Imaging optical systems have been widely used in industrial, medical, and military fields. With the rapid development of modern science and technology, the increasing requirements for imaging optical systems in various applications have promoted the development of optical systems towards higher performance, better imaging quality, smaller volume, lighter weight, and more novel and richer functions. However, it is difficult for traditional spherical and aspheric imaging optical systems to meet these requirements. Therefore, optical freeform surfaces are used in imaging optical systems.

Optical freeform surface is a kind of surface without rotational symmetry. Compared with spherical and aspheric surfaces, freeform surfaces have more degrees of freedom for design and stronger aberration correction ability. Thanks to these excellent characteristics of freeform surfaces, high-performance, compact, lightweight, and novel optical systems have been successfully designed and applied in astronomical telescopes, spectral analysis, remote sensing, and virtual reality. The application of freeform surfaces brings many benefits to optical systems, but it also inevitably destroys the rotational symmetry of the optical systems, which poses a great challenge to traditional optical system design methods.

In order to realize the efficient design of freeform optical systems, people not only extend the design methods based on traditional aberration theory but also propose some direct design methods which can directly obtain the parameters of freeform surfaces according to the system specifications. These methods can provide favorable initial structures for further optimization. In addition, some automatic design methods for optical systems whose imaging quality is close to the diffraction limit are proposed. But these design methods still have some limitations. Therefore, it is necessary to summarize the existing design methods of freeform surface imaging optical systems to better guide the future development of this field. At the same time, the progress of freeform surface design methods can promote the design of freeform surface optical systems. Therefore, it is very important to classify and summarize the existing applications of freeform surface imaging optical systems to better help people optimize their designs.

This paper introduces various design methods of freeform surface imaging optical systems in detail. Firstly, the limitations of designing optical systems based on traditional aberration theory are discussed. The nodal aberration theory proposed by Shack and Thompson is introduced. The advantages of nodal aberration theory in aberration analysis and optimization guidance of freeform surface imaging optical systems are summarized. Then the research progress and technical characteristics of various direct design methods are introduced in detail, including differential equation design method, simultaneous multiple surface design method, construction-iteration design method, series expansion design method, and design method based on machine learning. The differential equation design method is used to design the freeform surface, and its mathematical theory is relatively complete. The simultaneous multiple surface design method can directly design multiple surfaces at the same time. The number of surfaces usually equals that of fields of view that can be controlled by this method. The construction-iteration design method is highly versatile, and the light rays from multiple fields of view and different pupil coordinates are considered (Fig. 4). The series expansion design method uses power series to solve functional differential equations, which has a fast calculation speed. The design method based on machine learning combines the machine learning algorithm with the process of optical design, which can obtain a large number of design results with little manual participation. Then the automatic design methods of the optical systems which can directly and automatically obtain the imaging quality close to the diffraction limit are introduced. Finally, the advantages and limitations of different freeform surface imaging system design methods are compared (Table 1).

This paper also summarizes the applications of freeform surface imaging optical systems with high performance, as well as new structures and functions in various fields. Firstly, the early applications of freeform surfaces in imaging optical systems are reviewed, including the deformable lens designed by Chretien, the Alvarez lens with variable focal length, and the viewfinder of the Polaroid SX-70 camera. Then several high-performance freeform surface imaging optical systems are introduced, including off-axis reflection systems with a large field of view, low F number, or small volume (Fig. 8), imaging spectroscopy systems with higher spectral resolution and better imaging quality (Fig. 11), and extreme ultraviolet lithography projection systems with extremely high imaging quality and extremely low distortion (Fig. 13). Then, several kinds of freeform surface imaging optical systems with new structures and functions are introduced, including off-axis reflection systems with special structures such as primary mirror and tertiary mirror integration (Fig. 14), visual optical systems for head-mounted display and head-up display (Fig. 15), optical systems with local optical properties (Fig. 16), and freeform optical systems with lateral image translation or other special functions.

Design methods of freeform surface imaging optical systems have made great breakthroughs and may develop towards the following directions in the future, including automatic design, rapid design, combination with machine learning, and consideration of both design and process. With the continuous research and improvement of the design methods of freeform surface imaging optical systems, more and more freeform surface imaging optical systems have been designed. At present, the specific applications of freeform surface imaging optical systems mainly have the following development directions, such as higher performance, more novel functions, and newer structures. At the same time, these freeform surface optical systems with high performance and new functions will put forward higher requirements for the processing, metrology, and alignment technologies of freeform surfaces. The progress of these technologies will also continue to promote the development of freeform surface imaging optical systems.