Focusing of Waves in Ducts

Abstract

This paper presents an investigation of the propagation of cylindrical waves (the field being independent of x) along the z direction inducts described by where c(y, z) is the wave velocity, C0 and ϵ0 are constants, ϵ2(z) is an arbitrary function of z, and ϵ corresponds to the dielectric constant for electromagnetic waves. It has been shown that the intensity A02 of the wave is in general given by where E0 is a constant, yo is a constant, F is an arbitrary function of [y/yoƒ(z)L], and ƒ(z) is a dimensionless beam‐width parameter given by in the geometrical optics approximation. The nature of the variations of ƒ with z has been discussed for some simple profiles of ϵz(z) in the geometrical optics approximation; diffraction has also been taken into account when the beam (F) is Gaussian.

Analysis of propagation in a SELFOC fibre: The axial field component and the effect of multiple reflections

“…Equation 4 can be solved by a procedure similar to that adopted for solving Equation 3 by Sodha et al [5,6]. For a Gaussian beam the solution for H, inside the medium comes out to be…”

“…To solve these equations we specify the geometry of the system: take all variations in the x direction as zero, the axis of the beam along the z axis and the dielectric constant as = Eo -E2y~; E2y ~ < Eo (2) V~E --0 c 2 0t 2 and its solutions have been obtained in earlier investigations [1][2][3][4][5][6]. For the second case of polarization in the yz plane, since Ez is not independent of Ey it is easier to solve for the magnetic vector, which in the present case has only the x component finite (see Equation 1.)…”

Propagation of a Gaussian Beam in Planar and Cylindrically Inhomogeneous Media: Comparison of Modal Analysis and Approximate Approaches

“…Considering the electric field to have only x-components and assuming time dependence of the form exp (-icut) and the dielectric constant given by e = Ep -Ely 2-E3x 2, one obtains aEX + as 2 + aaE2 + k2 C1-E2 Y2-E3 x21 E~=0, we can expand equation (8) in a binomial series and E x can be written as a product of two separate infinite series We first consider the propagation of cylindrical electromagnetic waves through a planar duct described by equation (1) . We assume the fields to be independent of x and that the incident plane wave has electric field distribution given by…”

“…In both the cases, the WKB approximation was employed and it was shown that, corresponding to equations (1) and (2) the intensity distribution at any point z would be given by If F ( a ) andrespectively, where f(z) was to be deduced from an ordinary second-order differential equation . The form of F is to be determined from the intensity distribution atz=O .…”

“…Sodha et al [1] have recently presented an investigation of the propagation of cylindrical waves, having an intensity distribution along their wavefront, in a planar duct, described by E = ED -E2 (z)y 2, (1 where e is the dielectric constant and the wave is assumed to propagate along the z axis . In another paper Sodha et al [2] have considered the electromagnetic wave propagation in radially and axially non-uniform media in which the dielectric constant is given by…”