Abstract

A model of the 2μm amplified spontaneous emission (ASE) generation in the thulium-doped silica fibers pumped at 1575 nm is presented. Both Al-codoped and Al/Ge-codoped fiber core compositions are studied. The results show that the composition affects the relative slope efficiency of 10% and the bandwidth of 19% of the output ASE. Our results predict that the backward ASE is more powerful and spectrally broader compared to the forward ASE, which is in agreement with previous experiments. Using an asymmetric cavity feedback, 98% of the total output power can be directed in the backward ASE, but with the consequence of losing 50% of the bandwidth. Such sources are expected to deliver single-mode output with more than 70% slope and 39% power conversion efficiency.

© 2012 Optical Society of America

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References

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  1. T. F. Morse, K. Oh, and L. J. Reinhart, “A new gas detection technique utilizing amplified spontaneous emission light source from a co-doped Tm3+/Ho3+ silica fibre in the 2 μm region,” Meas. Sci. Technol. 9, 1409–1412 (1998).
    [CrossRef]
  2. B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
    [CrossRef]
  3. K. Oh, A. Kilian, L. Reinhart, Q. Zhang, T. F. Morse, and P. M. Weber, “Broadband superfluorescent emission of the H43→H63 transition in a Tm-doped multicomponent silicate fiber,” Opt. Lett. 19, 1131–1133 (1994).
    [CrossRef]
  4. S. D. Jackson and T. A. King, “Theoretical modeling of Tm-doped silica fiber lasers,” J. Lightwave Technol. 17, 948–956(1999).
    [CrossRef]
  5. P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
    [CrossRef]
  6. J. Xu, M. Prabhu, J. Lu, K. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with ring cavity,” Appl. Opt. 40, 1983–1988 (2001).
    [CrossRef]
  7. 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, 85–91 (2009).
    [CrossRef]
  8. J. Geng, Q. Wang, T. Luo, S. Jiang, and F. Amzajerdian, “Single-frequency narrow-linewidth Tm-doped fiber laser using silicate glass fiber,” Opt. Lett. 34, 3493–3495 (2009).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  12. Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2011

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photon. Technol. Lett. 23, 893–895 (2011).
[CrossRef]

Q. Xiao, P. Yan, Y. Wang, J. Hao, and M. Gong, “High-power all-fiber superfluorescent source with fused angle-polished side-pumping configuration,” Appl. Opt. 50, 1164–1169(2011).
[CrossRef]

2010

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

2009

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, 85–91 (2009).
[CrossRef]

J. Geng, Q. Wang, T. Luo, S. Jiang, and F. Amzajerdian, “Single-frequency narrow-linewidth Tm-doped fiber laser using silicate glass fiber,” Opt. Lett. 34, 3493–3495 (2009).
[CrossRef]

2008

2007

2006

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

Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
[CrossRef]

2005

Y. H. Tsang, A. F. El-Sherif, and T. A. King, “Broadband amplified spontaneous emission fibre source near 2 μm using resonant in-band pumping,” J. Mod. Opt. 52, 109–118 (2005).
[CrossRef]

2004

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[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]

2001

1999

1998

T. F. Morse, K. Oh, and L. J. Reinhart, “A new gas detection technique utilizing amplified spontaneous emission light source from a co-doped Tm3+/Ho3+ silica fibre in the 2 μm region,” Meas. Sci. Technol. 9, 1409–1412 (1998).
[CrossRef]

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

1994

Agger, S. D.

Amzajerdian, F.

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]

Blanc, W.

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[CrossRef]

Bouma, B. E.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

Brezinski, M. E.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

Carter, A. L. G.

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, 85–91 (2009).
[CrossRef]

Chang, J.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

Clarkson, W. A.

Dussardier, B.

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[CrossRef]

El-Sherif, A. F.

Y. H. Tsang, A. F. El-Sherif, and T. A. King, “Broadband amplified spontaneous emission fibre source near 2 μm using resonant in-band pumping,” J. Mod. Opt. 52, 109–118 (2005).
[CrossRef]

Engelbrecht, M.

Faure, B.

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[CrossRef]

Frith, G.

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, 85–91 (2009).
[CrossRef]

Fujimoto, J. G.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

Geng, J.

Gong, M.

Hao, J.

Haxsen, F.

Huang, Q.-J.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

Jackson, S. D.

Jiang, M.

Jiang, S.

Jones, D. J.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

Karasek, M.

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[CrossRef]

Kilian, A.

King, T. A.

Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
[CrossRef]

Y. H. Tsang, A. F. El-Sherif, and T. A. King, “Broadband amplified spontaneous emission fibre source near 2 μm using resonant in-band pumping,” J. Mod. Opt. 52, 109–118 (2005).
[CrossRef]

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

Ko, D.-K.

Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
[CrossRef]

Kracht, D.

Lee, J.

Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
[CrossRef]

Li, F.

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photon. Technol. Lett. 23, 893–895 (2011).
[CrossRef]

Liu, Z.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

Lu, J.

Luo, T.

Morse, T. F.

T. F. Morse, K. Oh, and L. J. Reinhart, “A new gas detection technique utilizing amplified spontaneous emission light source from a co-doped Tm3+/Ho3+ silica fibre in the 2 μm region,” Meas. Sci. Technol. 9, 1409–1412 (1998).
[CrossRef]

K. Oh, A. Kilian, L. Reinhart, Q. Zhang, T. F. Morse, and P. M. Weber, “Broadband superfluorescent emission of the H43→H63 transition in a Tm-doped multicomponent silicate fiber,” Opt. Lett. 19, 1131–1133 (1994).
[CrossRef]

Moulton, P. F.

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, 85–91 (2009).
[CrossRef]

Nelson, L. E.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

Oh, K.

T. F. Morse, K. Oh, and L. J. Reinhart, “A new gas detection technique utilizing amplified spontaneous emission light source from a co-doped Tm3+/Ho3+ silica fibre in the 2 μm region,” Meas. Sci. Technol. 9, 1409–1412 (1998).
[CrossRef]

K. Oh, A. Kilian, L. Reinhart, Q. Zhang, T. F. Morse, and P. M. Weber, “Broadband superfluorescent emission of the H43→H63 transition in a Tm-doped multicomponent silicate fiber,” Opt. Lett. 19, 1131–1133 (1994).
[CrossRef]

Pearson, L.

Peterka, P.

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[CrossRef]

Povlsen, J. H.

Prabhu, M.

Reinhart, L.

Reinhart, L. J.

T. F. Morse, K. Oh, and L. J. Reinhart, “A new gas detection technique utilizing amplified spontaneous emission light source from a co-doped Tm3+/Ho3+ silica fibre in the 2 μm region,” Meas. Sci. Technol. 9, 1409–1412 (1998).
[CrossRef]

Rines, G. A.

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, 85–91 (2009).
[CrossRef]

Sahu, J. K.

Samson, B.

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, 85–91 (2009).
[CrossRef]

Shen, D. Y.

Slobodtchikov, E. V.

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, 85–91 (2009).
[CrossRef]

Tang, Y.

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photon. Technol. Lett. 23, 893–895 (2011).
[CrossRef]

Tayebati, P.

Tearney, G. J.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

Tsang, Y. H.

Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
[CrossRef]

Y. H. Tsang, A. F. El-Sherif, and T. A. King, “Broadband amplified spontaneous emission fibre source near 2 μm using resonant in-band pumping,” J. Mod. Opt. 52, 109–118 (2005).
[CrossRef]

Ueda, K.

Wall, K. F.

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, 85–91 (2009).
[CrossRef]

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]

Wandt, D.

Wang, P.

Wang, Q.

Wang, Q.-P.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

Wang, Y.

Weber, P. M.

Xiao, Q.

Xing, D.

Xu, J.

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photon. Technol. Lett. 23, 893–895 (2011).
[CrossRef]

J. Xu, M. Prabhu, J. Lu, K. Ueda, and D. Xing, “Efficient double-clad thulium-doped fiber laser with ring cavity,” Appl. Opt. 40, 1983–1988 (2001).
[CrossRef]

Yan, P.

Yu, G.-Y.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

Zhang, Q.

Zhang, X.-Y.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

Appl. Opt.

Appl. Phys. B

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]

IEEE J. Sel. Top. Quantum Electron.

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, 85–91 (2009).
[CrossRef]

IEEE Photon. Technol. Lett.

Y. Tang, F. Li, and J. Xu, “High peak-power gain-switched Tm3+-doped fiber laser,” IEEE Photon. Technol. Lett. 23, 893–895 (2011).
[CrossRef]

J. Biomed. Opt.

B. E. Bouma, L. E. Nelson, G. J. Tearney, D. J. Jones, M. E. Brezinski, and J. G. Fujimoto, “Optical coherence tomographic imaging of human tissue at 1.55 μm and 1.81 μm using Er- and Tm-doped fiber sources,” J. Biomed. Opt. 3, 76–79 (1998).
[CrossRef]

J. Lightwave Technol.

J. Mod. Opt.

Y. H. Tsang, A. F. El-Sherif, and T. A. King, “Broadband amplified spontaneous emission fibre source near 2 μm using resonant in-band pumping,” J. Mod. Opt. 52, 109–118 (2005).
[CrossRef]

Y. H. Tsang, T. A. King, D.-K. Ko, and J. Lee, “Broadband amplified spontaneous emission double-clad fibre source with central wavelengths near 2 μm,” J. Mod. Opt. 53, 991–1001 (2006).
[CrossRef]

Meas. Sci. Technol.

T. F. Morse, K. Oh, and L. J. Reinhart, “A new gas detection technique utilizing amplified spontaneous emission light source from a co-doped Tm3+/Ho3+ silica fibre in the 2 μm region,” Meas. Sci. Technol. 9, 1409–1412 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

P. Peterka, B. Faure, W. Blanc, M. Karasek, and B. Dussardier, “Theoretical modelling of S-band thulium-doped silica fibre amplifiers,” Opt. Quantum Electron. 36, 201–212 (2004).
[CrossRef]

Optoelectron. Lett.

G.-Y. Yu, J. Chang, Q.-P. Wang, X.-Y. Zhang, Z. Liu, and Q.-J. Huang, “A theoretical model of thulium-doped silica fiber’s ASE in the 1900 nm waveband,” Optoelectron. Lett. 6, 45–47 (2010).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Absorption and emission cross-sections for Tm1 and Tm2 fiber core compositions [16]. Inset shows diode pumping (solid black line), inband pumping (blue dashed), and ASE (red dotted) transitions in Tm.

Fig. 2.
Fig. 2.

Power distribution along the Tm1 fiber length for the pump, the forward ASE, and the backward ASE.

Fig. 3.
Fig. 3.

Intensity spectra of the forward and backward ASE for Tm1 and Tm2 fibers. All are at 900 mW input pump power, except the Tm1 backward ASE is at 750 mW input pump power for better comparison.

Fig. 4.
Fig. 4.

ASE output power versus input pump power for both directions and both fiber compositions with no feedback.

Fig. 5.
Fig. 5.

Forward and backward ASE power versus length of the Tm1 fiber at 500 mW input pump power, and the backward/forward ratio of power.

Fig. 6.
Fig. 6.

ASE output power versus input pump power for both directions and both fiber compositions with R b = 4 % broadband reflection. Inset shows the backward/forward ratio of power for Tm1 fiber without and with 4% feedback.

Fig. 7.
Fig. 7.

Bandwidth of forward and backward ASE for Tm1 fiber, and the effect of the feedback via R b = 4 % broadband reflection on the backward ASE for Tm2 fiber.

Equations (7)

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

N ( z , t ) t = 0 I f ( ν , z , t ) + I b ( ν , z , t ) h ν Γ [ σ a ( N 0 N ( z , t ) ) σ e N ( z , t ) ] d ν N ( z , t ) τ ,
N ( z ) = 0 I f ( ν , z ) + I b ( ν , z ) h ν Γ σ a d ν 0 I f ( ν , z ) + I b ( ν , z ) h ν Γ [ σ a + σ e ] d ν + 1 / τ .
I f / b ( ν , z ) z = I f / b ( ν , z ) [ Γ σ a ( N 0 N ( z ) ) + Γ σ e N ( z ) γ ( ν ) ] + ϱ ( ν ) N ( z ) τ .
ϱ ( ν ) = h ν g ( ν ) η 1 2 [ 1 ( 1 N A 2 ) 1 / 2 ] .
Γ ( ν ) = 1 exp [ 2 ( a / w ( ν ) ) 2 ] ,
N ( z ) = 0 , I f ( ν , z ) = 0 , I b ( ν , z ) = 0 .
I f ( ν , z = 0 ) = R b I b ( ν , z = 0 ) + I p ( ν = ν p ) , I b ( ν , z = L ) = R f I f ( ν , z = L ) ,

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