Abstract

Tm3+Ho3+ codoped lanthanum aluminum germanate glasses are prepared by a simple melt-quenching method. The 2μm emission characteristic and energy transfer from Tm3+ to Ho3+ upon excitation of a conventional 808nm laser diode is investigated. The spectroscopic properties are estimated by using Judd–Ofelt parameters, branching ratios, and lifetimes. The microscopic energy transfer parameters of each process are also calculated. The large energy transfer coefficient of Tm3+ to Ho3+ and the little back energy transfer from Ho3+ to Tm3+ indicate that an efficient energy transfer process occurs.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  21. P. R. Watekar, S. M. Ju, S. J. Boo, and W. T. Han, “Linear and non-linear optical properties of Yb3+∕Tm3+ co-doped alumino-silicate glass prepared by sol-gel method,” J. Non-Cryst. Solids 351, 2446–2452 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010

Q. Zhang, G. Chen, G. Zhang, J. Qiu, and D. Chen, “Infrared luminescence of Tm3+/Yb3+ codoped lanthanum aluminum germanate glasses,” J. Appl. Phys. 107, 023102 (2010).
[CrossRef]

2009

C. Gheorghe, A. Lupei, V. Lupei, L. Gheorghe, and A. Ikesue, “Spectroscopic properties of Ho3+ doped Sc2O3 transparent ceramic for laser materials,” J. Appl. Phys. 105, 123110 (2009).
[CrossRef]

2008

P. R. Watekar, S. Ju, and W. T. Han, “Optical properties of Ho-doped alumino-germano-silica glass optical fiber,” J. Non-Cryst. Solids 354, 1453–1459 (2008).
[CrossRef]

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

2007

2006

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. B. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352, 5344–5352 (2006).
[CrossRef]

2005

P. R. Watekar, S. M. Ju, S. J. Boo, and W. T. Han, “Linear and non-linear optical properties of Yb3+∕Tm3+ co-doped alumino-silicate glass prepared by sol-gel method,” J. Non-Cryst. Solids 351, 2446–2452 (2005).
[CrossRef]

2004

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq, “Dynamics of Tm–Ho energy transfer and deactivation of the F43 low level of thulium in fluorozirconate glasses,” J. Appl. Phys. 95, 5451–5463 (2004).
[CrossRef]

2003

S. B. Rai, A. K. Singh, and S. K. Singh, “Spectroscopic properties of Ho3+ ions doped in tellurite glass,” Spectrochim. Acta, Part A 59, 3221–3226 (2003).
[CrossRef]

S. D. Jackson and S. Mossman, “Diode-cladding-pumped Yb3+, Ho3+-doped silica fiber laser operating at 2.1-μm,” Appl. Opt. 42, 3546–3549 (2003).
[CrossRef] [PubMed]

2002

J. Ganem, J. Crawford, P. Schmidt, N. W. Jenkins, and S. R. Bowman, “Thulium cross-relaxation in a low phonon energy crystalline host,” Phys. Rev. B 66, 245101 (2002).
[CrossRef]

2001

G. Ozen and B. DiBartolo, “The microscopic interaction parameter for Tm-to-Ho resonant energy transfer in LiYF4,” J. Phys. Condens. Matter 13, 195–202 (2001).
[CrossRef]

M. Shojiya, Y. Kawamoto, and K. Kadono, “Judd–Ofelt parameters and multiphonon relaxation of Ho3+ ions in ZnCl2-based glass,” J. Appl. Phys. 89, 4944–4950 (2001).
[CrossRef]

1998

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

1997

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344–14351 (1997).
[CrossRef]

1995

1991

1970

T. Miyakawa and D. L. Dexter, “Phonon sidebands, multiphonon relaxation of excited states, and phonon-assisted energy transfer between ions in solids,” Phys. Rev. B 1, 2961–2969 (1970).
[CrossRef]

1966

W. F. Krupke, “Optical absorption and fluorescence intensities in several rare-earth-doped Y2O3 and LaF3 single crystals,” Phys. Rev. 145, 325–337 (1966).
[CrossRef]

Barnes, N. P.

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. B. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352, 5344–5352 (2006).
[CrossRef]

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

Bigot, L.

B. Jacquier, L. Bigot, S. Guy, and A. M. Jurdyc, “Rare earth doped confined structures for lasers and amplifiers,” in Spectroscopic properties of rare earths in optical materials, GuokuiLiu and BernardJacquier, eds. (Springer, 2005), pp. 433–437.

Boo, S. J.

P. R. Watekar, S. M. Ju, S. J. Boo, and W. T. Han, “Linear and non-linear optical properties of Yb3+∕Tm3+ co-doped alumino-silicate glass prepared by sol-gel method,” J. Non-Cryst. Solids 351, 2446–2452 (2005).
[CrossRef]

Boulon, G.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Bowman, S. R.

J. Ganem, J. Crawford, P. Schmidt, N. W. Jenkins, and S. R. Bowman, “Thulium cross-relaxation in a low phonon energy crystalline host,” Phys. Rev. B 66, 245101 (2002).
[CrossRef]

Brenier, A.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Chen, D.

Q. Zhang, G. Chen, G. Zhang, J. Qiu, and D. Chen, “Infrared luminescence of Tm3+/Yb3+ codoped lanthanum aluminum germanate glasses,” J. Appl. Phys. 107, 023102 (2010).
[CrossRef]

Chen, G.

Q. Zhang, G. Chen, G. Zhang, J. Qiu, and D. Chen, “Infrared luminescence of Tm3+/Yb3+ codoped lanthanum aluminum germanate glasses,” J. Appl. Phys. 107, 023102 (2010).
[CrossRef]

Chen, G. X.

G. X. Chen, Q. Y. Zhang, G. F. Yang, and Z. H. Jiang, “Mid-infrared emission characteristic and energy transfer of Ho3+-doped tellurite glass sensitized by Tm3+,” J. Fluoresc. 17, 301–307 (2007).
[CrossRef] [PubMed]

Crawford, J.

J. Ganem, J. Crawford, P. Schmidt, N. W. Jenkins, and S. R. Bowman, “Thulium cross-relaxation in a low phonon energy crystalline host,” Phys. Rev. B 66, 245101 (2002).
[CrossRef]

da Vila, L. D.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq, “Dynamics of Tm–Ho energy transfer and deactivation of the F43 low level of thulium in fluorozirconate glasses,” J. Appl. Phys. 95, 5451–5463 (2004).
[CrossRef]

Dexter, D. L.

T. Miyakawa and D. L. Dexter, “Phonon sidebands, multiphonon relaxation of excited states, and phonon-assisted energy transfer between ions in solids,” Phys. Rev. B 1, 2961–2969 (1970).
[CrossRef]

Di Bartolo, B.

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

DiBartolo, B.

G. Ozen and B. DiBartolo, “The microscopic interaction parameter for Tm-to-Ho resonant energy transfer in LiYF4,” J. Phys. Condens. Matter 13, 195–202 (2001).
[CrossRef]

Fukuda, K.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Fukudat, T.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Ganem, J.

J. Ganem, J. Crawford, P. Schmidt, N. W. Jenkins, and S. R. Bowman, “Thulium cross-relaxation in a low phonon energy crystalline host,” Phys. Rev. B 66, 245101 (2002).
[CrossRef]

Geng, J. H.

Gheorghe, C.

C. Gheorghe, A. Lupei, V. Lupei, L. Gheorghe, and A. Ikesue, “Spectroscopic properties of Ho3+ doped Sc2O3 transparent ceramic for laser materials,” J. Appl. Phys. 105, 123110 (2009).
[CrossRef]

Gheorghe, L.

C. Gheorghe, A. Lupei, V. Lupei, L. Gheorghe, and A. Ikesue, “Spectroscopic properties of Ho3+ doped Sc2O3 transparent ceramic for laser materials,” J. Appl. Phys. 105, 123110 (2009).
[CrossRef]

Gomes, L.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq, “Dynamics of Tm–Ho energy transfer and deactivation of the F43 low level of thulium in fluorozirconate glasses,” J. Appl. Phys. 95, 5451–5463 (2004).
[CrossRef]

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344–14351 (1997).
[CrossRef]

Guy, S.

B. Jacquier, L. Bigot, S. Guy, and A. M. Jurdyc, “Rare earth doped confined structures for lasers and amplifiers,” in Spectroscopic properties of rare earths in optical materials, GuokuiLiu and BernardJacquier, eds. (Springer, 2005), pp. 433–437.

Guyot, Y.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Han, W. T.

P. R. Watekar, S. Ju, and W. T. Han, “Optical properties of Ho-doped alumino-germano-silica glass optical fiber,” J. Non-Cryst. Solids 354, 1453–1459 (2008).
[CrossRef]

P. R. Watekar, S. M. Ju, S. J. Boo, and W. T. Han, “Linear and non-linear optical properties of Yb3+∕Tm3+ co-doped alumino-silicate glass prepared by sol-gel method,” J. Non-Cryst. Solids 351, 2446–2452 (2005).
[CrossRef]

Heo, J.

Ikesue, A.

C. Gheorghe, A. Lupei, V. Lupei, L. Gheorghe, and A. Ikesue, “Spectroscopic properties of Ho3+ doped Sc2O3 transparent ceramic for laser materials,” J. Appl. Phys. 105, 123110 (2009).
[CrossRef]

Jackson, S. D.

Jacquier, B.

B. Jacquier, L. Bigot, S. Guy, and A. M. Jurdyc, “Rare earth doped confined structures for lasers and amplifiers,” in Spectroscopic properties of rare earths in optical materials, GuokuiLiu and BernardJacquier, eds. (Springer, 2005), pp. 433–437.

Jang, J. N.

Jenkins, N. W.

J. Ganem, J. Crawford, P. Schmidt, N. W. Jenkins, and S. R. Bowman, “Thulium cross-relaxation in a low phonon energy crystalline host,” Phys. Rev. B 66, 245101 (2002).
[CrossRef]

Jiang, S. B.

Jiang, Z. H.

G. X. Chen, Q. Y. Zhang, G. F. Yang, and Z. H. Jiang, “Mid-infrared emission characteristic and energy transfer of Ho3+-doped tellurite glass sensitized by Tm3+,” J. Fluoresc. 17, 301–307 (2007).
[CrossRef] [PubMed]

Jouini, A.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Ju, S.

P. R. Watekar, S. Ju, and W. T. Han, “Optical properties of Ho-doped alumino-germano-silica glass optical fiber,” J. Non-Cryst. Solids 354, 1453–1459 (2008).
[CrossRef]

Ju, S. M.

P. R. Watekar, S. M. Ju, S. J. Boo, and W. T. Han, “Linear and non-linear optical properties of Yb3+∕Tm3+ co-doped alumino-silicate glass prepared by sol-gel method,” J. Non-Cryst. Solids 351, 2446–2452 (2005).
[CrossRef]

Jurdyc, A. M.

B. Jacquier, L. Bigot, S. Guy, and A. M. Jurdyc, “Rare earth doped confined structures for lasers and amplifiers,” in Spectroscopic properties of rare earths in optical materials, GuokuiLiu and BernardJacquier, eds. (Springer, 2005), pp. 433–437.

Kadono, K.

M. Shojiya, Y. Kawamoto, and K. Kadono, “Judd–Ofelt parameters and multiphonon relaxation of Ho3+ ions in ZnCl2-based glass,” J. Appl. Phys. 89, 4944–4950 (2001).
[CrossRef]

Kanamori, T.

Kawamoto, Y.

M. Shojiya, Y. Kawamoto, and K. Kadono, “Judd–Ofelt parameters and multiphonon relaxation of Ho3+ ions in ZnCl2-based glass,” J. Appl. Phys. 89, 4944–4950 (2001).
[CrossRef]

Kitagawa, T.

Krupke, W. F.

W. F. Krupke, “Optical absorption and fluorescence intensities in several rare-earth-doped Y2O3 and LaF3 single crystals,” Phys. Rev. 145, 325–337 (1966).
[CrossRef]

Lupei, A.

C. Gheorghe, A. Lupei, V. Lupei, L. Gheorghe, and A. Ikesue, “Spectroscopic properties of Ho3+ doped Sc2O3 transparent ceramic for laser materials,” J. Appl. Phys. 105, 123110 (2009).
[CrossRef]

Lupei, V.

C. Gheorghe, A. Lupei, V. Lupei, L. Gheorghe, and A. Ikesue, “Spectroscopic properties of Ho3+ doped Sc2O3 transparent ceramic for laser materials,” J. Appl. Phys. 105, 123110 (2009).
[CrossRef]

Messaddeq, Y.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq, “Dynamics of Tm–Ho energy transfer and deactivation of the F43 low level of thulium in fluorozirconate glasses,” J. Appl. Phys. 95, 5451–5463 (2004).
[CrossRef]

Miyakawa, T.

T. Miyakawa and D. L. Dexter, “Phonon sidebands, multiphonon relaxation of excited states, and phonon-assisted energy transfer between ions in solids,” Phys. Rev. B 1, 2961–2969 (1970).
[CrossRef]

Moncorge, R.

R. Moncorge, “Current topics in rare-earth lasers,” in Spectroscopic properties of rare earths in optical materialsGuokuiLiu and BernardJacquier, eds. (Springer, 2005), pp. 332–339.

Mossman, S.

Ohishi, Y.

Ozen, G.

G. Ozen and B. DiBartolo, “The microscopic interaction parameter for Tm-to-Ho resonant energy transfer in LiYF4,” J. Phys. Condens. Matter 13, 195–202 (2001).
[CrossRef]

Qiu, J.

Q. Zhang, G. Chen, G. Zhang, J. Qiu, and D. Chen, “Infrared luminescence of Tm3+/Yb3+ codoped lanthanum aluminum germanate glasses,” J. Appl. Phys. 107, 023102 (2010).
[CrossRef]

Rai, S. B.

S. B. Rai, A. K. Singh, and S. K. Singh, “Spectroscopic properties of Ho3+ ions doped in tellurite glass,” Spectrochim. Acta, Part A 59, 3221–3226 (2003).
[CrossRef]

Ranieri, I. M.

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344–14351 (1997).
[CrossRef]

Reichle, D. J.

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. B. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352, 5344–5352 (2006).
[CrossRef]

Ribeiro, S. J. L.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq, “Dynamics of Tm–Ho energy transfer and deactivation of the F43 low level of thulium in fluorozirconate glasses,” J. Appl. Phys. 95, 5451–5463 (2004).
[CrossRef]

Sato, H.

A. Jouini, A. Brenier, Y. Guyot, G. Boulon, H. Sato, A. Yoshikawa, K. Fukuda, and T. Fukudat, “Spectroscopic and laser properties of the near-infrared tunable laser material Yb3+-doped CaF2 crystal,” Cryst. Growth Des. 8, 808–811 (2008).
[CrossRef]

Schmidt, P.

J. Ganem, J. Crawford, P. Schmidt, N. W. Jenkins, and S. R. Bowman, “Thulium cross-relaxation in a low phonon energy crystalline host,” Phys. Rev. B 66, 245101 (2002).
[CrossRef]

Shin, Y. B.

Shojiya, M.

M. Shojiya, Y. Kawamoto, and K. Kadono, “Judd–Ofelt parameters and multiphonon relaxation of Ho3+ ions in ZnCl2-based glass,” J. Appl. Phys. 89, 4944–4950 (2001).
[CrossRef]

Sigel, G. H.

Singh, A. K.

S. B. Rai, A. K. Singh, and S. K. Singh, “Spectroscopic properties of Ho3+ ions doped in tellurite glass,” Spectrochim. Acta, Part A 59, 3221–3226 (2003).
[CrossRef]

Singh, S. K.

S. B. Rai, A. K. Singh, and S. K. Singh, “Spectroscopic properties of Ho3+ ions doped in tellurite glass,” Spectrochim. Acta, Part A 59, 3221–3226 (2003).
[CrossRef]

Snitzer, E.

Takahashi, S.

Tarelho, L. V. G.

L. D. da Vila, L. Gomes, L. V. G. Tarelho, S. J. L. Ribeiro, and Y. Messaddeq, “Dynamics of Tm–Ho energy transfer and deactivation of the F43 low level of thulium in fluorozirconate glasses,” J. Appl. Phys. 95, 5451–5463 (2004).
[CrossRef]

L. V. G. Tarelho, L. Gomes, and I. M. Ranieri, “Determination of microscopic parameters for nonresonant energy-transfer processes in rare-earth-doped crystals,” Phys. Rev. B 56, 14344–14351 (1997).
[CrossRef]

Walsh, B. M.

B. M. Walsh, N. P. Barnes, D. J. Reichle, and S. B. Jiang, “Optical properties of Tm3+ ions in alkali germanate glass,” J. Non-Cryst. Solids 352, 5344–5352 (2006).
[CrossRef]

B. M. Walsh, N. P. Barnes, and B. Di Bartolo, “Branching ratios, cross sections, and radiative lifetimes of rare earth ions in solids: application to Tm3+ and Ho3+ ions in LiYF4,” J. Appl. Phys. 83, 2772–2787 (1998).
[CrossRef]

Watekar, P. R.

P. R. Watekar, S. Ju, and W. T. Han, “Optical properties of Ho-doped alumino-germano-silica glass optical fiber,” J. Non-Cryst. Solids 354, 1453–1459 (2008).
[CrossRef]

P. R. Watekar, S. M. Ju, S. J. Boo, and W. T. Han, “Linear and non-linear optical properties of Yb3+∕Tm3+ co-doped alumino-silicate glass prepared by sol-gel method,” J. Non-Cryst. Solids 351, 2446–2452 (2005).
[CrossRef]

Wu, J. F.

Yang, G. F.

G. X. Chen, Q. Y. Zhang, G. F. Yang, and Z. H. Jiang, “Mid-infrared emission characteristic and energy transfer of Ho3+-doped tellurite glass sensitized by Tm3+,” J. Fluoresc. 17, 301–307 (2007).
[CrossRef] [PubMed]

Yao, Z.

Yoshikawa, A.

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Figures (3)

Fig. 1
Fig. 1

Absorption spectrum of LAGTm5Ho. The inset shows the absorption and emission cross section of Tm 3 + and Ho 3 + .

Fig. 2
Fig. 2

Emission spectra of LAGTm, LAGTm2.5Ho, LAGTm5Ho, and LAGTm10Ho under the excitation of an 808 nm laser diode.

Fig. 3
Fig. 3

Energy transfer processes in Tm 3 + Ho 3 + codoped LAG glass.

Tables (4)

Tables Icon

Table 1 Measured and Calculated Values for the Line Strength in LAGTm5Ho

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Table 2 Judd–Ofelt Parameters of Ho 3 + in Various Glasses

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Table 3 Calculated Branching Ratios and Transition Probabilities of Ho 3 + in LAG Glass

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Table 4 Energy Transfer Coefficient and Critical Radius of Each Energy Transfer Process

Equations (16)

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F exp = 2.303 m c 2 π e 2 N d λ 2 OD ( λ ) d λ ,
F theor = 8 π 2 m υ 3 h ( 2 J + 1 ) [ ( n 2 + 2 ) 2 9 n S ED + n 3 S MD ] ,
S ED = λ = 2 , 4 , 6 Ω λ | S , L , J U ( λ ) S , L , J | 2 ,
S MD = ( 2 m c ) 2 | S , L , J L + 2 S S , L , J | 2 ,
| S , L , J L + 2 S S , L , J 1 | 2 = [ ( S + L + 1 ) 2 J 2 ] [ J 2 ( L S ) 2 4 J ] ,
| S , L , J L + 2 S S , L , J + 1 | 2 = [ ( S + L + 1 ) 2 ( J + 1 ) 2 ] [ ( J + 1 ) 2 ( L S ) 2 4 ( J + 1 ) ] .
A rad ( S , L , J | S , L , J ) = 64 π 4 e 2 3 h ( 2 J + 1 ) λ 3 [ n ( n 2 + 1 ) 2 9 S ED + n 3 S MD ] .
τ i = 1 A ( S , L , J | S , L , J ) ,
β S , L , J | S , L , J = A ( S , L , J | S , L , J ) A ( S , L , J | S , L , J ) .
σ abs = 2.303 OD ( λ ) N d .
σ em = λ 4 A rad 8 π c n 2 × λ I ( λ ) λ I ( λ ) d λ .
W D - A m = ( 2 π ) | H DA | 2 S DA m ,
S D - A ( m , E ) = g emis ( m - phonon ) D ( E ) g abs A ( E ) d E = S 0 m m ! e S 0 S D - A ( 0 , E ) = [ S 0 m m ! e S 0 g emis D ( E Δ E ) ] g abs A ( E ) d E ,
W D - A ( R ) = 6 c g low D ( 2 π ) 4 n 2 R 6 g up D m = 0 e ( 2 n ¯ + 1 ) S 0 S 0 m m ! ( n ¯ + 1 ) m σ emis D ( λ m + ) σ abs A ( λ ) d λ = C D - A R 6 ,
C D - A = 6 c g low D ( 2 π ) 4 n 2 g up D m = 0 e ( 2 n ¯ + 1 ) S 0 S 0 m m ! ( n ¯ + 1 ) m σ emis D ( λ m + ) σ abs A ( λ ) d λ .
R C 6 = C D - A τ D .

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