Abstract

A coupled-mode formalism, earlier used to describe transverse mode instabilities in single-pass optical fiber amplifiers is extended to the case of double-pass amplifiers. Contrary to the single-pass case, it is shown that the thermo-optic nonlinearity can couple light at the same frequency between the LP01 and LP11 modes, leading to a static deformation of the output beam profile. This novel phenomenon is caused by the interaction of light propagating in either direction with thermo-optic index perturbations caused by light propagating in the opposite direction. The threshold power for the static deformation is found to be several times lower than what is typically found for the dynamic modal instabilities observed in single-pass amplifiers.

© 2016 Optical Society of America

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  7. C. Jauregui, H.-J. Otto, F. Stutzki, J. Limpert, and A. Tünnermann, “Simplified modelling the mode instability threshold of high power fiber amplifiers in the presence of photodarkening,” Opt. Express 23, 20203–20218 (2015).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  17. O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
    [Crossref]
  18. K. Hejaz, A. Norouzey, R. Poozesh, A. Heidariazar, A. Roohforouz, R. R. Nasirabad, N. T. Jafari, A. H. Golshan, A. Babazadeh, and M. Lafouti, “Controlling mode instability in a 500 W ytterbium-doped fiber laser,” Laser Physics 24, 025102 (2014).
    [Crossref]

2016 (2)

J. Lægsgaard, “Optimizing Yb concentration of fiber amplifiers in the presence of transverse modal instabilities and photodarkening,” Appl. Opt. 55, 1966–1970 (2016).
[Crossref] [PubMed]

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

2015 (1)

2014 (2)

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

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

2013 (4)

2012 (1)

2011 (2)

2006 (1)

J. R. Marciante and J. D. Zuegel, “High-gain, polarization-preserving, Yb-doped fiber amplifier for low-duty-cycle pulse amplification,” Appl. Optics 45, 6798–6804 (2006).
[Crossref]

2001 (1)

1987 (1)

Alekseev, D.

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

Alkeskjold, T. T.

Antipov, O.

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

Babazadeh, A.

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

Broeng, J.

Chang, C.-L.

P.-Y. Lai, C.-L. Chang, S.-L. Huang, and S.-H. Chen, “Effective suppression of stimulated Raman scattering in high power fiber amplifiers using double-pass scheme,” in “FIBER LASERS XI: TECHNOLOGY, SYSTEMS, AND APPLICATIONS,”, vol. 8961 of Proceedings of SPIE, S Ramachandran, ed., SPIE; NKT Photon A S; PolarOnyx, Inc (SPIE-INT SOC OPTICAL ENGINEERING, 1000 20TH ST, PO BOX 10, BELLINGHAM, WA 98227-0010 USA, 2014), vol. 8961 of Proceedings of SPIE. Conference on Fiber Lasers XI - Technology, Systems, and Applications, San Francisco, CA, FEB 03–06, 2014.

Chen, S.-H.

P.-Y. Lai, C.-L. Chang, S.-L. Huang, and S.-H. Chen, “Effective suppression of stimulated Raman scattering in high power fiber amplifiers using double-pass scheme,” in “FIBER LASERS XI: TECHNOLOGY, SYSTEMS, AND APPLICATIONS,”, vol. 8961 of Proceedings of SPIE, S Ramachandran, ed., SPIE; NKT Photon A S; PolarOnyx, Inc (SPIE-INT SOC OPTICAL ENGINEERING, 1000 20TH ST, PO BOX 10, BELLINGHAM, WA 98227-0010 USA, 2014), vol. 8961 of Proceedings of SPIE. Conference on Fiber Lasers XI - Technology, Systems, and Applications, San Francisco, CA, FEB 03–06, 2014.

Cho, G.

Dong, L.

Eidam, T.

Fermann, M.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, Cambridge, 2001).

Galvanauskas, A.

Golshan, A. H.

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

Hansen, K. R.

Hariharan, A.

Harter, D.

Heidariazar, A.

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

Hejaz, K.

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

Huang, S.-L.

P.-Y. Lai, C.-L. Chang, S.-L. Huang, and S.-H. Chen, “Effective suppression of stimulated Raman scattering in high power fiber amplifiers using double-pass scheme,” in “FIBER LASERS XI: TECHNOLOGY, SYSTEMS, AND APPLICATIONS,”, vol. 8961 of Proceedings of SPIE, S Ramachandran, ed., SPIE; NKT Photon A S; PolarOnyx, Inc (SPIE-INT SOC OPTICAL ENGINEERING, 1000 20TH ST, PO BOX 10, BELLINGHAM, WA 98227-0010 USA, 2014), vol. 8961 of Proceedings of SPIE. Conference on Fiber Lasers XI - Technology, Systems, and Applications, San Francisco, CA, FEB 03–06, 2014.

Jafari, N. T.

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

Jansen, F.

Jauregui, C.

Kuznetsov, M.

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

Lægsgaard, J.

Lafouti, M.

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

Lai, P.-Y.

P.-Y. Lai, C.-L. Chang, S.-L. Huang, and S.-H. Chen, “Effective suppression of stimulated Raman scattering in high power fiber amplifiers using double-pass scheme,” in “FIBER LASERS XI: TECHNOLOGY, SYSTEMS, AND APPLICATIONS,”, vol. 8961 of Proceedings of SPIE, S Ramachandran, ed., SPIE; NKT Photon A S; PolarOnyx, Inc (SPIE-INT SOC OPTICAL ENGINEERING, 1000 20TH ST, PO BOX 10, BELLINGHAM, WA 98227-0010 USA, 2014), vol. 8961 of Proceedings of SPIE. Conference on Fiber Lasers XI - Technology, Systems, and Applications, San Francisco, CA, FEB 03–06, 2014.

Limpert, J.

Marciante, J. R.

J. R. Marciante and J. D. Zuegel, “High-gain, polarization-preserving, Yb-doped fiber amplifier for low-duty-cycle pulse amplification,” Appl. Optics 45, 6798–6804 (2006).
[Crossref]

Nasirabad, R. R.

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

Norouzey, A.

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

Otto, H. J.

C. Jauregui, H. J. Otto, J. Limpert, and A. Tünnermann, “Mode instabilities in high-power bidirectional fiber amplifiers and lasers,” in “Advanced Solid State Lasers,” (Optical Society of America, 2015), p. ATh2A.24.
[Crossref]

Otto, H.-J.

Poozesh, R.

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

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, Cambridge, 2001).

Roohforouz, A.

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

Sakuda, K.

Schmidt, O.

Schreiber, T.

Smith, A. V.

Smith, J. J.

Stutzki, F.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, Cambridge, 2001).

Tünnermann, A.

Tyrtyshnyy, V.

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

Vershinin, O.

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, Cambridge, 2001).

Ward, B. G.

Wirth, C.

Yamada, M.

Zuegel, J. D.

J. R. Marciante and J. D. Zuegel, “High-gain, polarization-preserving, Yb-doped fiber amplifier for low-duty-cycle pulse amplification,” Appl. Optics 45, 6798–6804 (2006).
[Crossref]

Appl. Opt. (2)

Appl. Optics (1)

J. R. Marciante and J. D. Zuegel, “High-gain, polarization-preserving, Yb-doped fiber amplifier for low-duty-cycle pulse amplification,” Appl. Optics 45, 6798–6804 (2006).
[Crossref]

Laser Physics (1)

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

Opt. Express (8)

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

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

T. Eidam, C. Wirth, C. Jauregui, F. Stutzki, F. Jansen, H.-J. Otto, O. Schmidt, T. Schreiber, J. Limpert, and A. Tünnermann, “Experimental observations of the threshold-like onset of mode instabilities in high power fiber amplifiers,” Opt. Express 19, 13218–13224 (2011).
[Crossref] [PubMed]

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

B. G. Ward, “Modeling of transient modal instability in fiber amplifiers,” Opt. Express 21, 12053–12067 (2013).
[Crossref] [PubMed]

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

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] [PubMed]

C. Jauregui, H.-J. Otto, F. Stutzki, J. Limpert, and A. Tünnermann, “Simplified modelling the mode instability threshold of high power fiber amplifiers in the presence of photodarkening,” Opt. Express 23, 20203–20218 (2015).
[Crossref] [PubMed]

Opt. Lett. (2)

Proc. SPIE (1)

O. Antipov, M. Kuznetsov, V. Tyrtyshnyy, D. Alekseev, and O. Vershinin, “Low-threshold mode instability in yb3+-doped few-mode fiber amplifiers: influence of a backward reflection,” Proc. SPIE 9728, 97280A (2016).
[Crossref]

Other (3)

C. Jauregui, H. J. Otto, J. Limpert, and A. Tünnermann, “Mode instabilities in high-power bidirectional fiber amplifiers and lasers,” in “Advanced Solid State Lasers,” (Optical Society of America, 2015), p. ATh2A.24.
[Crossref]

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in Fortran 77 (Cambridge University, Cambridge, 2001).

P.-Y. Lai, C.-L. Chang, S.-L. Huang, and S.-H. Chen, “Effective suppression of stimulated Raman scattering in high power fiber amplifiers using double-pass scheme,” in “FIBER LASERS XI: TECHNOLOGY, SYSTEMS, AND APPLICATIONS,”, vol. 8961 of Proceedings of SPIE, S Ramachandran, ed., SPIE; NKT Photon A S; PolarOnyx, Inc (SPIE-INT SOC OPTICAL ENGINEERING, 1000 20TH ST, PO BOX 10, BELLINGHAM, WA 98227-0010 USA, 2014), vol. 8961 of Proceedings of SPIE. Conference on Fiber Lasers XI - Technology, Systems, and Applications, San Francisco, CA, FEB 03–06, 2014.

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Figures (5)

Fig. 1
Fig. 1 Fraction of HOM output power versus pump power for the solutions of Eqs. (1013) (filled black/blue/green circles), as well as the approximate transfer-matrix solution (solid red curve).
Fig. 2
Fig. 2 Modal power distribution along z for forward- and backward-propagating fields in four selected cases: (a) 60 W pump, high HOM fraction, (b) 60 W pump, low HOM fraction, (c) 132 W pump high HOM fraction, (d) 198 W pump.
Fig. 3
Fig. 3 Spatial output power distribution for a selection of the solutions with high HOM fraction shown in Fig. 1: (a) 40 W pump, (b) 50 W pump, (c) 85 W pump, (d) 90 W pump. The black circle indicates the core boundary. Contour levels are on a linear scale.
Fig. 4
Fig. 4 Fraction of HOM output power versus pump power for the ’reference’ fiber, as well as for a fiber with doubled Yb concentration and pump cladding area (’High Yb’) and one with a V-parameter of 2.5.
Fig. 5
Fig. 5 Fraction of HOM output power versus pump power for the ’reference’ configuration, as well as for systems with 100 mW seed power, 1 mW HOM seed (corresponding to 0.1% of the total seed), or a π phase shift between FM and HOM in the reflection.

Equations (47)

Equations on this page are rendered with MathJax. Learn more.

E + ( r , t ) = 1 2 [ a 1 + ( z ) e i ( ω t β 1 z ) Ψ 1 ( r ) + a 2 + ( z ) e i ( ω t β 2 z ) Ψ 2 ( r ) + c . c . ]
E ( r , t ) = 1 2 [ a 1 ( z ) e i ( ω t + β 1 z ) Ψ 1 ( r ) + a 2 ( z ) e i ( ω t + β 2 z ) Ψ 2 ( r ) + c . c . ]
I ( r ) = I + ( r ) + I ( r ) = | a 1 + ( z ) | 2 | Ψ 1 ( r ) | 2 + | a 2 + ( z ) | 2 | Ψ 2 ( r ) | 2 + 2 Re [ a 1 + * ( z ) a 2 + ( z ) e i Δ β z Ψ 1 * ( r ) Ψ 2 ( r ) ] + | a 1 ( z ) | 2 | Ψ 1 ( r ) | 2 + | a 2 ( z ) | 2 | Ψ 2 ( r ) | 2 + 2 Re [ a 1 * ( z ) a 2 ( z ) e i Δ β z Ψ 1 * ( r ) Ψ 2 ( r ) ]
Q ( r ) = ( ω p ω 1 ) I ( r ) g ( r ) = I ( r ) g 0 ( z ) 1 + I ( r ) I sat ( z )
g 0 ( z ) = N Yb I p ( z ) ( σ a p σ es σ as σ e p ) P τ σ as I p ( z ) ( σ a p + σ e p ) + P τ
I sat ( z ) = ω ω p I p ( z ) ( σ a p + σ e p ) + P τ σ as + σ es
P τ = h ¯ ω p τ ; I p ( z ) = P p ( z ) A p
Δ ε ( r ) = η κ d r G ( r , r ) Q ( r , z ) + i g 0 ( z ) 1 + I ( r ) I sat ( z ) n 0 k 0
G ( r , r ) = 1 2 π m = G m ( r , r ) e i m ( ϕ ϕ )
d a 1 + d z = i k 0 2 n 0 [ a 1 + ( z ) Δ ε 11 ( z ) + a 2 + ( z ) e i Δ β z Δ ε 12 ( z ) ]
d a 2 + d z = i k 0 2 n 0 [ a 2 + ( z ) Δ ε 22 ( z ) + a 1 + ( z ) e i Δ β z Δ ε 12 ( z ) ]
d a 1 d z = i k 0 2 n 0 [ a 1 ( z ) Δ ε 11 ( z ) + a 2 ( z ) e i Δ β z Δ ε 12 ( z ) ]
d a 2 d z = i k 0 2 n 0 [ a 2 ( z ) Δ ε 22 ( z ) + a 1 ( z ) e i Δ β z Δ ε 12 ( z ) ]
Δ ε m n ( z ) = d r Ψ m * ( r ) Δ ε ( r ) Ψ n ( r )
d P p d z = P p N Yb A p A d d r [ σ a p n 2 ( r , z ) ( σ a p + σ e p ) ]
n 2 ( r , z ) = g 0 ( z ) N Yb ( σ as + σ es ) ( 1 + I s ( r , z ) I sat ( z ) ) + σ as σ as + σ es
a m ( L ) = a m + ( L ) R m ; R m = r m e i ϕ m e 2 i β m L , m = 1 , 2
Ψ 1 ( r ) = 1 2 π R 1 ( r ) ; Ψ 2 ( r ) = 1 τ R 2 ( r ) x r
d a 1 + d z = [ g 11 ( z ) i 2 ( | a 1 + ( z ) | 2 + | a 1 ( z ) | 2 ) G 11 ( z ) ] a 1 + ( z )
d a 1 d z = [ g 11 ( z ) i 2 ( | a 1 + ( z ) | 2 + | a 1 ( z ) | 2 ) G 11 ( z ) ] a 1 ( z )
g 11 ( z ) = g 0 0 r d d r r R 1 2 ( r ) 1 + I 0 ( r , z ) I sat ( z )
I 0 ( r , z ) = ( | a 1 + ( z ) | 2 + | a 1 ( z ) | 2 ) R 1 2 ( r ) 2 π
G 11 ( z ) = η k 0 ( ω p ω 1 ) 2 π n 0 κ g 0 ( z ) d r r R 1 2 ( r ) d r r G 0 ( r , r ) R 1 2 ( r ) 1 + I 0 ( r , z ) I sat ( z )
I ( r ) I 0 ( r , z ) + I 1 ( r ) = I 0 ( r , z ) + 2 Re [ a 1 + * ( z ) a 2 + ( z ) e i Δ β z + a 1 * ( z ) e i Δ β z ] R 1 ( r ) R 2 ( r ) x r
g 0 ( z ) 1 + I ( r ) I sat ( z ) g 0 ( z ) 1 + I 0 ( r , z ) I sat ( z ) ( 1 I 1 ( r ) I sat ( z ) + I 0 ( r , z ) )
Δ ε 12 ( z ) ( G 12 ( z ) i n 0 k 0 g 12 ( z ) ) 2 Re [ a 1 + * ( z ) a 2 + ( z ) e i Δ β z + a 1 * ( z ) a 2 ( z ) e i Δ β z ]
Δ ε 22 ( z ) i n 0 k 0 g 22 ( z ) + G 22 ( z ) [ | a 1 + ( z ) | 2 + | a 1 ( z ) | 2 ]
g 12 ( z ) = g 0 ( z ) 0 r d d r r R 1 2 ( r ) R 2 2 ( r ) I sat ( z ) + 2 I 0 ( r , z ) + I 0 2 ( r , z ) I sat ( z )
G 12 ( z ) = η ( ω p 1 2 π κ ) 2 π κ g 0 ( z ) 0 z d r r R 1 ( r ) R 2 ( r ) 0 r d d r r R 1 ( r ) R 2 ( r ) G 1 ( r , r ) ( 1 + I 0 ( r , z ) I sat ( z ) ) 2
g 22 ( z ) = g 0 0 r d d r r R 2 2 ( r ) 1 + I 0 ( r , z ) I sat ( z )
G 22 ( z ) = η ( ω p 1 2 π κ ) 2 π κ g 0 ( z ) 0 d r r R 2 2 ( r ) 0 r d d r r R 1 2 ( r ) R 2 ( r ) G 0 ( r , r ) ( 1 + I 0 ( r , z ) I sat ( z ) )
d a 2 + d z = γ + ( z ) a 2 + ( z ) + δ ( z ) a 2 * ( z )
d a 2 * d z = γ ( z ) a 2 * ( z ) δ * ( z ) a 2 + ( z )
γ + ( z ) = 1 2 { g 22 ( z ) i k 0 n 0 [ | a 1 + ( z ) | 2 + | a 1 ( z ) | 2 ] G 22 ( z ) [ g 12 ( z ) + i k 0 n 0 G 12 ( z ) ] | a 1 + ( z ) | 2 }
γ ( z ) = 1 2 { g 22 ( z ) i k 0 n 0 [ | a 1 + ( z ) | 2 + | a 1 ( z ) | 2 ] G 22 ( z ) [ g 12 ( z ) + i k 0 n 0 G 12 ( z ) ] | a 1 ( z ) | 2 }
δ ( z ) = 1 2 [ i k 0 n 0 G 12 ( z ) + g 12 ( z ) ] a 1 + ( z ) a 1 ( z )
a 2 + ( z + Δ z ) = t 11 ( z ) a 2 + ( z ) + t 12 ( z ) a 2 * ( z )
a 2 * ( z + Δ z ) = t 12 ( z ) a 2 + a 2 ( z ) + t 22 ( z ) a 2 * ( z )
t 11 ( z ) = 1 + γ + ( z ) Δ z ; t 12 ( z ) = δ ( z ) Δ z
t 21 ( z ) = δ * ( z ) Δ z ; t 22 ( z ) = 1 γ ( z ) Δ z
a 2 + ( L ) = T 11 a 2 + ( 0 ) + T 12 a 2 * ( 0 )
a 2 * ( L ) = T 21 a 2 + ( 0 ) + T 22 a 2 * ( 0 )
T = = i = 1 N t = ( z i )
a 2 * ( L ) = a 2 * ( L ) R 2 * a 2 + ( 0 ) T 21 + a 2 * ( 0 ) T 22 = R 2 * ( a 2 + * ( 0 ) T 11 * + a 2 ( 0 ) T 12 *
a 2 ( L ) = a 2 + ( L ) R 2 a 2 + * ( 0 ) T 21 * + a 2 ( 0 ) T 22 * = R 2 ( a 2 + ( 0 ) T 11 + a 2 * ( 0 ) T 12 )
a 2 + ( L ) = a 2 + ( 0 ) ( T 11 T 12 T 21 T 22 ) + a 2 + * ( 0 ) R 2 * ( T 11 * ( T 12 T 21 T 22 ) * ) T 12 T 22 1 | T 12 | 2 | T 22 | 2 r 2 2
a 2 ( 0 ) = R 2 a 2 + ( L ) T 21 * a + * ( 0 ) T 22 *

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