Abstract

A general theoretical model which permits theoretical predictions of the total transfer efficiency and hence identification of the appropriate energy transfer mechanisms responsible for gain enhancement in energy transfer dye lasers is developed. This model takes into account contributions due to both radiative and nonradiative transfer as well as contributions due to direct excitation of the acceptor. The theoretical predictions derived are compared with the experimental results of a laser dye mixture composed of coumarin 1 (donor) and acriflavine (acceptor) and are shown to be in excellent agreement with experimental results. The method developed is shown to be effective for predicting laser gain line shapes (and hence tunability) as well as fluorescence emission spectra of dye mixtures.

© 1990 Optical Society of America

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References

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  1. T. Forster, “Transfer Mechanisms of Electronics Excitation,” Discuss. Faraday Soc. 27, No. 1, 7–18 (1959).
  2. T. Forster, “Excitation Transfer,” Ann. Phys. 2, 12–19 (1948).
  3. D. L. Dexter, “A Theory of Sensitized Luminescence,” J. Chem. Phys. 21, 836–850 (1953).
    [CrossRef]
  4. M. D. Galanin, “The Problem of the Effect of Concentrations on the Luminescence of Solutions,” Sov. Phys. JETP 1, 317–325 (1955).
  5. J. B. Birks, “Energy Transfer in Organic Systems I,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 946–955 (1948).
  6. J. B. Birks, S. Georghious, “Energy Transfer in Organic Systems VI,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 958–965 (1968).
  7. J. B. Birks, M. S. S. C. P. Leite, “Effects of Diffusion on Transfer Efficiency,” J. Phys. B 3, 513–525 (1970).
    [CrossRef]
  8. J. B. Birks, M. S. S. C. P. Leite, “Energy Transfer in Organic Systems VII,” J. Phys. B 3, 24, 417–424 (1970).
    [CrossRef]
  9. N. J. Turro, Modern Molecular Photochemistry (Benjamin, New York, 1978).
  10. C. Lin, A. Dienes, “Study of Excitation Transfer in Laser Dye Mixtures by Direct Measurements of Fluorescence Lifetime,” J. Appl. Phys. 44, 5050–5052 (1973).
    [CrossRef]
  11. N. V. Unnikrishnan, H. S. Bhatth, R. D. Singh, “Energy Transfer in Dye Mixtures Studied by Laser Fluorimetry,” Opt. Acta 31, 983–978 (1984).
    [CrossRef]
  12. V. V. Rodchenkova, M. G. Reva, B. M. Uzhinov, “Use of Electron-Excitation Energy Transfer in Dye Laser Active Media,” Sov. J. Quantum Electron. 14, 48–51 (1984).
    [CrossRef]
  13. P. J. Sebastain, K. Sathianandan, “Energy Transfer Rhodamine-6G—Safranin-T Dye Laser,” Opt. Commun. 32, 422–424 (1980).
    [CrossRef]
  14. S. A. Ahmed, J. S. Gergely, “Energy Transfer Organic Dye Mixture Laser,” J. Chem. Phys. 61, 1584–1585 (1974).
    [CrossRef]
  15. A. Dienes, M. Madden, “Study of Excitation Transfer in Dye Mixtures by Measurements of Gain Spectra,” J. Appl. Phys. 44, 4161–4164 (1973).
    [CrossRef]
  16. P. Burlamacchi, X. Cutter, “Energy Transfer in Flashlamp Pumped Organic Dye Lasers,” Opt. Commun. 22, 283–287 (1977).
    [CrossRef]
  17. C. E. Moeller, C. M. Verber, A. H. Adelman, “Laser Pumping by Excitation Transfer in Dye Mixtures,” Appl. Phys. Lett. 18, 278–280 (1971).
    [CrossRef]
  18. M. A. Ali, S. A. Ahmed, K. Mitwally, “Fluorescence and Gain Predictions in Laser Dye Mixtures,” Appl. Opt. 28, 3708–3712 (1989).
    [CrossRef] [PubMed]
  19. J. B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970).
  20. M. A. Ali, S. A. Ahmed, “Comprehensive Examination of Radiationless Energy Transfer Models in Dyes: Comparisons of Theory and Experiment,” J. Chem. Phys. 90, 1484–1491 (1989).
    [CrossRef]
  21. M. A. Ali, S. A. Ahmed, A. S. Chokhavatia, “Examination of a Generalized Model for Radiationless Energy Transfer in Dyes,” J. Chem. Phys. 91, 2892–2897 (1989).
    [CrossRef]
  22. G. Jones, W. R. Jackson, W. R. Bergmark, “Solvents Effects on Emission Yield and Lifetime for Coumarin Laser Dyes,” J. Phys. Chem. 89, 294–299 (1985).
    [CrossRef]
  23. I. Kechkemeti, L. Koxma, M. M. Loiko, “Stimulated Emission from Binary Solutions of Coumarin 1 and Acriflavine Pumped with Laser Pulses,” Sov. J. Quantum Electron. 6, 1242–1243 (1976).
    [CrossRef]

1989 (3)

M. A. Ali, S. A. Ahmed, “Comprehensive Examination of Radiationless Energy Transfer Models in Dyes: Comparisons of Theory and Experiment,” J. Chem. Phys. 90, 1484–1491 (1989).
[CrossRef]

M. A. Ali, S. A. Ahmed, A. S. Chokhavatia, “Examination of a Generalized Model for Radiationless Energy Transfer in Dyes,” J. Chem. Phys. 91, 2892–2897 (1989).
[CrossRef]

M. A. Ali, S. A. Ahmed, K. Mitwally, “Fluorescence and Gain Predictions in Laser Dye Mixtures,” Appl. Opt. 28, 3708–3712 (1989).
[CrossRef] [PubMed]

1985 (1)

G. Jones, W. R. Jackson, W. R. Bergmark, “Solvents Effects on Emission Yield and Lifetime for Coumarin Laser Dyes,” J. Phys. Chem. 89, 294–299 (1985).
[CrossRef]

1984 (2)

N. V. Unnikrishnan, H. S. Bhatth, R. D. Singh, “Energy Transfer in Dye Mixtures Studied by Laser Fluorimetry,” Opt. Acta 31, 983–978 (1984).
[CrossRef]

V. V. Rodchenkova, M. G. Reva, B. M. Uzhinov, “Use of Electron-Excitation Energy Transfer in Dye Laser Active Media,” Sov. J. Quantum Electron. 14, 48–51 (1984).
[CrossRef]

1980 (1)

P. J. Sebastain, K. Sathianandan, “Energy Transfer Rhodamine-6G—Safranin-T Dye Laser,” Opt. Commun. 32, 422–424 (1980).
[CrossRef]

1977 (1)

P. Burlamacchi, X. Cutter, “Energy Transfer in Flashlamp Pumped Organic Dye Lasers,” Opt. Commun. 22, 283–287 (1977).
[CrossRef]

1976 (1)

I. Kechkemeti, L. Koxma, M. M. Loiko, “Stimulated Emission from Binary Solutions of Coumarin 1 and Acriflavine Pumped with Laser Pulses,” Sov. J. Quantum Electron. 6, 1242–1243 (1976).
[CrossRef]

1974 (1)

S. A. Ahmed, J. S. Gergely, “Energy Transfer Organic Dye Mixture Laser,” J. Chem. Phys. 61, 1584–1585 (1974).
[CrossRef]

1973 (2)

A. Dienes, M. Madden, “Study of Excitation Transfer in Dye Mixtures by Measurements of Gain Spectra,” J. Appl. Phys. 44, 4161–4164 (1973).
[CrossRef]

C. Lin, A. Dienes, “Study of Excitation Transfer in Laser Dye Mixtures by Direct Measurements of Fluorescence Lifetime,” J. Appl. Phys. 44, 5050–5052 (1973).
[CrossRef]

1971 (1)

C. E. Moeller, C. M. Verber, A. H. Adelman, “Laser Pumping by Excitation Transfer in Dye Mixtures,” Appl. Phys. Lett. 18, 278–280 (1971).
[CrossRef]

1970 (2)

J. B. Birks, M. S. S. C. P. Leite, “Effects of Diffusion on Transfer Efficiency,” J. Phys. B 3, 513–525 (1970).
[CrossRef]

J. B. Birks, M. S. S. C. P. Leite, “Energy Transfer in Organic Systems VII,” J. Phys. B 3, 24, 417–424 (1970).
[CrossRef]

1968 (1)

J. B. Birks, S. Georghious, “Energy Transfer in Organic Systems VI,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 958–965 (1968).

1959 (1)

T. Forster, “Transfer Mechanisms of Electronics Excitation,” Discuss. Faraday Soc. 27, No. 1, 7–18 (1959).

1955 (1)

M. D. Galanin, “The Problem of the Effect of Concentrations on the Luminescence of Solutions,” Sov. Phys. JETP 1, 317–325 (1955).

1953 (1)

D. L. Dexter, “A Theory of Sensitized Luminescence,” J. Chem. Phys. 21, 836–850 (1953).
[CrossRef]

1948 (2)

J. B. Birks, “Energy Transfer in Organic Systems I,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 946–955 (1948).

T. Forster, “Excitation Transfer,” Ann. Phys. 2, 12–19 (1948).

Adelman, A. H.

C. E. Moeller, C. M. Verber, A. H. Adelman, “Laser Pumping by Excitation Transfer in Dye Mixtures,” Appl. Phys. Lett. 18, 278–280 (1971).
[CrossRef]

Ahmed, S. A.

M. A. Ali, S. A. Ahmed, A. S. Chokhavatia, “Examination of a Generalized Model for Radiationless Energy Transfer in Dyes,” J. Chem. Phys. 91, 2892–2897 (1989).
[CrossRef]

M. A. Ali, S. A. Ahmed, “Comprehensive Examination of Radiationless Energy Transfer Models in Dyes: Comparisons of Theory and Experiment,” J. Chem. Phys. 90, 1484–1491 (1989).
[CrossRef]

M. A. Ali, S. A. Ahmed, K. Mitwally, “Fluorescence and Gain Predictions in Laser Dye Mixtures,” Appl. Opt. 28, 3708–3712 (1989).
[CrossRef] [PubMed]

S. A. Ahmed, J. S. Gergely, “Energy Transfer Organic Dye Mixture Laser,” J. Chem. Phys. 61, 1584–1585 (1974).
[CrossRef]

Ali, M. A.

M. A. Ali, S. A. Ahmed, A. S. Chokhavatia, “Examination of a Generalized Model for Radiationless Energy Transfer in Dyes,” J. Chem. Phys. 91, 2892–2897 (1989).
[CrossRef]

M. A. Ali, S. A. Ahmed, K. Mitwally, “Fluorescence and Gain Predictions in Laser Dye Mixtures,” Appl. Opt. 28, 3708–3712 (1989).
[CrossRef] [PubMed]

M. A. Ali, S. A. Ahmed, “Comprehensive Examination of Radiationless Energy Transfer Models in Dyes: Comparisons of Theory and Experiment,” J. Chem. Phys. 90, 1484–1491 (1989).
[CrossRef]

Bergmark, W. R.

G. Jones, W. R. Jackson, W. R. Bergmark, “Solvents Effects on Emission Yield and Lifetime for Coumarin Laser Dyes,” J. Phys. Chem. 89, 294–299 (1985).
[CrossRef]

Bhatth, H. S.

N. V. Unnikrishnan, H. S. Bhatth, R. D. Singh, “Energy Transfer in Dye Mixtures Studied by Laser Fluorimetry,” Opt. Acta 31, 983–978 (1984).
[CrossRef]

Birks, J. B.

J. B. Birks, M. S. S. C. P. Leite, “Effects of Diffusion on Transfer Efficiency,” J. Phys. B 3, 513–525 (1970).
[CrossRef]

J. B. Birks, M. S. S. C. P. Leite, “Energy Transfer in Organic Systems VII,” J. Phys. B 3, 24, 417–424 (1970).
[CrossRef]

J. B. Birks, S. Georghious, “Energy Transfer in Organic Systems VI,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 958–965 (1968).

J. B. Birks, “Energy Transfer in Organic Systems I,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 946–955 (1948).

J. B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970).

Burlamacchi, P.

P. Burlamacchi, X. Cutter, “Energy Transfer in Flashlamp Pumped Organic Dye Lasers,” Opt. Commun. 22, 283–287 (1977).
[CrossRef]

Chokhavatia, A. S.

M. A. Ali, S. A. Ahmed, A. S. Chokhavatia, “Examination of a Generalized Model for Radiationless Energy Transfer in Dyes,” J. Chem. Phys. 91, 2892–2897 (1989).
[CrossRef]

Cutter, X.

P. Burlamacchi, X. Cutter, “Energy Transfer in Flashlamp Pumped Organic Dye Lasers,” Opt. Commun. 22, 283–287 (1977).
[CrossRef]

Dexter, D. L.

D. L. Dexter, “A Theory of Sensitized Luminescence,” J. Chem. Phys. 21, 836–850 (1953).
[CrossRef]

Dienes, A.

C. Lin, A. Dienes, “Study of Excitation Transfer in Laser Dye Mixtures by Direct Measurements of Fluorescence Lifetime,” J. Appl. Phys. 44, 5050–5052 (1973).
[CrossRef]

A. Dienes, M. Madden, “Study of Excitation Transfer in Dye Mixtures by Measurements of Gain Spectra,” J. Appl. Phys. 44, 4161–4164 (1973).
[CrossRef]

Forster, T.

T. Forster, “Transfer Mechanisms of Electronics Excitation,” Discuss. Faraday Soc. 27, No. 1, 7–18 (1959).

T. Forster, “Excitation Transfer,” Ann. Phys. 2, 12–19 (1948).

Galanin, M. D.

M. D. Galanin, “The Problem of the Effect of Concentrations on the Luminescence of Solutions,” Sov. Phys. JETP 1, 317–325 (1955).

Georghious, S.

J. B. Birks, S. Georghious, “Energy Transfer in Organic Systems VI,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 958–965 (1968).

Gergely, J. S.

S. A. Ahmed, J. S. Gergely, “Energy Transfer Organic Dye Mixture Laser,” J. Chem. Phys. 61, 1584–1585 (1974).
[CrossRef]

Jackson, W. R.

G. Jones, W. R. Jackson, W. R. Bergmark, “Solvents Effects on Emission Yield and Lifetime for Coumarin Laser Dyes,” J. Phys. Chem. 89, 294–299 (1985).
[CrossRef]

Jones, G.

G. Jones, W. R. Jackson, W. R. Bergmark, “Solvents Effects on Emission Yield and Lifetime for Coumarin Laser Dyes,” J. Phys. Chem. 89, 294–299 (1985).
[CrossRef]

Kechkemeti, I.

I. Kechkemeti, L. Koxma, M. M. Loiko, “Stimulated Emission from Binary Solutions of Coumarin 1 and Acriflavine Pumped with Laser Pulses,” Sov. J. Quantum Electron. 6, 1242–1243 (1976).
[CrossRef]

Koxma, L.

I. Kechkemeti, L. Koxma, M. M. Loiko, “Stimulated Emission from Binary Solutions of Coumarin 1 and Acriflavine Pumped with Laser Pulses,” Sov. J. Quantum Electron. 6, 1242–1243 (1976).
[CrossRef]

Leite, M. S. S. C. P.

J. B. Birks, M. S. S. C. P. Leite, “Energy Transfer in Organic Systems VII,” J. Phys. B 3, 24, 417–424 (1970).
[CrossRef]

J. B. Birks, M. S. S. C. P. Leite, “Effects of Diffusion on Transfer Efficiency,” J. Phys. B 3, 513–525 (1970).
[CrossRef]

Lin, C.

C. Lin, A. Dienes, “Study of Excitation Transfer in Laser Dye Mixtures by Direct Measurements of Fluorescence Lifetime,” J. Appl. Phys. 44, 5050–5052 (1973).
[CrossRef]

Loiko, M. M.

I. Kechkemeti, L. Koxma, M. M. Loiko, “Stimulated Emission from Binary Solutions of Coumarin 1 and Acriflavine Pumped with Laser Pulses,” Sov. J. Quantum Electron. 6, 1242–1243 (1976).
[CrossRef]

Madden, M.

A. Dienes, M. Madden, “Study of Excitation Transfer in Dye Mixtures by Measurements of Gain Spectra,” J. Appl. Phys. 44, 4161–4164 (1973).
[CrossRef]

Mitwally, K.

Moeller, C. E.

C. E. Moeller, C. M. Verber, A. H. Adelman, “Laser Pumping by Excitation Transfer in Dye Mixtures,” Appl. Phys. Lett. 18, 278–280 (1971).
[CrossRef]

Reva, M. G.

V. V. Rodchenkova, M. G. Reva, B. M. Uzhinov, “Use of Electron-Excitation Energy Transfer in Dye Laser Active Media,” Sov. J. Quantum Electron. 14, 48–51 (1984).
[CrossRef]

Rodchenkova, V. V.

V. V. Rodchenkova, M. G. Reva, B. M. Uzhinov, “Use of Electron-Excitation Energy Transfer in Dye Laser Active Media,” Sov. J. Quantum Electron. 14, 48–51 (1984).
[CrossRef]

Sathianandan, K.

P. J. Sebastain, K. Sathianandan, “Energy Transfer Rhodamine-6G—Safranin-T Dye Laser,” Opt. Commun. 32, 422–424 (1980).
[CrossRef]

Sebastain, P. J.

P. J. Sebastain, K. Sathianandan, “Energy Transfer Rhodamine-6G—Safranin-T Dye Laser,” Opt. Commun. 32, 422–424 (1980).
[CrossRef]

Singh, R. D.

N. V. Unnikrishnan, H. S. Bhatth, R. D. Singh, “Energy Transfer in Dye Mixtures Studied by Laser Fluorimetry,” Opt. Acta 31, 983–978 (1984).
[CrossRef]

Turro, N. J.

N. J. Turro, Modern Molecular Photochemistry (Benjamin, New York, 1978).

Unnikrishnan, N. V.

N. V. Unnikrishnan, H. S. Bhatth, R. D. Singh, “Energy Transfer in Dye Mixtures Studied by Laser Fluorimetry,” Opt. Acta 31, 983–978 (1984).
[CrossRef]

Uzhinov, B. M.

V. V. Rodchenkova, M. G. Reva, B. M. Uzhinov, “Use of Electron-Excitation Energy Transfer in Dye Laser Active Media,” Sov. J. Quantum Electron. 14, 48–51 (1984).
[CrossRef]

Verber, C. M.

C. E. Moeller, C. M. Verber, A. H. Adelman, “Laser Pumping by Excitation Transfer in Dye Mixtures,” Appl. Phys. Lett. 18, 278–280 (1971).
[CrossRef]

Ann. Phys. (1)

T. Forster, “Excitation Transfer,” Ann. Phys. 2, 12–19 (1948).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

C. E. Moeller, C. M. Verber, A. H. Adelman, “Laser Pumping by Excitation Transfer in Dye Mixtures,” Appl. Phys. Lett. 18, 278–280 (1971).
[CrossRef]

Discuss. Faraday Soc. (1)

T. Forster, “Transfer Mechanisms of Electronics Excitation,” Discuss. Faraday Soc. 27, No. 1, 7–18 (1959).

J. Appl. Phys. (2)

A. Dienes, M. Madden, “Study of Excitation Transfer in Dye Mixtures by Measurements of Gain Spectra,” J. Appl. Phys. 44, 4161–4164 (1973).
[CrossRef]

C. Lin, A. Dienes, “Study of Excitation Transfer in Laser Dye Mixtures by Direct Measurements of Fluorescence Lifetime,” J. Appl. Phys. 44, 5050–5052 (1973).
[CrossRef]

J. Chem. Phys. (4)

D. L. Dexter, “A Theory of Sensitized Luminescence,” J. Chem. Phys. 21, 836–850 (1953).
[CrossRef]

M. A. Ali, S. A. Ahmed, “Comprehensive Examination of Radiationless Energy Transfer Models in Dyes: Comparisons of Theory and Experiment,” J. Chem. Phys. 90, 1484–1491 (1989).
[CrossRef]

M. A. Ali, S. A. Ahmed, A. S. Chokhavatia, “Examination of a Generalized Model for Radiationless Energy Transfer in Dyes,” J. Chem. Phys. 91, 2892–2897 (1989).
[CrossRef]

S. A. Ahmed, J. S. Gergely, “Energy Transfer Organic Dye Mixture Laser,” J. Chem. Phys. 61, 1584–1585 (1974).
[CrossRef]

J. Phys. B (2)

J. B. Birks, M. S. S. C. P. Leite, “Effects of Diffusion on Transfer Efficiency,” J. Phys. B 3, 513–525 (1970).
[CrossRef]

J. B. Birks, M. S. S. C. P. Leite, “Energy Transfer in Organic Systems VII,” J. Phys. B 3, 24, 417–424 (1970).
[CrossRef]

J. Phys. B (Proc. Phys. Soc.) (2)

J. B. Birks, “Energy Transfer in Organic Systems I,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 946–955 (1948).

J. B. Birks, S. Georghious, “Energy Transfer in Organic Systems VI,” J. Phys. B (Proc. Phys. Soc.)Ser. 2 1, 958–965 (1968).

J. Phys. Chem. (1)

G. Jones, W. R. Jackson, W. R. Bergmark, “Solvents Effects on Emission Yield and Lifetime for Coumarin Laser Dyes,” J. Phys. Chem. 89, 294–299 (1985).
[CrossRef]

Opt. Acta (1)

N. V. Unnikrishnan, H. S. Bhatth, R. D. Singh, “Energy Transfer in Dye Mixtures Studied by Laser Fluorimetry,” Opt. Acta 31, 983–978 (1984).
[CrossRef]

Opt. Commun. (2)

P. Burlamacchi, X. Cutter, “Energy Transfer in Flashlamp Pumped Organic Dye Lasers,” Opt. Commun. 22, 283–287 (1977).
[CrossRef]

P. J. Sebastain, K. Sathianandan, “Energy Transfer Rhodamine-6G—Safranin-T Dye Laser,” Opt. Commun. 32, 422–424 (1980).
[CrossRef]

Sov. J. Quantum Electron. (2)

I. Kechkemeti, L. Koxma, M. M. Loiko, “Stimulated Emission from Binary Solutions of Coumarin 1 and Acriflavine Pumped with Laser Pulses,” Sov. J. Quantum Electron. 6, 1242–1243 (1976).
[CrossRef]

V. V. Rodchenkova, M. G. Reva, B. M. Uzhinov, “Use of Electron-Excitation Energy Transfer in Dye Laser Active Media,” Sov. J. Quantum Electron. 14, 48–51 (1984).
[CrossRef]

Sov. Phys. JETP (1)

M. D. Galanin, “The Problem of the Effect of Concentrations on the Luminescence of Solutions,” Sov. Phys. JETP 1, 317–325 (1955).

Other (2)

N. J. Turro, Modern Molecular Photochemistry (Benjamin, New York, 1978).

J. B. Birks, Photophysics of Aromatic Molecules (Wiley, New York, 1970).

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Figures (3)

Fig. 1
Fig. 1

Fluorescence spectra of 2 × 10−3-M/liter coumarin (donor) and 5 × 10−5-M/liter acriflavine (acceptor) in ethanol: —, experimental; – – – – – – – – –, theoretical.

Fig. 2
Fig. 2

Predicted fluorescence, absorption, and gain of the dye mixture: 2 × 10−3-M/liter coumarin (donor) and 5 × 10−5-M/liter acriflavine (acceptor) in ethanol and the observed laser spectrum: —, absorption; – – – – – – fluorescence; ..........., observed laser output spectrum; – . – . – . –, fluorescence minus absorption (equivalent to the gainlike shape of the laser dye mixture).

Fig. 3
Fig. 3

Predicted fluorescence, absorption, and gain of a dye mixture: 2 × 10−3-M/liter coumarin (donor) and 1 × 10−3-M/liter acriflavine (acceptor) in ethanol and the observed laser spectrum: —, absorption; – – – – – –, fluorescence; ..........., observed laser output spectrum; – . – . – . –, fluorescence minus absorption (equivalent to the gainlike shape of the laser dye mixture).

Equations (19)

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α ( μ 1 ) = F 0 d ( μ 1 ) { 1 - exp [ - 2.3 [ A ] × E a ( μ 1 ) ] } .
α = 0 F 0 d ( μ ) { 1 - exp [ - 2.3 [ A ] × E a ( μ ) ] } d μ
α = 1 - 0 F 0 d ( μ ) exp [ - 2.3 [ A ] × E a ( μ ) ] d μ ,
f r = ( 1 - f n r ) α rad Φ 0 d ,
f n r = 1 - K 0 K + K 0 K 3 / 2 π 1 / 2 A exp ( A 2 / K ) [ 1 - erf ( A / K 1 / 2 ) ] ,
K = K 0 + 4 π D R eff N ˜ [ A ] , A = 4 π 1 / 2 D 1 / 2 R eff 2 N ˜ [ A ] , K 0 = 1 / T 0 d .
f = f n r + f r .
F 0 d = I 0 { 1 - exp [ - 2.3 E d ( μ e x ) [ D ] X ] } Φ 0 d = Q 1 Φ 0 d ,
Q 1 = I 0 { 1 - exp [ - 2.3 E d [ D ] X ] } ,
F d = I 0 ( 1 - exp { - 2.3 E d ( μ ex ) [ D ] X - 2.3 E a ( μ ex ) [ A ] X } ) [ E d ( μ ex ) [ D ] E d ( μ ex ) [ D ] + E a ( μ ex ) [ A ] ] [ ( 1 - f n r ) Φ 0 d - f r ] ;
F d = Q 2 [ E d ( μ ex ) [ D ] E d [ D ] + E a [ A ] ] ( 1 - f n r ) ( 1 - α rad ) Φ 0 d ,
Q 2 = I 0 { 1 - exp [ - 2.3 E d ( μ ex ) [ D ] X - 2.3 E a ( μ ex ) [ A ] X ] } .
F d = F 0 d ( Q 2 Q 1 ) ( 1 - f n r ) ( 1 - α rad ) ( E d [ D ] E d [ D ] + E a [ A ] ) ,
Q 2 / Q 1 = 1 - exp [ - 2.3 E d ( μ ex ) [ D ] X - 2.3 E a ( μ ex ) [ A ] X ] 1 - exp [ - 2.3 E d ( μ ex ) [ D ] X ] .
F a = Q 2 [ E d ( μ ex ) [ D ] E d [ D ] + E a [ A ] ] f n r Φ 0 a + Q 2 [ E a ( μ ex ) [ A ] E d [ D ] + E a [ A ] ] Φ 0 a + Q 2 [ E d [ D ] E d [ D ] + E a [ A ] ] f r Φ 0 a
= [ E d [ D ] ( f n r + r r ) + E a [ A ] E d [ D ] + E a [ A ] ] Q 2 Φ 0 a ,
F a = F 0 d [ E d [ D ] ( f n r + f r ) + E a [ A ] E d [ D ] + E a [ A ] ] ( Q 2 Q 1 ) ( Φ 0 a Φ 0 d ) .
F t = F a + F d .
f = 1 - ( I d / I 0 d ) .

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