Semiclassical approaches to below-threshold harmonics

We study the generation of below-threshold harmonics in a model atom by extending the three-step semiclassical model of harmonic generation to include effects of the atomic potential. We explore the generalization of semiclassical trajectories of the electron in the presence of the combined laser-atom potential and calculate the intensity-dependent dipole phase associated with these trajectories. Our results are in good agreement with fully quantum mechanical calculations, as well as with recent experimental observations. We show that the so-called long trajectory readily generalizes to below-threshold harmonic generation and is relatively insensitive to the choice of initial conditions. We also find that the short trajectory can only lead to low-energy harmonics for electrons that have been released close to the ion core in a process that is closer to multiphoton than to tunnel ionization.

Figure 1

Quantum path distributions for harmonics 7 through 13 of a 1.07-$\mu$m laser in hydrogen. The mostly vertical (colored) strips indicate $\alpha$ values of the dominant contributions to the dipole phase.

Figure 2

Visual representation of the initial conditions with atomic potential included. The red and black arrows represent the uphill and downhill OTB trajectories, respectively.

Figure 3

(a) Travel time versus return energy and (b) phase versus return energy, both for an intensity of ${10}^{14}$ W/${\text{cm}}^2$. Results from the tunneling trajectories are shown with (green) diamonds. Results from the OTB trajectories are shown with (red) squares for uphill trajectories and (blue) circles for downhill trajectories. With the thin solid line we show the result of the standard SCM. (c) Phase coefficient versus return energy calculated within the tunneling model, using Eq. (11), for intensities of ${10}^{14}$ W/${\text{cm}}^2$ (green diamonds) and $0.3\times {10}^{14}$ W/${\text{cm}}^2$ (orange circles).

Figure 4

Long trajectory dynamics in the tunneling-initiated generalized SCM. (a) Position versus time. The dashed (red) line represents the force from the laser field. (b) Energy versus time of the identical trajectory. The intensity is ${10}^{14}$ W/cm${}^2$ and the return energy is $-0.25$ a.u., which corresponds to $-0.58$ $U_p$, or approximately the seventh harmonic.

Figure 5

Dynamics of the short trajectory initiated by tunneling in the standard (solid black line) or generalized (dashed black line) SCM. The red dashed line represents the force from the laser field. (Inset) Position vs time for an uphill OTB trajectory, at an intensity of ${10}^{14}$ W/cm${}^2$ and a return energy of $-0.2$ a.u., or close to the energy of harmonic 7.

Figure 6

Semiclassical path distributions calculated from the generalized SCM, initiated by tunneling.

Figure 7

Semiclassical path distributions calculated from the generalized SCM, using the OTB model with downhill trajectories only.

Figure 8

Semiclassical path distributions calculated from the generalized SCM, using the OTB model with uphill trajectories only.