Abstract

A scheme based on conjugate Fourier series is proposed for the determination of the dissipative mode of an optical constant from the corresponding dispersive mode, and vice versa. The connection between the conjugate Fourier series and Fourier integral method is discussed. The advantages of the method in relation to the Kramers-Kronig approach are outlined.

© 1978 Optical Society of America

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  1. P. O. Nilsson, "Optical Properties of metals and alloys," Solid State Physics, Vol. 29, Edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1974), pp. 139–234.
  2. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 256.
  3. C. W. Peterson and B. W. Knight, "Causality calculations in the time domain: An efficient alternative to the Kramers- Kronig method, " J. Opt. Soc. Am. 63, 1238–1242 (1973).
  4. E. C. Titchmarsh, "Conjugate trigonometrical integrals," Proc. Lond. Math. Soc. 24, 109–130 (1925).
  5. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Clarendon, Oxford, 1948), Chap. 5.
  6. C. Brot, "Correlation functions in dipolar absorption-dispersion," in Dielectric and Related Molecular Processes, edited by M. Davies (Chemical Society Specialist Periodical Reports, 1973), Vol. 2, p. 7.
  7. B. K. P. Scaife, "The theory of the macroscopic properties of isotropic dielectrics, " in Dielectric and Related Molecular Processes, edited by M. Davies (The Chemical Society Specialist Periodical Reports, 1972), Vol. 1, pp. 1–20, Eqs. (32) and (44).
  8. K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics. II. Direct current characteristics, " J. Chem. Phys. 10, 98–105 (1942).
  9. B. Gross, "On the theory of dielectric loss, " Phys. Rev. 59, 748–750 (1941).
  10. M. Reisz, "Les fonctions conjuguées et les séries de Fourier, " C. R. Acad. Sci. Paris 178, 1464–1467 (1924).
  11. M. Reisz, "Sur les fonctions conjugées, " Math. Z. 27, 218–244 (1927).
  12. G. H. Hardy and J. E. Littlewood, "A point in the theory of conjugate functions," J. Lond. Math. Soc. 4, 242–245 (1929).
  13. A. Papoulis, The Fourier Integral and its Applications (Mc- Graw-Hill, New York, 1962), Chap. 10.
  14. H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.
  15. M. Altarelli and D. Y. Smith, "Superconvergence and sum rules for the optical constants: Physical meaning, comparison with experiment, and generalization," Phys. Rev. B 9, 1290–1298 (1974).
  16. J. Roth, B. Rao, and M. J. Dignam, "Application of the causality condition to thin film spectroscopy. A method for the evaluation of the thickness and optical constants, " J. Chem. Soc. Faraday Trans. 2, 71, 86–94 (1975).
  17. F. W. King, "Sum rules for the optical constants, " J. Math. Phys. 17, 1509–1514 (1976).
  18. M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).
  19. D. W. Johnson, "A Fourier series method for numerical Kramers-Kronig analysis, " J. Phys. A 8, 490–495 (1975).

1976 (1)

F. W. King, "Sum rules for the optical constants, " J. Math. Phys. 17, 1509–1514 (1976).

1975 (2)

D. W. Johnson, "A Fourier series method for numerical Kramers-Kronig analysis, " J. Phys. A 8, 490–495 (1975).

J. Roth, B. Rao, and M. J. Dignam, "Application of the causality condition to thin film spectroscopy. A method for the evaluation of the thickness and optical constants, " J. Chem. Soc. Faraday Trans. 2, 71, 86–94 (1975).

1974 (1)

M. Altarelli and D. Y. Smith, "Superconvergence and sum rules for the optical constants: Physical meaning, comparison with experiment, and generalization," Phys. Rev. B 9, 1290–1298 (1974).

1973 (1)

1972 (1)

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).

1942 (1)

K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics. II. Direct current characteristics, " J. Chem. Phys. 10, 98–105 (1942).

1941 (1)

B. Gross, "On the theory of dielectric loss, " Phys. Rev. 59, 748–750 (1941).

1929 (1)

G. H. Hardy and J. E. Littlewood, "A point in the theory of conjugate functions," J. Lond. Math. Soc. 4, 242–245 (1929).

1927 (1)

M. Reisz, "Sur les fonctions conjugées, " Math. Z. 27, 218–244 (1927).

1925 (1)

E. C. Titchmarsh, "Conjugate trigonometrical integrals," Proc. Lond. Math. Soc. 24, 109–130 (1925).

1924 (1)

M. Reisz, "Les fonctions conjuguées et les séries de Fourier, " C. R. Acad. Sci. Paris 178, 1464–1467 (1924).

Altarelli, M.

M. Altarelli and D. Y. Smith, "Superconvergence and sum rules for the optical constants: Physical meaning, comparison with experiment, and generalization," Phys. Rev. B 9, 1290–1298 (1974).

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).

Brot, C.

C. Brot, "Correlation functions in dipolar absorption-dispersion," in Dielectric and Related Molecular Processes, edited by M. Davies (Chemical Society Specialist Periodical Reports, 1973), Vol. 2, p. 7.

Cole, K. S.

K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics. II. Direct current characteristics, " J. Chem. Phys. 10, 98–105 (1942).

Cole, R. H.

K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics. II. Direct current characteristics, " J. Chem. Phys. 10, 98–105 (1942).

Dexter, D. L.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).

Dignam, M. J.

J. Roth, B. Rao, and M. J. Dignam, "Application of the causality condition to thin film spectroscopy. A method for the evaluation of the thickness and optical constants, " J. Chem. Soc. Faraday Trans. 2, 71, 86–94 (1975).

Gross, B.

B. Gross, "On the theory of dielectric loss, " Phys. Rev. 59, 748–750 (1941).

Hardy, G. H.

G. H. Hardy and J. E. Littlewood, "A point in the theory of conjugate functions," J. Lond. Math. Soc. 4, 242–245 (1929).

Johnson, D. W.

D. W. Johnson, "A Fourier series method for numerical Kramers-Kronig analysis, " J. Phys. A 8, 490–495 (1975).

King, F. W.

F. W. King, "Sum rules for the optical constants, " J. Math. Phys. 17, 1509–1514 (1976).

Knight, B. W.

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 256.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 256.

Littlewood, J. E.

G. H. Hardy and J. E. Littlewood, "A point in the theory of conjugate functions," J. Lond. Math. Soc. 4, 242–245 (1929).

Nilsson, P. O.

P. O. Nilsson, "Optical Properties of metals and alloys," Solid State Physics, Vol. 29, Edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1974), pp. 139–234.

Nussenzveig, H. M.

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.

Papoulis, A.

A. Papoulis, The Fourier Integral and its Applications (Mc- Graw-Hill, New York, 1962), Chap. 10.

Peterson, C. W.

Rao, B.

J. Roth, B. Rao, and M. J. Dignam, "Application of the causality condition to thin film spectroscopy. A method for the evaluation of the thickness and optical constants, " J. Chem. Soc. Faraday Trans. 2, 71, 86–94 (1975).

Reisz, M.

M. Reisz, "Sur les fonctions conjugées, " Math. Z. 27, 218–244 (1927).

M. Reisz, "Les fonctions conjuguées et les séries de Fourier, " C. R. Acad. Sci. Paris 178, 1464–1467 (1924).

Roth, J.

J. Roth, B. Rao, and M. J. Dignam, "Application of the causality condition to thin film spectroscopy. A method for the evaluation of the thickness and optical constants, " J. Chem. Soc. Faraday Trans. 2, 71, 86–94 (1975).

Scaife, B. K. P.

B. K. P. Scaife, "The theory of the macroscopic properties of isotropic dielectrics, " in Dielectric and Related Molecular Processes, edited by M. Davies (The Chemical Society Specialist Periodical Reports, 1972), Vol. 1, pp. 1–20, Eqs. (32) and (44).

Smith, D. Y.

M. Altarelli and D. Y. Smith, "Superconvergence and sum rules for the optical constants: Physical meaning, comparison with experiment, and generalization," Phys. Rev. B 9, 1290–1298 (1974).

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).

Titchmarsh, E. C.

E. C. Titchmarsh, "Conjugate trigonometrical integrals," Proc. Lond. Math. Soc. 24, 109–130 (1925).

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Clarendon, Oxford, 1948), Chap. 5.

C. R. Acad. Sci. Paris (1)

M. Reisz, "Les fonctions conjuguées et les séries de Fourier, " C. R. Acad. Sci. Paris 178, 1464–1467 (1924).

J. Chem. Phys. (1)

K. S. Cole and R. H. Cole, "Dispersion and absorption in dielectrics. II. Direct current characteristics, " J. Chem. Phys. 10, 98–105 (1942).

J. Chem. Soc. Faraday Trans. 2 (1)

J. Roth, B. Rao, and M. J. Dignam, "Application of the causality condition to thin film spectroscopy. A method for the evaluation of the thickness and optical constants, " J. Chem. Soc. Faraday Trans. 2, 71, 86–94 (1975).

J. Lond. Math. Soc. (1)

G. H. Hardy and J. E. Littlewood, "A point in the theory of conjugate functions," J. Lond. Math. Soc. 4, 242–245 (1929).

J. Math. Phys. (1)

F. W. King, "Sum rules for the optical constants, " J. Math. Phys. 17, 1509–1514 (1976).

J. Opt. Soc. Am. (1)

J. Phys. A (1)

D. W. Johnson, "A Fourier series method for numerical Kramers-Kronig analysis, " J. Phys. A 8, 490–495 (1975).

Math. Z. (1)

M. Reisz, "Sur les fonctions conjugées, " Math. Z. 27, 218–244 (1927).

Phys. Rev. (1)

B. Gross, "On the theory of dielectric loss, " Phys. Rev. 59, 748–750 (1941).

Phys. Rev. B (2)

M. Altarelli and D. Y. Smith, "Superconvergence and sum rules for the optical constants: Physical meaning, comparison with experiment, and generalization," Phys. Rev. B 9, 1290–1298 (1974).

M. Altarelli, D. L. Dexter, H. M. Nussenzveig, and D. Y. Smith, "Superconvergence and sum rules for the optical constants, " Phys. Rev. B 12, 4502–4509 (1972).

Proc. Lond. Math. Soc. (1)

E. C. Titchmarsh, "Conjugate trigonometrical integrals," Proc. Lond. Math. Soc. 24, 109–130 (1925).

Other (7)

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Clarendon, Oxford, 1948), Chap. 5.

C. Brot, "Correlation functions in dipolar absorption-dispersion," in Dielectric and Related Molecular Processes, edited by M. Davies (Chemical Society Specialist Periodical Reports, 1973), Vol. 2, p. 7.

B. K. P. Scaife, "The theory of the macroscopic properties of isotropic dielectrics, " in Dielectric and Related Molecular Processes, edited by M. Davies (The Chemical Society Specialist Periodical Reports, 1972), Vol. 1, pp. 1–20, Eqs. (32) and (44).

A. Papoulis, The Fourier Integral and its Applications (Mc- Graw-Hill, New York, 1962), Chap. 10.

H. M. Nussenzveig, Causality and Dispersion Relations (Academic, New York, 1972), Chap. 1.

P. O. Nilsson, "Optical Properties of metals and alloys," Solid State Physics, Vol. 29, Edited by H. Ehrenreich, F. Seitz, and D. Turnbull (Academic, New York, 1974), pp. 139–234.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, Oxford, 1960), p. 256.

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