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Journal of the Optical Society of America A

Fourier-optics approach to the symplectic group

Mark H. Kauderer

Published Feb 1, 1990• 7 citations
Publisher

Abstract

The n-dimensional symplectic group is discussed and classified in terms of operators for the first time to my knowledge. The symplectic embedding theorem is proved by using Kronecker block multiplication. In addition, the fundamental relation of Fourier optics is proved. It follows that all operator relations have been proved for Fourier optics. In combination with a previous result, the decomposition theorem for nonsingular matrices [ J. Math. Phys.12, 1772 ( 1971)], which is extended here somewhat, it follows that Fourier optics (excluding the phase-conjugate operator) is isomorphic to the symplectic group, the group of linear canonical transforms, and the group of first-order systems. This implies that every mathematical result concerning the symplectic group can be understood simply as a property of ideal cylindrical lenses under the paraxial approximation.

DOI
10.1364/josaa.7.000231

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The n-dimensional symplectic group is discussed and classified in terms of operators for the first time to my knowledge. The symplectic embedding theorem is proved by using Kronecker block multiplication. In addition, the fundamental relation of Fourier optics is proved. It follows that all operator relations have been proved for Fourier optics. In combination with a previous result, the decomposition theorem for nonsingular matrices [ J. Math. Phys.12, 1772 ( 1971)], which is extended here somewhat, it follows that Fourier optics (excluding the phase-conjugate operator) is isomorphic to the symplectic group, the group of linear canonical transforms, and the group of first-order systems. This implies that every mathematical result concerning the symplectic group can be understood simply as a property of ideal cylindrical lenses under the paraxial approximation.
This thesis concerns a general theory of beam propa-gation in optics.Well-known basic ideas are critically reviewed, and the analogy between quantum mechanics and paraxial optics is established through a number of n~w refinements.A parabolic equation for the electrodynamics wave relation is derived using the relativistic front-description of paraxial optics.The •packet-like beam solutions to this equation are shown to be hermite functions of complex argument.1 I The role and applications of canonical transformations in paraxial optics is investigated, attention is focused on some extensions of linear transformations mapping the position and momentum operators "q, '~ to a pair sands of canonically conjugate but not necessarily Hermitian operators.It is shown that for a linear transformation a related space 'jt. of entire functions exists with an inner product over the complex plane, such that the pair s, scan be realized in the Schrbdinger representation.A unitary mapping onto the ordinary Hilbert space of square-integrable functions over :l,t through an integral transform is provided.The well known Fresnel transform is a special case when sands are the harmonic oscillator raising and lowering operators, whose eigen-' functions are the Hermite functions of complex argument.With these functions the use of mathematical distributions in Fourier optics is extended.A generalization of the Cauchy problem is presented, which expresses the field in the plane l = o in terms of the field in any plane \ =~1 > o.The solution is obtained through an extensive use of the Hermite expansion of a distribution.While this expansion is not new, the fact that is not made in the context of the Hilbert space it is.The results obtained establish the applicability of special functions in optics, as well as a precise knowledge of the meaning and accuracy of paraxial wave optics.Applications of the theory to homogeneous and inhomogeneous medium are presented in particular in a selfcontained chapter about relativistic self-focussing in a non-linear medium.This work is primarily concerned with describing 1 1 1 an alternative approach to the study of a pair of canonical operators and it provides new insight over the role of symplectic geometry in quantum mechanics and optics.Chapter I canon i ea 1 tr ans for•ms.Const rue t ion an<1 app l i ea ti ons.< 19) .
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.
Optics has served as the expermental basis and model for much of science since Galielo. The minds at work in mechanics were so too in optics, in its early geometric aspects through Snell up to Newton, and its wave aspects since Huygens through Fourier and including Sommerfeld. Hamilton’s original application of generating functions for canonical transformations was in optics; only later were they recognized as a sharp tool for abstract mechanics through the introduction of the concept of a phase space with a symplectic metric [ 1 ]. This concept in embrionary form, served Heisenberg [ 2 ] and Weyl [ 3 ] for the introduction of quantum mechanics, where the Hamiltonian is quadratic in momentum. The history and scope of Sophus Lie’s work is well known to this audience, as is the fact that its manifold applications have been most successful when applied to systems with quadratic Hamiltonians [ 4 ]. Almost invariably, the kinetic energy term is p 2 /2m. This Hamiltonian is indeed also useful in optics, albeit in its Gaussian approximation, i.e., first-order optics which deals with linear transformations of phase space. The associated methods have been under research since the early seventies from the perspective of quantum mechanics [ 5 ] and groups of integral transforms [ 6 , 7 ].
We consider pseudounitary quantum systems and discuss various properties of pseudounitary operators. In particular we prove a characterization theorem for block-diagonalizable pseudounitary operators with finite-dimensional diagonal blocks. Furthermore, we show that every pseudounitary matrix is the exponential of i=−1 times a pseudo-Hermitian matrix, and determine the structure of the Lie groups consisting of pseudounitary matrices. In particular, we present a thorough treatment of 2×2 pseudounitary matrices and discuss an example of a quantum system with a 2×2 pseudounitary dynamical group. As other applications of our general results we give a proof of the spectral theorem for symplectic transformations of classical mechanics, demonstrate the coincidence of the symplectic group Sp(2n) with the real subgroup of a matrix group that is isomorphic to the pseudounitary group U(n,n), and elaborate on an approach to second quantization that makes use of the underlying pseudounitary dynamical groups.
Based on the eigenvalues of the real symplectic ABCD-matrix that characterizes the linear canonical integral transformation, a classification of this transformation and the associated ABCD-system is proposed and some nuclei (i.e. elementary members) in each class are described. In the one-dimensional case, possible optical nuclei are the magnifier, the lens, and the fractional Fourier transformer; in the two-dimensional case, we have - in addition to the obvious concatenations of one-dimensional nuclei - the four combinations of a magnifier or a lens with a rotator or a shearing operator, where the rotator and the shearer are obviously inherently two-dimensional. Any ABCD-system belongs to one of the classes described in this paper and is similar (in the sense of similarity of the respective symplectic matrices) to the corresponding nucleus.