Abstract

The dispersion theory for reflectivity of s- and p-polarized light is considered. The problem of phase retrieval from reflectance of p-polarized light, in the presence of complex zeros of the reflectivity in the upper half of complex frequency plane, is addressed. In such a case the traditional Kramers–Kronig relations are not valid, but the maximum-entropy method will provide the required solution.

© 2006 Optical Society of America

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  1. G. Kortüm, Reflectance Spectroscopy Principles, Methods, and Applications (Springer, 1969).
  2. J. Räty, K.-E. Peiponen, and T. Asakura, UV-Visible Reflection Spectroscopy of Liquids (Springer, 2004).
  3. R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).
  4. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).
  5. D. E. Aspnes, "The accurate determination of optical properties by ellipsometry," in Handbook of Optical Constants of Solids, E.Palik, ed. (Academic, 1995) pp. 69-112.
  6. K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
    [CrossRef]
  7. M. H. Lee, "Solving certain principal value integrals by reduction to the dilogarithm," Physica A 234, 581-588 (1996).
    [CrossRef]
  8. M. H. Lee and O. I. Sindoni, "Kramers-Kronig relations with logarithmic cernel and applications to the phase spectrum in the Drude model," Phys. Rev. E 56, 3891-3896 (1997).
    [CrossRef]
  9. F. W. King, "Efficient numerical approach to the evaluation of Kramers-Kronig transforms," J. Opt. Soc. Am. B 19, 2427-2436 (2002).
    [CrossRef]
  10. K. F. Palmer, M. Z. Williams, and B. A. Budde, "Multiply subtractive Kramers-Kronig analysis of optical data," Appl. Opt. 37, 2660-2673 (1998).
    [CrossRef]
  11. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibility," Opt. Commun. 218, 409-414 (2003).
    [CrossRef]
  12. V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility of polysilane," J. Chem. Phys. 119, 11095-11098 (2003).
    [CrossRef]
  13. K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. M. Vartiainen, "Kramers-Kronig relations and sum rules in nonlinear optics," Appl. Spectrosc. 58, 499-509 (2004).
    [CrossRef] [PubMed]
  14. P. Grosse and V. Offermann, "Analysis of reflectance data using the Kramers-Kronig relations," Appl. Phys. A 52, 138-144 (1991).
    [CrossRef]
  15. J. E. Toll, "Causality and the dispersion relations: logical foundations," Phys. Rev. 104, 1760-1770 (1956).
    [CrossRef]
  16. E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, "Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra," Appl. Spectrosc. 50, 1283-1289 (1996).
    [CrossRef]
  17. J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
    [CrossRef]
  18. R. M. A. Azzam, "Direct relations between Fresnel's interface reflection coefficients for the parallel and perpenticular polarizations," J. Opt. Soc. Am. 69, 1007-1011 (1979).
    [CrossRef]
  19. L. D. Landau and E. M. Lifshitz, Electrodynamics of Continuous Media (Pergamon, 1960).
  20. C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J. 27, 379-423 (1948).
  21. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies (Cambridge U. Press, 1988).
  22. F. Remacle and R. D. Levine, "Time domain information from resonant Raman excitation profiles: a direct inversion by maximim entropy," J. Chem. Phys. 99, 4908-4925 (1993).
    [CrossRef]
  23. P. K. Yang and J. Y. Huang, "Model-independent maximum-entropy method for the analysis of sum-frequency vibrational spectroscopy," J. Opt. Soc. Am. B 17, 1216-1222 (2000).
    [CrossRef]
  24. J. P. Burg, "Maximum entropy spectral analysis," presented at 37th Annual Meeting of the Society of Explor. Geophysics, Oklahoma City, Okla., October 31, 1967.
  25. E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, "Maximum entropy model in reflection spectra analysis," Opt. Commun. 89, 37-40 (1992).
    [CrossRef]
  26. 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]
  27. E. M. Vartiainen, "Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis," J. Opt. Soc. Am. B 9, 1209-1214 (1992).
    [CrossRef]
  28. E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, "Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the ∣chi(3)∣ spectra of polysilane," J. Opt. Soc. Am. B 13, 2106-2114 (1996).
    [CrossRef]

2004 (1)

2003 (3)

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibility," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility of polysilane," J. Chem. Phys. 119, 11095-11098 (2003).
[CrossRef]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

2002 (1)

2000 (1)

1998 (1)

1997 (1)

M. H. Lee and O. I. Sindoni, "Kramers-Kronig relations with logarithmic cernel and applications to the phase spectrum in the Drude model," Phys. Rev. E 56, 3891-3896 (1997).
[CrossRef]

1996 (3)

1993 (2)

F. Remacle and R. D. Levine, "Time domain information from resonant Raman excitation profiles: a direct inversion by maximim entropy," J. Chem. Phys. 99, 4908-4925 (1993).
[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]

1992 (2)

E. M. Vartiainen, "Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis," J. Opt. Soc. Am. B 9, 1209-1214 (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]

1991 (2)

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[CrossRef]

P. Grosse and V. Offermann, "Analysis of reflectance data using the Kramers-Kronig relations," Appl. Phys. A 52, 138-144 (1991).
[CrossRef]

1979 (1)

1956 (1)

J. E. Toll, "Causality and the dispersion relations: logical foundations," Phys. Rev. 104, 1760-1770 (1956).
[CrossRef]

1948 (1)

C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J. 27, 379-423 (1948).

Asakura, T.

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, "Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra," Appl. Spectrosc. 50, 1283-1289 (1996).
[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]

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

J. Räty, K.-E. Peiponen, and T. Asakura, UV-Visible Reflection Spectroscopy of Liquids (Springer, 2004).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

Aspnes, D. E.

D. E. Aspnes, "The accurate determination of optical properties by ellipsometry," in Handbook of Optical Constants of Solids, E.Palik, ed. (Academic, 1995) pp. 69-112.

Azzam, R. M. A.

Bashara, N. M.

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

Budde, B. A.

Burg, J. P.

J. P. Burg, "Maximum entropy spectral analysis," presented at 37th Annual Meeting of the Society of Explor. Geophysics, Oklahoma City, Okla., October 31, 1967.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies (Cambridge U. Press, 1988).

Grosse, P.

P. Grosse and V. Offermann, "Analysis of reflectance data using the Kramers-Kronig relations," Appl. Phys. A 52, 138-144 (1991).
[CrossRef]

Huang, J. Y.

King, F. W.

Kishida, H.

Koda, T.

Kortüm, G.

G. Kortüm, Reflectance Spectroscopy Principles, Methods, and Applications (Springer, 1969).

Landau, L. D.

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

Lee, M. H.

M. H. Lee and O. I. Sindoni, "Kramers-Kronig relations with logarithmic cernel and applications to the phase spectrum in the Drude model," Phys. Rev. E 56, 3891-3896 (1997).
[CrossRef]

M. H. Lee, "Solving certain principal value integrals by reduction to the dilogarithm," Physica A 234, 581-588 (1996).
[CrossRef]

Levine, R. D.

F. Remacle and R. D. Levine, "Time domain information from resonant Raman excitation profiles: a direct inversion by maximim entropy," J. Chem. Phys. 99, 4908-4925 (1993).
[CrossRef]

Lifshitz, E. M.

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

Lucarini, V.

K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. M. Vartiainen, "Kramers-Kronig relations and sum rules in nonlinear optics," Appl. Spectrosc. 58, 499-509 (2004).
[CrossRef] [PubMed]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibility," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility of polysilane," J. Chem. Phys. 119, 11095-11098 (2003).
[CrossRef]

Offermann, V.

P. Grosse and V. Offermann, "Analysis of reflectance data using the Kramers-Kronig relations," Appl. Phys. A 52, 138-144 (1991).
[CrossRef]

Palmer, K. F.

Peiponen, K.-E.

K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. M. Vartiainen, "Kramers-Kronig relations and sum rules in nonlinear optics," Appl. Spectrosc. 58, 499-509 (2004).
[CrossRef] [PubMed]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibility," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility of polysilane," J. Chem. Phys. 119, 11095-11098 (2003).
[CrossRef]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, "Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the ∣chi(3)∣ spectra of polysilane," J. Opt. Soc. Am. B 13, 2106-2114 (1996).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, "Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra," Appl. Spectrosc. 50, 1283-1289 (1996).
[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]

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 and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

J. Räty, K.-E. Peiponen, and T. Asakura, UV-Visible Reflection Spectroscopy of Liquids (Springer, 2004).

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies (Cambridge U. Press, 1988).

Räty, J.

J. Räty, K.-E. Peiponen, and T. Asakura, UV-Visible Reflection Spectroscopy of Liquids (Springer, 2004).

Remacle, F.

F. Remacle and R. D. Levine, "Time domain information from resonant Raman excitation profiles: a direct inversion by maximim entropy," J. Chem. Phys. 99, 4908-4925 (1993).
[CrossRef]

Saarinen, J. J.

K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. M. Vartiainen, "Kramers-Kronig relations and sum rules in nonlinear optics," Appl. Spectrosc. 58, 499-509 (2004).
[CrossRef] [PubMed]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility of polysilane," J. Chem. Phys. 119, 11095-11098 (2003).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibility," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

Shannon, C. E.

C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J. 27, 379-423 (1948).

Sindoni, O. I.

M. H. Lee and O. I. Sindoni, "Kramers-Kronig relations with logarithmic cernel and applications to the phase spectrum in the Drude model," Phys. Rev. E 56, 3891-3896 (1997).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies (Cambridge U. Press, 1988).

Toll, J. E.

J. E. Toll, "Causality and the dispersion relations: logical foundations," Phys. Rev. 104, 1760-1770 (1956).
[CrossRef]

Vartiainen, E. M.

K.-E. Peiponen, V. Lucarini, J. J. Saarinen, and E. M. Vartiainen, "Kramers-Kronig relations and sum rules in nonlinear optics," Appl. Spectrosc. 58, 499-509 (2004).
[CrossRef] [PubMed]

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, and T. Asakura, "Phase retrieval in optical spectroscopy: Resolving optical constants from power spectra," Appl. Spectrosc. 50, 1283-1289 (1996).
[CrossRef]

E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, "Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the ∣chi(3)∣ spectra of polysilane," J. Opt. Soc. Am. B 13, 2106-2114 (1996).
[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]

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

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies (Cambridge U. Press, 1988).

Williams, M. Z.

Yang, P. K.

Appl. Opt. (1)

Appl. Phys. A (1)

P. Grosse and V. Offermann, "Analysis of reflectance data using the Kramers-Kronig relations," Appl. Phys. A 52, 138-144 (1991).
[CrossRef]

Appl. Phys. Lett. (1)

J. J. Saarinen, E. M. Vartiainen, and K.-E. Peiponen, "Retrieval of the complex permittivity of spherical nanoparticles in a liquid host material from a spectral surface plasmon resonance measurement," Appl. Phys. Lett. 83, 893-895 (2003).
[CrossRef]

Appl. Spectrosc. (2)

Bell Syst. Tech. J. (1)

C. E. Shannon, "A mathematical theory of communications," Bell Syst. Tech. J. 27, 379-423 (1948).

J. Chem. Phys. (2)

F. Remacle and R. D. Levine, "Time domain information from resonant Raman excitation profiles: a direct inversion by maximim entropy," J. Chem. Phys. 99, 4908-4925 (1993).
[CrossRef]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive generalized Kramers-Kronig relations: application on third-harmonic generation susceptibility of polysilane," J. Chem. Phys. 119, 11095-11098 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (4)

Opt. Commun. (3)

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]

V. Lucarini, J. J. Saarinen, and K.-E. Peiponen, "Multiply subtractive Kramers-Krönig relations for arbitrary-order harmonic generation susceptibility," Opt. Commun. 218, 409-414 (2003).
[CrossRef]

Phys. Rev. (1)

J. E. Toll, "Causality and the dispersion relations: logical foundations," Phys. Rev. 104, 1760-1770 (1956).
[CrossRef]

Phys. Rev. B (1)

K.-E. Peiponen and E. M. Vartiainen, "Kramers-Kronig relations in optical data inversion," Phys. Rev. B 44, 8301-8303 (1991).
[CrossRef]

Phys. Rev. E (1)

M. H. Lee and O. I. Sindoni, "Kramers-Kronig relations with logarithmic cernel and applications to the phase spectrum in the Drude model," Phys. Rev. E 56, 3891-3896 (1997).
[CrossRef]

Physica A (1)

M. H. Lee, "Solving certain principal value integrals by reduction to the dilogarithm," Physica A 234, 581-588 (1996).
[CrossRef]

Other (8)

G. Kortüm, Reflectance Spectroscopy Principles, Methods, and Applications (Springer, 1969).

J. Räty, K.-E. Peiponen, and T. Asakura, UV-Visible Reflection Spectroscopy of Liquids (Springer, 2004).

R. M. A. Azzam and N. M. Bashara, Ellipsometry and Polarized Light (North-Holland, 1977).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, Dispersion, Complex Analysis and Optical Spectroscopy (Springer, 1999).

D. E. Aspnes, "The accurate determination of optical properties by ellipsometry," in Handbook of Optical Constants of Solids, E.Palik, ed. (Academic, 1995) pp. 69-112.

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

J. P. Burg, "Maximum entropy spectral analysis," presented at 37th Annual Meeting of the Society of Explor. Geophysics, Oklahoma City, Okla., October 31, 1967.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipies (Cambridge U. Press, 1988).

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Figures (3)

Fig. 1
Fig. 1

Real and imaginary parts of the Lorentzian relative permittivity, ϵ = ϵ + i ϵ , that was used in the modeling of the complex reflectivity. The parameters are ϵ = 1.0 , ω p = 3.0 eV , ω 0 = 2.0 eV , and Γ = 0.1 eV .

Fig. 2
Fig. 2

Real and imaginary parts of complex reflectivity, r p = r p + i r p , used in the ME analysis. The angle of incidence is taken to be α = 50 ° .

Fig. 3
Fig. 3

The true phase function of the complex reflectivity θ (thick curve) and the true error phase φ (thin curve). The estimate for φ (open circles) is computed with the aid of two anchor points, whose locations are indicated by the arrows. The thereby obtained estimate for θ is shown by square dots. The ME model parameters are K = 1 and M = 450 .

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

θ ( ω ) = 2 ω π P 0 ln r ( ω ) ω 2 ω 2 d ω ,
ln r ( ω ) ln r ( ω ) = 2 π P 0 ω θ ( ω ) ( 1 ω 2 ω 2 1 ω 2 ω 2 ) d ω ,
r ( ω ) = r ( ω ) exp i θ ( ω ) .
R s ( ω ) = cos α ( N 2 sin 2 α ) 1 2 cos α + ( N 2 sin 2 α ) 1 2 2 ,
R p ( ω ) = N 2 cos α ( N 2 sin 2 α ) 1 2 N 2 cos α + ( N 2 sin 2 α ) 1 2 2 ,
cos α ( N 2 sin 2 α ) 1 2 = 0 .
r p = cos α ( ϵ 1 ϵ 2 sin 2 α ) 1 2 cos α + ϵ 1 ϵ 2 sin 2 α = cos α ( A + i B ) cos α + ( A + i B ) = C + i D ,
ϵ = 1 2 ( A + i B ) 2 ± 1 2 [ ( A + i B ) 4 4 ( A + i B ) 2 sin 2 α ] 1 2 .
r p = r s cos 2 α 1 r s cos 2 α .
N 2 cos α ( N 2 sin 2 α ) 1 2 = 0 .
ϵ tan 2 α ϵ static ,
B ( ω ) = j = 1 J ω ω j ω ω j * ,
r true ( ω ) = r K K ( ω ) j = 1 J ω ω j ω ω j * .
r p ( z ) 2 = k = K K a k z k l = L L b l z l 2
r p ( z ) 2 1 l = L L b l z l 2 = z L 2 b L ( 1 + + b L b L z 2 L ) 2 .
r p ( z ) 2 d 0 2 1 + m = 1 M d m z m 2 ,
ν = ω ω 1 ω 2 ω 1 .
k = 1 M d k C ( n k ) = { d 0 2 , n = 0 0 , n = 1 , , M } ,
C ( m ) = 0 1 r p ( ν ) 2 exp ( 2 π i m ν ) d ν .
r p ( ν ) d 0 exp [ i θ ( ν ) ] m = 0 M d m exp ( i 2 π m ν ) ,
r p ( ν ) d 0 exp { i [ θ ( ν ) ψ ( ν ) ] } m = 0 M d m exp ( i 2 π m ν ) exp [ i ψ ( ν ) ] = d 0 exp [ i ϕ ( ν ) ] m = 0 M d m * exp ( i 2 π m ν ) ,
ϕ ( ν ) = k = 0 L B k ν k ,
( 1 ν 0 ν 0 L 1 ν 1 ν 1 L 1 ν L ν L L ) ( B 0 B 1 B L ) = ( ϕ ( ν 0 ) ϕ ( ν 1 ) ϕ ( ν L ) ) .
r p ( ν ) 2 r p ( ν ; K ) 2 = { r p ( ω 1 ) 2 , 0 ν < w K ( ω 1 ) r p ( ω ) 2 , w K ( ω 1 ) ν w K ( ω 2 ) r p ( ω 2 ) 2 , w K ( ω 2 ) ν < 1 } ,
w K ( ω ) = 1 2 K + 1 ( ω ω 1 ω 2 ω 1 + K ) ,
ν = w K ( ω ) ω K ( ω 1 ) w K ( ω 2 ) w K ( ω 1 ) ,
ϵ ( ω ) = ϵ + ω p 2 ω 0 2 ω 2 i Γ ω ,

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