  The recent history of optical design saw a progressive trend of also
designing without rotational symmetry, especially spectrometers due to the use
of reflective and diffractive elements in their designs. A freeform
hyperspectral imager design in CubeSat format is presented in this work, which
has large deviation from rotational symmetry. Based on a generalization of
paraxial optics and wavefront aberration expansion applicable to
plane-symmetric systems, in this dissertation we derived the aberration
coefficients of aberration types in the third group for plane-symmetric systems
that include the second order effects of the light beam footprint on optical
surfaces. We also expanded the theory to include the contributions from
freeform surfaces and induced aberrations. For the application to
plane-symmetric spectrometers, the aberration coefficients related to
distortion are of special interest. Comparisons between the distortion results
predicted by the aberration coefficients and simulated from real raytracing
show good consistency in example systems of a Dyson spectrometer and freeform
spectrometers that share the three-mirror double-pass structure, which is
similar to the spectrometer component of the hyperspectral imager we designed.
In comparison to real raytracing, results of the theory applied to the
spectrometer of the hyperspectral imager we designed show the limits of the
theory that is limited to the third group of aberrations, while in this design,
high-order groups also significantly contribute. In all cases, the analytical
formulae of the aberration coefficients contain the information on surface
contributions, induced aberrations, and the relation between the aberration
behavior and system parameters, which provide valuable insights in the analysis
of plane-symmetric systems.
