Nonadiabatic effect in high order harmonic generation revealed by a fully analytical method

TL;DR

The paper addresses analytical modeling of high-order harmonic generation (HHG) by introducing a fully analytical framework based on the strong-field approximation (SFA) with a perturbation expansion in the Keldysh parameter $\gamma$. It derives third-order ($\tau^{(3)}$) and fifth-order ($\Theta^{(5)}$) corrections for key trajectory quantities, including the excursion time, return energy, and phase, enabling a clear separation of adiabatic and nonadiabatic effects. A major contribution is the analytic calculation of nonadiabatic initial conditions at the tunnel exit, namely the exit velocity $\mathrm{Re}[v_{ex}]$ and position $\mathrm{Re}[x_{ex}]$, which can seed Coulomb-corrected HHG models. The results show that nonadiabatic corrections, especially the fifth-order term, can significantly enhance HHG yields at shorter wavelengths and align closely with NSFA, offering a path to fully analytical, Coulomb-included HHG modeling and faster macroscopic propagation simulations.

Abstract

We propose a fully analytical method for describing high-order harmonic generation (HHG). This method is based on the strong-field approximation (SFA) and electron-trajectory theory, but utilizes the perturbation expansion on the Keldysh parameter $γ$. This expansion allows us to clearly differentiate the nonadiabatic and adiabatic effects on HHG. We show that the nonadiabatic effect relating to high-order expansion depends on the laser wavelength and remarkably enhances the HHG yields for cases of short wavelengths, providing deeper insights into wavelength-dependent HHG yields which are important in producing attosecond pulses. Especially, our method provides the analytical and accurate descriptions of nonadiabatic exit velocity and position of the tunneling electron at the tunnel exit. These descriptions are meaningful for constructing a fully analytical and quantitative Coulomb-included HHG model, which is crucial in HHG-based attosecond measurement.

Figures (7)

• Figure 1: (a) The returned electron kinetic energy $E_{re}^{(3)}$ for $\gamma=0.5$ and (b) parameter-independent exponential factor coefficients $\theta_3$ as functions of the electron excursion time $\tau_0$.
• Figure 2: The higher-order factor $\mathrm{exp} (2\Theta_5)$ of HHG in Eq. (\ref{['eq:ADK']}) for different parameters. (a) $\mathrm{exp}(2\Theta_5)$ as a function of $\tau_0$ for various driving laser wavelength $\lambda$, with $I_0=\mathrm{4.9 \times 10^{14}~W/cm^2}$. (b) $\mathrm{exp}(2\Theta_5)$ as a function of $\tau_0$ for various laser intensity $I_0$, with $\lambda \mathrm{=1150~nm}$. The target atom is helium and the black solid lines, red dashed lines, blue short-dashed lines represent the results for $\gamma=0.25$, $0.45$, $0.65$ respectively.
• Figure 3: Error rates of TAE and FAE methods for short and long trajectories of the first return as a function of the adiabatic parameter $\gamma$. The laser wavelength is $\lambda\mathrm{=1600~nm}$, and the intensity ranges from $\mathrm{1.05\times 10^{14}~W/cm^2}$ to $\mathrm{1.28\times 10^{15}~W/cm^2}$, corresponding to the range of $0.7\geqslant \gamma \geqslant 0.2$.
• Figure 4: The exponential factor $\mathrm{Im}[ \Theta]$ as a function of electron return kinetic energy $E_{re}$ obtained using different methods when $\gamma=0.5$. The results for the S1 and L1 trajectories obtained by the NSFA method are shown in black solid and red long-dashed lines, while the results from of Eq. (\ref{['eq:ADK32']}) are represented by blue dotted and purple dash-dotted lines for the S1 and L1 trajectories, respectively. The results obtained by the Eq. (\ref{['eq:ADK52']}) are shown in green dash-dot-dotted and brown short-dashed lines for the S1 and L1 trajectories, respectively.
• Figure 5: Contributions of long (L) and short (S) trajectories of the first return (a), the second return (b) and the third return (c) to the HHG obtained with Eq. (\ref{['eq:frequency-dependent dipole moment2']}), and the HHG spectrum (d) involving contributions of all trajectories of these three returns obtained with Eq. (\ref{['eq:P']}). The results are for the helium atom driven by a laser field with intensity of $\mathrm{I=5\times 10^{14}~W/cm^2}$ and wavelength of $\lambda$ = 790 nm, corresponding to the parameter $\gamma=0.65$.