A theoretical treatment of two approaches to SBS mitigation with two-tone amplification

Iyad Dajani; Clint Zeringue; T. Justin Bronder; Thomas Shay; Athanasios Gavrielides; Craig Robin

doi:10.1364/OE.16.014233

A theoretical treatment of two approaches to SBS mitigation with two-tone amplification

Iyad Dajani, Clint Zeringue, T. Justin Bronder, Thomas Shay, Athanasios Gavrielides, and Craig Robin

Optics Express
Vol. 16,
Issue 18,
pp. 14233-14247
(2008)
•https://doi.org/10.1364/OE.16.014233

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Iyad Dajani, Clint Zeringue, T. Justin Bronder, Thomas Shay, Athanasios Gavrielides, and Craig Robin, "A theoretical treatment of two approaches to SBS mitigation with two-tone amplification," Opt. Express 16, 14233-14247 (2008)

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Related Topics
Table of Contents Category
- Nonlinear Optics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Lasers, fiber (140.3510)
- Nonlinear optics, four-wave mixing (190.4380)
- Scattering, stimulated Brillouin (290.5900)

About this Article
History
- Original Manuscript: July 9, 2008
- Revised Manuscript: August 20, 2008
- Manuscript Accepted: August 20, 2008
- Published: August 27, 2008

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Open Access

Abstract

A technique that employs two seed signals for the purpose of mitigating stimulated Brillouin scattering (SBS) effects in narrow-linewidth Yb-doped fiber amplifiers is investigated theoretically by constructing a self-consistent model that incorporates the laser gain, SBS, and four-wave mixing (FWM). The model reduces to solving a two-point boundary problem consisting of an 8x8 system of coupled nonlinear differential equations. Optimal operating conditions are determined by examining the interplay between the wavelength separation and power ratio of the two seeds. Two variants of this ‘two-tone’ amplification are considered. In one case the wavelength separation is precisely twice the Brillouin shift, while the other case considers a greater wavelength separation. For the former case, a two-fold increase in total output power over a broad range of seed power ratios centered about a ratio of approximately 2 is obtained, but with fairly large FWM. For the latter case, this model predicts an approximately 100% increase in output power (at SBS threshold with no signs of FWM) for a ‘two-tone’ amplifier with seed signals at 1064nm and 1068nm, compared to a conventional fiber amplifier with a single 1068nm seed. More significantly for this case, it is found that at a wavelength separation greater than 10nm, it is possible to appreciably enhance the power output of one of the laser frequencies.

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D. P. Machewirth, Q. Wang, B. Samson, K. Tankala, M. O’Connor, and M. Alam, “Current developments in high-power, monolithic, polarization maintaining fiber amplifiers for coherent beam combining applications,” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE 6453, 64531F (2007).
J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B. Roh, “Comparison of Stimulated Brillouin Scattering thresholds and spectra in non-polarization maintaining and polarization-maintaining passive fibers,” Fiber Lasers II: Technology, Systems, and Applications, Proc. SPIE 5709, 147–156 (2005).
B. G. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers.” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE 6453, 645307 (2007).
M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “ Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15, 8290–8299 (2007).
[Crossref] [PubMed]
S. Gray, A. Liu, D. T. Walton, J. Wang, M. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15, 17044–17050 (2007).
[Crossref] [PubMed]
M. D. Mermelstein, M. J. Andrejco, J. Fini, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Mode Area Yb-Doped Optical Fiber,” Fiber Lasers V: Technology, Systems, and Applications, Proc. SPIE 6873, 68730N (2008).
P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]
J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]
T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 24, 12188–12195 (2006).
[Crossref]
J. E. Kansky, C. X. Yu, D. V. Murphy, S. R. Shaw, R. C. Lawrence, and C. Higgs, “Beam control for a 2D polarization maintaining fiber optic phased array with a high-fiber count,” Proc. SPIE. 6306, 63060G (2006).
[Crossref]
V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, “Spectral beam combining of a broad-stripe diode laser array in an external cavity,” Opt. Lett. 25, 405–407 (2000).
[Crossref]
T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “ Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[Crossref]
F. Patel, Ph. D. Dissertation, University of California, Davis (2000).
L. G. Cohen, “Comparison of single-mode fiber dispersion measurement techniques,” J. Lightwave Technol. LT-3, 958 (1985).
[Crossref]
R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[Crossref] [PubMed]
B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer- Verlag,, Berlin, 1985).

2008 (1)

M. D. Mermelstein, M. J. Andrejco, J. Fini, C. Headley, and D. J. DiGiovanni, “11.2 dB SBS Gain Suppression in a Large Mode Area Yb-Doped Optical Fiber,” Fiber Lasers V: Technology, Systems, and Applications, Proc. SPIE 6873, 68730N (2008).

2007 (5)

T. H. Loftus, A. M. Thomas, P. R. Hoffman, M. Norsen, R. Royse, A. Liu, and E. C. Honea, “ Spectrally beam-combined fiber lasers for high-average-power applications,” IEEE J. Sel. Top. Quantum Electron. 13, 487–497 (2007).
[Crossref]

D. P. Machewirth, Q. Wang, B. Samson, K. Tankala, M. O’Connor, and M. Alam, “Current developments in high-power, monolithic, polarization maintaining fiber amplifiers for coherent beam combining applications,” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE 6453, 64531F (2007).

B. G. Ward, C. Robin, and M. Culpepper, “Photonic crystal fiber designs for power scaling of single-polarization amplifiers.” Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE 6453, 645307 (2007).

M. Li, X. Chen, J. Wang, S. Gray, A. Liu, J. Demeritt, A. B. Ruffin, A. M. Crowley, D. T. Walton, and L. A. Zenteno, “ Al/Ge co-doped large mode area fiber with high SBS threshold,” Opt. Express 15, 8290–8299 (2007).
[Crossref] [PubMed]

S. Gray, A. Liu, D. T. Walton, J. Wang, M. Li, X. Chen, A. B. Ruffin, J. A. Demeritt, and L. A. Zenteno, “502 Watt, single transverse mode, narrow linewidth, bidirectionally pumped Yb-doped fiber amplifier,” Opt. Express 15, 17044–17050 (2007).
[Crossref] [PubMed]

2006 (3)

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 24, 12188–12195 (2006).
[Crossref]

J. E. Kansky, C. X. Yu, D. V. Murphy, S. R. Shaw, R. C. Lawrence, and C. Higgs, “Beam control for a 2D polarization maintaining fiber optic phased array with a high-fiber count,” Proc. SPIE. 6306, 63060G (2006).
[Crossref]

2005 (1)

J. B. Spring, T. H. Russell, T. M. Shay, R. W. Berdine, A. D. Sanchez, B. G. Ward, and W. B. Roh, “Comparison of Stimulated Brillouin Scattering thresholds and spectra in non-polarization maintaining and polarization-maintaining passive fibers,” Fiber Lasers II: Technology, Systems, and Applications, Proc. SPIE 5709, 147–156 (2005).

2004 (1)

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

2000 (1)

V. Daneu, A. Sanchez, T. Y. Fan, H. K. Choi, G. W. Turner, and C. C. Cook, “Spectral beam combining of a broad-stripe diode laser array in an external cavity,” Opt. Lett. 25, 405–407 (2000).
[Crossref]

1985 (1)

L. G. Cohen, “Comparison of single-mode fiber dispersion measurement techniques,” J. Lightwave Technol. LT-3, 958 (1985).
[Crossref]

1972 (1)

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[Crossref] [PubMed]

Adel, P.

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

Alam, M.

Anderegg, J.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Andrejco, M. J.

Auerbach, M.

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

Berdine, R. W.

Brosnan, S.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Chen, X.

Cheung, E.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Choi, H. K.

Cohen, L. G.

L. G. Cohen, “Comparison of single-mode fiber dispersion measurement techniques,” J. Lightwave Technol. LT-3, 958 (1985).
[Crossref]

Cook, C. C.

Crowley, A. M.

Culpepper, M.

Daneu, V.

Demeritt, J.

Demeritt, J. A.

DiGiovanni, D. J.

Epp, P.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Fallnich, C.

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

Fan, T. Y.

Fini, J.

Gray, S.

Hammons, D.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Headley, C.

Higgs, C.

Hoffman, P. R.

Honea, E. C.

Kansky, J. E.

Komine, H.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Lawrence, R. C.

Li, M.

Liu, A.

Loftus, T. H.

Machewirth, D. P.

Mermelstein, M. D.

Murphy, D. V.

Norsen, M.

O’Connor, M.

Patel, F.

F. Patel, Ph. D. Dissertation, University of California, Davis (2000).

Pilipetsky, N. F.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer- Verlag,, Berlin, 1985).

Robin, C.

Roh, W. B.

Royse, R.

Ruffin, A. B.

Russell, T. H.

Samson, B.

Sanchez, A.

Sanchez, A. D.

Shaw, S. R.

Shay, T. M.

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 24, 12188–12195 (2006).
[Crossref]

Shkunov, V. V.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer- Verlag,, Berlin, 1985).

Smith, R. G.

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[Crossref] [PubMed]

Spring, J. B.

Tankala, K.

Thomas, A. M.

Turner, G. W.

Walton, D. T.

Wandt, D.

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

Wang, J.

Wang, Q.

Ward, B. G.

Weber, M.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Wessels, P.

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

Wickham, M.

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Yu, C. X.

Zel’dovich, B. Ya.

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer- Verlag,, Berlin, 1985).

Zenteno, L. A.

Appl. Opt. (1)

R. G. Smith, “Optical power handling capacity of low loss optical fibers as determined by stimulated Raman and Brillouin scattering,” Appl. Opt. 11, 2489–2494 (1972).
[Crossref] [PubMed]

Fiber Lasers II: Technology, Systems, and Applications, Proc. SPIE (1)

Fiber Lasers IV: Technology, Systems, and Applications, Proc. SPIE (2)

Fiber Lasers V: Technology, Systems, and Applications, Proc. SPIE (1)

IEEE J. Sel. Top. Quantum Electron. (1)

J. Lightwave Technol. (1)

L. G. Cohen, “Comparison of single-mode fiber dispersion measurement techniques,” J. Lightwave Technol. LT-3, 958 (1985).
[Crossref]

Opt. Express (4)

T. M. Shay, “Theory of electronically phased coherent beam combination without a reference beam,” Opt. Express 24, 12188–12195 (2006).
[Crossref]

P. Wessels, P. Adel, M. Auerbach, D. Wandt, and C. Fallnich, “ Novel suppression scheme for Brillouin scattering,” Opt. Express 12, 4443–4448 (2004).
[Crossref] [PubMed]

Opt. Lett. (1)

Proc. SPIE (1)

J. Anderegg, S. Brosnan, E. Cheung, P. Epp, D. Hammons, H. Komine, M. Weber, and M. Wickham, “Coherently coupled high power fiber arrays,” Proc. SPIE 6102, 61020U (2006).
[Crossref]

Proc. SPIE. (1)

Other (2)

F. Patel, Ph. D. Dissertation, University of California, Davis (2000).

B. Ya. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer- Verlag,, Berlin, 1985).

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Fig. 1.

The sensitivity of SBS to the ratio of the input signals in ‘two-tone’ amplification. The x-axis shows the ratio of power in one input signal (1068nm) to the total input power. The y-axis represents percentage of Stokes power.

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Fig. 2.

The output powers of the two amplified signals in a simulated ‘two-tone’ amplifier and a typical single seed fiber amplifier versus propagation distance along fiber

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Fig. 3.

Power of backward travelling Stokes light signals corresponding to 1068 nm and 1064 nm.

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Fig. 4.

The first two FWM-generated sidebands in the case of ‘two-tone’ amplification of a 1064nm seed and 1068nm seed as described above. Note that the power in these sidebands is one order of magnitude lower than the SBS power.

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Fig. 5.

The FWM behavior of SBS-suppressed two-tone amplifiers with a) 350 W (top) and b) 550 W (bottom) of pump power. The wavelength separation in this case is 4 nm. The additional pumping has altered the coherence period due to SPM and XPM.

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Fig. 6.

The evolution of two laser signals along the direction of propagation. For comparison, the case of single tone seeding is plotted.

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Fig. 7.

Total Stokes gain for the 1064 nm light for the two tone and one tone cases. For the two tone case the pump power is approximately 38 W which generated an output 1064 nm power of approximately 28 W (equal to the output at threshold in the single tone case).

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Fig. 8.

The power output of a two tone amplifier. The two tones are separated by twice the Brillouin shift allowing Stokes light generated by seed light to transfer its energy to the second seed. λ=1068 (2ν_B/ν) nm.

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Fig. 9.

The FWM for the two tones separated by twice the Brillouin shift. Due to the small wavelength separation the FWM is the lowest threshold nonlinear process.

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Fig. 10.

The dependence of total output power for two-tone amplifier on seed power ratio. The two frequencies are separated by twice the Brillouin shift. Here, The seed power ratio is defined as the input power of the higher frequency wave divided by that of the lower frequency.

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Equations (21)

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\nabla^{2} E_{i} - \frac{n_{i}^{2}}{c^{2}} \frac{\partial^{2}}{\partial t^{2}} E_{i} = \frac{1}{ε_{0} c^{2}} \frac{\partial^{2}}{\partial t^{2}} {P_{i}}^{(nl)},

E_{i} (\vec{r}, t) = \sum_{j} (\frac{1}{2}) A_{i, j} (z) ϕ_{i, j} (x, y) \exp [i (β_{i, j} - ω_{i} t)] + c . c .

\frac{d A_{1}}{dz} = \frac{g_{1}}{2} A_{1} - \frac{g_{B, 1} ε_{0} {cn}_{1 S} κ_{ao}}{4} {∣ A_{1 S} ∣}^{2} A_{1} + \frac{i ω_{1} n_{1}^{(2)} κ_{pm}}{c} [\begin{matrix} ({∣ A_{1} ∣}^{2} + 2 \sum_{i \neq 1} {∣ A_{i} ∣}^{2}) A_{1} \\ + 2 {A_{1}}^{*} A_{2} A_{3} \exp (i Δ β_{1} z) \\ + 2 A_{2}^{*} A_{3} A_{4} \exp (i Δ β_{2} z) \\ + A_{2}^{2} A_{4}^{*} \exp (i Δ β_{3} z) \end{matrix}],

Δ β_{1} = β_{2} + β_{3} - 2 β_{1} = β^{(2)} {(Δ ω)}^{2},

Δ β_{3} = 2 β_{2} - β_{1} - β_{4} = - β^{(2)} {(Δ ω)}^{2},

Δ β_{2} = β_{3} + β_{4} - β_{1} - β_{2} = 2 β^{(2)} {(Δ ω)}^{2},

κ_{pm} = κ_{ao} = \frac{\iint {∣ ϕ ∣}^{4} dx dy}{\iint {∣ ϕ ∣}^{2} dx dy} .

g_{1} = \frac{\iint (N_{2} σ_{1}^{(e)} - N_{1} σ_{1}^{(a)}) {∣ ϕ ∣}^{2} dx dy}{\iint {∣ ϕ ∣}^{2} dx dy},

\frac{d A_{1 S}}{dz} = - \frac{g_{1 S}}{2} A_{1 S} - \frac{g_{B, 1} ε_{0} {cn}_{1} κ_{ao}}{4} {∣ A_{1} ∣}^{2} A_{1 S},

\frac{d A_{2}}{dz} = \frac{g_{2}}{2} A_{2} - \frac{g_{B, 2} ε_{0} {cn}_{2 S} κ_{ao}}{4} {∣ A_{2 S} ∣}^{2} A_{2} + \frac{i ω_{2} n_{2}^{(2)} κ_{pm}}{c} [\begin{matrix} ({∣ A_{2} ∣}^{2} + 2 \sum_{i \neq 2} {∣ A_{i} ∣}^{2}) A_{2} \\ + 2 {A_{2}}^{*} A_{1} A_{4} \exp (- i Δ β_{3} z) \\ + 2 A_{1}^{*} A_{3} A_{4} \exp (i Δ β_{2} z) \\ + A_{1}^{2} A_{3}^{*} \exp (- i Δ β_{1} z) \end{matrix}]

\frac{d A_{2 S}}{dz} = - \frac{g_{2 S}}{2} A_{2 S} - \frac{g_{B, 2} ε_{0} {cn}_{2} κ_{a 0}}{4} {∣ A_{2} ∣}^{2} A_{2 S},

\frac{d A_{3}}{dz} = \frac{g_{3}}{2} A_{3} + \frac{i ω_{3} n_{3}^{(2)} κ_{pm}}{c} [\begin{matrix} ({∣ A_{3} ∣}^{2} + 2 \sum_{i \neq 3} {∣ A_{i} ∣}^{2}) A_{3} + A_{1}^{2} A_{2}^{*} \exp (- i Δ β_{1} z) \\ + 2 A_{1} A_{2} A_{4}^{*} \exp (- i Δ β_{2} z) \end{matrix}],

\frac{d A_{4}}{dz} = \frac{g_{4}}{2} A_{4} + \frac{i ω_{4} n_{4}^{(2)} κ_{pm}}{c} [\begin{matrix} ({∣ A_{4} ∣}^{2} + 2 \sum_{i \neq 4} {∣ A_{i} ∣}^{2}) A_{4} + A_{2}^{2} A_{1}^{*} \exp (- i Δ β_{3} z) \\ + 2 A_{2} A_{1} A_{3}^{*} \exp (- i Δ β_{2} z) \end{matrix}] .

N_{0} = N_{1} (z) + N_{2} (z),

N_{2} (z) = \frac{\sum_{i = 1}^{4} \frac{τ σ_{i}^{(a)}}{ℏ ω_{i}} I_{i} + \sum_{i = 1}^{2} \frac{τ σ_{iS}^{(a)}}{ℏ ω_{iS}} I_{iS} + \frac{τ σ_{p}^{(a)}}{ℏ ω_{p}} I_{p}}{\sum_{i = 1}^{4} \frac{τ (σ_{i}^{(a)} + σ_{i}^{(e)})}{ℏ ω_{i}} I_{i} + \sum_{i = 1}^{2} \frac{τ (σ_{iS}^{(a)} + σ_{iS}^{(e)})}{ℏ ω_{iS}} I_{iS} + \frac{τ (σ_{p}^{(a)} + σ_{p}^{(e)})}{ℏ ω_{p}} I_{p} + 1} \cdot N_{0},

\frac{d I_{p}}{dz} = \frac{d_{core}^{2}}{d_{clad}^{2}} (N_{2} σ_{p}^{(e)} - N_{1} σ_{p}^{(a)}) I_{p}

\frac{d A_{1}}{dz} = \frac{g_{1}}{2} A_{1} - \frac{g_{B, 1} ε_{0} {cn}_{1 S} κ_{a 0}}{4} {∣ A_{1 S} ∣}^{2} A_{1} + \frac{g_{B, 1} ε_{0} {cn}_{1} κ_{ao}}{4} {{∣ A_{2 S} ∣}^{2} A}_{1}

+ \frac{i ω_{1} n_{1}^{(2)} κ_{pm}}{c} [\begin{matrix} ({∣ A_{1} ∣}^{2} + 2 \sum_{i \neq 1} {∣ A_{i} ∣}^{2}) A_{1} \\ + 2 A_{1}^{*} A_{2} A_{3} \exp (i Δ β_{1} z) \\ + 2 A_{2}^{*} A_{3} A_{4} \exp (i Δ β_{2} z) \\ + A_{2}^{2} A_{4}^{*} \exp (i Δ β_{3} z) \end{matrix}],

\frac{d A_{2 S}}{dz} = - \frac{g_{2 S}}{2} A_{2 S} - \frac{g_{B, 2} ε_{0} {cn}_{2} κ_{a 0}}{4} {∣ A_{2} ∣}^{2} A_{2 S} + \frac{g_{B, 1} ε_{0} {cn}_{1} κ_{ao}}{4} {{∣ A_{1} ∣}^{2} A}_{2 S} .

A_{1 S} = A_{1 S} (L) \exp [\frac{g_{B} ε_{0} cn κ_{ao} {∣ A_{2} ∣}^{2}}{4} (1 - r) (L - z)],

A_{1 S} = A_{1 S} (L) \exp [\frac{g_{B} ε_{0} cn κ_{a o} {∣ A_{2} ∣}^{2}}{4} r (L - z)] .