Abstract

In the present study, the multiphoton near-infrared downconversion quantum cutting luminescence phenomena of Tm3+ ion in (Y1-xTmx)3Al5O12 powder phosphor, which is currently a hot research topic throughout the world, is reported. The x-ray diffraction spectra, the visible to near-infrared excitation and emission spectra, and fluorescence lifetimes are measured. It is found that Tm:YAG powder phosphor has intense two-photon quantum cutting luminescence, and, for the first time, it is found that Tm:YAG powder phosphor has strong four-photon near-infrared quantum cutting luminescence of 1788 nm 3F4→3H6 fluorescence of Tm3+ ion. It is also found that the theoretical up-limit of four-photon near-infrared quantum cutting efficiency is about 282.12%, which results from both the {1D2→3F2, 3H6→3H4} and {3H4→3F4, 3H6→3F4} cross-energy transfers.

©2013 Optical Society of America

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    [Crossref]
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    [Crossref] [PubMed]
  4. B. Bitnar, “Silicon, germanium silicon/germanium photocells for thermo photovoltaics applications,” Semicond. Sci. Technol. 18(5), S221–S227 (2003).
    [Crossref]
  5. X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
    [Crossref] [PubMed]
  6. D. Q. Chen, Y. S. Wang, Y. L. Yu, P. Huang, and F. Y. Weng, “Near-infrared quantum cutting in transparent nanostructured glass ceramics,” Opt. Lett. 33(16), 1884–1886 (2008).
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    [Crossref] [PubMed]
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  21. D. L. Dexter, “Possibility of luminescent quantum yields greater than unity,” Phys. Rev. 108(3), 630–633 (1957).
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  22. T. Trupke, M. A. Green, and P. Wurfel, “Improving solar cell efficiencies by down-conversion of high-energy photons,” J. Appl. Phys. 92(3), 1668–1674 (2002).
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2013 (2)

2012 (1)

X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
[Crossref] [PubMed]

2010 (3)

Q. Y. Zhang and X. Y. Huang, “Recent progress in quantum cutting phosphors,” Prog. Mater. Sci. 55(5), 353–427 (2010).
[Crossref]

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

J. J. Zhou, Y. Teng, X. F. Liu, S. Ye, X. Q. Xu, Z. J. Ma, and J. R. Qiu, “Intense infrared emission of Er3+ in Ca8Mg(SiO4)4Cl2 phosphor from energy transfer of Eu2+ by broadband down-conversion,” Opt. Express 18(21), 21663–21668 (2010).
[Crossref] [PubMed]

2009 (4)

X. B. Chen, J. G. Wu, X. L. Xu, Y. Z. Zhang, N. Sawanobori, C. L. Zhang, Q. H. Pan, and G. J. Salamo, “Three-photon infrared quantum cutting from single species of rare-earth Er3+ ions in Er0.3Gd0.7VO4 crystalline,” Opt. Lett. 34(7), 887–889 (2009).
[Crossref] [PubMed]

B. M. van der Ende, L. Aarts, and A. Meijerink, “Near-infrared quantum cutting for photovoltaics,” Adv. Mater. 21(30), 3073–3077 (2009).
[Crossref]

S. V. Eliseeva and J. C. G. Bünzli, “Lanthanide luminescence for functional materials and bio-sciences,” Chem. Soc. Rev. 39(1), 189–227 (2009).
[Crossref] [PubMed]

B. M. van der Ende, L. Aarts, and A. Meijerink, “Lanthanide ions as spectral converters for solar cells,” Phys. Chem. Chem. Phys. 11(47), 11081–11095 (2009).
[Crossref] [PubMed]

2008 (2)

J. X. Chen, J. R. Qiu, S. Ye, and X. Wang, “Cooperative quantum cutting of nano-crystalline BaF2: Tb3+, Yb3+ in oxyfluoride glass ceramics,” Chin. Phys. Lett. 25(6), 2078–2080 (2008).
[Crossref]

D. Q. Chen, Y. S. Wang, Y. L. Yu, P. Huang, and F. Y. Weng, “Near-infrared quantum cutting in transparent nanostructured glass ceramics,” Opt. Lett. 33(16), 1884–1886 (2008).
[Crossref] [PubMed]

2006 (1)

B. S. Richards, “Luminescent layers for enhanced silicon solar cell performance: Down-conversion,” Sol. Energy Mater. Sol. Cells 90(9), 1189–1207 (2006).
[Crossref]

2005 (1)

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

2003 (1)

B. Bitnar, “Silicon, germanium silicon/germanium photocells for thermo photovoltaics applications,” Semicond. Sci. Technol. 18(5), S221–S227 (2003).
[Crossref]

2002 (1)

T. Trupke, M. A. Green, and P. Wurfel, “Improving solar cell efficiencies by down-conversion of high-energy photons,” J. Appl. Phys. 92(3), 1668–1674 (2002).
[Crossref]

1999 (1)

R. T. Wegh, H. Donker, K. D. Oskam, and A. Meijerink, “Visible quantum cutting in LiGdF4 : Eu3+ through downconversion,” Science 283(5402), 663–666 (1999).
[Crossref] [PubMed]

1957 (1)

D. L. Dexter, “Possibility of luminescent quantum yields greater than unity,” Phys. Rev. 108(3), 630–633 (1957).
[Crossref]

1948 (1)

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenz,” Ann. Phys. 437(1–2), 55–75 (1948).
[Crossref]

Aarts, L.

B. M. van der Ende, L. Aarts, and A. Meijerink, “Near-infrared quantum cutting for photovoltaics,” Adv. Mater. 21(30), 3073–3077 (2009).
[Crossref]

B. M. van der Ende, L. Aarts, and A. Meijerink, “Lanthanide ions as spectral converters for solar cells,” Phys. Chem. Chem. Phys. 11(47), 11081–11095 (2009).
[Crossref] [PubMed]

Biner, D.

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

Bitnar, B.

B. Bitnar, “Silicon, germanium silicon/germanium photocells for thermo photovoltaics applications,” Semicond. Sci. Technol. 18(5), S221–S227 (2003).
[Crossref]

Bünzli, J. C. G.

S. V. Eliseeva and J. C. G. Bünzli, “Lanthanide luminescence for functional materials and bio-sciences,” Chem. Soc. Rev. 39(1), 189–227 (2009).
[Crossref] [PubMed]

Chen, D. Q.

Chen, J. X.

J. X. Chen, J. R. Qiu, S. Ye, and X. Wang, “Cooperative quantum cutting of nano-crystalline BaF2: Tb3+, Yb3+ in oxyfluoride glass ceramics,” Chin. Phys. Lett. 25(6), 2078–2080 (2008).
[Crossref]

Chen, Q. Q.

Chen, X. B.

Chiu, C. H.

Chou, C. L.

Chou, W. C.

Chu, Y.

Dai, N.

den Hertog, M. I.

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

Dexter, D. L.

D. L. Dexter, “Possibility of luminescent quantum yields greater than unity,” Phys. Rev. 108(3), 630–633 (1957).
[Crossref]

Donker, H.

R. T. Wegh, H. Donker, K. D. Oskam, and A. Meijerink, “Visible quantum cutting in LiGdF4 : Eu3+ through downconversion,” Science 283(5402), 663–666 (1999).
[Crossref] [PubMed]

Eilers, J. J.

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

Eliseeva, S. V.

S. V. Eliseeva and J. C. G. Bünzli, “Lanthanide luminescence for functional materials and bio-sciences,” Chem. Soc. Rev. 39(1), 189–227 (2009).
[Crossref] [PubMed]

Förster, T.

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenz,” Ann. Phys. 437(1–2), 55–75 (1948).
[Crossref]

Green, M. A.

T. Trupke, M. A. Green, and P. Wurfel, “Improving solar cell efficiencies by down-conversion of high-energy photons,” J. Appl. Phys. 92(3), 1668–1674 (2002).
[Crossref]

Gudel, H. U.

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

Han, S. Y.

X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
[Crossref] [PubMed]

Huang, P.

Huang, W.

X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
[Crossref] [PubMed]

Huang, X. Y.

X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
[Crossref] [PubMed]

Q. Y. Zhang and X. Y. Huang, “Recent progress in quantum cutting phosphors,” Prog. Mater. Sci. 55(5), 353–427 (2010).
[Crossref]

Jian, H. T.

Kox, M. H. F.

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

Kramer, K.

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

Kuo, H. C.

Li, J. Y.

Lin, J. Y.

Liu, X. F.

Liu, X. G.

X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
[Crossref] [PubMed]

Liu, Z. J.

Ma, Z. J.

Meijerink, A.

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

B. M. van der Ende, L. Aarts, and A. Meijerink, “Lanthanide ions as spectral converters for solar cells,” Phys. Chem. Chem. Phys. 11(47), 11081–11095 (2009).
[Crossref] [PubMed]

B. M. van der Ende, L. Aarts, and A. Meijerink, “Near-infrared quantum cutting for photovoltaics,” Adv. Mater. 21(30), 3073–3077 (2009).
[Crossref]

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

R. T. Wegh, H. Donker, K. D. Oskam, and A. Meijerink, “Visible quantum cutting in LiGdF4 : Eu3+ through downconversion,” Science 283(5402), 663–666 (1999).
[Crossref] [PubMed]

Oskam, K. D.

R. T. Wegh, H. Donker, K. D. Oskam, and A. Meijerink, “Visible quantum cutting in LiGdF4 : Eu3+ through downconversion,” Science 283(5402), 663–666 (1999).
[Crossref] [PubMed]

Pan, Q. H.

Qiu, J. R.

J. J. Zhou, Y. Teng, X. F. Liu, S. Ye, X. Q. Xu, Z. J. Ma, and J. R. Qiu, “Intense infrared emission of Er3+ in Ca8Mg(SiO4)4Cl2 phosphor from energy transfer of Eu2+ by broadband down-conversion,” Opt. Express 18(21), 21663–21668 (2010).
[Crossref] [PubMed]

J. X. Chen, J. R. Qiu, S. Ye, and X. Wang, “Cooperative quantum cutting of nano-crystalline BaF2: Tb3+, Yb3+ in oxyfluoride glass ceramics,” Chin. Phys. Lett. 25(6), 2078–2080 (2008).
[Crossref]

Richards, B. S.

B. S. Richards, “Luminescent layers for enhanced silicon solar cell performance: Down-conversion,” Sol. Energy Mater. Sol. Cells 90(9), 1189–1207 (2006).
[Crossref]

Salamo, G. J.

Sawanobori, N.

Shen, J. L.

Shu, G. W.

Teng, Y.

Trupke, T.

T. Trupke, M. A. Green, and P. Wurfel, “Improving solar cell efficiencies by down-conversion of high-energy photons,” J. Appl. Phys. 92(3), 1668–1674 (2002).
[Crossref]

van der Eerden, J. P. J. M.

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

van der Ende, B. M.

B. M. van der Ende, L. Aarts, and A. Meijerink, “Near-infrared quantum cutting for photovoltaics,” Adv. Mater. 21(30), 3073–3077 (2009).
[Crossref]

B. M. van der Ende, L. Aarts, and A. Meijerink, “Lanthanide ions as spectral converters for solar cells,” Phys. Chem. Chem. Phys. 11(47), 11081–11095 (2009).
[Crossref] [PubMed]

van Wijngaarden, J. T.

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

Vergeer, P.

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

Vlugt, T. J. H.

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

Wang, S. C.

Wang, X.

J. X. Chen, J. R. Qiu, S. Ye, and X. Wang, “Cooperative quantum cutting of nano-crystalline BaF2: Tb3+, Yb3+ in oxyfluoride glass ceramics,” Chin. Phys. Lett. 25(6), 2078–2080 (2008).
[Crossref]

Wang, Y. S.

Wegh, R. T.

R. T. Wegh, H. Donker, K. D. Oskam, and A. Meijerink, “Visible quantum cutting in LiGdF4 : Eu3+ through downconversion,” Science 283(5402), 663–666 (1999).
[Crossref] [PubMed]

Weng, F. Y.

Wu, C. H.

Wu, J. G.

Wurfel, P.

T. Trupke, M. A. Green, and P. Wurfel, “Improving solar cell efficiencies by down-conversion of high-energy photons,” J. Appl. Phys. 92(3), 1668–1674 (2002).
[Crossref]

Xu, X. L.

Xu, X. Q.

Yang, L. Y.

Ye, S.

J. J. Zhou, Y. Teng, X. F. Liu, S. Ye, X. Q. Xu, Z. J. Ma, and J. R. Qiu, “Intense infrared emission of Er3+ in Ca8Mg(SiO4)4Cl2 phosphor from energy transfer of Eu2+ by broadband down-conversion,” Opt. Express 18(21), 21663–21668 (2010).
[Crossref] [PubMed]

J. X. Chen, J. R. Qiu, S. Ye, and X. Wang, “Cooperative quantum cutting of nano-crystalline BaF2: Tb3+, Yb3+ in oxyfluoride glass ceramics,” Chin. Phys. Lett. 25(6), 2078–2080 (2008).
[Crossref]

Yu, Y. L.

Zhang, C. L.

Zhang, Q. Y.

Q. Y. Zhang and X. Y. Huang, “Recent progress in quantum cutting phosphors,” Prog. Mater. Sci. 55(5), 353–427 (2010).
[Crossref]

Zhang, Y. Z.

Zhou, J. J.

Adv. Mater. (1)

B. M. van der Ende, L. Aarts, and A. Meijerink, “Near-infrared quantum cutting for photovoltaics,” Adv. Mater. 21(30), 3073–3077 (2009).
[Crossref]

Ann. Phys. (1)

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenz,” Ann. Phys. 437(1–2), 55–75 (1948).
[Crossref]

Appl. Phys. Lett. (1)

J. J. Eilers, D. Biner, J. T. van Wijngaarden, K. Kramer, H. U. Gudel, and A. Meijerink, “Efficient visible to infrared quantum cutting through downconversion with the Er3+-Yb3+ couple in Cs3Y2Br9,” Appl. Phys. Lett. 96(15), 151106 (2010).
[Crossref]

Chem. Soc. Rev. (2)

S. V. Eliseeva and J. C. G. Bünzli, “Lanthanide luminescence for functional materials and bio-sciences,” Chem. Soc. Rev. 39(1), 189–227 (2009).
[Crossref] [PubMed]

X. Y. Huang, S. Y. Han, W. Huang, and X. G. Liu, “Enhancing solar cell efficiency: the search for luminescent materials as spectral converters,” Chem. Soc. Rev. 42(1), 173–201 (2012).
[Crossref] [PubMed]

Chin. Phys. Lett. (1)

J. X. Chen, J. R. Qiu, S. Ye, and X. Wang, “Cooperative quantum cutting of nano-crystalline BaF2: Tb3+, Yb3+ in oxyfluoride glass ceramics,” Chin. Phys. Lett. 25(6), 2078–2080 (2008).
[Crossref]

J. Appl. Phys. (1)

T. Trupke, M. A. Green, and P. Wurfel, “Improving solar cell efficiencies by down-conversion of high-energy photons,” J. Appl. Phys. 92(3), 1668–1674 (2002).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Phys. Chem. Chem. Phys. (1)

B. M. van der Ende, L. Aarts, and A. Meijerink, “Lanthanide ions as spectral converters for solar cells,” Phys. Chem. Chem. Phys. 11(47), 11081–11095 (2009).
[Crossref] [PubMed]

Phys. Rev. (1)

D. L. Dexter, “Possibility of luminescent quantum yields greater than unity,” Phys. Rev. 108(3), 630–633 (1957).
[Crossref]

Phys. Rev. B (1)

P. Vergeer, T. J. H. Vlugt, M. H. F. Kox, M. I. den Hertog, J. P. J. M. van der Eerden, and A. Meijerink, “Quantum cutting by cooperative energy transfer in YbxY1-xPO4: Tb3+,” Phys. Rev. B 71(1), 014119 (2005).
[Crossref]

Prog. Mater. Sci. (1)

Q. Y. Zhang and X. Y. Huang, “Recent progress in quantum cutting phosphors,” Prog. Mater. Sci. 55(5), 353–427 (2010).
[Crossref]

Science (1)

R. T. Wegh, H. Donker, K. D. Oskam, and A. Meijerink, “Visible quantum cutting in LiGdF4 : Eu3+ through downconversion,” Science 283(5402), 663–666 (1999).
[Crossref] [PubMed]

Semicond. Sci. Technol. (1)

B. Bitnar, “Silicon, germanium silicon/germanium photocells for thermo photovoltaics applications,” Semicond. Sci. Technol. 18(5), S221–S227 (2003).
[Crossref]

Sol. Energy Mater. Sol. Cells (1)

B. S. Richards, “Luminescent layers for enhanced silicon solar cell performance: Down-conversion,” Sol. Energy Mater. Sol. Cells 90(9), 1189–1207 (2006).
[Crossref]

Other (3)

R. Reisfeld, Lasers and Excited States of Rare-Earth (Springer-Verlag, 1977).

G. X. Xu, Rare Earth (Metallurgical Industry, 1995) (in Chinese).

M. A. Green, Third Generation Photovoltaics: Advanced Solar Energy Conversion (Springer-Verlag, 2003).

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Figures (8)

Fig. 1
Fig. 1 XRD pattern of the (Y0.800Tm0.200)3Al5O12 powder phosphor sample.

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Fig. 2
Fig. 2 The absorption spectrum of the (Y0.800Tm0.200)3Al5O12 powder phosphor sample.

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Fig. 3
Fig. 3 The visible excitation spectra of (A) (Y0.800Tm0.200)3Al5O12 and (B) (Y0.995Tm0.005)3Al5O12 powder phosphor, when the fluorescence received wavelength is positioned at 800 nm.

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Fig. 4
Fig. 4 The excitation spectra of (A) (Y0.800Tm0.200)3Al5O12 and (B) (Y0.995Tm0.005)3Al5O12 powder phosphor, when the fluorescence received wavelength is positioned at 1788 nm near-infrared wavelength.

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Fig. 5
Fig. 5 The luminescence spectra of (A) (Y0.800Tm0.200)3Al5O12 and (B) (Y0.995Tm0.005)3Al5O12 powder phosphor, when the 357.0 nm 3H6→1D2 excitation peak is selected as the excitation wavelength.

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Fig. 6
Fig. 6 The luminescence spectra of (A) (Y0.800Tm0.200)3Al5O12 and (B) (Y0.995Tm0.005)3Al5O12 powder phosphor, when the 680.0 nm 3H6→3F3 excitation peak is selected as the excitation wavelength.

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Fig. 7
Fig. 7 The fluorescence lifetime of the 800.0 nm (left) and 460.0 nm (right) visible fluorescence of (A) (Y0.800Tm0.200)3Al5O12 (red) and (B) (Y0.995Tm0.005)3Al5O12 (blue) powder phosphor, when excited by 680.0 nm (left) and 368.0 nm (right) pulsed light respectively.

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Fig. 8
Fig. 8 The schematic diagram of energy level structure and quantum cutting process.

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Equations (5)

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η tr,x%Tm ≈1− ∫ I x%Tm dt ∫ I 0.5%Tm dt .
η CR,x%Tm ( 3 H 4 )= η 3 H 4 ·[1− η tr,x%Tm ( 3 H 4 )]+2 η 3 F 4 η tr,x%Tm ( 3 H 4 ),
η CR,x%Tm ( 3 H 4 )=1+ η tr,x%Tm ( 3 H 4 ).
η CR,x%Tm ( 1 D 2 )={ η 1 D 2 ·[1− η tr,x%Tm ( 1 D 2 )]+2 η 3 H 4 η tr,x%Tm ( 1 D 2 )} ·{[ η 3 H 4 ·[1− η tr,x%Tm ( 3 H 4 )]+2 η 3 F 4 η tr,x%Tm ( 3 H 4 )},
η CR,x%Tm ( 1 D 2 )=[1+ η tr,x%Tm ( 1 D 2 )]·[1+ η tr,x%Tm ( 3 H 4 )].

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