Open Access
Add to BibSonomy
Abstract
X-rays have developed into an essential tool in variety of fields, such as biology, materials, chemistry, and physics etc. Numerous X-ray types, including the orbital angular momentum (OAM), the Laguerre–Gauss, and the Hermite–Gauss states, have been proposed. This greatly enhances the depth of application of X-ray. The X-ray states described above are mostly produced by binary amplitude diffraction elements. In light of this, this paper proposes a flat X-ray diffraction grating based on caustic theory to generate Airy-type X-ray. It is proved by the simulation of multislice method that the proposed grating can generate the Airy beam in the X-ray field. The results show that the generated beams have a secondary parabolic trajectory deflection with the propagation distance, which is consistent with the theory. Inspired by the success of Airy beam in light-sheet microscope, the Airy-type X-ray can be anticipated to enable novel image capability for bio or nanoscience.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
X-rays have emerged as a crucial waveband due to their penetration, fluorescence, and other properties. Different X-ray technologies, such as medical research [1–3], imaging [4], environmental analysis and protection [5–7], have been widely applied. Synchrotron radiation—X-rays created when high-energy electrons move with variable speed in a magnetic field—has the advantages of great brightness, broad spectrum, and high collimation. Since the birth of synchrotron radiation sources in 1947, series of synchrotron radiation facilities (SRF) have been established to produce high-energy X-rays, including ESRF in France, APS in the United States, SPring-8 in Japan, and Petra III in Germany. In addition to the most typical area of focusing, research is also being done on the production of orbital angular momentum (OAM) in the X-ray range. Researchers have studied and proposed several methods and techniques to generate OAM beams in free-electron laser (FEL) with helical undulators [8,9]. Additionally, X-ray OAM may be created using diffractive optics. For example, in 2019, Lee et al. fabricated a binary amplitude fork grating by focusing ion beam etching [10]. The grating was obtained by etching a 1$\mathrm{\mu}$m thick gold substrate attached to a 200 nm thick $Si_{3} N_{4}$ film. The Laguerre-Gaussian diffraction pattern of X-ray is successfully realized at the energy of 500 eV. In this paper, an Airy-type X-ray beam is proposed and explored numerically.
Airy beam is an analytically diffraction-free solution of the nonpotential Schrodinger equation proposed by Berry and Balazs in 1979 [11] and introduced in optical field by Siviloglou in 2007 [12]. The unique properties of Airy beam, such as non-diffraction, self-healing and parabola trajectory, have enabled many applications in a variety of fields, such as light trapping [13], particle manipulation [14,15], and microscopy [16–18]. Inspired by the success in optical field, various wave packets of Airy type are also realized, such as water [19], sound [20], electron beam [21], plasmon [22] and spin wave [23] etc. Even in electromagnetic spectrum, the Airy beam has been extended to THz wave from visible band [24]. Our interest is to develop the Airy-type beam in short wave, such as X-ray. By combining the figure of merits of Airy beam and X-ray, the Airy-type X-ray would inspired more applications in X-ray imaging and processing.
In visible spectrum, it is easy to generate an Airy beam by Fourier lens and cubic phase modulation, which is realized with inserting a programmable spatial light modulator (SLM) into the propagation path of light. Benefiting from the feasibility of SLM, variety types of Airy beam have been explored, such as circular self-focusing Airy beam [25], superimposed Airy beam [26] and Airy vortex beam [27], etc. However, it is not an easy task to modulate the X-ray with pixels as visible light, i.e., the SLM for the X-ray field is also absent. Therefore, in 2023, Shi et al. proposed a method to shape the X-rays directly at the source [28]. They found that free electrons interacting with bending van der Waals (vdW) materials can produce caustic X-rays including X-ray Airy beam. The characteristics of the produced Airy beam can be adjusted by varying the vdW material strain geometry and the electron energy. Besides, as aforementioned, binary element is adopted to modulate or focus X-ray. Generating Airy-type X-ray, therefore could also be realized with binary grating with special phase modulation. In this paper, a 3/2 binary flat diffraction grating is proposed to modulate the X-ray to generate 1+1D Airy-type X-ray. The simulation results show that the Airy beam appears in the process of propagation, and the deflection trajectory is consistent with the theoretical designed trajectory. Benefitting from the peculiar properties of Airy function, this X-ray wavefront can be anticipated promising applications in bioimaging. Our method may pave a way for the generation of special X-ray beam, such as Airy-type X-ray, Mathieu X-ray, Ince X-ray etc.
2. Theory and method
2.1 Caustic theory to design the grating structure for generating Airy-type X-ray
The theory of optical caustic was first proposed by Greenfield [29], in order to find self-accelerating non-diffracting beams with more trajectory. Considering that the non-diffracting beam is directly generated after incident beam passing through the phase mask, the integral form of Fresnel diffraction in the real space is
where P is the amplitude part, the exponential part represents the phase, $k$ is the wave number, $x$ is the lateral coordinates in the propagation process, $\xi$ is the coordinates of the phase mask, $\varphi (\xi )$ is the phase distribution on the mask, and $z$ is the propagation distance. Applying the stationary-phase approximation [30] to Eq. (1). The phase oscillation is slow near the stationary point, and the region near the point will provide principle value on the diffraction integral, while the other regions will cancel each other out due to the rapid phase oscillation. Applying first-order stationary-phase approximation to diffraction integral, the Eq. (2) can be obtained by setting the derivative of the phase part be zeroEquation (2) is a linear equation about the spatial coordinate $x$ and the propagation distance $z$. From the perspective of geometric optics, this equation represents a geometric ray, which indicates that the optical field can be represented by ray clusters. Further, the second-order stationary-phase approximation is considered, obtaining Eq. (3) by setting the derivative of Eq. (2) zero
By setting the first and second derivatives of the phase part zero, the optical field has the maximum intensity around the optical caustics (the envelope formed by the ray cluster). Now assuming that the main lobe of non-diffracting beam has a trajectory of $x=f(z)=a^{m} z^{m}$. By solving Eq. (2) and Eq. (3), a curve $f(z)$ can be obtained, and ${f}' (z)=(x-\xi )/z$. Solving this equation with designed trajectory, the phase distribution on the phase mask can be obtained as Eq. (4)
which is consistent with Ref. [31]. When $m=2$, the trajectory is $x=a^{2}z^{2}$, which represents a typical case, the Airy beam. The phase distribution on the phase mask isEquation (5) is the $3/2$ phase mask which can directly generate Airy beam. Compared with the traditional experimental device that generates Airy beam, this flat optical element can generate the Airy-type beam directly instead of the Fourier transform of cubic phase, which need more elements in optical system. Fig. 1 $(a)$ shows the $3/2$ phase distribution. For X-ray applications, this phase mask is binarized to an amplitude grating with Eq. (6)
Fig. 1. 3/2 phase mask and binary amplitude grating; ($a$) phase diagram that can be directly encoded on SLM to directly generate the Airy beam in the visible spectrum; ($b$) The $3/2$ grating described by Eq. (6).
2.2 Multislice method to simulate the X-ray propagation
Due to the very short wavelength and intense photon-matter coupling in X-ray region, it’s a challenge to simulate the X-ray propagation and modulation. The traditional scalar diffraction is no longer applicable to thick gratings or zone plates for modulating X-ray because the zone structures are approaching the X-ray wavelength. Therefore, approaches such as coupled wave theory (CWT) has been proposed for the calculation of thicker plates such as multilayer Laue lenses fabricated using thin film deposition techniques. However, CWT can only calculate structures that are geometrically periodic or approximately locally periodic. For more complex structures, such as our proposed $3/2$ grating, numerical methods, such as finite element analysis, must be used to obtain a realistic image.
We used the multislice methods that have been applied to Fresnel zone plate simulations [32]. As illustrated in Fig. 2, the grating is discretized into a set of slices with a thickness $\Delta z$, and the slices are filled with a medium with a complex refractive index of
Where $\delta$ and $\beta$ are the phase shift part and absorptive part of the X-ray refractive index.When the X-ray with complex amplitude $\psi$ propagates through the slice with a thickness $\Delta z$, the medium modulates it by
where $\psi ^{'}$ is the complex amplitude of the modulated wavefield. Short distance propagation $\Delta z$ between slices within the grating was accomplished by angle spectrum propagation methodsIs the Fresnel transfer function. For the propagation of the X-ray in free space after leaving the grating, we use
Subscript $d$ represents the observation plane and $z$ is the distance from the back of the grating to the observation plane. We choose the appropriate wavefield propagation algorithms depending on the propagation distance, which effectively avoids the aliasing effects.
3. Results and discussion
3.1 Progation properties of Airy-tpye X-ray generated with the 3/2 grating
The number of slices is determined by
where $\varepsilon _{1}=\varepsilon _{2}=0.1$ [33], $\Delta x$ is the pixel size. Other parameters we used in the simulation are given in Table 1.
Table 1. Parameters used for $3/2$ grating calculations
In simulation, the grating is composed of $WSi_{2} /Si$ structure, which presents high X-ray absorption contrast and outstanding stress performance and has been applied comprehensively for Laue lens to focus X-rays [34]. The phase shift and absorption coefficients of these two materials at 8 Kev energy are choosed from IMD [35], a software for modeling the optical properties-reflectance, transmittance, absorptance of X-ray multilayer films which includes an optical-constant database for over 150 materials, spanning the X-ray to the far-infrared spectrum region.
Figure 3 shows the wavefield intensity distribution at different distances produced by the $3/2$ grating. For more clarity, the region of $5\times 5\mu m$ is taken.
Fig. 3. Intensity of the wavefield created by the $3/2$ grating for parameter $a=0.03mm^{-0.5}$, and propagation distance ($a$) $z=5mm$; ($b$) $z=6mm$; ($c$) $z=7mm$;($d$) $z=7.6mm$. The inset sub-images show the corresponding intensity cross-section curves.
The Airy beam appears and gradually deflects to the lower left of the image during the propagation. The inset images in Fig. 3$(a\sim c)$ indicate that the profile of the cross-section of X-ray maintains Airy function profile with the propagation distance rising from $5mm$ to $7mm$ and the FWHM of the main lobe is almost one constant which is about $0.05\mu m$. It proves the diffraction-free property of Airy-type X-ray. However, the property is broken up at a propagation distance of $z =7.6mm$ as the FWHM of the main lobe rises to $0.095\mu m$, which means the approximation condition is destroyed.
This can be explained by the caustic theory, the wavefield is composed of geometric ray clusters emanating from the grating. However, there is a maximum size of the grating, which leads to a maximum non-diffracting distance. The maximum non-diffracting distance of the $f(z)=a^{m} z^{m}$ beam generated by the grating of size $L$ can be obtained from the geometric relation as
It can be calculated that the maximum non-diffracting distance is $7.5mm$ with the parameters in Table 1, which is consistent with numerical simulation results.
To further verify the validity of Eq. (13), the deflection of the generated beam at the propagation distance of $4mm$ to $9mm$ is simulated. Fig. 4 shows the curves of horizontal deflection (in microns) as a function of the propagation distance (in millimeters) with $L=50 \mu m$ and different $a$.
The origin of the deflection is taken at the top left corner of the grating. When $a=\left \{ 0.03,0.035,0.04\right \}mm^{-0.5}$, the maximum non-diffracting distance caculated by Eq. (13) are respectively $z=\left \{ 7.5,6.4,5.6\right \}mm$. Fig. 4 shows that after these three distances, the deflection of the beam gradually deviates from the theoretical trajectory due to the diffraction spread of the main lobe.
3.2 Self-healing property of Airy-type X-ray
A typical property of Airy type beam is the self-healing, i.e. the blocked part can be regenerated with proper propagation distance.
As shown in Fig. 5$(a)$, The dark area is marked as obstacle. Then the intensity distribution at various distance is simulated to characterize the self-healing properties.
Figure 6 shows the intensity distribution at $z=\left \{5.5,6,6.5\right \}mm$. The main lobe of the resulting Airy beam is missing at $z=5.5mm$ due to the blockage, and the energy is primarily dispersed at the side lobes. As the propagation distance increases, at $z=6mm$, the energy is gradually transferred from the side lobes to the main lobe, which is gradually formed. At $z=6.5mm$, the main lobe is fully formed.
Fig. 6. Intensity of the Airy X-ray with central blocked; ($a$) $z=5.5mm$; ($b$) $z=6mm$; ($c$) $z=6.5mm$.
This demonstrates that the Airy X-ray generated by the designed $3/2$ grating owns self-healing property can be used, for example, for X-ray sheet imaging. The self-healing property is also beneficial for the device fabrication. As shown in Fig. 1, the thickest part of the grating is in the top left corner. This part would be very challenge for fabrication. Because the common method to fabrication the grating is to deposit the materials layer by layer with sputtering or other nano/micro fabrication methods, a layer with larger thickness would spend more time and require higher stability when depositing it. Therefore, if we can delete the thick layers of the grating without decreasing beam quality dramatically, it will be very beneficial for device fabrication. As shown in Fig. 5$(b)$, we kept only the central part of the grating, i.e. the green wire frame. The results are shown in Fig. 7, which shows the intensity of the wavefield at propagation distances ranging from $5mm$ to $7mm$.
Fig. 7. Intensity of the wavefield created by the partial grating for parameter $a=0.03mm^{-0.5}$, and propagation distance ($a$) $z=5mm$; ($b$) $z=6mm$; ($c$) $z=7mm$; The inset sub-images show the corresponding intensity cross-section curves.
Compared to Fig. 3, partial grating decreases the beam size and intensity and after $z=5mm$ but has tiny effect on the beam shape which illustrate that the form of the beam is mainly modulated by the central part of the grating. However, we observe that the beam has undergone diffraction broadening at $z = 7mm$, which is shorten than $7.5mm$ calculated by Eq. (13) before.
Figure 8 shows the deflection of the Airy X-ray produced by the complete and partial grating from $4mm$ to $9mm$ with $a=0.03mm^{-0.5}$. The results show that the actual deflection after the partial grating deviates from theory at around $z =6.8 mm$, shorten than $7.5 mm$. This illustrates that the reduction of grating size reduces the maximum non-diffracting distance as predicted by Eq. (13).
4. Conclusion
In this paper, the Airy-type X-ray state is proposed and investigated. The Airy-type X-ray can be generated with $3/2$ grating, which is composed of $WSi_{2} /Si$ structures. The propagation properties of Airy-type X-ray are simulated numerically with multislice method. The simulated results show that the trajectory of Airy-type X-ray keeps the similar properties with its counterparts in other wave form, such as visible light, acoustic wave, water etc.. The self-healing property is also verified with numerical simulation. Meanwhile, this property reduces the difficulty to fabrication $3/2$ grating. This special X-ray beam may unlock more powerful applications of X-ray, such as X-ray sheet imaging as Airy beam in visible range.
Funding
National Natural Science Foundation of China (11875204, 61975148, U1932167).
Acknowledgments
This work was supported by the National Natural Science Foundation of China (NSFC) [grant numbers 61975148, 11875204, U1932167].
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. R. Mizutani and Y. Suzuki, “X-ray microtomography in biology,” Micron 43(2-3), 104–115 (2012). [CrossRef]
2. L. Marco, E. Greaves, and J. Alvarado, “Analysis of human blood serum and human brain samples by total reflection X-ray fluorescence spectrometry applying compton peak standardization,” Spectrochim. Acta, Part B 54(10), 1469–1480 (1999). [CrossRef]
3. M. B. Altman, M. J. Flynn, R. M. Nishikawa, I. J. Chetty, K. N. Barton, B. Movsas, J. H. Kim, and S. L. Brown, “The potential of iodine for improving breast cancer diagnosis and treatment,” Med. Hypotheses 80(1), 94–98 (2013). [CrossRef]
4. S.-A. Zhou and A. Brahme, “Development of phase-contrast X-ray imaging techniques and potential medical applications,” Phys. Med. 24(3), 129–148 (2008). [CrossRef]
5. M. Bernick, D. Kalnicky, G. Prince, and R. Singhvi, “Results of field-portable X-ray fluorescence analysis of metal contaminants in soil and sediment,” J. Hazard. Mater. 43(1-2), 101–110 (1995). [CrossRef]
6. C. Parsons, E. M. Grabulosa, E. Pili, G. H. Floor, G. Roman-Ross, and L. Charlet, “Quantification of trace arsenic in soils by field-portable X-ray fluorescence spectrometry: considerations for sample preparation and measurement conditions,” J. Hazard. Mater. 262, 1213–1222 (2013). [CrossRef]
7. M. Kaniu, K. Angeyo, A. Mwala, and M. Mangala, “Direct rapid analysis of trace bioavailable soil macronutrients by chemometrics-assisted energy dispersive X-ray fluorescence and scattering spectrometry,” Anal. Chim. Acta 729, 21–25 (2012). [CrossRef]
8. E. Hemsing, A. Knyazik, M. Dunning, D. Xiang, A. Marinelli, C. Hast, and J. B. Rosenzweig, “Coherent optical vortices from relativistic electron beams,” Nat. Phys. 9(9), 549–553 (2013). [CrossRef]
9. P. R. Ribič, B. Rösner, D. Gauthier, E. Allaria, F. Döring, L. Foglia, L. Giannessi, N. Mahne, M. Manfredda, C. Masciovecchio, R. Mincigrucci, N. Mirian, E. Principi, E. Roussel, A. Simoncig, S. Spampinati, C. David, and G. De Ninno, “Extreme-ultraviolet vortices from a free-electron Laser,” Phys. Rev. X 7(3), 031036 (2017). [CrossRef]
10. J. T. Lee, S. Alexander, S. Kevan, S. Roy, and B. McMorran, “Laguerre–Gauss and Hermite–Gauss soft X-ray states generated using diffractive optics,” Nat. Photonics 13(3), 205–209 (2019). [CrossRef]
11. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47(3), 264–267 (1979). [CrossRef]
12. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007). [CrossRef]
13. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36(15), 2883–2885 (2011). [CrossRef]
14. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2(11), 675–678 (2008). [CrossRef]
15. D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003). [CrossRef]
16. N. R. Subedi, P. S. Jung, E. L. Bredeweg, S. Nemati, S. E. Baker, D. N. Christodoulides, and A. E. Vasdekis, “Integrative quantitative-phase and airy light-sheet imaging,” Sci. Rep. 10(1), 20150 (2020). [CrossRef]
17. H. Kafian, M. Lalenejad, S. Moradi-Mehr, S. A. Birgani, and D. Abdollahpour, “Light-sheet fluorescence microscopy with scanning non-diffracting beams,” Sci. Rep. 10(1), 8501 (2020). [CrossRef]
18. T. Vettenburg, H. I. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11(5), 541–544 (2014). [CrossRef]
19. S. Fu, Y. Tsur, J. Zhou, L. Shemer, and A. Arie, “Propagation dynamics of Airy water-wave pulses,” Phys. Rev. Lett. 115(3), 034501 (2015). [CrossRef]
20. F. Mitri, “Airy acoustical–sheet spinner tweezers,” J. Appl. Phys. 120(10), 104901 (2016). [CrossRef]
21. N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494(7437), 331–335 (2013). [CrossRef]
22. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35(12), 2082–2084 (2010). [CrossRef]
23. H. Dai, Z. A. Mohamed, A. Xiao, Y. Xue, Z. Guo, Y. Zhang, X. Zhang, and C. Liu, “Magnetically tunable Airy-like beam of magnetostatic surface spin waves,” Chin. Opt. Lett. 19(1), 013501 (2021). [CrossRef]
24. R. Ivaškevičiūtė-Povilauskienė, P. Kizevičius, E. Nacius, D. Jokubauskis, K. Ikamas, A. Lisauskas, N. Alexeeva, I. Matulaitienė, V. Jukna, S. Orlov, L. Minkeviǎ, and G. Valušis, “Terahertz structured light: nonparaxial Airy imaging using silicon diffractive optics,” Light: Sci. Appl. 11(1), 326 (2022). [CrossRef]
25. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, “Observation of abruptly autofocusing waves,” Opt. Lett. 36(10), 1842–1844 (2011). [CrossRef]
26. F. Lu, L. Tan, Z. Tan, H. Wu, and Y. Liang, “Dynamical power flow and trapping-force properties of two-dimensional Airy-beam superpositions,” Phys. Rev. A 104(2), 023526 (2021). [CrossRef]
27. H. Dai, Y. Liu, D. Luo, and X. Sun, “Propagation properties of an optical vortex carried by an Airy beam: experimental implementation,” Opt. Lett. 36(9), 1617–1619 (2011). [CrossRef]
28. X. Shi, M. Shentcis, Y. Kurman, L. J. Wong, F. J. G de Abajo, and I. Kaminer, “Free-electron-driven X-ray caustics from strained van der Waals materials,” Optica 10(2), 292–301 (2023). [CrossRef]
29. E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106(21), 213902 (2011). [CrossRef]
30. R. Wong, Asymptotic approximations of integrals (SIAM, 2001).
31. L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. Lacourt, and J. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19(17), 16455–16465 (2011). [CrossRef]
32. S. Ali and C. Jacobsen, “Effect of tilt on circular zone plate performance,” J. Opt. Soc. Am. A 37(3), 374–383 (2020). [CrossRef]
33. K. Li, M. Wojcik, and C. Jacobsen, “Multislice does it all—calculating the performance of nanofocusing X-ray optics,” Opt. Express 25(3), 1831–1846 (2017). [CrossRef]
34. H. Yan, R. Conley, N. Bouet, and Y. Chu, “Hard X-ray nanofocusing by multilayer Laue lenses,” J. Phys. D: Appl. Phys. 47(26), 263001 (2014). [CrossRef]
35. D. L. Windt, “Imd—software for modeling the optical properties of multilayer films,” Comput. Phys. 12(4), 360–370 (1998). [CrossRef]
