Abstract

We consider a model of a mode-locked fiber ring laser for which the evolution of a propagating pulse in a birefringent optical fiber is periodically perturbed by rotation of the polarization state owing to the presence of a passive polarizer. The stable modes of operation of this laser that correspond to pulse trains with uniform amplitudes are fully classified. Four parameters, i.e., polarization, phase, amplitude, and chirp, are essential for an understanding of the resultant pulse-train uniformity. A reduced set of four coupled nonlinear differential equations that describe the leading-order pulse dynamics is found by use of the variational nature of the governing equations. Pulse-train uniformity is achieved in three parameter regimes in which the amplitude and the chirp decouple from the polarization and the phase. Alignment of the polarizer either near the slow or the fast axis of the fiber is sufficient to establish this stable mode locking.

© 2002 Optical Society of America

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References

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  1. A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers,” Appl. Phys. Lett. 23, 142–170 (1973).
    [CrossRef]
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).
  3. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electro-magnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).
  4. S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
    [CrossRef]
  5. L. F. Mollenauer, P. V. Mamyshev, J. Gripp, M. J. Neubelt, N. Mamysheva, L. Grüner-Nielsen, and T. Veng, “Demonstration of massive wavelength-division multiplexing over transoceanic distances by use of dispersion-managed solitons,” Opt. Lett. 25, 704–706 (2000).
    [CrossRef]
  6. K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. 28, 2226–2227 (1992).
    [CrossRef]
  7. H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
    [CrossRef]
  8. M. Fermann, M. J. Andrejco, Y. Silverberg, and M. L. Stock, “Passive mode locking by using nonlinear polarization evolution in a polarizing-maintaining erbium-doped fiber laser,” Opt. Lett. 18, 894–896 (1993).
    [CrossRef]
  9. M. Hofer, M. E. Fermann, F. Haberl, M. H. Ober, and A. J. Schmidt, “Mode locking with cross-phase and self-phase modulation,” Opt. Lett. 16, 502–504 (1991).
    [CrossRef] [PubMed]
  10. B. C. Collings, S. T. Cundiff, N. N. Akhmediev, T. J. Soto-Crespo, K. Bergman, and W. H. Knox, “Polarization-locked temporal vector solitons in a fiber lasers: experiment,” J. Opt. Soc. Am. B 17, 354–365 (2000).
    [CrossRef]
  11. A. D. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465–471 (2000).
    [CrossRef]
  12. G. Sucha, S. R. Bolton, S. Weiss, and D. S. Chemla, “Period doubling and quasi-periodicity in additive-pulse mode-locked lasers,” Opt. Lett. 20, 1794–1796 (1995).
    [CrossRef] [PubMed]
  13. D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53–64 (1991).
    [CrossRef]
  14. J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
    [CrossRef]
  15. P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modelling dispersion managed breathers,” SIAM J. Appl. Math. 59, 1288–1302 (1999).
    [CrossRef]
  16. J. N. Kutz, C. Hile, W. L. Kath, R. D. Li, and P. Kumar, “Pulse propagation in nonlinear optical fiber lines that employ phase-sensitive parametric amplifiers,” J. Opt. Soc. Am. B 11, 2112–2123 (1994).
    [CrossRef]
  17. J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM J. Appl. Math. 56, 611–626 (1996).
    [CrossRef]
  18. A. D. Kim, C. G. Goedde, and W. L. Kath, “Stabilizing dark solitons by using periodic phase-sensitive amplifications,” Opt. Lett. 21, 465–467 (1996).
    [CrossRef] [PubMed]

2000

1999

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modelling dispersion managed breathers,” SIAM J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

1998

1996

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

A. D. Kim, C. G. Goedde, and W. L. Kath, “Stabilizing dark solitons by using periodic phase-sensitive amplifications,” Opt. Lett. 21, 465–467 (1996).
[CrossRef] [PubMed]

1995

1994

1993

1992

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. 28, 2226–2227 (1992).
[CrossRef]

1991

1989

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
[CrossRef]

1974

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electro-magnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

1973

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers,” Appl. Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Akhmediev, N. N.

Andrejco, M. J.

Bergman, K.

Bolton, S. R.

Chemla, D. S.

Collings, B. C.

Cundiff, S. T.

Evangelides, S. G.

Fermann, M.

Fermann, M. E.

Goedde, C. G.

Gordon, J. P.

Gripp, J.

Grüner-Nielsen, L.

Haberl, F.

Hasegawa, A.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers,” Appl. Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Haus, H. A.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. 28, 2226–2227 (1992).
[CrossRef]

Hile, C.

Hofer, M.

Holmes, P.

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modelling dispersion managed breathers,” SIAM J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

Ippen, E. P.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. 28, 2226–2227 (1992).
[CrossRef]

Kath, W. L.

Kim, A. D.

A. D. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465–471 (2000).
[CrossRef]

A. D. Kim, C. G. Goedde, and W. L. Kath, “Stabilizing dark solitons by using periodic phase-sensitive amplifications,” Opt. Lett. 21, 465–467 (1996).
[CrossRef] [PubMed]

Knox, W. H.

Kumar, P.

Kutz, J. N.

A. D. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465–471 (2000).
[CrossRef]

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modelling dispersion managed breathers,” SIAM J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

J. N. Kutz, P. Holmes, S. G. Evangelides, and J. P. Gordon, “Hamiltonian dynamics of dispersion-managed breathers,” J. Opt. Soc. Am. B 15, 87–96 (1998).
[CrossRef]

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

J. N. Kutz, C. Hile, W. L. Kath, R. D. Li, and P. Kumar, “Pulse propagation in nonlinear optical fiber lines that employ phase-sensitive parametric amplifiers,” J. Opt. Soc. Am. B 11, 2112–2123 (1994).
[CrossRef]

Li, R. D.

Mamyshev, P. V.

Mamysheva, N.

Manakov, S. V.

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electro-magnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Mollenauer, L. F.

Muraki, D.

A. D. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465–471 (2000).
[CrossRef]

Muraki, D. J.

D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53–64 (1991).
[CrossRef]

Neubelt, M. J.

Ober, M. H.

Schmidt, A. J.

Silverberg, Y.

Soto-Crespo, T. J.

Stegeman, G. I.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
[CrossRef]

Stock, M. L.

Sucha, G.

Tamura, K.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. 28, 2226–2227 (1992).
[CrossRef]

Tappert, F.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers,” Appl. Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Trillo, S.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
[CrossRef]

Veng, T.

Wabnitz, S.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
[CrossRef]

Weiss, S.

Wright, E. M.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
[CrossRef]

Appl. Phys. Lett.

A. Hasegawa and F. Tappert, “Transmission of stationary nonlinear optical pulses in dispersive dielectric fibers,” Appl. Phys. Lett. 23, 142–170 (1973).
[CrossRef]

Electron. Lett.

K. Tamura, H. A. Haus, and E. P. Ippen, “Self-starting additive pulse mode-locked erbium fiber ring laser,” Electron. Lett. 28, 2226–2227 (1992).
[CrossRef]

IEEE J. Quantum Electron.

H. A. Haus, E. P. Ippen, and K. Tamura, “Additive-pulse modelocking in fiber lasers,” IEEE J. Quantum Electron. 30, 200–208 (1994).
[CrossRef]

A. D. Kim, J. N. Kutz, and D. Muraki, “Pulse-train uniformity in optical fiber lasers passively mode-locked by nonlinear polarization rotation,” IEEE J. Quantum Electron. 36, 465–471 (2000).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Commun.

S. Trillo, S. Wabnitz, E. M. Wright, and G. I. Stegeman, “Polarized soliton instability and branching in birefringent fibers,” Opt. Commun. 70, 166–172 (1989).
[CrossRef]

Opt. Lett.

Physica D

D. J. Muraki and W. L. Kath, “Hamiltonian dynamics of solitons in optical fibers,” Physica D 48, 53–64 (1991).
[CrossRef]

SIAM J. Appl. Math.

J. N. Kutz and W. L. Kath, “Stability of pulses in nonlinear optical fibers using phase-sensitive amplifiers,” SIAM J. Appl. Math. 56, 611–626 (1996).
[CrossRef]

P. Holmes and J. N. Kutz, “Dynamics and bifurcations of a planar map modelling dispersion managed breathers,” SIAM J. Appl. Math. 59, 1288–1302 (1999).
[CrossRef]

Sov. Phys. JETP

S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electro-magnetic waves,” Sov. Phys. JETP 38, 248–253 (1974).

Other

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1989).

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Figures (12)

Fig. 1
Fig. 1

Schematic of the ring cavity laser that includes a polarizer and polarization controllers in a birefringent optical fiber. A small portion of the fiber is erbium-doped fiber, which is pumped via a wavelength-division multiplexed (WDM) coupler at 980 nm and provides gain to the cavity. The mode-locked soliton pulse stream is coupled out through a 90/10 coupler; i.e., 10% of the pulse energy is coupled out. The two primary experimental parameters that can easily be adjusted are the polarizer angle relative to the fast and slow axes of the birefringent fiber and the birefringence strength, which is determined by the polarization controllers.

Fig. 2
Fig. 2

Stable mode-locking evolution of the governing CNLSs with a polarizer. In this case we consider the parameters Θ=0.45π and K=0.1. Although the amplitude appears to lock to a constant value, a detailed look after each round trip of the cavity shows amplitude fluctuations on the order of 4% (see Fig. 7 below).

Fig. 3
Fig. 3

Amplitude, chirp, polarization, and phase evolution over 100 round trips of the fiber cavity near the polarization-locked solution P=0. K=0.10, and the evolution without the polarizer has a single unstable eigenvalue leading to growth. This solution blows up just after z=150. The addition of polarizer ensures that P remains near zero, so a uniform pulse train results.

Fig. 4
Fig. 4

Pulse-train uniformity (η fluctuations) over 200 round trips of the fiber cavity with K=0.10 and near the polarization-locked solution P=0. As the polarizer is detuned from Θ=0, the amplitude fluctuations increase and the pulse-train uniformity is destroyed.

Fig. 5
Fig. 5

Comparison of the exact numerical solution for Pπ/2 with the perturbation expansion results for =0.1. Analytic data generated for phase (ψ) and polarization (P) include the leading-order solution and the first correction. The analytic curve of amplitude (η) and chirp (β) contains the leading-order solution and two correction terms. As the phase grows monotonically, it is more enlightening to examine sin2 ψ than ψ itself. Note that the perturbation results are in excellent agreement with the numerical results, except for a small phase slip observed in all quantities. In practice, this phase slip is unimportant, as we are concerned only with the size of the parameter fluctuations.

Fig. 6
Fig. 6

Amplitude, chirp, polarization, and phase evolution over 100 round trips of the fiber cavity near the polarization-locked solution P=π/2. K=0.10 and the evolution without the polarizer is a stable center. The addition of polarizer ensures that P remains near π/2, so the uniform pulse train is maintained.

Fig. 7
Fig. 7

Pulse-train uniformity (η fluctuations) over 200 round trips of the fiber cavity with K=0.10 and near the polarization-locked solution P=π/2. As the polarizer is detuned from Θ=π/2, the amplitude fluctuations increase and the pulse train’s uniformity is destroyed. However, the uniformity here is not so severely affected as is detuning from Θ=0. (See Fig. 12 below).

Fig. 8
Fig. 8

Eigenvalues of the linearized equations for the phase-locked solution ψ0=π/2 as a function of birefringence K. Three distinct regions of operation are predicted; two of them are physically realizable. For 0<K<0.14 there are two unstable and two stable eigenvalues, implying unstable growth. For 0.14<K<0.22 the eigenvalues are all purely imaginary, and the solution is stable. The region to the right of the dashed line is not relevant physically, as it gives rise to imaginary values of P0.

Fig. 9
Fig. 9

Amplitude, chirp, polarization, and phase evolution over several hundred round trips of the fiber cavity near the phase-locked solution ψ0=π/2. K=0.10, and the evolution without the polarizer has two unstable eigenvalues that lead to growth. The addition of the polarizer enhances the instability.

Fig. 10
Fig. 10

Amplitude, chirp, polarization, and phase evolution over several hundred round trips of the fiber cavity near the phase-locked solution ψ0=π/2. K=0.18, and the evolution without the polarizer has purely imaginary eigenvalues, leading to oscillations without growth. The addition of the polarizer destabilizes a uniform pulse train.

Fig. 11
Fig. 11

Amplitude, chirp, and phase evolution over 100 round trips of the fiber cavity near the phase- and polarization-locked solution with Θ=0.0305π, P(0)=0.0305π, η(0)=1.0, ψ(0)=β(0)=0, and K=0.10. The addition of the polarizer stabilizes the otherwise unstable solution and quickly evolves to the predicted values of η, β, and ψ given by Eqs. (16).

Fig. 12
Fig. 12

Performance of fiber ring laser for Θ[0, π/2] with K=0.1. Vertical axis, passive polarizer angle Θ; horizontal axis, initial starting amplitude η(0). Near Θ=0 and Θ=π/2 the laser produces uniform amplitude pulse solutions. Black, no pulse fluctuations; white, fluctuations greater than 20%. B=1/3, I=1, β(0)=0, L=1, and ψ(0)=0. The fluctuation strength is calculated by consideration of minimum and maximum values of η fluctuations with its average value for values of η between round trips 50 and 100. Dashed curve, the family of solutions (16). Solutions away from P=0 settle to this curve after many round trips until its termination at Θ0.1518π. Average values of η<0.4 are considered to be operationally unstable.

Equations (39)

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i Uz+12 2Ut2-KU+(|U|2+A|V|2)U+BV2U*=0,
i Vz+12 2Vt2+KV+(A|U|2+|V|2)V+BU2V*=0.
U=Iη sech(ηt)cos P exp(-iψ/2+iβt2+iϕ/2),
V=Iη sech(ηt)sin P exp(+iψ/2+iβt2+iϕ/2).
dPdz=BIη sin(2P)sin(2ψ),
dψdz=-43BIη cos(2P)sin2(ψ)+2K,
dηdz=-2βη,
π2 dβdz=η4-π2β2-Iη3[1-B sin2(2P)sin2(ψ)].
P+=tan-1[α tan(P--Θ)]+Θ,
C(P, ψ)=sin2[2P(z)]sin2[ψ(z)]=constant.
sin 2P0 sin 2ψ0=0,
cos 2P0 sin2 ψ0=3K2BIη0,
η0=I(1-B sin2 2P0 sin2 ψ0).
P(z)=π/2,
ψ(z)=tan-13Kσ tan2z3Kσ3,
η(z)=I,
β(z)=0,
η1(z)=η1(0)cos2I2zπ-πIβ1(0)sin2I2zπ,
β1(z)=β1(0)cos2I2zπ+Iπη1(0)sin2I2zπ,
P1(z)=P1(0)|cos Λz|3Kσ tan2 Λz+11/2,
ψ1(z)=BKπI sin2I2zπ[η1(0)(4Kπ2σ-3I4+3I4 cos 2Λz)+2π2IKσβ1(0)sin 2Λz]+π cos2I2zπ[β1(0)(4Kπ2σ-3I4+3I4 cos 2Λz)-2I3Kση1(0)sin 2Λz]-4Kπ3σβ1(0)I2(4Kπ2σ-3I4)×(σ-2BI2 sin2 Λz),
dP˜dz=2/3BI2 sin 2P0ψ˜,
dψ˜dz=0,
dη˜dz=-2Iβ˜,
π22 dβ˜dz=I3η˜.
(η˜,β˜)=a1 cos2I2πz+a2,a3 cos2I2πz+a4,
P˜=[2/3BK˜I2C sin 2P0]z2,
η0=I(1-B sin2 2P0),
cos 2P0=3K2BI/η0.
η03-I(1-B)η02-9K24BI=0.
dP˜dz=-BIη0 sin 2P0ψ˜,
dψ˜dz=-43BI(cos 2P0η˜-2η0 sin 2P0P˜),
dη˜dz=-2η0β˜,
π22 dβdz=η03η˜+2IBη03 sin 4P0P˜.
λ2(λ2+4η04π2)+1681η02-4K2
×λ2+4η04π2+54π2η0K2=0,
β=0,
η=I2 1+1-6KI sin(2Θ)tan(2Θ)1/2,
ψ=±sin-13K2BIη cos(2Θ)1/2.

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