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, 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, T. Kobayashi, “Subpicosecond molecular dynamics studied by degenerate four-wave mixing with incoherent light,” Phys. Rev. 36, 1298–1304 (1987).
    [CrossRef]
  3. M. Tomita, 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, T. Yajima, “Coherent propagation effect of incoherent light,” J. Opt. Soc. Am. B 3, 548–553 (1986).
    [CrossRef]
  5. H. Nakatsuka, Y. Katashima, 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, 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, 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, A. C. Balant, “Observation of the formation of the 0π pulse,” Phys. Rev. Lett. 53, 552–555 (1984).
    [CrossRef]
  10. E. P. Ippen, 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, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.
  12. H. J. Eichler, U. Klein, 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, 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, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
    [CrossRef]
  15. S. L. McCall, E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [CrossRef]
  16. M. D. Havey, L. C. Balling, J. J. Wright, “Measurement of the 3P1lifetime in Sr,” Phys. Rev. A 13, 1269–1270 (1976).
    [CrossRef]
  17. M. Abramowitz, 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, 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, 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, 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, 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)

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

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, 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, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

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

1988 (1)

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

1987 (5)

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]

T. Hattori, 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, 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, P. Ye, “Fourth-order coherence-function of laser induced molecular grating reorientational grating and population grating,” J. Phys. (Paris) 48, 2089–2096 (1987).
[CrossRef]

1986 (3)

R. Trebino, E. K. Gustafson, 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]

N. Morita, K. Torizuka, T. Yajima, “Coherent propagation effect of incoherent light,” J. Opt. Soc. Am. B 3, 548–553 (1986).
[CrossRef]

M. Tomita, M. Matsuoka, “Ultrafast pump–probe measurement using intensity correlation of incoherent light,” J. Opt. Soc. Am. B 3, 560–563 (1986).
[CrossRef]

1984 (1)

J. E. Rothenberg, D. Grischkowsky, 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, 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, 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, E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Aaviksoo, J.

A. Rebane, J. Aaviksoo, 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, 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, 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, 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, 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, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.

Débarre, A.

Eichler, H. J.

H. J. Eichler, U. Klein, 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, 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, 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, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

Griffiths, P. R.

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

Grischkowsky, D.

J. E. Rothenberg, D. Grischkowsky, 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, 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, E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Hattori, T.

T. Hattori, 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, 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, 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, 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, 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, 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, D. Langhans, “Coherence time measurement of picosecond pulses by light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
[CrossRef]

Kobayashi, T.

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

Kuhl, J.

A. Rebane, J. Aaviksoo, 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, 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, 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, E. L. Hahn, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Mi, X.

P. Fu, Z. Yu, X. Mi, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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, 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)

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

A. S. L. Gomes, L. H. Acioli, C. de Araúijo, 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, P. Tchénio, “An amplitude correlator for broadband laser source characterization,” Opt. Commun. 73, 309–313 (1989).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (2)

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

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

Phys. Rev. A (2)

M. D. Havey, L. C. Balling, 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, 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)

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

E. P. Ippen, 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, J. A. de Haseth, Fourier Transform Infrared Spectroscopy (Wiley, New York, 1984), Chap. 11.

M. Abramowitz, 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.

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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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