Low-frequency approximation for high-order harmonic generation by a bicircular laser field

We present low-frequency approximation (LFA) for high-order harmonic generation (HHG) process. LFA represents the lowest-order term of an expansion of the final-state interaction matrix element in powers of the laser-field frequency $\omega$. In this approximation the plane-wave recombination matrix element which appears in the strong-field approximation is replaced by the exact laser-free recombination matrix element calculated for the laser-field dressed electron momenta. First, we have shown that the HHG spectra obtained using the LFA agree with those obtained solving the time-dependent Schrödinger equation. Next, we have applied this LFA to calculate the HHG rate for inert gases exposed to a bicircular field. The bicircular field, which consists of two coplanar counter-rotating fields having different frequencies (usually $\omega$ and $2\omega$), is presently an important subject of scientific research since it enables efficient generation of circularly polarized high-order harmonics (coherent soft x rays). Analyzing the photorecombination matrix element we have found that the HHG rate can efficiently be calculated using the angular momentum basis with the states oriented in the direction of the bicircular field components. Our numerical results show that the HHG rate for atoms having $p$ ground state, for higher high-order harmonic energies, is larger for circularly polarized harmonics having the helicity $-1$. For lower energies the harmonics having helicity $+1$ prevails. The transition between these two harmonic energy regions can appear near the Cooper minimum, which, in the case of Ar atoms, makes the selection of high-order harmonics having the same helicity much easier. This is important for applications (for example, for generation of attosecond pulse trains of circularly polarized harmonics).

Figure 1

Comparison of the harmonic intensity obtained using the LFA (red curves) and the TDSE (blue curves) for HHG by hydrogen atom and a linearly polarized few-cycle laser pulse with a sine-squared envelope, wavelength 800 nm, intensity $3\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$, carrier-envelope phase ${105}^{\circ }$, and total pulse duration of four optical cycles (upper panel) and seven optical cycles (lower panel).

Figure 2

Logarithm of the differential photorecombination cross sections for Ar, $|D_{\pm }^j{|}^2$ (in a.u.), with $j=x$ ($y$) for upper (lower) panels and “$-$” (“$+$”) for left (right) panels, presented in false color in the momentum plane.

Figure 3

Logarithm of the differential photorecombination cross sections for Ar as a function of the electron energy $E_q$ for the $f_{-}$ state and ${\phi }_q=0$ (see the upper left panel of Fig. 2).

Figure 4

Logarithm of the ratio of the differential photorecombination cross sections for exact and plane-wave continuum states, presented similarly as in Fig. 2.

Figure 5

Harmonic intensity for Ar and bicircular field with component wavelengths 800 and 400 nm and equal component intensities $4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The two upper curves are obtained using the strong-field approximation, while the two lower curves are calculated in the LFA.

Figure 6

Same as in Fig. 5 but for component wavelengths 1300 and 650 nm and equal component intensities $2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 7

Focal-averaged harmonic intensity for Ar and bicircular field with component wavelengths 1300 and 650 nm and equal component peak intensities $I_0=2\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$.

Figure 8

Same as in Fig. 4 but for Ne atoms.

Figure 9

Harmonic intensity for Ne and bicircular field with component wavelengths 1300 and 650 nm and equal component intensities $4\times {10}^{14}\mathrm{W}/{\mathrm{cm}}^2$. The two upper curves are obtained using the strong-field approximation, while the two lower curves are calculated in the LFA.