Coupled kinetic Boltzmann electromagnetic approach for intense ultrashort laser excitation of plasmonic nanostructures

We propose a multiphysical computational approach that allows for efficient coupling of full-vector Maxwell-based propagation codes with kinetic Boltzmann equations to investigate the spatial dynamics of non-equilibrium processes in plasmonic nanostructures upon intense laser excitation. Accessing the energy-resolved electron distribution provides a direct path towards multidimensional modeling of transient optical, electron emission, and electron transport processes. Simulations are performed for a gold nanoparticle upon infrared ultrashort-pulse excitation close to the melting threshold, evidencing the interplay between strong intrinsic and (non)thermal nonlinearities and accessing simultaneously the non-equilibrium thermal and propagation dynamics. While delivering the results within a reasonable simulation time and while being open to further extensions, the proposed approach can serve as a reliable compromise between point quantum and space-dimensional classical models.

Figure 1

Schematics of three main subparts in the proposed non-equilibrium model (Maxwell, kinetic Boltzmann, and macroscopic subparts). Involved couplings are indicated by the blue arrows, whereas the model additions and extensions are shown inside redline circles.

Figure 2

(a) Spatial distribution of fluence as the integral of the intensity over the pulse duration (inside the particle and in air), and (b) the free-carrier density distribution after the electron emission from the metal surface upon intense $200\mathrm{fs}$ ultrashort laser-pulse excitation with laser fluence of $0.1\mathrm{J}/{\mathrm{cm}}^2$. [(c),(d)] Time evolution of the distribution function at the chosen hot spot (c) in the front side, (d) in the rear side on the surface of the metal nanoparticle.

Figure 3

Energy-resolved non-equilibrium electron dynamics for (a) $50\mathrm{fs}$, (b) $200\mathrm{fs}$, and (c) $1\mathrm{ps}$ pulse excitations with $0.1\mathrm{J}/{\mathrm{cm}}^2$ fluence. The nonthermal electron and hole distributions $\mathrm{\Delta }{f}_{NT}=f\left(E\right)-f\left(T_e\right)$ are shown at the top of each subfigure. The dashed horizontal black line indicates the Fermi energy level $E_F$. The dashed vertical blue line shows the time delay for the establishment of quasi-Fermi-Dirac distribution within the electronic subsystem.

Figure 4

(a) Electron-ion temperature dynamics. The equilibrium temperature values are evaluated from kinetic Boltzmann approach. The black line indicates the melting temperature for gold $T_{\mathrm{melt}}\approx 1337$ K $(0.115$ eV). (b) Dynamics of the energy-averaged electron-electron or electron-surface collision frequencies. (c) Emitted carriers by photo- and thermionic emission. The results are shown for $50\mathrm{fs}$, $200\mathrm{fs}$, and $1\mathrm{ps}$ pulse durations with a fixed fluence $0.1\mathrm{J}/{\mathrm{cm}}^2$.

Figure 5

Fourier transforms from the normalized backward reflected electric field, showing the spectral nonlinearity. The responses near the third harmonic and central wavelength $\lambda =800$ nm are magnified and shown for $50\mathrm{fs}$, $200\mathrm{fs}$, and $1\mathrm{ps}$ pulses with $0.1\mathrm{J}/{\mathrm{cm}}^2$.

Figure 6

Fourier transform from the normalized electric field on the surface of the nanoparticle. [(a)–(c)] The optical nonlinearity for $F_0=0.1\mathrm{J}/{\mathrm{cm}}^2$, $F_0/4$, and $F_0/20$ intensities. [(d)–(f)] Optical nonlinearity from only thermal response (red), only intrinsic nonlinearity (blue), and total response (green) for $F_0$. Frequency dependencies near the third and the second harmonics are magnified in (b), (c), (e), and (f). Pulse duration is fixed to be $200\mathrm{fs}$.

Figure 7

(a) Fluence in case of linear energy deposition (without any thermal nonlinearity). The maximum amplitude inside the particle is indicated. (b) The difference between nonlinear and linear fluence inside the particle and in air. Pulse duration here is $200\mathrm{fs}$. Fluence is calculated as the integral of the intensity over the pulse duration.

Figure 8

Averaged non-equilibrium electron temperature distributions inside Au metal nanoparticle upon pulse excitation by $200\mathrm{fs}$ pulse and fluence $0.1\mathrm{J}/{\mathrm{cm}}^2$ [(a),(c)] $-50\mathrm{fs}$ before the pulse peak and [(b),(d)] $+50\mathrm{fs}$ after the pulse peak. Transport term is off for (a) and (b) and included for (c) and (d).