Abstract
A small dielectric object with positive permittivity may resonate when the free-space wavelength is large in comparison with the object dimensions if the permittivity is sufficiently high. We show that these resonances are described by the magnetoquasistatic approximation of the Maxwell's equations in which the normal component of the displacement current density field vanishes on the surface of the particle. They are associated to values of permittivities and frequencies for which source-free quasistatic magnetic fields exist, which are connected to the eigenvalues of a magnetostatic integral operator. We present the general physical properties of magnetoquasistatic resonances in dielectrics with arbitrary shape. They arise from the interplay between the polarization energy stored in the dielectric and the energy stored in the magnetic field. Our findings improve the understanding of resonances in high-permittivity dielectric objects and provide a powerful tool that greatly simplifies the analysis and design of high index resonators.
Highlights
Antonello Tamburrino Department of Electrical and Information Engineering, Università di Cassino e del Lazio Meridionale, Cassino, 03043, Italy and Department of Electrical and Computer Engineering, Michigan State University, East Lansing, Michigan 48824, USA
These resonances can be predicted by the electroquasistatic approximation of the Maxwell equations, and they are associated to the values of permittivity for which source-free electrostatic fields exist [1,2]
We investigate a finite-size cylinder of radius R and height h = R, which is very common among nanofabricated structures, because it is compatible with planar nanofabrication processes
Summary
Let us consider a homogeneous and isotropic dielectric object with a bounded arbitrary shape and relative permittivity εR. The mathematical structure of the integral operator (4) does not depend on the linear characteristic dimension of the dielectric object lc; namely it is scale invariant This fact combined with Eq (7) leads to an important property of the magnetoquasistatic resonances: For any given shape of the object, the proportional to resonance both lc and f√reεqRu.eFncuirethsearmreoraelw, tahyes inversely resonance frequencies accumulate at infinity. The modes of the second set are div-free within the object and have a vanishing normal component on the object surface and nonzero curl The latter set is solution of the eigenvalue problem (5) and the corresponding resonance frequencies ωn are given by (7). The eigenvalue problem (5) can be numerically solved through a finite-element approach, briefly outlined in Appendix C, where, unlike differential formulations, only the spatial domain is discretized and the radiation condition is automatically satisfied. Where ris the mirror image of rwith respect to the substrate plane
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