Compared with the traditional optical elements, diffractive optical elements (DOEs) have many advantages, such as light weight, flexible design, and unique dispersion. Greatly promoting the miniaturization and lightweight of optical systems, DOEs are widely used in laser communication, laser radar, space imaging, high-precision optical testing, and other fields. The high-precision analysis of light field modulation by DOEs is a key point in the application of DOEs. However, most optical design software calculates the deflection of light by DOEs with the grating equation model, ignoring the physical structure of DOEs. The quantitative analysis of the comprehensive performance of the system has defects, especially the analysis of energy utilization, stray light, and other performance. The thin element approximation (TEA) model based on the scalar diffraction theory is widely employed to quantitatively analyze the complex-amplitude transmittance, diffraction efficiency, and other properties of DOEs. When the microstructure size of a DOE is larger than 10 times the wavelength, the TEA model can obtain comparatively accurate results. However, the error increases rapidly as the microstructure size gradually approaches the order of magnitude of wavelength. The vector diffraction theory using mathematical tools to strictly solve Maxwell equations is a perfect solution for high-precision analysis of DOEs. Unfortunately, strict vector diffraction calculation involves a large amount of data and is thus generally applicable to small-size elements only. When the diffraction phenomenon of the microstructures is analyzed by the strict vector diffraction theory, violent oscillation of the light field is only observed at the positions with abrupt changes in height. Due to this phenomenon, the paper equates the effect of step microstructures on light wave modulation with step response functions and synthesizes the modulation effect of the DOE as the coherent superposition of multiple step response functions (STEP-RFs). When the incident light wave and the step microstructure have space-invariant characteristics, the modulation of the incident light wave by a large-size element can be quickly obtained by this method. The proposed method is expected to serve as a practical solution for high-precision and rapid analysis of large-aperture DOEs and promote the engineering application of such DOEs.

The surface microstructures of most DOEs can be decomposed into multiple step structures. For example, the boss structure in Fig. 1 can be decomposed into a rising-edge step and a descending-edge step. Then, the modulation of the incident light field by the step structures is calculated by the vector diffraction theory and solidified into a step response function. Finally, the step response functions of all steps are synthesized into the response of the DOE according to the principle of coherent combination of the light fields. For large-aperture diffractive elements, this paper only needs to find out the characteristic step structures and conduct vector analysis of the characteristic steps by the strict vector theory to obtain the response functions before synthesizing the light field distribution after the incident light field is modulated by the DOE by the above method. Sub-window splicing is the most direct and effective strategy for light field synthesis for large-aperture DOEs. To ensure the influence of a step structure on the light field of adjacent sub-windows, the overlapping area of the sub-windows must be larger than half of the range of the response function.

Specifically, the three main factors affecting calculation accuracy, i.e., the minimum linewidth of microstructures, the range of the response function, and the step positioning error, are studied. To quantitatively evaluate the calculation accuracy of the proposed response function method, this paper uses the results calculated by the finite-element method (FEM) as a reference to calculate the relative errors in amplitude and phase point by point and takes the root-mean-square (RMS) value of the relative errors as the quantitative evaluation index. When the range of the response function is larger than 7λ, the relative-error RMS of the light field tends to stabilize with small fluctuations (Table 1). Even when the minimum linewidth of the microstructure reaches one time the wavelength, the combined relative-error RMS of the amplitude and phase is kept below 6.1% (Fig. 3). The relative-error RMS increases linearly with the positioning error approximately (Fig. 5). When the positioning error is kept below 30 nm, the relative-error RMS increment of light field amplitude remains smaller than 5%, and the relative-error RMS of phase remains below 2%. For the further evaluation of the far-field characteristics of this method, the far-field distribution of the 2-level Fresnel lens is calculated by the Kirchhoff diffraction integral formula on the basis of the near-field obtained by the STEP-RFs. The diffraction efficiency and the characteristics of the point spread function are analyzed, as shown in Table 3. Compared with the results calculated by the FEM, the maximum relative light intensity error of the lens calculated by the proposed response function method is 1.8%, and the maximum relative error in diffraction efficiency is 1.64%. In addition, the calculation efficiency of the STEP-RFs is at least 1500 times higher than that of the COMSOL software (Table 5).

This paper proposes a fast and high-precision analysis method for large-aperture DOEs with variable periods. The minimum linewidth of the microstructure, the range of the response function, and the step positioning error are three main factors that affect calculation accuracy. The numerical calculation results show that a smaller minimum characteristic size of the microstructure corresponds to a larger calculation error. When the minimum linewidth of the microstructure is close to one time the wavelength, the relative-error RMS is smaller than 6.1%, and the accuracy is still high. When the range of the response function is larger than 7λ, the relative-error RMS of the light field tends to stabilize with small fluctuations. The relative-error RMS increases linearly with the positioning error approximately. When the positioning error is kept smaller than 30 nm, the relative-error RMS increment of the light field amplitude remains below 5%, and the relative-error RMS of phase remains smaller than 2%. The positioning error is inevitable in the calculation of large-aperture diffractive elements. Nevertheless, even if a positioning error smaller than 25 nm is present, the differences between the maximum light intensity and diffraction efficiency of the far field and the results of strict vector theory analysis are still smaller than 2%, and the accuracy is still high. At last, the computational efficiency of the proposed method is at least 1500 times higher than that of the COMSOL software, and the efficiency improvement effect is more salient for larger apertures.