XXVII Workshop on Geometric Methods in Physics 29.06-05.07.2008

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, finite-dimensional systems based on the Lie algebra su(2). Wavefunctions are N-point 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^2-4 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.