Data collected from the phase-contrast instrument was comparably normalized by respective x-ray–only shots, which mitigated propagated lens defects. In the center of each image, a bright spot appeared; this is the third harmonic of the x-ray free-electron laser, which focuses differently from the fundamental wavelength because of its higher photon energy. This feature is most obvious in Fig. 2 spatial scan images, as no x-ray–only background images were saved and were consequently not subtracted. These images were not used for any quantitative purposes. The phase-contrast images in Figs. 2 to 4 were processed with ImageJ to their current color scheme, but all quantitative data were collected with images processed only by the aforementioned x-ray–only normalization. The color scheme was applied evenly, without bias or alteration, to the entire data image.

To calculate the compression across the two elastic waves, the visibility of the two features was measured from their lineouts according to Eq. 3. V is the visibility, Imax is the maximum intensity, and Imin is the minimum intensity across a feature.V=ImaxIminImax+Imin(3)

Each value of visibility corresponds to a phase shift (Δϕ) created by infinitesimal changes in density over the elastic shocks. To quantify the relationship between phase shift and visibility, a simulation propagated an 8.2-keV probe x-ray beam with a specified phase shift. This was repeated across 10,000 shocks of increasing phase shift to create Fig. 5.

This visibility corresponds to a change in phase induced in the propagated electromagnetic wave as it passes through a density discontinuity in the material. The red dashed line shows the maximum visibility.

This change in density can be calculated by comparing the difference in x-ray path length (Δl) across each individual shock.Δl=d[n1n2](meters)(4)

n1 and n2 are the complex indices of refraction before and after the elastic shock, respectively, and d is the propagation distance. In this work, d is the thickness of the material along the propagation direction. A general complex index of reflection can be expressed asn=1δ+iβ(5)where δ is the dispersive term and β is the absorptive term.δ=nareλ22πf1(6)β=nareλ22πf2(7)

na is the number density of the material. re is the classical electron radius (2.8179 × 10−15 m). λ is the wavelength of the probe x-rays (1.5120 × 10−10 m for 8.2-keV radiation). f1 and f2 are the silicon atomic scattering factors. f1 = 14.261519 e/atom and f2 = 0.3190926 e/atom at 8.2 keV.

Therefore, n1 and n2 across an elastic shock can be written asn1=1δ1+iβ1=1na1reλ22πf1+ina1reλ22πf2(8)n2=1δ2+iβ2=1na2reλ22πf1+ina2reλ22πf2(9)

Substituting n1 and n2 into the equation for x-ray path difference results in the followingΔl=d[n1n2](10)Δl=d[(1δ1+iβ1)(1δ2+iβ2)](11)

This simplifies toΔl=d[reλ22π][(na2na1)f1i(na2na1)f2](12)

When solved for the difference in number density (na2na1), this results in(na2na1)=Δld[reλ22π][f1if2](atoms/m3)(13)

The difference in the optical path length can be calculated from the collected data by multiplying the determined phase shift across an elastic wave and multiplying it by the probe x-ray wavelength.Δl=λΔϕ2π(14)

With the system fully constrained, the difference in the density and the corresponding compression of the material can be explicitly calculated. The minimum-resolvable change in phase with this experimental setup is approximately 2π/10, indicating that a density change of 0.025 g/cm3 in silicon will generate a phase-contrast fringe. For the data presented in Fig. 3A, the change in phase across the primary elastic feature (E1) lies between π and 2π; as the visibility decreases, the compression increases. The change in phase across the secondary elastic feature (E2) lies between zero and π; as the visibility increases, the compression increases.

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