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International Symposium on Signal Processing and Its Applications (ISSPA)

Classification of the linear canonical transformation and its associated real symplectic matrix

Martin J. Bastiaans, Tatiana Alieva

Published Feb 1, 2007• 1 citation
Publisher

Abstract

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.

DOI
10.1109/isspa.2007.4555352

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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.
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 is concerned with the properties of dynamical systems with Hamiltonians which are general quadratic combinations of classical or quantum canonical variables or of Boson or Fermion annihilation and creation operators.Chapter I, consisting of six sections, deals mainly with sets of canonical forms for quadratic Hamiltonians under the action of the group of linear canonical transformations.By viewing the group of C.C.R.Bogoliubov transformations as Sp(2tf, i?) , a full set of canonical Boson Hamiltonians, labelled by the invariants for symplectic conjugacy classes, is obtained.Since the conjugacy class of a real orthogonal matrix is determined by its eigenvalues, there is only one type of canonical form for quadratic Fermion Hamiltonians.Unlike the Fermion case, indefinite Boson Hamiltonians can not in general be reduced to a sum of quasi-particle number operators.However, it is shown how spectral theory in Krein space rigorizes the full reduction of a strictly positive quadratic Boson Hamiltonian for which the single particle space is infinite dimensional.In the case of infinite degrees of freedom, there are various existence criteria for a unitary operator which implements a given Bogoliubov transformation and the equivalence of these is discussed in Section 6.Cahpter II, consisting of seven sections, applies the techniques of Chapter I to algebraic quantization.The Boson commutation relations , among formal mode operators obtained in heuristic quantization, are shown to be incompatible with the canonical commutation relations among conjugate variables unless the motion is stable.Segal quantization of any real
Enlightened by the algorithmic isomorphism between the rotation of the Wigner distribution function (WDF) and the αth fractional Fourier transform, we show that the optical Fresnel transform performed on the input through an ABCD system makes the output naturally adapting to the associated Wigner transform, i.e. there exists algorithmic isomorphism between ABCD transformation of the WDF and the optical Fresnel transform. We prove this adaption in the context of operator language. Both the single-mode and the two-mode Fresnel operators as the image of classical Fresnel transform are introduced in our discussions, while the two-mode Wigner operator in the entangled state representation is introduced for fitting the two-mode Fresnel operator.
Abstract The main purpose of this work is to identify the general quadratic operator which is invariant under the action of Linear Canonical Transformations (LCTs). LCTs are known in signal processing and optics as the transformations which generalize certain useful integral transforms. In quantum theory, they can be identified as the linear transformations which keep invariant the canonical commutation relations characterizing the coordinates and momenta operators. In this paper, LCTs corresponding to a general pseudo-Euclidian space are considered. Explicit calculations are performed for the monodimensional case to identify the corresponding LCT invariant operator then multidimensional generalizations of the obtained results are deduced. It was noticed that the introduction of a variance-covariance matrix, of coordinate and momenta operators, and a pseudo-orthogonal representation of LCTs facilitate the identification of the invariant quadratic operator. According to the calculations carried out, the LCT invariant operator is a second order polynomial of the coordinates and momenta operators. The coefficients of this polynomial depend on the mean values and the statistical variances-covariances of these coordinates and momenta operators themselves. The eigenstates of the LCT invariant operator are also identified with it and some of the main possible applications of the obtained results are discussed.
We show that the group of linear canonical transformations in a 2N-dimensional phase space is the real symplectic group Sp(2N), and discuss its unitary representation in quantum mechanics when the N coordinates are diagonal. We show that this Sp(2N) group is the well-known dynamical group of the N-dimensional harmonic oscillator. Finally, we study the case of n particles in a q-dimensional oscillator potential, for which N = nq, and discuss the chain of groups Sp(2nq)⊃Sp(2n)×O(q). An application to the calculation of matrix elements is given in a following paper.
Corresponding to the Fresnel transform there exists a unitary operator in quantum optics theory, which could be known the Fresnel operator (FO). We show that the multiplication rule of the FO naturally leads to the quantum optical ABCD law. The canonical operator methods as mapping of ray-transfer ABCD matrix is explicitly shown by the normally ordered expansion of the FO through the coherent state representation and the technique of integration within an ordered product of operators. We show that time evolution of the damping oscillator embodies the quantum optical ABCD law.