Abstract

Observation of 0π-pulse formation with incoherent light through field cross-correlation is theoretically and experimentally demonstrated. The experiment illuminates some of the properties of this new incoherent light technique, which is used to analyze the fast temporal variations of the signal. In particular, the technique’s time resolution, signal-to-noise ratio, and sensivity to dispersion are investigated.

© 1992 Optical Society of America

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References

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  1. T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacethylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett 133, 230–234 (1987).
    [Crossref]
  2. K. Kurokawa, T. Hattori, and T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
    [Crossref]
  3. M. Tomita and M. Matsuoka, “Ultrafast pump–probe measurement using intensity correlation of incoherent light,” J. Opt. Soc. Am. B 3, 560–563 (1986).
    [Crossref]
  4. N. Morita, K. Torizuka, and T. Yajima, “Coherent propagation effect of incoherent light,” J. Opt. Soc. Am. B 3, 548–553 (1986).
    [Crossref]
  5. H. Nakatsuka, Y. Katashima, and K. Inouye, “Incoherent light gated optical Kerr shutter for fluorescence lifetime measurement,” Opt. Commun. 69, 169–172 (1988).
    [Crossref]
  6. S. Saikan, T. Kishida, A. Imaoka, K. Ichikawa, A. Furusawa, and H. Oasawa, “Optical memory based on heterodyne-detected accumulated photon echoes,” Opt. Lett. 14, 841–843 (1989).
    [Crossref] [PubMed]
  7. A. Débarre, J.-C. Keller, J.-L. Le Gouët, and P. Tchénio, “Field cross-correlation retrieval of optically stored data,” J. Opt. Soc. Am. B 8, 153–159 (1991).
    [Crossref]
  8. M. D. Crisp, “Propagation of small area pulses of coherent light through a resonant medium,” Phys. Rev. A 6, 1604–1611 (1970).
    [Crossref]
  9. J. E. Rothenberg, D. Grischkowsky, and A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
    [Crossref]
  10. E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.
    [Crossref]
  11. P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.
  12. H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
    [Crossref]
  13. A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
    [Crossref]
  14. A. Débarre, J.-C. Keller, J.-L. Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
    [Crossref]
  15. S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [Crossref]
  16. M. D. Havey, L. C. Balling, and J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
    [Crossref]
  17. M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 298.
  18. N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981), Chap. 11.
  19. R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am B 3, 1295–1304 (1986).
    [Crossref]
  20. P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
    [Crossref]
  21. W. E. Moerner, ed., Persistent Spectral Hole-Burning. Science and Applications (Springer-Verlag, Berlin, 1988).
    [Crossref]
  22. A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole-burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
    [Crossref]
  23. A. Débarre, J.-C. Keller, J.-L. Le Gouët, P. Tchénio, and J.-P. Galaup, “Optical information storage in condensed matter with stochastic excitation,” J. Opt. Soc. Am. B 8, 2529–2536 (1991).
    [Crossref]
  24. B. Blümich, “White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy,” Prog. Nucl. Magn. Reson. Spectrosc. 19, 331–417 (1987).
    [Crossref]
  25. B. Blümich, “Two dimensional interferometry,” Rev. Sci. Instrum. 58, 911–919 (1987).
    [Crossref]

1991 (2)

1989 (4)

S. Saikan, T. Kishida, A. Imaoka, K. Ichikawa, A. Furusawa, and H. Oasawa, “Optical memory based on heterodyne-detected accumulated photon echoes,” Opt. Lett. 14, 841–843 (1989).
[Crossref] [PubMed]

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[Crossref]

A. Débarre, J.-C. Keller, J.-L. Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[Crossref]

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole-burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[Crossref]

1988 (1)

H. Nakatsuka, Y. Katashima, and K. Inouye, “Incoherent light gated optical Kerr shutter for fluorescence lifetime measurement,” Opt. Commun. 69, 169–172 (1988).
[Crossref]

1987 (5)

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacethylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett 133, 230–234 (1987).
[Crossref]

K. Kurokawa, T. Hattori, and T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
[Crossref]

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[Crossref]

B. Blümich, “White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy,” Prog. Nucl. Magn. Reson. Spectrosc. 19, 331–417 (1987).
[Crossref]

B. Blümich, “Two dimensional interferometry,” Rev. Sci. Instrum. 58, 911–919 (1987).
[Crossref]

1986 (3)

1984 (1)

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

1980 (1)

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[Crossref]

1976 (1)

M. D. Havey, L. C. Balling, and J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
[Crossref]

1970 (1)

M. D. Crisp, “Propagation of small area pulses of coherent light through a resonant medium,” Phys. Rev. A 6, 1604–1611 (1970).
[Crossref]

1969 (1)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

Aaviksoo, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole-burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[Crossref]

Acioli, L. H.

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[Crossref]

Balant, A. C.

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

Balling, L. C.

M. D. Havey, L. C. Balling, and J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
[Crossref]

Blümich, B.

B. Blümich, “White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy,” Prog. Nucl. Magn. Reson. Spectrosc. 19, 331–417 (1987).
[Crossref]

B. Blümich, “Two dimensional interferometry,” Rev. Sci. Instrum. 58, 911–919 (1987).
[Crossref]

Crisp, M. D.

M. D. Crisp, “Propagation of small area pulses of coherent light through a resonant medium,” Phys. Rev. A 6, 1604–1611 (1970).
[Crossref]

de Araúijo, C.

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[Crossref]

de Haseth, J. A.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.

Débarre, A.

Eichler, H. J.

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[Crossref]

Fu, P.

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[Crossref]

Furusawa, A.

Galaup, J.-P.

Gomes, A. S. L.

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[Crossref]

Gouët, J.-L.

A. Débarre, J.-C. Keller, J.-L. Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[Crossref]

Griffiths, P. R.

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.

Grischkowsky, D.

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

Gustafson, E. K.

R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am B 3, 1295–1304 (1986).
[Crossref]

Hahn, E. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

Hattori, T.

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacethylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett 133, 230–234 (1987).
[Crossref]

K. Kurokawa, T. Hattori, and T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
[Crossref]

Havey, M. D.

M. D. Havey, L. C. Balling, and J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
[Crossref]

Ichikawa, K.

Imaoka, A.

Inouye, K.

H. Nakatsuka, Y. Katashima, and K. Inouye, “Incoherent light gated optical Kerr shutter for fluorescence lifetime measurement,” Opt. Commun. 69, 169–172 (1988).
[Crossref]

Ippen, E. P.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.
[Crossref]

Katashima, Y.

H. Nakatsuka, Y. Katashima, and K. Inouye, “Incoherent light gated optical Kerr shutter for fluorescence lifetime measurement,” Opt. Commun. 69, 169–172 (1988).
[Crossref]

Keller, J.-C.

Kishida, T.

Klein, U.

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[Crossref]

Kobayashi, T.

K. Kurokawa, T. Hattori, and T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
[Crossref]

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacethylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett 133, 230–234 (1987).
[Crossref]

Kuhl, J.

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole-burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[Crossref]

Kurokawa, K.

K. Kurokawa, T. Hattori, and T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
[Crossref]

Langhans, D.

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[Crossref]

Le Gouët, J.-L.

Matsuoka, M.

McCall, S. L.

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

Mi, X.

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[Crossref]

Morita, N.

Nakatsuka, H.

H. Nakatsuka, Y. Katashima, and K. Inouye, “Incoherent light gated optical Kerr shutter for fluorescence lifetime measurement,” Opt. Commun. 69, 169–172 (1988).
[Crossref]

Oasawa, H.

Rebane, A.

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole-burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[Crossref]

Richard, A.

A. Débarre, J.-C. Keller, J.-L. Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[Crossref]

Rio Leite, J. R.

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[Crossref]

Rothenberg, J. E.

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

Saikan, S.

Shank, C. V.

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.
[Crossref]

Siegman, A. E.

R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am B 3, 1295–1304 (1986).
[Crossref]

Tchénio, P.

Tomita, M.

Torizuka, K.

Trebino, R.

R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am B 3, 1295–1304 (1986).
[Crossref]

Van Kampen, N. G.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981), Chap. 11.

Wright, J. J.

M. D. Havey, L. C. Balling, and J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
[Crossref]

Yajima, T.

Ye, P.

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[Crossref]

Yu, Z.

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[Crossref]

Appl. Phys. (1)

H. J. Eichler, U. Klein, and D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[Crossref]

Appl. Phys. Lett. (1)

A. Rebane, J. Aaviksoo, and J. Kuhl, “Storage and time reversal of femtosecond light signals via persistent spectral hole-burning holography,” Appl. Phys. Lett. 54, 93–95 (1989).
[Crossref]

Chem. Phys. Lett (1)

T. Hattori and T. Kobayashi, “Femtosecond dephasing in a polydiacethylene film measured by degenerate four-wave mixing with an incoherent nanosecond laser,” Chem. Phys. Lett 133, 230–234 (1987).
[Crossref]

J. Opt. Soc. Am B (1)

R. Trebino, E. K. Gustafson, and A. E. Siegman, “Fourth-order partial-coherence effects in the formation of integrated-intensity gratings with pulsed light sources,” J. Opt. Soc. Am B 3, 1295–1304 (1986).
[Crossref]

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

J. Phys. (Paris) (1)

P. Fu, Z. Yu, X. Mi, and P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[Crossref]

Opt. Commun. (3)

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, and J. R. Rio Leite, “Dispersion of coherence spikes of incoherent broadband dye lasers,” Opt. Commun. 73, 475–478 (1989).
[Crossref]

A. Débarre, J.-C. Keller, J.-L. Gouët, A. Richard, and P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[Crossref]

H. Nakatsuka, Y. Katashima, and K. Inouye, “Incoherent light gated optical Kerr shutter for fluorescence lifetime measurement,” Opt. Commun. 69, 169–172 (1988).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (2)

S. L. McCall and E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[Crossref]

K. Kurokawa, T. Hattori, and T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
[Crossref]

Phys. Rev. A (2)

M. D. Havey, L. C. Balling, and J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
[Crossref]

M. D. Crisp, “Propagation of small area pulses of coherent light through a resonant medium,” Phys. Rev. A 6, 1604–1611 (1970).
[Crossref]

Phys. Rev. Lett. (1)

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

Prog. Nucl. Magn. Reson. Spectrosc. (1)

B. Blümich, “White noise nonlinear system analysis in nuclear magnetic resonance spectroscopy,” Prog. Nucl. Magn. Reson. Spectrosc. 19, 331–417 (1987).
[Crossref]

Rev. Sci. Instrum. (1)

B. Blümich, “Two dimensional interferometry,” Rev. Sci. Instrum. 58, 911–919 (1987).
[Crossref]

Other (5)

E. P. Ippen and C. V. Shank, “Techniques for measurement,” in Ultrashort Light Pulses Picosecond Techniques and Applications, S. L. Shapiro, ed. (Springer-Verlag, Berlin, 1977), pp. 83–122.
[Crossref]

P. R. Griffiths and J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions (Dover, New York, 1972), pp. 298.

N. G. Van Kampen, Stochastic Processes in Physics and Chemistry (North-Holland, Amsterdam, 1981), Chap. 11.

W. E. Moerner, ed., Persistent Spectral Hole-Burning. Science and Applications (Springer-Verlag, Berlin, 1988).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1

Experimental arrangement. (a) General diagram. (b) The field cross-correlator with the delay line.

Fig. 2
Fig. 2

Autocorrelation measurements. Dashed curve, autocorrelation function of the exciting pulse. Solid curve, autocorrelation function of the propagated pulse with optical density α0L ≃ 190.

Fig. 3
Fig. 3

Measurements of the square of the field cross-correlation function 〈|C(T)|2〉 between the propagated pulse (signal) and the reference pulse as a function of their relative delay. The optical density (α0L) is (a) 7, (b) 13, (c) 25, (d) 39, (e) 68, (f) 104, (g) 177, (h) 192.

Fig. 4
Fig. 4

Experimental (solid curve) and theoretical (small crosses) profiles of the modulus of the field cross-correlation 〈|C(T)|〉 between the propagated pulse (signal) and the reference pulse as a function of their relative delay. The optical density (α0L) is (a) 7, (b) 13, (c) 25, (d) 39, (e) 68, (f) 104, (g) 177, (h) 192.

Fig. 5
Fig. 5

Solid curve, experimental field cross-correlation profile for an optical density (α0L = 192). Dotted curve, the corresponding theoretical profile when only dispersion is taken into account.

Equations (41)

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

d θ ( z ) d z = - ( α 0 / 2 ) θ ( z ) .
E ( r , t ) = E ( L , t ) exp [ i ( ω t - k · r ) ] + c . c . ,
E ( L , t ) = - + d τ T ( τ ) E ( 0 , t - τ ) ,
E ( L , t ) = - + d ν exp ( 2 i π ν t ) T ^ ( ν ) E ^ ( 0 , ν ) ,
T ^ ( ν ) = exp [ - α 1 ( ν ) + i α 2 ( ν ) ] ,
α 1 ( ν ) = ( α 0 L / 2 ) V ( ν ) ,
V ( ν ) = exp [ - π ( ν / a ) 2 ] ,             a = [ ( 2 π k θ ) / ( M Sr λ 0 2 ) ] 1 / 2 ,
α 2 ( ν ) = π P [ - + d ν α 1 ( ν ) / ( ν - ν ) ] = ( α 0 L ) / π 1 / 2 Daws ( π 1 / 2 ν / a ) ,
ν a ,             α 2 ( ν ) [ ( α 0 L ) / ( 2 π ) ] ( a / ν ) .
Δ ν disp ( 2 a α 0 L ) / ( 2 π ) .
Δ ν abs ( 2 a ) [ π - 1 L n ( α 0 L / 2 ) ] 1 / 2 .
I ( t ) = E ( L , t ) 2 = | - + d τ T ( τ ) E ( 0 , t - τ ) | 2 ,
T ( τ ) - + d ν exp ( 2 i π ν τ ) exp [ ( i α 0 L a ) / ( 2 π ν ) ] δ ( τ ) - H ( τ ) [ ( α 0 L ) a / τ ] 1 / 2 J 1 ( 2 [ ( α 0 L ) a τ ) ] 1 / 2 ,
H ( t ) = 0 if t < 0 , H ( t ) = 1 if t > 0 ,
E e ( r , t ) = E ( 0 , t ) exp [ i ( ω t - k · r ) + c . c . ,
E ( 0 , t ) = E 0 ( t ) u ( t ) ,
u ( t ) = 0 , u ( t ) u ( t + τ ) = 0 , u ( t ) u * ( t - τ ) = g ( τ ) ,
u ( t 1 ) u ( t n ) u * ( t 1 ) u * ( t m ) = δ n m P u ( t 1 ) u * ( t P ( 1 ) ) u ( t n ) u * ( t P ( n ) ) ,
2 π / ω τ c ( 2 a ) - 1 ,             τ c , ( 2 a ) - 1 , τ sh τ e ,
E ( r , t ) = E ( L , t ) exp [ i ( ω t - k · r ) ] + c . c . ,
E ( L , t ) = - + d τ T ( τ ) E ( 0 , t - τ ) ,
I ( t ) = E ( L , t ) 2 = - + d τ E 0 2 ( t - τ ) φ ( τ ) ,
φ ( τ ) = - + d τ 1 T ( τ ) T * ( τ 1 ) g ( τ - τ 1 ) .
E r = ( r , t ) = E ( 0 , t - T ) exp [ i ( ω t - k r · r ) ] + c . c . ,
C ( T ) = - + d t E ( L , t ) E * ( 0 , t - T ) .
C ( T ) 2 = | - + d t - + d τ T ( τ ) E * ( 0 , t - T ) E ( 0 , t - τ ) | 2 .
C ( T ) 2 = | - + d t E 0 2 ( t ) | 2 | - + d τ T ( τ ) u * ( t - T ) u ( t - τ ) | 2 | - + d τ T ( τ ) g ( T - τ ) | 2 .
I ( r ) - + d t E r ( r , t ) + E ( r , t ) 2 .
E r ( r , t ) = E ( 0 , t - T ) exp [ i ϕ 0 ( T ) ] exp [ i ( ω t - k r r ) ] + c . c . ,
m ( r ) C ( T ) exp [ i ( k r - k ) · r - i ϕ 0 ] + c . c . ,
m ( r ) = C ( T ) exp ( - i ϕ 0 ) exp [ i ( k r - k ) · r ] + c . c .
C ( T ) 2 = - + d t 1 - + d t 2 - + d τ 1 - + d τ 2 [ T ( τ 1 ) T ( τ 2 ) × E ( 0 , t 1 - τ 1 ) E * ( 0 , t 1 - T ) E * ( 0 , t 2 - τ 2 ) × E ( 0 , t 2 - T ) ] .
E ( 0 , t 1 - τ 1 ) E * ( 0 , t 1 - T ) E * ( 0 , t 2 - τ 2 ) E ( 0 , t 2 - T ) = E ( 0 , t 1 - τ 1 ) E * ( 0 , t 1 - T ) E * ( 0 , t 2 - τ 2 ) E ( 0 , t 2 - T ) + E ( 0 , t 1 - τ 1 ) E * ( 0 , t 2 - τ 2 ) E * ( 0 , t 1 - T ) × E ( 0 , t 2 - T ) .
C ( T ) 2 = B I ( T ) + B II ( T ) ,
B I ( T ) = C ( T ) 2 | - + d t E 0 2 ( t ) | 2 | - + d τ T ( τ ) g ( T - τ ) | 2 , B II ( T ) [ - + d t E 0 2 ( t - T ) E 0 2 ( t ) ] × [ - + d u - + d τ 1 g ( - u + τ 1 ) T ( τ 1 ) × - + d τ 2 g * ( - u + τ 2 ) T * ( τ 2 ) ]
B I max | - + d t E 0 2 ( t ) | 2 g 2 ( 0 ) , B II [ - + d t E 0 4 ( t ) ] | - + d τ g 2 ( τ ) | 2 .
B II [ - + d t E 0 4 ( t ) ] [ - + d u G ( u ) 2 ] ,
G ( u ) = ( T g ) ( u ) ,
B II [ - + d t E 0 4 ( t ) ] [ - + d ν G ^ ( ν ) 2 ] [ - + d t E 0 4 ( t ) ] [ - + d ν T ^ ( ν ) 2 S 2 ( ν ) ] ,
S ( ν ) = - + d τ g ( τ ) exp ( 2 i π ν τ ) .
g t ( τ ) = - + d ν T ^ ( ν ) 2 S ( ν ) exp ( 2 i π ν τ ) .

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