Abstract

We performed theoretical calculations of the relative diffraction efficiency of partially coherent light-induced integrated-intensity gratings using pulsed sources, paying particular attention to thermal gratings. We provided a simple intuitive picture of the phenomenon and then calculated exact expressions that, unlike instantaneous-intensity-grating results, necessarily require the use of fourth-order coherence functions. Assuming several radiation models, we evaluated these expressions and found that the results proved to be insensitive to the specific radiation model assumed. The application of these results to a previously performed pulsed-laser experiment yielded a better fit to the data than an expression involving only second-order coherence, which is valid only in the cw limit. We included the effects of grating decay and, in addition, compared the use of integrated-intensity gratings for ultrashort-pulse-length measurement with standard techniques. Finally, we calculated expressions for the relative diffraction efficiency of integrated-intensity gratings created with excitation beams from two separate and independent sources of different frequency, and we report an experiment whose results were found to agree with this theory.

© 1986 Optical Society of America

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  1. G. Martin, R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index changes,” Appl. Phys. Lett. 34, 371–373 (1979).
    [CrossRef]
  2. R. K. Jain, M. B. Klein, “Degenerate four-wave mixing in semiconductors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 365–366.
  3. J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
    [CrossRef]
  4. K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
    [CrossRef]
  5. J. R. Salcedo, A. E. Siegman, “Laser induced photoacoustic grating effects in molecular crystals,” IEEE J. Quant. Electron. QE-15, 250–256 (1979).
    [CrossRef]
  6. V. S. Idiatulin, Yu. N. Teryaev, “Transient gratings in a nonlinear medium,” Opt. Quantum Electron. 14, 51–56 (1982).
    [CrossRef]
  7. H. J. Eichler, U. Klein, D. Langhans, “Coherence time measurement of picosecond pulses by a light-induced grating method,” Appl. Phys. 21, 215–219 (1980).
    [CrossRef]
  8. R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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]
  9. J. R. Andrews, R. M. Hochstrasser, “Transient grating studies of energy deposition in radiationless processes,” Chem. Phys. Lett. 76, 207–212, (1980).
    [CrossRef]
  10. Z. Vardeny, J. Tauc, “Picosecond coherence coupling in the pump and probe technique,” Opt. Commun. 39, 396–400 (1981).
    [CrossRef]
  11. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  12. R. Trebino, “Subpicosecond-relaxation studies using tunable-laser-induced-grating techniques,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1983).
  13. B. S. Wherrett, A. L. Smirl, T. F. Boggess, “Theory of degenerate four-wave mixing in picosecond excitation-probe experiments,” IEEE J. Quantum Electron. QE-19, 680–689 (1983).
    [CrossRef]
  14. C. V. Shank, D. H. Auston, “Parametric coupling in an optically excited plasma in Ge,” Phys. Rev. Lett. 34, 479 (1975).
    [CrossRef]
  15. A. von Jena, H. E. Lessing, “Coherent coupling effects in picosecond absorption experiments,” Appl. Phys. 19, 131–144 (1979).
    [CrossRef]
  16. T. F. Heinz, S. L. Palfrey, K. B. Eisenthal, “Coherent coupling effects in pump-probe measurements with collinear, copropagating beams,” Opt. Lett. 9, 359–361 (1984).
    [CrossRef] [PubMed]
  17. W. M. Grossman, D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916. (1980).
    [CrossRef]
  18. M. Schubert, K.-E. Süsse, W. Vogel, “Influence of chaotic pump radiation with finite bandwidth on the intensity correlation of resonance fluorescence radiation,” Opt. Commun. 30, 275–278 (1979).
    [CrossRef]
  19. H. J. Kimble, L. Mandel, “Resonance fluorescence with excitation of finite bandwidth,” Phys. Rev. A 15, 689–699 (1977).
    [CrossRef]
  20. It appears, from our discussion, that in the cw limit the diffraction efficiencies in both cases go to infinity. This is, of course, not the case because material relaxation prevents the buildup of infinite grating strength. When the pulse is longer than the material relaxation time, the relaxation time then replaces the pulse length in the above argument. Section 6 treats this effect in greater detail. In any event, the argument presented in the text is intended to give relative grating strengths only.
  21. Equations (11) and (12) explain why thermal gratings have obscured population gratings in grating experiments on dyes dissolved in ethanol but not in experiments using water as a solvent. The solvent-dependent factors in Eq. (11) result in a thermal-grating diffraction efficiency proportional to [(dn/dT)/ρcυ]2, with other factors depending on the solute, laser light, or beam geometry or exhibiting little variation.9The value of [(dn/dT)/ρcυ]2for ethanol is more than 100 times that for water.
  22. R. C. Desai, M. D. Levenson, J. A. Baker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate wave-front generation,” Phys. Rev. A 27, 1968–1976 (1983).
    [CrossRef]
  23. A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1955).
    [CrossRef]
  24. B. Picinbono, E. Boileau, “Higher-order coherence functions of optical fields and phase fluctuations,” J. Opt. Soc. Am. 58, 784–789 (1968).
    [CrossRef]
  25. J. J. Song, J. H. Lee, M. D. Levenson, “Picosecond relaxation measurements by polarization spectroscopy in condensed phases,” Phys. Rev. A 17, 1439–1447 (1978).
    [CrossRef]
  26. J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
    [CrossRef]
  27. R. Trebino, A. E. Siegman, C. L. Ladera, “Suppression of thermal gratings in polarization spectroscopy,” J. Opt. Soc. Am. B 1, 549–550 (1984).

1984

R. Trebino, A. E. Siegman, C. L. Ladera, “Suppression of thermal gratings in polarization spectroscopy,” J. Opt. Soc. Am. B 1, 549–550 (1984).

T. F. Heinz, S. L. Palfrey, K. B. Eisenthal, “Coherent coupling effects in pump-probe measurements with collinear, copropagating beams,” Opt. Lett. 9, 359–361 (1984).
[CrossRef] [PubMed]

1983

B. S. Wherrett, A. L. Smirl, T. F. Boggess, “Theory of degenerate four-wave mixing in picosecond excitation-probe experiments,” IEEE J. Quantum Electron. QE-19, 680–689 (1983).
[CrossRef]

R. C. Desai, M. D. Levenson, J. A. Baker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate wave-front generation,” Phys. Rev. A 27, 1968–1976 (1983).
[CrossRef]

1982

V. S. Idiatulin, Yu. N. Teryaev, “Transient gratings in a nonlinear medium,” Opt. Quantum Electron. 14, 51–56 (1982).
[CrossRef]

R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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

K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
[CrossRef]

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

1980

W. M. Grossman, D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916. (1980).
[CrossRef]

J. R. Andrews, R. M. Hochstrasser, “Transient grating studies of energy deposition in radiationless processes,” Chem. Phys. Lett. 76, 207–212, (1980).
[CrossRef]

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

1979

J. R. Salcedo, A. E. Siegman, “Laser induced photoacoustic grating effects in molecular crystals,” IEEE J. Quant. Electron. QE-15, 250–256 (1979).
[CrossRef]

G. Martin, R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index changes,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

M. Schubert, K.-E. Süsse, W. Vogel, “Influence of chaotic pump radiation with finite bandwidth on the intensity correlation of resonance fluorescence radiation,” Opt. Commun. 30, 275–278 (1979).
[CrossRef]

A. von Jena, H. E. Lessing, “Coherent coupling effects in picosecond absorption experiments,” Appl. Phys. 19, 131–144 (1979).
[CrossRef]

1978

J. J. Song, J. H. Lee, M. D. Levenson, “Picosecond relaxation measurements by polarization spectroscopy in condensed phases,” Phys. Rev. A 17, 1439–1447 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

1977

H. J. Kimble, L. Mandel, “Resonance fluorescence with excitation of finite bandwidth,” Phys. Rev. A 15, 689–699 (1977).
[CrossRef]

1975

C. V. Shank, D. H. Auston, “Parametric coupling in an optically excited plasma in Ge,” Phys. Rev. Lett. 34, 479 (1975).
[CrossRef]

1968

1955

Andrews, J. R.

J. R. Andrews, R. M. Hochstrasser, “Transient grating studies of energy deposition in radiationless processes,” Chem. Phys. Lett. 76, 207–212, (1980).
[CrossRef]

Auston, D. H.

C. V. Shank, D. H. Auston, “Parametric coupling in an optically excited plasma in Ge,” Phys. Rev. Lett. 34, 479 (1975).
[CrossRef]

Baker, J. A.

R. C. Desai, M. D. Levenson, J. A. Baker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate wave-front generation,” Phys. Rev. A 27, 1968–1976 (1983).
[CrossRef]

Baltrameyunas, R.

R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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.

B. S. Wherrett, A. L. Smirl, T. F. Boggess, “Theory of degenerate four-wave mixing in picosecond excitation-probe experiments,” IEEE J. Quantum Electron. QE-19, 680–689 (1983).
[CrossRef]

Boileau, E.

Dannelyus, R.

R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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]

Desai, R. C.

R. C. Desai, M. D. Levenson, J. A. Baker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate wave-front generation,” Phys. Rev. A 27, 1968–1976 (1983).
[CrossRef]

Dlott, D. D.

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

Eichler, H. J.

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

Eisenthal, K. B.

Fayer, M. D.

K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Grossman, W. M.

W. M. Grossman, D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916. (1980).
[CrossRef]

Heinz, T. F.

Hellwarth, R. W.

G. Martin, R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index changes,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Hochstrasser, R. M.

J. R. Andrews, R. M. Hochstrasser, “Transient grating studies of energy deposition in radiationless processes,” Chem. Phys. Lett. 76, 207–212, (1980).
[CrossRef]

Idiatulin, V. S.

V. S. Idiatulin, Yu. N. Teryaev, “Transient gratings in a nonlinear medium,” Opt. Quantum Electron. 14, 51–56 (1982).
[CrossRef]

Jain, R. K.

R. K. Jain, M. B. Klein, “Degenerate four-wave mixing in semiconductors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 365–366.

Kimble, H. J.

H. J. Kimble, L. Mandel, “Resonance fluorescence with excitation of finite bandwidth,” Phys. Rev. A 15, 689–699 (1977).
[CrossRef]

Klein, M. B.

R. K. Jain, M. B. Klein, “Degenerate four-wave mixing in semiconductors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 365–366.

Klein, U.

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

Ladera, C. L.

R. Trebino, A. E. Siegman, C. L. Ladera, “Suppression of thermal gratings in polarization spectroscopy,” J. Opt. Soc. Am. B 1, 549–550 (1984).

Langhans, D.

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

Lee, J. H.

J. J. Song, J. H. Lee, M. D. Levenson, “Picosecond relaxation measurements by polarization spectroscopy in condensed phases,” Phys. Rev. A 17, 1439–1447 (1978).
[CrossRef]

Lessing, H. E.

A. von Jena, H. E. Lessing, “Coherent coupling effects in picosecond absorption experiments,” Appl. Phys. 19, 131–144 (1979).
[CrossRef]

Levenson, M. D.

R. C. Desai, M. D. Levenson, J. A. Baker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate wave-front generation,” Phys. Rev. A 27, 1968–1976 (1983).
[CrossRef]

J. J. Song, J. H. Lee, M. D. Levenson, “Picosecond relaxation measurements by polarization spectroscopy in condensed phases,” Phys. Rev. A 17, 1439–1447 (1978).
[CrossRef]

Lutz, D. R.

K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
[CrossRef]

Madison, L.

K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
[CrossRef]

Mandel, L.

H. J. Kimble, L. Mandel, “Resonance fluorescence with excitation of finite bandwidth,” Phys. Rev. A 15, 689–699 (1977).
[CrossRef]

Martin, G.

G. Martin, R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index changes,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Nelson, K. A.

K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
[CrossRef]

Palfrey, S. L.

Picinbono, B.

Piskarskas, A.

R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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. Dannelyus, 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]

Salcedo, J. R.

J. R. Salcedo, A. E. Siegman, “Laser induced photoacoustic grating effects in molecular crystals,” IEEE J. Quant. Electron. QE-15, 250–256 (1979).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

Schubert, M.

M. Schubert, K.-E. Süsse, W. Vogel, “Influence of chaotic pump radiation with finite bandwidth on the intensity correlation of resonance fluorescence radiation,” Opt. Commun. 30, 275–278 (1979).
[CrossRef]

Shank, C. V.

C. V. Shank, D. H. Auston, “Parametric coupling in an optically excited plasma in Ge,” Phys. Rev. Lett. 34, 479 (1975).
[CrossRef]

Shemwell, D. M.

W. M. Grossman, D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916. (1980).
[CrossRef]

Siegman, A. E.

R. Trebino, A. E. Siegman, C. L. Ladera, “Suppression of thermal gratings in polarization spectroscopy,” J. Opt. Soc. Am. B 1, 549–550 (1984).

J. R. Salcedo, A. E. Siegman, “Laser induced photoacoustic grating effects in molecular crystals,” IEEE J. Quant. Electron. QE-15, 250–256 (1979).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

A. E. Siegman, “Bragg diffraction of a Gaussian beam by a crossed-Gaussian volume grating,” J. Opt. Soc. Am. 67, 545–550 (1955).
[CrossRef]

Smirl, A. L.

B. S. Wherrett, A. L. Smirl, T. F. Boggess, “Theory of degenerate four-wave mixing in picosecond excitation-probe experiments,” IEEE J. Quantum Electron. QE-19, 680–689 (1983).
[CrossRef]

Song, J. J.

J. J. Song, J. H. Lee, M. D. Levenson, “Picosecond relaxation measurements by polarization spectroscopy in condensed phases,” Phys. Rev. A 17, 1439–1447 (1978).
[CrossRef]

Süsse, K.-E.

M. Schubert, K.-E. Süsse, W. Vogel, “Influence of chaotic pump radiation with finite bandwidth on the intensity correlation of resonance fluorescence radiation,” Opt. Commun. 30, 275–278 (1979).
[CrossRef]

Tauc, J.

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

Teryaev, Yu. N.

V. S. Idiatulin, Yu. N. Teryaev, “Transient gratings in a nonlinear medium,” Opt. Quantum Electron. 14, 51–56 (1982).
[CrossRef]

Trebino, R.

R. Trebino, A. E. Siegman, C. L. Ladera, “Suppression of thermal gratings in polarization spectroscopy,” J. Opt. Soc. Am. B 1, 549–550 (1984).

R. Trebino, “Subpicosecond-relaxation studies using tunable-laser-induced-grating techniques,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1983).

Vaitkus, Yu.

R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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]

Vardeny, Z.

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

Vogel, W.

M. Schubert, K.-E. Süsse, W. Vogel, “Influence of chaotic pump radiation with finite bandwidth on the intensity correlation of resonance fluorescence radiation,” Opt. Commun. 30, 275–278 (1979).
[CrossRef]

von Jena, A.

A. von Jena, H. E. Lessing, “Coherent coupling effects in picosecond absorption experiments,” Appl. Phys. 19, 131–144 (1979).
[CrossRef]

Wherrett, B. S.

B. S. Wherrett, A. L. Smirl, T. F. Boggess, “Theory of degenerate four-wave mixing in picosecond excitation-probe experiments,” IEEE J. Quantum Electron. QE-19, 680–689 (1983).
[CrossRef]

Appl. Phys.

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

A. von Jena, H. E. Lessing, “Coherent coupling effects in picosecond absorption experiments,” Appl. Phys. 19, 131–144 (1979).
[CrossRef]

Appl. Phys. Lett.

G. Martin, R. W. Hellwarth, “Infrared-to-optical image conversion by Bragg reflection from thermally induced index changes,” Appl. Phys. Lett. 34, 371–373 (1979).
[CrossRef]

Chem. Phys. Lett.

J. R. Andrews, R. M. Hochstrasser, “Transient grating studies of energy deposition in radiationless processes,” Chem. Phys. Lett. 76, 207–212, (1980).
[CrossRef]

IEEE J. Quant. Electron.

J. R. Salcedo, A. E. Siegman, “Laser induced photoacoustic grating effects in molecular crystals,” IEEE J. Quant. Electron. QE-15, 250–256 (1979).
[CrossRef]

IEEE J. Quantum Electron.

B. S. Wherrett, A. L. Smirl, T. F. Boggess, “Theory of degenerate four-wave mixing in picosecond excitation-probe experiments,” IEEE J. Quantum Electron. QE-19, 680–689 (1983).
[CrossRef]

J. Appl. Phys.

W. M. Grossman, D. M. Shemwell, “Coherence lengths and phase conjugation by degenerate four-wave mixing,” J. Appl. Phys. 51, 914–916. (1980).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

R. Trebino, A. E. Siegman, C. L. Ladera, “Suppression of thermal gratings in polarization spectroscopy,” J. Opt. Soc. Am. B 1, 549–550 (1984).

Opt. Commun.

M. Schubert, K.-E. Süsse, W. Vogel, “Influence of chaotic pump radiation with finite bandwidth on the intensity correlation of resonance fluorescence radiation,” Opt. Commun. 30, 275–278 (1979).
[CrossRef]

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

Opt. Lett.

Opt. Quantum Electron.

V. S. Idiatulin, Yu. N. Teryaev, “Transient gratings in a nonlinear medium,” Opt. Quantum Electron. 14, 51–56 (1982).
[CrossRef]

Phys. Rev. A

H. J. Kimble, L. Mandel, “Resonance fluorescence with excitation of finite bandwidth,” Phys. Rev. A 15, 689–699 (1977).
[CrossRef]

R. C. Desai, M. D. Levenson, J. A. Baker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate wave-front generation,” Phys. Rev. A 27, 1968–1976 (1983).
[CrossRef]

J. J. Song, J. H. Lee, M. D. Levenson, “Picosecond relaxation measurements by polarization spectroscopy in condensed phases,” Phys. Rev. A 17, 1439–1447 (1978).
[CrossRef]

Phys. Rev. B

K. A. Nelson, D. R. Lutz, M. D. Fayer, L. Madison, “Laser-induced phonon spectroscopy, optical generation of ultrasonic waves and investigation of electronic excited-state interactions in solids,” Phys. Rev. B 24, 3261–3275, (1981).
[CrossRef]

Phys. Rev. Lett.

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

J. R. Salcedo, A. E. Siegman, D. D. Dlott, M. D. Fayer, “Dynamics of energy transport in molecular crystals: the picosecond transient grating method,” Phys. Rev. Lett. 41, 131–134 (1978).
[CrossRef]

C. V. Shank, D. H. Auston, “Parametric coupling in an optically excited plasma in Ge,” Phys. Rev. Lett. 34, 479 (1975).
[CrossRef]

Sov. J. Quantum Electron.

R. Baltrameyunas, Yu. Vaitkus, R. Dannelyus, 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]

Other

R. K. Jain, M. B. Klein, “Degenerate four-wave mixing in semiconductors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983), pp. 365–366.

It appears, from our discussion, that in the cw limit the diffraction efficiencies in both cases go to infinity. This is, of course, not the case because material relaxation prevents the buildup of infinite grating strength. When the pulse is longer than the material relaxation time, the relaxation time then replaces the pulse length in the above argument. Section 6 treats this effect in greater detail. In any event, the argument presented in the text is intended to give relative grating strengths only.

Equations (11) and (12) explain why thermal gratings have obscured population gratings in grating experiments on dyes dissolved in ethanol but not in experiments using water as a solvent. The solvent-dependent factors in Eq. (11) result in a thermal-grating diffraction efficiency proportional to [(dn/dT)/ρcυ]2, with other factors depending on the solute, laser light, or beam geometry or exhibiting little variation.9The value of [(dn/dT)/ρcυ]2for ethanol is more than 100 times that for water.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

R. Trebino, “Subpicosecond-relaxation studies using tunable-laser-induced-grating techniques,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1983).

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Figures (5)

Fig. 1
Fig. 1

Experimental arrangement for the study of integrated-intensity gratings. Two excitation pulses experience a variable delay with respect to each other and excite the grating. The probe pulse, further delayed, probes the induced grating, and a detector detects the diffracted beam.

Fig. 2
Fig. 2

Plot of theoretical diffraction efficiency versus delay between excitation beams for the case of τcτp.

Fig. 3
Fig. 3

Experimental integrated-intensity-grating diffraction efficiency versus relative delay in an experiment of Eichler et al.7 The dashed curve represents the best fit to a (cw-limit) second-order theory, while the solid curve represents the best fit to an approximation to the fourth-order theory developed here. Note that the fourth-order theory yields a much better fit in the wings of the data, where only incomplete washout of the grating occurs, and poor pulse-overlap limits the diffraction efficiency.

Fig. 4
Fig. 4

Experimental apparatus for the study of integrated-intensity gratings formed with two independent excitation sources.

Fig. 5
Fig. 5

Two-excitation-laser integrated-intensity-grating diffraction efficiency versus Δω. Also shown is a dye-laser line shape.

Equations (59)

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A = m = 1 N exp ( i ϕ m ) ,
η | m = 1 N exp ( i ϕ m ) | 2 .
E 1 ( r , t ) = Re E 1 ( t ) exp i ( ω t k 1 · r ) e ˆ , E 2 ( r , t ) = Re E 2 ( t ) exp i ( ω t k 2 · r ) e ˆ ,
I ( r , t ) = I 1 ( t ) + I 2 ( t ) + Re 2 c 8 π ( / μ ) 1 / 2 E 1 ( t ) E 2 * ( t ) × exp ( i k g r · r ) ,
W ( r ) = α τ p / 2 τ p / 2 I ( r , t ) d t ,
T ( r ) = T 0 + ( Φ ρ c υ ) W ( r ) ,
T ( r ) = T 0 + Δ T 0 + Re Δ T g r exp ( i k g r · r ) ,
Δ T 0 = Φ α ρ c υ τ p / 2 τ p / 2 [ I 1 ( t ) + I 2 ( t ) ] d t
Δ T g r = Φ α ρ c υ 2 c 8 π ( / μ ) 1 / 2 τ p / 2 τ p / 2 E 1 ( t ) E 2 * ( t ) d t .
Δ n g r = | ( d n d T ) Φ α ρ c υ 2 c 8 π ( / μ ) 1 / 2 τ p / 2 τ p / 2 E 1 ( t ) E 2 * ( t ) d t | ,
η = K 2 | τ p / 2 τ p / 2 E 1 ( t ) E 2 * ( t ) d t | 2 ,
K = ( d n d T ) Φ α ρ c υ 2 c 8 π ( / μ ) 1 / 2 k p r L .
η = K 2 | E 1 | 2 | E 2 | 2 τ p 2 .
η = K 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 E 1 ( t 1 ) E 2 * ( t 1 ) E 1 * ( t 2 ) E 2 ( t 2 ) d t 1 d t 2 .
η = K 2 | E 1 | 2 | E 2 | 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 u 1 ( t 1 ) u 2 * ( t 1 ) u 1 * ( t 2 ) u 2 ( t 2 ) d t 1 d t 2 ,
η ˜ = 1 τ p 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 u 1 ( t 1 ) u 2 * ( t 1 ) u 1 * ( t 2 ) u 2 ( t 2 ) d t 1 d t 2 .
η ˜ = 1 τ p 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 u 1 ( t 1 ) u 2 ( t 2 ) u 1 * ( t 2 ) u 2 * ( t 1 ) d t 1 d t 2 ,
η ˜ = 1 τ p 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 u 1 ( t 1 ) u 2 ( t 2 ) u 1 * ( t 2 ) u 2 * ( t 1 ) d t 1 d t 2 ,
η ˜ = 1 τ p 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 Γ ( 4 ) ( t 1 , t 2 + τ d ; t 2 , t 1 + τ d ) d t 1 d t 2 .
Γ ( 4 ) ( t 1 , t 2 ; t 3 , t 4 ) = Γ ( 2 ) ( t 1 t 3 ) Γ ( 2 ) ( t 2 t 4 ) × | Γ ( 2 ) ( t 1 t 4 ) Γ ( 2 ) ( t 2 t 3 ) Γ ( 2 ) ( t 1 t 2 ) Γ ( 2 ) ( t 3 t 4 ) | ,
Γ ( 4 ) ( t 1 , t 2 ; t 3 , t 4 ) = Γ ( 2 ) ( t 1 t 3 ) Γ ( 2 ) ( t 2 t 4 ) + Γ ( 2 ) ( t 1 t 4 ) Γ ( 2 ) ( t 2 t 3 ) .
η ˜ τ c τ p + exp ( 2 | τ d / τ c | ) = τ c τ p + | Γ ( 2 ) ( τ d ) | 2 .
η ˜ = 1 τ p 2 A ( t 1 ) A ( t 2 + τ d ) A ( t 1 + τ d ) A ( t 2 ) × Γ ( 4 ) ( t 1 , t 2 + τ d ; t 1 + τ d , τ 2 ) d t 1 d t 2 .
η ˜ τ c τ p I ( t ) I ( t + τ d ) d t + | Γ ( 2 ) ( τ d ) | 2 ,
η ˜ ζ τ c τ p [ ζ τ d / τ p cosh ( ζ τ d / τ p ) sinh ( ζ τ d / τ p ) sinh 3 ( ζ τ d / τ p ) ] + exp ( 2 | τ d / τ c | ) ,
η ˜ SHG = | A ( t ) A ( t + τ d ) u ( t ) u ( t + τ d ) | 2 d t ,
η ˜ SHG = A 2 ( t ) A 2 ( t + τ d ) Γ ( 4 ) ( t , t + τ d ; t , t + τ d ) d t .
η ˜ TG τ c τ p I ( t ) I ( t + τ d ) d t + | Γ ( 2 ) ( τ d ) | 2
η ˜ SHG I ( t ) I ( t + τ d ) d t + | Γ ( 2 ) ( τ d ) | 2 ,
η ˜ ( t ) = t t A ( t 1 ) A ( t 2 + τ d ) A ( t 2 ) A ( t 1 + τ d ) × Γ ( 4 ) ( t 1 , t 2 + τ d ; t 2 , t 1 + τ d ) h ( t t 1 ) h ( t t 2 ) d t 1 d t 2 ,
η ˜ ( t ) = exp ( 2 t + τ d τ t h ) [ τ c τ t h sinh ( τ p | τ d | τ t h ) ] + 4 τ t h 2 | Γ ( 2 ) ( τ d ) | 2 sinh 2 ( τ p | τ d | 2 τ t h ) ] .
η ˜ ( t ) τ c 4 τ t h sinh ( τ p / τ t h ) sinh 2 ( τ p / 2 τ t h ) + | Γ ( 2 ) ( τ d ) | 2 .
η ˜ ( t ) τ c 2 τ t h + | Γ ( 2 ) ( τ d ) | 2 .
η ˜ ( t ) τ c τ p + | Γ ( 2 ) ( τ d ) | 2 .
η ˜ ( t ) τ c τ p [ 1 + 1 12 ( τ p τ t h ) 2 ] + | Γ ( 2 ) ( τ d ) | 2 .
η ˜ ( Δ ω ) = 1 τ p 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 Γ 1 ( 2 ) ( t 1 ; t 2 ) Γ 2 ( 2 ) * ( t 1 ; t 2 ) d t 1 d t 2 ,
η ˜ ( Δ ω ) = 1 τ p 2 τ p / 2 τ p / 2 τ p / 2 τ p / 2 Γ 1 ( 2 ) ( t 1 t 2 ) × | Γ 2 ( 2 ) * ( t 1 t 2 ) d t 1 d t 2 .
η ˜ ( Δ ω ) = 1 2 τ p τ p τ p ( 1 | t / τ p | ) Γ 1 ( 2 ) ( t ) Γ 2 ( 2 ) * ( t ) d t ,
η ˜ ( Δ ω ) 1 τ p Γ 1 ( 2 ) ( t ) Γ 2 ( 2 ) * ( t ) d t ,
η ˜ ( Δ ω ) 1 τ p I 1 ( ω ) I 2 ( ω ) d ω ,
E ( t ) = A 0 exp ( i ω 0 t + ϕ ( t ) + Φ 0 ) ,
I G ( ω ) = 4 π ln 2 δ ω exp [ 4 ln 2 ( ω ω 0 δ ω ) 2 ]
I L ( ω ) = 4 / δ ω 1 + 4 ( ω ω 0 δ ω ) 2 ,
Γ ( 2 ) ( t ) = exp ( i ω 0 t ) exp ( | t / τ c | )
Γ ( 2 ) ( t ) = exp ( i ω 0 t ) exp ( π t 2 / 2 τ c 2 )
τ c = | Γ ( 2 ) ( t ) | 2 d t .
η ˜ = [ τ c τ p 1 2 τ c 2 τ p 2 ] + [ 1 2 τ d τ p + τ d 2 τ p 2 + τ d τ c τ p 2 + 1 2 τ c 2 τ p 2 τ c τ p ] exp ( 2 | τ d / τ c | ) ,
η ˜ τ c τ p + [ 1 2 τ d τ p τ c τ p ] exp ( 2 | τ d / τ c | ) .
η ˜ = [ τ c τ p 1 2 τ c 2 τ p 2 ] + [ 1 + τ c 2 τ p 2 ] exp ( 2 | τ d / τ c | ) ,
η ˜ τ c τ p + exp ( 2 | τ d / τ c | )
η ˜ = 2 τ c τ p erf ( 2 τ p τ c ) + exp ( π τ d 2 / τ c 2 ) 1 π τ c 2 τ p 2 [ 1 exp ( π τ p 2 / τ c 2 ) ] ,
η ˜ 2 τ c τ p + exp ( π τ d 2 / τ c 2 ) ,
A ( t ) = ( 4 ln 2 π τ p ) 1 / 4 exp [ 2 ln 2 ( t 2 / τ p 2 ) ] ,
A 4 ( t ) d t = I 2 ( t ) d t = 1 ,
η ˜ = exp [ 2 ln 2 ( τ d 2 / τ p 2 ] { exp ( τ p 2 / τ c 2 2 ln 2 ) × [ 1 erf ( τ p / τ c 2 ln 2 ) ] + exp ( 2 | τ d / τ c | ) } .
η ˜ 2 ln 2 π ( τ c τ p ) exp [ 2 ln 2 ( τ d 2 / τ p 2 ) ] + exp ( 2 | τ d / τ c | ) .
η ˜ = exp [ 2 ln 2 ( τ d 2 / τ p 2 ) ] { 2 ln 2 π ( τ c τ p ) × ( 1 + 2 ln 2 π τ c 2 τ p 2 ) 1 / 2 + exp ( π τ d 2 / τ c 2 ) } ,
η ˜ 2 ln 2 π ( τ c τ p ) exp [ 2 ln 2 ( τ d 2 / τ p 2 ) ] + exp ( π τ d 2 / τ c 2 ) .
η ˜ ζ τ c τ p [ ζ τ d / τ p cosh ( ζ τ d / τ p ) sinh ( ζ τ d / τ p ) sinh 3 ( ζ τ d / τ p ) ] + exp ( 2 | τ d / τ c | ) ,

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