Terahertz near-field microscopy of metallic circular split ring resonators with graphene in the gap

Optical resonators are fundamental building blocks of photonic systems, enabling meta-surfaces, sensors, and transmission filters to be developed for a range of applications. Sub-wavelength size (< λ/10) resonators, including planar split-ring resonators, are at the forefront of research owing to their potential for light manipulation, sensing applications and for exploring fundamental light-matter coupling phenomena. Near-field microscopy has emerged as a valuable tool for mode imaging in sub-wavelength size terahertz (THz) frequency resonators, essential for emerging THz devices (e.g. negative index materials, magnetic mirrors, filters) and enhanced light-matter interaction phenomena. Here, we probe coherently the localized field supported by circular split ring resonators with single layer graphene (SLG) embedded in the resonator gap, by means of scattering-type scanning near-field optical microscopy (s-SNOM), using either a single-mode or a frequency comb THz quantum cascade laser (QCL), in a detectorless configuration, via self-mixing interferometry. We demonstrate deep sub-wavelength mapping of the field distribution associated with in-plane resonator modes resolving both amplitude and phase of the supported modes, and unveiling resonant electric field enhancement in SLG, key for high harmonic generation.

Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays

We present a novel multipole formulation for computing the band structures of two-dimensional arrays of cylindrical Helmholtz resonators. This formulation is derived by combining existing multipole methods for arrays of ideal cylinders with the method of matched asymptotic expansions. We construct asymptotically close representations for the dispersion equations of the first band surface, correcting and extending an established lowest-order (isotropic) result in the literature for thin-walled resonator arrays. The descriptions we obtain for the first band are accurate over a relatively broad frequency and Bloch vector range and not simply in the long-wavelength and low-frequency regime, as is the case in many classical treatments. Crucially, we are able to capture features of the first band, such as low-frequency anisotropy, over a broad range of filling fractions, wall thicknesses and aperture angles. In addition to describing the first band we use our formulation to compute the first band gap for both thin- and thick-walled resonators, and find that thicker resonator walls correspond to both a narrowing of the first band gap and an increase in the central band gap frequency.

Tailored acoustic metamaterials. Part II. Extremely thick-walled Helmholtz resonator arrays

We present a solution method which combines the technique of matched asymptotic expansions with the method of multipole expansions to determine the band structure of cylindrical Helmholtz resonator arrays in two dimensions. The resonator geometry is considered in the limit as the wall thickness becomes very large compared with the aperture width (the extremely thick-walled limit). In this regime, the existing treatment in Part I (Smith & Abrahams, 2022 Tailored acoustic metamaterials. Part I. Thin- and thick-walled Helmholtz resonator arrays), with updated parameters, is found to return spurious spectral behaviour. We derive a regularized system which overcomes this issue and also derive compact asymptotic descriptions for the low-frequency dispersion equation in this setting. We find that the matched-asymptotic system is able to recover the first few bands over the entire Brillouin zone with ease, when suitably truncated. A homogenization treatment is outlined for describing the effective bulk modulus and effective density tensor of the resonator array for all wall thicknesses. We demonstrate that extremely thick-walled resonators are able to achieve exceptionally low Helmholtz resonant frequencies, and present closed-form expressions for determining these explicitly. We anticipate that the analytical expressions and the formulation outlined here may prove useful in designing metamaterials for industrial and other applications.

Metamaterial-Based Sub-Microwave Electromagnetic Field Energy Harvesting System

This paper presents the comprehensive analysis of the sub-microwave, radio frequency band resonant metastructures’ electromagnetic properties with a particular emphasis on the possibility of their application in energy harvesting systems. Selected structures based on representative topologies of metamaterials have been implemented in the simulation environment. The models have been analyzed and their substitute average electromagnetic parameters (absorption, reflection, transmission and homogenized permeability coefficients) have been determined. On the basis of simulation research, prototypes of electromagnetic field two-dimensional absorbers have been manufactured and verified experimentally in the proposed test system. The absorber has been implemented as a component of the low-cost energy harvesting system with a high-frequency rectifier and a voltage multiplier, obtaining usable DC energy from the electromagnetic field in certain frequency bands. The energy efficiency of the system has been determined and the potential application in energy harvesting technology has been assessed.

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.

Asymptotics for metamaterials and photonic crystals

Metamaterial and photonic crystal structures are central to modern optics and are typically created from multiple elementary repeating cells. We demonstrate how one replaces such structures asymptotically by a continuum, and therefore by a set of equations, that captures the behaviour of potentially high-frequency waves propagating through a periodic medium. The high-frequency homogenization that we use recovers the classical homogenization coefficients in the low-frequency long-wavelength limit. The theory is specifically developed in electromagnetics for two-dimensional square lattices where every cell contains an arbitrary hole with Neumann boundary conditions at its surface and implemented numerically for cylinders and split-ring resonators. Illustrative numerical examples include lensing via all-angle negative refraction, as well as omni-directive antenna, endoscope and cloaking effects. We also highlight the importance of choosing the correct Brillouin zone and the potential of missing interesting physical effects depending upon the path chosen.