[B._Beckhoff,_et_al.]_Handbook_of_Practical_X-Ray_(b-ok.org).pdf

Advantage of the model is that it gives an intensity

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advantage of the model is that it gives an intensity distribution, including auxiliary maxima and background radiation in the focal position. The intensity distribution in the focal plane of a zone plate for a point source can be calculated analytically. 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 res- olution defined by the zone plate aperture, the so-called diffraction limited resolution. 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, rep- resented in the following parts of the paper.
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174 A. Erko 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 partially absorbed by zone plate mater- ial. 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: E lost = E abs + m = 1 m = −∞ E m + m = m =5 E m (3.126) for all m < 0 and m > 5. If the ray is still considered 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 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 (3.126) are: E 1 , E 3 , E 5 . 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 the (3.120–3.123). The definition of the RAY coordinate system is shown in Fig. 3.57. r Circle of the angular radius dx Y X Z F Y X Focal plane Zone plate dx dj dy Fig. 3.57. 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
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X-Ray Optics 175 The ray is deflected randomly by the angle of 0 δξ δξ max . According to Fig. 3.57 the values of the diffraction angle δϕ and δψ are defined in small angle approximation by the expression: ( δξ ) 2 = ( δξ ) 2 + ( δϕ ) 2 . (3.127) For each ray the values δψ and δϕ are randomly selected within the angular range of 0 δψ, δϕ 0 . 707 δξ max . Than values of δξ calculated according to (3.127) pass through the probability generator in accordance to (3.121) and (3.123). Only those values of angles which passed the filter and have a Bessel
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