1. Introduction

X-rays are electromagnetic waves of short wavelengths, of the order of angstrom (~0.1 nm). The corresponding refractive index in matter is n=1-δ-iβ, where the real part of the refractive index correction δ is about 10⁻⁵~10⁻⁷ and the imaginary part is β of one or two order of magnitude smaller than δ. The fact that the refractive index is less than and very close to unity makes x-ray focusing difficult [1, 2]. In fact, x-ray focusing is always a central theme in x-ray optics for diffraction/scattering, spectroscopy, imaging, and microscopy studies in condensed matter physics, materials science, and life science [2–9]. The success of devising x-ray consecutive compound refractive lenses (CRL) [3–5] is an example, along with many research activities of pursuing new focusing mechanisms [3–11].

Here we propose a new x-ray focusing scheme by using the recently reported micron sized x-ray cavity [12, 13] combined with compound refractive lenses [3–5]. Due to the successive back diffraction occurring in the cavity, the forward transmitted and the back reflected beams have many more chances going through the lenses so the focusing effect is expected to enhance. In order to see this possible focusing effect, we have designed crystal devices, composed of the CRL of multiple curved crystal plates and yet satisfying x-ray back diffraction. Diffraction experiments from these devices with successive forward transmission and backward reflection are then performed and the beam size of the transmitted beam through the devices is measured. Indeed, in addition to the usual focusing due to refraction in CRL’s, focusing enhancement resulting from diffraction is observed, that is reported in this paper.

2. Optical design and diffraction experiments

The curved multi-plate crystal cavities consisting of compound refractive lenses are manufactured on (001) silicon wafers by lithographic techniques. Several holes with parabolic cross section are made on the wafers such that a series of concave lenses are lined up along the [310] direction. The (12 4 0) reflection is used as the mirror reflection to generate successive reflections within the lenses for 14.4388 keV x-rays [12, 13]. Figure 1 shows the layout of the crystal device composed of parabolic holes. The radius of curvature is R and the distance between two adjacent holes is d.

 

Fig. 1 Schematic of compound refractive lenses with the (12 4 0) satisfying back reflection for E=14.4388 keV: Each hole serves as a crystal cavity of a Fabry-Perot type. The crystal between two adjacent holes serves as an x-ray lens. The total length along [12,4 0] is 1.2 mm.

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To avoid either the incident beam or the reflected beam from hitting the Si wafer surface, we first identified the minimal miscut direction of a 4-inch Si wafer by measuring miscut using reflectivity and (004) Bragg diffraction measurements. The desired [310] for the (12 4 0) back diffraction was chosen with the minimal miscut about 0.02°. The height of 250 μm for the CRL was selected to ensure the x-ray beam was not blocked by the wafer surface. The total length of the crystal device was also kept as short as possible. Considering possible misalignment between the [310] direction and the direction lining up the manufactured curved crystal plates, three sets of CRL lenses, each tilted ±0.5° from the nominal [310] direction, were prepared. We then chose an appropriate set of crystal devices for the diffraction experiment.

The lens formula F=R/(2Nδ) [3] for a CRL was used to determine the radius R of the lens and the number N of lenses for a selected focal length F. Because the distance between the crystal device and the detector was about 900 mm, the focal length F=715 mm was selected to facilitate the measurement of beam size. The corrections in the refractive index are δ=2.33x10⁻⁶ and β=1.72x10⁻⁸ for the 14.4388 keV (0.8605Å) x-ray photons, at which a 24-beam diffraction, including the back reflection (12 4 0), of silicon takes place [12].

The total crystal thickness of the device, excluding the diameters of the holes, was optimized to be about 120 μm, at which the reflectivity of the (12 4 0) reflection from the crystal device is equal to the transmissivity of the (000) reflection through the crystal device. This implies that the overall transmitted and reflected amplitudes of the (000) and (12 4 0) beams are equal so that the interference between the two beams due to cavity resonance is maximized [12]. Moreover, the absorption effect due to crystal thickness was also considered. We also tried to optimize the gain of the devised CRL by considering the real gain of a usual CRL for an ideally coherent incident beam as the guideline for preparing our crystal device. The real gain g defined as g=aGi(σ_f /σ₁) is adopted, where the absorption a=exp[-μNd] with μ=4πβ/λ. Gi is the ideal gain given by Gi={A /σ_f }[1+{r_f /r₀}]), σ_f is the diffraction-limited resolution of the lens, σ₁ is the real focus size, and A is the effective lens aperture defined as A=2R[2/(μRN)]^-0.5 [3–5]. Based on these considerations, we prepared a CRL by etching 12 equally spaced parabolic holes, each having a radius of R=40 μm and the distance between the adjacent holes is d=10 μm on silicon (001) wafers using x-ray lithographic techniques (see Fig. 1). This produced a crystal device consisting of 12 lenses, namely 12 crystal plates (N=12). The estimated real gain is g=46.4 for μ=25.2 (1/cm), Gi=168, a=0.68, A=102 μm, and σ_f=0.6 μm. The real focus size σ₁ is 1.5 μm according to the relation, σ₁=σ₀ (r_f /r₀ ), for the (incident) source size σ₀ (about 120 μm), the image distance r_f (about 0.724 m), and the source distance r₀ (about 58.4 m) from the crystal device for beam line BL12XU at SPring-8. r_f is estimated from the definition r_f=F[r₀ /(r₀ -F)]. The detail of the CRL crystal orientation is shown in Fig. 1.

The experiment was carried out at the Taiwan undulator beamline BL12XU at the SPring-8 synchrotron facility in Japan. The experimental setup and conditions were the same as that reported in Ref [12]. The incident radiation was monochromatized first by a Si (111) double-crystal and then by a four-crystal ultra-high resolution monochromator [14], to give the energy resolution ΔE=0.36 meV at 14.4388 keV. The CRL crystal device, placed at the center of an 8-circle diffractometer, was aligned for the (12 4 0) back diffraction. The horizontal tilt, Δθ, of the transmitted (000) beam shows an intensity dip at Δθ=0° (marked C) in Fig. 2(a) , at which the 24-beam diffraction occurs exactly for the photon energy E_o=14.4388 keV. The broad dip is due to the (12 4 0) back reflection because the wide width of the diffraction profile is proportional to $2\sqrt{\left|{\chi }_G\right|}$, while the sharp dip results from the 24-beam diffraction, which depends on the multi-beam interaction and whose diffraction width is smaller than the width related to the first power of the magnitude of the electric susceptibility χ_G [12, 13, 15]. Namely, the 2-beam (12 4 0) back reflection occurs within −0.1°<Δθ<0.1°, the angular range of the Darwin width, and the 24-beam diffraction takes place at Δθ=0°.

 

Fig. 2 (a) Angular Δθ scans of the transmitted (000) beam of the (12 4 0) back diffraction (A, B, and C correspond to Δθ=−0,053°, 0.02°, and 0°);(b) Horizontal transmitted beam sizes through the cavity crystal measured at the exact position for the 2-beam back reflection, Δθ=−0,053° (open circles), at the 24-beam diffraction position Δθ=0° (curve with solid squares), and for refractive focusing (triangles) as the distance from the exit end of the cavity crystal along the transmitted-beam direction (errors in beam-size~1.0 μm).

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To investigate how the 2-beam back reflection and the 24-beam diffraction affect the beam focusing, we measured the horizontal size of the transmitted beam at Δθ=−0.053°, 0.02°, and Δθ=0° (marked as A, B, and C in Fig. 2(a)). Since points A and B lie within the Darwin width (the broad dip), at points A and B the two-beam back reflection of (12 4 0) takes place. Point C indicates the exact position of the 24-beam diffraction. At each of the angular positions, we then scanned a knife-edge of surface roughness 250 Å across the transmitted beam along [-1 3 0] in front of the detector at various distance from the exit end of the CRL cavity along the [000] direction to measure the size of the transmitted beam through the compound refractive lenses. Figure 2(b) shows such measurements. If the crystal device is tilted to Δθ=0.15° and the photon energy is shifted 3 eV below the exact energy 14.4388 keV for the two-beam back reflection, then there is no diffraction taking place. The transmitted x-ray beam will be affected only due to refraction when transmitting through the crystal device. Under this condition, the horizontal size of the transmitted beam versus the distance from the exit end of the crystal device was measured, which is shown in Fig. 2(b) as the curve marked with triangles. This situation without diffraction was indeed verified experimentally as follows: A crystal device with the lens axis off 10° from the [12,4 0] gave the same curve as the one with triangles in Fig. 2(b). Since points A and B are within the Darwin width of the two-beam back reflection of (12 4 0) and the measurements at Δψ=−0.053° and 0.02° are almost the same, only one set of measurements (the curve marked with open circles) is shown. As can be seen, the transmitted beam was therefore focused to the point 800 mm from the end of the cavity along the incident beam direction due to refraction, and to 650 mm (18% reduction) for Δθ=−0.053° due to the 2-beam back reflection, while the beam size measurements (the curve with solid squares) at Δθ=0° give the focal length of about 356 mm (56% reduction) owing to the 24-beam diffraction. The horizontal beam sizes at the focal point are 6.2, 7.3 and 7.4 μm for Δθ=0°, Δθ=−0.053°, and refraction, respectively. The vertical beam size about 50μm defined by slits is unchanged. This reflects strong horizontal focusing for the 24-beam diffraction and medium strong focusing for the 2-beam back reflection and weaker focusing for refraction. The difference in the focal length due to refraction between the measured 800 mm and the designed 715 mm is because the designed value is for ideal situation. In reality, the measured focal length is longer.

We have also measured the intensity of the focused transmitted beam, which is about 18% of that of the direct incident beam. This value is comparable with that of existing focusing devices.

3. Theoretical considerations

This beam focusing effect can be understood from the x-ray wavevectors of the transmitted beams determined by the excitation of the dispersion surface according to the dynamical diffraction theory of back diffraction [15–18]: The Darwin width of the (12 4 0) back diffraction, proportional to $2\sqrt{\left|{\chi }_G\right|}$, is equal to 0.071°, where the electric susceptibility χ_G=−3.927x10⁻⁷+i1.8x10⁻⁸ and G=(12 4 0). The angle of refraction for one crystal plate is about 0.0002° and 0.0028° for 12 crystal plates (See Fig. 1), much smaller than the Darwin width. However, the excitation of the dispersion surface for the CRL's parabolic surface due to dynamical effects could be quite different even for a minuscule change in the beam direction. Also, the wavevectors of the excited waves inside the first crystal plate could change when the transmitted x-rays enter the second crystal plate and there are 12 plates involved in the diffraction process. Similar situation occurs for the back-reflected beam. Numerical calculation based on the dynamical theory of the complete transmission and reflection of x-rays among the total 12 crystal lenses is a formidable task and the detail is out of the scope of this paper. As an alternative, we first consider the dynamical back diffraction from a single crystal lens and then extend it to the 12 lenses. Figure 3 shows schematically the excitation of the dispersion surface for the two-beam and 24-beam diffraction. In real space (Fig. 3(a)), the incident beam of the wavevector k_o0 with a zero angular deviation Δθ ~0 from the exactly parallel direction, impinges on the concave crystal lens. The incident beam hits the upper part of the curved entrance surface at the contact point C₁ and generates a forward transmitted beam K_o1 inside the crystal and a backward reflected beam (not shown) in vacuum. Then the forward transmitted beam K_o1 propagates inside the crystal and generates another pair of back reflected (not shown) and forward transmitted beam k_o1 when going through the exit curved crystal surface at point C₂. Here k’s and K’s stand for the wavevectors outside and inside the crystal lens, respectively. The vectors n₁₁ and n₁₂ are the normals to the entrance and the exit surface at points C₁ and C₂, respectively, where the first index equal to 1 indicates the first lens. The second index equal to 1 means the entrance surface and 2 the exit surface. The corresponding wavevectors and surface normals for the lower-half crystal surface are labeled with “prime’.

 

Fig. 3 Schematic of the focusing of the transmitted beams: (a) Ray-tracing in a single crystal lens under back diffraction condition with Δθ ~0 in real space; (b) Projection of the dispersion surface of the 2-beam case on the plane of incidence of (12 4 0) in reciprocal space; (c) Projected dispersion surface of the 24-beam case on the plane of incidence of (12 4 0).

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In the reciprocal space (Fig. 3(b)), the reciprocal lattice points O and G represent the incident (000) and the (12 4 0) back reflection. The vertical line denoted as Σ_o represents the wavefront of the incident beam k_o0 with a negligibly small beam divergence. The central part of the dispersion surface of an exact back reflection G projected onto the plane of incidence of (12 4 0) is the dotted line nearly perpendicular to OG (See also Fig. 5.17a in Ref [15].). While the dispersion surface for refraction is schematically shown as the dashed grey line. In the two-beam case, since Δθ~0, the wavevector of the incident beam k_o0 starts from the entrance point E₀ to point O. The corresponding forward transmitted beam inside the crystal is T₁₁O=K_o1, where T₁₁ is the excitation point on the dispersion surface along the normal n₁₁. (Here the continuity of the tangential components of the wavevectors inside and outside the crystal at the crystal boundary is fulfilled.) This crystal wave T₁₁O then generates the beam k_o1 = E₁O by exciting the incident wavefront Σ_o at E₁ along the normal n₁₂. This excitation of the dispersion surface repeats when the beam k_o1 reaches the second crystal at a lower contact position closer to the central part C_oC_o of the crystal than that for k_o0 (Fig. 3(a)). The corresponding surface normals n ₂₁ and n ₂₂ become more parallel to OG (Fig. 3(b)). At the end of the 12th crystal lens, the exit beams k_o12 and k’_o12 (not shown) are focused onto point O just like k_o2 and k’_o2 shown in Fig. 3(b), but with stronger focusing. Similarly, the exit beam due to refraction will be also focused onto point O but with less focusing effect, because the excitation points on the grey line are closer to the line OG. If the incident parallel beam hits the upper part of the first curved crystal surface at a point, say C”, in between C_o and C₁, then the similar focusing takes place. This is because all the wavevectors of the forward transmitted beams, inside and outside the crystal, point towards the reciprocal lattice point O.

The focusing effect of the CRL crystal due to the 24-beam diffraction can be understood from the dispersion surface shown in Fig. 3(c). There are 96(=24x4) dispersion sheets involved [18]. For simplicity, only two sheets are schematically shown as the dotted lines projected onto the plane of incidence. Starting from the entrance point E₀, the incident beam k_o0 excites two points, T₁₁ and T₁₂, on the dispersion surface along the entrance surface normal n ₁₁, thus generating two transmitted beams inside the crystal. These two beams, in turn, excite the Σ_o surface at E₁₁ and E₁₂ along the two slightly different exit surface normals, n₁₂₁ and n₁₂₂, to generate two exit beams from the first lens along k_o11=E₁₁O and k_o12.=E₁₂O. Clearly, additional focusing comes into play due to the excitation of more dispersion sheets. This explains why the 24-beam diffraction shows strong focusing effect, compared to the 2-beam and the refractive focusing.

Moreover, if the angle of incidence is exactly equal to the Bragg angle (90°) of the (12 4 0), i.e., Δθ=0, the upper half and the lower half of the parallel incident beam hitting on the curved crystal surface are symmetric with respect to the C_oC_o line in Fig. 3(a). This incident beam satisfying either the two-beam or the 24-beam diffraction condition after transmitting through the 12 lenses will be certainly focused onto a focal point on the C_oC_o line. This focusing situation is shown schematically in the upper most figure of Fig. 4(b) . If the angle of incidence is greater than the Bragg angle, the angle Δ θ between the incident beam k_o0 and GO is larger than zero (Δθ >0) and the incident wavefront Σ_o is also titlted by Δθ from the vertical (see, Fig. 4(a)). In this case, the excitation of the dispersion surface for the two-beam case in the reciprocal space shown in Fig. 4(a) is very similar to that shown in Fig. 3(b) for the situation with Δθ ~0, except that the transmitted beams k_o1 and k_o2 through the upper parts of the first and second lenses are more focused than the transmitted beams k’_o1 and k’_o2 through the lower parts of the first and second lenses, respectively. This is because the angle between k_o1 and GO is larger than the angle between k’_o1 and GO. The same is true for k_o2 and k’_o2. The ray tracing for focusing with Δθ >0 is shown in the middle of Fig. 4(b). Similarly, the ray tracing for focusing with Δθ <0 has a similar behavior, except that the lower part of the transmitted is more focused than the upper part of the beam (See, the lower figure of Fig. 4(b)). Since the 24-beam diffraction occurs at Δθ=0 and the two-beam case takes place at Δθ >0 and Δθ<0, the measured focal length and beam size may be affected the fact just discussed.

 

Fig. 4 (a) In the reciprocal space, the projection of the dispersion surface of the 2-beam case on the plane of incidence of (12 4 0) with Δθ>0; (b) Ray tracing of focusing for the incident beam with Δθ~0, Δθ>0, and Δθ<0.

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4. Conclusion

In conclusion, we have observed unusual optical effects in curved multi-plate x-ray crystal cavities consisting of compound refractive lenses. That is, the focusing effect from the CRL is further enhanced by the back diffraction and the 24-beam diffraction. Thus, a small sized x-ray beam is produced. This enhanced beam-focusing, governed by the curvature of the curved CRL, is due to the change of the direction of the wavevector of the transmitted beam during the excitation of the dispersion surface of the diffractions involved. This focusing mechanism may find usage in producing small sized x-ray beams in very low-emittance synchrotron facilities.