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Journal of the Optical Society of America A

Finite signals in planar waveguides

Juvenal Rueda-Paz, Kurt Bernardo Wolf

Published Mar 23, 2011• 3 citations
Publisher

Abstract

We examine the evolution in phase space of an N-point signal, produced and sensed at finite arrays transverse to a planar waveguide within the framework of the finite quantization of geometric optics. We use the Kravchuk coherent states provided by the finite oscillator model to evince the nonlinear transformations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.

DOI
10.1364/josaa.28.000641

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We examine the evolution in phase space of an N-point signal, produced and sensed at finite arrays transverse to a planar waveguide within the framework of the finite quantization of geometric optics. We use the Kravchuk coherent states provided by the finite oscillator model to evince the nonlinear transformations that elliptic-profile waveguides produce on phase space by means of the SO(3) Wigner function.
We examine the evolution of an N -point signal produced and sensed at finite arrays of points transverse to a planar waveguide, within the framework of the finite quantization of geometric optics. In contradistinction to the common mechanical Hamiltonians (kinetic plus potential energy terms) the classical waveguide Hamiltonian is the square root of a difference of squares of the refractive index profile minus the optical momentum. The finitely quantized model requires the solution of the square eigenvalue and eigenfunction problem which leads to a step-two difference equation that contains two solutions and two signs of energy. We find the proper linear combinations to fit the Kravchuk functions of the finite oscillator model.
We introduce a model of multimodal waveguides with a finite number of sensor points. This is a finite oscillator whose eigenstates are Kravchuk functions, which are orthonormal on a finite set of points and satisfy a physically important difference equation. The fractional finite Fourier–Kravchuk transform is defined to self-reproduce these functions. The analysis of finite signal processing uses the representations of the ordinary rotation group SO(3). This leads naturally to a phase space for finite optics such that the continuum limit (N→∞) reproduces Fourier paraxial optics.
We construct a bilinear form with the properties of the Wigner distribution function for a model of finite optics: the multimodal linear waveguide. This is a guide that can carry a finite number of oscillator modes, and sends/reads the data by an equal number of sensors. The Wigner distribution function is a function of the classical observables of position and momentum, as well as the mode content; it provides a visual image corresponding to the (`musical') score of the signal. The dynamical group for this model is SU(2) and the wavefunctions span the space of a finite-dimensional irreducible representation of this group. Phase space is a sphere and the linear optical transformations are: translations along the waveguide, refractive wedges and inclined slabs, which correspond to rotations around the 3-, 1-, and 2-axes, respectively. Coherent and Schrodinger cat states are readily identified.
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.
The finite q -oscillator is a model that obeys the dynamics of the harmonic oscillator, with the operators of position, momentum and Hamiltonian being functions of elements of the q -algebra su q (2). The spectrum of position in this discrete system, in a fixed representation j , consists of 2 j + 1 'sensor'-points , and similarly for the momentum observable. The spectrum of energies is finite and equally spaced, so the system supports coherent states. The wavefunctions involve dual q -Kravchuk polynomials, which are solutions to a finite-difference Schrödinger equation. Time evolution (times a phase) defines the fractional Fourier– q -Kravchuk transform. In the classical limit as q → 1 we recover the finite oscillator Lie algebra, the N = 2 j → ∞ limit returns the Macfarlane–Biedenharn q -oscillator and both limits contract the generators to the standard quantum-mechanical harmonic oscillator.
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.