Launching surface plasmon waves via vanishingly small periodic gratings.

The scattering of electromagnetic waves by periodic layered media plays a crucial role in many applications in optics and photonics, in particular in nanoplasmonics for topics as diverse as extraordinary optical transmission, photonic crystals, metamaterials, and surface plasmon resonance biosensing. With these applications in mind, we focus on surface plasmon resonances excited in the context of insulator-metal structures with a periodic, corrugated interface. The object of this contribution is to study the geometric limits required to generate these fundamentally important phenomena. For this we use the robust, rapid, and highly accurate field expansions method to investigate these delicate phenomena and demonstrate how very small perturbations (e.g., a 5 nm deviation on a 530 nm period grating) can generate strong (in this instance 20%) plasmonic absorption, and vanishingly small perturbations (e.g., a 1 nm deviation on a 530 nm period grating) can generate nontrivial (in this instance 1%) plasmonic absorption.

Field enhancement and saturation of millimeter waves inside a metallic nanogap.

This paper investigates the millimeter electromagnetic waves passing through a metal nanogap. Based upon the study of a perfect electrical conductor model, we show that the electric field enhancement inside the gap saturates as the gap size approaches zero, and the ultimate enhancement strength is inversely proportional to the thickness of the metal film. In addition, no significant enhancement can be gained by decreasing the gap size further if the aspect ratio between the dimensions of the underlying geometric structure exceeds approximately 100.

Fast high-order perturbation of surfaces methods for simulation of multilayer plasmonic devices and metamaterials.

The scattering of time-harmonic linear waves by periodic media arises in a wide array of applications from materials science and nondestructive testing to remote sensing and oceanography. In this work we have in mind applications in optics, more specifically plasmonics, and the surface plasmon polaritons that are at the heart of remarkable phenomena such as extraordinary optical transmission, surface-enhanced Raman scattering, and surface plasmon resonance biosensing. In this paper we develop robust, highly accurate, and extremely rapid numerical solvers for approximating solutions to grating scattering problems in the frequency regime where these are commonly used. For piecewise-constant dielectric constants, which are commonplace in these applications, surface formulations are clearly advantaged as they posit unknowns supported solely at the material interfaces. The algorithms we develop here are high-order perturbation of surfaces methods and generalize previous approaches to take advantage of the fact that these algorithms can be significantly accelerated when some or all of the interfaces are trivial (flat). More specifically, for configurations with one nontrivial interface (and one trivial interface) we describe an algorithm that has the same computational complexity as a two-layer solver. With numerical simulations and comparisons with experimental data, we demonstrate the speed, accuracy, and applicability of our new algorithms.

A fast and high-order accurate surface perturbation method for nanoplasmonic simulations: basic concepts, analytic continuation and applications.

In this paper we demonstrate that rigorous high-order perturbation of surfaces (HOPS) methods coupled with analytic continuation mechanisms are particularly well-suited for the assessment and design of nanoscale devices (e.g., biosensors) that operate based on surface plasmon resonances generated through the interaction of light with a periodic (metallic) grating. In this connection we explain that the characteristics of the latter are perfectly aligned with the optimal domain of applicability of HOPS schemes, as these procedures can be shown to be the methods of choice for low to moderate wavelengths of radiation and grating roughness that is representable by a few (e.g., tens of) Fourier coefficients. We argue that, in this context, the method can, for instance, produce full and precise reflectivity maps in computational times that are orders of magnitude faster than those of alternative numerical schemes (e.g., the popular "C-method," finite differences, integral equations or finite elements). In this initial study we concentrate on the description of the basic principles that underlie the solution scheme, including those that relate to analytic continuation procedures. Within this framework, we explain how, in spite of conventional wisdom to the contrary, the resulting perturbative techniques can provide a most valuable tool for practical investigations in plasmonics. We demonstrate this with some examples that have been previously discussed in the literature (including treatments of the reflectivity and band gap structure of some simple geometries) and extend this to demonstrate the wider applicability of the proposed approach.

Boundary perturbation methods for high-frequency acoustic scattering: shallow periodic gratings.

Despite significant recent advances in numerical methodologies for simulating rough-surface acoustic scattering, their applicability has been constrained by the limitations of state-of-the-art computational resources. This has been particularly true in high-frequency applications where the sheer size of the full-wave simulations render them impractical, and engineering processes must therefore rely on asymptotic models [e.g., Kirchhoff approximation (KA)]. However, the demands for high precision can make the latter inappropriate, thus efficient, error-controllable methodologies must be devised. This paper presents a computational strategy that combines the virtues of rigorous solvers (error control) with those of high-frequency asymptotic models (frequency-independent computational costs). These methods are based on high-order "boundary perturbations," which display high precision and unparalleled efficiency. This is accomplished by incorporating asymptotic phase information to effect a significant decrease in computational effort, simultaneously retaining the full-wave nature of the approach. The developments of this contribution are constrained to configurations that preclude multiple scattering; it is further explained how the schemes can be made applicable to general scattering scenarios, though implementation details are left for future work. Even for single-scattering configurations, the approach presented here gives significant gains in accuracy when compared to asymptotic theories (e.g., KA) with modest additional computational cost.

An efficient high-order algorithm for acoustic scattering from penetrable thin structures in three dimensions.

This paper presents a high-order accelerated algorithm for the solution of the integral-equation formulation of volumetric scattering problems. The scheme is particularly well suited to the analysis of "thin" structures as they arise in certain applications (e.g., material coatings); in addition, it is also designed to be used in conjunction with existing low-order FFT-based codes to upgrade their order of accuracy through a suitable treatment of material interfaces. The high-order convergence of the new procedure is attained through a combination of changes of parametric variables (to resolve the singularities of the Green function) and "partitions of unity" (to allow for a simple implementation of spectrally accurate quadratures away from singular points). Accelerated evaluations of the interaction between degrees of freedom, on the other hand, are accomplished by incorporating (two-face) equivalent source approximations on Cartesian grids. A detailed account of the main algorithmic components of the scheme are presented, together with a brief review of the corresponding error and performance analyses which are exemplified with a variety of numerical results.

Shape deformations in rough-surface scattering: cancellations, conditioning, and convergence.

We analyze the conditioning properties of classical shape-perturbation methods for the prediction of scattering returns from rough surfaces. A central observation relates to the identification of significant cancellations that are present in the recurrence relations satisfied by successive terms in a perturbation series. We show that these cancellations are precisely responsible for the observed performance of shape-deformation methods, which typically deteriorates with decreasing regularity of the scattering surfaces. We further demonstrate that the cancellations preclude a straightforward recursive estimation of the size of the terms in the perturbation series, which, in turn, has historically prevented the derivation of a direct proof of its convergence. On the other hand, we also show that such a direct proof can be attained if a simple change of independent variables is effected in advance of the derivation of the perturbation series. Finally, we show that the relevance of these observations goes beyond the theoretical, as we explain how they provide definite guiding principles for the design of new, stabilized implementations of methods based on shape deformations.

Shape deformations in rough-surface scattering: improved algorithms.

We present new, stabilized shape-perturbation methods for calculations of scattering from rough surfaces. For practical purposes, we present new algorithms for both low- (first- and second-) and high-order implementations. The new schemes are designed with guidance from our previous results that uncovered the basic mechanism behind the instabilities that can arise in methods based on shape perturbations [D. P. Nicholls and F. Reitich, J. Opt. Soc. Am. A 21, 590 (2004)]. As was shown there, these instabilities stem from significant cancellations that are inevitably present in the recursions underlying these methods. This clear identification of the source of instabilities resulted also in a collection of guiding principles, which we now test and confirm. As predicted, improved low-order algorithms can be attained from an explicit consideration of the recurrence. At high orders, on the other hand, the complexity of the formulas precludes an explicit account of cancellations. In this case, however, the theory suggests a number of alternatives to implicitly mollify them. We show that two such alternatives, based on a change of independent variables and on Dirichlet-to-interior-derivative operators, respectively, successfully resolve the cancellations and thus allow for very-high-order calculations that can significantly expand the domain of applicability of shape-perturbation approaches.

Prescribed error tolerances within fixed computational times for scattering problems of arbitrarily high frequency: the convex case.

We present a new algorithm for the numerical solution of problems of electromagnetic or acoustic scattering by large, convex obstacles. This algorithm combines the use of an ansatz for the unknown density in a boundary-integral formulation of the scattering problem with an extension of the ideas of the method of stationary phase. We include numerical results illustrating the high-order convergence of our algorithm as well as its asymptotically bounded computational cost as the frequency increases.