# Improved Exponential Phase Mask for Generating Defocus Invariance of Wavefront Coding Systems
## 2 Improved Exponential Phase Mask and Optimization Method  
The traditional surface equation of exponential phase mask is [1]  
\[f(u,v)=\alpha ue^{\beta u^{2}}+\alpha ve^{\beta v^{2}} \tag{1}\]  
where \(u\) and \(v\) are the normalized pupil plane coordinates in \(x\) and \(y\) directions, respectively. The range of normalized pupil coordinates is \(-1\) to \(1\), which equals the true pupil coordinates, which are split by the greatest pupil coordinates. The following description is  
Figure 1: Scheme of wavefront coding technology. condensed to a one-dimensional situation because the two directions are independent of each other and have the same change condition. Figure 2 shows its PSF in one dimension. The peak position of the point spread function of the exponential phase mask has an obvious translation on the image plane, as shown by the red dotted line in Figure 2, and its maximum value is not at zero, resulting in a mismatch between the object shooting range and the imaging range during the imaging process, resulting in the imaging error. The exponential phase mask equation's Taylor expansion is shown as  
\[f(u)=\alpha ue^{\beta u^{2}}=\alpha u(1+\frac{\beta u^{2}}{1!}+\frac{\beta^{2} u^{4}}{2!}+\frac{\beta^{3}u^{6}}{3!}+\cdots) \tag{2}\]  
Term \(\alpha u\) in Equation (2) is the first-order expansion of the pupil coordinate. The first-order term of the coordinate corresponds to the optical system's translational aberration from the standpoint of wave aberration. As a result, by adding one term to the equation to eliminate the first-order component, the phase mask shape may be created, which weakens the picture plane translation of the point spread function. As the Taylor expansion of \(\alpha ue^{-\beta u^{2}}\) is  
\[\alpha ue^{-\beta u^{2}}=\alpha u(1-\frac{\beta u^{2}}{1!}+\frac{\beta^{2}u^{ 4}}{2!}-\frac{\beta^{3}u^{6}}{3!}+\cdots) \tag{3}\]  
The first-order term has been removed when Equation (2) is subtracted from Equation (3). As a result, the improved exponential phase mask equation given in this research is as follows:  
\[f_{improved-exponential}(u,v)=\alpha u(e^{\beta u^{2}}-e^{-\beta u^{2}})+\alpha v(e^ {\beta v^{2}}-e^{-\beta v^{2}}) \tag{4}\]  
The surface parameters of the phase mask must be optimized after getting the phase mask's surface equation. Fisher information is chosen as the phase mask optimization approach here [1]. When a phase mask function \(f(u)\) is added to the pupil of the optical system without aberrations, the exit pupil equation of the system is  
\[p(u)=\left\{ \begin{array}{ll}\frac{1}{\sqrt{2}}e^{jf(u)}&\quad\text{for}\,|u |\leq 1\\ 0&\quad\text{for}\,|u|>1\end{array} \right. \tag{5}\]  
The optical transfer function (OTF) of the optical system with different defocusing coefficients can be calculated by the exit pupil equation of the system, as  
\[H(\rho,\psi)=\int_{-\infty}^{\infty}p(u+\frac{\rho}{2})e^{j\psi(u+\rho/2)^{2} }\times p^{*}(u-\frac{\rho}{2})e^{-j\psi(u-\rho/2)^{2}}du \tag{6}\]  
Figure 2: PSF in one-dimension of exponential phase mask. where \(\rho\) is the normalized spatial frequency from \(-2\) to \(2\), which equals the normalized frequencies are also the real frequencies divided by the maximum frequencies. \(\psi\) represents the phase difference caused by defocusing, and its calculation formula is [2]  
\[\psi=\frac{\pi D^{2}}{4\lambda}(\frac{1}{f}-\frac{1}{d_{0}}-\frac{1}{d_{i}+ \Delta d}) \tag{7}\]  
where \(D\) is the pupil diameter, \(\lambda\) is the working wavelength, \(f\) is the focal length, \(d_{0}\) and \(d_{i}\) are the object distance and image distance of the optical system when focusing, and \(\Delta d\) is the difference between the image distance of the optical system when focusing and the image distance of the actual imaging. MTF can be used to assess the optical system's imaging quality. The greater the region between the MTF function and the coordinate axis, the higher the optical system's imaging quality.