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Journal of Physics A Mathematical and Theoretical

Linear transformations and aberrations in continuous and finite systems

Kurt Bernardo Wolf

Published Jul 15, 2008• 10 citations
Publisher

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 N 2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.

DOI
10.1088/1751-8113/41/30/304026

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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 N 2 − 4 aberrations. This offers a new parametrization of based on a chosen subgroup.
In finite Hamiltonian systems, the operators of position, momentum, and energy have a finite number N of eigenvalues. These operators can be naturally realized as generators of the Lie algebra su(2), in a representation of spin j, of dimension N = 2j+1. Time evolution is rotation of a phase space sphere, whose projections perform the harmonic motion of an oscillator. The (centrally extended) group of rigid—linear—motions of this phase space is then U(2). On the other hand, N‐point wavefunctions—signals—can be subject to a U(N) group of unitary matrices, containing the linear U(2); aberrations are transformations outside that subgroup. As in geometric optics, we classify the aberration multiplets by order and weight. Their action on phase space is displayed by means of a Wigner function on the sphere, to be compared with the corresponding geometric canonical transformations.
Finite Hamiltonian systems contain operators of position, momentum, and energy, having a finite number N of equally-spaced eigenvalues. Such systems are under the æis of the algebra su(2), and their phase space is a sphere. Rigid motions of this phase space form the group SU(2); overall phases complete this to U(2). But since N-point states can be subject to U(N) ⊃U(2) transformations, the rest of the generators will provide all N 2 unitary transformations of the states, which appear as nonlinear transformations—aberrations—of the system phase space. They are built through the “finite quantization” of a classical optical system.
In this essay we finitely quantize the Hamiltonian system of geometric optics to a finite system that is also Hamiltonian, but where signals are described by complex N-vectors, which are subject to unitary transformations that form the group U(N). This group can be decomposed into U(2)-paraxial and aberration transformations. Proper irreducible representation bases are thus provided by quantum angular momentum theory. For one-dimensional systems we have waveguide models. For two-dimensional systems we can have Cartesian or polar sensor arrays, where digital images are subject to unitary rotation, gyration or asymmetric Fourier transformations, as well as a unitary map between the two arrays.
Squeezing and its inverse magnification form a one-parameter group of linear canonical transformations of continuous signals in paraxial optics. We search for corresponding unitary matrices to apply on signal vectors in N-point finite Hamiltonian systems. The analysis is extended to the phase space representation by means of Wigner quasi-probability distribution functions on the discrete torus and on the sphere. Together with two previous studies of the fractional Fourier and Fresnel transforms, we complete the finite counterparts of the group of linear canonical transformations.
The group of symplectic linear canonical transformations has an integral kernel which has quadratic and linear phases, and which is realized by the geometric paraxial optical model. The discrete counterpart of this model is a finite Hamiltonian system that acts on N -point signals through N × N matrices whose elements also have a constant absolute value, although they do not form a representation of that group. Those matrices that are also unitary are Hadamard matrices. We investigate the manifolds of these N × N matrices under the equivalence imposed by the model, and find them to be on two-sided cosets. By means of an algorithm we determine representatives that lead to collections of mutually unbiased bases.
Both geometric and wave optical models, as well as classical and quantum mechanics, realize linear transformations with matrices; for plane optics, these are 2×2 and of unit determinant. Students and some researchers could assume that the structure of this matrix group is fairly evident and hardly interesting. However, the properties and applications even of this lowest 2×2 case are already unexpectedly rich. While in mechanics they cover classical angular momentum, quantum spin, and represent ‘2+1’ relativity, in optical models they lead from the geometrical description of light propagation in the paraxial regime to wave optics via linear canonical transforms requiring a more penetrating view of their manifold structure and multiple covers. The purpose of this review article is to highlight the topological space of 2×2 matrices as it applies to classical versus quantum and wave models, to underline how the latter requires the double cover of the former, thus using 2×2 matrices as an alternative viewpoint of the quantization process, beside the traditional characterization by commutation and non-commutations of position and momentum.