Alexander I. Nosich

The method of analytical regularization in wave-scattering and eigenvalue problems: foundations and review of solutions

We discuss the foundations and state-of-the-art of the method of analytical regularization (MAR) (also called the semi-inversion method). This is a collective name for a family of methods based on conversion of a first-kind or strongly-singular second-kind integral equation to a second-kind integral equation with a smoother kernel, to ensure point-wise convergence of the usual discretization schemes. This is done using analytical inversion of a singular part of the original equation; discretization and semi-inversion can be combined in one operation. Numerous problems being solved today with this approach are reviewed, although in some of them, MAR comes in disguise.

Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization

A fast and accurate method is developed to compute the natural frequencies and scattering characteristics of arbitrary-shape two-dimensional dielectric resonators. The problem is formulated in terms of a uniquely solvable set of second-kind boundary integral equations and discretized by the Galerkin method with angular exponents as global test and trial functions. The log-singular term is extracted from one of the kernels, and closed-form expressions are derived for the main parts of all the integral operators. The resulting discrete scheme has a very high convergence rate. The method is used in the simulation of several optical microcavities for modern dense wavelength-division-multiplexed systems.

Plane wave scattering and absorption by resistive-strip and dielectric-strip periodic gratings

The problems of plane wave scattering by resistive- and dielectric-strip gratings are considered. The formulation involves a set of resistive-type boundary conditions that characterizes nonzero jumps in tangential field components. The method of solution is based on analytical inversion of the static part of the full-wave equations and results in a rapidly convergent numerical algorithm. The dependences of the transmitted, reflected, and absorbed power fractions on the electrical and material parameters are presented.

Periodicity-induced effects in the scattering and absorption of light by infinite and finite gratings of circular silver nanowires

We study numerically the effect of periodicity on the plasmon-assisted scattering and absorption of visible light by infinite and finite gratings of circular silver nanowires. The infinite grating is a convenient object of analysis because of the possibility to reduce the scattering problem to one period. We use the well-established method of partial separation of variables however make an important improvement by casting the resulting matrix equation to the Fredholm second-kind type, which guarantees convergence. If the silver wires have sub-wavelength radii, then two types of resonances co-exist and may lead to enhanced reflection and absorption: the plasmon-type and the grating-type. Each type is caused by different complex poles of the field function. The low-Q plasmon poles cluster near the wavelength where dielectric function equals -1. The grating-type poles make multiplets located in close proximity of Rayleigh wavelengths, tending to them if the wires get thinner. They have high Q-factors and, if excited, display intensive near-field patterns. A similar interplay between the two types of resonances takes place for finite gratings of silver wires, the sharpness of the grating-type peak getting greater for longer gratings. By tuning carefully the grating period, one can bring together two resonances and enhance the resonant scattering of light per wire by several times.

Optical Theorem Helps Understand Thresholds of Lasing in Microcavities With Active Regions

Within the framework of the recently proposed approach to view the lasing in open microcavities as a linear eigenproblem for the Maxwell equations with exact boundary and radiation conditions, we study the correspondence between the modal thresholds and field overlap coefficients. Macroscopic gain is introduced into the cavity material within the active region via the “active” imaginary part of the refractive index. Each eigenvalue is constituted of two positive numbers, namely, the lasing wavenumber and the threshold value of material gain. This approach yields clear insight into the lasing thresholds of individual modes. The Optical Theorem, if applied to the lasing-mode field, puts the familiar “” condition on firm footing. It rigorously quantifies the role of the spatial overlap of the mode E-field with the active region, whose shape and location are efficient tools of the threshold manipulation. Here, the effective mode volume in open resonator is introduced from first principles. Examples are given for the 1-D cavities equipped with active layers and distributed Bragg reflectors and 2-D cavities with active disks and annular Bragg reflectors.

Directional Emission, Increased Free Spectral Range, and Mode

Achieving single-mode operation and highly directional (preferably unidirectional) in-plane light output from whispering-gallery (WG)-mode semiconductor microdisk resonators without seriously degrading the mode Q-factor challenges designers of low-threshold microlasers. To address this problem, basic design rules to tune the spectral and emission characteristics of microscale optical cavity structures with nanoscale features by tailoring their geometry are formulated and discussed in this paper. The validity and usefulness of these rules is demonstrated by reviewing a number of previously studied cavity shapes with global and local deformations. The rules provide leads to novel improved WG-mode cavity designs, two of which are presented: a cross-shaped photonic molecule with introduced asymmetry and a photonic-crystal-assisted microdisk resonator. Both these designs yield degenerate-mode splitting, as well as Q-factor enhancement and directional light output of one of the split modes

Q

We assess the accuracy and relevance of the numerical algorithms based on the principles of geometrical optics (GO) and physical optics (PO) in the analysis of reduced-size homogeneous dielectric lenses prone to behave as open resonators. As a benchmark solution, we use the Muller boundary integral equations (MBIEs) discretized with trigonometric Galerkin scheme that has guaranteed and fast convergence as well as controllable accuracy. The lens cross-section is chosen typical for practical applications, namely an extended hemiellipse whose eccentricity satisfies the GO focusing condition. The analysis concerns homogeneous lenses made of rexolite, fused quartz, and silicon with the size varying between 3 and 20 wavelengths in free space. We consider the 2D case with both - and -polarized plane waves under normal and oblique incidence, and compare characteristics of the near fields.

-Factors in 2-D Wavelength-Scale Optical Microcavity Structures

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.

Small Hemielliptic Dielectric Lens Antenna Analysis in 2-D: Boundary Integral Equations Versus Geometrical and Physical Optics

Our objective is the assessment of the accuracy of a conventional finite-difference time-domain (FDTD) code in the computation of the near- and far-field scattering characteristics of a circular dielectric cylinder. We excite the cylinder with an electric or magnetic line current and demonstrate the failure of the two-dimensional FDTD algorithm to accurately characterize the emission rate and the field patterns near high-Q whispering-gallery-mode resonances. This is proven by comparison with the exact series solutions. The computational errors in the emission rate are then studied at the resonances still detectable with FDTD, i.e., having Q-factors up to 10(3).

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

Lasing modes in cyclic photonic molecules (CPMs) composed of several identical thin semiconductor microdisks in free space are studied in a linear approximation. Maxwells equations with exact boundary conditions and the radiation condition at infinity are considered as a specific eigenvalue problem that enables one to find natural frequencies and threshold gains. It is demonstrated that careful tuning of the distance between the disks in CPMs is able to drastically reduce the lasing thresholds of the whispering-gallery modes having small azimuth indices.