Few-femtosecond resolved imaging of laser-driven nanoplasma expansion

From 1.3 × 10⁶ recorded scattering images we found 24 500 clearly identifiable hits. After removing faint images, multiparticle hits that display Newton rings [40, 41], and hits from clustered nanoparticles, a dataset of 5700 bright single-shot images with clear fringe structures has been systematically analyzed as described below. Out of that, a subset of 1070 images was assigned to the high intensity region of the x-ray focus which is associated with homogeneous NIR pumping. It should be emphasized that the latter discrimination step cannot be made based on the scatting pattern alone but requires the quantitative fit of the image with our model to determine the associated incident x-ray fluence (see appendix E.6). The distributions of classified images versus pulse delay are displayed in figure 1(c).

In the small-angle scattering limit, the far-field scattering pattern encode a two-dimensional projection of the particle density. Nevertheless, for three-dimensional targets with spherical symmetry, the unique and unrestricted reconstruction of the radial density profile would still be possible if the complete scattering image can be recorded. In real-world experiments, however, a substantial part of information is lost due to the unavoidable detector gaps in pump-probe experiments and limited signal at larger scattering angles. For the quantitative reconstruction of the density profile from the available information contained in the single-shot scattering images an appropriate shape model is imperative. In our analysis we use a simulation-guided few-parameter density profile [28] and retrieve the profile parameters by systematically fitting simulated scattering patterns to the individual measured scattering images. The employed shape model is based on an electron density profile described by a modified Fermi function with

$\begin{equation}n(\vec{r})=\frac{{n}_{0}}{{\left[\mathrm{exp}\left(\frac{\kappa (\vec{r})}{\lambda \sigma }\right)+1\right]}^{\sigma }}.\end{equation} \tag{ 1 }$

Here n₀ is the core electron density, κ is the local surface offset of the sampling point $\vec{r}$, σ is the edge softness parameter, and λ is the scale length of the density decay. A sharp density edge corresponds to the limit σ → 0. For a spherical system with a reference surface at radius R, the surface offset is κ = r − R. The variation of radius, scale length, and softness are connected with specific signatures in the scattering pattern, as illustrated in figure 2 for a spherical system. For a fixed softness parameter, the radius and scale length are almost independently coupled to the fringe spacing and angular decay of the radial scattering profile (cf figures 2(a)–(d)), respectively. As the underlying mapping between these profile parameters and the signatures in the scattering images is a function of the softness, the latter must be known or has to be determined independently for a unique parameter reconstruction. This requirement is illustrated in figures 2(e) and (f), where the variation of the edge softness is shown to also modify the fringe spacing and the angular decay of the scattering profile. It should be noted that in previous work, the problem was usually circumvented by assuming an unphysical sharp edge (σ → 0). In our analysis we determine the finite edge softness using an additional constraint, as described in detail below.

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In the description of the scattering process, both plasma electrons as well as atomically bound electrons with binding energies smaller than the x-ray photon energy can be treated as free electrons. In contrast, the scattering contribution of strongly bound electrons is negligible, such as in the case of the silicon K-shell electrons in our scenario. It should further be noted, that the K-shell electrons of Si are effectively unreachable for the NIR-driven ionization. Hence the Si K-shell can be neglected for both ionization and the scattering process. As a result and assuming that the fraction of emitted electrons is negligible for the large particles considered here, the density of actively scattering electrons is representative for the atomic density and their number can be assumed to be constant.

In our analysis the slight ellipsoidal deformation of the nanospheres resulting from the particle synthesis must be included for high accuracy fits (see appendix E). For the resulting ellipsoidal reference surface with direction dependent radius, the local surface offset is defined as $\kappa =\left[\vec{r}-\vec{\rho }\,\right]\cdot \vec{n}$, where $\vec{\rho }(\vec{r})$ specifies the closest point on the reference surface and $\vec{n}$ is the associated unit vector of the surface normal. The neglect of directional variations of the remaining relevant parameters σ and λ is justified by the small anisotropy predicted by microscopic calculations [28].

For a given value of the edge softness parameter σ, high quality fits of individual images can be performed and yield the scale length λ and the minimal and maximal core radii ρ_min and ρ_max that characterize the minor and major axis of the projected density associated with the ellipsoidal control surface. For further analysis we use the effective radius ${R}_{\text{eff}}=\frac{{\rho }_{\mathrm{min}}+{\rho }_{\mathrm{max}}}{2}$. So far, however, the remaining dependence of the scale length and core radius on the edge softness results in an apparent critical ambiguity for the analysis. We resolve this ambiguity by exploiting the conservation of the number of scattering electrons (i.e. conservation of mass in the case of charge neutrality), as described in the following.

The treatment of the scattering problem via the few-parameter profile enables systematic forward fitting of a large dataset with thousands of scattering images. Figures 3(a) and (b) displays fit results for the delay dependent evolution of scale length and effective radius when assuming a fixed very sharp edge (σ = 0.1). In this case, the associated total mass (number of electrons contributing to the scattering as given by the integral of the electron density) would increase by almost 90% during the first 200 fs of the expansion (figure 3(c)), which is obviously unphysical. The profile parameters retrieved assuming a very soft edge (σ = 0.9) differ substantially (figure 3(d) and (e)) and, in turn, would correspond to a mass decrease by more than 20% (figure 3(f)). The mass mismatch evolution map resulting from fits with systematically varied softness parameters is shown in false color representation in figure 3(j), where the above discussed representative cases are indicated by horizontal lines (top and bottom). While the impact of the edge softness on the mismatch is little for unpumped particles as its effect vanishes in the limit of small scale length (sharp density step), the map shows that mass conservation during expansion requires a dynamical change of the softness parameter, expressed by the path of mass conservation (white color) in figure 3(j). The delay dependent softness associated with conserved total mass (black curve) defines the final and physically consistent evolution of all parameters and their mean values, i.e. softness, scale length and core radius, see figures 3(g)–(j). Only these parameters have been used for further analysis and in figure 1.

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The NIR pump laser and the FEL are independent sources, and as such their relative timing can only be controlled at the accuracy level of the radio-frequency synchronization system. At the LCLS, this synchronization is typically on the order of a 150 fs rms and must be corrected by the use of a single-shot x-ray optical cross correlation method [37, 38]. The cross correlation method used here is based on the idea of encoding time onto the spectrum of an optical reference [50, 51]. This so-called 'spectral encoding' allows for relative measurements with precision below 10 fs [39].

The basic observable of the timing tool is the change in the frequency resolved transmission of a super-continuum, generated from a replica of the NIR pulse, through a dielectric plate with (signal) and without (reference) the co-propagating x-ray pulse, see figures D1(a) and (b) for exemplary raw signals and the corresponding relative x-ray induced transmission change, respectively. Due to a spectral chirp of the super-continuum, the spectral transmission change exhibits a step that is associated with the pulse overlap and can be translated into a relative pulse delay. To this end, a systematic scan of the relative pulse delay, e.g. via electronic delay settings, is performed that yields a map for the correlation of the edge position and pulse delay, still including the jitter. A second order polynomial fit to that map provides a mapping function for position and relative delay, see figure D1(c).

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In order to identify the absolute time overlap between the NIR and the x-ray pulse, we used the method described in [52] whereby the photo-dissociation of quasi-bound ${\mathrm{N}}_{2}^{++}$ dications into N⁺ fragment ions [53] serves as an indication of the x-ray induced appearance of ${\mathrm{N}}_{2}^{++}$ via the x-ray core ionization and subsequent normal Auger decay. In particular the delay dependent abundance of the resulting low and high energy cations serves as a measure for the pulse order. Due to the high ionization potential of N₂, the attenuated NIR-pulse is unable to excite or ionize the neutral molecule, leaving the ionization by the subsequent x-ray pulse and the resulting population of quasi-bound ${\mathrm{N}}_{2}^{++}$ almost unchanged. However, initial irradiation of the neutral molecules by the x-ray generates a population of quasi-bound ${\mathrm{N}}_{2}^{++}$ that, after further excitation by the NIR pulse, results in dissociation into a pair of N⁺ with corresponding recoil energy due to Coulomb repulsion.

Figure D2(a) presents averaged time-of-flight spectra for shots with the x-ray pulse first (blue) or the NIR pulse first (purple). Note that area A indicates the recoil-free signal from ${\mathrm{N}}_{2}^{++}$ that exhibits higher yield if the NIR arrives first. Area B is representative for N⁺ signal with recoil and shows increased yield when the NIR is late. The ratio of the signals integrated within these areas is used to trace the pulse order as function of the (not offset corrected) delay. Figure D2(b) shows the distribution of the single-shot yield ratios from a systematic delay scan. For each delay bin, the distribution can be described well by a Gaussian envelope. The resulting delay-dependent center position (width) of the Gaussian fits is shown as cyan (orange) curve. The step around −160 fs is indicative of pulse overlap, i.e. zero delay.

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Assuming no-depletion and linear response for both the x-ray and NIR excitations processes, the derivative of the delay-dependent ratio signal directly reflects the cross correlation of the pulses envelopes

$\begin{equation}\frac{\mathrm{d}}{\mathrm{d}\tau }\frac{A(\tau )}{B(\tau )}\propto ({I}_{\mathrm{x}-\mathrm{r}\mathrm{a}\mathrm{y}}\ast {I}_{\text{NIR}})(\tau ).\end{equation} \tag{ D.1 }$

A fit of the step feature in the evolution of the delay-dependent centers in figure D2(b) with an error function (black) and taking the derivative of the latter yields the cross-correlation in figure D2(c), whose peak is located at Δt₀ = −160 fs, defining zero delay. Assuming Gaussian pulses the FWHM-width of the cross correlation of 67 fs can be used to estimate the mean x-ray pulse duration to about 38 fs from the knowledge of the NIR pulse duration of 50 fs (see inset).

To derive the formalism used to describe the scattering of ellipsoids in our study we depart from the general solution to the classical scattering problem in the limit of weak scattering, i.e. neglecting multiple scattering (Born approximation). Assuming a linearly polarized incident electric field ${\vec{E}}_{\mathrm{i}\mathrm{n}\mathrm{c}}={\vec{E}}_{0}\,{\text{e}}^{-\text{i}{\vec{k}}_{\mathrm{i}\mathrm{n}}\vec{r}}$ with wave vector ${\vec{k}}_{\text{in}}$ and an arbitrarily shaped scattering object defined via the scattering electron density $n(\vec{r})$ we can calculate the scattered field amplitude from

$\begin{equation}E(\vec{q})={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}\iiint n(\vec{r}){\text{e}}^{-\text{i}\vec{q}\cdot \vec{r}}\,{\mathrm{d}}^{3}r\end{equation} \tag{ E.1 }$

with momentum transfer vector $\vec{q}$ defined as the difference between incoming and outgoing wave vector according to $\vec{q}={\vec{k}}_{\text{out}}-{\vec{k}}_{\text{in}}$, classical electron radius r_e and detector distance l_det. Note that the angular dependence of the scattered field amplitude due to polarization effects can be neglected for small scattering angles.

For the following discussion we consider, without loss of generality, an incident wave vector along the z-axis ${\vec{k}}_{\mathrm{i}\mathrm{n}}=k{\vec{e}}_{z}$. We further limit the discussion to a small angle scattering scenario with |q| ≪ |k_in|) such that we can now assume q_z ≃ 0 and rewrite equation (E.1) as follows:

$\begin{align}\hfill E(\vec{q})& ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }n(\vec{r}){\text{e}}^{-\text{i}({q}_{x}x+{q}_{y}y+{q}_{z}z)}\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }\underset{{n}_{\mathrm{2}\mathrm{D}}(x,y)}{\underbrace{{\int }_{-\infty }^{\infty }n(\vec{r})\mathrm{d}z}}\,{\text{e}}^{-\text{i}({q}_{x}x+{q}_{y}y)}\mathrm{d}x\,\mathrm{d}y\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{n}_{\mathrm{2}\mathrm{D}}(x,y){\text{e}}^{-\text{i}({q}_{x}x+{q}_{y}y)}\mathrm{d}x\,\mathrm{d}y\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}\mathcal{F}\left[{n}_{\mathrm{2}\mathrm{D}}\right]({q}_{x},{q}_{y}),\hfill \end{align} \tag{ E.2 }$

which reflects that the scattering intensity in the small angle scattering limit is determined by the Fourier transform of the projected density n_2D.

If we are only interested in a line profile instead of the full 2D scattering image, e.g. for q_y = 0, we can further simplify the expression and write:

$\begin{align}\hfill E({q}_{x},{q}_{y}=0)& ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{n}_{\mathrm{2}\mathrm{D}}(x,y){\text{e}}^{-\text{i}({q}_{x}x+{q}_{y}y)}\mathrm{d}x\,\mathrm{d}y\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }\underset{{n}_{1d}(x)}{\underbrace{{\int }_{-\infty }^{\infty }{n}_{\mathrm{2}\mathrm{D}}(x,y)\mathrm{d}y}}\,{\text{e}}^{-\text{i}({q}_{x}x)}\mathrm{d}x\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }{n}_{\mathrm{1}\mathrm{D}}(x){\text{e}}^{-\text{i}({q}_{x}x)}\mathrm{d}x\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}\mathcal{F}\left[{n}_{\mathrm{1}\mathrm{D}}\right]({q}_{x})\hfill \end{align} \tag{ E.3 }$

which relates the scattered field along a specific polar angle with the Fourier transform of the density projected onto the corresponding axis. Based on the above equation we will now derive solution strategies for the density distributions that are most relevant to our work, i.e. that of homogeneous spheres and ellipsoids with sharp and soft surfaces.

As the first step we will revisit the well-known solution for a homogeneous sphere with a sharp surface described by the density profile

$\begin{equation*}n(r)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill r > R\hfill \\ \hfill {n}_{0}\hfill & \hfill r\leqslant R\hfill \end{array}\right.\end{equation*}$

with the homogeneous electron density n₀. Due to the radial symmetry we only need to calculate the scattering for a single scattering direction, i.e. we are only interested in the double projected density n_1D(x). The corresponding integral can be calculated in polar coordinates

$\begin{equation*}{n}_{\mathrm{1}\mathrm{D}}(x)={n}_{0}{\int }_{0}^{2\pi }{\int }_{0}^{{\rho }_{\mathrm{max}}}\rho \,\mathrm{d}\rho \,\mathrm{d}\varphi ={n}_{0}\pi {\rho }_{\mathrm{max}}^{2},\end{equation*}$

where ρ_max(x) describes the radial extension of the sphere for the given position on the projection axis x with

$\begin{equation*}{\rho }_{\mathrm{max}}(x)=\left\{\begin{array}{cc}\hfill \sqrt{{R}^{2}-{x}^{2}}\hfill & \hfill x\leqslant R\hfill \\ \hfill 0\hfill & \hfill x > R.\hfill \end{array}\right.\end{equation*}$

Inserting this result yields

$\begin{equation}{n}_{\mathrm{1}\mathrm{D}}(x)=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill x > R\hfill \\ \hfill {n}_{0}\pi ({R}^{2}-{x}^{2})\hfill & \hfill x\leqslant R\hfill \end{array}\right.\end{equation} \tag{ E.4 }$

and allows to obtain the scattered field by solving the corresponding Fourier integral

$\begin{align*}\hfill E({q}_{x})& ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-\infty }^{\infty }{n}_{\mathrm{1}\mathrm{D}}(x){\text{e}}^{-\text{i}({q}_{x}x)}\mathrm{d}x\hfill \\ \hfill & ={n}_{0}{E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{\int }_{-R}^{R}\pi ({R}^{2}-{x}^{2}){\text{e}}^{-\text{i}({q}_{x}x)}\mathrm{d}x\hfill \\ \hfill & ={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{n}_{0}4\pi {R}^{3}\left(\frac{\mathrm{sin}({q}_{x}R)-{q}_{x}R\,\mathrm{cos}({q}_{x}R)}{{q}_{x}^{3}{R}^{3}}\right).\hfill \end{align*}$

It is often beneficial to work with the normalized scattered field E_n (E_n (0) = 1) which reads

$\begin{equation*}{E}_{n}(q)=3\left(\frac{\mathrm{sin}(qR)-qR\,\mathrm{cos}(qR)}{{q}^{3}{R}^{3}}\right).\end{equation*}$

Note, that we can drop the subscript x of the momentum transfer vector due to the radial symmetry.

The doubly-projected density n_1D that is needed to evaluate the scattering integral in equation (E.3) for an object is also an important quantity in the field of reconstruction algorithms in computer tomography, where it is known as the Radon transform. In particular, the Radon transform of arbitrarily shaped and oriented ellipsoids are of wider interest and are analytically known [54]

$\begin{equation}{n}_{\mathrm{1}\mathrm{D}}(\rho ,\vec{n})=\left\{\begin{array}{cc}\hfill 0\hfill & \hfill \vert \rho \vert \geqslant \zeta \hfill \\ \hfill {n}_{0}\pi \frac{abc}{{\zeta }^{3}}\left({\zeta }^{2}-{\rho }^{2}\right)\hfill & \hfill \vert \rho \vert < \zeta ,\hfill \end{array}\right.\end{equation} \tag{ E.5 }$

where $\vec{n}={(\mathrm{sin}\,\theta \,\mathrm{cos}\,\varphi ,\mathrm{sin}\,\theta \,\mathrm{sin}\,\varphi ,\mathrm{cos}\,\theta )}^{\mathrm{t}}$ is the normalized vector characterizing the direction the density is projected on and $\rho =\vec{r}\vec{n}$ is the remaining coordinate along this direction. The parameters a, b and c determine the ellipsoids length along the x, y and z direction for an unrotated ellipsoid. The effects of the dimension and orientation of the ellipsoid are now encapsulated in the factor ζ:

$\begin{align*}\hfill {\zeta }^{2}& ={a}^{2}\left(\mathrm{cos}\,\theta \,\mathrm{sin}\,{\psi }_{2}\,\mathrm{sin}\,{\psi }_{3}+\mathrm{sin}\,\theta \,\mathrm{sin}\,\varphi (\mathrm{cos}\,{\psi }_{3}\,\mathrm{sin}\,{\psi }_{1}+\mathrm{cos}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{sin}\,{\psi }_{3})\right.\hfill \\ \hfill & \quad {\left.+\mathrm{cos}\,\varphi \,\mathrm{sin}\,\theta (\mathrm{cos}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{3}-\mathrm{cos}\,{\psi }_{2}\,\mathrm{sin}\,{\psi }_{1}\,\mathrm{sin}\,{\psi }_{3})\right)}^{2}+{b}^{2}\left(\mathrm{cos}\,\theta \,\mathrm{sin}\,{\psi }_{3}\,\mathrm{cos}\,{\psi }_{3}\right.\hfill \\ \hfill & {\left.\quad -\mathrm{sin}\,\theta \,\mathrm{cos}\,\varphi (\mathrm{cos}\,{\psi }_{1}\,\mathrm{sin}\,{\psi }_{3}+\mathrm{sin}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{cos}\,{\psi }_{3})+\mathrm{sin}\,\varphi \,\mathrm{sin}\,\theta (\mathrm{cos}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{cos}\,{\psi }_{3}-\mathrm{sin}\,{\psi }_{1}\,\mathrm{sin}\,{\psi }_{3})\right)}^{2}\hfill \\ \hfill & \quad +{c}^{2}{(\mathrm{cos}\,\theta \,\mathrm{cos}\,{\psi }_{2}-\mathrm{sin}\,\theta \,\mathrm{sin}\,(\varphi -{\psi }_{1})\mathrm{sin}\,{\psi }_{2})}^{2}\hfill \end{align*}$

which gives us the effective dimension of the projected ellipsoid along the direction of the projection vector $\vec{n}$. The angles ψ₁, ψ₂ and ψ₃ are Euler angles in x-convention for the rotation of the ellipsoid (rotation first by angle ψ₁ about the z-axis, secondly by angle 0 ⩾ ψ₂ ⩽ π about the x-axis, and finally by angle ψ₃ about the z-axis again). For our purposes we can, without loss of generality, assume a detector in the x–y plane (our propagation direction is z), i.e. θ = π/2 such that the angle φ is essentially the polar angle defining the line-out from the 2D scattering pattern. Doing this already simplifies the above expressions considerably:

$\begin{align*}\hfill {\zeta }^{2}& ={a}^{2}{(\mathrm{sin}\,\varphi (\mathrm{cos}\,{\psi }_{3}\,\mathrm{sin}\,{\psi }_{1}+\mathrm{cos}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{sin}\,{\psi }_{3})+\mathrm{cos}\,\varphi (\mathrm{cos}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{3}-\mathrm{cos}\,{\psi }_{2}\,\mathrm{sin}\,{\psi }_{1}\,\mathrm{sin}\,{\psi }_{3}))}^{2}\hfill \\ \hfill & \quad +{b}^{2}{(-\mathrm{cos}\,\varphi (\mathrm{cos}\,{\psi }_{1}\,\mathrm{sin}\,{\psi }_{3}+\mathrm{sin}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{cos}\,{\psi }_{3})+\mathrm{sin}\,\varphi (\mathrm{cos}\,{\psi }_{1}\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{cos}{\psi }_{3}-\mathrm{sin}\,{\psi }_{1}\,\mathrm{sin}\,{\psi }_{3}))}^{2}\hfill \\ \hfill & \quad +{c}^{2}{(-\mathrm{sin}(\varphi -{\psi }_{1})\mathrm{sin}\,{\psi }_{2})}^{2}.\hfill \end{align*}$

For any given projection direction $\vec{n}$ the expression for the double projected density is structure-wise identical to that for a homogeneous sphere, cf equation (E.4). However, the value of the radius equivalent ζ changes as a function of the projection direction. Before we discuss the implications of this aspect, we further reduce the complexity of the expression for ζ with the help of trigonometrical identities such that

$\begin{equation*}{\zeta }^{2}={a}^{2}{(\mathrm{sin}\,{\varphi }^{\prime }\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{sin}\,{\psi }_{3}+\mathrm{cos}\,{\varphi }^{\prime }\,\mathrm{cos}\,{\psi }_{3})}^{2}+{b}^{2}{(\mathrm{sin}\,{\varphi }^{\prime }\,\mathrm{cos}\,{\psi }_{2}\,\mathrm{cos}\,{\psi }_{3}-\mathrm{cos}\,{\varphi }^{\prime }\,\mathrm{sin}\,{\psi }_{3})}^{2}+{c}^{2}{(\mathrm{sin}\,{\varphi }^{\prime }\,\mathrm{sin}\,{\psi }_{2})}^{2},\end{equation*}$

where φ' = φ − ψ₁. It can be shown, that for any given set of parameters a, b, c, ψ₁, ψ₂, ψ₃ and φ the description of ζ² can be reduced to an even simpler functional form:

$\begin{equation*}{\zeta }^{2}={a}_{\mathrm{e}\mathrm{ff}}^{2}\,\mathrm{cos}{({\varphi }^{\prime }+{\Delta})}^{2}+{b}_{\mathrm{e}\mathrm{ff}}^{2}\,\mathrm{sin}{({\varphi }^{\prime }+{\Delta})}^{2},\end{equation*}$

where the effective parameters a_eff and b_eff describe the dimensions of the projection and Δ describes its orientation in the x–y plane.

Now we can solve the Fourier integral for each projection direction individually in full analogy to the spherical case discussed above. The result is a scattered field that resembles the scattering of a sphere for any given direction φ'' = φ' + Δ, but with a direction dependent radius R(φ'') ≡ ζ. The result for the scattered field amplitude reads

$\begin{equation*}E(q,{\varphi }^{{\prime\prime}})={E}_{0}\frac{{r}_{\mathrm{e}}}{{l}_{\mathrm{det}}}{n}_{0}4\pi abc\left(\frac{\mathrm{sin}(qR({\varphi }^{{\prime\prime}}))-qR({\varphi }^{{\prime\prime}})\mathrm{cos}(qR({\varphi }^{{\prime\prime}}))}{{q}^{3}R{({\varphi }^{{\prime\prime}})}^{3}}\right).\end{equation*}$

The corresponding normalized fields (E_n (0, φ'') = 1) can be written as

$\begin{equation}{E}_{n}(q,{\varphi }^{\prime })=3\left(\frac{\mathrm{sin}(qR({\varphi }^{{\prime\prime}}))-qR({\varphi }^{{\prime\prime}})\mathrm{cos}(qR({\varphi }^{{\prime\prime}}))}{{q}^{3}R{({\varphi }^{{\prime\prime}})}^{3}}\right)\end{equation} \tag{ E.6 }$

which resembles the solution of a sharp sphere, but with an angular dependent radius.

Next, we will consider spherical objects with a soft surface density profile, akin to the one used in the main manuscript, i.e.

$\begin{equation*}n(r)=\frac{{n}_{0}}{{\left[\mathrm{exp}\left(\frac{r-R}{\lambda \sigma }\right)+1\right]}^{\sigma }},\end{equation*}$

where n₀ is the core density, R is the core radius, σ is the edge softness parameter, and λ is the scale length of the density decay. Again, due to the radial symmetry we only need to calculate the scattering for a single scattering direction, i.e. we are only interested in the double projected density n_1D(x). The corresponding integral can be calculated in polar coordinates

$\begin{align*}\hfill {n}_{\mathrm{1}\mathrm{D}}(x)& ={\int }_{0}^{2\pi }{\int }_{0}^{\infty }n(r)\rho \,\mathrm{d}\rho \,\mathrm{d}\varphi \hfill \\ \hfill & =2\pi {\int }_{0}^{\infty }n(\sqrt{{x}^{2}+{\rho }^{2}})\rho \,\mathrm{d}\rho .\hfill \end{align*}$

In contrast to the scenarios with a sharp surface discussed above, this integral can in general not be solved analytically. However, since only a single one-dimensional integral has to be evaluated to calculate the 2D scattering pattern, a direct numerical integration is a fast and viable approach.

Eventually we are looking for an efficient way to compute the scattering pattern for an arbitrarily dimensioned and oriented ellipsoid with a softened density profile given by

$\begin{equation}n(\vec{r})=\frac{{n}_{0}}{{\left[\mathrm{exp}\left(\frac{\kappa (\vec{r})}{\lambda \sigma }\right)+1\right]}^{\sigma }},\end{equation} \tag{ E.7 }$

where n₀ is the core density, κ is the local surface offset of the sampling point $\vec{r}$ (i.e. its distance to the closest point on the ellipsoidal control surface, see main text), σ is the edge softness parameter, and λ is the scale length of the density decay. For this type of profile no general solutions for the singly- or doubly-projected densities n_2D and n_1D are known. An apparent alternative way to obtain these projections involves the three-dimensional sampling of the density on a grid and subsequent numerical projection. This requires the evaluation of the perpendicular distance from the control ellipsoid surface κ for all given sampling points. In the limit of a sphere this turns into the radial distance from the surface, i.e. κ = r − R, which is easily calculated. However, in the general case of an ellipsoid, the evaluation of the perpendicular (or closest) distance is not trivial. In fact, its computation involves numerical root finding procedures [55], which practically precludes application of this method in forward fitting approaches.

To solve this problem, we propose an approximation that is based on a combination of the solution for a homogeneous ellipsoid with sharp surface and the solution for a soft sphere. To recall, for an ellipsoid with sharp surface the scattering profile for any given scattering direction is given by that of sharp sphere, the ellipsoidal character enters the description in the form of an angular dependent effective sphere radius. Here, for the ellipsoid with a soft surface, we use a similar approach by replacing the solution of a sharp sphere with that of a soft sphere for a given scattering direction. This approximation will work best for small eccentricities and even becomes exact for a sphere. A systematic analysis of the approximation quality is shown in figure E1. Note that the eccentricity of the particles used in figure E1 has been chosen larger than the average eccentricity of the particle in the experiment to show the outstanding quality of the approximation in the considered parameter regime.

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The main tool for the fitting procedure will be the above approximation of the scattering pattern of a soft ellipsoid by that of a soft sphere with angle dependent effective radius. In a first step the recorded experimental scattering pattern is transformed into a polar representation. Polar coordinates are the natural coordinates of the numerical scheme for ellipsoid scattering described above and are therefore optimally suited for a comparison of experiment and theory. Furthermore, for properly chosen resolution this step intrinsically applies a sub-sampling of the scattering signal at large scattering angles where the signal strength and therefore the signal-to-noise ratio naturally decreases.

In our approach a simulated scattering image is fully defined by a set of seven parameters. Besides the parameters defined above, i.e. the density projection dimensions a_eff and b_eff, the surface scale length λ, the projections orientation Δ and the edge softness σ that are necessary to compute the normalized scattering pattern I_n, we consider the incident field intensity I₀ and a constant background signal I_bg.

$\begin{equation}I={I}_{0}{I}_{\mathrm{n}}({a}_{\text{eff}},{b}_{\text{eff}},\lambda ,{\Delta},\sigma )+{I}_{\text{bg}}.\end{equation} \tag{ E.8 }$

In our forward fitting procedure parameter optimization has been achieved using the 'trust-region-reflective' algorithm embedded in MATLAB's optimization toolbox. Figure E2 shows two representative exemplary optimization results in comparison with the experimental reference pattern for a sharp surface (top panels) and a soft surface (bottom panels).

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