# Improved Exponential Phase Mask for Generating Defocus Invariance of Wavefront Coding Systems
## 1 Introduction  
The wavefront coding technology proposed in 1995 has become an effective way for increasing the depth of field of an optical system [1]. The wavefront coding system is to add a designed phase mask to the pupil plane of the traditional optical system as Figure 1 shows. The phase mask is used to make the point spread function of the optical system have the characteristics of defocus invariance. However, the size of the point spread function will get larger and imaging results will be blurred. So, it requires an image restoration process to make blurred images clear. The higher the degree of defocus invariance, the better the image restoration result of the system is. When we need a system with a large depth of focus such as long-distance photography, we can add a designed phase mask to the aperture of the traditional phase mask. The phase mask function which can generate the defocus invariance is from the mathematical derivation based on the Fourier optics. We want that there is no defocus variable in the MTF function, and then solve the equation to get the phase mask function. It may effectively ease the conflict between the improvement of the depth of field and the decline of imaging resolution by integrating the optical imaging method with an image restoration algorithm. The defocusing invariance character of an optical system can be obtained by adding an appropriate phase mask to the aperture. The corresponding image restoration algorithm is designed to restore the blurred image taken by optical systems [2] based on defocusing invariance, and the image in a wide range of depth of field may be obtained in the end. As a result, designing a phase mask with a suitable shape to modify the optical system is one of the key points of wavefront coding technology. Many new phase masks have been proposed since 1995, including extended cubic phase mask [3], logarithmic phase mask [4; 5; 6], exponential phase mask [7], fractional power phase mask [8], extended polynomial phase mask [9], sinusoidal phase mask [10], tangentphase mask [11], square root phase mask [12], anti-sinusoidal phase mask [13; 14] and Jacobi-Fourier phase mask [15]. The most common non-rotationally symmetric phase mask is the cubic phase mask. The numerical analytical solutions of point spread function and optical transfer function can be calculated by using the stable phase method [2] or Fresnel integral method [16], which is convenient for theoretical analysis. The cubic phase mask characteristics have been studied extensively in the space domain [17] and frequency domain [18] and applied to the iris recognition system [19]. The sharpness and clarity of the restored images of cubic wavefront mask systems will, however, be lowered in practice. In the extended depth of field imaging, [PERSON] et al. examined the performance of phase masks that were optimized using an image quality criterion after deconvolution, finding that the exponential phase mask and fractional power phase mask outperform the cubic phase mask and logarithmic phase mask [20]. Sinusoidal phase masks have smaller imaging translations than other phase masks, although the modulation transfer function (MTF) fluctuates substantially with defocusing [10]. The logarithmic phase mask and its improved functions have low MTF defocusing invariance [7]. The change in MTF with defocusing is one of the most critical elements in determining the imaging results under the same reconstruction settings. Although the MTF of an exponential phase mask has a high invariance, the position translation of its PSF on the image plane is considerable, leading the mask to leave the imaging area and resulting in imaging findings that are inappropriate. The wavefront coding optical system with higher invariance of defocus has better defocus imaging capability. Due to the high MTF defocusing invariance of the exponential phase mask, the phase mask with better defocusing invariance can be obtained if the image plane translation of the PSF is improved.