Resonance frequencies of arbitrarily shaped dielectric cylinders

Abstract

We consider eigenvalue problems for dielectric cylindrical scatterers of arbitrary cross with generalized conditions at infinity that enable one to take into account complex eigenvalues. The existence of resonance (scattering) frequencies associated with these eigenvalues is proved. The technique involves the determination of characteristic numbers (CNs) of the Fredholm operator-valued functions of the problems constructed using Green's potentials. Separating principal parts in the form of meromorphic operator pencils, we apply the operator generalization of Rouché's theorem to verify the occurrence of CNs in close proximities of the pencil poles. The results are illustrated in detail using the case of a dielectric cylinder of circular cross section.

High-Precision Calculation Using the Method of Analytical Regularization for the Cut-Off Wave Numbers for Waveguides of Arbitrary Cross Sections with Inner Conductors

“…This is a Fredholm matrix equation of second kind. More precisely, one can show using the definitions and results of [1,2] that the left-hand sides of ISLAE (23), (24) 23) and ( 24) form a Fredholm infinite-matrix (summation) equation system of the second kind equivalent to the initial homogeneous boundary IE (7). This system, written as H(k) = 0, constitutes, in turn, the problem on CNs for OVF H(k).…”

“…The determination of the cut-off frequencies (or cut-off wave numbers) in waveguides with (or possibly without) inner conductors is closely related to the solution of a purely mathematical problem: the finding of non-trivial solutions of a homogeneous boundary value problem (BVP) for the Helmholtz equation. This problem, in turn, can be reduced to a homogeneous boundary integral equation (IE); the latter is treated following [1,2] as a problem of characteristic numbers (CNs) for an integral operator-valued function (OVF) of the (complex) frequency spectral parameter defined in terms of the boundary IE and for the equivalent infinite-matrix (summation) OVF.…”

TE-Polarized Electromagnetic Wave Diffraction by a Circular Slotted Cylinder

“…. Using the technique of operator-valued functions (OVFs) [38] developed in [39][40][41], reducing BEP to a boundary integral equation (BIE) with integration over the fictitious boundary, a circular arc replacing the slot N , and separating the principal parts (in the form of meromorphic operator pencils) of the obtained finite-meromorphic boundary integral OVFs with respect to both the spatial variables and spectral parameter, one can show that, at least for a sufficiently narrow slot, BEP (1)-( 5) has isolated complex eigenvalues with the negative imaginary parts situated in small proximities of κ nm = µ nm . Finally, one can verify that the eigenvalue spectrum K(N ) = ∪ n,m κ nm (N ) of BEP (1)-( 5) can be considered as a regular perturbation of the 'non-perturbed' real spectrum K of the internal BEP in the partial domain, the disk r = r 0 , with respect to the diameter of the slot.…”