Numerical computation of the scattering matrix of an electromagnetic resonator

Electromagnetic scattering by a partially graphene-coated dielectric cylinder: Efficient computation and multiple plasmonic resonances

We present a numerical approach for the solution of electromagnetic scattering from a dielectric cylinder partially covered with graphene. It is based on a classical Fourier-Bessel expansion of the fields inside and outside the cylinder to which we apply ad hoc boundary conditions in the presence of graphene. Due to the singular nature of the electric field at the edges of the graphene sheet, we introduce auxiliary boundary conditions. The result is a particularly simple and efficient method allowing the study of diffraction from such structures. We also highlight the presence of multiple plasmonic resonances that we ascribe to the surface modes of the coated cylinder.

Planar acoustic scattering by a multi-layered split ring resonator

The problem of two-dimensional acoustic scattering of time-harmonic plane waves by a multi-ringed cylindrical resonator is considered. The resonator is made up of an arbitrary number of concentric sound-hard split rings with zero thickness. Each ring opening is oriented in any direction. The acoustics pressure field in each layered region enclosed between adjacent rings is described by an eigenfunction expansion in polar coordinates. An integral equation/Galerkin method is used to relate the unknown coefficients of the expansions between adjacent regions separated by a ring. The multiple scattering problem is then formulated as a reflection/transmission problem between the layers, which is solved using an efficient iterative scheme. An exploration of the parameter space is conducted to determine first, the conditions under which the lowest resonant frequency can be minimised, and second, how non-trivial resonances of the multi-ring resonators can be explained from those of simpler arrangements, such as a single-ring resonator. It is found here that increasing the number of rings while alternating the orientation lowers the first resonant frequency, and exhibits a dense and nearly regular resonant structure that is analogous to the rainbow trapping effect.

The split ring resonator

We obtain band diagrams for a two-dimensional periodic structure consisting of an infinite square array of infinitely thin concentric circles (split rings) with narrow gaps. Our approach exploits the narrowness of the gaps and yields algebraic equations relating the frequency to the Bloch wavenumber and geometric properties of the array. Further asymptotic analysis indicates that the gravest mode has a frequency that scales in an inverse logarithmic fashion with the size of the gap and that exhibits anomalous dispersion. Near the origin of the Brillouin zone this ‘acoustic’ mode is dispersionless. Numerical solution of the eigenvalue problem in the single-gap case confirms these conclusions. The two lowest modes of the split ring can be interpreted as a splitting of the gravest propagating Rayleigh mode.

Study of electromagnetic diffraction by curved strip gratings by use of the C-method

The C-method is one of the most efficient and versatile methods designed for diffraction gratings. Its strength lies in the use of a coordinate system in which the surface of the grating coincides with a coordinate surface. The result is a great simplification in writing the boundary conditions. We exploit this simplification to treat the problem of diffraction from curved strip gratings, and we use the combined boundary conditions method that has been introduced for planar strip gratings and proved to be very efficient.