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

? of a multilayer mirror can be estimated as 44 ? ?

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λ of a multilayer mirror can be estimated as [44] λ λ mN eff , (3.77) where m is the diffraction order. The effective full width of a half maximum (FWHM) of the multilayer mirror rocking curve is approximated by θ 2 λ λ . (3.78) For a real multilayer mirror, the angle of peak reflectivity is greater than that predicted by the conventional Bragg formula. The formula which takes into account the average refraction index ˜ δ of the multilayer materials is given by equation sin θ m = 2 d + 2 d ˜ δ , (3.79) where ˜ δ = (1 γ opt ) δ l + γ opt δ h . Quasiperiodic Designs To make a more precise optimization of a multilayer reflectivity one has to take into account the fact that all materials in the X-ray range have consider- able absorption. The optimum design which maximizes the peak reflectivity with the minimum number of layers is quasiperiodic. The thickness of a heavy material should be decreased from the bottom to the top layer. Practically for the X-ray multilayers with a large number of periods the difference between
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X-Ray Optics 139 optimum quasiperiodic design and simple periodic design is very small. The quasiperiodic design affects only the bottom layers whose contribution in the final efficiency is relatively small. This was shown for example using mini- mization programs in [103]. Aperiodic Designs In the case of transparent materials direct methods exploring Fourier trans- form are useful to find quickly the best solution [104]. Computer algorithms for the inverse problem solution, which have been developed for visible light, in some cases can be applied to design an X-ray mirror. However, the design free- dom is much more limited due to the considerable absorption in all materials in the X-ray range. The most important application of these methods is the design of aperiodic structures to increase the energy bandwidth of reflection. The method supposes to divide a multilayer stack on several period regions in depth. Each region has a period which corresponds to a defined reflected energy at Bragg angle. A superposition of several regions with slightly differ- ent periods will provide an enlarged bandwidth of reflected light in comparison with a perfect periodical structure. In this way minimization programs can enlarge the bandwidth keeping the reflectance value as high as possible. In addition, the total external reflection phenomena can be used to reflect low- energy photons. The case of aperiodic design is called “supermirror” as opposed to the mirrors corresponding to periodic stack and which can give a high reflectivity at a total reflectance angle for low energy photons. Such a supermirror has been fabricated for the BESSY II beamline. The results of the mirror tests are shown in Fig. 3.35. A multilayer mirror with 50 W/Si bilayers of different 5 10 15 20 25 30 35 40 0.01 0.1 1 0.4 8 0.5 8 Reflectance Energy (keV) Fig. 3.35. Spectral characteristics of the BESSY “supermirror” measured at two different grazing incidence angles
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140 B. Vidal j =1 j = m R m -1 R m d m d m -1 d 1 Substrate Fig. 3.36.
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