Modern Developments in X-Ray and Neutron Optics Episode 12 pps

440 J. Hrd´y and J. Hrd´a Fig. 26.1. Symmetric Bragg diffraction on a crystal with a lattice plane spacing d like a mirror. Thus, from the X-ray optics point of view the refraction is not too interesting. The situation is different if the crystal surface is not parallel with the diffracting crystallographic planes. In this case the crystal does not behave exactly as a mirror any more. The incident and diffracted beams are not sym- metrical with respect to the surface normal (which is trivial) but they are also not symmetrical with respect to the normal to the diffracting crystallographic planes. There are two limiting cases. The asymmetric diffraction corresponds to the situation when the surface normal lies in the plane of diffraction, i.e., the plane determined by the incident beam and the normal to the diffracting planes. In this case we will call the crystal asymmetric crystal. The diffracted beam lies in the plane of diffraction; thus the diffraction is still coplanar. The inclined diffraction occurs when the plane determined by the surface nor- mal and the normal to the diffracting planes is perpendicular to the plane of diffraction. Here we will call the crystal inclined crystal. As will be shown later, the diffraction is noncoplanar. The general asymmetric diffraction is the combination of the asymmetric and the inclined diffraction. 26.1.1 Asymmetric Diffraction For the asymmetric diffraction the values ω 0 and Δθ 0 for an incident beam and the values ω h and Δθ h for a diffracted beam are not identical and are different from the values ω s and Δθ s for the symmetric diffraction. The following set of relations holds [4]: ω 0 = ω s b −1/2 , ω s =(2r e λ 2 P |F hr |e −M )/πV sin 2θ B , Δθ 0 =(1/2)(1 + 1/b)Δθ s , Δθ s = r e λ 2 F 0r /πV sin 2θ B , ω h = ω s (b) 1/2 , (26.2) Δθ h =(1/2)(1 + b)Δθ s , θ 0 = θ B +Δθ 0 , θ h = θ B +Δθ h , b =sin(θ B − α)/ sin(θ B + α). 26 Diffractive-Refractive Optics 441 Here V is the unit-cell volume, r e = e 2 /mc 2 (classical electron radius), F hr is the real part of the structure factor F h (h stands here for M¨uller indices hkl), P is the polarization factor, and e −M is the temperature factor. The index s stands for the symmetrical diffraction. The angle α is the angle between the diffracting planes and the surface and is taken as positive for grazing incidence. The asymmetry index, b, is defined according to Matsushita and Hashizume [4]. The typical values of Δθ s and ω s are from fractions to tens of angular seconds. The angle θ B is the Bragg angle calculated from the Bragg law (1). For the cross sections CS 0 And CS h of the incident and the diffracted beams the following holds: CS h = CS 0 /b (26.3) and together with (26.2), ω h CS h = ω 0 CS 0 . (26.4) The consequence of the above relations may be demonstrated with the help of the DuMond graph (Fig. 26.2a). The real situation is shown in Fig. 26.2b. Let us suppose that a parallel and polychromatic beam is impinging on an asymmetrically cut (= asymmetric) crystal with some angle, θ, between the beam and the lattice planes. The deviation from a mirror-like behavior may be described by the quantities δ and Δδ. It obviously holds (for one harmonic): δ(α)=Δθ 0 − Δθ h , Δδ(α)=|ω 0 − ω h |. (26.5) The deviation δ and the spread Δδ, which may be changed by changing α resembles the refraction of light on a prism. The only difference is that the Fig. 26.2. (a) DuMond diagram of an asymmetric Bragg diffraction showing that as compared with a symmetric diffraction the diffracted beam is deviated and spread. Figure 26.2b shows the asymmetric Bragg diffraction of a polychromatic pencil beam in real space 442 J. Hrd´y and J. Hrd´a prism accepts a broad range of wavelengths, whereas the crystal accepts only narrow wavelength interval ω 0 (dλ/dθ) and thus the interval Δδ is narrow. Nevertheless, it exhibits a wavelength dispersion, as in the case of a prism. One may deduce that this refraction effect (δ, Δδ) may play an important role when the surface of crystal is curved (not bent), as in the case of refraction lenses in classical optics. The asymmetric diffraction with flat crystals is used to either compress or to extend the diffracted beam. This will be treated elsewhere in this book. (see the Chap. 29) 26.1.2 Inclined Diffraction The inclined Bragg diffraction is a noncoplanar diffraction. The behavior of the diffracted beam may be seen in Fig. 26.3a, which shows the wave vectors and the dispersion surfaces in reciprocal space [5,6]. The points P 0,1 and P 0,2 are the origins of the impinging vectors, directed into the origin, O,ofthe reciprocal space, points P h,1 and P h,2 are the origins of diffracted vectors for symmetric Bragg diffraction (β =0).ThepointsP h,1,β and P h,2,β are the origins of diffracted vectors for the inclined diffraction. The indices 1 and 2 represent the limiting beams within the diffraction region, ω. It is seen that if the impinging monochromatic and parallel beam is scanned through the diffraction region, ω 0 (ω 0 = ω s ), then the diffracted beam is deviated from the plane of diffraction, and this deviation grows during the scan. The consequence of this is demonstrated in Fig. 26.3b. It shows that if a parallel and polychromatic beam impinges on an inclined crystal with an inclination angle β, then the diffracted beam is deviated from the plane of diffraction in a sagittal direction (perpendicular to the plane of diffraction) and the beam Fig. 26.3. Wave vectors in reciprocal space for an inclined diffraction (a). Inclined diffraction in a real space (b) 26 Diffractive-Refractive Optics 443 is sagittally spread. The deviation, δ, of the central beam from the plane of diffraction is δ = K tan β (26.6) where K =(2r e F 0 /πV )d hkl λ. (26.7) For Si crystals K =1.256 × 10 −3 d hkl (nm) λ (nm). The inclined crystal monochromators based on the inclined diffraction are used to decrease the impinging radiation power density of synchrotron radiation. Here the devia- tion and the spread of the diffracted beam is the manifestation of refraction. As in the asymmetric diffraction, here the beam spread is also limited by the wavelength acceptance of a crystal for a given incidence angle. From the above it is clear that, as in the asymmetric case, interesting applications may be expected if the diffracting surface is machined into a suitable shape. The refraction effect exists also in general asymmetric diffraction (the com- bination of asymmetric and inclined diffraction) and in Laue diffraction. These will be discussed later. X-ray refractive lenses are now commonly known and are successfully used for focusing synchrotron radiation [7]. The aim of our work was to study the diffraction on crystals with curved diffracting surface and to investigate the possible applications of effects based on refraction described earlier. The idea of a crystal monochromator with a curved (not bent) diffracting surface is not new. The Johansson spectrometer [8,9] is a Bragg crystal with a circular profile machined into the working surface of the crystal. The crystal is then bent. Such a crystal focuses the monochromatic radiation on the Row- land circle. Spieker [10], designed a channel-cut crystal monochromator with profiled working surfaces such that the position of the exit beam remains fixed when tuning the wavelength. These two methods are based only on geometry; the refraction effect is completely neglected. 26.2 Bragg Diffraction on a Transverse Groove (Meridional Focusing) From what was explained in section 26.1.1 and from Fig. 26.2b it may be deduced that the radiation diffracted on properly designed transverse groove may be meridionally focused. This is demonstrated in Fig. 26.4a. The diffracted beam 1 is deviated to the right due to the asymmetric diffraction and it is spread. The beam 3 is also spread and is deviated in the opposite direction, i.e., to the left. The beam 2, which is diffracted from the bottom of the groove where the diffraction is symmetrical, is neither deviated nor spread. The prob- lem is to find the function g(x) describing the shape of the groove, such that the centers of all diffracted beams (i.e., centers of the fans) will be concentrated into one point, the focus. Substituting (26.2) into (26.5) we obtain δ =2Δθ s tan θ B tan α/(tan 2 θ B − tan 2 α). (26.8) 444 J. Hrd´y and J. Hrd´a Fig. 26.4. Bragg diffraction on a transverse groove machined into a symmetric crystal (a). The diffracted beam is convergent. Figure 26.4b shows image of an X-ray beam diffracted on a crystal with a transverse groove Let the profile of the groove be described by a function y = g(x)(see Fig. 26.4a). Let us suppose that the impinging radiation is parallel. In order that the beam impinging on the surface of the groove at a certain point, A(x, y), be diffracted to the focus, the deviation, δ, must be [11] δ =[−x sin(θ B +Δθ 0 )+y cos(θ B +Δθ 0 )]/f, (26.9) where f is the focal distance. Taking into account that tan α = −g  (x)(g  =dg/dx) and neglecting Δθ 0 in (26.9), then (26.8) and (26.9) gives the differential equation [11] [x sin θ B − g(x)cosθ B ]/f =2Δθ s tan θ B g  (x)/{tan 2 θ B − [g  (x)] 2 }, (26.10) which describes approximately the shape of the transverse groove. In [12] this equation was further modified to include the finite divergence of the impinging radiation. The shape of the groove obviously depends on the wavelength, λ,the focusing distance, f, and the source–crystal distance, S.Inorderthatsuch a focusing monochromator could be used for a broad wavelength region, it is necessary to produce either several parallel grooves for various λs or only one groove whose shape changes along the groove axis. The focusing conditions could be then adjusted by a translation of the crystal. We have demonstrated this kind of focusing by an experiment performed in ESRF at the BM5 beamline [12]. The transverse groove, machined into a Si(111) crystal, was calculated for λ =0.15 nm,S=40m,andf =2m. Figure 26.4b shows the image of the diffracted radiation at the distance of 2 m from the crystal. The figure shows, in the upper and lower parts, the image of the radiation diffracted from the flat part of the crystal and between them 26 Diffractive-Refractive Optics 445 there is the image of the radiation which is diffracted on the groove and is concentrated into a narrow bright line. The width of the groove was about 2.5 mm. Because of the spread, Δδ, of the diffracted beam the focus cannot be sharp. Even without any refraction effect the diffracted radiation would be concentrated at the right side of the groove because of the asymmetric diffrac- tion. It makes sense to compare the peak intensity in the focal plane after diffraction from the groove and from a flat asymmetric crystal with the asymmetry corresponding to the right side of the groove. The ray-tracing simulation of the experiment showed that the groove would give about 3.3 times higher intensity in the peak at the focal plane than an asymmetric concentrator. 26.3 Harmonics Free Channel-Cut Crystal Monochromator with Profiled Surface Another application of diffraction on the meridionally profiled (curved) sur- face may be a channel-cut crystal monochromator which suppresses higher harmonics in the broad region of the Bragg angles. Let us suppose that the first diffracting surface (the first wall of the channel) is flat and symmetrically cut (its surface is parallel with diffracting crystallographic planes). If the sec- ond wall is also flat but asymmetrically cut, then Δθ h and ω h for the first wall (Δθ h =Δθ s and ω h = ω s )isnotequaltoΔθ 0 and ω 0 for the second wall, and only a part (or none) of the radiation diffracted from the first wall is diffracted from the second wall: i.e., diffraction is detuned. This depends on the degree of overlap of the corresponding Darwin–Prins (DP) curves. The values Δω and ω decrease with the order of diffraction k. This means that for a certain asymmetry of the second wall, the DP curves for higher harmonics do not overlap any more but the overlapping of DP curves for the fundamen- tal harmonic is still sufficient. The radiation diffracted from the channel-cut crystal is then practically free of higher harmonics. This way of obtaining harmonics rejection, which is valid for one λ and its close neighborhood was suggested by Matsushita and Hashizume [4]. The mathematical description of this situation is following: (P |F (k) hr |e −M(k) )/|F 0r | =( 1 / 2 )|1/b 1/2 − 1| (26.11) where b corresponds to the second surface and k (>1) stands for the order of diffraction. As the left part of (26.11) is independent of θ, b must also be independent of θ. This implies that α must change with θ, which means that the second wall must be curved in order that the channel-cut crystal monochromator rejects higher harmonics in the whole region of θ or λ [13]. 446 J. Hrd´y and J. Hrd´a Fig. 26.5. Harmonics-free channel-cut crystal monochromator: the derivation of the shape of the second diffracting surface (a). The second diffracting surface may be convex (b)orconcave(c) The equation for b (see (26.2)) may by rewritten as follows: tan α =[(1−b)/(1 + b)] tan θ = B tan θ. (26.12) Let us introduce the axes of the coordinates with the origin on the first wall, such that the X-ray beam is impinging at the origin on the first wall (Fig. 26.5a). The axis of rotation of the monochromator also passes through the origin. Let the profile of the second wall be described by the function f(x). Then (26.12) may be rewritten in the following form: df(x)/dx = Bf(x)/x. (26.13) The angle α is here taken as negative. Thus b>1andB<0. The solution of this differential equation is f(x)=Cx B (26.14) where B is negative and the second wall of the crystal is convex. It is obvious, that by a similar consideration as above, the second wall of the channel-cut crystal may be cut so that the angle, α, is positive [14]. In this case the DP acceptance curve for the diffraction on the second wall is shifted toward higher angles, θ. The condition when the DP curves just touch each other is the same as in the previous case, however, |1/b 1/2 − 1| =1/b 1/2 − 1, because b<1. This leads to (26.14) where B is positive and the second wall is concave. Both kinds of channel-cut crystal monochromators are schematically shown in Fig. 26.5b, c. It is obvious that the monochromator with a convex wall concentrates the diffracted beam but slightly increases its divergence, as follows from para- graph 26.2 (or section 26.2 or 26.2). The monochromator with concave beam creates a broad beam and slightly decreases its divergence or may even cre- ate a slightly convergent beam if the impinging beam is almost parallel, as it is in the case of synchrotron radiation. This has been discussed in detail in [14]. Obviously, the harmonics rejection here is the consequence of the 26 Diffractive-Refractive Optics 447 dependence of refraction on α and the order of diffraction. The harmonics- free channel-cut crystal monochromator discussed above has not yet been tested experimentally. The width of Darwin–Prins function for the π polarization is cos 2θ times smaller than for the σ polarization component. As was shown by Hart and Rodrigues [15], a double crystal monochromator in a nondispersive (+, −)set- ting which is detuned may reject the π polarization component similarly as it rejects higher harmonics [16, 17]. Only the degree of detuning is different. It is obvious that there should exist a channel-cut crystal monochromator with a suitably curved diffracting surface such that it rejects π polariza- tion components for a broad region of θ. This will be treated in detail elsewhere [18]. 26.4 Bragg Diffraction on a Longitudinal Groove (Sagittal Focusing) In section 26.1.2 it was shown that in the case of an inclined diffraction the diffracted beam is deviated sagittally (perpendicularly to the plane of diffrac- tion). Let us suppose that a longitudinal groove is produced in the diffracting surface of a crystal, as shown in Fig. 26.6a. The opposite walls of the groove deviate the beam in opposite directions. It is clear, that a properly designed shape of the groove may sagittally concentrate the diffracted beam at cer- tain distance, f, from the crystal. The geometry of the diffraction is shown in Fig. 26.6b. For the determination of the shape of the groove we will suppose Fig. 26.6. Bragg diffraction on a crystal with a longitudinal groove (a ). The diffrac- ted beam is convergent. Geometry of the sagittal focusing due to the longitudinal groove (top view)(b) 448 J. Hrd´y and J. Hrd´a that the distance of the grooved crystal monochromator from a point source is S and the focal length is f. Let the shape of the groove be described by a function y(x). For the groove to act as a lens, it is necessary that the beam impinging on the crystal (groove) at a distance x from the longitudinal axis of the groove, be deviated by an angle δ ∼ = tan δ ∼ = [x(S + f )/S]/f = xR/f. (26.15) Equation (26.6) may be rewritten in the following way: tan δ = K(dy/dx). (26.16) Equations (26.15) and (26.16) give a differential equation with the solution [6] y =(R/2Kf)x 2 + constant. (26.17) The meaning of the above result is that the longitudinal parabolic groove focuses the radiation and thus acts as a sagittally focusing lens. For syn- chrotron radiation two crystals in a parallel, nondispersive (+, −) orientation are commonly used. The parabolic longitudinal groove may be then produced in both crystals. If only one crystal or more crystals in a nondispersive posi- tion are used, then the advantage of the sagittal focusing is deteriorated by two effects. The first one is shown in Fig. 26.7a. The vertical size of the beam increases after each diffraction. This depends on the depth of the groove. The second effect is the sagittal spread of the deviated beam which prevents the focus from being sharp. Both effects mentioned above (aberrations) may be canceled by using a dispersive arrangement of crystals. From Fig. 26.7b it is clearly seen that the vertical broadening which appears after diffraction from first two crys- tals is completely canceled after diffraction on the following two crystals. The dispersion arrangement also cancels the sagittal spread seen in Fig. 26.3b. Fig. 26.7. A longitudinal groove broadens the diffracted beam vertically (a). Dispersive four crystal arrangement with longitudinal grooves cancels the vertical broadening of a diffracted beam originating from the first two crystals (b) 26 Diffractive-Refractive Optics 449 The nature of the sagittal spread is shown in Fig. 26.3a. When the imping- ing (monochromatic) beam spans the diffraction region, ω 0 , from smaller to higher θ, then the sagittal deviation grows. For example the beam correspond- ing to a smaller θ at the beginning of the diffraction region leavesthecrystal with minimal sagittal deviation. Let us suppose that there is another crys- tal adjusted in dispersion position with respect to the first crystal. This beam impinges on the second crystal at the end of the diffraction region correspond- ingtoahigherθ and the sagittal deviation is maximal. The resulting deviation after diffraction on both crystals is 2δ for any beam impinging on the crys- tals within the region ω 0 . The angle δ is the average deviation as shown in Fig. 26.3b. This holds for any θ within the region ω.Thisveryimportantresult shows that the (−, +, +, − ) arrangement, shown in Fig. 26.7b, is ideal [19]. The second and the third crystals cancel the aberrations discussed above and the first and fourth crystals keep the direction of impinging and exit beams the same, which is important for synchrotron radiation. The position of the exit beam remains independent of θ. Moreover, the dispersion arrangement is the high resolution one. This arrangement should provide practically point-to- point focusing, which means that we may expect a sharp focus. The practical expressions important for the design of the parabolic groove are following y(mm) = a(mm −1 )(x(mm)) 2 , (26.18) a =(S + f)/2NKfS, (26.19) where f (mm) is the focusing distance, S (mm) is the monochromator-source distance and N is the number of diffraction events on the grooves. For the four crystal arrangement shown in Fig. 26.7b, N = 4 provided that the beam is diffracted only once on each crystal. The focusing distance f may be determined from f = S/(2aNKS − 1). (26.20) The parabolic groove may also be cut into an asymmetrically cut crystal. This is treated in detail by Hrd´y [20]. It was shown there that for this case all the above formulae may be used. Only K must be replaced by K  = K [(2 + b +1/b)/4cosα]. (26.21) The difference between the function of the symmetrically and asymmetri- cally cut grooved crystals may be seen in Fig. 26.8. It shows a dependence of the focusing distance, f, on the Bragg angle θ for a four-crystal (+, −, −, +) monochromator with the same crystals and grooves. Crystals cut symmetri- cally is compared with asymmetric crystals with α =12.38 ◦ . The monochro- mator with the asymmetric crystals gives a shorter focusing distance. In the angular region around θ =22 ◦ the focusing distance is almost constant and for the highly asymmetric case, close to θ = α, the focusing distance is very small. The above expressions enable one to be able to design the crystals, the grooves and the asymmetry angle to meet the experimental requirements. [...]... grating is a basic optical element for the construction of a variety of X-ray optical devices including fast X-ray modulators Understanding its properties is essential for effectively designing high resolution, focusing dispersive X-ray optics 28.2 Static Volume Grating Properties One can consider two types of volume gratings: gratings etched into a multilayer/crystal mirror, called etched gratings, and. .. X-Ray Optics A Erko, A Firsov, D.V Roshchoupkin, and I Schelokov Abstract Systematic experimental and theoretical investigations of different types of Bragg-Fresnel gratings, both static and dynamic, are discussed Static gratings are produced by etching in a multilayer or by evaporating gold or nickel masks on the surfaces of symmetric or asymmetric Si [111] crystals These have been used to obtain X-ray. .. radiation sources in the nanometer range, synchrotrons, storage rings and X-ray lasers in the future, and on the methods of microelectronics technology that enable fabrication of structures of X-Ray electronic devices with submicron or nanometer element sizes, i.e micro-photonic devices As in other fields of engineering associated with receiving, transmission, processing and storage of information, the... primary and its higher order reflections, i.e., (g 1 · u) = 0 and the corresponding integrated reflectivity is independent of the deformation represented by the displacement u [18] This is also valid in our case of cylindrical bending and symmetric transmission geometry On the other hand (g 2 · u) = −(g 3 · u) need not be zero and the deformation can bring about a large increase of the MBR-effect keeping... simulation and the testing of the particular case of lamellar gratings with a rectangular groove profile The gratings are investigated in two different experimental geometries: sagittal diffraction and meridional diffraction The definitions of a sagittal and of a meridional grating are given in Figs 28.1 and 28.3 28 Volume Modulated Diffraction X-Ray Optics 473 Fig 28.1 Definitions for a sagittal grating 28.2.1... Bragg–Fresnel Gratings In sagittal geometry, the grating grooves are aligned parallel to the optical plane, as defined by the ingoing and outgoing X-ray beams In this case diffraction takes place perpendicular to the optical plane In meridional geometry, the grating grooves are aligned perpendicular to the optical plane and diffraction takes place in the optical plane Surface Sagittal Grating The simplest... microphotonics, since they enable transformation of electrical, optical or acoustic signals into X-ray beam modulation [4] In this chapter we report on systematic theoretical and experimental investigations of volume (Bragg–Fresnel) gratings: (a) static, made with etching technology including a metallic structure on the surface of multilayers and crystals, and (b) dynamic, produced by surface and volume... reflector, and the depth of profile must be equal to the X-ray penetration depth in the multilayer, the so-called extinction depth In this case ordinary diffraction on a thin grating takes place, yielding many diffraction orders simultaneously in the detector (2θ) scan mode similar to Fig 28.2 Combining (28.1) for the optimal depth of profile with the average value of the multilayer refractive index δ= δ1... resolution of some scattering devices installed usually at steady state sources [1, 2] An increase of the luminosity is carried out by focusing in real space, while a higher resolution can be achieved by focusing in momentum space and rather small effective mosaicity of the BPCs However, in the case of TOF scattering devices, the BPC elements practically have not been used and with respect to the TOF... sagittal and meridional diffraction gratings are discussed Dynamic diffraction gratings are produced by propagating a surface acoustic wave along a piezoelectric crystal Experimental data are compared with the theoretical calculations 28.1 Introduction Experimental and theoretical data point to a new application of nanometer radiation in diagnostics, transmission and processing of information: X-ray electronics . reciprocal space [5,6]. The points P 0,1 and P 0,2 are the origins of the impinging vectors, directed into the origin, O,ofthe reciprocal space, points P h,1 and P h,2 are the origins of diffracted vectors. Bragg diffraction (β =0).ThepointsP h,1,β and P h,2,β are the origins of diffracted vectors for the inclined diffraction. The indices 1 and 2 represent the limiting beams within the diffraction region,. to increase luminosity and angular/energy resolution of some scattering devices installed usually at steady state sources [1, 2]. An increase of the luminosity is car- ried out by focusing in