Abstract

A phase-retrieval procedure based on the maximum-entropy method is applied to infrared–visible sum-frequency generation spectroscopy. Several typical effects on the error phase are also examined with the aid of a known theoretical model. The interference between the resonant and the nonresonant parts changes the behavior of the error phase. Even in some cases when the nonresonant part is complex, the error phase becomes like a step function. This result contradicts the smoothness assumption for the error phase, and the whole phase-retrieval procedure breaks down in these cases. A comparison of the results of phase-retrieval procedures between infrared–visible sum-frequency generation and coherent anti-Stokes Raman scattering spectra is made. Some ideas that worked well in previous analyses of coherent anti-Stokes Raman scattering spectra become inapplicable in the infrared–visible sum-frequency generation spectra in spite of the resemblance of their line shape functions.

© 1997 Optical Society of America

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References

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  1. R. M. Bevensee, Maximum Entropy Solutions toScientific Problems (Prentice-Hall, Englewood Cliffs, N.J., 1993).
  2. N. L. Bonavito, J. E. Dorband, and T. Busse, “Maximum entropy restoration of blurred and oversaturated Hubble telescopeimagery,” Appl. Opt. 32, 5768–5774 (1993).
    [CrossRef] [PubMed]
  3. E. Prince, “Construction of maximum-entropy density maps, and their use in phasedetermination and extension,” Acta Crystallogr. 49, 61–66 (1993).
  4. E. M. Vartiainen and K.-E. Peiponen, “Meromorphic degenerate nonlinear susceptibility: phase retrieval fromthe amplitude spectrum,” Phys. Rev. B 50, 1941–1944 (1994).
    [CrossRef]
  5. E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropymethod: an application to the |χ(3)| spectraof polysilane,” J. Opt. Soc. Am. B 13, 2106–2114 (1996).
    [CrossRef]
  6. E. M. Vartiainen, “Phase retrieval approach for coherent anti-Stokes Raman scatteringspectrum analysis,” J. Opt. Soc. Am. B 9, 1209–1214 (1992).
    [CrossRef]
  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 forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
    [CrossRef]
  9. J. Y. Huang and Y. R. Shen, “Sum-frequency generationas a surface probe,” in Laser Spectroscopy and Photochemistryon Metal Surfaces, H. L. Dai and W. Ho, eds. (World Scientific, Singapore, 1995), Vol. 1, pp. 5–53.
  10. S. H. Lin and A. A. Villaeys, “Theoretical description of steady-state sum-frequency generation inmolecular adsorbates,” Phys. Rev. A 50, 5134–5144 (1994).
    [CrossRef] [PubMed]
  11. P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
    [CrossRef]
  12. 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).
    [CrossRef]
  13. Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 272–275.
  14. D. C. Duffy, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied usingsum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
    [CrossRef]
  15. T. H. Ong, P. B. Davies, and C. D. Bain, “Sum-frequency spectroscopy of alkoxy-terminated alkanethiols in contactwith liquid,” Langmuir 9, 1836–1845 (1993).
    [CrossRef]
  16. R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared–visible sum-frequencygeneration,” Opt. Lett. 15, 1276–1278 (1990).
    [CrossRef] [PubMed]
  17. 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 phase measurementstudy,” Chem. Phys. Lett. 172, 303–306 (1990).
    [CrossRef]
  18. J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
    [CrossRef]

1996 (1)

1995 (1)

D. C. Duffy, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied usingsum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

1994 (3)

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

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

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
[CrossRef]

1993 (4)

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

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

T. H. Ong, P. B. Davies, and C. D. Bain, “Sum-frequency spectroscopy of alkoxy-terminated alkanethiols in contactwith liquid,” Langmuir 9, 1836–1845 (1993).
[CrossRef]

E. Prince, “Construction of maximum-entropy density maps, and their use in phasedetermination and extension,” Acta Crystallogr. 49, 61–66 (1993).

1992 (2)

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

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).
[CrossRef]

1990 (2)

R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared–visible sum-frequencygeneration,” 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 phase measurementstudy,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

1988 (1)

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
[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]

Bain, C. D.

T. H. Ong, P. B. Davies, and C. D. Bain, “Sum-frequency spectroscopy of alkoxy-terminated alkanethiols in contactwith liquid,” Langmuir 9, 1836–1845 (1993).
[CrossRef]

Bonavito, N. L.

Buck, M.

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).
[CrossRef]

Busse, T.

Byers, J. D.

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
[CrossRef]

Chin, R. P.

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).
[CrossRef]

Chuang, T. J.

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).
[CrossRef]

Creeth, A. M.

D. C. Duffy, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied usingsum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

Davies, P. B.

D. C. Duffy, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied usingsum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

T. H. Ong, P. B. Davies, and C. D. Bain, “Sum-frequency spectroscopy of alkoxy-terminated alkanethiols in contactwith liquid,” Langmuir 9, 1836–1845 (1993).
[CrossRef]

Dorband, J. E.

Duffy, D. C.

D. C. Duffy, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied usingsum-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/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
[CrossRef]

Hasegawa, T.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

Hicks, J. M.

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
[CrossRef]

Huang, J. Y.

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).
[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 phase measurementstudy,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

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

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/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
[CrossRef]

Iwasa, Y.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

Kishida, H.

E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropymethod: an application to the |χ(3)| spectraof polysilane,” 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 forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

Koda, T.

E. M. Vartiainen, K.-E. Peiponen, H. Kishida, and T. Koda, “Phase retrieval in nonlinear optical spectroscopy by the maximum-entropymethod: an application to the |χ(3)| spectraof polysilane,” 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 forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

Lin, S. H.

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

Ong, T. H.

T. H. Ong, P. B. Davies, and C. D. Bain, “Sum-frequency spectroscopy of alkoxy-terminated alkanethiols in contactwith liquid,” Langmuir 9, 1836–1845 (1993).
[CrossRef]

Peiponen, K.-E.

Petralli-Mallow, T.

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
[CrossRef]

Prince, E.

E. Prince, “Construction of maximum-entropy density maps, and their use in phasedetermination and extension,” Acta Crystallogr. 49, 61–66 (1993).

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]

Seki, H.

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).
[CrossRef]

Shen, Y. R.

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).
[CrossRef]

R. Superfine, J. Y. Huang, and Y. R. Shen, “Phase measurement for surface infrared–visible sum-frequencygeneration,” 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 phase measurementstudy,” 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/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
[CrossRef]

Superfine, R.

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 phase measurementstudy,” Chem. Phys. Lett. 172, 303–306 (1990).
[CrossRef]

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

P. Guyot-Sionnest, R. Superfine, J. H. Hunt, and Y. R. Shen, “Vibrational spectroscopy of a silane monolayer at air/solid and liquid/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
[CrossRef]

Tokura, Y.

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

Vartiainen, E. M.

Villaeys, A. A.

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

Yee, H. I.

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
[CrossRef]

Acta Crystallogr. (1)

E. Prince, “Construction of maximum-entropy density maps, and their use in phasedetermination and extension,” Acta Crystallogr. 49, 61–66 (1993).

Appl. Opt. (1)

Chem. Phys. Lett. (2)

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 phase measurementstudy,” 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/solidinterfaces using sum-frequency generation,” Chem. Phys. Lett. 144, 1–5 (1988).
[CrossRef]

J. Opt. Soc. Am. B (2)

Langmuir (2)

D. C. Duffy, P. B. Davies, and A. M. Creeth, “Polymer-surfactant aggregates at a hydrophobic surface studied usingsum-frequency vibrational spectroscopy,” Langmuir 11, 2931–2937 (1995).
[CrossRef]

T. H. Ong, P. B. Davies, and C. D. Bain, “Sum-frequency spectroscopy of alkoxy-terminated alkanethiols in contactwith liquid,” Langmuir 9, 1836–1845 (1993).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. A (1)

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

Phys. Rev. B (4)

J. D. Byers, H. I. Yee, T. Petralli-Mallow, and J. M. Hicks, “Second-harmonic generation circular-dichroism spectroscopy from chiralmonolayers,” Phys. Rev. B 49, 14, 643–14, 647 (1994).
[CrossRef]

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

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).
[CrossRef]

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

Phys. Rev. Lett. (1)

H. Kishida, T. Hasegawa, Y. Iwasa, T. Koda, and Y. Tokura, “Dispersion relation in the third-order electric susceptibility forpolysilane film,” Phys. Rev. Lett. 14, 3724–3727 (1993).
[CrossRef]

Other (3)

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

Y. R. Shen, The Principles of Nonlinear Optics (Wiley, New York, 1984), pp. 272–275.

R. M. Bevensee, Maximum Entropy Solutions toScientific Problems (Prentice-Hall, Englewood Cliffs, N.J., 1993).

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Figures (5)

Fig. 1
Fig. 1

Effect on the error phase of changing nonresonant strength, illustrated with different |χNR(2)|/|Aq| values. The values of |χNR(2)|/|Aq| are (a) 1, (b) 0.5, (c) 0.1, (d) 0.05, and (e) 0.01. The filled circles in the left-hand figures denote the simulated squared modulus |χ(2)|2, and the corresponding MEM approximations are shown by the solid curves. The nonresonant strength is decreasing from the top to the bottom. The figures at the right show the corresponding error phases with (K=1) and without (K=0) the frequency-squeezing procedure. SFG, sum-frequency generation.

Fig. 2
Fig. 2

Effect on the error phase of changing nonresonant phase, illustrated with the values of nonresonant phase of (a) 45°, (b) 90°, (c) 180°, (d) 225°, and (e) 270° and the value of |χNR(2)|/|Aq| fixed at 0.6. The filled circles in the left-hand figures denote the simulated squared modulus |χ(2)|2, and the corresponding MEM approximations are shown by the solid curves. The nonresonant phase is increasing from the top to the bottom. The figures at the right shows the corresponding error phases with (K=1) and without (K=0) the frequency-squeezing procedure.

Fig. 3
Fig. 3

Effect of peak overlapping on the error phase. The filled circles in the left-hand figures denote the simulated squared modulus |χ(2)|2, and the corresponding MEM approximations are shown by the solid curves. The two spectral peaks are getting progressively closer from (a) to (d), and the corresponding error phases with (K=1) and without (K=0) the frequency-squeezing procedure are shown at the right.

Fig. 4
Fig. 4

Effect on the error phase of inhomogeneous broadening for an isolated spectral peak. The filled circles in the left-hand figures denote the simulated squared modulus |χ(2)|2, and the corresponding MEM approximations are shown by the solid curves. The values of σ/γ are 0.1, 0.2, 1, and 2 in (a), (b), (c), and (d), respectively. The figures at the right show the corresponding error phases with (K=1) and without (K=0) the frequency-squeezing procedure.

Fig. 5
Fig. 5

(a) Measured IVSFG spectrum (filled squares) of the CH3 symmetric stretch of the pentadecanoic acid monolayer on the water surface and its MEM interpolation (solid curve). (b)–(d) Measured phase values (filled squares) and the estimated phases (solid curves) from the MEPRP. The error phase in (b) is estimated by a constant value that is the error-phase value corresponding to the measured phase value indicated by an arrow. The error phases in (c) and (d) are estimated by linear interpolation of two error-phase values corresponding to the measured phase values shown by the arrows.

Equations (13)

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

I(ωs)|χs(2)(-ωs; ωv, ωIR)|2=|χNR(2)+χR(2)|=χNR(2)+q Aqγq(ωq-ωIR-iγq)2,
χNR(2)=|χNR(2)|exp(iθ).
μggQq00,αgQq00,
hf1f2 log S(f)df.
Sˆ(v)=|β|21+k=1Mak exp(i2πkv)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(v)exp(-i2πmv)dv.
χˆ(2)(v)=|β|exp[iϕ(v)]1+k=1Mak exp(i2πkv).
|χK(2)(v)||χ(2)(ω1)|0v<K2K+1|χ(2)(ω1+(ω2-ω1)[(2K+1)v-K])|K2K+1vK+12K+1|χ(2)(ω2)|K+12K+1<v1.
χs,eff(2)(ωIR, ωq0)=-χs(2)(ωIR, ωq) 12πσ×exp-12σ2(ωq-ωq0)2dωq.
v{Im[χ(v)]}v=vR1=0,
v{Im[χ(v)]}v=vR2=0.

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