Linear ultrafast dynamics of plasmon and magnetic resonances in nanoparticles

In this study we present an analytical description of the ultrafast localized surface plasmon and magnetic resonance dynamics in a single nanoparticle (Ag or Si), driven by an ultrashort (fs time scale) Gaussian pulse. Three possible scenarios have been found depending on the incident field, i.e., pulse duration much shorter than, similar to, and much longer than the localized surface plasmon resonance (LSPR) lifetime. A rich physics arises for ${\tau }_{\mathrm{pulse}}<{\tau }_{\mathrm{LSPR}}$, even in the linear regime. The surface plasmon dynamics is manifested as (i) a temporal delay of the surface plasmon excitation with regard to the freely propagating pulse and as (ii) a negative exponential tail after the exciting pulse is over. In addition, for sub-fs pulses clear oscillations in the near-field decay have been observed. A similar scenario has been observed considering a nonabsorbing Si sphere. Nanoparticle resonance dynamics may lead to a wealth of new phenomena and applications in nanophotonics such as multipole order resonance interference, pulse-induced delay or temporal shaping on the fs scale, high harmonic generation, attosecond near-field pulse sources, and electron acceleration from metasurface or 3D engineered nanostructures.

Figure 1

(a) Scheme of the idea: a 40 nm diameter Ag nanoparticle placed in the beam waist of an incident Fourier-limited Gaussian pulse in the long-wavelength limit ($\lambda \gg d=2a$) under the paraxial approximation. In the beam waist the electric field is polarized along the $x$ axis. (b) Plane wave extinction efficiency of a 40 nm silver particle, computed considering radiative correction effect. (c), (d), (e) Normalized spectra of the laser pulse ($|E_{\mathrm{inc}}|$ in red) centered at ${\lambda }_{\mathrm{LSPR}}=369$ nm having a bandwidth bigger, similar, or smaller compared to the LSPR one, considering a pulse duration respectively of 0.5 fs, 5.4 fs (${\tau }_{\mathrm{LSPR}})$, and 50 fs.

Figure 2

Color maps showing the $E$-field enhancement ($|{\mathbf{E}}_1|/E_0$) inside the 40 nm Ag sphere on an ultrafast time scale. The plasmon resonance is excited with a Gaussian Fourier-limited pulse centered at ${\lambda }_0$ with a pulse duration of (a) 0.5 fs, (b) 5.4 fs, and (c) 50 fs. The dashed lines represent the resonance wavelength ${\lambda }_{\mathrm{LSPR}}$, while the solid white lines show the pulse bandwidths.

Figure 3

$E$ field induced inside the 40 nm (diameter) Ag sphere (solid black line) by a laser pulse centered at the LSPR wavelength, ${\lambda }_{\mathrm{LSPR}}=369$ nm, for different pulse durations: (a) ${\tau }_{\mathrm{pulse}}=0.5$ fs, (b) 5.4 fs, and (c) 50 fs. The absolute values of the electric fields have been normalized and plotted in a semilog scale in order to observe the shift of the LSPR-induced field with respect to the incoming one (dashed red line). For short pulses (${\tau }_{\mathrm{pulse}}\le {\tau }_{\mathrm{LSPR}}=5.4$ fs) it is clearly visible that the near field induced in the sphere is delayed, because of the resonant excitation, and decays as a Lorentz mode slower than the driving Gaussian pulse, showing ultrafast oscillations only for pulses much shorter than ${\tau }_{\mathrm{LSPR}}$ [panel (a)].

Figure 4

Difference in exciting the LSPR with (a) a 0.5 fs ultrashort pulse and (b) a 5.4 fs pulse. In both cases we show the $E_1\left(t\right)$ field inside the Ag particle excited with a pulse matching the resonance (solid black curve) and with a pulse of the same duration but out of resonance (dashed blue curve). The insets show the corresponding spectra of incoming pulses at and out of resonance compared to the resonance extinction curve (red). When pulses are really short (broadband), they are still able to excite the resonance as shown by the overlap of black and blue curves in (a). Vice versa when they have nonoverlapping bandwidth comparable to the resonance one; the $E$ field inside the sphere out of resonance, blue curve in (b), remains unchanged. (c) Delay of the $E$ field inside the sphere with respect to the incoming pulse as a function of pulse duration.

Figure 5

(a), (b), (c) Spectral Gaussian bandwidths (dashed black curves), compared to electric (red) and magnetic (green) dipole resonance contributions to the scattering cross section for a 460 nm Si sphere, computed by Mie theory, of the following pulses: (a) 50 fs centered at 1678 nm (centered at the magnetic dipole resonance), (b) 50 fs at 1262 nm (centered at the electric dipole resonance), and (c) 5.4 fs pulse at 1470 nm (between the two dipoles). (d), (e), (f) Dipole moments $p\left(t\right)$ and $m\left(t\right)$ computed by the inverse Fourier transform, with the incident pulses respectively as in (a), (b), (c). Clearly visible are the delays of the dipole resonance contributions with respect to the driving pulse in case (d) and (e) where the resonances are selectively excited, while in case (f) the pulse is so broadband that even if centered between the two dipolar resonances, it still excites both with a similar delay.

Figure 6

Delay of the maximum of the electric (left columns) and magnetic (right column) dipole moments excited inside the sphere as a function of the incoming pulse duration for different sphere diameters [(a), (b)] and as a function of the sphere diameter for different pulse durations [(c), (d)]. The Si refractive index is fixed at $n_{\mathrm{Si}}=3.5$.

Figure 7

Electric dipole moment induced in a single 40 nm Ag particle, solved in time by inverse Fourier transform with (continuous red line) and without (dashed red) considering radiation effect, considering a driving pulse ($E_{\mathrm{inc}}$) of time duration (FWHM) of (a) 0.5 fs, (b) $5.4\mathrm{fs}={\tau }_{\mathrm{LSPR}}$, and (c) 50 fs. All moments have been calculated by using Mie coefficients and the modified Drude-Sommerfeld model for the dielectric constant.