# Improved Exponential Phase Mask for Generating Defocus Invariance of Wavefront Coding Systems
## 3 Simulated Results  
The parameters of the improved exponential phase mask and other phase masks are optimized in one dimension. The optimum threshold value is \(th=0.3\), with a defocus range of 0 to 60, to achieve both defocusing imaging invariance and imaging quality. Table 1 shows the optimization results; the first parameter is alpha, and the second parameter is beta. According to the surface equations of the phase masks and their corresponding optimization parameters in Table 1, the phase profiles of the above six kinds of phase masks are drawn in Figure 3a. Figure 3b shows the two-dimensional phase profile of the improved exponential phase mask at \(\alpha=22.744\), \(\beta=1.645\). The invariance of MTF curves is examined first to verify the defocusing imaging performance of the improved exponential phase mask proposed in this paper. Figure 4 shows the MTF curves of the optimized six types of phase masks with defocusing phase parameters of 0, 20, 40, and 60. The improved exponential phase mask has the highest curve invariance under different defocusing values, as evidenced in the defocusing MTF curves of six different phase masks. The Hilbert angle is used to calculate the similarity level of two functions in Hilbert space, which can be calculated as  
\[\theta(\psi)=\cos^{-1}(\frac{\langle MTF(u,\psi),MTF(u,0)\rangle}{\|MTF(u,\psi )\|\|MTF(u,0)\|}) \tag{11}\]  
The symbol \(\langle\cdot\rangle\) stands for calculating the inner product of two vectors, and the symbol \(\|\cdot\|\) stands for calculating the 2-norm of vectors. Equation (11) shows that when the Hilbert angle is less, the difference between the two functions is smaller, and the phase mask's defocus invariance is stronger. The appropriate Hilbert space angles for the MTF curves of six types of phase masks are determined in the defocusing range [0, 60], and the results are displayed in Figure 5. The angle of the improved exponential phase mask is bigger than that of the exponential phase mask and the tangent phase mask when the defocusing value is less than 30, but its value is still low, so the invariance is maintained. When the defocus value is more than 30, however, the Hilbert angle of the improved exponential phase mask is much lower than the other phase masks, demonstrating that the improved exponential phase mask can perform better when the defocus value is large. In addition, the system \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Function** & **Parameters** \\ \hline Improved exponential & \(\alpha u(e^{bu^{2}}-e^{-\beta u^{2}})\) & 22.744, 1.645 \\ Exponential & \(\alpha e^{\beta u^{2}}\) & 29.924, 1.832 \\ Cubic & \(\alpha a^{3}\) & 89.757 \\ Sinusoidal & \(\alpha u^{4}\sin(\beta u)\) & 211.174, 1.856 \\ Improved logarithmic & \(sgn(u)\alpha u^{4}\log(\|u\|+b\|)\) & 162.286, 1.990 \\ Tangent & \(\alpha u^{2}\tan(\beta u)\) & 40.593, 1.342 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Optimization results of six kinds of phase masks. Figure 3: Phase profile (**a**) Phase profiles of improved exponential, exponential, cubic, sinusoidal, improved logarithmic, and tangent phase masks. (**b**) The two-dimensional phase profile of improved exponential phase mask. should have similar imaging quality under different defocusing parameters to facilitate image restoration, so the area between the MTF curve and coordinate axis under different defocusing parameters is also one of the parameters to consider when evaluating phase mask imaging performance. Figure 4: Defocusing MTF 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. The region between the MTF and the two coordinate axes can be utilized to assess the system's imaging quality. The following formula can be used to compute it:  
\[\text{Area}(\psi)=\int_{-2}^{2}\text{MTF}(\rho,\psi)d\rho \tag{12}\]  
The larger the area is, the better the imaging quality is.