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Abstract
A coherent X-ray beam expander based on a multilens interferometer is proposed in this paper. The multilens interferometer allows efficient generation of a highly diverging coherent beam up to several milliradians in the hard X-ray energy range. The optical properties of the interferometer were experimentally demonstrated at the ESRF ID13 undulator beamline (Grenoble, France), using 12.4 keV X-rays. The beam expander allowed us to control the angular size and photon flux density of the formed beam and enabled operation in both coherent and incoherent modes. The experimental results were fully consistent with the theoretical concepts and appropriate computer simulations. Future design improvements and related applications are also discussed.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The recent trend toward fourth-generation synchrotron radiation sources has led to dramatic increases in the brightness and spatial coherence of the generated X-ray beams, compared with older designs. The transition from astigmatic to slightly stretched sources with sizes not exceeding several tens of micrometers in both the vertical and horizontal directions stimulated the development of novel and existing coherence-related experimental techniques, such as coherent correlation spectroscopy, X-ray microscopy, interferometry, phase-contrast, and diffraction imaging [1,2]. In addition, this has triggered the development of X-ray optics capable of fully utilizing the laser-like properties of novel sources; the most promising among these is refractive optics, the youngest of the existing X-ray optics. X-ray compound refractive lenses (CRLs) have become a major tool in modern X-ray beamlines, owing to their low sensitivity to shape errors, overall ease of use, and high versatility. By modifying the shape, composition, and number of individual lenses, CRLs can be adapted to photon energies in the 2–200 keV range, enabling flexible adjustment of focal lengths for a wide range of applications. CRLs can provide beam-conditioning functions such as condensers, collimators, beam-shapers, or higher harmonics suppressors [3–9]. Moreover, CRLs are extensively used in X-ray imaging and microscopy, interferometry, Fourier optics, and spectroscopy [10–22].
Intelligent preparation and special formation of X-ray beams in advance allows to fully realize the capabilities of modern X-ray techniques, and even more, permits the creation of the necessary conditions for their effective use. At the same time, ensuring the possibility of varying the beam’s transverse size, as well as controlling the photon flux density, is a desirable challenge in beam shaping and beam conditioning goals. It should be emphasized that there are well-developed and highly demanded X-ray techniques that require a high degree of spatial coherence and a large illumination area at the sample position. For example, phase-contrast imaging and spectroscopy methods require the vertical scanning of samples for imaging their full extent. The ability to control the photon flux density enables to study objects that are sensitive to radiation loads, for example, various polymers or biological samples.
In the X-ray range, a beam expansion can be realized using various porous materials or polydisperse structures, with the particle and/or pore sizes in the 0.1–100 µm range [23]. X-ray radiation passing through such materials is practically not absorbed, but scattered, forming a diverging incoherent beam with an angular size of up to several tens of microradians. Asymmetrically cut single crystals might be applied for beam expansion [24,25], but at higher energies (>30 keV) the incident angle on the crystal becomes too shallow in the Bragg geometry. This can be overcome by matching the geometric and polychromatic focal lengths of two cylindrically bent crystals in the Laue diffraction mode [26].
Beam expansion for larger angular sizes can be realized using various focusing optical elements, including focusing mirrors, Fresnel zone plates, and X-ray refractive parabolic lenses. Focusing yields a diverging beam behind the focus while maintaining the coherent properties of the original incoming beam. It should be noted that diffractive optics cannot operate at high energies of X-ray radiation (above 20 keV), and they are also susceptible to radiation-related damage. Curved or bent mirrors are difficult to manufacture and require precise adjustment, which significantly complicates the optical scheme.
Refractive parabolic lenses are capable of operating in the hard X-ray energy range, but the angular sizes of the created expanded beams are determined by the lenses’ effective aperture, which depends on the absorption of X-ray radiation in the lens material. Silicon nanofocusing lenses with extremely short focal distances can provide expanded milliradian-scale divergent beams; however, in this case, the aperture of such a lens will be only a few micrometers. The use of multiple arrays of CRLs located parallel to each other can significantly increase the acceptance of the incoming beam.
This paper proposes a beam expander based on a multilens system representing a silicon structure of 100 parallel identical planar CRLs. Under coherent illumination, a multilens system generates many diverging beams that interfere in the area where they overlap, with periodic patterns of interference fringes formed at certain distances, called Talbot distances. The optical properties of such a 100-lens interferometer, as well as the ability to control the angular size and photon flux density of the expanding beam, were experimentally demonstrated at the European Synchrotron Radiation Facility (ESRF) in Grenoble, for 12.4-keV-energy X-rays. A theoretical analysis of the expanded beam propagation and formation of the Talbot interference patterns is presented, and the corresponding computer simulation based on a fast Fourier transform computation for wave optics equations was performed [27]. The experimental results agree quite well with the calculated results.
2. Theory
The X-ray multilens interferometer is an array of M parallel identical planar CRLs stacked across their optical axes at the same distance d from each other. Each CRL has a physical aperture A and consists of N individual double concave parabolic lenses with the curvature radius R at the parabola apex. It should be noted that the physical aperture A of the CRLs must be less or equal to the distance d between them. In practice, achieving equality of these parameters becomes possible by using a special interferometer design in which the double concave elements of the CRLs are staggered in the interferometer structure. For clarity, the multilens interferometer considered in this paper is shown in Fig. 1, where its main parameters are marked.
Fig. 1. Scanning electron microscopy images of the planar 100-lens interferometer, with the indication of its main geometric parameters such as the curvature radius R, physical aperture A, thin part b between two refractive surfaces of the individual lenses, as well as distance d between adjacent CRLs.
The lens system is capable of transforming a plane monochromatic X-ray beam with wavelength λ to a set of M mutually coherent beams, where each beam is focused at a distance f = R / 2Nδ, where δ is the decrement of the refractive index n of the interferometer material. The formed foci are spaced apart in the transverse direction by a distance d relative to each other, while the size of the focal spots is diffraction-limited and several orders of magnitude smaller than the distance between them. Previously, this feature was considered in detail on the example of the bilens system (M = 2), which allowed us to describe the further propagation and interaction of the beams downstream of the foci in the framework of the classical theory of in-line interferometry [28].
Thus, in the case of the multilens interferometer (M > 2), the corresponding optical transformation can be presented as the result of the interference of the beams divergent from the extremely small foci in the area where they overlap. A full overlap of all M beams occurs at a distance l = (M – 1) d / α from the foci, where α = Aeff / f is the angular size of the focused beam, and Aeff is the effective aperture of a single CRL. The effective aperture Aeff ≈ (λfδ / 2β)1/2 depends on the absorption in the interferometer material, which is described by the imaginary part β of the refraction index n = 1 – δ + iβ [29,30]. It is assumed here that the effective aperture Aeff is comparable to or smaller than the physical aperture A. Because the effective aperture Aeff of a typical X-ray CRL is small compared with its focal distance f, the distance l is always much larger than the full aperture of the multilens interferometer Afull = AM = (M – 1) d + A. Therefore, the paraxial approximation can be used for describing the interference of the beams.
Let us consider a system of mutually coherent line sources that generate M divergent beams, with adjacent beams separated by a distance d in the transverse direction. In the paraxial approximation, the amplitude of the wavefield at the point with a transverse coordinate x in the observation plane placed at a distance z from the sources (so that z > l ≫ x and Afull), is proportional to the sum of the waves coming from each source:
In the general case, at an arbitrary distance z, the interference field described by Eq. (1) has a complex spatial structure. However, for some distances, the condition of constructive interference is satisfied, in which the last exponential term with an argument independent of x takes serially repeated values with the enumeration of the index m. As a result, the sum of the waves is a periodic distribution of the wavefield amplitude. For example, the periodic system of the line sources is reproduced in the form of interference fringes observed at the fundamental Talbot distance z = ZT = 2d2 / λ. The corresponding sequence of the exponent values is {1, i, 1, i, …}, while the expression under consideration can be represented as two sums of geometric progressions related to sub-sequences of units and imaginary units given by even and odd m, respectively. Hence, the intensity distribution at the fundamental Talbot distance is
Halfway to the interference pattern, a similar secondary Talbot image is formed but shifted by half a period in the space of the fringes. It is also worth noting the infinite family of fractional Talbot images, observed at the corresponding distances zn = ZT / 2n, where n > 1 takes integer values. The intensity distributions at these distances, as well as at the secondary Talbot distance with n = 1, are described as follows:
A careful analysis of Eq. (3) shows that the period of the interference fringes is Λ = d / n, while the intensity at their maximum is proportional to M2. The fringe width σ, defined as the full width at half maximum (FWHM), does not exceed Λ / M; however, according to Eq. (2), this value is twice as large as the fundamental Talbot image.
As for the intensity distribution formed by a line source I0(x,z), in the optical system based on a multilens interferometer, the intensity distribution of a Gaussian beam diverging from the focus produced by a single CRL should be considered:
This expression describes an envelope of the intensity maxima of the interference pattern structure, including the outside Talbot distances. The transverse size ω of the envelope is (8ln2)1/2c ≈ α·z (FWHM), which as can be seen, increases linearly with the distance z. This allows us to propose a more effective technique for beam expansion based on the multilens interferometer, compared with a similar approach in which a single CRL is used. Indeed, the angular size α of the beam is the same in both cases. However, the X-ray transmission factor for the multilens system determined as
3. Manufacturing
Planar multilens interferometers have been manufactured using the process involving electron beam lithography and deep etching into silicon [31–33]. An interferometer is a structure consisting of M = 100 parallel planar CRLs with an etching depth of approximately 50 µm, which are transversally separated by a distance d equal to 10 µm. Each CRL is composed of biconcave individual parabolic lenses with a physical aperture A of 10 µm, and the curvature radius R in the apexes of its parabolic surfaces is 1.25 µm. Different from the bilens interferometer [11], the equality between the distance d and the physical aperture A of the CRLs in a 100-lens interferometer was achieved using a specially developed design of its structure, in which adjacent CRLs are displaced relative to each other by half the longitudinal size of the biconcave parabolic lenses. Scanning electron microscopy images of the 100-lens interferometer are shown in Fig. 1. The thinnest part b between two adjacent refractive surfaces of the individual lenses in the CRLs is 2 µm. The physical aperture Afull of the interferometer is 1 mm (10 µm × 100), which matches the beam size 100 m downstream from the source, for most undulator beamlines at the third-generation synchrotron. It should be noted that the estimated roughness of the lens surface is approximately 20 nm, which does not significantly affect the formation of interference patterns.
To expand the applicability of the multilens system for a wide range of energies, five 100-lens interferometers were fabricated on the same silicon chip. They were formally designed for the 10–50 keV energy range, with 10 keV steps so that the focal distance f of the CRLs in the interferometers for the corresponding energy was fixed at approximately 40 mm by varying the number of biconcave individual lenses N in their CRL arrays. The selection of the proper interferometer, which is suitable for the desired energy, can be performed by a parallel displacement across the beam of the chip. Table 1 summarizes the main characteristics of the interferometers.
Table 1. Main characteristics of the 100-lens interferometers
As shown in Table 1, the beam transformations performed by each interferometer at the designed energy have high efficiency or, in other words, a high transmission factor T. This is because at the intended energies, the absorption of X-ray radiation is quite low; consequently, the effective aperture Aeff of the CRLs in the interferometers noticeably exceeds the physical aperture A. Thus, the angular size α of the beams diverging from the foci is entirely determined by the physical aperture A as α = A / f and is approximately 250 µrad. Here and below, the transmittance T given by Eq. (5) is calculated with the additional multiplier exp[–4πβb(N – 1) / λ], which considers X-ray absorption at the webs between two adjacent refractive surfaces of the individual lenses in the CRLs.
It should be noted that although interferometers have been designed for certain energies, they can be used over a wide range of energies, with only the CRL focal length f changing. Therefore, the angular size of the diverging beams can be significantly increased by decreasing the focal length f, which is achieved when using interferometers at lower energies than was assumed according to their design. This possibility is clearly illustrated in Fig. 2, where the angular size of the generated beam is presented for each interferometer over a wide range of energies. The dots mark the angular size of the beam formed by each interferometer at the designed X-ray energy, as indicated in Table 1. In addition, in Fig. 2, some values of the transmission factor T are shown schematically by dashed lines.
Fig. 2. Theoretical dependence of the angular size α of the beam formed by each 100-lens interferometer on the X-ray energy.
Owing to the higher absorption of X-rays at low energies, the effective aperture Aeff of the CRLs also decreases, approaching the physical aperture. The inflection area of the lines (plots) in Fig. 2 corresponds to the moment when the effective aperture becomes comparable to the physical aperture. In this case, its influence on the angular size of the beam becomes decisive. Our estimates show that the transmission factor T at such energies takes on a value of approximately 70%. A further decrease in energy is accompanied by a decrease in both the effective aperture and transmittance. However, decreasing the aperture also minimizes the influence of possible diffraction effects arising at the edges of the physical aperture of the CRLs on the Talbot interference pattern formation.
4. Experiment
The experimental test of the 100-lens interferometers was carried out at the ESRF ID13 undulator beamline at an X-ray energy of 12.4 keV (wavelength, ∼1 Å). This energy is well suited for studying the optical properties of interferometers and for considering the possibility of their use as beam expanders. To adjust the selected X-ray energy, a liquid nitrogen cooled double-crystal Si-111 monochromator was used. The effective source size s0 in the vertical direction, measured by the interference technique using a boron fiber, was approximately 30 µm [34].
A silicon chip with 100-lens interferometers was installed on a stage with all the necessary rotational and translational movements, which was located at a distance z0 = 96 m from the source. To select the desired interferometer, the chip was moved vertically across the optical axis. A two-coordinate slit was installed in front of the interferometer at a distance of 30 mm, limiting the aperture of the incident beam. The beam formed by the lens system was recorded at different z distances using a high-resolution X-ray charge-coupled device (CCD) detector (Rigaku XSigh Micron LC). The input field of view of this detector was 1.375 mm × 1.815 mm (V × H), while its spatial resolution defined by the based scintillator optical system was approximately 1.3 µm (effective pixel size was 0.55 µm). When the expanded beam was wider than the field of view of the detector, the detector position was scanned along the expanded beam. The general experimental layout for the study of the 100-lens interferometers’ optical properties is depicted in Fig. 3.
Fig. 3. The general experimental layout of the study of the 100-lens interferometers’ optical properties and its test as the beam expander.
4.1 Test of the interferometer optical properties
First, we tested the optical properties of each 100-lens interferometer by recording the interference patterns at different Talbot distances, using the CCD detector. Considering the vertical source size s0 of 30 µm, the theoretical spatial coherence length of the beam at the interferometer position, defined as
was 320 µm, which allowed to coherently illuminate 32 rows of the interferometer CRLs. Thus, to fully utilize the coherent properties of the beam, a vertical slit with a wider gap of 400 µm was used. Considering the distant location of the source (96 m), according to the thin lens formula, the theoretical imaging distance of the CRLs of each interferometer practically does not differ from the focal distance f. Nevertheless, to calculate the Talbot distances, which are much greater than f, accounting for the finite distance z0 between the source and interferometer is still necessary, which in the first approximation yields the following:The experimental interference patterns observed at the secondary Talbot distance 1/2 ZT, which was the same for all interferometers, are presented in Fig. 4(a). As expected, the larger the number of individual lenses N in the CRLs of an interferometer, the larger the size of the generated interference pattern. For example, the fifth interferometer with 81 lenses in the CRLs yielded the width ω of the expanded that was approximately 1.4 mm (FWHM), whereas for the first interferometer with only three lenses in the CRLs, the distance was comparable to the slit size of 0.4 mm. It is worth noting that typical exposure times were in the 10–150 ms range during the 7/8 electron beam bunch mode (current, 200 mA), depending on the interferometer. The choice of the collecting time for each interferometer was dictated by the desire to achieve almost complete loading of the 24-bit CCD sensor of the detector.
Fig. 4. Experimental results of the 100-lens interferometer test. (a) Experimental interference patterns, recorded at the secondary Talbot distance 1/2 ZT for all 5 interferometers.
(b) Experimental and simulated interference patterns, obtained at various Talbot distances for the third CRL set consisting of 29 individual lenses, and (c) an intensity distribution profile taken in a vertical cross-section of the interference pattern, obtained at the secondary Talbot distance 1/2 ZT for the third interferometer.
A more detailed study of the optical characteristics of the interferometers was carried out using the third interferometer consisting of 29 individual lenses in the CRLs. The top row of images in Fig. 4(b) shows the fragments of experimental interference patterns observed at the secondary Talbot distance 1/2 ZT, as well as at the two distances 1/4 ZT = z2 and 1/6 ZT = z3 corresponding to the second and third fractional Talbot images. The periods of the interference fringes obtained at the 1/4 ZT and 1/6 ZT distances are, respectively, two and three times smaller than the period of the interference pattern recorded at the secondary Talbot distance 1/2 ZT, which is fully consistent with the theoretical description. This result is shown by the dashed lines in Fig. 4(b) for clarity. The observed distortion of the intensity distribution of the fringes in the horizontal direction is associated with the diffraction of X-rays at the edges of the entrance to the 20-µm-wide horizontal slit, which limits the interferometer aperture. A computer simulation was carried out using the parameters of the considered optical layout, considering the sizes of the source and the horizontal slit that limit the interferometer’s aperture. The simulated interference patterns are shown in the bottom row of the images in Fig. 4(b). It is clearly seen that these computer-simulated results are in a good agreement with the experimental images.
It should be noted that, for the selected X-ray energy of 12.4 keV, the theoretical effective aperture Aeff of the interferometer CRLs consisting of 29 individual lenses is approximately 5.9 µm, which is noticeably smaller than their physical aperture A (which is equal to 10 µm). Therefore, the interferometer generates an interference pattern without significant distortions caused by the diffraction on the edges of the multilens structure. The intensity distribution profile obtained for the line passing through the center of the interference pattern recorded at the secondary Talbot distance 1/2 ZT is shown in Fig. 4(c). The measured period of the interference fringes was 10.1 ± 0.01 µm, exactly matching the theoretical result; this was estimated by considering the finite distance z0 as Λ = Cd.
The experimentally measured fringe width σ was 1.4 ± 0.1 µm (FWHM), significantly exceeding the theoretical minimal value of approximately 0.4 µm, which was calculated as
However, with a more careful analysis of Eq. (8) describing the influence of all of the above factors on the measured fringe width, the estimation of the effective source size may be possible. Despite that the error of such a measurement is quite large, the obtained result allows a rough estimation of the effective source size, which was approximately 35 ± 15 µm. It should be noted that a more accurate measurement can be performed if the spatial resolution of the detector is comparable to or less than the true fringe width.
4.2 Test of the interferometer-based beam expander
The same experimental optical layout was used for testing the 100-lens interferometers as beam expanders (Fig. 3). The images of the expanded beam, formed by the third 100-lens interferometer consisting of 29 individual lenses in the CRLs, were obtained at a distance z ≫ f equal to 3.9 m. During the experiment, the gap sizes of the vertical slit, which limits the input aperture of the multilens system, varied from 0.2 mm to 1 mm. Figure 5(a) shows images of a typical expanded beam and a typical direct beam that passed through the aperture slit with a vertical gap of 0.5 mm. To estimate the efficiency of the beam transformation T, the ratio of the integral intensities recorded at the detector corresponding to the expanded and direct beams was calculated. The measured value was 44 ± 5%, quite close to the theoretical transmission factor of 45%, as determined by Eq. (5), considering the X-ray absorption at the CRL webs.
Fig. 5. Experimental results of the interferometer-based beam expander test. (a) Images of the direct beam transmitted through a vertical aperture slit with the size of 0.5 mm and an expanded beam formed by the third interferometer consisting of 29 individual lenses in its CRLs, obtained at a distance z of 3.9 m. (b) Amplified fragment of the expanded beam image. (c) Intensity distribution profiles of the expanded beams obtained for the different vertical slit sizes: 0.2, 0.5, and 1.0 mm. (d) Comparison of the two dependencies of the expanded beam’s angular size on the X-ray energy, obtained by analytical estimation and numerical simulation.
Note that the expanded beam was registered at a distance other than the Talbot distance. In this case, a complex interference field was formed, which in the first approximation could be described by Eq. (1). Hence, it follows that at a high degree of spatial coherence of the incoming X-ray beam, the corresponding intensity distribution represents a fine interference structure. Figure 5(b) shows an amplified fragment of the expanded beam image, revealing that the structure of the intensity distribution has a characteristic size of approximately 2 µm, much less than the period of the previously considered Talbot’s interference patterns. Such a fine structure and contrast of the observed interference pattern are associated not only with the spatial coherence of the incoming beam but are also defined by the influence of the detector's spatial resolution.
Expanded beam images obtained using the aperture slit with the vertical gaps smaller than 0.2 mm and larger than 1.0 mm had similar structures. The intensity distribution profiles obtained for the line through the center of each expanded beam image are shown in Fig. 5(c). Evidently, the beam intensity is directly proportional to the slit size, which fully agrees with the theoretical prediction. The measured size of the expanded beams ω was the same for all slit sizes, and was 3.1 ± 0.2 mm (FWHM), more than six times the size of the direct beam. Thus, by varying the slit gap’s width, it is possible to control the flux density of the photon beam, while its size remains unchanged.
The angular size α of the expanded beam calculated as ω / z was approximately 790 ± 50 µrad, quite different from the theoretical estimation of 860 µrad given by the already known ratio Aeff / f. This was owing to a rather rough assumption that the transmitted beam was abruptly limited by the effective aperture Aeff of each CRL of the interferometer. Although more than 98% of the radiation passed through the effective aperture, it was not a substitute for the physical aperture; therefore, in this case, its value could be used only for qualitative estimates.
However, it should be noted that the computer simulation of the process of the expanding beam propagation estimated its angular size as 793 ± 10 µrad, which was fully consistent with the experimental result. In addition, during the numerical experiment, measurements of the angular size of the expanded beam’s dependence on the X-ray energy were carried out. A comparison of the two corresponding plots of the angular size obtained by the analytical calculation and numerical simulation is shown in Fig. 5(d). Evidently, the difference between the curves decreases with increasing energy, and is less than 10% of the analytical estimation for all considered energy values. For example, at the chosen energy of 12.4 keV, this difference is approximately 8.1% (70 µrad / 860 µrad). The experimental results are also represented by the dot mark in Fig. 5(d) for clarity.
As noted above, the fine interference structure of the expanded beam is determined by the high spatial coherence of the incident radiation. However, under incoherent X-ray illumination, when the spatial coherence length of the incident radiation lcoh at the interferometer position is comparable to or less than the distance d between its CRLs, the fine interference structure is absent, and the expanded beam has a Gaussian intensity distribution described by Eq. (4). According to Eq. (6), it can be achieved by increasing the effective source size s0 or by decreasing the distance z0 to the interferometer. For modern synchrotron radiation sources of the 3rd and fourth generations, the possibility of directly varying these parameters over a wide range is severely limited. Nevertheless, a significant reduction in the degree of spatial coherence of radiation can be obtained using a well-scattering porous material (decoherer) placed in front of the interferometer. Multiple scattering of X-rays by a porous structure leads to a strong, almost random distortion of the wavefront of the transmitted beam, resulting in the suppression of its speckle structure [35].
To experimentally demonstrate this effect, a 10-mm-thick plate made of polydisperse porous beryllium was used as the decoherer [23]. The pore size varied from 0.1 µm to 50 µm, with an average of several micrometers. The decoherer was installed between the aperture slit and the interferometer. The vertical size of the slit, limiting the interferometer aperture, was 200 µm. The second 100-lens interferometer, containing 13 biconcave elements in each of its CRLs, was chosen as the beam expander. In this case, the detector could accept the entire expanded beam without changing its position when capturing images. To avoid the formation of speckles, which arise in the far-field owing to the X-ray scattering on the static porous structure of the decoherer, the decoherer was subjected to linear oscillations in the vertical direction, at a frequency of 10 kHz. The optical layout is shown in Fig. 6.
Fig. 6. The experimental layout for testing the interferometer-based beam expander in the incoherent operation mode.
The images of the expanded beam were observed at a distance of approximately one meter, corresponding to the secondary Talbot distance 1/2 ZT. The recorded image of the beam without the decoherer represents a bright periodic interference fringe pattern (left image in Fig. 7(a)). The measured period of the interference pattern was 10.1 ± 0.03 µm, consistent with previous studies of the optical properties of interferometers. Figure 7(b) shows the intensity distribution profile obtained for the line through the center of the expanded beam. A high contrast (approximately 92%) of the fringes can be observed.
Fig. 7. Experimental result of the incoherent operation mode test of the interferometer-based beam expander. (a) Images of an expanded beam formed by the second 100-lens interferometer containing 13 biconcave elements in each of its CRLs, obtained without using the decoherer (left image) and with the decoherer (right image). The intensity distribution profiles in vertical cross-section of the images of the expanded beam (b) without and (c) with the decoherer.
When the decoherer was used, the interference fringes completely disappeared, and the intensity distribution of the expanded beam became homogeneous. This indicates the absence of coherent properties. The observed image of the expanded beam is shown on the right in Fig. 7(a). Note that the scattering of X-ray radiation on the porous structure of the decoherer led to the broadening of the beam in the horizontal direction, and this effect was manifested in the formation of a halo in the image of the expanded beam. According to our estimations, the main scattering of radiation in the horizontal direction occurred within an angle not exceeding 50 µrad.
In the vertical direction, the expansion was mainly caused by the divergent beams formed by the interferometer CRLs. The measured size of the expanded beam, ω, was 510 ± 20 µm (FWHM). The related angular size α of 505 ± 20 µrad was in a good agreement with the numerically simulated result of 514 ± 20 µrad. Figure 7(c) shows the intensity profile in the vertical cross-section of the beam image, as well as the theoretical curve, obtained as the convolution of the simulated Talbot image and the Gaussian function whose width corresponds to the experimentally defined angular size of the beam scattered on the decoherer. It should be noted that, at the center of the expanded beam, the theoretical intensity distribution is slightly different from the experimental distribution. This is caused by the influence of the third harmonic of the undulator radiation transmitted directly through the 200-µm-wide aperture slit.
5. Conclusion
In this paper, we proposed a beam expander based on a multilens interferometer, for controlling the angular size and photon flux density of incident hard X-rays. The considered multilens system, representing a silicon structure of 100 parallel planar CRLs, can generate coherent beams with large divergences, up to several milliradians, with significantly higher transmission efficiency compared with individual CRLs, where only a small part of the incident beam corresponding to the lens effective aperture is expanded.
The expanded beam profile was studied theoretically and experimentally at various distances, and the corresponding computer simulations were presented. Under coherent illumination, the 100-lens interferometers formed periodic interference patterns at the Talbot distance, where the conditions of constructive interference were fulfilled. The analysis of the periodic interference patterns was performed at the secondary Talbot distance, as well as at the second and third fractional Talbot distances. It was shown that a flexible variation in the size of the interference pattern over a wide range is possible by changing the number of individual lenses in each CRL of the interferometer. Such periodic interference patterns with an adjustable period and size can be used as special raster illumination for accurate (fine) radiation therapy [36–38]. Raster-based diagnostic techniques in materials science and biology can also be developed.
It should be noted that, with an optimal experimental setup and a high-resolution detector, the fringe width is determined mainly by the projection of the X-ray source, allowing accurate measurements of its size [39]. Considering the ongoing transition of modern synchrotrons to diffraction-limited X-ray sources with laser-like optical properties (MAX IV, ESRF-EBS, PETRA-IV), the development of source diagnostics including wavefront sensors is particularly relevant [1,2,40–42].
The multilens interferometer, which forms a set of periodically spaced, mutually coherent focused beams, can be considered as an analog of a grating interferometer with extremely small slits [43–46]. This can dramatically improve the spatial resolution of X-ray phase-contrast methods [28]. In addition, because of the high transmission efficiency and the ability to control the angular sizes of beams diverging from the foci, the reconstruction of the phase-shift profile of complex-structured weakly absorbing objects becomes possible. Moreover, the multilens system can be used at X-ray energies higher than 30 keV, while manufacturing an interferometer with a thick micro-pitch grating is problematic.
A multilens interferometer can transform a collimated beam into an expanding beam with an angular size independent of the transverse size of the incident beam with equal efficiency, preserving its original coherence properties. This issue was investigated at large distances (> 1 m) from the interferometer, in contrast to the Talbot positions. The measured angular size of the formed beam and the observed transmission efficiency were in full agreement with the calculated results. The possibility of controlling the photon flux density at a constant angular size of the expanded beam was also demonstrated. It was shown experimentally that decreasing the size of the slit limiting the interferometer aperture led to a proportional decrease in the intensity of the expanded beam, allowing to study bulky radiation-sensitive objects in medical or biological research.
As previously mentioned, the expanded beam has a fine structure owing to the high coherence of the incident radiation. However, for research techniques that require a large illumination area and lower resolution, this is undetectable. The fine structure can be easily avoided if needed, provided that the spatial coherence length of the radiation in front of the interferometer is reduced to a value smaller than the space between adjacent CRLs. For most modern X-ray sources, this condition can be achieved at a distance of no more than 10 m from the source to the interferometer. In this case, new-generation synchrotrons can conveniently set up beam-shaping optics close to the source. In contrast, the fine structure of the expanded beam can be controlled by an improved interferometer design by varying the space between the CRLs and the number of individual lenses in the lens array. Such an aperiodic structure of the lens array can significantly reduce the effect of constructive interference on the contrast of the forming interference patterns.
Under incoherent X-ray illumination, a multilens interferometer can generate an entirely homogeneous expanded beam. To experimentally demonstrate such behavior, a decoherer was used in front of the interferometer. Evidently, the Talbot interference fringes, observed earlier, were completely washed out. Moreover, by changing the volume of the decoherer material, the degree of coherence of the generated beam can be controlled, which significantly extends the capabilities of the proposed approach. A uniform expanded beam can be used in diffraction, spectroscopy, and radiography experiments, where coherence is not required, but a large illumination area is indispensable. Such easy switching between coherent and incoherent illumination allows the combination of both phase and absorption contrast imaging in the same experiment. Another interesting option for fine-tuning the spatial coherence might be realized by positioning the decoherer in the interferometer foci.
It should be noted that the use of two multilens interferometers mounted in cross-geometry (perpendicular to each other) allows the generation of expanded beams in two directions. Modern MEMS technology allows the production of Si structures with a depth of a few hundred microns [47]. Stacking via bonding of several interferometers is also possible when using Silicon On Insulator (SOI) wafers and special technological approaches. Given the quite high thermal loads of modern synchrotrons and X-ray free-electron lasers, the use of diamond optics is quite promising [48–50]. Recently proposed two-dimensional microlenses made of diamond by ion-beam lithography [51] can be used for designing novel two-dimensional interferometers.
In conclusion, a beam expander based on a multilens interferometer is quite promising for use in modern X-ray radiation sources, for providing beam-conditioning and beam-transport optical functions. The ability to generate stable and easily controlled expanded beams makes these interferometers versatile research tools in medical, biological, and materials sciences.
Funding
Russian Science Foundation (Project No. 19-72-30009).
Acknowledgments
The authors are very grateful to M. Burghammer and M. Rosenthal for their help during the experiments at the ID 13 beamline. The authors thank the Russian Academic Excellence Project at the Immanuel Kant Baltic Federal University for funding travel and accommodation for X-ray tests.
Disclosures
The authors declare no conflicts of interest.
Data availability
The 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.
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