Theory of high-order harmonic generation from molecules by intense laser pulses

We show that high-order harmonics generated from molecules by intense laser pulses can be expressed as the product of a returning electron wave packet and the photo-recombination cross section where the electron wave packet can be obtained from the simple strong-field approximation (SFA) or from a companion atomic target. Using these wave packets but replacing the photo-recombination cross sections obtained from SFA or from the atomic target by the accurate cross sections from molecules, the resulting high-order harmonic spectra are shown to agree well with the benchmark results from direct numerical solution of the time-dependent Schrödinger equation, for the case of H2 in laser fields. The result illustrates that these powerful theoretical tools can be used for obtaining high-order harmonic spectra from molecules. More importantly, the results imply that the photo-recombination cross section extracted from laser-induced high-order harmonic spectra can be used for time-resolved dynamic chemical imaging of transient molecules with temporal resolutions down to a few femtoseconds. (Some figures in this article are in colour only in the electronic version) High-order harmonic generation (HHG) is one of the most studied nonlinear phenomena in intense laser–matter interaction [1]. HHG is most easily understood using the three-step model [2–4]: first, the electron is released by tunnel ionization; second, it is accelerated by the oscillating electric field of the laser and later driven back to the target ion; and third, the electron recombines with the ion to emit a highenergy photon. A semiclassical formulation of the threestep model based on the strong-field approximation (SFA) is given by Lewenstein et al [4]. In this model (often called the Lewenstein model), the liberated continuum electron experiences the full effect from the laser field, but the effect of the target potential is neglected. Despite this limitation, the Lewenstein model has been widely used for studying HHG from atoms and molecules, and for characterizing the nature of the harmonics generated. Since the continuum electron needs to come back to revisit the parent ion in order to emit radiation, the neglect of the electron–ion interaction in the SFA model is rather questionable. In the past few years, various efforts have been made to improve upon the SFA model, i.e., by including the Coulomb distortion [5–8] or by using the Ehrenfest theorem [9]. All of these improvements can still be considered insufficient since they fail to accurately include the scattering of the continuum electrons with the parent ions. 0953-4075/08/081002+06$30.00 1 © 2008 IOP Publishing Ltd Printed in the UK J. Phys. B: At. Mol. Opt. Phys. 41 (2008) 081002 Fast Track Communication With moderate effort, direct numerical solution of the time-dependent Schrödinger equation (TDSE) for atomic targets can be accurately carried out, at least within the single active electron (SAE) approximation. The TDSE methods are computationally more demanding for molecular targets. Even for simplest diatomic molecules, one has to solve a threedimensional time-dependent Schrödinger equation provided the molecular axis has an arbitrary orientation with respect to the polarization of the laser field. Such calculations require fast CPU and large memory computers and have been accomplished only recently [10–12]. In order to compare with experimental results, thus most of the existing calculations for HHG from molecules were carried out using the SFA model [13–15]. Based on the experience from high harmonics generated by atomic targets, such SFA model is not expected to offer accurate predictions. The lack of a reliable theory for describing HHG from molecules has prevented the possible exploitation of molecular structure using infrared laser pulses. This is unfortunate since it has been well recognized that infrared lasers have the potential for probing time-resolved molecular dynamics, with temporal resolutions down to subten femtoseconds. In view of the difficulty in solving the TDSE as a practical theoretical tool for predicting the nonlinear interactions between molecules with intense lasers, recently Le et al [16] and Morishita et al [17] have developed an alternative approach for calculating HHG and have tested the model for atomic targets. The theory is based on the three-step model but without the approximations that lead to the SFA (or the Lewenstein model). The HHG is viewed as resulting from the photo-recombination of the returning electrons generated by the laser pulses. The HHG yield is expressed as the product of a returning wave packet with the photo-recombination cross section. The shape of the wave packet has been shown to be largely independent of the targets for the same laser pulse and can thus be obtained from a reference atom or from the SFA. Thus, there are two ways to obtain HHG spectra without actually solving the TDSE. The first method is called SWSFA model where the HHG spectra are obtained by replacing the dipole matrix element using scattering waves for the continuum electrons instead of the plane waves used in the SFA, but retaining the electron wave packet from the SFA. The other method is to compare the HHG spectra with another reference atom where the HHG spectra have been obtained by accurate calculations or by experiment. If the accurate photorecombination cross sections for both the reference atom and the target are known, then the HHG spectra can be obtained from the HHG spectra of the other atom. The validity of the two methods have been established in Le et al [16] and Morishita et al [17] for atomic systems where the results from the models are compared to accurate results from the TDSE. While the model has been tested only for atoms so far, there is a general belief that the same models would apply to molecular targets also. In this communication, our goal is to apply the models to H2 molecules where accurate HHG spectra have been calculated by solving the TDSE [10–12]. The results from this comparison confirm that the models indeed work well and the method can be extended further as a general tool for calculating HHG from molecular targets. First let us be specific about the two methods for obtaining HHG spectra. The first method is called the scattering-wavebased strong-field approximation (SW-SFA) [16]. The HHG yield S(θ, ω) for a molecule whose internuclear axis makes an alignment angle θ with respect to the laser polarization direction can be obtained from the SFA result SSFA(θ, ω) by S(θ, ω) = ∣∣∣∣ T (θ,E) T PWA(θ, E) ∣∣∣∣ 2 S(θ, ω) (1) where T and T PWA are the exact transition dipole matrix element and its plane-wave approximation (PWA), respectively. Here the electron energy E is related to the emitted photon energy ω by E = k2/2 = ω−Ip, with Ip being the ionization potential of the target (atomic units are used throughout the communication unless otherwise indicated). As only the ratio of the transition dipoles enters equation (1), one can formulate the SW-SFA model by using the above equation with the photoionization or its time-reversed process, i.e., the photo-recombination cross sections. In this communication we refer to the photoionization process, as it is more widely available theoretically and experimentally. Note that one can identify, up to some factor, SSFA/|T PWA|2 as the flux of the returning electron, which we will call a ‘wave packet’. The returning electron that contributes most to the HHG is the one that propagates along the laser polarization direction. Therefore, the relevant differential cross section is for k that is parallel to the laser polarization axis, that is, for θk = 0 and π . Note that for asymmetric molecules, the two contributions from θk = 0 and π differ, in general. Clearly, the formulation of the SW-SFA given in equation (1) is not unique. In fact, one can also use the ‘wave packet’ from a reference atom (i.e., atom with the same Ip as for the target). In other words, the HHG yield can also be