Abstract

We developed a generalized field-propagating model for active optical fibers that takes into account mode beating and mode coupling through the amplifying medium. We applied the model to the particular case of a few-mode erbium doped fiber amplifier. Results from the model predict that mode coupling mediated by the amplifying medium is very low. Furthermore, we applied the model to a typical amplifier configuration. In this particular configuration, the new model predicts much lower differential modal gain than that predicted by a classical intensity model.

© 2017 Optical Society of America

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References

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  1. D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nature Photon. 7, 354–362 (2013).
    [Crossref]
  2. J.-B. Trinel, G. Le Cocq, E.R. Andresen, Y. Quiquempois, and L. Bigot, “Latest results and future perspectives on Few-Mode Erbium Doped Fiber Amplifiers,” Opt. Fiber Technol., in press (2016).
    [Crossref]
  3. C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271–283 (1991).
    [Crossref]
  4. 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, 3236 (2007).
    [Crossref] [PubMed]
  5. Z. Jiang and J. Marciante, “Impact of transverse spatial-hole burning on beam quality in large-mode-area Yb-doped fibers,” J. Opt. Soc. Am. B 25, 247–254 (2008).
    [Crossref]
  6. Q. Kang, E.-L. Lim, Y. Jung, J. Sahu, F. Poletti, C. Baskiotis, S.-U. Alam, and D. Richardson, “Accurate modal gain control in a multimode erbium doped fiber amplifier incorporating ring doping and a simple LP01 pump configuration,” Opt. Express 20, 20835–20843 (2012).
    [Crossref] [PubMed]
  7. G. Le Cocq, L. Bigot, A. Le Rouge, M. Bigot-Astruc, P. Sillard, C. Koebele, M. Salsi, and Y. Quiquempois, “Modeling and characterization of a few-mode EDFA supporting four mode groups for mode division multiplexing,” Opt. Express 20, 27051–27061 (2012).
    [Crossref] [PubMed]
  8. E.-L. Lim, Q. Kang, M. Gecevicius, F. Poletti, S. Alam, and D. Richardson, “Vector Mode effects in Few Moded Erbium Doped Fiber Amplifiers,” in OFC Technical Digest (2013), paper OTu3G.2.
  9. S. Akhtari and P. Krummrich, “Impact of mode beating effects in optical multi-mode amplifiers for space division multiplexing,” IEEE Photon. Technol. Lett. 25, 2482–2485 (2013).
    [Crossref]
  10. S. Akhtari, M. Finkenbusch, R. Winfield, and P. Krummrich, “Experimental analysis of the impact of beating between signal modes on few-mode erbium doped fiber amplifier performance,” in OFC Technical Digest (2015), paper Tu3C.4.
  11. P. Stremplewski and C. Koepke, “Up-converted emission and mode beating in Er^3+- doped fibers,” Opt. Express 23, 28288 (2015).
    [Crossref] [PubMed]
  12. R. Nasiri Mahalati, D. Askarov, and J. Kahn, “Adaptive modal gain equalization techniques in multi-mode erbium-doped fiber amplifiers,” J. Lightwave Technol. 32, 2133–2143 (2014).
    [Crossref]
  13. B. Sévigny, G. Le Cocq, C.C.C. Carrero, C. Valentin, P. Sillard, G. Bouwmans, L. Bigot, and Y. Quiquempois, “Advanced S2 imaging: Application of multivariate statistical analysis to spatially and spectrally resolved datasets,” J. Lightwave Technol. 32, 4004–4010 (2014).
    [Crossref]

2015 (1)

2014 (2)

2013 (2)

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nature Photon. 7, 354–362 (2013).
[Crossref]

S. Akhtari and P. Krummrich, “Impact of mode beating effects in optical multi-mode amplifiers for space division multiplexing,” IEEE Photon. Technol. Lett. 25, 2482–2485 (2013).
[Crossref]

2012 (2)

2008 (1)

2007 (1)

1991 (1)

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271–283 (1991).
[Crossref]

Akhtari, S.

S. Akhtari and P. Krummrich, “Impact of mode beating effects in optical multi-mode amplifiers for space division multiplexing,” IEEE Photon. Technol. Lett. 25, 2482–2485 (2013).
[Crossref]

S. Akhtari, M. Finkenbusch, R. Winfield, and P. Krummrich, “Experimental analysis of the impact of beating between signal modes on few-mode erbium doped fiber amplifier performance,” in OFC Technical Digest (2015), paper Tu3C.4.

Alam, S.

E.-L. Lim, Q. Kang, M. Gecevicius, F. Poletti, S. Alam, and D. Richardson, “Vector Mode effects in Few Moded Erbium Doped Fiber Amplifiers,” in OFC Technical Digest (2013), paper OTu3G.2.

Alam, S.-U.

Andresen, E.R.

J.-B. Trinel, G. Le Cocq, E.R. Andresen, Y. Quiquempois, and L. Bigot, “Latest results and future perspectives on Few-Mode Erbium Doped Fiber Amplifiers,” Opt. Fiber Technol., in press (2016).
[Crossref]

Askarov, D.

Baskiotis, C.

Bigot, L.

Bigot-Astruc, M.

Bouwmans, G.

Carrero, C.C.C.

Desurvire, E.

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271–283 (1991).
[Crossref]

Fini, J.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nature Photon. 7, 354–362 (2013).
[Crossref]

Finkenbusch, M.

S. Akhtari, M. Finkenbusch, R. Winfield, and P. Krummrich, “Experimental analysis of the impact of beating between signal modes on few-mode erbium doped fiber amplifier performance,” in OFC Technical Digest (2015), paper Tu3C.4.

Gecevicius, M.

E.-L. Lim, Q. Kang, M. Gecevicius, F. Poletti, S. Alam, and D. Richardson, “Vector Mode effects in Few Moded Erbium Doped Fiber Amplifiers,” in OFC Technical Digest (2013), paper OTu3G.2.

Giles, C.

C. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271–283 (1991).
[Crossref]

Gong, M.

Jiang, Z.

Jung, Y.

Kahn, J.

Kang, Q.

Koebele, C.

Koepke, C.

Krummrich, P.

S. Akhtari and P. Krummrich, “Impact of mode beating effects in optical multi-mode amplifiers for space division multiplexing,” IEEE Photon. Technol. Lett. 25, 2482–2485 (2013).
[Crossref]

S. Akhtari, M. Finkenbusch, R. Winfield, and P. Krummrich, “Experimental analysis of the impact of beating between signal modes on few-mode erbium doped fiber amplifier performance,” in OFC Technical Digest (2015), paper Tu3C.4.

Le Cocq, G.

Le Rouge, A.

Li, C.

Liao, S.

Lim, E.-L.

Marciante, J.

Nasiri Mahalati, R.

Nelson, L.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nature Photon. 7, 354–362 (2013).
[Crossref]

Poletti, F.

Quiquempois, Y.

Richardson, D.

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nature Photon. 7, 354–362 (2013).
[Crossref]

Q. Kang, E.-L. Lim, Y. Jung, J. Sahu, F. Poletti, C. Baskiotis, S.-U. Alam, and D. Richardson, “Accurate modal gain control in a multimode erbium doped fiber amplifier incorporating ring doping and a simple LP01 pump configuration,” Opt. Express 20, 20835–20843 (2012).
[Crossref] [PubMed]

E.-L. Lim, Q. Kang, M. Gecevicius, F. Poletti, S. Alam, and D. Richardson, “Vector Mode effects in Few Moded Erbium Doped Fiber Amplifiers,” in OFC Technical Digest (2013), paper OTu3G.2.

Sahu, J.

Salsi, M.

Sévigny, B.

Sillard, P.

Stremplewski, P.

Trinel, J.-B.

J.-B. Trinel, G. Le Cocq, E.R. Andresen, Y. Quiquempois, and L. Bigot, “Latest results and future perspectives on Few-Mode Erbium Doped Fiber Amplifiers,” Opt. Fiber Technol., in press (2016).
[Crossref]

Valentin, C.

Winfield, R.

S. Akhtari, M. Finkenbusch, R. Winfield, and P. Krummrich, “Experimental analysis of the impact of beating between signal modes on few-mode erbium doped fiber amplifier performance,” in OFC Technical Digest (2015), paper Tu3C.4.

Yan, P.

Yuan, Y.

Zhang, H.

IEEE Photon. Technol. Lett. (1)

S. Akhtari and P. Krummrich, “Impact of mode beating effects in optical multi-mode amplifiers for space division multiplexing,” IEEE Photon. Technol. Lett. 25, 2482–2485 (2013).
[Crossref]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. B (1)

Nature Photon. (1)

D. Richardson, J. Fini, and L. Nelson, “Space-division multiplexing in optical fibres,” Nature Photon. 7, 354–362 (2013).
[Crossref]

Opt. Express (4)

Other (3)

S. Akhtari, M. Finkenbusch, R. Winfield, and P. Krummrich, “Experimental analysis of the impact of beating between signal modes on few-mode erbium doped fiber amplifier performance,” in OFC Technical Digest (2015), paper Tu3C.4.

J.-B. Trinel, G. Le Cocq, E.R. Andresen, Y. Quiquempois, and L. Bigot, “Latest results and future perspectives on Few-Mode Erbium Doped Fiber Amplifiers,” Opt. Fiber Technol., in press (2016).
[Crossref]

E.-L. Lim, Q. Kang, M. Gecevicius, F. Poletti, S. Alam, and D. Richardson, “Vector Mode effects in Few Moded Erbium Doped Fiber Amplifiers,” in OFC Technical Digest (2013), paper OTu3G.2.

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Figures (9)

Fig. 1
Fig. 1 Transverse intensity and electric field distribution at a time t for (a). HE 31 e v e n mode, (b). EH 11 e v e n mode, and their combination when: (c). Θ ( z ) = 0 [ 2 π ] ( LP 21 b , y ), (d). Θ(z) = 0.5π [2π], (e). Θ ( z ) = π [ 2 π ] ( LP 21 b , y ), (f). Θ(z) = 1.5π [2π].
Fig. 2
Fig. 2 Longitudinal evolution of the term gp,i,j presented in Eq. (6b) at pump wavelength (λp = 980 nm) for modes TE01 (a) and HE 21 o d d (b). Only modes n corresponding to higher values of this term are identified. It has to be noted that some curves are superposed and are identified together with the symbol “&”.
Fig. 3
Fig. 3 Longitudinal evolution of the term gs,i,j presented in Eq. (6b) for signal wavelength (λs = 1550 nm) for modes i: (a) HE 11 o d d, (b) HE 11 e v e n, (c) TE01, (d) TM01, (e) HE 21 o d d and (f) HE 21 e v e n. Only modes n corresponding to higher values of this term are identified. It has to be noted that some curves are superposed and are identified together with the symbol “&”.
Fig. 4
Fig. 4 Longitudinal modal power evolution at λp = 980 nm obtained with field model, only some modes are identified. It has to be noted that some curves are superposed and are identified together with the symbol “&”.
Fig. 5
Fig. 5 Longitudinal modal power evolution at 1550 nm obtained with field model. It has to be noted that some curves are superposed and are identified together with the symbol “&”.
Fig. 6
Fig. 6 Longitudinal evolution of normalized average N2 population calculated with intensity model using LP modes (solid) and field model using vector modes (dashed). For the first beat length, resultant intensity profile of pump beam is depicted at specific z positions.
Fig. 7
Fig. 7 Longitudinal evolution of modal power at 980 nm calculated with intensity LP model (solid) and field model using vector modes(dashed)
Fig. 8
Fig. 8 Longitudinal evolution of modal power at 2 signal wavelengths : (a) 1530 nm, (b) 1560 nm. Calculated with intensity LP model (solid) and with field model using vector modes (dashed)
Fig. 9
Fig. 9 Modal and spectral gain provided by the FM-EDFA simulated with (a) intensity LP code and (b) field model using vector modes.

Tables (2)

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Table 1 DSG calculated with intensity model and advanced field model

Tables Icon

Table 2 DMG calculated with intensity model and advanced field model

Equations (16)

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ϕ k , i , ± ( z ) z = ± ϕ k , i , ± ( z ) σ e , k I k , i ( x , y ) N 2 ( x , y , z ) d S ϕ k , i , ± ( z ) σ a , k I k , i ( x , y ) ( N ( x , y , z ) N 2 ( x , y , z ) ) d S ± Δ v I k , i ( x , y ) N 2 ( x , y , z ) d S
N 2 ( x , y , z ) = N ( x , y , z ) × k i ± ϕ k , i , ± ( z ) I k , i ( x , y ) σ a , k A 21 + k i ± ϕ k , i , ± ( z ) I k , i ( x , y ) ( σ a , k + σ e , k )
E k , i ( x , y , z ) = F k , i ( x , y ) A k , i ( z ) exp ( ± j β k , i z )
k ( x , y , z ) = i E k , i ( x , y , z ) i E * k , i ( x , y , z )
k ( x , y , z ) = F 1 ( x , y ) 2 | A 1 ( z ) | 2 + F 2 ( x , y ) 2 | A 2 ( z ) | 2 + F 1 ( x , y ) F 2 ( x , y ) A 1 ( z ) A 2 * ( z ) exp ( j Θ k , 1 , 2 ( z ) ) + F 1 ( x , y ) F 2 ( x , y ) A 1 * ( z ) A 2 ( z ) exp ( j Θ k , 1 , 2 ( z ) )
A k , i , ± ( z ) z = j = 1 n 1 2 A k , j , ± ( z ) g k , i , j exp ( ± j Θ k , i , j ( z ) )
g k , i , j = [ σ e , k N 2 ( x , y , z ) σ a , k N 1 ( x , y , z ) ] F k , i ( x , y ) F k , j ( x , y ) d S
I k , i A S E ( x , y ) = F k , i ( x , y ) F k , i ( x , y )
ϕ k , i , ± ( z ) z = ± ϕ k , i , ± ( z ) σ e , k I k , i A S E ( x , y ) N 2 ( x , y , z ) d S ϕ k , i , ± ( z ) σ a , k I k , i A S E ( x , y ) ( N ( x , y , z ) N 2 ( x , y , z ) ) d S ± Δ v I k , i A S E ( x , y ) N 2 ( x , y , z ) d S
N 2 ( x , y , z ) = N ( x , y , z ) × K num p ( x , y , z ) + K num s ( x , y , z ) + K num ASE ( x , y , z ) A 21 + K denom p ( x , y , z ) + K denom s ( x , y , z ) + K denom ASE ( x , y , z )
K num p ( x , y , z ) = ± p , ± ( x , y , z ) σ a , p
K num s ( x , y , z ) = k = 1 Ω s ± k , ± ( x , y , z ) σ a , k
K num ASE ( x , y , z ) = k = 1 Ω e i = 1 n ( λ e ) ± ϕ k , i , ± ( z ) I k , i A S E ( x , y ) σ a , k
K denom p ( x , y , z ) = ± p , ± ( x , y , z ) ( σ a , p + σ e , p )
K denom s ( x , y , z ) = k = 1 Ω s ± k , ± ( x , y , z ) ( σ a , k + σ e , k )
K denom ASE ( x , y , z ) = k = 1 Ω e i = 1 n ( λ e ) ± ϕ k , i , ± ( z ) I k , i A S E ( x , y ) ( σ a , k + σ e , k )

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