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

# A b g b fig 320 a rectangular profile of a laminar

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a b g b Fig. 3.20. (a) Rectangular profile of a laminar grating, (b) Blazed grating

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118 A. Erko where t is a grating bars height. The diffraction eﬃciency in the m th order E m is proportional to the Fourier coeﬃcients of the diffraction integral in Fraunhofer approximation: E m = 4 ( ) 2 sin 2 m π g d sin 2 π t λ (cos α + cos β m ) , (3.35) where g is the grove width and β m is the angle of the m th diffraction order. For any given incidence angle α the groove depth can be optimized, making the 0th diffraction order as small as possible and maximizing the higher diffraction orders. In this case the groove depth should be t = λ 2 cos α + cos a sin sin α d . (3.36) Using (3.29) it is possible to estimate the maximum diffraction-tolerable roughness σ diff of a grating applying the λ/ 4 criterion: σ diff = λ min 8 cos α c , (3.37) where λ min is the minimum wavelength of used radiation and α c is the critical angle. The mean of reﬂectivity compared to that from a smooth surface is given by the so-called Debye–Waller factor. The maximum value of a surface roughness parameter can be calculated by the equation: σ < λ ln( R f /R r ) 4 π cos α , (3.38) where R r is the demanded reﬂectivity of a “rough” surface, R f is the ﬂat surface reﬂectivity. In a real situation the σ diff criterion is not strong enough in order to reach a good reﬂectivity and σ should be much less than the diffraction-limited value of (3.37). Blazed grating can be realized by the use of a wedge-shaped profile, shown in Fig. 3.20b. For a grating operated so that the angle θ = 0 . 5 ( α β ) is fixed, the grating equation may be written as = 2 d sin γ cos θ. (3.39) The blazed condition is optimal for only a small wavelength range for a given grating and diffraction order. The eﬃciency of a blazed grating in an ideal case may be estimated for some integer order m k when the maximum of the diffraction intensity does not correspond to the zero order. In particular, the diffraction eﬃciency can be written as: E m = R r sin 2 [ π 2 ( m k m ) 2 ] π 2 ( m k m ) 2 , (3.40) where R r is the surface reﬂectivity at the used wavelength and k is the opti- mized order of diffraction.
X-Ray Optics 119 Transmission Diffraction Grating Eﬃciency The transmission gratings at normal incidence are used for low-energy X-ray beam monochromatization. They are produced by evaporation of a phase- shifting material on a thin film or Si 3 N 4 membrane. There is also the pos- sibility to produce a free-standing structure to reduce absorption in the grating. To produce an optimum phase-shift of ∆Φ between adjacent zones (i.e., to ensure the maximum diffraction eﬃciency) for a transmission grating at normal incidence a structure with thickness t opt is required t opt = Φ opt λ 2 π δ , (3.41) where λ is the wavelength and δ is the refractive index decrement of the zone plate material. The complex refractive index is ˜ n = 1 δ i β (3.42) χ = β δ , (3.43) with β the absorption index and χ the optical characteristic of the material.

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