Dispersion relations and sum rules for magnetoreflectivity
JOSA, Vol. 66, Issue 6, pp. 547-554 (1976)
http://dx.doi.org/10.1364/JOSA.66.000547
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Abstract
Dispersion relations and sum rules for the dichroic reflectivity and phase shifts of circularly polarized modes are developed for the magneto-optical case. The reduction in crossing-relation symmetry arising from the presence of a magnetic field and the consequent non-Kramers-Kronig form of the dichroism dispersion relations are discussed in terms of the analyticity of the amplitude reflectivity. Sum rules are derived from the low- and high-frequency limits of the dichroism dispersion relations. These rules include the general results that ∫^{∞}_{0} ω^{-1} ln[r+(ω)/r_{-}(ω)]dω = 0 and ∫^{∞}_{0}[θ+(ω)-θ_{-}(ω)]dω = πω_{c}, where r_{±}(ω) and θ±(ω) are the amplitude and phase of the amplitude reflectivity for the circular modes and ω_{c} is the cyclotron frequency. Approximate finite-energy dispersion relations and sum rules are developed and their range of validity examined.
© 1976 Optical Society of America
Citation
David Y. Smith, "Dispersion relations and sum rules for magnetoreflectivity," J. Opt. Soc. Am. 66, 547-554 (1976)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-66-6-547
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References
- M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, Phys. Rev. B 6, 4502 (1972).
- A. Villani and A. H. Zimmerman, Phys. Lett. A 44, 295 (1973); and Phys. Rev. B 8, 3914 (1973).
- M. Altarelli and D. Y. Smith, Phys. Rev. B 9, 1290 (1974).
- F. C. Jahoda, thesis (Cornell University, 1957) (unpublished); and Phys, Rev. 107, 1261 (1957).
- B. Velický, Czech. J. Phys. B 11, 541 (1961).
- M. Gottlieb, Ph. D. thesis (University of Pennsylvania, 1959) (unpublished).
- See, J. S. Toll, thesis (Princeton University, 1952) (unpublished); and Phys. Rev. 104, 1760 (1956).
- H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972).
- I. M. Boswarva, R. E. Howard, and A. B. Lidiard, Proc. R. Soc. London Ser. A 269, 125 (1962).
- D. Y. Smith, Proceedings of the International Symposium on Color Centers in Alkali Halides, 1968, Rome (unpublished), p. 252; Bull. Am. Phys. Soc. 19, 93 (1974); and 19, 259 (1974).
- D. Y. Smith, J. Opt. Soc. Am. 66, 454 (1976).
- In this expression η must be restricted to nonzero values in the case of conductors. In these materials the dielectric function has an ω^{-1} singularity at ω = 0 which leads to an ω^{½} behavior in ln ˜γ(ω) near ω = 0. To ensure that V(ω) is square integrable, it is therefore necessary to require η ≠ 0 in Eq. (2) and in its analogue for circular modes. In insulators lim_{ω→0} lñγ(ω) is a nonzero constant and no such restriction applies.
- The restriction ω ≠ 0 in Eq. (12) also occurs for the derivation from the quotient function Q. It arises from having to avoid improper rearrangement under the integral sign when going from the integral from - ∞ to ∞ one from 0 to ∞.
- S. E. Schnatterly, Phys. Rev. 183, 664 (1969).
- D. Y. Smith, Phys. Rev. B 13, (1976) (to be published).
- F. Stern, in Solid State Physics, edited by F. Seitz and D. Turnbull (Academic, New York, 1963), Vol. 15.
- A second low-frequency sum rule may be obtained from the expression for ln[γ¯_{+}(ω)/γ¯_{→}(ω)], Eq. (12), if a specific model is assumed. For example, if an insulator consists of a single magnetically active Lorentzian absorption with plasma frequency ω_{p} centered at ω_{0}, the low-frequency limit of ln[γ¯_{+}(ω)/γ¯_(ω)] is[Equation] where n_{0} is the static index (1 + ω^{2}_{p}/ω^{2}_{0})^{½}. Combining this with Eq. (12) yields [Equation]. For more complicated models of the absorption spectrum similar forms hold but with ω_{0}, ω_{p}, etc., replaced with average values.
- H. Becquerel, C. R. Acad. Sci. (Paris) 125, 679 (1897).
- C. G. Darwin and W. H. Watson, Proc. R. Soc. London 114, 474 (1927).
- The dielectric function for this system is ∊±(ω) = ∊_{b} - ω^{2}_{p} / (ω^{2} - ω^{2}_{0} ± ωωc + iωγ), with the background dielectric function ∊_{b} equal unity for vacuum.
- See, for example, R. K. Ahrenkiel, T. H. Lee, S. L. Lyu, and F. Moser, Solid State Commun. 12, 1113 (1973).
- An alternative approximation is to employ the inverse-second-moment phase sum rule discussed in Ref. 17 rather than the zeroth moment phase sum rule to evaluating the left-hand side of Eq. (31). This leads to a result similar to Eq. (32), but with ω¯^{2} replaced n_{0} ω¯^{2}_{0}, where n_{0} is the contribution to the static index from transitions at energies ω >β.
- As in the vacuum background case a second sum rule derived from the low-frequency limit holds for insulators, but its form depends on the model assumed. For the single Lorentzian absorption treated in Ref. 17 the low-frequency limit of the quotent yields [Equation] where now ń_{0} = (ε_{∞} + ω^{2}_{p}/ω^{2}_{0})^{½}. This leads to [Equation].
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