOEs with small feature size, where it is difficult to accurately measure the etching depth with step meter or white light interferometer. In contrast, zero-order intensity will always increase when the depth deviates from the target for the traditional 0-\(\pi\) DOE, as shown in Figure 5. The theoretical depth of this 0-\(\pi\) DOE was d = \(\lambda/(2(\text{n }-1))\) = 712 nm. Figure 4: Simulated reconstructed intensity distribution of the designed non 0-\(\pi\) DOE for 1 \(\times\) 7 beam splitting with etching depth of (**a**) 530 nm; (**b**) 560 nm; (**c**) 590 nm.