Time-resolved dynamical Franz-Keldysh effect produced by an elliptically polarized laser

An analytical formula is reported describing the time-resolved dynamical Franz-Keldysh effect (Tr-DFKE) produced by an elliptically polarized laser at subfemtosecond timescale. The Houston function is assumed as the time-dependent wave function of the parabolic two-band system. For elliptical polarization, the resulting formula exhibits subcycle changes in the optical properties; the modulation of the dielectric function is smaller than that for linear polarization. In contrast, the subcycle modulation of the dielectric function, a significant feature of the Tr-DFKE, disappears for a circularly polarized laser. This analytical formula shows good qualitative agreement with the first-principle calculation employing the time-dependent density functional theory for diamond.

Figure 1

Changes in $\mathrm{Im}\left[\varepsilon (T_p,\omega )\right]$ $\left(\mathrm{\Delta }\mathrm{Im}\left[\varepsilon (T_p,\omega )\right]\right)$ under circularly [(b) and (e)], elliptically [(c) and (f)], and linearly [(d) and (g)] polarized lasers. (b)–(d) present the numerical results from the real-time TDDFT, and (e)–(g) are the results from analytical formulas Eqs. (8), (14), and (15), respectively. Panel (a) plots the electric field of the pump laser against the probe time, the range of which corresponds to that for the real-time TDDFT simulation.

Figure 2

Time average of $\mathrm{\Delta }\mathrm{Im}\left[\varepsilon (T_p,\omega )\right]$ for a laser field with frequency 0.4 eV and a field amplitude 10 MV/cm calculated from Eqs. (8), (14), and (15). For reference, the static FKE using the Tharmalingam formula [11] is represented (the black dashed line).