APPLICATION OF DIFFRACTIVE OPTICS TO THE RAY-TRACING CODE RAY FOR FRESNEL ZONE PLATES

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APPLICATION OF DIFFRACTIVE OPTICS TO THE RAY-TRACING CODE RAY FOR FRESNEL ZONE PLATES
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  E:\E_\Articles\ICXOM\ZP_Ray\Text+buld.doc 1 APPLICATION OF DIFFRACTIVE OPTICS TO THE RAY-TRACING CODE  RAY   FOR FRESNEL ZONE PLATES.  N. Artemiev a , F. Schäfers  b  and A. Erko  b   a    Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221  Praha 8, Czech Republic. b  BESSY GmbH, Albert-Einstein-Str.15, 12489 Berlin, 1. INTRODUCTION. Ray-tracing is an indispensable tool for the designing optical systems for synchrotron radiation sources, and various programs have been developed during the last decades [1-2]. By using a general purpose ray-trace 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 ray-trace 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 ray-tracing code for zone plate 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 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. AXIS DEFINITION Using the Fresnel – Kirchhoff diffraction integral one may obtain the forms of diffraction  patterns [4]. Referring to Figure 1, this gives the complex disturbance at  P   as ∫∫ −⋅+⋅−= dS  sr  sr ik iA P U  )cos(cos)](exp[)2/()(  β α λ   (1) where  A  is the amplitude at unit radius from  P  0 , k   is the wave number ( k = 2 π / λ ), α , β  are the angles between r  , s  and the normal to the aperture, dS   is an element of area of the aperture, and the integral is carried out over the whole aperture.  Figure 1. The coordinate system used in the calculation of the diffraction pattern of a circular aperture. In the practice in order to get the intensity distribution behind a zone plate one must take the double integral Eq. (1) for each point P(x',y',z') on the screen and that would be just the distribution for a point source at P 0 (x,y,z). In its turn, if the source is not just a point, this  E:\E_\Articles\ICXOM\ZP_Ray\Text+buld.doc 2  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. However, for the purposes of ray-tracing we do not need to do it or even can not do that  because the ray-tracing method does not give us a correct description of the incident wave front on a zone plate. Using the conditions in which this is to be applied, the diffraction angles are small, and the diffracting aperture dimensions (the zone widths) are large compared to the wavelength but small compared to the arms of the optical system. A further simplification appears when calculating only a diffraction pattern from an axial point source. Each point of the source is considered as a point source, which creates exactly the same diffraction pattern as a point source laying on the principal optical axis. Diffraction patterns from different point sources are shifted along the image plane in according with the actual position of the source point and corresponding magnification (See Fig. 2).  Figure 2. Axial and off-axial point sources and corresponding diffraction patterns, which are identical to each other but separated in according with distance between the sources and magnification of the optical scheme. The next step towards the simplification of the calculations or in other words, decreasing the calculation time, is taken when discussing the diffraction of a “ray” on a zone plate. Obviously, we cannot speak about diffraction of a ray. It would be more correctly to say, that we calculate the diffraction of a plane incident wave, the wave vector of which is represented  by the ray. 3. INTEGRAL DIFFRACTION EFFICIENCIES Consider a zone plate in which odd zones are transparent and even zones are covered by a material with the rectangular form of grooves (phase-amplitude zone plate). The phase shift and attenuation of amplitude produced by the even zones are given by [5] ∆φ = 2πδ t /  λ ; (2a) ∆β = 2πβ t /  λ − χ∆φ ; (2b) where χ = β/α ; and δ   are the absorption and refraction indexes accordingly. Combining the amplitudes of the waves, transmitted through the transparent and covered zone, one can calculate the diffraction efficiency in the m-th order: Ε m  = I  m  / I  in  = [1/(  π 2 m 2  )]*[1 + exp(-2 χ∆φ  ) - 2cos(  ∆φ  ) - exp(- χ∆φ  )]; (3) for the of orders number m = ±1; ±3; ±5;.. ±(2j-1) . and  j = 0; ±1; ±2; ... ±J  . 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  using differential of the Eq. (3).  E:\E_\Articles\ICXOM\ZP_Ray\Text+buld.doc 3 sin(  ∆φ opt   ) + χ  [(cos(  ∆φ opt   ) exp(- χ∆φ opt   )] = 0; (4) where the value of ∆φ opt   is the optimal phase shift in zone plate material. This equation can be solved numerically and an optimal phase shift can be fitted by the sum of two exponential curves: ∆φ opt   = 1.69 + 0.71 exp(- χ /0.55) + 0.74 exp(- χ /2.56) (5) The optimal thickness of the zone plate can be calculated using the expression Eq.(4) as: t  opt   = ∆φ opt    λ  / (  2πδ) ; (6) Analogous to Eq.(6), the efficiencies of the zero order diffraction E 0 and relative part of an intensity, absorbed in the material of a zone plate E abs  can be calculated using expressions:  E  0  = 0.25 [1 + exp(-2 χ∆φ  ) + 2cos(  ∆φ  ) exp(- χ∆φ  )]; (7) and  E  abs  = 0.5 [1 - exp(-2 χ∆φ  )]. (8) 3. DIFFRACTION LIMITED RESOLUTION Although the above analysis indicates the positions and diffraction efficiency of the foci of a zone plate, it does not give the form of diffraction maxima, which can be obtained using the Fresnel Kirchhoff diffraction integral. The solution of the Fresnel Kirchhoff diffraction integral can be found in a form of the Bessel function of first order with an argument [4]: ν m  = rN k (r´/F  m  ); (9) where r´   the radial distance between the optical axis and an arbitrary point in the image  plane. In our case r´ = x´  . . If the number of zones is large enough (N > 100) the radial intensity distribution at the focus of a 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) 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 ). For a zone plate there is always a low-intensity background caused by the zero order and high diffraction orders. 4. RAY-TRACING MODEL. 4.1 Ray propagation probabilities. Tracing the zone plate the program first solves the standard ray-tracing task of the ray surviving probability. The ray, which falls into the aperture of the zone plate is considered to be absorbed by opaque zones. Together with the rays diffracted to the negative ( m < 0 ) and high ( m > 5 ) orders this ray is considered as the "lost" ray because the intensity created by those orders at the place of the positive first order focus is infinitesimally weak. So, the ray must be thrown away with the probability:  E:\E_\Articles\ICXOM\ZP_Ray\Text+buld.doc 4 ∑∑ ∞==−=−∞= ++= mmmmmmabslost   E  E  E  E  51  (11)  for all m < 0  and m > 5. If the ray is still considered as a survived one, then its destiny has also two ways. 1). The ray is not diffracted (zero order) and its angle ξ  to the optical axis remain unchanged with the probability of 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 2  , E 3  . .  Figure 3. 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. 4.2 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 ´(v m  )  and calculated by the Eq.(10). The ray is deflected randomly by the angle of 0 <=   δξ   <= δξ max  . The definition of the diffraction angle δϕ  and δψ  can be done using the expression for directing cosines (Figure 3): cos  δϕ)  = [1 - cos(  δψ) 2  - cos(  δξ) 2  ]  1/2   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 ; where  y  zp  and  x   zp  are the coordinates, where the ray hit the zone plate. 5. TEST OF THE COMPUTER CODE The results, obtained with the help of the RAY software are compared with calculations of the program KRGF which have been used for the calculation of x-ray holograms [6]. The KRGF deals with wave-fronts and phases and calculates a diffraction pattern using the Fresnel - Kirchhoff integral. In the figures 4(a) and 4(b) shown below, are presented the calculations of the RAY ray-trace software (upper plots) and their comparison with the results of the test program KRGF (bottom plots). The calculations are performed for a monochromatic point source, located on the optical axes of the zone plates. A comparison of the intensity profile in the focal plane for the both programs are shown in figure 5 for the linear zone plates.  E:\E_\Articles\ICXOM\ZP_Ray\Text+buld.doc 5 -100 -60 -20 20 60 100-80-4004080 RAY KRGFµm   µ  m 0 40 80 120 160 20004080120160200 RAYµm   µ  m -100 -80 -60 -40 -20 0 20 40 60 80 100-80-4004080   RAY KRGFµm   µ  m 40 80 120 160 2004080120160200     µ  m (a)(b) -100 -50 0 50 10010 -4 10 -3 10 -2 10 -1 10 0      N  o  r  m .   i  n   t  e  n  s   i   t  y Radius ( µ m)    KRGF RAY    Figure 4. A focal plane of circular (a) and linear (b)  zone plates calculated using programs RAY (Up) and  KRGF (bottom).  Figure 5. Intensity profile in the focal plane of the  zone plate calculated by the RAY and KRGF programs   6. CONCLUSIONS For the development of a ray-tracing 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 the result, correct intensity distribution is calculated only in a close vicinity of the positive orders foci of a zone plate. A zone plate could only be the final element of the traced optical system. One more task, which is still to be solved, consists in correct ray-tracing description of a zone plate 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. 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. REFERENCES 1. C. Welnak, G.J.Chen, F. Cerrina Nucl. Inst. and Meth. A347 , (1994), 344-347 2. T. Yamada, N. Kawada, M. Doi, T. Shoji, N. Tsuruoka, H. Iwasaki, J. Synchrotron Rad. 8 , (2001), 1047-1050 3. F. Schäfers "RAY - the BESSY raytrace program to calculate synchrotron radiation  bamlines", Technischer Bericht, BESSY TB 202 , (1996), 1-37 4. M. Born and E Wolf, Principles of Optics, 5th ed., Pergamon Press, Elmsford, N.Y. (1975) 378 - 386 5. A. I. Erko, V.V.Aristov, B.Vidal, "  Diffraction X-Ray Optics " IOP Publishing, Bristol, (1996) 26-36 6. A. Firsov, A. Svintsov, S.I. Zaitsev, A. Erko, V. Aristov "The first synthetic X-ray hologram: results", Optics Communications, 202,  (2002), 55-59,
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