Abstract

We have developed and applied a maximum-entropy phase-retrieval procedure to analyze sum-frequency vibrational spectra from a CCl4/octadecyl tricholosilane/silica interface and a hydrogen-terminated diamond C(111) surface. Some a priori knowledge of a nonlinear optical spectrum was employed for determining the phase of nonlinear optical susceptibility, and therefore the requirement for experimental phase measurement can be avoided. The results agree well with those from the Lorentzian line-shape model and justify the applicability of the a priori constraints employed in our phase-retrieval procedure.

© 2000 Optical Society of America

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References

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  1. E. M. Vartiainen, “Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis,” J. Opt. Soc. Am. B 9, 1209–1214 (1992).
    [CrossRef]
  2. 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]
  3. E. M. Vartiainen, K.-E. Peiponen, and H. Kishida, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the |χ(3)| spectra of polysilane, T. Koda,” J. Opt. Soc. Am. B 13, 2106–2114 (1996).
    [CrossRef]
  4. 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]
  5. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory and phase retrieval of meromorphic total susceptibility,” J. Phys.: Condens. Matter 9, 8937–8943 (1997).
  6. K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory of effective meromorphic nonlinear susceptibility of nanocomposites,” J. Phys.: Condens. Matter 10, 2483–2488 (1998).
  7. F. L. Ridener and R. H. Good, “Dispersion relations for nonlinear systems of arbitrary degree,” Phys. Rev. B 11, 2768–2770 (1975).
    [CrossRef]
  8. H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility for polysilane films,” Phys. Rev. Lett. 70, 3724–3727 (1993).
    [CrossRef] [PubMed]
  9. P.-K. Yang and J. Y. Huang, “Phase-retrieval problems in infrared-visible sum-frequency generation spectroscopy by the maximum-entropy method,” J. Opt. Soc. Am. B 14, 2443–2448 (1997).
    [CrossRef]
  10. R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared-visible sum-frequency generation,” Opt. Lett. 15, 1276–1278 (1990).
    [CrossRef] [PubMed]
  11. R. Superfine, J. Y. Huang, and Y. R. Shen, “Experimental determination of the sign of molecular dipole derivatives: an infrared-visible sum frequency generation absolute measurement study,” Chem. Phys. Lett. 172, 303–306 (1990).
    [CrossRef]
  12. J. Y. Huang and Y. R. Shen, “Sum-frequency as a surface probe,” in Laser Spectroscopy and Photochemistry on Metal Surfaces, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.
  13. S. H. Lin and A. A. Villaeys, “Theoretical description of steady-state sum-frequency generation in molecular adsorbates,” Phys. Rev. A 50, 5134–5144 (1994).
    [CrossRef] [PubMed]
  14. P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
    [CrossRef]
  15. R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, H. Seki, and M. Buck, “Vibrational spectra of hydrogen on diamond C(111)-(1×1),” Phys. Rev. B 45, 1522–1524 (1992); R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, “Interaction of atomic hydrogen with the diamond C(111) surface studied by infrared-visible sum-frequency generation spectroscopy,” Phys. Rev. B 52, 5985–5995 (1995).
    [CrossRef]
  16. T. H. Ong, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied using sum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
    [CrossRef]
  17. P.-K. Yang and J. Y. Huang, “Linewidth-deduction method for nonlinear optical spectroscopy with transform-limited light pulses,” J. Opt. Soc. Am. B 15, 1130–1134 (1998).
    [CrossRef]
  18. Van den Bos, “Alternative interpretation of maximum entropy spectral analysis,” IEEE Trans. Inf. Theory IT-17, 493–494 (1971).
    [CrossRef]
  19. T. J. Ulrych and M. Ooe, “Autoregressive and mixed autoregressive-moving average models and spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, ed. (Springer-Verlag, Berlin, 1983), Chap. 3, pp. 73–125.
  20. J. Ihm, S. G. Louie, and M. L. Cohen, “Self-consistent pseudopotential calculations for Ge and diamond (111) surfaces,” Phys. Rev. B 17, 769–775 (1978).
    [CrossRef]

1998

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory of effective meromorphic nonlinear susceptibility of nanocomposites,” J. Phys.: Condens. Matter 10, 2483–2488 (1998).

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
[CrossRef]

P.-K. Yang and J. Y. Huang, “Linewidth-deduction method for nonlinear optical spectroscopy with transform-limited light pulses,” J. Opt. Soc. Am. B 15, 1130–1134 (1998).
[CrossRef]

1997

P.-K. Yang and J. Y. Huang, “Phase-retrieval problems in infrared-visible sum-frequency generation spectroscopy by the maximum-entropy method,” J. Opt. Soc. Am. B 14, 2443–2448 (1997).
[CrossRef]

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory and phase retrieval of meromorphic total susceptibility,” J. Phys.: Condens. Matter 9, 8937–8943 (1997).

1996

1995

T. H. Ong, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied using sum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

1994

S. H. Lin and A. A. Villaeys, “Theoretical description of steady-state sum-frequency generation in molecular adsorbates,” Phys. Rev. A 50, 5134–5144 (1994).
[CrossRef] [PubMed]

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]

1993

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

1992

1990

R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared-visible sum-frequency generation,” Opt. Lett. 15, 1276–1278 (1990).
[CrossRef] [PubMed]

R. Superfine, J. Y. Huang, and Y. R. Shen, “Experimental determination of the sign of molecular dipole derivatives: an infrared-visible sum frequency generation absolute measurement study,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

1978

J. Ihm, S. G. Louie, and M. L. Cohen, “Self-consistent pseudopotential calculations for Ge and diamond (111) surfaces,” Phys. Rev. B 17, 769–775 (1978).
[CrossRef]

1975

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

1971

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

Asakura, T.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory of effective meromorphic nonlinear susceptibility of nanocomposites,” J. Phys.: Condens. Matter 10, 2483–2488 (1998).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory and phase retrieval of meromorphic total susceptibility,” J. Phys.: Condens. Matter 9, 8937–8943 (1997).

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]

Cohen, M. L.

J. Ihm, S. G. Louie, and M. L. Cohen, “Self-consistent pseudopotential calculations for Ge and diamond (111) surfaces,” Phys. Rev. B 17, 769–775 (1978).
[CrossRef]

Creeth, A. M.

T. H. Ong, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied using sum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

Davies, P. B.

T. H. Ong, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied using sum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

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]

Guyot-Sionnest, P.

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
[CrossRef]

Hasegawa, T.

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

Huang, J. Y.

Hunt, J. H.

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
[CrossRef]

Ihm, J.

J. Ihm, S. G. Louie, and M. L. Cohen, “Self-consistent pseudopotential calculations for Ge and diamond (111) surfaces,” Phys. Rev. B 17, 769–775 (1978).
[CrossRef]

Iwasa, Y.

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

Kishida, H.

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

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

Koda, T.

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

Lin, S. H.

S. H. Lin and A. A. Villaeys, “Theoretical description of steady-state sum-frequency generation in molecular adsorbates,” Phys. Rev. A 50, 5134–5144 (1994).
[CrossRef] [PubMed]

Louie, S. G.

J. Ihm, S. G. Louie, and M. L. Cohen, “Self-consistent pseudopotential calculations for Ge and diamond (111) surfaces,” Phys. Rev. B 17, 769–775 (1978).
[CrossRef]

Ong, T. H.

T. H. Ong, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied using sum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

Peiponen, K.-E.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory of effective meromorphic nonlinear susceptibility of nanocomposites,” J. Phys.: Condens. Matter 10, 2483–2488 (1998).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory and phase retrieval of meromorphic total susceptibility,” J. Phys.: Condens. Matter 9, 8937–8943 (1997).

E. M. Vartiainen, K.-E. Peiponen, and H. Kishida, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the |χ(3)| spectra of polysilane, T. Koda,” 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 and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval from the amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
[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]

Shen, Y. R.

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, “Experimental determination of the sign of molecular dipole derivatives: an infrared-visible sum frequency generation absolute measurement study,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared-visible sum-frequency generation,” Opt. Lett. 15, 1276–1278 (1990).
[CrossRef] [PubMed]

Superfine, R.

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, “Experimental determination of the sign of molecular dipole derivatives: an infrared-visible sum frequency generation absolute measurement study,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared-visible sum-frequency generation,” Opt. Lett. 15, 1276–1278 (1990).
[CrossRef] [PubMed]

Tokura, Y.

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

Van den Bos,

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

Vartiainen, E. M.

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory of effective meromorphic nonlinear susceptibility of nanocomposites,” J. Phys.: Condens. Matter 10, 2483–2488 (1998).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory and phase retrieval of meromorphic total susceptibility,” J. Phys.: Condens. Matter 9, 8937–8943 (1997).

E. M. Vartiainen, K.-E. Peiponen, and H. Kishida, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropy method: an application to the |χ(3)| spectra of polysilane, T. Koda,” 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 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, “Phase retrieval approach for coherent anti-Stokes Raman scattering spectrum analysis,” J. Opt. Soc. Am. B 9, 1209–1214 (1992).
[CrossRef]

Villaeys, A. A.

S. H. Lin and A. A. Villaeys, “Theoretical description of steady-state sum-frequency generation in molecular adsorbates,” Phys. Rev. A 50, 5134–5144 (1994).
[CrossRef] [PubMed]

Yang, P.-K.

Appl. Spectrosc.

Chem. Phys. Lett.

R. Superfine, J. Y. Huang, and Y. R. Shen, “Experimental determination of the sign of molecular dipole derivatives: an infrared-visible sum frequency generation absolute measurement study,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solid interfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1998).
[CrossRef]

IEEE Trans. Inf. Theory

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

J. Opt. Soc. Am. B

J. Phys.: Condens. Matter

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory and phase retrieval of meromorphic total susceptibility,” J. Phys.: Condens. Matter 9, 8937–8943 (1997).

K.-E. Peiponen, E. M. Vartiainen, and T. Asakura, “Dispersion theory of effective meromorphic nonlinear susceptibility of nanocomposites,” J. Phys.: Condens. Matter 10, 2483–2488 (1998).

Langmuir

T. H. Ong, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied using sum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

Opt. Lett.

Phys. Rev. A

S. H. Lin and A. A. Villaeys, “Theoretical description of steady-state sum-frequency generation in molecular adsorbates,” Phys. Rev. A 50, 5134–5144 (1994).
[CrossRef] [PubMed]

Phys. Rev. B

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]

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

J. Ihm, S. G. Louie, and M. L. Cohen, “Self-consistent pseudopotential calculations for Ge and diamond (111) surfaces,” Phys. Rev. B 17, 769–775 (1978).
[CrossRef]

Phys. Rev. Lett.

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

Other

J. Y. Huang and Y. R. Shen, “Sum-frequency as a surface probe,” in Laser Spectroscopy and Photochemistry on Metal Surfaces, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.

R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, H. Seki, and M. Buck, “Vibrational spectra of hydrogen on diamond C(111)-(1×1),” Phys. Rev. B 45, 1522–1524 (1992); R. P. Chin, J. Y. Huang, Y. R. Shen, T. J. Chuang, and H. Seki, “Interaction of atomic hydrogen with the diamond C(111) surface studied by infrared-visible sum-frequency generation spectroscopy,” Phys. Rev. B 52, 5985–5995 (1995).
[CrossRef]

T. J. Ulrych and M. Ooe, “Autoregressive and mixed autoregressive-moving average models and spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, ed. (Springer-Verlag, Berlin, 1983), Chap. 3, pp. 73–125.

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Figures (6)

Fig. 1
Fig. 1

Application of the phase-retrieval procedure in a SFG spectrum of OTS/CCl4 interface. The solid curve in (a) denotes the maximum-entropy estimate. The real and imaginary parts obtained from the MEPRP are shown by the solid curves in (b) and (c). For comparison, the results from the Lorentzian line-shape model are presented with dashed curves.

Fig. 2
Fig. 2

Phase-retrieval results of IVSFG spectra of H/C(111) diamond (1×1) surfaces with different hydrogen coverages: (a) 100% monolayer (ML), (b) 49% ML, (c) 29% ML, and (d) 14% ML. The real and imaginary parts obtained from the MEPRP are shown on the right side. The error phase was approximated with a linear function of frequency.

Fig. 3
Fig. 3

Schematic diagram illustrating the constraint used in the phase retrieval of Fig. 2, which depicts the area marked by vertical lines to be equal to that marked by horizontal lines.

Fig. 4
Fig. 4

Comparison of the results with the Lorentzian line-shape model (dashed curves) and the MEPRP (solid curves) for the spectrum shown in Fig. 2(c).

Fig. 5
Fig. 5

Real and imaginary parts of nonresonant background as a function of surface hydrogen coverage (ΘH) deduced with (a) the MEPRP and (b) the Lorentzian line-shape model. The solid curves in (a) are drawn for eye guiding, while the straight lines in (b) are a fit to A+B(1-ΘH) for ΘH30%.

Fig. 6
Fig. 6

Ball-and-stick model of the top view of C(111)-(1×1) surface.

Equations (17)

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

Pi(ωs)=χijk(2)Ej(ωvis)Ek(ωir).
I(ωs)|χs(2)(-ωs; ωvis, ωir)|2
=|χNR(2)+χR(2)|=χNR(2)+qAq(ωir-ωq+iγq)2,
χNR(2)=|χNR(2)|exp(iθ).
hf1f2 log S(f)df.
S(ν)=|β|21+k=1Mak exp(i2πkν)2,
R(0)R(-1)R(-M)R(1)R(0)R(1-M)R(M)R(M-1)R(0)1a1aM
=|β|200.
R(m)=01S(ν)exp(-i2πmν)dν.
xm=-k=1Makxm-k+em.
X(z)=-(a1z-1+a2z-2++aMz-M)X(z)+E(z).
X(ν)=E(ν)1+k=1Mak exp(-i2πkν).
E(ν)=|β| exp[iϕ(ν)],
χ(2)(ν)=|β| exp[iϕ(ν)]1+k=1Mak exp(i2πkν).
ϕ(ν)=B0+B1ν++BLνL=l=0LBlνl.
1ν0ν0L1ν1ν1L1νLνLLB0B1BL=ϕ(ν0)ϕ(ν1)ϕ(νL).
|χK(2)(ν)||χ(2)(ω1)|0ν<K2K+1|χ(2){ω1+(ω2-ω1)[(2K+1)ν-K]}|K2K+1νK+12K+1|χ(2)(ω2)|K+12K+1<ν1.

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