Influence of orbital symmetry on diffraction imaging with rescattering electron wave packets

The ability to directly follow and time-resolve the rearrangement of the nuclei within molecules is a frontier of science that requires atomic spatial and few-femtosecond temporal resolutions. While laser-induced electron diffraction can meet these requirements, it was recently concluded that molecules with particular orbital symmetries (such as πg) cannot be imaged using purely backscattering electron wave packets without molecular alignment. Here, we demonstrate, in direct contradiction to these findings, that the orientation and shape of molecular orbitals presents no impediment for retrieving molecular structure with adequate sampling of the momentum transfer space. We overcome previous issues by showcasing retrieval of the structure of randomly oriented O2 and C2H2 molecules, with πg and πu symmetries, respectively, and where their ionization probabilities do not maximize along their molecular axes. While this removes a serious bottleneck for laser-induced diffraction imaging, we find unexpectedly strong backscattering contributions from low-Z atoms.

The steps taken to arrive at the C 2 H 2 + bond length spectrum from the experimental data. The same steps are taken to analyse the O2 data. See the text for more details.

Isolation of molecular information from backscattering electron distributions.
A crucial step in extracting molecular structure imprinted in experimentally measured molecular backscattered electron distributions (ρ E ) the superimposed modulations need to be isolated from the overall distribution by subtracting the monotonically decreasing 'background'. We therefore outline our procedure before applying it to experimental results. The resulting curve can be called the interference signal and an analogous parameter is used in conventional electron diffraction [1]. In principle, the interference signal can be calculated by subtracting either theoretical simulations or the backscattered electron distributions from atoms with similar ionisation potentials that were measured under the same experimental conditions. Here, and for the first time, we calculate the interference signal from LIED by subtracting an empirically determined background (ρ B ) from the logarithm of ρ E , ρ = log 10 (ρ E ) -log 10 (ρ B ) = log 10 (ρ E /ρ B ).
We test our empirical background fitting procedure on independent atom model based simulations that were calculated following procedures that have been outlined previously [11,23].
Since the independent atom model does not need to take into account the laser-field, we emulate experimental backscattered electron distributions by simply calculating differential cross-sections (DCS) as a function of returning electron energy at a constant scattering angle of θ r = 180°. The calculated molecular DCS (grey, solid) of C 2 H 2 is observed to modulate about the corresponding atomic DCS (red, solid) in Supplementary Fig. 1a. These modulations are due to the coherent interference of scattered waves from each atom in the molecule. The bottom two panels show zoomed in regions where the modulations are more visible. The atomic term can be considered equivalent to the background ρ B mentioned above. Also shown is a fourth order polynomial fit (black dashed) to the molecular DCS that very closely follows the atomic term. The interference signals resulting from the subtraction of the atomic DCS (red, solid) and the empirical fit (black, dashed) are presented in Supplementary Fig. 1b. The main features of the atomic subtracted interference signal are reproduced by the empirical fit. The normalised Fourier transforms of these signals are presented in Supplementary Fig. 1c alongside the results for when third (grey, dotted) and fifth (grey, dot-dashed) order polynomials are utilised. The most important observation is that the peak of the fundamental frequency component is within 0.05 Å for all curves. This shows that empirical background subtraction based on polynomial fitting is a robust and accurate method to isolate molecular modulations in FT-LIED. We also tested this polynomial background subtraction procedure on the data presented in Fig. 1a & Supplementary Fig. 3c of Ref.
[14] and our results were consistent with those published.

Background subtraction
The robustness of the background subtraction methodology is presented in Supplementary Fig. 2. Thorough tests were performed on the experimental data to ensure that the results of the background fitting algorithm were consistent. In Supplementary Fig. 2a the experimentally measured backscattered electron distribution from C 2 H 2 is presented alongside lines of best fit for 4 th , 5 th and 6 th order polynomials. A zoomed in version between 8-12 Å -1 shows that these fits lie mostly on top of each other. It was found that these polynomial orders gave almost identical results for both O 2 and C 2 H 2 , which highlights the reliability of the background subtraction procedure. The resultant Fourier transformed spectra are presented in Supplementary Fig. 2b where almost identical results are observed. We find that polynomial fits below 3 rd order do not adequately remove the 'background' signal and fits above 6 th order start to remove the real signal.

Experimental data analysis
The procedure to get from the raw experimental data to the bond length spectrum is outlined in Supplementary Fig. 3. We focus on the C 2 H 2 case here but the method is the same for O 2 .
Step 1: The ionic time-of-flight shows a variety of positively charged particles detected after single and multiple ionisation as well as fragmentation. The inset shows a zoomed in view of the C 2 H 2 cation peak.
Step 2: The electrons associated with the C 2 H 2 cation peak are extracted as a function of their final detected momentum. Only electrons that scatter within a ±4° cone around the laser polarisation are used in the data analysis.