Abstract

We investigate analytically the occurrence of modulation instability in doped fiber lasers and amplifiers using a Maxwell–Bloch description for the dopants and without making the usual parabolic-gain approximation. We find a new modulation instability occurring near the Rabi frequency, which is not predicted by the conventional complex Ginzburg–Landau model. We discuss the implications of this new instability for fiber amplifiers and lasers and analyze the effects of the saturable host absorption on the laser instabilities. Atomic detuning is shown to significantly enhance the new modulation instability, in both the normal- and the anomalous-dispersion regimes.

© 1997 Optical Society of America

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References

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  1. C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991).
  2. G. H. M. van Tartwijk and G. P. Agrawal, “Nonlinear dynamics in the generalized Lorenz–Haken model,” Opt. Commun. 133, 565–577 (1997).
    [CrossRef]
  3. G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
    [CrossRef] [PubMed]
  4. G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).
  5. F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
    [CrossRef]
  6. S. Colin, E. Contesse, P. Le Boudec, G. Stephan, and F. Sanchez, “Evidence of a saturable-absorption effect in heavily erbium-doped fibers,” Opt. Lett. 21, 1987–1989 (1996).
    [CrossRef] [PubMed]
  7. E. Lacot, F. Stoeckel, and M. Chenevier, “Self pulsing, chaos and antiphase dynamics in an Er3+ doped fiber laser,” J. Phys. (France) III 5, 269–279 (1995).
    [CrossRef]
  8. Q. L. Williams and R. Roy, “Fast polarization dynamics of an erbium-doped fiber ring laser,” Opt. Lett. 21, 1478–1480 (1996).
    [CrossRef] [PubMed]
  9. S. Bielawski, D. Derozier, and P. Glorieux, “Antiphase dynamics and polarization effects in the Nd-doped fiber laser,” Phys. Rev. A 46, 2811–2822 (1992).
    [CrossRef] [PubMed]
  10. H. Zeghlache and A. Boulnois, “Polarization instability in lasers. I. Model and steady states of neodymium-dopedfiber lasers,” Phys. Rev. A 52, 4229–4242 (1995); “Polarization instability in lasers: II: Influence of the pump polarizationon the dynamics,” Phys. Rev. A 52, 4243–4254 (1995).
    [CrossRef] [PubMed]
  11. G. P. Agrawal, “Modulation instability in erbium-doped fiber amplifiers,” IEEE Photonics Technol. Lett. 4, 562–564 (1992).
    [CrossRef]
  12. C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Self-starting of passively mode-locked lasers with fast saturable absorbers,” Opt. Lett. 20, 350–352 (1995).
    [CrossRef] [PubMed]
  13. M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
    [CrossRef]

1997

G. H. M. van Tartwijk and G. P. Agrawal, “Nonlinear dynamics in the generalized Lorenz–Haken model,” Opt. Commun. 133, 565–577 (1997).
[CrossRef]

1996

1995

C.-J. Chen, P. K. A. Wai, and C. R. Menyuk, “Self-starting of passively mode-locked lasers with fast saturable absorbers,” Opt. Lett. 20, 350–352 (1995).
[CrossRef] [PubMed]

E. Lacot, F. Stoeckel, and M. Chenevier, “Self pulsing, chaos and antiphase dynamics in an Er3+ doped fiber laser,” J. Phys. (France) III 5, 269–279 (1995).
[CrossRef]

1992

M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
[CrossRef]

S. Bielawski, D. Derozier, and P. Glorieux, “Antiphase dynamics and polarization effects in the Nd-doped fiber laser,” Phys. Rev. A 46, 2811–2822 (1992).
[CrossRef] [PubMed]

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

1991

G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
[CrossRef] [PubMed]

Agrawal, G. P.

G. H. M. van Tartwijk and G. P. Agrawal, “Nonlinear dynamics in the generalized Lorenz–Haken model,” Opt. Commun. 133, 565–577 (1997).
[CrossRef]

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

G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
[CrossRef] [PubMed]

Bielawski, S.

S. Bielawski, D. Derozier, and P. Glorieux, “Antiphase dynamics and polarization effects in the Nd-doped fiber laser,” Phys. Rev. A 46, 2811–2822 (1992).
[CrossRef] [PubMed]

Chen, C.-J.

Chenevier, M.

E. Lacot, F. Stoeckel, and M. Chenevier, “Self pulsing, chaos and antiphase dynamics in an Er3+ doped fiber laser,” J. Phys. (France) III 5, 269–279 (1995).
[CrossRef]

Colin, S.

Contesse, E.

Derozier, D.

S. Bielawski, D. Derozier, and P. Glorieux, “Antiphase dynamics and polarization effects in the Nd-doped fiber laser,” Phys. Rev. A 46, 2811–2822 (1992).
[CrossRef] [PubMed]

Glorieux, P.

S. Bielawski, D. Derozier, and P. Glorieux, “Antiphase dynamics and polarization effects in the Nd-doped fiber laser,” Phys. Rev. A 46, 2811–2822 (1992).
[CrossRef] [PubMed]

Kimura, Y.

M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
[CrossRef]

Kubota, H.

M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
[CrossRef]

Lacot, E.

E. Lacot, F. Stoeckel, and M. Chenevier, “Self pulsing, chaos and antiphase dynamics in an Er3+ doped fiber laser,” J. Phys. (France) III 5, 269–279 (1995).
[CrossRef]

Le Boudec, P.

Menyuk, C. R.

Nakazawa, M.

M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
[CrossRef]

Roy, R.

Sanchez, F.

Stephan, G.

Stoeckel, F.

E. Lacot, F. Stoeckel, and M. Chenevier, “Self pulsing, chaos and antiphase dynamics in an Er3+ doped fiber laser,” J. Phys. (France) III 5, 269–279 (1995).
[CrossRef]

Suzuki, K.

M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
[CrossRef]

van Tartwijk, G. H. M.

G. H. M. van Tartwijk and G. P. Agrawal, “Nonlinear dynamics in the generalized Lorenz–Haken model,” Opt. Commun. 133, 565–577 (1997).
[CrossRef]

Wai, P. K. A.

Williams, Q. L.

IEEE Photonics Technol. Lett.

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

J. Phys. (France) III

E. Lacot, F. Stoeckel, and M. Chenevier, “Self pulsing, chaos and antiphase dynamics in an Er3+ doped fiber laser,” J. Phys. (France) III 5, 269–279 (1995).
[CrossRef]

Opt. Commun.

G. H. M. van Tartwijk and G. P. Agrawal, “Nonlinear dynamics in the generalized Lorenz–Haken model,” Opt. Commun. 133, 565–577 (1997).
[CrossRef]

Opt. Lett.

Phys. Rev. A

M. Nakazawa, K. Suzuki, Y. Kimura, and H. Kubota, “Coherent π-pulse propagation with pulse breakup in an erbium-dopedfiber waveguide amplifier,” Phys. Rev. A 45, R2682–R2685 (1992).
[CrossRef]

S. Bielawski, D. Derozier, and P. Glorieux, “Antiphase dynamics and polarization effects in the Nd-doped fiber laser,” Phys. Rev. A 46, 2811–2822 (1992).
[CrossRef] [PubMed]

G. P. Agrawal, “Optical pulse propagation in doped fiber amplifiers,” Phys. Rev. A 44, 7493–7501 (1991).
[CrossRef] [PubMed]

Phys. Rev. E

F. Sanchez and G. Stephan, “General analysis of instabilities in erbium-doped fiber lasers,” Phys. Rev. E 53, 2110–2122 (1996).
[CrossRef]

Other

C. O. Weiss and R. Vilaseca, Dynamics of Lasers (Weinheim, New York, 1991).

G. P. Agrawal, Nonlinear Fiber Optics, 2nd ed. (Academic, New York, 1995).

H. Zeghlache and A. Boulnois, “Polarization instability in lasers. I. Model and steady states of neodymium-dopedfiber lasers,” Phys. Rev. A 52, 4229–4242 (1995); “Polarization instability in lasers: II: Influence of the pump polarizationon the dynamics,” Phys. Rev. A 52, 4243–4254 (1995).
[CrossRef] [PubMed]

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Figures (10)

Fig. 1
Fig. 1

Modulation instability spectrum for an erbium-doped fiber amplifier at various values of T2 (indicated in the figure). Parameters are g0=6.91 L-1, P0=1 mW, T1=0.1 ms, β2 =-20 ps2/L, γ=3W-1 L-1, and Psat=1 mW when T2 =0.1 ps. For undoped fibers, MI occurs up to Ωcrit/2π =3.9 GHz

Fig. 2
Fig. 2

MI spectrum for the amplifier of Fig. 1, for even longer dephasing times T2 (indicated in the figure). In the range 21 <T2<80 ps, MI is totally quenched. When T2 approaches 100 ps, the MI spectrum starts to show a narrow, weak peak around the Rabi frequency ΩRabi/Ωc=0.0013.

Fig. 3
Fig. 3

MI analysis for a figure-eight laser. Solid curves indicate the results of the full model, while dashed curves show those of the CGL model. Top figure shows the net MI gain spectra, while the bottom figure shows the corresponding trajectory of the eigenvalue K on the complex plane. Parameters are α =0.4 L-1, g0=6 L-1, β2=-0.09 ps2 L-1, θ=0.1 W-1 L-1, γ =0.008 W-1 L-1, T2=1.27 ps, T1=108 ps, and Psat=10 mW.

Fig. 4
Fig. 4

MI analysis for a dye laser. Similar as in Fig. 3, except for the parameters: α=0.1 L-1, g0=3 L-1, β2=-0.09 ps2 L-1, θ=0.001 W-1 L-1, γ=0.008 W-1 L-1, T2 =2.45 ps, T1 =103 ps, and Psat=1 mW.

Fig. 5
Fig. 5

MI threshold as a function of saturable absorption θ for the fiber laser of Fig. 3.

Fig. 6
Fig. 6

MI threshold as a function of saturable absorption θ for the dye laser of Fig. 4.

Fig. 7
Fig. 7

Comparison of net MI gain spectra in the absence of saturable absorption (θ=0), for the cases of normal and anomalous dispersion. Other parameters of the fiber laser are the same as in Fig. 3.

Fig. 8
Fig. 8

New MI band at normal dispersion as a function of population relaxation time T1 (indicated). Other parameters of the fiber laser are the same as those in Fig. 3.

Fig. 9
Fig. 9

Effect of detuning on the new MI in the normal-dispersion regime. Parameters identical to those in Fig. 7.

Fig. 10
Fig. 10

Similar to Fig. 9, except that the laser now operates in the anomalous-dispersion regime.

Equations (48)

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E(x, y, z, t)=12xˆF(x, y)A(z, t)×exp[i(β0z-ω0t)]+c.c.,
P(x, y, z, t)=12xˆF(x, y)B(z, t)×exp[i(β0z-ω0t)]+c.c.,
Az=i2B-12αA-iβ222At2+(θ+iγ)|A|2A,
T2 dBdt=(iδ-1)B-iAg,
T1 dgdt=g0-g+Im(A*B)/Psat,
n2340n2cRe[χ˜(3)(ω0)],
Psat2cn0Aeff2µ2T1T2,
B˜(Δω)A˜(Δω)=-igs1-i(ΔωT2+δ),
Δβ1(δ)=12g2T2 1(1-iδ)2,
Δβ2(δ)=gsT22 i(1-iδ)3.
Az=12(g-α)A+12(b-iβ2) 2At2+(θ+iγ)|A|2A,
T1 dgdt=g0-g-g|A|2Psat.
As(z)=[PA(z)]1/2 exp[iφs(z)],
Bs(z)=As(z)gs(z)δ+i,
gs(z)=g01+PA(z)Psat(1+δ2)-1.
φs(z)=γ0zdzPA(z)+δ/21+δ20zdzgs(z).
dfdz=g01+δ2ff+1-αf+2θPsat(1+δ2)f2.
2θPsat(1+δ2)z
=-CC++1-CC-lnPA(z)P0+CC+lnPA(z)-C+Psat(1+δ2)P0-C+Psat(1+δ2)+1-CC-lnPA(z)-C-Psat(1+δ2)P0-C-Psat(1+δ2),
C=1+C+C+-C-,
2C±=c2-1±[(1-c2)2-4(c1-c2)]1/2,
c1=g02θPsat(1+δ2)2,c2=α2θPsat(1+δ2).
g01+δ2-αz
=lnPA(z)P0-g0α(1+δ2)×lnPA(z)-Psat(1+δ2)g0α(1+δ2)-1P0-Psat(1+δ2)g0α(1+δ2)-1,
lnPA(z)P0+PA(z)-P0Psat=g0z1+δ2.
A(z, t)=[(P0)1/2+u(z, t)+iv(z, t)]×PA(z)P01/2 exp[iφs(z)],
B(z, t)=gs(z)δ+i[(P0)1/2+p(z, t)+iq(z, t)]×PA(z)P01/2 exp[iφs(z)],
g(z, t)=[g0+x(z, t)]1+PA(z)Psat(1+δ2)-1,
y(z, t)=y0 expidzK(z)-iΩt,
y=u, v, p, q, x,
{[2iK(z)+gs(z)-4θPA(z)][2iK(z)+gs(z)]
+β22Ω2[Ω2+sgn(β2)Ωc2(z)]}(1-iΩT2)×[(1-iΩT1)(1-iΩT2)+I(z)]-gs(z)[2iK(z)+gs(z)](1-iΩT2)×[1-iΩT1-I(z)]-gs(z)×[2iK(z)+gs(z)-4θPA(z)]×[(1-iΩT1)(1-iΩT2)+I(z)]+gs2(z)[1-iΩT1-I(z)]=0,
ΩRabi(z)PA(z)PsatT1T21/2.
h(Ω)-20Ldz Im[K(Ω, z)],
As(z)=(P0)1/2 exp[iφs(z)],
Bs(z)=δ-i1+δ2Asgs,
gs=g01+P0Psat(1+δ2)-1.
gs=(α-2θP0)(1+δ2),
dφsdz=γP0+12δgs1+δ2-α.
A(z, t)=[(P0)1/2+u(z, t)+iv(z, t)]exp[iφs(z)],
B(z, t)=δ-i1+δ2[(P0)1/2+p(z, t)+iq(z, t)]gs exp[iφs(z)],
g(z, t)=[g0+x(z, t)]1+P0Psat(1+δ2)-1,
y(z, t)=y0 exp[i(Kz-Ωt)],
y=u, v, p, q, x,
{[2iK+α-6θP0](2iK+α-2θP0)
+β22Ω2[Ω2+sgn(β2)Ωc2]}(1-iΩT2)×[(1-iΩT1)(1-iΩT2)+I0]-gs(2iK+α-2θP0)(1-iΩT2)×[1-iΩT1-I0]-gs(2iK+α-6θP0)×[(1-iΩT1)(1-iΩT2)+I0]+gs2(1-iΩT1-I0)=0.
ΩRabiP0PsatT1T21/2.
2iK+gs1+δ2-4θP02iK+gs1+δ2+β2Ω2-δgs1+δ2β2Ω2+4γP0-δgs1+δ2×{(1-iΩT2)[(1-iΩT1)(1-iΩT2)+I0]+δ2(1-iΩT1)}+-gs1+δ22iK+gs1+δ2×(1-iΩT1)(1-iΩT2)-1-iΩT21+δ2I0+δ2(1-iΩT1)+-gs1+δ22iK+gs1+δ2-4θP0×(1-iΩT1)(1-iΩT2)+1-iΩT2δ21+δ2I0+δ2(1-iΩT1)+gs21+δ2×[(1+δ2)(1-iΩT1)-I0]+2iΩT2gs δ21+δ22iK+gs1+δ2-2θP0×1-iΩT1-I0(1+δ2)+δgs1+δ2β2Ω2-δgs1+δ2×[(1+δ2)(1-iΩT1)-I0]+δgs1+δ2β2Ω2+4γP0-δgs1+δ2×[(1+δ2)(1-iΩT1)+(1-iΩT2)I0]=0,

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