The depth-of-focus of traditional infrared optical system always has the limited value. Installation errors and environmental temperature changes can cause defocus, degrading the image quality of infrared optical system. Therefore, infrared system imaging with extending depth-of-focus has been a popular research area in optical design. Currently, the depth-of focus expansion methods that researchers usually use have certain drawbacks. As wavefront coding technology gradually rises, it provides a new way to effectively expand the depth-of-focus of infrared optical system.Common infrared detectors are mainly divided into two types uncooled and cooled models. Compared to them, cooled infrared detectors can effectively eliminate the influence of stray light and thermal noise, and have many advantages in applications. Cooled infrared optical system refers to an optical system that matches a cooled infrared detector. The characteristic is that the cold stop of the detector is required as the aperture stop of the infrared optical system. Due to the fact that the phase mask in wavefront coding system needs to be placed at the position of the aperture stop, in order to avoid structure contradiction and reduce design complexity, infrared optical system based on wavefront coding generally matches an uncooled detector, and hardly matches a cooled detector.Therefore, this paper proposes an optical system that matches with a cooled infrared detector. The system uses a secondary imaging structure. It uses the cold stop of the detector as the aperture stop, and makes the cold stop efficiency achieve 100%. At the same time, the parallel plate is also similar to the effect of an aperture stop by controlling the direction of the light. This structure solves the structure conflict between the required position of the phase mask and the position of the cold stop. Finally, the parallel plate is changed to the extended polynomial surface type. And the phase mask surface is determined by modifying the polynomial coefficients, successfully introducing the wavefront coding technology into the cooled infrared optical system.In terms of optimizing phase mask parameter, the method of direct iterative calculation by normally combining optical software with evaluation algorithms is not adopted. Instead, the appropriate value is determined by analyzing the different effects of phase mask parameter values. The specific process is to connect the optical design software with Matlab. Firstly, we use optical design software to track the light rays. Secondly, we use Matlab to read the different MTF values corresponding to the cut-off frequency as the phase mask values change, and draw the relationship curve between them. From this, it can be concluded that the phase mask value cannot be too large. Finally, we select four defocus image plane positions within a distance range of 20 times the depth-of-focus of the original system. Taking cosine similarity as the evaluation standard and using Matlab to read and calculate the data in optical software, we can get the similarity degrees between MTF curves with different phase mask values. The relationship curve between the phase mask values and the MTF similarity of the wavefront coding systems at different defocus positions can be drawn. Based on the above conclusions, the critical value that can maintain consistency of MTF is determined as the appropriate phase mask value. After selecting the phase mask value, through comparison, it is found that the MTF of normal infrared optical system gradually decreases with the increase of defocus, and many zero positions appear within the cut-off frequency. After adding a phase mask, the MTF of wavefront coding system keeps good consistency at different defocus positions, which shows that the system is insensitive to defocus.The blurred images of the wavefront coding system are obtained through software simulation experiments. And then PSF is used as a filter in Lucy-Richardson algorithm to restore the images. The lines and characters in the restored images are clearly visible, which verifies that the cooled infrared optical system based on wavefront coding successfully expands the depth-of-focus range to 20 times that of the original system. Considering that the image results are mainly observed directly by human eyes, we not only use the common evaluation method based on root mean square error of image pixels, but also introduce an evaluation index based on human visual perception to comprehensively evaluate the qualities of the restored images. Through the comprehensive analysis of the MSE and MSSIM results, it is concluded that the qualities of the restored images are good. The main factors that cause the slight differences between the original images and the restored images are noise, artifacts, ringing, and match errors between coding and decoding. In addition, the results also reflect that the wavefront coding system can break through the diffraction limit of traditional infrared optical system to some extent, and will have more research space and prospects.