Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Solving the Scattering Problem for Open Wave-Guide Networks, I Fundamental Solutions and Integral Equations

Charles L. Epstein

TL;DR

This work develops a layer-potential framework for the scalar, time-harmonic scattering problem of two open dielectric wave-guide channels in the plane. By constructing outgoing Green’s functions via Fourier analysis in the axial direction and decomposing the perturbed Green’s function into continuous-spectrum and guided-mode parts, the authors reduce the transmission problem to a system of Fredholm integral equations on the axis $\{x_1=0\}$. They establish precise boundary-kernel estimates and show the integral equations are Fredholm of index zero on Banach spaces with decay, laying groundwork for guaranteed solvability (subject to a trivial null-space) and explicit representations. The approach yields explicit projections onto wave-guide modes and provides a foundation for accurate numerical methods, with Part II addressing radiation asymptotics and Part III establishing radiation conditions and uniqueness in general settings.

Abstract

We introduce a layer potential representation for the solution of the transmission problem defined by two dielectric channels, or open wave-guides, meeting along the straight-line interface, $\{x_1=0\}.$ The main observation is that the outgoing fundamental solution for the operator $Δ+k_1^2+q(x_2),$ acting on functions defined in ${\mathbb R}^2,$ is easily constructed using the Fourier transform in the $x_1$-variable and the elementary theory of ordinary differential equations. These fundamental solutions can then be used to represent the solution to the transmission problem in half planes. The transmission boundary conditions lead to integral equations along the intersection of the half planes, which, in our normalization, is the $x_2$-axis. We show that, in appropriate Banach spaces, these integral equations are Fredholm equations of second kind, which are therefore generically solvable. We analyze the representation of the guided modes in our formulation.

Solving the Scattering Problem for Open Wave-Guide Networks, I Fundamental Solutions and Integral Equations

TL;DR

This work develops a layer-potential framework for the scalar, time-harmonic scattering problem of two open dielectric wave-guide channels in the plane. By constructing outgoing Green’s functions via Fourier analysis in the axial direction and decomposing the perturbed Green’s function into continuous-spectrum and guided-mode parts, the authors reduce the transmission problem to a system of Fredholm integral equations on the axis . They establish precise boundary-kernel estimates and show the integral equations are Fredholm of index zero on Banach spaces with decay, laying groundwork for guaranteed solvability (subject to a trivial null-space) and explicit representations. The approach yields explicit projections onto wave-guide modes and provides a foundation for accurate numerical methods, with Part II addressing radiation asymptotics and Part III establishing radiation conditions and uniqueness in general settings.

Abstract

We introduce a layer potential representation for the solution of the transmission problem defined by two dielectric channels, or open wave-guides, meeting along the straight-line interface, The main observation is that the outgoing fundamental solution for the operator acting on functions defined in is easily constructed using the Fourier transform in the -variable and the elementary theory of ordinary differential equations. These fundamental solutions can then be used to represent the solution to the transmission problem in half planes. The transmission boundary conditions lead to integral equations along the intersection of the half planes, which, in our normalization, is the -axis. We show that, in appropriate Banach spaces, these integral equations are Fredholm equations of second kind, which are therefore generically solvable. We analyze the representation of the guided modes in our formulation.
Paper Structure (18 sections, 14 theorems, 280 equations, 5 figures, 1 table)

This paper contains 18 sections, 14 theorems, 280 equations, 5 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scattering problems for wave-guide networks
  3. The simplest non-trivial case
  4. Material in Parts II and III
  5. A Layer Potential Approach
  6. The Structure of the Perturbed Green's Function
  7. Estimates for the Boundary Kernel
  8. The Integral Equations
  9. Admissible Data
  10. The Projections onto Wave-Guide Modes
  11. Some Concluding Remarks
  12. The Bi-infinite Case
  13. Proof of Theorem \ref{['thm0']}
  14. Proof of Theorem \ref{['thm00']}
  15. Estimates on the Wronskian and Basic Solutions of $L_{\xi}$
  16. ...and 3 more sections

Key Result

Theorem 1

The kernels, $\mathfrak w^{[j]}(x_2,y_2),\, j=0,1,2,$ are infinitely differentiable outside of $B_d=[-d,d]\times [-d,d].$ Within $B_d$ they are singular along the diagonal, where the kernel $\mathfrak w^{[j]}(x_2,y_2)$ has an $|x_2-y_2|^{2-j}\log|x_2-y_2|$-singularity.

Figures (5)

  • Figure 1: Two dielectric channels meeting along a straight interface. The $x_3$-axis is orthogonal to the plane of the image.
  • Figure 2: The contour $\Gamma^+_{\nu}$ shown in blue. The roots of Wronskian $\{\pm\xi_n\}$ are shown as red asterisks.
  • Figure 3: Three dielectric channels meeting in a compact interaction zone, $D,$ showing sectors $S_1, S_2,S_3.$
  • Figure 4: Zeros of the Wronskian with $k_1=16, k_2=18.$
  • Figure 5: The blue contour is $\Gamma^+_{\nu,\epsilon}$ showing the smooth curves replacing intervals $[-\epsilon-k_1,\epsilon-k_1]\cup[-\epsilon+k_1,\epsilon+k_1].$ The roots of the Wronskian are shown as asterisks, and $\pm k_1$ as diamonds.

Theorems & Definitions (35)

  • Remark 1
  • Theorem 1
  • Remark 2
  • Theorem : Theorem 3.2 (b), Chapter 5 of CoddingtonLevinson
  • Definition 1
  • Proposition 1
  • proof
  • Proposition 2
  • Lemma 1
  • proof : Proof of Lemma
  • ...and 25 more