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

X y z and a x y z are the amplitude in the object and

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x, y, z ) and A ( x , y , z ) are the amplitude in the object and image planes, k is the wave number ( k = 2 π/λ ), ¯ r and ¯ s are the propagation vectors, d x d y d z is an element of area of the aperture, and the integral is carried out over the whole object aperture. In practice, in order to get the intensity distribution behind a zone plate one must take the double integral in (3.13) for each point A ( x , y , z ) on the screen and that would be just the distribution for a point source at A 0 ( x, y, z ). In turn, if the source is not just a point, this procedure must be repeated for each point of the source. This method can give good results but the time needed for such calculations is too long. Here we are supposing that the wavelength λ is much smaller than object and image sizes and therefore one can use a so-called “small angle approxi- mation” that corresponds to cos( Zr ) 1 (see Fig. 3.18). Equation (3.13) can not be solved analytically in the general form. To simplify the task one can use two approximations of the diffraction equation: in the Fresnel (near-field) and Fraunhofer (far-field) forms described below. r A 0 ( x,y,z ) Z R 0 X Y Image plane Object plane Y X A 0 ( x , y , z ) r 0 Fig. 3.18. The coordinate system used in the calculation of the diffraction pattern of a circular aperture
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X-Ray Optics 113 A simple equation, which corresponds to typical diffraction conditions, can be derivated on the basis of Kirchhoff formula. Let us take the value of the radius in Cartesian coordinates as: r 2 = R 2 0 + ( x x ) 2 + ( y y ) 2 . (3.14) The different approximations of the r value are used to define different types of diffraction. 3.3.2 Fraunhofer Approximation In an approximation of the Kirchhoff formula which defines conditions of Fraunhofer diffraction, the overall dimensions of the object are much smaller than the distances to the source or to the point of observation. Supposing the distances in (3.14) having the following values: R 0 x , y x, y, (3.15) which corresponds to the far-field approximation. In this case the value of the radius-vector will be: r r 0 x r 0 x y r 0 y (3.16) and defining the diffraction angles as sin ϕ = x r 0 and sin ψ = y r 0 , one gets the amplitude in the image plane as a function of angles for the plane parallel incident wave: A ( ϕ, ψ ) = i exp( i kr 0 ) r 0 λ A ( x, y, 0) exp[i k ( x sin ϕ + y sin ψ )] d x d y. (3.17) This equation shows that the amplitude in the image plane corresponds to the Fourier transform of the amplitude in the object plane in angular coordinates. From this equation one can easily derive well-known diffraction properties of gratings, slits, and apertures in Fraunhofer approximation. 3.3.3 Fresnel Approximation Fresnel diffraction normally corresponds to the diffraction phenomena ob- served close to two-dimensional objects illuminated by plane parallel incident wave. As a criterion of this type of diffraction one can use the following relation: R 0 x , y x, y, (3.18)
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114 A. Erko which corresponds to the near-field approximation. In this case the value of the radius-vector in (3.14) can be approximated: r R 0 ( x x ) 2 ( y y ) 2 2 R 0 . (3.19)
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  • Spring '14
  • MichaelDudley

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