Optimization of resonant effects in nanostructures via Weierstrass factorization

Abstract

International audienceExtensive numerical computations are often required to optimize the optical properties of nanostructures. Here, we use the Weierstrass factorization theorem to express the scattering matrix in terms of spectral singularities. We show that the location of poles and zeros of the scattering matrix fully determines all scattering properties, and that the scattering spectra of nanostructures can be decomposed into Lorentzian resonances over an arbitrary range of frequencies. This technique is applied to design nanoshell particles with enhanced absorption

Optimizing Nanoparticle Designs for Ideal Absorption of Light

“…These results were computed by taking advantage of the newly developed Weierstrass factorization for resonant photonic structures. 13,14 One readily remarks that this solution tends to the well-known quasi-static dipolar plasmon resonance at ε ̅ s = −2 when kR → 0. When only this IA mode is of interest, one can try to describe IA with a "point-like" model aimed at providing approximate descriptions of the lowest {ε ̅ s } electric dipole resonance.…”

“…18 The S-matrix of a lossless scatterer is characterized by zeros (absorbing modes) in the upper-half complex frequency plane, and poles (emitting modes) in the lower-half plane frequency plane. 13,14 Adding absorption to the particle causes the absorbing modes to descend toward the lower complex plane with IA occurring at those values of ε ̅ s for which a zero of the S-matrix lies on the real frequency axis, that is, there exists an n such that…”

“…Finding these solutions is quite difficult when using commonly employed techniques involving Cauchy integrals in the complex plane or conjugate gradient methods, but they can be readily solved using techniques based on Weierstrass factorization, 13,14 which exploit the meromorphic expansion of eq 14.…”

“…Since these models are limited to treatments of fundamental electric modes and make rather inaccurate predictions for all but the smallest particle sizes, we derive significantly improved analytic formulas in eq 16 predicting the required electric and magnetic fundamental dipole IA solutions in homogeneous scatterers. Section 3 shows numerical results using a Weierstrass factorization method 13,14 that determines fundamental and higher order IA modes to arbitrary accuracy. After some discussions and calculations concerning multimode absorption and IA bandwidth, Section 4 illustrates some predictions for IA in homogeneous spheres with gold and silver at specific frequencies and sizes.…”

Purcell factor of spherical Mie resonators

“…The normal modes of the system are solutions of Maxwell equations in the absence of sources, with outgoing radiation conditions, and with a certain normalization [26,[28][29][30]. Due to the outgoing radiation conditions, the Hamiltonian of the system is non-hermitian and the normal modes of the system have complex eigenfrequencies ω µ [26,31,32]. Let us notice that a normalization condition of the kind V |E µ (r, ω µ )| 2 dV cannot be applied to normal modes with resonant complex frequencies ω µ because E µ (|r| → ∞, ω µ ) → ∞ [20,[26][27][28][29]33].…”