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

Zone plate aberrations the aberrations of optical

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Zone Plate Aberrations The aberrations of optical elements are conventionally analysed in terms of L : the optical path difference: L = R 2 1 + r 2 n + R 2 2 + r 2 n R 2 1 + R 2 2 . (3.107) These aberrations become noticeable when the term ∆ L is comparable with λ/ 4. (a) Spherical aberrations: For the regular zone plate constructed in accordance with (3.105), the term responsible for the spherical aberration is: 1 8 n 2 λ 2 1 R 3 1 + R 3 2 R 1 + R 2 < λ 4 . (3.108) One finds that the spherical aberration is a maximum for R 1 = 2 F , i.e., for the unit magnification. The maximum zone number N is limited by the value of: N < 8 F 3 λ . (3.109) (b) Chromatic aberrations Since the focal length of a zone plate is a function of wavelength, the chro- matic aberrations are very strong. In some schemes of X-ray microscopes [170] holographically made zone plates in combination with pinhole in focal plane are used as the monochromators. Chromatic aberration is significant for: L = 2 ± λ 4 . (3.110) The maximum number of zones for which chromatic aberration can still be neglected is given by: N < λ λ . (3.111) This equation is also valid for the calculation of an energy resolution when the pinhole of the diffraction-limited size β is used in a focal plane. (c) Off-axis aberrations These aberrations arise due to off-axis source position or ellipticity in zone- plate structure. If the primary beam is parallel and α is a beam incidence
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170 A. Erko angle (angle to the optical axes), then the optical pass difference will have aberration terms, as: L = r 3 n α 4 F 2 3 r 2 n a 2 4 F + 2 . (3.112) Using (3.112), one can estimate tolerance on zone plate orientation for the definite resolution in microprobe application. The first term in (3.112) is responsible for the coma aberration and second for the astigmatism and field curvature. Applying the λ/ 4 criteria for the above mentioned equation, one can write alignment requirements. In case of a zone plate with a number of zones N > 100, coma is the main aberration: -coma: α < N 1 . 5 F λ . (3.113) In case of a zone plate with a number of zones N < 100, astigmatism and field curvature are the main aberrations: -astigmatism and field curvature: α 1 3 N . (3.114) In comparison with a conventional lens a zone plate forms a distortion- free image. Another technological requirement to the optics designed for high- resolution image transmission is that it should be free of wave-front phase distortions caused by zone plate surface roughness and refraction index fluc- tuations within the medium. The approximate maximum value of the surface roughness could be roughly estimated by equation: ς < 0 . 1 t opt . (3.115) For example, the surface distortions for transparent silicon zone plates can practically reach ς = 0 . 1–0.3 µ m without affecting the image quality substantially.
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