Abstract

We present an analytical model of the self- and cross-phase-modulation-induced modulation instabilities in nonlinear, dispersive, homogeneously broadened, unsaturable fiber amplifiers. The rigorous treatment of the gain dispersion and of the gain-induced, near-resonance refractive-index dispersion permits description of the general case of arbitrary detuning between the laser frequency and the resonance and arbitrary bandwidth of the amplifying transition. Simple closed-form expressions are obtained for the combined gain coefficients and, in the self-phase-modulation-induced case only, for the integrated gain. A qualitatively new modulation-instability regime of an isolated wave in the normal group-velocity-dispersion region is described.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Hasegawa and W. Brinkman, “Tunable coherent IR and FIR source utilizing modulation instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
    [CrossRef]
  2. G. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).
  3. G. Agrawal, “Modulation instability in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 4, 562–564 (1992).
    [CrossRef]
  4. S. Murdoch, J. Harvey, and N. Christensen, “The effect of resonant dispersion and gain on modulation instability,” Opt. Commun. 121, 3–18 (1995).
    [CrossRef]
  5. W. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991).
    [CrossRef]
  6. S. Fleming and T. Whitley, “Measurement and analysis of pump-dependent refractive index and dispersion effects in erbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 32, 1113–1121 (1996).
    [CrossRef]
  7. E. Desurvire, Erbium-Doped Fiber Amplifiers, Principles and Applications (Wiley, New York, 1994).
  8. A. Siegman, Lasers (University Science, Mill Valley, Calif. 1986).
  9. W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quantum Electron. 16, 541–550 (1984).
    [CrossRef]
  10. J. Herrmann and B. Wilhelmi, Laser für ultrakurze lichtimpulse (Akademie, Berlin, 1984).
  11. R. Hickernell, K. Takada, M. Yamada, M. Shimizu, and M. Horiguchi, “Pump-induced dispersion of erbium-doped fiber measured by Fourier-transform spectroscopy,” Opt. Lett. 18, 19–21 (1993).
    [CrossRef] [PubMed]
  12. G. Agrawal, D. Baldeck, and R. Alfano, “Modulation instability induced by cross phase-modulation in optical fibers,” Phys. Rev. A 39, 3406–413 (1989).
    [CrossRef] [PubMed]
  13. E. Greer, D. Patrick, P. Wigley, and J. Taylor, “Picosecond pulse generation from a continuous-wave diode laser through cross-phase modulation in an optical fiber,” Opt. Lett. 15, 851–854 (1990).
    [CrossRef] [PubMed]

1996 (1)

S. Fleming and T. Whitley, “Measurement and analysis of pump-dependent refractive index and dispersion effects in erbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 32, 1113–1121 (1996).
[CrossRef]

1995 (1)

S. Murdoch, J. Harvey, and N. Christensen, “The effect of resonant dispersion and gain on modulation instability,” Opt. Commun. 121, 3–18 (1995).
[CrossRef]

1993 (1)

1992 (1)

G. Agrawal, “Modulation instability in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 4, 562–564 (1992).
[CrossRef]

1991 (1)

W. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991).
[CrossRef]

1990 (1)

1989 (1)

G. Agrawal, D. Baldeck, and R. Alfano, “Modulation instability induced by cross phase-modulation in optical fibers,” Phys. Rev. A 39, 3406–413 (1989).
[CrossRef] [PubMed]

1984 (1)

W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quantum Electron. 16, 541–550 (1984).
[CrossRef]

1980 (1)

A. Hasegawa and W. Brinkman, “Tunable coherent IR and FIR source utilizing modulation instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

Agrawal, G.

G. Agrawal, “Modulation instability in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 4, 562–564 (1992).
[CrossRef]

G. Agrawal, D. Baldeck, and R. Alfano, “Modulation instability induced by cross phase-modulation in optical fibers,” Phys. Rev. A 39, 3406–413 (1989).
[CrossRef] [PubMed]

Alfano, R.

G. Agrawal, D. Baldeck, and R. Alfano, “Modulation instability induced by cross phase-modulation in optical fibers,” Phys. Rev. A 39, 3406–413 (1989).
[CrossRef] [PubMed]

Baldeck, D.

G. Agrawal, D. Baldeck, and R. Alfano, “Modulation instability induced by cross phase-modulation in optical fibers,” Phys. Rev. A 39, 3406–413 (1989).
[CrossRef] [PubMed]

Brinkman, W.

A. Hasegawa and W. Brinkman, “Tunable coherent IR and FIR source utilizing modulation instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

Christensen, N.

S. Murdoch, J. Harvey, and N. Christensen, “The effect of resonant dispersion and gain on modulation instability,” Opt. Commun. 121, 3–18 (1995).
[CrossRef]

Fleming, S.

S. Fleming and T. Whitley, “Measurement and analysis of pump-dependent refractive index and dispersion effects in erbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 32, 1113–1121 (1996).
[CrossRef]

Greer, E.

Harvey, J.

S. Murdoch, J. Harvey, and N. Christensen, “The effect of resonant dispersion and gain on modulation instability,” Opt. Commun. 121, 3–18 (1995).
[CrossRef]

Hasegawa, A.

A. Hasegawa and W. Brinkman, “Tunable coherent IR and FIR source utilizing modulation instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

Hickernell, R.

Horiguchi, M.

Miniscalco, W.

W. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991).
[CrossRef]

Murdoch, S.

S. Murdoch, J. Harvey, and N. Christensen, “The effect of resonant dispersion and gain on modulation instability,” Opt. Commun. 121, 3–18 (1995).
[CrossRef]

Patrick, D.

Rudolph, W.

W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quantum Electron. 16, 541–550 (1984).
[CrossRef]

Shimizu, M.

Takada, K.

Taylor, J.

Whitley, T.

S. Fleming and T. Whitley, “Measurement and analysis of pump-dependent refractive index and dispersion effects in erbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 32, 1113–1121 (1996).
[CrossRef]

Wigley, P.

Yamada, M.

IEEE J. Quantum Electron. (2)

A. Hasegawa and W. Brinkman, “Tunable coherent IR and FIR source utilizing modulation instability,” IEEE J. Quantum Electron. 16, 694–697 (1980).
[CrossRef]

S. Fleming and T. Whitley, “Measurement and analysis of pump-dependent refractive index and dispersion effects in erbium-doped fiber amplifiers,” IEEE J. Quantum Electron. 32, 1113–1121 (1996).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

G. Agrawal, “Modulation instability in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 4, 562–564 (1992).
[CrossRef]

J. Lightwave Technol. (1)

W. Miniscalco, “Erbium-doped glasses for fiber amplifiers at 1500 nm,” J. Lightwave Technol. 9, 234–250 (1991).
[CrossRef]

Opt. Commun. (1)

S. Murdoch, J. Harvey, and N. Christensen, “The effect of resonant dispersion and gain on modulation instability,” Opt. Commun. 121, 3–18 (1995).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

W. Rudolph, “Calculation of pulse shaping in saturable media with consideration of phase memory,” Opt. Quantum Electron. 16, 541–550 (1984).
[CrossRef]

Phys. Rev. A (1)

G. Agrawal, D. Baldeck, and R. Alfano, “Modulation instability induced by cross phase-modulation in optical fibers,” Phys. Rev. A 39, 3406–413 (1989).
[CrossRef] [PubMed]

Other (4)

J. Herrmann and B. Wilhelmi, Laser für ultrakurze lichtimpulse (Akademie, Berlin, 1984).

E. Desurvire, Erbium-Doped Fiber Amplifiers, Principles and Applications (Wiley, New York, 1994).

A. Siegman, Lasers (University Science, Mill Valley, Calif. 1986).

G. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, Calif., 1989).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

Typical gain-coefficient—scaled detuning map of the MI process in the anomalous group-velocity-dispersion region. A filled point in the g0>0 area indicates satisfied MI conditions for at least one Ω, -Ω pair; in the g0<0 area also the requirement for positive effective gain coefficient is added.

Fig. 2
Fig. 2

Contour map of the areas with negative β2Ω2-Im(G) in the scaled frequency-detuning coordinates for g0=1 m-1 and β2/τ2=1.10-2 m-1.

Fig. 3
Fig. 3

Typical gain-coefficient—scaled detuning map of the MI process in the normal group-velocity-dispersion region. Filled points denote the same conditions as in Fig. 1.

Fig. 4
Fig. 4

Evolution of the gain-coefficient spectrum in a fiber amplifier with g0=0.5 m-1, β2/τ2=1 m-1, γ=0.02 W-1 m-1, and α=0.25 dB/km. Input power is P0=1 mW, and detuning is Δτ=0.7.

Fig. 5
Fig. 5

Asymmetrical spectra corresponding to symmetrical detunings Δτ=+0.5 (solid curve) and Δτ=-0.5 (dashed curve). The other parameters are g0=0.5 m-1, β2/τ2=-1.10-2 m-1, γ=0.02 W-1 m-1, α=0.25 dB/km, l=25 m, and P0=1 mW. Note the shift of the sidebands in the case Δτ=-0.5.

Fig. 6
Fig. 6

Model of the MI-spectrum evolution in a typical aluminogermanosilicate EDFA. The stimulated-emission spectrum of the real amplifier, given in Ref. 7, is modeled with three Lorentzians. The parameters are λL=1.555 μm, P0=1 mW, β2=-15 ps2/km, γ=0.02 W-1 m-1, and α=0.25 dB/km.

Fig. 7
Fig. 7

Evolution of typical XPM-induced MI spectra in a fiber amplifier. The parameters characterizing (a) the first wave and (b) the second wave are given in the text.

Equations (53)

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

gse(ω)=Lg01-iLτ(ω-ωL),
gse(ω)=g0Lm=0[iLτ(ω-ωL)]m.
Az+i2 β2 2At2=iγ|A|2A+12 g0×m=0Lm+1τ tmA-12 αA,
A(z, g0, τ, Δ)=(P0+u+iv)×expiγP00Z exp(g0LRz)dz-i2 g0Llz+12 (g0LR-α)z,
uz-12 β2 2vt2=12 g0m=1 ReLm+1τ tmu+12 g0m=1 ImLm+1τ tmv,
vz+12 β2 2ut2=2γP0u exp(g0LRz)+12 g0m=1 ReLm+1τ tmv-12 g0m=1 ImLm+1τ tmu.
u(z, t)=u0 exp-iΩt+i0Zk(z)dz,
v(z, t)=v0 exp-iΩt+i0Zk(z)dz,
g0Lm=1Lmτ tm(u, v)=g0Lm=1(-iτLΩ)m(u, v)=G(g0, τ, Δ, Ω)(u, v).
Re(G)
=-g0 τΩ(2τΔ+τΩ)(1+τ2Δ2)[1+(τΔ+τΩ)2],
Im(G)
=g0 (τΩ)2τΔ[3-(τΔ)2+(τΩ)2](1+τ2Δ2)[1+(τΔ-τΩ)2][1+(τΔ+τΩ)2].
ik-12 Re(G)u+β22 Ω2-12 Im(G)v=0,
β22 Ω2-12 Im(G)+2γP0u exp(g0LRz)u
-ik-12 Re(G)v=0.
k(z, g0, τ, Δ, Ω)=-i Re(G)2±β22 Ω2-12 Im(G)×1+2γP0 exp(g0LRz)β22 Ω2-12 Im(G).
β2Ω2-Im(G)<0,
4γP0 exp(g0LRz)>|β2Ω2-Im(G)|.
g(z, g0, τ, Δ, Ω)=(g0LR-α)2-Im(k)=(g0LR-α)2+Re(G)2-β22 Ω2-12 Im(G)×4γP0 exp(g0LRz-α)β2Ω2-Im(G)-1.
g¯=(g0LR-α)l+Re(G)l+8flLNL ln C C-f2-f×arccos fC-C*-f2+f arccos fC*,
Ωmax4γP0 exp(g0LRz)/|β2Ω2-Im(G)|.
A1z+i2 β21 2A1t2=iγ1(|A1|2+2|A2|2)A1+12 g01×m=0L1m+1τ1 tm×A1-12 α1A1,
A2z+i2 β22 2A2t2=iγ2(|A2|2+2|A1|2)A2+12 g02×m=0L2m+1τ2 tm×A2-12 α2A2.
A1(z)=(P01+u1+iv1)×expiγ1P010z exp(g01L1Rz)dz+2iγ2P020z exp(g02L2Rz)dz-i2 g01L1Iz+12 (g01-α1)L1Rz,
A2(z)=(P02+u2+iv2)×expiγ2P020z exp(g02L2Rz)dz+2iγ1P010z exp(g01L1Rz)dz-i2 g02L2Iz+12 (g02L2R-α2)z,
ujz-12 β2j 2vjt2=12 g0jm=1 ReLjm+1τj tmuj+12 g0jm=1 ImLjm+1τj tmvj,
vjz+12 β2j 2ujt2=2γjP0juj exp(g0jLjRz)+4γjγ(3-j)P0jP0(3-j)u(3-j)×exp[g0(3-j)L(3-j)Rz]-12 g0j×m=1 ImL1m+1τj tm×uj+12 g0j×m=1 ReLjm+1τj tmvj.
uj(z, t)=xj(z)exp(-iΩjt),
vj(z, t)=yj(z)exp(-iΩjt),
xj(z)=θj(z)exp12 Re[Gj(Δj, Ωj)]z,
yj(z)=φj(z)exp12 Re[Gj(Δj, Ωj)]z.
Gj=g0jLj m=1(-iτjLjΩj)m.
θj(z)=θj0 expi0zk(z)dz,
φj(z)=φj0 expi0zk(z)dz,
ik(z)θ1+β212 Ω12-12 (Im G1)φ1=0,
-β212 Ω12-12 (Im G1)+2γ1P01×exp(g01L1Rz)θ1+ik(z)φ1-4γ1γ2P01P02 exp(g02L2Rz)exp[i(Ω1-Ω2)t]
×exp-12 [Re(G1)-Re(G2)]zθ2=0,
ik(z)θ2+β222 Ω22-12 (Im G2)φ2=0,
-β222 Ω22-12 (Im G2)+2γ2P02×exp(g02L2Rz)θ2+ik(z)φ2-4γ1γ2P01P02×exp(g01L1Rz)exp[i(Ω1-Ω2)t]
×exp-12 [Re(G1)-Re(G2)]zθ1=0.
gj=Ωj22 [β2j-Im(Gj)/Ωj2],
hj=1+4γjP0j exp(g0jLjRz-αj)Ωj2[β2j-Im(Gj)Ωj2],
q=16γ1γ2P01P02×exp[(g01L1R+g02L2R)z],
k(z)
=±(g12h1+g22h2)±[(g12h1-g22h2)2+4g1g2q]1/221/2.
(g12h1-g22h2)2+4g1g2q>0,
(g12h1+g22h2)<[(g12h1-g22h2)2+4g1g2q]1/2.
gj(z)
=(g0jLjR-αj)2+Re(Gj)2+[(g12h1-g22h2)2+4g1g2q]1/2-(g12h1+g22h2)21/2.
(g12h1-g22h2)2<4|g1g2|q,
sign(g1g2)<0.
gj(z)=(g0jLjR-αj)2+Re(Gj)2+g12h1g22h2+|g1g2|q×sinarctg4|g1g2|q-(g12h1-g22h2)22(g12h1+g22h2).

Metrics