Strong curvature effects in Neumann wave problems
- Mads Clausen Institute, University of Southern Denmark, Alsion 2, DK-6400 Sonderborg (Denmark)
- Department of Mathematics, Technical University of Denmark, Matematiktorvet, DK-2800 Kgs. Lyngby (Denmark)
Waveguide phenomena play a major role in basic sciences and engineering. The Helmholtz equation is the governing equation for the electric field in electromagnetic wave propagation and the acoustic pressure in the study of pressure dynamics. The Schroedinger equation simplifies to the Helmholtz equation for a quantum-mechanical particle confined by infinite barriers relevant in semiconductor physics. With this in mind and the interest to tailor waveguides towards a desired spectrum and modal pattern structure in classical structures and nanostructures, it becomes increasingly important to understand the influence of curvature effects in waveguides. In this work, we demonstrate analytically strong curvature effects for the eigenvalue spectrum of the Helmholtz equation with Neumann boundary conditions in cases where the waveguide cross section is a circular sector. It is found that the linear-in-curvature contribution originates from parity symmetry breaking of eigenstates in circular-sector tori and hence vanishes in a torus with a complete circular cross section. The same strong curvature effect is not present in waveguides subject to Dirichlet boundary conditions where curvature contributions contribute to second-order in the curvature only. We demonstrate this finding by considering wave propagation in a circular-sector torus corresponding to Neumann and Dirichlet boundary conditions, respectively. Results for relative eigenfrequency shifts and modes are determined and compared with three-dimensional finite element method results. Good agreement is found between the present analytical method using a combination of differential geometry with perturbation theory and finite element results for a large range of curvature ratios.
- OSTI ID:
- 22093687
- Journal Information:
- Journal of Mathematical Physics, Vol. 53, Issue 8; Other Information: (c) 2012 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA); ISSN 0022-2488
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
GENERAL PHYSICS
72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
BOUNDARY CONDITIONS
CROSS SECTIONS
DIFFERENTIAL GEOMETRY
DIRICHLET PROBLEM
EIGENFREQUENCY
EIGENFUNCTIONS
EIGENSTATES
EIGENVALUES
ELECTROMAGNETIC RADIATION
FINITE ELEMENT METHOD
NANOSTRUCTURES
PERTURBATION THEORY
QUANTUM MECHANICS
SCHROEDINGER EQUATION
SEMICONDUCTOR MATERIALS
SPECTRA
SYMMETRY BREAKING
WAVE PROPAGATION
WAVEGUIDES