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Modal bases of coaxial electromagnetic step index fibers

Martin Halla

Abstract

We consider the eigenvalue problem to find the modes of an electromagnetic coaxial step index fiber. More specific, we consider a closed (meaning PEC boundary conditions) cylindrical waveguide with circular cross section $Γ$, wave propagation modeled by the time-harmonic Maxwell's equations with frequency $ω$, the permeability $μ$ and the permittivity $ε$ being scalar, uniformly positive, piece-wise constant and depending only on the radial variable of the cross section. We prove that if the deviation from the homogeneous case is small, i.e., $δ_{ε,μ}:=\|ε-ε_0\|_{L^\infty}+\|μ-μ_0\|_{L^\infty}\ll1$, then the tangential electric (magnetic) fields of the modes form a Riesz basis in $\mathbf{H}_{0}(\operatorname{curl}_Γ;Γ)$ ($\mathbf{H}(\operatorname{curl}_Γ;Γ)$). For a constant permeability (permittivity) the Riesz basis property for the tangential electric (magnetic) fields holds also in the natural trace space $\mathbf{H}_{0}^{-1/2}(\operatorname{curl}_Γ;Γ)$ ($\mathbf{H}^{-1/2}(\operatorname{curl}_Γ;Γ)$). These results hold also for complex frequencies $ω$. In addition, if $ω\in\mathbb{R}$, then for small enough $δ_{ε,μ}$ all wavenumbers are located on the axes and there exist no backward modes. Key tools in the analysis are a particular reformulation of the eigenvalue problem, the perturbation theory for selfadjoint operators under a local subordinate condition and uniform properties of Bessel functions.

Modal bases of coaxial electromagnetic step index fibers

Abstract

We consider the eigenvalue problem to find the modes of an electromagnetic coaxial step index fiber. More specific, we consider a closed (meaning PEC boundary conditions) cylindrical waveguide with circular cross section , wave propagation modeled by the time-harmonic Maxwell's equations with frequency , the permeability and the permittivity being scalar, uniformly positive, piece-wise constant and depending only on the radial variable of the cross section. We prove that if the deviation from the homogeneous case is small, i.e., , then the tangential electric (magnetic) fields of the modes form a Riesz basis in (). For a constant permeability (permittivity) the Riesz basis property for the tangential electric (magnetic) fields holds also in the natural trace space (). These results hold also for complex frequencies . In addition, if , then for small enough all wavenumbers are located on the axes and there exist no backward modes. Key tools in the analysis are a particular reformulation of the eigenvalue problem, the perturbation theory for selfadjoint operators under a local subordinate condition and uniform properties of Bessel functions.
Paper Structure (20 sections, 24 theorems, 188 equations)

This paper contains 20 sections, 24 theorems, 188 equations.

Table of Contents

  1. Introduction
  2. The modal eigenvalue problem
  3. Setting and assumptions
  4. The modal eigenvalue problem
  5. The tangential electric field formulation
  6. Coaxial separation of modes
  7. Properties of Bessel functions
  8. Asymptotic approximations
  9. Gaps of roots and extrema
  10. Boundedness of Bessel function products
  11. The selfadjoint perturbation
  12. Reduction of $A_{11}\mathbf{v}=\nu^2 K_{11}\mathbf{v}$ to a scalar problem
  13. Reduction of Laplace eigenvalue problems
  14. Computation of $\det\ul{A}^\mathrm{Neu}_{N_{\Gamma}}(\nu)$ and $\det\ul{A}^\mathrm{Dir}_{N_{\Gamma}}(\nu)$
  15. Eigenvalue gaps
  16. ...and 5 more sections

Key Result

Lemma 2.2

Let the assumptions formulated in subsec:setting be satisfied. Then there exists $\delta_0>0$ such that $\omega$ is not a cut-off frequency (see def:cut-off) for each $\epsilon,\mu$ that satisfy $\delta_{\epsilon,\mu}<\delta_0$.

Theorems & Definitions (53)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Lemma 2.6
  • proof
  • ...and 43 more