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[18178.pdf](https://mdr.nims.go.jp/filesets/81d2628e-928c-4e8d-ba28-740f2b9a6432/download)

## Creator

[Yuta Ishii](https://orcid.org/0000-0002-8957-5833), [Yuichi Yamasaki](https://orcid.org/0000-0002-8560-3462)

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©2026 The Physical Society of Japan[In Copyright](http://rightsstatements.org/vocab/InC/1.0/)

## Other metadata

[Evaluation of Topological Charge Distribution of Structured Soft X-ray Beam](https://mdr.nims.go.jp/datasets/c98bc996-c3fc-43e5-b9c0-adae03c0bdd9)

## Fulltext

Evaluation of Topological Charge Distribution ofStructured Soft X-ray BeamYuta Ishii*1 and Yuichi Yamasaki†1,2,31Center for Basic Research on Materials (CBRM), National Institute for MaterialsScience (NIMS), Tsukuba 305-0047, Japan2RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan3International Center for Synchrotron Radiation Innovation Smart, TohokuUniversity, Sendai 980-8577, JapanAbstractOptical structured beams, such as Laguerre-Gaussian and Hermite-Gaussian modes, arecharacterized by their topological charge, associated with phase rotation and photon orbital an-gular momentum. Here, we present a quantitative evaluation of highly coherent structured softX-ray beams generated by diffraction gratings with use of the inline holography technique. Therecorded holographic images show good agreement with theoretical calculations based on thescaled fast Fourier transformation. Spatial frequency filtering enables visualization of the phasedistribution and reveals the radial distribution of the topological charge intrinsic to the struc-tured beams. This work provides a foundation for quantitatively characterizing structured softX-ray beams and highlights their potential for probing and manipulating magnetic propertiesof materials.Optical Laguerre-Gaussian (LG) beam [?], characterized by a helical wavefront, has been thesubject of extensive investigation for several decades owing to their rich physical properties, includ-ing finite orbital angular momentum (OAM) and topological characteristics arising from azimuthalphase rotation [?, ?, ?]. An LG beam can be characterized by the topological charge (TC), definedasℓ =12π∮C∇θ(τ ) · dτ , (1)where θ(τ ) denotes the phase of the photon, and C represents a closed circle encircling the beam’scenter. The TC corresponds to the number of phase rotations around the beam axis and is directlyrelated to the photon’s OAM, expressed as L = ℏℓ. Such structured optical beams have attractedconsiderable interest due to their potential applications in super-resolution microscopy[?], opticaltweezers [?], and quantum information processing [?].*ISHII.Yuta@nims.go.jp†YAMASAKI.Yuichi@nims.go.jp1LG beams can be generated through topological distortions, such as dislocation defects in crys-talline materials [?, ?]. Moreover, recent studies have explored the interaction between opticalvortex beams and magnetic materials, demonstrating their potential for probing and manipulat-ing magnetic properties [?, ?, ?]. In the soft X-ray regime, previous experimental studies havedemonstrated that dislocations in both ferromagnetic and antiferromagnetic lattices can produceLG beams [?, ?, ?]. Furthermore, soft X-ray LG beams hold considerable promise for character-izing topological dislocations in spin textures by exploiting the topological nature of their phasedistribution, as proposed in earlier studies [?, ?]. Soft X-rays are widely employed to investigatemagnetic ordering via resonant techniques, such as X-ray magnetic circular dichroism (XMCD)and resonant X-ray scattering (RXS), owing to their sensitivity to spin, orbital, and anisotropicmagnetic moments [?, ?, ?]. Therefore, the visualization of the phase structure of LG beams andthe quantitative evaluation of their TCs are essential for the application of soft X-ray LG beams toa broad range of magnetic materials. Especially, the TC distribution in LG beams scattered frommaterials reflects the spatial distribution of topological dislocations in magnetic materials. Thus,determining the spatial distribution of the TC is crucial not only for visualizing and evaluatingdislocations in magnetic textures, but also for investigating the interaction between LG beams andmagnetic materials through the OAM carried by the beams.For the quantitative evaluation of TC, we have recently developed an inline holography tech-nique based on coherent soft x-ray microscopy [?, ?]. In this method, an interference pattern formedby the reference beam and the waves diffracted from the sample is detected using a two-dimensionaldetector. The phase distribution of the diffracted wave is encoded within the interference patternand can be retrieved through Fourier analysis. In our previous work, we successfully visualizedthe rotational phase distribution of an LG soft X-ray beam generated by a fork grating using thistechnique. However, the holographic patterns lack sufficient precision for a quantitative evaluationof TC due to the limited coherence of the incident X-rays. Consequently, quantitative analyzes,such as determining the spatial distribution of the photons’ TCs, remain challenging.Here, we report inline holography measurements of highly coherent LG and Hermite-Gaussianmodes (HG) soft X-ray beams generated using two types of diffraction gratings. Inline holog-raphy measurements with highly coherent incident soft X-rays, as available at the synchrotronfacility NanoTerasu in Japan, enable the quantitative visualization of their phase distributions andthe determination of the radial distribution of TC. This approach provides a direct and quantitativecharacterization of the phase structure inherent in structured soft X-ray beams.Inline holography measurements were performed at BL14U, NanoTerasu, using the soft X-ray microscope [?]. The experimental setup is depicted in Fig. 1(a). The incident X-ray beamwas focused using a Fresnel zone plate (FZP) with outer and center beam-stopper radii of 60 and30 µm, respectively, and the number of zones N = 231. The wavelength of the X-ray is tunedto λ = 1.6 nm, which ideally results in a focal length of 1.5 mm via an FZP. The higher orderdiffraction waves from the FZP were sorted by an order sorting aperture (OSA) with a diameter of20 µm. The gratings with a circular outline for generating structured soft X-rays were placed 900µm downstream from the first focal point of the FZP. The pitch and the radius of the gratings are200 nm and 2.5 µm, respectively. The diffracted X-ray wave from the grating interferes with thetransmitted direct X-ray beam through the surroundings of the grating. The interference patternsare recorded using a Charge Coupled Device (CCD) camera, which is positioned 0.345 m awayfrom the sample.Here, two types of gratings are used to generate two types of structured soft X-ray beams: LG2Figure 1: (Color online) (a) Experimental setup of the inline holography based on soft x-ray mi-croscopy equipment. (b) Schematic view of double fork-shaped grating with topological numberb = 2. The black and white regions represent the amplitude of the grating’s transmittance. (c)Holographic image of the n = ±1 diffraction waves from the fork grating with b = 2. Inset showsthe enlarged view of the n = +1 diffraction waves. (d) Simulation results obtained by scaled fastFourier transformation (FFT) calculation. (e)-(g) Experimentally obtained amplitude and phasedistributions of the n = ±1 soft x-ray diffraction waves.and HG modes. These gratings were made from Ta metal with a thickness of 300 nm depositedonto a membrane of Si3N4, and were fabricated by NTT-AT, Japan. Incident X-rays can transmitaround the gratings, which interferes with the diffracted soft x-rays from the grating.Figure 1(c) presents the holographic image of the first-order diffracted soft x-ray waves (n = ±1)from a double-fork-shaped grating characterized by a topological number b = 2, whose schematicview of the grating is shown in Fig. 1 (b). The transmittance of the fork grating is expressed as [?]t(ρ, ϕ) =12(1 + sgn[sin(2πdρ cos ϕ + bϕ)]), (2)with two-dimensional polar coordinates (ρ, ϕ), where d is the pitch of the grating far from thecenter. The vortex beam with OAM ℓ = nb is generated as the n-th Bragg diffraction from thegrating. Distinct stripe patterns superimposed on a reference beam pattern are observed in bothdiffracted waves, accompanied by double fork-shaped structures near the center, as shown in theinset of Fig. 1(c). These features signify interference between LG beams with ℓ = ±2 and thereference beam transmitted around the grating. The results of a numerical simulation based on3Figure 2: (Color online) Distributions of topological charge of the diffraction waves (a) n = 1 and(b) n = −1 along the radial direction ρn.a scaled fast Fourier transform (FFT) [?] are shown in Fig. 1(d). The experimentally obtainedholographic image exhibits excellent agreement with the simulated pattern. It is noted that theweaker intensity of the n = 0 diffraction wave compared with the n = ±1 diffraction waves can bereproduced by including the imaginary part of the complex refractive index of the Ta-based FZP,which can reduce the intensity through interference with the direct beam.The interference intensity of the n-th Bragg diffraction generated by a fork grating with thetopological number b is expressed as [?, ?]Iinternb ∝J ′nb(ρn) sin[kR2 + nbϕn + αnb + n(b − 1)π2], (3)where (ρn,ϕn) denote the local cylindrical coordinates defined with respect to the center of the n-thBragg wave, and k represents the photon wavenumber. The parameter R2 characterizes the phasedifference between the reference wave and the diffracted wave. The function J ′nb(ρn) is given asJ ′nb(ρn) =∫ a0ei f0ρ2ρJnb(ρnρ)dρ = |J ′nb(ρn)|eiαnb , (4)with the nb-th Bessel function of the first kind Jnb. Consequently, both the amplitude and phaseinformation of the scattered wave are encoded within the interference intensity pattern and canbe retrieved through Fourier-transform analysis. The amplitude An(r, ϕ) and phase θn(r, ϕ) distri-butions of the n = ±1 diffracted waves, fn(r, ϕ) = An(r, ϕ) exp[iθn(r, ϕ)], are reconstructed usingfrequency-space filtering, as shown in Figs. 1 (e)-(h). Detailed descriptions of the analysis proce-dure are provided in our previous reports [?, ?]. Experimentally, the reconstructed amplitude andphase exhibit annular and double-spiral phase structures, respectively. Furthermore, the rotationaldirection of the spiral phase is inverted between the n = ±1 waves, demonstrating characteristics ofthe LG beam with ℓ = ±2.The high coherence of the incident soft X-rays, together with the precise interference patterns,enables the quantitative analysis of the scattered structured soft X-rays, including the determinationof the photon TC distribution along the radial direction ρn. The diffracted waves are decomposed4Figure 3: (Color online) (a) Schematic of grating producing HG0,1 mode waves as n = ±1 diffrac-tion. (b) Holographic image of direct beam and n = ±1 diffracted waves. Experimentally obtained(c) amplitude and (d) phase distribution of n = 1 HG waves. (e) Radial distributions of TC forn = 1 HG wave.into the components with ℓ using the following equation;Lℓn(ρn) =∫ 2π0f expn (ρn, ϕn) exp(−iℓϕn)dϕn. (5)with experimentally obtained wave f expn (ρn, ϕn). This equation provides the radial distribution ofeach TC component. Figures 2(a) and (b) show the radial TC distributions Lℓn(ρn), for the n = ±1diffracted waves. Pronounced peaks are observed at ℓ = ±2 and around ρn ∼ 500 µm for then = ±1 waves, respectively, while contributions from other components are negligible. Theseresults demonstrate that the rotational phase structure and the TC distribution of the LG soft X-raybeams are quantitatively determined with high fidelity.While the above results demonstrate the applicability of the present technique to LG modeswith helical phase structures, it is also important to verify its capability for beams with fundamen-tally different symmetries. In this context, we extend the analysis to the HG modes, which arecharacterized by Cartesian symmetry and lack orbital angular momentum [?]. The transmittance ofthe grating employed in this study is schematically illustrated in Fig. 3 (a), where the phase of thegrating is shifted by π between its upper and lower regions. Such grating generates an HG0,1 modebeam, in which the photon phase features a π shift between the upper and lower regions relativeto the beam center. The holographic image obtained through inline holography measurements ispresented in Fig. 3 (b).Diffracted wave patterns accompanied by interference fringes are clearly observed for the n =±1 diffraction waves. For both waves, the stripe patterns in the upper half of the diffraction patternare out of phase by π with respect to those in the lower half. The amplitude An(r, ϕ) and phase5θn(r, ϕ) distributions for the n = 1 diffraction wave are extracted as shown in Figs. 3(c) and (d),respectively. These results confirm that phase inversion occurs along the horizontal central axis forthe HG0,1 mode.Applying the same analysis method as described in Eq. ??, Fig. 3 (e) presents the TC distri-bution of the n = 1 diffracted wave. Pronounced TC components are observed at ℓ = ±1. AnHG-mode beam can be expressed as a superposition of two LG modes; specifically, the HG0,1mode corresponds to LG modes with ℓ = ±1. These results demonstrate that the present techniqueenables the quantitative determination of the TC components intrinsic to HG-mode beams. In ad-dition, minor contributions from several other TC components are also detected. These are likelyattributed to limitations in the fabrication accuracy of the grating or to residual contamination fromthe n = 0 diffraction beam. This observation further suggests that the present technique is suffi-ciently sensitive to resolve TC components even in soft X-ray waves generated from imperfect orarbitrary defect structures.In this study, we have demonstrated the quantitative evaluation of structured soft X-ray beams,specifically LG and HG mode waves. By employing highly coherent soft X-rays, the interfer-ence patterns obtained through inline holography measurements exhibit excellent agreement withtheoretical simulations. Further analysis allows for the quantitative determination of the TC dis-tributions intrinsic to the structured beams. These results pave the way for the development ofstructured soft X-ray beams for probing and manipulating magnetic materials.AcknowledgmentsWe thank Tetsuya Nakamura, Yoshinori Kotani, Akiho Sumiyoshiya, Yusuke Tanimoto, and TaishiKawabata at PhoSIC for supporting us with the CDI experiments at NanoTerasu BL14U, andTaka-hisa Arima and Yusuke Wakabayashi for fruitful discussions. This work was partially sup-ported by JSPS KAKENHI (Project Nos. JP19K23590, JP19H04399, JP20K20107, JP23K17145,JP24K03205, JP24H01685, JP24K17603, JP25K03387), by PRESTO (JPMJPR2102) and CREST(JPMJCR1861 and JPMJCR2435) Japan Science and Technology Agency (JST). 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