Linear transformations and aberrations in continuous and finite systems

Abstract

In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of position and momentum in the phase space. In the Lie-theoretical presentation, linear inhomogeneous transformations are generated by linear and quadratic functions of the phase space, while aberrations of increasing order are generated by homogeneous functions with higher powers of these coordinates. In a way parallel but distinct from the Schrödinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finite-dimensional systems based on the Lie algebra , whose wavefunctions are N-point signals, phase space is a sphere and transformations are represented by the N × N unitary matrices that form the group . We factor this group into -linear and nonlinear unitary transformations of phase space and classify all its N2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.