Abstract
We shall first derive general results, that is those valid for an arbitrary interfacial profile (Sections 2-1 to 2-3); some of these results will be restricted to non-absorbing interfaces and substrates. The remainder of the chapter is devoted to important special profiles for which the reflection amplitude may be obtained exactly. Both the general and the specific exact results are useful in testing approximate theories and numerical calculations.
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References
References quoted in text
G. G. Stokes (1849) “On the perfect blackness of the central spot in Newton’s rings, and on the verification of Fresnel’s formulae for the intensities of the reflected and refracted rays”, Cambridge and Dublin Mathematical Journal 4, 1–15.
F. A. Jenkins and H. E. White (1950) “Fundamentals of optics”, McGraw-Hill, Sections 13.8 and 28. 10.
S. N. Jasperson and S. E. Schnatterly (1969). “An improved method for high reflectivity ellipsometry based on a new polarization modulation technique”, Rev. Sci. Instr. 40, 761–767.
D. Beaglehole (1980) “Ellipsometric study of the surface of simple liquids”, Physica 100B, 163–174.
G. Birkhoff and G. -C. Rota (1969) “Ordinary differential equations”, Blaisdell ( Waltham, Massachusetts).
J. Heading (1965) “Refractive index profiles based on the hypergeometric equation and the confluent hypergeometric equation”, Proc. Camb. Phil. Soc. 61, 897–913.
A. K. Bose (1964) “A class of solvable potentials”, Nuovo Cimento 32, 679–688.
R. L. Deavenport (1966) “A normal mode theory of an underwater acoustic duct by means of Green’s function”, Radio Science 1, 709–724.
R. Vasudevan, K. Venkatesan, and G. Jagannathan (1967) “Construction of solvable potentials and some aspects of regularization of singular potentials”, Nuovo Cimento (suppl.) 5, 621–643.
B. S. Westcott (1969) “Exact solutions for vertically polarized electromagnetic waves in horizontally stratified media”, Proc. Camb. Phil. Soc. 66, 675–684.
J. Heading (1970) “The equality of the moduli of certain ratios occurring in the connexion formulae of solutions of some transcendental differential equations”, Proc. Camb. Phil. Soc. 67, 347–361.
C. Eckart (1930) “The penetration of a potential barrier by electrons”, Phys. Rev. 35, 1303–1309.
P. S. Epstein (1930) “Reflection of waves in an inhomogeneous absorbing medium”, Proc. Nat. Acad. Sci. 16, 627–637.
L. D. Landau and E. M. Lifshitz (1965) “Quantum mechanics”, Pergamon.
E. T. Whittaker and G. N. Watson (1927) “A course of modern analysis”, Cambridge.
F. Oberhettinger (1964) “Hypergeometric functions”, Chapter 15 of Handbook of Mathematical Functions, edited by M. Abramowitz and I. A. Stegun, NBS Applied Mathematics Series No. 55.
J. G. Galejs (1961) “ELF waves in the presence of exponential ionospheric conductivity profiles”, IRE Trans. Ant. Prop. 9, 554–562.
R. Burman and R. N. Gould (1963) “On the propagation of vertically polarized electromagnetic waves in a horizontally stratified medium”, J. Atm. Terr. Phys. 25, 543–549.
F. Abelès (1964) “Optical properties of inhomogeneous films”, NBS Misc. Publ. 265, 41–58.
F. W. J. Olver (1964) “Bessel functions of integer order” (chapter 9 of “Handbook of mathematical functions”, edited by M. Abramowitz and I. A. Stegun, NBS Appl. Math. Series 55).
J. W. S. Rayleigh (1880) “On the reflection of vibrations at the confines of two media between which the transition is gradual”, Proc. Lond. Math. Soc. 11, 51–56.
J. Lekner (1982) “Exact reflection amplitudes for the Rayleigh profile”, Physica 116A, 235–247.
The derivation of conservation laws and reciprocity relations from comparison identities (Section 2-1) is based on
J. Lekner (1982) “Reflection of long waves by interfaces”, Physica 112A, 544–556.
For alternative derivations, see Landau and Lifshitz (1965), Section 25 and
J. Heading (1975) “Ordinary differential equations, theory and practice”, Elek Science, Chapter 4.
Extension of Stokes’ idea of reversing the wavemotions is discussed by gantavÿ and Knittl:
U. gantavÿ (1961) “On the reversibility of light beams in conducting media”, Optica Acta 8, 301–307;
Z. Knittl (1962) “The principle of reversibility and thin film optics”, Optica Acta 9, 33–45;
Z. Knittl (1976) “Optics of thin films”, Wiley, Chapter 6.
Sections 2-2 and 2-3 are based on the author’s paper on the Rayleigh profile (quoted above) and on
J. Lekner (1985) “Reflection at oblique incidence and the existence of a Brewster angle”, J. Opt. Soc. Amer. A2, 186–188.
The uniform layer between two bulk media (Section 2-4) is considered (for example), in
M. Born and E. Wolf (1965) “Principles of optics” (3rd edition), Pergamon, Section 1.6.4, and R. M. Azzam and N. M. Bashara (1977) “Ellipsometry and polarized light”, North Holland.
Exactly solvable profiles (Section 2-5) are listed in the texts by
K. G. Budden (1961) “Radio waves in the ionosphere”, Cambridge;
V. L. Ginzburg (1964) “The propagation of electromagnetic waves in plasmas”, Pergamon;
J. R. Wait (1970) “Electromagnetic waves in stratified media”, (second edition), Pergamon; and
L. M. Brekhovskikh (1980) “Waves in layered media” (second edition), Academic Press.
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Lekner, J. (1987). Exact results. In: Theory of Reflection of Electromagnetic and Particle Waves. Developments in Electromagnetic Theory and Applications, vol 3. Springer, Dordrecht. https://doi.org/10.1007/978-94-015-7748-9_2
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DOI: https://doi.org/10.1007/978-94-015-7748-9_2
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