Modeling the time-dependent electron dynamics in dielectric materials induced by two-color femtosecond laser pulses: Applications to material modifications

Controlling the electron dynamics during laser-matter interactions is a key factor to control the energy deposition and subsequent material modifications induced by femtosecond laser pulses. One way to achieve this goal is to use two-color femtosecond laser pulses. In this paper, the electron dynamics in dielectric materials induced by two-color femtosecond laser pulses is studied by solving dedicated optical Bloch equations. This model includes photo- and impact ionization, the laser heating of conduction electrons, their recombination to the valence band, and their collisions with phonons. The influence of photon energies, laser intensities, and pulse-to-pulse delay is analyzed. Depending on the interaction process, colors cooperate to excite electrons or drive them independently. For the given laser parameters, an optimal pulse-to-pulse delay is found which enhances significantly the energy deposition into the material, in agreement with experimental observations.

Figure 1

Resonant ionization paths made of $\hslash {\omega }_1$ photons (red shortest arrow) and $\hslash {\omega }_2$ photons (blue longest arrow). The energy gap is $E_g=7.5\mathrm{eV}$. (a) Two incommensurable photon energies with a unique five-photon resonant combination. (b) Two commensurable photon energies with five- and three-photon resonant combinations. (c) Two commensurable photon energies with five-, four-, and three-photon resonant combinations. (d) Two commensurable photon energies with six-, five-, four-, and three-photon resonant combinations.

Figure 2

Ionization degree given by the two-level Bloch model at the end of the laser pulse as a function of $I$ and $r$: (a) case of Fig. 1 and (d) case of Fig. 1. Ionization degree as a function of $I$ and for several values of $r$ (same legend for these four subplots): (b) case of Fig. 1, (c) case of Fig. 1, (e) case of Fig. 1, and (f) case of Fig. 1. For the printed grayscale version, the following information allows one to distinguish different curves. (b), (c), (e), (f) The lines from the top to bottom at $I={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ correspond to $r=1$, $r=0.75$, $r=0.5$, $r=0.25$, and $r=0$, respectively.

Figure 3

(a) Ionization degree as a function of $r$ at $I=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, for the cases of Fig. 1 (see legend). (b) Number of absorbed photons as a function of $r$ at $I=5\times {10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, for the cases of Fig. 1 (see legend).

Figure 4

Energy density gained by CB electrons, at the end of the laser pulse, for the two-color laser of Fig. 1, with ${\tau }_{\mathrm{coh}}=10$ fs. (a) $U_{\mathrm{CB}}$ as a function of $I$ and $r$. (b) $U_{\mathrm{CB}}$ as a function of $I$ and several values of $r$. The horizontal dotted line represents the energy density at the equilibrium state given by Eq. (9), towards which $U_{\mathrm{CB}}$ saturates at high intensities in the presence of collisions. (c) $U_{\mathrm{CB}}$ as a function of $r$ and several values of $I$. For the printed grayscale version, the following information allows one to distinguish different curves. (b) The lines from the bottom to top at $I={10}^{10}\mathrm{W}/{\mathrm{cm}}^2$ correspond to $r=1$, $r=0.75$, $r=0.5$, $r=0.25$, and $r=0$, respectively. (c) The lines from the bottom to top for all $r$ correspond to $I={10}^{10}\mathrm{W}/{\mathrm{cm}}^2$, $I={10}^{11}\mathrm{W}/{\mathrm{cm}}^2$, $I={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, $I={10}^{13}\mathrm{W}/{\mathrm{cm}}^2$, and $I={10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, respectively.

Figure 5

$U_{\mathrm{CB}}$ as a function of $I$ and several values of $r$: without collisions (solid lines), with weak collisions ${\tau }_{\mathrm{coh}}=10$ fs (dashed lines), and with strong collisions ${\tau }_{\mathrm{coh}}=1$ fs [dotted lines, which are the same curves plotted in Fig. 4]. The horizontal dotted line in black represents the energy density of Eq. (9). The maximum energy that the electron subsystem can achieve is depicted by the horizontal dashed line in gray. For the printed grayscale version, the following information allows one to distinguish different curves. For each bunch of curves, the lines from the bottom to top at $I={10}^{10}$ $\mathrm{W}/{\mathrm{cm}}^2$ correspond to $r=1$, $r=0.75$, $r=0.5$, $r=0.25$, and $r=0$, respectively.

Figure 6

Time evolution of the ionization degree $Z$ induced by the two-color laser of Fig. 1 with $I={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ and $r=0.5$, calculated according to different right-hand-side terms in Eq. (1). The simulation starts at $t_1=t_2=0$ fs. For the printed grayscale version, the following information allows one to distinguish different curves. The lines from the bottom to top at $t=300$ fs correspond to the model version $\mathcal{L}$, $\mathcal{L}+{\mathcal{G}}_{\text{coh}}+{\mathcal{G}}_{\text{imp}}+{\mathcal{G}}_{\text{rec}}$, $\mathcal{L}+{\mathcal{G}}_{\text{coh}}$, and $\mathcal{L}+{\mathcal{G}}_{\text{coh}}+{\mathcal{G}}_{\text{imp}}$, respectively.

Figure 7

Ionization degree $Z$ as a function of the overall intensity $I$ of the two-color laser of Fig. 1, calculated according to different right-hand-side terms in Eq. (1) and several values of $r$: (a) $r=0$ (pure infrared), (b) $r=0.5$, and (c) $r=1$ (pure ultraviolet). For the printed grayscale version, the following information allows one to distinguish different curves. (a)–(c) The lines from the bottom to top at $I={10}^{11}\mathrm{W}/{\mathrm{cm}}^2$ correspond to the model version $\mathcal{L}$, $\mathcal{L}+{\mathcal{G}}_{\text{coh}}+{\mathcal{G}}_{\text{imp}}+{\mathcal{G}}_{\text{rec}}$, $\mathcal{L}+{\mathcal{G}}_{\text{coh}}$, and $\mathcal{L}+{\mathcal{G}}_{\text{coh}}+{\mathcal{G}}_{\text{imp}}$, respectively.

Figure 8

Energy gained by CB electrons $U_{\mathrm{CB}}$ as a function of the overall intensity $I$ of the two-color laser of Fig. 1, calculated according to different right-hand-side terms in Eq. (1) and several values of $r$: (a) $r=0$ (pure infrared), (b) $r=0.5$, and (c) $r=1$ (pure ultraviolet). The horizontal dotted line in black represents the energy density of Eq. (9). The maximum energy that the electron subsystem can achieve is depicted by the horizontal dashed line in gray. For the printed grayscale version, the following information allows one to distinguish different curves. (a)–(c) The lines from the bottom to top at $I={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$ correspond to the model version $\mathcal{L}$, $\mathcal{L}+{\mathcal{G}}_{\text{coh}}+{\mathcal{G}}_{\text{imp}}+{\mathcal{G}}_{\text{rec}}$, $\mathcal{L}+{\mathcal{G}}_{\text{coh}}$, and $\mathcal{L}+{\mathcal{G}}_{\text{coh}}+{\mathcal{G}}_{\text{imp}}$, respectively.

Figure 9

Ionization degree $Z$ (top row) and energy gained by CB electrons $U_{\mathrm{CB}}$ (bottom row), calculated at the end of the laser pulse, as a function of the time delay between the IR and UV colors $\mathrm{\Delta }{t}_{1,2}$. Several IR intensities $I_1$ (see legend) are added to the UV intensity $I_2$: (a), (d) $I_2={10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, (b), (e) $I_2=5\times {10}^{12}\mathrm{W}/{\mathrm{cm}}^2$, and (c), (f) $I_2={10}^{13}\mathrm{W}/{\mathrm{cm}}^2$. In (d)–(f) the horizontal dotted line in black represents the energy density of Eq. (9), at which the system saturates. For the printed grayscale version, the following information allows one to distinguish different curves. (a)–(f) The lines from the bottom to top at $\mathrm{\Delta }{t}_{1,2}=-300$ fs correspond to $I_1=0$, $I_1=5\mathrm{TW}/{\mathrm{cm}}^2$, $I_1=10$ $\mathrm{TW}/{\mathrm{cm}}^2$, $I_1=20$ $\mathrm{TW}/{\mathrm{cm}}^2$, $I_1=30\mathrm{TW}/{\mathrm{cm}}^2$, $I_1=40\mathrm{TW}/{\mathrm{cm}}^2$, and $I_1=50\mathrm{TW}/{\mathrm{cm}}^2$, respectively.

Figure 10

Top row: Electric field $E\left(t\right)$ given by Eq. (5) (gray curves) with the photon energies of Fig. 1 and the intensities $I_2={10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (UV color) and $I_1=2\times {10}^{13}\mathrm{W}/{\mathrm{cm}}^2$ (infrared). The related square-sine-like slowly varying envelopes $a_l{sin}^2[\pi (t-t_l)/{\tau }_l]$ are plotted for each color (see legend). Bottom row: Corresponding electron population in CB levels ${\rho }_{j,j}$, averaged in time intervals of 50 fs and calculated without recombination. (a), (d) UV comes 50 fs before IR (we consider $t_2=-50$ fs and $t_1=0$ fs). (b), (e) There is no time delay between the two colors (we consider $t_2=t_1=0$ fs). (c), (f) IR comes 50 fs before UV (we consider $t_2=0$ fs and $t_1=50$ fs). For the printed grayscale version, the following information allows one to distinguish different curves. (d)–(f) The lines from the top to bottom at $t=0$ correspond to ${\rho }_{1,1},{\rho }_{2,2},...,{\rho }_{11,11}$, respectively.

Figure 11

Equation (A10) applied to the two-level Bloch system: Allowed values of $|{\rho }_{0,1}{|}^2$ and ${\rho }_{1,1}$ according to the value of $C$.

Figure 12

(a) Example of the energy density gained by CB electrons in the Bloch model (solid curves) as a function of time, calculated following the conditions of Sec. 4 with ${\tau }_{\mathrm{coh}}=10$ fs. We consider a single-color laser of $a=0.1$ GV/m ($I\approx 2\times {10}^9\mathrm{W}/{\mathrm{cm}}^2$): The red curve accounts for pure infrared ($r=0$), while the blue curve accounts for pure ultraviolet ($r=1$). The derivative at half maximum (dotted curves), calculated with least squares, is taken as the heating rate (here, ${\mathcal{N}}_0{W}_{\mathrm{CB}}\approx 0.62\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^3$ and thus $\sigma \approx 621$ mho/cm for infrared; ${\mathcal{N}}_0{W}_{\mathrm{CB}}\approx 0.31\times {10}^{15}\mathrm{W}/{\mathrm{cm}}^3$ and $\sigma \approx 314$ mho/cm for ultraviolet). Simulation starts at $t_1=t_2=0$ fs. (b) Heating rate as a function of laser intensity for ${\tau }_{\mathrm{coh}}=10$ fs (solid lines) and ${\tau }_{\mathrm{coh}}=1$ fs (dashed lines). Same color legend. (c) Corresponding conductivity of CB electrons. For the printed grayscale version, the following information allows one to distinguish different curves. For every above-mentioned configuration at low intensity, the curve corresponding to $r=0$ is always above the curve corresponding to $r=1$.