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Optical systems that do not have axial symmetry can provide useful and unique solutions to certain imaging problems. However, the complexity of the optical design task grows as the degrees of symmetry are reduced and lost: there are more aberration terms to control, and achieving a sharp image over a wide field-of-view at fast optical speeds becomes challenging. Plane-symmetric optical systems represent a large family of practical non-axially symmetric systems that are simple enough to be easily described and thus are well understood. Design methodologies and aberration theory of plane-symmetric optical systems have been discussed in the literature, and various interesting solutions have been reported [1-4]. The little discussed in the literature technique of confocal systems is effective for the design of unsymmetrical optics. A confocal unsymmetrical system is constructed in such a way that there is sharp image along a given ray (called the optical axis ray (OAR)) surface after surface. It is possible to show that such a system can have a reduced number of field aberrations, and that the system will behave closer to an axially symmetric system [5-6]. In this paper, we review a methodology for the design of unsymmetrical optical systems. We utilize an aspherical/freeform surface constructed by superposition of a conic expressed in a coordinate system that is centered on the off-axis surface segment rather than centered on the axis of symmetry, and an XY polynomial. The conic part of the aspherical/freeform surface describes the base shape that is required to achieve stigmatic imaging surface after surface along the OAR. The XY polynomial adds a more refined shape description to the surface sag and provides effective degrees of freedom for higher-order aberration correction. This aspheric/freeform surface profile is able to best model the ideal reflective surface and to allow one to intelligently approach the optical design. Examples of two- and threemirror unobscured wide field-of-view reflective systems are provided to show how the methods and corresponding aspheric/freeform surface are applied. We also demonstrate how the method can be extended to design a monolithic freeform objective.
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