General constrained lens or reflector equations with relation to the Abbe sine condition

Various constrained lenses have been used since the 1940s for beam scanning applications or as an alternative to conventional lenses. The optical path within such lenses is generally constrained to follow a waveguide or other transmission line. A famous example is the Rotman lens, which in 2D space transforms waves from three point foci into three corresponding plane waves. For a given focal length, a reflector antenna can be designed that can have a significantly larger aperture than is possible with a spherical Abbe surface. The resultant optical aberration is as good as or better than that for the corresponding Abbe system for scan angles out to several beamwidths.