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Acta Optica Sinica

Exact analytic solutions of high-order Gaussian beam modes in media with quadratic radial refractive index and gain profiles

R Zhang

Published Jan 1, 1986• 0 citations
OpenAlex

Abstract

The exact analytic and steady-state solutions are derived for high-order Gaussian beam propagation modes in media with quadratic radial refractive index and gain (loss) profiles based on the slowly varying envelope approximation (SVEA). Explicit expressions of the solutions are given for some useful specific cases. Possible applications of the present theory to optical cavities and waveguides with such media and to problems of optical bistability in cavities having media with quadratic radial index and absorption profile are discussed.

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The exact analytic and steady-state solutions are derived for high-order Gaussian beam propagation modes in media with quadratic radial refractive index and gain (loss) profiles based on the slowly varying envelope approximation (SVEA). Explicit expressions of the solutions are given for some useful specific cases. Possible applications of the present theory to optical cavities and waveguides with such media and to problems of optical bistability in cavities having media with quadratic radial index and absorption profile are discussed.
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.
In a recent paper1 I formulated a method for calculating propagation through nonlinear media for a number of nonlinear optical phenomena. This method does not use the SVEA and can therefore solve nonlinear propagation problems when refraction and reflection at the boundary interfaces are important. Here, I report the first calculation using this exact (albeit numerical) method. For this first test of the utility of the method, I have chosen to study optical bistability in a thin film whose index of refraction is intensity dependent.
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 present a theory of optical bistability for a Fabry-Perot cavity containing a medium with nonlinear refraction but linear absorption with only plane-wave and slowly varying envelope approximations. An analytic expression for the critical intensity Icfor the onset of bistability is derived and used to compare cavity designs. It is shown that 1) the important material parameter for minimum Icis the ration_{2}/\alpha 2) the limit to Icis set by limitations on finesse due to inhomogeneities rather than the absorption itself, and 3) the cavity design which gives lowest Icfor a given finesse is that for which the mirror transmissivity equals the absorption per pass; at high finesse this design leads to a total peak cavity transmission of 1/4 when mirror reflectivities are equal.
We derive the propagation equation for the slowly varying envelope (SVE) of the electric field in a homogeneous dispersive nonisotropic medium. Terms in the propagation equation of the field envelope that originate because of finite beam size are obtained and explained, including generalized Fresnel diffraction terms and a term that rotates the SVE wave packet in a uniaxial crystal about an axis perpendicular to the optical plane. Numerical estimates of these terms in the propagation of short pulses in the uniaxial crystal β-barium borate are presented. We obtain an analytic expression for the SVE for an initial Gaussian wave packet by keeping terms up to second order in the wave equation for the SVE.
Resonant absorption in a laser pulse can be reduced to zero if the trailing part induces stimulated emission and returns the absorbed energy back to the pulse. This kind of effect is called Self-Induced Transparency (SIT). Analytic descriptions of short pulse propagation in a resonant medium, based on Slowly Varying Envelope Approximation (SVEA) and in linear media, have been extended in studies of SIT along two principal directions: First, to improve the SVEA by introducing Short Pulse Approximation (SPA); and second, to consider the effect of nonlinear refractive index.