Abstract

We report on the high-power amplification of a 1064 nm linearly polarized laser in an all-fiber polarization-maintained master oscillator power amplifier, which can operate at an output power level of 1.3 kW. The beam quality (M2) was measured to be <1.2 at full power operation. The polarization extinction rate of the fiber amplifier was measured to be above 94% before mode instabilities (MIs) set in, which reduced to about 90% after the onset of MI. The power scaling capability of strategies for suppressing MI is analyzed based on a semianalytical model, the theoretical results of which agree with the experimental results. It shows that mitigating MI by coiling the gain fiber is an effective and practical method in standard double-cladding large mode area fiber, and, by tight coiling of the gain fiber to the radius of 5.5 cm, the MI threshold can be increased to three times higher than that without coiling or loose coiling. Experimental studies have been carried out to verify the idea, which has proved that MI was suppressed successfully in the amplifier by proper coiling.

© 2015 Chinese Laser Press

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References

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    [Crossref]

2014 (7)

J. Liu, H. Shi, K. Liu, Y. Hou, and P. Wang, “210  W single-frequency, single-polarization, thulium-doped all-fiber MOPA,” Opt. Express 22, 13572–13578 (2014).
[Crossref]

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Experimental study on mode instabilities in all-fiberized high-power fiber amplifiers,” Chin. Opt. Lett. 12, 020603 (2014).
[Crossref]

K. R. Hansen, T. T. Alkeskjold, and J. Lægsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
[Crossref]

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Fiber amplifiers under thermal loads leading to transverse mode instability,” Proc. SPIE 8961, 89612P (2014).
[Crossref]

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

2013 (10)

A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
[Crossref]

S. Mo, S. Xu, X. Huang, W. Zhang, Z. Feng, D. Chen, T. Yang, and Z. Yang, “A 1014  nm linearly polarized low noise narrow linewidth single-frequency fiber laser,” Opt. Express 21, 12419–12423 (2013).
[Crossref]

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Modal instability of rod fiber amplifiers: a semi-analytic approach,” Proc. SPIE 8601, 860123 (2013).
[Crossref]

W.-W. Ke, X.-J. Wang, X.-F. Bao, and X.-J. Shu, “Thermally induced mode distortion and its limit to power scaling of fiber lasers,” Opt. Express 21, 14272–14281 (2013).
[Crossref]

C. Jauregui, H. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Passive mitigation strategies for mode instabilities in high-power fiber laser systems,” Opt. Express 21, 19375–19386 (2013).
[Crossref]

H.-J. Otto, C. Jauregui, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Controlling mode instabilities by dynamic mode excitation with an acousto-optic deflector,” Opt. Express 21, 17285–17298 (2013).
[Crossref]

K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high power fiber amplifiers,” Opt. Express 21, 1944–1971 (2013).
[Crossref]

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
[Crossref]

A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
[Crossref]

2012 (8)

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

M. Karow, H. Tunnermann, J. Neumann, D. Kracht, and P. Wessels, “Beam quality degradation of a single frequency Yb-doped photonic crystal fiber amplifier with low mode instability threshold power,” Opt. Lett. 37, 4242–4244 (2012).
[Crossref]

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

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012).
[Crossref]

J. Wang, J. Hu, L. Zhang, X. Gu, J. Chen, and Y. Feng, “A 100  W all-fiber linearly-polarized Yb-doped single-mode fiber laser at 1120  nm,” Opt. Express 20, 28373–28378 (2012).
[Crossref]

H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20, 15710–15722 (2012).
[Crossref]

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292  W with improved mode stability,” Opt. Express 20, 5742–5753 (2012).
[Crossref]

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

2011 (3)

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

B. Samson, A. Carter, and K. Tankala, “Rare-earth fibres power up,” Nat. Photonics 5, 466–467 (2011).
[Crossref]

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

2010 (3)

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngsø, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167  W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178  nm,” Opt. Express 18, 16345–16352 (2010).
[Crossref]

B. Samson and A. Carter, “Recent progress on power scaling narrow linewidth fiber amplifiers and their applications,” Rev. Laser Eng. 41, 714–717 (2010).

2008 (1)

2007 (1)

R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

2006 (1)

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

2004 (1)

2000 (1)

1976 (1)

Afzal, R.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Alkeskjold, T. T.

Babazadeh, A.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Bao, X.-F.

Becker, F.

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

Belke, S.

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

Bjarklev, A.

Brar, K.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Broeng, J.

Carter, A.

B. Samson, A. Carter, and K. Tankala, “Rare-earth fibres power up,” Nat. Photonics 5, 466–467 (2011).
[Crossref]

B. Samson and A. Carter, “Recent progress on power scaling narrow linewidth fiber amplifiers and their applications,” Rev. Laser Eng. 41, 714–717 (2010).

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

B. Samson, G. Frith, A. Carter, and K. Tankala, “High-power large-mode area optical fibers for fiber lasers and amplifiers,” in OFC/NFOEC (2008).

Chen, D.

Chen, J.

Chen, M.

Cole, J.

R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

Courtney, S.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Dajani, I.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

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

de Vries, O.

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012).
[Crossref]

C. Jauregui, H.-J. Otto, N. Modsching, O. de Vries, J. Limpert, and A. Tünnermann, “The impact of photodarkening on mode instabilities in high power fiber laser systems,” presented at Advanced Solid State Lasers, Shanghai, China, 2014, paper ATh2A.1.

Dilley, C.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Dong, L.

L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
[Crossref]

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

T.-W. Wu, L. Dong, and H. Winful, “Bend performance of leakage channel fibers,” Opt. Express 16, 4278–4285 (2008).
[Crossref]

Eberhardt, R.

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012).
[Crossref]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

Edgecumbe, J.

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

Eidam, T.

Feng, Y.

Feng, Z.

Foy, P.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Frith, G.

B. Samson, G. Frith, A. Carter, and K. Tankala, “High-power large-mode area optical fibers for fiber lasers and amplifiers,” in OFC/NFOEC (2008).

Galipeau, J.

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

Galvanauskas, A.

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

Geng, J.

Goldberg, L.

Gu, G.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Gu, X.

Haarlammert, N.

Hamedani Golshan, A.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Hansen, K. P.

Hansen, K. R.

Hawkins, T.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Hefter, U.

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

Heidariazar, A.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Hejaz, K.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Henrie, J.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Honea, E.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Hou, Y.

Hu, I.-N.

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

Hu, J.

Hu, Y.

Huang, X.

Jansen, F.

Jauregui, C.

Jiang, S.

Jørgensen, M. M.

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Fiber amplifiers under thermal loads leading to transverse mode instability,” Proc. SPIE 8961, 89612P (2014).
[Crossref]

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Modal instability of rod fiber amplifiers: a semi-analytic approach,” Proc. SPIE 8601, 860123 (2013).
[Crossref]

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292  W with improved mode stability,” Opt. Express 20, 5742–5753 (2012).
[Crossref]

Kaneda, Y.

Karow, M.

Ke, W.-W.

Kliner, A.

Kliner, D. A. V.

Kong, F.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Koplow, J. P.

Kracht, D.

Kuznetsov, A.

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

Lægsgaard, J.

Lafouti, M.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Lanari, A.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

Laurila, M.

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Fiber amplifiers under thermal loads leading to transverse mode instability,” Proc. SPIE 8961, 89612P (2014).
[Crossref]

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Modal instability of rod fiber amplifiers: a semi-analytic approach,” Proc. SPIE 8601, 860123 (2013).
[Crossref]

M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292  W with improved mode stability,” Opt. Express 20, 5742–5753 (2012).
[Crossref]

Liem, A.

Limpert, J.

Liu, C.-H.

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

Liu, J.

Liu, K.

Liu, Z.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Experimental study on mode instabilities in all-fiberized high-power fiber amplifiers,” Chin. Opt. Lett. 12, 020603 (2014).
[Crossref]

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of mode instabilities in high power fiber amplifiers by detecting scattering light,” presented at International Photonics and OptoElectronics Meetings, Wuhan, China, 2014, paper FTh2F.2.

Lyngsø, J. K.

Ma, P.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Experimental study on mode instabilities in all-fiberized high-power fiber amplifiers,” Chin. Opt. Lett. 12, 020603 (2014).
[Crossref]

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of mode instabilities in high power fiber amplifiers by detecting scattering light,” presented at International Photonics and OptoElectronics Meetings, Wuhan, China, 2014, paper FTh2F.2.

Machewirth, D. P.

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

Marcuse, D.

Mcclane, D.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Mo, S.

Modsching, N.

C. Jauregui, H.-J. Otto, N. Modsching, O. de Vries, J. Limpert, and A. Tünnermann, “The impact of photodarkening on mode instabilities in high power fiber laser systems,” presented at Advanced Solid State Lasers, Shanghai, China, 2014, paper ATh2A.1.

Neumann, B.

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

Neumann, J.

Norouzey, A.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

O’Connor, M.

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

Olausson, C. B.

Otto, H.

Otto, H.-J.

Ozisik, M. N.

M. N. Ozisik, Heat Conduction, 2nd ed. (Wiley, 1993).

Peschel, T.

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012).
[Crossref]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

Peyghambarian, N.

Poozesh, R.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Rekas, M.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

Rezaei Nasirabad, R.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Robin, C.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

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

Roohforouz, A.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Ruppik, S.

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

Saitoh, K.

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

Samson, B.

B. Samson, A. Carter, and K. Tankala, “Rare-earth fibres power up,” Nat. Photonics 5, 466–467 (2011).
[Crossref]

B. Samson and A. Carter, “Recent progress on power scaling narrow linewidth fiber amplifiers and their applications,” Rev. Laser Eng. 41, 714–717 (2010).

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

B. Samson, G. Frith, A. Carter, and K. Tankala, “High-power large-mode area optical fibers for fiber lasers and amplifiers,” in OFC/NFOEC (2008).

Savage-Leuchs, M.

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

Schermer, R.

R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
[Crossref]

Schreiber, T.

N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012).
[Crossref]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

Shi, H.

Shirakawa, A.

Shu, X.-J.

Smith, A.

Smith, A. V.

Smith, J.

Smith, J. J.

Sosnowski, T.

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

Spiegelberg, C.

Stock, M. L.

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

Stutzki, F.

Tabatabaei Jafari, N.

K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. Rezaei Nasirabad, N. Tabatabaei Jafari, A. Hamedani Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500  W ytterbium-doped fiber laser,” Laser Phys. 24, 025102 (2014).
[Crossref]

Tankala, K.

B. Samson, A. Carter, and K. Tankala, “Rare-earth fibres power up,” Nat. Photonics 5, 466–467 (2011).
[Crossref]

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

B. Samson, G. Frith, A. Carter, and K. Tankala, “High-power large-mode area optical fibers for fiber lasers and amplifiers,” in OFC/NFOEC (2008).

Tao, R.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Experimental study on mode instabilities in all-fiberized high-power fiber amplifiers,” Chin. Opt. Lett. 12, 020603 (2014).
[Crossref]

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of mode instabilities in high power fiber amplifiers by detecting scattering light,” presented at International Photonics and OptoElectronics Meetings, Wuhan, China, 2014, paper FTh2F.2.

Thomas, A.

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

Tsybin, I.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

Tudury, G.

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

Tunnermann, H.

Tünnermann, A.

Ueda, K.

Wang, J.

Wang, P.

Wang, X.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Experimental study on mode instabilities in all-fiberized high-power fiber amplifiers,” Chin. Opt. Lett. 12, 020603 (2014).
[Crossref]

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of mode instabilities in high power fiber amplifiers by detecting scattering light,” presented at International Photonics and OptoElectronics Meetings, Wuhan, China, 2014, paper FTh2F.2.

Wang, X.-J.

Ward, B.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

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

Wessels, P.

Winful, H.

Wirth, C.

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

Wu, T.-W.

Xu, S.

Yang, T.

Yang, Z.

Zeringue, C.

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

Zhang, C.

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

Zhang, L.

Zhang, W.

Zhou, P.

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Experimental study on mode instabilities in all-fiberized high-power fiber amplifiers,” Chin. Opt. Lett. 12, 020603 (2014).
[Crossref]

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of mode instabilities in high power fiber amplifiers by detecting scattering light,” presented at International Photonics and OptoElectronics Meetings, Wuhan, China, 2014, paper FTh2F.2.

Zhu, C.

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

Chin. Opt. Lett. (1)

IEEE J. Quantum Electron. (1)

R. Schermer and J. Cole, “Improved bend loss formula verified for optical fiber by simulation and experiment,” IEEE J. Quantum Electron. 43, 899–909 (2007).
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J. Lightwave Technol. (1)

J. Opt. Soc. Am. (1)

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[Crossref]

Nat. Photonics (1)

B. Samson, A. Carter, and K. Tankala, “Rare-earth fibres power up,” Nat. Photonics 5, 466–467 (2011).
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C. B. Olausson, A. Shirakawa, M. Chen, J. K. Lyngsø, J. Broeng, K. P. Hansen, A. Bjarklev, and K. Ueda, “167  W, power scalable ytterbium-doped photonic bandgap fiber amplifier at 1178  nm,” Opt. Express 18, 16345–16352 (2010).
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N. Haarlammert, O. de Vries, A. Liem, A. Kliner, T. Peschel, T. Schreiber, R. Eberhardt, and A. Tünnermann, “Build up and decay of mode instability in a high power fiber amplifier,” Opt. Express 20, 13274–13283 (2012).
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H.-J. Otto, F. Stutzki, F. Jansen, T. Eidam, C. Jauregui, J. Limpert, and A. Tünnermann, “Temporal dynamics of mode instabilities in high-power fiber lasers and amplifiers,” Opt. Express 20, 15710–15722 (2012).
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L. Dong, “Stimulated thermal Rayleigh scattering in optical fibers,” Opt. Express 21, 2642–2656 (2013).
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A. V. Smith and J. J. Smith, “Increasing mode instability thresholds of fiber amplifiers by gain saturation,” Opt. Express 21, 15168–15182 (2013).
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H.-J. Otto, C. Jauregui, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Controlling mode instabilities by dynamic mode excitation with an acousto-optic deflector,” Opt. Express 21, 17285–17298 (2013).
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A. V. Smith and J. J. Smith, “Steady-periodic method for modeling mode instability in fiber amplifiers,” Opt. Express 21, 2606–2623 (2013).
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C. Jauregui, H. Otto, F. Stutzki, F. Jansen, J. Limpert, and A. Tünnermann, “Passive mitigation strategies for mode instabilities in high-power fiber laser systems,” Opt. Express 21, 19375–19386 (2013).
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K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Theoretical analysis of mode instability in high power fiber amplifiers,” Opt. Express 21, 1944–1971 (2013).
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S. Mo, S. Xu, X. Huang, W. Zhang, Z. Feng, D. Chen, T. Yang, and Z. Yang, “A 1014  nm linearly polarized low noise narrow linewidth single-frequency fiber laser,” Opt. Express 21, 12419–12423 (2013).
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J. Liu, H. Shi, K. Liu, Y. Hou, and P. Wang, “210  W single-frequency, single-polarization, thulium-doped all-fiber MOPA,” Opt. Express 22, 13572–13578 (2014).
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M. Laurila, M. M. Jørgensen, K. R. Hansen, T. T. Alkeskjold, J. Broeng, and J. Lægsgaard, “Distributed mode filtering rod fiber amplifier delivering 292  W with improved mode stability,” Opt. Express 20, 5742–5753 (2012).
[Crossref]

B. Ward, C. Robin, and I. Dajani, “Origin of thermal modal instabilities in large mode area fiber amplifiers,” Opt. Express 20, 11407–11422 (2012).
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W.-W. Ke, X.-J. Wang, X.-F. Bao, and X.-J. Shu, “Thermally induced mode distortion and its limit to power scaling of fiber lasers,” Opt. Express 21, 14272–14281 (2013).
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A. Smith and J. Smith, “Mode instability in high power fiber amplifiers,” Opt. Express 19, 10180–10192 (2011).
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J. Wang, J. Hu, L. Zhang, X. Gu, J. Chen, and Y. Feng, “A 100  W all-fiber linearly-polarized Yb-doped single-mode fiber laser at 1120  nm,” Opt. Express 20, 28373–28378 (2012).
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K. R. Hansen, T. T. Alkeskjold, and J. Lægsgaard, “Impact of gain saturation on the mode instability threshold in high-power fiber amplifiers,” Opt. Express 22, 11267–11278 (2014).
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Proc. SPIE (10)

M. L. Stock, C.-H. Liu, A. Kuznetsov, G. Tudury, A. Galvanauskas, and T. Sosnowski, “Polarized, 100  kW peak power, high brightness nanosecond lasers based on 3C optical fiber,” Proc. SPIE 7914, 79140U (2011).
[Crossref]

L. Dong, K. Saitoh, F. Kong, P. Foy, T. Hawkins, D. Mcclane, and G. Gu, “All-solid photonic bandgap fibers for high power lasers (Invited Paper),” Proc. SPIE 8547, 85470J (2012).
[Crossref]

K. Brar, M. Savage-Leuchs, J. Henrie, S. Courtney, C. Dilley, R. Afzal, and E. Honea, “Threshold power and fiber degradation induced modal instabilities in high power fiber amplifiers based on large mode area fibers,” Proc. SPIE 8961, 89611R (2014).
[Crossref]

C. Wirth, T. Schreiber, M. Rekas, I. Tsybin, T. Peschel, R. Eberhardt, and A. Tünnermann, “High-power linear-polarized narrow linewidth photonic crystal fiber amplifier,” Proc. SPIE 7580, 75801H (2010).
[Crossref]

C. Robin, I. Dajani, C. Zeringue, B. Ward, and A. Lanari, “Gain-tailored SBS suppressing photonic crystal fibers for high power applications,” Proc. SPIE 8237, 82371D (2012).
[Crossref]

A. Carter, J. Edgecumbe, D. P. Machewirth, J. Galipeau, B. Samson, K. Tankala, and M. O’Connor, “Recent progress in the development of kW-level monolithic PM-LMA fiber amplifiers,” Proc. SPIE 6344, 6344F (2006).

S. Belke, F. Becker, B. Neumann, S. Ruppik, and U. Hefter, “Completely monolithic linearly polarized high-power fiber laser oscillator,” Proc. SPIE 8961, 896124 (2014).
[Crossref]

I.-N. Hu, C. Zhu, C. Zhang, A. Thomas, and A. Galvanauskas, “Analytical time-dependent theory of thermally-induced modal instabilities in high power fiber amplifiers,” Proc. SPIE 8601, 860109 (2013).
[Crossref]

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Modal instability of rod fiber amplifiers: a semi-analytic approach,” Proc. SPIE 8601, 860123 (2013).
[Crossref]

M. M. Jørgensen, K. R. Hansen, M. Laurila, T. T. Alkeskjold, and J. Lægsgaard, “Fiber amplifiers under thermal loads leading to transverse mode instability,” Proc. SPIE 8961, 89612P (2014).
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Rev. Laser Eng. (1)

B. Samson and A. Carter, “Recent progress on power scaling narrow linewidth fiber amplifiers and their applications,” Rev. Laser Eng. 41, 714–717 (2010).

Other (5)

A. V. Smith and J. J. Smith, “Maximizing the mode instability threshold of a fiber amplifier,” arXiv:1301.3489 (2013).

R. Tao, P. Ma, X. Wang, P. Zhou, and Z. Liu, “Study of mode instabilities in high power fiber amplifiers by detecting scattering light,” presented at International Photonics and OptoElectronics Meetings, Wuhan, China, 2014, paper FTh2F.2.

C. Jauregui, H.-J. Otto, N. Modsching, O. de Vries, J. Limpert, and A. Tünnermann, “The impact of photodarkening on mode instabilities in high power fiber laser systems,” presented at Advanced Solid State Lasers, Shanghai, China, 2014, paper ATh2A.1.

B. Samson, G. Frith, A. Carter, and K. Tankala, “High-power large-mode area optical fibers for fiber lasers and amplifiers,” in OFC/NFOEC (2008).

M. N. Ozisik, Heat Conduction, 2nd ed. (Wiley, 1993).

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Figures (7)

Fig. 1.
Fig. 1. Architecture of the all-fiber PM amplifier.
Fig. 2.
Fig. 2. Output characteristics of the main amplifier. (a) Output power. (b) Output spectrums. (c) PER.
Fig. 3.
Fig. 3. Fluctuation characteristics of the output beam. (a) Time traces. (b) Frequency distribution.
Fig. 4.
Fig. 4. Threshold calculation of the fiber amplifier.
Fig. 5.
Fig. 5. Threshold pump power for different fiber cladding diameters.
Fig. 6.
Fig. 6. Effect of coiling on MI threshold power.
Fig. 7.
Fig. 7. Fluctuation characteristics of the output beam after tight coiling. (a) Time trace (b) Frequency distribution.

Tables (1)

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Table 1. Parameters of Test Amplifier

Equations (22)

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E(r,ϕ,z,t)=m=0l=1Aml(z,t)ψml(r,ϕ)ej(βmlzωmlt)+c.c.,
Is(r,ϕ,z,t)=2n0ε0cE(r,ϕ,z,t)E(r,ϕ,z,t)*=I0+I˜
I0=I11(z,t)ψ1(r,ϕ)ψ1(r,ϕ)+I22(z,t)ψ2(r,ϕ)ψ2(r,ϕ),
I˜=I12(z,t)ψ1(r,ϕ)ψ2(r,ϕ)ej(qzΩt)+I21(z,t)ψ1(r,ϕ)ψ2(r,ϕ)ej(qzΩt),
Ikl(z,t)=4n0ε0cAk(z,t)Al*(z,t),
q=β1β2,Ω=ω1ω2.
2T(r,ϕ,z,t)+Q(r,ϕ,z,t)κ=1αT(r,ϕ,z,t)t,
Q(r,ϕ,z,t)g(r,ϕ,z,t)(vpvsvs)Is(r,ϕ,z,t),
g(r,ϕ,z,t)=[(σsa+σse)nu(r,ϕ,z,t)σsa]NYb(r,ϕ),
κTr+hqT=0,
T(r,ϕ,z,t)=1πασηvm=1Jv(δmr)H(δm)t=0t[B11(ϕ,z)I11(z,t)+B22(ϕ,z)I22(z,t)+B12(ϕ,z)I12(z,t)ej(qzΩt)+B12(ϕ,z)I12*(z,t)ej(qzΩt)]eαδm2(tt)dt
Bkl(ϕ,z)={02πdϕ0Rg0Jv(δm,r)cosv(ϕϕ)ψk(r,ϕ)ψk(r,ϕ)1+I0/Isaturationdr,k=l02πdϕ0Rg0Jv(δm,r)cosv(ϕϕ)ψk(r,ϕ)ψl(r,ϕ)(1+I0/Isaturation)2dr,kl,
H(δm)=0RrJv2(δm,r)dr,σ=ηκ(vpvsvs),
n2=(n0+ng+nNL)2n02jg(r,ϕ,z,t)n0k0+2n0nNL,
nNL(r,ϕ,z,t)=ηT(r,ϕ,z,t)=h11(r,ϕ,z,t)+h22(r,ϕ,z,t)+h12(r,ϕ,z,t)ejqz+h21(r,ϕ,z,t)ejqz
hkl(r,ϕ,z,t)={ασπvm=1Jv(δmr)H(δm)0tBkk(ϕ,z)Ikk(z,t)eαδm2(tt)dt,k=lασπvm=1Jv(δmr)H(δm)0tBkl(ϕ,z)Ikl(z,t)eαδm2(tt)jΩtdt,kl.
|A1|2z=g(r,ϕ,z)ψ1ψ1rdrdϕ|A1|2,
|A2|2z=[g(r,ϕ,z)ψ2ψ2rdrdϕ+|A1|2χ(Ω,t)]|A2|2
χ(Ω)=2n0ω22c2β2Im(h¯12ψ1ψ2rdrdϕ),
h¯kl(r,ϕ,z)=ασπvm=1Jv(δmr)H(δm)Bkl(ϕ,z)αδm2jΩ.
ξ(L)ω0P1(L)2π0LP1(z)|χ(Ω0)|dzexp{0L[g(r,ϕ,z)ψ2ψ2rdrdϕ]dz+0LP1(z)χ(Ω0)dz},
ξ(L)ξ0exp[0Ldzg(r,ϕ,z)(ψ2ψ2ψ1ψ1)rdrdϕαcoilLcoil]+ξ042π0LP1(z)|χ(Ω0)|dzRN(Ω0)×exp[0Ldzg(r,ϕ,z)(ψ2ψ2ψ1ψ1)rdrdϕ+0LP1(z)χ(Ω0)dzαcoilLcoil],

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