Olga V. Shapoval

Integral Equation Analysis of Plane Wave Scattering by Coplanar Graphene-Strip Gratings in the THz Range

The plane wave scattering and absorption by finite and infinite gratings of free-space standing infinitely long graphene strips are studied in the THz range. A novel numerical approach, based on graphene surface impedance, hyper-singular integral equations, and the Nystrom method, is proposed. This technique guarantees fast convergence and controlled accuracy of computations. Reflectance, transmittance, and absorbance are carefully studied as a function of graphene and grating parameters, revealing the presence of surface plasmon resonances. Specifically, larger graphene relaxation times increases the number of resonances in the THz range, leading to higher wave transmittance due to the reduced losses; on the other hand an increase of graphene chemical potential up-shifts the frequency of plasmon resonances. It is also shown that a relatively low number of graphene strips ( >10) are able to reproduce Rayleigh anomalies. These features make graphene strips good candidates for many applications, including tunable absorbers and frequency selective surfaces.

THz wave scattering by a graphene strip and a disk in the free space: integral equation analysis and surface plasmon resonances

The excitation of the surface plasmon resonances on a graphene strip and a disk in free space is studied numerically as a 2D and 3D electromagnetic wave-scattering problem, respectively. The associated mathematical model is based on the Maxwell equations with resistive boundary conditions on the surface of a zero-thickness strip or disk, where the graphene electron conductivity is included as a parameter and determined from the Kubo formalism. It is shown that plasmon resonance frequencies in the terahertz range shift with variation of the chemical potential of the graphene. Far-field and near-field patterns are plotted at several resonance frequencies.

Modeling of Plasmon Resonances of Multiple Flat Noble-Metal Nanostrips With a Median-Line Integral Equation Technique

The surface plasmon and the periodicity-induced resonances in the scattering and absorption of light by multiple flat nanosize noble-metal strips are investigated using a new efficient model. It exploits the fact that the nanostrip thickness is a small fraction of the wavelength in the visible range. This justifies shrinking the strip cross section to its median line and using the generalized boundary conditions on that line, with the strip thickness entering the coefficients. As a result, the scattering problem is reduced to the singular and hypersingular integral equations. We discretize them using quadrature formulas of interpolation type and build an algorithm having guaranteed convergence and controlled accuracy of computations. It enables fast simulation of structures consisting of many noble-metal strips. Near- and far-field characteristics for finite flat grating of silver and gold nanostrips are presented.

Finite gratings of many thin silver nanostrips: Optical resonances and role of periodicity

We study numerically the optical properties of the periodic in one dimension flat gratings made of multiple thin silver nanostrips suspended in free space. Unlike other publications, we consider the gratings that are finite however made of many strips that are well thinner than the wavelength. Our analysis is based on the combined use of two techniques earlier verified by us in the scattering by a single thin strip of conventional dielectric: the generalized (effective) boundary conditions (GBCs) imposed on the strip median lines and the Nystrom-type discretization of the associated singular and hyper-singular integral equations (IEs). The first point means that in the case of the metal strip thickness being only a small fraction of the free-space wavelength (typically 5 nm to 50 nm versus 300 nm to 1 μm) we can neglect the internal field and consider only the field limit values. In its turn, this enables reduction of the integration contour in the associated IEs to the strip median lines. This brings significant simplification of the scattering analysis while preserving a reasonably adequate modeling. The second point guarantees fast convergence and controlled accuracy of computations that enables us to compute the gratings consisting of hundreds of thin strips, with total size in hundreds of wavelengths. Thanks to this, in the H-polarization case we demonstrate the build-up of sharp grating resonances (a.k.a. as collective or lattice resonances) in the scattering and absorption cross-sections of sparse multi-strip gratings, in addition to better known localized surface-plasmon resonances on each strip. The grating modes, which are responsible for these resonances, have characteristic near-field patterns that are distinctively different from the plasmons as can be seen if the strip number gets larger. In the E-polarization case, no such resonances are detectable however the build-up of Rayleigh anomalies is observed, accompanied by the reduced scattering and absorption.

Validity and Limitations of the Median-Line Integral Equation Technique in the Scattering by Material Strips of Sub-Wavelength Thickness

Considered is the 2-D scattering of a plane wave by a thin flat material strip. The data obtained by using the empirical method of generalized boundary conditions and singular integral equations on the strip median line are compared with the results of solving the Muller boundary integral equation that takes full account of strip thickness. Discretization of integral equations in both cases is performed using the Nystrom methods that lead to convergent algorithms. Numerical results cover E and H polarizations and two types of thin strips: conventional dielectric and metal in the optical range. The validity and limitations of approximate model are established and discussed.

Resonance effects in the optical antennas shaped as finite comb-like gratings of noble-metal nanostrips

Active research into nanoscience and nanotechnologies that are available for nano fabrication have lead to considerable progress in the understanding of the optical properties of metals on nanometer scale. Here, noble-metal strip-like nanostructures are attractive objects of research. Indeed, they can be easily manufactured and serve as building blocks of optical nanoantennas and sensors with unique geometry-dependent optical properties. This is because they display intensive localized surface-plasmon resonances in the visible and far-infrared ranges that lead to near- and far-field enhancement effects. Thanks to surface-plasmon resonances, multi-element finite gratings have attractive properties of extraordinarily large reflection, absorption, and transmission, depending on the arrangement of the elementary cell of the grating. All these phenomena are greatly influenced by the so-called grating resonances which appear due to periodicity. The 2D modeling of electromagnetic wave scattering by thin noble-metal nanosize strips and their finite-periodical ensembles arranged in comb-like gratings is considered. Our analysis is carried out using new efficient, convergent and accurate method. It is based, first, on the use of the generalized boundary conditions (GBC) valid for a thin and highcontrast material layer; they allow us to consider only the limit values of the field components and reduce integration contour to the collection of corresponding strip median lines. Second, for the building of discrete model of the obtained singular integral equations, we use very efficient Nystrom-type algorithm with quadrature formulas of interpolation type. We study the SPRs of the finite periodic comb-like strip ensembles versus the incidence angle of the plane electromagnetic wave and the strip characteristics; both near-field and far-field properties of the associated surfaceplasmon resonances and especially local field enhancements or focusing effects are analyzed. Moreover, we investigate the periodicity-induced properties such as the grating resonances in the context of the development of optimal design strategies for efficient multi-strip optical nanoantennas

Comparison of Refractive-Index Sensitivities of Optical-Mode Resonances on a Finite Comb-Like Grating of Silver Nanostrips

We investigate the interplay of several different types of resonances in the scattering of light by finite comb-like nanogratings made of silver strips, in the H- and E-polarization cases. The resonances studied correspond to the localized surface plasmon modes, the periodicity-induced grating mode, and the cavity modes. They show up as Fano shapes in the total scattering cross sections and absorption cross sections. We find that the grating-mode and the cavity-mode resonances have higher values of both bulk refractive-index sensitivity and figure-of-merit than the localized-surface-plasmon resonances, in the visible band.

Bulk refractive-index sensitivities of the THz-range plasmon resonances on a micro-size graphene strip

We studied numerically the potential use of a micro-size graphene strip as a surface plasmon (SP) resonance-based bulk refractive-index sensor in the THz frequency regime. Our accurate computational instrument was an in-house algorithm based on integral equations (IEs) and Nystrom discretization. The refractive-index sensitivities and figure-of-merit (FOM) values of the associated plasmon resonances were calculated. It was found that the primary plasmon mode P 1 is more sensitive to the refractive-index changes than plasmons of higher orders, although the latter demonstrated much larger FOM values explained by the higher Q-factors. The FOM values of the higher-order resonances on a graphene strip in the THz range are at a level similar to the FOM values of the localized SP resonances on a noble-metal strip in the optical range.

Electromagnetic Engineering of a Single-Mode Nanolaser on a Metal Plasmonic Strip Placed into a Circular Quantum Wire

We investigate the emission of waves by a thin silver nanostrip placed into the center of a circular quantum wire (QWR), in the visible-light range. Our analysis uses the mathematically grounded approach called lasing eigenvalue problem (LEP). Keeping in mind that at the threshold the lasing-mode frequency is real valued (does not attenuate in time), the LEP is formulated as a boundary-value problem for the Maxwell equations with exact boundary conditions and the Sommerfeld radiation condition. The eigenvalues are pairs of real numbers, where the first is the emission wavelength and the second is the associated threshold value of material gain in the QWR. Due to the twofold symmetry of the cross-sectional geometry, we split the studied problem into four different independent classes of symmetry and derive four symmetry-adapted Greens functions of the QWR without strip. On imposing the generalized boundary conditions and taking into account these Greens functions, we obtain four independent integral equations (IEs) at strips median line. We discretize these IEs with the Nystrom-type schemes and further look for the eigenvalues of each class separately with the aid of iterative search algorithm. Our analysis shows that such a plasmonic-strip-based nanolaser can emit visible light on the localized surface plasmon modes and also on the shell modes or QWR polariton modes perturbed by the strip. Single-mode operation is apparently possible provided that the QWR diameter is small, and hence, the first shell mode is blue-shifted.

Plasmon resonances in the H-wave scattering by a nanosize thin flat silver strip

Nanosize thin flat silver strip in the air is considered as an optical resonator. The scattering characteristics of the H-polarized electromagnetic wave diffraction are analyzed using the decoupled singular and hyper-singular integral equations (IEs) for the electric and magnetic surface currents on the strip. IEs are discretized with the aid of the interpolation type quadrature formulas. This algorithm, also known as the method of discrete singularities (MDS), is numerically efficient and guarantees fast convergence and controlled accuracy of computations.