Berry-Phase Translation of X Rays by a Deformed Crystal

We experimentally demonstrated an enhanced translation of an x-ray beam nearly parallel to the diffracting planes over millimeter distances in a deformed silicon crystal. This effect is a consequence of the Berry-phase effect in phase space [K. Sawada et al., Phys. Rev. Lett. 96, 154802 (2006)], which enables an interplay between the gap in the dispersion surface in momentum space and the atomic displacements in real space. Such an interplay in phase space enhances the beam translation by some 5 orders of magnitude, leading to the macroscopic effect.

Figure 1

(a) Dispersion curves in a two-dimensional momentum space with a gap $\Delta k$ at the Bragg condition. The two curved lines are constant-frequency contours on the 2D dispersion surface $\omega (\overrightarrow{k})$. They represent the allowed propagating modes of x rays with $\omega (\overrightarrow{k})=\mathrm{const}$ inside a perfect crystal. The offset angle $\Delta \theta$ from the Bragg angle ${\theta }_B$ satisfies $\Delta \theta /{\theta }_B\simeq \Delta k/|\overrightarrow{k}|$ for near-Bragg conditions. (b) Crystal deformation in real space with an atomic displacement $\overrightarrow{r}\to \overrightarrow{r}+\overrightarrow{u}(\overrightarrow{r})$. Dashed lines represent crystal planes for a perfect crystal. $u_0$ is an amount of atomic displacement at the bottom surface layer.

Figure 2

(a) Schematic illustration of the discovered phenomena. The beam exhibits a translation inside the deformed region, reaches the sample edge, and exits without angular divergence. (b) Schematic diagram of the designed crystal curved to observe the phenomena. The incident x rays are composed of several wave packets that experience three distinct varieties of propagation (i)–(iii), depending on the incident angle. (i) On-Bragg condition, the total reflection occurs (paths 1 and 2). (ii) Near-Bragg condition, the enhanced beam translation occurs (paths 3–6). At the edge of the deformed region, the exit beam travels parallel to the incident one and forms a peak at the detector (paths 5 and 6). (iii) Far beyond the Bragg condition, the beams propagate through the crystal without any large beam translation (paths 7 and 8).

Figure 3

(a) Surface profile of the sample crystal, $14\mathrm{mm}\times 11\mathrm{mm}$, measured with a Fizeau interferometer GPI XPHR by Zygo Corp. The green ovals show the size (FWHM) and the location of the illuminating beam spots (BS). The concave surface was set to the incident beam. (b) Vertical profile of (a) along $y_s$ direction passing through the BSs with the residual from a linear function, where $y_s$ is a position coordinate on the sample surface.

Figure 4

Experimental setup. The channel cut and the monochromator crystals were set in (++) arrangement to reduce the energy bandwidth and the angular divergence of the output beam. A phosphor-coupled x-ray camera with $4.55\mu \mathrm{m}$ effective pixel size was used as a detector. The Bragg reflection was also confirmed by an independent counter in the $\theta \mathrm{\text{−}}2\theta$ reflection geometry. The coordinate $y_d\equiv y_s\sin \theta$ represents a position on the detector. The polarization state of the beam was switched from $s$ polarization to $p$ polarization by using a diamond crystal phase retarder with the thickness of 2 mm.

Figure 5

Intensity distributions of the crystal transmitted image in a common logarithmic scale with (a) off-Bragg condition and (b),(c) near-Bragg condition on two different beam spots BS1 and BS2, respectively. The crystal was set to be $700\mu \mathrm{m}$ higher in the case of (c) than in (b). The sharp peaks due to the enhanced translation were observed in the two images indicated by arrows. (d) Vertical intensity projection analyzed from the images in (a)–(c). The displacement of the peaks with arrows in (b) for BS1 and in (c) for BS2, from $y_d=1500$ to $800\mu \mathrm{m}$ away from the beam center, coincided with the crystal movement of $700\mu \mathrm{m}$.

Figure 6

(a) Schematic illustration of the beam exiting at the edge of deformed region. (b) Vertical intensity projections of the transmitted images, for three different glazing incidence angles with $s$-polarized x rays, and for one angle with $p$-polarized x rays, on a common logarithmic scale. The peak was clearly observed only at the incident angles of $-2\mathrm{arcsec}$, not at large offset angles of $-24$ and $+25\mathrm{arcsec}$. The peak intensity was 1.4 times higher with $s$-polarized x rays than with $p$-polarized x rays, consistent with the Darwin width ratio.