Boundary Integral Operators in Linear and Second-order Nonlinear Nano-optics

Recent advances in the fabrication of nanoscale structures have enabled the production of almost arbitrarily shaped nanoparticles and so-called optical metamaterials. Such materials can be designed to have optical properties not found in nature, such as negative index of refraction. Noble metal nanostructures can enhance the local electric field, which is beneficial for nonlinear optical effects. The study of nonlinear optical properties of nanostructures and metamaterials is becoming increasingly important due to their possible uses in nanoscale optical switches, frequency converters and many other devices.The responses of nanostructures depend heavily on their geometry, which calls for versatile modeling methods. In this work, we develop a boundary element method for the modeling of surface second-harmonic generation from isolated nanoparticles of very general shape. The method is also capable of modeling spatially periodic structures by the use of appropriate Green’s function. We further show how to utilize geometrical symmetries to lower the computational time and memory requirements in the boundary element method even in cases where the incident field is not symmetrical.We validate the boundary element approach by the calculation of second-harmonic scattering from gold spheres of different radii. Comparison to analytical solution reveals that under one percent relative error is easily achieved. The method is then applied to model second-harmonic microscopy of single gold nanodots and second-harmonic generation from arrays of L- and T-shaped gold particles. The agreement between the calculations and measurements is shown to be excellent.To provide a more intuitive understanding of the optical response of nanostructures, we develop a full-wave spectral approach, which is based on boundary integral operators. We present a theory which proves that the resonances of a smooth scatterer are isolated poles that occur at complex frequencies. Other types of singularities, such as branch-cuts, may occur only via the fundamental Green function or material dispersion. We propose a definition of an eigenvalue problem at fixed real frequencies which gives rise to modes defined over the surface of the scatterer. We illustrate that these modes accurately describe the optical responses that are usually seen for certain particle shapes when using plane-wave excitations. With the spectral approach, the resonance frequencies and the modal responses of a scatterer can be found as intrinsic properties independent of any incident field. We show that the spectral theory is compatible with the Mie theory for pherical particles and with a previously studied quasi-static theory in the limit of zero frequency

Boundary Integral Operators in Linear and Second-order Nonlinear Nano-optics

Recent advances in the fabrication of nanoscale structures have enabled the production of almost arbitrarily shaped nanoparticles and so-called optical metamaterials. Such materials can be designed to have optical properties not found in nature, such as negative index of refraction. Noble metal nanostructures can enhance the local electric field, which is beneficial for nonlinear optical effects. The study of nonlinear optical properties of nanostructures and metamaterials is becoming increasingly important due to their possible uses in nanoscale optical switches, frequency converters and many other devices.The responses of nanostructures depend heavily on their geometry, which calls for versatile modeling methods. In this work, we develop a boundary element method for the modeling of surface second-harmonic generation from isolated nanoparticles of very general shape. The method is also capable of modeling spatially periodic structures by the use of appropriate Green’s function. We further show how to utilize geometrical symmetries to lower the computational time and memory requirements in the boundary element method even in cases where the incident field is not symmetrical.We validate the boundary element approach by the calculation of second-harmonic scattering from gold spheres of different radii. Comparison to analytical solution reveals that under one percent relative error is easily achieved. The method is then applied to model second-harmonic microscopy of single gold nanodots and second-harmonic generation from arrays of L- and T-shaped gold particles. The agreement between the calculations and measurements is shown to be excellent.To provide a more intuitive understanding of the optical response of nanostructures, we develop a full-wave spectral approach, which is based on boundary integral operators. We present a theory which proves that the resonances of a smooth scatterer are isolated poles that occur at complex frequencies. Other types of singularities, such as branch-cuts, may occur only via the fundamental Green function or material dispersion. We propose a definition of an eigenvalue problem at fixed real frequencies which gives rise to modes defined over the surface of the scatterer. We illustrate that these modes accurately describe the optical responses that are usually seen for certain particle shapes when using plane-wave excitations. With the spectral approach, the resonance frequencies and the modal responses of a scatterer can be found as intrinsic properties independent of any incident field. We show that the spectral theory is compatible with the Mie theory for pherical particles and with a previously studied quasi-static theory in the limit of zero frequency

Modes and resonances of plasmonic scatterers

We present a rigorous full-wave electromagnetic approach to analyze the modes and resonances of dielectric and plasmonic nanoparticles of practically any geometry. Using boundary integral operators, we identify the resonances as inherent properties of the particles and propose a modal expansion for their optical response. We show that the resonance frequencies are isolated points on the complex plane. The approach allows the particles to be analyzed without specifying an incident field, which can be separately tailored for the desired interaction with the modes.We also connect the general theory to the Mie theory in spherical geometry and provide a connection to the quasistatic theory. In comparison to earlier work on modes and resonances of scatterers, our approach has the benefit that modes are defined entirely over a compact boundary surface of the scatterer. Furthermore, the boundary integral operator is of second-kind Fredholm type, enabling the rigorous characterization of the resonances.Peer reviewe

Boundary element method for surface nonlinear optics of nanoparticles

We present the frequency-domain boundary element formulation for solving surface second-harmonic generation from nanoparticles of virtually arbitrary shape and material. We use the Rao-Wilton-Glisson basis functions and Galerkin’s testing, which leads to very accurate solutions for both near and far fields. This is verified by a comparison to a solution obtained via multipole expansion for the case of a spherical particle. The frequency-domain formulation allows the use of experimentally measured linear and nonlinear material parameters or the use of parameters obtained using ab-initio principles. As an example, the method is applied to a non-centrosymmetric L-shaped gold nanoparticle to illustrate the formation of surface nonlinear polarization and the second-harmonic radiation properties of the particle. This method provides a theoretically well-founded approach for modelling nonlinear optical phenomena in nanoparticles.Peer reviewe

Multipolar nonlinear light-matter interactions with Gaussian vector beams

We show that surface second-harmonic generation (SHG) with focused Gaussian vector beams can be described in terms of effective Mie-type multipolar contributions to the SHG signal even in the electric dipole approximation of constitutive relations. Traditionally, Mie-type multipoles arise from field retardation across nanoparticles. In our case, the multipolar light-matter interaction is due to excitation with Gaussian vector beams and the tensorial properties of the SH response. As different multipoles have different radiative properties, we demonstrate the presence of multipoles by measuring strongly asymmetric SH emission into reflected and transmitted directions from a nonlinear thin film with isotropic surface symmetry, where symmetric emission is expected using traditional formalisms based on plane-wave excitation. The proposed multipole approach provides a convenient way to explain the measured asymmetric emission. Secondly, we generalize the treatment beyond the electric dipole approximation and propose that analogous vector excitation-induced multipolar effects could also occur in the microscopic light-matter interaction. Our results may allow new possibilities to designing confined and thin nonlinear sources with desired radiation patterns.Peer reviewe

Multipolar second-harmonic emission with focused Gaussian beams

We show that electric-dipole-allowed surface second-harmonic (SH) generation with focused Gaussian beams can be described in terms of Mietype multipolar contributions to the SH signal. In contrast to the traditional case, where Mie multipoles arise from field retardation across nanoparticles, the multipoles here arise from the confined source volume and the tensorial properties of the SH response. We demonstrate this by measuring strongly asymmetric SH emission into reflected and transmitted directions from a nonlinear thin film with isotropic surface symmetry, where symmetric emission is expected using traditional formalisms based on plane-wave excitation. The proposed multipole approach provides a convenient way to explain the measured asymmetric emission. Our results suggest that the separation of surface and bulk responses, which have dipolar and higher-multipolar character, respectively, may be even more difficult than thought. On the other hand, the multipolar approach may allow tailoring of focal conditions in order to design confined and thin nonlinear sources with desired radiation patterns.Peer reviewe

Particle plasmon resonances in L-shaped gold nanoparticles

We present an extensive experimental and theoretical study of the particle plasmon resonances of L-shaped gold nanoparticles. For the small characteristic size of the particles, we observe more higher-order resonances than previously from related shapes, and show that a short-wavelength resonance arises from the particle arm width and is not the suggested volume plasmon. We interpret the resonances through the local vector electric field in the structure and by fully taking into account the particle symmetry.Peer reviewe