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Application of the WKB method to X-ray diffraction by one dimensional periodic structures
Abstract
The problem of X-ray diffraction by one dimensional periodic structures is solved by the WKB method. It is shown that this approach is equivalent to the usual thin film theory if the effect of multiple reflections may be ignored. The WKB method permits analytical solution for a large class of periodic structures, in particular, all Fourier representable profiles may now be solved.
Recent advances in thin film fabrication techniques have made multilayer
mirrors an important new addition in the field of soft X-ray diffraction
optics. A multilayer is a one-dimensional periodic structure that
consists of an alternating sequence of high and low atomic number
elements evaporated or sputtered onto a substrate. Characterization of
these devices has been carried out on a variety of X-ray sources,
including synchrotron radiation, over a large range of parameters.
Extensive modeling of the diffraction characteristics of multilayers has
shown that the theory of optics of thin films gives good agreement
between measured and calculated diffraction profiles. Application of
multilayer mirrors as normal incidence X-ray collectors, monochromators,
and beam splitters is being explored.
Two types of reflectors, selective multilayer mirrors and broadband concave mirrors with sliding modes, are considered for use in XUV cavities. Both of these reflector types can be used with back-reflection of XUV beams to yield efficiencies of tens of percent. An optic which is an analogue of the ideal optical concentrators used in solar physics can be employed in optical pumping schemes to increase the flux density of the pumping radiation directed onto the active medium. For some materials, efficiencies of 10-40 percent have been reported in the 40-400 A wavelength range using these optics.
A general theory is developed for the scattering of x rays from a single nonideal interface between two dielectric media. It is then extended to describe the scattering of x rays from a multilayer structure composed of many nonideal interfaces. The most unique feature of this theory is that there are no constraints on the physical structure of the interfaces; the interfaces can have any form of roughness or compositional inhomogeneity. A simple analytical expression is derived for both the near and far radiation field to first order, assuming that the scattering is weak. The theory is valid for arbitrary polarization and at all angles of incidence (measured from the normal) less than the critical angle for total external reflection. Finally, the results are applied to study the effect of different interface structures on the performance of multilayer x‐ray optics.
The possibility of making multilayer reflecting coatings by using materials whose dielectric susceptibility ∊ is very near to unity at the frequency under consideration is discussed. For the pair of substances with a given Re∊ and Im∊ the explicit expressions are obtained for the value of reflection coefficient and for the optimum thickness of layers. The published data on Re∊ and Im∊ allowed us to choose the pairs of substances for which one may hope to create (in principle) the mirrors for normal incidence with R ~ 50% at λ ≳ 30 Å.
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Topics covered include: sub-kilovolt sources of x-ray emission; detectors for sub-kilovolt radiation; atomic scattering coefficients and calculations; reflective and diffractive x-ray imaging; coherence, interference and future possibilities. (GHT)
Multilayer coatings consisting of very smooth ReW and carbon films used as near‐normal‐incidence reflectors show theoretical performance in the 150–200‐Å wavelength region and should allow the fabrication of useful normal‐incidence mirrors for wavelengths as short as 50 Å.
Diffraction of X-rays in multilayer systems such as the molecular and the evaporated/sputtered multi-layers have been studied based upon the optics of thin films. General solutions for the intensity profiles of x-ray diffraction from multi-layers of finite thickness have been obtained for angles of incidence outside the total reflection region. It is also shown that the optical expression for diffraction in the x-ray region by periodic systems agree with the conventional crystal diffraction theories.
Proc. Topical Conf. on Low energy X-ray diagnostics
- Underwood
Proc. Topical Conf. on Low energy X-ray diagnostics
- Lee