Abstract

The coherent propagation phenomenon of incoherent light through a resonant medium was experimentally demonstrated in the subpicosecond time region, using Na atoms and the light from an imperfectly mode-locked cw dye laser. The coherence spike with a 0.35-psec width in the cross-correlation profile between the propagated incoherent light and the initial light develops the same specific oscillatory structure as seen in the propagation of a weak coherent subpicosecond pulse. A theoretical analysis was also performed, and the experimental results are in considerably good agreement with the calculated ones.

© 1986 Optical Society of America

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  1. S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
    [CrossRef]
  2. N. Morita, T. Yajima, Y. Ishida, “Coherent transient spectroscopy with ultra-high time-resolution using incoherent light,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 239–241.
    [CrossRef]
  3. R. Beach, S. R. Hartman, “Incoherent photon echoes,” Phys. Rev. Lett. 53, 663–666 (1984).
    [CrossRef]
  4. N. Morita, T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A. 30, 2525–2536 (1984).
    [CrossRef]
  5. H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
    [CrossRef]
  6. See, for example, L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), pp. 78–129.
  7. M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
    [CrossRef]
  8. H. P. Grieneisen, J. Goldhar, N. A. Kurnit, “Observation of zero-degree pulse propagation in a resonant medium,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds (Plenum, New York, 1973), pp. 5–21.
    [CrossRef]
  9. M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
    [CrossRef]
  10. H. S. Kwok, E. Yablonovitch, “30-psec CO2laser pulses generated by optical free induction decay,” Appl. Phys. Lett. 30, 158–160 (1977).
    [CrossRef]
  11. H.-J. Hartmann, A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117–122 (1983); “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
    [CrossRef]
  12. J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
    [CrossRef]
  13. J. E. Rothenberg, D. Grischkowsky, “Measurement of the phase of a frequency-swept ultrashort optical pulse,” J. Opt. Soc. Am. B 2, 626–633 (1985).
    [CrossRef]
  14. See, for example, E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–121.
    [CrossRef]
  15. See, for example, J. Perina, in Coherence of Light (Van Nostrand, London, 1971), p. 139.
  16. C. R. Vidal, F. B. Haller, “Heat pipe oven applications. I. Isothermal heater of well defined temperature. II. Production of metal vapor–gas mixtures,” Rev. Sci. Instrum. 42, 1779–1784 (1971).
    [CrossRef]
  17. J. E. Rothenberg, D. Grischkowsky, “Observation of a 1.9-psec polarization beat,” Opt. Lett. 10, 22–24 (1985).
    [CrossRef] [PubMed]
  18. Y. E. Lee, Statistical Theory of Communication (Wiley, New York, 1960).
  19. D. L. Price, K. Skold, “A detailed evaluation of the mechanical correlation chopper for neutron time-of-flight spectroscopy,” Nucl. Instrum. Methods 82, 208–222 (1970).
    [CrossRef]
  20. C. V. Nowikow, R. Grice, “Cross-correlation time-of-flight analysis of molecular beam scattering,” J. Phys. E 12, 515–521 (1979).
    [CrossRef]
  21. See, for example, Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, New York, 1970), Chap. 16.
  22. N. Takeuchi, N. Sugimoto, H. Baba, K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
    [CrossRef] [PubMed]
  23. B. Blumich, “Stochastic time resolved CIDNP spectroscopy: suggestion of an experimental method,” Mol. Phys. 51, 1283–1291 (1984).
    [CrossRef]

1985

1984

B. Blumich, “Stochastic time resolved CIDNP spectroscopy: suggestion of an experimental method,” Mol. Phys. 51, 1283–1291 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

R. Beach, S. R. Hartman, “Incoherent photon echoes,” Phys. Rev. Lett. 53, 663–666 (1984).
[CrossRef]

N. Morita, T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A. 30, 2525–2536 (1984).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
[CrossRef]

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

1983

N. Takeuchi, N. Sugimoto, H. Baba, K. Sakurai, “Random modulation cw lidar,” Appl. Opt. 22, 1382–1386 (1983).
[CrossRef] [PubMed]

H.-J. Hartmann, A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117–122 (1983); “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[CrossRef]

1979

C. V. Nowikow, R. Grice, “Cross-correlation time-of-flight analysis of molecular beam scattering,” J. Phys. E 12, 515–521 (1979).
[CrossRef]

1977

H. S. Kwok, E. Yablonovitch, “30-psec CO2laser pulses generated by optical free induction decay,” Appl. Phys. Lett. 30, 158–160 (1977).
[CrossRef]

1974

M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[CrossRef]

1971

C. R. Vidal, F. B. Haller, “Heat pipe oven applications. I. Isothermal heater of well defined temperature. II. Production of metal vapor–gas mixtures,” Rev. Sci. Instrum. 42, 1779–1784 (1971).
[CrossRef]

1970

D. L. Price, K. Skold, “A detailed evaluation of the mechanical correlation chopper for neutron time-of-flight spectroscopy,” Nucl. Instrum. Methods 82, 208–222 (1970).
[CrossRef]

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

Allen, L.

See, for example, L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), pp. 78–129.

Asaka, S.

H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Baba, H.

Balant, A. C.

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Beach, R.

R. Beach, S. R. Hartman, “Incoherent photon echoes,” Phys. Rev. Lett. 53, 663–666 (1984).
[CrossRef]

Blumich, B.

B. Blumich, “Stochastic time resolved CIDNP spectroscopy: suggestion of an experimental method,” Mol. Phys. 51, 1283–1291 (1984).
[CrossRef]

Crisp, M. D.

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

Eberly, J. H.

See, for example, L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), pp. 78–129.

Fujiwara, M.

H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Goldhar, J.

M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[CrossRef]

H. P. Grieneisen, J. Goldhar, N. A. Kurnit, “Observation of zero-degree pulse propagation in a resonant medium,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds (Plenum, New York, 1973), pp. 5–21.
[CrossRef]

Grice, R.

C. V. Nowikow, R. Grice, “Cross-correlation time-of-flight analysis of molecular beam scattering,” J. Phys. E 12, 515–521 (1979).
[CrossRef]

Grieneisen, H. P.

H. P. Grieneisen, J. Goldhar, N. A. Kurnit, “Observation of zero-degree pulse propagation in a resonant medium,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds (Plenum, New York, 1973), pp. 5–21.
[CrossRef]

Grischkowsky, D.

Haller, F. B.

C. R. Vidal, F. B. Haller, “Heat pipe oven applications. I. Isothermal heater of well defined temperature. II. Production of metal vapor–gas mixtures,” Rev. Sci. Instrum. 42, 1779–1784 (1971).
[CrossRef]

Hamadani, M.

M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[CrossRef]

Hartman, S. R.

R. Beach, S. R. Hartman, “Incoherent photon echoes,” Phys. Rev. Lett. 53, 663–666 (1984).
[CrossRef]

Hartmann, H.-J.

H.-J. Hartmann, A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117–122 (1983); “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[CrossRef]

Ippen, E. P.

See, for example, E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–121.
[CrossRef]

Ishida, Y.

N. Morita, T. Yajima, Y. Ishida, “Coherent transient spectroscopy with ultra-high time-resolution using incoherent light,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 239–241.
[CrossRef]

Javan, A.

M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[CrossRef]

Kurnit, N. A.

M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[CrossRef]

H. P. Grieneisen, J. Goldhar, N. A. Kurnit, “Observation of zero-degree pulse propagation in a resonant medium,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds (Plenum, New York, 1973), pp. 5–21.
[CrossRef]

Kwok, H. S.

H. S. Kwok, E. Yablonovitch, “30-psec CO2laser pulses generated by optical free induction decay,” Appl. Phys. Lett. 30, 158–160 (1977).
[CrossRef]

Laubereau, A.

H.-J. Hartmann, A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117–122 (1983); “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[CrossRef]

Lee, Y. E.

Y. E. Lee, Statistical Theory of Communication (Wiley, New York, 1960).

Matsuoka, M.

S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Morita, N.

N. Morita, T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A. 30, 2525–2536 (1984).
[CrossRef]

N. Morita, T. Yajima, Y. Ishida, “Coherent transient spectroscopy with ultra-high time-resolution using incoherent light,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 239–241.
[CrossRef]

Nakatsuka, H.

S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
[CrossRef]

Nowikow, C. V.

C. V. Nowikow, R. Grice, “Cross-correlation time-of-flight analysis of molecular beam scattering,” J. Phys. E 12, 515–521 (1979).
[CrossRef]

Perina, J.

See, for example, J. Perina, in Coherence of Light (Van Nostrand, London, 1971), p. 139.

Price, D. L.

D. L. Price, K. Skold, “A detailed evaluation of the mechanical correlation chopper for neutron time-of-flight spectroscopy,” Nucl. Instrum. Methods 82, 208–222 (1970).
[CrossRef]

Rothenberg, J. E.

Sakurai, K.

Shank, C. V.

See, for example, E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–121.
[CrossRef]

Skold, K.

D. L. Price, K. Skold, “A detailed evaluation of the mechanical correlation chopper for neutron time-of-flight spectroscopy,” Nucl. Instrum. Methods 82, 208–222 (1970).
[CrossRef]

Sugimoto, N.

Takeuchi, N.

Tomita, M.

H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
[CrossRef]

Vidal, C. R.

C. R. Vidal, F. B. Haller, “Heat pipe oven applications. I. Isothermal heater of well defined temperature. II. Production of metal vapor–gas mixtures,” Rev. Sci. Instrum. 42, 1779–1784 (1971).
[CrossRef]

Yablonovitch, E.

H. S. Kwok, E. Yablonovitch, “30-psec CO2laser pulses generated by optical free induction decay,” Appl. Phys. Lett. 30, 158–160 (1977).
[CrossRef]

Yajima, T.

N. Morita, T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A. 30, 2525–2536 (1984).
[CrossRef]

N. Morita, T. Yajima, Y. Ishida, “Coherent transient spectroscopy with ultra-high time-resolution using incoherent light,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 239–241.
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

M. Hamadani, J. Goldhar, N. A. Kurnit, A. Javan, “Coherent optical pulse reshaping in a resonant molecular absorber,” Appl. Phys. Lett. 25, 160–163 (1974).
[CrossRef]

H. S. Kwok, E. Yablonovitch, “30-psec CO2laser pulses generated by optical free induction decay,” Appl. Phys. Lett. 30, 158–160 (1977).
[CrossRef]

J. Opt. Soc. Am. B

J. Phys. E

C. V. Nowikow, R. Grice, “Cross-correlation time-of-flight analysis of molecular beam scattering,” J. Phys. E 12, 515–521 (1979).
[CrossRef]

Mol. Phys.

B. Blumich, “Stochastic time resolved CIDNP spectroscopy: suggestion of an experimental method,” Mol. Phys. 51, 1283–1291 (1984).
[CrossRef]

Nucl. Instrum. Methods

D. L. Price, K. Skold, “A detailed evaluation of the mechanical correlation chopper for neutron time-of-flight spectroscopy,” Nucl. Instrum. Methods 82, 208–222 (1970).
[CrossRef]

Opt. Commun.

H.-J. Hartmann, A. Laubereau, “Coherent pulse propagation in the infrared on the picosecond time scale,” Opt. Commun. 47, 117–122 (1983); “Transient infrared spectroscopy on the picosecond time-scale by coherent pulse propagation,” J. Chem. Phys. 80, 4663–4670 (1984).
[CrossRef]

H. Nakatsuka, M. Tomita, M. Fujiwara, S. Asaka, “Subpicosecond photon echoes by using nanosecond laser pulses,” Opt. Commun. 52, 150–152 (1984).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. D. Crisp, “Propagation of small-area pulses of coherent light through a resonant medium,” Phys. Rev. A 1, 1604–1611 (1970).
[CrossRef]

S. Asaka, H. Nakatsuka, M. Fujiwara, M. Matsuoka, “Accumulated photon echoes with incoherent light in Nd+3-doped silicate glass,” Phys. Rev. A 29, 2286–2289 (1984).
[CrossRef]

Phys. Rev. A.

N. Morita, T. Yajima, “Ultrahigh-time-resolution coherent transient spectroscopy with incoherent light,” Phys. Rev. A. 30, 2525–2536 (1984).
[CrossRef]

Phys. Rev. Lett.

R. Beach, S. R. Hartman, “Incoherent photon echoes,” Phys. Rev. Lett. 53, 663–666 (1984).
[CrossRef]

J. E. Rothenberg, D. Grischkowsky, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
[CrossRef]

Rev. Sci. Instrum.

C. R. Vidal, F. B. Haller, “Heat pipe oven applications. I. Isothermal heater of well defined temperature. II. Production of metal vapor–gas mixtures,” Rev. Sci. Instrum. 42, 1779–1784 (1971).
[CrossRef]

Other

Y. E. Lee, Statistical Theory of Communication (Wiley, New York, 1960).

See, for example, Radar Handbook, M. I. Skolnik, ed. (McGraw-Hill, New York, 1970), Chap. 16.

See, for example, E. P. Ippen, C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–121.
[CrossRef]

See, for example, J. Perina, in Coherence of Light (Van Nostrand, London, 1971), p. 139.

N. Morita, T. Yajima, Y. Ishida, “Coherent transient spectroscopy with ultra-high time-resolution using incoherent light,” in Ultrafast Phenomena IV, D. H. Auston, K. B. Eisenthal, eds. (Springer-Verlag, Berlin, 1984), pp. 239–241.
[CrossRef]

H. P. Grieneisen, J. Goldhar, N. A. Kurnit, “Observation of zero-degree pulse propagation in a resonant medium,” in Coherence and Quantum Optics, L. Mandel, E. Wolf, eds (Plenum, New York, 1973), pp. 5–21.
[CrossRef]

See, for example, L. Allen, J. H. Eberly, Optical Resonance and Two-Level Atoms (Wiley, New York, 1975), pp. 78–129.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup. PM, photomultiplier.

Fig. 2
Fig. 2

Autocorrelation profile of the input incoherent light by second-harmonic (SH) generation.

Fig. 3
Fig. 3

Cross-correlation profiles by SH generation between the incoherent light pulses before and after passing through the Na vapor, corresponding to the cases in which the values of the mean number density N ¯ of Na atoms are (a) 0, (b) 1.8 × 1012, (c) 3.0 × 1012, (d) 4.3 × 1012, and (e) 9.7 × 1012 cm−3. The effective medium length l is 30 cm.

Fig. 4
Fig. 4

Magnified cross-correlation profiles around the coherence peaks. The condition for each figure is the same as in Fig. 3. The dashed curves represent the corresponding theoretical profiles.

Equations (13)

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

| E * ( t ) E ( t + τ ) | 2 ,
μ i t t + t c d t E ˜ ( t ) 1             ( i = 1 , 2 )
| μ i - t d t E ˜ ( t ) | 1             ( i = 1 , 2 )
E ( z , t ) = Re { E ˜ ( z , t ) exp [ - i ω 0 ( t - z / c ) ] } .
E ˜ ( z , t ) = E ^ ( z , s ) = - d s E ^ ( 0 , s - s ) F ( z , s ) ,
F ( z , s ) = 1 2 π - d ω exp [ - 0 z d z A ( z , ω ) - i ω s ] ,
A ( z , ω ) = ( α 1 + α 2 ) / 2 - i ( δ n 1 + δ n 2 ) ω / c .
0 z d z A ( z , ω ) = B ( ω ) × 0 z d z N ( z ) = B ( ω ) × N ¯ × l ,
E ^ ( 0 , s ) = V ( s ) R ( s ) ,
J ( τ ) - d s E ^ * ( z , s ) E ^ * ( 0 , s - τ ) E ^ ( z , s ) E ^ ( 0 , s - τ ) - d s V ( s ) 2 V ( s - τ ) 2 K ( z , τ ) ,
K ( z , τ ) = - d s - d s F * ( z , s ) F ( z , s ) × R * ( - s ) R * ( - τ ) R ( - s ) R ( - τ ) .
K ( z , τ ) = | - d s f ( τ - s ) F ( z , s ) | 2 + - d s - d s F * ( z , s ) F ( z , s ) f ( s - s ) f ( 0 ) .
K ( z , τ ) = | 1 2 π - d ω P ( ω ) exp [ - 0 z d z A ( z , ω ) - i ω τ ] | 2 + ( 1 2 π ) 2 - d ω P ( ω ) exp { - 2 0 z d z Re [ A ( z , ω ) ] } × - d ω P ( ω ) ,

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