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 su(2), 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 U(N ). We factor this group into SU(2)-linear and nonlinear unitary transformations of phase space and classify all its N 2 − 4 aberrations. This offers a new parametrization of U(N ) based on a chosen SU(2) subgroup.

Group Theory in Finite Hamiltonian Systems

“…Its expectation value in a state yields a quasiprobability distribution that is the SU(2)-covariant Wigner function. We can thus see the music sheet of finite signals on a sphere (best projected on a rectangle), where we can classify the N 2 unitary U(N ) transformations that linearly map and aberrate signals [21], in correspondence with geometric optical models. In the contraction limit to the HW-Wigner function, unitarity somehow becomes canonicity.…”

The harmonic oscillator behind all aberrations

“…Moreover, when we recall the development from classical to quantum mechanical understanding, it may be that geometric optics can be sucessfully deformed into a model for nano-optical devices, to process light signals produced by a finite number of phase-controlled light-emiting dots, and registered by the same number of sensors. In Section 5 we condense some considerations on our recent work in that direction [8,9] and review the aims of the Óptica Matemática project, saluting the memory of Marcos Moshinsky for baring before us the symmetries of the harmonic oscillator.…”

Royal road from geometric to discrete optics

“…Since there are no more than N 2 independent self-adjoint complex matrices that exponentiate to the group U(N 2 ), no more than N 2 -4 aberrations exist in the discrete model (removing the 3 linear transformations and the overall phase). The correspondence between the two models has been defined through the matrices {P r Q s } Weyl , using the Weyl ordering of the factors (which preserves self-adjointness) -but now we also have a second set of aberration generators {P r Q s H} Weyl that include a harmonic oscillator-like factor [8]. We can check by counting that these exhaust the Lie algebra u(N 2 ).…”

“…It serves to visualize the action of aberrations on signals [8], and to identify translated coherent states with their rotated discrete counterparts [10]. Here also, when su(2) contracts to the Heisenberg-Weyl algebra, the radius of the sphere grows and the Wigner function becomes the standard one on the phase plane.…”

“…With these tools, in Ref. [8] we pictured the linear transformations and aberrations acting on a "rectangle" signal of 0's and 1's. For 2D it does not seem convenient to interpret a pixelated image on a 4-sphere within an ambient 6D space; although the direct product nature of su x (2)×su y (2) can be exploited, we have not yet studied this possibility.…”