Cesaro summation by spheres of lattice sums and Madelung constants

<p style='text-indent:20px;'>We study convergence of 3D lattice sums via expanding spheres. It is well-known that, in contrast to summation via expanding cubes, the expanding spheres method may lead to formally divergent series (this will be so e.g. for the classical NaCl-Madelung constant). In the present paper we prove that these series remain convergent in Cesaro sense. For the case of second order Cesaro summation, we present an elementary proof of convergence and the proof for first order Cesaro summation is more involved and is based on the Riemann localization for multi-dimensional Fourier series.</p>

Simple Solutions of Lattice Sums for Electric Fields Due to Infinitely Many Parallel Line Charges

We present surprisingly simple closed-form solutions for electric fields and electric potentials at arbitrary position ( x , y ) within a plane crossed by infinitely long line charges at regularly repeating positions using angular or elliptic functions with complex arguments. The lattice sums for the electric-field components and the electric potentials could be exactly solved, and the duality symmetry of trigonometric and lemniscate functions occurred in some solutions. The results may have relevance in calculating field configurations with rectangular boundary conditions. Several series related to Gauss’s constant are presented, established either as corollary results or via parallel investigations conducted in the spirit of experimental mathematics.

Analysis of Scattering by Plasmonic Gratings of Circular Nanorods Using Lattice Sums Technique

A self-contained formulation for analyzing electromagnetic scattering by a significant class of planar gratings composed of plasmonic nanorods, which were infinite length along their axes, is presented. The procedure for the lattice sums technique was implemented in a cylindrical harmonic expansion method based on the generalized reflection matrix approach for full-wave scattering analysis of plasmonic gratings. The method provided a high computational efficiency and can be considered as one of the best-suited numerical tools for the optimization of plasmonic sensors and plasmonic guiding devices both having a planar geometry. Although the proposed formalism can be applied to analyze a wide class of plasmonic gratings, three configurations were studied in the manuscript. Firstly, a multilayered grating of silver nanocylinders formed analogously to photonic crystals was considered. In the region far from the resonances of a single plasmonic nanocylinder, the structure showed similar properties compared to conventional photonic crystals. When one or a few nanorods were periodically removed from the original crystal, thus forming a crystal with defects, a new band was formed in the spectral responses because of the resonant tunneling through the defect layers. The rigorous formulation of plasmonic gratings with defects was proposed for the first time. Finally, a plasmonic planar grating of metal-coated dielectric nanorods coupled to the dielectric slab was investigated from the viewpoint of design of a refractive index sensor. Dual-absorption bands attributable to the excitation of the localized surface plasmons were studied, and the near field distributions were given in both absorption bands associated with the resonances on the upper and inner surfaces of a single metal-coated nanocylinder. Resonance in the second absorption band was sensitive to the refractive index of the background medium and could be useful for the design of refractive index sensors. Also analyzed was a phase-matching condition between the evanescent space-harmonics of the plasmonic grating and the guided modes inside the slab, leading to a strong coupling.

Efficient analysis method of light scattering by a grating of plasmonic nanorods

Purpose The purpose of this paper is to develop a rigorous self-contained formulation for analyzing electromagnetic scattering by grating of plasmonic nanorods. The authors investigate this structure from the viewpoint of the practical application as a refractive index plasmonic sensor. The presented rigorous formulation is accompanied with a neat implementation providing a high computation efficiency and could be considered as an important tool for designing and optimizing compact sensors. Design/methodology/approach Scattering of an incident plane wave by grating made of a periodic arrangement of metal-coated dielectric nanocylinders on a dielectric slab is rigorously investigated using the recursive algorithm combined with the lattice sums technique. As a dielectric slab, the authors consider glass material, which is widely used in experiments, whereas silver (Ag) is used as a low loss metal suitable to excite plasmon resonances. The main advantage of the developed self-contained formulation is that first the authors extract the reflection and transmission matrices of a single planar array from a separate analysis of the grating and the slab and then obtain the scattering characteristics of the whole structure by using a recursive formula. The method is computationally fast. Findings Dependence of the surface plasmon resonance wavelength on the refractive index of the surrounding medium is carefully investigated. The resonance peaks are red-shifted with respect to an increasing refractive index of surrounding medium showing an almost linear behavior. Near field distributions are analyzed at the resonance wavelengths of the spectral responses. Special attention is paid to the formation of the dual-absorption bands because of the excitation of the localized surface plasmons. The authors give physical insight to the coupling between grating and the glass slab. The authors found that a strong enhancement of the field inside the slab occurs when the scattered wave of the grating is phase-matched to the guided modes supported by the slab. Originality/value In the authors’ formulation, they do not use any approximation and it is rigorous and accurate. The authors use their original method. The method is based on the lattice sums technique and uses the recursive algorithm to calculate the generalized reflection and transmission characteristics by grating. Such fast and accurate method is an effective tool apt for designing and optimizing tailored sensors, for e.g. advanced biomedical applications.