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| Format: | Preprint |
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2026
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| Online Access: | https://arxiv.org/abs/2603.16716 |
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| _version_ | 1866914402454732800 |
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| author | Halla, Martin |
| author_facet | Halla, Martin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_16716 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Modal bases of coaxial electromagnetic step index fibers Halla, Martin Analysis of PDEs 35P05, 35R05, 78M35 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. |
| title | Modal bases of coaxial electromagnetic step index fibers |
| topic | Analysis of PDEs 35P05, 35R05, 78M35 |
| url | https://arxiv.org/abs/2603.16716 |