Abstract

In this study, we take the pump rate into consideration for the first time to give a theoretical description of radiation trapping in three-level systems. We numerically verify that under strong pumping, the population of the ground state is depleted, which leads to saturation of the radiation trapping within the pumped region. This saturation inevitably clamps the lifetime lengthening that is experimentally verified on a 0.05 at% thin ruby crystal based on the axial pinhole method. Our model is confirmed to be valid in lifetime measurement when the ruby fluorescence is collected from both the pumped and the unpumped regions.

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  1. E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc.1(1), 40–51 (1926).
    [CrossRef]
  2. T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev.72(12), 1212–1233 (1947).
    [CrossRef]
  3. S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B73(14), 144101 (2006).
    [CrossRef]
  4. H. Kühn, S. T. Fredrich-Thornton, C. Kränkel, R. Peters, and K. Petermann, “Model for the calculation of radiation trapping and description of the pinhole method,” Opt. Lett.32(13), 1908–1910 (2007).
    [CrossRef] [PubMed]
  5. G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B106(1), 63–71 (2012).
    [CrossRef]
  6. T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett.4(11), 564–566 (1960).
    [CrossRef]
  7. W. A. Shurcliff and R. C. Jones, “The trapping of fluorescence light produced within objects of high geometrical symmetry,” J. Opt. Soc. Am.39(11), 912–916 (1949).
    [CrossRef]
  8. N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999).
    [CrossRef]
  9. M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
    [CrossRef]
  10. Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993).
    [CrossRef] [PubMed]
  11. W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
    [CrossRef]

2012

G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B106(1), 63–71 (2012).
[CrossRef]

2007

2006

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B73(14), 144101 (2006).
[CrossRef]

1999

N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999).
[CrossRef]

1993

Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993).
[CrossRef] [PubMed]

1990

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
[CrossRef]

1960

T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett.4(11), 564–566 (1960).
[CrossRef]

1949

1947

T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev.72(12), 1212–1233 (1947).
[CrossRef]

1926

E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc.1(1), 40–51 (1926).
[CrossRef]

Barnes, N. P.

N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999).
[CrossRef]

Ding, L.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Fields, R. A.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
[CrossRef]

Fincher, C. L.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
[CrossRef]

Fredrich-Thornton, S. T.

Grattan, K. T. V.

Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993).
[CrossRef] [PubMed]

Guy, S.

S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B73(14), 144101 (2006).
[CrossRef]

Holstein, T.

T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev.72(12), 1212–1233 (1947).
[CrossRef]

Innocenzi, M. E.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
[CrossRef]

Jones, R. C.

Kränkel, C.

Kühn, H.

Li, W.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Lu, W.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Maiman, T. H.

T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett.4(11), 564–566 (1960).
[CrossRef]

Milne, E. A.

E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc.1(1), 40–51 (1926).
[CrossRef]

Palmer, A. W.

Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993).
[CrossRef] [PubMed]

Pan, H.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Petermann, K.

Peters, R.

Shurcliff, W. A.

Su, L.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Toci, G.

G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B106(1), 63–71 (2012).
[CrossRef]

Walsh, B. M.

N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999).
[CrossRef]

Xu, J.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Yan, C.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Yura, H. T.

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
[CrossRef]

Zeng, H.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Zhang, Z.

Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993).
[CrossRef] [PubMed]

Zhao, G.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

Appl. Phys. B

G. Toci, “Lifetime measurements with the pinhole method in presence of radiation trapping: I-theoretical model,” Appl. Phys. B106(1), 63–71 (2012).
[CrossRef]

Appl. Phys. Lett.

W. Li, H. Pan, L. Ding, H. Zeng, W. Lu, G. Zhao, C. Yan, L. Su, and J. Xu, “Efficient diode-pumped Yb:Gd2SiO5 laser,” Appl. Phys. Lett.88(22), 221117 (2006).
[CrossRef]

M. E. Innocenzi, H. T. Yura, C. L. Fincher, and R. A. Fields, “Thermal modeling of continuous-wave end-pumped solid-state lasers,” Appl. Phys. Lett.56(19), 1831–1833 (1990).
[CrossRef]

IEEE J. Quantum Electron.

N. P. Barnes and B. M. Walsh, “Amplified spontaneous emission-application to Nd:YAG lasers,” IEEE J. Quantum Electron.35(1), 101–109 (1999).
[CrossRef]

J. Lond. Math. Soc.

E. A. Milne, “The diffusion of imprisoned radiation through a gas,” J. Lond. Math. Soc.1(1), 40–51 (1926).
[CrossRef]

J. Opt. Soc. Am.

Opt. Lett.

Phys. Rev.

T. Holstein, “Imprisonment of resonance radiation in gases,” Phys. Rev.72(12), 1212–1233 (1947).
[CrossRef]

Phys. Rev. B

S. Guy, “Modelization of lifetime measurement in the presence of radiation trapping in solid-state materials,” Phys. Rev. B73(14), 144101 (2006).
[CrossRef]

Phys. Rev. B Condens. Matter

Z. Zhang, K. T. V. Grattan, and A. W. Palmer, “Temperature dependences of fluorescence lifetimes in Cr3+-doped insulating crystals,” Phys. Rev. B Condens. Matter48(11), 7772–7778 (1993).
[CrossRef] [PubMed]

Phys. Rev. Lett.

T. H. Maiman, “Optical and microwave-optical experiments in ruby,” Phys. Rev. Lett.4(11), 564–566 (1960).
[CrossRef]

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Figures (8)

Fig. 1
Fig. 1

(left) The energy diagram of the theoretical three-level system. Level 1 is the ground state. (right) The geometry of our pinhole method. D is the pumped region and U is the unpumped region.

Fig. 2
Fig. 2

(a) The time evolutions of the population density of ND2, NU2, and ND1 within one period of 50 ms at Pth. The green and pink curves are the fitted results using single and double exponential decay, respectively. (b) The evolution of the dominant reabsorption factor for pumps of 0.1Pth, Pth, 10Pth, and 50Pth.

Fig. 3
Fig. 3

(a) The population decay time τ' versus the normalized incident pump for three parameters a . (b) τ' vs. a for the high pump of 10Pth (squares) and low pump of 0.1Pth (triangles).

Fig. 4
Fig. 4

The composite population decay time versus the collected light ratio for two pumps with a = 1.0 mm.

Fig. 5
Fig. 5

The experimental apparatus of our pinhole method.

Fig. 6
Fig. 6

The experimental measured lifetime as a function of the incident pump power. The label on the top x axis is the ratio of the absorbed power to the saturation power. The saturation power in experiment is 8 mW. The error bar is for the maximal and minimal data.

Fig. 7
Fig. 7

The experimental measured lifetime (solid) and the collected power (open) versus the transverse shift. The distance between the crystal and the pinhole are 0.5 mm (black) and 5 mm (red).

Fig. 8
Fig. 8

The experimental measured lifetime versus the distance between the crystal and the pinhole. The pinhole is on the pump axis.

Equations (11)

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n ( r , t ) t = n ( r , t ) τ 21 + W r n ( r ' , t ) G ( r ' , r ) d 3 r ' ,
G ( r , r ' ) = 1 4 π | r - r ' | 2 | r - r ' | exp [ ( r - r ' ) σ N D 1 ] .
d N D d t = N D τ 21 + W r N D ( f V D V s ) × 1 + W r N U ( f V U V s ) × 1 ,
d N U d t = N U τ 21 + W r N U ( f V U V s ) × 1 + W r N D ( f V D V s ) × 1 ,
d N D d t = N D τ 21 + N D τ 21 σ N D 1 a σ N D 1 a + 1 ,
d N D 3 d t = R p N D 3 τ 31 N D 3 τ 32 ,
d N D 2 d t = N D 3 τ 32 N D 2 τ 21 + N D 2 τ 21 σ N D 1 a σ N D 1 a + 1 + N U 2 τ 21 σ N D 1 b σ N D 1 b + 1 ,
d N D 1 d t = R p + N D 3 τ 31 + N D 2 τ 21 N D 2 τ 21 σ N D 1 a σ N D 1 a + 1 N U 2 τ 21 σ N D 1 b σ N D 1 b + 1 ,
d N U 2 d t = N U 2 τ 21 + N U 2 τ 21 ( σ N U 1 c 1 σ N U 1 c 1 + 1 + σ N U 1 c 2 σ N U 1 c 2 + 1 ) + N D 2 τ 21 σ N U 1 d σ N U 1 d + 1 ,
d N U 1 d t = N U 2 τ 21 N U 2 τ 21 ( σ N U 1 c 1 σ N U 1 c 1 + 1 + σ N U 1 c 2 σ N U 1 c 2 + 1 ) N D 2 τ 21 σ N U 1 d σ N U 1 d + 1 ,
ΔT=(α P ph e α /2π K c ) r r b (1 e 2 r 2 / w p 2 )dr/r  ,

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