High-order harmonics and supercontinua formed by a weak optical pump in the presence of an extreme terahertz field

We show that in the presence of an extreme terahertz (THz) field an additional weak field in the optical frequency range can modulate ionization probability and thereby generate high-order harmonics. The extreme THz pump suppresses recollisions, thereby also fully suppressing the recollision-based harmonics. Nevertheless, high-order harmonics are effectively generated via an alternative mechanism, the so-called Brunel mechanism, based on ionization and subsequent acceleration in the driving field (but not on reabsorption). Using ab initio simulations for the hydrogen atom and short few-cycle optical driver, we show the appearance of a coherent carrier-envelope-phase-insensitive supercontinuum formed by such harmonics, compressible into an isolated pulse with 100-attosecond-scale duration.

Figure 1

Nonlinear response of argon to a weak optical probe pulse in the presence of a strong THz field—a simple-man picture. (a) Exemplary electric field (upper blue line) consisting of a strong THz field ($E_{\mathrm{THz}}=0.06$ a.u.) centered at 100-$\mu \mathrm{m}$ wavelength and a weaker fundamental harmonic at 800-nm wavelength ($E_{\mathrm{opt}}=0.018$ a.u.), together with the ionization rate $W\left(t\right)$ (lower red curve) induced by such a wave shape in argon according to the simple-man tunnel model. The lower inset shows the composition of the full driver signal from strong THz $E_{\mathrm{THz}}$ and weak optical $E_{\mathrm{opt}}$ field. The upper inset shows the schematics of the basic mechanism. (b) Corresponding Brunel harmonic response as given by Eq. (1) (blue solid line) and Brunel harmonics created by the optical field alone with the amplitude $E_{\mathrm{opt}}=0.078$ a.u. [corresponding to $E_{\mathrm{opt}}+E_{\mathrm{THz}}$ from (a); black dashed line] and $E_{\mathrm{opt}}=0.018$ a.u. [corresponding to $E_{\mathrm{opt}}$ from (a); green dashed line]. The orange dotted line shows the impact from bound-bound transitions for the two-color case. The inset shows the intensity of $n\mathrm{th}$ harmonics (for several values of $n$) depending on $E_{\mathrm{opt}}$ (solid lines) as well as the impact of bound-bound transitions (dotted lines).

Figure 2

Slope $l$ in the expression $I_r\left(n\right)\sim {10}^{nl}$ as a function of the ionization energy with simultaneously rescaled frequencies, pump intensities, and durations as described in the text (red solid curve) and for fixed frequencies, pump durations, and intensities (black dashed line). Nearly constant slope indicates equivalent combs, that is, the same number of effectively radiated harmonics. Vertical lines exemplify ionization energies and band gaps of several materials (black labels) and the THz field strengths used for the simulations (red labels).

Figure 3

Radiation in the dipole approximation and beyond from a classical electron born at $t=0$ with zero initial velocity. (a) Nonzero components of the second derivatives of the electric $\mathbf{d}$ and magnetic $\mathbf{m}$ dipoles as a function of time for the parameters in Fig. 1. The other components related to the dipole and quadrupole electric and dipole magnetic moments are negligible. (b) Ratios of energy radiated by the electric dipole component $d_z$ in the $z$ direction ${\mathcal{E}}_{dz}$ (red dashed line) and of the energy radiated by the dipole magnetic momentum ${\mathcal{E}}_m$ (blue solid line) to the energy ${\mathcal{E}}_{dx}$ radiated by the electric dipole component $d_x$.

Figure 4

The response of the hydrogen atom according to TDSE simulations [Eqs. (5) and (6)] with $E_{\mathrm{opt}}$ given by a 5-fs-long pulse (a) and (b) with and (c) and (d) without a strong THz field. (a) and (c) The harmonic spectrum of the atomic response ${\mathbf{E}}_r$ according to Eq. (7). Spectral amplitudes for CEO phases of 0 (red solid lines) and $\pi$ (blue dashed line) and spectral phases (black solid and dashed lines for CEO phases of 0 and $\pi$, respectively). (b) and (d) Corresponding XFROG traces (see text for details). Additional parameters are given in the text.

Figure 5

(a) and (c) Spatiotemporal dynamics of the electron wave packet ${\left|\psi (x,t)\right|}^2$ simulated by 1D TDSE with the soft-core potential (a) for the $\mathrm{THz}+\mathrm{optical}$ field and (c) for only the optical pump. (b) and (d) The corresponding PESs calculated after the end of the pulse. The inset in (b) shows the corresponding PESs on a larger energy scale. The parameters are the same as in Fig. 4.

Figure 6

The square of the electric field depending on time for the Fourier-limited pulse obtained from the spectrum in Fig. 4 in the range between $\omega =1.5{\omega }_0$ and $\omega =10{\omega }_0$ (solid blue line). The FWHM duration of the resulting pulse is 120 as. To demonstrate the CEO insensitivity, the dashed orange and dotted green lines show the cases of the different CEO phases of the optical driver, compressed with the phase mask for the zero CEO phase. The red dash-dotted line shows the compressed pulse with the same lower but no upper frequency filtering limit. In this case the FWHM duration is 85 fs.