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.

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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. Express19(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. Express20(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. Express19(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. Express19(24), 23965–23980 (2011).
    [CrossRef] [PubMed]
  8. A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(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. Express20(10), 11407–11422 (2012).
    [CrossRef] [PubMed]
  10. 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]
  11. 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]
  12. 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]
  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]
  16. 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]
  17. N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A3(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. A27(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. B3(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.
  24. M. K. Davis, M. J. F. Digonnet, and R. H. Pantell, “Thermal effects in doped fibers,” J. Lightwave Technol.16(6), 1013–1023 (1998).
    [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. Express20(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. Express20(22), 24545–24558 (2012).
    [CrossRef] [PubMed]

2012

2011

2010

1998

1986

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. B3(2), 253–273 (1986).
[CrossRef]

1983

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A27(4), 1968–1976 (1983).
[CrossRef]

1973

1971

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A3(1), 404–412 (1971).
[CrossRef]

1969

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

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

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. A27(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. A3(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. A27(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 phase conjugation by transient real-time holography: a review,” J. Opt. Soc. Am. B3(2), 253–273 (1986).
[CrossRef]

H. J. Hoffman, “Thermally induced degenerate four-wave mixing,” IEEE J. Quantum Electron.22(4), 552–562 (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. A27(4), 1968–1976 (1983).
[CrossRef]

Limpert, J.

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. Express20(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. Express20(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. Express19(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. Express19(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. A3(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. A3(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]

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]

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. Express20(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. Express20(12), 12912–12925 (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. Express19(14), 13218–13224 (2011).
[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]

C. Jauregui, T. Eidam, J. Limpert, and A. Tünnermann, “The impact of modal interference on the beam quality of high-power fiber amplifiers,” Opt. Express19(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]

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]

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]

Wirth, C.

Wong, C. S.

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A3(1), 404–412 (1971).
[CrossRef]

Appl. Opt.

IEEE J. Quantum Electron.

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

J. Lightwave Technol.

J. Opt. Soc. Am.

J. Opt. Soc. Am. B

Opt. Express

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express19(11), 10180–10192 (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. Express19(14), 13218–13224 (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. Express19(24), 23965–23980 (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. Express20(1), 440–451 (2012).
[CrossRef] [PubMed]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express20(10), 11407–11422 (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. Express20(12), 12912–12925 (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. Express19(4), 3258–3271 (2011).
[CrossRef] [PubMed]

A. V. Smith and J. J. Smith, “Influence of pump and seed modulation on the mode instability thresholds of fiber amplifiers,” Opt. Express20(22), 24545–24558 (2012).
[CrossRef] [PubMed]

Opt. Lett.

Phys. Rev.

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

N. Bloembergen, W. H. Lowdermilk, M. Matsuoka, and C. S. Wong, “Theory of stimulated concentration scattering,” Phys. Rev. A3(1), 404–412 (1971).
[CrossRef]

R. C. Desai, M. D. Levenson, and J. A. Barker, “Forced Rayleigh scattering: thermal and acoustic effects in phase-conjugate,” Phys. Rev. A27(4), 1968–1976 (1983).
[CrossRef]

Phys. Rev. Lett.

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

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 (10)

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

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
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 )
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
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
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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