Abstract

Raman fiber lasers have been proposed as potential candidates for scaling beyond the power limitations imposed on near diffraction-limited rare-earth doped fiber lasers. One limitation is the modal instability (MI) and we explore the physics of this phenomenon in Raman fiber amplifiers (RFAs). By utilizing the conservation of number of photons and conservation of energy in the absence of loss, the 3 × 3 governing system of nonlinear equations describing the pump and the signal modal content are decoupled and solved analytically for cladding-pumped RFAs. By comparing the extracted signal at MI threshold for the same step index-fiber, it is found that the MI threshold is independent of the length of the amplifier or whether the amplifier is co-pumped or counter-pumped; dictated by the integrated heat load along the length of fiber. We extend our treatment to gain-tailored RFAs and show that this approach is of limited utility in suppressing MI. Finally, we formulate the physics of MI in core-pumped RFAs where both pump and signal interferences participate in writing the time-dependent index of refraction grating.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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  1. J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
    [Crossref] [PubMed]
  2. J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010).
    [Crossref]
  3. A. Yusim, “Recent progress in scaling of high power fiber lasers at IPG Photonics,” presented at Solid State and Diode Laser Technology Review,(2009).
  4. T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
    [Crossref] [PubMed]
  5. B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
    [Crossref] [PubMed]
  6. C. A. Codemard, P. Dupriez, Y. Jeong, J. K. Sahu, M. Ibsen, and J. Nilsson, “High-power continuous-wave cladding-pumped Raman fiber laser,” Opt. Lett. 31(15), 2290–2292 (2006).
    [Crossref] [PubMed]
  7. J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010).
    [Crossref] [PubMed]
  8. B. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19(12), 11852–11866 (2011).
    [Crossref] [PubMed]
  9. H. J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015).
    [Crossref] [PubMed]
  10. Y. Feng, L. R. Taylor, and D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express 17(21), 19021–19026 (2009).
    [Crossref] [PubMed]
  11. C. Vergien, I. Dajani, and C. Robin, “18 W single-stage single-frequency acoustically tailored Raman fiber amplifier,” Opt. Lett. 37(10), 1766–1768 (2012).
    [Crossref] [PubMed]
  12. L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014).
    [Crossref]
  13. V. R. Supradeepa and J. W. Nicholson, “Power scaling of high-efficiency 1.5 μm cascaded Raman fiber lasers,” Opt. Lett. 38(14), 2538–2541 (2013).
    [Crossref] [PubMed]
  14. L. Zhang, C. Liu, H. Jiang, Y. Qi, B. He, J. Zhou, X. Gu, and Y. Feng, “Kilowatt Ytterbium-Raman fiber laser,” Opt. Express 22(15), 18483–18489 (2014).
    [Crossref] [PubMed]
  15. Q. Xiao, P. Yan, D. Li, J. Sun, X. Wang, Y. Huang, and M. Gong, “Bidirectional pumped high power Raman fiber laser,” Opt. Express 24(6), 6758–6768 (2016).
    [Crossref] [PubMed]
  16. J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
    [Crossref]
  17. C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
    [Crossref]
  18. R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic, 2008).
  19. S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013).
    [Crossref] [PubMed]
  20. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermally induced mode coupling in rare-earth doped fiber amplifiers,” Opt. Lett. 37(12), 2382–2384 (2012).
    [Crossref] [PubMed]
  21. C. Jauregui, H. J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
    [Crossref] [PubMed]
  22. Raman Amplification in Fiber Optical Communication Systems, edited by C. Headley and G. P. Agrawal (Elsevier, 2005).
  23. A. V. Smith and J. J. Smith, “Spontaneous Rayleigh seed for stimulated Rayleigh scattering in high power fiber amplifiers,” IEEE Photonics J. 5(5), 7100807 (2013).
    [Crossref]
  24. C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).
  25. H. Otto, C. Jauregui, F. Stutzki, J. Limpert, and A. Tünnermann, “Dependence of mode instabilities on the extracted power of fiber laser systems,” in Advanced Solid-State Lasers Congress, G. Huber and P. Moulton, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper ATu3A.02.
    [Crossref]
  26. C. Robin, I. Dajani, and B. Pulford, “Modal instability-suppressing, single-frequency photonic crystal fiber amplifier with 811 W output power,” Opt. Lett. 39(3), 666–669 (2014).
    [Crossref] [PubMed]

2016 (2)

2015 (1)

2014 (3)

2013 (3)

2012 (3)

2011 (2)

2010 (3)

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
[Crossref]

J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010).
[Crossref] [PubMed]

2009 (2)

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

Y. Feng, L. R. Taylor, and D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express 17(21), 19021–19026 (2009).
[Crossref] [PubMed]

2008 (1)

2006 (1)

Alkeskjold, T. T.

Barty, C. P. J.

Beach, R. J.

Breitkopf, S.

Broeng, J.

Bullington, A. L.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

Calia, D. B.

Codemard, C. A.

C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
[Crossref]

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

C. A. Codemard, P. Dupriez, Y. Jeong, J. K. Sahu, M. Ibsen, and J. Nilsson, “High-power continuous-wave cladding-pumped Raman fiber laser,” Opt. Lett. 31(15), 2290–2292 (2006).
[Crossref] [PubMed]

Cui, S.

L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014).
[Crossref]

Dajani, I.

Dawson, J. W.

Dubinskii, M.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

Dupriez, P.

Eidam, T.

Feng, Y.

Gong, M.

Gu, X.

Hansen, K. R.

He, B.

Heebner, J. E.

Hu, J.

L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014).
[Crossref]

Huang, Y.

Ibsen, M.

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

C. A. Codemard, P. Dupriez, Y. Jeong, J. K. Sahu, M. Ibsen, and J. Nilsson, “High-power continuous-wave cladding-pumped Raman fiber laser,” Opt. Lett. 31(15), 2290–2292 (2006).
[Crossref] [PubMed]

Jansen, F.

Jauregui, C.

Jeong, Y.

Ji, J.

C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
[Crossref]

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

Jiang, H.

L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014).
[Crossref]

L. Zhang, C. Liu, H. Jiang, Y. Qi, B. He, J. Zhou, X. Gu, and Y. Feng, “Kilowatt Ytterbium-Raman fiber laser,” Opt. Express 22(15), 18483–18489 (2014).
[Crossref] [PubMed]

Lægsgaard, J.

Li, D.

Limpert, J.

Liu, C.

Madden, T.

Messerly, M. J.

Modsching, N.

Naderi, S.

Nicholson, J. W.

Nilsson, J.

C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
[Crossref]

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

C. A. Codemard, P. Dupriez, Y. Jeong, J. K. Sahu, M. Ibsen, and J. Nilsson, “High-power continuous-wave cladding-pumped Raman fiber laser,” Opt. Lett. 31(15), 2290–2292 (2006).
[Crossref] [PubMed]

Otto, H. J.

Pax, P. H.

Pulford, B.

Qi, Y.

Robin, C.

Sahu, J. K.

C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
[Crossref]

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

C. A. Codemard, P. Dupriez, Y. Jeong, J. K. Sahu, M. Ibsen, and J. Nilsson, “High-power continuous-wave cladding-pumped Raman fiber laser,” Opt. Lett. 31(15), 2290–2292 (2006).
[Crossref] [PubMed]

Schmidt, O.

Schreiber, T.

Shverdin, M. Y.

Siders, C. W.

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

Smith, A. V.

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

Smith, J. J.

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

Sridharan, A. K.

Stappaerts, E. A.

Stutzki, F.

Sun, J.

Supradeepa, V. R.

Taylor, L. R.

Tünnermann, A.

Vergien, C.

Wang, X.

Ward, B.

Wirth, C.

Xiao, Q.

Yan, P.

Zhang, L.

L. Zhang, C. Liu, H. Jiang, Y. Qi, B. He, J. Zhou, X. Gu, and Y. Feng, “Kilowatt Ytterbium-Raman fiber laser,” Opt. Express 22(15), 18483–18489 (2014).
[Crossref] [PubMed]

L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014).
[Crossref]

Zhou, J.

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

J. Ji, C. A. Codemard, M. Ibsen, J. K. Sahu, and J. Nilsson, “Analysis of the conversion to the first Stokes in cladding-pumped fiber Raman amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15(1), 129–139 (2009).
[Crossref]

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(5), 7100807 (2013).
[Crossref]

Laser Photonics Rev. (1)

L. Zhang, H. Jiang, S. Cui, J. Hu, and Y. Feng, “Versatile Raman fiber laser for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014).
[Crossref]

Opt. Express (11)

S. Naderi, I. Dajani, T. Madden, and C. Robin, “Investigations of modal instabilities in fiber amplifiers through detailed numerical simulations,” Opt. Express 21(13), 16111–16129 (2013).
[Crossref] [PubMed]

L. Zhang, C. Liu, H. Jiang, Y. Qi, B. He, J. Zhou, X. Gu, and Y. Feng, “Kilowatt Ytterbium-Raman fiber laser,” Opt. Express 22(15), 18483–18489 (2014).
[Crossref] [PubMed]

H. J. Otto, N. Modsching, C. Jauregui, J. Limpert, and A. Tünnermann, “Impact of photodarkening on the mode instability threshold,” Opt. Express 23(12), 15265–15277 (2015).
[Crossref] [PubMed]

Q. Xiao, P. Yan, D. Li, J. Sun, X. Wang, Y. Huang, and M. Gong, “Bidirectional pumped high power Raman fiber laser,” Opt. Express 24(6), 6758–6768 (2016).
[Crossref] [PubMed]

C. Jauregui, H. J. Otto, S. Breitkopf, J. Limpert, and A. Tünnermann, “Optimizing high-power Yb-doped fiber amplifier systems in the presence of transverse mode instabilities,” Opt. Express 24(8), 7879–7892 (2016).
[Crossref] [PubMed]

J. W. Dawson, M. J. Messerly, R. J. Beach, M. Y. Shverdin, E. A. Stappaerts, A. K. Sridharan, P. H. Pax, J. E. Heebner, C. W. Siders, and C. P. J. Barty, “Analysis of the scalability of diffraction-limited fiber lasers and amplifiers to high average power,” Opt. Express 16(17), 13240–13266 (2008).
[Crossref] [PubMed]

Y. Feng, L. R. Taylor, and D. B. Calia, “25 W Raman-fiber-amplifier-based 589 nm laser for laser guide star,” Opt. Express 17(21), 19021–19026 (2009).
[Crossref] [PubMed]

J. E. Heebner, A. K. Sridharan, J. W. Dawson, M. J. Messerly, P. H. Pax, M. Y. Shverdin, R. J. Beach, and C. P. J. Barty, “High brightness, quantum-defect-limited conversion efficiency in cladding-pumped Raman fiber amplifiers and oscillators,” Opt. Express 18(14), 14705–14716 (2010).
[Crossref] [PubMed]

B. Ward, “Solid-core photonic bandgap fibers for cladding-pumped Raman amplification,” Opt. Express 19(12), 11852–11866 (2011).
[Crossref] [PubMed]

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H. J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20(10), 11407–11422 (2012).
[Crossref] [PubMed]

Opt. Lett. (5)

Proc. SPIE (2)

J. W. Dawson, M. J. Messerly, J. E. Heebner, P. H. Pax, A. K. Sridharan, A. L. Bullington, R. J. Beach, C. W. Siders, C. P. J. Barty, and M. Dubinskii, “Power scaling analysis of fiber lasers and amplifiers based on non-silica materials,” Proc. SPIE 7686, 768611 (2010).
[Crossref]

C. A. Codemard, J. Ji, J. K. Sahu, and J. Nilsson, “100-W CW cladding-pumped Raman fiber laser at 1120 nm,” Proc. SPIE 7580, 75801N (2010).
[Crossref]

Other (5)

R. W. Boyd, Nonlinear Optics, 3rd Ed. (Academic, 2008).

Raman Amplification in Fiber Optical Communication Systems, edited by C. Headley and G. P. Agrawal (Elsevier, 2005).

A. Yusim, “Recent progress in scaling of high power fiber lasers at IPG Photonics,” presented at Solid State and Diode Laser Technology Review,(2009).

C. A. Balanis, Advanced Engineering Electromagnetics, (John Wiley & Sons, 1989).

H. Otto, C. Jauregui, F. Stutzki, J. Limpert, and A. Tünnermann, “Dependence of mode instabilities on the extracted power of fiber laser systems,” in Advanced Solid-State Lasers Congress, G. Huber and P. Moulton, eds., OSA Technical Digest (online) (Optical Society of America, 2013), paper ATu3A.02.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Gain-Tailored design: Region with a Raman gain of g 1 in blue and region with a Raman gain of g 2 in red.
Fig. 2
Fig. 2 Evolution of total signal and HOM powers as a function of position along the fiber for a co-pumped and a counter-pumped RFA. In both cases the fiber length was chosen to provide an amplifier efficiency of 70%. The dashed horizontal line (here and in some of the figures below) represents the 5% level that defines the MI threshold.
Fig. 3
Fig. 3 Heat load vs. longitudinal position along the fiber for the co-pumped and counter-pumped configurations. The fiber lengths were chosen to provide in both cases 70% efficiency.
Fig. 4
Fig. 4 MI signal threshold as a function of seed power showing a linear dependence.
Fig. 5
Fig. 5 Higher order mode content as a function of longitudinal position for co-pumped RFAs with different lengths. For these cases, the pump powers were set at approximately 800, 900, 1000, 1100, 1200 W, but the MI threshold occurred at the same signal power level (556W).
Fig. 6
Fig. 6 Higher-order mode content in a cladding-pumped RFA as a function of signal power for various inner core radii. The Raman gain coefficients are 5× 10 13 m/W and 10 13 m/W for the inner and outer regions, respectively.
Fig. 7
Fig. 7 Evolution of total signal and HOM powers as a function of position along the direction of propagation for a co-propagating, core-pumped, 50 m long RFA.
Fig. 8
Fig. 8 MI threshold in a co-propagating, core-pumped RFA as a function of an artificially induced Δ(Δβ) .
Fig. 9
Fig. 9 Evolution of total signal and HOM powers as a function of position along a fiber for a co-propagating, core-pumped 10 m long RFA. Here, the MI threshold is lower than what was computed for a 50 m long RFA (see Fig. 7).

Equations (31)

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

E s (x,y,z)= 1 2 k A sk (z) φ sk (x,y) e i( β sk z ω sk t ) +c.c.,
Q ˜ = n 0 (α1) 2 μ 0 c ( k | A sk | 2 | φ sk | 2 + kl A sk * A sl φ sk * φ sl exp[ i( Δ β skl zΔ ω kl t ) ]+c.c. ) g R P P A clad  ,
2 ΔTqΔT=Q/ κ th ,
ΔT( r ,Ω)= 1 κ th G( r , r ,q(Ω))Q( r ,Ω) d 2 r ,
2 G( r , r ,q(Ω))q(Ω)G( r , r ,q(Ω))=δ( r r ).
ΔT( r ,Δ ω smj )+ΔT( r ,Δ ω smj )= g R P p A clad mj A sm * A sj Γ mj ( r ,Δ ω smj ) e i( Δ β smj zΔ ω smj t ) ,
Γ mj ( r ,Δ ω smj )= n 0 (α1) 2 μ 0 c κ th φ sm ( r ' ),G( r , r ' ,q(Δ ω smj ) ) φ sj ( r ' ) , φ sm ( r ' ),G( r , r ' ,q(Δ ω smj ) ) φ sj ( r ' ) = φ sm * ( r ' )G( r , r ' ,q(Δ ω smj ) ) φ sj ( r ' ) d 2 r ' .
d A sj dz = g R P p A clad [ 1 2 +i k 0 ( m | A sm | 2 κ jjmm (0) + mj | A sm | 2 κ jmjm ( (1) j1 Δω ) ) ] A sj ,
κ jmkl (Δω)= dn dT φ sj * ( r ) Γ kl ( r ,Δω) φ sm ( r ) d 2 r .
d P sj dz = g R P sj [ 1+ (1) j jm γ jm (Δω) P sm ] P P A clad  ,
γ jm (Δω)=4 k 0 μ 0 c n 0 Im( κ jmjm ( Δω ) ),
γ jm (Δω)= γ jm (Δω).
d P p dz =α g R ( j P sj ) P p A clad  .
d P tot dz = g R P tot P P A clad  .
P p (z)=α P tot (z)+ P p ( 0 )+α P tot (0).
P tot (z)= exp( C 1 A clad g R z ) P tot (0) C 1 [ P p (0)+αexp( C 1 A clad g R z ) P tot (0) ] ,
P sj ( z )= P tot ( z ) 1+ P skj ( 0 ) P sj ( 0 ) [ exp[ (1) j1 γ 12 ( P tot ( z ) P tot ( 0 ) ) ] ] ,
P p ( 0 ) α γ 12,max ln( P s1 ( 0 ) 19 P s2 ( 0 ) ),
P tot ( z )= exp( C 2 A clad g R z ) P tot (0) C 2 [ C 2 +α P tot (0)( 1exp( C 2 A clad g R z ) ) ] ,
d P sj dz = P sj [ g sj (eff) + (1) j jm γ ˜ jm (Δω) P sm ] P p A clad  .
g sj (eff) = [ g 1 H( R 1 r )+ g 2 H( r R 1 ) ] | φ sj | 2 d 2 r ,
γ ˜ jm (Δω)=4 k 0 μ 0 c n 0 Im (   κ ˜ jmjm ( Δω ) ),
κ ˜ jmjm (Δω)= dn dT φ sj * ( r ) Γ ˜ jm ( r ,Δω) φ sm ( r ) d 2 r ,
Γ ˜ jm ( r ,Δω)= n 0 (α1) 2 μ 0 c κ th φ sm ( r ' ),[ g 1 H( R 1 r )+ g 2 H( r R 1 ) ]G( r , r ' ,q( Δω ) ) φ sj ( r ' ) .
  d P p dz α( j g eff sj P sj ) P p A clad  .
G(r; r , θ )= 1 4 { m= [ J m ( q r ) Y m ( q r clad ) J m ( q r clad ) Y m ( q r ) ] J m ( q r ) J m ( q r clad ) e im( θ θ ) for r< r m= [ J m ( q r ) Y m ( q r clad ) J m ( q r clad ) Y m ( q r ) ] J m ( q r ) J m ( q r clad ) e im( θ θ ) for r> r ,
Q ˜ (z) t = Q ˜ dxdy= (α1) g R A clad P tot P p .
Q ˜ CorePumped = g R,0 ( α1 ) ( n 0 2c μ 0 ) 2 ( k | A sk | 2 | φ k | 2 + kl A sk * A sl φ k * φ l exp[ i(Δ β s zΔωt) ]+c.c. ) ×( k | A pk | 2 | φ k | 2 + kl A pk * A pl φ k * φ l exp( i(Δ β p zΔωt) )+c.c. ),
d A sj dz =i k 0s A sj [ m l | A sm | 2 | A pl | 2 ν jjmlml ( r ,0 ) +2Re( A s1 * A p1 A s2 A p2 * ν jj1212 ( r ,0 ) e i(Δ β s Δ β p )z ) + m lj | A sl | 2 | A pm | 2 ν jlmlmj ( r , (1) j1 Δω ) i n 0 4c k 0s μ 0 m | A pm | 2 Λ jmjm ] +i k 0s ( A pj e (1) j1 i(Δ β s Δ β p )z m lj A pl * A sl | A sm | 2 ν jlmlmj ( r , (1) j1 Δω ) ),
d A pj dz =i k 0p A pj [ m l | A pm | 2 | A sl | 2 ν jjmlml ( r ,0 ) +2Re( A p1 * A s1 A p2 A s2 * ν jj1212 ( r ,0 ) e i(Δ β s Δ β p )z ) + m lj | A pl | 2 | A sm | 2 ν jlmlmj ( r , (1) j1 Δω )+ iα n 0 4c k 0p μ 0 m | A sm | 2 Λ jmjm ] +i k 0p ( A sj e (1) j i(Δ β s Δ β p )z m lj A sl * A pl | A pm | 2 ν jlmlmj ( r , (1) j1 Δω ) ),
ν ijklmn ( r ,Δω )=ξ φ i * ( r ) φ k ( r ) φ l ( r )G( r , r ,q(Δω)) φ m ( r ) φ n ( r ) φ j ( r ) d 2 r . Λ ijkl ( r )= g R,0 φ i * ( r ) φ j * ( r ) φ k ( r ) φ l ( r ) d 2 r . ξ= ( n 0 2 μ 0 c ) 2 dn dT g R,0 κ th (α1).

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