Abstract

A model of degenerate four-wave mixing (DFWM) in Cr4+:YAG saturable absorbers is given in the transient regime. Because of the presence of absorbing dipoles oriented along specific directions, the DFWM process exhibits an anisotropic behavior. The effects of pump beam fluences, relative polarization of the beams, crystal orientation, absorbance, excited-state absorption, and nonresonant losses are analyzed. The theoretical results are supported by DFWM experiments in Cr4+:YAG performed at 1.06 µm with nanosecond pulses.

© 1996 Optical Society of America

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    [CrossRef] [PubMed]
  2. H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
    [CrossRef]
  3. P. F. Liao and D. M. Bloom, “Continuous-wave backward-wave generation by degenerate four-wave mixing in ruby,” Opt. Lett. 3, 4–6 (1978).
    [CrossRef] [PubMed]
  4. A. M. Ghazzawi, J. K. Tyminski, and R. C. Powell, “Four-wave mixing in alexandrite crystals,” Phys. Rev. B 30, 7182–7186 (1984).
    [CrossRef]
  5. T. Catunda, A. M. Cansian, and J. C. Castro, “Saturation effects in degenerate four-wave mixing in ruby and GdAlO3:Cr3+,” J. Opt. Soc. Am. B 8, 820–823 (1991).
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  6. R. W. Boyd, M. T. Gruneisen, P. Narum, D. J. Simkin, B. Dunn, and D. L. Yang, “Saturated absorption and degenerate four-wave mixing in Nd3+ beta alumina,” Opt. Lett. 11, 162–164 (1986).
    [CrossRef]
  7. R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
    [CrossRef]
  8. D. E. Watkins, J. F. Figueira, and S. J. Thomas, “Observation of resonantly enhanced degenerate four-wave mixing in doped alkali halides,” Opt. Lett. 5, 169–171 (1980).
    [CrossRef] [PubMed]
  9. T. T. Basiev, P. G. Zverev, S. B. Mirov, and S. Pal, “Phase conjugation in LiF and NaF color center crystals,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers and T. Tschudi, eds., Proc. SPIE1500, 65–71 (1991).
    [CrossRef]
  10. A. Brignon and J.-P. Huignard, “Transient analysis of degenerate four-wave mixing in saturable absorbers: application to Cr4+:GSGG at 1.06 µm,” Opt. Commun. 110, 717–726 (1994).
    [CrossRef]
  11. G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
    [CrossRef]
  12. A. Brignon and J.-P. Huignard, “Phase conjugation in Cr4+:YAG at 1.06 µm,” Opt. Lett. 21, 1126–1128 (1996).
    [CrossRef] [PubMed]
  13. L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).
  14. M. I. Demchuk, V. P. Mikhailov, N. I. Zhavoronkov, N. V. Kuleshov, P. V. Prokoshin, K. V. Yumashev, M. G. Livshits, and B. I. Minkov, “Chromium-doped forsterite as a solid-state saturable absorber,” Opt. Lett. 17, 929–930 (1992).
    [CrossRef] [PubMed]
  15. H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
    [CrossRef]
  16. K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A. V. Shestakov, “Dual Q switching and laser action at 1.06 and 1.44 µm in a Nd3+:YAG-Cr4+:YAG oscillator at 300 K,” Opt. Lett. 18, 814–816 (1993).
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  17. P. Yankov, “Cr4+:YAG Q-switching of Nd:host laser oscillators,” J. Phys. D 27, 1118–1120 (1994).
    [CrossRef]
  18. H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
    [CrossRef]
  19. N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
    [CrossRef]
  20. A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
    [CrossRef]
  21. A. Brignon, J. Raffy, and J.-P. Huignard, “Transient degenerate four-wave mixing in a saturable Nd:YAG amplifier: effect of pump beam propagation,” Opt. Lett. 19, 865–867 (1994).
    [CrossRef] [PubMed]
  22. A. Brignon, “Temporal analysis of pulsed phase conjugation in saturable amplifiers: application to Nd:YVO4,” J. Opt. Soc. Am. B 13, 1748–1757 (1996).
    [CrossRef]
  23. S. Miyanaga, H. Ohtateme, K. Kawano, and H. Fujiwara, “Excited-state absorption and pump propagation effects on optical phase conjugation in a saturable absorber,” J. Opt. Soc. Am. B 10, 1069–1076 (1993).
    [CrossRef]
  24. M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
    [CrossRef]
  25. H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 51–54 and 57–58.
  26. R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); errata, Opt. Lett. 3, 205 (1978).
    [CrossRef] [PubMed]
  27. J. C. Diels, I. McMichael, and H. Vanherzeele, “Degenerate four-wave mixing of picosecond pulses in the saturable amplification of a dye laser,” IEEE J. Quantum Electron. QE-20, 630–636 (1984).
    [CrossRef]
  28. R. J. Knize, “Efficiency of degenerate four-wave mixing in a two-level saturable absorbing medium,” Opt. Lett. 18, 1606–1608 (1993).
    [CrossRef] [PubMed]
  29. A. Brignon and J.-P. Huignard, “Energy efficiency of phase conjugation by saturable-gain degenerate four-wave mixing in Nd:YAG amplifiers,” Opt. Commun. 119, 171–177 (1995).
    [CrossRef]
  30. W. P. Brown, “Absorption and depletion effects on degenerate four-wave mixing in homogeneously broadened absorbers,” J. Opt. Soc. Am. 73, 629–634 (1983).
    [CrossRef]
  31. R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. 102, 288–292 (1993).
    [CrossRef]
  32. W. R. Tompkin, M. S. Malcuit, and R. W. Boyd, “Polarization properties of phase conjugation by degenerate four-wave mixing in a medium of rigidly held dye molecules,” J. Opt. Soc. Am. B 6, 757–760 (1989).
    [CrossRef]
  33. H. J. Eichler, A. Haase, R. Menzel, and A. Siemoneit, “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier,” J. Phys. D 26, 1884–1891 (1993).
    [CrossRef]
  34. Ref. 25, pp. 84–86.

1996 (2)

1995 (3)

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

A. Brignon and J.-P. Huignard, “Energy efficiency of phase conjugation by saturable-gain degenerate four-wave mixing in Nd:YAG amplifiers,” Opt. Commun. 119, 171–177 (1995).
[CrossRef]

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

1994 (6)

A. Brignon and J.-P. Huignard, “Transient analysis of degenerate four-wave mixing in saturable absorbers: application to Cr4+:GSGG at 1.06 µm,” Opt. Commun. 110, 717–726 (1994).
[CrossRef]

P. Yankov, “Cr4+:YAG Q-switching of Nd:host laser oscillators,” J. Phys. D 27, 1118–1120 (1994).
[CrossRef]

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
[CrossRef]

A. Brignon, J. Raffy, and J.-P. Huignard, “Transient degenerate four-wave mixing in a saturable Nd:YAG amplifier: effect of pump beam propagation,” Opt. Lett. 19, 865–867 (1994).
[CrossRef] [PubMed]

1993 (5)

1992 (2)

M. I. Demchuk, V. P. Mikhailov, N. I. Zhavoronkov, N. V. Kuleshov, P. V. Prokoshin, K. V. Yumashev, M. G. Livshits, and B. I. Minkov, “Chromium-doped forsterite as a solid-state saturable absorber,” Opt. Lett. 17, 929–930 (1992).
[CrossRef] [PubMed]

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

1991 (1)

1990 (1)

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
[CrossRef]

1989 (1)

1987 (1)

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

1986 (1)

1984 (2)

A. M. Ghazzawi, J. K. Tyminski, and R. C. Powell, “Four-wave mixing in alexandrite crystals,” Phys. Rev. B 30, 7182–7186 (1984).
[CrossRef]

J. C. Diels, I. McMichael, and H. Vanherzeele, “Degenerate four-wave mixing of picosecond pulses in the saturable amplification of a dye laser,” IEEE J. Quantum Electron. QE-20, 630–636 (1984).
[CrossRef]

1983 (1)

1980 (1)

1979 (1)

H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
[CrossRef]

1978 (2)

1971 (1)

Abrams, R. L.

Basiev, T. T.

T. T. Basiev, P. G. Zverev, S. B. Mirov, and S. Pal, “Phase conjugation in LiF and NaF color center crystals,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers and T. Tschudi, eds., Proc. SPIE1500, 65–71 (1991).
[CrossRef]

Birnbaum, M.

Bloom, D. M.

Boyd, R. W.

Brignon, A.

A. Brignon, “Temporal analysis of pulsed phase conjugation in saturable amplifiers: application to Nd:YVO4,” J. Opt. Soc. Am. B 13, 1748–1757 (1996).
[CrossRef]

A. Brignon and J.-P. Huignard, “Phase conjugation in Cr4+:YAG at 1.06 µm,” Opt. Lett. 21, 1126–1128 (1996).
[CrossRef] [PubMed]

A. Brignon and J.-P. Huignard, “Energy efficiency of phase conjugation by saturable-gain degenerate four-wave mixing in Nd:YAG amplifiers,” Opt. Commun. 119, 171–177 (1995).
[CrossRef]

A. Brignon and J.-P. Huignard, “Transient analysis of degenerate four-wave mixing in saturable absorbers: application to Cr4+:GSGG at 1.06 µm,” Opt. Commun. 110, 717–726 (1994).
[CrossRef]

A. Brignon, J. Raffy, and J.-P. Huignard, “Transient degenerate four-wave mixing in a saturable Nd:YAG amplifier: effect of pump beam propagation,” Opt. Lett. 19, 865–867 (1994).
[CrossRef] [PubMed]

Brown, W. P.

Bufetova, G. A.

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

Cansian, A. M.

Castro, J. C.

Catunda, T.

Chase, L. L.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
[CrossRef]

Chen, W.

Coïc, H.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Crofts, G. J.

R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. 102, 288–292 (1993).
[CrossRef]

Damzen, M. J.

R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. 102, 288–292 (1993).
[CrossRef]

Demchuk, M. I.

Dennis, W. M.

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Diels, J. C.

J. C. Diels, I. McMichael, and H. Vanherzeele, “Degenerate four-wave mixing of picosecond pulses in the saturable amplification of a dye laser,” IEEE J. Quantum Electron. QE-20, 630–636 (1984).
[CrossRef]

Dunn, B.

Eichler, H. J.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

H. J. Eichler, A. Haase, R. Menzel, and A. Siemoneit, “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier,” J. Phys. D 26, 1884–1891 (1993).
[CrossRef]

H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
[CrossRef]

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 51–54 and 57–58.

Eichler, J.

H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
[CrossRef]

Eilers, H.

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Féru, P.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Figueira, J. F.

Fujiwara, H.

Ghazzawi, A. M.

A. M. Ghazzawi, J. K. Tyminski, and R. C. Powell, “Four-wave mixing in alexandrite crystals,” Phys. Rev. B 30, 7182–7186 (1984).
[CrossRef]

Green, R. P. M.

R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. 102, 288–292 (1993).
[CrossRef]

Gruneisen, M. T.

Gulyamova, E. S.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Günter, P.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 51–54 and 57–58.

Haase, A.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

H. J. Eichler, A. Haase, R. Menzel, and A. Siemoneit, “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier,” J. Phys. D 26, 1884–1891 (1993).
[CrossRef]

Hill, K. O.

Hoffman, K. R.

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Huignard, J.-P.

A. Brignon and J.-P. Huignard, “Phase conjugation in Cr4+:YAG at 1.06 µm,” Opt. Lett. 21, 1126–1128 (1996).
[CrossRef] [PubMed]

A. Brignon and J.-P. Huignard, “Energy efficiency of phase conjugation by saturable-gain degenerate four-wave mixing in Nd:YAG amplifiers,” Opt. Commun. 119, 171–177 (1995).
[CrossRef]

A. Brignon and J.-P. Huignard, “Transient analysis of degenerate four-wave mixing in saturable absorbers: application to Cr4+:GSGG at 1.06 µm,” Opt. Commun. 110, 717–726 (1994).
[CrossRef]

A. Brignon, J. Raffy, and J.-P. Huignard, “Transient degenerate four-wave mixing in a saturable Nd:YAG amplifier: effect of pump beam propagation,” Opt. Lett. 19, 865–867 (1994).
[CrossRef] [PubMed]

Il’ichev, N. N.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Jacobsen, S. M.

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Jolly, A.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Kawano, K.

Kir’yanov, A. V.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Klimov, I. V.

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

Knize, R. J.

Knof, J.

H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
[CrossRef]

Kokta, M. R.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Krutova, L. I.

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

Kuleshov, N. V.

Le Nevé, M.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Liao, P. F.

Lind, R. C.

Livshits, M. G.

Lukin, A. V.

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

Malcuit, M. S.

Maruenda, P.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

McMichael, I.

J. C. Diels, I. McMichael, and H. Vanherzeele, “Degenerate four-wave mixing of picosecond pulses in the saturable amplification of a dye laser,” IEEE J. Quantum Electron. QE-20, 630–636 (1984).
[CrossRef]

Menzel, R.

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

H. J. Eichler, A. Haase, R. Menzel, and A. Siemoneit, “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier,” J. Phys. D 26, 1884–1891 (1993).
[CrossRef]

Mikhailov, V. P.

Minkov, B. I.

Mirov, S. B.

T. T. Basiev, P. G. Zverev, S. B. Mirov, and S. Pal, “Phase conjugation in LiF and NaF color center crystals,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers and T. Tschudi, eds., Proc. SPIE1500, 65–71 (1991).
[CrossRef]

Miyanaga, S.

Narum, P.

Nikolaev, D. A.

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

Noack, C.

H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
[CrossRef]

Ohtateme, H.

Okhrimchuk, A. G.

A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
[CrossRef]

Pal, S.

T. T. Basiev, P. G. Zverev, S. B. Mirov, and S. Pal, “Phase conjugation in LiF and NaF color center crystals,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers and T. Tschudi, eds., Proc. SPIE1500, 65–71 (1991).
[CrossRef]

Pashinin, P. P.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Payne, S. A.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
[CrossRef]

Pohl, D. W.

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 51–54 and 57–58.

Powell, R. C.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
[CrossRef]

A. M. Ghazzawi, J. K. Tyminski, and R. C. Powell, “Four-wave mixing in alexandrite crystals,” Phys. Rev. B 30, 7182–7186 (1984).
[CrossRef]

Prokoshin, P. V.

Raffy, J.

Sandulenko, V. A.

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

Shcherbakov, I. A.

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

Shestakov, A. V.

Shpuga, S. M.

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Sidorova, E. A.

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

Siemoneit, A.

H. J. Eichler, A. Haase, R. Menzel, and A. Siemoneit, “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier,” J. Phys. D 26, 1884–1891 (1993).
[CrossRef]

Simkin, D. J.

Solntsev, V. M.

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

Spariosu, K.

Stultz, R.

Thomas, S. J.

Tompkin, W. R.

Tsvetkov, V. B.

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

Tyminski, J. K.

A. M. Ghazzawi, J. K. Tyminski, and R. C. Powell, “Four-wave mixing in alexandrite crystals,” Phys. Rev. B 30, 7182–7186 (1984).
[CrossRef]

Vanherzeele, H.

J. C. Diels, I. McMichael, and H. Vanherzeele, “Degenerate four-wave mixing of picosecond pulses in the saturable amplification of a dye laser,” IEEE J. Quantum Electron. QE-20, 630–636 (1984).
[CrossRef]

Vial, J. C.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Vicrey, J.

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Watkins, D. E.

Wilke, G. D.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
[CrossRef]

Yang, D. L.

Yankov, P.

P. Yankov, “Cr4+:YAG Q-switching of Nd:host laser oscillators,” J. Phys. D 27, 1118–1120 (1994).
[CrossRef]

Yen, W. M.

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

Yumashev, K. V.

Zhavoronkov, N. I.

Zverev, P. G.

T. T. Basiev, P. G. Zverev, S. B. Mirov, and S. Pal, “Phase conjugation in LiF and NaF color center crystals,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers and T. Tschudi, eds., Proc. SPIE1500, 65–71 (1991).
[CrossRef]

Ann. Phys. Fr. (1)

M. Le Nevé, J. C. Vial, A. Jolly, H. Coïc, P. Féru, P. Maruenda, and J. Vicrey, “Etude de la saturation du Cr4+:YAG pompé à 1.06 µm,” Ann. Phys. Fr. 20, 623–624 (1995).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

H. J. Eichler, A. Haase, M. R. Kokta, and R. Menzel, “Cr4+:YAG as passive Q-switch for a Nd:YALO oscillator with an average repetition rate of 2.7 kHz, TEM00 mode and 13 W output,” Appl. Phys. B 58, 409–411 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

H. Eilers, K. R. Hoffman, W. M. Dennis, S. M. Jacobsen, and W. M. Yen, “Saturation of 1.064 µm absorption in Cr,Ca:Y3Al5O12 crystals,” Appl. Phys. Lett. 61, 2958–2960 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

J. C. Diels, I. McMichael, and H. Vanherzeele, “Degenerate four-wave mixing of picosecond pulses in the saturable amplification of a dye laser,” IEEE J. Quantum Electron. QE-20, 630–636 (1984).
[CrossRef]

J. Opt. Soc. Am. (1)

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

J. Phys. D (2)

P. Yankov, “Cr4+:YAG Q-switching of Nd:host laser oscillators,” J. Phys. D 27, 1118–1120 (1994).
[CrossRef]

H. J. Eichler, A. Haase, R. Menzel, and A. Siemoneit, “Thermal lensing and depolarization in a highly pumped Nd:YAG laser amplifier,” J. Phys. D 26, 1884–1891 (1993).
[CrossRef]

Opt. Commun. (3)

R. P. M. Green, G. J. Crofts, and M. J. Damzen, “Phase conjugate reflectivity and diffraction efficiency of gain gratings in Nd:YAG,” Opt. Commun. 102, 288–292 (1993).
[CrossRef]

A. Brignon and J.-P. Huignard, “Energy efficiency of phase conjugation by saturable-gain degenerate four-wave mixing in Nd:YAG amplifiers,” Opt. Commun. 119, 171–177 (1995).
[CrossRef]

A. Brignon and J.-P. Huignard, “Transient analysis of degenerate four-wave mixing in saturable absorbers: application to Cr4+:GSGG at 1.06 µm,” Opt. Commun. 110, 717–726 (1994).
[CrossRef]

Opt. Lett. (9)

A. Brignon and J.-P. Huignard, “Phase conjugation in Cr4+:YAG at 1.06 µm,” Opt. Lett. 21, 1126–1128 (1996).
[CrossRef] [PubMed]

M. I. Demchuk, V. P. Mikhailov, N. I. Zhavoronkov, N. V. Kuleshov, P. V. Prokoshin, K. V. Yumashev, M. G. Livshits, and B. I. Minkov, “Chromium-doped forsterite as a solid-state saturable absorber,” Opt. Lett. 17, 929–930 (1992).
[CrossRef] [PubMed]

K. Spariosu, W. Chen, R. Stultz, M. Birnbaum, and A. V. Shestakov, “Dual Q switching and laser action at 1.06 and 1.44 µm in a Nd3+:YAG-Cr4+:YAG oscillator at 300 K,” Opt. Lett. 18, 814–816 (1993).
[CrossRef] [PubMed]

R. W. Boyd, M. T. Gruneisen, P. Narum, D. J. Simkin, B. Dunn, and D. L. Yang, “Saturated absorption and degenerate four-wave mixing in Nd3+ beta alumina,” Opt. Lett. 11, 162–164 (1986).
[CrossRef]

D. E. Watkins, J. F. Figueira, and S. J. Thomas, “Observation of resonantly enhanced degenerate four-wave mixing in doped alkali halides,” Opt. Lett. 5, 169–171 (1980).
[CrossRef] [PubMed]

P. F. Liao and D. M. Bloom, “Continuous-wave backward-wave generation by degenerate four-wave mixing in ruby,” Opt. Lett. 3, 4–6 (1978).
[CrossRef] [PubMed]

R. L. Abrams and R. C. Lind, “Degenerate four-wave mixing in absorbing media,” Opt. Lett. 2, 94–96 (1978); errata, Opt. Lett. 3, 205 (1978).
[CrossRef] [PubMed]

R. J. Knize, “Efficiency of degenerate four-wave mixing in a two-level saturable absorbing medium,” Opt. Lett. 18, 1606–1608 (1993).
[CrossRef] [PubMed]

A. Brignon, J. Raffy, and J.-P. Huignard, “Transient degenerate four-wave mixing in a saturable Nd:YAG amplifier: effect of pump beam propagation,” Opt. Lett. 19, 865–867 (1994).
[CrossRef] [PubMed]

Opt. Mater. (1)

A. G. Okhrimchuk and A. V. Shestakov, “Performance of YAG:Cr4+ laser crystal,” Opt. Mater. 3, 1–13 (1994).
[CrossRef]

Opt. Spectrosc. (USSR) (1)

L. I. Krutova, A. V. Lukin, V. A. Sandulenko, E. A. Sidorova, and V. M. Solntsev, “Phototropic centers in chromium-doped garnets,” Opt. Spectrosc. (USSR) 63, 693–695 (1987).

Phys. Rev. B (2)

A. M. Ghazzawi, J. K. Tyminski, and R. C. Powell, “Four-wave mixing in alexandrite crystals,” Phys. Rev. B 30, 7182–7186 (1984).
[CrossRef]

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Four-wave mixing of Nd3+-doped crystals and glasses,” Phys. Rev. B 41, 8593–8602 (1990).
[CrossRef]

Phys. Status Solidi (1)

H. J. Eichler, J. Eichler, J. Knof, and C. Noack, “Lifetimes of laser-induced population density gratings in ruby,” Phys. Status Solidi 52, 481–486 (1979).
[CrossRef]

Quantum Electron. (2)

G. A. Bufetova, I. V. Klimov, D. A. Nikolaev, V. B. Tsvetkov, and I. A. Shcherbakov, “Laser with an adaptive loop cavity,” Quantum Electron. 25, 760–761 (1995).
[CrossRef]

N. N. Il’ichev, A. V. Kir’yanov, P. P. Pashinin, S. M. Shpuga, and E. S. Gulyamova, “Changes in the profile and state of polarisation of a short light pulse (λ ∼ 1.06 µm) during propagation in a YAG:Cr4+ crystal,” Quantum Electron. 24, 771–776 (1994).
[CrossRef]

Other (3)

H. J. Eichler, P. Günter, and D. W. Pohl, Laser-Induced Dynamic Gratings (Springer-Verlag, Berlin, 1986), pp. 51–54 and 57–58.

Ref. 25, pp. 84–86.

T. T. Basiev, P. G. Zverev, S. B. Mirov, and S. Pal, “Phase conjugation in LiF and NaF color center crystals,” in Innovative Optics and Phase Conjugate Optics, R. Ahlers and T. Tschudi, eds., Proc. SPIE1500, 65–71 (1991).
[CrossRef]

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Figures (12)

Fig. 1
Fig. 1

Interaction of a resonant light pulse at λ ∼ 1.06 µm with a Cr4+:YAG crystal. x, y, and z are, respectively, the directions of the [100], [010], and [001] crystallographic axes of YAG; the unit vectors ej (j = x, y, z) give the positions of the three groups of resonantly absorbing dipoles; the orientation of the electric vector E at the entry of the crystal relative to the x axis is given by the angle ϕ.

Fig. 2
Fig. 2

Energy-level diagram of the relevant excitation and relaxation processes in Cr4+:YAG.

Fig. 3
Fig. 3

Energy transmission of Cr4+:YAG at λ = 1.064 µm as a function of the crystal rotation angle ϕ, for linearly polarized radiation and various input fluences U(0): (a) U(0) = 2.8 J/cm2, (b) U(0) = 0.42 J/cm2, (c) U(0) = 0.21 J/cm2, (d) U(0) = 0.079 J/cm2, (e) U(0) = 0.011 J/cm2. The data points are the measured values, and the solid curves are the theoretical plots calculated with the material parameters in Table 1.

Fig. 4
Fig. 4

Energy transmission of Cr4+:YAG at λ = 1.064 µm as a function of the input fluence U(0) for a linearly polarized radiation. The data points were recorded with crystal rotations corresponding to the maxima (ϕ = 0°) and the minima (ϕ = 45°) shown in Fig. 3. The solid curve is a fit to the experimental points obtained for ϕ = 0°, and the dashed curve is the theoretical plot corresponding to the ϕ = 45° case.

Fig. 5
Fig. 5

Schematics of the various configurations for DFWM in Cr4+:YAG at λ = 1.06 µm studied in this paper: ①, DFWM with copolarized beams; ② DFWM with an orthogonally polarized signal beam; ③, ④, DFWM with orthogonally polarized pump beams; ⑤, two-pass retroreflecting DFWM scheme employing a high-reflecting mirror (M).

Fig. 6
Fig. 6

Energy reflectivity R as a function of the input forward-pump fluence normalized to the saturation fluence (Usat) for various small-intensity absorbances (α0 + α)L. These curves are plotted for configuration ①, ϕ = 0°, the experimental parameters of Cr4+:YAG (see Table 1), δ = 0.2, and equal-input pump fluence U1(0) = U2(L).

Fig. 7
Fig. 7

Schematic diagram of the experiment of DFWM in Cr4+:YAG at λ = 1.064 µm.

Fig. 8
Fig. 8

Experimental (symbols) and theoretical (curves) plots of the energy reflectivity versus the input forward pump fluence [U1(0) = U2(L)] normalized to the saturation fluence (Usat) for a weak-signal beam (β = 15) and various DFWM configurations: (a) configuration ①, ϕ = 0°; (b) configuration ①, ϕ = 45°; (c) configuration ⑤, ϕ = 0°.

Fig. 9
Fig. 9

Experimental (symbols) plots of the energy reflectivity versus the input forward-pump beam fluence [U1(0) = U2(L)] for a weak-signal beam (β = 15), ϕ = 45°, and various DFWM configurations: (a) configuration ①; (b) configuration ②; (c) configuration ③; (d) configuration ④. The theoretical plot (solid curves) is the same for both figures.

Fig. 10
Fig. 10

(Top) Experimental plots of the energy reflectivity versus crystal rotation angle ϕ for an input pump beam fluence of U1(0) = U2(L) = 1.4 × Usat, the weak-signal beam (β = 15) and various DFWM configurations. (Bottom) Theoretical curves plotted with the experimental parameters: (a) configuration ①; (b) configuration ②; (c) configuration ③.

Fig. 11
Fig. 11

Same as Fig. 10 but for an input pump fluence of U1(0) = U2(L) = 6.4 × Usat. (a) Configuration ①, (b) configuration ②, (c) configuration ③, (d) configuration ④.

Fig. 12
Fig. 12

Phase-conjugate energy reflectivity Rth as a function of the pump beam fluence [U1(0) = U2(L)] in Cr4+:YAG when only the thermal grating contribution is considered. The curves are plotted for the material parameters of Table 1 (α0 L = 4.17, α/α0 = 0.092, Usat = 31 mJ/cm2), f = 0.3, and the DFWM configurations of Figs. 8 and 9: (a) configuration ①, ϕ = 0°; (b) configurations ① – ④, ϕ = 45°; (c) configuration ⑤, ϕ = 0°.

Tables (2)

Tables Icon

Table 1 Material Parameters for the Cr4+:YAG Sample Given the Best Fit to the Experimental Plots of Fig. 4, ϕ = 0°

Tables Icon

Table 2 Polarization of the Phase Conjugate Beam for Various DFWM Configurations and Crystal Orientations

Equations (37)

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N 0 / 3 = N 1 j ( z , t ) + N 2 j ( z , t ) ( j = x   or   y ) ,
d N 1 j ( z , t ) d t = σ 0 | E j ( z , t ) | 2 h ν N 1 j ( z , t ) ,
E = E x e x + E y e y ,
E j = | A j | exp ( i φ j ) exp ( i k z + i ω t ) ( j = x   or   y ) ,
N 1 j ( z , t ) = N 0 3 exp 0 t | E j ( z , t ) | 2 d t / U sat ,
χ j ( z , t ) = i k [ ( 1 i δ ) σ 0 N 1 j ( z , t ) + σ ESA N 2 j ( z , t ) ] ,
χ j ( z , t ) = i α 0 k [ G + ( 1 G i δ ) n j ( z , t ) ] ,
d 2 E j d z 2 + k 2 E j = k 2 χ j i α k E j
d | A j | d z = α 0 2 G + α α 0 + ( 1 G ) n j ( z , t ) | A j | ,
d φ j d z = α 0 δ 2 n j ( z , t ) ,
T = U x ( L ) + U y ( L ) U x ( 0 ) + U y ( 0 ) ,
U j ( z ) = 0 t p | A j ( z , t ) | 2 d t ( j = x   or   y ) .
α 0 L = | ln   T 0 ( 1 ρ ) ln   T S | ,
G = ρ   ln   T S α 0 L ,
α L = ( 1 ρ ) ln   T S .
E = E 0 + δ E ,
E 0 = ( A 1 x e x + A 1 y e y ) exp ( i k z + i ω t ) + ( A 2 x e x + A 2 y e y ) exp ( + i k z + i ω t ) ,
δ E = ( A 3 x e x + A 3 y e y ) exp ( i k z + i ω t ) + ( A 4 x e x + A 4 y e y ) exp ( + i k z + i ω t ) ,
A m j = A m · e j = | A m j | exp ( i φ m j ) ( m = 1 , 2 , 3 , 4 ) , ( j = x , y ) ,
n j ( z , t ) = exp 0 t | E ( z , t ) · e j | 2 d t / U sat .
n j ( z , t ) = 1 0 t [ ( E 0 · e j ) ( δ E * · e j ) + ( E 0 * · e j ) ( δ E · e j ) ] d t U sat ,
( z , t ) = exp 0 t | E 0 ( z , t ) · e j | 2 U sat d t .
= n = + ( 1 ) | n | γ | n | exp [ 2 i k n z + i n ( φ 1 j φ 2 j ) ] ,
γ n ( z , t ) = exp ( U S ) J n ( U M ) ,
U S ( z , t ) = 0 t ( | A 1 j | 2 + | A 2 j | 2 ) d t / U sat ,
U M ( z , t ) = 2 0 t | A 1 j A 2 j | d t / U sat .
d A 1 j d z = Γ A 1 j + α 0 Δ 2 γ 0 γ 1 | A 2 j | | A 1 j | A 1 j ,
d A 2 j d z = Γ A 2 j + α 0 Δ 2 γ 0 γ 1 | A 1 j | | A 2 j | A 2 j ,
d A 3 j d z = Γ A 3 j + α 0 Δ 2 ( γ 0 A 3 j κ 31 A 1 j κ 32 A 2 j ) ,
d A 4 j d z = Γ A 4 j + α 0 Δ 2 ( γ 0 A 4 j κ 41 A 1 j κ 42 A 2 j ) ,
κ p , m ( z , t ) = γ 0 ( f p , m + f k , n * ) γ 1 ( f p , n + f k , m * )
f p , m ( z , t ) = 1 U sat 0 t A p j ( z , t ) A m j * ( z , t ) d t ( p = 3 , 4   and   m = 1,2 ) .
R = 0 t p [ | A 4 x ( 0 , t ) | 2 + | A 4 y ( 0 , t ) | 2 ] d t 0 t p [ | A 3 x ( 0 , t ) | 2 + | A 3 y ( 0 , t ) | 2 ] d t .
τ D = γ C p Λ 2 4 π 2 κ 1.3   µ s ,
Δ T ( z , t ) = K γ C p 0 t A 1 A 3 * d t ,
K ( z , t ) = f α 0 γ 0 ( z , t ) + α ,
d A 4 d z = α + α 0 γ 0 2 A 4 i π λ Δ n A 2 ,

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