Spectral caustic in two-color high-order harmonic generation: Role of Coulomb effects

The high-order harmonic spectrum generated in a two-color intense laser field under certain conditions has a pronounced maximum caused by the so-called spectral caustic. Using a numerical solution of the three-dimensional time-dependent Schrödinger equation we study the width of the maximum and the degree of the generation enhancement due to the caustic as a function of the fundamental intensity, wavelength, and atomic ionization potential. It is shown that the degree of this enhancement can be quantitatively characterized well by a single parameter: the ratio of the radius of the free electronic oscillation in the laser field to the atomic size. This behavior can be explained by the Coulomb attraction of the photoelectron to the parent ion, whereas the effect of this attraction on the width of the maximum is negligible. Choosing the field parameters providing a pronounced enhancement, we calculate the high-order harmonic macroscopic response taking into account the transient phase-matching. We show that the caustic feature can be used to provide a quasimonochromatic XUV source with an almost linear frequency vs time dependence. Such source can be used, in particular, for time-resolved single-shot XUV imaging.

Figure 1

Generating two-color field $E$ (black dashed line) and normalized kinetic energy of the recombining electron $W$ (red solid line) as a function of the ionization time $t_i$. The generating field is given by expression (1) with $a=0.44,\phi =-1.04$ in (a) and (b), and $a=0.65,\phi =-0.98$ in (c) an (d). The figures in the left and right columns differ only in the range of ${\omega }_0{t}_i$ shown.

Figure 2

Total energy of the high-order harmonics calculated numerically via 3D TDSE solution for xenon using Eq. (2) for different parameters of the two-color generating field $a$ and $\phi$. The line shows the parameters for which the two maxima of the classical kinetic energy of the returning electron have the same value (see Fig. 1 and text for more details). The fundamental intensity is $0.5\times {10}^{14}\mathrm{W}{\mathrm{cm}}^{-2}$, the fundamental wavelength is (a) $\lambda =1.83\mu \mathrm{m}$, which corresponds to the Keldysh parameter $\gamma =0.62$, (b) $\lambda =1.52\mu \mathrm{m}$ ($\gamma =0.75$), and (c) $\lambda =1.30\mu \mathrm{m}$ ($\gamma =0.87$).

Figure 3

Normalized spectra calculated via 3D TDSE numerical solution for neon and xenon in the two-color field with parameters $a=0.44,\phi =-1.04$, and the Keldysh parameter $\gamma =0.75$. The spectra for (a) the same wavelength and different intensities and atomic species and (b) neon at two different wavelengths and intensities.

Figure 4

Degree of the HHG enhancement due to the spectral caustic [given by the expression (6)] calculated for neon and xenon as a function of (a) the Keldysh parameter and (b) the parameter $r$ given by Eq. (8). Fundamental intensities and corresponding values of the $\beta$ parameter are presented in the graphs. The black dotted line in (b) shows approximations of this value by Eq. (7).

Figure 5

Normalized width of the maximum due to the spectral caustic for neon and xenon as a function of the parameter $U_p/{\omega }_0$. Fundamental intensities are the same as in Fig. 4. Black dotted line shows the approximation of the width by Eq. (12).

Figure 6

(a) Gabor transform of the microscopic high-frequency response calculated for neon in the two-color field. (b) Gabor transform of the macroscopic response calculated taking into account the transient phase-matching. The peak fundamental intensity is $2.0\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, the wavelength is $1.3\mu \mathrm{m}$, the full pulse duration $2{\tau }_{\mathrm{edge}}$ is 56 fundamental cycles (duration at the half of the peak intensity level is 90 fs), the gas density is $N=5.0\times {10}^{17}{\mathrm{cm}}^{-3}$, and the target thickness is $z_{\mathrm{full}}=0.1$ cm.

Figure 7

Same as Fig. 6, but for the two times shorter pulse and the two times higher gas density.