Bessel Functions and Analysis of Circular Waveguides

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.

💡 Research Summary

This research paper presents a rigorous mathematical and numerical framework for analyzing the propagation characteristics of circularly coiled waveguides, with applications spanning both the optical and acoustical domains. The primary challenge addressed is the complex nature of wave propagation in curved geometries, which necessitates the use of Bessel functions characterized by complex orders and complex arguments. To tackle this, the authors implement a sophisticated approach involving a change of variables combined with the classical Frobenius method. This allows for the precise computation of these complex Bessel functions, which are essential for determining the eigenvalue spectrum of the waveguide.

A significant technical contribution of this work is the integration of the Perfectly Matched Layer (PML) technique. By employing PML, the researchers are able to effectively simulate the unbounded nature of the surrounding medium within a finite computational domain, thereby solving the Bessel eigenvalue problem for a three-layer optical slab waveguide structure with high precision. This methodology enables the derivation of highly accurate loss factors for the eigensolutions, providing a critical metric for understanding energy attenuation in coiled waveguides. These results serve as a vital benchmark for verifying the implementation of other numerical models in the field.

Furthermore, the study provides crucial numerical evidence for the Glazman criterion. This criterion is fundamental to the mathematical theory of wave propagation, as it relates to the well-posedness and stability of solutions in homogeneous circular waveguides subject to impedance boundary conditions. By verifying this criterion, the paper reinforces the mathematical foundation required to ensure that physical models of such waveguides are stable and predictable. Ultimately, the paper bridges the gap between advanced mathematical theory and practical engineering applications, providing a robust toolset for the design and analysis of complex waveguide structures.