Advancements in the design, manufacturing and testing of optical systems have created the need for new functional representations for aspheric surfaces. The representations must define surfaces that can compensate for a high degree of wavefront asphericity and represent steeply sloped surfaces as the surface normal becomes perpendicular to the optical axis. As the standard asphere is explicitly defined, the range of surfaces that it can properly describe is limited. This work develops both a parametrically defined surface approach and an implicitly defined surface approach. Whereas the surface sag of an explicit surface is defined directly using one equation, the sag of a parametric surface is defined using at least two equations. The sag of an implicit surface is defined indirectly using a surface function. The utility of these novel approaches is demonstrated using examples of current interest. Specifically, a truncated parametric Taylor surface and an implicit xyz-polynomial surface are shown to be more general definitions that represent highly aspheric surfaces better the standard explicit asphere. Ray tracing and optimization strategies for parametric and implicit surface representations are discussed. Additionally, this work shows that a Fourier series is not a useful optical surface and introduces the explicit superconic surface, which is a redefinition of the standard superconic surface. Finally, we compare the surface types discussed for ray tracing speed, optimization complexity, and ability to represent highly aspheric surfaces.
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