Abstract

We have studied the room temperature behavior of the Er3+:4S3/2 fluorescence of three Er3+-doped crystals: Er3+:YLF with 30% and 8.5% doping and Er3+:BaYF doped at 20%. We studied the fluorescence by pumping with an 800-nm laser beam sinusoidally modulated in intensity in the range 10–1000 Hz, and we observed the presence of a second harmonic of modulation in the time evolution of the intensity of fluorescence. Second-harmonic intensity was also predicted by means of numerical integration of the rate equations that describe our system. Both experimental data and theoretical prediction show that the intensity reaches a maximum value at a modulation frequency that depends on the pump rate and on upconversion parameter α. This dependence allows us to obtain the value of α. To estimate this value we also measured the lifetimes of the various manifolds involved in the upconversion process.

© 2001 Optical Society of America

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    [CrossRef]
  14. G. S. Maciel, C. B. de Araùjo, Y. Messaddeq, and M. A. Aegerter, “Frequency up-conversion in Er3+-doped fluoroindate glasses pumped at 1.48 μm,” Phys. Rev. B 55, 6335–6342 (1997).
    [CrossRef]
  15. B. R. Reddy and P. Venkateswarlu, “Infrared to visible energy upconversion in Er3+-doped oxide glass,” Appl. Phys. Lett. 64, 1327–1329 (1994).
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
  20. D. A. Zubenko, M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Different mechanisms of nonlinear quenching of luminescence,” Phys. Rev. B 55, 8881–8886 (1997).
    [CrossRef]
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    [CrossRef]
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2000 (2)

R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er3+:Y2O3,” Opt. Lett. 5, 338–430 (2000).
[CrossRef]

M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Phys. Rev. B 61, 3337–3346 (2000).
[CrossRef]

1999 (2)

S. R. Lüthi, M. Pollnau, H. U. Güdel, and M. P. Hehlen, “Near-infrared to visible upconversion in Er3+-doped Cs3Lu2Cl9, Cs3Lu2Br9, and Cs3Y2I9 excited at 1.54 μm,” Phys. Rev. B 60, 162–178 (1999).
[CrossRef]

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, “Upconversion mechanism in Er3+-doped fluorozirconate glasses under 800 nm excitation,” J. Appl. Phys. 85, 29–37 (1999).
[CrossRef]

1998 (1)

C. T. M. Ribeiro, A. R. Zanatta, L. A. O. Nunes, Y. Messaddeq, and M. A. Aegerter, “Optical spectroscopy of Er3+ and Yb3+ co-doped fluoroindate glasses,” J. Appl. Phys. 83, 2256–2260 (1998).
[CrossRef]

1997 (4)

G. S. Maciel, C. B. de Araùjo, Y. Messaddeq, and M. A. Aegerter, “Frequency up-conversion in Er3+-doped fluoroindate glasses pumped at 1.48 μm,” Phys. Rev. B 55, 6335–6342 (1997).
[CrossRef]

M. Pollnau, “The route toward a diode-pumped 1-W erbium 3-μm fiber laser,” IEEE J. Quantum Electron. 33, 1982–1990 (1997).
[CrossRef]

D. A. Zubenko, M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Different mechanisms of nonlinear quenching of luminescence,” Phys. Rev. B 55, 8881–8886 (1997).
[CrossRef]

A. Toncelli, A. Di Lieto, P. Minguzzi, and M. Tonelli, “Discovering energy paths in laser crystals,” Opt. Lett. 22, 1165–1167 (1997).
[CrossRef] [PubMed]

1996 (1)

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

1995 (2)

M. Pollnau, W. Luthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

J. Koetke and G. Huber, “Infrared excited-state absorption and stimulated-emission cross sections of Er3+-doped crystals,” Appl. Phys. B 61, 151–158 (1995).
[CrossRef]

1994 (3)

R. Micheletti, P. Minguzzi, M. A. Noginov, and M. Tonelli, “Upconversion luminescence of Er and Cr in YSGG laser crystals,” J. Opt. Soc. Am. B 11, 2095–2099 (1994).
[CrossRef]

B. R. Reddy and P. Venkateswarlu, “Infrared to visible energy upconversion in Er3+-doped oxide glass,” Appl. Phys. Lett. 64, 1327–1329 (1994).
[CrossRef]

M. Pollnau, Th. Graf, E. Balmer, W. Luthy, and H. P. Weber, “Explanation of the cw operation of the Er+3 3-μm crystal laser,” Phys. Rev. A 49, 3990–3996 (1994).
[CrossRef] [PubMed]

1992 (1)

D. S. Knowles and H. P. Jenssen, “Upconversion versus Pr-deactivation for efficient 3 μm laser operation in Er,” IEEE J. Quantum Electron. 28, 1197–1207 (1992).
[CrossRef]

1990 (1)

M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Nonlinear population processes of Er laser levels in chromium-doped laser crystals,” Opt. Quantum Electron. 22, S61–S74 (1990).
[CrossRef]

1973 (1)

F. E. Auzel, “Materials and devices using double-pumped phosphorus with energy transfer,” Proc. IEEE 61, 758–786 (1973).
[CrossRef]

1966 (1)

Acioli, L. H.

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Aegerter, M. A.

C. T. M. Ribeiro, A. R. Zanatta, L. A. O. Nunes, Y. Messaddeq, and M. A. Aegerter, “Optical spectroscopy of Er3+ and Yb3+ co-doped fluoroindate glasses,” J. Appl. Phys. 83, 2256–2260 (1998).
[CrossRef]

G. S. Maciel, C. B. de Araùjo, Y. Messaddeq, and M. A. Aegerter, “Frequency up-conversion in Er3+-doped fluoroindate glasses pumped at 1.48 μm,” Phys. Rev. B 55, 6335–6342 (1997).
[CrossRef]

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Auzel, F. E.

F. E. Auzel, “Materials and devices using double-pumped phosphorus with energy transfer,” Proc. IEEE 61, 758–786 (1973).
[CrossRef]

Balmer, E.

M. Pollnau, Th. Graf, E. Balmer, W. Luthy, and H. P. Weber, “Explanation of the cw operation of the Er+3 3-μm crystal laser,” Phys. Rev. A 49, 3990–3996 (1994).
[CrossRef] [PubMed]

Biswas, A.

R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er3+:Y2O3,” Opt. Lett. 5, 338–430 (2000).
[CrossRef]

de Araùjo, C. B.

G. S. Maciel, C. B. de Araùjo, Y. Messaddeq, and M. A. Aegerter, “Frequency up-conversion in Er3+-doped fluoroindate glasses pumped at 1.48 μm,” Phys. Rev. B 55, 6335–6342 (1997).
[CrossRef]

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Di Lieto, A.

Florez, A.

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Friend, C. S.

R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er3+:Y2O3,” Opt. Lett. 5, 338–430 (2000).
[CrossRef]

Gamelin, D. R.

M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Phys. Rev. B 61, 3337–3346 (2000).
[CrossRef]

Gomes, A. S. L.

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Graf, Th.

M. Pollnau, Th. Graf, E. Balmer, W. Luthy, and H. P. Weber, “Explanation of the cw operation of the Er+3 3-μm crystal laser,” Phys. Rev. A 49, 3990–3996 (1994).
[CrossRef] [PubMed]

Güdel, H. U.

M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Phys. Rev. B 61, 3337–3346 (2000).
[CrossRef]

S. R. Lüthi, M. Pollnau, H. U. Güdel, and M. P. Hehlen, “Near-infrared to visible upconversion in Er3+-doped Cs3Lu2Cl9, Cs3Lu2Br9, and Cs3Y2I9 excited at 1.54 μm,” Phys. Rev. B 60, 162–178 (1999).
[CrossRef]

Hehlen, M. P.

M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Phys. Rev. B 61, 3337–3346 (2000).
[CrossRef]

S. R. Lüthi, M. Pollnau, H. U. Güdel, and M. P. Hehlen, “Near-infrared to visible upconversion in Er3+-doped Cs3Lu2Cl9, Cs3Lu2Br9, and Cs3Y2I9 excited at 1.54 μm,” Phys. Rev. B 60, 162–178 (1999).
[CrossRef]

Huber, G.

J. Koetke and G. Huber, “Infrared excited-state absorption and stimulated-emission cross sections of Er3+-doped crystals,” Appl. Phys. B 61, 151–158 (1995).
[CrossRef]

Inoue, H.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, “Upconversion mechanism in Er3+-doped fluorozirconate glasses under 800 nm excitation,” J. Appl. Phys. 85, 29–37 (1999).
[CrossRef]

Jenssen, H. P.

D. S. Knowles and H. P. Jenssen, “Upconversion versus Pr-deactivation for efficient 3 μm laser operation in Er,” IEEE J. Quantum Electron. 28, 1197–1207 (1992).
[CrossRef]

Kapoor, R.

R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er3+:Y2O3,” Opt. Lett. 5, 338–430 (2000).
[CrossRef]

Knowles, D. S.

D. S. Knowles and H. P. Jenssen, “Upconversion versus Pr-deactivation for efficient 3 μm laser operation in Er,” IEEE J. Quantum Electron. 28, 1197–1207 (1992).
[CrossRef]

Koetke, J.

J. Koetke and G. Huber, “Infrared excited-state absorption and stimulated-emission cross sections of Er3+-doped crystals,” Appl. Phys. B 61, 151–158 (1995).
[CrossRef]

Kogelnik, H.

Li, T.

Lüthi, S. R.

M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Phys. Rev. B 61, 3337–3346 (2000).
[CrossRef]

S. R. Lüthi, M. Pollnau, H. U. Güdel, and M. P. Hehlen, “Near-infrared to visible upconversion in Er3+-doped Cs3Lu2Cl9, Cs3Lu2Br9, and Cs3Y2I9 excited at 1.54 μm,” Phys. Rev. B 60, 162–178 (1999).
[CrossRef]

Luthy, W.

M. Pollnau, W. Luthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

M. Pollnau, Th. Graf, E. Balmer, W. Luthy, and H. P. Weber, “Explanation of the cw operation of the Er+3 3-μm crystal laser,” Phys. Rev. A 49, 3990–3996 (1994).
[CrossRef] [PubMed]

Maciel, G. S.

G. S. Maciel, C. B. de Araùjo, Y. Messaddeq, and M. A. Aegerter, “Frequency up-conversion in Er3+-doped fluoroindate glasses pumped at 1.48 μm,” Phys. Rev. B 55, 6335–6342 (1997).
[CrossRef]

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Makishima, A.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, “Upconversion mechanism in Er3+-doped fluorozirconate glasses under 800 nm excitation,” J. Appl. Phys. 85, 29–37 (1999).
[CrossRef]

Menezes, L. S.

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Messaddeq, Y.

C. T. M. Ribeiro, A. R. Zanatta, L. A. O. Nunes, Y. Messaddeq, and M. A. Aegerter, “Optical spectroscopy of Er3+ and Yb3+ co-doped fluoroindate glasses,” J. Appl. Phys. 83, 2256–2260 (1998).
[CrossRef]

G. S. Maciel, C. B. de Araùjo, Y. Messaddeq, and M. A. Aegerter, “Frequency up-conversion in Er3+-doped fluoroindate glasses pumped at 1.48 μm,” Phys. Rev. B 55, 6335–6342 (1997).
[CrossRef]

C. B. de Araùjo, L. S. Menezes, G. S. Maciel, L. H. Acioli, A. S. L. Gomes, Y. Messaddeq, A. Florez, and M. A. Aegerter, “Infrared-to-visible cw frequency upconversion in Er3+-doped fluoroindate glasses,” Appl. Phys. Lett. 68, 602–604 (1996).
[CrossRef]

Micheletti, R.

Minguzzi, P.

Noginov, M. A.

D. A. Zubenko, M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Different mechanisms of nonlinear quenching of luminescence,” Phys. Rev. B 55, 8881–8886 (1997).
[CrossRef]

R. Micheletti, P. Minguzzi, M. A. Noginov, and M. Tonelli, “Upconversion luminescence of Er and Cr in YSGG laser crystals,” J. Opt. Soc. Am. B 11, 2095–2099 (1994).
[CrossRef]

M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Nonlinear population processes of Er laser levels in chromium-doped laser crystals,” Opt. Quantum Electron. 22, S61–S74 (1990).
[CrossRef]

Nunes, L. A. O.

C. T. M. Ribeiro, A. R. Zanatta, L. A. O. Nunes, Y. Messaddeq, and M. A. Aegerter, “Optical spectroscopy of Er3+ and Yb3+ co-doped fluoroindate glasses,” J. Appl. Phys. 83, 2256–2260 (1998).
[CrossRef]

Pollnau, M.

M. Pollnau, D. R. Gamelin, S. R. Lüthi, H. U. Güdel, and M. P. Hehlen, “Power dependence of upconversion luminescence in lanthanide and transition-metal-ion systems,” Phys. Rev. B 61, 3337–3346 (2000).
[CrossRef]

S. R. Lüthi, M. Pollnau, H. U. Güdel, and M. P. Hehlen, “Near-infrared to visible upconversion in Er3+-doped Cs3Lu2Cl9, Cs3Lu2Br9, and Cs3Y2I9 excited at 1.54 μm,” Phys. Rev. B 60, 162–178 (1999).
[CrossRef]

M. Pollnau, “The route toward a diode-pumped 1-W erbium 3-μm fiber laser,” IEEE J. Quantum Electron. 33, 1982–1990 (1997).
[CrossRef]

M. Pollnau, W. Luthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

M. Pollnau, Th. Graf, E. Balmer, W. Luthy, and H. P. Weber, “Explanation of the cw operation of the Er+3 3-μm crystal laser,” Phys. Rev. A 49, 3990–3996 (1994).
[CrossRef] [PubMed]

Prasad, P. N.

R. Kapoor, C. S. Friend, A. Biswas, and P. N. Prasad, “Highly efficient infrared-to-visible energy upconversion in Er3+:Y2O3,” Opt. Lett. 5, 338–430 (2000).
[CrossRef]

Reddy, B. R.

B. R. Reddy and P. Venkateswarlu, “Infrared to visible energy upconversion in Er3+-doped oxide glass,” Appl. Phys. Lett. 64, 1327–1329 (1994).
[CrossRef]

Ribeiro, C. T. M.

C. T. M. Ribeiro, A. R. Zanatta, L. A. O. Nunes, Y. Messaddeq, and M. A. Aegerter, “Optical spectroscopy of Er3+ and Yb3+ co-doped fluoroindate glasses,” J. Appl. Phys. 83, 2256–2260 (1998).
[CrossRef]

Shcherbakov, I. A.

D. A. Zubenko, M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Different mechanisms of nonlinear quenching of luminescence,” Phys. Rev. B 55, 8881–8886 (1997).
[CrossRef]

M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Nonlinear population processes of Er laser levels in chromium-doped laser crystals,” Opt. Quantum Electron. 22, S61–S74 (1990).
[CrossRef]

Smirnov, V. A.

D. A. Zubenko, M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Different mechanisms of nonlinear quenching of luminescence,” Phys. Rev. B 55, 8881–8886 (1997).
[CrossRef]

M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Nonlinear population processes of Er laser levels in chromium-doped laser crystals,” Opt. Quantum Electron. 22, S61–S74 (1990).
[CrossRef]

Soga, K.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, “Upconversion mechanism in Er3+-doped fluorozirconate glasses under 800 nm excitation,” J. Appl. Phys. 85, 29–37 (1999).
[CrossRef]

Toncelli, A.

Tonelli, M.

Tsuda, M.

M. Tsuda, K. Soga, H. Inoue, and A. Makishima, “Upconversion mechanism in Er3+-doped fluorozirconate glasses under 800 nm excitation,” J. Appl. Phys. 85, 29–37 (1999).
[CrossRef]

Venkateswarlu, P.

B. R. Reddy and P. Venkateswarlu, “Infrared to visible energy upconversion in Er3+-doped oxide glass,” Appl. Phys. Lett. 64, 1327–1329 (1994).
[CrossRef]

Weber, H. P.

M. Pollnau, W. Luthy, and H. P. Weber, “Population mechanisms of the green Er3+:LiYF4 laser,” J. Appl. Phys. 77, 6128–6134 (1995).
[CrossRef]

M. Pollnau, Th. Graf, E. Balmer, W. Luthy, and H. P. Weber, “Explanation of the cw operation of the Er+3 3-μm crystal laser,” Phys. Rev. A 49, 3990–3996 (1994).
[CrossRef] [PubMed]

Zanatta, A. R.

C. T. M. Ribeiro, A. R. Zanatta, L. A. O. Nunes, Y. Messaddeq, and M. A. Aegerter, “Optical spectroscopy of Er3+ and Yb3+ co-doped fluoroindate glasses,” J. Appl. Phys. 83, 2256–2260 (1998).
[CrossRef]

Zubenko, D. A.

D. A. Zubenko, M. A. Noginov, V. A. Smirnov, and I. A. Shcherbakov, “Different mechanisms of nonlinear quenching of luminescence,” Phys. Rev. B 55, 8881–8886 (1997).
[CrossRef]

Appl. Opt. (1)

Appl. Phys. B (1)

J. Koetke and G. Huber, “Infrared excited-state absorption and stimulated-emission cross sections of Er3+-doped crystals,” Appl. Phys. B 61, 151–158 (1995).
[CrossRef]

Appl. Phys. Lett. (2)

B. R. Reddy and P. Venkateswarlu, “Infrared to visible energy upconversion in Er3+-doped oxide glass,” Appl. Phys. Lett. 64, 1327–1329 (1994).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Er3+ energy levels: α, the upconversion process; η, cross relaxation from  4S3/2; Γ, the pump laser.

Fig. 2
Fig. 2

Theoretical prediction of second-harmonic intensity as a function of modulation frequency for several values of the pump rate (increasing from bottom to top). Arrows, νmax.

Fig. 3
Fig. 3

νmax values obtained from numerical integration of Eqs. (2): (a) νmax versus Γ0 for various values of α (increasing from bottom to top). (b) νmax versus α for various values of Γ0 (increasing from bottom to top). Curves, νmax=A(αΓ0)1/2 for the values of A.

Fig. 4
Fig. 4

Experimental result of second-harmonic intensity for Er3+:YLF at 30% doping with various values of pump intensity, namely, 13, 21, 31, 42, and 58 mW (bottom to top). Arrows, νmax.

Fig. 5
Fig. 5

Experimental maximum frequency for Er3+:YLF at 8.5%.

Tables (3)

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Table 1 Absorption and Waist Measurementsa

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Table 2 Lifetime Measurements, Effective Branching Ratios β˜52, and A Values for the Crystals under Investigation

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Table 3 Upconversion Parameter α Compared with Values Already Present in the Literature

Equations (12)

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N˙0=-Γ(t)+i=15 βi0 Niτi+αN22-ηN0N5-δN0N3,
N˙1=-N1τ1+i=25 βi1 Niτi+ηN0N5+2δN0N3,
N˙2=-N2τ2+i=35 βi2 Niτi-2αN22-σESAΓ(t)N2,
N˙3=Γ(t)-N3τ3+i=45 βi3 Niτi+ηN0N5-δN0N3,
N˙4=-N4τ4+β54 N5τ5,
N˙5=-N5τ5+αN22-ηN0N5+σESAΓ(t)N2,
N˙2=-N2τ2+β˜52 N5τ5-2αN22+β˜32Γ0[1+ cos(2πνt)],
N˙5=-N5τ5+αN22,
νmax=A(αΓ0)1/2,
β˜52lin=τ5τ5R(β53+β54β43)β˜32,
β˜52cr=1-τ5τ5Rβ˜32,
β˜52=τ5τ5R(β53+β54β43)β˜32+1-τ5τ5Rβ˜32.

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