The Oscillator Representation of the Metaplectic Group Applied to Quantum Electronics and Computerized Tomography

Abstract

Starting from two basic postulates which are motivated by the radial symmetry of the transverse energy patterns of graded-index optical fibers and the quantum mechanical treatment of the laser oscillations, this paper deals with a unified approach to the quantized transverse eigenmode spectrum of circularly symmetric and rectangular multimode parabolic-index optical waveguides. The approach is based on the topologically irreducible, continuous, unitary, linear representations of the diamond solvable Lie group D(R) which forms an extension of the real Heisenberg two-step nilpotent Lie group A(R) by the one-dimensional compact torus group T. As an application, the coupling coefficients of transverse eigenmodes in coaxial circular and rectangular laser resonators and optical waveguides are computed in terms of Krawtchouk polynomials via the smallest real three-step nilpotent Lie group B(R). Finally, as an application to computerized axial tomography, an extension of the group theoretical method is indicated to establish a singular value decomposition for the classical Radon transform ℜ in n-dimensional Euclidean space R n (n ≥ 2) via the discrete spectrum of the reductive dual pair (O(n,R), Sp(l, R)). As a consequence, the symbolic calculus furnishes a new group theoretical proof of the inversion formula for ℜ and the dual inversion formula for the back-projector tℜ