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Abstract
The distribution of topological charges on X-ray vortices was measured by differential Fourier space filtering microscope, differential radial Hilbert transform microscope. It was experimentally verified for the first time using a Spiral Fresnel zone plate objective lens. This X-ray microscope is highly sensitive to X-ray topological defects, such as edges and vortices, at the exit-face wave field of objects. Its efficient use is also discussed.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Artificially structured light is extensively explored especially in the visible wavelength region, and has led to various revolutionary applications in modern science and technology. Structured light with the topological defects in the wave fields, such as edges and vortices, are analogous to phase defects in various physical phenomena [1]. Optical vortex is realized in a wide range of frequencies [2–6] and carries the orbital angular momentum (OAM) on their helical wave fronts [7,8]. The optical vortex is utilized in terms of its unique properties of an integer number of helical circulations of the wave front, the topological charge, and zero intensity at the center in super-resolution microscopes [9] and in nano-fabrication beyond the diffraction limit [10,11]. The topological charge of an optical vortex is diagnosed by various methods especially in the visible region but seldom in the X-ray region [12,13].
In this paper, we characterize the X-ray structured light containing multiple topological charges due to OAMs carried by multiple X-ray vortices in the beam. We used a Fourier space filtering microscope with a spiral phase filter based on radial Hilbert transform (RHT) principle [14,15]. The differential RHT microscope with a reversed spiral phase filter is predicted to provide information on the rotation of probability current density in electron beam [16]. We here verified that this microscope is capable of visualizing the distribution of topological charge of X-ray vortices for the first time and moved on to prove that this topological charge is proportional to the Berry curvature, which is the rotation of generalized vector potential on the parameter space.
2. Principle of radial Hilbert transform microscope
The RHT is an extension of the Hilbert transformation developed for signal processing of complex-valued data such as to calculate the derivative of one-dimensional time series of complex amplitudes [Fig. 1(a)]. The RHT instead transforms the input complex amplitudes of two-dimensional array of specimen’s complex transmissivity. Three operations are needed to perform RHT on the specimen’s complex transmissivity: (i) Fourier transformation ($F$) of the specimen’s complex transmissivity, (ii) multiplication of the output of (i) and a phase filter in the reciprocal space, and (iii) inverse Fourier transformation ($F^{-1}$). The resulting array of complex amplitudes equals to the convolution of the specimen’s complex transmissivity and the inverse Fourier transformation of the phase filter [Figs. 1(b) and 1(c)].
Fig. 1. (a) The schematics of Hilbert transformation. The phase filter in the angular frequency domain, $G(\omega )$, is a step function of $\pi$ with its inverse Fourier transformation corresponding to $g(t)$. The complex amplitude of signal in the time & angular frequency domain are expressed as $\psi _{1} (t)$ & $\Psi (\omega )$. The edge of the signal $\psi _{1}(t)$ is enhanced in the output $\psi _{2}(t)$, convolution of $\psi _{1} (t)$ and $g(t)$. (b) The schematics of radial Hilbert transformation. Using a spiral phase filter with multiple of 2$\pi$ phase jump in the reciprocal space, $G({\vec k})$, the edge of the objects is enhanced in the output image in the real space. The function of SFZP is summarized in the purple dotted box. (c) The method of orbital angular momentum imaging using differential radial Hilbert transform with images taken with the reversed orientations of SFZP. It shows a distinct enhancement or decrement of intensity at the center of spiral phase anomalies. This visualizes the topological charge which is proportional to the Berry curvature $\Omega _{z}$ as shown in Eq. (3). Amplitude & phase are expressed by brightness & hue, respectively.
More detail on the property of the RHT microscope is described as follows. We put $\psi _{1}({\vec r})$ to be the complex transmissivity of a specimen. The Fourier transformation of $\psi _{1}({\vec r})$ is $\Psi ({\vec k})$. A spiral phase filter with the jump of $2l \pi$, expressed as
is multiplied on $\Psi ({\vec k})$ in the reciprocal space [Figs. 1(b) and 1(c)], where $l$ is an integer topological charge, $\phi _{k}$ is the azimuthal angle in the reciprocal space. $G({\vec k})$ is the Fourier transformation of $g({{\vec r}}) \propto \textrm {e}^{il\phi }$ with the azimuthal angle, $\phi$, in the real space [16]. Finally, the output complex amplitude is obtained by the inverse Fourier transformation of $\Psi ({\vec k}) G({\vec k})$, which is the convolution of $\psi _{1}({\vec r})$ and $g({{\vec r}})$. In our RHT microscope experiment, the functions of a spiral phase filter and of an objective lens to perform the inverse Fourier transformation are simultaneously realized by using a Spiral Fresnel zone plate (SFZP) [15] [Figs. 1(b) and (c)]. This RHT microscope, Fourier space filtering microscope with a spiral phase filter, is known to provide an edge enhanced image of the specimen [15].3. Imaging of orbital angular momentum using differential radial Hilbert transform microscope
We now explain the mechanism of the differential RHT microscope with a spiral phase filter and its flipped one to provide X-ray OAM imaging downstream of chiral materials. We assume that a chiral specimen has a spiral phase anomaly with the exit-face wave field of $\psi _{1}({\vec r}) \propto \textrm {e}^{im\phi }$ where $m$ is an integer topological charge. The exit-face wave field of this specimen hosts an OAM and accompanies a complete circulation of the Poynting vector ${\vec S}$. The orientation and the number of this circulation correspond to the sign and the topological charge of the formed optical vortex.
We then take two RHT microscope images of the specimen ($I^{l=1}$ and $I^{l=-1}$) by reversing the spiral phase filter or the orientations of the SFZP objective lens. The differential image between these two images is proportional to the rotation of the Poynting vector ${\vec S}$ at the exit-face of the specimen [16] as follows,
4. Experimental setup and result
Hereafter we show how to realize the edge-enhanced X-ray imaging and X-ray OAM imaging downstream of chiral materials using an X-ray RHT microscope. This experiment was performed at Experimental Hutch 2 along BL29XUL [23], an undulator beamline of SPring-8 facility in Japan, using 7.71 keV X-rays from a double crystal monochromator. Its higher order reflection light was reduced by a pair of total reflection mirrors. The gap of the undulator source was set to 12.207 mm to equalize the peak energy of 1st order harmonics to 7.71 keV. An objective SFZP lens was placed at the focal plane of the upstream Fresnel Zone Plate (FZP) with the focal length of 1.72 m while the X-rays transmitted through the specimen between the FZP and the SFZP [Fig. 2(a)]. The unwanted Fresnel diffraction term in the propagated wave field between the specimen and the SFZP is cancelled by this setup. Thereby, the complex amplitude at the SFZP plane becomes equal to the Fourier transformation of the specimen’s complex transmissivity [24].
Fig. 2. (a) Setup of RHT microscope with an SFZP objective lens placed at the focal plane of the Fresnel Zone Plate. (b) Microscope image of the SFZP used in the experiment. (c) Edge enhanced image of a copper mesh by the RHT microscope exhibiting the rims of the aperture. The exposure time was 5 seconds.
The specifications of the FZP and the SFZP were the diameter of 2$r_{N}$= 664 & 648 $\mu$m, the first zone radius of $r_{1}$=16.6 & 10.8 $\mu$m, the outermost zone width of $\Delta r_{N}$= 0.415 & 0.18 $\mu$m, number of zones of 400 & 900, respectively. The zone depth of 1.84 $\mu$m was chosen for the tantalum FZP & SFZP to realize $\pi$-phase shift at 7.71 keV. The microscope image of the specimen was taken with the magnification factor of 1.09 by selecting the distances between sample-SFZP and SFZP-detector to be 1.40 m and 1.52 m, both around twice the focal lengths of the SFZP of 0.73 m to meet the lens formula. The X-ray images were taken with an X-ray image sensor composed of a scintillator and a lens-coupled CMOS camera with the effective pixel size of 325 nm [25]. This RHT microscope enables us to observe objects over a wide range of spatial length scales. The field of view was reduced to approximately the radius of the SFZP, around 324 $\mu$m, in one dimension and larger in the other dimension due to the order selection of light. This field of view is defined by the two blades placed in front of the specimen and of the image sensor[Fig. 2(a)].
We experimentally verified that the RHT microscope, a Fourier space filtering microscope with a spiral phase filter, provides an edge-enhanced imaging with a wide field of view. For this experiment, we observed a copper mesh with the pitch of 12.5 $\mu$m[Fig. 1(b)]. As shown in Fig. 2(c), an edge-enhanced image with the bright open squares at the rims of the aperture of mesh was observed as was expected. This result shows that the edge-enhanced imaging of various objects will be promising with RHT microscope with SFZP.
We then confirmed the theoretical prediction that a differential RHT microscope with reversed spiral phase filter visualizes the topological charge of the OAM downstream of chiral materials. For this purpose, we fabricated a specimen composed of spiral phase plates on a silicon substrate with the maximum depth of 19.5 $\mu$m, corresponding to the phase shift of $2 \pi$ for 7.71 keV X-rays. The radii of spiral phase plates and the distances between the neighbors were set to be 34 $\mu$m and 80 $\mu$m, respectively. The thickness gradients of the upper and lower spiral phase plates [Fig. 3(a)] were set to decrease in an anti-clockwise and in a clockwise orientation around the center. By transmitting X-ray through these two plates and by setting the sample towards downstream, phase delay occurred in the same orientations and generated X-ray vortices with the topological charge of $m = 1$ and $m = -1$, where $m$ is an integer topological charge. Thereby X-ray vortices with the reversed OAMs were realized at the exit-face. The FZP, the SFZP and the specimen were fabricated on a 500 $\mu$mt silicon substrate, back edged to have the thickness of $\simeq 80~\mu$m by the Shinko-Seiki Corp. (Kobe, Japan).
Fig. 3. (a) Image of two spiral phase plates with reversed orientations of thickness decrement observed by Scanning Electron Microscope (SEM). $m = \pm 1$ is the topological charge of exit-face X-ray vortices when these two plates are set towards downstream. RHT microscope images of the specimen in (a) are shown in (b) $l=1$ (anti-clockwise from downstream) and (c) $l=-1$ (clockwise) where $l$ is the setting of the SFZP objective lens. The exposure time were 5 seconds. (d) Differential image $I^{l=1}-I^{l=-1}$ of images in (b) and (c). The resultant image is proportional to the topological charge of X-ray vortices at the exit-face of specimen and to the Berry curvature. (e) Distribution of $I^{l=1}-I^{l=-1}$ calculated by Eq. (2). The scale bars shown in (b)-(e) represent 20 $\mu$m.
The images of this specimen were obtained for two settings of the SFZP objective lens [Fig. 2(a)], one in an anti-clockwise ($l = 1$) and the other in a clockwise ($l = -1$) orientation from downstream, where $l$ corresponds to an integer topological charge of the SFZP [see Figs. 3(b) and 3(c)]. The microscope image showed that the center of the spiral phase anomalies (specimen) exhibited a distinct bright spot when $(l,m)= (1, -1)$ and $(-1, 1)$, where the topological charges of the SFZP and of the specimen’s chiral phase anomaly are cancelled. Such intensity enhancements were not observed when $(l,m)= (1, 1)$ and $(-1, -1)$. The distribution of the topological charge was derived by differentiating two images in Figs. 3(b) and 3(c) and is shown in Fig. 3(d), where a remarkable pair of bright and dark spots was exhibited at the centers of the spiral phase anomalies. Similar pair was also clearly observed in the theoretical calculation shown in Fig. 3(e). The ratios of maximum/minimum values at the image of $m=\pm 1$ spiral phase anomalies were -1.2 and -1.1 for the experiment [Fig. 3(d)] and for the theoretical calculation [Fig. 3(e)] which agreed fairy well. This result appears consistent with the ratio, -1, of the topological charge in Eq. (3) (for $m= 1$ and $m= -1$). The slight discrepancy of the observed ratio is likely caused by the spatial dependence.
5. Summary and future prospect
To summarize, we experimentally verified that the topological charges on the X-ray vortices are visualized by a differential RHT microscope with a reversed spiral phase filter. This is the first result to determine the distribution of topological charges of X-ray OAMs downstream of chiral materials. We also demonstrated that edge enhanced imaging is enabled by an X-ray RHT microscope.
We demonstrated that RHT microscope provides us a high sensitivity to detect topological defects, both edges and vortices, in the wave-field downstream of specimens. The Bragg reflection at crystal containing threading spiral dislocations will possibly transfer and induce topological defects in the reflected wave-field [Ohwada et al. in preparation] [26]. The differential X-ray RHT microscope in Bragg reflection geometry will enable us to purely visualize the screw dislocations inside nano-crystalline specimens. It will be a completely new alternative to electron microscope which uses mapping of atoms to find screw dislocations [27]. X-ray RHT microscope will also enable us to provide an edge enhanced image in every in-plane direction and enable us to directly observe the domain wall [28], e.g. in crystalline ferroelectric and magnetic materials. The X-ray RHT microscope in Bragg reflection geometry will further play a key role in investigating the property and the dynamical interactions of dislocations, which deeply affect various properties of materials, such as the strength of metals [29], the efficiency of light emitting diodes [30] and of electronic devices [31].
Acknowledgments
Experiments at SPring-8 BL29XU have been performed with the approval of RIKEN (under proposal numbers of 20170065). Y. K. expresses sincere gratitude to Drs. L. Szyrwiel and A. Takeuchi for fruitful discussions.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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