Abstract

A theoretical model is proposed to study a passive fiber laser coherent beam combination (CBC) with an all-optical ring-cavity feedback loop. The dynamic evolution of a frequency filter is carefully studied and is shown to be sensitive to stimulated Brillouin scattering (SBS) and phase noise. SBS plays an important role in the process of mode selection or CBC because the linewidth of the longitudinal mode is narrowed continuously in the ring-cavity feedback. Numerical studies show that the SBS process is significantly influenced by phase noise, optical path length difference, and four-wave mixing. Our results indicate that SBS would be a remarkable problem in this type of CBC unless the phase noise causes a rapid mode jump in the frequency domain. Since the mode jump worsens the efficiency of the coherent combination or beam quality, a trade-off consideration is necessary for the SBS, phase noise, and beam quality. The maximal SBS-allowed CBC efficiency is obtained numerically for output laser power of 700 W–3.5 kW.

© 2014 Optical Society of America

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2013

2012

2011

C. Zeringue, C. Vergien, and I. Dajani, “Pump-limited, 203  W, single-frequency monolithic fiber amplifier based on laser gain competition,” Opt. Lett. 36, 618–620 (2011).
[CrossRef]

B. Wang and A. Sanchez, “All-fiber passive coherent combining of high power lasers,” Opt. Eng. 50, 111606 (2011).
[CrossRef]

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

2010

2009

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15, 320–327 (2009).
[CrossRef]

I. Dajani, C. Zeringue, and T. M. Shay, “Investigation of nonlinear effects in multitone-driven narrow-linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15, 406–414 (2009).
[CrossRef]

2008

2007

2005

2002

2001

D. C. Brown and H. J. Hoffman, “Thermal, stress and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[CrossRef]

1999

1995

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

1990

R. W. Boyd, K. Rzawski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Aceves, A. B.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Augst, S. J.

Bao, X.-F.

Barber, P. R.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Barthelemy, A.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Barty, C. P. J.

Beach, R. J.

Bochove, E. J.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15, 320–327 (2009).
[CrossRef]

Boyd, R. W.

R. W. Boyd, K. Rzawski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Braiman, Y.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Bronder, T. J.

Brown, D. C.

D. C. Brown and H. J. Hoffman, “Thermal, stress and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[CrossRef]

Bruesselbach, H.

Carman, R. J.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Chowdhury, D.

Colet, P.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Culver, W.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Dajani, I.

Davidson, N.

Dawes, J. M.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Dawson, J. W.

Deiterding, R.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Desfarges-Berthelemot, A.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Fan, T. Y.

Fitelson, M.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Fridman, M.

Friesem, A. A.

Gavrielides, A.

I. Dajani, C. Zeringue, T. J. Bronder, T. Shay, A. Gavrielides, and C. Robin, “A theoretical treatment of two approaches to SBS mitigation with two-tone amplification,” Opt. Express 16, 14233–14247 (2008).
[CrossRef]

T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007).
[CrossRef]

Goodno, G. D.

Hanna, D. C.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Harrison, R. G.

Heebner, J. E.

Hendow, S.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Henry, L. J.

Hernandez-Cordero, J.

Hoffman, H. J.

D. C. Brown and H. J. Hoffman, “Thermal, stress and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[CrossRef]

Honea, E.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Hult, D. W.

Jacobo, A.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Jones, D. C.

Jones, P.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Ke, W.-W.

Kermene, Y.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Kobyakov, A.

Kovalev, V. I.

Kozlov, V. A.

Lefort, L.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Liu, A. P.

Loftus, T. H.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Mackechnie, C. J.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Mahodaux, C.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Mangir, M. S.

McComb, T. S.

McNaught, S. J.

Messerly, M. J.

Miller, C.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Minden, M.

Minelly, J.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Morse, T. F.

Narum, P.

R. W. Boyd, K. Rzawski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Nelson, B.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Newell, T. C.

T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007).
[CrossRef]

Nixon, M.

Norsen, M.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Pask, H. M.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Pax, P. H.

Peterson, P.

T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007).
[CrossRef]

Pureur, D.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Ranka, J. K.

Rhodes, C.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Robin, C.

Rogers, J. L.

Rothenberg, J. E.

Rowland, K. B.

Rzawski, K.

R. W. Boyd, K. Rzawski, and P. Narum, “Noise initiation of stimulated Brillouin scattering,” Phys. Rev. A 42, 5514–5521 (1990).
[CrossRef]

Sabourdy, D.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

Saitou, T.

Sanchez, A.

B. Wang and A. Sanchez, “All-fiber passive coherent combining of high power lasers,” Opt. Eng. 50, 111606 (2011).
[CrossRef]

S. J. Augst, J. K. Ranka, T. Y. Fan, and A. Sanchez, “Beam combining of ytterbium fiber amplifiers,” J. Opt. Soc. Am. B 24, 1707–1715 (2007).
[CrossRef]

Sauer, M.

Sekiguchi, T.

Shakir, S. A.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

E. J. Bochove and S. A. Shakir, “Analysis of a spatial-filtering passive fiber laser beam combining system,” IEEE J. Sel. Top. Quantum Electron. 15, 320–327 (2009).
[CrossRef]

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Sharma, M. P.

T. C. Newell, P. Peterson, A. Gavrielides, and M. P. Sharma, “Temperature effects on the emission properties of Yb-doped optical fibers,” Opt. Commun. 273, 256–259 (2007).
[CrossRef]

Shay, T.

Shay, T. M.

L. J. Henry, T. M. Shay, D. W. Hult, and K. B. Rowland, “Enhancement of output power from narrow linewidth amplifiers via two-tone effect–high power experimental results,” Opt. Express 18, 23939–23947 (2010).
[CrossRef]

I. Dajani, C. Zeringue, and T. M. Shay, “Investigation of nonlinear effects in multitone-driven narrow-linewidth high-power amplifiers,” IEEE J. Sel. Top. Quantum Electron. 15, 406–414 (2009).
[CrossRef]

Shirakawa, A.

Shu, X.-J.

Shverdin, M. Y.

Siders, C. W.

Sridharan, A. K.

Stappaerts, E. A.

Starcher, Y.

E. J. Bochove, S. A. Shakir, Y. Starcher, A. Jacobo, P. Colet, A. B. Aceves, Y. Braiman, R. Deiterding, C. Miller, and C. Rhodes, “Simple model to explain instabilities in passively-phased highpower fiber laser arrays,” Proc. SPIE 8080, 808009 (2011).
[CrossRef]

Thielen, P. A.

Thomas, A. M.

T. H. Loftus, A. M. Thomas, M. Norsen, J. Minelly, P. Jones, E. Honea, S. A. Shakir, S. Hendow, W. Culver, B. Nelson, and M. Fitelson, “Four-channel, high power passively phase locked fiber array,” Advanced Solid-State Photonics 2008 (Optical Society of America, 2008), paper WA4.

Tropper, A. C.

H. M. Pask, R. J. Carman, D. C. Hanna, A. C. Tropper, C. J. Mackechnie, P. R. Barber, and J. M. Dawes, “Ytterbium-doped silica fiber lasers: versatile sources for the 1–1.2  um region,” IEEE J. Sel. Top. Quantum Electron. 1, 2–13 (1995).
[CrossRef]

Ueda, K.

Vergien, C.

Wang, B.

B. Wang and A. Sanchez, “All-fiber passive coherent combining of high power lasers,” Opt. Eng. 50, 111606 (2011).
[CrossRef]

Wang, X.-J.

Weber, M. E.

Weiss, S. B.

Wickham, M. G.

Zeringue, C.

Adv. Opt. Photon.

Electron. Lett.

D. Sabourdy, Y. Kermene, A. Desfarges-Berthelemot, L. Lefort, A. Barthelemy, C. Mahodaux, and D. Pureur, “Power scaling of fibre lasers with all-fibre interferometric cavity,” Electron. Lett. 38, 692–693 (2002).
[CrossRef]

IEEE J. Quantum Electron.

D. C. Brown and H. J. Hoffman, “Thermal, stress and thermal-optic effects in high average power double-clad silica fiber lasers,” IEEE J. Quantum Electron. 37, 207–217 (2001).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron.

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Figures (11)

Fig. 1.
Fig. 1.

Ring laser configuration for the passive phasing of N fiber amplifiers.

Fig. 2.
Fig. 2.

Output spectrum of a high-power amplifier seeded by a flat spectrum in the range of 1000–1100 nm.

Fig. 3.
Fig. 3.

Phase noise yielded by an index noise. (a) PSD of the index noise with a two-peak Gaussian function. (b) OPD fluctuation for a 20 m fiber amplifier. (c) Corresponding phase noise of 1070 nm signal. (d) Phase difference between 1065 and 1075 nm signals. (e) PSD. (f) PSF of the phase noise.

Fig. 4.
Fig. 4.

Phase-locking process affected by the gain profile and SBS: (a) output signal spectrum and (b) the evolution of the CBC efficiency in the phaselocking process.

Fig. 5.
Fig. 5.

Statistical study on the time-averaged SBS power and the CBC efficiency influenced by the phase noise, which is characterized by CPN for a given OPLD. Condition of Pb+Ps=100W is kept in each of the sampled points.

Fig. 6.
Fig. 6.

Output signal spectra integrating over time for (a) CPN=0, (b) CPN=0.15rad, and (c) CPN=0.3rad. Solid lines in (b) and (c) are corresponding envelopes fitting to the Lorentzian function.

Fig. 7.
Fig. 7.

Statistical study on SBS influenced by OPLD distributions for the static case (a) and for the case of incorporating with the phase noise (b). Ten random OPLD distributions are evaluated for each value of the σOPLD in (a), while five solid lines in (b) denote five different sets of the phase-noise spectra.

Fig. 8.
Fig. 8.

Statistical study on SBS influenced by the FWM process. 15 sets of random samples of OPLD and phase-noise spectra are used in the comparison.

Fig. 9.
Fig. 9.

Statistical results on the SBS power versus the CBC efficiency around SBST for (a) the output signal power of 100 and 200 W in each channel and (b) of 300, 400, and 500 W in each channel. The black and red shadow regions in (a) denote SBS-allowed regions for Pth=100 and 200 W, respectively. Similarly, the black, red and blue shadow regions in (b) denote SBS-allowed regions for Pth=300, 400, and 500 W, respectively.

Fig. 10.
Fig. 10.

Typical plots on the integrated signal spectra at SBST of (a) 100 W, (b) 300 W, and (c) 500 W. Solid lines are corresponding envelopes fitting to the Gaussian function.

Fig. 11.
Fig. 11.

Maximal values of ηCBC and the corresponding minimal signal linewidths limited by SBS.

Tables (1)

Tables Icon

Table 1. Parameters Used in the Numerical Calulations

Equations (31)

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S(ν)=1N2|j=1Nexp(2πiνdi/c)|2,
ν0di/c=mi+ξi,Δνdi/c=qiξi+δ,(i=1,2,,N),
qi=ξi+(q1+ξ1)d2(q2+ξ2)d1d2d1+di(q2q1)d2d1,(i=3,4,,N).
S(ν)=|dsρ(s)e2πiνs/c|2=e2π2ν2σ2/c2,
CFB(λ)=|d2x⃗uf(x⃗,λ)u01*(x⃗,λ)dx⃗2|u01(x⃗,λ)|2|2.
uf(x⃗,λ)=ifλRm=1Nd2x⃗um(x⃗,λ)eiπx⃗·x⃗/fλ,
ηCBC=dνS(ν)1/N11/N.
dps(z,νs)dz=Γs(νs)[σse(νs)N2σsa(νs)(N0N2)ls]ps(z,νs)ps(z,νs)AaodνbgSBS(νs,νb)pb(z,νb),dpb(z,νb)dz=Γs(νb)[σse(νb)N2σsa(νb)(N0N2)ls]pb(z,νb)1Aaodνs[gSBS(νs,νb)pb(z,νb)+Q(νs,νb)]ps(z,νs),dPpf(z)dz=Γp[σpeN2σpa(N0N2)lp]Ppf(z),dPpb(z)dz=Γp[σpeN2σpa(N0N2)lp]Ppb(z).
N2(z)N0=k=s,p,bdνpk(z,ν)Γk(ν)σak(ν)/νhAeffτf+k=s,p,bdννpk(z,ν)Γk(ν)(σek(ν)+σak(ν)),
gSBS(νs,νb,z)=g0ΓSBS2(FcoFcl)[tanFcoΔν+CTΔT(z)ΓSBS/2tanFclΔν+CTΔT(z)ΓSBS/2],
Fco=2ncovsoundνsc,Fcl=2nclvsoundνsc,
Q(νs,νb,z)=gSBS(νs,νb)ΓSBSckBT(z)8ncovsoundAao,
T(z)=T0+ηhαp(z)Pp(z)2π[1ρthw+14κs]+ΔTt+ΔTcl,
ΔTt=12πκcoatlnρtρclρcl,ΔTcl=12πκslnρclρcoρco,
Δn(t,z)=dωA(ω,z)cos(ωt).
Δn¯(t)=1L0LdzΔn(t,z)A¯(ω)=1L0LdzA(ω,z).
S(ω)=1T|A¯(ω)|=1πkake(ωbk)2/μk2,
PSF(τ)=[limT1T0Tdt(ϕ(t+τ)ϕ(t))2]1/2.
Ei=kuk(x⃗)Aik(z)ei(βikzωit)+c.c.,
dPmdz=iγ[Pm2+2Pmijlm(Pi+Pj+Pl)+2ijlmPiPjPlPmeiΔkijmlz],
Δkijml=βi+βjβmβl,
γ=2πn2/λ¯Aeff,
Pths=21GAaog0Leff,
Leff=G1lnGLm+Ld
eν2σ12e(ν+Δν)2σ22=e(νΔν)2σ32Δν2σ12+σ22,
σ32=σ12σ22σ12+σ22,Δν=σ12Δνσ12+σ22.
RPN=IPN(ωH)IPN(0),IPN(ω)=[ωPSD2(w)dw]1/2,
CPN=[i=1NPSFi2(τr)RPN,i]1/2.
Δϕi(t)=|ϕ(t+τr)ϕ(t)|,(i=1,2,,N).
ηstat=Δϕ(ΔϕΔϕ)2=3,
δν(t)=dϕ(t)dt.

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