Abstract

Bistable dark solitary-wave solutions (bistable holes) to the generalized nonlinear Schrödinger equation are shown to exist in the normal dispersion regime for nonlinearities that are Kerr-like at low intensities, rise sufficiently rapidly at intermediate intensities, and become Kerr-like again or approach a constant value at large intensities. The bistable nature and soliton character of the holes are confirmed through numerical switching simulations. The concept of asymptotic pinning (of the x-dependent part) of the phase is used to explain the resultant velocities of the output solitons and the observed asymmetry in the emitted radiation.

© 1989 Optical Society of America

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References

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  1. A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
    [CrossRef]
  2. V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).
  3. A. Hasegawa, Y. Kodama, Opt. Lett. 7, 285 (1982).
    [CrossRef] [PubMed]
  4. L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
    [CrossRef]
  5. D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
    [CrossRef] [PubMed]
  6. A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
    [CrossRef] [PubMed]
  7. R. H. Enns, S. S. Rangnekar, A. E. Kaplan, Phys. Rev. A 35, 466 (1987); Phys. Rev. A 36, 1270 (1987).
    [CrossRef] [PubMed]
  8. R. H. Enns, Phys. Rev. A 36, 5441 (1987).
    [CrossRef] [PubMed]
  9. R. H. Enns, S. S. Rangnekar, Opt. Lett. 12, 108 (1987); IEEE J. Quantum Electron. QE-23, 1199 (1987).
    [CrossRef] [PubMed]
  10. L. J. Mulder, R. H. Enns, IEEE J. Quantum Electron. QE-24, 1567 (1988).
    [CrossRef]

1988 (2)

D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
[CrossRef] [PubMed]

L. J. Mulder, R. H. Enns, IEEE J. Quantum Electron. QE-24, 1567 (1988).
[CrossRef]

1987 (3)

R. H. Enns, S. S. Rangnekar, A. E. Kaplan, Phys. Rev. A 35, 466 (1987); Phys. Rev. A 36, 1270 (1987).
[CrossRef] [PubMed]

R. H. Enns, Phys. Rev. A 36, 5441 (1987).
[CrossRef] [PubMed]

R. H. Enns, S. S. Rangnekar, Opt. Lett. 12, 108 (1987); IEEE J. Quantum Electron. QE-23, 1199 (1987).
[CrossRef] [PubMed]

1985 (1)

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
[CrossRef] [PubMed]

1982 (1)

1980 (1)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

1973 (1)

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

1972 (1)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Enns, R. H.

L. J. Mulder, R. H. Enns, IEEE J. Quantum Electron. QE-24, 1567 (1988).
[CrossRef]

R. H. Enns, S. S. Rangnekar, Opt. Lett. 12, 108 (1987); IEEE J. Quantum Electron. QE-23, 1199 (1987).
[CrossRef] [PubMed]

R. H. Enns, S. S. Rangnekar, A. E. Kaplan, Phys. Rev. A 35, 466 (1987); Phys. Rev. A 36, 1270 (1987).
[CrossRef] [PubMed]

R. H. Enns, Phys. Rev. A 36, 5441 (1987).
[CrossRef] [PubMed]

Giuliani, G.

D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
[CrossRef] [PubMed]

Gordon, J. P.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Grischkowsky, D.

D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
[CrossRef] [PubMed]

Halas, N. J.

D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
[CrossRef] [PubMed]

Hasegawa, A.

A. Hasegawa, Y. Kodama, Opt. Lett. 7, 285 (1982).
[CrossRef] [PubMed]

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Kaplan, A. E.

R. H. Enns, S. S. Rangnekar, A. E. Kaplan, Phys. Rev. A 35, 466 (1987); Phys. Rev. A 36, 1270 (1987).
[CrossRef] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
[CrossRef] [PubMed]

Kodama, Y.

Krökel, D.

D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
[CrossRef] [PubMed]

Mollenauer, L. F.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Mulder, L. J.

L. J. Mulder, R. H. Enns, IEEE J. Quantum Electron. QE-24, 1567 (1988).
[CrossRef]

Rangnekar, S. S.

R. H. Enns, S. S. Rangnekar, A. E. Kaplan, Phys. Rev. A 35, 466 (1987); Phys. Rev. A 36, 1270 (1987).
[CrossRef] [PubMed]

R. H. Enns, S. S. Rangnekar, Opt. Lett. 12, 108 (1987); IEEE J. Quantum Electron. QE-23, 1199 (1987).
[CrossRef] [PubMed]

Shabat, A. B.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Stolen, R. H.

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

Tappert, F.

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

Zakharov, V. E.

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

Appl. Phys. Lett. (1)

A. Hasegawa, F. Tappert, Appl. Phys. Lett. 23, 142, 171 (1973).
[CrossRef]

IEEE J. Quantum Electron. (1)

L. J. Mulder, R. H. Enns, IEEE J. Quantum Electron. QE-24, 1567 (1988).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. A (2)

R. H. Enns, S. S. Rangnekar, A. E. Kaplan, Phys. Rev. A 35, 466 (1987); Phys. Rev. A 36, 1270 (1987).
[CrossRef] [PubMed]

R. H. Enns, Phys. Rev. A 36, 5441 (1987).
[CrossRef] [PubMed]

Phys. Rev. Lett. (3)

L. F. Mollenauer, R. H. Stolen, J. P. Gordon, Phys. Rev. Lett. 45, 1095 (1980).
[CrossRef]

D. Krökel, N. J. Halas, G. Giuliani, D. Grischkowsky, Phys. Rev. Lett. 60, 29 (1988).
[CrossRef] [PubMed]

A. E. Kaplan, Phys. Rev. Lett. 55, 1291 (1985); IEEE J. Quantum Electron. QE-21, 1538 (1985).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

V. E. Zakharov, A. B. Shabat, Sov. Phys. JETP 34, 62 (1972).

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Figures (4)

Fig. 1
Fig. 1

(a) Energy surface P derived from Eq. (1) for Kerr–step–Kerr model for α = 1 and β = 0.1. (b) The energy curves from (a) for B = 0.45 and 0.6. The arrows refer to switching runs discussed in the text when a source term is added to Eq. (1).

Fig. 2
Fig. 2

Numerical run for the downswitching transition c(Ac = 0.79) to d(Ad = 0.486) of Fig. 1(b). (a) The modulus |E|, (b) the phase factor θ+. Imax = 1, z0 = 0.3, λ = 2, G = 3, grid spacing Δx = 0.086, r ≡ Δz/2(Δx)2 = 0.1, and xboundary = ±150.

Fig. 3
Fig. 3

(a) Input (z = 0) profile c and output (z = 30) profile d. The output profile is exactly fitted (dashed curve) by the analytic one-soliton formula for A = 0.486. (b) The contour plot for the transition cd.

Fig. 4
Fig. 4

Upswitching transition e(Ae = 0.40) → f and g of Fig. 1(b). (a) The final profile |E| (solid curve, numerical; dashed curve, calculated) and the initial (solid curve) and z = 8 (dashed curve) phases. (b) The contour plot showing soliton tracks for ef, g.

Equations (3)

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i E z ± ½ E x x + f ( E 2 ) E = 0 ,
U y y + 2 [ δ ¯ - f ( U 2 ) ] U - C 2 / U 3 = 0 ,
f = { β I I < I 0 α I I > I 0 ,

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