Quantum and Semiclassical Physics behind Ultrafast Optical Nonlinearity in the Midinfrared: The Role of Ionization Dynamics within the Field Half Cycle

Ultrafast ionization dynamics within the field half cycle is shown to be the key physical factor that controls the properties of optical nonlinearity as a function of the carrier wavelength and intensity of a driving laser field. The Schrödinger-equation analysis of a generic hydrogen quantum system reveals universal tendencies in the wavelength dependence of optical nonlinearity, shedding light on unusual properties of optical nonlinearities in the midinfrared. For high-intensity low-frequency fields, free-state electrons are shown to dominate over bound electrons in the overall nonlinear response of a quantum system. In this regime, semiclassical models are shown to offer useful insights into the physics behind optical nonlinearity.

Figure 1

Time-resolved radiation amplitudes $a_{\mathrm{bb}}$ (blue) and $a_{\mathrm{ff}}$ (red) of bound-state and free electrons (a),(b) and their spectra (c),(d) for a laser driver (purple dashed line) with $I_0=110\mathrm{TW}/{\mathrm{cm}}^2$, ${\tau }_0=10/{\omega }_0$, and ${\lambda }_0=0.8$ (a),(c) and $4.0\mu \mathrm{m}$ (b),(d). Results of calculations using the semiclassical model for free electrons are given with a green line. (e) The spectra of $a_{\mathrm{ff}}$ (green) and $a_{\mathrm{bf}}$ (blue and purple) calculated using the TDSE for a laser driver pulse with $I_0=110\mathrm{TW}/{\mathrm{cm}}^2$, ${\tau }_0=10/{\omega }_0$, and ${\lambda }_0=0.8$ (purple) and $4.0\mu \mathrm{m}$ (blue and green).

Figure 2

Relative peak spectral intensity of the third (top box) and fifth (bottom box) harmonics generated by bound-state (blue) and free (green) electrons calculated as a function of the driver field intensity for ${\lambda }_0=0.8$ (a) and $4.0\mu \mathrm{m}$ (b).

Figure 3

The maps of the probability density ${|\psi (\overrightarrow{r},t=0)|}^2$ for a driver field with $I_0=110\mathrm{TW}/{\mathrm{cm}}^2$, ${\tau }_0=10/{\omega }_0$, and ${\lambda }_0=0.8$ (a) and $4.0\mu \mathrm{m}$ (b).

Figure 4

Continuum population $C(t)$ calculated as a function of time using the TDSE (green), the density of free electrons calculated using the Yudin-Ivanov formula (blue), and the free-electron current density calculated using the TDSE (red) and the semiclassical model of $j_{\mathrm{pl}}$ (blue) for a driver field (dashed line) with $I_0=110$ (a),(b) and $50\mathrm{TW}/{\mathrm{cm}}^2$ (c),(d), ${\lambda }_0=0.8\mu \mathrm{m}$ (a),(c) and ${\lambda }_0=4.0\mu \mathrm{m}$ (b),(d), ${\tau }_0=10/{\omega }_0$.

Figure 5

(a) The continuum population as a function of time calculated within the field half cycle (dashed line) using the TDSE for ${\lambda }_0=0.8\mu \mathrm{m}$ (dash-dotted line) and ${\lambda }_0=4.0\mu \mathrm{m}$ (solid line) at $I_0=100\mathrm{TW}/{\mathrm{cm}}^2$. The inset shows the ratio $\eta$ of the continuum population after a field half cycle $C(T/2)$ for given ${\lambda }_0$ and $I_0$, normalized to $C(T/2)$ for ${\lambda }_0=0.8\mu \mathrm{m}$ and the same $I_0$, plotted as a function of ${\lambda }_0$ for $I_0=50$ (blue line) and $110\mathrm{TW}/{\mathrm{cm}}^2$ (green line). Also shown is the $\xi$ ratio as a function of ${\lambda }_0$ for $I_0=75\mathrm{TW}/{\mathrm{cm}}^2$ (orange line). (b) The ratio $\sigma$ calculated for $k=1$ (third harmonic) as a function of ${\lambda }_0$ using the TDSE (solid green line) versus two asymptotic dependences corresponding to the ${\omega }^{-2}$ and ${\omega }^{-3}$ scaling laws (dashed lines) for $I_0=110\mathrm{TW}/{\mathrm{cm}}^2$. The inset shows the Keldysh parameter $\gamma$ (dashed lines) and the ratio $\xi$ (solid lines) calculated using the TDSE as a function of the field intensity for ${\lambda }_0=0.8$ (green line), 4.0 (orange line), and $10\mu \mathrm{m}$ (blue line).