Abstract

We present an extension to our earlier proposed statistical model [Phys. Rev. B 62, 15628 (2002)] for studying migration-assisted homogeneous upconversion in erbium-doped fibers. The extension takes into account minimum proximity distance between erbium ions randomly distributed in the host material and the nonuniformity of the excitation distribution among them. We derive a transcendental equation for the population inversion and find the dependence of the upconversion rate on the population inversion and the pump power for the entire range of feasible Er concentrations. We verify the validity and accuracy of the model by means of time-resolved Monte Carlo simulations.

© 2005 Optical Society of America

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  1. L. H. Spiekman and D. Zimmerman, "Optical amplification for metro: EDFA/EDWA Amplets and semiconductor technologies," in Optical Fiber Communications Conference (Optical Society of America, Washington, D.C., 2003), paper ThC5.
  2. P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
    [CrossRef]
  3. P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
    [CrossRef]
  4. S. Sergeyev, "Model of high-concentration erbium-doped fibre amplifier: effects of migration and upconversion processes," Electron. Lett. 39, 511-512 (2003).
    [CrossRef]
  5. J. Nilsson, B. Jaskorzynska, and P. Blixt, "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching," IEEE Photonics Technol. Lett. 5, 1427-1429 (1993).
    [CrossRef]
  6. J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
    [CrossRef]
  7. S. Tammela, P. Kiiveri, S. Sarkilahti, M. Hotoleanu, H. Valkonen, M. Rajala, J. Kurki, and K. Janka, "Direct nanoparticle deposition process for manufacturing very short high gain Er-doped silica glass fibers," in Proceedings of European Conference on Optical Communication , P. Danielson, ed. (Technical University of Denmark, Copenhagen, Denmark, 2002), p. 9.4.2.
  8. J. L. Philipsen and A. Bjarklev, "Monte Carlo simulation of homogeneous upconversion in erbium-doped silica glasses," IEEE J. Quantum Electron. 33, 845-854 (1997).
    [CrossRef]
  9. J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
    [CrossRef]
  10. D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
    [CrossRef]
  11. A. K. Przevuskii and N. V. Nikonorov, "Monte Carlo simulation of upconversion processes in erbium-doped materials," Opt. Mater. 21, 729-741 (2003).
    [CrossRef]
  12. V. A. Gaisenok and A. I. Slobodyanyuk, "Effect of energy cumulation of singlet-excited molecules on luminescence of dye solutions," Opt. Spektrosk. 65, 39-41 (1988).
  13. S. V. Sergeyev and B. Jaskorzynska, "Statistical model for energy-transfer induced upconversion in Er-doped glasses," Phys. Rev. B 62, 15628 (2000).
    [CrossRef]
  14. S. Sergeyev, D. Khoptyar, and B. Jaskorzynska, "Upconversion and migration in erbium-doped waveguides in the continuous-wave excitation switch-off regime," Phys. Rev. B 65, 233104-1-23104-4 (2002).
    [CrossRef]
  15. D. Khoptyar and B. Jaskorzynska, "Experimental verification of the statistical model for migration enhanced upconversion in Er-doped silica," in Proceedings of European Conference in Optical Communication , P. Danielson, ed. (Technical University of Denmark, Copenhagen, Denmark, 2002), p. 2.2.6.
  16. S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
    [CrossRef]
  17. V. P. Gaponsev and N. S. Platonov, "Migration accelerated quenching of luminescence in glasses activated by rare-earth ions," in Dynamical Process in Disordered Systems, W. M. Yen, ed., Vol. 50 of Material Science Forum, 1st ed. (Trans. Tech Publications, Adermannsdorf, Switzerland, 1989), pp. 165-222.
  18. A few comments should be added to Eq. (3). In the early work on the subject 12 the upconversion contribution was modeled as nk n_k Sigma^N _i not = k P_ki n¯. Such an assumption unavoidably leads to conclusion that for small population inversion upconversion grows as n, which is erroneous. Besides, migration may be expressed in the form symmetrical to the upconversion [the third term in Eq. (3)]: n_k Sigma^N(1-n) _i not = k W_ki + (1-n_k) Sigma^Nn _i not = k W_ki, which does not change the final results of our calculations.
  19. V. M. Agranovich and M. D. Galanin, Electronic Excitation Energy Transfer in Condensed Matter (North-Holland, Amsterdam, 1982).
  20. A. I. Burstein, "Concentration quenching of noncoherent excitation in solutions," Sov. Phys. Usp. 143, 553-600 (1984).
    [CrossRef]
  21. E. N. Bodunov, "Approximate methods in the theory of nonradiative energy transfer of localized excitations in disordered media: a review." Opt. Spektrosk. 74, 518-551 (1993).

2003 (2)

S. Sergeyev, "Model of high-concentration erbium-doped fibre amplifier: effects of migration and upconversion processes," Electron. Lett. 39, 511-512 (2003).
[CrossRef]

A. K. Przevuskii and N. V. Nikonorov, "Monte Carlo simulation of upconversion processes in erbium-doped materials," Opt. Mater. 21, 729-741 (2003).
[CrossRef]

2002 (1)

S. Sergeyev, D. Khoptyar, and B. Jaskorzynska, "Upconversion and migration in erbium-doped waveguides in the continuous-wave excitation switch-off regime," Phys. Rev. B 65, 233104-1-23104-4 (2002).
[CrossRef]

2000 (1)

S. V. Sergeyev and B. Jaskorzynska, "Statistical model for energy-transfer induced upconversion in Er-doped glasses," Phys. Rev. B 62, 15628 (2000).
[CrossRef]

1999 (4)

S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
[CrossRef]

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

1997 (1)

J. L. Philipsen and A. Bjarklev, "Monte Carlo simulation of homogeneous upconversion in erbium-doped silica glasses," IEEE J. Quantum Electron. 33, 845-854 (1997).
[CrossRef]

1995 (1)

J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
[CrossRef]

1993 (2)

J. Nilsson, B. Jaskorzynska, and P. Blixt, "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching," IEEE Photonics Technol. Lett. 5, 1427-1429 (1993).
[CrossRef]

E. N. Bodunov, "Approximate methods in the theory of nonradiative energy transfer of localized excitations in disordered media: a review." Opt. Spektrosk. 74, 518-551 (1993).

1991 (1)

P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
[CrossRef]

1988 (1)

V. A. Gaisenok and A. I. Slobodyanyuk, "Effect of energy cumulation of singlet-excited molecules on luminescence of dye solutions," Opt. Spektrosk. 65, 39-41 (1988).

1984 (1)

A. I. Burstein, "Concentration quenching of noncoherent excitation in solutions," Sov. Phys. Usp. 143, 553-600 (1984).
[CrossRef]

Babonas, J.

J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
[CrossRef]

Bjarklev, A.

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

J. L. Philipsen and A. Bjarklev, "Monte Carlo simulation of homogeneous upconversion in erbium-doped silica glasses," IEEE J. Quantum Electron. 33, 845-854 (1997).
[CrossRef]

Blixt, P.

J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
[CrossRef]

J. Nilsson, B. Jaskorzynska, and P. Blixt, "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching," IEEE Photonics Technol. Lett. 5, 1427-1429 (1993).
[CrossRef]

P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
[CrossRef]

Bodunov, E. N.

E. N. Bodunov, "Approximate methods in the theory of nonradiative energy transfer of localized excitations in disordered media: a review." Opt. Spektrosk. 74, 518-551 (1993).

Bremberg, D.

S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
[CrossRef]

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

Broeng, J.

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

Bruce, A. J.

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

Burstein, A. I.

A. I. Burstein, "Concentration quenching of noncoherent excitation in solutions," Sov. Phys. Usp. 143, 553-600 (1984).
[CrossRef]

Carlnäs, T.

P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
[CrossRef]

DiGiovanni, D. J.

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

Gaisenok , V. A.

V. A. Gaisenok and A. I. Slobodyanyuk, "Effect of energy cumulation of singlet-excited molecules on luminescence of dye solutions," Opt. Spektrosk. 65, 39-41 (1988).

Helmfrid, S.

S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
[CrossRef]

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

Jaskorzynska, B.

S. Sergeyev, D. Khoptyar, and B. Jaskorzynska, "Upconversion and migration in erbium-doped waveguides in the continuous-wave excitation switch-off regime," Phys. Rev. B 65, 233104-1-23104-4 (2002).
[CrossRef]

S. V. Sergeyev and B. Jaskorzynska, "Statistical model for energy-transfer induced upconversion in Er-doped glasses," Phys. Rev. B 62, 15628 (2000).
[CrossRef]

S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
[CrossRef]

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
[CrossRef]

J. Nilsson, B. Jaskorzynska, and P. Blixt, "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching," IEEE Photonics Technol. Lett. 5, 1427-1429 (1993).
[CrossRef]

P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
[CrossRef]

Khoptyar, D.

S. Sergeyev, D. Khoptyar, and B. Jaskorzynska, "Upconversion and migration in erbium-doped waveguides in the continuous-wave excitation switch-off regime," Phys. Rev. B 65, 233104-1-23104-4 (2002).
[CrossRef]

Myslinski, P.

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

Nikonorov, N. V.

A. K. Przevuskii and N. V. Nikonorov, "Monte Carlo simulation of upconversion processes in erbium-doped materials," Opt. Mater. 21, 729-741 (2003).
[CrossRef]

Nilsson, J.

J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
[CrossRef]

J. Nilsson, B. Jaskorzynska, and P. Blixt, "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching," IEEE Photonics Technol. Lett. 5, 1427-1429 (1993).
[CrossRef]

P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
[CrossRef]

Palsdonir, B.

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

Palsdottir, B.

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

Pálsdóttir, B.

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

Philipsen, J. L.

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
[CrossRef]

Philipsen , J. L.

J. L. Philipsen and A. Bjarklev, "Monte Carlo simulation of homogeneous upconversion in erbium-doped silica glasses," IEEE J. Quantum Electron. 33, 845-854 (1997).
[CrossRef]

Przevuskii , A. K.

A. K. Przevuskii and N. V. Nikonorov, "Monte Carlo simulation of upconversion processes in erbium-doped materials," Opt. Mater. 21, 729-741 (2003).
[CrossRef]

Sergeyev, S.

S. Sergeyev, "Model of high-concentration erbium-doped fibre amplifier: effects of migration and upconversion processes," Electron. Lett. 39, 511-512 (2003).
[CrossRef]

S. Sergeyev, D. Khoptyar, and B. Jaskorzynska, "Upconversion and migration in erbium-doped waveguides in the continuous-wave excitation switch-off regime," Phys. Rev. B 65, 233104-1-23104-4 (2002).
[CrossRef]

Sergeyev , S. V.

S. V. Sergeyev and B. Jaskorzynska, "Statistical model for energy-transfer induced upconversion in Er-doped glasses," Phys. Rev. B 62, 15628 (2000).
[CrossRef]

Slobodyanyuk, A. I.

V. A. Gaisenok and A. I. Slobodyanyuk, "Effect of energy cumulation of singlet-excited molecules on luminescence of dye solutions," Opt. Spektrosk. 65, 39-41 (1988).

Swillo, M.

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

Szubert, C.

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

Electron. Lett. (3)

S. Sergeyev, "Model of high-concentration erbium-doped fibre amplifier: effects of migration and upconversion processes," Electron. Lett. 39, 511-512 (2003).
[CrossRef]

D. Bremberg, S. Helmfrid, B. Jaskorzynska, M. Swillo, J. L. Philipsen, and B. Pálsdóttir, "Observation of energy-distribution-dependent homogeneous upconversion in erbium-doped silica glass fibers," Electron. Lett. 14, 1189-1191 (1999).
[CrossRef]

S. Helmfrid, D. Bremberg, B. Jaskorzynska, and J. L. Philipsen, "Spatial holeburning in second order excitation probability distribution for densely erbium-doped fibers," Electron. Lett. 14, 1191-1193 (1999).
[CrossRef]

IEEE J. Quantum Electron. (2)

J. L. Philipsen and A. Bjarklev, "Monte Carlo simulation of homogeneous upconversion in erbium-doped silica glasses," IEEE J. Quantum Electron. 33, 845-854 (1997).
[CrossRef]

J. L. Philipsen, J. Broeng, A. Bjarklev, S. Helmfrid, D. Bremberg, B. Jaskorzynska, and B. Palsdonir, "Observation of strongly nonquadratic homogeneous upconversion in Er3+-doped silica fibers and reevaluation of the degree of clustering," IEEE J. Quantum Electron. 35, 1741-1749 (1999).
[CrossRef]

IEEE Photonics Technol. Lett. (3)

P. Blixt, J. Nilsson, T. Carlnäs, and B. Jaskorzynska, "Concentration dependent upconversion in Er3+-doped fiber amplifiers: experiments and modeling," IEEE Photonics Technol. Lett. 3, 996-998 (1991).
[CrossRef]

P. Myslinski, C. Szubert, A. J. Bruce, D. J. DiGiovanni, and B. Palsdottir, "Performance of high-concentration erbium-doped fiber amplifiers," IEEE Photonics Technol. Lett. 11, 973-975 (1999).
[CrossRef]

J. Nilsson, B. Jaskorzynska, and P. Blixt, "Performance reduction and design modification of erbium-doped fiber amplifiers resulting from pair-induced quenching," IEEE Photonics Technol. Lett. 5, 1427-1429 (1993).
[CrossRef]

J. Lightwave Technol. (1)

J. Nilsson, P. Blixt, B. Jaskorzynska, and J. Babonas, "Evaluation of parasitic upconversion mechanisms Er3+-doped silica glass fibers by analysis of fluorescence at 980 nm," J. Lightwave Technol. 13, 341-349 (1995).
[CrossRef]

Opt. Mater. (1)

A. K. Przevuskii and N. V. Nikonorov, "Monte Carlo simulation of upconversion processes in erbium-doped materials," Opt. Mater. 21, 729-741 (2003).
[CrossRef]

Opt. Spektrosk. (2)

V. A. Gaisenok and A. I. Slobodyanyuk, "Effect of energy cumulation of singlet-excited molecules on luminescence of dye solutions," Opt. Spektrosk. 65, 39-41 (1988).

E. N. Bodunov, "Approximate methods in the theory of nonradiative energy transfer of localized excitations in disordered media: a review." Opt. Spektrosk. 74, 518-551 (1993).

Phys. Rev. B (2)

S. V. Sergeyev and B. Jaskorzynska, "Statistical model for energy-transfer induced upconversion in Er-doped glasses," Phys. Rev. B 62, 15628 (2000).
[CrossRef]

S. Sergeyev, D. Khoptyar, and B. Jaskorzynska, "Upconversion and migration in erbium-doped waveguides in the continuous-wave excitation switch-off regime," Phys. Rev. B 65, 233104-1-23104-4 (2002).
[CrossRef]

Sov. Phys. Usp. (1)

A. I. Burstein, "Concentration quenching of noncoherent excitation in solutions," Sov. Phys. Usp. 143, 553-600 (1984).
[CrossRef]

Other (6)

L. H. Spiekman and D. Zimmerman, "Optical amplification for metro: EDFA/EDWA Amplets and semiconductor technologies," in Optical Fiber Communications Conference (Optical Society of America, Washington, D.C., 2003), paper ThC5.

D. Khoptyar and B. Jaskorzynska, "Experimental verification of the statistical model for migration enhanced upconversion in Er-doped silica," in Proceedings of European Conference in Optical Communication , P. Danielson, ed. (Technical University of Denmark, Copenhagen, Denmark, 2002), p. 2.2.6.

V. P. Gaponsev and N. S. Platonov, "Migration accelerated quenching of luminescence in glasses activated by rare-earth ions," in Dynamical Process in Disordered Systems, W. M. Yen, ed., Vol. 50 of Material Science Forum, 1st ed. (Trans. Tech Publications, Adermannsdorf, Switzerland, 1989), pp. 165-222.

A few comments should be added to Eq. (3). In the early work on the subject 12 the upconversion contribution was modeled as nk n_k Sigma^N _i not = k P_ki n¯. Such an assumption unavoidably leads to conclusion that for small population inversion upconversion grows as n, which is erroneous. Besides, migration may be expressed in the form symmetrical to the upconversion [the third term in Eq. (3)]: n_k Sigma^N(1-n) _i not = k W_ki + (1-n_k) Sigma^Nn _i not = k W_ki, which does not change the final results of our calculations.

V. M. Agranovich and M. D. Galanin, Electronic Excitation Energy Transfer in Condensed Matter (North-Holland, Amsterdam, 1982).

S. Tammela, P. Kiiveri, S. Sarkilahti, M. Hotoleanu, H. Valkonen, M. Rajala, J. Kurki, and K. Janka, "Direct nanoparticle deposition process for manufacturing very short high gain Er-doped silica glass fibers," in Proceedings of European Conference on Optical Communication , P. Danielson, ed. (Technical University of Denmark, Copenhagen, Denmark, 2002), p. 9.4.2.

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Figures (9)

Fig. 1
Fig. 1

Comparison between the models. Unphysical growth of the UCR in the initial model from Refs. 13 and 14 (dashed line) is removed by taking into account the finite proximity of the Er ions (dashed-dotted curve). The model correspondence to MC results (squares) is further improved if nonuniform excitation distribution is accounted for (solid curve). Er concentration 1026 m-3 Rm=20 Å, Ru=10 Å,R0=3.5Å (UCR is in units of the spontaneous relaxation rate 1/τ2).

Fig. 2
Fig. 2

Radial distribution for Er ions (solid line), radial excitation distribution (gray tone), and its approximation (dotted line).

Fig. 3
Fig. 3

Illustration to the choice of the event during MC simulations. All of the events probabilities Pi are mapped to the segment (0, Ptot), where Ptot is the sum of the all events probabilities i NPi. A random number p is generated in (0, Ptot). An event m is chosen for which i m-1Pi<p<i mPi.

Fig. 4
Fig. 4

Illustration to the generation of the Poisson time intervals. Random numbers p are generated in [0, 1]. Number of trials is counted until first successful kth trial occurs, i.e., pk<1/M. Then stochastic values qk=k/M have Poisson distribution. In order to obtain kth time step during simulations one multiplies qk by the total event probabilities Ptot.

Fig. 5
Fig. 5

Upconversion rate (in the units of the spontaneous relaxation rate 1/τ2) versus population inversion in EDF. Monte Carlo simulations are shown by: circles (static upconversion at 980-nm pumping), squares (migration-assisted upconversion at 980-nm pumping), and diamonds (migration-assisted upconversion at 1480-nm pumping). Solid and dashed curves represent the corresponding characteristics calculated according to Eqs. (4) and (8). The dotted curve represents the kinetic limit.

Fig. 6
Fig. 6

Low-pumping upconversion constant CUCLP (normalized to the metastable lifetime, τ2) versus Rm/Ru ratio, Ru=10 Å. cER=1026 m-3. Monte Carlo simulations are represented by squares, and the model predictions [Eq. (10)] are represented by the solid curve. Estimation by use of Eq. (11) (dashed-dotted curve) where minimum proximity radius R0=3.5 Å is disregarded.

Fig. 7
Fig. 7

Low-pumping upconversion constant CUCLP (normalized to the metastable lifetime, τ2) versus Er concentration cER. Monte Carlo simulations include static upconversion (circles) Ru=10 Å and migration-assisted upconversion (squares) Rm=20 Å. Kinetic limit, static upconversion, and strong migration asymptotes are given by Eqs. (2), (12), and (13), respectively. The solid line is given by Eq. (11).

Fig. 8
Fig. 8

Donor fluorescence quenching by acceptors. The excitation originating at donor 1 is delayed at donors 3 and 4 on its way to donor 5, where it is quenched by the acceptor (shown by a black circle).

Fig. 9
Fig. 9

Migration-assisted HUC. The excitation originating at donor 1 is delayed at donors 3 and 4, where it has an extra probability to be quenched by excitation that came from donor 6.

Equations (46)

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h(x)=4πVx2H(x-R0),
Pik=Ru6xik6,Wik=Rm6xik6.
W=limNk=1Nnkl=1NnPklk=1Nnknknl=1NnPkl=n4π3cERRu6R03.
(1-nkβ)α-nk-nkiknNPki-nkikNWki+nikNWki
=0,
(1-nβ)α-n-nW(n, α)=0.
W(n, α)=α(1-βn)n-1.
n(α, k, r)=α1+βα0e-tG1(t)G2(t)dt1+0e-tG1(t)tG2(t)dt,
G1(t)=exp-nkt1+αβgt1+αβ, w,
G2(t)=exp-krt1+αβgt1+αβ, wr,
g(t, w)=erf(tw)-1-exp(-w2t)wπt,
w=Ru3/R03,r=Rm6/Ru6,
n(α, k, r)=α1+βα(n+r)Fk(n+r)21+βαn+rFk(n+r)21+βα,
n(α, k)=α1+βα0e-tG1(t)dt.
W(n, α)α0n0nCUCLP,
CUCLP=k20e-tG2(t)erf(wt)t dt0e-tG2(t)dt.
CUCLPw1-F(kr/2)rF(kr/2),
CUCLPkr083π3cER2Ru3Rm3.
CUCLPr02π23Ru3cER.
n(r)11+Ru6r6nk1+αβ n().
R0Pik(r)n(r)h(r)dr=ReffPik(r)h(r)dr
Reff=Ruα1+αβarctgRu3R03α1+αβ-11/3
Reff=Ru3α1+αβ0Z(1+b1+z2+b2/2)dz(1+z2+b1+z2+b2/2)-11/3,
b=πkr1+βα,Z=α1+αβRu3R03.
pi=0m-1Pi, i=0mPi.
pk1/M,
pi>1/M,i<k,
n(α)=nk(α, ri1riN)ri1riN.
nk(α)=α1+βα0e-t exp-iknNPkit1+βα×exp-ikNWkit1+βαdt-n0e-t×exp-tiknNPki1+βαt exp-tikNWki1+βαdt.
G1(t)=exp-t1+βαiknNPkiri1rinN,
G2(t)=exp-t1+βαikNWkitri1riN.
G1(t)=exp-t1+βαP(rki)rkinN.
G1(t)=limV0RH(r)exp-t1+αβRur6drcERnV.
G1(t)
=exp-4πcERnR01-exp-t1+αβRur6r2dr.
G1(t)=exp{-nkt/1+αβg[t/(1+αβ), w]},
g(t, w)=erf(tw)-[1-exp(-tw2)]/(πtw),
w=Ru3/R03.
G2(t)=exp{-krt/1+αβg[t/(1+αβ), wr]}
n(α)=α1+βα0e-tG1(t)G2(t)dt-n0e-tG1(t)tG2(t)dt,
0=-nII+(1-nIIβ)α-nIIP21(x)n-nIIS+nS.
P21(x)=Ru6/x6,S=ikNWki.
f(S)=kr2πS3/2 exp-k2r4S
nII(x)=α1+αβ1+P21(x)n1+αβ[1-F(·)]1+P21(x)n1+αβ ,
F(·)=Fkr2[1+βα+P21(x)n]1/2.
F(x)=1+πx+π2x2-1

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