Abstract

For the inter-modal four-wave mixing (IMFWM) in high-power fiber lasers, the phase-matching frequency shift and coherence length are calculated to determine the fiber mode combinations corresponding to IMFWM peaks; then, the parameters of the laser are optimized accordingly to suppress the IMFWM. On the basis of laser parameter optimization, the fiber coiling method is applied to further suppress the IMFWM. To validate this method, a master oscillator power amplifier was configured with 20/400 fiber to produce IMFWM peaks at 1108 and 1071.6 nm in the output spectrum, and then those peaks were removed by reducing the bending radius of the fiber.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

2016 (3)

2015 (5)

2014 (1)

2013 (1)

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

2012 (1)

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
[Crossref] [PubMed]

2011 (1)

J. Nilsson and D. N. Payne, “Physics. High-power fiber lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

2010 (1)

2009 (1)

M. A. Lapointe and M. Piché, “Linewidth of high-power fiber lasers,” Proc. SPIE 7386, 73860S (2009).
[Crossref]

2008 (1)

2007 (2)

M. Gong, Y. Yuan, C. Li, P. Yan, H. Zhang, and S. Liao, “Numerical modeling of transverse mode competition in strongly pumped multimode fiber lasers and amplifiers,” Opt. Express 15(6), 3236–3246 (2007).
[Crossref] [PubMed]

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

2000 (1)

1987 (1)

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5(12), 1712–1715 (1987).
[Crossref]

1981 (1)

C. Lin and M. A. Bösch, “Large-Stokes-shift stimulated four-photon mixing in optical fibers,” Appl. Phys. Lett. 38(7), 479–481 (1981).
[Crossref]

1976 (1)

1975 (1)

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11(3), 100–103 (1975).
[Crossref]

1974 (1)

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

1971 (1)

1965 (1)

Agrawal, G. P.

Alfano, R. R.

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5(12), 1712–1715 (1987).
[Crossref]

Alkeskjold, T. T.

Ashkin, A.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Baldeck, P. L.

P. L. Baldeck and R. R. Alfano, “Intensity effects on the stimulated four photon spectra generated by picosecond pulses in optical fibers,” J. Lightwave Technol. 5(12), 1712–1715 (1987).
[Crossref]

Barthelemy, A.

Begleris, I.

Bendahmane, A.

Bjorkholm, J. E.

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Bösch, M. A.

C. Lin and M. A. Bösch, “Large-Stokes-shift stimulated four-photon mixing in optical fibers,” Appl. Phys. Lett. 38(7), 479–481 (1981).
[Crossref]

Charan, K.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
[Crossref] [PubMed]

Chen, J.

Chen, X. L.

Cheng, J.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
[Crossref] [PubMed]

Chraplyvy, A. R.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Clarkson, W. A.

Cole, J. H.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

Couderc, V.

Desgroseilliers, M.

Dupiol, R.

Essiambre, R. J.

Y. Xiao, R. J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers,” Opt. Express 22(26), 32039–32059 (2014).
[Crossref] [PubMed]

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Fabert, M.

Fang, Q.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Y. Xu, Q. Fang, Y. Qin, X. Meng, and W. Shi, “2 kW narrow spectral width monolithic continuous wave in a near-diffraction-limited fiber laser,” Appl. Opt. 54(32), 9419–9421 (2015).
[Crossref] [PubMed]

Feng, Y. J.

Y. J. Feng, X. J. Wang, W. W. Ke, Y. H. Sun, K. Zhang, Y. Ma, T. L. Li, Y. S. Wang, and J. Wu, “Numerical analysis to four-wave mixing induced spectral broadening in high power fiber lasers,” Proc. SPIE 9255, 92550Q (2015).
[Crossref]

Friis, S. M. M.

Gloge, D.

Gnauck, A. H.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Goldberg, L.

Gong, M.

Grüner-Nielsen, L.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
[Crossref] [PubMed]

Guo, S.

He, B.

Horak, P.

Hu, M.

Huang, B.

Huang, Z. H.

Jakobsen, D.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
[Crossref] [PubMed]

Jiang, X. L.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Jing, F.

Jung, Y.

Ke, W. W.

M. Hu, W. W. Ke, Y. F. Yang, M. Lei, K. Liu, X. L. Chen, C. Zhao, Y. F. Qi, B. He, X. J. Wang, and J. Zhou, “Low threshold Raman effect in high power narrowband fiber amplifier,” Chin. Opt. Lett. 14(1), 011901 (2016).
[Crossref]

Y. J. Feng, X. J. Wang, W. W. Ke, Y. H. Sun, K. Zhang, Y. Ma, T. L. Li, Y. S. Wang, and J. Wu, “Numerical analysis to four-wave mixing induced spectral broadening in high power fiber lasers,” Proc. SPIE 9255, 92550Q (2015).
[Crossref]

Kibler, B.

Kliner, D. A. V.

Koplow, J. P.

Krupa, K.

Labruyère, A.

Lægsgaard, J.

Lapointe, M. A.

M. A. Lapointe and M. Piché, “Linewidth of high-power fiber lasers,” Proc. SPIE 7386, 73860S (2009).
[Crossref]

Lei, M.

Leng, J.

Leproux, P.

Li, C.

Li, J. H.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Li, T. L.

Y. J. Feng, X. J. Wang, W. W. Ke, Y. H. Sun, K. Zhang, Y. Ma, T. L. Li, Y. S. Wang, and J. Wu, “Numerical analysis to four-wave mixing induced spectral broadening in high power fiber lasers,” Proc. SPIE 9255, 92550Q (2015).
[Crossref]

Li, Z. B.

Liang, X. B.

Liao, S.

Lin, C.

C. Lin and M. A. Bösch, “Large-Stokes-shift stimulated four-photon mixing in optical fibers,” Appl. Phys. Lett. 38(7), 479–481 (1981).
[Crossref]

Lin, H. H.

Lingle, R.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Liu, K.

Lv, H.

Ma, Y.

Y. J. Feng, X. J. Wang, W. W. Ke, Y. H. Sun, K. Zhang, Y. Ma, T. L. Li, Y. S. Wang, and J. Wu, “Numerical analysis to four-wave mixing induced spectral broadening in high power fiber lasers,” Proc. SPIE 9255, 92550Q (2015).
[Crossref]

Mafi, A.

Malitson, I. H.

Marcuse, D.

Martin, A.

Meng, X.

Meng, X. J.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Mestre, M. A.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Millot, G.

Mumtaz, S.

Nazemosadat, E.

Nilsson, J.

Norwood, R. A.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Olausson, C. B.

Parmigiani, F.

Payne, D. N.

J. Nilsson and D. N. Payne, “Physics. High-power fiber lasers,” Science 332(6032), 921–922 (2011).
[Crossref] [PubMed]

Pedersen, M. E.

J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
[Crossref] [PubMed]

Petersen, S. R.

Petropoulos, P.

Peyghambarian, N.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Piché, M.

M. A. Lapointe and M. Piché, “Linewidth of high-power fiber lasers,” Proc. SPIE 7386, 73860S (2009).
[Crossref]

Pourbeyram, H.

Qi, Y. F.

Qin, Y.

Qin, Y. G.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Richardson, D. J.

Rottwitt, K.

Ryf, R.

Y. Xiao, R. J. Essiambre, M. Desgroseilliers, A. M. Tulino, R. Ryf, S. Mumtaz, and G. P. Agrawal, “Theory of intermodal four-wave mixing with random linear mode coupling in few-mode fibers,” Opt. Express 22(26), 32039–32059 (2014).
[Crossref] [PubMed]

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Schermer, R. T.

R. T. Schermer and J. H. Cole, “Improved Bend Loss Formula Verified for Optical Fiber by Simulation and Experiment,” IEEE J. Quantum Electron. 43(10), 899–909 (2007).
[Crossref]

Shi, W.

Q. Fang, J. H. Li, W. Shi, Y. G. Qin, Y. Xu, X. J. Meng, R. A. Norwood, and N. Peyghambarian, “5kW Near-diffraction-limited and 8kW High-brightness Monolithic Continuous Wave Fiber Lasers Directly Pumped by Laser Diodes,” IEEE Photonics J. 9(5), 1506107 (2017).
[Crossref]

Y. Xu, Q. Fang, Y. Qin, X. Meng, and W. Shi, “2 kW narrow spectral width monolithic continuous wave in a near-diffraction-limited fiber laser,” Appl. Opt. 54(32), 9419–9421 (2015).
[Crossref] [PubMed]

Stolen, R. H.

R. H. Stolen, “Phase-matched-stimulated four-photon mixing in silica-fiber waveguides,” IEEE J. Quantum Electron. 11(3), 100–103 (1975).
[Crossref]

R. H. Stolen, J. E. Bjorkholm, and A. Ashkin, “Phase-matched three-wave mixing in silica fiber optical waveguides,” Appl. Phys. Lett. 24(7), 308–310 (1974).
[Crossref]

Sun, Y.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Sun, Y. H.

Y. J. Feng, X. J. Wang, W. W. Ke, Y. H. Sun, K. Zhang, Y. Ma, T. L. Li, Y. S. Wang, and J. Wu, “Numerical analysis to four-wave mixing induced spectral broadening in high power fiber lasers,” Proc. SPIE 9255, 92550Q (2015).
[Crossref]

Sylvestre, T.

Tkach, R. W.

R. J. Essiambre, M. A. Mestre, R. Ryf, A. H. Gnauck, R. W. Tkach, A. R. Chraplyvy, Y. Sun, X. L. Jiang, and R. Lingle, “Experimental investigation of inter-modal four-wave mixing in few-mode fibers,” IEEE Photonics Technol. Lett. 25(6), 539–542 (2013).
[Crossref]

Tonello, A.

Traynor, N.

Tulino, A. M.

Wabnitz, S.

Wang, J.

Wang, J. J.

Wang, K.

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Wang, Y. S.

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Y. J. Feng, X. J. Wang, W. W. Ke, Y. H. Sun, K. Zhang, Y. Ma, T. L. Li, Y. S. Wang, and J. Wu, “Numerical analysis to four-wave mixing induced spectral broadening in high power fiber lasers,” Proc. SPIE 9255, 92550Q (2015).
[Crossref]

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J. Cheng, M. E. Pedersen, K. Charan, K. Wang, C. Xu, L. Grüner-Nielsen, and D. Jakobsen, “Intermodal four-wave mixing in a higher-order-mode fiber,” Appl. Phys. Lett. 101(16), 161106 (2012).
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Figures (8)

Fig. 1
Fig. 1 Schematic diagram of a comparison between the FWM and IMFWM in a high-power fiber laser.
Fig. 2
Fig. 2 Phase-matching diagrams of the IMFWM for the 25/400 fiber: (a) phase-matching diagram for Case 1 of the fiber mode combinations; and (b) phase-matching diagram for Case 2 of the fiber mode combinations.
Fig. 3
Fig. 3 Effect of the core radius and numerical aperture on the IMFWM: (a) the relationship between the core radius, numerical aperture, and the number of pairs of IMFWM peaks that can be produced in high-power fiber lasers; and (b) the relationship between the core radius, numerical aperture, and the phase-matching frequency shift of (11,01→11,01).
Fig. 4
Fig. 4 Effect of the light wavelength on the IMFWM: (a) the relationship between the light wavelength and number of pairs of IMFWM peaks that can be produced in high-power fiber lasers; (b) the relationship between the light wavelength and the phase-matching frequency shift of three fiber mode combinations for cases in which the number of pairs of IMFWM peaks is greater than 1; and (c) the relationship between the light wavelength and the phase-matching frequency shift of (11,01→11,01) for cases in which the number of pairs of IMFWM peaks is 1.
Fig. 5
Fig. 5 Relationship between the output power of each mode and the bending radius for a 20/400 Yb-doped fiber amplifier.
Fig. 6
Fig. 6 Schematic of a fiber laser based on the MOPA scheme.
Fig. 7
Fig. 7 Output characteristics of the power amplifier when the bending radius is 15 cm: (a) the output spectra at different pump powers; and (b) the output power versus the pump power.
Fig. 8
Fig. 8 Output characteristics of the power amplifier when the bending radius was 7.5 cm: (a) output spectra at different pump powers; (b) output power versus the pump power.

Tables (2)

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Table 1 Phase-matching Frequency Shift and Coherence Length for the 25/400 Fiber

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Table 2 Parameters Used for Simulations of a 20/400 Yb-doped Fiber Amplifier

Equations (26)

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κ=Δ k M +Δ k W +Δ k NL =0.
Δ k M =( n 3 ω 3 + n 4 ω 4 n 1 ω 1 n 2 ω 2 )/c Δ k W =(Δ n 3 ω 3 +Δ n 4 ω 4 Δ n 1 ω 1 Δ n 2 ω 2 )/c,
Δ k M =β( ω 3 )+β( ω 4 )β( ω 1 )β( ω 2 ).
Δ k M Ω s ( β 1 ( ω 2 )- β 1 ( ω 1 ))+ Ω s 2 ( β 2 ( ω 1 )+ β 2 ( ω 2 ))/2+ Ω s 4 ( β 3 ( ω 2 )- β 3 ( ω 1 ))/6 + Ω s 4 ( β 4 ( ω 1 )+ β 4 ( ω 2 ))/24 .
Δ k M Ω s ( β 1 ( ω 2 )- β 1 ( ω 1 ))+ Ω s 2 ( β 2 ( ω 1 )+ β 2 ( ω 2 ))/2.
Δ k M =2πc( β 1 ( ν ¯ 2 )- β 1 ( ν ¯ 1 ))Δ ν ¯ +2 π 2 c 2 ( β 2 ( ν ¯ 1 )+ β 2 ( ν ¯ 2 ))Δ ν ¯ 2 .
Δ k W =2π(Δ n 3 ν ¯ 3 +Δ n 4 ν ¯ 4 Δ n 1 ν ¯ 1 Δ n 2 ν ¯ 2 ).
b= (β/k) 2 n 2 n c 2 n 2 β/kn n c n = n ˜ n n c n ,
Δ k W =2π( n c n)( b 3 ν ¯ 3 + b 4 ν ¯ 4 b 1 ν ¯ 1 b 2 ν ¯ 2 ).
b= β 2 n 2 k 2 n c 2 k 2 n 2 k 2 = W 2 V 2 =1 U 2 V 2 .
Δ k W =2π( n c n)( b 11 ν ¯ 3 + b 01 ν ¯ 4 b 11 ν ¯ 1 b 01 ν ¯ 2 ).
b 11 ν ¯ 3 b 11 ν ¯ 1 + d( b 11 ν ¯ ) d ν ¯ | ν= ν ¯ 1 ( ν ¯ 3 ν ¯ 1 )+ 1 2 d 2 ( b 11 ν ¯ ) d ν ¯ 2 | ν= ν ¯ 1 ( ν ¯ 3 ν ¯ 1 ) 2 .
d(b ν ¯ ) d ν ¯ = 1 a 2Δ n c d(Vb) dV dV dk .
b 11 ν ¯ 3 = b 11 ν ¯ 1 d( b 11 V) dV Δ ν ¯ +πa 2Δ n c d 2 ( b 11 V) d V 2 Δ ν ¯ 2 .
b 01 ν ¯ 4 = b 01 ν ¯ 2 + d( b 01 V) dV Δ ν ¯ +πa 2Δ n c d 2 ( b 01 V) d V 2 Δ ν ¯ 2 .
Δ k W =2π( n c n)( ( d( b 01 V) dV d( b 11 V) dV )Δ ν ¯ +πa 2Δ n c ( d 2 ( b 01 V) d V 2 + d 2 ( b 11 V) d V 2 )Δ ν ¯ 2 ).
Δ k W =2π( n c n)( b 01 ν ¯ 3 + b 11 ν ¯ 4 b 01 ν ¯ 1 b 01 ν ¯ 2 ) =2π( n c n)( ( b 11 b 01 ) ν ¯ 2 +( d( b 11 V) dV d( b 01 V) dV )Δ ν ¯ +πa 2Δ n c ( d 2 ( b 01 V) d V 2 + d 2 ( b 11 V) d V 2 )Δ ν ¯ 2 ).
L coh =2π/|δκ|=2π/|δ(Δ k M (Δ ν ¯ )+Δ k W (Δ ν ¯ ))|,
δκ=2π( n c n)Δ ν ¯ δa( ( d 2 ( b 01 V) d V 2 d 2 ( b 11 V) d V 2 ) V a +π 2Δ n c ( d 2 ( b 11 V) d V 2 + d 2 ( b 01 V) d V 2 )Δ ν ¯ ) +2π( n c n)π 2Δ n c V( d 3 ( b 11 V) d V 3 + d 3 ( b 01 V) d V 3 )Δ ν ¯ 2 δa ,
δκ=2π( n c n) V a δa( d( b 11 b 01 ) dV ν ¯ 2 +( d 2 ( b 11 V) d V 2 d 2 ( b 01 V) d V 2 )Δ ν ¯ ) +2π( n c n)π 2Δ n c Δ ν ¯ 2 δa( ( d 2 ( b 01 V) d V 2 + d 2 ( b 11 V) d V 2 )+V( d 3 ( b 01 V) d V 3 + d 3 ( b 11 V) d V 3 ) ).
Δ k M =4 π 2 c 2 β 2 ( ν ¯ 1 )Δ ν ¯ 2 ,
2α= π U 2 exp(2 W 3 R eff /3 a 3 β 2 ) e m a R eff W 3/2 V 2 K m1 (W) K m+1 (W) .
N 2 (r,φ,z) N 1 (r,φ,z) = [ P p + (z)+ P p (z)] σ ap Γ p (r,φ) h ν p + i P si + (z) σ as Γ si (r,φ) h ν s [ P p + (z)+ P p (z)] σ ep Γ p (r,φ) h ν p + 1 τ + i P si + (z) σ es Γ si (r,φ) h ν s ,
± d P p ± (z) dz ={ 0 2π 0 a [ σ ep N 2 (r,φ,z) σ ap N 1 (r,φ,z) ] Γ p (r,φ)rdrdφ } P p ± (z) α p P p ± (z),
d P si + (z) dz ={ 0 2π 0 a [ σ es N 2 (r,φ,z) σ as N 1 (r,φ,z) ] Γ si (r,φ)rdrdφ } P si + (z) [ α s + α si bend (R)] P si + (z) j d ij [ P si + (z) P sj + (z)] ,
P p + (0)= P p f , P p (L)= P p b , P si + (0)= P si in .

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