Coulomb-corrected molecular orbital tomography of nitrogen

High-order harmonic generation (HHG) from aligned molecules has provided a promising way to probe the molecular orbital with an Ångström resolution. This method, usually called molecular orbital tomography (MOT) replies on a simple assumption of the plane-wave approximation (PW), which has long been questioned due to that PW approximation is known to be valid in the keV energy region. However, the photon energy is usually no more than 100 eV in HHG. In this work, we experimentally reconstruct the highest occupied molecular orbital (HOMO) of nitrogen (N2) by using a Coulomb-corrected MOT (CCMOT) method. In our scheme, the molecular continuum states are described by a Coulomb wave function instead of the PW approximation. With CCMOT, the reconstructed orbital is demonstrated to agree well with the theoretical prediction and retain the main features of the HOMO of N2. Compared to the PW approximation method, the CCMOT shows a significant improvement in eliminating the artificial structures caused by PW approximation.

Scientific RepoRts | 6:23236 | DOI: 10.1038/srep23236 To overcome the defect of PW approximation, much effort has been expended to correct the SFA by including Coulomb effects in strong-field ionization 22,23 . For MOT, it's also desirable to take the Coulomb effects of the parent ion on the continuum electrons into account. In this work, we demonstrate a Coulomb-corrected MOT (CCMOT) method to reconstruct the HOMO of N 2 in experiment. By using a two-center Coulomb wave to describe the molecular continuum states, we successfully reconstruct the HOMO of N 2 . The reconstructed orbital matches well with the theoretical result and retains the main features of the Hartree-Fock orbital. Compared to the reconstruction based on PW approximation, the CCMOT can effectively remove the additional structures introduced by PW approximation in the reconstructed orbital. Our result shows a significant improvement over the PW result and provides a more accurate method for MOT.

Results
In our experiment, we have measured the harmonic emission from N 2 (I p = 15.58 eV) and its reference atom Ar (I p = 15.76 eV) with the same laser conditions (see Methods). Figure 1(a,b) display the spatially resolved harmonic spectra generated from isotropically aligned (without pump pulse) N 2 and Ar respectively and their corresponding spatially integrated HHG signals are presented as the solid (N 2 ) and dashed (Ar) lines in Fig. 1(c). One can see that, with the increase of harmonic order, the HHG yield from Ar (dashed line) decreases rapidly. While for N 2 , it presents a minimum at 25th harmonic in the harmonic spectrum.
To achieve the MOT, one needs to align the molecules in the laboratory frame. In our experiment, this is performed by using a pump pulse with its polarization rotated by a half-wave plate, and the harmonic generation is driven by a probe pulse that has variable time delay with respect to the pump pulse (see Methods). In Fig. 2(a), we present the measured HHG yield (circles) of the 17th harmonic from aligned N 2 as a function of the pump-probe delay (τ). Here, the pump pulse has parallel polarization with respect to the probe pulse, and the HHG yield has been normalized to that measured without pump pulse (isotropic alignment). The HHG yield is significantly modulated with a period given by the rotational period of N 2 (T rot = 8.4 ps). The calculated evolution of the alignment degree 〈 cos 2 α〉 (calculation details can be found in previous reports 24,25 ), where α is the angle between the molecule axis and the pump polarization, is also displayed as the solid line. The temporal evolution of HHG yield is in good agreement with that of 〈 cos 2 α〉 . At τ = 4.1 ps, where the 〈 cos 2 α〉 reaches the maximum, we have measured the HHG signals with alignment angle (the angle between the pump and probe polarization directions) rotated from 0° to 90° with a step of 10°. Corresponding results are presented in Fig. 2(b). The strength of each harmonic is demonstrated to decrease monotonously as the alignment angle increases. This result is consistent with the previous experiments 9, 10 . Figure 2(c) shows the harmonic yield ratios between N 2 and Ar (i.e., A A / mo ref 2 2 with A mo and A ref being the amplitude of the HHG of molecules and reference atoms.) at different alignment angles. In the range from 25th to 31st harmonic, the ratio increases rapidly due to the decrease of the HHG yield from Ar. To ensure the spatial quality of the orbital reconstruction, the experimental data has been further extrapolated up to 360° by imposing the assumed symmetry of the HOMO of N 2 . To determine the recombination dipole d mo (k), the harmonic phases (φ mo , φ ref ) are also required. In this work, we utilize both the experimental and theoretical phases to reconstruct the molecule orbital. The experimental phases (EPs) are obtained from Haessler's experiment 10 . The theoretical phases (TPs) are calculated by the quantitative rescattering theory (QRS) 21 with the same parameters as in experiment.
With the recombination dipole determined, we next perform the molecule orbital reconstruction. In Fig. 3, we show the HOMO images of N 2 reconstructed with the CCMOT (see Methods). Figure 3(c,d) are the results reconstructed by using the TCC wave in combination with the theoretical (TCC+TP) and experimental (TCC+EP) phases, respectively. Here, we must emphasize that the harmonic bandwidth in our experiment is limited from 17th to 31st harmonics. Therefore for comparison, we have calculated the Hartree-Fock (HF) orbital filtered for experimental samplings as a benchmark in Fig. 3(a). To obtain the filtered HF orbital, we first calculate the recombination dipole d HF (k) by using the HF HOMO [ψ HF (r)] in terms of ψ ⋅ e r r ( ) HF ik r , then the dipole d HF (k) is filtered for the experimental harmonics (17th-31st harmonics) and orientation (0°-90°) samplings, and projected along the driving laser polarization direction. Finally the filtered HF orbital is obtained by performing the inverse Fourier transform of the filtered dipole. One can see that, our experimental reconstructions (both the cases of TCC+TP and TCC+EP) agree well with the simulated one, which exhibits alternating positive and negative lobes and two nodal planes along the y direction. Due to the restricted harmonic bandwidth, the spatial frequency of the orbital is filtered and therefore the reconstructed orbital is elongated. In Fig. 3(b), it shows the PW result reconstructed by using the experimental data in ref. 10. Compared to the simulated orbital, the reconstruction based on the PW approximation introduces some artificial structures in the spatial range of |x|, |y| > 5 a.u. These structures can be effectively eliminated by considering the Coulomb effects as shown in Fig. 3(c,d). Note that our PW result in Fig. 3(b) and Haessler's result 10 are different from that by Itatani et al. 9 , especially in the spatial range of |x|, |y| > 5 a.u., which is due to the difference in the samplings of harmonic range. Moreover, by using the CCMOT method, we also successfully reproduce the orbitals of other non-polar molecules, e.g., O 2 and CO 2 . The CCMOT can also be extended to heteronuclear molecules (e.g., CO) in combination with other techniques that can decouple the HHG with the recollision from the head or end of the molecule 15,16,26 , such as decoding odd-even harmonics 15 or using two-color method 16 .

Discussion
For a quantitative comparison, we have also plotted the slices (along the internuclear axis) of the above four reconstructed orbitals in Fig. 4. It's obvious that the two line profiles based on the CCMOT method [i.e., TCC+TP (green dash-dotted line) and TCC+EP (red dotted line)] are essentially consistent with the theoretical result (black solid line). Besides, the internuclear distances (defined as the distance between the nodes of the lobes along the molecular axis) reconstructed by TCC+TP and TCC+EP are 1.92 a.u. and 2.02 a.u., which are both very close to the exact value of 2.06 a.u. For the case of PW approximation (blue dashed line), the profile line agrees well with the simulated result in spatial range of |x| < 5 a.u. Beyond this range, it shows deep modulations, which arise from the additional structures in Fig. 3(b) that do not exist in the exact HOMO image. In contrast, the CCMOT provides a more accurate reproduction of the molecule orbital.
In summary, we have experimentally demonstrated the molecule orbital reconstruction by using a CCMOT method. With this method, the molecule orbital can be reproduced from the continuum wave functions without the PW approximation. By employing the two-center Coulomb wave which includes the main Coulomb effects to describe the continuum states, we successfully reproduce the HOMO of N 2 . The reconstructed orbital retains the main features of the Hartree-Fock orbital and shows good agreement with the theoretical result. Compared to PW result, the reconstruction with Coulomb corrections effectively eliminates the artificial structures induced by the PW approximation. Our result provides a more accurate method for molecule orbital reconstruction and is conducive to clarify the long-standing controversy in the original MOT theory. Methods Experimental methods. Our experiment is preformed by using a commercial Ti:sapphire laser system (Legend Elite-Duo, Coherent, Inc.), which delivers 30-fs, 800-nm pulses at a repetition rate of 1 kHz. Figure 5 shows a schematic layout of the experiment. The output laser is spilt into two beams. One is extended to 50-fs for aligning the molecules (pump pulse) and the other is for generating harmonics (probe pulse). The polarization of the pump pulse is rotated by a half-wave plate. Iris diaphragms is used to independently adjust the laser beam size of the pump and probe pulses. The delay between the pump and probe pulses can be changed by the delay line. The two beams are collinearly focused on a pulsed supersonic gas jet emitted from a 100-μm diameter nozzle. The stagnation pressure of the gases is maintained at 2 bars and the gas jet is placed 2 mm after the laser focus. The temperature of the gas in the interaction region is estimated to be about 70 K 27,28 , which ensures a high degree of molecular alignment. Throughout our experiment, the laser energies of the pump and probe pulses are kept constant and the corresponding intensities are estimated to be 5 × 10 13 W/cm 2 and 2 × 10 14 W/cm 2 , respectively. The generated high-order harmonic spectrum is detected by a home-made flat-field soft x-ray spectrometer 29 .
Theoretical methods. According to the SFA theory, the induced dipole moment for HHG can be given as a factorized expression 9,10,12 : where θ is the angle between the driving-field polarization and the molecular axis. The first factor a ion (ω, θ) represents the tunnel ionization amplitude, a acc (ω, θ) describes the propagation amplitude of the recolliding electron wave packet in the continuum, and d (k) is the recombination matrix element between the initial orbital and the continuum wave function. In experiment, to extract the recombination dipole d (k) from the measured harmonic signals, one should determine the first two factors. This can be achieved by measuring the harmonic emission from a reference atom that has the same ionization potential (I p ) with the molecule. By measuring the amplitude A(ω, θ) and phase φ(ω, θ) of the generated harmonics, the dipole moment D(ω, θ) can be determined as: D (ω, θ) = A(ω, θ)e iφ(ω, θ) . The recombination dipole matrix element d ref (k) of the reference atom can be known from theory, then the molecular dipole can be obtained by:   Fig. 3. The black solid, blue dashed, green dash-dotted, and red dotted lines are for cases of theoretical filtered HF orbital [ Fig. 3(a)], PW result reproduced by using the experimental data 10 [ Fig. 3(b)], TCC+TP [ Fig. 3(c)], and TCC+EP [ Fig. 3(d)], respectively.