Reduction of SAXS Signal due to Doppler Broadening Induced Loss of Coherence

Abstract

We present an analytical and numerical study of how Doppler-induced spectral broadening in laser-heated plasmas degrades the coherence of small-angle X-ray scattering (SAXS) signals, and show that the resulting loss of temporal coherence reduces the SAXS intensity. Applying this formalism to two benchmark geometries - single density steps (wires) and periodic gratings -- we obtain analytic estimates. For gratings, finite coherence simultaneously lowers Bragg-peak heights and broadens their widths, whereas for isolated steps only the overall scaling with q affected. We map the parameter space relevant to current SASE and self-seeded XFELs, revealing that Doppler effects remain managable for the trieval of geometry parameters (less than few 10 % error) for SASE bandwidths but become the dominant error source in seeded configurations or above-keV temperatures. Practical consequences for density-gradient retrieval and interface-sharpness measurements are quantified. The results supply clear criteria for when Doppler broadening must be included in SAXS data analysis and offer a route to infer electron temperature directly from coherence-loss signatures.

Figures (25)

• Figure 1: Definition of directions.
• Figure 2: Doppler broadening FWHM from Eqn. (\ref{['eqn:doppler_shift']}) (solid lines with black dots) and the exact numerical solution including finite $\gamma$, electron recoil and exact Lorentz transformation of the observation angle $\theta$ into the electron rest frame, as well as Maxwell-Juettner diustributed electrons (filled color plot).
• Figure 3: Visualizing the coordinate transform from $x_1,y_1$ to $X,Y$ and the integrand $M(X,Y)$ (Eqn. (\ref{['eqn:integration']})). Light gray: illuminated area; orange: coherence area; blue: area filled by the wire; dark gray: area $I(q)$.
• Figure 4: Effect of finite coherence on wire scattering on the example of $\sigma = 5\,\mathrm{nm}$. Here, we neglect the Doppler effect to keep $a=1$. $E_0$ is set to $8\text{\,}\mathrm{keV}$. Note that the energy spread is extremely large ($\Delta E = 400\text{\,}\mathrm{eV},\,\Delta E/E_0=5\%$) in order to emphasize the effective intensity increase.
• Figure 5: Reduction of scattering signal $I\propto a=\sigma_{\omega,0}/\sigma_\omega$ of a step-target (cp. Eqn. (\ref{['eqn:grating']})), or peak integrated fluence $\int_{-\infty}^{+\infty} I_l(k)dk/\int_{-\infty}^{+\infty} I_{l,\omega_0}|_{T=0}dk=a$ of a grating target (cp Eqn. (\ref{['eqn:normalization']}), in dependence of the three parameters electron temperature $T$, X-ray bandwidth FWHM $\Delta E = 2\sqrt{2\ln{2}}\,\sigma_{E,0}$, and wave number $q$. The bandwidth was chosen representative for SASE beams. The respective plots for parameter ranges relevant for seeded beams can be found in Appendix D for reference.