Equilateral triangular waveguides, modal structure, attenuation characteristics and quality factor

TL;DR

This paper delivers fully analytic closed-form expressions for the attenuation constants and quality factors of equilateral triangular waveguides (ETWs) by leveraging plane-wave superpositions, symmetry properties, and Eisenstein integers to order and classify the eigenvalues. The modal analysis covers both TM and TE families, with spectra organized by triples $(l,m,n)$ satisfying $l+m+n=0$ and eigenvalues $k_ot^2 = k_0^2 E_{mn}$ where $E_{mn}=m^2+mn+n^2$, and $E_{mn}$ is represented in the Eisenstein form $E_{mn}=(m+n\Omega)(m+n\bar{\Omega})$ ( $\Omega=\tfrac{1}{2}+i\tfrac{\sqrt{3}}{2}$ ). The authors derive normalized attenuation constants $\alpha'^{(TM)}$ and $\alpha'^{(TE)}$, and quality factors $Q_p^{(TM)}$ and $Q_p^{(TE)}$ in terms of explicit contour/area integrals over the mode fields, providing complete analytic expressions and validating them with attenuation plots and field visualizations. They also reveal symmetry properties, such as index interchanges and sign changes, that govern degeneracies and propagation, and show how Eisenstein-prime factorization clarifies spectral structure and accidental degeneracies. The results extend beyond ETWs to general cylindrical cavities, offering practical formulas for designing high-Q devices in nano-photonics and informing related studies of triangular billiards and polygonal waveguides. Overall, the work fuses classical waveguide theory with number-theoretic techniques to produce rigorously tractable, widely applicable results for ETWs.

Abstract

Equilateral triangular waveguides are one of the very few special kind of waveguides, whose field solutions can be constructed without necessarily solving the Maxwell's equations. Solutions can be obtained simply by superposing some plane wave solutions and requiring the solutions to obey the necessary boundary conditions for each modes, as would be expected of any Maxwell equation's solution or that of any Helmholtz equation resulting from the problem of Heat flow, membrane vibrations, etc. In fact, this was how one of the earlier solutions or eigen functions found by Gabriel Lamé, were obtained. Julian Schwinger, performed the same analysis, but used the superposition of complex exponential functions, in order to obtain the eigen functions of an equilateral triangular waveguide. The solutions exhibit other special symmetric properties that leave their solutions invariant and these special symmetries and applications, are what we investigate in this work, particularly as it relates to their attenuation characteristics and quality factors. Finally, by employing the number theory of Eisenstein integers or primes we have obtained a well ordered and more rigorous eigenvalues of the equilateral triangular waveguides, which is definitely something missing in the literature.

Figures (24)

• Figure 1: An Equilateral Triangular Waveguide Cross Section
• Figure 2: figure
• Figure 3: $m=n=1$
• Figure 4: $m=n=1$
• Figure 5: $m=n=1$