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We show momentum-space characteristics of X-rays affected by Berry’s phase in a deformed crystal, allowing a 15 keV beam inside a silicon crystal to be translated parallel to its optical axis while retaining its angular divergence and wave front. This data is the first evidence supporting the whole theoretical picture of Sawada et al., Phys. Rev. Lett. 96, 154802 (2006), consisting of two equations of motion about the X-ray propagation. An output beam was as much as 3.3% of the incident after propagating through 1.3 mm silicon along a lateral direction of the chip inclined at 17.722°. As its initial practical application we further utilized the device as an X-ray intensity modulator. Our results revealed a new aspect of the Berry phase and lead to an X-ray waveguide that can enhance the flexibility of future high-energy experiments.
© 2016 Optical Society of America
1. Introduction
Manipulation of light using an artificial periodic structure is one of the most challenging topics in modern physics and its applications. Successful outcomes have been achieved in metamaterials and photonic crystals, yielding novel optical fibers [1], light-trapping cavities [2], and future visions of an invisibility cloak [3, 4]. However, manipulation of X-rays is difficult because of a weak interaction with materials. Here a new way of X-ray control is realized by millimeter-scale translation of X-rays due to a nanoscale atomic displacement in a crystal [5,6]. The next question arises whether this huge translation affects momentum characteristics or not. Theoretically the translated X-rays should possess the angular characteristics that are almost independent of the output position. For adequate understanding of this phenomenon, we start with a brief introduction of the X-ray diffraction and its theoretical background.
2. Theoretical background
X-ray propagation in a single crystal is explained by the wave dynamics, and further by the dynamical diffraction theory [7,8]. The theory takes into account multiple scattering of X-rays, and provides exact solutions to the wave field in a lattice. The numerical solutions of the wave functions are applicable to even complicated distortions in crystals [9–13]. On the other hand analytical representation of an X-ray trajectory is also useful to grasp a macroscopic effect by crystal distortions.
To address the point, Sawada et al. takes into account the dynamical scatterings in equations of motion of an X-ray wave packet [14]. This approach describes the contribution of deformation as the correction to the X-ray propagation in a perfect crystal. A wave packet in a perfect crystal propagates parallel to the group velocity v⃗g = ∂ω(k⃗)/∂k⃗, where ω(k⃗) is a dispersion relation between frequency ω and the wave vector k⃗. This is naturally not the case if some perturbation such as crystal deformation is present. In a deformed crystal, Bragg’s condition depends on the position, and correspondingly the group velocity of X-rays is also subject to the atomic displacement. Therefore the additional contribution of the deformation should depend on incident angles and positions of X-rays, and is correspondingly described in terms of both momentum and real spaces, namely phase space. Such correction terms are written in terms of the Berry curvatures defined in phase space that relates to the Berry phase [15].
Propagation of a wave packet in phase space is described both by its center position and direction, and follows equations of motion written as [14]
where the first terms represent the common properties on transmitted light, and the second terms are the corrections due to the crystal deformation. Ωkr and Ωrk are Berry curvatures that represent the profiles of the wave functions in both momentum and real spaces as we discussed before. Ωrk is defined by a periodic part of the electromagnetic Bloch function Ũ(r⃗) as 〈∂rŨ(r⃗)|i∂kŨ(r⃗)〉 − 〈∂kŨ(r⃗)|i∂rŨ(r⃗)〉, while Ωkr is similar in form. In the X-ray regime, Ωkr in Eq. (1) becomes large in the vicinity of Bragg’s condition, and enhances the wave-packet translation. This effect was experimentally verified by Kohmura et al. [5,6]. However the other equation about the dynamics of a wave front, Eq. (2), has not yet been examined.In order to analyze a role of Eq. (2), we first consider an electric susceptibility in Fourier series written as χ(r⃗) = ∑G χGeiG⃗·r⃗, where G⃗ is a reciprocal lattice vector. An effect of crystal deformation is simply taken into account by displacing an atomic position from r⃗ to r⃗+ u⃗(r⃗). The response function is then replaced by χ(r⃗+ u⃗(r⃗)) = ∑G χGeiG⃗·(r⃗+u⃗(r⃗)) = ∑G[χGeiG⃗·u⃗(r⃗)]eiG⃗·r⃗. This means χG for a perfect crystal is replaced by χGeiG⃗·u⃗(r⃗) for a deformed crystal. In the dispersion relation, the Fourier components appear as |χG|, and thus the effect of the deformation gives no contribution within the replacement. The dispersion relation has no position dependence, and the first term of Eq. (2) is −∂ω/∂r⃗ = 0. The solution of Eq. (2) is accordingly dk⃗/dt = 0, namely the wave vector k⃗ is constant. This means that the angular divergence of a translated beam is not affected by the crystal deformation. Therefore the wave-front direction of an output beam should be the same as that of the original beam.
Our focus in this paper is to investigate this theoretical expectation by experiments. A notable point here is that the unique translation produces a large change of a beam trajectory (Eq. (1)) with negligible change of a wave-front direction (Eq. (2)). The combination of the two equations expects X-ray outputs of any trajectory through a deformed crystal to be unidirectional (Fig. 1).
Fig. 1 Schematic views of the X-ray translation: (a) X-rays penetrate through a distortion-free silicon crystal with no macroscopic shift. (b) X-rays show parallel translation due to atomic displacement that can be controlled by the piezoelectric device.
3. Experiment
An experiment on the angular divergences of the translated X-rays was performed with BL29XUL in SPring-8 [16, 17]; see Fig. 2 for details of the setup. We arranged a spatially-and-spectrally narrow 15 keV reference beam using a double-crystal Si(111) monochromator, three slits, and 400 reflections of a channel-cut Si(100) crystal located upstream of the test optics. The energy dispersion of the reference beam is expected to be ΔE/E ∼ 3 × 10−5 based on the experimental setup. The angular divergence of the reference beam is about 2.5 arcsec in FWHM, which is estimated from iterative comparisons of observed and simulated rocking curves of 111 reflections of a Si(111) analyzer crystal. The test optics consists of a 100 μm-thick square Si(100) chip. The amount of crystal deformation was about 5 μm that is controlled by our piezoelectric bending machine to maximize the target signals. The reference beam was incident at the vicinity of Bragg’s condition of 400 reflections on a Si(100) chip surface near an edge. The output beams were confirmed by a phosphor CCD camera, then the angular divergences were observed by introducing an analyzer Si(111) crystal and a photodiode. Data acquisitions were performed in one day, when fluctuation of the beam currents of the electron storage ring was maintained at less than 1%.
Fig. 2 Schematic view of the experiment. The original beam of 15 keV X-rays was first shaped by the 0.5 mm-square Front-End (FE) slit. The double-crystal Si(111) monochromator was introduced to make an energy dispersion sufficiently small. Then the X-ray beam passed through the 2 mm-square Transport Channel (TC) slit, and is collimated by Slit A of (Vertical, Horizontal) = (10, 500) μm. The channel-cut Si(100) crystal was further placed to collimate the reference beam. The test optics and a combination of the phosphor screen, the 3% transmission neutral density filter, the optical zoom lens, and the CCD detector were finally arranged downstream. The sample crystal was a 100 μm-thick well-polished 20×20 mm Si(100) chip, and the beam incidence angle was aimed at near Bragg’s condition of 17.722° for 400 reflection. X-ray fluxes were monitored by the ionization chambers (ICs), and a part of the Bragg reflection was observed by a photodiode (PD). The rocking curves were measured using the blade (for blocking non-target signals), the analyzer Si(111) crystal, and another PD. The bottom ruler and the labels indicate the distances from the light source.
4. Analysis and result
The resulting CCD images are shown in Fig. 3. Two separated signals were clearly present only when we adjusted both the beam incident angle and the amount of crystal deformation to induce the X-ray translation (Fig. 3(c)). One of the output comes out from the crystal edge (Beam-I), and the other was near the reference beam (Beam-II; see Figs. 3(c), 3(d) and 4(a)). The silicon refraction was negligible at such short distance away from the test optics (Fig. 3(b)), indicating that the translation and splitting into Beam-I and -II are the pure response of the Berry-phase effect. The splitting here is explained by the angular coverage of Bragg’s condition. The intrinsic Darwin width of 400 reflection of Si(100) is 1.8 arcsec by the XOP software [18], indicating that the acceptance of a main component is only about 60% of the incident beam with the angular divergence of 2.5 arcsec. We therefore blocked Beam-II as a residual part by a blade located downstream of the test optics, and then conducted systematic surveys to find out how Beam-I is affected by changing experimental parameters.
Fig. 3 Comparison of transmission images taken by the CCD camera: (a) The reference beam without the test optics. (b) The signals through the crystal chip with the incidence at an off-Bragg angle, confirming that the silicon refraction was negligible. (c) The output of the Berry-phase translation, yielding the splitting into Beam-I and Beam-II. (d) An exposure-corrected absorption map constructed by an X-axis raw-beam scan with a 0.5 mm step size, showing the positions of the crystal and the output beams. The red and green contours indicate the shapes/positions of (c) and an X-ray beam without the narrow Slit A, respectively. Beam-I was from the side surface of the crystal, and Beam-II was near from the position of the reference beam. The bottom labels of each panel show the combinations of optical elements for taking respective images.
The angular dependence of the X-ray beam intensities, namely the rocking curves of the reference beam and Beam-I, is illustrated in Fig. 4(b). The Darwin width of Beam-I was almost comparable to that of the reference beam of about 5 arcsec in FWHM. The angular divergence was preserved in the translated beam. On the other hand the peak angles of the rocking curves appear to be consistent within experimental uncertainty of about 2 arcsec, which is a selection bias expected from the intrinsic Darwin width of 400 reflection of the Si(100) sample. In conclusion, our observations revealed that X-rays preserved their angular characteristics in the shift, such that incident and output beams were almost parallel at the level of order arcsec. This is a remarkable contrast with the huge translation along the crystal which was inclined at 17.722°. These results are consistent with the theoretical expectation from Eqs. (1) and (2).
Fig. 4 Summary of the observed signals: (a) The vertical projection profiles of Figs. 3(a), 3(b), and 3(c) are shown color-coded. (b) The rocking curves are by the same colors as (a); the red line indicates the signal of Beam-I. (c) The oscilloscope measurements of the Beam-I fluxes using IC3. The red and green lines show background-subtracted and background profiles, respectively. (d) Comparison of the vertical projection profiles of Beam-I taken by Z-axis scanning of the test optics. The red solid line shows the same of that in (a), and the dashed-and-dotted lines show data observed at other positions. The peak values are plotted in black, showing the steep decline in the output flux at 0.4 mm in distance from the position of the reference beam.
5. Discussion
The X-ray translation is enhanced by a small change of atomic displacement (Eq. (1)). We thus considered the device applicable for an intensity modulator. For example, a piezo oscillation was generated by a function generator with a frequency of 500 Hz and a swinging voltage of 2.0 V, corresponding to a small vibration of order 100 nm. A simple demonstration of our device as an on-off switch of the translated X-rays is shown in Fig. 4(c). The temporal variation due to the active control of distortion was clearly found in the Beam-I flux by IC3.
The output efficiency of this device was estimated by Z-axis scanning of the sample crystal. Figure 4(d) shows Beam-I projection profiles at different distances from the crystal edge. The line chart of the peak values clearly indicates the detection of Beam-I in Z ≲ 0.4 mm from the reference beam. Here using another material with small absorption is useful to improve the maximum translation length and its efficiency. A remarkable point is that 3.3% of the reference beam appeared at about 0.4 mm away, corresponding to a 1.3 mm translation through the inclined silicon crystal at 17.722°. A transmission coefficient of 1.3 mm silicon is 5.3% [19], and so the observed value is comparable but slightly lower than that expected from absorption. This is because the Beam-I acceptance of 60% described above; at face value the ratio provides a plausible match to the amount of the missing flux of Beam-I.
Regarding the Beam-II branch, the local peak is also due to the translation but its angular characteristics are slightly different from Bragg’s condition. Thus the enhancement of the translation is suppressed. Accordingly, the wave packet propagates through another region characterized by different deformation of the crystal. The existence of Beam-II results from the differences both in the beam angle and path with those of Beam-I, yielding the different amount of translation. If the angular divergence of the incident beam is much smaller, then Beam-II will disappear due to the total acceptance of Bragg’s condition. If the angular divergence is in contrast much larger, then Beam-II will become broader. In our experiment the crystal deformation was optimized only for obtaining a large flux of Beam-I, and resulted in the clear peak of Beam-II fortunately. Future studies of interest are to deform a crystal more precisely. That enables us to control the whole properties of both Beam-I and -II, and further to merge all the signals at the same point.
An advantage of this device is that the large parallel shift of an optical axis in X-rays can be produced, just by using a single piece of a crystal. Moreover, the translation does not diffuse the wave front, suggesting a potential means for an interference device. A natural solution making an effective use of all these properties would be downsizing an X-ray interferometer that is a long-awaited technique, for example required for opening up the next generation of space telescopes [20, 21]. Another possible clue to future prospect is an application to a high-speed X-ray switch that is also needed to expand the availability of X-ray free-electron lasers. Future improvement in such fields would be of interest with much room for discussion.
In summary we unveiled the Berry-phase effect by observing the small angular divergence of the translated X-rays by a deformed crystal. Our results pointed out the utility of the Berry curvatures in both momentum and real spaces. Such an approach is not only useful in X-ray optics but gives us a general view of science over various research fields including condensed matter physics. Moreover in a practical viewpoint, we found applications of this response as an X-ray waveguide and/or an intensity modulator. The resulting optics secured reasonable efficiency in the translation as much as 3.3% at 0.4 mm away. The understanding of the geometrical phase concept makes an ordinary silicon chip a unique optical device that would open up a future technique of X-ray manipulation.
Funding
JSPS/KAKENHI: Numbers 26610051 (D. T.), 25286092 (Y. K.), 26287065 and 26289120 (K. S.).
Acknowledgments
This research has made use of data obtained by BL29XUL in SPring-8. The authors express gratitude to K. Sunouchi, H. Ohmori, and RIKEN manufacturing teams for developing devices. D. T. thanks H. Osawa for advice on measurements, and acknowledges the support from the RIKEN/SPDR program.
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