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

The spatial resolution of a standard zone plate is

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of volume diffraction/refraction in the structure of the optical element. The spatial resolution ∆ of a standard zone plate is estimated in scalar diffraction theory by the Rayleigh criterion [50] which leads to: ∆ = 1 . 22( δr ) m , (3.124) where ( δr ) is the width of the outermost zone and m is the diffraction order. The resolution can be further improved by using curved zone profiles [171], but in practice even if zone plates with very small outer zone widths (say 10 nm) could be fabricated, it will be not possible to reach the resolution expected from (3.124) even by using a high-diffraction order. In a real zone plate the thickness of material limits the validity of the (3.124) through, as in the case of waveguides, a volume diffraction phenomenon. To produce an optimum phase- shift of ∆Φ opt between adjacent zones (to ensure the maximum diffraction efficiency) requires a structure with thickness t opt , (3.41). Unfortunately, for X-rays with wavelengths less than 0.2 nm the value of optimal thickness t opt is of the order of several µ m for all materials. Thus, zone plate technology in this energy range is complicated even at a sub- µ m spatial resolution, due to the high-aspect ratios (line height/width) required.
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X-Ray Optics 173 An expression for the minimum possible zone width, introduced as the validity criterion of scalar diffraction theory is [7]: ( δr ) min = m λt opt . (3.125) This approximation is in good agreement with the rigorous electromagnetic theory and with the theory of volume holograms [172]. For zones with spacings less than ( δr ) min the scalar diffraction theory is not valid due to multiple diffraction of radiation at the zone plate structure. For high-energy X-rays greater than 1000 eV, the value of t opt is of the order of several microns. A small focal spot size down to 0 . 1 µ m × 0 . 1 µ m can be achieved at a focal distance of 10–100 mm using a so-called modified zone plate [173] exploiting the first and higher orders of diffraction simultaneously. Ray-Tracing Model of a Zone Plate Ray tracing is an indispensable tool for the design of optical systems for syn- chrotron radiation sources, and various programs have been developed during the last decades [174, 175]. 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 microfocusing with diffraction limited imaging. In the paper [176], a ray-tracing code for zone plates incorporated into the program RAY [164] 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
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