Mode instability thresholds for Tm-doped fiber amplifiers pumped at 790 nm

Arlee V. Smith; Jesse J. Smith

doi:10.1364/OE.24.000975

Mode instability thresholds for Tm-doped fiber amplifiers pumped at 790 nm

Arlee V. Smith and Jesse J. Smith

Optics Express
Vol. 24,
Issue 2,
pp. 975-992
(2016)
•https://doi.org/10.1364/OE.24.000975

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Arlee V. Smith and Jesse J. Smith, "Mode instability thresholds for Tm-doped fiber amplifiers pumped at 790 nm," Opt. Express 24, 975-992 (2016)

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Related Topics
Table of Contents Category
- Lasers and Laser Optics
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Fiber optics amplifiers and oscillators (060.2320)
- Nonlinear optics, fibers (060.4370)
- Thermal effects (140.6810)
- Stimulated scattering, modulation, etc. (190.2640)

About this Article
History
- Original Manuscript: November 19, 2015
- Revised Manuscript: January 5, 2016
- Manuscript Accepted: January 7, 2016
- Published: January 12, 2016

Accessible

Open Access

Abstract

We use a detailed numerical model of stimulated thermal Rayleigh scattering to compute mode instability thresholds in Tm³⁺-doped fiber amplifiers. The fiber amplifies 2040 nm light using a 790 nm pump. The cross-relaxation process is strong, permitting power efficiencies of 60%. The predicted instability thresholds are compared with those in similar Yb³⁺-doped fiber amplifiers with 976 nm pump and 1060 nm signal, and are found to be higher, even though the heat load is much higher in Tm-doped amplifiers. The higher threshold in the Tm-doped fiber is attributed to its longer signal wavelength, and to stronger gain saturation, due in part to cross-relaxation heating.

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References

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Publication

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tunnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
[Crossref] [PubMed]
A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
[Crossref] [PubMed]
A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
[Crossref] [PubMed]
A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quant. Electron. 20, 3000112 (2014).
[Crossref]
G. D. Goodno, L. D. Book, J. E. Rothenberg, M. E. Weber, and S. B. Weiss, “Narrow linewidth power scaling and phase stabilization of 2-μm thulium fiber lasers,” Opt. Eng. 50, 111608 (2011).
[Crossref]
T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.
G. D. Goodno, L. D. Book, and J. E. Rothenberg, “Low-phase-noise, single-frequency, single-mode 608 W thulium fiber amplifier,” Opt. Lett. 34, 1204–1206 (2009).
[Crossref] [PubMed]
C. Gaida, M. Kienel, M. Muller, A. Klenke, M. Gebhardt, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Coherent combination of two Tm-doped fiber amplifiers,” Opt. Lett. 40, 2301–2304 (2015).
[Crossref] [PubMed]
K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Trans. 128, 706–716 (2006).
[Crossref]
K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]
B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
[Crossref] [PubMed]
L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
[Crossref] [PubMed]
R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).
B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
[Crossref] [PubMed]
P. Peterka, I. Kasik, A. Dhar, B. Dussardier, and W. Blanc, “Theoretical modeling of fiber laser at 810 nm based on thulium-doped silica fibers with enhanced 3H4 level lifetime,” Opt. Express 19, 2773–2781 (2011).
[Crossref] [PubMed]
A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[Crossref] [PubMed]
K. R. Hansen and J. Laegsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
[Crossref] [PubMed]
B. G. Ward, “Maximizing power output from continuous-wave single-frequency fiber amplifiers,” Opt. Lett. 40, 542–545 (2015).
[Crossref] [PubMed]
Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]
Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]
S. D. Jackson and S. Mossman, “Efficiency dependence on the Tm3+ and Al3+ concentrations for Tm3+- doped silica double-clad fiber lasers,” Appl. Opt. 42, 2702–2707 (2003).
[Crossref] [PubMed]
S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express 14, 50–57 (2006).
[Crossref] [PubMed]
S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17, 948–956 (1999).
[Crossref]
B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm:silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B. 78, 325–333 (2004).
[Crossref]
G. Turri, V. Sudesh, M. Richardson, M. Bass, A. Toncelli, and M. Tonelli, “Temperature-dependent spectroscopic properties of Tm3+ in germanate, silica, and phosphate glasses: a comparative study,” J. Appl. Phys. 103, 093104 (2008).
[Crossref]
P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15, 85–92 (2009).
[Crossref]
A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19, 11318–11329 (2011).
[Crossref] [PubMed]
A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5, 7100807 (2013).
[Crossref]
A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20, 24545–24558 (2012).
[Crossref] [PubMed]
D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. (Academic, 1991).
J. Ballato and P. Dragic, “Materials development for next generation optical fiber,” Materials 7, 4411–4430 (2014).
[Crossref]
A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref]
M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]
H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]
M. M. Broer, D. M. Krol, and D. J. DiGiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18, 799–801 (1993).
[Crossref] [PubMed]

2015 (5)

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]

B. G. Ward, “Maximizing power output from continuous-wave single-frequency fiber amplifiers,” Opt. Lett. 40, 542–545 (2015).
[Crossref] [PubMed]

C. Gaida, M. Kienel, M. Muller, A. Klenke, M. Gebhardt, F. Stutzki, C. Jauregui, J. Limpert, and A. Tunnermann, “Coherent combination of two Tm-doped fiber amplifiers,” Opt. Lett. 40, 2301–2304 (2015).
[Crossref] [PubMed]

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

2014 (3)

K. R. Hansen and J. Laegsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quant. Electron. 20, 3000112 (2014).
[Crossref]

J. Ballato and P. Dragic, “Materials development for next generation optical fiber,” Materials 7, 4411–4430 (2014).
[Crossref]

2013 (8)

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref]

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5, 7100807 (2013).
[Crossref]

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
[Crossref] [PubMed]

B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[Crossref] [PubMed]

2012 (3)

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20, 24545–24558 (2012).
[Crossref] [PubMed]

2011 (5)

P. Peterka, I. Kasik, A. Dhar, B. Dussardier, and W. Blanc, “Theoretical modeling of fiber laser at 810 nm based on thulium-doped silica fibers with enhanced 3H4 level lifetime,” Opt. Express 19, 2773–2781 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19, 11318–11329 (2011).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tunnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
[Crossref] [PubMed]

G. D. Goodno, L. D. Book, J. E. Rothenberg, M. E. Weber, and S. B. Weiss, “Narrow linewidth power scaling and phase stabilization of 2-μm thulium fiber lasers,” Opt. Eng. 50, 111608 (2011).
[Crossref]

2009 (2)

P. F. Moulton, G. A. Rines, E. V. Slobodtchikov, K. F. Wall, G. Frith, B. Samson, and A. L. G. Carter, “Tm-doped fiber lasers: fundamentals and power scaling,” IEEE J. Sel. Top. Quantum Electron. 15, 85–92 (2009).
[Crossref]

G. D. Goodno, L. D. Book, and J. E. Rothenberg, “Low-phase-noise, single-frequency, single-mode 608 W thulium fiber amplifier,” Opt. Lett. 34, 1204–1206 (2009).
[Crossref] [PubMed]

2008 (1)

G. Turri, V. Sudesh, M. Richardson, M. Bass, A. Toncelli, and M. Tonelli, “Temperature-dependent spectroscopic properties of Tm3+ in germanate, silica, and phosphate glasses: a comparative study,” J. Appl. Phys. 103, 093104 (2008).
[Crossref]

2006 (2)

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Trans. 128, 706–716 (2006).
[Crossref]

S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express 14, 50–57 (2006).
[Crossref] [PubMed]

2004 (1)

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm:silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B. 78, 325–333 (2004).
[Crossref]

Alkeskjold, T. T.

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]

Ballato, J.

J. Ballato and P. Dragic, “Materials development for next generation optical fiber,” Materials 7, 4411–4430 (2014).
[Crossref]

Barnes, N. P.

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm:silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B. 78, 325–333 (2004).
[Crossref]

Bass, M.

Blanc, W.

Book, L. D.

G. D. Goodno, L. D. Book, and J. E. Rothenberg, “Low-phase-noise, single-frequency, single-mode 608 W thulium fiber amplifier,” Opt. Lett. 34, 1204–1206 (2009).
[Crossref] [PubMed]

Broeng, J.

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]

Broer, M. M.

M. M. Broer, D. M. Krol, and D. J. DiGiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18, 799–801 (1993).
[Crossref] [PubMed]

Carter, A. L. G.

Chen, J.-Z.

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

Cho, C.-H.

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

Cole, K. D.

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Trans. 128, 706–716 (2006).
[Crossref]

Dajani, I.

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
[Crossref] [PubMed]

Dhar, A.

DiGiovanni, D. J.

M. M. Broer, D. M. Krol, and D. J. DiGiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18, 799–801 (1993).
[Crossref] [PubMed]

Dong, L.

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
[Crossref] [PubMed]

Dragic, P.

J. Ballato and P. Dragic, “Materials development for next generation optical fiber,” Materials 7, 4411–4430 (2014).
[Crossref]

Dussardier, B.

Ehrenreich, T.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

Eidam, T.

Frith, G.

Gaida, C.

Gebhardt, M.

Goodno, G. D.

G. D. Goodno, L. D. Book, and J. E. Rothenberg, “Low-phase-noise, single-frequency, single-mode 608 W thulium fiber amplifier,” Opt. Lett. 34, 1204–1206 (2009).
[Crossref] [PubMed]

Hansen, K. R.

K. R. Hansen and J. Laegsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]

Jackson, S. D.

S. D. Jackson and S. Mossman, “Efficiency dependence on the Tm3+ and Al3+ concentrations for Tm3+- doped silica double-clad fiber lasers,” Appl. Opt. 42, 2702–2707 (2003).
[Crossref] [PubMed]

S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17, 948–956 (1999).
[Crossref]

Jansen, F.

Jauregui, C.

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

Jiang, S.

Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

Jorgensen, M. M.

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

Kasik, I.

Kienel, M.

King, T. A.

S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17, 948–956 (1999).
[Crossref]

Klenke, A.

Krol, D. M.

M. M. Broer, D. M. Krol, and D. J. DiGiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18, 799–801 (1993).
[Crossref] [PubMed]

Laegsgaard, J.

K. R. Hansen and J. Laegsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
[Crossref] [PubMed]

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]

Laurila, M.

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

Lee, Y.-W.

Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

Leveille, R.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

Limpert, J.

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

Lin, Y.-H.

Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]

Ling, H.-Y.

Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]

Liu, Z.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

Ma, P.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

Majid, I.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

Marcuse, D.

D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. (Academic, 1991).

Modsching, N.

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

Mossman, S.

S. D. Jackson and S. Mossman, “Efficiency dependence on the Tm3+ and Al3+ concentrations for Tm3+- doped silica double-clad fiber lasers,” Appl. Opt. 42, 2702–2707 (2003).
[Crossref] [PubMed]

Moulton, P.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

Moulton, P. F.

Muller, M.

Noordegraaf, D.

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

Otto, H.-J.

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

Peterka, P.

Povlsen, J. H.

S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express 14, 50–57 (2006).
[Crossref] [PubMed]

Richardson, M.

Rines, G.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

Rines, G. A.

Robin, C.

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
[Crossref] [PubMed]

Rothenberg, J. E.

G. D. Goodno, L. D. Book, and J. E. Rothenberg, “Low-phase-noise, single-frequency, single-mode 608 W thulium fiber amplifier,” Opt. Lett. 34, 1204–1206 (2009).
[Crossref] [PubMed]

Samson, B.

Schmidt, O.

Schreiber, T.

Slobodtchikov, E. V.

Smith, A. V.

A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quant. Electron. 20, 3000112 (2014).
[Crossref]

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5, 7100807 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20, 24545–24558 (2012).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19, 11318–11329 (2011).
[Crossref] [PubMed]

Smith, J. J.

A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quant. Electron. 20, 3000112 (2014).
[Crossref]

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5, 7100807 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20, 24545–24558 (2012).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19, 11318–11329 (2011).
[Crossref] [PubMed]

Stutzki, F.

Sudesh, V.

Tankala, K.

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

Tao, R.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

Toncelli, A.

Tonelli, M.

Tseng, H.-W.

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

Tunnermann, A.

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

Turri, G.

Wall, K. F.

Walsh, B. M.

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm:silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B. 78, 325–333 (2004).
[Crossref]

Wang, X.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

Ward, B.

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
[Crossref] [PubMed]

Ward, B. G.

B. G. Ward, “Maximizing power output from continuous-wave single-frequency fiber amplifiers,” Opt. Lett. 40, 542–545 (2015).
[Crossref] [PubMed]

B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
[Crossref] [PubMed]

Weber, M. E.

Weiss, S. B.

Wirth, C.

Zhou, P.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

Appl. Opt. (1)

S. D. Jackson and S. Mossman, “Efficiency dependence on the Tm3+ and Al3+ concentrations for Tm3+- doped silica double-clad fiber lasers,” Appl. Opt. 42, 2702–2707 (2003).
[Crossref] [PubMed]

Appl. Phys. B. (1)

B. M. Walsh and N. P. Barnes, “Comparison of Tm:ZBLAN and Tm:silica fiber lasers; spectroscopy and tunable pulsed laser operation around 1.9 μm,” Appl. Phys. B. 78, 325–333 (2004).
[Crossref]

Fibers (1)

Y.-W. Lee, H.-W. Tseng, C.-H. Cho, J.-Z. Chen, and S. Jiang, “Heavily Tm3+-doped silicate fiber for high-gain fiber amplifiers,” Fibers 1, 82–92 (2013).
[Crossref]

IEEE J. Quantum. Electron. (1)

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of wavelength dependence of mode instability based on a semi-analytical model,” IEEE J. Quantum. Electron. 51, 1600106 (2015).

IEEE J. Sel. Top. Quant. Electron. (1)

A. V. Smith and J. J. Smith, “Overview of a steady-periodic model of modal instability in fiber amplifiers,” IEEE J. Sel. Top. Quant. Electron. 20, 3000112 (2014).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

IEEE Photonics J. (1)

A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5, 7100807 (2013).
[Crossref]

J. Appl. Phys. (1)

J. Heat Trans. (1)

K. D. Cole, “Steady-periodic Green’s functions and thermal-measurement applications in rectangular coordinates,” J. Heat Trans. 128, 706–716 (2006).
[Crossref]

J. Lightwave Technol. (1)

S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17, 948–956 (1999).
[Crossref]

Materials (1)

J. Ballato and P. Dragic, “Materials development for next generation optical fiber,” Materials 7, 4411–4430 (2014).
[Crossref]

Opt. Eng. (1)

Opt. Express (13)

S. D. Agger and J. H. Povlsen, “Emission and absorption cross section of thulium doped silica fibers,” Opt. Express 14, 50–57 (2006).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20, 24545–24558 (2012).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
[Crossref] [PubMed]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
[Crossref] [PubMed]

B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[Crossref] [PubMed]

K. R. Hansen and J. Laegsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode competition in high power fiber amplifiers,” Opt. Express 19, 11318–11329 (2011).
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
[Crossref] [PubMed]

H.-J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tunnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23, 15265–15277(2015).
[Crossref] [PubMed]

Opt. Lett. (5)

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Laegsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37, 2382–2384 (2012).
[Crossref] [PubMed]

B. G. Ward, “Maximizing power output from continuous-wave single-frequency fiber amplifiers,” Opt. Lett. 40, 542–545 (2015).
[Crossref] [PubMed]

G. D. Goodno, L. D. Book, and J. E. Rothenberg, “Low-phase-noise, single-frequency, single-mode 608 W thulium fiber amplifier,” Opt. Lett. 34, 1204–1206 (2009).
[Crossref] [PubMed]

M. M. Broer, D. M. Krol, and D. J. DiGiovanni, “Highly nonlinear near-resonant photodarkening in a thulium-doped aluminosilicate glass fiber,” Opt. Lett. 18, 799–801 (1993).
[Crossref] [PubMed]

Opt. Mat. Express (1)

Y.-W. Lee, H.-Y. Ling, Y.-H. Lin, and S. Jiang, “Heavily Tm3+-doped silicate fiber with high gain per unit length,” Opt. Mat. Express 5, 549–557 (2015).
[Crossref]

Proc. SPIE (2)

A. V. Smith and J. J. Smith, “Mode instability thresholds of fiber amplifiers,” Proc. SPIE 8601, 860108 (2013).
[Crossref]

M. M. Jorgensen, M. Laurila, D. Noordegraaf, T. T. Alkeskjold, and J. Laegsgaard, “Thermal-recovery of modal instability in rod fiber amplifiers,” Proc. SPIE 8601, 86010U (2013).
[Crossref]

Other (2)

D. Marcuse, Theory of dielectric optical waveguides, 2nd ed. (Academic, 1991).

T. Ehrenreich, R. Leveille, I. Majid, K. Tankala, G. Rines, and P. Moulton, “1-kW, all-glass Tm:fiber laser,” in SPIE Photonics West (SPIE, 2010), pp. 1–15.

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Fig. 1 Energy levels for Tm³⁺ in silicate glass [15]. The pump wavelength is 790 nm and the signal wavelength is 2040 nm. The double arrows indicate the 2-for-1 cross-relaxation process.

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Fig. 2 Signal emission cross sections for the ³H₆ − ³F₄ (1↔2) transition in thulium-doped aluminum silicate fiber, from [15, 19, 22–24].

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Fig. 3 Signal absorption cross sections for the ³H₆ − ³F₄ (1↔2) transition in thulium-doped aluminum silicate fiber, from [15, 19, 22].

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Fig. 4 Walsh and Barnes [24] absorption cross section for Tm³⁺ ions in silica fiber.

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Fig. 5 Power curves generated by our simplified model for Yb- and Tm-doped fibers in the bidirectionally-pumped case with total pump power of 1000 W, and the parameters given in Table 1. Approximately 100 W of power is lost from the pump and signal in the Yb-doped fiber and 325 W in the Tm-doped fiber.

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Fig. 6 Power curves generated by our full STRS model for operation at threshold of bidirectionally pumped Yb- and Tm-doped fiber amplifiers using the parameters listed in Table 1.

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Fig. 7 Thermal lensing of fundamental mode for bidirectionally-pumped Yb-doped and Tm-doped fiber amplifiers near STRS threshold. These curves correspond to the powers plotted in Fig. 6, and use parameters in Table 1. The thresholds are given in Table 2. The unlensed area is 380 μm².

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Fig. 8 Computed total heat profiles for cuts through the fiber center of the 25 μm diameter core Yb- and Tm-doped fibers of Table 1 with each end of the fiber pumped with 500 pump. More than three times more heat is deposited in Tm-doped fiber than in Yb-doped fiber. Note the different vertical scales for the two plots.

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Fig. 9 Cuts through the fiber center of the total computed antisymmetric parts of the heat profiles for the bidirectionally pumped 25 μm diameter core Yb- and Tm-doped fibers of Table 1, with each end of the fiber pumped with 500 W.

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Fig. 10 Cuts through the fiber center of the computed antisymmetric parts of heating described by the three terms of Eq. (13) in the Tm-doped fiber using parameters given in Table 1. Here, the fiber is bidirectionally pumped with 500 W pump input in each end. The left plot shows heating due to nonradiative decay from level 2; the middle plot shows heating due to nonradiative decay from level 4; the right plot show heating due to the cross relaxation process. The plot axes are the same as in Fig. 9, except the vertical axis ranges from −20 to 20 GW/m³. The sum of these three terms comprises the righthand plot of Fig. 9.

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Fig. 11 Cuts through the fiber center of the computed antisymmetric parts of heating described by the three terms of Eq. (13) for the 50 μm core diameter (short-fat) fiber pumped with 500 W from each end. The vertical scale is the same as in Fig. 10. The length scale range is 0 to 1.2 m. The transverse scale range is −75 μm to 75 μm.

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Tables (3)

Table 1 Parameters used in comparing similar Yb- and Tm-doped fiber amplifiers.

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Table 2 Powers (in watts) at mode instability thresholds computed for similar Yb- and Tm-doped fiber amplifiers with the parameters given in Table 1. Input signal power is 6 W. Left-behind power is the difference between the absorbed pump power and the increase in signal power. Most of this left-behind power is heat.

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Table 3 Signal power at mode instability thresholds for short-fat and long-skinny Tm-doped fiber amplifiers. Ratio is the ratio of long-skinny threshold to short-fat threshold.

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Equations (16)

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

\begin{array}{l} \frac{d n_{1}}{d t} & = & A_{21} n_{2} + A_{41} n_{4} + (σ_{p}^{e} n_{4} - σ_{p}^{a} n_{1}) I_{p} / h ν_{p} \\ - k_{41}^{22} N_{○} n_{4} n_{1} + k_{22}^{41} N_{○} n_{2}^{2} + (σ_{s}^{e} n_{2} - σ_{s}^{a} n_{1}) I_{s} / h ν_{s}, \end{array}

\begin{array}{l} \frac{d n_{2}}{d t} & = & A_{42} n_{4} - A_{21} n_{2} \\ + 2 k_{41}^{22} N_{○} n_{4} n_{1} - 2 k_{22}^{41} N_{○} n_{2}^{2} - (σ_{s}^{e} n_{2} - σ_{s}^{a} n_{1}) I_{s} / h ν_{s}, \end{array}

\begin{array}{l} \frac{d n_{4}}{d t} & = & - A_{41} n_{4} - A_{42} n_{4} + (σ_{p}^{a} n_{1} - σ_{p}^{e} n_{4}) I_{p} / h ν_{p} \\ - k_{41}^{22} N_{○} n_{4} n_{1} + k_{22}^{41} N_{○} n_{2}^{2}, \end{array}

1 = n_{1} + n_{2} + n_{4} .

\frac{d I_{s}}{d z} = N_{○} (σ_{s}^{e} n_{2} - σ_{s}^{a} n_{1}) I_{s},

\frac{d I_{p}^{\pm}}{d z} = \pm N_{○} (σ_{p}^{e} n_{4} - σ_{p}^{a} n_{1}) I_{p}^{\pm},

\begin{array}{l} 0 & = & A_{21} n_{2} + A_{41} n_{4} + (σ_{p}^{e} n_{4} - σ_{p}^{a} n_{1}) I_{p} / h ν_{p} \\ - k_{41}^{22} N_{○} n_{4} n_{1} + k_{22}^{41} N_{○} n_{2}^{2} + (σ_{s}^{e} n_{2} - σ_{s}^{a} n_{1}) I_{s} / h ν_{s}, \end{array}

\begin{array}{l} 0 & = & A_{42} n_{4} - A_{21} n_{2} + 2 k_{41}^{22} N_{○} n_{4} n_{1} - 2 k_{22}^{41} N_{○} n_{2}^{2} \\ - (σ_{s}^{e} n_{2} - σ_{s}^{a} n_{1}) I_{s} / h ν_{s}, \end{array}

\begin{array}{l} 0 & = & - A_{41} n_{4} - A_{42} n_{4} + (σ_{p}^{a} n_{1} - σ_{p}^{e} n_{4}) I_{p} / h ν_{p} \\ - k_{41}^{22} N_{○} n_{4} n_{1} + k_{22}^{41} N_{○} n_{2}^{2}, \end{array}

1 = n_{1} + n_{2} + n_{4} .

Q = - \frac{d I_{p}^{+}}{d z} + \frac{d I_{p}^{-}}{d z} - \frac{d I_{s}}{d z} .

\begin{array}{l} Q & = & N_{○} n_{4} (A_{42} [h ν_{p} - h ν_{s}] + A_{41} h ν_{p}) + N_{○} n_{2} A_{21} h ν_{s} \\ + N_{○}^{2} (k_{41}^{22} n_{1} n_{4} - k_{22}^{41} n_{2}^{2}) (h ν_{p} - 2 h ν_{s}) . \end{array}

\begin{array}{l} Q^{'} & = & N_{○} n_{4} (A_{42}^{'} [h ν_{p} - h ν_{s}] + A_{41}^{'} h ν_{p}) + N_{○} n_{2} A_{21}^{'} h ν_{s} \\ + N_{○}^{2} (k_{41}^{22} n_{1} n_{4} - k_{22}^{41} n_{2}^{2}) (h ν_{p} - 2 h ν_{s}) . \end{array}

g_{1} g_{4} exp [- E_{4} / k T] k_{41}^{22} = g_{2} g_{2} exp [- 2 E_{2} / k T] k_{22}^{41},

k_{22}^{41} = k_{41}^{22} \frac{g_{1} g_{4}}{g_{2}^{2}} exp [(2 E_{2} - E_{4}) / k T] .

k_{22}^{41} = 0.05 k_{41}^{22} .

Parameter	Yb-doped	Tm-doped

λ_p	976 nm	790 nm
λ_s	1060 nm	2040 nm
L	4 m	4 m
P_p	500 & 500 W	500 & 500 W
d_core	25 μm	25 μm
d_clad	400 μm	400 μm
n_core	1.451	1.4537
n_clad	1.45	1.45
V	4.0	4.0
N_dope	1.2 × 10²⁶ m⁻³	5.0 × 10²⁶ m⁻³
d_dope	25 μm	25 μm
$σ_{p}^{a}$	2.47 × 10⁻²⁴ m²	6.0 × 10⁻²⁵ m²
$σ_{p}^{e}$	2.44 × 10⁻²⁴ m²	0.5 × 10⁻²⁵ m²
$σ_{s}^{a}$	6.0 × 10⁻²⁷ m²	0.1 × 10⁻²⁵ m²
$σ_{p}^{e}$	3.58 × 10⁻²⁵ m²	1.5 × 10⁻²⁵ m²
LP₀₁ Power	6 W	6 W
A₂₁(A′₂₁)	1.1 × 10³(1.1 × 10³) s⁻¹	2.0 × 10³(1.8 × 10³) s⁻¹
A₄₁(A′₄₁)		1.5 × 10³(0) s⁻¹
A₄₂(A′₄₂)		1.6 × 10⁴(1.6 × 10⁴) s⁻¹
$k_{41}^{22}$		20 × 10⁻²³ m³ s⁻¹
$k_{22}^{41}$		2.0 × 10⁻²³ m³ s⁻¹
ρ	2201 kg/m³	2201 kg/m³
κ	1.38 W/m·K	1.38 W/m·K
dn/dT	1.2 × 10⁻⁵ K⁻¹	1.2 × 10⁻⁵ K⁻¹
C	703 J/kg·K	703 J/kg·K

	Co-pumped		Counter-pumped		Bidirectionally-pumped
	Yb	Tm	Yb	Tm	Yb	Tm

Signal out	1451	2046	2233	3947	2235	3210
Pump in	1600	3100	2410	6585	2503	4906
Pump abs.	1570	3005	2362	6007	2456	4686
Left-behind	127	959	191	2066	198	1483

L	d_core	V	Co-pumped	Counter-pumped	Bidirectionally-pumped

4 m	25 μm	4.0	2046 W	3947 W	3210 W
1.2 m	50 μm	8.0	899 W	1244 W	1433 W

Ratio			2.28	3.17	2.24

Parameter	Yb-doped	Tm-doped

λ_p	976 nm	790 nm
λ_s	1060 nm	2040 nm
L	4 m	4 m
P_p	500 & 500 W	500 & 500 W
d_core	25 μm	25 μm
d_clad	400 μm	400 μm
n_core	1.451	1.4537
n_clad	1.45	1.45
V	4.0	4.0
N_dope	1.2 × 10²⁶ m⁻³	5.0 × 10²⁶ m⁻³
d_dope	25 μm	25 μm
$σ_{p}^{a}$	2.47 × 10⁻²⁴ m²	6.0 × 10⁻²⁵ m²
$σ_{p}^{e}$	2.44 × 10⁻²⁴ m²	0.5 × 10⁻²⁵ m²
$σ_{s}^{a}$	6.0 × 10⁻²⁷ m²	0.1 × 10⁻²⁵ m²
$σ_{p}^{e}$	3.58 × 10⁻²⁵ m²	1.5 × 10⁻²⁵ m²
LP₀₁ Power	6 W	6 W
A₂₁(A′₂₁)	1.1 × 10³(1.1 × 10³) s⁻¹	2.0 × 10³(1.8 × 10³) s⁻¹
A₄₁(A′₄₁)		1.5 × 10³(0) s⁻¹
A₄₂(A′₄₂)		1.6 × 10⁴(1.6 × 10⁴) s⁻¹
$k_{41}^{22}$		20 × 10⁻²³ m³ s⁻¹
$k_{22}^{41}$		2.0 × 10⁻²³ m³ s⁻¹
ρ	2201 kg/m³	2201 kg/m³
κ	1.38 W/m·K	1.38 W/m·K
dn/dT	1.2 × 10⁻⁵ K⁻¹	1.2 × 10⁻⁵ K⁻¹
C	703 J/kg·K	703 J/kg·K

	Co-pumped		Counter-pumped		Bidirectionally-pumped
	Yb	Tm	Yb	Tm	Yb	Tm

Signal out	1451	2046	2233	3947	2235	3210
Pump in	1600	3100	2410	6585	2503	4906
Pump abs.	1570	3005	2362	6007	2456	4686
Left-behind	127	959	191	2066	198	1483

Abstract

References

2015 (5)

2014 (3)

2013 (8)

2012 (3)

2011 (5)

2009 (2)

2008 (1)

2006 (2)

2004 (1)

2003 (1)

1999 (1)

1993 (1)

Agger, S. D.

Alkeskjold, T. T.

Ballato, J.

Barnes, N. P.

Bass, M.

Blanc, W.

Book, L. D.

Broeng, J.

Broer, M. M.

Carter, A. L. G.

Chen, J.-Z.

Cho, C.-H.

Cole, K. D.

Dajani, I.

Dhar, A.

DiGiovanni, D. J.

Dong, L.

Dragic, P.

Dussardier, B.

Ehrenreich, T.

Eidam, T.

Frith, G.

Gaida, C.

Gebhardt, M.

Goodno, G. D.

Hansen, K. R.

Jackson, S. D.

Jansen, F.

Jauregui, C.

Jiang, S.

Jorgensen, M. M.

Kasik, I.

Kienel, M.

King, T. A.

Klenke, A.

Krol, D. M.

Laegsgaard, J.

Laurila, M.

Lee, Y.-W.

Leveille, R.

Limpert, J.

Lin, Y.-H.

Ling, H.-Y.

Liu, Z.

Ma, P.

Majid, I.

Marcuse, D.

Modsching, N.

Mossman, S.

Moulton, P.

Moulton, P. F.

Muller, M.

Noordegraaf, D.

Otto, H.-J.

Peterka, P.

Povlsen, J. H.

Richardson, M.

Rines, G.

Rines, G. A.

Robin, C.

Rothenberg, J. E.

Samson, B.

Schmidt, O.

Schreiber, T.

Slobodtchikov, E. V.

Smith, A. V.