By Hans J. Krappe, Reinhard Lipperheide, Institut für Kernphysik
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At a simple plane surface with air on one side and material with base index ns on the other. Here the reflectivity is given by the well-known Fresnel formulas sm2((Aa + i/O and * 2 =; an ;;t a it s i tan2((^a + i/O for the TE-wave and the TM-wave respectively. With the incident beam in air ns sin \j/s = sin \j/a = s. 29) At normal incidence (s = 0), for the case with the transition layer present, Eq. 30) for both waves, and so also for natural light. With 5 = 0, Eqs. 4, the value of JR given by Eq.
Further more, the reflectivity is wavelength dependent, whereas this is not appreciably so in the case of a gradient transition layer. As with interference coatings, gradient layers can be used to increase reflectivity instead of decreasing it. In this case the index n0 of the layer at its outer surface should be large instead of small. If the base material is metallic, its refractive index is a complex number, so that a true transition layer cannot be made of a transparent material. However, a dielectric gradient layer can still serve to reduce the reflectivity substantially.
3. ^o? 'o)> s* = — n0 0'/ Spherical gradient and related vectors (from Marchand, 1970). 11) where a and ß are coefficients to be determined. 12) showing that cosö = a + ßcos\j/0, cos(\j/0 — Θ) = a cos i/f0 + /}. 13) It follows that /? = sinö/sini^ 0 , a = cosö — ßcosil/0. 14) Writing Eq. 11) in component form gives I X 0 r\0L . n Ρθ hp - r0 n0 + iß-K ^V (3-15) z z = r\ a o «o/ Since a and /? are given in terms of Θ by Eqs. 14) and Θ in terms of r by Eq. 8), Eqs. 15) represent the essential ray-tracing equations relative to the untilted system.
Advanced Methods in the Evaluation of Nuclear Scattering Data by Hans J. Krappe, Reinhard Lipperheide, Institut für Kernphysik