Coherent diffractive imaging of single helium nanodroplets with a high harmonic generation source

Coherent diffractive imaging of individual free nanoparticles has opened routes for the in situ analysis of their transient structural, optical, and electronic properties. So far, single-shot single-particle diffraction was assumed to be feasible only at extreme ultraviolet and X-ray free-electron lasers, restricting this research field to large-scale facilities. Here we demonstrate single-shot imaging of isolated helium nanodroplets using extreme ultraviolet pulses from a femtosecond-laser-driven high harmonic source. We obtain bright wide-angle scattering patterns, that allow us to uniquely identify hitherto unresolved prolate shapes of superfluid helium droplets. Our results mark the advent of single-shot gas-phase nanoscopy with lab-based short-wavelength pulses and pave the way to ultrafast coherent diffractive imaging with phase-controlled multicolor fields and attosecond pulses.


Supplementary Note 1) Multicolor Mie fits using the literature values of bulk Helium
The input parameters for the multidimensional Mie-based optimization are the particle size, the refractive indices at the wavelengths of the contributing harmonics, the relative intensities of the harmonics, and the intensity of the XUV pulse. We find that using the literature values for the refractive indices of bulk liquid helium at the harmonic wavelengths as fixed input parameters does not allow fitting the measured diffraction patterns. This claim is supported by the results of two attempts to fit the data. First, only the particle size and the total XUV intensity were varied, while the relative harmonic intensities were fixed to the average measured spectrum (cf. Fig 3d, main manuscript) and the literature values of the refractive indices at the harmonic wavelengths (n11=0.97 + 0.0i; n13=1.14+0.032i; n15=1.03+0.029i; n17=0.9964 + 0.041i; see Methods) were used. As a second test, we also allowed the optimization of the intensities of the individual harmonics, as discussed below.
Typical results for the first scenario are shown in Supplementary Figure 1 for three measured profiles (black) derived from a representative ring-type scattering pattern. The optimization with fixed refractive indicies and fixed relative harmonic intensities (green) is unable to provide a reasonable fit of the measured data. Particularly large discrepancies to the measured profile remain at high scattering angles. For comparison, all qualitative features of the profiles can be fitted when the refractive indicies of the 13 th and 15 th harmonic are included in the optimization (purple curve).
For the second test, the refractive indices were kept fixed to the literature values but the intensity ratios of the harmonics were freely varied in addition to the optimization parameters droplet size and total XUV intensity. The resulting fits are displayed in Supplementary Figure 2. A better match between the measured profiles (black) and the calculated profiles (purple) can be obtained compared to the first test shown in Supplementary Figure 1. However, the residual deviations remain substantially larger when compared to the results obtained with optimized optical parameters and fixed intensity ratios, compare black and red/blue symbols in Supplementary Figure 3b. Moreover, the intensity ratios between the harmonics retrieved from the fitting procedure (see Supplementary Figure 3a) are in stark contrast to the measured spectrum (cf. Fig. 3d of the main manuscript), where the 13 th and 15 th harmonic have the strongest contributions (blue and red). Here, the most intense contribution given from the fits originates from the 17 th harmonic that has been measured to be the weakest. We can therefore conclude that the refractive indices for Helium nanodroplets are different from the reported values of bulk liquid helium.

Supplementary Note 2) Solutions with comparable residuals
As described in the main manuscript, best fits were achieved when varying the refractive indices of the most intense harmonics 13 and 15, while the relative harmonic intensities were set to the average measured values and the refractive indices at the wavelengths of the 11 th and 17 th harmonics were fixed to the literature values.
Under these conditions, the majority of the best fits for the 18 recorded scattering profiles leads to refractive indices that lie around an average pair of values: n13 =0.925+ 0.018i; n15 = 1.269+0.042i, see Supplementary Figure 4a. In these fits, the 13 th harmonic gives the dominant contribution to the diffraction image (Supplementary Figure 5). However, for all analyzed patterns, a second solution is found by the algorithm where the dominant contribution systematically comes from the 15 th harmonic (Supplementary Figure 6). The refractive indices obtained by the second solution are shown in Supplementary Figure 4b, here the results cluster around another value pair (n13 =1.2827+0.0464i; n15 = 0.9518+0.0269i). The straight lines indicate the corresponding value pairs of single fit results. The fitting errors for these two systematic solutions are compared in Supplementary Figure 3b to the results obtained for open intensity ratios and refractive indices fixed to the literature bulk values. While it is clear that varying the refractive indices leads to a substantial reduction of the residual errors, the quality of the two systematic solutions is similarly good. We note that the refractive indices for the solution with dominant 15 th harmonic lie closer to the literature values. However, considering the signal to noise ratio in our experiment, we cannot fully exclude one of the two solutions. Additional experiments using a XUV pulses with larger distances between the harmonics and/or with a better signal to noise ratio are required to resolve this ambiguity.
Supplementary Figure 7 displays histograms of the resulting droplet sizes from the optimization. Comparing Supplementary Figure 7a and b shows that the algorithm, while interchanging the roles of 13 th and 15 th harmonic between the two solutions, also adjusts the droplet size accordingly to account for the different wavelength of the dominant harmonic (the distributions are shifted by approximately 50 nm). We note that the size distribution of the droplet jet is probably broader than indicated by the histogram, because droplets with sizes outside the observed range (250 nm -550 nm) cannot be fitted. Smaller droplets produce a too dark diffraction pattern and very large droplets have a very small fringe separation such that minima are more likely contaminated by noise fluctuations.
However, despite the remaining ambiguity, the results support that the multicolor approach in principle allows the systematic characterization of the properties of individual droplets within a wide size distribution.