Modelling of the ultrafast dynamics and surface plasmon properties of silicon upon irradiation with mid-IR femtosecond laser pulses

We present a theoretical investigation of the ultrafast processes and dynamics of the produced excited carriers upon irradiation of silicon with femtosecond pulsed lasers in the mid-infrared (mid-IR) spectral region. The evolution of the carrier density and thermal response of the electron-hole and lattice subsystems are analyzed for various wavelengths ${\lambda }_L$ in the range between 2.2 and 3.3 μm, where the influence of two- and three-photon absorption mechanisms is explored. The role of induced Kerr effect is highlighted and it manifests a more pronounced influence at smaller wavelengths in the mid-IR range. Elaboration on the conditions that lead to surface plasmon (SP) excitation indicate the formation of weakly bound SP waves on the material surface. The lifetime of the excited SP is shown to rise upon increasing wavelength, yielding a larger one than that predicted for higher laser frequencies. The calculation of damage thresholds for various pulse durations ${\tau }_p$ shows that they rise according to a power law $(\sim {\tau }_p^{\zeta \left({\lambda }_L\right)})$ where the increasing rate is determined by the exponent $\zeta \left({\lambda }_L\right)$. Investigation of the multiphoton absorption rates and impact ionization contribution at different ${\tau }_p$ manifests a lower damage for ${\lambda }_L=2.5\mu \mathrm{m}$ compared to that for ${\lambda }_L=2.2\mu \mathrm{m}$ for long ${\tau }_p$.

Figure 1

(a) Computation of $\mathrm{Re}\left({\chi }^{{\left(3\right)}^{\prime }}\right)$ as a function of the wavelength [35, 49, 54]. The dependence of the Kerr coefficient on ${\lambda }_L$ and $N_e$ is illustrated for (b) low and (c) high carrier densities.

Figure 2

(a) Dependence of reflectivity on the Kerr effect. The laser intensity is sketched in arbitrary units $({\lambda }_L=2.2\mu \mathrm{m})$. (b) Effect of the Kerr effect on the intensity transmitted through the surface $({\lambda }_L=2.2\mu \mathrm{m})$ (normalized to 1). (c) Maximum change of reflectivity (for $n_2=0$ and $n_2\ne 0$) as a function of the wavelength. The solid line corresponds to variance of $max\left(\mathrm{\Delta }R\right)$ in the whole range of timepoints, while the dashed line represent $max\left(\mathrm{\Delta }R\right)$ up to the timepoint where the laser intensity is maximum ($E_p=0.1\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$ and ${\tau }_p=100\mathrm{fs}$, at $x=y=z=0$).

Figure 3

(a) Evolution of $T_e$, $T_L$, $N_e$ for $n_2\ne 0$ (at $x=y=z=0$). (b) Absorption coefficient evolution. The laser intensity in both graphs is sketched in arbitrary units ($E_p=0.1\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$, ${\tau }_p=100\mathrm{fs}$, ${\lambda }_L=2.2\mu \mathrm{m}$).

Figure 4

(a) Dependence of evolution of $N_e$ on the Kerr effect. The laser intensity is sketched in arbitrary units (${\lambda }_L=2.2\mu \mathrm{m}$). (b) Percentage of change of $max\left(N_e\right)$ (for $n_2=0$ and $n_2\ne 0$) as a function of the wavelength. The maximum $N_e$ is also illustrated ($E_p=0.1\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$ and ${\tau }_p=100\mathrm{fs}$ at $x=y=z=0$).

Figure 5

(a) Dependence of evolution $T_e$ and $T_L$ on the Kerr effect. (b) Percentage change of $max\left(T_L\right)$ (for $n_2=0$ and $n_2\ne 0$) as a function of the wavelength ($E_p=0.1\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$, ${\lambda }_L=2.2\mu \mathrm{m}$, and ${\tau }_p=100\mathrm{fs}$ at $x=y=z=0$).

Figure 6

Surface plasmon periodicity as a function of the (a) carrier density and (b) fluence at different wavelengths. $({\tau }_p=100\mathrm{fs})$.

Figure 7

Surface plasmon's (a) damping length, (b) decay length, and (c) lifetime at different laser wavelengths $({\tau }_p=100\mathrm{fs})$.

Figure 8

(a) Damage threshold fluences $E_{\mathrm{DT}}$ vs pulse duration. The inset shows an enhanced view of $E_{\mathrm{DT}}$ for ${\tau }_p$ in the range [100 fs, 500 fs] (points represent the simulated data values, while lines are derived after fitting using a power law ${\tau }_p^{\zeta \left({\lambda }_L\right)}$, i.e., thick dashed line for ${\lambda }_L=2.2\mu \mathrm{m}$, solid line for ${\lambda }_L=2.5\mu \mathrm{m}$, and thin dashed line for ${\lambda }_L=3.3\mu \mathrm{m}$). (b) Lattice temperature for two- and three-photon absorption for ${\lambda }_L=2.2\mu \mathrm{m}$ and ${\lambda }_L=2.5\mu \mathrm{m}$ at $E_p=0.1\mathrm{J}/\mathrm{c}{\mathrm{m}}^2$ ($x=y=z=0$).

Figure 9

Theoretical predictions of damage thresholds against experimental results for $s$- and $p$-polarized beams [64, 71] (${\tau }_p=90\mathrm{fs}$, at ${45}^{\circ }$ incident angle).