Diffractive optics is the most dynamic and potential branch of micro-optics based on diffraction theory. Diffractive optical element (DOE) is widely used in the design of optical systems as an element that modulates light waves through optimized structures. As DOE has the characteristics of light weight, easy replication, and high degree of freedom, and can achieve wavefront conversion, spectral modulation, array generation, and other functions that are difficult to be achieved by traditional refraction and reflection elements, it has become one of the research hotspots in modern optics. The characteristics of DOE can be widely combined with optical systems due to its compliance with the development trend of miniaturization and functional integration of modern optical systems. In addition, it has played an important role in modern industrial and national defense fields such as information processing, optical fiber communication, biomedicine, and space technology, and has shown broad application prospects.

DOE based on scalar diffraction theory is the most widely used, and it has the characteristics of reasonable calculation and wide application ranges. In the process of combining with practical optical systems such as laser shaping, micro-measurement, and advanced processing, its design method has made significant progress in design theory, design process, optimization algorithm, and auxiliary design tools. The modeling, design optimization, pre-processing optimization, and evaluation analysis of DOE in typical optical systems can be realized.

In recent years, there are a large number of interests in DOE designs and some reviews of DOE with a specific function or purpose, and the latest DOE design methods are required to be summarized. In order to promote the further development of DOE design methods based on scalar diffraction theory and better serve the development requirements of modern optical systems for structural compactness and functional integration, it is necessary to summarize the research progress of existing DOE design methods, discuss the problems restricting their further development, and prospect the future development trend, so as to provide reference and inspiration for the future research on DOE design methods.

This paper summarizes the design methods of DOE based on scalar diffraction theory. The basic principle of DOE design is reviewed, and the existing DOE design methods based on the diffraction principle and interference principle are briefly described. The specific DOE design methods and their applicability are described through several typical applications, and the technical difficulties in DOE design and the possible application direction in future science and technology are predicted.

The scalar diffraction theory is applicable to the case where the feature size of DOE is much larger than the wavelength. The commonly used formulas include Kirchhoff diffraction integral formula based on point source and plane wave angular spectrum theory based on plane wave source. The scalar diffraction theory only considers the paraxial approximation of a single linearly polarized light but fails to discuss the vectorization and polarization coupling of the wave. It has the advantages of small calculation amounts, fast calculation speeds and can obtain the design results that meet the requirements of the optical system when the feature size is more than ten times larger than the wavelength.

The design of DOE based on the diffraction principle is an inverse design problem. Since there is generally no analytical solution to this problem, it is necessary to solve the optimal solution through an optimization algorithm based on iteration, search, or deep learning (Table 1). The characteristics of the optimization algorithm in terms of convergence, initial value sensitivity, calculation speed, and whether the solution is the global optimum are critical, which largely determines the design efficiency and the proximity of the design solution to the real solution.

The design of DOE based on the interference principle is a problem of obtaining an analytical solution through inverse decomposition. In terms of design, it can be summarized as the inverse decomposition problem of solving the interference sources with known interference light field. In terms of processing, it can be summarized as the inverse decomposition problem of solving the interference sources with known encoded light intensity distribution, where the interference source distribution can be any of amplitude, phase, or complex amplitude. DOE following this principle is often processed by holographic interference lithography.

DOE is designed to be applied to practical optical systems, so the development of DOE design methods is closely related to the requirements of applications. Considering the functional requirements of DOE in the fields of light field regulation, wavefront modulation, spectral modulation, and imaging, this paper discusses the new development of DOE design methods in traditional application scenarios such as beam shaping and array generation, and summarizes the integration and development of DOE design methods and new system requirements in cutting-edge directions such as all-optical diffraction neural networks and extreme ultraviolet lithography masks. In addition, the main design methods of complex amplitude DOE and dynamic DOE are summarized from the perspective of the development direction of DOE as an optical element.

After years of theoretical design and practical requirements of DOE for optical systems, DOE design methods based on scalar diffraction theory have made important progress in the theoretical model establishment, optimization algorithm development, and joint optimization with processing technology. However, in the face of higher diffraction efficiency, higher modulation accuracy, wider spectrum and temperature range, and more diverse functions of optical systems, the existing design methods still have problems of slow design speed, complex design process, and limited design freedom. In the future, the universality, accuracy, and applicability of DOE design methods can be improved by innovating physical models, learning from other fields, and integrating the advantages of existing optimization methods. It is expected that the well-designed DOE will play an increasingly important role in fields such as biomedicine, AR display, and space technology.