Stimulated thermal Rayleigh scattering in optical fibers

Liang Dong

doi:10.1364/OE.21.002642

Stimulated thermal Rayleigh scattering in optical fibers

Liang Dong

Optics Express
Vol. 21,
Issue 3,
pp. 2642-2656
(2013)
•https://doi.org/10.1364/OE.21.002642

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Liang Dong, "Stimulated thermal Rayleigh scattering in optical fibers," Opt. Express 21, 2642-2656 (2013)

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Related Topics
Table of Contents Category
- Fiber Optics and Optical Communications
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)
- Lasers, fiber (060.3510)

About this Article
History
- Original Manuscript: November 29, 2012
- Revised Manuscript: December 27, 2012
- Manuscript Accepted: January 17, 2013
- Published: January 28, 2013

Accessible

Open Access

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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References

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Publication

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).
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[Crossref] [PubMed]
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[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]
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]
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]
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]
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]
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]
L. M. Peterson and T. A. Wiggins, “Forward stimulated thermal Rayleigh scattering,” J. Opt. Soc. Am. 63(1), 13–16 (1973).
[Crossref]
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]
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]
R. W. Boyd, “Nonlinear Optics,” third edition, Elsevier, 2008.
A. W. Snyder and J. D. Love, “Optical Waveguide Theory,” Chapman and Hall, 1983.
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]
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]

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)

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

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.

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]

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.

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]

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.

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]

Davis, M. K.

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]

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.

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]

Eidam, T.

Fabelinskii, I. L.

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]

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.

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]

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.

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]

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.

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.

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]

Peterson, L. M.

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

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.

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]

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.

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]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

Smith, J. J.

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]

A. V. Smith and J. J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19(11), 10180–10192 (2011).
[Crossref] [PubMed]

Starunov, V. S.

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]

Steinmetz, A.

Stutzki, F.

Tünnermann, A.

Ward, B.

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]

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)

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]

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)

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]

J. Opt. Soc. Am. (1)

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

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

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]

Opt. Express (8)

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]

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]

Opt. Lett. (4)

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]

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)

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]

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]

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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Fig. 1 (a) Simulated nonlinear coupling coefficient amplitude χ_mn_l and damping factor Γ_m_l 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.

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Fig. 2 (a) Simulated nonlinear coupling coefficient amplitude χ_mn_l and damping factor Γ_m_l 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.

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Fig. 3 Nonlinear coupling coefficient χ at the peak of real part of χ and the corresponding frequency f_max 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.

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Fig. 4 Nonlinear coupling coefficient χ at the peak of real part of χ and the corresponding frequency f_max 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).

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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).

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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.

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Tables (2)

Table 1 Coefficients of Silica Used in This Work

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

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Equations (43)

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

e_{m n} (r, ϕ) = \frac{f_{m n} (r)}{\sqrt{2 n ε_{0} c N_{m n}}} \cos (m ϕ)

f_{m n} (r) = \frac{J_{m} (U_{m n} r / a)}{J_{m} (U_{m n})} a \geq r \geq 0

f_{m n} (r) = \frac{K_{m} (W_{m n} r / a)}{K_{m} (W_{m n})} r > a

N_{0 n} = \int_{0}^{\infty} r d r \int_{0}^{2 π} f_{0 n}^{2} (r) d ϕ = 2 π \int_{0}^{\infty} f_{0 n}^{2} (r) r d r for m = 0

N_{m n} = \int_{0}^{\infty} r d r \int_{0}^{2 π} f_{m n}^{2} (r) \cos^{2} (m ϕ) d ϕ = π \int_{0}^{\infty} f_{m n}^{2} (r) r d r for m > 0

E (r, ϕ, z) = \sum_{m = 0}^{\infty} \sum_{n = 1}^{\infty} E_{m n} (r, ϕ, z) e^{i (β_{m n} z - ω_{m n} t)}

E_{m n} (r, ϕ, z) = \sqrt{P_{m n} (z)} e_{m n} (r, ϕ)

\begin{array}{l} I \approx \sum_{m = 0}^{\infty} \sum_{n = 1}^{\infty} P_{m n} (z) \frac{f_{m n}^{2} (r)}{N_{m n}} \cos^{2} (m ϕ) + \\ 2 \sum_{m = 1}^{\infty} \sum_{n = 1}^{\infty} \sqrt{P_{01} (z) P_{m n} (z)} \frac{f_{01} (r) f_{m n} (r)}{\sqrt{N_{01} N_{m n}}} \cos (m ϕ) \cos [(β_{m n} - β_{01}) z - (ω_{m n} - ω_{01}) t] \end{array}

T (r, ϕ, z, t) = T_{0} (r, ϕ, z) + \tilde{T} (r, ϕ) e^{i (q z - Ω t)}

Ω = ω_{m n} - ω_{01}

q = β_{m n} - β_{01}

\begin{array}{l} ρ C \frac{\partial \tilde{T} (r, ϕ) e^{i (q z - Ω t)}}{\partial t} - κ \nabla^{2} \tilde{T} (r, ϕ) e^{i (q z - Ω t)} = \\ \frac{1}{2} (\frac{λ_{s}}{λ_{p}} - 1) \sqrt{\frac{P_{01} (z) P_{m n} (z)}{N_{01} N_{m n}}} g (r) f_{01} (r) f_{m n} (r) e^{i m ϕ} e^{i (q z - Ω t)} \end{array}

\tilde{T} (r, ϕ) = T_{m} (r) e^{i m ϕ}

\begin{array}{l} \frac{\partial^{2} T_{m} (r)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T_{m} (r)}{\partial r} + (i \frac{Ω ρ C}{κ} - q^{2} - \frac{m^{2}}{r^{2}}) T_{m} (r) = \\ - \frac{1}{2 κ} (\frac{λ_{s}}{λ_{p}} - 1) \sqrt{\frac{P_{01} (z) P_{m n} (z)}{N_{01} N_{m n}}} g (r) f_{01} (r) f_{m n} (r) \end{array}

\frac{\partial^{2} T_{m l} (r)}{\partial r^{2}} + \frac{1}{r} \frac{\partial T_{m l} (r)}{\partial r} + (q_{m l}^{2} + i \frac{Ω ρ C}{κ} - \frac{m^{2}}{r^{2}}) T_{m l} (r) = 0

T_{m l} (r) = J_{m} (r \sqrt{q_{m l}^{2} + i \frac{Ω ρ C}{κ}})

T_{m l} (b) = 0

q_{m l}^{2} \approx \frac{π^{2}}{16 b^{2}} {(4 l - 1 + 2 m)}^{2} - i \frac{Ω ρ C}{κ}

T_{m l} (r) \approx J_{m} [\frac{π}{4 b} (4 l - 1 + 2 m) r]

T_{m} (r) = \sum_{l = 1}^{\infty} a_{l} T_{m l} (r)

\int_{0}^{b} T_{m l_{1}} (r) T_{m l_{2}} (r) r d r = 0 when l_{1} \neq l_{2}

\begin{array}{l} \int_{0}^{b} T_{m l_{1}} (r) T_{m l_{2}} (r) r d r = - \frac{b^{2}}{2} J_{m - 1} [\frac{π}{4} (4 l - 1 + 2 m)] J_{m + 1} [\frac{π}{4} (4 l - 1 + 2 m)] \\ when l_{1} = l_{2} = l \end{array}

\begin{array}{l} a_{l} = \frac{1}{2 κ} (\frac{λ_{s}}{λ_{p}} - 1) \sqrt{\frac{P_{01} (z) P_{m n} (z)}{N_{01} N_{m n}}} \\ \frac{\int_{0}^{b} g (r) f_{01} (r) f_{m n} (r) T_{m l} (r) r d r}{[q^{2} + \frac{π^{2}}{16 b^{2}} {(4 l - 1 + 2 m)}^{2} - i \frac{Ω ρ C}{κ}] \int_{0}^{b} T_{m l}^{2} (r) r d r} \end{array}

\tilde{n} = k_{T} \tilde{T} (r, ϕ) e^{i (q z - Ω t)} = k_{T} T_{m} (r) e^{i m ϕ} e^{i (q z - Ω t)}

\tilde{ε} \approx 2 n \tilde{n} = 2 n k_{T} \tilde{T} (r, ϕ) e^{i (q z - Ω t)} = 2 n k_{T} T_{m} (r) e^{i m ϕ} e^{i (q z - Ω t)}

{\tilde{p}}^{N L} = ε_{0} \tilde{ε} E = 2 n ε_{0} k_{T} T_{m}^{*} (r) e^{i m ϕ} e^{i (β_{01} z - ω_{01} t)} E_{m n} + 2 n ε_{0} k_{T} T_{m} (r) e^{i m ϕ} e^{i (β_{m n} z - ω_{m n} t)} E_{01}

\nabla^{2} E - {(\frac{n}{c})}^{2} \frac{\partial^{2} E}{\partial t^{2}} = \frac{1}{ε_{0} c^{2}} \frac{\partial^{2} {\tilde{p}}^{N L}}{\partial t^{2}}

\frac{\partial P_{01} (z)}{\partial z} = - g_{01} r e a l (χ_{m n}^{*}) P_{01} (z) P_{m n} (z) + g_{01} P_{01} (z)

\frac{\partial P_{m n} (z)}{\partial z} = g_{01} r e a l (χ_{m n}) P_{01} (z) P_{m n} (z) + (g_{m n} - α_{m n}) P_{m n} (z)

g_{01} = \frac{1}{N_{01}} \int_{0}^{\infty} r d r \int_{0}^{2 π} g (r) f_{01}^{2} (r) d ϕ = \frac{\int_{0}^{\infty} g (r) f_{01}^{2} (r) r d r}{\int_{0}^{\infty} f_{01}^{2} (r) r d r}

g_{m n} = \frac{1}{N_{m n}} \int_{0}^{\infty} r d r \int_{0}^{2 π} g (r) f_{m n}^{2} (r) \cos^{2} (m ϕ) d ϕ = \frac{\int_{0}^{\infty} g (r) f_{m n}^{2} (r) r d r}{\int_{0}^{\infty} f_{m n}^{2} (r) r d r}

\begin{array}{l} g_{01} χ_{m n} = g_{01} (χ_{m n}^{r} + i χ_{m n}^{i}) = g_{01} \sum_{l = 1}^{\infty} \frac{2 (\frac{2 Ω}{Γ_{m l}} - i)}{1 + {(\frac{2 Ω}{Γ_{m l}})}^{2}} χ_{m n l} = \frac{2 π k k_{T}}{ρ C} (\frac{λ_{s}}{λ_{p}} - 1) \\ \sum_{l = 1}^{\infty} \frac{2 (\frac{2 Ω}{Γ_{m l}} - i)}{1 + {(\frac{2 Ω}{Γ_{m l}})}^{2}} \frac{\int_{0}^{d} g (r) f_{01} (r) f_{m n} (r) T_{m l} (r) r d r \int_{0}^{b} f_{01} (r) f_{m n} (r) T_{m l} (r) r d r}{N_{01} N_{m n} Γ_{m l} \int_{0}^{b} T_{m l}^{2} (r) r d r} \end{array}

Γ_{m l} = \frac{2 κ}{ρ C} [q^{2} + \frac{π^{2}}{16 b^{2}} {(4 l - 1 + 2 m)}^{2}]

\frac{\partial P_{01}^{N} (z)}{\partial z} = - g_{01} χ_{m n}^{r} e^{(g_{m n} - α_{m n}) z} P_{01}^{N} (z) P_{m n}^{N} (z)

\frac{\partial P_{m n}^{N} (z)}{\partial z} = g_{01} χ_{m n}^{r} e^{g_{01} z} P_{01}^{N} (z) P_{m n}^{N} (z)

P_{01}^{N} (z) = P_{01} (z) e^{- g_{01} z}

P_{m n}^{N} (z) = P_{m n} (z) e^{- (g_{m n} - α_{m n}) z}

P_{m n} (L) \approx P_{m n} (0) e^{(g_{m n} - g_{01} - α_{m n}) L} e^{χ_{m n}^{r} P_{01} (L)}

P_{01}^{t h} \approx \frac{1}{χ_{m n}^{r}} [\ln (x \frac{P_{01} (0)}{P_{m n} (0)}) - (g_{m n} - g_{01} - α_{m n}) L]

P_{01}^{t h} \approx \frac{1}{χ_{m n}^{r}} \ln (x \frac{P_{01} (0)}{P_{m n} (0)})

P_{01}^{t h} \approx \frac{1}{χ_{m n}^{r}} \ln (x \frac{P_{01} (0)}{P_{m n} (0)}) e^{1.25 / e^{g} 01^{L}}

\begin{array}{l} \frac{\partial \sqrt{S_{01}^{N} (ω_{01}, z)}}{\partial z} \approx - \frac{1}{2} g_{01} χ_{m n}^{r} e^{(g_{01} \frac{Γ_{m n}}{Γ_{01}} - α_{m n}) z} \\ \int_{0}^{\infty} \sqrt{2 S_{01}^{N} (ω_{01}, z) S_{m n}^{N} (ω_{01} - Ω, z)} d ω_{01} \sqrt{S_{m n}^{N} (ω_{m n}, z)} \end{array}

\begin{array}{l} \frac{\partial \sqrt{S_{m n}^{N} (ω_{m n}, z)}}{\partial z} \approx \frac{1}{2} g_{01} χ_{m n}^{r} e^{g_{01} z} \\ \int_{0}^{\infty} \sqrt{2 S_{01}^{N} (ω_{01}, z) S_{m n}^{N} (ω_{01} - Ω, z)} d ω_{01} \sqrt{S_{01}^{N} (ω_{01}, z)} \end{array}

Coefficient	Symbol	Value	Unit
Thermal optics coefficient	K_T	1.1 × 10⁻⁵	K⁻¹
Density	ρ	2.2 × 10³	kg/m³
Specific heat	C	741	J/kg/K
Thermal conductivity	κ	1.38	w/m/K
Thermal diffusivity	D = κ/ρC	8.46 × 10⁻⁷	m²/s

	Hansen [10]	Equation (27) K_T = 1.2 × 10⁻⁵ K⁻¹	Equation (27) K_T = 1.1 × 10⁻⁵ K⁻¹
a = d = 10μm, V = 3	0.068W⁻¹ @ 5.2KHz	0.066W⁻¹@ 5.2KHz	0.060W⁻¹@ 5.2KHz
a = d = 20μm, V = 3	0.068W¹ @ 1.4KHz	0.068W⁻¹ @ 1.4KHz	0.063W⁻¹ @ 1.4KHz
a = d = 40μm, V = 3	0.068W¹ @ 0.4KHz	0.066W⁻¹ @ 0.4KHz	0.060W⁻¹ @ 0.4KHz
a = d = 20μm, V = 2.5	0.039W⁻¹ @ 1.1KHz	0.039W⁻¹@ 1.1KHz	0.036W⁻¹@ 1.1KHz
a = d = 20μm, V = 3.5	0.079W⁻¹ @ 1.6KHz	0.080W⁻¹@ 1.6KHz	0.073W⁻¹@ 1.6KHz
a = 20μm, d = 15μm, V = 3	0.043W⁻¹ @ 1.6KHz	0.044W⁻¹@ 1.6KHz	0.040W⁻¹@ 1.6KHz
a = 20μm, d = 10μm, V = 3	0.015W⁻¹ @ 2.0KHz	0.015W⁻¹@ 2.0KHz	0.013W⁻¹@ 2.0KHz

Coefficient	Symbol	Value	Unit
Thermal optics coefficient	K_T	1.1 × 10⁻⁵	K⁻¹
Density	ρ	2.2 × 10³	kg/m³
Specific heat	C	741	J/kg/K
Thermal conductivity	κ	1.38	w/m/K
Thermal diffusivity	D = κ/ρC	8.46 × 10⁻⁷	m²/s

	Hansen [10]	Equation (27) K_T = 1.2 × 10⁻⁵ K⁻¹	Equation (27) K_T = 1.1 × 10⁻⁵ K⁻¹
a = d = 10μm, V = 3	0.068W⁻¹ @ 5.2KHz	0.066W⁻¹@ 5.2KHz	0.060W⁻¹@ 5.2KHz
a = d = 20μm, V = 3	0.068W¹ @ 1.4KHz	0.068W⁻¹ @ 1.4KHz	0.063W⁻¹ @ 1.4KHz
a = d = 40μm, V = 3	0.068W¹ @ 0.4KHz	0.066W⁻¹ @ 0.4KHz	0.060W⁻¹ @ 0.4KHz
a = d = 20μm, V = 2.5	0.039W⁻¹ @ 1.1KHz	0.039W⁻¹@ 1.1KHz	0.036W⁻¹@ 1.1KHz
a = d = 20μm, V = 3.5	0.079W⁻¹ @ 1.6KHz	0.080W⁻¹@ 1.6KHz	0.073W⁻¹@ 1.6KHz
a = 20μm, d = 15μm, V = 3	0.043W⁻¹ @ 1.6KHz	0.044W⁻¹@ 1.6KHz	0.040W⁻¹@ 1.6KHz
a = 20μm, d = 10μm, V = 3	0.015W⁻¹ @ 2.0KHz	0.015W⁻¹@ 2.0KHz	0.013W⁻¹@ 2.0KHz

Abstract

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