Wave Propagation in a 3-D Optical Waveguide

Abstract

In this paper we study the problem of wave propagation in a 3-D optical fiber. The goal is to obtain a solution for the time-harmonic field caused by a source in a cylindrically symmetric waveguide. The geometry of the problem, corresponding to an open waveguide, makes the problem challenging. To solve it, we construct a transform theory which is a nontrivial generalization of a method for solving a 2-D version of this problem given by Magnanini and Santosa. 3 The extension to 3-D is made complicated by the fact that the resulting eigenvalue problem defining the transform kernel is singular both at the origin and at infinity. The singularities require the investigation of the behavior of the solutions of the eigenvalue problem. Moreover, the derivation of the transform formulas needed to solve the wave propagation problem involves nontrivial calculations.The paper provides a complete description on how to construct the solution to the wave propagation problem in a 3-D optical waveguide with cylindrical symmetry. A follow-up article will study the particular cases of a step-index fiber and of a coaxial waveguide. In those cases we will obtain concrete formulas for the field and numerical examples.

The uniqueness and existence of solutions for the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation

“…In this section, we recall the construction of the Green's function for the 3D homogeneous Helmholtz equation in a step-index waveguide without perturbation, which has been done in [4] and [5]. This Green's function will be used to construct the solution of the perturbed 3D Helmholtz equation (1.1) later.…”

“…We will view (2.4) as an eigenvalue problem in l 2 C and call it the associated eigenvalue problem to (2.1) (see [6] or [7]). From [4] and [5], we can obtain the Green's function G.r, Â , z; r 0 , Â 0 , / of (2.1),…”

“…These conditions are physically motivated, we refer to [4], [5], and references therein for a more detailed description.…”

A spectral approach to a constrained optimization problem for the Helmholtz equation in unbounded domains

“…As far as the authors know, there are not results in this direction for the full Maxwell's equations. A challenging issue in computational studies related to (1) is how to deal numerically with the unbounded domain. Usually, one has to introduce a (bounded) computational domain Ω and then prescribe boundary conditions on ∂Ω which approximate the problem in the whole space.…”

Analytical results for 2‐D non‐rectilinear waveguides based on a Green's function

“…Such a choice of n corresponds to an index of refraction depending only on the transversal coordinate and, thus, (1) describes the electromagnetic wave propagation in a rectilinear open waveguide. By using the approach proposed in [7], the results in [6] have been generalized in [8] to the case in which the index of refraction is not necessarily decreasing along the positive direction. The use of a rigorous transform theory guarantees that the superposition of guided, radiation and evanescent modes is complete.…”