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Wave Propagation in a 3-D Optical Waveguide

Oleg Alexandrov, Giulio Ciraolo

TL;DR

This work addresses time-harmonic wave propagation in a cylindrically symmetric 3-D optical waveguide by formulating a transform theory that extends a 2-D approach to the 3-D setting. The core challenge is a singular self-adjoint eigenvalue problem arising from separation of variables in cylindrical coordinates, which the authors tackle to obtain a complete Green's-function representation. They develop a spectral framework with fundamental radial solutions, compute spectral measures, and prove a finite, structured decomposition of fields into guided modes plus a radiation part, enabling practical computation for arbitrary core profiles. The results lay a rigorous foundation for analyzing and numerically evaluating fields in open optical fibers and set the stage for explicit treatments of step-index and coaxial geometries in follow-up work.

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.\cite{MS} 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.

Wave Propagation in a 3-D Optical Waveguide

TL;DR

This work addresses time-harmonic wave propagation in a cylindrically symmetric 3-D optical waveguide by formulating a transform theory that extends a 2-D approach to the 3-D setting. The core challenge is a singular self-adjoint eigenvalue problem arising from separation of variables in cylindrical coordinates, which the authors tackle to obtain a complete Green's-function representation. They develop a spectral framework with fundamental radial solutions, compute spectral measures, and prove a finite, structured decomposition of fields into guided modes plus a radiation part, enabling practical computation for arbitrary core profiles. The results lay a rigorous foundation for analyzing and numerically evaluating fields in open optical fibers and set the stage for explicit treatments of step-index and coaxial geometries in follow-up work.

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.\cite{MS} 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.

Paper Structure

This paper contains 9 sections, 15 theorems, 208 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Eigenvalue Problem
  3. A Study of the Solutions of the Eigenvalue Problem
  4. Classification of the Solutions
  5. The Theory of Eigenvalue Problems
  6. Computing the Measures
  7. Computing the Transform
  8. Finding Green's Function
  9. Conclusion

Key Result

Lemma 1

There exists a solution $j_m(r, l)$$(r > 0,\, l\in\mathbb C,\, m\in\mathbb Z)$ of 2.5 such that The functions $j_m(r, l)$ and $j_m'(r, l)$ are analytic in $l$ as $r$ is fixed. There exists another solution $y_m(r,l)$ of 2.5 such that

Figures (3)

  • Figure 1: An optical fiber.
  • Figure 2: The function q(r).
  • Figure 3: The function $\chi_m(\lambda)$.

Theorems & Definitions (15)

  • Lemma 1
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 1
  • Lemma 5
  • Lemma 6
  • Theorem 2
  • Corollary 1
  • Theorem 3
  • ...and 5 more