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European Journal of Physics

The harmonic oscillator problem and the parabolic index optical waveguide: I. Classical and ray optic analysis

A. K. Ghatak, Elmar Sauter

Published Apr 1, 1989• 11 citations
Publisher

Abstract

The authors discuss the harmonic oscillator problem in the domains of classical mechanics and the parabolic index optical waveguide problem in the domains of ray optics. While discussing the classical mechanics and the ray optics of the two problems the authors have used the Lagrangian formalism which allows them to see the close relationship between the two problems. In part II of the paper the authors study the corresponding quantum mechanical and wave optical problems. In particular whilst looking at the time evolution of a coherent state the authors demonstrate that it has a close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide.

DOI
10.1088/0143-0807/10/2/012

Related papers

The harmonic oscillator problem and the parabolic index optical waveguide: II. Quantum mechanical and wave optical analyses

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For pt.I see ibid., vol.10, p.136-43 (1989). In the preceding paper the authors used the Lagrangian formalism to show the close relationship between the classical mechanics of the harmonic oscillator and the ray optics of the parabolic index optical waveguide. The authors have now studied the corresponding quantum mechanical and wave optical problems and, in particular, have studied the time evolution of a coherent state and have shown its close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide. The corresponding transitions to classical mechanics and ray optics have been shown. The authors have also presented an analysis of the recently developed squeezed states and have shown its correspondence to the waveguide problem. They have demonstrated that the relationship between classical and quantum mechanics is the same as between ray optics and wave optics. All such analyses should be of considerable pedagogical value.

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The authors discuss the harmonic oscillator problem in the domains of classical mechanics and the parabolic index optical waveguide problem in the domains of ray optics. While discussing the classical mechanics and the ray optics of the two problems the authors have used the Lagrangian formalism which allows them to see the close relationship between the two problems. In part II of the paper the authors study the corresponding quantum mechanical and wave optical problems. In particular whilst looking at the time evolution of a coherent state the authors demonstrate that it has a close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide.
For pt.I see ibid., vol.10, p.136-43 (1989). In the preceding paper the authors used the Lagrangian formalism to show the close relationship between the classical mechanics of the harmonic oscillator and the ray optics of the parabolic index optical waveguide. The authors have now studied the corresponding quantum mechanical and wave optical problems and, in particular, have studied the time evolution of a coherent state and have shown its close relationship to the propagation of a Gaussian beam in a parabolic index optical waveguide. The corresponding transitions to classical mechanics and ray optics have been shown. The authors have also presented an analysis of the recently developed squeezed states and have shown its correspondence to the waveguide problem. They have demonstrated that the relationship between classical and quantum mechanics is the same as between ray optics and wave optics. All such analyses should be of considerable pedagogical value.
It has been found a solution of the equation for classical nonstationary harmonic oscillator, with time dependent frequency in integral form. It has been shown, that the quantum optical problem for propagation of electromagnetic waves in nonlinear waveguide is related to the solutions of classical nonstationary harmonic oscillator. This method could be used for obtaining other solutions of the classical harmonic oscillator, with other time dependent frequency, if we have already determined the fluctuations σq, σp and covariance cqp by some different methods. There have been found the substitutions, which allow to establish a connection between the general method of linear invariants [Malkin et.al., Phys Rev.2D (1970) 1371] and some particular cases in later papers. This gives the opportunity to present the problem with equation of two- dimensional nonstationary harmonic oscillator, which is physically more informative. 1
This contribution has two main purposes. First, using classical optics we show how to model two coupled quantum harmonic oscillators and two interacting quantized fields. Second, we present classical analogs of coupled harmonic oscillators that correspond to anisotropic quadratic graded indexed media in a rotated reference frame, and we use operator techniques, common to quantum mechanics, to solve the propagation of light through a particular type of graded indexed medium. Additionally, we show that the system presents phase transitions.
The choice of the representation (the complete set of functions) is very important for describing a field in many problems of wave propagation. A representation proved to be convenient here depends on excitation conditions and waveguide properties. It was shown [ 1 ] that the coherent states (CS) representation and the integrals of motion method turn to be extremely convenient for investigating Gaussian beam propagation in quadratic-index media. In [ 2 ] the correlated coherent states (CCS) were used to describe arbitrary Gaussian beam propagations in such media. The coherent state representation turns to be also convenient for description of nonparaxial propagation of both coherent and partially coherent radiation in the gradient medium [ 3 , 4 ]. Here we show that CCS turn to be also more convenient representation for investigating wave propagation in optic active quadratic media. CCS in such media describe a Gaussian wave packets with constant parameters during beam propagation. These states permit the concept of geometric ray width and wave front curvature to be introduced by natural manner and the relationship between wave and ray optics description to be clearly visualized for quadratic media with loss or gain.