A raytracing code for zone plates

Transcription

1 A raytracing code for zone plates Alexei Erko *, Franz Schaefers, Nikolay Artemiev a BESSY GmbH, Albert-Einstein-Str.15, Berlin, Germany a Laboratoire d'optique Appliquee ENSTA Ecole Polytechnique Chemin de la Huniere F Palaiseau cedex, France. ABSTRACT A raytracing code for zone plates incorporated in the BESSY raytracing program RAY is described. This option allows one to calculate intensity distributions in a focal plane of circular or linear zone plates considering diffraction limited resolution. Zone plate material properties are also taking into account using optical constants data tables. The complete code is available as PC-Windows version. Keywords: raytracing, x-ray optics, zone plates 1. INTRODUCTION Raytracing is an indispensable tool for the design of optical systems for synchrotron radiation sources, and various programs have been developed during the last decades [1-2]. By using a general purpose raytrace program, it is possible to obtain detailed information about the overall performance of the beamline optical system. Usually, the optical elements that are treated by a raytrace program are slits and screens, mirrors and gratings, Bragg crystals and multilayers. Modifications of the wave front of light produced by these optical elements are described in the frame of geometrical optics and analytical equations rather well. However, the weak point of the ray-optics is micro-focusing with diffraction limited imaging. In this paper a raytracing code for zone plates incorporated into the program RAY [3] which is extensively used for beamline performance calculations at BESSY is described. The mathematical model allows one to follow a chromatic blurring of the focal spot as well as the smearing of the focus due to an unevenness of the incident wave front (described by rays). Another advantage of the model is that it gives an intensity distribution, including auxiliary maxima and background radiation in the focal position. 2. DIFFRACTION ON A ZONE PLATE Using the Fresnel Kirchhoff diffraction integral one may obtain the form of the diffraction pattern [4]. Referring to figure 1, this gives the complex amplitude A(x',y',z') as exp[ ik( r + s)] A( x', y', z' ) = ( i / 2λ) A( x, y, z) dxdydz ( r + s) (1) where A(x,y,z) and A(x',y',z') are the amplitudes in the object and image planes, k is the wave number (k = 2π / λ), r and s are the propagation vectors, dx x dy x dz is an area element of the aperture, and the integral is carried out over the whole object aperture. In practice in order to get the intensity distribution behind a zone plate one must take the double integral Eq. (1) for each point A(x',y',z') on the screen and that would be just the distribution for a point source at A 0 (x,y,z). In turn, if the source is not just a point, this procedure must be repeated for each point of the source. This method can give a good result but the time needed for such calculations is awfully long. * erko@bessy.de ; phone , fax

2 A 0 (x,y,z) Y Object plane r Zone plate A 0 (x',y',z') Y' Focal plane Z X s R 1 X' Figure 1. The coordinate system used in the calculation of the diffraction pattern of a circular aperture On the other side the intensity distribution in the focal plane of a zone plate for the point source can be calculated analytically [4]. Supposing a non-coherent irradiation, that is valid for all existing x-ray sources, one can reconstruct the image as a superposition of the images of the point sources, distributed in the object plane. Each point source will be transferred through a zone plate with a resolution defined by the zone plate aperture, the so-called diffraction limited resolution. One can use an approximated analytical solution of the Eq. 1 for the point source to define the position of the image for each object point. At this step one can replace a wave front presentation with a ray-presentation. In the case of ray transmittance each point of the source (or preceding element on an optical arrangement) produces a ray with defined parameters: spatial coordinates and energy. After interaction with a zone plate the ray angular coordinates are changed according to the defined probability. The probability distribution can be calculated using analytical formulas, represented in the following parts of the paper. 3. INTEGRAL DIFFRACTION EFFICIENCIES Consider a zone plate in which odd zones are transparent and even zones are covered by a material with a rectangular form of grooves (phase-amplitude zone plate) [5]. The phase shift and attenuation of the amplitude produced by the even zones are given by φ = 2πδt /λ; (2a) R 2 β = 2πβt /λ χ φ; (2b) where χ = β/δ; β and δ are the absorption and refraction indexes respectively. Combining the amplitudes of the waves, transmitted through the transparent and covered zone, one can calculate the diffraction efficiency in the m th order [6]: Ε m = I m / I in = [1/(π 2 m 2 )][1 + exp(-2χ φ) - 2cos( φ) exp(-χ φ)]; (3) for the odd orders m = ±1; ±3; ±5;... Even orders do not exist. The I in is the incoming wave integral intensity and I m the integral intensity in the m-th order of diffraction. The maximum of the function I m /I in can be found taking the derivative of the Eq. (3): sin( φ opt ) + χ [(cos( φ opt )exp(-χ φ opt )] = 0; (4) 2

3 where the value of φ opt is the optimum phase shift in the zone plate material. This equation can be solved numerically and an optimum phase shift can be approximated by the sum of two exponential curves: The optimum thickness of the zone plate can be calculated using Eq.(4) as: φ opt ~ exp(-χ/0.55) exp(-χ/2.56) (5) t opt = φ opt λ / (2πδ); (6) Analogous to Eq.(3), the efficiencies of the zero order diffraction E 0 and the part absorbed in the material of a zone plate E abs can be calculated using expressions: and E 0 = 0.25 [1 + exp(-2χ φ) + 2cos( φ) exp(-χ φ)]; (7) E abs = 0.5 [1 - exp(-2χ φ)]. (8) Depending of the value of parameter χ all materials can be called as a "phase" material or an "amplitude" one. Usually effective phase-shifting materials could be characterized by the value of χ < DIFFRACTION LIMITED RESOLUTION Although the above analysis indicates the positions of the foci and the diffraction efficiency of a zone plate, it does not give the form of the diffraction maxima, which can be obtained using the Fresnel - Kirchhoff diffraction integral. The solution of the two-dimensional Fresnel - Kirchhoff diffraction integral can be found in a form of the Bessel or sin(x)/(x) function of first order with an argument [4]: ν m = r N k (r'/f m ); (9) where r' the radial distance between the optical axis and an arbitrary point in the image plane. The radial intensity distribution at the focus of a circular zone plate is well described by an Airy pattern analogous to a perfect thin lens: I m '(ν m ) ~ [2J 1 (ν m )/ν m ] 2 (10) The solution of the one-dimensional diffraction integral can be found in a form of the sinus function of first order with the argument [4]: ν m = r N k (x'/f m ); (9') where x or y the linear distance between the optical axis and an arbitrary point in the image plane. The linear intensity distribution at the focus of a linear zone plate is well described by a pattern analogous to a perfect thin lens: I m ' (ν m ) ~ [2sin(ν m )/ν m ] 2 (10') There is, however, a significant difference in the intensity distribution at the focus of a zone plate compared to the focus of a perfect refractive lens, which is not shown up by Eq. (10) and (10'). For a zone plate there is always a low-intensity background caused by zero order light and high diffraction orders. 3

4 5. RAY-TRACING MODEL 5.1 Ray propagation probabilities Tracing the zone plate the program first solves the standard raytracing task of the ray surviving probability. The ray, which falls into the aperture of the zone plate, is considered to be partially absorbed by the zone plate material. Together with the rays diffracted to negative (m < 0) and high (m > 5) orders this ray is considered as "lost" because its intensity at the first order focus is infinitesimally weak. So, the ray must be thrown away with the probability: for all m < 0 and m > 5. E lost = E abs + m = 1 m = E m + m = m = 5 E m (11) If the ray is still considered as a survived one, then its destiny has also two ways: 1). A ray is not diffracted (zero order) and its angle ξ to the optical axis remains unchanged with probability E 0. 2). The remaining probabilities for the ray to be diffracted into first, third and fifth positive orders according to Eq.(3) are: E 1, E 3, E Diffraction limited resolution For the diffracted ray the probability to be deflected by the diffraction angles δϕ, δψ and δξ to the X, Y and Z axis respectively, is defined by I m ' (ν m ) and calculated by Eq.(10). The definition of the RAY-coordinate system is shown in figure 2. Y Y' Zone plate r δψ δξ δϕ Focal plane Z X X' Circle of the angular radius δξ F Figure 2. The reference frame of the program, the angles of diffraction of a ray and the circle of the angular radius δξ at the position of the first order maximum of a zone plate The ray is deflected randomly by the angle of 0 <= δξ <= δξ max. According to figure 2 the values of the diffraction angle δϕ and δψ are defined in small angle approximation by the expression: (δξ) 2 = (δψ) 2 + (δϕ) 2 (12) For each ray the values δψ and δϕ are randomly selected within the angular range of 0 <= δψ, δϕ <= δξ max. Then the value of δξ calculated according to Eq.12 passes through the probability generator in accordance to Eq. 10. Only those angles which pass this filter and have a Bessel-function-like probability distribution are used in the further procedure. Respecting the real intersection point of the ray with the zone plate, the real angles of its deflection are calculated as follows: ϕ = δϕ - y zp / F m ; and ψ = δψ - x zp / F m ; (13) where y zp and x zp are the coordinates, where the ray hits the zone plate. 4

5 6. TEST OF THE COMPUTER CODE Diffraction limited resolution has been checked using a point source and a divergent beam. An image aperture was placed in a distance calculated by the thin-lens formula. The results, obtained with the RAY program are shown in figure 3 for a linear (left) and circular (right) zone plate, respectively. The zone plate parameters are listed in the table. The same zone plate was calculated using a diffraction program used for x-ray holograms [7]. The results were found to be identical Figure 3. Diffraction limited resolution for a linear (left) and a circular (right) zone plate The calculations are performed for a monochromatic point source at the energy of 100 ev, located on the optical axes of the zone plate. The zone plate material is a carbon with an optimum thickness of 171 nm. The two-dimensional intensity profile in the focal plane for the zone plates, shown in figure 3, was obtained by raytracing of 10 8 rays. Table 1. Parameters of the zone plates used for the diffraction limited resolution calculations. Zone plate type First order focal length F 1 Aperture R1 / R2 Linear 1000 mm 1 mm x 1 mm 2000 mm / 2000 mm Circular 1000 mm 1 mm 2000 mm / 2000 mm The integral efficiency of zone plates made with different materials is shown in figure 4. In this case, to demonstrate the RAY performance in different energy ranges, a parallel beam with photon energy of 8500 ev have been chosen. The linear zone plate focuses in horizontal (X) direction. The outer zone width is equal to 0.3. At the focal distance of 1000 mm the corresponding lens aperture is 500. Three materials were chosen: gold with an optimum thickness of nm, carbon with optimum thickness of 11.1 and finally a non-transparent material for the simulation of an amplitude zone plate. Zone plate s parameters are shown in table 2. Table 2. Parameters of zone plates used for the integral efficiency calculations. Zone plate type: linear; first order focal length F 1 =1000 mm; aperture: 500 x 500 Zone plate material / thickness Flux density gain, rays/ 2 First order integral efficiency % Carbon / Gold / Absorbing material RAY presents the results in form of a 100 x 100 pixel array of a multi-channel analyzer. The size of each channel depends on the aperture, covered with scattered rays. This aperture can be defined by a slit placed in the image plane, as was done for calculations shown in figure 4. Figure 4a represents the number of rays, collected in channels in X direction, integrated vertically, for three zone plate materials (bottom scale). The slit is placed in the focal plane, the 5

6 corresponding width of the channel is Shown also is the direct beam, measured in the plane of the optical element (upper scale). Zone plate aperture is 500 and the corresponding width of each channel in horizontal (X) direction is The area covered by each curve in figure 4a corresponds to the integral diffraction efficiency. In the table 1 these areas are normalized to the incident number of rays of In figure 4b the corresponding flux density in rays / 2. is shown. A flux density gain in rays / 2 can be obtained using the data in figure 4b. Namber of rays / channel (x10 6 ) Gold 1.63 Absorbing zone plate Carbon 11.1 Direct beam on ZP (top scale) Flux density (rays/ 2 ) Gold 1.63 Absorbing zone plate Carbon 11.1 Direct beam on ZP (top scale) Figure 4. Number of rays per channel (a) and flux density (b) in the focal plane of a zone plate made with different materials (bottom scale). Shown also is the direct beam flux (a) and flux density (b) on a zone plate (top scale). The total number of rays is Image transfer has been examined using an object, consisting of four squire frames with 1 x 1 in size (figure 5a). The object is located at a distance of 2000 mm from the zone plate. The zone plate focal length is 1000 mm and the minimum zone width is a) b) Figure 5. An object consisting of four squire frames with 1 x 1 size (a) and its raytracing image obtained with a gold zone plate at a distance of 2000 mm (b). 6

7 7. ANGULAR SPECTRA ANALYSIS Conventional zone plates as well as reflection or Bragg-Fresnel zone plates can be described as a superposition of diffraction gratings with different periods (spatial frequencies). This model is very common in the mathematical analysis of holograms and diffraction images [8]. An illustration of this model is shown in figure 6. A zone plate irradiated with a parallel beam produces diffraction limited images of different orders F 1 F 3 F 5 etc. at distances of F 1, F 2, F 3 from the optical element. Each grating, located on the radius r i from the optical axis has a local period of d and directs the diffraction orders to point F n. In figure 6 three periods d 1 d 2 and d 3 are shown, but in reality we have a continuous change of the period within the zone plate radius. d 1 d 2 α 1 r 3 d 3 α 2 Slit r 2 α 3 r 1 F 5 F 3 F 1 Figure 6. Representation of a zone plate as a superposition of diffraction gratings with variable period Using a raytracing presentation, one can define each direction of diffraction as a ray, deflected by the optical element toward to the focal points of the diffraction orders. Therefore the angular spectra of these rays will correspond to the angular spectra of diffracted beams in diffraction theory. According to general principles of diffraction optics this spectra can be used for the calculation of the focal spot size by Fourier transform of the angular distribution. In figure 7 an angular spectrum calculated with RAY for a zone plate with parameters as listed in table 2 is shown. Relative angular flux density (rays/mrad) First order spectra in image plane (slit limited) 1 st order Diffraction on a ZP 3 d order 5 th order Angle (mrad) Figure 7 Angular spectra calculated just behind a ZP (solid line) and in the focal plane limited by a slit of 10 (dashed line) The raytrace program RAY was used for the development of a practical method for measurements of zone plate parameters: efficiency and resolution. In figure 7 an angular spectra of the rays is shown, diffracted on a zone plate and calculated just behind an optical element. One can evidently see different angular regions, responsible for focusing in 7

8 different diffraction orders: +1; +3; +5. These spectra can be used for a direct measurement of zone plate resolution and efficiency. With the help of a Fast Fourier Transform program the spatial frequency spectrum can be converted into an intensity distribution in the focal plane. Our calculations show identical results obtained by Fourier transformation of the spectra and by raytracing. The parameters measured by this method are free from experimental errors, caused by a source size. A parallel beam spatially filtered through a pinhole is used in this experiment. Measurements of zone plate spatial frequency spectra where done at the BESSY beamline KMC-2 using highly a collimated x-ray beam at an energy of 8500 ev. A linear meridional (lines perpendicular to beam direction) Bragg- Fresnel reflection zone plate made on the surface of a Si(111) crystal has been used. The zone plane parameters are listed in table 3. The optimum thickness of the phase-shifting material was calculated taking into account the Bragg angle for Si 111 reflection at 8500 ev [9]. In the case of gold at 8500 ev the optimum thickness was found to be 190 nm. Table 3. Parameters of the Bragg-Fresnel zone plate used for the measurements t= 0.5 t opt sin θ B (14) Energy range 8500 ev Bragg angle for Si deg Minimum zone width 1.47 Optimal gold thickness 190 nm (1630 nm at normal incidence) Lens aperture along the crystal surface perpendicular beam direction 740 (170 projection on the beam direction) 600 The angular spectra measurements were done using the experimental arrangement shown in figure 8. The monochromatic beam with λ/λ ~ 5000 was collimated by a toroidal mirror and filtered by a pinhole of 100 in diameter. An angular spectrum of the zone plate was measured at a distance of ~ 1000 mm with one-crystal Si 111 Bragg spectrometer in non-dispersive geometry. The incident beam divergence after the pinhole, measured with a crystal spectrometer, was of the order of 4.5 arcsec. Due to the small beam size the sample was scanned along the beam by a translation stage. The spectra from the different parts of the zone plate have been measured and composed on the same graph to build a complete angular spectrum of the Bragg-Fresnel lens. The reflection on the free crystal surface also has been measured to produce a reference spectrum of the direct beam. Crystal analyser (Si111) Θ SR - beam Pinhole 100 Detector 2 Θ Bragg-Fresnel zone plate Figure 8. Experimental setup for the measurements of the angular diffraction spectra The diffraction spectrum of the lens is shown in figure 9. All the measured spectra overlap on one graph and are normalized to the reflection from the free crystal surface. On the same figure 9 the calculated spectrum obtained by 8

9 raytracing the zone plate with an input beam divergence of 4.5 arcsec is also shown. A minimum zone width for the best fit was found to be 0.34 and a gold thickness of 1350 nm. These values are in good correspondence to the projections on the beam direction of the outer zone with and gold layer thickness of the BFL. Normalized angular flux density Free surface reflection Diffraction on a zone structure raytracing Angle θ (arcsec) Normalized flux FWHM = 0.3 Raytracing Experiment Figure 9. The measured angular diffraction spectra in comparison with raytracing calculations (left) and a reconstructed flux distribution in the focal plane (right) In figure 9 (right) is shown a reconstructed flux distribution in the focal plane obtained by Fourier transformation of the calculated raytracing spectra and the shape of the experimental spectra. Both curves are in very good agreement and have the same Full Width Half Maximum (FWHM) of 0.3. To prove the resolution in the meridional direction the performance of a meridional Bragg-Fresnel grating has been measured. A 2 period, 20 nm thick nickel grating was evaporated on the surface of an asymmetrically cut Si (111) crystal with asymmetry angle of 13 deg. A diffraction efficiency of 1% was obtained at 8.45 kev at a grazing angle of 0.48 deg, corresponding to an effective period for a transmission grating at normal incidence of 16.8 nm or, for a zone plate, an outer zone width of 8.4 nm. 8. CONCLUSIONS For the development of a raytracing procedure for such an optical element like a Fresnel zone plate one has to combine two opposite descriptions of the nature of light, which usually exclude each other: wave theory and geometrical optics. As a result, a correct intensity distribution is calculated in a close vicinity of the positive order foci. Material properties of a zone plate are taken into account using a transformation of integral diffraction efficiencies to a probability of ray deflection or absorption. The same principle is used for the calculation of a diffraction limited resolution. Using the RAY cod a zone plate can be calculate also as an imaging device. The main advantage of this model is that we avoid double integration over the area of a zone plate as well as over the area of its focal plane. Analysis of the angular distribution of diffracted rays leads to the possibility to fit experimental spectra and find zone plate characteristics, such as the outer zone width and the thickness of a phase-shift layer. These characteristics can be obtained almost independently from the experimental conditions, taking into account only beam divergence. Using this method a considerable resolution improvement for the meridional Bragg-Fresnel lens in comparison with a normalincidence zone plate with the same outer zone width could be shown. The resolution enhancement of 4.3 times was measured experimentally. The measured efficiency of the Bragg Fresnel zone plate with a thickness of 190 nm corresponds to the efficiency of a transmission zone plate with a thickness of 1350 nm. ACKNOWLEDGEMENTS. This work has been done in the frame for the COST 7th frame program X-ray and neutron optics during the Short Scientific Mission to BESSY GmbH, Berlin. 9

10 REFERENCES 1 C. Welnak, G.J.Chen, F. Cerrina Nucl. Inst. and Meth. A347, (1994), T. Yamada, N. Kawada, M. Doi, T. Shoji, N. Tsuruoka, H. Iwasaki, J. Synchrotron Rad. 8, (2001), F. Schaefers RAY - the BESSY raytrace program to calculate synchrotron radiation beamlines, Technischer Bericht, BESSY TB 202, (1996), M. Born and E Wolf, Principles of Optics, 5th ed., Pergamon Press, Elmsford, N.Y. (1975) Erko, V.V.Aristov, B.Vidal, Diffraction X-Ray Optics IOP Publishing, Bristol, (1996) J. Kirz, J. Opt Soc Am., 64, (1974), Firsov, A. Svintsov, S.I. Zaitsev, A. Erko, V. Aristov The first synthetic X-ray hologram: results, Optics Communications, 202, (2002), 55-59, 8 Collier R. J., Burckrardt Ch. B., Lin L. H., Optical Holography, Academic Press, New York, London, (1971) 9 Erko, F. Schaefers, A. Firsov, W. B. Peatman, W. Eberhardt, R. Signorato The BESSY X-Ray Microfocus Beamline Project Spectrochimica Acta Part B: Atomic Spectroscopy (2004) to be published. 10