Applied Physics A

, Volume 40, Issue 3, pp 151–158 | Cite as

Application of Fourier's allied integrals to the Kramers-Kronig transformation of reflectance data

  • B. Harbecke
Contributed Papers


Since the fast Fourier algorithm is available, the representation of the Kramers-Kronig transformation by Fourier's allied integrals offers the possibility of a quick and comfortable calculation. Further advantages of the procedure are explained and demonstrated, e.g. the possibility to examine the “response function” after the first Fourier transformation and to extract physical information; the feature of numerical self-consistency by obtaining back the original spectrum after the second transformation. We present Fourier's allied integrals for the complex-amplitude reflection coefficient and describe a procedure which allows an application of the method to the reflectance of semiinfinite media. Extrapolations of the spectra to low and high frequencies are recommended; thus the range of integration is sufficiently large that the phase needs no correction. The performance of the transformations together with technical details is exposed for four measured spectra as examples: the reflectance of NaCl, InSb, and MnSe at room temperature, together with the reflectance of MnSe at liquid helium temperature.


42.10 42.20 78.20 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    R.J. Bell:Introductory Fourier Transform Spectroscopy (Academic, New York 1972) Chap. 8Google Scholar
  2. 2.
    B.C. Tichmarch:Introduction to the Theory of Fourier Integrals (University Press, Oxford 1962) pp. 119–120Google Scholar
  3. 3.
    B. Gross: Phys. Rev.59, 748 (1941) especially the statement between Eqs. (9) and (10)Google Scholar
  4. 4.
    J.W. Cooley, J.W. Tukey: Math. Comput.19, 297 (1965)Google Scholar
  5. 5.
    C.W. Peterson, B.W. Knight: J. Opt. Soc. Am.63, 1238 (1973)Google Scholar
  6. 6.
    M.G. Sceats, G.C. Morris: Phys. Stat. Solidi (a)14, 643 (1972)Google Scholar
  7. 7.
    F.W. King: J. Opt. Soc. Am.68, 996 (1978)Google Scholar
  8. 8.
    F.W. King: J. Chem. Phys.71, 4726 (1979)Google Scholar
  9. 9.
    W.N. Hansen, W.A. Abdou: J. Opt. Soc. Am.67, 1537 (1977)Google Scholar
  10. 10.
    D.A. Aspnes: Surf. Sci.135, 284 (1983)Google Scholar
  11. 11.
    J.A. Bardwell, M.J. Dignam: J. Chem. Phys.83, 5468 (1985)Google Scholar
  12. 12.
    J.A. Bardwell, M.J. Dignam: Analytica Chimica Acta172, 101 (1985)Google Scholar
  13. 13.
    D.W. Johnson: J. Phys. A8, 490 (1975)Google Scholar
  14. 14.
    F.W. King: J. Phys. C10, 3199 (1977)Google Scholar
  15. 15.
    J.S. Toll: Phys. Rev.104, 1760 (1956)Google Scholar
  16. 16.
    F. Stern: Elementary Theory of the Optical Properties of Solids, inSolid State Physics 15, 334 (Academic, New York 1963)Google Scholar
  17. 17.
    D.C. Champeney:Fourier Transforms and Their Physical Application (Academic, New York 1973) App. UGoogle Scholar
  18. 18.
    P. Antosik, J. Mikusinski, R. Sikorski:Theory of Distributions (Elsevier, Amsterdam 1973) Chap. 13Google Scholar
  19. 19.
    M.J. Lighthill:Introduction to Fourier Analysis and Generalized Functions (Cambridge U. Press, London 1970)Google Scholar
  20. 20.
    L.D. Landau, E.M. Lifshitz:Statistical Physics and Electrodynamics of Continuous Media (Pergamon, Oxford 1960)Google Scholar
  21. 21.
    H.J. Nussbaumer:Fast Fourier Transform and Convolution Algorithmes, 2nd ed., Springer Ser. Inf. Sci. 2 (Springer, Berlin, Heidelberg 1982)Google Scholar
  22. 22.
    E.O. Brigham:The Fast Fourier Transform (Prentice-Hall, Englewood Cliffs NJ 1974) pp. 163–167Google Scholar
  23. 23.
    D.L. Decker, R.L. Wild: Phys. Rev. B4, 3425 (1971)Google Scholar
  24. 24.
    R.J. Pollard, V.H. McCann, J.B. Ward: J. Phys. C16, 345 (1983)Google Scholar

Copyright information

© Springer-Verlag 1986

Authors and Affiliations

  • B. Harbecke
    • 1
  1. 1.I. Physikalisches Institut der Rheinisch-Westfälischen Technischen Hochschule AachenAachenFed. Rep. Germany

Personalised recommendations