Lattice Resonances Excited by Finite-Width Light Beams

Abstract

Periodic arrays of metallic nanostructures support collective lattice resonances, which give rise to optical responses that are, at the same time, stronger and more spectrally narrow than those of the localized plasmons of the individual nanostructures. Despite the extensive research effort devoted to investigating the optical properties of lattice resonances, the majority of theoretical studies have analyzed them under plane-wave excitation conditions. Such analysis not only constitutes an approximation to realistic experimental conditions, which require the use of finite-width light beams, but also misses a rich variety of interesting behaviors. Here, we provide a comprehensive study of the response of periodic arrays of metallic nanostructures when excited by finite-width light beams under both paraxial and nonparaxial conditions. We show how as the width of the light beam increases, the response of the array becomes more collective and converges to the plane-wave limit. Furthermore, we analyze the spatial extent of the lattice resonance and identify the optimum values of the light beam width to achieve the strongest optical responses. We also investigate the impact that the combination of finite-size effects in the array and the finite width of the light beam has on the response of the system. Our results provide a solid theoretical framework to understand the excitation of lattice resonances by finite-width light beams and uncover a set of behaviors that do not take place under plane-wave excitation.

Lattice Resonances for Thermoplasmonics

“…On the other hand, once the light beam is wide enough to fully excite the lattice resonance, the response of the array does not improve by further increasing w inc . At this point, the system reaches the limiting case of plane-wave excitation, as we have shown recently . For this reason, we can understand the collectiveness of a lattice resonance from the way that the optical response of the array evolves with w inc .…”

“…In this situation, a significant asymmetry appears, with the slice along the y -axis becoming remarkably wider than the one along the x -axis. The explanation for this behavior is that, as we have recently shown in ref , when w inc is not large enough for the response of the array to be converged to the plane-wave excitation limit, the spatial extension of the lattice resonance exceeds that of the incident light beam. However, this only happens along the direction of propagation of the lattice resonance, which is always perpendicular to the polarization of the light beam that excites it.…”

“…As a consequence, Re{α –1 } takes increasingly large positive values, thus forcing the cancellation condition to shift closer to the Rayleigh anomaly, where Re { G xx false( boldk ∥ false) } is also increasingly large. Since the lattice sum represents the strength of the interactions between the nanoparticles of the array, a larger value corresponds to a lattice resonance for which these interactions play a bigger role and, therefore, is more collective in nature …”

“…Importantly, as a lattice resonance becomes more collective, it demands that a larger number of nanoparticles be uniformly excited in order to sustain it. This means that, if the light beam that is used to excite the array is not wide enough to uniformly illuminate a sufficiently large number of nanoparticles, the lattice resonance will not be efficiently excited, thus resulting in a weaker absorbance than could otherwise be obtained . On the other hand, once the light beam is wide enough to fully excite the lattice resonance, the response of the array does not improve by further increasing w inc .…”

“…We assume the arrays to be excited by a mainly x -polarized beam of finite width and a Gaussian spatial profile. The electric field of this beam at position R i is given by boldE i = 1 4 π 2 ∫ false| boldk ∥ false| ≤ n k normald boldk ∥ boldE ( k ∥ ) e i boldk ∥ · boldR i where E ( k ∥ ) = E inc [ x̂ – ẑ k x / k z ] f (| k ∥ |), f (| k ∥ |) = 2π w inc 2 exp[− w inc 2 | k ∥ | 2 /2], E inc is a constant, and w inc is a parameter that controls the width of the beam. Note that, for E i to satisfy Maxwell’s equations, E ( k ∥ ) · [ k ∥ + k z ẑ ] = 0, where k z = n 2 k 2 − false| boldk ∥ false| 2 .…”