Abstract
We have seen in Sect. 2.2 that the reflection amplitudes of an arbitrary profile tend to the Fresnel values as the thickness \( \Delta z \) of the profile tends to zero. An equivalent limit to consider is that of reflection by a profile of fixed extent, as the wavelength increases. We might expect the reflection amplitudes to be well represented, in the long wave limit, by the first few terms of a series in the ratio of the interface thickness to the wavelength.
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References
Lekner J (1984) Invariant formulation of the reflection of long waves by interfaces. Physica 128A:229–252
Lekner J (1982a) Second-order ellipsometric coefficients. Physica 113A:506–520
Further Readings
The perturbation theory of Sect. 3.1 is based on
Lekner J (1982b) Reflection of long waves by interfaces. Physica 112A:544–556
The s wave Green’s function for reflection at an inhomogeneity between like media (leading to (3.14)) was given by
Morse PM, Feshbach H (1953) Methods of theoretical physics. McGraw-Hill, New York, p 1071
Perturbation theory for reflection of the s wave at an interface between unlike media has previously been developed by
Triezenberg TG (1973) Capillary waves in a diffuse liquid-gas interface. Ph.D thesis, University of Maryland (unpublished)
There have been several (mainly formal) derivations of expansions for reflection properties in powers of the interface thickness (Sects. 3.2 and 3.4)
Maclaurin RC (1905) Theory of the reflection of light near the polarising angle. Proc Roy Soc A76:49–65
Rayleigh JWS (1912) On the propagation of waves through a stratified medium, with special reference to the question of reflection. Proc Roy Soc A86:207–266
Abelès F (1950) Recherches sur la propagation des ondes électromagnétiques sinusoïdales dans les milieux stratifiés. Application aux couches minces. Ann de Phys 5:596–640
Drazin PG (1963) On one-dimensional propagation of long waves. Proc Roy Soc A273:400–411
The formula (3.46) for the first order contribution to \( r_{p} /r_{s} \) is more than a century old. It is often attributed to Drude, but was, in essence, first obtained by L. Lorenz in 1860. Rayleigh (1912, quoted above) derives this result, and gives references to earlier work of Lorenz, Van Ryn, Drude, Schott and Maclaurin
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Lekner, J. (2016). Reflection of Long Waves. In: Theory of Reflection. Springer Series on Atomic, Optical, and Plasma Physics, vol 87. Springer, Cham. https://doi.org/10.1007/978-3-319-23627-8_3
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DOI: https://doi.org/10.1007/978-3-319-23627-8_3
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