Stigmatic X-ray imaging using a single spherical Laue crystal M. Sanchez Del Rio, D. Bianchi, T. A. Pikuz, A. Ya Faenov, S. A. Jr Pikuz, L. Delgado-Aparicio, N. Pablant, M. Bitter, K. Hill To cite this version: M. Sanchez Del Rio, D. Bianchi, T. A. Pikuz, A. Ya Faenov, S. A. Jr Pikuz, et al.. Stigmatic X-ray imaging using a single spherical Laue crystal. 11th International Conference on Synchrotron Radiation Instrumentation (SRI), Jul 2012, Lyon, France. 425, 4 p., 2013, Journal of Physics Conference Series. <10.1088/1742-6596/425/19/192021>. <hal-01572786> HAL Id: hal-01572786 https://hal.archives-ouvertes.fr/hal-01572786 Submitted on 8 Aug 2017 HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
Stigmatic X-ray imaging using a single spherical Laue crystal M.Sanchez del Rio [1], D. Bianchi [2], T.A. Pikuz [3,4], A. Ya Faenov [3,4], S.A. Pikuz, Jr [4], L. Delgado-Aparicio [5], N. Pablant [5], M. Bitter [5] and K. Hill [5] [1] European Synchrotron Radiation Facility, BP 220, 38043-Grenoble Cedex, France. [2] Technische Universitaet Wien, 1040 Vienna, Austria. [3] Quantum Beam Science Directorate, Japan Atomic Energy Agency, Kizugawa, Kyoto 619-0215, Japan. [4] Joint Institute for High Temperatures, Russian Academy of Sciences, Moscow 125412, Russia. [5] Princeton Plasma Physics Laboratory, Princeton, New Jersey 08543, USA srio@esrf.eu Abstract. We propose a crystal configuration using a single Laue spherical crystal for imaging applications. A crystal in Laue geometry set to focus a divergent beam in the meridional (diffraction) plane, but does not focus in the sagittal plane. A transmission object placed in the beam is imaged with different horizontal and vertical aspect ratio, but it is possible to find a configuration with similar aspect ratio. This system is studied using ray tracing, which permit to reproduce preliminary experimental data [1]. The concept of a stigmatic focusing by a single optical element may have applications in imaging, like in transmission microscopy or for hard X-ray backlighting and self-imaging in high energy density plasma experiments. Inertial confinement fusion experiments and large tokamak projects such as ITER may benefit from this optical configuration. 1. Introduction High-energy x-ray imaging methods suitable for recording and analyzing plasma compression with high spatial resolution are sorely needed for plasma diagnostics. To determine the plasma conditions in the compressed region requires using high spatial resolution imaging instruments with simultaneous high spectral resolution. Crystals provide a compromise of performances and a simplicity of design, in contrast to multilayers which have poor spectral resolution and small solid angle acceptance; and Fresnel zone plates that suffer from higher order contributions and chromatic aberrations. We discuss an imaging method using spherically bent crystals that image objects in transmission. 2. Point to point focusing by Bragg and Laue crystals When using a single spherical crystal in symmetric Bragg (reflection) configuration, the position of the foci (meridional and sagittal) is given in paraxial approximation by the equations: 1 1 2 1 1 2sinθB + = ; + = (1) p qt R sinθ B p qs R Where p is the source-crystal distance, R is the crystal radius, θ B is the incident grazing angle (which in this configuration is equivalent to the Bragg angle) and q t and q s are the distances from the crystal to the tangential and sagittal foci, respectively. A stigmatic focus (q s = q t ) is only possible in normal Published under licence by Ltd 1
incidence (sin θ B =1), or by replacing the sphere by a toroid, in this case the crystal present different radii to in meridional and sagittal planes (R t R s ). If q t =p =Rsin θ B we are in the so-called Rowland configuration, which presents a magnification one (M t = q t /p) for the Bragg symmetric case (the diffraction planes are parallel to the crystal surface). The Rowland configuration is particularly interesting because it minimizes the diffracted energy bandwidth or, in other words, optimizes the energy resolution. It is noticeable that the Bragg symmetric crystals is a non-dispersive system, therefore it is possible to make a point-to-point focusing independently of the photon energy. The crystal is concave (R>0) as p and q t have the same sign (they lie on the same side of the crystal). Asymmetric crystals are dispersive, and produce two distinct foci, one for monochromatic radiation and another for polychromatic radiation [2,3]. In symmetric Bragg reflection, both monochromatic and polychromatic foci at found at the same point. For crystals in Laue (transmission) geometry the situation is more complex. We consider the case of symmetric Laue, where the Bragg planes are perpendicular to the crystal surface. Thus, the Bragg angle is the complementary of the grazing angle, and Eq. (1) can be written as: 1 1 2 + = (2) p qt R cosθ B If we consider that p>0 and q t <0 (we want a real image on the other side of the crystal), we find that the crystal is concave (R>0) for p < q t and convex (R<0) for p > q t. We are interested in the last case, and consider, for example, R=-30 cm, p=200 cm, a silicon 400 crystal reflecting photons of 22160 ev, thus θ B =11.89 deg. Eq. (2) gives a focal position of q t =-13.67 cm. Remarkably, the Laue crystal is always a dispersive system (also for symmetric Laue) thus the evolution of the beam after diffraction, and in consequence the focal position and focal dimension, depend on the energy bandwidth after diffraction. In fact, Eq (2) is related to the "crossing rays" focusing, or polychromatic focusing. Monochromatic focusing may not exist, or may exist in another position. Because of the rays dispersion, a point-to-point focusing is not possible for polychromatic focusing, and there is always a limit in the image size. To check that, ray tracing experiments have been done using the SHADOW package [4] for the parameters defined before, using a point source with a divergence 0.005 rad illuminating a crystal of 1 cm diameter. The beam evolution in the diffracting plane close to the focal position q t is shown in Fig. 1. The results confirm that: i) Focusing is produced for polychromatic beam, and a monochromatic beam is not focused, and ii) The polychromatic beam size at q t is not zero. Its value cannot be smaller than the value for a monochromatic component. Figure 1. a) Beam evolution close to the focal position (q-q t =0) for the beam diffracted by a Si 400 spherical crystal (R=-30cm). Solid line: incident beam is polychromatic, dotted line: incident beam is monochromatic (E=22160 ev) and dashed line: beam evolution without crystal. b) Crystal setup [1] in the meridional plane, including object and image. 2
3. Imaging objects using a spherical Laue crystal We discus here the image produced by an absorption object placed between the source and the crystal. The object produces an intensity modulation of the beam. The beam is recorded after diffraction by an area detector (e.g., photographic film) retrieving an image of the object. The interest for microscopy applications is to produce images with high magnification and constant aspect-ration (horizontal vs. vertical). As mentioned before, the evolution of the beam in horizontal and vertical is different using Bragg spherical crystals (astigmatism), thus tangential (M t ) and sagittal (Ms) magnifications are different. In previous works [5] we proposed a microscopy configuration using a crystal in Bragg configuration. The adequate positioning of the object and detector with respect to the crystal guarantees a constant aspect ratio with variable magnification. In this paragraph we apply similar concepts but using Laue crystals. The crystal can be considered a lens-type magnifier of focal distance F 1 1 1 + = (3) p qt Ft For p=200 cm and q t =-13,67cm the focal distance F t =12.795 cm. If an object is placed at a distance a upstream from the crystal, an image can be recorded at a distance b downstream from the crystal (Fig. 1b). Using the lens equation: 1 1 1 + = (4) a b F t the detector will show an image of the object magnified by a factor M t =b/a. We first study two magnifications M t =1 and M t =2. The corresponding a and b distances are reported in Table 1. In the sagittal direction the beam evolves like if there is no crystal, so the magnifications is given by the geometrical ratio (see Fig. 1b): p + b M s = (5) p a Obviously, M s =1 is only obtained in the unrealistic case that a=b=0, i.e., the object, image and crystal are at the same point. See Table 1 for the values of the two examples Table 1. Values of the analyzed cases CASE F t [cm] M t p [cm] q t [cm] a [cm] b [cm] M s 1 12.795 1 200-13.67 2F t =25.591 2F t =25.591 1.29 2 12.795 2 200-13.67 3F t /2=19.193 3F t =38.386 1.32 3 12.795 1.3 200-13.67 22.645 19.439 1.3 4 9.63 2 56-11.63 14.327 29.371 2 Ray tracing results starting from a point source and using a grid that covers the full crystal (diameter 1 cm) are performed for cases 1-3 (Table 1). The energy bandwidth is large (~ 3 kev, Fig. 2a) and is modulated because as a function of the variation in divergence induced by the grid pattern. Fig. 2b shows the image at the focal position, confirming a good focusing in the horizontal (tangential) direction, and no focusing in vertical. Fig 3a shows the grid pattern and its image after diffraction for case 1. In meridional (horizontal) plane a magnification one is found, as expected. Fig 3b (case 2) shows much smaller demagnification in the sagittal than in the meridional planes. In general, for a given pair of a and b that verify Eq. 4, the resulting M t and M s are different. However, for each value of F t, there is a pair of a and b values that give the same horizontal and vertical magnifications. This is shown in Fig. 3c (case 3). In the last case (case 4, Fig. 3d), the focal distance has been changed to obtain a constant magnification M t =M s =2 demonstrating that this setup can be adapted for given specifications. These simulations confirm that the Laue spherical crystal can be used as a stigmatic imager for objects in transmission, thus explaining the experimental results in [1] 3
Figure 2. a) Energy bandwidth (FWHM is 3000 ev) produced by a bent (R=-30 cm) Si 400 crystal at E=22160 ev. b) Image at focal point (q t ). Horizontal corresponds to diffraction plane. Units are cm a) b) c) d) Figure 3. Ray tracing results showing the grid (object) and the image of the grid after diffraction for the different cases in table 1 a) case 1 (M t =1, Ms =1.29), b) case 2 (M t =2, M s =1.32), c) case 3 (M t = M s =1.3), and d) case 4 (M t = M s =2). Units are cm. References [1] Faenov, A.Y., T.A. Pikuz, V. Avrutin, N. Izyumskaya, L. Shabelnikov, E. Shulakov, and G.A. Kyrala, Review of Scientific Instruments, 2003. 74(3): p. 2224-2227. [2] Chukhovskii, F.N. and M. Krisch, Journal of Applied Crystallography, 1992. 25(2): p. 211-213. [3] Sanchez del Rio, M., SPIE Proceedings, 1998. 3448: p. 230-245. [4] Sanchez del Rio, M., N. Canestrari, F. Jiang, and F. Cerrina, Journal of Synchrotron Radiation, 2011. 18(5): p. 708-716. [5] Sanchez del Rio, M., L. Alianelli, T.A. Pikuz, and A.Y. Faenov, Review of Scientific Instruments, 2001. 72(8): p. 3291-3303. 4