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Soviet Journal of Quantum Electronics

Coherent states and light propagation in inhomogeneous media

Sergei G. Krivoshlykov, I N Sisakyan

Published Mar 31, 1980• 12 citations

Abstract

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.

DOI: 10.1070/qe1980v010n03abeh009979

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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.

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.

We study the propagation of quantum states of light in separable longitudinally inhomogeneous waveguides based on the momentum operator. This is carried out in the optical field-strength space E by means of the optical propagator obtained by the path integral formalism. We analyze virtual squeezing appearing in these media and its effect on quantum states of light, that is, Gouy’s quantum phases depending on both the longitudinal inhomogeneity and the quantum state. In addition, it is justified that these media perform unitary Infeld–Plebanski transformations, thus avoiding real squeezing. Likewise, WKB and exact results are presented for the propagation of coherent and squeezed states of light and, additionally, Bessel–Gauss quantum states of light propagating in a power-law parametric family of longitudinally inhomogeneous waveguides are studied in detail.

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.

Using quantum theory, we study the propagation of an optical field in an inhomogeneous dielectric, and apply this scheme to traveling optical fields in a waveguide. We introduce a field-atom interaction Hamiltonian and derive the refractive index using quantum optics. We show that the transmission and reflection of optical fields at an interface between different materials can be described with normalized Fresnel coefficients and that this representation is related to the beam splitter operator. We then study the propagation properties of the optical fields for two types of slab waveguides: step-index and graded-index. The waveguides are divided into multiple layers to represent the spatial dependence of the optical field. We can evaluate the number of photons in an arbitrary volume in the waveguide using this procedure. Using the present method, the quantum properties of weak optical fields in a waveguide are revealed, while coherent states with higher amplitudes reduces to representation of classical waveguide optics.

Based on the equation for the second-order coherence function, the propagation of partially coherent radiation is studied under the combined action of diffraction and refraction caused by the inhomogeneous distribution of the real and imaginary parts of the perturbation of the permittivity of a medium in the case of inhomogeneous absorption (amplification). The limits of application of the method of geometrical optics for inhomogeneously absorbing (amplifying) media are studied, as well as of the methods that neglect refraction caused by the inhomogeneous distribution of the imaginary part of the permittivity of the medium.