Abstract

The soliton-dragging logic gate is based on the soliton-dragging effect, which occurs when pulses of opposite polarization states in a birefringent optical fiber collide. The central frequencies of the two pulses and hence their velocities are shifted. The soliton-dragging effect is studied here for two solitons of the same amplitude, two solitons of different amplitudes, and two pulses that are not solitons. The dependences of the effect on the initial pulse separation, the initial pulse amplitudes, and the initial pulse profiles are all determined. The pulse interaction is studied by means of several different methods: (1) a complete solution of the coupled nonlinear Schrödinger equation, (2) a solution of ordinary differential equations that includes the effect of frequency chirp, (3) a solution of ordinary differential equations that neglects the effect of frequency chirp, and (4) the Born approximation. We compare the last three approaches with the first. We find that the second approach accurately predicts the velocity and the time shifts for the larger amplitude pulse in all cases of practical interest that we studied. Thus the second approach is useful in the design of soliton logic-based networks. The third approach is significantly less accurate, while the fourth approach is the least accurate of all.

© 1993 Optical Society of America

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References

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  1. M. N. Islam, “All-optical cascadable NOR gate with gain,” Opt. Lett. 15, 417–419 (1990).
    [CrossRef] [PubMed]
  2. M. N. Islam, C. E. Soccolich, D. A. B. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990).
    [CrossRef] [PubMed]
  3. M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
    [CrossRef]
  4. M. N. Islam, J. R. Sauer, “GEO-modules as a natural basis for all-optical fiber logic systems,” IEEE J. Quantum Electron. 27, 843–848 (1991).
    [CrossRef]
  5. J. R. Sauer, M. N. Islam, S. P. Dijaili, “A soliton ring network,” J. Lightwave Technol. (to be published).
  6. M. N. Islam, Department of Electrical Engineering and Computer Science, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109 (personal communication).
  7. M. N. Islam, C. R. Menyuk, C.-J. Chen, C. E. Soccolich, “Chirp mechanisms in soliton-dragging logic gates,” Opt. Lett. 16, 214–216 (1991).
    [CrossRef] [PubMed]
  8. Q. Wang, P. K. A. Wai, C.-J. Chen, C. R. Menyuk, “Soliton shadows in birefringent optical fibers,” Opt. Lett. 17, 1265–1267 (1992).
    [CrossRef] [PubMed]
  9. T. Ueda, W. L. Kath, “Dynamics of coupled solitons in nonlinear otpical fibers,” Phys. Rev. A 42, 563–571 (1990).
    [CrossRef] [PubMed]
  10. C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” IEEE J. Quantum Electron. 25, 2674–2682 (1989).
    [CrossRef]
  11. C. R. Menyuk, “Nonlinear pulse propagation in birefringent optical fibers,” IEEE J. Quantum Electron. QE-23, 174–176 (1987).
    [CrossRef]
  12. C. R. Menyuk, “Stability of solitons in birefringent optical fibers. II. Arbitrary amplitudes,” J. Opt. Soc. Am. B 5, 392–402 (1988).
    [CrossRef]
  13. Y. S. Kivshar, “Soliton stability in birefringent optical fibers: analytical approach,” J. Opt. Soc. Am. B 7, 2204–2209 (1990).
    [CrossRef]
  14. S. V. Manakov, “On the theory of two-dimensional stationary self-focusing of electromagnetic waves,” Sov. Phys. JEPT 38, 248–253 (1974).
  15. E. Caglioti, S. Trillo, S. Wabnitz, B. Crosignani, P. Di Porto, “Finite-dimensional description of nonlinear pulse propagation in optical-fiber couplers with applications to soliton switching,” J. Opt. Soc. Am. B 7, 374–385 (1990).
    [CrossRef]
  16. C. Paré, M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
    [CrossRef] [PubMed]

1992

1991

M. N. Islam, C. R. Menyuk, C.-J. Chen, C. E. Soccolich, “Chirp mechanisms in soliton-dragging logic gates,” Opt. Lett. 16, 214–216 (1991).
[CrossRef] [PubMed]

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
[CrossRef]

M. N. Islam, J. R. Sauer, “GEO-modules as a natural basis for all-optical fiber logic systems,” IEEE J. Quantum Electron. 27, 843–848 (1991).
[CrossRef]

1990

1989

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

1988

1987

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

1974

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

Caglioti, E.

Chen, C.-J.

Crosignani, B.

Di Porto, P.

Dijaili, S. P.

J. R. Sauer, M. N. Islam, S. P. Dijaili, “A soliton ring network,” J. Lightwave Technol. (to be published).

Florjanczyk, M.

C. Paré, M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef] [PubMed]

Islam, M. N.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
[CrossRef]

M. N. Islam, C. R. Menyuk, C.-J. Chen, C. E. Soccolich, “Chirp mechanisms in soliton-dragging logic gates,” Opt. Lett. 16, 214–216 (1991).
[CrossRef] [PubMed]

M. N. Islam, J. R. Sauer, “GEO-modules as a natural basis for all-optical fiber logic systems,” IEEE J. Quantum Electron. 27, 843–848 (1991).
[CrossRef]

M. N. Islam, “All-optical cascadable NOR gate with gain,” Opt. Lett. 15, 417–419 (1990).
[CrossRef] [PubMed]

M. N. Islam, C. E. Soccolich, D. A. B. Miller, “Low-energy ultrafast fiber soliton logic gates,” Opt. Lett. 15, 909–911 (1990).
[CrossRef] [PubMed]

M. N. Islam, Department of Electrical Engineering and Computer Science, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109 (personal communication).

J. R. Sauer, M. N. Islam, S. P. Dijaili, “A soliton ring network,” J. Lightwave Technol. (to be published).

Kath, W. L.

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

Kim, K. S.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
[CrossRef]

Kivshar, Y. S.

Manakov, S. V.

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

Menyuk, C. R.

Miller, D. A. B.

Paek, U. C.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
[CrossRef]

Paré, C.

C. Paré, M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef] [PubMed]

Sauer, J. R.

M. N. Islam, J. R. Sauer, “GEO-modules as a natural basis for all-optical fiber logic systems,” IEEE J. Quantum Electron. 27, 843–848 (1991).
[CrossRef]

J. R. Sauer, M. N. Islam, S. P. Dijaili, “A soliton ring network,” J. Lightwave Technol. (to be published).

Simpson, J. R.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
[CrossRef]

Soccolich, C. E.

Trillo, S.

Ueda, T.

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

Wabnitz, S.

Wai, P. K. A.

Wang, Q.

Electron. Lett.

M. N. Islam, C. E. Soccolich, C.-J. Chen, K. S. Kim, J. R. Simpson, U. C. Paek, “All-optical inverter with one picojoule switching energy,” Electron. Lett. 27, 130–131 (1991).
[CrossRef]

IEEE J. Quantum Electron.

M. N. Islam, J. R. Sauer, “GEO-modules as a natural basis for all-optical fiber logic systems,” IEEE J. Quantum Electron. 27, 843–848 (1991).
[CrossRef]

C. R. Menyuk, “Pulse propagation in an elliptically birefringent Kerr medium,” 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. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. A

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

C. Paré, M. Florjańczyk, “Approximate model of soliton dynamics in all-optical couplers,” Phys. Rev. A 41, 6287–6295 (1990).
[CrossRef] [PubMed]

Sov. Phys. JEPT

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

Other

J. R. Sauer, M. N. Islam, S. P. Dijaili, “A soliton ring network,” J. Lightwave Technol. (to be published).

M. N. Islam, Department of Electrical Engineering and Computer Science, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109 (personal communication).

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Figures (9)

Fig. 1
Fig. 1

Evolution of (a) the velocity shift, (b) the time shift, (c) the width, and (d) the coefficient of frequency chirp of the soliton in the u polarization along a fiber of length of 20Z0, with δ = 1.0, B = 2/3, A0 = 1.0, and s0 = 0, where τ = 1.763 is the FWHM of the soliton and z0 = π/2 is a soliton period. Results are shown for the complete model (dotted curves), the variational method with chirp (solid curves), the variational method without chirp (dashed curves), and the Born approximation (dashed–dotted curves).

Fig. 2
Fig. 2

(a) Velocity shift and (b) time shift of the soliton in the u polarization versus the initial separation, with A0 = 1.0 and δ = 1.0, at a distance of 20Z0, where τ = 1.763 is the FWHM of the soliton. The shifts for the soliton in the υ polarization have the same magnitude but opposite sign.

Fig. 3
Fig. 3

(a) Velocity shift and (b) time shift of the soliton in the u polarization versus the initial amplitude of the soliton, with s0 = 0, δ = 1.0, and τ = 1.763, at a distance of 20Z0. The shifts for the soliton in the υ polarization have the same magnitude but opposite sign.

Fig. 4
Fig. 4

Relative error of the time shifts, ρ = (treducedtcomplete)/tcomplete, at a distance of 20Z0 obtained from the three reduced models, versus the initial amplitude, with s0 = 0 and δ = 1.0.

Fig. 5
Fig. 5

(a) Velocity shift and (b) time shift of the soliton in the u polarization and (c) velocity shift and (d) time shift of the soliton in the υ polarization, versus the initial separation. The initial parameters are A10 = 1.5, A20 = 1.0, and δ = 1.0, while τ1 = 1.175 and τ2 = 1.763 are the FWHM values for the solitons. The distance is 20Z0.

Fig. 6
Fig. 6

(a) Velocity shift and (b) time shift of the soliton in u polarization and (c) velocity shift and (d) time shift of the soliton in the υ polarization, versus the initial amplitude of the soliton in the υ polarization. The initial parameters are A10/A20 = 1.5, s0 = 0, and δ = 1.0, while τ1 = 1.175 and τ2 = 1.763 are the FWHM values for the solitons. The distance is 20Z0.

Fig. 7
Fig. 7

Ratio of the velocity shifts of the solitons versus the ratio of the initial amplitudes at a distance of 20Z0 with s0 = 0 and δ = 1.0. Results obtained from all three reduced models are shown as a solid line. The diamonds, triangles, and circles are from the complete model, with A20 = 0.5, 0.8, and 1.0, respectively.

Fig. 8
Fig. 8

(a) Velocity shift and (b) time shift of the soliton in u polarization and (c) velocity shift and (d) normalized time shift of the soliton in the υ polarization, versus the initial separation at a distance of 20Z0. The initial parameter values are A10 = 1.11, A20 = 0.48, W10 = W20 = 1.0, and δ = 0.894. The FWHM value of both pulses is τ = 1.763.

Fig. 9
Fig. 9

Parameter values are the same as those in Fig. 8, but there is an abrupt loss in the gate. After propagating a length of 1.3Z0, the pulses undergo an abrupt intensity loss of 25% and then propagate a further 35Z0.

Equations (19)

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i u ξ + i δ u s + 1 2 2 u s 2 + ( | u | 2 + B | υ | 2 ) u = 0 , i υ ξ i δ υ s + 1 2 2 υ s 2 + ( B | u | 2 + | υ | 2 ) υ = 0 ,
u ( ξ = 0 , s ) = A 10 sech [ ( s + s 0 ) / W 10 ] , υ ( ξ = 0 , s ) = A 20 sech [ ( s s 0 ) / W 20 ] ,
L = L ( u , u * , υ , υ * ) d s ,
L = i ( u u ξ * u ξ u * ) + i ( υ υ ξ * υ ξ υ * ) + i δ ( u u s * u s u * ) i δ ( υ υ s * υ s υ * ) + ( | u s | 2 | u | 4 ) + ( | υ s | 2 | υ | 4 ) 2 B | u | 2 | υ | 2 ,
u = A 1 sech ( s s 1 W 1 ) × exp { i [ V 1 ( s s 1 ) + b 1 2 W 1 ( s s 1 ) 2 + σ 1 ] } , υ = A 2 sech ( s s 2 W 2 ) × exp { i [ V 2 ( s s 2 ) + b 2 2 W 2 ( s s 2 ) 2 + σ 2 ] } ,
d d ξ L q ξ L q = 0 ,
d s i d ξ = ( 1 ) i 1 δ + V i ,
d V i d ξ = B K j 2 W 1 W 2 sech 2 ( s s j W j ) × s [ sech 2 ( s s i W i ) ] d s ,
d W i d ξ = b i ,
d b i d ξ = 4 K j π 2 W i 3 { W i 1 K i + 3 B K j W i 2 K i W j sech 2 ( s s j W j ) × s [ ( s s i ) sech 2 ( s s i W i ) ] d s } ,
d t d ξ = = δ + V ,
d V d ξ = 2 K B W 2 α [ F ( α ) ] ,
d W d ξ = b ,
d b d ξ = 4 π 2 W 2 { 1 W K 3 B K α [ α F ( α ) ] } ,
V 1 = ω cent u = i d s ( u * u s ) / d s | u | 2 , V 2 = ω cent υ = i d s ( υ * υ s ) / d s | υ | 2 .
d V 1 d ξ = B d s | υ | 2 s | u | 2 / d s | u | 2 , d V 2 d ξ = B d s | u | 2 s | υ | 2 / d s | υ | 2 .
| u ( ξ , s ) | = A 10 sech [ ( s + s 0 δ ξ ) / W 10 ] , | υ ( ξ , s ) | = A 20 sech [ ( s + s 0 + δ ξ ) / W 20 ] ,
Δ V 1 ( ξ ) = B A 2 2 δ { F [ ( 2 s 0 2 δ ξ ) / W ] F [ 2 s 0 / W ] } , Δ V 2 ( ξ ) = B A 1 2 δ { F [ ( 2 s 0 2 δ ξ ) / W ] F [ 2 s 0 / W ] } ,
u = α u , υ = α υ , s = s / α , ξ = ξ / α 2 , δ = α δ ,

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