Abstract

Recently, mode instability was observed in optical fiber lasers at high powers, severely limiting power scaling for single-mode outputs. Some progress has been made towards understanding the underlying physics. A thorough understanding of the effect is critical for continued progress of this very important technology area. Mode instability in optical fibers is, in fact, a manifestation of stimulated thermal Rayleigh scattering. In this work, a quasi-closed-form solution for the nonlinear coupling coefficient is found for stimulated thermal Rayleigh scattering in optical fibers. The results help to significantly improve understanding of mode instability.

© 2013 OSA

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References

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  1. T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010).
    [Crossref] [PubMed]
  2. F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011).
    [Crossref] [PubMed]
  3. 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 laser amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
    [Crossref] [PubMed]
  4. F. Stutzki, H. J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36(23), 4572–4574 (2011).
    [Crossref] [PubMed]
  5. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20(1), 440–451 (2012).
    [Crossref] [PubMed]
  6. C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
    [Crossref] [PubMed]
  7. K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19(24), 23965–23980 (2011).
    [Crossref] [PubMed]
  8. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
    [Crossref] [PubMed]
  9. 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]
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    [Crossref]
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    [Crossref]
  13. D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
    [Crossref]
  14. I. L. Fabelinskii and V. S. Starunov, “Some studies of the spectra of thermal and stimulated molecular scattering of light,” Appl. Opt. 6(11), 1793–1804 (1967).
    [Crossref] [PubMed]
  15. C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
    [Crossref]
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    [Crossref]
  17. N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
    [Crossref]
  18. L. M. Peterson and T. A. Wiggins, “Forward stimulated thermal Rayleigh scattering,” J. Opt. Soc. Am. 63(1), 13–16 (1973).
    [Crossref]
  19. R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A 27(4), 1968–1976 (1983).
    [Crossref]
  20. H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron. 22(4), 552–562 (1986).
    [Crossref]
  21. H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B 3(2), 253–273 (1986).
    [Crossref]
  22. R. W. Boyd, “Nonlinear Optics,” third edition, Elsevier, 2008.
  23. A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” Chapman and Hall, 1983.
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    [Crossref]
  25. C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
    [Crossref] [PubMed]
  26. A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express 20(22), 24545–24558 (2012).
    [Crossref] [PubMed]

2012 (5)

2011 (6)

2010 (1)

1998 (1)

1986 (2)

H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron. 22(4), 552–562 (1986).
[Crossref]

H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B 3(2), 253–273 (1986).
[Crossref]

1983 (1)

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A 27(4), 1968–1976 (1983).
[Crossref]

1973 (1)

1971 (1)

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
[Crossref]

1969 (1)

W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 22(18), 915–918 (1969).
[Crossref]

1968 (1)

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
[Crossref]

1967 (4)

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. 18(4), 107–109 (1967).
[Crossref]

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
[Crossref]

I. L. Fabelinskii and V. S. Starunov, “Some studies of the spectra of thermal and stimulated molecular scattering of light,” Appl. Opt. 6(11), 1793–1804 (1967).
[Crossref] [PubMed]

Alkeskjold, T. T.

Andersen, T. V.

Barker, J. A.

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A 27(4), 1968–1976 (1983).
[Crossref]

Bloembergen, N.

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
[Crossref]

Broeng, J.

Cho, C. W.

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
[Crossref]

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. 18(4), 107–109 (1967).
[Crossref]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
[Crossref]

Dajani, I.

Davis, M. K.

Desai, R. C.

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A 27(4), 1968–1976 (1983).
[Crossref]

Digonnet, M. J. F.

Eidam, T.

Fabelinskii, I. L.

Foltz, N. D.

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
[Crossref]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
[Crossref]

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. 18(4), 107–109 (1967).
[Crossref]

Gabler, T.

Gaida, C.

Gray, M. A.

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

Hanf, S.

Hansen, K. R.

Herman, R. M.

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

Hoffman, H. J.

H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron. 22(4), 552–562 (1986).
[Crossref]

H. J. Hoffman, “Thermally induced phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B 3(2), 253–273 (1986).
[Crossref]

Jansen, F.

Jauregui, C.

Kaiser, W.

W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 22(18), 915–918 (1969).
[Crossref]

Lægsgaard, J.

Levenson, M. D.

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A 27(4), 1968–1976 (1983).
[Crossref]

Limpert, J.

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20(1), 440–451 (2012).
[Crossref] [PubMed]

F. Stutzki, H. J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36(23), 4572–4574 (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 laser amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011).
[Crossref] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010).
[Crossref] [PubMed]

Lowdermilk, W. H.

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
[Crossref]

Matsuoka, M.

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
[Crossref]

Otto, H. J.

Pantell, R. H.

Peterson, L. M.

Pohl, D.

W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 22(18), 915–918 (1969).
[Crossref]

Rank, D. H.

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
[Crossref]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
[Crossref]

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. 18(4), 107–109 (1967).
[Crossref]

Robin, C.

Rother, W.

W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 22(18), 915–918 (1969).
[Crossref]

Schmidt, O.

Schreiber, T.

Seise, E.

Smith, A. V.

Smith, J. J.

Starunov, V. S.

Steinmetz, A.

Stutzki, F.

Tünnermann, A.

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20(1), 440–451 (2012).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (2012).
[Crossref] [PubMed]

F. Stutzki, H. J. Otto, F. Jansen, C. Gaida, C. Jauregui, J. Limpert, and A. Tünnermann, “High-speed modal decomposition of mode instabilities in high-power fiber lasers,” Opt. Lett. 36(23), 4572–4574 (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 laser amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

F. Stutzki, F. Jansen, T. Eidam, A. Steinmetz, C. Jauregui, J. Limpert, and A. Tünnermann, “High average power large-pitch fiber amplifier with robust single-mode operation,” Opt. Lett. 36(5), 689–691 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
[Crossref] [PubMed]

T. Eidam, S. Hanf, E. Seise, T. V. Andersen, T. Gabler, C. Wirth, T. Schreiber, J. Limpert, and A. Tünnermann, “Femtosecond fiber CPA system emitting 830 W average output power,” Opt. Lett. 35(2), 94–96 (2010).
[Crossref] [PubMed]

Ward, B.

Wiggins, T. A.

L. M. Peterson and T. A. Wiggins, “Forward stimulated thermal Rayleigh scattering,” J. Opt. Soc. Am. 63(1), 13–16 (1973).
[Crossref]

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
[Crossref]

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. 18(4), 107–109 (1967).
[Crossref]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
[Crossref]

Wirth, C.

Wong, C. S.

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
[Crossref]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron. 22(4), 552–562 (1986).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

Opt. Express (8)

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Physical origin of mode instabilities in high-power fiber laser systems,” Opt. Express 20(12), 12912–12925 (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(22), 24545–24558 (2012).
[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 laser amplifiers,” Opt. Express 19(14), 13218–13224 (2011).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, H. J. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Temperature-induced index gratings and their impact on mode instabilities in high-power fiber laser systems,” Opt. Express 20(1), 440–451 (2012).
[Crossref] [PubMed]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express 19(4), 3258–3271 (2011).
[Crossref] [PubMed]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Thermo-optical effects in high-power ytterbium-doped fiber amplifiers,” Opt. Express 19(24), 23965–23980 (2011).
[Crossref] [PubMed]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (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. (4)

Phys. Rev. (1)

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. 175(1), 271–274 (1968).
[Crossref]

Phys. Rev. A (2)

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A 27(4), 1968–1976 (1983).
[Crossref]

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A 3(1), 404–412 (1971).
[Crossref]

Phys. Rev. Lett. (4)

W. Rother, D. Pohl, and W. Kaiser, “Time and frequency dependence of stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 22(18), 915–918 (1969).
[Crossref]

C. W. Cho, N. D. Foltz, D. H. Rank, and T. A. Wiggins, “Stimulated Rayleigh scattering,” Phys. Rev. Lett. 18(4), 107–109 (1967).
[Crossref]

R. M. Herman and M. A. Gray, “Theoretical prediction of the stimulated thermal Rayleigh scattering in liquid,” Phys. Rev. Lett. 19(15), 824–828 (1967).
[Crossref]

D. H. Rank, C. W. Cho, N. D. Foltz, and T. A. Wiggins, “Stimulated thermal Rayleigh scattering,” Phys. Rev. Lett. 19(15), 828–830 (1967).
[Crossref]

Other (2)

R. W. Boyd, “Nonlinear Optics,” third edition, Elsevier, 2008.

A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” Chapman and Hall, 1983.

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Figures (6)

Fig. 1
Fig. 1

(a) Simulated nonlinear coupling coefficient amplitude χmnl and damping factor Γml for each l, and (b) real, imaginary and phase of χ for a step index fiber with NA = 0.06, 2a = 2d = 30μm, 2b = 400μm and V = 5.3348. Simulations of 4 fibers with core diameters 2a = 30μm, 25μm, 20μm, and 15μm, with rest of the parameters kept constant are summarized in Fig. 2.

Fig. 2
Fig. 2

(a) Simulated nonlinear coupling coefficient amplitude χmnl and damping factor Γml for each l, and (b) real, imaginary and phase of χ for step index fibers with NA = 0.06, 2b = 400μm, 2a = 2d = 30μm, 25μm, 20μm, and 15μm respectively, V = 5.3348, 4.4456, 3.5565 and 2.6674 respectively.

Fig. 3
Fig. 3

Nonlinear coupling coefficient χ at the peak of real part of χ and the corresponding frequency fmax for a step index fiber with 2b = 400μm, 2a = 2d = 30μm, dependence on (a) V (NA is varied to change V) and (b) fraction of the doped radius d/a at NA = 0.06.

Fig. 4
Fig. 4

Nonlinear coupling coefficient χ at the peak of real part of χ and the corresponding frequency fmax for step index fibers with 2b = 400μm and 2a = 2d, core diameter is varied while V is kept constant for each lines (NA is varied to keep V constant).

Fig. 5
Fig. 5

Simulated threshold powers (a) at x = 1% and various input conditions with P11(0)/P01(0) = 10−5, 10−10, 10−15, 10−20, 10−25, and 10−30 and (b) x = 0.0526, i.e. 5% of total power in LP11 mode, for P11(0)/P01(0) = 10−2, 10−3, and 10−4. The fiber parameters are NA = 0.06, 2b = 400μm, 2a = 2d = 30μm, V = 5.3348 and α11 = 0. The dashed red lines in (a) are obtained from Eq. (32) and solid black lines in both figures are from Eq. (35).

Fig. 6
Fig. 6

Simulated LP01 and LP11 powers along a fiber amplifier, (a) without re-seeding and (b) with re-seeding. The fiber parameters are NA = 0.06, 2b = 400μm, 2a = 2d = 30μm, V = 5.3348 and α11 = 0. Total amplifier gain is 13.5dB at the STRS threshold. Input power P01(0) = 19.2W and P11(0) = 10−25 × P01(0). Threshold power is 428.3W.

Tables (2)

Tables Icon

Table 1 Coefficients of Silica Used in This Work

Tables Icon

Table 2 Benchmarking of Maximum Nonlinear Coupling Coefficient to These in [10]

Equations (43)

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

e mn ( r,ϕ )= f mn ( r ) 2n ε 0 c N mn cos( mϕ )
f mn ( r )= J m ( U mn r/a ) J m ( U mn ) ar0
f mn ( r )= K m ( W mn r/a ) K m ( W mn ) r>a
N 0n = 0 rdr 0 2π f 0n 2 ( r ) dϕ=2π 0 f 0n 2 ( r )rdr for m=0
N mn = 0 rdr 0 2π f mn 2 ( r ) cos 2 ( mϕ )dϕ=π 0 f mn 2 ( r )rdr for m>0
E( r,ϕ,z )= m=0 n=1 E mn ( r,ϕ,z ) e i( β mn z ω mn t )
E mn ( r,ϕ,z )= P mn (z) e mn ( r,ϕ )
I m=0 n=1 P mn (z) f mn 2 ( r ) N mn cos 2 ( mϕ ) + 2 m=1 n=1 P 01 ( z ) P mn ( z ) f 01 ( r ) f mn ( r ) N 01 N mn cos( mϕ )cos[ ( β mn β 01 )z( ω mn ω 01 )t ]
T( r,ϕ,z,t )= T 0 ( r,ϕ,z )+ T ˜ ( r,ϕ ) e i( qzΩt )
Ω= ω mn ω 01
q= β mn β 01
ρC T ˜ ( r,ϕ ) e i( qzΩt ) t κ 2 T ˜ ( r,ϕ ) e i( qzΩt ) = 1 2 ( λ s λ p 1 ) P 01 ( z ) P mn ( z ) N 01 N mn g( r ) f 01 ( r ) f mn ( r ) e imϕ e i( qzΩt )
T ˜ ( r,ϕ )= T m ( r ) e imϕ
2 T m ( r ) r 2 + 1 r T m ( r ) r +( i ΩρC κ q 2 m 2 r 2 ) T m ( r )= 1 2κ ( λ s λ p 1 ) P 01 ( z ) P mn ( z ) N 01 N mn g( r ) f 01 ( r ) f mn ( r )
2 T ml ( r ) r 2 + 1 r T ml ( r ) r +( q ml 2 +i ΩρC κ m 2 r 2 ) T ml ( r )=0
T ml ( r )= J m ( r q ml 2 +i ΩρC κ )
T ml ( b )=0
q ml 2 π 2 16 b 2 ( 4l1+2m ) 2 i ΩρC κ
T ml ( r ) J m [ π 4b ( 4l1+2m )r ]
T m ( r )= l=1 a l T ml ( r )
0 b T m l 1 ( r ) T m l 2 ( r )rdr=0 when l 1 l 2
0 b T m l 1 ( r ) T m l 2 ( r )rdr= b 2 2 J m1 [ π 4 ( 4l1+2m ) ] J m+1 [ π 4 ( 4l1+2m ) ] when l 1 = l 2 =l
a l = 1 2κ ( λ s λ p 1 ) P 01 ( z ) P mn ( z ) N 01 N mn 0 b g( r ) f 01 ( r ) f mn ( r ) T ml ( r )rdr [ q 2 + π 2 16 b 2 ( 4l1+2m ) 2 i ΩρC κ ] 0 b T ml 2 ( r )rdr
n ˜ = k T T ˜ ( r,ϕ ) e i( qzΩt ) = k T T m ( r ) e imϕ e i( qzΩt )
ε ˜ 2n n ˜ =2n k T T ˜ ( r,ϕ ) e i( qzΩt ) =2n k T T m ( r ) e imϕ e i( qzΩt )
p ˜ NL = ε 0 ε ˜ E=2n ε 0 k T T m * ( r ) e imϕ e i( β 01 z ω 01 t ) E mn +2n ε 0 k T T m ( r ) e imϕ e i( β mn z ω mn t ) E 01
2 E ( n c ) 2 2 E t 2 = 1 ε 0 c 2 2 p ˜ NL t 2
P 01 ( z ) z = g 01 real( χ mn * ) P 01 ( z ) P mn ( z )+ g 01 P 01 ( z )
P mn ( z ) z = g 01 real( χ mn ) P 01 ( z ) P mn ( z )+( g mn α mn ) P mn ( z )
g 01 = 1 N 01 0 rdr 0 2π g( r ) f 01 2 ( r )dϕ= 0 g( r ) f 01 2 ( r )rdr 0 f 01 2 ( r )rdr
g mn = 1 N mn 0 rdr 0 2π g( r ) f mn 2 ( r ) cos 2 ( mϕ )dϕ= 0 g( r ) f mn 2 ( r )rdr 0 f mn 2 ( r )rdr
g 01 χ mn = g 01 ( χ mn r +i χ mn i )= g 01 l=1 2( 2Ω Γ ml i ) 1+ ( 2Ω Γ ml ) 2 χ mnl = 2πk k T ρC ( λ s λ p 1 ) l=1 2( 2Ω Γ ml i ) 1+ ( 2Ω Γ ml ) 2 0 d g( r ) f 01 ( r ) f mn ( r ) T ml ( r )rdr 0 b f 01 ( r ) f mn ( r ) T ml ( r )rdr N 01 N mn Γ ml 0 b T ml 2 ( r )rdr
Γ ml = 2κ ρC [ q 2 + π 2 16 b 2 ( 4l1+2m ) 2 ]
P 01 N ( z ) z = g 01 χ mn r e ( g mn α mn )z P 01 N ( z ) P mn N ( z )
P mn N ( z ) z = g 01 χ mn r e g 01 z P 01 N ( z ) P mn N ( z )
P 01 N ( z )= P 01 ( z ) e g 01 z
P mn N ( z )= P mn ( z ) e ( g mn α mn )z
P mn ( L ) P mn ( 0 ) e ( g mn g 01 α mn )L e χ mn r P 01 ( L )
P 01 th 1 χ mn r [ ln( x P 01 ( 0 ) P mn ( 0 ) )( g mn g 01 α mn )L ]
P 01 th 1 χ mn r ln( x P 01 ( 0 ) P mn ( 0 ) )
P 01 th 1 χ mn r ln( x P 01 ( 0 ) P mn ( 0 ) ) e 1.25/ e g 01 L
S 01 N ( ω 01 ,z ) z 1 2 g 01 χ mn r e ( g 01 Γ mn Γ 01 α mn )z 0 2 S 01 N ( ω 01 ,z ) S mn N ( ω 01 Ω,z ) d ω 01 S mn N ( ω mn ,z )
S mn N ( ω mn ,z ) z 1 2 g 01 χ mn r e g 01 z 0 2 S 01 N ( ω 01 ,z ) S mn N ( ω 01 Ω,z ) d ω 01 S 01 N ( ω 01 ,z )

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