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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 65, Iss. 2 — Feb. 1, 1975
  • pp: 146–149

Dispersion relations and complex reflectivity

G. H. Goedecke  »View Author Affiliations

JOSA, Vol. 65, Issue 2, pp. 146-149 (1975)

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Inclusion of the conventional classical theory of radiation reaction in the standard classical microscopic model of a material medium induces negligible corrections to the Kramers—Kronig dispersion relations for the complex dielectric constant, and predicts a new ultrahigh-frequency behavior, the same for all media, for any optical property associated with the medium, dependent only on the electron-number density in the medium. One conventional application of dispersion relations to the logarithm of the normal incidence complex reflectivity at a plane interface is shown to be incorrect in principle, regardless of whether radiation reaction is considered. New, correct dispersion relations, to which the conventional ones may be very good approximations in some cases, are given for this quantity. Quantum-field-theoretic considerations also show that the new relations are valid.

G. H. Goedecke, "Dispersion relations and complex reflectivity," J. Opt. Soc. Am. 65, 146-149 (1975)

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  1. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, London, 1960), Ch. 9, contains a thorough discussion of the requisite properties. Later in this paper we give a brief discussion.
  2. See, for example, H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, Princeton, N. J., 1945), Ch. 14.
  3. See, for example, J. R. Taylor, Scattering Theory (Wiley, New York, 1972), Ch. 15.
  4. W. K. H. Panofsky and M. Phillips, Classical Electricity and Magnetism (Addison–Wesley, Cambridge, Mass., 1955), Ch. 21.
  5. J. D. Jackson, Classical Electrodynamics (Wiley, New York, 1962), Ch. 17. The integro-differential form of this equation eliminates runaway complementary solutions, but does not eliminate weak violation of causality.
  6. Many authors [e. g., M. Born and E. Wolf, Principles of Optics, 4th ed. (Pergamon, London, 1970), Chs. 1 and 13; F. A. Jenkins and H. E. White, Fundamentals of Physical Optics (McGraw-Hill, New York, 1937), Ch. 18], derive the Fresnel relations for oblique incidence in such a way that in the limit of normal incidence, the complex reflectivity for polarization parallel to the plane of incidence appears to have a sign opposite to that for perpendicular polarization, even though the two polarizations are physically indistinguishable at normal incidence. Born and Wolf point out that the distinction is "rather formal. " A careful analysis shows that the ratio of the tangential components of the complex amplitudes of the electric fields does represent the complex reflectivity, in that the squared magnitude of this ratio is the spectral reflectance, at any angle of incidence, for either polarization, and this ratio has the same value for either polarization at normal incidence. The minus sign in our definition is unimportant; it was chosen as a matter of taste.
  7. Reference 1, Ch. 9, Sec. 59.
  8. For metals, the complex dielectric constant ε(ω) has a pole at ω = 0; for simplicity, we have omitted consideration of functions that have poles on the real axis, because this property is immaterail to our main results. See Ref. 1 for the dispersion relations for such functions.
  9. Reference 1, Ch. 9, Sec. 62. Landau and Lifshitz (Ref. 1) use a time dependence exp(–iωt), whereas we have chosen exp(iωt), the usual choice in optics literature. This difference is responsible for our requirement that a causal ε (ω) should have no singularities in the lower half-plane, rather than in the upper.
  10. M. R. Querry, C. R. Waring, W. E. Holland, L. M. Earls, M. D. Herrman, W. F. Nijm, and G. M. Hale, J. Opt. Soc. Am. 64, 39 (1974).
  11. For example, G. Andermann, A. Caron, and D. A. Dows, J. Opt. Soc. Am. 55, 1210 (1965).
  12. J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison–Wesley, Reading, Mass., 1955), Ch. 11.
  13. W. L. Burke, Phys. Rev. A 2, 1501 (1970); J. Math. Phys. 12, 401 (1971).

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