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

Intensity gain intensity gain is defined as the ratio

Info icon This preview shows pages 191–194. Sign up to view the full content.

View Full Document Right Arrow Icon
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.
Image of page 191

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
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)
Image of page 192
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
Image of page 193

Info icon This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 194
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern