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Bidimensional focusing of x rays by refraction in an edge Werner Jark1,* and Gianluca Grenci2 1 2
Elettra—Sincrotrone Trieste S.c.p.A., S.S. 14 km 163.5, I-34149 Basovizza (TS), Italy
CNR-IOM, TASC laboratory, S.S. 14 km 163.5, I-34149 Basovizza (TS), Italy and Mechanobiology Institute, Singapore (MBI), National University of Singapore, T-Lab, #10-01 5A Engineering Drive 1, Singapore 117411 *Corresponding author: [email protected] Received December 13, 2013; accepted January 12, 2014; posted January 22, 2014 (Doc. ID 203031); published February 25, 2014 When an x-ray beam passes through the tip of a triangular prism, i.e., an edge, it undergoes two consecutive refraction processes. This will also happen when the incident beam is not perpendicular to the tip but when the beam progresses at a very small inclination to it. It will be shown that in such a condition, when both interfaces adjacent to the tip have concave surfaces, decoupled focusing in two orthogonal directions can be introduced in the transmitted x-ray beam. The limitations for this application are discussed, and focusing of x rays to spots with diffraction limited sizes of the order of 100 nanometers is found to be feasible. The feasibility of bidimensional focusing by use of such a device was experimentally verified. © 2014 Optical Society of America OCIS codes: (340.0340) X-ray optics; (180.7460) X-ray microscopy; (120.5710) Refraction; (120.7000) Transmission. http://dx.doi.org/10.1364/OL.39.001250
The first successful bidimensional x-ray focusing was achieved through reflection optics by Kirkpatrick and Baez [1] by use of a pair of two crossed mirrors in a tandem configuration in which the focusing in the two orthogonal directions was completely decoupled. This decoupling scheme for focusing by use of two optical components in tandem is pursued and proposed frequently when the production of a bidimensionally focusing device is technologically extremely difficult or impossible. The first to repeat the tandem concept was Kirkpatrick [2], who proposed it for bidimensional x-ray focusing by use of single refracting interfaces. He showed that the refraction upon transmission through a concave interface can also focus x rays in one dimension [2]. This focusing will be found in both possible orientations, when the incident beam hits a concave interface beyond the critical angle for total reflection as well as when it exits through such an interface into the laboratory environment [2]. Presently, bidimensional focusing by refractive optics is mostly obtained by use of stacks of radially symmetric concave lenses [3]. However the smallest focii with refractive optics still required the operation of a tandem pair [4]. Another notable concept providing ultra-small x-ray focii is the use of Laue lenses [5], which once more need to be operated in the crossed tandem configuration. Montel [6] has shown that the drawback of the tandem concept, i.e., the focus being stigmatic due to the need to operate the optics with two different focal lengths, can be overcome in the case of reflective optics. In his solution, the “catamegonic roof-shaped objective” [6], the incident beam hits successively in any order the two reflecting surfaces, which are angled at 90° with respect to each other at the same position. This study will discuss that an analogous solution can also be applied to the refracting tandem. In fact, in an appropriately inclined 90° edge an incident beam can undergo two successive refractions in two orthogonal directions. The path of such an x-ray beam passing through two orthogonal interfaces, which need to be concave for focusing, is shown in Fig. 1. In this figure, the incident x-ray beam follows a trajectory in the x direction. It is focused in the y direction 0146-9592/14/051250-04$15.00/0
upon external grazing incidence with an angle θy onto a surface, which has a concave curvature in the plane of incidence, i.e., in the x-y plane. Successively, the beam hits internally the top interface with an angle of grazing incidence θz . This interface is once more concave in the direction of the beam trajectory, and it thus focuses the beam in the orthogonal z direction. Sanchez del Rio and Alianelli [7] discuss the surface shapes required for the focusing in this condition. It is now assumed that both interfaces remain uncurved in the direction perpendicular to the edge. Then for the refocusing of an incident divergent x-ray beam according to Sanchez del Rio and Alianelli [7], one would need to shape both refracting interfaces ideally to Cartesian ovals. For an incident plane wave, the first interface would instead need a hyperbolic interface, while the exit interface would have to have an elliptical profile. The latter shape was derived already earlier by Evans-Lutterodt et al. [8]. For the latter plane wave case, the edge discussed here needs then to follow the intersection line between a vertically oriented hyperbolic cylinder opening into the direction of the source and a horizontally lying elliptical cylinder whose first vertex is closer to the source than the hyperbola vertex. Such an intersection line is shown in Fig. 2. In this figure, the curved edge is positioned in the octant with x > 0, y > 0, and z < 0. In order to facilitate the discussion here, both interfaces will be described with parabolas, which is a possible approximation for the ideal shape [7].
Fig. 1. Schematical illustration of an x-ray beam path following the arrow through an edge, in which the beam is subject to two consecutive refraction processes in orthogonal directions, i.e., upward and toward the reader. The drawing is not made to scale, and the angles are drawn exaggeratedly steep. © 2014 Optical Society of America
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Fig. 2. Intersection line between a vertically orientated hyperbolic cylinder with focal length f h , which opens toward the left (decreasing x), and a horizontally lying elliptical cylinder (focal length f e ), which opens to the right (increasing x), where its second vertex will be found. The double curved edge is positioned in the octant with x > 0, y > 0, and z < 0.
Then, as shown to the left in Fig. 3, the interface for the incident beam, i.e., a hyperbola with the vertex at x 0 and extending to x < 0, follows in the x-y plane the curve (the index h stands for hyperbola) yh
pp 2δ −xh f h :
(1)
In this equation, δ is the refractive index decrement from unity (n 1 − δ), and f h is the distance of the focal plane from the point xh 0, which lies between the beam footprint and the focus. In this case, yh > 0. The profile of the second interface (to the right in Fig. 3) is now considered to have no variation depending on the y coordinate. The interface for the exiting beam, an ellipse with one vertex at xe 0, then follows for x > 0 the curve (index e for ellipse) pp ze − 2δ x0e f e
(2)
with ze < 0. Obviously, when these two profiles are now adjacent to each other in an edge, as in Fig. 2, one needs to use f h ≠ f e for coinciding focii. The derivatives ∂yh ∕∂xh and ∂ze ∕∂xe of Eqs. (1) and (2) provide directly the local angles of grazing incidence θh and θe onto the refracting interfaces, while the reciprocal of the second derivatives ∂y2h ∕∂x2h and ∂z2e ∕∂x2e provide the local radii of curvature Ri . For some rough prediction of the focusing properties one can derive by use of them f h Rh θ3h ∕δ and f e Re θ3e ∕δ. At external incidence, the refraction into the edge will only be observed when the angle of grazing incidence θh is larger than the critical angle for total reflection [2], i.e., for θh > 2δ0.5 . For the internal incidence, such a limitation does not exist [2]. The related beam deflection angles Δi are given by Δh θh − θ2h − 2δ0.5 and Δe θ2e 2δ0.5 − θe . Then both
Fig. 3. Schematical illustration of the curvature of the two consecutively refracting interfaces (in white) in two independent devices with the refracting interfaces oriented as in Fig. 2. Both vertices are at x 0.
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deflection angles are limited to Δi ≤ 2δ0.5 , i.e., to the critical angle, when θh approaches the critical angle and when θe becomes extremely grazing, respectively. Ignoring now reflection losses at the interfaces, the transmission through the edge is efficient as long as the beam trajectories in the material remain shorter than or are of the order of the attenuation length AL of the edge material. Interestingly, in Fig. 1 one sees that for refraction at grazing incidence onto a straight edge the optical path in the material is increasing with increasing distance in z from the edge while it does not vary with the position in y. Then the related increasing absorption at larger distances in z from the edge limits the effective geometrical aperture A in this z direction but not in the y direction. This latter effective aperture is the edge transmission function integrated over the height of the slice [3]. Formally, the result is the size of a linear slit providing the same exit photon flux. Instead, in the y direction the effective geometrical aperture is limited by the active edge length L projected onto the plane perpendicular to the beam trajectory. For rough estimates, one can use Z Az ≈
z0 z−∞
z dz θe AL exp − θe AL
(3)
and Ay θh L:
(4)
The more important parameter for obtaining high spatial resolution is the numerical aperture NA of the focusing objective. As observed already in [8] NA is limited here to 2δ0.5 , as this is the limit for the deflection as discussed above. The most deflected beam in the y direction is impinging externally onto the first interface with the critical angle. By use of the derivative of Eq. (1) this leads to θh;max 0.52δf h 0.5 −xh;max −0.5 θcrit 2δ0.5 : In this condition, one then finds xh;max −f h ∕4: The absolute value of xh;max is then roughly the active length of the edge L. When this length L is now made identical to the attenuation length of the edge material AL one obtains from Eq. (1) a geometrical aperture of roughly Ah yh;max 22δ0.5 AL and a limitation for the minimum focal length of f h 4AL. The length limitation to AL applies now also to the refracting interface in the other direction, which leads then for coinciding focii positions to the limitation f e f h AL 1.25f h 5AL: These are feasible geometrical parameters for the edge. As a single edge now presents a half lens, the obtainable diffraction limit r for the spatial resolution can be estimated to be r ≈ λ∕NA, where NA is limited to NA ≤ 2δ0.5 . As δ ∝ λ2 [1] the diffraction limit for the spatial resolution is then independent of the chosen photon energy, and for lighter low absorbing materials like plastics or Be, the estimate by use of tabulated data for δ [9] is then for r ≈ 60 nm. The experiment was performed at a thin slice of SU-8, which is an epoxy-based negative tone resist of density 1.2 g∕cm3 with a composition in terms of weight fractions of C(0.727)O(0.182)H(0.069)Sb(0.014)F(0.006)S(0.002)
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[10]. This slice was spin coated onto a silicon wafer to a thickness of ∼200 μm. It was then patterned by exposure to UV light through a quartz mask, developed with PGMEA, and finally released from the substrate. Because of stress introduced into the surface after removal the slice deformed such that the surface but also the side wall resulted to be concave of approximately spherical shape. The slice length in beam direction is L 8.52 mm, while it had a thickness of 0.18 mm. This object was positioned at the SYRMEP beamline at the Elettra synchrotron radiation laboratory [11] behind a double crystal x-ray monochromator at 22.6 mm distance from a source measuring 70 μm in the vertical direction and 465 μm in the horizontal direction (2.36 GeV electron energy). The photon energy was set to 10 keV and a CCD camera downstream of a fluorescence screen registered the shadow image of the sample at a distance of 1.6 m from it with 4.5 μm equivalent pixel size. The sample was oriented such that the first refraction takes place at grazing incidence onto the curved side wall. The transmitted intensity was brought to a focus at the detector by adjusting both angles θh and θe independently. Figure 4 shows to the left the orientation and the bending of the sample as seen by an observer from the detector position in three different angular positions of θe in a scan around the horizontal axis. The center orientation is assumed to provide focusing in both directions. Note that the related intersection line in Fig. 2 lies in front of the emphasized curved edge in the octant with x > 0, y < 0, and z < 0. In the three images, the expected intensity distributions at the detector position caused by refraction at different features of the sample are superimposed as lighter gray areas. In correspondence to the edge of the side wall, closer to the camera one will always find a horizontally focused line extending over the entire thickness of the slice. The intensity refracted twice at the edge should then form long vertical lines at the same horizontal position as the previous line. For the case of bidimensional focusing, this refracted intensity should form a point. The intensity exiting after only a single refraction upon internal incidence onto the curved top surface will be found in horizontal lines above the shadow. In focusing condition, the narrow horizontal line has minimum height; otherwise in the defocused cases a larger distribution overlaps partly with the shadow of the
Fig. 4. Schematical illustration of the refracting edge, i.e., the right sample border, as an observer at the detector position would see it in a scan of the angle θe , which is increasing from the left to the right. In the three figures to the left, those areas are superimposed as lighter gray lines, in which refracted intensities will be registered at the detector position for an edge with optimum curvature. In the center image, the beam is focused in both directions. In the figure to the right it is assumed that the edge is incorrectly shaped with spherical curvature.
sample or it is completely detached from it. The latter case is observed for more grazing angles of incidence when according to f e ∝ θ3e the focused image would have been found already between the sample and the detector. Instead the overlap is found for larger angles of grazing incidence and the corresponding image position far behind the detector. Now the side wall with spherical curvature does not have the correct curvature over the entire edge length. This will introduce significant aberrations, and thus the doubly refracted beam will not be refracted onto straight lines or into a point as indicated but some deformation will be observed. An example of a related simulation is shown to the right in Fig. 4. In this simulation, roughly the first half of the side wall closer to the source refracts the beam to the same y coordinate, while the remaining part of the edge will refract it more. Then in focusing condition, one expects to find a spot with some tail extending sideways to the right. Depending on the aberrations in the top surface the tail can be inclined downward or upward. The just discussed features are now observed in the registered intensity distributions, as shown in Fig. 5. In the focusing condition (center image), one sees that the singly refracted beam in the top surface forms two narrow parallel lines. This is due to an additional deformation of the top surface close to the border. In this image, the focused spot is found, and the aberrated intensity is refracted to the left and downward. This indicates that the angles of incidence onto the part of the surface closer to the observer are too steep. This provides insufficient beam deflection in this area. In this case, the spherical surface is overbent compared to the smaller curvature in the better suited parabolic or elliptical profile. The right registration corresponds rather well to the simulation to the right in Fig. 4. In this case roughly 1∕3 of the edge extending from the center toward the source refracts the beam to similar y coordinates. The part of the edge closer to the observer is underbent and thus deflects too much. In systematic studies, it was verified that for the photon energy of 10 keV the tabulated optical properties for the SU-8 with the abovementioned composition [10] of δ 2.67 × 10−6 and AL 1.8 mm [9] consistently describe the refraction and the transmission of the slice in different orientations. For both refracting interfaces, the angle of grazing incidence onto the well focusing area is about 1.6° (28 mrad) and the radius of curvature in these areas is in both cases R 0.2 m. Then, according to Eqs. (3) and (4), the effective aperture of the edge for focusing in both directions is about 50 μm for the vertical direction and 70 μm in the orthogonal horizontal direction. The related geometrical acceptance does not have a rectangular shape, but it is trapezium shaped, which is caused by the edge inclination. Ideally the focus should have measured 5 μm by 33 μm, or 1 pixel by 7 pixels. The first is smaller than the point spread function (PSF) of the detector, for which a FWHM of 3.7 pixels was measured. The observed FWHM in this direction in the bidimensional focus is measured to be 5 pixels and it is thus slightly larger than the PSF. On the other hand, the width of the refracted distorted line in the right image, when the beam is only horizontally focused, is 7 pixels, which is the expected size for the re-imaged horizontal source at
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Fig. 6. Detached and connected assemblies of four curved edges in a symmetric configuration, which will provide fourfold increased geometrical aperture to the incident x rays. Fig. 5. Registered shadow images and refracted intensities at 1.6 m downstream from a curved edge for three different positions in angle of grazing incidence θe , in a scan, as shown in Fig. 4. Each single picture covers at the detector 0.56 mm × 0.9 mm (125 pixels × 200 pixels).
the detector position. In the bidimensional focus, the horizontal focus size is slightly larger with 8 pixels, which is smaller than the related geometrical aperture of the edge. Obviously, this latter broadening is due to the easily recognizable aberrations. Similar aberrations will also broaden the focus size in the vertical direction. Nevertheless, in order to judge the perfection of the refracting surfaces at this point, the blurring will be assigned to slope errors in these surfaces. For an rms slope error of σ rms , the blurred focus s after a single refraction at larger angles of grazing incidence will measure s ≈ 2.4σ rms f . Then one derives from a blurred image size of about 4 pixels a value for this surface error of the order of σ rms ≈ 5 μrad, which is a surprisingly small value. The gain in photon flux density in the focus was measured to be threefold, while with reference to the measured focus size one would have expected a gain of about 4.3-fold. Then only about 30% of the transmittable intensity is lost by other effects, e.g., in defects in the tip of the edge. The present study shows that a curved edge can focus x rays bidimensionally. UV lithography in combination with chemical etching was found to provide sufficiently small roughness in the etched walls, such that the refraction will be blurred only insignificantly. The test object had spherical curvature, which introduced severe aberrations into the focus. These aberrations were rather visible as a particularly long focal length was chosen. At shorter focal length, they will be less severe. When the goal is focusing to the diffraction limit, then the curvature needs to follow the required curvature very precisely and the focal length needs to be significantly shorter with roughly fourfold the attenuation length of the edge material. This latter number could have been here as small as 7.2 mm, which is more than two orders of magnitude shorter than the chosen image distance. Then with Ah 22δ0.5 AL the optimum geometrical acceptance would be sixfold smaller. In Fig. 2 one finds four sections of the curve that have equivalent curvature and focus thus at the same distance. The related focusing object could be formed by bending a very elongated elliptical hole in a thin slice symmetrically around a hyperbolic cylinder. Such a symmetric arrangement as presented in Fig. 6 can collect fourfold
more flux in a larger aperture, which could also permit to obtain a twofold smaller diffraction limit for the focus size. This would result in a simple single refractive lens, which can formally achieve the diffraction limited spatial resolution, that otherwise, as far as refracting devices are concerned, only more complicated stacks of refractive lenses can provide [3,4]. The most suitable materials for the edge production are plastics, synthetic diamonds, and Be. Edges in the latter two are particularly suited for operation in high-power beams, as the absorbed power can efficiently be dissipated into the unexposed bulk adjacent to the edge. We grateful acknowledge the help of Luigi Rigon from Trieste University during the experiment. We thank Massimo Tormen (CNR-IOM) and Diane Eichert (Elettra) for logistics support. The project was supported by the EU within the I3 action: European Light Sources Activities (ELISA). References 1. P. Kirkpatrick and A. Baez, J. Opt. Soc. Am. 38, 766 (1948). 2. P. Kirkpatrick, J. Opt. Soc. Am. 39, 796 (1949). 3. B. Lengeler, C. G. Schroer, J. Tuemmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, J. Synchrotron Rad. 6, 1153 (1999). 4. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Küchler, Appl. Phys. Lett. 87, 124103 (2005). 5. H. Yan, V. Rose, D. Shu, E. Lima, H. C. Kang, R. Conley, C. Liu, N. Jahedi, A. T. Macrander, G. B. Stephenson, M. Holt, Y. S. Chu, M. Lu, and J. Maser, Opt. Express 19, 15069 (2011). 6. M. Montel, X-Ray Microscopy and Microradiography (Academic, 1957), pp. 177–185. 7. M. Sanchez del Rio and L. Alianelli, J. Synchrotron Rad. 19, 366 (2012). 8. K. Evans-Lutterodt, J. M. Ablett, A. Stein, C.-C. Kao, D. M. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. L. Gammel, H. Huggins, S. Ustin, G. Bogart, and L. Ocola, Opt. Express 11, 919 (2003). 9. B. L. Henke, E. M. Gullickson, and J. C. Davis, At. Data Nucl. Data Tables 54, 181 (1993), http://www‑cxro.lbl.gov/ optical_constants/. 10. M. Simon, “Roentgenlinsen mit grosser Apertur (X-ray lenses with large aperture),” Ph.d. thesis (Karlsruher Institut für Technologie, 2010). 11. L. Rigon, Z. Zhong, F. Arfelli, R.-H. Menk, and A. Pillon, Proc. SPIE 4682, 255 (2002).
OPTICS LETTERS / Vol. 39, No. 5 / March 1, 2014
Bidimensional focusing of x rays by refraction in an edge Werner Jark1,* and Gianluca Grenci2 1 2
Elettra—Sincrotrone Trieste S.c.p.A., S.S. 14 km 163.5, I-34149 Basovizza (TS), Italy
CNR-IOM, TASC laboratory, S.S. 14 km 163.5, I-34149 Basovizza (TS), Italy and Mechanobiology Institute, Singapore (MBI), National University of Singapore, T-Lab, #10-01 5A Engineering Drive 1, Singapore 117411 *Corresponding author: [email protected] Received December 13, 2013; accepted January 12, 2014; posted January 22, 2014 (Doc. ID 203031); published February 25, 2014 When an x-ray beam passes through the tip of a triangular prism, i.e., an edge, it undergoes two consecutive refraction processes. This will also happen when the incident beam is not perpendicular to the tip but when the beam progresses at a very small inclination to it. It will be shown that in such a condition, when both interfaces adjacent to the tip have concave surfaces, decoupled focusing in two orthogonal directions can be introduced in the transmitted x-ray beam. The limitations for this application are discussed, and focusing of x rays to spots with diffraction limited sizes of the order of 100 nanometers is found to be feasible. The feasibility of bidimensional focusing by use of such a device was experimentally verified. © 2014 Optical Society of America OCIS codes: (340.0340) X-ray optics; (180.7460) X-ray microscopy; (120.5710) Refraction; (120.7000) Transmission. http://dx.doi.org/10.1364/OL.39.001250
The first successful bidimensional x-ray focusing was achieved through reflection optics by Kirkpatrick and Baez [1] by use of a pair of two crossed mirrors in a tandem configuration in which the focusing in the two orthogonal directions was completely decoupled. This decoupling scheme for focusing by use of two optical components in tandem is pursued and proposed frequently when the production of a bidimensionally focusing device is technologically extremely difficult or impossible. The first to repeat the tandem concept was Kirkpatrick [2], who proposed it for bidimensional x-ray focusing by use of single refracting interfaces. He showed that the refraction upon transmission through a concave interface can also focus x rays in one dimension [2]. This focusing will be found in both possible orientations, when the incident beam hits a concave interface beyond the critical angle for total reflection as well as when it exits through such an interface into the laboratory environment [2]. Presently, bidimensional focusing by refractive optics is mostly obtained by use of stacks of radially symmetric concave lenses [3]. However the smallest focii with refractive optics still required the operation of a tandem pair [4]. Another notable concept providing ultra-small x-ray focii is the use of Laue lenses [5], which once more need to be operated in the crossed tandem configuration. Montel [6] has shown that the drawback of the tandem concept, i.e., the focus being stigmatic due to the need to operate the optics with two different focal lengths, can be overcome in the case of reflective optics. In his solution, the “catamegonic roof-shaped objective” [6], the incident beam hits successively in any order the two reflecting surfaces, which are angled at 90° with respect to each other at the same position. This study will discuss that an analogous solution can also be applied to the refracting tandem. In fact, in an appropriately inclined 90° edge an incident beam can undergo two successive refractions in two orthogonal directions. The path of such an x-ray beam passing through two orthogonal interfaces, which need to be concave for focusing, is shown in Fig. 1. In this figure, the incident x-ray beam follows a trajectory in the x direction. It is focused in the y direction 0146-9592/14/051250-04$15.00/0
upon external grazing incidence with an angle θy onto a surface, which has a concave curvature in the plane of incidence, i.e., in the x-y plane. Successively, the beam hits internally the top interface with an angle of grazing incidence θz . This interface is once more concave in the direction of the beam trajectory, and it thus focuses the beam in the orthogonal z direction. Sanchez del Rio and Alianelli [7] discuss the surface shapes required for the focusing in this condition. It is now assumed that both interfaces remain uncurved in the direction perpendicular to the edge. Then for the refocusing of an incident divergent x-ray beam according to Sanchez del Rio and Alianelli [7], one would need to shape both refracting interfaces ideally to Cartesian ovals. For an incident plane wave, the first interface would instead need a hyperbolic interface, while the exit interface would have to have an elliptical profile. The latter shape was derived already earlier by Evans-Lutterodt et al. [8]. For the latter plane wave case, the edge discussed here needs then to follow the intersection line between a vertically oriented hyperbolic cylinder opening into the direction of the source and a horizontally lying elliptical cylinder whose first vertex is closer to the source than the hyperbola vertex. Such an intersection line is shown in Fig. 2. In this figure, the curved edge is positioned in the octant with x > 0, y > 0, and z < 0. In order to facilitate the discussion here, both interfaces will be described with parabolas, which is a possible approximation for the ideal shape [7].
Fig. 1. Schematical illustration of an x-ray beam path following the arrow through an edge, in which the beam is subject to two consecutive refraction processes in orthogonal directions, i.e., upward and toward the reader. The drawing is not made to scale, and the angles are drawn exaggeratedly steep. © 2014 Optical Society of America
March 1, 2014 / Vol. 39, No. 5 / OPTICS LETTERS
Fig. 2. Intersection line between a vertically orientated hyperbolic cylinder with focal length f h , which opens toward the left (decreasing x), and a horizontally lying elliptical cylinder (focal length f e ), which opens to the right (increasing x), where its second vertex will be found. The double curved edge is positioned in the octant with x > 0, y > 0, and z < 0.
Then, as shown to the left in Fig. 3, the interface for the incident beam, i.e., a hyperbola with the vertex at x 0 and extending to x < 0, follows in the x-y plane the curve (the index h stands for hyperbola) yh
pp 2δ −xh f h :
(1)
In this equation, δ is the refractive index decrement from unity (n 1 − δ), and f h is the distance of the focal plane from the point xh 0, which lies between the beam footprint and the focus. In this case, yh > 0. The profile of the second interface (to the right in Fig. 3) is now considered to have no variation depending on the y coordinate. The interface for the exiting beam, an ellipse with one vertex at xe 0, then follows for x > 0 the curve (index e for ellipse) pp ze − 2δ x0e f e
(2)
with ze < 0. Obviously, when these two profiles are now adjacent to each other in an edge, as in Fig. 2, one needs to use f h ≠ f e for coinciding focii. The derivatives ∂yh ∕∂xh and ∂ze ∕∂xe of Eqs. (1) and (2) provide directly the local angles of grazing incidence θh and θe onto the refracting interfaces, while the reciprocal of the second derivatives ∂y2h ∕∂x2h and ∂z2e ∕∂x2e provide the local radii of curvature Ri . For some rough prediction of the focusing properties one can derive by use of them f h Rh θ3h ∕δ and f e Re θ3e ∕δ. At external incidence, the refraction into the edge will only be observed when the angle of grazing incidence θh is larger than the critical angle for total reflection [2], i.e., for θh > 2δ0.5 . For the internal incidence, such a limitation does not exist [2]. The related beam deflection angles Δi are given by Δh θh − θ2h − 2δ0.5 and Δe θ2e 2δ0.5 − θe . Then both
Fig. 3. Schematical illustration of the curvature of the two consecutively refracting interfaces (in white) in two independent devices with the refracting interfaces oriented as in Fig. 2. Both vertices are at x 0.
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deflection angles are limited to Δi ≤ 2δ0.5 , i.e., to the critical angle, when θh approaches the critical angle and when θe becomes extremely grazing, respectively. Ignoring now reflection losses at the interfaces, the transmission through the edge is efficient as long as the beam trajectories in the material remain shorter than or are of the order of the attenuation length AL of the edge material. Interestingly, in Fig. 1 one sees that for refraction at grazing incidence onto a straight edge the optical path in the material is increasing with increasing distance in z from the edge while it does not vary with the position in y. Then the related increasing absorption at larger distances in z from the edge limits the effective geometrical aperture A in this z direction but not in the y direction. This latter effective aperture is the edge transmission function integrated over the height of the slice [3]. Formally, the result is the size of a linear slit providing the same exit photon flux. Instead, in the y direction the effective geometrical aperture is limited by the active edge length L projected onto the plane perpendicular to the beam trajectory. For rough estimates, one can use Z Az ≈
z0 z−∞
z dz θe AL exp − θe AL
(3)
and Ay θh L:
(4)
The more important parameter for obtaining high spatial resolution is the numerical aperture NA of the focusing objective. As observed already in [8] NA is limited here to 2δ0.5 , as this is the limit for the deflection as discussed above. The most deflected beam in the y direction is impinging externally onto the first interface with the critical angle. By use of the derivative of Eq. (1) this leads to θh;max 0.52δf h 0.5 −xh;max −0.5 θcrit 2δ0.5 : In this condition, one then finds xh;max −f h ∕4: The absolute value of xh;max is then roughly the active length of the edge L. When this length L is now made identical to the attenuation length of the edge material AL one obtains from Eq. (1) a geometrical aperture of roughly Ah yh;max 22δ0.5 AL and a limitation for the minimum focal length of f h 4AL. The length limitation to AL applies now also to the refracting interface in the other direction, which leads then for coinciding focii positions to the limitation f e f h AL 1.25f h 5AL: These are feasible geometrical parameters for the edge. As a single edge now presents a half lens, the obtainable diffraction limit r for the spatial resolution can be estimated to be r ≈ λ∕NA, where NA is limited to NA ≤ 2δ0.5 . As δ ∝ λ2 [1] the diffraction limit for the spatial resolution is then independent of the chosen photon energy, and for lighter low absorbing materials like plastics or Be, the estimate by use of tabulated data for δ [9] is then for r ≈ 60 nm. The experiment was performed at a thin slice of SU-8, which is an epoxy-based negative tone resist of density 1.2 g∕cm3 with a composition in terms of weight fractions of C(0.727)O(0.182)H(0.069)Sb(0.014)F(0.006)S(0.002)
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[10]. This slice was spin coated onto a silicon wafer to a thickness of ∼200 μm. It was then patterned by exposure to UV light through a quartz mask, developed with PGMEA, and finally released from the substrate. Because of stress introduced into the surface after removal the slice deformed such that the surface but also the side wall resulted to be concave of approximately spherical shape. The slice length in beam direction is L 8.52 mm, while it had a thickness of 0.18 mm. This object was positioned at the SYRMEP beamline at the Elettra synchrotron radiation laboratory [11] behind a double crystal x-ray monochromator at 22.6 mm distance from a source measuring 70 μm in the vertical direction and 465 μm in the horizontal direction (2.36 GeV electron energy). The photon energy was set to 10 keV and a CCD camera downstream of a fluorescence screen registered the shadow image of the sample at a distance of 1.6 m from it with 4.5 μm equivalent pixel size. The sample was oriented such that the first refraction takes place at grazing incidence onto the curved side wall. The transmitted intensity was brought to a focus at the detector by adjusting both angles θh and θe independently. Figure 4 shows to the left the orientation and the bending of the sample as seen by an observer from the detector position in three different angular positions of θe in a scan around the horizontal axis. The center orientation is assumed to provide focusing in both directions. Note that the related intersection line in Fig. 2 lies in front of the emphasized curved edge in the octant with x > 0, y < 0, and z < 0. In the three images, the expected intensity distributions at the detector position caused by refraction at different features of the sample are superimposed as lighter gray areas. In correspondence to the edge of the side wall, closer to the camera one will always find a horizontally focused line extending over the entire thickness of the slice. The intensity refracted twice at the edge should then form long vertical lines at the same horizontal position as the previous line. For the case of bidimensional focusing, this refracted intensity should form a point. The intensity exiting after only a single refraction upon internal incidence onto the curved top surface will be found in horizontal lines above the shadow. In focusing condition, the narrow horizontal line has minimum height; otherwise in the defocused cases a larger distribution overlaps partly with the shadow of the
Fig. 4. Schematical illustration of the refracting edge, i.e., the right sample border, as an observer at the detector position would see it in a scan of the angle θe , which is increasing from the left to the right. In the three figures to the left, those areas are superimposed as lighter gray lines, in which refracted intensities will be registered at the detector position for an edge with optimum curvature. In the center image, the beam is focused in both directions. In the figure to the right it is assumed that the edge is incorrectly shaped with spherical curvature.
sample or it is completely detached from it. The latter case is observed for more grazing angles of incidence when according to f e ∝ θ3e the focused image would have been found already between the sample and the detector. Instead the overlap is found for larger angles of grazing incidence and the corresponding image position far behind the detector. Now the side wall with spherical curvature does not have the correct curvature over the entire edge length. This will introduce significant aberrations, and thus the doubly refracted beam will not be refracted onto straight lines or into a point as indicated but some deformation will be observed. An example of a related simulation is shown to the right in Fig. 4. In this simulation, roughly the first half of the side wall closer to the source refracts the beam to the same y coordinate, while the remaining part of the edge will refract it more. Then in focusing condition, one expects to find a spot with some tail extending sideways to the right. Depending on the aberrations in the top surface the tail can be inclined downward or upward. The just discussed features are now observed in the registered intensity distributions, as shown in Fig. 5. In the focusing condition (center image), one sees that the singly refracted beam in the top surface forms two narrow parallel lines. This is due to an additional deformation of the top surface close to the border. In this image, the focused spot is found, and the aberrated intensity is refracted to the left and downward. This indicates that the angles of incidence onto the part of the surface closer to the observer are too steep. This provides insufficient beam deflection in this area. In this case, the spherical surface is overbent compared to the smaller curvature in the better suited parabolic or elliptical profile. The right registration corresponds rather well to the simulation to the right in Fig. 4. In this case roughly 1∕3 of the edge extending from the center toward the source refracts the beam to similar y coordinates. The part of the edge closer to the observer is underbent and thus deflects too much. In systematic studies, it was verified that for the photon energy of 10 keV the tabulated optical properties for the SU-8 with the abovementioned composition [10] of δ 2.67 × 10−6 and AL 1.8 mm [9] consistently describe the refraction and the transmission of the slice in different orientations. For both refracting interfaces, the angle of grazing incidence onto the well focusing area is about 1.6° (28 mrad) and the radius of curvature in these areas is in both cases R 0.2 m. Then, according to Eqs. (3) and (4), the effective aperture of the edge for focusing in both directions is about 50 μm for the vertical direction and 70 μm in the orthogonal horizontal direction. The related geometrical acceptance does not have a rectangular shape, but it is trapezium shaped, which is caused by the edge inclination. Ideally the focus should have measured 5 μm by 33 μm, or 1 pixel by 7 pixels. The first is smaller than the point spread function (PSF) of the detector, for which a FWHM of 3.7 pixels was measured. The observed FWHM in this direction in the bidimensional focus is measured to be 5 pixels and it is thus slightly larger than the PSF. On the other hand, the width of the refracted distorted line in the right image, when the beam is only horizontally focused, is 7 pixels, which is the expected size for the re-imaged horizontal source at
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Fig. 6. Detached and connected assemblies of four curved edges in a symmetric configuration, which will provide fourfold increased geometrical aperture to the incident x rays. Fig. 5. Registered shadow images and refracted intensities at 1.6 m downstream from a curved edge for three different positions in angle of grazing incidence θe , in a scan, as shown in Fig. 4. Each single picture covers at the detector 0.56 mm × 0.9 mm (125 pixels × 200 pixels).
the detector position. In the bidimensional focus, the horizontal focus size is slightly larger with 8 pixels, which is smaller than the related geometrical aperture of the edge. Obviously, this latter broadening is due to the easily recognizable aberrations. Similar aberrations will also broaden the focus size in the vertical direction. Nevertheless, in order to judge the perfection of the refracting surfaces at this point, the blurring will be assigned to slope errors in these surfaces. For an rms slope error of σ rms , the blurred focus s after a single refraction at larger angles of grazing incidence will measure s ≈ 2.4σ rms f . Then one derives from a blurred image size of about 4 pixels a value for this surface error of the order of σ rms ≈ 5 μrad, which is a surprisingly small value. The gain in photon flux density in the focus was measured to be threefold, while with reference to the measured focus size one would have expected a gain of about 4.3-fold. Then only about 30% of the transmittable intensity is lost by other effects, e.g., in defects in the tip of the edge. The present study shows that a curved edge can focus x rays bidimensionally. UV lithography in combination with chemical etching was found to provide sufficiently small roughness in the etched walls, such that the refraction will be blurred only insignificantly. The test object had spherical curvature, which introduced severe aberrations into the focus. These aberrations were rather visible as a particularly long focal length was chosen. At shorter focal length, they will be less severe. When the goal is focusing to the diffraction limit, then the curvature needs to follow the required curvature very precisely and the focal length needs to be significantly shorter with roughly fourfold the attenuation length of the edge material. This latter number could have been here as small as 7.2 mm, which is more than two orders of magnitude shorter than the chosen image distance. Then with Ah 22δ0.5 AL the optimum geometrical acceptance would be sixfold smaller. In Fig. 2 one finds four sections of the curve that have equivalent curvature and focus thus at the same distance. The related focusing object could be formed by bending a very elongated elliptical hole in a thin slice symmetrically around a hyperbolic cylinder. Such a symmetric arrangement as presented in Fig. 6 can collect fourfold
more flux in a larger aperture, which could also permit to obtain a twofold smaller diffraction limit for the focus size. This would result in a simple single refractive lens, which can formally achieve the diffraction limited spatial resolution, that otherwise, as far as refracting devices are concerned, only more complicated stacks of refractive lenses can provide [3,4]. The most suitable materials for the edge production are plastics, synthetic diamonds, and Be. Edges in the latter two are particularly suited for operation in high-power beams, as the absorbed power can efficiently be dissipated into the unexposed bulk adjacent to the edge. We grateful acknowledge the help of Luigi Rigon from Trieste University during the experiment. We thank Massimo Tormen (CNR-IOM) and Diane Eichert (Elettra) for logistics support. The project was supported by the EU within the I3 action: European Light Sources Activities (ELISA). References 1. P. Kirkpatrick and A. Baez, J. Opt. Soc. Am. 38, 766 (1948). 2. P. Kirkpatrick, J. Opt. Soc. Am. 39, 796 (1949). 3. B. Lengeler, C. G. Schroer, J. Tuemmler, B. Benner, M. Richwin, A. Snigirev, I. Snigireva, and M. Drakopoulos, J. Synchrotron Rad. 6, 1153 (1999). 4. C. G. Schroer, O. Kurapova, J. Patommel, P. Boye, J. Feldkamp, B. Lengeler, M. Burghammer, C. Riekel, L. Vincze, A. van der Hart, and M. Küchler, Appl. Phys. Lett. 87, 124103 (2005). 5. H. Yan, V. Rose, D. Shu, E. Lima, H. C. Kang, R. Conley, C. Liu, N. Jahedi, A. T. Macrander, G. B. Stephenson, M. Holt, Y. S. Chu, M. Lu, and J. Maser, Opt. Express 19, 15069 (2011). 6. M. Montel, X-Ray Microscopy and Microradiography (Academic, 1957), pp. 177–185. 7. M. Sanchez del Rio and L. Alianelli, J. Synchrotron Rad. 19, 366 (2012). 8. K. Evans-Lutterodt, J. M. Ablett, A. Stein, C.-C. Kao, D. M. Tennant, F. Klemens, A. Taylor, C. Jacobsen, P. L. Gammel, H. Huggins, S. Ustin, G. Bogart, and L. Ocola, Opt. Express 11, 919 (2003). 9. B. L. Henke, E. M. Gullickson, and J. C. Davis, At. Data Nucl. Data Tables 54, 181 (1993), http://www‑cxro.lbl.gov/ optical_constants/. 10. M. Simon, “Roentgenlinsen mit grosser Apertur (X-ray lenses with large aperture),” Ph.d. thesis (Karlsruher Institut für Technologie, 2010). 11. L. Rigon, Z. Zhong, F. Arfelli, R.-H. Menk, and A. Pillon, Proc. SPIE 4682, 255 (2002).
