Abstract

We define a complex refractive-index profile for the pumped erbium-doped fiber that depends on radial distance, pump and signal powers, and erbium-doping profile to obtain a modal gain and loss of the propagating signal and pump power by Rayleigh–Ritz variational analysis. This profile provides a novel way of looking at the gain characteristics of erbium-doped fiber amplifiers. The advantage of this approach is that it gives the actual modal gain and also eliminates the need to approximate the modal fields and can easily take into account any dopant and index profile.

© 1999 Optical Society of America

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References

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  1. J. E. Sader, “Method for analysis of complex refractive-index profile fibers,” Opt. Lett. 15, 107 (1990).
    [CrossRef]
  2. M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fiber using matrix approach,” J. Lightwave Technol. 6, 1285 (1988).
    [CrossRef]
  3. E. K. Sharma, M. P. Singh, and A. Sharma, “Variational analysis of optical fibers with loss or gain,” Opt. Lett. 18, 2096 (1993).
    [CrossRef] [PubMed]
  4. E. Desurvire, Erbium Doped Fiber Amplifiers (Wiley, New York, 1994).
  5. F. F. Ruhl, “Accurate analytical formulas for gain-optimised EDFAs,” Electron. Lett. 28, 312 (1992).
    [CrossRef]
  6. F. F. Ruhl, “Implicit analytical solution for erbium doped fiber amplifiers,” Electron. Lett. 2, 465 (1992).
    [CrossRef]
  7. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.
  8. W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
    [CrossRef]
  9. J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation: constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 538 (1981).
    [CrossRef]

1993 (1)

1992 (2)

F. F. Ruhl, “Accurate analytical formulas for gain-optimised EDFAs,” Electron. Lett. 28, 312 (1992).
[CrossRef]

F. F. Ruhl, “Implicit analytical solution for erbium doped fiber amplifiers,” Electron. Lett. 2, 465 (1992).
[CrossRef]

1991 (1)

W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
[CrossRef]

1990 (1)

J. E. Sader, “Method for analysis of complex refractive-index profile fibers,” Opt. Lett. 15, 107 (1990).
[CrossRef]

1988 (1)

M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fiber using matrix approach,” J. Lightwave Technol. 6, 1285 (1988).
[CrossRef]

1981 (1)

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation: constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 538 (1981).
[CrossRef]

Barnes, W. L.

W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
[CrossRef]

Ghatak, A. K.

M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fiber using matrix approach,” J. Lightwave Technol. 6, 1285 (1988).
[CrossRef]

Laming, R. I.

W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
[CrossRef]

Massot, J. N.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation: constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 538 (1981).
[CrossRef]

Meunier, J. P.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation: constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 538 (1981).
[CrossRef]

Morkel, P. R.

W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
[CrossRef]

Pigeon, J.

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation: constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 538 (1981).
[CrossRef]

Ruhl, F. F.

F. F. Ruhl, “Accurate analytical formulas for gain-optimised EDFAs,” Electron. Lett. 28, 312 (1992).
[CrossRef]

F. F. Ruhl, “Implicit analytical solution for erbium doped fiber amplifiers,” Electron. Lett. 2, 465 (1992).
[CrossRef]

Sader, J. E.

J. E. Sader, “Method for analysis of complex refractive-index profile fibers,” Opt. Lett. 15, 107 (1990).
[CrossRef]

Sharma, A.

Sharma, E. K.

Shenoy, M. R.

M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fiber using matrix approach,” J. Lightwave Technol. 6, 1285 (1988).
[CrossRef]

Singh, M. P.

Tarbox, E. J.

W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
[CrossRef]

Thyagarajan, K.

M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fiber using matrix approach,” J. Lightwave Technol. 6, 1285 (1988).
[CrossRef]

Electron. Lett. (2)

F. F. Ruhl, “Accurate analytical formulas for gain-optimised EDFAs,” Electron. Lett. 28, 312 (1992).
[CrossRef]

F. F. Ruhl, “Implicit analytical solution for erbium doped fiber amplifiers,” Electron. Lett. 2, 465 (1992).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. L. Barnes, R. I. Laming, E. J. Tarbox, and P. R. Morkel, “Absorption and emission cross-section of Er3+ doped silica fibers,” IEEE J. Quantum Electron. 27, 1004 (1991).
[CrossRef]

J. Lightwave Technol. (1)

M. R. Shenoy, K. Thyagarajan, and A. K. Ghatak, “Numerical analysis of optical fiber using matrix approach,” J. Lightwave Technol. 6, 1285 (1988).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

J. P. Meunier, J. Pigeon, and J. N. Massot, “A general approach to the numerical determination of modal propagation: constants and field distribution of optical fibers,” Opt. Quantum Electron. 13, 538 (1981).
[CrossRef]

Other (2)

E. Desurvire, Erbium Doped Fiber Amplifiers (Wiley, New York, 1994).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, New York, 1983), Chap. 18.

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Figures (5)

Fig. 1
Fig. 1

Radial profile of the imaginary part of the pump index. Pump power levels, q, are marked.

Fig. 2
Fig. 2

Radial profile of the imaginary part of the signal index at λs=1.53 µm. Pump power levels, q, are marked.

Fig. 3
Fig. 3

Variation of signal gain with wavelength at pump power levels q=10 and q=100 (for uniform doping in the entire core). Comparisons with perturbation and with the calculations of Ref. 4 are is also shown; v, p, and d correspond to a pump power level q=100; v, p, and d correspond to a relatively small pump power level q=10.

Fig. 4
Fig. 4

Variational calculation of signal gain with wavelength, showing saturation at high pump power levels q from 5 to 150 for uniform doping in the entire core.

Fig. 5
Fig. 5

Perturbation calculation of signal gain with wavelength, showing saturation at high pump power levels q from 5 to 150 for uniform doping in the entire core.

Tables (1)

Tables Icon

Table 1 Convergence of Results of Gain and Loss Fibers at q=15 and λs=1.53 µm

Equations (24)

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dIsdz=σas(ηsN2-N1)Is,
dIpdz=(-σapN1)Ip,
N1(r, z)
=ρ(r) 1+ηs1+ηsPs(z)Psat(λs)|ψs(r)|21+Pp(z)Psat(λp)|ψp(r)|2+Ps(z)Psat(λs)|ψs(r)|2,
N2(r, z)
=ρ(r) Pp(z)Psat(λp)|ψp(r)|2+11+ηsPs(z)Psat(λs)|ψs(r)|21+Pp(z)Psat(λp)|ψp(r)|2+Ps(z)Psat(λs)|ψs(r)|2.
dIsdz=ρ(r)σas[ηsq|ψp(r)|2-1]1+q|ψp(r)|2+p|ψs(r)|2Is,
dIpdz=-ρ(r)σap1+ηs1+ηsp|ψs(r)|21+q|ψp(r)|2+p|ψs(r)|2Ip,
wp,s2=20|ψp,s(r)|2rdr.
dIdz=4πλ0niI,
ni(signal)=ρ(r)σasλs[ηsq|ψp(r)|2-1]4π[1+q|ψp(r)|2+p|ψs(r)|2],
ni(pump)=-ρ(r)σapλp[1+(ηs)/(1+ηs)p|ψs(r)|2]4π[1+q|ψp(r)|2+p|ψs(r)|2].
n(r)=n1(r)+ini(r)r<a,
=n2(cladding)r>a.
ψ(r)=m=0N-1amϕm(r),
ϕm(r)=12αexp-αr22a2Lmαr2a2,
0ϕnϕmRdR=δnm.
[H][A]=β2[A],
Hnmr=(α/a2)Tnm+k02n22δnm+k0201[n12(r)-n22]ϕnϕmRdR,
Hnmi=2k0201n1niϕnϕmRdR,
Tnm=-(2m+1)δnm-(m+1)δm+1,n-mδm-1,n,
Δβ2=ik022n10ni(r)ψp,s2(r)rdr0ψp,s2(r)rdr=2iβiβr,
dpdz=n1nes 2σasωs2×0 ψs2(r) ρ(r)[ηsqψp2(r)-1)rdr][1+qψp2(r)+pωs(r)2]p,
dqdz=-n1nep 2σapωp2×0ψp2(r) ρ(r)1+ηs1+ηspψs(r)2rdr[1+qψp2(r)+pψs(r)2]q.

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