Abstract

We describe how optical beams from a mode-locked laser will couple in a photorefractive crystal. We show that the two-beam-coupling gain coefficient is proportional to the square of the electric field correlation of the incoming light pulses. Consequently we show that two-beam-coupling experiments can measure the average coherence length of mode-locked laser pulses. We also describe how the temporal envelopes of the mode-locked optical pulses distort as they couple and propagate through the photorefractive crystal, and we give examples of how pulses can be shaped by using photorefractive coupling.

© 1990 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. L. Solymar, “Theory of volume holographic formation in photorefractive crystals,” in Electro-Optic and Photorefractive Materials, P. Günter, ed. (Springer-Verlag, Berlin, 1987), pp. 229–245.
    [CrossRef]
  2. A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μ m,” IEEE J. Quantum Electron. 24, 289–302 (1988).
    [CrossRef]
  3. A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, T. F. Boggess, “Formation, decay, and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606–615 (1989).
    [CrossRef]
  4. A. L. Smirl, G. C. Valley, R. A. Mullen, K. Bohnert, C. D. Mire, T. F. Boggess, “Picosecond photorefractive effect in BaTiO3,” Opt. Lett. 12, 501–503 (1987).
    [CrossRef] [PubMed]
  5. A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.
  6. A. M. Johnson, A. M. Glass, W. M. Simpson, D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.
  7. A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.
  8. V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
    [CrossRef]
  9. F. P. Strohkendl, J. M. C. Jonathan, R. W. Hellwarth, “Hole–electron competition in photorefractive gratings,” Opt. Lett. 11, 312–314 (1986).
    [CrossRef]
  10. Here we use the fact that the ensemble average of the derivative of a function equals the derivative of the ensemble average of the function. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Process (McGraw-Hill, New York, 1965), Chap. 9, pp. 314–318.
  11. J. Feinberg, “Optical phase conjugation in photorefractive materials,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 417–443.
    [CrossRef]
  12. 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]
  13. J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, pp. 63–68.
  14. For example, if both of the optical fields are derived from the same mode-locked laser, then Eq. (6) will hold as long as the laser exhibits no long-term changes in the statistics of its phase fluctuations during an experiment. However, if the alignment of the mirrors of the laser should drift with time, for example, then the laser’s pulse statistics might change, and Eq. (6) would no longer follow from Eq. (5).
  15. The coupling coefficients η12and η21in the coupled-wave equations (21a) and (21b) will be sightly different if the term ∇(∇·E)is retained. Namely, cos θ will be replaced by cos θ− Re[(zˆ·êi)(êi·êi*)] with i= 1 in Eq. (21a) and i= 2 in Eq. (21b), where s1and s2are the unit vectors along k1and k2, respectively, z is a unit vector in the z direction, and we assume that the optical field amplitudes vary only along the z direction.
  16. A. V. Alekseev-Popov, A. V. Knyaz’kov, A. S. Saikin, “Recording volume amplitude-phase holograms in a lead-lanthanum zirconate-titanate ceramic,” Sov. Tech. Phys. Lett 9, 475–477 (1983).
  17. K. Walsh, T. J. Hall, R. E. Burge, “Influence of polarization state and absorption gratings on photorefractive two-wave mixing in GaAs,” Opt. Lett. 12, 1026–1028 (1987).
    [CrossRef] [PubMed]
  18. Y. Lee, “Studies of the photorefractive effect in barium titanate: higher-order spatial harmonics and two-beam energy coupling,” (University of Southern California, Los Angeles, Calif., 1989).
  19. R. M. Pierce, R. S. Cudney, G. D. Bacher, J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414–416 (1990).
    [CrossRef] [PubMed]
  20. 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]
  21. Z. Vardeny, J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
    [CrossRef]
  22. R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
    [CrossRef]
  23. S. L. Palfrey, T. F. Heinz, “Coherent interactions in pump-probe absorption measurements: the effect of phase gratings,” J. Opt. Soc. Am. B 2, 674–678 (1985).
    [CrossRef]
  24. W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
    [CrossRef]
  25. In Ref. 8 we defined the coherence time of the pulse as one-half of the FWHM of the field correlation function. This definition leads to a coherence time = τc= (2 ln 2)/(πΔυ) for an optical field having a Gaussian power spectral density with a FWHM of Δυ(Hz). With this definition the coherence time τc equals the pulse duration τp for a transform-limited pulse. A more common definition12 of τc for cw beams is τc=∫−∞+∞|Γ(τd)|2dτd. In this case τc= [(2 ln 2)/π]1/21/(Δυ) for an optical field having a Gaussian spectrum with a FWHM of Δν(Hz).
  26. D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
    [CrossRef] [PubMed]
  27. A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
    [CrossRef] [PubMed]
  28. E. D. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
    [CrossRef]

1990

V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

R. M. Pierce, R. S. Cudney, G. D. Bacher, J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414–416 (1990).
[CrossRef] [PubMed]

1989

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

A. L. Smirl, K. Bohnert, G. C. Valley, R. A. Mullen, T. F. Boggess, “Formation, decay, and erasure of photorefractive gratings written in barium titanate by picosecond pulses,” J. Opt. Soc. Am. B 6, 606–615 (1989).
[CrossRef]

1988

D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μ m,” IEEE J. Quantum Electron. 24, 289–302 (1988).
[CrossRef]

1987

1986

1985

1983

A. V. Alekseev-Popov, A. V. Knyaz’kov, A. S. Saikin, “Recording volume amplitude-phase holograms in a lead-lanthanum zirconate-titanate ceramic,” Sov. Tech. Phys. Lett 9, 475–477 (1983).

1982

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

1981

Z. Vardeny, J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

1980

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]

1969

E. D. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

Alekseev-Popov, A. V.

A. V. Alekseev-Popov, A. V. Knyaz’kov, A. S. Saikin, “Recording volume amplitude-phase holograms in a lead-lanthanum zirconate-titanate ceramic,” Sov. Tech. Phys. Lett 9, 475–477 (1983).

Bacher, G. D.

Baltrameyunas, R.

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

Boggess, T. F.

Bohnert, K.

Bohnert, K. M.

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μ m,” IEEE J. Quantum Electron. 24, 289–302 (1988).
[CrossRef]

Burge, R. E.

Bylsma, R. B.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

Cudney, R. S.

Danelyus, R.

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

Dominic, V.

V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

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]

Fauchet, P. M.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Feinberg, J.

V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

R. M. Pierce, R. S. Cudney, G. D. Bacher, J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414–416 (1990).
[CrossRef] [PubMed]

J. Feinberg, “Optical phase conjugation in photorefractive materials,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 417–443.
[CrossRef]

Giuliani, G.

D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Glass, A. M.

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

A. M. Johnson, A. M. Glass, W. M. Simpson, D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.

Gong, T.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, pp. 63–68.

Grischkowsky, D.

D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Gustafson, E. K.

Halas, N. J.

D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

Hall, T. J.

Hawkins, R. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Heinz, T. F.

Hellwarth, R. W.

Heritage, J. P.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Johnson, A. M.

A. M. Johnson, A. M. Glass, W. M. Simpson, D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

Jonathan, J. M. C.

Kirschner, E. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Klein, M. B.

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

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]

Knyaz’kov, A. V.

A. V. Alekseev-Popov, A. V. Knyaz’kov, A. S. Saikin, “Recording volume amplitude-phase holograms in a lead-lanthanum zirconate-titanate ceramic,” Sov. Tech. Phys. Lett 9, 475–477 (1983).

Krokel, D.

D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

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]

Leaird, D. E.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Lee, Y.

Y. Lee, “Studies of the photorefractive effect in barium titanate: higher-order spatial harmonics and two-beam energy coupling,” (University of Southern California, Los Angeles, Calif., 1989).

Liou, L.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Mire, C. D.

Mullen, R. A.

Nighan, W. L.

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Olson, D. H.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

A. M. Johnson, A. M. Glass, W. M. Simpson, D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.

Palfrey, S. L.

Papoulis, A.

Here we use the fact that the ensemble average of the derivative of a function equals the derivative of the ensemble average of the function. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Process (McGraw-Hill, New York, 1965), Chap. 9, pp. 314–318.

Pierce, R. M.

V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

R. M. Pierce, R. S. Cudney, G. D. Bacher, J. Feinberg, “Measuring photorefractive trap density without the electro-optic effect,” Opt. Lett. 15, 414–416 (1990).
[CrossRef] [PubMed]

Piskarskas, A.

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

Pyatrauskas, M.

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

Rytz, D.

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

Saikin, A. S.

A. V. Alekseev-Popov, A. V. Knyaz’kov, A. S. Saikin, “Recording volume amplitude-phase holograms in a lead-lanthanum zirconate-titanate ceramic,” Sov. Tech. Phys. Lett 9, 475–477 (1983).

Siegman, A. E.

Simpson, W. M.

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

A. M. Johnson, A. M. Glass, W. M. Simpson, D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.

Smirl, A. L.

Solymar, L.

L. Solymar, “Theory of volume holographic formation in photorefractive crystals,” in Electro-Optic and Photorefractive Materials, P. Günter, ed. (Springer-Verlag, Berlin, 1987), pp. 229–245.
[CrossRef]

Strohkendl, F. P.

Tauc, J.

Z. Vardeny, J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

Thurston, R. N.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Tomlinson, W. J.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Treacy, E. D.

E. D. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

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]

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

Vaitkus, Yu.

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

Valley, G. C.

Vardeny, Z.

Z. Vardeny, J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

Walsh, K.

Weiner, A. M.

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Yao, X. S.

V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

Appl. Phys.

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.

V. Dominic, X. S. Yao, R. M. Pierce, J. Feinberg, “Measuring the coherence length of mode-locked laser pulses in real time,” Appl. Phys. Lett. 56, 521–523 (1990).
[CrossRef]

IEEE J. Quantum Electron.

A. L. Smirl, G. C. Valley, K. M. Bohnert, T. F. Boggess, “Picosecond photorefractive and free-carrier transient energy transfer in GaAs at 1 μ m,” IEEE J. Quantum Electron. 24, 289–302 (1988).
[CrossRef]

E. D. Treacy, “Optical pulse compression with diffraction gratings,” IEEE J. Quantum Electron. QE-5, 454–458 (1969).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

Z. Vardeny, J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
[CrossRef]

W. L. Nighan, T. Gong, L. Liou, P. M. Fauchet, “Self-diffraction: a new method for characterization of ultrashort laser pulses,” Opt. Commun. 69, 339–344 (1989).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

D. Krokel, N. J. Halas, G. Giuliani, D. Grischkowsky, “Dark-pulse propagation in optical fibers,” Phys. Rev. Lett. 60, 29–32 (1988).
[CrossRef] [PubMed]

A. M. Weiner, J. P. Heritage, R. J. Hawkins, R. N. Thurston, E. M. Kirschner, D. E. Leaird, W. J. Tomlinson, “Experimental observation of the fundamental dark soliton in optical fibers,” Phys. Rev. Lett. 61, 2445–2448 (1988).
[CrossRef] [PubMed]

Sov. J. Quantum Electron.

R. Baltrameyunas, Yu. Vaitkus, R. Danelyus, M. Pyatrauskas, A. Piskarskas, “Applications of dynamic holography in determination of coherence times of single picosecond light pulses,” Sov. J. Quantum Electron. 12, 1252–1254 (1982).
[CrossRef]

Sov. Tech. Phys. Lett

A. V. Alekseev-Popov, A. V. Knyaz’kov, A. S. Saikin, “Recording volume amplitude-phase holograms in a lead-lanthanum zirconate-titanate ceramic,” Sov. Tech. Phys. Lett 9, 475–477 (1983).

Other

Here we use the fact that the ensemble average of the derivative of a function equals the derivative of the ensemble average of the function. See, for example, A. Papoulis, Probability, Random Variables, and Stochastic Process (McGraw-Hill, New York, 1965), Chap. 9, pp. 314–318.

J. Feinberg, “Optical phase conjugation in photorefractive materials,” in Optical Phase Conjugation, R. Fisher, ed. (Academic, New York, 1983), pp. 417–443.
[CrossRef]

Y. Lee, “Studies of the photorefractive effect in barium titanate: higher-order spatial harmonics and two-beam energy coupling,” (University of Southern California, Los Angeles, Calif., 1989).

In Ref. 8 we defined the coherence time of the pulse as one-half of the FWHM of the field correlation function. This definition leads to a coherence time = τc= (2 ln 2)/(πΔυ) for an optical field having a Gaussian power spectral density with a FWHM of Δυ(Hz). With this definition the coherence time τc equals the pulse duration τp for a transform-limited pulse. A more common definition12 of τc for cw beams is τc=∫−∞+∞|Γ(τd)|2dτd. In this case τc= [(2 ln 2)/π]1/21/(Δυ) for an optical field having a Gaussian spectrum with a FWHM of Δν(Hz).

A. M. Johnson, A. M. Glass, W. M. Simpson, R. B. Bylsma, D. H. Olson, “Microwatt picosecond pulse autocorrelator using photorefractive GaAs:Cr,” in OSA Annual Meeting, Vol. 11 of 1988 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1988), p. 128.

A. M. Johnson, A. M. Glass, W. M. Simpson, D. H. Olson, “Infrared picosecond pulse diagnostics using photorefractive beam coupling,” in Conference on Lasers and Electro-Optics, Vol. II of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 226.

A. M. Johnson, W. M. Simpson, A. M. Glass, M. B. Klein, D. Rytz, R. Trebino, “Infrared picosecond pulse correlation measurements using photorefractive beam coupling and harmonic generation in KNbO3and BaTiO3,” in OSA Annual Meeting, Vol. 18 of 1989 Optical Society of America Technical Digest Series (Optical Society of America, Washington, D.C., 1989), p. 53.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 3, pp. 63–68.

For example, if both of the optical fields are derived from the same mode-locked laser, then Eq. (6) will hold as long as the laser exhibits no long-term changes in the statistics of its phase fluctuations during an experiment. However, if the alignment of the mirrors of the laser should drift with time, for example, then the laser’s pulse statistics might change, and Eq. (6) would no longer follow from Eq. (5).

The coupling coefficients η12and η21in the coupled-wave equations (21a) and (21b) will be sightly different if the term ∇(∇·E)is retained. Namely, cos θ will be replaced by cos θ− Re[(zˆ·êi)(êi·êi*)] with i= 1 in Eq. (21a) and i= 2 in Eq. (21b), where s1and s2are the unit vectors along k1and k2, respectively, z is a unit vector in the z direction, and we assume that the optical field amplitudes vary only along the z direction.

L. Solymar, “Theory of volume holographic formation in photorefractive crystals,” in Electro-Optic and Photorefractive Materials, P. Günter, ed. (Springer-Verlag, Berlin, 1987), pp. 229–245.
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

(a) A strong optical beam leads a weak beam by 30 psec. (b) While these optical pulses are present in the crystal, the space-charge field momentarily deviates from its steady-state value: It initially decays and then recovers.

Fig. 2
Fig. 2

(a) A strong optical beam follows a weak beam by 70 psec. (b) While these pulses are present, the space-charge field grows and then decays, but it never deviates by even 1 part in 106 from its steady-state value.

Fig. 3
Fig. 3

Schematic illustrating pulse shaping by two-beam coupling in a photorefractive crystal. The square temporal profiles of the two incident pulses are altered after coupling in the crystal. [A steady-state grating has already been built up in the crystal, and each new light pulse barely perturbs this grating (see Figs. 1 and 2)]. Note that when the two pulses overlap in time the transmitted field of pulse 2 constructively interferes with the diffracted field of pulse 1, while the transmitted field of pulse 1 destructively interferes with the diffracted field of pulse 2. When the two pulses do not overlap, there is no interference.

Fig. 4
Fig. 4

Pulse shaping by using Gaussian pulses with the same spectral width. The peaks of both beams reach the entrance face of the crystal at the same time (τd = 0), but the duration of beam 1 is 10 times greater than that of beam 2. (a) The temporal profile of beam 1 before (dotted curve) and after (solid curve) the photorefractive crystal. Note that beam 2 couples energy out of beam 1. We assume an incident average intensity ratio of I2/I1 = 100 and a coupling strength ηL = −1.23. (b) The temporal profile of beam 1 before (dotted curve) and after (solid curve) coupling in the crystal. Here we reverse the direction of energy coupling in the crystal, so that beam 2 couples energy into beam 1. We use an incident average intensity ratio of I2/I1 = 100 and a couping strength ηL = +1.23.

Fig. 5
Fig. 5

Pulse shaping in a photorefractive crystal by using equal-width, transform-limited Gaussian pulses (FWHM intensity = 70 psec). The curves show the temporal envelope of incident beam 1 before the crystal (dotted curves) and after the crystal (solid curves). (a) Beam 1 arrives 50 psec before beam 2. The coupling strength is set at ηL = −2 (so that beam 1 loses energy), and the average intensity ratio is set at I2/I1 = 10. (b) Beam 1 arrives 85 psec before beam 2. The coupling strength is set at ηL = +1.5 (so that beam 1 gains energy), and the average intensity ratio is set at I2/I1 = 100.

Fig. 6
Fig. 6

Pulse shaping in a photorefractive crystal by using Gaussian pulses with the same temporal width (FWHM intensity 70 psec). The temporal envelope of the incident beam (1) is shown before the crystal (dotted curve) and after the crystal (solid curve). Here beam 1 is transform limited, but beam 2 is chirped and has a spectral width 8 times that of beam 1. The time delay between the two beams is zero, and the coupling strength is ηL = 1.5 (beam 1 gains energy).

Equations (62)

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

E sc ( r , t ) t + a I 0 ( t ) E sc ( r , t ) = i b E 1 ( r , t ) E 2 * ( r , t ) ,
E sc ( r , t ) ¯ t + a I 0 ( t ) E sc ( r , t ) ¯ = i b E 1 ( r , t ) E 2 * ( r , t ) ¯ ,
E sc ¯ = i b a E 1 ( r , t ) E 2 * ( r , t ) ¯ | E 1 ( r , t ) | 2 + | E 2 ( r , t ) | 2 ,
F ( t ) = 1 T T / 2 T / 2 F ( τ ) d τ .
E 1 ( r , t ) = A 1 ( r , t ) u 1 ( r , t ) ,
E 2 ( r , t ) = A 2 ( r , t τ d ) u 2 ( r , t τ d ) ,
G ( τ d ) E 1 ( r , t ) E 2 * ( r , t ) ¯ = A 1 ( r , t ) A 2 * ( r , t τ d ) u 1 ( r , t ) u 2 * ( r , t τ d ) ¯ .
G ( τ d ) = A 1 ( r , t ) A 2 * ( r , t τ d ) γ ( 2 ) ( τ d ) ,
γ ( 2 ) ( τ d ) u 1 ( r , t ) u 2 * ( r , t τ d ) ¯ .
E sc ¯ = i b a A 1 ( r , t ) A 2 * ( r , t τ d ) γ ( 2 ) ( τ d ) I 0 .
2 E ( r , t ) μ 0 2 D ( r , t ) t 2 = 0 ,
E ( r , t ) = Re { [ e ˆ 1 E 1 ( r , t ) exp ( i k 1 · r ) + e ˆ 2 E 2 ( r , t ) exp ( i k 2 · r ) ] exp ( i ω 0 t ) } ,
D ( r , t ) = Re { D 1 ( r , t ) exp ( i k 1 · r ) + D 2 ( r , t ) exp ( i k 2 · r ) ] exp ( i ω 0 t ) } ,
D i ( r , t ) ε 0 ε ( r ) · e ˆ i E i ( r , t ) , i = 1 , 2.
ε ( r ) = ε r + Re [ Δ ε r · exp ( i k g · r ) ] ,
( z + 1 υ t ) E 1 ( z , t ) = i ω 4 n 1 c cos θ ( e ˆ 1 * · Δ ε r · e ˆ 2 ) E 2 ( z , t ) ,
( z + 1 υ t ) E 2 ( z , t ) = i ω 4 n 2 c cos θ ( e ˆ 2 * · Δ ε r · e ˆ 1 ) E 1 ( z , t ) .
υ ( c / n ) cos θ .
| 2 E i z 2 | k i | E i z | ,
| 2 ( ε · e ˆ i E i ) t 2 | ω | ( ε · e ˆ i E i ) t | ,
| ( Δ ε r · e ˆ E i ) t | ω | Δ ε ˆ r · e ˆ i E i | .
Δ ε ˆ r ε ˆ r ,
Δ ε ˆ r = ε ˆ r · ( R · k ˆ g ) · ε r E sc ¯ ,
( z + 1 υ t ) E 1 ( z , t ) = η 12 G ( z , τ d ) I 0 E 2 ( z , t ) ,
( z + 1 υ t ) E 2 ( z , t ) = η 21 * G * ( z , τ d ) I 0 E 1 ( z , t ) ,
η 12 = ω 4 n 1 c cos θ b a [ e ˆ 1 * · ε r · ( R · k ˆ g ) · ε r · e ˆ 2 ] ,
η 12 * = ω 4 n 2 c cos θ b a * [ e ˆ 2 * · ( ε r · ( R · k ˆ g ) · ε r ) * · e ˆ 1 ] .
| E 1 ( z , t ) | 2 ¯ z = + 2 Re ( η 12 ) | G ( z , τ d ) | 2 I 0 ,
| E 2 ( z , t ) | 2 ¯ z = 2 Re ( η 21 ) | G ( z , τ d ) | 2 I 0 ,
G ( z , τ d ) z = G ( z , τ d ) I 0 ( η 12 | E 2 | 2 ¯ η 21 | E 1 | 2 ¯ ) .
t F ( z , t ) = 0
G ( z , τ d ) = G ( 0 , τ d ) exp ( η 12 z ) ,
| E 1 ( z , t ) | 2 ¯ | E 1 ( 0 , t ) | 2 ¯ = | G ( 0 , τ d ) | 2 exp [ ( g 12 z ) 1 ] I 0 ,
g 12 2 Re ( η 12 ) .
A i ( 0 , t ) = E i 0 I i ( t ) , i = 1 , 2.
Γ ( τ d ) I 1 ( t ) I 2 * ( t τ d ) [ | I 1 ( t ) | 2 | I 2 ( t ) | 2 ] 1 / 2 γ ( 2 ) ( τ d ) .
| E 1 ( z , t ) | 2 ¯ | E 1 ( 0 , t ) | 2 ¯ | E 1 ( 0 , t ) | 2 ¯ = [ exp ( g 12 z ) 1 ] | Γ ( τ d ) | 2 .
| E 1 ( z , t ) | 2 ¯ = | E 1 ( 0 , t ) | 2 ¯ exp ( g 12 z ) .
γ ( 2 ) ( τ d ) = exp [ ( τ d 2 / τ s 2 ) ln 2 ] ,
E ( 0 , t ) = E 0 exp [ i φ ( t ) ] exp [ 2 ( t 2 / τ p 2 ) ln 2 ] ,
Γ ( τ d ) = exp [ τ d 2 ( 1 τ p 2 + 1 τ s 2 ) ln 2 ] ,
γ ( 2 ) ( τ d ) = exp [ ( τ d 2 / τ s 2 ) ln 2 ] ,
FWHM = 2 τ p τ s ( τ p 2 + τ s 2 ) 1 / 2 .
E 1.2 ( z , t ) = 1 2 π E 1 , 2 ( z , ω ) e i ω t d ω .
F 1 ( z , ω ) z = + η 12 G ( z , τ d ) I 0 F 2 ( z , ω ) ,
F 2 ( z , ω ) z = η 21 * G * ( z , τ d ) I 0 F 1 ( z , ω ) ,
F 1 ( z , ω ) exp ( i ω z υ ) E 1 ( z , ω ) ,
F 2 ( z , ω ) exp ( i ω z υ ) E 2 ( z , ω ) .
F 1 ( 0 , ω ) = E 1 0 ,
F 2 ( 0 , ω ) = E 2 0 ,
F 1 ( z , ω ) z | z = 0 = + η 12 G ( 0 , τ d ) I 0 E 2 0 ,
F 2 ( z , ω ) z | z = 0 = η 21 * G * ( 0 , τ d ) I 0 E 1 0
E 1 ( z , t ) = E 1 ( 0 , t z υ ) cos | β ( z , τ d ) | + Γ ( τ d ) | Γ ( τ d ) | E 2 ( 0 , t z υ ) sin | β ( z , τ d ) | ,
E 2 ( z , t ) = E 2 ( 0 , t z υ ) cos | β ( z , τ d ) | + Γ * ( τ d ) | Γ ( τ d ) | E 1 ( 0 , t z υ ) sin | β ( z , τ d ) | ,
β ( z , τ d ) G ( 0 , τ d ) I 0 ( e η z 1 ) = [ | E 1 ( 0 , t ) | 2 ¯ | E 2 ( 0 , t ) | 2 ¯ ] 1 / 2 Γ ( τ d ) ( e η z 1 ) .
I 1 ( t ) = exp [ 2 ln 2 ( 1 + i q 2 1 ) t 2 τ p 1 2 ] ,
I 2 ( t τ d ) = exp [ 2 ( t τ d ) 2 τ p 2 2 ln 2 + i ω 0 τ d ] ,
Γ ( τ d ) = 2 exp [ i 2 tan 1 ( q 2 1 q 2 + 1 ) i ω 0 τ d ] ( q 2 + 3 ) 1 / 4 × exp [ 2 ln 2 q 2 + 3 ( 2 + i q 2 1 ) τ d 2 τ p 2 2 ] .
E 1 ( z , t ) = E 10 { I 1 [ t ( z / υ ) ] + q Γ ( τ d ) ( e η z 1 ) I 2 [ t ( z / υ ) τ d ] } .
I 1 ( t ) = exp ( 2 ln 2 t 2 τ p 2 ) ,
I 2 ( t τ d ) = exp [ 2 ln 2 × ( 1 + i m 2 1 ) ( t τ d ) 2 τ p 2 + i ω 0 τ d ] .
Γ ( τ d ) = 2 exp [ i 2 tan 1 ( m 2 1 2 ) i ω 0 τ d ] ( m 2 + 3 ) 1 / 4 × [ 2   ln   2 m 2 + 3 ( 1 + m 2 + i m 2 1 ) τ d 2 τ p 2 ] .

Metrics