Abstract

When the real (imaginary) part of the transfer function of a causal linear system is known for all frequencies, the imaginary (real) part may be calculated for all frequencies from the Kramers-Kronig relations. For physical systems, the real part could be the refractive index or the amplitude, and the imaginary part could be the extinction coefficient or the phase. Experimentally these quantities are known only for limited-frequency intervals. This paper presents generalized Kramers-Kronig relations, from which the real and imaginary parts may be calculated for all frequencies from knowledge of these parts for at least partly overlapping frequency intervals. When the procedure is applied to experimental data, errors are introduced. Certain types of errors of the known real and imaginary parts completely destroy the possibility of calculating the unknown parts, whereas others give negligible errors. The existence of a filter with the property of allowing the restoration of a truncated spectrum is established. The transfer function and the impulse response function of this filter are given.

© 1982 Optical Society of America

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  1. R. de L. Kronig, "On the theory of dispersion of x-rays," J. Opt. Soc. Am. 12, 547–557 (1926).
  2. H. A. M. Kramers, "La diffusion de la lumiere par les atomes," in Atti del Congresso Internazionale dei Fisici, Settembre 1927 (N. Zanichelli, Bologna, Italy, 1928), pp. 545–557.
  3. H. A. Kramers, "Die Dispersion und Absorption von Röntgenstralen," Phys. Z. 30, 522–523 (1929).
  4. J. Tauc, "Optical properties of semiconductors" in Proceedings of the International School of Physics "Enrico Fermi," The Optical Properties of Solids, J. Tauc, ed. (Academic, New York, London, 1966), pp. 63–89.
  5. T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.
  6. H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. XIV.
  7. R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Tokyo, 1960), Chap. 11.
  8. D. J. Patil, "Representation of HP-functions," Bull. Am. Math. Soc. 78, 617–620 (1972).
  9. D. J. Patil, "Recapturing H2-functions on a polydisc," Trans. Am. Math. Soc. 188, 97–103 (1974).
  10. R. N. Mukherjee, "Representations of H2-functions on the real line," Boll. U.M.I. 10, 666–671 (1974).
  11. R. Z. Bachrach and F. C. Brown, "Exciton-optical properties of TlBr and TlCl," Phys. Rev. B 1, 818–831 (1970).
  12. R. K. Ahrenkiel, "Modified Kramers-Kronig analysis of optical spectra," J. Opt. Soc. Am. 61, 1651–1655 (1971).
  13. E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford U. Press, London, 1959), Chap. V (especially Theorem 95).
  14. G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.
  15. A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).
  16. H. Riesel, Department of Numerical and Computing Science, Royal Institute of Technology, Stockholm, Sweden (personal communication).
  17. R. W. Schafer, R. M. Mersereau, and M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).
  18. I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, Scotland, 1956), Chap. IV.
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.
  20. 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).
  21. See Ref. 13, especially Chap. V, Sec. 1.

1974 (2)

D. J. Patil, "Recapturing H2-functions on a polydisc," Trans. Am. Math. Soc. 188, 97–103 (1974).

R. N. Mukherjee, "Representations of H2-functions on the real line," Boll. U.M.I. 10, 666–671 (1974).

1973 (1)

1972 (1)

D. J. Patil, "Representation of HP-functions," Bull. Am. Math. Soc. 78, 617–620 (1972).

1971 (1)

1970 (1)

R. Z. Bachrach and F. C. Brown, "Exciton-optical properties of TlBr and TlCl," Phys. Rev. B 1, 818–831 (1970).

1929 (1)

H. A. Kramers, "Die Dispersion und Absorption von Röntgenstralen," Phys. Z. 30, 522–523 (1929).

1926 (1)

Ahrenkiel, R. K.

Bachrach, R. Z.

R. Z. Bachrach and F. C. Brown, "Exciton-optical properties of TlBr and TlCl," Phys. Rev. B 1, 818–831 (1970).

Björck, Å.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.

Bode, H. W.

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. XIV.

Brown, F. C.

R. Z. Bachrach and F. C. Brown, "Exciton-optical properties of TlBr and TlCl," Phys. Rev. B 1, 818–831 (1970).

Burrell, G. J.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

Churchill, R. V.

R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Tokyo, 1960), Chap. 11.

Dahlquist, G.

G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.

Ellis, B.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

Gradshteyn, I. S.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.

Knight, B. W.

Kramers, H. A.

H. A. Kramers, "Die Dispersion und Absorption von Röntgenstralen," Phys. Z. 30, 522–523 (1929).

Kramers, H. A. M.

H. A. M. Kramers, "La diffusion de la lumiere par les atomes," in Atti del Congresso Internazionale dei Fisici, Settembre 1927 (N. Zanichelli, Bologna, Italy, 1928), pp. 545–557.

Kronig, R. de L.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, and M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).

Moss, T. S.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

Mukherjee, R. N.

R. N. Mukherjee, "Representations of H2-functions on the real line," Boll. U.M.I. 10, 666–671 (1974).

Patil, D. J.

D. J. Patil, "Recapturing H2-functions on a polydisc," Trans. Am. Math. Soc. 188, 97–103 (1974).

D. J. Patil, "Representation of HP-functions," Bull. Am. Math. Soc. 78, 617–620 (1972).

Peterson, C. W.

Richards, M. A.

R. W. Schafer, R. M. Mersereau, and M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).

Riesel, H.

H. Riesel, Department of Numerical and Computing Science, Royal Institute of Technology, Stockholm, Sweden (personal communication).

Ryzhik, I. M.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, and M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).

Secrest, D.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).

Sneddon, I. N.

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, Scotland, 1956), Chap. IV.

Stroud, A. H.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).

Tauc, J.

J. Tauc, "Optical properties of semiconductors" in Proceedings of the International School of Physics "Enrico Fermi," The Optical Properties of Solids, J. Tauc, ed. (Academic, New York, London, 1966), pp. 63–89.

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford U. Press, London, 1959), Chap. V (especially Theorem 95).

Boll. U.M.I. (1)

R. N. Mukherjee, "Representations of H2-functions on the real line," Boll. U.M.I. 10, 666–671 (1974).

Bull. Am. Math. Soc. (1)

D. J. Patil, "Representation of HP-functions," Bull. Am. Math. Soc. 78, 617–620 (1972).

J. Opt. Soc. Am. (3)

Phys. Rev. B (1)

R. Z. Bachrach and F. C. Brown, "Exciton-optical properties of TlBr and TlCl," Phys. Rev. B 1, 818–831 (1970).

Phys. Z. (1)

H. A. Kramers, "Die Dispersion und Absorption von Röntgenstralen," Phys. Z. 30, 522–523 (1929).

Trans. Am. Math. Soc. (1)

D. J. Patil, "Recapturing H2-functions on a polydisc," Trans. Am. Math. Soc. 188, 97–103 (1974).

Other (13)

H. A. M. Kramers, "La diffusion de la lumiere par les atomes," in Atti del Congresso Internazionale dei Fisici, Settembre 1927 (N. Zanichelli, Bologna, Italy, 1928), pp. 545–557.

J. Tauc, "Optical properties of semiconductors" in Proceedings of the International School of Physics "Enrico Fermi," The Optical Properties of Solids, J. Tauc, ed. (Academic, New York, London, 1966), pp. 63–89.

T. S. Moss, G. J. Burrell, and B. Ellis, Semiconductor Opto-Electronics (Butterworth, London, 1973), Chap. 2.

H. W. Bode, Network Analysis and Feedback Amplifier Design (Van Nostrand, New York, 1945), Chap. XIV.

R. V. Churchill, Complex Variables and Applications (McGraw-Hill, Tokyo, 1960), Chap. 11.

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford U. Press, London, 1959), Chap. V (especially Theorem 95).

G. Dahlquist and Å. Björck, Numerical Methods (Prentice Hall, Englewood Cliffs, N.J., 1974), Chaps. 4 and 5.

A. H. Stroud and D. Secrest, Gaussian Quadrature Formulas (Prentice Hall, Englewood Cliffs, N.J., 1966).

H. Riesel, Department of Numerical and Computing Science, Royal Institute of Technology, Stockholm, Sweden (personal communication).

R. W. Schafer, R. M. Mersereau, and M. A. Richards, "Constrained iterative restoration algorithms," Proc. IEEE 69, 432–450 (1981).

I. N. Sneddon, Special Functions of Mathematical Physics and Chemistry (Oliver and Boyd, Edinburgh, Scotland, 1956), Chap. IV.

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Entry 3.911.1.

See Ref. 13, especially Chap. V, Sec. 1.

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