Kurt Bernardo Wolf
Linear transformations and aberrations in continuous and in finite systems
In geometric optics there is a natural distinction between the paraxial and aberration regimes, which contain respectively the linear and nonlinear canonical transformations of phase space. In a way parallel to but distinct from the Schroedinger quantization of continuous classical systems, we quantize the geometric optical model into discrete, finitedimensional systems based on the Lie algebra su(2). Wavefunctions are Npoint signals, whose phase space is a sphere, and whose transformations are represented by NxN unitary matrices. We factor this group into U(2)linear ("paraxial") rotations of the phase space sphere plus a global phase, while the rest of the N^2 transformations of U(N)are nonlinear "aberrations" of its surface. These N^24 aberrations are classified following the geometric optical model; they are unitary, so no information from the signal is lost. Their visual characterization can be seen nicely on phase space.
