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

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

Export Citation
- BibTex
- Endnote (RIS)
- HTML
- Plain Text
Get PDF (1499 KB)
Set citation alerts
Save article

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.

Full Article | PDF Article

References

View by:
Article Order
|
Year
|
Author
|
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).
[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, 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, 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, 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]
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.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.

Click here to see a list of articles that cite this paper

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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

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).

Download Full Size | PPT Slide | PDF

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).

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

Tables (2)

Table 1 Coefficients of Silica Used in This Work

View Table | View all tables in this article

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

View Table | View all tables in this article

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

References

2012 (5)

2011 (6)

2010 (1)

1998 (1)

1986 (2)

1983 (1)

1973 (1)

1971 (1)

1969 (1)

1968 (1)

1967 (4)

Alkeskjold, T. T.

Andersen, T. V.

Barker, J. A.

Bloembergen, N.

Broeng, J.

Cho, C. W.

Dajani, I.

Davis, M. K.

Desai, R. C.

Digonnet, M. J. F.

Eidam, T.

Fabelinskii, I. L.

Foltz, N. D.

Gabler, T.

Gaida, C.

Gray, M. A.

Hanf, S.

Hansen, K. R.

Herman, R. M.

Hoffman, H. J.

Jansen, F.

Jauregui, C.

Kaiser, W.

Lægsgaard, J.

Levenson, M. D.

Limpert, J.

Lowdermilk, W. H.

Matsuoka, M.

Otto, H. J.

Pantell, R. H.

Peterson, L. M.

Pohl, D.

Rank, D. H.

Robin, C.

Rother, W.

Schmidt, O.

Schreiber, T.

Seise, E.

Smith, A. V.

Smith, J. J.

Starunov, V. S.

Steinmetz, A.

Stutzki, F.

Tünnermann, A.

Ward, B.

Wiggins, T. A.

Wirth, C.

Wong, C. S.

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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

Opt. Express (8)

Opt. Lett. (4)

Phys. Rev. (1)

Phys. Rev. A (2)

Phys. Rev. Lett. (4)

Other (2)

Cited By

Figures (6)

Tables (2)

Equations (43)

Metrics

Login or Create Account