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

# The amplitude of the m th fourier component of the

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The amplitude of the m th Fourier component of the wave in the plane of a diffraction grating c m can be written in a simple form, following from the Fourier analysis of a plane diffraction grating: c m = 1 m π . (3.44) Therefore, the amplitude coeﬃcient of the wave, coming through the “free”, “positive” zones at the output surface of a diffraction grating can be written as: S m 1 = 1 m π exp( i kt ) . (3.45) The amplitude coeﬃcient of the wave coming though a phase shifting material in negative zones can be written as: S m 2 = 1 m π exp( i kt ˜ n ) . (3.46) Taking into account (3.45) and (3.46), one can write an equation for the amplitude coeﬃcient on the output surface of a m -order S m as: S m ( r ) = 1 m π exp ( i kt ) if r r 2 n r 2 n 1 1 m π exp ( i kt + i Φ χ Φ ) if r r 2 n 1 r 2 n 2 . (3.47) For r and r n see equations (3.19) to (3.23).

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120 A. Erko where χ = β/δ . Combining the amplitudes of the waves, transmitted through the transparent and covered zone, one can calculate the intensity E m diffrac- tion eﬃciency in the m th order [94]: E m = 1 ( m π ) 2 [1 + exp ( 2 χ Φ ) 2 cos (∆ Φ ) exp ( χ Φ )] (3.48) for the odd orders m = ± 1; ± 3; ± 5; . . . Even orders do not exist. The maximum of the function E m can be found taking derivative of the (3.48): sin (∆ Φ opt ) + χ [cos (∆ Φ opt ) exp ( χ Φ opt )] = 0 , (3.49) where the value of ∆ Φ opt is the optimum phase shift in the diffraction grating material. This equation can be solved numerically and an optimum phase shift can be approximated by the sum of two exponential curves: Φ opt 1 . 69 + 0 . 71 exp χ 0 . 55 + 0 . 74 exp χ 2 . 56 . (3.50) The optimum thickness of the zone plate can be calculated using (3.41). Anal- ogous to (3.48), the eﬃciencies of the zero order diffraction E 0 and the part absorbed in the material of a diffraction grating E abs can be calculated using expressions: E 0 = 0 . 25 [1 + exp( 2 χ Φ ) + 2 cos (∆ Φ ) exp( χ Φ )] (3.51) and E abs = 0 . 5 [1 exp ( 2 χ Φ )] . (3.52) Depending on the value of parameter χ all materials can be called as a “phase” material or an “amplitude” one. Usually effective phase-shifting materials could be characterized by the value of χ < 0 . 1. Bragg–Fresnel Grating The properties of a diffraction grating fabricated on a total external reﬂection surface are almost isotropic in respect to the angle between ingoing beam direction and groove orientation. The depth of profile for such a grating is of the order of several nanometers and shadowing effect is important only at very small grazing angles. For such a case the rigorous theory is developed, which takes into account the grating material and grove depth [95]. In comparison with a conventional grating, the diffraction grating evapo- rated on a surface of a Bragg reﬂector, i.e. a Bragg–Fresnel grating, has a very remarkable anisotropy to the ingoing beam direction. One can define two main cases with different diffraction properties: sagittal and meridional directions.
X-Ray Optics 121

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• Spring '14
• MichaelDudley

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