# Alternative Design of Binary Phase Diffractive Optical Element with Non-\(\pi\) Phase Difference
## 2 Method  
Before giving the proposed design method, we first analyze the influence of phase difference change on the output light field for the conventional 0-\(\pi\) DOE. Figure 1a is the phase distribution of a binary DOE used to generate the 3 \(\times\) 7 spot array with equal intensity. Figure 1b-d are the output light fields obtained by performing two-dimensional Fourier transform operation of the phase distribution in Figure 1a with phase difference of \(\pi\), 0.9\(\pi\), and 1.1\(\pi\), respectively. The insets in the bottom are the cross-section intensity distribution. It can be seen that the reconstructed 3 \(\times\) 7 spots are very uniform when phase difference is \(\pi\). Then, when phase difference is changed to 0.9\(\pi\) or 1.1\(\pi\), the spots are no longer uniform, which is mainly because the zero-order intensity is significantly enhanced compared to other diffraction orders. However, it should be noted that the uniformity of other diffraction orders except the zero order is still very good, which means that the change in phase difference does not affect the uniformity of other diffraction orders. The above analysis indicates that the 0-\(\pi\) DOE has the following interesting properties: when the phase difference is varied from \(\pi\) to other values, the zero-order light intensity will be enhanced, and the uniformity of other diffraction orders can be retained. Our novel idea of the non 0-\(\pi\) DOE design is based on such special properties of the 0-\(\pi\) DOE. The detailed design process can be divided into three steps as shown in Figure 2. To begin with, the zero order of the target pattern is set to zero, resulting in a modified target pattern. Thereafter, the modified IFTA named three-step iterative soft quantization algorithm proposed by [PERSON] [26; 27; 28] is taken to design the 0-\(\pi\) DOE for this modified pattern. By iteratively applying the two-dimensional Fourier transform and inverse Fourier transform, and setting constrains at the DOE plane and output plane, the phase-only distribution can be obtained. In the DOE plane, the amplitude is removed and the phase is quantized progressively. In the output plane, the region of desired orders is  
Figure 1: (**a**) Phase distribution of the calculated binary diffractive optical element (DOE) and the reconstructed intensity distribution with phase difference of (**b**) \(\pi\), (**c**) 0.9\(\pi\), (**d**) 1.1\(\pi\). defined as the signal window and the rest as the noise window. In the iteration, the complex amplitude in the noise area and the phase in the signal area are not constrained, which are referred to as amplitude freedom and phase freedom, respectively. Only the amplitude in the signal area is replaced by the target distribution multiplied by a scale factor, which is used to adjust the power repartition between the signal and noise areas. A more detailed flowchart can be found in [28]. When the iteration is terminated, phase distribution of the DOE can be obtained. As mentioned before, changing the phase difference of the 0-\(\pi\) DOE can enhance the zero-order intensity while maintaining the uniformity of other diffraction orders. Therefore, in the last step, we keep the phase distribution unchanged, then vary the phase difference from 0 to 2\(\pi\), and calculate the corresponding output light field. For each value, the corresponding output light field is calculated with two-dimensional Fourier transform operation and the diffraction efficiency and uniformity are evaluated and plotted with respect to the phase difference. Finally, the optimal phase difference can be found intuitively from the curve. Diffraction efficiency \(\eta\) and uniformity \(\sigma\) are defined as follows:  
\[\eta=\sum Is/\sum I, \tag{1}\]  
\[\sigma=1-\frac{Is_{\text{max}}-Is_{\text{min}}}{Is_{\text{max}}+Is_{\text{min }}}, \tag{2}\]  
where \(Is\) and \(I\) denote the reconstructed intensity distribution in the desired diffraction orders and the whole output plane, respectively. Subscript max and min mean the maximum and minimum value.