Two-Photon X-Ray Diffraction

The interference pattern of a circular photon source has long been used to define the optical diffraction limit. Here we show the breakdown of conventional x-ray diffraction theory for the fundamental case of a “source,” consisting of a back-illuminated thin film in a circular aperture. When the conventional spontaneous x-ray scattering by atoms in the film is replaced at high incident intensity by stimulated resonant scattering, the film becomes the source of cloned photon twins and the diffraction pattern becomes self-focussed beyond the diffraction limit. The case of cloned x-ray biphotons is compared to and distinguished from the much studied case of entangled optical biphotons.

Figure 1

Historical and projected future increase in average brightness of storage rings (blue) and peak brightness of XFELs (red). On the right we have also given in black the number of photons per coherence volume or degeneracy parameter, calculated from the peak brightness with Eq. (1) for a wavelength of $\lambda =0.1\mathrm{nm}$. In practice, the achievable degeneracy parameter is lowered by beam line losses.

Figure 2

Assumed diffraction geometry. The incident fields $E({\mathit{r}}_1,0)$ and $E({\mathit{r}}_2,0)$ in the sample plane constitute a two-photon amplitude. The diffracted intensities $I({\mathit{\rho }}_{\mathit{1}},z_0)$ and $I({\mathit{\rho }}_{\mathit{2}},z_0)$ are observed in coincidence in the distant detector plane.

Figure 3

(a) Airy diffraction intensity as a function of normalized momentum transfer $q/q_0$, where $q_0=1.22\pi /R$ is the first node of the Airy pattern. We have assumed $\lambda =1.6\mathrm{nm}$ (Co $L_3$ resonance) and a circular hole of radius $R=725\mathrm{nm}$ in an aperture. The pattern of the hole (black) is given by Eq. (12), the spontaneous diffraction pattern for an inserted 20 nm thick Co film (blue) is given by Eq. (13), and the stimulated two-photon diffraction pattern (red) is given by Eq. (14). (b) The enlarged pattern around the optical axis ($q=0$), revealing the reduction of the width of the stimulated pattern.

Figure 4

Airy diffraction pattern, intensity versus momentum transfer, calculated with Eq. (15) for a circular hole of area $A_{\mathrm{s}}=\pi R^2$ with $R=725\mathrm{nm}$ containing a 20 nm thick Co film and assuming Co $L_3$ ($\lambda =1.6\mathrm{nm}$) resonant excitation. The blue curve is the same as in Fig. 3, and the other curves are for different intensities of a 50 fs SASE pulse consisting of two 20 fs coherent subpulses. The momentum transfer is scaled relative to the first node of the Airy pattern at $q_0=1.22\pi /R$. The inset shows the enlarged pattern around the optical axis ($q=0$), revealing the intensity change in the central Airy disk.

Figure 5

Comparison of the observed Airy diffraction contrast relative to spontaneous contrast (red circles) as a function of incident intensity for monochromatized 50 fs SASE pulses [8] and calculated change (thick gray line) upon stimulation using Eq. (15). The vertical lines indicate the three intensity values used in Fig. 4.

Figure 6

(a) Airy diffraction intensity, scaled to unity at $q=0$ for better comparison, versus momentum transfer for a circular hole, using the same parameters as for Fig. 3. We compare the one-photon (black) and biphoton (green) hole patterns to the stimulated Co film (red) pattern. (b) Enlarged distribution around the optical axis. The hole pattern has a FWHM $0.844q_0$ (black), the stimulated film pattern $0.606q_0$ (red), and the biphoton pattern $0.422q_0$ (green).