Modeling population and thermal lenses in the presence of Auger Upconversion for Nd<sup>3+</sup> doped materials

E. C. Ximendes; W. F. Silva; M. V. D. Vermelho; N. G. C. Astrat; L. C. Malacarne; M. L. Baesso; T. Catunda; C. Jacinto

doi:10.1364/OE.23.015983

Modeling population and thermal lenses in the presence of Auger Upconversion for Nd³⁺ doped materials

E. C. Ximendes, W. F. Silva, M. V. D. Vermelho, N. G. C. Astrat, L. C. Malacarne, M. L. Baesso, T. Catunda, and C. Jacinto

Optics Express
Vol. 23,
Issue 12,
pp. 15983-15991
(2015)
•https://doi.org/10.1364/OE.23.015983

Get Citation
Copy Citation Text
E. C. Ximendes, W. F. Silva, M. V. D. Vermelho, N. G. C. Astrat, L. C. Malacarne, M. L. Baesso, T. Catunda, and C. Jacinto, "Modeling population and thermal lenses in the presence of Auger Upconversion for Nd³⁺ doped materials," Opt. Express 23, 15983-15991 (2015)

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (1336 KB)
Set citation alerts
Save article

Related Topics
Table of Contents Category
- Materials
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Laser materials (140.3380)
- Thermal effects (140.6810)
- Optical properties (160.4760)
- Nonlinear optics, materials (190.4400)

About this Article
History
- Original Manuscript: February 4, 2015
- Revised Manuscript: May 23, 2015
- Manuscript Accepted: May 28, 2015
- Published: June 9, 2015

Accessible

Open Access

Abstract

We present a theoretical model to thermal (TL) and population (PL) lenses effects in the presence of Auger upconversion (AU) for analysis of Nd³⁺ doped materials. The model distinguishes and quantifies the contributions from TL and PL. From the experimental and theoretical results, the AU cannot be neglected because it plays an important role on the excited state population and therefore on the temperature and polarizability difference between excited and ground states. Considering the extensive use of these techniques, the model presented here could be useful for the investigation of materials and also to avoid misleading analysis of lenses transients.

Full Article | PDF Article

OSA Recommended Articles

Fluorescence quantum efficiency measurements in the presence of Auger upconversion by the thermal lens method

Viviane Pilla, Tomaz Catunda, Hans P. Jenssen, and Arlete Cassanho
Opt. Lett. 28(4) 239-241 (2003)

Thermal lens and Auger upconversion losses' effect on the efficiency of Nd³⁺-doped lead lanthanum zirconate titanate transparent ceramics

Andrea S. S. de Camargo, Carlos Jacinto, Tomaz Catunda, Luiz Antonio de O. Nunes, Ducinei Garcia, and José Antonio Eiras
J. Opt. Soc. Am. B 23(10) 2097-2106 (2006)

Energy transfer upconversion determination by thermal-lens and Z-scan techniques in Nd³⁺-doped laser materials

C. Jacinto, D. N. Messias, A. A. Andrade, and T. Catunda
J. Opt. Soc. Am. B 26(5) 1002-1007 (2009)

References

View by:
Article Order
|
Year
|
Author
|
Publication

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: thermal and optical properties of laser glasses - a review,” J. Non-Cryst. Solids 352(32–35), 3582–3597 (2006).
[Crossref]
J. Shen, R. D. Lowe, and R. D. Snook, “A Model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165(2–3), 385–396 (1992).
[Crossref]
S. M. Lima and T. Catunda, “Discrimination of resonant and nonresonant contributions to the nonlinear refraction spectroscopy of ion-doped solids,” Phys. Rev. Lett. 99(24), 243902 (2007).
[Crossref] [PubMed]
D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]
O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorobev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39(7), 910–918 (2003).
[Crossref]
A. Rodenas, C. Jacinto, L. R. Freitas, D. Jaque, and T. Catunda, “Nonlinear refraction and absorption through phase transition in a Nd:SBN laser crystal,” Phys. Rev. B 79(3), 033108 (2009).
[Crossref]
R. C. Powell, Physics of Solid-State Laser Materials (Springer, 1998).
L. R. Freitas, C. Jacinto, A. Rodenas, D. Jaque, and T. Catunda, “Time-resolved study electronic and thermal contributions to the nonlinear refractive index of Nd3+:SBN laser crystals,” J. Lumin. 128(5–6), 1013–1015 (2008).
[Crossref]
A. A. Andrade, E. Tenorio, T. Catunda, M. L. Baesso, A. Cassanho, and H. P. Jenssen, “Discrimination between electronic and thermal contributions to the nonlinear refractive index of SrAlF5:Cr+3,” J. Opt. Soc. Am. B 16(3), 395–400 (1999).
[Crossref]
O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31(6), 763–765 (2006).
[Crossref] [PubMed]
J. R. Silva, L. C. Malacarne, M. L. Baesso, S. M. Lima, L. H. C. Andrade, C. Jacinto, M. P. Hehlen, and N. G. C. Astrath, “Modeling the population lens effect in thermal lens spectrometry,” Opt. Lett. 38(4), 422–424 (2013).
[PubMed]
C. Jacinto, T. Catunda, D. Jaque, and J. G. Sole, “Fluorescence quantum efficiency and Auger upconversion losses of the stoichiometric laser crystal NdAl3(BO3)4,” Phys. Rev. B 72(23), 235111 (2005).
[Crossref]
S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]
C. Jacinto, S. Oliveira, T. Catundab, A. Andrade, J. Myers, and M. Myers, “Upconversion effect on fluorescence quantum efficiency and heat generation in Nd3+-doped materials,” Opt. Express 13(6), 2040–2046 (2005).
[Crossref] [PubMed]
V. Ostroumov, T. Jensen, J. P. Meyn, G. Huber, and M. A. Noginov, “Study of luminescence concentration quenching and energy transfer upconversion in Nd-doped LaSc3(BO3)4 and GdVO4 laser crystals,” J. Opt. Soc. Am. B 15(3), 1052–1060 (1998).
[Crossref]
A. S. S. de Camargo, C. Jacinto, T. Catunda, A. O. Nunes, D. Garcia, and J. A. Eiras, “Thermal lens and Auger upconversion losses' effect on the efficiency of Nd3+-doped lead lanthanum zirconate titanate transparent ceramics,” J. Opt. Soc. Am. B 23(10), 2097–2106 (2006).
[Crossref]
V. Pilla, T. Catunda, H. P. Jenssen, and A. Cassanho, “Fluorescence quantum efficiency measurements in the presence of Auger upconversion by the thermal lens method,” Opt. Lett. 28(4), 239–241 (2003).
[Crossref] [PubMed]
C. Jacinto, D. N. Messias, A. A. Andrade, and T. Catunda, “Energy transfer upconversion determination by thermal-lens and Z-scan techniques in Nd3+-doped laser materials,” J. Opt. Soc. Am. B 26(5), 1002–1007 (2009).
T. P. Rodrigues, V. S. Zanuto, R. A. Cruz, T. Catunda, M. L. Baesso, N. G. C. Astrath, and L. C. Malacarne, “Discriminating the role of sample length in thermal lensing of solids,” Opt. Lett. 39(13), 4013–4016 (2014).
[PubMed]
J. M. Jewell and I. D. Aggarwal, “Thermal lensing in heavy-metal fluoride glasses,” J. Non-Crystal. Solids 142(1–3), 260–268 (1992).
L. G. Hwa, “Rayleigh-Brillouin scattering in calcium aluminosilicate glasses,” J. Raman Spectrosc. 29(4), 269–272 (1998).
[Crossref]
http://kigre.com/files/q98data.pdf , (In January 2015).
R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]
J. R. Silva, L. C. Malacarne, M. L. Baesso, S. M. Lima, L. H. C. Andrade, C. Jacinto, M. P. Hehlen, and N. G. C. Astrath, “Modeling the population lens effect in thermal lens spectrometry,” Opt. Lett. 38(4), 422–424 (2013).
[Crossref] [PubMed]

A. Rodenas, C. Jacinto, L. R. Freitas, D. Jaque, and T. Catunda, “Nonlinear refraction and absorption through phase transition in a Nd:SBN laser crystal,” Phys. Rev. B 79(3), 033108 (2009).
[Crossref]

2008 (1)

L. R. Freitas, C. Jacinto, A. Rodenas, D. Jaque, and T. Catunda, “Time-resolved study electronic and thermal contributions to the nonlinear refractive index of Nd3+:SBN laser crystals,” J. Lumin. 128(5–6), 1013–1015 (2008).
[Crossref]

2007 (2)

S. M. Lima and T. Catunda, “Discrimination of resonant and nonresonant contributions to the nonlinear refraction spectroscopy of ion-doped solids,” Phys. Rev. Lett. 99(24), 243902 (2007).
[Crossref] [PubMed]

D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]

2006 (3)

C. Jacinto, D. N. Messias, A. A. Andrade, S. M. Lima, M. L. Baesso, and T. Catunda, “Thermal lens and Z-scan measurements: thermal and optical properties of laser glasses - a review,” J. Non-Cryst. Solids 352(32–35), 3582–3597 (2006).
[Crossref]

O. L. Antipov, D. V. Bredikhin, O. N. Eremeykin, A. P. Savikin, E. V. Ivakin, and A. V. Sukhadolau, “Electronic mechanism for refractive-index changes in intensively pumped Yb:YAG laser crystals,” Opt. Lett. 31(6), 763–765 (2006).
[Crossref] [PubMed]

A. S. S. de Camargo, C. Jacinto, T. Catunda, A. O. Nunes, D. Garcia, and J. A. Eiras, “Thermal lens and Auger upconversion losses' effect on the efficiency of Nd3+-doped lead lanthanum zirconate titanate transparent ceramics,” J. Opt. Soc. Am. B 23(10), 2097–2106 (2006).
[Crossref]

2005 (2)

C. Jacinto, T. Catunda, D. Jaque, and J. G. Sole, “Fluorescence quantum efficiency and Auger upconversion losses of the stoichiometric laser crystal NdAl3(BO3)4,” Phys. Rev. B 72(23), 235111 (2005).
[Crossref]

C. Jacinto, S. Oliveira, T. Catundab, A. Andrade, J. Myers, and M. Myers, “Upconversion effect on fluorescence quantum efficiency and heat generation in Nd3+-doped materials,” Opt. Express 13(6), 2040–2046 (2005).
[Crossref] [PubMed]

2003 (2)

O. L. Antipov, O. N. Eremeykin, A. P. Savikin, V. A. Vorobev, D. V. Bredikhin, and M. S. Kuznetsov, “Electronic changes of refractive index in intensively pumped Nd:YAG laser crystals,” IEEE J. Quantum Electron. 39(7), 910–918 (2003).
[Crossref]

V. Pilla, T. Catunda, H. P. Jenssen, and A. Cassanho, “Fluorescence quantum efficiency measurements in the presence of Auger upconversion by the thermal lens method,” Opt. Lett. 28(4), 239–241 (2003).
[Crossref] [PubMed]

2002 (1)

S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]

1999 (1)

A. A. Andrade, E. Tenorio, T. Catunda, M. L. Baesso, A. Cassanho, and H. P. Jenssen, “Discrimination between electronic and thermal contributions to the nonlinear refractive index of SrAlF5:Cr+3,” J. Opt. Soc. Am. B 16(3), 395–400 (1999).
[Crossref]

1998 (2)

V. Ostroumov, T. Jensen, J. P. Meyn, G. Huber, and M. A. Noginov, “Study of luminescence concentration quenching and energy transfer upconversion in Nd-doped LaSc3(BO3)4 and GdVO4 laser crystals,” J. Opt. Soc. Am. B 15(3), 1052–1060 (1998).
[Crossref]

L. G. Hwa, “Rayleigh-Brillouin scattering in calcium aluminosilicate glasses,” J. Raman Spectrosc. 29(4), 269–272 (1998).
[Crossref]

1992 (2)

J. Shen, R. D. Lowe, and R. D. Snook, “A Model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165(2–3), 385–396 (1992).
[Crossref]

J. M. Jewell and I. D. Aggarwal, “Thermal lensing in heavy-metal fluoride glasses,” J. Non-Crystal. Solids 142(1–3), 260–268 (1992).

1989 (1)

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]

Aggarwal, I. D.

J. M. Jewell and I. D. Aggarwal, “Thermal lensing in heavy-metal fluoride glasses,” J. Non-Crystal. Solids 142(1–3), 260–268 (1992).

Andrade, A.

Andrade, A. A.

Andrade, L. H. C.

Antipov, O. L.

Astrath, N. G. C.

Baesso, M. L.

Bredikhin, D. V.

Cassanho, A.

Catunda, T.

D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]

S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]

Catundab, T.

Chase, L. L.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]

Cruz, R. A.

de Camargo, A. S. S.

Eiras, J. A.

Eremeykin, O. N.

Freitas, L. R.

Garcia, D.

Hehlen, M. P.

Huber, G.

Hwa, L. G.

L. G. Hwa, “Rayleigh-Brillouin scattering in calcium aluminosilicate glasses,” J. Raman Spectrosc. 29(4), 269–272 (1998).
[Crossref]

Ivakin, E. V.

Jacinto, C.

Jaque, D.

Jensen, T.

Jenssen, H. P.

Jewell, J. M.

J. M. Jewell and I. D. Aggarwal, “Thermal lensing in heavy-metal fluoride glasses,” J. Non-Crystal. Solids 142(1–3), 260–268 (1992).

Jiao, H.

S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]

Kuznetsov, M. S.

Lima, S. M.

S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]

Lowe, R. D.

J. Shen, R. D. Lowe, and R. D. Snook, “A Model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165(2–3), 385–396 (1992).
[Crossref]

Malacarne, L. C.

Messias, D. N.

D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]

Meyn, J. P.

Myers, J.

Myers, J. D.

D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]

Myers, M.

Myers, M. J.

D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]

Noginov, M. A.

Nunes, A. O.

Nunes, L. A. O.

S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]

Oliveira, S.

Ostroumov, V.

Payne, S. A.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]

Pilla, V.

Powell, R. C.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]

Rodenas, A.

Rodrigues, T. P.

Savikin, A. P.

Shen, J.

J. Shen, R. D. Lowe, and R. D. Snook, “A Model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165(2–3), 385–396 (1992).
[Crossref]

Silva, J. R.

Snook, R. D.

J. Shen, R. D. Lowe, and R. D. Snook, “A Model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165(2–3), 385–396 (1992).
[Crossref]

Sole, J. G.

Sukhadolau, A. V.

Tenorio, E.

Vorobev, V. A.

Wilke, G. D.

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]

Zanuto, V. S.

Chem. Phys. (1)

J. Shen, R. D. Lowe, and R. D. Snook, “A Model for cw laser-induced mode-mismatched dual-beam thermal lens spectrometry,” Chem. Phys. 165(2–3), 385–396 (1992).
[Crossref]

IEEE J. Quantum Electron. (1)

J. Lumin. (1)

J. Non-Cryst. Solids (1)

J. Non-Crystal. Solids (1)

J. M. Jewell and I. D. Aggarwal, “Thermal lensing in heavy-metal fluoride glasses,” J. Non-Crystal. Solids 142(1–3), 260–268 (1992).

J. Opt. Soc. Am. B (4)

J. Raman Spectrosc. (1)

L. G. Hwa, “Rayleigh-Brillouin scattering in calcium aluminosilicate glasses,” J. Raman Spectrosc. 29(4), 269–272 (1998).
[Crossref]

Opt. Express (1)

Opt. Lett. (8)

S. M. Lima, H. Jiao, L. A. O. Nunes, and T. Catunda, “Nonlinear refraction spectroscopy in resonance with laser lines in solids,” Opt. Lett. 27(10), 845–847 (2002).
[Crossref] [PubMed]

D. N. Messias, T. Catunda, J. D. Myers, and M. J. Myers, “Nonlinear electronic line shape determination in Yb3+-doped phosphate glass,” Opt. Lett. 32(6), 665–667 (2007).
[Crossref] [PubMed]

R. C. Powell, S. A. Payne, L. L. Chase, and G. D. Wilke, “Index-of-refraction change in optically pumped solid-state laser materials,” Opt. Lett. 14(21), 1204–1206 (1989).
[Crossref] [PubMed]

Phys. Rev. B (2)

Phys. Rev. Lett. (1)

Other (2)

R. C. Powell, Physics of Solid-State Laser Materials (Springer, 1998).

http://kigre.com/files/q98data.pdf , (In January 2015).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

Fig. 1 (a) Radial dependence of the laser induced temperature profile,

T (r, 0, t)

, for different exposure times. Dashed and solid lines are analytical solutions, Eq. (4), without and considering AU process, respectively. Open circle are the exact numerical solution from the diffusion Eq. (1) with AU process.

β = 10

and the other parameters listed in Table 1 (Part 1) were used in simulations. I(t) dependence on

θ_{e l}

for (b) diverging TL (ZBLAN:Nd fluoride glass) and (c) converging TL (Q-98:Nd phosphate glass).

θ_{e l}

was artificially varied from 0 to 4. All these results of the right figure are for β = 0 (without AU).

Download Full Size | PPT Slide | PDF

Fig. 2 [(a), (b), (c), and (d)] - Normalized TL signal transients for a typical value of

θ_{e l} = 4

, Auger upconversion parameter varying from β = 0 up to β = 20, and thermal diffusivity of [(a) and (b)]

D = 3 \times 10^{- 7} m^{2} / s

and [(c) and (d)]

D = 3 \times 10^{- 7} m^{2} / s

, typics of glasses and crystals, respectively. In [(a) and (c)] and [(b) and (d)] were used other representative parameters given in Table 1 (Part 1), which lead to positive and negative TL signals. [(e) and (f)] - Typical normalized experimental lensing signals for (e) ZBLAN:Nd (1.0 mol%) and (f) Q-98:Nd (1.0 wt.%) samples. Continuous lines denotes least-squares curves fit using I(t) equation. The excitation power for these transients was fixed at

P_{i n}

= 200 mW. The obtained parameters are given in Table 1 (Part 2).

Download Full Size | PPT Slide | PDF

Fig. 3 θ_th versus the excitation power P_e for ZBLAN:Nd sample obtained from the experimental data fitting with β = 0, 2, and 5. Dashed lines are guides for the eye. The fitting parameters obtained for β = 2 were θ_th/P_e = (810 ± 50) × 10⁶ K/W,

〈 θ_{e l} 〉 = (1.5 \pm 0.1)

, and D = (2.7 ± 0.3) × 10⁻⁷ m²/s.

Download Full Size | PPT Slide | PDF

Tables (1)

Table 1 Part 1: Physical properties of the samples. Parameters for ZBLAN and Q-98 are from references [1,17–22]. Part 2: Parameters obtained from the computational fits for ZBLAN:Nd and Q-98:Nd samples.

View Table

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

\partial T (r, z, t) / \partial t - D \nabla^{2} T (r, z, t) = Q_{0} φ e^{- 2 r^{2} / w_{o e}^{2}} e^{- A_{e} z}

\frac{\partial N_{e}}{\partial t} = R_{P} N_{T} - (R_{P} + \frac{1}{τ}) N_{e} - γ N_{e}^{2}

N_{e} (r, z, t) = N_{T} {\frac{2 S Tan h (\frac{\sqrt{Δ (S, β)}}{2 τ} t)}{(S + 1) Tan h (\frac{\sqrt{Δ (S, β)}}{2 τ} t) + \sqrt{Δ (S, β)}}} e^{- A_{e} z}

T (r, z, t) = 2 \frac{P_{e} A_{e}}{π ρ C ω_{0 e}^{2}} φ (S_{0}, β) e^{- A_e z} \int_{0}^{t} \frac{e^{- \frac{2 r^{2} / ω_{o e}^{2}}{1 + 2 ξ / t_{c}}}}{1 + 2 ξ / t_{c}} d ξ

ϕ_{T L} (g, t) = \frac{2 π}{λ_{p}} \int_{0}^{L} [S (r, z, t) - S (0, z, t)] d z

ϕ_{P L} (g, t) = \frac{2 π}{λ_{p}} \int_{o}^{L} c_{k} N_{e} (g, z, t) d z

ϕ_{T L} = θ_{t h} φ (S_{0}, β) \int_{0}^{\infty} χ (α, L) e^{- ω_{0 e}^{2} α^{2} / 8} \times (1 - e^{- \frac{1}{4} ω_{o e}^{2} α^{2} \frac{t}{t_{c}}}) [J_{0} (α ω_{0 e} \sqrt{m g}) - 1] α^{- 1} d α

ϕ_{P L} = θ_{e l} (\frac{2 S Tan h (\frac{\sqrt{Δ (S, β)}}{2 τ} t)}{(S + 1) Tan h (\frac{\sqrt{Δ (S, β)}}{2 τ} t) + \sqrt{Δ (S, β)}})

χ (α, L) = \frac{\partial n}{\partial T} + \frac{4}{L α} (n - 1) (1 + ν) α_{T} h (α, L) + \frac{n^{3} E_{α T}}{4 (1 - ν)} [(q_{⊥} + q_{∥}) - (\frac{4 [q_{∥} ν + q_{⊥} (2 + ν)] h (α, L)}{L α})]

PART 1					PART 2
Parameter	Units	ZBLAN		Q-98	Parameter	Units	ZBLAN	Q-98
$ρ$	$k g m^{- 3}$	4330		3099	$β$		$2.0 \pm 0.4$	$1.2 \pm 0.4$
$n$		1.50		1.55
$c$	$J k g^{- 1} k^{- 1}$	520		800	$θ_{t h} / P_{e}$	$10^{3} K / W$	$506 \pm 40$	$753 \pm 50$
$\partial n / \partial T$	$10^{- 6} K^{- 1}$		−14.8	−4.5
$α_{T}$	$10^{- 6} K^{- 1}$	17.1		9.9	$θ_{e l}$		$1.5 \pm 0.1$	$0.45 \pm 0.09$
$ν$		0.31		0.24
$Y$	$10^{9} P a$	53.8		72.1	D	$10^{- 7} m^{2} / s$	$2.7 \pm 0.3$	$3.4 \pm 0.3$
$q_{∥}$	$10^{- 12} P a^{- 1}$	1.79		0.35
$q_{⊥}$	$10^{- 12} P a^{- 1}$	1.80		1.95	$Δ α$	$10^{- 32} m^{3}$	$15.4 \pm 0.8$	$3.3 \pm 0.4$
$η_{o}$		0.98		0.90
$L$	mm	0.7		1.9	$k_{T}$	$W / K . m$	$0.61 \pm 0.07$	$0.84 \pm 0.08$
$τ (F_{3 / 2}^{4})$	ms	0.460		0.346
$σ_{a b s}$	$10^{- 24} m^{2}$	2		3	$A_{e}$	$m^{- 1}$	$312 \pm 40$	$297 \pm 30$
$N_{T}$	$10^{26} m^{- 3}$	1.90		1.11
$λ_{e x}$	nm	796		802
$< λ_{e m} >$	nm	1060		1060

Abstract

References

2014 (1)

2013 (2)

2009 (2)

2008 (1)

2007 (2)

2006 (3)

2005 (2)

2003 (2)

2002 (1)

1999 (1)

1998 (2)

1992 (2)

1989 (1)

Aggarwal, I. D.

Andrade, A.

Andrade, A. A.

Andrade, L. H. C.

Antipov, O. L.

Astrath, N. G. C.

Baesso, M. L.

Bredikhin, D. V.

Cassanho, A.

Catunda, T.

Catundab, T.

Chase, L. L.

Cruz, R. A.

de Camargo, A. S. S.

Eiras, J. A.

Eremeykin, O. N.

Freitas, L. R.

Garcia, D.

Hehlen, M. P.

Huber, G.

Hwa, L. G.

Ivakin, E. V.

Jacinto, C.

Jaque, D.

Jensen, T.

Jenssen, H. P.

Jewell, J. M.

Jiao, H.

Kuznetsov, M. S.

Lima, S. M.

Lowe, R. D.

Malacarne, L. C.

Messias, D. N.

Meyn, J. P.

Myers, J.

Myers, J. D.

Myers, M.

Myers, M. J.

Noginov, M. A.

Nunes, A. O.

Nunes, L. A. O.

Oliveira, S.

Ostroumov, V.

Payne, S. A.

Pilla, V.

Powell, R. C.

Rodenas, A.

Rodrigues, T. P.

Savikin, A. P.

Shen, J.

Silva, J. R.

Snook, R. D.

Sole, J. G.

Sukhadolau, A. V.

Tenorio, E.

Vorobev, V. A.

Wilke, G. D.

Zanuto, V. S.

Chem. Phys. (1)

IEEE J. Quantum Electron. (1)

J. Lumin. (1)

J. Non-Cryst. Solids (1)

J. Non-Crystal. Solids (1)

J. Opt. Soc. Am. B (4)

J. Raman Spectrosc. (1)