Abstract

A method of phase retrieval from the experimental modulus |χ(3)| spectra of third-order nonlinear optical susceptibility is presented on the basis of the maximum-entropy model. This method enables one to derive the real and the imaginary parts of χ(3) without using the nonlinear Kramers–Kronig calculation, which is usually hazardous in nonlinear optical spectroscopy because of the limited range of measurements. Theoretical and experimental modulus spectra of polysilane are analyzed. The results from the experimental modulus are compared with the Kramers–Kronig calculations as well as with the measured phase values. It is also shown how the method can be optimized to reduce the distortions that result from noise in the calculated spectra.

© 1996 Optical Society of America

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  1. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass, 1960).
  2. D. Y. Smith, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 35–68.
    [CrossRef]
  3. H. Ehrenreich, H. R. Phillip, and B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
    [CrossRef]
  4. Sh. M. Kogan, “On the electrodynamics of weakly nonlinear media,” Sov. Phys. JETP 16, 217–220 (1963).
  5. P. J. Price, “Theory of quadratic response functions,” Phys. Rev. 130, 1792–1797 (1963).
    [CrossRef]
  6. W. L. Caspers, “Dispersion relations for nonlinear response,” Phys. Rev. 133, A1249–A1251 (1964).
    [CrossRef]
  7. F. L. Ridener and R. H. Good, “Dispersion relations for third-degree nonlinear systems,” Phys. Rev. B 10, 4980–4987 (1974).
    [CrossRef]
  8. F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
    [CrossRef]
  9. K.-E. Peiponen, “Nonlinear susceptibilities as a function of several complex angular-frequency variables,” Phys. Rev. B 37, 6463–6467 (1988).
    [CrossRef]
  10. H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
    [CrossRef] [PubMed]
  11. P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
    [CrossRef]
  12. E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
    [CrossRef]
  13. D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
    [CrossRef]
  14. E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
    [CrossRef]
  15. E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Comparison between the optical constants obtained by the Kramers–Kronig analysis and the maximum entropy method: infrared optical properties of orthorhombic sulfur,” Appl. Opt. 32, 1126–1129 (1993).
    [CrossRef] [PubMed]
  16. E. M. Vartiainen, T. Asakura, and K.-E. Peiponen, “Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy,” Opt. Commun. 104, 149–156 (1993).
    [CrossRef]
  17. E. M. Vartiainen, “Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis,” J. Opt. Soc. Am. B 9, 1209–1214 (1992).
    [CrossRef]
  18. T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
    [CrossRef] [PubMed]
  19. R. D. Miller and J. Michl, “Polysilane high polymers,” Chem. Rev. 89, 1359–1410 (1989).
    [CrossRef]
  20. F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
    [CrossRef]
  21. C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
    [CrossRef]
  22. E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957).
    [CrossRef]
  23. E. T. Jaynes, “On the rationale of maximum-entropy methods,” Proc. IEEE 70, 939–952 (1982).
    [CrossRef]
  24. S. F. Gull and G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
    [CrossRef]
  25. R. K. Bryan and J. Skilling, “Maximum entropy image reconstruction from phaseless Fourier data,” Opt. Acta 33, 287–299 (1986).
    [CrossRef]
  26. N. L. Bonavito, J. E. Dorband, and T. Busse, “Maximum entropy restoration of blurred and oversaturated Hubble Space Telescope imagery,” Appl. Opt. 32, 5768–5774 (1993).
    [CrossRef] [PubMed]
  27. D. M. Collins, “Electron density images from imperfect data by iterative entropy maximization,” Nature 298, 49(1982).
    [CrossRef]
  28. J. Navaza, “On the maximum-entropy estimate of the electron density function,” Acta Crystallogr. Sect. A 41, 232–244 (1985).
    [CrossRef]
  29. M. C. Kemp, “Maximum entropy reconstruction in emission tomography,” in Medical Radionucleaic Imaging 1980, (IAEA, Vienna, 1981), Vol. 1, pp. 128–141.
  30. T. Matsumoto, T. Tanji, and A. Tonomura, “Application of maximum entropy method (MEM) to in-line Fresnel electron holography,” Optik 97, 169–173 (1994).
  31. J. P. Burg, “Maximum entropy spectral analysis,” in Modern Spectrum Analysis, D. G. Childers, ed. (Institute of Electrical and Electronics Engineers, New York, 1978), pp. 34–41.
  32. S. Haykin and S. Kesler, “Prediction-error filtering and maximum-entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.
  33. A. van den Bos, “Alternative interpretation of maximum entropy spectral analysis,” IEEE Trans. Inf. Theory IT-17, 493–494 (1971).
    [CrossRef]
  34. J. K. Kauppinen, D. F. Moffat, M. R. Hollberg, and H. H. Mantch, “A new line-narrowing procedure based on Fourier self-deconvolution, maximum entropy, and linear prediction,” Appl. Spectrosc. 45, 411–416 (1991).
    [CrossRef]
  35. W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989).
  36. D. M. Roesller, “Kramers–Kronig analysis of reflection data,” Brit. J. Appl. Phys. 16, 1119–1123 (1965).
    [CrossRef]
  37. A. L. Harris, C. E. D. Chidsey, N. J. Levinson, and D. N. Loiacono, “Monolayer vibrational spectroscopy by infrared–visible sum generation at metal and semiconductor surfaces,” Chem. Phys. Lett. 141, 350 (1987).
    [CrossRef]

1994 (3)

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
[CrossRef]

T. Matsumoto, T. Tanji, and A. Tonomura, “Application of maximum entropy method (MEM) to in-line Fresnel electron holography,” Optik 97, 169–173 (1994).

1993 (4)

E. M. Vartiainen, T. Asakura, and K.-E. Peiponen, “Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy,” Opt. Commun. 104, 149–156 (1993).
[CrossRef]

N. L. Bonavito, J. E. Dorband, and T. Busse, “Maximum entropy restoration of blurred and oversaturated Hubble Space Telescope imagery,” Appl. Opt. 32, 5768–5774 (1993).
[CrossRef] [PubMed]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Comparison between the optical constants obtained by the Kramers–Kronig analysis and the maximum entropy method: infrared optical properties of orthorhombic sulfur,” Appl. Opt. 32, 1126–1129 (1993).
[CrossRef] [PubMed]

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

1992 (4)

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

E. M. Vartiainen, “Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis,” J. Opt. Soc. Am. B 9, 1209–1214 (1992).
[CrossRef]

1991 (1)

1989 (1)

R. D. Miller and J. Michl, “Polysilane high polymers,” Chem. Rev. 89, 1359–1410 (1989).
[CrossRef]

1988 (1)

K.-E. Peiponen, “Nonlinear susceptibilities as a function of several complex angular-frequency variables,” Phys. Rev. B 37, 6463–6467 (1988).
[CrossRef]

1987 (1)

A. L. Harris, C. E. D. Chidsey, N. J. Levinson, and D. N. Loiacono, “Monolayer vibrational spectroscopy by infrared–visible sum generation at metal and semiconductor surfaces,” Chem. Phys. Lett. 141, 350 (1987).
[CrossRef]

1986 (2)

R. K. Bryan and J. Skilling, “Maximum entropy image reconstruction from phaseless Fourier data,” Opt. Acta 33, 287–299 (1986).
[CrossRef]

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

1985 (1)

J. Navaza, “On the maximum-entropy estimate of the electron density function,” Acta Crystallogr. Sect. A 41, 232–244 (1985).
[CrossRef]

1982 (2)

D. M. Collins, “Electron density images from imperfect data by iterative entropy maximization,” Nature 298, 49(1982).
[CrossRef]

E. T. Jaynes, “On the rationale of maximum-entropy methods,” Proc. IEEE 70, 939–952 (1982).
[CrossRef]

1978 (1)

S. F. Gull and G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[CrossRef]

1975 (1)

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

1974 (1)

F. L. Ridener and R. H. Good, “Dispersion relations for third-degree nonlinear systems,” Phys. Rev. B 10, 4980–4987 (1974).
[CrossRef]

1971 (1)

A. van den Bos, “Alternative interpretation of maximum entropy spectral analysis,” IEEE Trans. Inf. Theory IT-17, 493–494 (1971).
[CrossRef]

1965 (1)

D. M. Roesller, “Kramers–Kronig analysis of reflection data,” Brit. J. Appl. Phys. 16, 1119–1123 (1965).
[CrossRef]

1964 (1)

W. L. Caspers, “Dispersion relations for nonlinear response,” Phys. Rev. 133, A1249–A1251 (1964).
[CrossRef]

1963 (3)

H. Ehrenreich, H. R. Phillip, and B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Sh. M. Kogan, “On the electrodynamics of weakly nonlinear media,” Sov. Phys. JETP 16, 217–220 (1963).

P. J. Price, “Theory of quadratic response functions,” Phys. Rev. 130, 1792–1797 (1963).
[CrossRef]

1957 (1)

E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957).
[CrossRef]

1948 (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
[CrossRef]

Abe, S.

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Asakura, T.

E. M. Vartiainen, T. Asakura, and K.-E. Peiponen, “Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy,” Opt. Commun. 104, 149–156 (1993).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Comparison between the optical constants obtained by the Kramers–Kronig analysis and the maximum entropy method: infrared optical properties of orthorhombic sulfur,” Appl. Opt. 32, 1126–1129 (1993).
[CrossRef] [PubMed]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

Bonavito, N. L.

Bryan, R. K.

R. K. Bryan and J. Skilling, “Maximum entropy image reconstruction from phaseless Fourier data,” Opt. Acta 33, 287–299 (1986).
[CrossRef]

Burg, J. P.

J. P. Burg, “Maximum entropy spectral analysis,” in Modern Spectrum Analysis, D. G. Childers, ed. (Institute of Electrical and Electronics Engineers, New York, 1978), pp. 34–41.

Busse, T.

Caspers, W. L.

W. L. Caspers, “Dispersion relations for nonlinear response,” Phys. Rev. 133, A1249–A1251 (1964).
[CrossRef]

Chidsey, C. E. D.

A. L. Harris, C. E. D. Chidsey, N. J. Levinson, and D. N. Loiacono, “Monolayer vibrational spectroscopy by infrared–visible sum generation at metal and semiconductor surfaces,” Chem. Phys. Lett. 141, 350 (1987).
[CrossRef]

Collins, D. M.

D. M. Collins, “Electron density images from imperfect data by iterative entropy maximization,” Nature 298, 49(1982).
[CrossRef]

Daniel, G. J.

S. F. Gull and G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[CrossRef]

Dorband, J. E.

Ehrenreich, H.

H. Ehrenreich, H. R. Phillip, and B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989).

Good, R. H.

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

F. L. Ridener and R. H. Good, “Dispersion relations for third-degree nonlinear systems,” Phys. Rev. B 10, 4980–4987 (1974).
[CrossRef]

Gull, S. F.

S. F. Gull and G. J. Daniel, “Image reconstruction from incomplete and noisy data,” Nature 272, 686–690 (1978).
[CrossRef]

Hadjichristov, G. B.

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

Hagan, D. J.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

Harris, A. L.

A. L. Harris, C. E. D. Chidsey, N. J. Levinson, and D. N. Loiacono, “Monolayer vibrational spectroscopy by infrared–visible sum generation at metal and semiconductor surfaces,” Chem. Phys. Lett. 141, 350 (1987).
[CrossRef]

Hasegawa, T.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Haykin, S.

S. Haykin and S. Kesler, “Prediction-error filtering and maximum-entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.

Hollberg, M. R.

Hutchings, D. C.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

Iwasa, Y.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Jaynes, E. T.

E. T. Jaynes, “On the rationale of maximum-entropy methods,” Proc. IEEE 70, 939–952 (1982).
[CrossRef]

E. T. Jaynes, “Information theory and statistical mechanics,” Phys. Rev. 106, 620–630 (1957).
[CrossRef]

Kajzar, F.

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

Kauppinen, J. K.

Kemp, M. C.

M. C. Kemp, “Maximum entropy reconstruction in emission tomography,” in Medical Radionucleaic Imaging 1980, (IAEA, Vienna, 1981), Vol. 1, pp. 128–141.

Kesler, S.

S. Haykin and S. Kesler, “Prediction-error filtering and maximum-entropy spectral estimation,” in Nonlinear Methods of Spectral Analysis, 2nd ed., S. Haykin, ed. (Springer-Verlag, Berlin, 1983), pp. 9–72.

Kircheva, P. P.

P. P. Kircheva and G. B. Hadjichristov, “Kramers–Kronig relations in FWM spectroscopy,” J. Phys. B 27, 3781–3793 (1994).
[CrossRef]

Kishida, H.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

Koda, T.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Kogan, Sh. M.

Sh. M. Kogan, “On the electrodynamics of weakly nonlinear media,” Sov. Phys. JETP 16, 217–220 (1963).

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass, 1960).

Levinson, N. J.

A. L. Harris, C. E. D. Chidsey, N. J. Levinson, and D. N. Loiacono, “Monolayer vibrational spectroscopy by infrared–visible sum generation at metal and semiconductor surfaces,” Chem. Phys. Lett. 141, 350 (1987).
[CrossRef]

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Addison-Wesley, Reading, Mass, 1960).

Loiacono, D. N.

A. L. Harris, C. E. D. Chidsey, N. J. Levinson, and D. N. Loiacono, “Monolayer vibrational spectroscopy by infrared–visible sum generation at metal and semiconductor surfaces,” Chem. Phys. Lett. 141, 350 (1987).
[CrossRef]

Mantch, H. H.

Matsumoto, M.

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Matsumoto, T.

T. Matsumoto, T. Tanji, and A. Tonomura, “Application of maximum entropy method (MEM) to in-line Fresnel electron holography,” Optik 97, 169–173 (1994).

Messier, J.

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

Michl, J.

R. D. Miller and J. Michl, “Polysilane high polymers,” Chem. Rev. 89, 1359–1410 (1989).
[CrossRef]

Miller, R. D.

R. D. Miller and J. Michl, “Polysilane high polymers,” Chem. Rev. 89, 1359–1410 (1989).
[CrossRef]

Moffat, D. F.

Navaza, J.

J. Navaza, “On the maximum-entropy estimate of the electron density function,” Acta Crystallogr. Sect. A 41, 232–244 (1985).
[CrossRef]

Peiponen, K.-E.

E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Comparison between the optical constants obtained by the Kramers–Kronig analysis and the maximum entropy method: infrared optical properties of orthorhombic sulfur,” Appl. Opt. 32, 1126–1129 (1993).
[CrossRef] [PubMed]

E. M. Vartiainen, T. Asakura, and K.-E. Peiponen, “Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy,” Opt. Commun. 104, 149–156 (1993).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

K.-E. Peiponen, “Nonlinear susceptibilities as a function of several complex angular-frequency variables,” Phys. Rev. B 37, 6463–6467 (1988).
[CrossRef]

Phillip, H. R.

H. Ehrenreich, H. R. Phillip, and B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989).

Price, P. J.

P. J. Price, “Theory of quadratic response functions,” Phys. Rev. 130, 1792–1797 (1963).
[CrossRef]

Ridener, F. L.

F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
[CrossRef]

F. L. Ridener and R. H. Good, “Dispersion relations for third-degree nonlinear systems,” Phys. Rev. B 10, 4980–4987 (1974).
[CrossRef]

Roesller, D. M.

D. M. Roesller, “Kramers–Kronig analysis of reflection data,” Brit. J. Appl. Phys. 16, 1119–1123 (1965).
[CrossRef]

Rosilio, C.

F. Kajzar, J. Messier, and C. Rosilio, “Nonlinear optical properties of thin films of polysilane,” J. Appl. Phys. 60, 3040–3044 (1986).
[CrossRef]

Segall, B.

H. Ehrenreich, H. R. Phillip, and B. Segall, “Optical properties of aluminum,” Phys. Rev. 132, 1918–1928 (1963).
[CrossRef]

Shannon, C.

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
[CrossRef]

Sheik-Bahae, M.

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

Skilling, J.

R. K. Bryan and J. Skilling, “Maximum entropy image reconstruction from phaseless Fourier data,” Opt. Acta 33, 287–299 (1986).
[CrossRef]

Smith, D. Y.

D. Y. Smith, in Handbook of Optical Constants of Solids, E. D. Palik, ed. (Academic, New York, 1985), pp. 35–68.
[CrossRef]

Sunamura, H.

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Tachibana, H.

T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

Tanji, T.

T. Matsumoto, T. Tanji, and A. Tonomura, “Application of maximum entropy method (MEM) to in-line Fresnel electron holography,” Optik 97, 169–173 (1994).

Teukolsky, S. A.

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H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
[CrossRef] [PubMed]

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[CrossRef] [PubMed]

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T. Matsumoto, T. Tanji, and A. Tonomura, “Application of maximum entropy method (MEM) to in-line Fresnel electron holography,” Optik 97, 169–173 (1994).

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E. M. Vartiainen, T. Asakura, and K.-E. Peiponen, “Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy,” Opt. Commun. 104, 149–156 (1993).
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[CrossRef]

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes (Cambridge U. Press, Cambridge, UK, 1989).

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E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, “Maximum entropy model in reflection spectra analysis,” Opt. Commun. 89, 37–40 (1992).
[CrossRef]

E. M. Vartiainen, T. Asakura, and K.-E. Peiponen, “Generalized noniterative maximum entropy procedure for phase retrieval problems in optical spectroscopy,” Opt. Commun. 104, 149–156 (1993).
[CrossRef]

Opt. Quantum Electron. (1)

D. C. Hutchings, M. Sheik-Bahae, D. J. Hagan, and E. W. van Stryland, “Kramers–Krönig relations in nonlinear optics,” Opt. Quantum Electron. 24, 1–30 (1992).
[CrossRef]

Optik (1)

T. Matsumoto, T. Tanji, and A. Tonomura, “Application of maximum entropy method (MEM) to in-line Fresnel electron holography,” Optik 97, 169–173 (1994).

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[CrossRef]

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T. Hasegawa, Y. Iwasa, H. Sunamura, T. Koda, Y. Tokura, H. Tachibana, M. Matsumoto, and S. Abe, “Nonlinear optical spectroscopy on one-dimensional excitons in silicon polymer, polysilane,” Phys. Rev. Lett. 69, 668–671 (1992).
[CrossRef] [PubMed]

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relations in the third-order electric susceptibility for polysilane film,” Phys. Rev. Lett. 70, 3724–3727 (1993).
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Figures (10)

Fig. 1
Fig. 1

(a) Theoretical noise-free (solid curve) and noisy (dots; RMS error is 8.3%) |χ(3)(−3ω;, ω, ω, ω)| as a function of ω ∈ [0.5 eV, 2.2 eV]. (b) Experimental modulus |χ(3)(−3ω; ω, ω, ω)| of the PDHS polysilane (dots) and its averaged spline fit (solid curve).

Fig. 2
Fig. 2

RMS error versus the order of M in the MEM estimates of theoretical spectra |χ(3)(−3ω; ω, ω, ω)|2 with and without noise (amount given by RMS error): (a) ERMS = 0 and (b) ERMS = 8.3% (SNR = 12).

Fig. 3
Fig. 3

RMS error in the MEM estimates of theoretical noisy |χ(3)|2 compared with the noisy original spectrum (curve 1) and with the noiseless spectrum (curve 2).

Fig. 4
Fig. 4

Theoretical noisy modulus |χ(3)| and its MEM estimates with different M values: Mmax = 1700 and Mopt = 91.

Fig. 5
Fig. 5

(a) Phase of the theoretical χ(3) (solid curve) and the corresponding experimental phase values (dots). (b) Theoretical (noise-free) error-phase curve obtained with (dots) and without (circles) the squeezing procedure; solid and dotted lines correspond to linear-regression fits. (c) (Squeezed) error phase values in the noisy case with M = Mmax (dots) and with M = Mopt (circles) and their linear-regression fit (solid line).

Fig. 6
Fig. 6

(a) Real and (b) imaginary parts of χ(3), whose modulus is shown in Fig. 1 (the topmost curves), and the corresponding MEM estimates (M = Mopt) with L = 1 and L = 3. The additional information used for the MEM estimates is the phase values at energies indicated by the arrows.

Fig. 7
Fig. 7

RMS error versus the order of M in the MEM estimates of the experimental squared modulus |χ(3)|2 in the range M ∈ (10, 110).

Fig. 8
Fig. 8

Experimental modulus |χ(3)| of PDHS (dots) and its spline fit (topmost solid curve) along with the MEM estimates with different M values: Mmax = 1500 and Mopt = 72.

Fig. 9
Fig. 9

Error-phase values of the MEM estimate with M = Mmax (dots) and with M = Mopt (circles) for the experimental susceptibility χ(3). The solid line is the linear-regression fit.

Fig. 10
Fig. 10

(a) Experimental real and (b) imaginary part values (dots and open circles) of the susceptibility χ(3) of PDHS and the corresponding curves obtained by the Kramers–Kronig analysis (dotted curves) and the MEPR procedure (solid curves) with Mopt = 72. The additional information used for the MEM estimates with L = 1 is the experimental phase values at energies pointed by the arrows. The undermost MEM curves (with L = 3) are computed with all the available experimental phase data.

Equations (24)

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x m x ( t m ) , 0 m M ,
R ( m ) = E [ x n * x n + m ] ,
S ( f ) = Δ t m = R ( m ) exp ( i 2 π m f Δ t ) ,
R ( m ) = 0 1 / Δ t S ( f ) exp ( i 2 π m f Δ t ) d f ,
S ( ν ) = m = R ( m ) exp ( i 2 π m ν ) ,
R ( m ) = 0 1 S ( ν ) exp ( i 2 π m ν ) d ν ,
h 0 1 log   S ( ν ) d ν ,
R ( m ) 0 1 log   S ˆ ( ν ) d ν =0, | m | > M .
S ˆ ( ν ) = | β | 2 1 + k = 1 M a k   exp ( i 2 π k ν ) 2 ,
R ( 0 ) R ( 1 ) R ( M ) R ( 1 ) R ( 0 ) R ( 1 M ) R ( M ) R ( M 1 ) R ( 0 )   1 a 1 a M = | β | 2 0 0 .
x m = k = 1 M a k x m k + e m ,
x ( t ) = a ( τ ) x ( t τ ) d τ + e ( t ) = a ( t ) * x ( t ) + e ( t ) ,
X ( f ) = { x ( t ) } = { a ( t ) } X ( f ) + E ( f ) ,
X ( f ) = E ( f ) 1 + { a ( t ) } .
f ( ν ) = E ( ν ) 1 + k = 1 M a k   exp ( i 2 π k ν ) .
E ( ν ) = | β | exp [ i ϕ ( ν ) ] ,
χ ˆ ( 3 ) ( ν ) = | β | exp [ i ϕ ( ν ) ] 1 + k = 1 M a k   exp ( i 2 π k ν ) .
ϕ ( ν ) = B 0 + B 1 ν + + B L ν L = l = 0 L B l ν l ,
1 ν 0 ν 0 L 1 ν 1 ν 1 1 ν L ν L L   B 0 B 1 B L = ϕ ( ν 0 ) ϕ ( ν 1 ) ϕ ( ν L ) .
| χ K ( ν ) | | χ ( 3 ) ( ω 1 ) | , 0 ν < z K ( ω 1 ) | χ ( 3 ) ( ω ) | , z K ( ω 1 ) ν z K ( ω 2 ) , | χ ( 3 ) ( ω 2 ) | , z K ( ω 2 ) < ν 1
z K ( ω ) = ( 2 K + 1 ) 1 ω ω 1 ω 2 ω 1 + K ,
ν = z K ( ω ) z K ( ω 1 ) z K ( ω 2 ) z K ( ω 1 ) .
R ( m ) = N 1 k = 0 N 1 | χ k ( 3 ) | 2   exp ( i 2 π k m / N ) ,
E RMS 2 ( M ) = k = 0 N 1 | S ˆ ( ν k ; M ) S 0 ( ν k ) | 2 k = 0 N 1 | S 0 ( ν k ) | 2 ,

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