In the area of computational electromagnetics, there is an extensive literature on broadband solvers that were developed to analyze multiscale objects [1-11]. Some of these structures involved small details, the numerical solutions to which with conventional elements - such as triangles - required dense discretizations with respect to wavelength. Some other objects may have needed dense discretizations to accurately model equivalent currents at critical locations, even if their geometric features allowed larger elements. In any case, development and implementation of a broadband solver to handle such relatively large objects with dense discretizations are often associated with maintaining "low-frequency" stability [12-30], since the conventional methods tend to break down when discretization elements become small in comparison to the operating wavelength. Accuracy and efficiency are sought in terms of two components: formulation/ discretization and solution algorithms. In the context of formulation/discretization, alternative formulations have been developed, e.g., the augmented electric-field integral equation [14, 19], potential integral equations (PIEs) [23-26], and other formulations incorporating electric charges, to name a few for perfect electric conductors (PECs). In terms of solution algorithms, low-frequency-stable methods havebeencontinuouslyproposedand implemented. Diverse implementations of the low-frequency Multilevel Fast Multipole Algorithm (MLFMA) using multipoles [1,4], inhomogeneous plane waves [3, 12], or other expansion techniques [9, 11, 28-30] merely form one track on the development ofbroadband solution algorithms.

In the previous issue of SOLBOX (SOLBOX-20), we discussed periodic arrangements of dielectric rods with interesting electromagnetic properties [1]. As computational simulations demonstrate, when material and geometric properties — such as the shapes, sizes, and periodicities of the rods — are properly selected, these arrangements can behave like homogeneous objects with near-zero refractive indices. In SOLBOX-20, this phenomenon was shown by constructing triangular prisms and investigating the refraction of electromagnetic beams through them. Using cylindrical rods with 3.75 mm radius, 17 mm center-to-center periodicity, and 8.8 relative permittivity (corresponding to alumina), near-zero refractive index values could effectively be obtained at 10.3 GHz. As also shown in the same numerical set, modifying the material of the rods, i.e., using 4.0 relative permittivity, dramatically changed the behaviors of the structures, which seemed to possess ordinary refractive index values just above unity. Specifically, without using an appropriate material, such an arrangement of rods did not provide near-zero-index (NZI) characteristics, while the effective relative permittivity induced in the environment simply corresponded to that of a diluted material in vacuum.

With their exotic behaviors that can be useful in a plethora of applications, near-zero-index (NZI) materials have been topics of many recent studies in electromagnetics [1–11]. Specifically, by possessing refractive index values close to zero, such materials can be employed for electromagnetic tunneling, coupling, beam steering and splitting, to name a few [1–8]. Manipulations of electromagnetic waves become possible by NZI structures as they stretch the wavelength to infinity. For example, once electromagnetic waves enter an NZI object located in an ordinary medium (e.g., vacuum), they tend to leave the object in the directions perpendicular to the surface. By then shaping the NZI object, one can create directive beams in the desired angles, even using isotopic sources as the main excitations.

The design and analysis of nano-optical structures involving nanoparticles has been discussed several times in previous issues of this column, e.g., see SOLBOX-08 [1] and SOLBOX-16 [2] for nano-optical couplers, and SOLBOX-13 [3] for nano-arrays. Given a grid of nanoparticles, the general aim is to reach a configuration by keeping/deleting nanoparticles such that the final structure operates as desired. In the case of nano-optical couplers, the designed configurations provide efficient transmission of electromagnetic waves through sharp bends and corners, while they also allow for diverse transmission options at junctions. Despite their geometric simplicity, such couplers can hence enable the construction of complex nano-optical networks. In the case of nano-arrays, similar nanoparticle configurations are used for beam shaping, particularly to obtain directional radiation from isotropic sources. On the other hand, in all these cases easy-to-define geometrical properties and design specifications turn out to be remarkably challenging computational problems. Using surface integral equations, the numbers of unknowns can be maintained at reasonable levels; however, large numbers of possible configurations make it impossible to test each candidate design to find the best design. For this reason, an in-house implementation of genetic algorithms was used in [1–3] to limit the number of simulations, while reaching satisfactory results. It was shown that successful coupler and array designs can be obtained via several thousands (instead of billions) of trials per optimization.

High-frequency techniques [1] are excellent tools for analyzing electrically large structures, especially if their geometries do not possess complex interactions, e.g., those caused by resonances. However, as the technology (both in terms of software and hardware) develops, it becomes possible to solve increasingly large problems via full-wave solvers without resorting to approximations and assumptions. Specifically, full-wave methods enable the analysis of electrically large objects with more direct applications of Maxwell's equations, while fundamental mechanisms, such as diffractions, reflections, and ray-like behaviors, are expected to be revealed as simulation results rather than being introduced as initial assumptions. Such full-wave solvers not only supply reliable simulations for arbitrarily complex structures, but also support high-frequency techniques by providing reference results for further analyses.

Zero-index and near-zero-index (NZI) materials are artificial structures that have recently attracted great interest in the electromagnetics and optics communities, thanks to their potentials in manipulating electromagnetic waves for a plethora of applications [1-5]. As with all other types of metamaterials, NZI structures are built of small unit cells, such as dielectric, metallic, and plasmonic particles, although they are often homogenized and represented by using effective electromagnetic properties in their numerical simulations. In fact, when accurately performed, homogenization can provide excellent analysis abilities, particularly when these structures are integrated with other devices to construct larger systems. At the same time, even when an NZI structure is greatly simplified by using a homogeneous model, the resulting electromagnetic problem may still possess several numerical challenges, considering that the conventional implementations are developed for ordinary materials. Some of these difficulties, e.g., handling numerically small or large values in the computer environment, may be easier to overcome, while there are also challenges that arise due to inherent properties of the developed methods and algorithms and that need more elaborate techniques to solve.

The topic of SOLBOX-08 in the September 2017 issue of the URSI Radio Science Bulletin designed effective nano-couplers that consisted of nanoparticles to improve electromagnetic power transmission along bent nanowire systems [1]. It was shown that by finding optimal nanoparticle configurations and arrangements it was possible to significantly improve transmission, even for nanowires with sharp corners. For this purpose, genetic algorithms were employed to perform on/off (1/0) optimization on given nanoparticle grids. Even though they were compact, the designed nano-couplers could be so effective that they mitigated the need for smoothly curved bends that often wasted the available physical space. We note that nano-couplers also reduce cross-talk between different nanowire transmission lines, as they suppress diffracted waves at the bending locations [2].

From computational point of view, frequency-selective structures are challenging to analyze since they involve resonating elements. In fact, resonances are essential for the operation of these structures, i.e., they generally provide the required responses (shadowing, focusing, full transmission, etc.) when their elements resonate. These resonances can be studied well via analytical approaches, e.g., by using circuit representations to model unit cells. However, as for other structures, numerical simulations — particularly those based on three-dimensional models — can be extremely useful for complete analyses of electromagnetic characteristics of complex frequency-selective structures before their realizations. The challenge is that when there are resonating elements, linear systems constructed in numerical simulations tend to become very ill-conditioned. For example, when using iterative solvers, iteration counts can be extremely large, while a convergence to a given error threshold may not guarantee an accurate result.

In three-dimensional electromagnetic solvers, extreme values for electrical parameters typically lead to instability, inaccuracy, and/or inefficiency issues. Despite using the term “extreme,” such relatively large or small values of conductivity, permittivity, permeability, wavenumber, intrinsic impedance, and other electrical parameters are commonly observed in natural cases. Computational electromagnetic solvers adapt themselves to handle challenging cases by replacing exact models with approximate models, while minimizing the modeling error due to these transformations. For example, most metals with high conductivity values are assumed to be perfectly conducting, especially if the considered structure is comparable to the wavelength. This is very common for practical devices, such as antennas, metamaterials, filters, etc., at radio and microwave frequencies. In some cases — e.g., when the overall structure is small in terms of a wavelength — even a full-wave solver may not be required to analyze the underlying phenomena. Examples are circuit theory based on lumped elements and transmission-line modeling. On the other side, penetrable models are commonly used to represent dielectric and magnetic materials, when their electrical parameters (specifically, permittivity and permeability) have numerically “reasonable” values that facilitate their full-wave solutions without a fundamental issue. As the electrical parameters become extreme and other conditions (sizes, excitations, geometric properties) are satisfied, numerical approximations may again become useful, leading to the well-known implementations such as those based on impedance boundary conditions and physical optics.

In computational electromagnetics, nonuniform discretizations with a large variety in the sizes of the discretization elements have always been challenging to handle. Such problems are inherently multi-scale, where different regimes coexist, as small elements are used to model tiny details while large elements are used on suitable parts comparable to the wavelength. When the variety in the element size is large, conventional implementations often suffer from inaccuracy, instability, and/or inefficiency issues due to numerical breakdowns in the context of discretization, expansion, and/or matrix solution. As a natural consequence, there is an enormous collective effort in the literature [1–9] to develop accurate, stable, and efficient numerical solvers for multi-scale problems involving nonuniform discretizations.

Periodicity is a major property of many structures in electromagnetic applications, such as antenna arrays [1], frequency-selective surfaces, metamaterials [2], and photonic crystals [3]. Array elements (unit cells, antennas, etc.) are periodically arranged in order to enhance the response of a single element or to gain new capabilities due to interactions between them. Naturally, electromagnetic simulations of periodic structures have a long history in the literature. This is still an active area as the structures become more complicated and require more sophisticated (accurate, efficient, stable) tools for precise analysis. When the structure is infinitely large, and particularly when the periodicity is sub-wavelength, there are excellent tools based on Fourier series and finite-diff erence models, e.g., see [4] and [5]. Similarly, when the elements can be represented pointwise with or without interactions, there are analytical (e.g., using array factor) and semi-analytical (e.g., using lumped elements and/or transmission-line models) techniques. On the other hand, challenges arise when each element is a complex electromagnetic problem itself while it needs to interact with other complex elements. In addition, if the finiteness is an important geometric parameter of a periodic structure, its full-wave models may lead to large-scale problems that can be difficult to solve numerically. In such cases, it is tempting to substitute high-order mathematical models for the elements [6], specifically by isolating their inner dynamics to derive the required equivalence [7] within the decomposed models. A well-known method in this path is called the Equivalence-Principle Algorithm (EPA), which has been used and improved (e.g., hybridized with or used in other methods) by numerous researchers [7–11].

Nanowires are popular components of nano-optical systems because they can be useful in many related applications [1], such as optical transmission [2-4], sub-wavelength imaging [5, 6], and energy harvesting [7]. These structures are usually made of silver (Ag) or gold (Au), which are active at optical frequencies with strong plasmonic responses and which provide the favorable characteristics of nanowires. For example, by using a transmission line involving an arrangement of nanowires, electromagnetic energy can be carried to distances long with respect to wavelength. As the technology in this area develops, nanowires with improved geometric properties [8] – such as regularity, cross-sectional preciseness, and surface smoothness – become available, further expanding their usage. Naturally, electromagnetic simulations of nanowires [9-11], especially using their three-dimensional full-wave models, are essential to studying and understanding these important structures, as well as to designing them. In this issue of Solution Box, an optimization problem involving a nanowire transmission line to be improved by a coupler is presented (SOLBOX-08). Specifically, a pair of nanowires with a sharp 90° bend, which leads to significant deteriorations in the power transmission capability due to reflections and diffractions, is considered. The purpose was to design an efficient coupler in a limited space around the corner, in order to improve the transmission as much as possible. In the sample solution that is also considered in this issue, cubic nanoparticles were used to reduce the reflections and to improve the power transmission. Starting from a full grid, the existence and absence of each nanocube was decided based on an optimization via genetic algorithms (GAs). The trials during the optimization were efficiently performed by using the Multilevel Fast Multipole Algorithm (MLFMA) [12, 13]. This has been designed for accurate analysis of plasmonic objects [14-16] without resorting to approximate and asymptotic techniques. Different numerical solutions, analysis methods, and optimization tools to design more efficient couplers, probably leading to better transmission capabilities, are welcome. We are also looking for alternative solutions to previous problems (SOLBOX-01 to SOLBOX-07), which can be found in earlier editions of this column. Please consider submitting your contributions to Ozgur Ergul (ozergul@metu.edu.tr).

Photonic crystals are well-known electronic devices that enable modification, transformation, and selection of light or, more generally, electromagnetic waves [1]. Even though they have a long history of more than 100 years, these structures are becoming increasingly popular, thanks to the rapid developments in nanotechnology and material science, enabling the design and fabrication of very small details [2]. As in all areas of electromagnetics, numerical simulations of photonic crystals are important, since they can provide the analysis and investigation of these structures before their fabrication [3–12]. Three-dimensional simulations, where the photonic crystals are modeled as finite, aperiodic, and possibly inhomogeneous structures, are especially essential to understanding the electromagnetic responses of various designs [8, 11].

The topic of this issue is an optimization problem (SOLBOX-06), involving nanocubes at optical frequencies. This type of array arrangement (namely, a nanoarray) is useful for controlling and directing optical waves in numerous applications, such as optical links, sensing, and energy harvesting [1–6]. A particular aim in SOLBOX-06 is to maximize the scattering or radiation of a given array structure in desired directions. Despite the number of unknowns being relatively small, the main challenge is the size of the optimization space, which is grows exponentially with the number of nanocubes. In addition, the nanocubes are made of silver (Ag). They must therefore be modeled as penetrable objects with negative real permittivity, at optical frequencies. The sample solutions also included in this issue use genetic algorithms, combined with a solver based on surface integral equations, and the Multilevel Fast Multipole algorithm. Alternative solutions of the same problems with other solvers, possibly using different and more-efficient optimization tools (heuristic or gradient-based) and solution methods (e.g., volume integral equations, other types of acceleration algorithms, iterative solvers, and discretization techniques), are welcome. Please submit your solutions to ozergul@metu.edu.tr.