Compared with visible light systems, cooled infrared imaging optical systems have better application effects in terrible climatic conditions. Compared with uncooled infrared imaging optical systems, they have higher detection sensitivity, longer viewing distances, and more excellent image quality. Therefore, the cooled infrared imaging optical systems are widely used in many fields, such as aerospace and military applications. Cooled infrared imaging optical systems with long focal lengths and large apertures have the problems of long barrel lengths, large volume, and high cost. To solve these problems and achieve a cold shield efficiency of 100%, the design of the catadioptric optical system is generally adopted, such as the Cassegrain-based catadioptric optical system. As sufficient theoretical guidance for determining the initial structure of such systems is lacking, we propose a method for optimal values of key parameters. We design a catadioptric cooled mid-wave infrared imaging optical system based on Cassegrain, which provides important theoretical guidance for the determination of the initial structure of this kind of system.

We derive the calculation formulas which are expressed by three key parameters: the shading coefficient α, magnification of the second mirror of Cassegrain β_sec, and the vertical magnification of relay mirror β_relay, including the initial structure parameters of the optical system, the T value of system length, the primary spherical aberration, and the primary coma aberration of the Cassegrain system. The variation of the difficulty in correcting aberration and compactness of the system with α, β_sec, and β_relay are analyzed through derived calculation formulas. Based on the contradictory relationship between the difficulty in correcting aberration and compactness of the system, the optimal value method of key parameters is proposed. The initial structure of the optical system is determined by the optimal value method, and the initial structure is further optimized through ZEMAX. A catadioptric cooled mid-wave infrared imaging optical system is designed, of which focal length is -600 mm and F number is 2. Finally, we finish the tolerance analysis on the optical system using the Mont Carlo statistical analysis method. The correctness of the theory and the machinability of the optical system are proved.

Combined with the derived calculation formulas, the T value of the optical system, the primary spherical aberration, and the primary coma of Cassegrain, the variation curves of S_Ⅰ, S_Ⅱ, and T value with α, β_sec, and β_relay are given (Figs. 4-6). We also analyze the change rules of the system compactness and difficulty in correcting aberration with α, β_sec, and β_relay. Based on the contradictory relationship between the difficulty in correcting aberration and compactness of the system, we propose the optimal value method of key parameters. The value of α should be as small as possible to ensure sufficient light intake and compactness of system structure and the value of β_sec should be as large as possible to reduce the difficulty in correcting aberrations. Considering the contradictory relationship between the difficulty of correcting aberrations and the compactness of the system, the value of β_relay should not be too large or too small. Based on the optimal value method, three key parameters are determined as α=0.3, β_sec=-3, and β_relay=-0.5. The initial structure of Cassegrain is determined through the value of α, β_sec, and β_relay and optimized slightly through ZEMAX. The design results show that the initial structure of Cassegrain determined according to the optimal value method only needs simple optimization to obtain better image quality (Fig. 7). The initial structure of the optical system is formed by connecting the relay mirror group and the small aberration Cassegrain (Fig. 8) and optimized further. We obtain the catadioptric cooled mid-wave infrared imaging optical system with a long focal length and a large aperture, which is composed of Cassegrain and a relay mirror group with 6 lenses (Fig. 9). The optical system is compact in structure with a total length of 428 mm. Compared with the initial structure, the value of β_relay decreases, which proves that the length of the barrel can be reduced by reducing the value of β_relay. Although the aberration of Cassegrain increases significantly, the residual aberration can be fully compensated by the relay mirror group. At 33 lp/mm, the modulation transfer function (MTF) value of each field of view is greater than 0.4 (Fig. 10), and the imaging quality of the optical system is ideal. The results of tolerance analysis of the system by Monte Carlo statistical analysis show that more than 98% of the samples have MTF values greater than 0.2 and more than 90% have values greater than 0.3. The imaging quality of the optical system meets the requirements and this system is machinable.

Aiming at the design of a catadioptric optical system based on the Cassegrain, we propose an optimal value method of key parameters. The method provides theoretical guidance for the selection of key parameters when determining the initial structure of this kind of optical system and solves the problems of long structure and correcting aberration hard caused by the improper value of key parameters. The initial structure of Cassegrain is slightly optimized by ZEMAX. The results show that the system obtained by this method can meet the design requirements of compactness and reduce the difficulty of aberration correction. After optimizing the initial structure, we design a catadioptric cooled mid-wave infrared imaging optical system with a long focal length and a large aperture, whose structure is compact with a total length of 428 mm. The MTF value of each field of view is greater than 0.4 at the Nyquist frequency, and the root mean square (RMS) radius of each field of view is less than 4 μm, indicating that the imaging quality of the optical system is ideal. The results of tolerance analysis of the system by Monte Carlo statistical analysis show that more than 98% of the samples have MTF values greater than 0.2 and more than 90% have values greater than 0.3. Therefore, the imaging quality of the optical system meets the requirements and this system is machinable. The design results show that when designing a catadioptric optical system based on Cassegrain, the initial structure of the system can be determined by the optimal value of key parameters that we proposed, and the optical system with ideal image quality and compact structure can be obtained by conventional optimization.