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

If both the materials are completely absorption free

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If both the materials are completely absorption free, the quarter wave stack gives the highest reflectivity approaching R = 100%. However, accord- ing to the scheme of Fig. 3.34a and b, the electric field has a maximum at the interfaces. Practically it means, in case of rough interfaces significant light scattering takes place, which leads to the damage of the mirror and consider- ably reduces multilayer reflectance. In a quarter wave stack the thickness of materials are of the order of half a period. In an ideal Bragg crystal, the atomic planes thickness is usually
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136 B. Vidal Electric field in dielectric layers Standing waves Electric field in absorber layers Standing waves H L L H H H H B H = absorbing layer B = dielectric layer (vacuum in a crystal) B B B H H H L L L L (a) (b) Fig. 3.34. Two basic designs for a high reflectivity mirror. ( a ) The quarter-wave stack. ( b ) Crystal lattice structure much thinner than a period value. The different planes are spaced λ/ 2 apart (for normal incidence) and contribute in phase to the reflected wave. The absorption losses can be minimized due to the fact that the atomic planes are located at nodes of standing waves generated by the superposition of incidence and reflected waves. Reducing the absorption layer thickness the absorption losses can be made arbitrarily small compared to the reflectivity. In the limit a very thin film of absorbing material is equivalent to an absorption-free film of low reflectivity. Therefore, the number of layers can be increased considerably in comparison to the number of layers of the dielectric quarter wave stack. This fact leads not only to a reflection efficiency comparable with a dielectric multilayer mirror, but also to higher spectral selectivity (energy resolution) of the mirror. However, the crystal-like multilayer design has important experimental limitations. First, it is impossible to fabricate infinitely thin layers and the second, the absorption coefficient β of the spacer layer, where the electric field is high, is not negligible. Practically, almost more than 200 layers are necessary to produce high- efficiency reflection with hard and middle X-rays, about 100 layers with soft X-rays and only around 10–20 layers with far UV at about 30 nm. Material Selection Rules The following rules can be used to select two materials of a bilayer stacking in order to have the highest possible reflectivity [101] Select a first material with the lowest possible absorption β as a spacer material. Find a second material with the largest possible reflection coefficient at the boundary with the first material. If several materials give similar reflection coefficients, choose the one with the smaller absorption coefficient.
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X-Ray Optics 137 In addition, the choice of materials depends on interface quality, roughness and small diffusion rate, as well as on the exclusion of hazardous materials.
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