Bessel Functions and Analysis of Circular Waveguides

TL;DR

This work addresses the numerical challenge of solving Bessel eigenproblems arising in circularly bent three-layer waveguides by combining a conformal mapping with a Frobenius-based evaluation of Bessel functions of large complex order and large argument. It develops a two-solution basis $V_{\mu}$ and $W_{\mu}$, derives order-derivative relations, and implements high-precision computation to obtain accurate eigenvalues $\lambda$ and propagation constants $\beta$. The study verifies the Glazman criterion for the impedance-boundary problem and extends to a three-layer bent waveguide, solving a nonlinear system via Newton methods with seeds from straight-waveguide modes. It further replaces the impedance boundary by a perfectly matched layer (PML) to impose radiation conditions more reliably, showing substantial differences in predicted losses and mode profiles, and benchmarks bend-loss predictions against classical formulas. The results provide high-accuracy spectral data and mode-loss benchmarks for validating 3D Maxwell simulations and offer reproducible open-source tools for the community.

Abstract

The paper is devoted to the study of circularly coiled optical slab waveguides, which is also applicable to acoustical waveguides. We use a change of variables and the classical Frobenius method to compute Bessel functions of complex order and complex argument, and combine it with a perfectly matched layer technique to solve the relevant Bessel eigenvalue problem and deliver accurate loss factors for eigensolutions to the three-layer optical slab waveguide problem. The solutions provide a benchmark for verifying model implementations of this problem and allow for a numerical verification of the Glazman criterion that provides a foundation for the well-posedness and stability analysis of homogeneous circular waveguides with impedance boundary conditions.

Figures (8)

• Figure 1: Geometry and boundary conditions for a circular waveguide with a core and a cladding with refractive indices $n_{\text{core}}$ and $n_{\text{clad}}$, respectively.
• Figure 1: The real (plot \ref{['plot:RealAlphaValues']}) and imaginary (plot \ref{['plot:ImagAlphaValues']}) components of the $\alpha_{n} = \alpha_{n}(\lambda_{n})$ function \ref{['eq:AlphaValue']} for the first 30 eigenvalues of \ref{['eq:Bessel_eqn_new']} using $n_{0} = 1$, $\kappa = 10$, $r_0 = 100$, $b = 0.5$, and $d = 1$. According to the theoretical results in Demkowicz_Gopalakrishnan_Heuer_24, in the limit of a straight waveguide ($r_0 \to \infty$), $\operatorname*{Re}(\alpha_{n}) \to n \pi$ and $\operatorname*{Im}(\alpha_n)\to-2bd\kappa/(\pi n)$ (see the black dashed lines in \ref{['plot:RealAlphaValues']} and \ref{['plot:ImagAlphaValues']}). In plot \ref{['plot:GlazmanFiniteSums']}, we present the finite sums $G_n$\ref{['eq:finite_Glazman']} associated with Glazman condition \ref{['eq:Glazman_condition']}, observing the convergence of the series.
• Figure 1: Real components of the mode profiles of the three-layer waveguide \ref{['eq:Bessel_eqn_new']} at $r_0 = 13000$ with impedance BC \ref{['eq:BC2']}, which are nearly indistinguishable from the straight waveguide case. Notice that the coordinate $r$ has been re-centered at zero. See Table \ref{['tab:eig13000']} for their corresponding eigenvalues.
• Figure 1: Passing from a segment in the real line to a path in the complex plane, inspired by the PML methodology. The goal of extending the solution to the complex plane is to capture the decay that the outgoing wave function is expected to present when evaluated on the (orange) curved path. Such decaying behavior is enforced through the constraint $u=0$ at $z=z_{\text{PML}}$.
• Figure 1: Comparison of the circularly curved three-layer waveguide's amplitude loss factors, $-{\operatorname*{Im}}(\beta)$, as a function of the bend radius ($r_0$) for the (a) first even mode, (b) first odd mode, and (c) second even mode. Results obtained with our numerical approach using the PML BC are compared with loss values obtained with formulas from the literature by Marcatili marcatili1969bends, Marcuse marcuse1982light, and Takuma et al. takuma1981bent.

Theorems & Definitions (4)

• Remark 1.1
• Remark 4.1
• Remark 5.1
• Remark 5.2