Abstract

The second-order nonlinear-optical response of a material is measured in two dimensions of frequency. This new spectroscopy technique, based on the use of Fourier-transform spectral interferometry, is a transposition to optics of two-dimensional nuclear magnetic resonance. Although we demonstrate the technique by measuring the second-order phase-matching map of a nonresonant nonlinear crystal, this method should find useful applications in the measurement of second-order nonlinear susceptibility and in photon-echo experiments.

© 1996 Optical Society of America

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  1. N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
    [CrossRef]
  2. P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
    [CrossRef] [PubMed]
  3. S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
    [CrossRef] [PubMed]
  4. D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
    [CrossRef]
  5. L. Müller, A. Kumar, R. R. Ernst, J. Chem. Phys. 63, 5490 (1975)R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
    [CrossRef]
  6. C. Froehly, A. Lacourt, J. C. Vienot, J. Opt. (Paris) 4, 183 (1973).
  7. F. Reynaud, F. Salin, A. Barthelemy, Opt. Lett 14, 275 (1989).
    [CrossRef] [PubMed]
  8. J.-P. Geindre, P. Audebert, A. Rousse, F. Falliès, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, A. Antonetti, Opt. Lett. 19, 1997 (1994).
    [CrossRef] [PubMed]
  9. K. Misawa, T. Kobayashi, Opt. Lett. 20, 1550 (1995).
    [CrossRef] [PubMed]
  10. L. Lepetit, G. Chériaux, M. Joffre, J. Opt. Soc. Am. B 12, 2467 (1995).
    [CrossRef]
  11. M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).
  12. L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.
  13. D. J. Kane, R. Trebino, Opt. Lett. 18, 823 (1993)R. Trebino, D.J. Kane, J. Opt. Soc. Am. A 10, 1101 (1993).
    [CrossRef] [PubMed]
  14. J. Paye, M. Ramaswamy, J. G. Fujimoto, E. P. Ippen, Opt. Lett. 18, 1946 (1993)J. Paye, IEEE J. Quantum Electron. 30, 2693 (1994).
    [CrossRef] [PubMed]
  15. Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
    [CrossRef]
  16. We use a Type II arrangement to obtain a nontrivial phase-matching map. Indeed, symmetry implies that phase matching depends only quadratically on the difference frequency in a Type I crystal, so the variation will be small within the relatively small spectral width of our laser source..
  17. P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis (Wiley, New York, 1986).

1995 (3)

K. Misawa, T. Kobayashi, Opt. Lett. 20, 1550 (1995).
[CrossRef] [PubMed]

L. Lepetit, G. Chériaux, M. Joffre, J. Opt. Soc. Am. B 12, 2467 (1995).
[CrossRef]

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

1994 (3)

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

J.-P. Geindre, P. Audebert, A. Rousse, F. Falliès, J. C. Gauthier, A. Mysyrowicz, A. Dos Santos, G. Hamoniaux, A. Antonetti, Opt. Lett. 19, 1997 (1994).
[CrossRef] [PubMed]

Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
[CrossRef]

1993 (2)

1992 (1)

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

1989 (1)

F. Reynaud, F. Salin, A. Barthelemy, Opt. Lett 14, 275 (1989).
[CrossRef] [PubMed]

1988 (1)

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

1975 (1)

L. Müller, A. Kumar, R. R. Ernst, J. Chem. Phys. 63, 5490 (1975)R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
[CrossRef]

1973 (1)

C. Froehly, A. Lacourt, J. C. Vienot, J. Opt. (Paris) 4, 183 (1973).

1964 (1)

N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
[CrossRef]

Abella, I. D.

N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
[CrossRef]

Antonetti, A.

Audebert, P.

Barthelemy, A.

F. Reynaud, F. Salin, A. Barthelemy, Opt. Lett 14, 275 (1989).
[CrossRef] [PubMed]

Becker, P. C.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Bigot, J.-Y.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

Brabec, T.

Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
[CrossRef]

Brito-Cruz, C. H.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Chemla, D. S.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

Chériaux, G.

L. Lepetit, G. Chériaux, M. Joffre, J. Opt. Soc. Am. B 12, 2467 (1995).
[CrossRef]

L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.

Cunningham, J. E.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Curley, P.F.

Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
[CrossRef]

de Haseth, J. A.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis (Wiley, New York, 1986).

Dos Santos, A.

Ernst, R. R.

L. Müller, A. Kumar, R. R. Ernst, J. Chem. Phys. 63, 5490 (1975)R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
[CrossRef]

Falliès, F.

Fork, R. L.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Fragnito, H. L.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Froehly, C.

C. Froehly, A. Lacourt, J. C. Vienot, J. Opt. (Paris) 4, 183 (1973).

Fujimoto, J. G.

Gauthier, J. C.

Gauvin, S.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

Geindre, J.-P.

Griffiths, P. R.

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis (Wiley, New York, 1986).

Hamoniaux, G.

Hartmann, S. R.

N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
[CrossRef]

Henry, J. E.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Hierle, R.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

Hulin, D.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.

Ippen, E. P.

Joffre, M.

L. Lepetit, G. Chériaux, M. Joffre, J. Opt. Soc. Am. B 12, 2467 (1995).
[CrossRef]

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.

Kane, D. J.

Kobayashi, T.

Krausz, F.

Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
[CrossRef]

Kumar, A.

L. Müller, A. Kumar, R. R. Ernst, J. Chem. Phys. 63, 5490 (1975)R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
[CrossRef]

Kurnit, N. A.

N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
[CrossRef]

Lacourt, A.

C. Froehly, A. Lacourt, J. C. Vienot, J. Opt. (Paris) 4, 183 (1973).

Lepetit, L.

L. Lepetit, G. Chériaux, M. Joffre, J. Opt. Soc. Am. B 12, 2467 (1995).
[CrossRef]

L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.

Migus, A.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.

Misawa, K.

Müller, L.

L. Müller, A. Kumar, R. R. Ernst, J. Chem. Phys. 63, 5490 (1975)R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
[CrossRef]

Mycek, M.-A.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

Mysyrowicz, A.

Paye, J.

Ramaswamy, M.

Reynaud, F.

F. Reynaud, F. Salin, A. Barthelemy, Opt. Lett 14, 275 (1989).
[CrossRef] [PubMed]

Rousse, A.

Salin, F.

F. Reynaud, F. Salin, A. Barthelemy, Opt. Lett 14, 275 (1989).
[CrossRef] [PubMed]

Schäfer, W.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

Schmitt-Rink, S.

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

Shank, C. V.

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

Spielmann, Ch.

Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
[CrossRef]

Toussaere, E.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

Trebino, R.

Vienot, J. C.

C. Froehly, A. Lacourt, J. C. Vienot, J. Opt. (Paris) 4, 183 (1973).

Weiss, S.

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

White, J. O.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

Zyss, J.

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

IEEE J. Quantum Electron. (1)

Ch. Spielmann, P.F. Curley, T. Brabec, F. Krausz, IEEE J. Quantum Electron. 30, 1100 (1994).
[CrossRef]

J. Chem. Phys. (1)

L. Müller, A. Kumar, R. R. Ernst, J. Chem. Phys. 63, 5490 (1975)R. R. Ernst, G. Bodenhausen, A. Wokaun, Principles of Nuclear Magnetic Resonance in One and Two Dimensions (Oxford U. Press, Oxford, 1987).
[CrossRef]

J. Opt. (1)

C. Froehly, A. Lacourt, J. C. Vienot, J. Opt. (Paris) 4, 183 (1973).

J. Opt. Soc. Am. B (1)

Nonlin. Opt. (1)

M. Joffre, J. O. White, D. Hulin, A. Migus, E. Toussaere, R. Hierle, S. Gauvin, J. Zyss, Nonlin. Opt. 11, 5 (1995).

Opt. Lett (1)

F. Reynaud, F. Salin, A. Barthelemy, Opt. Lett 14, 275 (1989).
[CrossRef] [PubMed]

Opt. Lett. (4)

Phys. Rev. B (1)

D. S. Chemla, J.-Y. Bigot, M.-A. Mycek, S. Weiss, W. Schäfer, Phys. Rev. B 50, 8439 (1994).
[CrossRef]

Phys. Rev. Lett. (3)

N. A. Kurnit, I. D. Abella, S. R. Hartmann, Phys. Rev. Lett. 13, 567 (1964).
[CrossRef]

P. C. Becker, H. L. Fragnito, C. H. Brito-Cruz, R. L. Fork, J. E. Cunningham, J. E. Henry, C. V. Shank, Phys. Rev. Lett. 61, 1647 (1988).
[CrossRef] [PubMed]

S. Weiss, M.-A. Mycek, J.-Y. Bigot, S. Schmitt-Rink, D. S. Chemla, Phys. Rev. Lett. 69, 2685 (1992)M. S. Pshenichnikov, K. Duppen, D. A. Wiersma, Phys. Rev. Lett. 74, 674 (1995)E. J. Mayer, G. O. Smith, V. Heuckeroth, J. Kuhl, K. Bott, A. Schulze, T. Meier, S. W. Koch, P. Thomas, R. Hey, K. Ploog, Phys. Rev. B 51, 10909 (1995).
[CrossRef] [PubMed]

Other (3)

L. Lepetit, M. Joffre, G. Chériaux, A. Migus, D. Hulin, in Quantum Electronics and Laser Science Conference, Vol. 16 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), paper QWG1.

We use a Type II arrangement to obtain a nontrivial phase-matching map. Indeed, symmetry implies that phase matching depends only quadratically on the difference frequency in a Type I crystal, so the variation will be small within the relatively small spectral width of our laser source..

P. R. Griffiths, J. A. de Haseth, Fourier Transform Infrared Spectrometry, Vol. 83 of Chemical Analysis (Wiley, New York, 1986).

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Figures (3)

Fig. 1
Fig. 1

Experimental setup for the measurement of the second-order nonlinear response function: BS’s, beam splitters; TS’s, translation stages; L’s, lenses; NLC, nonlinear crystal; S, sample; UVF, ultraviolet filter; IRF, infrared filter; SL, slit; SP, spectrometer; CCD, charge-coupled device array detector. A, B, and C are three independent detection channels. The thick dotted lines show the frequency-doubled light beams.

Fig. 2
Fig. 2

Fifty-one spectral interferograms obtained as a function of time delay τ in a 3-mm-thick Type II KDP crystal: (a) raw experimental data, (b) data corrected for optical-path fluctuations and defects.

Fig. 3
Fig. 3

Amplitude in arbitrary units of the 2D nonlinear response function Ξ(2)(ω, σ) as a function of the sum frequency ω and the difference frequency σ. The sample is a 3-mm-thick Type II KDP crystal. The internal angle of incidence with respect to the crystal axis is 74.8 (74.0°) for the ordinary (extraordinary) wave. (a) Experiment, (b) theory.

Equations (4)

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E ( 2 ) ( t ) = 1 4 π 2 Ξ ( 2 ) ( ω 1 , ω 2 ) E ( ω 1 ) E ( ω 2 ) × exp [ - i ( ω 1 + ω 2 ) t ] d ω 1 d ω 2 ,
E τ ( 2 ) ( ω ) = 1 2 π Ξ ( 2 ) ( ω 1 , ω 2 ) E ( ω 1 ) exp ( - i ω 1 τ ) E ( ω 2 ) × exp ( i ω 2 τ ) δ ( ω - ω 1 - ω 2 ) d ω 1 d ω 2 = 1 2 π Ξ ( 2 ) ( ω + σ 2 , ω - σ 2 ) × E ( ω + σ 2 ) E ( ω - σ 2 ) exp ( - i σ τ ) d σ .
Ξ ( 2 ) ( ω + σ 2 , ω - σ 2 ) = E τ ( 2 ) ( ω ) exp ( i σ τ ) d τ E ( ω + σ 2 ) E ( ω - σ 2 ) .
Ξ ( 2 ) ( ω 1 , ω 2 ) exp { i [ k e ( ω 1 ) + k o ( ω 2 ) - k e ( ω 1 + ω 2 ) ] L } - 1 k e ( ω 1 ) + k o ( ω 2 ) - k e ( ω 1 + ω 2 ) ,

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