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Physical Review A

Metaplectic group and Fourier optics

H. Bacry, Michel Cadilhac

Published May 1, 1981• 88 citations
Publisher

Abstract

The evolution of a light wave along an optical system, in paraxial approximation, can be described with the aid of a group of operators acting as canonical transformations in the framework of nonrelativistic quantum mechanics in two dimensions. This group is isomorphic to the ten-dimensional metaplectic group Mp(2). The homomorphism Mp(2)\ensuremath{\rightarrow}Sp(2) (symplectic group) gives a description of the geometrical-optics approximation.

DOI
10.1103/physreva.23.2533

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Physical Review A•Nov 1, 1983•52 citations
The evolution of a light wave along an optical system, in paraxial approximation, can be described with the aid of a group of operators acting as canonical transformations in the framework of nonrelativistic quantum mechanics in two dimensions. This group is isomorphic to the ten-dimensional metaplectic group Mp(2). The homomorphism Mp(2)\ensuremath{\rightarrow}Sp(2) (symplectic group) gives a description of the geometrical-optics approximation.
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.
The symplectic group Sp(n) acts on phase space while the unitary representation of its double cover, Mp(n), the metaplectic group, acts on functions defined on configuration space. We will construct an extension Mp(n) of Mp(n) acting on square integrable functions on phase space. This is performed using previous results of ours involving explicit expressions of the twisted Weyl symbols of metaplectic operators and Bopp pseudodifferential operators, which are phase space extensions of the usual Weyl operators.
The scalar wave equation is analyzed in the relativistic front form, appropriate for paraxial-wave optics. The group-theoretical basis for this treatment is uncovered. The formal similarity of the propagation of paraxial beams through optical systems to the quantum mechanics of particles in two dimensions subject to harmonic impulses, and the role of the metaplectic group of Bacry and Cadilhac, are both traced back to the structure of the Poincar\'e group. Light rays are defined in this context as in statistical-wave optics, and the laws for their free propagation as well as transmission through lenses are derived.
The Heisenberg–Weyl ring contains the metaplectic group of canonical transforms acting unitarily on ℒ 2(ℛ). These ring elements are characterized through (i) the integral transform kernels, (ii) coset distributions, and (iii) classical functions under any quantization scheme. The isomorphism under group composition leads to several new relations involving twisted products and quantization of Gaussian classical functions. The Wigner inversion operator is a special central group element. It is shown that the only quantization scheme invariant under metaplectic transformations is the Weyl scheme. The structure studied here appears to be relevant to the study of wave optics with aberration.
With the extension of the work of the preceding paper, the relativistic front form for Maxwell's equations for electromagnetism is developed and shown to be particularly suited to the description of paraxial waves. The generators of the Poincar\'e group in a form applicable directly to the electric and magnetic field vectors are derived. It is shown that the effect of a thin lens on a paraxial electromagnetic wave is given by a six-dimensional transformation matrix, constructed out of certain special generators of the Poincar\'e group. The method of construction guarantees that the free propagation of such waves as well as their transmission through ideal optical systems can be described in terms of the metaplectic group, exactly as found for scalar waves by Bacry and Cadilhac. An alternative formulation in terms of a vector potential is also constructed. It is chosen in a gauge suggested by the front form and by the requirement that the lens transformation matrix act locally in space. Pencils of light with accompanying polarization are defined for statistical states in terms of the two-point correlation function of the vector potential. Their propagation and transmission through lenses are briefly considered in the paraxial limit. This paper extends Fourier optics and completes it by formulating it for the Maxwell field. We stress that the derivations depend explicitly on the "henochromatic" idealization as well as the identification of the ideal lens with a quadratic phase shift and are heuristic to this extent.