Incoherent Diffractive Imaging via Intensity Correlations of Hard X Rays

Established x-ray diffraction methods allow for high-resolution structure determination of crystals, crystallized protein structures, or even single molecules. While these techniques rely on coherent scattering, incoherent processes like fluorescence emission—often the predominant scattering mechanism—are generally considered detrimental for imaging applications. Here, we show that intensity correlations of incoherently scattered x-ray radiation can be used to image the full 3D arrangement of the scattering atoms with significantly higher resolution compared to conventional coherent diffraction imaging and crystallography, including additional three-dimensional information in Fourier space for a single sample orientation. We present a number of properties of incoherent diffractive imaging that are conceptually superior to those of coherent methods.

Figure 1

(a) Phase acquired by a photon with wave vector ${\mathbf{k}}_{\mathrm{in}}$ upon coherent scattering by an atom at ${\mathbf{r}}_i$ into direction ${\mathbf{k}}_{\mathrm{out}}$, relative to scattering by an atom at the origin. (b) Corresponding Ewald sphere construction using the photon momentum transfer $\mathbf{q}={\mathbf{k}}_{\mathrm{out}}-{\mathbf{k}}_{\mathrm{in}}$. The black arcs in (a) and (b) represent the angular extent of a detector in the far field and the 2D Ewald sphere coverage in 3D Fourier space, respectively. (c),(d) Phase difference between two photons incoherently scattered by an atom at ${\mathbf{r}}_i$, one into direction ${\mathbf{k}}_2$ and the other one into direction ${\mathbf{k}}_1$, relative to an atom at the origin. The incoming wave does not transfer any photon momentum on the outgoing wave. However, intensity correlations between outgoing wave vectors $({\mathbf{k}}_1,{\mathbf{k}}_2)$, $({\mathbf{k}}_1^{\prime },{\mathbf{k}}_2^{\prime })$, etc. induce momentum transfers $\mathbf{q}={\mathbf{k}}_2-{\mathbf{k}}_1$, ${\mathbf{q}}^{\prime }={\mathbf{k}}_2^{\prime }-{\mathbf{k}}_1^{\prime }$, etc., which contain information about the object. As a consequence, a volumetric 3D coverage of $\mathbf{q}$ values in Fourier space builds up that reaches out twice as far as the Ewald sphere for common detector geometries.

Figure 2

Simulation of a monoatomic cubic crystal with $10\times 10\times 10$ unit cells and a single fluorescing atom per cell showing Bragg peaks in the IDI signal. (a) Intensity distribution on the detector resulting from a single $\le 2.6\mathrm{fs}$ XFEL excitation pulse. The inset shows random speckles with no indication that the object is periodic. (b),(c) Orthogonal slices through the $\mathbf{q}$-space intensity autocorrelation. The insets show Bragg peaks corresponding to the lattice constant of the crystal.