Extending nodal aberration theory to freeform surfaces

Freeform optical surfaces are creating exciting new opportunities in mathematics, mechanics, and optics for design, fabrication, assembly, and testing.1 While the term ‘freeform’ is currently applied to a broad range of surface shapes, in our research on imaging, a freeform is a surface whose sag varies with the radial and azimuthal (') components, and is neither off-axis conic, aspheric, nor purely anamorphic. Our latest research is based on this specific class of '-polynomial surfaces, which enables innovative package geometries, as size and weight are increasingly driving constraints in designing optical systems for imaging applications. In developing new 3D packages for imaging optical systems, we often break the rotational symmetry through tilts and decenters that degrade the imaging performance. Using freeform surfaces can directly compensate the resulting aberrations. To optimize the shapes of these surfaces, we must first understand the aberration effect they induce. A foundation that describes how the fields evolve when symmetry no longer exists is nodal aberration theory (NAT), discovered by Roland Shack2 and developed by Kevin Thompson.3 However, NAT has been limited to optical imaging systems made of rotationally symmetric components that are tilted and/or decentered. Our research presents a path to extend NAT to describe the aberration fields of freeform surfaces. The behavior of an aberration field in a '-polynomial surface is tied to its axial position relative to the aperture stop (or pupil) of the optical system. At the stop surface, the beam footprint is the same for all field points, so all field angles receive the same contribution from the freeform surface. In this case, the net aberration is field constant.4 Our work considers the most general case, where the freeform component may be anywhere in the optical system. For such a surface away from the stop, the beam Figure 1. A representative freeform optical surface. This particular shape is classified as a Zernike trefoil overlay, often encountered in the mounting of mirrors in large telescopes. P and V denote where the surface error is a peak or a valley, respectively. : Wavelength.