Depth-graded multilayer coating
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A multi-layer coating consisting of alternating layers with different optical properties and the same thickness will tend to have a narrow frequency response, getting narrower as more layers are added; for some applications such as precise focussing of a monochromatic laser light source this is exactly what's desired, but it is useless for astronomical optics where it is often required to detect a whole range of frequencies emitted by some source of interest.
The design of such coatings generally starts with an approximate analytical solution and then uses the simplex method of multi-variable optimisation to solve for optimal thicknesses of the layers. Typically the thin layers (to reflect high-energy X-rays) are on the inside since low-energy X-rays are absorbed more readily. One model used is a power law distribution of thicknesses, with the thickness of the ith bilayer as a/(b+i)^c for some optimised a, b, c.
An optimum multilayer design depends on the graze angle, so ideally a different prescription would be used on each shell of a multi-shell X-ray Wolter mirror; in practice the same prescription is used for about ten shells.
- Wang, Zhanshan. "Fabrication and Characterization of Depth-Graded X-Ray Multilayers" (PDF). Proceedings of 8th International Conference in X-ray Microscopy.
- Christensen, Finn. "Measured reflectance of graded multilayer mirrors designed for astronomical hard X-ray telescopes" (PDF).