Strong-field approximation for the wavelength scaling of high-harmonic generation

We derive analytic expressions for the spectral amplitude of high-order harmonic generation from a single atom using the strong-field approximation (SFA). We demonstrate good agreement with time-dependent Schrödinger equation calculations, including the dependence on the drive wavelength across the range 566–2260 nm. Previous claims of a discrepancy in the drive wavelength scaling ignore changes in the timing of trajectories corresponding to a fixed harmonic photon energy. Under this condition, tunnel ionization is shown to play the most important role for the short trajectories. The established agreement enables us to use the SFA to predict that the intensity at the classical cutoff scales inversely with the ninth power of the drive wavelength in the high photon energy limit when the Coulomb singularity dominates the recombination amplitude.

Figure 1

Classical electron position $x$ (bottom row) in sinusoidal electric fields (top row) of unit amplitude and frequencies ${\omega }_{\mathrm{L}}=1$ a.u. (blue dashed lines) and ${\omega }_{\mathrm{L}}=0.5$ a.u. (red solid lines). The left (right) column shows short (long) trajectories of the red field; both columns show the cutoff trajectory of the blue field. For the purposes of illustration, we choose the relative phase of the fields (whose periods differ by a factor of 2) so that the recombination times of the trajectories coincide. The trajectories of the red field are chosen to have recombination kinetic energy equal to the cutoff of the blue field. Electron birth events are indicated by dots.

Figure 2

Comparison of the various stationary phase approximations to the spectral intensity for ${\lambda }_{\mathrm{L}}=566$ nm. Below the classical cutoff at ${\omega }_{\mathrm{c}}=2.64$ a.u., second-order stationary phase approximations are shown for short and long trajectories (thin dark blue solid lines), sum of long and short contributions (thin dark blue dashed line), and uniform approximation (thin light red solid line). Above the cutoff, third-order stationary phase approximations are shown with (thick dark blue solid line) and without (thin light red dashed line) first-order behavior of the integrand. Inset: zoomed-in view around the cutoff.

Figure 3

(a) and (b) Harmonic spectrum computed using TDSE (dark blue lines) and SFA (light red lines, second-order SPA below cutoff, augmented third-order SPA above cutoff) with drive wavelength at (a) 2263 nm and (b) 566 nm. The contributions from the long and short trajectories are displayed separately below the cutoff.

Figure 4

(a) Dipole spectral intensity at harmonic frequency $\omega =4$ a.u. versus drive wavelength for a single laser half cycle of intensity $5\times {10}^{14}$ W/cm${}^2$ in helium; SFA (light red solid lines) and TDSE (dark blue dots) for short (S) and long (L) trajectories; power law fits to the TDSE results for ${\lambda }_{\mathrm{L}}>800$ nm (dashed dark blue lines). (b) Peak value of the dipole spectral intensity near the cutoff versus drive wavelength, otherwise identical parameters to (a).

Figure 5

Drive wavelength dependence of electron spatial and temporal spreading for short (S, dark blue solid lines), long (L, dark blue solid lines), and cutoff (C, light red solid lines) trajectories; parameters equal to Fig. 4a. (a) Trajectory duration; lines proportional to ${\lambda }_{\mathrm{L}}$ (light red dashed lines) are shown as a guide to the eye. (b) Attosecond linear temporal chirp. (c) Full scaling due to wave-packet spreading and attosecond chirp with power-law fits for ${\lambda }_{\mathrm{L}}>800$ nm (dashed dark blue lines).

Figure 6

Dependence on drive wavelength of (a) laser field phase at moment of electron birth (in units of $\pi$), and (b) amplitude squared of the zero-transverse-momentum component of the launched wave packet for short (S, dark blue solid lines), long (L, dark blue solid lines), and cutoff (C, light red solid lines) trajectories. The dark blue dashed lines show power-law fits for ${\lambda }_{\mathrm{L}}>800$ nm. (For the long trajectory, the power-law fit is indistinguishable from the data at this scale.) The parameters are equal to those of Fig. 4a.

Figure 7

(a) Normalized continuum wave-packet scaling factors for short (dark blue solid lines) and long (light red solid lines) trajectories (a) as a function of normalized frequency $\overline{\omega }$ and (b) as a function of the ratio of laser wavelength to that which results in cutoff, ${\epsilon }^{-1}={\lambda }_{\mathrm{L}}/{\lambda }_{\mathrm{LC}}$. Power-law fits are shown for $2<{\epsilon }^{-1}<10$ (dashed lines).