Optical Gain in Solids after Ultrafast Strong-Field Excitation

Multiphoton excitation of a solid by a few-cycle, intense laser pulse forms a very nonequilibrium distribution of charge carriers, where occupation probabilities do not necessarily decrease with energy. Within a fraction of the pulse, significant population inversion can emerge between pairs of valence-band states with a dipole-allowed transition between them. This population inversion leads to stimulated emission in a laser-excited solid at frequencies where the unperturbed solid is transparent. We establish the optimal conditions for observing this kind of strong-field-induced optical gain.

Figure 1

This schematic illustrates the ideal setting for the formation of population inversion between valence bands $v_1$ and $v_2$ at crystal momenta that form a star of $\mathbf{k}$. The left and right insets show the ${\mathbf{d}}_{c{v}_1}(\mathbf{k})$ and ${\mathbf{d}}_{c{v}_2}(\mathbf{k})$ vectors, respectively. These matrix elements control the probabilities of photoexcitation to the lowest conduction band. If the electric field of the photoexciting laser pulse is perpendicular to the vertical axis, then ${\mathbf{F}}_L·{\mathbf{d}}_{c{v}_2}(\mathbf{k})\equiv 0$, so that all the $\{v_2,\mathbf{k}\}\to \{c,\mathbf{k}\}$ transitions are strongly suppressed. At the same time, the laser pulse will efficiently drive some of the $\{v_1,\mathbf{k}\}\to \{c,\mathbf{k}\}$ transitions.

Figure 2

(a) The maximal population inversion in dependence of transition energy $\hslash \omega$ and peak laser field within the solid $F_L$. (b) Peak population differences responsible for absorption (see the text for details). The insets in (a) show the magnitudes and orientations of ${\mathbf{d}}_{cv}(\mathbf{k})$ for the two stars of $\mathbf{k}$ where population differences are particularly prominent; LH, SO, and CB denote the light-hole, split-off, and conduction bands.

Figure 3

The imaginary part of the effective refractive index for the two modes propagating along the [110] direction of the photoexcited GaN crystal. The insets display the polarization directions (${\mathbf{e}}_1$ and ${\mathbf{e}}_2$) relative to the crystal orientation. The blue and red circles represent the positions of nitrogen and gallium atoms, respectively.