Abstract

We study solitons (strictly speaking, solitary waves) in a model of two linearly coupled waveguides with the Kerr nonlinearity and resonant gratings, neglecting the material dispersion. An effective dispersion is induced by the linear couplings between the forward and backward Bragg-scattered waves and between the two cores. First, we consider a transition from the obvious symmetric solitons to nontrivial asymmetric ones for quiescent (standing) solitons. The solutions are found in an approximate analytical form by means of the variational approximation and, independently, by direct finite-difference numerical simulations. Results produced by the two methods are in a fairly good agreement. We further establish the stability of the asymmetric solitons by direct simulations, while showing that the symmetric solitons coexisting with the asymmetric ones are always unstable. Next, we consider traveling solitary waves. We fix the frequency detuning, while the strength of the coupling between the two cores and the velocity of the moving soliton are varied. In this case the solutions are found only by direct numerical methods, revealing that moving asymmetric solitons exist and are stable. Similar to the case of the quiescent solitary waves, the symmetric solitons coexisting with the asymmetric ones prove to be always unstable.

© 1998 Optical Society of America

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1997 (5)

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, J. Lightwave Technol. 15, 1303 (1997).
[CrossRef]

H. Hatami-Hanza, P. L. Chu, B. A. Malomed, and G. D. Peng, Opt. Commun. 134, 59 (1997).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, J. Lightwave Technol. 15, 1494 (1997).
[CrossRef]

B. J. Eggleton and C. M. de Sterke, J. Opt. Soc. Am. B 14, 2980 (1997).
[CrossRef]

B. J. Eggleton and C. M. de Sterke, J. Opt. Soc. Am. B 14, 2980 (1997).
[CrossRef]

1996 (4)

B. A. Malomed, I. M. Skinner, P. L. Chu, and G. D. Peng, Phys. Rev. E 53, 4084 (1996).
[CrossRef]

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, Opt. Fiber Technol. 2, 253 (1996).
[CrossRef]

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, IEEE Photonics Technol. Lett. 8, 944 (1996).
[CrossRef]

1995 (4)

1994 (4)

C. M. de Sterke and J. E. Sipe, Prog. Opt. 33, 203 (1994).
[CrossRef]

B. A. Malomed and R. S. Tasgal, Phys. Rev. E 49, 5787 (1994).
[CrossRef]

M. J. Steel and C. M. de Sterke, Phys. Rev. A 49, 5048 (1994).
[CrossRef] [PubMed]

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, Phys. Rev. A 50, 3447 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (4)

C. M. de Sterke, Opt. Lett. 17, 914 (1992).
[CrossRef] [PubMed]

A. B. Aceves, C. De Angelis, and S. Wabnitz, Opt. Lett. 17, 1566 (1992).
[CrossRef] [PubMed]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[CrossRef]

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuus, and J. M. Glinsky, Appl. Phys. Lett. 60, 404 (1992).
[CrossRef]

1991 (2)

P. St. J. Russel, J. Mod. Opt. 38, 1599 (1991).
[CrossRef]

H. G. Winful, R. Zamir, and S. Feldman, Appl. Phys. Lett. 58, 1001 (1991).
[CrossRef]

1990 (5)

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 42, 2858 (1990).
[CrossRef] [PubMed]

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, Electron. Lett. 26, 1459 (1990).
[CrossRef]

C. M. Ragdale, D. Reid, and I. Bennion, in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE 1171, 148 (1990).
[CrossRef]

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 42, 550 (1990).
[CrossRef]

J. E. Rothenberg, Opt. Lett. 15, 495 (1990).
[CrossRef] [PubMed]

1989 (2)

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

A. B. Aceves, and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

1988 (3)

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
[CrossRef] [PubMed]

J. E. Sipe and H. G. Winful, Opt. Lett. 13, 132 (1988).
[CrossRef]

B. Jaskorzynska and D. Schadt, IEEE J. Quantum Electron. 24, 2117 (1988).
[CrossRef]

1987 (2)

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

F. Ouellette, Opt. Lett. 12, 847 (1987).
[CrossRef] [PubMed]

1986 (1)

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, Appl. Phys. Lett. 29, 236 (1986).
[CrossRef]

1985 (1)

H. G. Winful, Appl. Phys. Lett. 46, 527 (1985).
[CrossRef]

1982 (1)

H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
[CrossRef]

1979 (1)

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

1977 (1)

D. J. Kaup and A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).
[CrossRef]

1972 (1)

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

1958 (1)

W. E. Thirring, Ann. Phys. (N.Y.) 3, 91 (1958).
[CrossRef]

Ann. Phys. (N.Y.) (1)

W. E. Thirring, Ann. Phys. (N.Y.) 3, 91 (1958).
[CrossRef]

Appl. Phys. Lett. (7)

H. G. Winful, Appl. Phys. Lett. 46, 527 (1985).
[CrossRef]

H. G. Winful and G. D. Cooperman, Appl. Phys. Lett. 40, 298 (1982).
[CrossRef]

H. G. Winful, R. Zamir, and S. Feldman, Appl. Phys. Lett. 58, 1001 (1991).
[CrossRef]

H. G. Winful, J. H. Marburger, and E. Garmire, Appl. Phys. Lett. 35, 379 (1979).
[CrossRef]

N. D. Sankey, D. F. Prelewitz, and T. G. Brown, Appl. Phys. Lett. 60, 1427 (1992).
[CrossRef]

M. Cada, J. He, B. Acklin, M. Proctor, D. Martin, F. Morier-Genoud, M. A. Dupertuus, and J. M. Glinsky, Appl. Phys. Lett. 60, 404 (1992).
[CrossRef]

K. Tai, A. Tomita, J. L. Jewell, and A. Hasegawa, Appl. Phys. Lett. 29, 236 (1986).
[CrossRef]

Electron. Lett. (2)

S. LaRochelle, Y. Hibino, V. Mizrahi, and G. I. Stegeman, Electron. Lett. 26, 1459 (1990).
[CrossRef]

P. A. Krug, T. Stephens, G. Dhosi, G. Yoffe, F. Ouellette, and P. Hill, Electron. Lett. 31, 1091 (1995).
[CrossRef]

IEEE J. Quantum Electron. (1)

B. Jaskorzynska and D. Schadt, IEEE J. Quantum Electron. 24, 2117 (1988).
[CrossRef]

IEEE Photonics Technol. Lett. (1)

W. H. Loh, R. I. Laming, N. Robinson, A. Cavaciuti, F. Vaninetti, C. J. Anderson, M. N. Zervas, and M. J. Cole, IEEE Photonics Technol. Lett. 8, 944 (1996).
[CrossRef]

J. Lightwave Technol. (2)

N. M. Litchinitser, B. J. Eggleton, and D. B. Patterson, J. Lightwave Technol. 15, 1303 (1997).
[CrossRef]

C. M. de Sterke, B. J. Eggleton, and P. A. Krug, J. Lightwave Technol. 15, 1494 (1997).
[CrossRef]

J. Mod. Opt. (1)

P. St. J. Russel, J. Mod. Opt. 38, 1599 (1991).
[CrossRef]

J. Opt. Soc. Am. B (3)

Lett. Nuovo Cimento (1)

D. J. Kaup and A. C. Newell, Lett. Nuovo Cimento 20, 325 (1977).
[CrossRef]

Opt. Commun. (1)

H. Hatami-Hanza, P. L. Chu, B. A. Malomed, and G. D. Peng, Opt. Commun. 134, 59 (1997).
[CrossRef]

Opt. Fiber Technol. (1)

C. M. de Sterke, N. G. R. Broderick, B. J. Eggleton, and M. J. Steel, Opt. Fiber Technol. 2, 253 (1996).
[CrossRef]

Opt. Lett. (7)

Phys. Lett. A (1)

A. B. Aceves, and S. Wabnitz, Phys. Lett. A 141, 37 (1989).
[CrossRef]

Phys. Rev. A (5)

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 38, 5149 (1988).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 42, 550 (1990).
[CrossRef]

M. J. Steel and C. M. de Sterke, Phys. Rev. A 49, 5048 (1994).
[CrossRef] [PubMed]

M. J. Steel, D. G. A. Jackson, and C. M. de Sterke, Phys. Rev. A 50, 3447 (1994).
[CrossRef] [PubMed]

C. M. de Sterke and J. E. Sipe, Phys. Rev. A 42, 2858 (1990).
[CrossRef] [PubMed]

Phys. Rev. E (2)

B. A. Malomed and R. S. Tasgal, Phys. Rev. E 49, 5787 (1994).
[CrossRef]

B. A. Malomed, I. M. Skinner, P. L. Chu, and G. D. Peng, Phys. Rev. E 53, 4084 (1996).
[CrossRef]

Phys. Rev. Lett. (4)

B. J. Eggleton, R. E. Slusher, C. M. de Sterke, P. A. Krug, and J. E. Sipe, Phys. Rev. Lett. 76, 1627 (1996).
[CrossRef] [PubMed]

W. Chen and D. L. Mills, Phys. Rev. Lett. 58, 160 (1987).
[CrossRef] [PubMed]

D. N. Christodoulides and R. I. Joseph, Phys. Rev. Lett. 62, 1746 (1989).
[CrossRef] [PubMed]

A. Kozhekin and G. Kurizki, Phys. Rev. Lett. 74, 5020 (1995).
[CrossRef] [PubMed]

Proc. SPIE (1)

C. M. Ragdale, D. Reid, and I. Bennion, in Fiber Laser Sources and Amplifiers, M. J. Digonnet, ed., Proc. SPIE 1171, 148 (1990).
[CrossRef]

Prog. Opt. (1)

C. M. de Sterke and J. E. Sipe, Prog. Opt. 33, 203 (1994).
[CrossRef]

Sov. Phys. JETP (1)

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

Other (10)

J. M. Bilbault and M. Remoissenet, J. Appl. Phys. 70, 4544 (1990); C. Martijn de Sterke and J. E. Sipe, Phys. Rev. A 43, 2467 (1990); S. John and N. Aközbek, Phys. Rev. Lett. PRLTAO 71, 1168 (1993); Yu. S. Kivshar, Phys. Rev. Lett. PRLTAO 70, 3055 (1993); N. G. Broderick and C. M. de Sterke, Phys. Rev. E PLEEE8 52, 4458 (1995).
[CrossRef] [PubMed]

S. M. Jensen, IEEE J. Quantum. Electron. QE-18, 1580 (1982); A. M. Maier, Kvant. Elektron. (Moscow) 9, 2996 (1982).
[CrossRef]

P. L. Chu, B. A. Malomed, and G. D. Peng, J. Opt. Soc. Am. B 10, 1379 (1993); N. N. Akhmediev and A. A. Ankiewicz, Phys. Rev. Lett. 70, 2395 (1993).
[CrossRef] [PubMed]

Y. Silberberg and I. Bar-Joseph, Phys. Rev. Lett. 48, 1541 (1982); A. Mecozzi, S. Trillo, and S. Wabnitz, Opt. Lett. 12, 1008 (1987).
[CrossRef] [PubMed]

Yu. K. Kishar, Phys. Rev. E 51, 1613 (1995); M. Picciau, G. Leo, and G. Assanto, J. Opt. Soc. Am. B 13, 661 (1996); S. Trillo, Opt. Lett. OPLEDP 21, 1732 (1996).
[CrossRef] [PubMed]

T. Peschel, U. Peschel, F. Lederer, and B. A. Malomed, Phys. Rev. E 55, 4730 (1997); C. Conti, S. Trillo, and G. Assanto, Phys. Rev. Lett. 78, 2341 (1997); H. He and P. D. Drummond, Phys. Rev. Lett. PRLTAO 78, 4311 (1997).
[CrossRef]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, San Diego, 1989).

B. J. Eggleton, T. Stephens, P. A. Krug, G. Dhosi, Z. Brodzeli, and F. Ouellette, in Optical Fiber Communication Conference, Vol. 2 of 1996 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1996), postdeadline paper PD5.

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, Boston, 1991), Chap. 3.

A. W. Snyder and J. D. Love, Optical Waveguide Theory, 1st ed. (Chapman & Hall, New York, 1983), Chaps. 27–29.

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Figures (8)

Fig. 1
Fig. 1

Schematic diagram of one implementation of the model using nonlinear dual-core fibers with gratings.

Fig. 2
Fig. 2

Bifurcation diagram for the quiescent soliton at ω=0.5. The solid and dashed curves depict, respectively, the numerical results and the variational approximation. The dot-dashed line is the branch corresponding to the symmetric solutions. The branch at Θ<0, which is a mirror image of the curve shown, is not displayed here.

Fig. 3
Fig. 3

Shapes of the larger of the two quiescent-soliton components in the two cores. The upper and lower graphs show the real and imaginary parts of the forward-propagating wave. As in Fig. 2, the solid and dashed curves represent the numerical and variational results. Here, ω=0.5 and λ=0.2.

Fig. 4
Fig. 4

Three-dimensional bifurcation diagram for the quiescent solitary wave, showing the effective soliton’s asymmetry Θ as a function of the frequency ω and coupling constant λ.

Fig. 5
Fig. 5

Set of the dispersion curves for the linearized equations (6)–(9) at three different values of the coupling constant: λ=0.02, λ=0.5, and λ=0.99.

Fig. 6
Fig. 6

Three-dimensional bifurcation diagram for the moving soliton, showing the asymmetry Θ versus the velocity c and the coupling constant λ.

Fig. 7
Fig. 7

Evolution of slightly perturbed symmetric (upper) and asymmetric (lower) quiescent solitons at ω=0.8 and λ=0.16.

Fig. 8
Fig. 8

Evolution of a slightly perturbed moving asymmetric soliton. Here, ω=0.5, λ=0.04, and c=0.8.

Equations (41)

Equations on this page are rendered with MathJax. Learn more.

E1(z, τ)=E1+(z, τ)exp[-i(ω0τ-k0z)]+E1-(z, τ)exp[-i(ω0τ+k0z)]+c.c.,
E2(z, τ)=E2+(z, τ)exp[-i(ω0τ-k0z)]+E2-(z, τ)exp[-i(ω0τ+k0z)]+c.c.
ω0=πc0nd¯,k0=πd,
+i E1+z+n¯c0 E1+τ+κE1-+Γ|E1+|2E1+
+2Γ|E1-|2E1++KE2+=0,
-i E1-z+n¯c0 E1-τ+κE1++Γ|E1-|2E1-
+2Γ|E1+|2E1-+KE2-=0,
+i E2+z+n¯c0 E2+τ+κE2-+Γ|E2+|2E2+
+2Γ|E2-|2E2++KE1+=0,
-i E2-z+n¯c0 E2-τ+κE2++Γ|E2-|2E2-
+2Γ|E2+|2E2-+KE1-=0.
E1,2+κ2Γ U1,2,E1,2-κ2Γ V1,2,
zxκ,τn¯κc0 t.
iU1t+iU1x+(σ|U1|2+|V1|2)U1+V1+λU2=0,
iV1t-iV1x+(σ|V1|2+|U1|2)V1+U1+λV2=0,
iU2t+iU2x+(σ|U2|2+|V2|2)U2+V2+λU1=0,
iV2t-iV2x+(σ|V2|2+|U2|2)V2+U2+λV1=0.
U1,2=exp(-iωt)u1,2(ξ),V1,2=exp(-iωt)v1,2(ξ),
ωu1+i(1-c)u1+(σ|u1|2+|v1|2)u1+v1+λu2=0,
ωv1-i(1+c)v1+(σ|v1|2+|u1|2)v1+u1+λv2=0,
ωu2+i(1-c)u2+(σ|u2|2+|v2|2)u2+v2+λu1=0,
ωv2-i(1+c)v2+(σ|v2|2+|u2|2)v2+u2+λv1=0.
v1,2=-u1,2*.
ωu1+iu1+(σ+1)|u1|2u1-u1*+λu2=0,
ωu2+iu2+(σ+1)|u2|2u2-u2*+λu1=0.
L=ω(u1u1*+u2u2*)+i2 (u1xu1*-u1x*u1)+i2 (u2xu2*-u2x*u2)+12 (σ+1)(|u1|4+|u2|4)-12 (u12+u1*2+u22+u2*2)+λ(u1u2*+u1*u2).
u1=A1 sech(μx)+iB1 sinh(μx)sech2(μx),
u2=A2 sech(μx)+iB2 sinh(μx)sech2(μx),
L-+Ldx=2ωμ (A12+A22)+23μ ω(B12+B22)-43 (A1B1+A2B2)+2(σ+1)3μ (A14+A24)-0.8571 σ+1μ (B14+B24)+4(σ+1)15μ (A12B12+A22B22)-2μ (A12+A22)+23μ (B12+B22)+4λμ A1A2+4λ3μ B1B2
λA2-A1+23 (σ+1)A13+215 (σ+1)A1B12-13 μB1
+ωA1=0,
λA1-A2+23 (σ+1)A23+215 (σ+1)A2B22-13 μB2
+ωA2=0,
13 λB2+13 B1-0.8571(σ+1)B13+215 (σ+1)A12B1
-13 μA1+13 ωB1=0,
13 λB1+13 B2-0.8571(σ+1)B23+215 (σ+1)A22B2
-13 μA2+13 ωB2=0,
2ω(A12+A22)+2ω3 (B12+B22)+2(σ+1)3 (A14+A24)-0.8571(σ+1)(B14+B24)+4(σ+1)15×(A12B12+A22B22)-2(A12+A22)+23 (B12+B22)
+4λA1A2+4λ3 B1B2=0.
Θ(u1m2-u2m2)/(u1m2+u2m2)
ω2=λ2+1+k2±2λ1+k2,

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