Computing X-Ray Reflectance Of Focusing Multilayer Films

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Abstract
F. Abeles developed an exact 2 X 2 matrix method for computing reflectance and transmittance of flat stratified structures of arbitrary complex refractive index profile. By distorting the coordinate system for his method from rectangular to confocal ellipsoids of revolution, the same method can be applied to focusing systems. The well-known 1:1 toroidal focusing geometry, corresponding to unit magnification, occurs at the equators of the ellipsoids. Near the equator, if the Bragg angle is not too small, and also close to the axis of foci, curved layers of uniform thickness are often close enough to the ideal profile for satisfactory spatial and spectral resolution. In very thick multilayers with small absorption, uniform layer thickness in the toroidal geometry results in some dispersion with interference fringes on one side of the maximum. Away from the equator or axis, and particularly at low Bragg angles, profiling or gradation of thickness may be necessary for satisfactory resolution. Examples of computations with Abeles's method are shown.

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