Surface Integral Method for the Second Harmonic Generation in Metal Nanoparticles

Second harmonic (SH) radiation in metal nanoparticles is generated by both nonlocal-bulk and local-surface SH sources, induced by the electromagnetic field at the fundamental frequency. We propose a surface integral equation (SIE) method for evaluating the SH radiation generated by metal nanoparticles with arbitrary shapes, considering all SH sources. We demonstrate that the contribution of the nonlocal-bulk SH sources to the SH electromagnetic field can be taken into account through equivalent surface electric and magnetic currents. We numerically solve the SIE problem by using the Galerkin method and the Rao-Wilton-Glisson basis functions in the framework of the distribution theory. The accuracy of the proposed method is verified by comparison with the SH-Mie analytical solution. As an example of a complex-shaped particle, we investigate the SH scattering by a triangular nano-prism. This method paves the way for a better understanding of the SH generation process in arbitrarily shaped nanoparticles and can also have a high impact in the design of novel nanoplasmonic devices with enhanced SH emission

Spectral theory of electromagnetic scattering by a coated sphere

In this paper, we introduce an alternative representation of the electromagnetic field scattered from a homogeneous sphere coated with a homogeneous layer of uniform thickness. Specifically, we expand the scattered field using a set of modes that are independent of the permittivity of the coating, while the expansion coefficients are simple rational functions of the permittivity. The theory we develop represents both a framework for the analysis of plasmonic and photonic modes and a straightforward methodology to design the permittivity of the coating to pursue a prescribed tailoring of the scattered field. To illustrate the practical implications of this method, we design the permittivity of the coating to zero either the backscattering or a prescribed multipolar order of the scattered field, and to maximize an electric field component in a given point of space

Resonance frequency and radiative Q-factor of plasmonic and dielectric modes of small objects

The electromagnetic scattering resonances of a non-magnetic object much smaller than the incident wavelength in vacuum can be either described by the electroquasistatic approximation of the Maxwell's equations if its permittivity is negative, or by the magnetoquasistatic approximation if its permittivity is positive and sufficiently high. Nevertheless, these two approximations fail to correctly account for the frequency shift and the radiative broadening of the resonances when the size of the object becomes comparable to the wavelength of operation. In this manuscript, the radiation corrections to the electroquasistatic and magnetoquasistatic resonances of arbitrarily-shaped objects are derived, which only depend on the quasistatic current modes. Then, closed form expressions of the frequency-shift and the radiative Q-factor of both plasmonic and dielectric modes of small objects are introduced, where the dependencies on the material and the size of the object are factorized. In particular, it is shown that the radiative Q-factor explicitly depends on the multipolar components of the quasistatic modes

Cloaking of Arbitrarily-Shaped Objects with Homogeneous Coatings

We present a theory for the cloaking of arbitrarily-shaped objects and demonstrate electromagnetic scattering-cancellation through designed homogeneous coatings. First, in the small-particle limit, we expand the dipole moment of a coated object in terms of its resonant modes. By zeroing the numerator of the resulting rational function, we accurately predict the permittivity values of the coating layer that abates the total scattered power. Then, we extend the applicability of the method beyond the small-particle limit, deriving the radiation corrections of the scattering-cancellation permittivity within a perturbation approach. Our method permits the design of invisibility cloaks for irregularly-shaped devices such as complex sensors and detectors

Full-wave electromagnetic modes and hybridization in nanoparticle dimers

The plasmon hybridization theory is based on a quasi-electrostatic approximation of the Maxwell's equations. It does not take into account magnetic interactions, retardation effects, and radiation losses. Magnetic interactions play a dominant role in the scattering from dielectric nanoparticles. The retardation effects play a fundamental role in the coupling of the modes with the incident radiation and in determining their radiative strength; their exclusion may lead to erroneous predictions of the excited modes and of the scattered power spectra. Radiation losses may lead to a significant broadening of the scattering resonances. We propose a hybridization theory for non-hermitian composite systems based on the full-Maxwell equations that, overcoming all the limitations of the plasmon hybridization theory, unlocks the description of dielectric dimers. As an example, we decompose the scattered field from silicon and silver dimers, under different excitation conditions and gap-sizes, in terms of dimer modes, pinpointing the hybridizing isolated-sphere modes behind them.Comment: Supplemental material available upon reques

Full-wave analytical solution of second-harmonic generation in metal nanospheres

We present a full-wave analytical solution for the problem of second-harmonic generation from spherical nanoparticles. The sources of the second-harmonic radiation are represented through an effective nonlinear polarization. The solution is derived in the framework of the Mie theory by expanding the pump field, the nonlinear sources and the second-harmonic fields in series of spherical vector wave functions. We use the proposed solution for studying the second-harmonic radiation generated from gold nanospheres as function of the pump wavelength and the particle size, in the framework of the Rudnick-Stern model. We demonstrate the importance of high-order multipolar contributions to the second-harmonic radiated power. Moreover, we investigate the p- and s- components of the SH radiation as the Rudnick-Stern parameters change, finding a strong variation. This approach provides a rigorous methodology to understand second-order optical processes in metal nanoparticles, and to design novel nanoplasmonic devices in the nonlinear regime.Comment: 16 pages, 10 figure

Electromagnetic modes and resonances of two-dimensional bodies

The electromagnetic modes and the resonances of homogeneous, finite size, two-dimensional bodies are examined in the frequency domain by a rigorous full wave approach based on an integro-differential formulation of the electromagnetic scattering problem. Using a modal expansion for the current density that disentangles the geometric and material properties of the body the integro-differential equation for the induced surface (free or polarization) current density field is solved. The current modes and the corresponding resonant values of the surface conductivity (eigen-conductivities) are evaluated by solving a linear eigenvalue problem with a non-Hermitian operator. They are inherent properties of the body geometry and do not depend on the body material. The material only determines the coefficients of the modal expansion and hence the frequencies at which their amplitudes are maximum (resonance frequencies). The eigen-conductivities and the current modes are studied in detail as the frequency, the shape and the size of the body vary. Open and closed surfaces are considered. The presence of vortex current modes, in addition to the source-sink current modes (no whirling modes), which characterize plasmonic oscillations, is shown. Important topological features of the current modes, such as the number of sources and sinks, the number of vortexes, the direction of the vortexes are preserved as the size of the body and the frequency vary. Unlike the source-sink current modes, in open surfaces the vortex current modes can be resonantly excited only in materials with positive imaginary part of the surface conductivity. Eventually, as examples, the scattering by two-dimensional bodies with either positive or negative imaginary part of the surface conductivity is analyzed and the contributions of the different modes are examined

Volume Integral Formulation for the Calculation of Material Independent Modes of Dielectric Scatterers

In the frame of volume integral equation methods, we introduce an alternative representation of the electromagnetic field scattered by a homogeneous object of arbitrary shape at a given frequency, in terms of a set of modes independent of its permittivity. This is accomplished by introducing an auxiliary eigenvalue problem, based on a volume integral operator. With this modal basis the expansion coefficients of the scattered field are simple rational functions of the permittivity of the scatterer. We show, by studying the electromagnetic scattering from a sphere and a cylinder of dimensions comparable to the incident wavelength, that only a moderate number of modes is needed to accurately describe the scattered far field. This method can be used to investigate resonant scattering phenomena, including plasmonic and photonic resonances, and to design the permittivity of the object to pursue a prescribed tailoring of the scattered field. Moreover, the presented modal expansion is computationally advantageous compared to direct solution of the volume integral equation when the scattered field has to be computed for many different values of the dielectric permittivity, given the size and shape of the dielectric body

Quantum Theory of Radiative Decay Rate and Frequency Shift of Surface Plasmon Modes

In this paper we study, in the time domain, the interaction between localized surface plasmons and photons in arbitrarily shaped metal nanoparticles, by using the Hopfield approach to quantize the plasmon modes, where the electron oscillations are represented by a harmonic matter field linearly coupled to the electromagnetic radiation. The plasmon - photon coupling gives rise to dressed plasmon modes. We have found that the radiation does not induce a significant coupling among the different quasi-electrostatic plasmon modes for particles of size up to the plasma wavelength, but causes a frequency shift and an exponential decay in time of the modes. By solving the equations governing the expectation values of the plasmon creation and annihilation operators, we obtain a new closed-form full-wave expression for the decay rate and for the frequency shift of the plasmon modes. It is non-perturbative and it only depends on the surface charge distribution of the quasi-electrostatic plasmon modes. We validate the expression against the Mie theory for a nano-sphere of radius comparable to the plasma wavelength. Eventually, we investigate the decay rate and the frequency shift of the plasmon modes in isolated and interacting nanoparticle of non-canonical shape, as their size increases up to the plasma wavelength