Aberration correction results and imaging performance of the optical systems are partly determined by the mathematical description method of the optical element surfaces. For example, a lens with the surface described by an even asphere equation can effectively correct the spherical aberration. Also, the mathematical expression determines the speed of ray tracing and optimization convergence affects the difficulty in surface processing and testing. Hence, it affects the overall development cycle and cost of the optoelectronic system. It is important to explore the novel optical surface description method to obtain an optical imaging system with higher performance. The optical surface should have more degrees of freedom to describe the complex shapes and correct the optical aberrations, and the optimization of the parameters should converge quickly with an acceptable result in the design process.

Since the 16th century, researchers have been investigating description methods of the optical surface. Especially since the end of the last century, various novel description methods have been proposed. They have played an active role in different optical systems and have effectively improved the system's performance. Novel optical surfaces are effective in practical applications such as periscopes, progressive addition lenses, viewfinders, remote sensing, projection, and lithography.

Optical surfaces can be divided into two categories: explicitly and implicitly defined surfaces according to the mathematical expression. In the process of optimization, it is necessary to calculate the intersection point of the rays with the surface and the first partial derivative of the surface at the intersection point to determine the propagation direction of the rays after passing through the optical surface. For surfaces with explicit definitions, it is easy to get the calculation result through explicit expressions. However, for surfaces with implicit definitions, finite difference or other methods are utilized to trace rays. Hence, the ray tracing speed of an explicitly defined surface is larger. Almost all optical surfaces are defined by explicit expressions. A typical implicitly defined surface is a non-uniform rational B-Spline surface.

Depending on the effect of parameters on the sag of the surface, mathematical descriptions of optical surfaces may be local or global. For the global descriptions, the sag and partial derivative on the whole surface will be changed when the arbitrary parameter of the optical surface is adjusted. The local descriptions have a more powerful ability to tune the local shape. Each parameter of local descriptions has a limited range of influence on the shape of the surface, so the local curvature of the surface can be adjusted without affecting the shape of the surface outside its area of action. Spline surface, Gaussian basis function surface, and wavelet function surface are local descriptions. Moreover, the stitched surface also can be considered a local description.

To control the surface shape effectively, designers would like to use the low-order parameters of the mathematical descriptions during optimization. However, from a macroscopic point of view, this approach reduces the degrees of freedom, which goes against the original intention of using a complex optical surface to be flexible and more descriptive. If the designer can control the surface shape strongly, it may be bold to optimize the optical system with high-order parameters. The surface shape of a complex optical surface can be constrained by controlling the first and second partial derivatives and the local Gaussian curvature of the optical surface.

To meet the design requirements of optical systems with high performance, it is necessary to extend the mathematical description and design methods of optical surfaces. Complex optical surfaces with additional degrees of freedom can effectively improve the performance of the optical system.

The developing status of mathematical description and design methods for complex optical surfaces is summarized. First, the mathematical description methods of optical surfaces are briefly discussed. Currently, most mathematical descriptions of optical surfaces are global and explicit. For optical surfaces with explicit definitions, Cartesian coordinates are generally applied to model them (Fig. 1). The mathematical expression of a complex optical surface can be decomposed into two main parts: the base term and the deformation term. Sphere, conic, and bi-conic are common basic terms, which mainly contain the second-order components and express the main shape of the surface. Deformation terms, such as polynomials, describe additional, and asymmetric surface shapes are responsible for the aberration correction. Then, the state of the art and progress in the mathematical description of optical surfaces are elaborated. With the increasing complexity of optical systems, conventional spherical and aspherical surfaces severely lack degrees of freedom and are hard to meet design requirements. Researchers have proposed various expressions to describe optical surfaces. The basic terms and complex optical surfaces are listed separately. Subsequently, methods on how to control the local or global shape of complex optical surfaces are introduced. Controlling the shape of complex optical surfaces is important. On the one hand, it ensures the manufacturability and detectability of optical components. On the other hand, a reasonable shape can save processing time and costs. In the end, the mathematical formalisms of the novel optical surfaces are given. Practical examples are listed to demonstrate the feasibility and effectiveness of these surfaces. The future research directions for the mathematical description method of the complex optical surface are discussed and analyzed.

Benefiting from the development of processing techniques, complex optical surfaces are widely applied in practice. In summary, the mathematical description and design methods of complex optical surfaces still need in-depth and detailed explorations to promote the achievement of high-performance optical systems.