# Improved Exponential Phase Mask for Generating Defocus Invariance of Wavefront Coding Systems
## 3 Simulated Results
As a result, it may be used to assess the imaging quality of phase masks using various defocus values. Figure 6 depicts the area change values of the MTF curve and the coordinate axis of six different types of phase masks in the [0, 60] defocusing range, which shows that the improved exponential phase mask's area change is very sluggish, and the overall area change is smaller than that of previous phase masks. As a result, the improved exponential phase mask provides a more stable imaging quality in the defocusing region. We also use the Fisher information to examine the OTF curve. The simulation results of six phase masks in the defocus range of 0 to 60 are shown in Figure 7.  
Figure 5: Hilbert space angle of defocus MTF curves of six kinds of phase masks. Figure 6: Difference area of MTF under different defocus parameters of improved exponential, exponential, cubic, sinusoidal, improved logarithmic, and tangent phase mask. Under varied defocus values, the improved exponential phase mask exhibits low Fisher information, as shown in Figure 7. It is just slightly higher than the sinusoidal phase mask, but it has a greater MTF invariance. As a result, the improved exponential phase mask outperforms the other five phase masks in terms of defocus invariance. The modified exponential phase mask's imaging invariance in the picture plane is also taken into account. As a result, the one-dimensional PSF curves of six distinct types of phase masks are simulated under various defocusing settings, with the defocusing parameters set to 0, 20, 40, and 60. PSF curves are chosen for comparison to observe the changes, and the results are displayed in Figure 8.  
Figure 8: Defocusing PSF curves of six kinds of phase mask (**a**) Improved exponential phase mask (**b**) Exponential phase mask (**c**) Cubic phase mask (**d**) Sinusoidal phase mask (**e**) Improved logarithmic phase mask (**f**) Tangent phase mask. Figure 3:0 Fisher information of the six phase masks. The improved exponential phase mask effectively suppressed the offset of the peak position, as shown in the simulation results in the red boxes in Figure 8a,b. The mismatch of imaging areas between objects and images caused by the offset of PSF's peak location results in a considerable mistake in the system's actual use. When compared to the other six phase masks, the improved phase mask's peak intensity is as stable as the cubic phase masks and more stable than the other four phase masks. The PSF will change by defocusing on the image plane for the non-rotationally symmetric phase mask, introducing artifacts for image restoration, and causing image quality reduction. As a result, the peak shift changes of these six types of phase masks are compared in Figure 3:1 The improved exponential phase mask's peak shift amount is second only to the cubic phase mask and better than the other four types of phase masks. As a result, the improved exponential phase mask exhibits strong defocus invariance in both spatial frequency domain and image plane analysis when compared to existing phase masks. As a result, it may perform better in extended depth of field imaging. Finally, we simulated the imaging process to evaluate the performance of phase masks. We calculate the 2D-PSF of the system in MATLAB, and downsample it to a 5 \(\times\) 5 template. The \"Image man\" has a size of 256 \(\times\) 256. We use this PSF to convolve the original image and add Gaussian noise with a variance of 0.0001 to this image to simulate the optical imaging result from the wavefront coding optical system. Then we use the Wiener filter to restore this image to get the final imaging result as Figure 10 shows. Gaussian noise with a variance of 0.0001 has been added to the simulation, and the SNR is 80 dB.