Abstract

We present an analytical solution for the coupled rate and propagation equations for a dynamic two-level homogeneously broadened system interacting with radiation and with constant population inversion along the longitudinal axis of the fiber, z. We derive an analytical solution for the z dependence of these equations, which greatly simplifies the numerical solution for the output powers’ time dependence. Amplified spontaneous emission and background loss influences are considered in the model, in contrast to the previous analytical solution presented by Y. Sun et al. The solution is derived, and the importance of each term for the dynamic modeling of typical erbium-doped fiber amplifiers is analyzed.

© 2004 Optical Society of America

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References

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  1. C. R. Giles and E. Desurvire, “Modeling erbium-doped fiber amplifiers,” J. Lightwave Technol. 9, 271–283 (1991).
    [CrossRef]
  2. A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
    [CrossRef]
  3. Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
    [CrossRef]
  4. Y. Sun, J. L. Zyskind, and A. K. Sristava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1006 (1997).
    [CrossRef]
  5. A. Bononi and L. A. Rusch, “Doped-fiber amplifier dynamics: a system perspective,” J. Lightwave Technol. 16, 945–956 (1998).
    [CrossRef]
  6. T. Georges and E. Delevaque, “Analytic modeling of high-gain erbium-doped fiber amplifiers,” Opt. Lett. 17, 1113–1115 (1992).
    [CrossRef] [PubMed]
  7. E. Desurvire, Erbium-Doped Fiber Amplifiers, 1st ed. (Wiley, New York, 1994), Section 5.5, p. 379.
  8. Photonic Modules Reference Manual, VPItransmissionMaker (Virtual Photonics, Inc., Holmdel, N.J., 2000), Vol. 1, pp. 8–46–8–54.
  9. Component Library, EDF Dynamic-Analytical Model, OptiSystem 3.0 (Optiwave Corporation, Ottawa, Ontario, Canada, 2003), pp. 289–294.
  10. C. Dimopoulos, “Study of dynamic phenomena in WDM optical fiber links and networks based on EDFA,” Ph.D. thesis (University of Essex, Colchester, UK, 2001).
  11. J. Laegsgaard and A. Bjarklev, “Photonic crystal fibers with large nonlinear coefficients,” J. Opt. A, Pure Appl. Opt. 6, 1–5 (2003).
    [CrossRef]
  12. C. Mazzali and H. L. Fragnito, “Analysis of background loss influence on EDFA gain dynamics,” Proceedings of the Twenty-Fourth European Coference on Optical Communicatios (Madrid, Spain, 1998), pp. 493–494.
  13. S. Novak and R. Gieske, “Simulink model for EDFA dynamics applied to gain modulation,” J. Lightwave Technol. 20, 986–992 (2002).
    [CrossRef]
  14. This code can be downloaded at www.ifi.unicamp.br/foton/DynamicEDFA.

2003 (1)

J. Laegsgaard and A. Bjarklev, “Photonic crystal fibers with large nonlinear coefficients,” J. Opt. A, Pure Appl. Opt. 6, 1–5 (2003).
[CrossRef]

2002 (1)

1998 (1)

1997 (1)

Y. Sun, J. L. Zyskind, and A. K. Sristava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1006 (1997).
[CrossRef]

1996 (1)

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

1992 (1)

1991 (1)

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

1990 (1)

A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
[CrossRef]

Aspell, J.

A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
[CrossRef]

Bjarklev, A.

J. Laegsgaard and A. Bjarklev, “Photonic crystal fibers with large nonlinear coefficients,” J. Opt. A, Pure Appl. Opt. 6, 1–5 (2003).
[CrossRef]

Bononi, A.

Delevaque, E.

Desurvire, E.

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

Evankow, J. D.

A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
[CrossRef]

Georges, T.

Gieske, R.

Giles, C. R.

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

Jopson, R. M.

A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
[CrossRef]

Laegsgaard, J.

J. Laegsgaard and A. Bjarklev, “Photonic crystal fibers with large nonlinear coefficients,” J. Opt. A, Pure Appl. Opt. 6, 1–5 (2003).
[CrossRef]

Luo, G.

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

Novak, S.

Rusch, L. A.

Saleh, A. A.

A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
[CrossRef]

Saleh, A. A. M.

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

Sristava, A. K.

Y. Sun, J. L. Zyskind, and A. K. Sristava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1006 (1997).
[CrossRef]

Srivastava, A. K.

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

Sulhoff, J. W.

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

Sun, Y.

Y. Sun, J. L. Zyskind, and A. K. Sristava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1006 (1997).
[CrossRef]

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

Zyskind, J. L.

Y. Sun, J. L. Zyskind, and A. K. Sristava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1006 (1997).
[CrossRef]

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

Electron. Lett. (1)

Y. Sun, G. Luo, J. L. Zyskind, A. A. M. Saleh, A. K. Srivastava, and J. W. Sulhoff, “Model for gain dynamics in erbium-doped fiber amplifiers,” Electron. Lett. 32, 1490–1491 (1996).
[CrossRef]

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

Y. Sun, J. L. Zyskind, and A. K. Sristava, “Average inversion level, modeling, and physics of erbium-doped fiber amplifiers,” IEEE J. Sel. Top. Quantum Electron. 3, 991–1006 (1997).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

A. A. Saleh, R. M. Jopson, J. D. Evankow, and J. Aspell, “Modeling of gain in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 2, 714–717 (1990).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. A, Pure Appl. Opt. (1)

J. Laegsgaard and A. Bjarklev, “Photonic crystal fibers with large nonlinear coefficients,” J. Opt. A, Pure Appl. Opt. 6, 1–5 (2003).
[CrossRef]

Opt. Lett. (1)

Other (6)

E. Desurvire, Erbium-Doped Fiber Amplifiers, 1st ed. (Wiley, New York, 1994), Section 5.5, p. 379.

Photonic Modules Reference Manual, VPItransmissionMaker (Virtual Photonics, Inc., Holmdel, N.J., 2000), Vol. 1, pp. 8–46–8–54.

Component Library, EDF Dynamic-Analytical Model, OptiSystem 3.0 (Optiwave Corporation, Ottawa, Ontario, Canada, 2003), pp. 289–294.

C. Dimopoulos, “Study of dynamic phenomena in WDM optical fiber links and networks based on EDFA,” Ph.D. thesis (University of Essex, Colchester, UK, 2001).

C. Mazzali and H. L. Fragnito, “Analysis of background loss influence on EDFA gain dynamics,” Proceedings of the Twenty-Fourth European Coference on Optical Communicatios (Madrid, Spain, 1998), pp. 493–494.

This code can be downloaded at www.ifi.unicamp.br/foton/DynamicEDFA.

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Figures (4)

Fig. 1
Fig. 1

Intrinsic erbium-doped fiber parameters used to simulate the graphs in Fig. 2. Absorption and emission constants and other parameter values are shown.

Fig. 2
Fig. 2

Each term contribution to dN2/dt (absolute values) as a function of N2 and of the input power at λ=1480 nm, P1480in. Observe that just the Y. Sun term contribution depends on the input power. Simulations are obtained with Eq. (8) and intrinsic EDFA parameters shown in Fig. 1. (a), (b), and (c) are for L=50, 20, and 10 m, respectively, which correspond to the amplifier gain at 1530 nm and N2=0.75 of 47.9, 19.2, and 9.59 dB, respectively. In each graph we indicate with horizontal arrows in the Y. Sun terms the corresponding P1480in used. The vertical arrows crossing each Y. Sun curve show the value of N2 at which dN2/dt=0 for this particular input power at 1480 nm, i.e., the value of N2 at which the sum of all term contributions is zero, and which corresponds to the N2 value given by the static solution of the equations when the respective input power at 1480 nm is entering the amplifier.

Fig. 3
Fig. 3

Each term contribution to dN2/dt absolute value as a function of N2. Simulations obtained with Eq. (13) and intrinsic EDFA parameters shown in Fig. 1. (a), (b), and (c) are for L=50, 20, and 10 m, respectively, which correspond to the amplifier gain at 1530 nm and N2=0.75 of 47.9, 19.2, and 9.59 dB, respectively. As in Fig. 2, the vertical arrows crossing each Y. Sun curve show the value of N2 at which dN2/dt=0 for this particular input power at 1480 nm, i.e., the value of N2 at which the sum of all term contributions is zero, and which corresponds to the N2 value given by the static solution of the equations when the respective input power at 1480 nm is entering the amplifier.

Fig. 4
Fig. 4

WDM surviving channel’s output power excursion when three of four channels are dropped at t=0 in a typical EDFA employing a 20-m doped fiber. We used for the simulations a pump power of 100 mW at 1480 nm, a constant surviving channel power of 600 µW at 1539.6 nm, and a dropped channel power of 1.8 mW at 1549.2 nm. The gain at 1549.2 nm before the drop operation is ∼10 dB in each one of the three curves. The lower curve is the simulation result obtained with Eqs. (11) and (13), and the other two curves are simulation results obtained when the background losses or the ASE contributions are neglected.

Equations (14)

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N2(z, t)t=-N2(z, t)τ0-1ρS n=1M[(γn+αn)×N2(z, t)-αn]Pn(z, t),
Pn(z, t)z=un{[(γn+αn)N2(z, t)-αn-αloss]×Pn(z, t)+2γnΔνN2(z, t)},
N2(z, t)t=-N2(z, t)τ0-1ρS n=1MunPn(z, t)z-2γnΔνN2(z, t).
Pnout(t)=Pnin(t)Gn(t)+2nnsp[Gn(t)-1]Δυ,
Gn(t)=exp{[(γn+αn)N2(t)-αn]L},
nnsp=N2(t)γn(γn+αn)N2(t)-αn,
dN2(t)dt=-N2(t)τ0-1ρSL n=1M[Pnout(t)-Pnin(t)-2γnΔυN2(t)L]
dN2(t)dt=-N2(t)τ0-1ρSL n=1M{Pnin(t)[Gn(t)-1]+2nnsp[Gn(t)-1]Δυ-2γnΔυN2(t)L}.
-1τ0-1ρS nM2γnΔυ
τm=τ01-τ0ρS nM2γnΔυ.
N2(z, t)t=-N2(z, t)τ0-1ρS n=1MunPn(z, t)z-2γnΔνN2(z, t)+αlossPn(z, t).
Gn(t)=exp{[(γn+αn)N2(t)-αn-αloss]L}.
dN2(t)dt=()-αlossρSL nM0LPn(z)dz,
dN2(t)dt=()-αlossρSL nMPnin(t)ln[Gn(t)][Gn(t)-1]L+2nnspΔνLGn(t)-1ln[Gn(t)]-1.

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