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Soviet Journal of Quantum Electronics

Coherent states and nonparaxial propagation of light in graded-index media

Sergei G. Krivoshlykov, I N Sisakyan

Published Apr 30, 1983• 4 citations
Publisher

Abstract

A study is made of the trajectories and widths of coherent states in nonparaxial propagation of light in a medium with a refractive index which differs slightly from a quadratic profile. Explicit expressions are obtained for the trajectories and widths and it is shown that they generally depend on the wavelength of light. In the limit of wavelengths tending to zero the expressions for the trajectories reduce to the corresponding expressions in ray optics and the ray widths decrease to zero. The coherent states thus make it possible to generalize the concept of a geometric ray and to allow for the wave properties of light in solutions obtained by ray optics methods.

DOI
10.1070/qe1983v013n04abeh004192

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A study is made of the trajectories and widths of coherent states in nonparaxial propagation of light in a medium with a refractive index which differs slightly from a quadratic profile. Explicit expressions are obtained for the trajectories and widths and it is shown that they generally depend on the wavelength of light. In the limit of wavelengths tending to zero the expressions for the trajectories reduce to the corresponding expressions in ray optics and the ray widths decrease to zero. The coherent states thus make it possible to generalize the concept of a geometric ray and to allow for the wave properties of light in solutions obtained by ray optics methods.
Methods of integrals of motion, coherent states, dynamical symmetry group, and density matrix formalism are used to describe the propagation of light in weakly inhomogeneous media, in particular, in a two-dimensional medium having a quadratic transverse distribution of the refractive index and an arbitrarily inhomogeneous longitudinal distribution. Expressions are obtained for the trajectories and widths of the modes and rays. The coefficients of coupling between the modes and rays due to the longitudinal inhomogeneity of the medium and distortion of the axis are discussed. It is shown that, for certain laws of longitudinal inhomogeneity, matching occurs when no energy redistribution takes place between the modes and mismatch is found when all the energy is coupled out of the medium. On the basis of these effects, methods of matched joining of two different graded-index optical waveguides and of coupling light out of a graded-index waveguide are proposed. It is shown that each initial mode of the medium is strongly coupled to three final modes. These results may be used in fiber-optic communication lines.
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.
A small-wavelength, quasi-geometric optics is developed. The approach is based on an amplitude wave function that represents both the probability amplitude of the geometrical rays and the physical-optics disturbance. All ray paths contribute phases to the amplitude function proportional to their optical paths, measured in units of normalized wavelength. However, the dominant contributions are due to those rays that satisfy Fermat’s principle. This formulation leads to an integral equation for the wave-amplitude function with a kernel function that is a path integral. The integral equation reduces, in weakly focusing paraxial media, to a solution of a Schrödinger type of partial differential equation. The classical ray variables are operators with expectation values satisfying the geometric-optics equations. The second moments of the ray operators, with minimum-uncertainty initial wave packet, define second-order optical parameters. A special case of the quasi-geometric-optics integral equation is the quasi-optical laser-mode equation, allowing the interpretation of the stationary modes as probability amplitudes. For media with arbitrary inhomogeneous index of refraction, a matrix wave equation with a two-component wave function is derived. The matrix ray operators also satisfy a correspondence principle. A physical interpretation of the new wave function is presented.
Abstract The present paper concerns a formalism aimed to describe the behavior and trajectories of complex scalar waves with space-varying amplitude in media characterized by variable refractive index. The classical wave equation is transformed into an eikonal-type equation, without imposing the geometrical optics limit, satisfied by an effective refractive index. The effective index combines the refractive properties of the medium with self-refraction phenomena generated by the space dependence of the wave amplitude. It also induces wave packet dispersion, leading to relations between the group and phase velocities that are particularly relevant in the context of de Broglie waves. The equations of motion are provided, and it is shown that the self-refraction term bends the trajectory of the wave and changes its velocity. The dynamics is dictated by the curvature of the trajectory expressed in terms of the gradient of the effective refractive index. The uncovered findings point toward a possible generalization of geometrical optics with relevant implications for its relation to quantum mechanics, general relativity and transformation optics.
The nonparaxial propagation of partially coherent Gaussian Schell-model (GSM) beams in a graded-index waveguide is studied using the coherent states and coherent modes decomposition methods, as well as perturbation theory. Long-scale decoherence and recoherence of the GSM beam occur due to the revival effect caused by mode interference. It is shown that the radius of coherence and the degree of coherence can be determined based on measurements of purely geometric characteristics of the GSM beam. The information entropy and quantum mechanical purity of GSM beams are evaluated.
A simple elegant expression of nonadiabatic light wave evolution is necessary in order to have a deeper insight for complicated optical phenomena in light science as well as in everyday life. Light wave propagation in linear media which have time-dependent electromagnetic parameters is investigated by utilizing a quadratic invariant of the system. The time behavior of the nonadiabatic geometric phase of the waves that yield a cyclic nonadiabatic evolution is analyzed in detail. Various quantum properties of light waves in this situation, such as variances of electric and magnetic fields, uncertainty product, coherent and squeezed states, and their classical limits, are developed. For better understanding of our research, we applied our analysis in a particular case. The variances of the fields D and B are illustrated and their time behaviors are addressed. Equivalent results for the corresponding classical systems are deduced from the study of the time evolution of the appropriate coherent and squeezed states.