Strong-field approximation for harmonic generation in diatomic molecules

The generation of high-order harmonics in diatomic molecules is investigated within the framework of the strong-field approximation. We show that the conventional saddle-point approximation is not suitable for large internuclear distances. An adapted saddle-point method that takes into account the molecular structure is presented. We analyze the predictions for the harmonic-generation spectra in both the velocity and length gauges. At large internuclear separations, we compare the resulting cutoffs with the predictions of the three-step semiclassical mechanism. Good agreement is obtained only by using the adapted saddle-point method combined with the velocity gauge.

Figure 1

(Color online) The absolute value of the projection on the internuclear axis of $\mathbf{\nabla }{\tilde{\psi }}_0\left(\mathbf{k}\right)$ is shown for the LCAO approximation (solid curve) and the exact ground state (dashed curve) of ${{\mathrm{H}}_2}^{+}$. The internuclear distance is $R=2$ atomic units.

Figure 2

(Color online) The maximal kinetic energy upon arriving at nucleus $B$ of an electron detached with zero initial kinetic energy from the nucleus $A$, as a function of the internuclear distance. Solid curve: result for a monochromatic laser field; see Ref. 24. Dashed curve: result for a trapezoidal envelope (see text).

Figure 3

Calculated harmonic spectra using the length gauge and the conventional saddle-point method (see text), for various internuclear distances in ${{\mathrm{H}}_2}^{+}$: (a) $R=0$ (the atomic case, $I_{\mathrm{p}}=1$), (b) $R=0.5\pi {\alpha }_0$, (c) $R=\pi {\alpha }_0$, and (d) $R=1.5\pi {\alpha }_0$. The laser and molecule parameters are described in the beginning of Sec. 4, and the arrows show the cutoffs of the transfer harmonics calculated from the simple-man’s model.

Figure 4

(Color online) Harmonic spectra (same parameters as in Fig. 3), but calculated by using the adapted saddle-point method and the length gauge. The lower curve in panel (d) shows the direct harmonics only. The arrows show the cutoffs of the transfer harmonics calculated from the simple-man’s model (see Fig. 2).

Figure 5

Calculated harmonic spectra using the velocity gauge and the conventional saddle-point method (see text), for various internuclear distances in ${{\mathrm{H}}_2}^{+}$: (a) $R=0$ (the atomic case, $I_{\mathrm{p}}=1$), (b) $R=0.5\pi {\alpha }_0$, (c) $R=\pi {\alpha }_0$, and (d) $R=1.5\pi {\alpha }_0$. The laser and molecule parameters are described in the beginning of Sec. 4. The arrows show the cutoffs of the transfer harmonics calculated from the simple-man’s model.

Figure 6

(Color online) Harmonic spectra (same parameters as in Fig. 3), but calculated by using the adapted saddle-point method and the velocity gauge. The lower curve in panel (d) shows the direct harmonics only. The arrows show the cutoffs of the transfer harmonics calculated from the simple-man’s model.

Figure 7

Transfer harmonics (same parameters as in Fig. 3), calculated using the adapted saddle-point method in the length gauge (left column) and the velocity gauge (right column). The arrows show the cutoffs of the transfer-harmonics spectrum, predicted by the simple-man’s model.