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Topological aspects of zero modes in cavity resonators

Osamu Kamigaito

TL;DR

The paper establishes a rigorous link between electromagnetic zero modes in cavity resonators and the cavity's topology by identifying zero modes with harmonic differential forms and showing their degeneracy equals the dimension of a corresponding homology group. It recasts Maxwell's equations as a Laplacian eigenproblem on forms with relative or absolute boundary conditions and uses the Hodge decomposition to derive isomorphisms between de Rham cohomology and harmonic spaces, then employs De Rham's theorem and Poincaré duality to connect zero modes to singular (co)homology and Euler characteristics. A key result is that the alternating sum of zero-mode dimensions relates to the Euler characteristic of the boundary, which in turn ties to the integral of curvature over the boundary; an explicit cavity example demonstrates how Betti numbers encode the degeneracies. Together, the work provides a topological framework for understanding cavity resonances with potential implications for cavity design and electromagnetic-field control.

Abstract

We discuss the relationship between the zero modes of electromagnetic fields in a cavity resonator and the cavity's topological characteristics. We show that the dimension of the electromagnetic zero-mode space coincides with the dimension of the corresponding homology group of the cavity. Moreover, we prove that the alternating sum of the dimensions of the electromagnetic zero-mode spaces is closely related to the Euler characteristic of the cavity boundary, and hence to the integral of the curvature.

Topological aspects of zero modes in cavity resonators

TL;DR

The paper establishes a rigorous link between electromagnetic zero modes in cavity resonators and the cavity's topology by identifying zero modes with harmonic differential forms and showing their degeneracy equals the dimension of a corresponding homology group. It recasts Maxwell's equations as a Laplacian eigenproblem on forms with relative or absolute boundary conditions and uses the Hodge decomposition to derive isomorphisms between de Rham cohomology and harmonic spaces, then employs De Rham's theorem and Poincaré duality to connect zero modes to singular (co)homology and Euler characteristics. A key result is that the alternating sum of zero-mode dimensions relates to the Euler characteristic of the boundary, which in turn ties to the integral of curvature over the boundary; an explicit cavity example demonstrates how Betti numbers encode the degeneracies. Together, the work provides a topological framework for understanding cavity resonances with potential implications for cavity design and electromagnetic-field control.

Abstract

We discuss the relationship between the zero modes of electromagnetic fields in a cavity resonator and the cavity's topological characteristics. We show that the dimension of the electromagnetic zero-mode space coincides with the dimension of the corresponding homology group of the cavity. Moreover, we prove that the alternating sum of the dimensions of the electromagnetic zero-mode spaces is closely related to the Euler characteristic of the cavity boundary, and hence to the integral of the curvature.
Paper Structure (17 sections, 127 equations, 1 figure, 1 table)

This paper contains 17 sections, 127 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Settings with differential forms
  3. Harmonic forms and harmonic fields
  4. De Rham cohomology
  5. Hodge isomorphism
  6. Poincaré duality on manifolds with boundary
  7. De Rham's theorem
  8. Relationship with singular homology
  9. Euler characteristics
  10. Discussions
  11. Electromagnetic fields in cavity resonators
  12. Mathematical formulas
  13. Riemannian metric
  14. Hodge star operator
  15. Integration of differential forms
  16. ...and 2 more sections

Figures (1)

  • Figure 1: Example of a resonant cavity. A cylindrical outer shell contains two drift tubes, and a metal sphere is floating in the vacuum. $S_1$ is the inner surface of the cavity body, and $S_2$ is the surface of the sphere. In (a) we show cycles $z_k^r$ representing classes in $H_k(M,\partial M)$, and in (b) we show cycles $z_k^a$ representing classes in $H_k(M)$.

Theorems & Definitions (3)

  • proof
  • proof
  • proof