We aim to address the inadequacies in traditional double-layer diffractive optical element (DLDOE) designs, which ignore the influence of environmental temperatures and incident angles on diffraction efficiency. To this end, we propose the temperature-angle-bandwidth integrated average diffraction efficiency (TABIADE) maximization method for DLDOE design that involves DLDOE optimization and analysis under wide temperature ranges and wide field of view (FOV) to determine the optimal design wavelength pairs and corresponding parameters. Our results demonstrate that the TABIADE maximization method significantly improves the diffraction efficiency of DLDOE over a wide temperature range and large FOV, thereby enhancing the imaging quality of refraction-diffraction hybrid systems. This approach enhances the theoretical foundation for DLDOE design and provides crucial guidance for the practical engineering applications.

We employ a mathematical model to analyze the effects of environmental temperatures and incident angles on the diffraction efficiency of DLDOE and propose the TABIADE maximization method for optimization. First, we derive the mathematical model of DLDOE diffraction efficiency, with the influence of temperature and incident angle variations considered. Subsequently, based on this mathematical model, we develop the TABIADE maximization method, which comprehensively considers the influences of temperatures, angles, and wavelengths to enhance the diffraction efficiency of DLDOE. Next, we conduct a case study by adopting DLDOE in the mid-wave infrared band and optimize the design wavelength pairs to achieve high diffraction efficiency over a wide temperature range. Finally, we apply the optimized DLDOE to design a mid-wave infrared refraction-diffraction hybrid system and conduct experimental validation, thereby demonstrating the superior performance of the optimized diffractive elements.

DLDOE designed by employing the TABIADE maximization method demonstrates significant advantages in response to variations in environmental temperatures and incident angles. Figures. 1(a) and (b) illustrate the effects of design wavelength pairs on diffraction characteristics under different environmental temperatures and incident angles. Table 1 compares the performance parameters of DLDOE designed based on the TABIADE maximization method and the traditional bandwidth integrated average diffraction efficiency (BIADE) maximization method. DLDOE designed via the TABIADE maximization method exhibits smaller diffraction orders and microstructure heights, with significantly improved diffraction efficiency under various environmental temperatures and incident angles. Figures. 1(c) and (d) further compare the diffraction efficiency difference between these two design methods. The comparison demonstrates that DLDOE designed by adopting the TABIADE maximization method has higher diffraction efficiency and TABIADE values over a wide temperature range and large FOV, which confirms the superiority of this method. Finally, the optimized DLDOE is applied to mid-wave infrared refraction-diffraction hybrid systems, yielding excellent imaging performance under different environmental temperatures.

Traditional methods for designing DLDOE often ignore the influence of environmental temperatures and incident angles on diffraction efficiency, resulting in suboptimal designs. We propose the TABIADE maximization method for optimizing DLDOE under wide temperature ranges and large FOV. By comparing DLDOE designed by utilizing the TABIADE maximization method with those designed via the traditional BIADE maximization method, we find that the former significantly improves diffraction efficiency and enhances the imaging quality of refraction-diffraction hybrid systems. Application of the optimized DLDOE to mid-wave infrared refraction-diffraction hybrid systems yields excellent imaging performance under various environmental temperatures. Therefore, our study provides novel insights and methods for the design and engineering application of DLDOE.