Dispersionseigene

Dispersionseigene is a term used to describe eigenvalues associated with a medium’s dispersion properties. In wave physics, these values encode the relationship between frequency and wavenumber that governs how different spectral components propagate. Although common in some literature, the term is not universally used; in many contexts these quantities are simply referred to as dispersion relations or eigenfrequencies. Dispersionseigene are obtained by solving a linear eigenvalue problem derived from the governing equations with appropriate boundary conditions.

Mathematical formulation: For a wavevector k in a medium, one often encounters an eigenproblem A(k) ψ = ω^2 B(k) ψ. The eigenvalues ω_n(k) are the dispersionseigene, giving the allowed frequencies for each mode n at k. In periodic media like photonic crystals, Bloch’s theorem yields a discrete set of ω_n(k) that form dispersion curves, revealing band gaps and anisotropic dispersion.

Applications and interpretation: The dispersionseigene determine phase and group velocities, v_p = ω/|k| and v_g = dω/d|k|, and underpin the design of optical fibers, acoustic filters, and metamaterials. Computational methods include plane-wave expansion, finite element, and finite-difference time-domain simulations.

Terminology note: The term “dispersionseigene” is not standardized; many sources use “dispersion relation” or “eigenfrequencies.” The concept, however, is widely used: eigenvalues that describe how waves propagate in a given medium.