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The propagation of electromagnetic waves in a stratified medium

  • D. R. Hartree (a1)

The equations of propagation of electromagnetic waves in a stratified medium (i.e. a medium in which the refractive index is a function of one Cartesian coordinate only—in practice the height) are obtained first from Maxwell's equations for a material medium, and secondly from the treatment of the refracted wave as the sum of the incident wave and the wavelets scattered by the particles of the medium. The equations for the propagation in the presence of an external magnetic field are also derived by a simple extension of the second method.

The significance of a reflection coefficient for a layer of stratified medium is discussed and a general formula for the reflection coefficient is found in terms of any two independent solutions of the equations of propagation in a given stratified medium.

Three special cases are worked out, for waves with the electric field in the plane of incidence, viz.

(1) A finite, sharply bounded, medium which is “totally reflecting” at the given angle of incidence.

(2) Two media of different refractive index with a transition layer in which μ2 varies linearly from the value in one to the value in the other.

(3) A layer in which μ2 is a minimum at a certain height and increases linearly to 1 above and below, at the same rate.

For cases (2) and (3) curves are drawn showing the variation of reflection coefficient with thickness of the stratified layer.

Case (3) may be of some importance as a first approximation to the conditions in the Heaviside layer.

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* Darwin, C. G., Trans. Camb. Phil. Soc., Vol. 23, p. 137 (1924).

[Note added in proof. Reference should also be made to a somewhat different treatment of this aspect of the optical properties of matter by Oséen, C. W. (Ann. der Phys. Vol. 48, p. 1 (1915) and Phys. Zeit., Vol. 16, p. 404 (1915)),Ewald, P. P. (Ann. der Phys., Vol. 49, pp. 1, 117 (1916));Lundblad, R. (Dissertation, Upsala (1920))].

For isotopic media the tensor suffix notation used by Darwin is not necessary; we shall write x, y, z for the Cartesian coordinates of a point instead of x 1, x 2, x 3, and (for example) E x, E y, E z for the components of the electric field strength.

Lorentz, , Theory of Electrons, p. 138.

* For a further discussion of the non-convergence of the integrals, see Darwin, , loc. cit., pp. 139, 144.

* If E is a constant vector and ø a variable scalar,

This result is obvious if the tensor suffix notation used by Darwin (loc. cit.) is adopted. The left-hand side is then and the right

* I.e. the x component of the vector σE is

and similarly for the y, z components, or in the abbreviated notation used by Darwin (loc. cit.), (σE)αβE β.

* The sign has been changed from the formula given by Darwin, as by σE we here mean σαβE β, while in Darwin's work the electric field has the same suffix as the first suffix of σ; the part of the tensor depending on H is antisymmetrical.

Indeed, if we take a plane parallel plate of a transparent medium, the variation in the intensity of the reflected beam with the thickness of the medium shows that the reflected beam does depend on the refracting medium at a distance from the boundary. This variation is usually expressed as the effect of interference between the set of waves formed by successive reflection at the front and back boundaries, but is equally well ascribed to the different thicknesses of medium over which the integration of scattered wavelets is carried out. [Cf. Ewald, P. P., Fortschritte der Chemie, Physik, und Phys. Chem., Series B, Vol. 18, Part 8 (1925).] This set of waves formed by multiple reflection is really only a device for evaluating the reflection coefficient; what we actually have is a wave system satisfying (22–24) (which reduce to (23) for normal incidence) with μ2 constant between two planes and 1 outside them, and corresponding to waves incident from one side only. There is no need to think of reflection back and forth between the surfaces.

* are n 1, n 2, n 3 of Darwin's paper.

* On the usual ideas derived from elementary optics, the reflection coefficient of the surface of a totally reflecting medium would be taken as 1, so that waves incident ou a parallel plate at an angle greater than the critical angle would be completely reflected however thin the plate.

* I am indebted to Dr L. Nordheim for calling my attention to this.

Tables for calculating J ± ⅓ are given in Watson's Theory of Bessel Functions, but the differential coefficients J′ ± ⅓ are not tabulated. It actually appeared easiest to use the series for G, G′, H, H′ up to ζ=2, and to integrate (43) as it stands from there to ζ=4, checking occasional values of G and H by interpolation from the tables. For ζ>4 it is simplest to use the asymptotic formulae. (48) is useful for checking calculated values of G, G′, H, H′.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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