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

# Intensity gain intensity gain is defined as the ratio

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Intensity Gain Intensity gain is defined as the ratio of the intensity in the focal spot of an optical element and the intensity behind a pinhole of size equals the spot size of the lens. Using the thin lens approximation, one can define the intensity gain G zp as: G zp = ε (2 r ( R 1 F ) /b s F ) 2 , (3.116) where b s is the source diameter, 2 r is the lens aperture and ε is the lens eﬃciency.

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X-Ray Optics 171 Zone Plate Aperture and Depth of Focus The zone plate aperture can be easily calculated from (3.106). Thus, the opening aperture is equal to: 2 r = ( δr ) , (3.117) where ( δr ) is the minimum zone width. The focus depth of a zone plate also has a strong wavelength dependence: F = ± 2 ( δr ) 2 λ . (3.118) It can be introduced for another definition of focus depth using a numerical aperture (NA): F = ± λ 2(NA) 2 , where NA = λ 2( δr ) . (3.119) 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). The phase shift and the amplitude attenuation of the input light in the even zones analogous to transmission diffraction gratings is given by the (3.41–3.52) [7]. 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 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: ν m r N k r F m , (3.120) where r = x 2 + y 2 , the radial distance between the optical axis and an arbitrary point in the image plane 2 r N is the aperture of a zone plate. 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 ) = 2 J ( ν m ) ν m 2 . (3.121)
172 A. Erko The solution of the one-dimensional diffraction integral can be found in a form of the sinus function of first order with an argument: ν m = x N k x F m , (3.122) where x or y are 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 ) = 2 sin( ν m ) ν m 2 . (3.123) 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 (3.121) and (3.123). For a zone plate there is always a low-intensity background caused by the zero order and high-diffraction orders. Thickness-limited Resolution of a Real 3-D Zone Plate High-resolution transmission zone plates widely used as focusing X-ray opti- cal elements have reached the theoretical limit of spatial resolution of about 15 nm [6]. This limit is defined not only by technological possibilities to pro- duce small outer zone widths or continuous thin layers, but by the effects of volume diffraction/refraction in the structure of the optical element. The

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• Spring '14
• MichaelDudley

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