Abstract

We describe several new aspects of light–matter interactions for solids in which interatomic coupling of impurity atoms plays a dominant role in population dynamics. We explore the implications of spatial coherence in such multiatom interactions by introducing a density-matrix theory of cooperative upconversion, focusing on pair systems for which analytic results can be obtained. We predict population pulsations in coherent cooperative upconversion, enhanced quantum efficiency, enhanced energy transfer, and pair-mediated instabilities, not only in cooperative upconversion media without external cavities but in upconversion lasers and conventional lasers in highly doped solids as well. These predictions are compared with rate equation solutions and observations in lasers with inversions sustained by cooperative processes, particularly the 2.8-μm Er laser. Rate equations fail to predict the observed steady-state instabilities of this laser, which are well reproduced by density-matrix theory, furnishing evidence of weak coherent delocalizations in a rare-earth system.

© 1994 Optical Society of America

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References

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  1. F. Auzel, Proc. IEEE 61, 758 (1973).
    [Crossref]
  2. S. Hufner, Optical Spectra of Transparent Rare Earth Compounds (Academic, New York, 1978), Chap. 5.
  3. O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.
  4. J. Chivian, W. Case, and D. Eden, Appl. Phys. Lett. 35, 124 (1979).
    [Crossref]
  5. H. Ni and S. C. Rand, Opt. Lett. 16, 1424 (1991).
    [Crossref] [PubMed]
  6. A comprehensive review of upconversion processes is given by J. C. Wright, “Up-conversion and excited state energy transfer in rare-earth doped materials,” in Topics in Applied Physics, F. K. Fong, ed. (Springer, New York, 1976), Vol. 15, p. 239.
    [Crossref]
  7. R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
    [Crossref]
  8. R. J. Thrash and L. F. Johnson, J. Opt. Soc. Am. B 11, 881 (1994).
    [Crossref]
  9. J. Y. Allain, M. Monerie, and H. Poignant, Electron. Lett. 26, 261 (1990).
    [Crossref]
  10. R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
    [Crossref]
  11. S. G. Grubb, K. W. Bennett, R. S. Cannon, and W. F. Humer, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 669–670.
  12. D. S. Funk, S. B. Stevens, and J. G. Eden, IEEE Photon. Technol. Lett. 5, 154 (1993).
    [Crossref]
  13. P. Xie and S. C. Rand, Opt. Lett. 17, 1116, 1822 (1992).
    [Crossref] [PubMed]
  14. P. Xie and S. C. Rand, Opt. Lett. 15, 848 (1990).
    [Crossref] [PubMed]
  15. P. Xie, “Continuous-wave cooperative upconversion lasers,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1992).
  16. H. Chou, “Upconversion processes and Cr-sensitization of Er- and Er, Ho-activated solid state laser materials,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).
  17. A. S. Davydov and A. A. Serikov, Phys. Status Solidi 51, 57 (1972).
    [Crossref]
  18. R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
    [Crossref]
  19. L. M. Hobrock, “Spectra of thulium in yttrium orthoaluminate crystals and its four-level laser operation in the mid-infrared,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1972). In the present paper we follow recent practice by exchanging the labels of the 3H4and the 3F4states of Tm with respect to Hobrock, designating 3F4as the first excited state. See, e.g., A. Brenier, J. Rubin, R. Moncorge, and C. Pedrini, J. Phys. (Paris) 50, 1463 (1989).
    [Crossref]
  20. See, e.g., R. Seydel, From Equilibrium to Chaos (Elsevier, New York, 1988).
  21. J. Rai and C. M. Bowden, in International Conference on Quantum Electronics, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QTuN3.
  22. D. M. Sinnett, J. Appl. Phys. 33, 1578 (1962).
    [Crossref]
  23. R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), pp. 146–152.
  24. See, e.g., Handbook of Mathematics, I. N. Bronshtein and K. A. Semendyayev, eds. (Van Nostrand Reinhold, New York, 1985), p. 419.
  25. P. Xie and S. C. Rand, Opt. Lett. 17, 1198 (1992); Opt. Lett. 17, 1822 (1992).
    [Crossref] [PubMed]
  26. P. Xie and S. C. Rand, Appl. Phys. Lett. 63, 3125 (1993).
    [Crossref]
  27. R. C. Stoneman and L. Esterowitz, Opt. Lett. 17, 816 (1992).
    [Crossref] [PubMed]
  28. See, e.g., Ref. 20, p. 73.
  29. Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
    [Crossref] [PubMed]

1994 (1)

1993 (2)

P. Xie and S. C. Rand, Appl. Phys. Lett. 63, 3125 (1993).
[Crossref]

D. S. Funk, S. B. Stevens, and J. G. Eden, IEEE Photon. Technol. Lett. 5, 154 (1993).
[Crossref]

1992 (4)

1991 (2)

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

H. Ni and S. C. Rand, Opt. Lett. 16, 1424 (1991).
[Crossref] [PubMed]

1990 (2)

P. Xie and S. C. Rand, Opt. Lett. 15, 848 (1990).
[Crossref] [PubMed]

J. Y. Allain, M. Monerie, and H. Poignant, Electron. Lett. 26, 261 (1990).
[Crossref]

1988 (1)

R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
[Crossref]

1986 (1)

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
[Crossref]

1979 (1)

J. Chivian, W. Case, and D. Eden, Appl. Phys. Lett. 35, 124 (1979).
[Crossref]

1973 (1)

F. Auzel, Proc. IEEE 61, 758 (1973).
[Crossref]

1972 (1)

A. S. Davydov and A. A. Serikov, Phys. Status Solidi 51, 57 (1972).
[Crossref]

1962 (1)

D. M. Sinnett, J. Appl. Phys. 33, 1578 (1962).
[Crossref]

Alimov, O. K.

O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.

Allain, J. Y.

J. Y. Allain, M. Monerie, and H. Poignant, Electron. Lett. 26, 261 (1990).
[Crossref]

Ashurov, M. Kh.

O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.

Auzel, F.

F. Auzel, Proc. IEEE 61, 758 (1973).
[Crossref]

Barthem, R. B.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
[Crossref]

Basiev, T. T.

O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.

Bennett, K. W.

S. G. Grubb, K. W. Bennett, R. S. Cannon, and W. F. Humer, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 669–670.

Bowden, C. M.

J. Rai and C. M. Bowden, in International Conference on Quantum Electronics, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QTuN3.

Buisson, R.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
[Crossref]

Cannon, R. S.

S. G. Grubb, K. W. Bennett, R. S. Cannon, and W. F. Humer, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 669–670.

Carter, S. F.

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

Case, W.

J. Chivian, W. Case, and D. Eden, Appl. Phys. Lett. 35, 124 (1979).
[Crossref]

Chivian, J.

J. Chivian, W. Case, and D. Eden, Appl. Phys. Lett. 35, 124 (1979).
[Crossref]

Chou, H.

H. Chou, “Upconversion processes and Cr-sensitization of Er- and Er, Ho-activated solid state laser materials,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

Davey, S. T.

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

Davydov, A. S.

A. S. Davydov and A. A. Serikov, Phys. Status Solidi 51, 57 (1972).
[Crossref]

Eden, D.

J. Chivian, W. Case, and D. Eden, Appl. Phys. Lett. 35, 124 (1979).
[Crossref]

Eden, J. G.

D. S. Funk, S. B. Stevens, and J. G. Eden, IEEE Photon. Technol. Lett. 5, 154 (1993).
[Crossref]

Esterowitz, L.

Funk, D. S.

D. S. Funk, S. B. Stevens, and J. G. Eden, IEEE Photon. Technol. Lett. 5, 154 (1993).
[Crossref]

Gills, Z.

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
[Crossref] [PubMed]

Grubb, S. G.

S. G. Grubb, K. W. Bennett, R. S. Cannon, and W. F. Humer, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 669–670.

Hanna, D. C.

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

Harmand, H.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
[Crossref]

Hobrock, L. M.

L. M. Hobrock, “Spectra of thulium in yttrium orthoaluminate crystals and its four-level laser operation in the mid-infrared,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1972). In the present paper we follow recent practice by exchanging the labels of the 3H4and the 3F4states of Tm with respect to Hobrock, designating 3F4as the first excited state. See, e.g., A. Brenier, J. Rubin, R. Moncorge, and C. Pedrini, J. Phys. (Paris) 50, 1463 (1989).
[Crossref]

Hufner, S.

S. Hufner, Optical Spectra of Transparent Rare Earth Compounds (Academic, New York, 1978), Chap. 5.

Humer, W. F.

S. G. Grubb, K. W. Bennett, R. S. Cannon, and W. F. Humer, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 669–670.

Iwata, C.

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
[Crossref] [PubMed]

Johnson, L. F.

Kirpichenkova, E. O.

O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.

Lenth, W.

R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
[Crossref]

Loudon, R.

R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), pp. 146–152.

Macfarlane, R. M.

R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
[Crossref]

Monerie, M.

J. Y. Allain, M. Monerie, and H. Poignant, Electron. Lett. 26, 261 (1990).
[Crossref]

Murav’ev, V. B.

O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.

Ni, H.

Poignant, H.

J. Y. Allain, M. Monerie, and H. Poignant, Electron. Lett. 26, 261 (1990).
[Crossref]

Rai, J.

J. Rai and C. M. Bowden, in International Conference on Quantum Electronics, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QTuN3.

Rand, S. C.

Roy, R.

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
[Crossref] [PubMed]

Schwartz, I. B.

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
[Crossref] [PubMed]

Serikov, A. A.

A. S. Davydov and A. A. Serikov, Phys. Status Solidi 51, 57 (1972).
[Crossref]

Seydel, R.

See, e.g., R. Seydel, From Equilibrium to Chaos (Elsevier, New York, 1988).

Silversmith, A. J.

R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
[Crossref]

Sinnett, D. M.

D. M. Sinnett, J. Appl. Phys. 33, 1578 (1962).
[Crossref]

Smart, R. G.

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

Stevens, S. B.

D. S. Funk, S. B. Stevens, and J. G. Eden, IEEE Photon. Technol. Lett. 5, 154 (1993).
[Crossref]

Stoneman, R. C.

Szebesta, D.

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

Thrash, R. J.

Tong, F.

R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
[Crossref]

Triandaf, I.

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
[Crossref] [PubMed]

Tropper, A. C.

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

Vial, J. C.

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
[Crossref]

Wright, J. C.

A comprehensive review of upconversion processes is given by J. C. Wright, “Up-conversion and excited state energy transfer in rare-earth doped materials,” in Topics in Applied Physics, F. K. Fong, ed. (Springer, New York, 1976), Vol. 15, p. 239.
[Crossref]

Xie, P.

P. Xie and S. C. Rand, Appl. Phys. Lett. 63, 3125 (1993).
[Crossref]

P. Xie and S. C. Rand, Opt. Lett. 17, 1116, 1822 (1992).
[Crossref] [PubMed]

P. Xie and S. C. Rand, Opt. Lett. 17, 1198 (1992); Opt. Lett. 17, 1822 (1992).
[Crossref] [PubMed]

P. Xie and S. C. Rand, Opt. Lett. 15, 848 (1990).
[Crossref] [PubMed]

P. Xie, “Continuous-wave cooperative upconversion lasers,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1992).

Appl. Phys. Lett. (3)

J. Chivian, W. Case, and D. Eden, Appl. Phys. Lett. 35, 124 (1979).
[Crossref]

R. M. Macfarlane, F. Tong, A. J. Silversmith, and W. Lenth, Appl. Phys. Lett. 52, 1300 (1988).
[Crossref]

P. Xie and S. C. Rand, Appl. Phys. Lett. 63, 3125 (1993).
[Crossref]

Electron. Lett. (2)

J. Y. Allain, M. Monerie, and H. Poignant, Electron. Lett. 26, 261 (1990).
[Crossref]

R. G. Smart, D. C. Hanna, A. C. Tropper, S. T. Davey, S. F. Carter, and D. Szebesta, Electron. Lett. 27, 1307 (1991).
[Crossref]

IEEE Photon. Technol. Lett. (1)

D. S. Funk, S. B. Stevens, and J. G. Eden, IEEE Photon. Technol. Lett. 5, 154 (1993).
[Crossref]

J. Appl. Phys. (1)

D. M. Sinnett, J. Appl. Phys. 33, 1578 (1962).
[Crossref]

J. Lumin. (1)

R. B. Barthem, R. Buisson, J. C. Vial, and H. Harmand, J. Lumin. 34, 295 (1986).
[Crossref]

J. Opt. Soc. Am. B (1)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

Z. Gills, C. Iwata, R. Roy, I. B. Schwartz, and I. Triandaf, Phys. Rev. Lett. 69, 3169 (1992).
[Crossref] [PubMed]

Phys. Status Solidi (1)

A. S. Davydov and A. A. Serikov, Phys. Status Solidi 51, 57 (1972).
[Crossref]

Proc. IEEE (1)

F. Auzel, Proc. IEEE 61, 758 (1973).
[Crossref]

Other (12)

S. Hufner, Optical Spectra of Transparent Rare Earth Compounds (Academic, New York, 1978), Chap. 5.

O. K. Alimov, M. Kh. Ashurov, T. T. Basiev, E. O. Kirpichenkova, and V. B. Murav’ev, in Selective Laser Spectroscopy of Activated Crystals and Glasses, V. V. Osiko, ed. (Nova Science, New York, 1988), Chap. 2.

P. Xie, “Continuous-wave cooperative upconversion lasers,” Ph.D. dissertation (University of Michigan, Ann Arbor, Mich., 1992).

H. Chou, “Upconversion processes and Cr-sensitization of Er- and Er, Ho-activated solid state laser materials,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1989).

S. G. Grubb, K. W. Bennett, R. S. Cannon, and W. F. Humer, in Conference on Lasers and Electro-Optics, Vol. 12 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992), pp. 669–670.

L. M. Hobrock, “Spectra of thulium in yttrium orthoaluminate crystals and its four-level laser operation in the mid-infrared,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1972). In the present paper we follow recent practice by exchanging the labels of the 3H4and the 3F4states of Tm with respect to Hobrock, designating 3F4as the first excited state. See, e.g., A. Brenier, J. Rubin, R. Moncorge, and C. Pedrini, J. Phys. (Paris) 50, 1463 (1989).
[Crossref]

See, e.g., R. Seydel, From Equilibrium to Chaos (Elsevier, New York, 1988).

J. Rai and C. M. Bowden, in International Conference on Quantum Electronics, Vol. 8 of 1990 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1990), paper QTuN3.

R. Loudon, The Quantum Theory of Light, 2nd ed. (Clarendon, Oxford, 1983), pp. 146–152.

See, e.g., Handbook of Mathematics, I. N. Bronshtein and K. A. Semendyayev, eds. (Van Nostrand Reinhold, New York, 1985), p. 419.

A comprehensive review of upconversion processes is given by J. C. Wright, “Up-conversion and excited state energy transfer in rare-earth doped materials,” in Topics in Applied Physics, F. K. Fong, ed. (Springer, New York, 1976), Vol. 15, p. 239.
[Crossref]

See, e.g., Ref. 20, p. 73.

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Figures (14)

Fig. 1
Fig. 1

Dynamics of cooperative upconversion in a system consisting of two four-level atoms initially occupying state |1〉, as indicated by filled circles. The system relaxes by promoting one atom to level |3〉 as the second decays to the ground state (curved arrows). In this fashion upconversion fluorescence from energy levels higher than the initial state becomes possible at several wavelengths (straight arrows). The correspondence between levels of the model and Er is as follows: |0〉 ↔ 4|15/2, |1〉 ↔ 4|13/2, |2〉 ↔ 4|11/2, |3〉 ↔ 4|F9/2.

Fig. 2
Fig. 2

Recycling by cooperative upconversion. The initial excitation of level |3〉, either by resonant optical excitation or by a cooperative process, occurs with quantum efficiency η0. This is followed by a cascade of nonradiative decay and emission processes with individual branching ratios of ηi. States |1a〉 and |1b〉 are different Stark components of level |1〉. Finally a long-lived level is reached in which cooperative upconversion can occur spontaneously without further absorption of photons from the pump field. The remnant excited-state population can be recycled to level |3〉 and additional photons may be emitted, thereby enhancing quantum efficiency. The radiative transition |2〉 → |1b〉 is the focus of quantum-efficiency considerations.

Fig. 3
Fig. 3

Numerical rate equation calculation of pulsations in the inversion of the pair-pumped laser after termination of steady-state excitation. There are no free parameters: intensity l = 153 W/cm2 (1.7 times threshold) in a cavity (τc = 1.5 ns) of dopant density 5.7 × 1020 cm−3 with γ2 = 0.0714 ms−1γ3 = 0.179 ms−1, γ31 = 0.0769 ms−1, γ32 = 0.103 ms−1, γ4 = 143 ms−1, B01 = 6.9 × 10−2 cm2/J, B12 = 2.4 × 10−10 cm3/s,15 and α = 3.2 × 10−17 cm3/s.16 (a) The initial part of postexcitation decay (inset, magnified trace with temporal resolution on the microsecond time scale showing quasi-periodicity). (b) Postexcitation decay at coarse temporal resolution [inset, fast-Fourier-transform (FFT) spectrum].

Fig. 4
Fig. 4

Energy-level diagram of a model pair system. Left, individual atoms A and B may be pictured as each absorbing one pump photon (thin vertical arrow) to reach the interaction level indicated by the filled circles. Pair upconversion then leads to upconversion fluorescence after accommodation of the energy mismatch Δ, as indicated by the thick vertical arrow. Right, an alternative representation shows the coupled pair system as a ladder of product states |A〉|B〉. Absorption of two pump photons is required for promotion of the pair system to state |11〉. Direct excitation to state |02〉 by two-photon absorption of a single atom is off resonant because of the energy defect Δ. Emission from state |02〉 occurs after spontaneous pair upconversion, which may be viewed as an internal relaxation from state |11〉 to |02〉 accompanied by emission of phonons for energy conservation.

Fig. 5
Fig. 5

Density-matrix calculation of excited pair state populations versus time in the presence of delocalization. The initial state is characterized by ρ11 = 1, ρ22 = 0, γ1 = γ2 = 1 kHz, and two values of interatomic coupling are considered, namely, (a) L = 3 kHz and (b) L = 1 kHz. Population pulsations are evident for the larger value of coupling.

Fig. 6
Fig. 6

Growth of quasi-periodic output from a constant steady-state condition of a pair-pumped laser (density-matrix calculation). (a) l2 = 2.6 × 104 kHz2, β = 900 kHz2, and γ2 = 100 cm−1, γ1 = 0. (b) Fourier spectrum of (a).

Fig. 7
Fig. 7

Response function for enhanced energy transfer resulting from modulation of the optical driving field for two values of the interatomic coupling strength L. (a) L = 5 kHz, (b) L = 0.5 kHz. Other parameters used are γ1 = 0.43 kHz (twice the measured19 decay rate), γ2 = 1.6 kHz, and γ12 = 1.02 kHz.

Fig. 8
Fig. 8

Output versus input power for the fourfold cw upconversion laser operating at 701.5 nm (λex = 1.5 μm). Inset, laser emission spectrum. (b) Energy diagram identifying the Er3+ levels involved in the pumping and the emission processes of the fourfold upconversion laser.

Fig. 9
Fig. 9

Oscillations in output power observed at (a) the leading and (b) the trailing edges of a 2.8-μm Er:LiYF4 laser pulse excited by a square pulse at λex = 1.5 μm of 10-ms duration. Excitation was gated acousto-optically from the output of a cw NaCl laser. The rise and fall times of the pump pulses were less than 50 ns. Insets, Fourier transforms showing dominant frequency components.

Fig. 10
Fig. 10

Traces of pair-pumped Er:CaF2 laser output at 2.8 μm at various pumping intensities. (a) The pump pulse. (b) Threshold operation. (c) True cw operation with leading-and trailing-edge transients (1.1 times threshold). (d) Cw operation at higher power (1.5 times threshold). (e) The growth of sustained oscillation from noise (two times threshold). (f) Unstable, quasi-periodic output (three times threshold).

Fig. 11
Fig. 11

Fourier transforms of the Fig. 10 time series, showing the presence of several modulation frequencies at high pumping intensity but no subharmonics. Leading- and trailing-edge transients were excluded from the analysis.

Fig. 12
Fig. 12

Experimental trace of the growth of the instability shown in Fig. 10(e) with high temporal resolution. Inset, Fourier-transform spectrum.

Fig. 13
Fig. 13

Phase space plots of unstable output from the 2.8-μm pair-pumped laser: (a) dx/dt versus t for the x(t) time sequence in the 3–6-ms interval shown in Fig. 10(e) at two times laser threshold (Er:CaF2), (b) dx/dt versus t at three times laser threshold (Er:CaF2), (c) dx/dt versus t at five times laser threshold (Er:LiYF4).

Fig. 14
Fig. 14

Output trace of the 2.8-μm Er:CaF2 laser pumped by a long square pulse in the red spectral region (λex = 652 nm). At this pump wavelength output is continuous even at three times above threshold, and the trailing-edge spike has no rapid underlying oscillations.

Equations (50)

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η eff = η 0 η 1 η 2 [ 1 + η 3 η up P ( 1 ) η 1 η 2 + η 3 η up P ( 1 ) η 1 η 2 η 3 η up P ( 2 ) η 1 η 2 + ] = n = 0 η 0 η 1 η 2 ( η 3 η up η 1 η 2 ) n = η 0 η 1 η 2 1 - η 1 η 2 η 3 η up .
M = η eff / η 0 η 1 η 2 = ( 1 - η 1 η 2 η 3 η up ) - 1 .
η E * = η eff × λ in / λ out .
d N 0 d t = γ 30 N 3 + γ 20 N 2 + γ 10 N 1 + α N 1 2 - B 01 I ( N 0 - N 1 ) ,
d N 1 d t = γ 31 N 3 + γ 21 N 2 - γ 10 N 1 - 2 α N 1 2 + B 01 I ( N 0 - N 1 ) + B 12 Q ( N 2 - N 1 ) ,
d N 2 d t = γ 32 N 3 - γ 2 N 2 - B 12 Q ( N 2 - N 1 ) ,
d N 3 d t = - γ 3 N 3 + α N 1 2 ,
d Q d t = B 12 Q ( N 2 - N 1 ) - Q τ c .
i d ρ d t = [ H , ρ ] - i γ ρ .
H = [ E - i L * i L E ]
E 1 , 2 = E ± ( L L * ) 1 / 2 ,
Ψ 1 = 11 - 02 2 ,             Ψ 2 = 11 + 02 2 .
d ρ 11 d t = L ρ 21 + L * ρ 12 - γ 1 ρ 11 ,
d ρ 22 d t = ( L ρ 21 + L * ρ 12 ) - γ 2 ρ 22 ,
d ρ 12 d t = L ( ρ 22 - ρ 11 ) - γ 12 ρ 12 .
ρ 11 ( t ) = exp ( - γ 12 t ) ( 2 L L * ω 0 2 - Δ 2 - ω 0 2 2 ω 0 2 cos ω 0 t - Δ ω 0 sin ω 0 t ) ,
ρ 22 ( t ) = exp ( - γ 12 t ) ( 2 L L * ω 0 2 ) ( 1 - cos ω 0 t ) ,
ρ 12 ( t ) = exp ( - γ 12 t ) ( 2 L L * ω 0 2 ) [ Δ ( 1 - cos ω 0 t ) - ω 0 sin ω 0 t ] .
Δ = ( γ 1 - γ 2 ) / 2 ,
ω 0 = ( 4 L L * - Δ 2 ) 1 / 2 ,
ρ 12 ( t ) = L ρ 21 + L * ρ 12 .
H ^ 0 = E 1 1 + E 2 2 + α 2 ω α α .
H ^ int = i L 1 2 + h . c . + f A f B * α 2 Δ ω 1 0 + h . c .
d ρ 00 d t = ρ 3 + γ 1 ρ 11 + γ 2 ρ 22 ,
d ρ 11 d t = ρ 3 + ρ 4 - γ 1 ρ 11 ,
d ρ 22 d t = - ρ 4 - γ 2 ρ 22 .
d ρ 3 d t = I 2 ( ρ 11 - ρ 00 ) - γ 01 ρ 3 ,
d ρ 4 d t = 2 β ( ρ 22 - ρ 11 ) - γ 12 ρ 4 ,
ρ 3 = ( f α ) 2 ρ 10 + ( f * α * ) 2 ρ 01 ,
ρ 4 = - L ( ρ 12 + ρ 21 ) ,
l = 2 f 2 α 2 ,
β = L 2 .
[ ρ ˙ 1 ρ ˙ 2 ρ ˙ 3 ρ ˙ 4 ] = [ - γ 1 0 1 - 1 0 - γ 2 0 - 1 2 I 2 I 2 - γ 01 0 - 2 β 2 β 0 - γ 12 ] [ ρ 1 ρ 2 ρ 3 ρ 4 ] + [ 0 0 I 2 0 ] .
I c = ( 16 β + 3 γ 2 2 3 ) 1 / 2 .
d ρ 22 d t = - ( L ρ 21 + L * ρ 12 ) - γ 2 ρ 22 + λ p .
[ ρ ˙ 11 ρ ˙ 22 ρ ˙ 12 ] = [ - γ 1 0 1 0 - γ 2 - 1 - 2 β 2 β - γ 12 ] [ ρ 11 ρ 22 ρ 12 ] + [ 0 λ p ( t ) 0 ] .
[ ρ 11 ( t ) ρ 22 ( t ) ρ 12 ( t ) ] = [ ρ 11 ( 0 ) ρ 22 ( 0 ) ρ 12 ( 0 ) ] exp ( i ω t ) ,
ρ 11 ( 0 ) = ( γ 2 + i ω ) ( γ 12 i ω ) + 2 β ( γ 12 + i ω ) [ ( γ 12 + i ω ) 2 + ω 0 2 ] λ p ( 0 ) ,
ρ 22 ( 0 ) = 2 β ( γ 12 + i ω ) [ ( γ 12 + i ω ) 2 + ω 0 2 ] λ p ( 0 ) ,
ρ 12 ( 0 ) = - 2 β ( γ 2 + i ω ) ( γ 12 + i ω ) [ ( γ 12 + i ω ) 2 + ω 0 2 ] λ p ( 0 ) .
ω 0 = [ 4 β - ( γ 1 - γ 2 2 ) 2 ] 1 / 2 .
R ( ω ) = | ρ 22 ( 0 ) ρ 22 ( 0 ) ω = 0 | = γ 12 ( γ 12 2 + ω 0 2 ) { ( ω 0 2 + γ 12 2 ) [ ( ω 0 2 - ω 2 + γ 12 2 ) 2 + 4 ω 2 γ 12 2 } 1 / 2 .
ω max 2 = 3 ω 0 2 - 4 γ 12 2 + ( ω 0 2 - 32 ω 0 2 γ 12 2 ) 1 / 2 4 .
[ n ˙ 0 n ˙ 1 n ˙ 2 q ˙ ] = [ - B 01 I - γ 30 B 01 I + 2 α N 1 s γ 1 - γ 30 γ 20 - γ 30 0 B 01 I - γ 31 - B 01 I - B 12 Q s - 4 α N 1 s - γ 1 - γ 31 [ B 12 Q s + γ 21 - γ 31 ] B 12 ( N 2 s - N 1 s ) - γ 32 B 12 Q s - γ 32 - B 12 Q s - γ 2 - γ 32 - B 12 ( N 2 s - N 1 s ) 0 - B 12 Q s B 12 Q s 0 ] [ n 0 n 1 n 2 q ] .
[ n 0 ( t ) n 1 ( t ) n 2 ( t ) q ( t ) ] = [ n 0 ( t ) n 1 ( t ) n 2 ( t ) q ( t ) ] exp ( λ t ) ,
D 1 = a 1 ,             D 2 = | a 1 a 0 a 3 a 2 | ,             D 3 = | a 1 a 2 0 a 3 a 2 a 1 0 a 4 a 3 | , D 4 = | a 1 a 0 0 0 a 3 a 2 a 1 0 0 a 4 a 3 a 2 0 0 0 a 4 | .
a 0 = a b c d + γ 3 [ 3 a b d + b c d + b d ( γ 10 + γ 20 ) ] + b c d γ 30 , a 1 = a b c + 3 a b d + 2 b c d + b d ( 2 γ 3 + γ 20 + γ 10 ) + 3 a b γ 3 + a c ( γ 2 + γ 32 ) + b c ( γ 3 + γ 30 ) + 2 a γ 2 γ 3 + b γ 3 ( γ 10 + γ 20 ) + c ( γ 2 γ 30 + γ 2 γ 3 + γ 20 γ 32 ) + γ 10 γ 2 γ 3 , a 2 = a c + 3 a b + 2 b c + 2 b d + 2 a ( γ 2 + γ 3 ) + b ( γ 10 + γ 20 + 2 γ 3 ) + c ( 2 γ 2 + 2 γ 30 + γ 31 + 2 γ 32 ) + [ γ 10 ( γ 2 + γ 3 ) + γ 2 γ 3 ] , a 3 = 2 ( a + b + c ) + γ 1 + γ 2 + γ 3 , a 4 = 1.
D 1 = a 1 ,
D 2 = a 1 a 2 - a 3 a 0 ,
D 3 = a 3 D 2 - a 4 a 1 2 .

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