Abstract

We examine analytically, numerically, and experimentally the phase shift incurred by a soliton pulse when it collides with a copropagating, orthogonally polarized soliton pulse in a highly birefringent optical fiber. Use of a well-known average variational principle and a Gaussian ansatz reduces the dynamics to a set of ordinary differential equations for which an approximate analytic solution is found in the case of highly birefringent fibers. The analytic approximation is shown to be in good agreement with the full numerical model and experimental data, allowing it to be used as an evaluation tool for the design of nonlinear optical loop mirror switches.

© 1997 Optical Society of America

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References

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  1. N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
    [Crossref] [PubMed]
  2. M. N. Islam, E. R. Sunderman, R. H. Stolen, W. Pleibel, and J. R. Simpson, “Soliton switching in a fiber nonlinear loop mirror,” Opt. Lett. 14, 811–813 (1989).
    [Crossref] [PubMed]
  3. B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
    [Crossref]
  4. N. A. Whitaker, M. C. Gabriel, H. Avramopoulos, and A. Huang, “All-optical, all-fiber circulating shift register with an inverter,” Opt. Lett. 16, 1999–2001 (1991).
    [Crossref] [PubMed]
  5. M. Jinno and T. Matsumoto, “Ultrafast all-optical logic operations in a nonlinear Sagnac interferometer with two control beams,” Opt. Lett. 16, 220–222 (1991).
    [Crossref] [PubMed]
  6. J. K. Lucek and K. Smith, “All-optical signal regenerator,” Opt. Lett. 18, 1226–1228 (1993).
    [Crossref] [PubMed]
  7. J. D. Moores, K. Bergman, H. A. Haus, and E. P. Ippen, “Demonstration of optical switching by means of solitary wave collisions in a fiber ring reflector,” Opt. Lett. 16, 138–140 (1991).
    [Crossref] [PubMed]
  8. J. D. Moores, K. Bergman, H. A. Haus, and E. P. Ippen, “Optical switching using fiber ring reflectors,” J. Opt. Soc. Am. B 8, 594–601 (1991).
    [Crossref]
  9. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [Crossref]
  10. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
    [Crossref]
  11. Q. Wang, P. K. A. Wai, C.-J. Chen, and C. R. Menyuk, “Numerical modeling of soliton-dragging logic gates,” J. Opt. Soc. Am. B 10, 2030–2039 (1993).
    [Crossref]
  12. A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985), Chap. 3, pp. 72–87.
  13. T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
    [Crossref] [PubMed]
  14. G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974), Chap. 14.
  15. D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
    [Crossref]
  16. D. J. Kaup, B. A. Malomed, and R. S. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
    [Crossref]
  17. H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 2.
  18. L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1976), Chap. 1.
  19. W. H. Beyer, ed., CRC Standard Mathematical Tables (CRC, Boca Raton, Fla., 1981), pp. 527–536.
  20. T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
    [Crossref]
  21. M. D. Feit and J. A. Fleck, “Light propagation in graded-index optical fibers,” Appl. Opt. 17, 3990–3998 (1978).
    [Crossref] [PubMed]
  22. T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
    [Crossref]

1994 (1)

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[Crossref]

1993 (3)

1991 (5)

1990 (1)

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[Crossref] [PubMed]

1989 (2)

M. N. Islam, E. R. Sunderman, R. H. Stolen, W. Pleibel, and J. R. Simpson, “Soliton switching in a fiber nonlinear loop mirror,” Opt. Lett. 14, 811–813 (1989).
[Crossref] [PubMed]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

1988 (1)

1987 (1)

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

1984 (1)

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[Crossref]

1983 (1)

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

1978 (1)

Ablowitz, M. J.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[Crossref]

Anderson, D.

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

Avramopoulos, H.

Bergman, K.

Blow, K. J.

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

Chen, C.-J.

Constantine, P. D.

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

Doran, N. J.

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

N. J. Doran and D. Wood, “Nonlinear-optical loop mirror,” Opt. Lett. 13, 56–58 (1988).
[Crossref] [PubMed]

Feit, M. D.

Fleck, J. A.

Gabriel, M. C.

Goldstein, H.

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 2.

Haus, H. A.

Hou, T. Y.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[Crossref]

Huang, A.

Ippen, E. P.

Islam, M. N.

Jinno, M.

Kath, W. L.

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[Crossref] [PubMed]

Kaup, D. J.

D. J. Kaup, B. A. Malomed, and R. S. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[Crossref]

Landau, L. D.

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1976), Chap. 1.

Lifshitz, E. M.

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1976), Chap. 1.

Lowengrub, J. S.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[Crossref]

Lucek, J. K.

J. K. Lucek and K. Smith, “All-optical signal regenerator,” Opt. Lett. 18, 1226–1228 (1993).
[Crossref] [PubMed]

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

Malomed, B. A.

D. J. Kaup, B. A. Malomed, and R. S. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[Crossref]

Marshall, I. W.

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

Matsumoto, T.

Menyuk, C. R.

Q. Wang, P. K. A. Wai, C.-J. Chen, and C. R. Menyuk, “Numerical modeling of soliton-dragging logic gates,” J. Opt. Soc. Am. B 10, 2030–2039 (1993).
[Crossref]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

Moores, J. D.

Nelson, B. P.

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

Nelson, K.

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

Newell, A. C.

A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985), Chap. 3, pp. 72–87.

Pleibel, W.

Shelley, M. J.

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[Crossref]

Simpson, J. R.

Smith, K.

Stolen, R. H.

Sunderman, E. R.

Taha, T. R.

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[Crossref]

Tasgal, R. S.

D. J. Kaup, B. A. Malomed, and R. S. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[Crossref]

Ueda, T.

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[Crossref] [PubMed]

Wai, P. K. A.

Wang, Q.

Whitaker, N. A.

Whitham, G. B.

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974), Chap. 14.

Wood, D.

Appl. Opt. (1)

Electron. Lett. (1)

B. P. Nelson, K. J. Blow, P. D. Constantine, N. J. Doran, J. K. Lucek, I. W. Marshall, and K. Nelson, “All-optical Gbit/s switching using nonlinear optical loop mirror,” Electron. Lett. 27, 704–705 (1991).
[Crossref]

IEEE J. Quantum Electron. (2)

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr media,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
[Crossref]

C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
[Crossref]

J. Comput. Phys. (2)

T. R. Taha and M. J. Ablowitz, “Analytical and numerical aspects of certain nonlinear evolution equations. II. Numerical, nonlinear Schrödinger equation,” J. Comput. Phys. 55, 203–230 (1984).
[Crossref]

T. Y. Hou, J. S. Lowengrub, and M. J. Shelley, “Removing the stiffness from interfacial flows with surface tension,” J. Comput. Phys. 114, 312–338 (1994).
[Crossref]

J. Opt. Soc. Am. B (2)

Opt. Lett. (6)

Phys. Rev. A (2)

T. Ueda and W. L. Kath, “Dynamics of coupled solitons in nonlinear optical fibers,” Phys. Rev. A 42, 563–571 (1990).
[Crossref] [PubMed]

D. Anderson, “Variational approach to nonlinear pulse propagation in optical fibers,” Phys. Rev. A 27, 3135–3145 (1983).
[Crossref]

Phys. Rev. E (1)

D. J. Kaup, B. A. Malomed, and R. S. Tasgal, “Internal dynamics of a vector soliton in a nonlinear optical fiber,” Phys. Rev. E 48, 3049–3053 (1993).
[Crossref]

Other (5)

H. Goldstein, Classical Mechanics (Addison-Wesley, Reading, Mass., 1980), Chap. 2.

L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, London, 1976), Chap. 1.

W. H. Beyer, ed., CRC Standard Mathematical Tables (CRC, Boca Raton, Fla., 1981), pp. 527–536.

G. B. Whitham, Linear and Nonlinear Waves (Wiley-Interscience, New York, 1974), Chap. 14.

A. C. Newell, Solitons in Mathematics and Physics (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1985), Chap. 3, pp. 72–87.

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Figures (4)

Fig. 1
Fig. 1

Comparison of the initial and the final pulse amplitudes that are due to a collision with a three-soliton pulse (left-axis label) along with a comparison of the collisional phase shifts calculated from the three models (right-axis label). Note that the final pulse is time shifted by 8 fs.

Fig. 2
Fig. 2

Comparison of the ODE solutions for the pulse parameters that represent the signal’s central position velocity (V1) and central position (t1) and its approximate solutions (dotted curves). Note that as A/δ increases, the agreement degrades.

Fig. 3
Fig. 3

Nonlinear phase shift computed by means of the full coupled NLS equations (PDE model, solid curves), the reduced ODE model (dotted curves), and the approximation to the ODE model (dashed curves) for A=1, 3, 5. Much of the error observed in the case A=1 arises from the Gaussian (as opposed to a hyperbolic secant) ansatz approximation.

Fig. 4
Fig. 4

Experimental, numerical, and analytic evaluation of the collisional phase shift of a first-order signal soliton caused by collision with a control pulse as a function of control pulse energy and peak power. The vertical lines depict the energies corresponding to integral soliton orders in the control pulse.

Equations (35)

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i uz+122ut2+(|u|2+B|v|2)u=0
i(vz-δ vt)+122vt2+(|v|2+B|u|2)v=0,
|E0|2=λ0Aeff2πn2ω0z0
z0=t01.762 2πcλ02|D|=0.84m.
δ=z0 1.76t01Vgu-1VgvΔnz0c1.76t0=12.38,
u=sech(t)exp(iz/2).
L=-L(u, u*, v, v*)dt,
L=iu u*z-u* uz+iv v*z-v* vz-iδv v*t-v* vt-43|u|2|v|2+ut2-|u|4+vt2-|v|4.
δLδp=zLpz+tLpt-Lp=0,
δLδρ=ddzLρz-Lρ=0,
u=η exp-t-t1W2exp{i[V1(t-t1)+ϕ1]},
v=Aη exp-t-t2W2exp{i[V2(t-t2)+ϕ2]},
|uhyperbolicsecant|2dt=2,
|uGaussian|2dt=(πW/2)1/2η2.
2πWη2=4.
dt1dz=V1,
dt2dz=V2-δ,
dV1dz=22A2η23W(t2-t1)exp-(t1-t2)2W,
dV2dz=22η23W(t1-t2)exp-(t1-t2)2W.
ϕ1=-zW+0zdz12V12+2W2(t-t1)2+η2 exp-2W(t-t1)2+23A2η2×exp-2W(t-t2)2-22A2η23W×(t-t1)(t2-t1)exp-(t1-t2)2W,
t2=-t1A2-δz+Δt,
V2=-V1A2,
dt1dz=V1,
dV1dz=-22A2η23W[t1(1+1/A2)+δz-Δt]×exp-[t1(1+1/A2)+δz-Δt]2W.
t1(1+1/A2)2t11,
dV1dz-22A2η23W(δz-Δt)exp-(δz-Δt)2W.
V12WA2η23δexp-(δz-Δt)2W,
t12WA2η23δ0L exp-(δz-Δt)2Wdz.
t12πWη23Aδ2=4W3Aδ2,
A/δ1.
ϕ1zη2-1W+2A2η230z exp-2W(δz-Δt)2dz.
η2-1W=12,
η±=2π(2±4-π)1/2.
ϕ1z2+4A26δ{erf[2/W(δz-Δt)]+1},
ϕcollision=4A23δ=2λ2|D|A23πΔn(t0/1.76),

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