Abstract

A nonlinear, coupled, magneto-optic waveguide system supporting bright and dark spatial solitons is investigated to determine the stability and possible application of the resulting bright–dark soliton dynamics. An extensive variational analysis is verified by full numerical simulations. It is discovered that the stability of the magneto-optic coupled states can be exploited to give guaranteed isolator functionality, which could lead to very robust component protection in monolithic integrated circuits.

© 2005 Optical Society of America

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  1. S. M. Jensen, "The nonlinear coherent coupler," IEEE J. Quantum Electron. 18, 1580-1583 (1982).
    [CrossRef]
  2. N. Finlayson and G. I. Stegeman, "Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler," Appl. Phys. Lett. 56, 2276-2278 (1990).
    [CrossRef]
  3. F. Dios, X. Nogues, and F. Canal, "Critical power in a symmetric nonlinear directional coupler," Opt. Quantum Electron. 24, 1191-1201 (1992).
    [CrossRef]
  4. D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).
  5. A. D. Boardman and M. Xie, "Spatial solitons in discontinuous magneto-optic waveguides," J. Opt. B Quantum Semiclassical Opt. 3, S244-S250 (2001).
    [CrossRef]
  6. A. D. Boardman and M. Xie, "Vector spatial solitons in complex magneto-optic waveguides," J. Opt. Soc. Am. B 19, 563-572 (2002).
    [CrossRef]
  7. A. D. Boardman and M. Xie, "Spatial solitons in modulatedmagneto-optic waveguides," in Soliton-Driven Photonics , A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, London, 2001).
  8. A. D. Boardman, M. Xie, and K. Xie, "Surface magneto-optic solitons," J. Phys. D 36, 2211-2217 (2003).
    [CrossRef]
  9. A. D. Boardman, P. Egan, T. Twardowski, and M. Wilkins, "Nonlinear surface-guided waves in self-focusing optical media," in Nonlinear Waves in Solid State Physics , A. D. Boardman, M. Bertolotti, and T. Twardowski, eds. (Plenum, New York, 1991).
  10. G. I. Stegeman, "Introduction to monlinear optics: a selected overview," in Beam Shaping and Control with Nonlinear Optics , F. Kajzar and R. Reinisch, eds. (Plenum, New York, 1997).
  11. K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
    [CrossRef]
  12. S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
    [CrossRef]
  13. Takenori Jekijima, Takashi Fujii, and Kikuo Wakino, "Optical Faraday rotator using Ce-substituted fibrous YIG single crystal grown by floating-zone method with YAG laser heating," IEEE Trans. Microwave Theory Tech. 47, 2294-2298 (1999).
    [CrossRef]
  14. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).
  15. A. D. Boardman and K. Xie, "Spatial bright-dark soliton steering through waveguide coupling," Opt. Quantum Electron. 30, 783-794 (1998).
    [CrossRef]

2003 (1)

A. D. Boardman, M. Xie, and K. Xie, "Surface magneto-optic solitons," J. Phys. D 36, 2211-2217 (2003).
[CrossRef]

2002 (1)

2001 (1)

A. D. Boardman and M. Xie, "Spatial solitons in discontinuous magneto-optic waveguides," J. Opt. B Quantum Semiclassical Opt. 3, S244-S250 (2001).
[CrossRef]

2000 (1)

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

1999 (1)

Takenori Jekijima, Takashi Fujii, and Kikuo Wakino, "Optical Faraday rotator using Ce-substituted fibrous YIG single crystal grown by floating-zone method with YAG laser heating," IEEE Trans. Microwave Theory Tech. 47, 2294-2298 (1999).
[CrossRef]

1998 (1)

A. D. Boardman and K. Xie, "Spatial bright-dark soliton steering through waveguide coupling," Opt. Quantum Electron. 30, 783-794 (1998).
[CrossRef]

1993 (1)

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

1992 (1)

F. Dios, X. Nogues, and F. Canal, "Critical power in a symmetric nonlinear directional coupler," Opt. Quantum Electron. 24, 1191-1201 (1992).
[CrossRef]

1990 (1)

N. Finlayson and G. I. Stegeman, "Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler," Appl. Phys. Lett. 56, 2276-2278 (1990).
[CrossRef]

1982 (1)

S. M. Jensen, "The nonlinear coherent coupler," IEEE J. Quantum Electron. 18, 1580-1583 (1982).
[CrossRef]

Boardman, A. D.

A. D. Boardman, M. Xie, and K. Xie, "Surface magneto-optic solitons," J. Phys. D 36, 2211-2217 (2003).
[CrossRef]

Boardman , A. D.

A. D. Boardman and M. Xie, "Vector spatial solitons in complex magneto-optic waveguides," J. Opt. Soc. Am. B 19, 563-572 (2002).
[CrossRef]

A. D. Boardman and M. Xie, "Spatial solitons in discontinuous magneto-optic waveguides," J. Opt. B Quantum Semiclassical Opt. 3, S244-S250 (2001).
[CrossRef]

A. D. Boardman and K. Xie, "Spatial bright-dark soliton steering through waveguide coupling," Opt. Quantum Electron. 30, 783-794 (1998).
[CrossRef]

Canal, F.

F. Dios, X. Nogues, and F. Canal, "Critical power in a symmetric nonlinear directional coupler," Opt. Quantum Electron. 24, 1191-1201 (1992).
[CrossRef]

Darwish, A. M.

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

Dios, F.

F. Dios, X. Nogues, and F. Canal, "Critical power in a symmetric nonlinear directional coupler," Opt. Quantum Electron. 24, 1191-1201 (1992).
[CrossRef]

Finlayson , N.

N. Finlayson and G. I. Stegeman, "Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler," Appl. Phys. Lett. 56, 2276-2278 (1990).
[CrossRef]

Fujii, Takashi

Takenori Jekijima, Takashi Fujii, and Kikuo Wakino, "Optical Faraday rotator using Ce-substituted fibrous YIG single crystal grown by floating-zone method with YAG laser heating," IEEE Trans. Microwave Theory Tech. 47, 2294-2298 (1999).
[CrossRef]

Hall, K. L.

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

Haugen, H. K.

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

Ippen, E. P.

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

Jekijima, Takenori

Takenori Jekijima, Takashi Fujii, and Kikuo Wakino, "Optical Faraday rotator using Ce-substituted fibrous YIG single crystal grown by floating-zone method with YAG laser heating," IEEE Trans. Microwave Theory Tech. 47, 2294-2298 (1999).
[CrossRef]

Jensen, S. M.

S. M. Jensen, "The nonlinear coherent coupler," IEEE J. Quantum Electron. 18, 1580-1583 (1982).
[CrossRef]

Koren, U.

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

Mascher, P.

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

Nogues, X.

F. Dios, X. Nogues, and F. Canal, "Critical power in a symmetric nonlinear directional coupler," Opt. Quantum Electron. 24, 1191-1201 (1992).
[CrossRef]

Raybon, G.

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

Robinson, B. J.

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

Stegeman, G. I.

N. Finlayson and G. I. Stegeman, "Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler," Appl. Phys. Lett. 56, 2276-2278 (1990).
[CrossRef]

Thompson, D. A.

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

Wakino, Kikuo

Takenori Jekijima, Takashi Fujii, and Kikuo Wakino, "Optical Faraday rotator using Ce-substituted fibrous YIG single crystal grown by floating-zone method with YAG laser heating," IEEE Trans. Microwave Theory Tech. 47, 2294-2298 (1999).
[CrossRef]

Wallance, S. G.

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

Xie, K.

A. D. Boardman, M. Xie, and K. Xie, "Surface magneto-optic solitons," J. Phys. D 36, 2211-2217 (2003).
[CrossRef]

A. D. Boardman and K. Xie, "Spatial bright-dark soliton steering through waveguide coupling," Opt. Quantum Electron. 30, 783-794 (1998).
[CrossRef]

Xie, M.

A. D. Boardman, M. Xie, and K. Xie, "Surface magneto-optic solitons," J. Phys. D 36, 2211-2217 (2003).
[CrossRef]

A. D. Boardman and M. Xie, "Vector spatial solitons in complex magneto-optic waveguides," J. Opt. Soc. Am. B 19, 563-572 (2002).
[CrossRef]

A. D. Boardman and M. Xie, "Spatial solitons in discontinuous magneto-optic waveguides," J. Opt. B Quantum Semiclassical Opt. 3, S244-S250 (2001).
[CrossRef]

Appl. Phys. Lett. (3)

K. L. Hall, A. M. Darwish, E. P. Ippen, U. Koren, and G. Raybon, "Femtosecond index nonlinearities in InGaAsP optical amplifiers," Appl. Phys. Lett. 62, 1320-1322 (1993).
[CrossRef]

S. G. Wallance, B. J. Robinson, P. Mascher, H. K. Haugen, and D. A. Thompson, "Refractive indices of InGaAsP lattice-matched to GaAs at wavelength relevant to device design," Appl. Phys. Lett. 76, 2791-2793 (2000).
[CrossRef]

N. Finlayson and G. I. Stegeman, "Spatial switching, instabilities, and chaos in a three-waveguide nonlinear directional coupler," Appl. Phys. Lett. 56, 2276-2278 (1990).
[CrossRef]

IEEE J. Quantum Electron. (1)

S. M. Jensen, "The nonlinear coherent coupler," IEEE J. Quantum Electron. 18, 1580-1583 (1982).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

Takenori Jekijima, Takashi Fujii, and Kikuo Wakino, "Optical Faraday rotator using Ce-substituted fibrous YIG single crystal grown by floating-zone method with YAG laser heating," IEEE Trans. Microwave Theory Tech. 47, 2294-2298 (1999).
[CrossRef]

J. Opt. B Quantum Semiclassical Opt. (1)

A. D. Boardman and M. Xie, "Spatial solitons in discontinuous magneto-optic waveguides," J. Opt. B Quantum Semiclassical Opt. 3, S244-S250 (2001).
[CrossRef]

J. Opt. Soc. Am. B (1)

J. Phys. D (1)

A. D. Boardman, M. Xie, and K. Xie, "Surface magneto-optic solitons," J. Phys. D 36, 2211-2217 (2003).
[CrossRef]

Opt. Quantum Electron. (2)

A. D. Boardman and K. Xie, "Spatial bright-dark soliton steering through waveguide coupling," Opt. Quantum Electron. 30, 783-794 (1998).
[CrossRef]

F. Dios, X. Nogues, and F. Canal, "Critical power in a symmetric nonlinear directional coupler," Opt. Quantum Electron. 24, 1191-1201 (1992).
[CrossRef]

Other (5)

D. L. Lee, Electromagnetic Principles of Integrated Optics (Wiley, New York, 1986).

A. D. Boardman and M. Xie, "Spatial solitons in modulatedmagneto-optic waveguides," in Soliton-Driven Photonics , A. D. Boardman and A. P. Sukhorukov, eds. (Kluwer, London, 2001).

A. D. Boardman, P. Egan, T. Twardowski, and M. Wilkins, "Nonlinear surface-guided waves in self-focusing optical media," in Nonlinear Waves in Solid State Physics , A. D. Boardman, M. Bertolotti, and T. Twardowski, eds. (Plenum, New York, 1991).

G. I. Stegeman, "Introduction to monlinear optics: a selected overview," in Beam Shaping and Control with Nonlinear Optics , F. Kajzar and R. Reinisch, eds. (Plenum, New York, 1997).

G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974).

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Figures (9)

Fig. 1
Fig. 1

Sketch of a possible coupled waveguide system, together with a cross section that shows magneto-optical material at the top and bottom of the guide.

Fig. 2
Fig. 2

Simulation of potential function U(x1, x2) plotted against x1 and x2. Typical data: q12=-0.058, q21=-0.056, A1=0.09, A2=0.08.

Fig. 3
Fig. 3

Variation of A1 with A2 for sgn(q12)(m1+m2)=1.1418; the curves show the stability thresholds. Typical data: q12=-0.0350204, q21=-0.018, β1=2.47, β2=3.05.

Fig. 4
Fig. 4

Bright–dark soliton interaction for A1A2=0.1. The scales are in dimensionless units but typically a unit will be a Rayleigh length in millimeters along z and a beam width of micrometers along x.

Fig. 5
Fig. 5

Bright–dark soliton interaction for A1=-0.17, A2=0.15.

Fig. 6
Fig. 6

Variation of A1 with A2 for sgn(q12)(m1+m2)=-1.1418; the curves show the stability thresholds.

Fig. 7
Fig. 7

“On” state with stable propagation of an overlapping bright–dark soliton for A1=0.15, A2=0.15. Here the beams are traveling parallel to the current supplying the magnetic field.

Fig. 8
Fig. 8

“On” state with unstable propagation of a bright–dark soliton combination antiparallel to the applied current direction at A1=-0.15, A2=-0.15. This simulation shows the nonreciprocal role of the magnetic field.

Fig. 9
Fig. 9

“Off” state at A1=A2=0 with full protection of the receiver resulting from the bright and dark solitons following a curved path, even though they are separating.

Equations (39)

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2Ex2+2Ey2+2Ez2+ω2c2 is1·E=0,
is1(y)=3,y<01,0<y<d14,y>d1.
E=τ1(x, z)[ξy1(y)yˆ+ξz1(y)zˆ]exp[-i(ω/c)β1z]exp(iωt),
2Ex2+2Ey2+2Ez2+ω2c2is1·E+PM0+PNL0=0
is1=3,y<01,0<y<d14,d1<y<d1+d22,d1+d2<y<d1+d2+d35,y>d1+d2+d3.
E=(ξy1yˆ+ξz1zˆ)τ1exp[-i(ω/c)β1z]exp(iωt)+(ξy2yˆ+ξz2zˆ)τ2exp[-i(ω/c)β2z]exp(iωt),
PM=00ixyixz-ixy0iyz-ixz-iyz0×0τ1ξy1 exp-iωcβ1z+τ2ξy2 exp-iωcβ2zτ1ξz1 exp-iωcβ1z+τ2ξz2 exp-iωcβ2z
×exp(iωt).
PNL=PNL1,0<y<d1PNL2,d1+d2<y<d1+d2+d30,otherwise.
Px=(3/4)0[2χxxyy(|Ex|2+|Ey|2+|Ez|2)Ex+χxyyx(Ex2+Ey2+Ez2)Ex*],
χxxyy=χxxyy1,guide1χxxyy2,guide2,
χxyyx=χxyyx1,guide1χxyyx2,guide2.
2¯yz11=+i-yz(ξy1*ξz1-ξy1ξz1*)dy-(|ξy1|2+|ξz1|2)dy,
2¯yz12=-i-yz(ξy1*ξz2+ξz1*ξy2)dy-(|ξy1|2+|ξz1|2)dy,
χ¯h1=3/4-χxxyy[(|ξy1|2+|ξz1|2)(|ξy2|2+|ξz2|2)+|ξy1ξy2*+ξz1ξz2*|2]dy+-χxyyx|ξy1ξy2+ξz1ξz2|2dy-(|ξy1|2+|ξz1|2)dy.
D0 xx,2(β,β2)1/2ωcD02zz,
τ1*=cωD02ψ1χ¯e1=g1ψ1, τ2*=cωD02ψ2χ¯e2=g2ψ2,
λ1=β1β21/2,λ2=β2β11/2,q12=1/4g22χ¯h1,
q12q211/4ψ2ψ2,q21q121/4ψ1ψ1,
iβ1β21/2ψ1z+2ψ1x2-2q12q211/2|ψ1|2ψ1+4 sgn(q12)(q12q21)1/2|ψ2|2ψ1+ν1ψ1=0,
iβ2β11/2ψ2z+2ψ2x2+2q21q121/2|ψ2|2ψ2+4 sgn(q12)(q12q21)1/2|ψ1|2ψ2+ν2ψ2=0.
ν1=2ω2c2D02 ¯yz11,ν2=2ω2c2D02 ¯yz22.
ψ1=q21q121/4η1 tanh[η1(x-x1)]×expiξ12(x-x1)+iθ1,
ψ2=q12q211/4η2 sech[η2(x-x2)]×expiξ22(x-x2)+iθ2,
m1=-η1β1β2q21q12,m2=η2β2β1q12q21.
U(x1, x2)=-η12q21q121/2-ν1 tanh2×η1(x-x1)dx+η22q12q211/2-ν2 sech2×η2(x-x2)dx+L12+const.,
ν1=0,x<-wA1,-w<x<w0,x>w;
ν2=0,x<-wA2,-w<x<w0,x>w,
U(x1, x2)=-2η12q21q121/2A1w-η1q21q121/2A1×[tanhη1(w-x1)-tanhη1(-w-x1)]+η2q12q211/2A2[tanhη2(w-x2)-tanhη2(-w-x2)]+L12+const.
U(x1, x2)=-(2η2(q21/q12)1/2A1w-η(q21/q12)1/2A1[tanhη(w-x1)-tanhη(-w-x1)]+η(q12/q21)1/2A2[tanhη(w-x2)-tanhη(-w-x2)]+8 sgn(q12)(q12q21)1/2×η3{1-2 csch2[2η(x2-x1)]×2η(x2-x1)c tanh[2η(x2-x1)]-1})+const.
12m1dx1dz2+12m2dx2dz2+ηq21q121/2×A1{tanh[η(w-x1)]+tanh[η(w+x1)]}-ηq12q211/2A2{tanh[η(w-x2)]+tanh[η(w+x2)]}+4η3 sgn(q12)×(q12q21)1/24sinh2[η(x2-x1)]×η(x2-x1)tanh[η(x2-x1)]-1=C,
2U(x0)x02=-2η3q21q121/2A1 cos2 α{sech2[η(w-x0 cos α)]tanh[η(w-x0 cos α)]+sech2[η(w+x0 cos α)]×tanh[η(w+x0 cos α)]}+2η3q12q211/2A2 sin2 α{sech2[η(w-x0 sin α)]tanh[η(w-x0 sin α)]+sech2[η(w+x0 sin α)]×tanh[η(w+x0 sin α)]}-64η5 sgn(q12)(q12q21)1/2(sin α-cos α)2sech2[η(sin α-cos α)x0]tanh3[η(sin α-cos α)x0]×2η(sin α-cos α)x0-tanh[η(sin α-cos α)x0]-3η(sin α-cos α)x0tanh2[η(sin α-cos α)x0]+3tanh[η(sin α-cos α)x0].
α2U(x0)x02x0=0=42q21q121/2A1 cos α sin α+2q12q211/2A2 cos α sin α×η3 sech2(ηw)tanh(ηw)-12815 sgn(q12)(q12q21)1/2η5×(sin α-cos α)(sin α+cos α).
m2 tan2 α+A tan α-m1=0,
A=-1516η2 sgn(q12)sech2(ηw)tanh(ηw)×m1A2|q21|+m2A1|q12|+m1-m2.
F1=-4(q21/q12)1/2A1 cos αη3 sech2(ηw)tanh(ηw)+(64/15) sgn(q12)(q12q21)1/2η5(sin α-cos α),
F2=4(q12/q21)1/2A2 sin αη3 sech2(ηw)tanh(ηw)-(64/15) sgn(q12)(q12q21)1/2η5(sin α-cos α),
F2ηβ1β2q21q121/2dx1dz+F1ηβ2β1q12q211/2dx2dz=0,
-F2m1x1+F1m2x2=const.,

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