Abstract

Dynamics of the gap 2π pulse dynamics in one-dimensional resonantly absorbing Bragg gratings are studied. A new family of stable oscillating and excited unstable gap 2π pulses is analytically and numerically described by a transition from the two-wave Maxwell–Bloch equation to the modified sine-Gordon equation and by direct integration of the two-wave Maxwell–Bloch equation.

© 2002 Optical Society of America

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References

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  1. C. M. Bowden, M. Bertolotti, and C. Sibilia, eds., Linear and Nonlinear Nanoscale Optics (American Institute of Physics, New York, 2001).
  2. B. I. Mantsyzov and R. N. Kuzmin, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP 64, 37–44 (1986).
  3. W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
    [CrossRef] [PubMed]
  4. C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
    [CrossRef]
  5. A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. 81, 3647–3650 (1998).
    [CrossRef]
  6. C. Conti, G. Assanto, and S. Trillo, “Self-sustained trapping mechanism of zero-velocity parametric gap-solitons,” Phys. Rev. E 59, 2467–2470 (1999).
    [CrossRef]
  7. B. I. Mantsyzov, “Gap 2π-pulse with an inhomogeneously broadened line and an oscillating solitary wave,” Phys. Rev. A 51, 4939–4943 (1995).
    [CrossRef] [PubMed]
  8. F. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
    [CrossRef]
  9. J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
    [CrossRef]
  10. A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
    [CrossRef]
  11. M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
    [CrossRef]
  12. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 16.

2000 (1)

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

1999 (2)

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

C. Conti, G. Assanto, and S. Trillo, “Self-sustained trapping mechanism of zero-velocity parametric gap-solitons,” Phys. Rev. E 59, 2467–2470 (1999).
[CrossRef]

1998 (2)

A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. 81, 3647–3650 (1998).
[CrossRef]

F. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

1997 (1)

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

1995 (1)

B. I. Mantsyzov, “Gap 2π-pulse with an inhomogeneously broadened line and an oscillating solitary wave,” Phys. Rev. A 51, 4939–4943 (1995).
[CrossRef] [PubMed]

1987 (1)

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

1986 (1)

B. I. Mantsyzov and R. N. Kuzmin, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP 64, 37–44 (1986).

1977 (1)

M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
[CrossRef]

Assanto, G.

C. Conti, G. Assanto, and S. Trillo, “Self-sustained trapping mechanism of zero-velocity parametric gap-solitons,” Phys. Rev. E 59, 2467–2470 (1999).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

Binder, R.

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Bishop, A. R.

M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
[CrossRef]

Chen, W.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Conti, C.

C. Conti, G. Assanto, and S. Trillo, “Self-sustained trapping mechanism of zero-velocity parametric gap-solitons,” Phys. Rev. E 59, 2467–2470 (1999).
[CrossRef]

F. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

De Rossi, F.

F. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

Donovan, M. E.

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Ell, C.

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

Fogel, M. B.

M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
[CrossRef]

Gibbs, H. M.

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Khitrova, G.

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Koch, S. W.

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

Kozhekin, A. E.

A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. 81, 3647–3650 (1998).
[CrossRef]

Krumhansl, J. A.

M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
[CrossRef]

Kurizki, G.

A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. 81, 3647–3650 (1998).
[CrossRef]

Kuzmin, R. N.

B. I. Mantsyzov and R. N. Kuzmin, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP 64, 37–44 (1986).

Lee, E. S.

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

Lindberg, M.

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Malomed, B.

A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. 81, 3647–3650 (1998).
[CrossRef]

Mantsyzov, B. I.

B. I. Mantsyzov, “Gap 2π-pulse with an inhomogeneously broadened line and an oscillating solitary wave,” Phys. Rev. A 51, 4939–4943 (1995).
[CrossRef] [PubMed]

B. I. Mantsyzov and R. N. Kuzmin, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP 64, 37–44 (1986).

Mills, D. L.

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

Peyghambarian, N.

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Prineas, J. P.

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

Schulzgen, A.

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Trillo, S.

C. Conti, G. Assanto, and S. Trillo, “Self-sustained trapping mechanism of zero-velocity parametric gap-solitons,” Phys. Rev. E 59, 2467–2470 (1999).
[CrossRef]

F. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

Trullinger, S. E.

M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
[CrossRef]

Wundke, K.

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

Phys. Rev. A (1)

B. I. Mantsyzov, “Gap 2π-pulse with an inhomogeneously broadened line and an oscillating solitary wave,” Phys. Rev. A 51, 4939–4943 (1995).
[CrossRef] [PubMed]

Phys. Rev. B (2)

J. P. Prineas, C. Ell, E. S. Lee, G. Khitrova, H. M. Gibbs, and S. W. Koch, “Exciton–polariton eigenmodes in light-coupled In0.04Ga0.96As/GaAs semiconductor multiple-quantum-well periodic structure,” Phys. Rev. B 61, 13 863–13 872 (2000).
[CrossRef]

M. B. Fogel, S. E. Trullinger, A. R. Bishop, and J. A. Krumhansl, “Dynamics of sine-Gordon solitons in the presence of perturbations,” Phys. Rev. B 15, 1578–1592 (1977).
[CrossRef]

Phys. Rev. E (1)

C. Conti, G. Assanto, and S. Trillo, “Self-sustained trapping mechanism of zero-velocity parametric gap-solitons,” Phys. Rev. E 59, 2467–2470 (1999).
[CrossRef]

Phys. Rev. Lett. (5)

A. Schulzgen, R. Binder, M. E. Donovan, M. Lindberg, K. Wundke, H. M. Gibbs, G. Khitrova, and N. Peyghambarian, “Direct observation of excitonic Rabi oscillations in semiconductors,” Phys. Rev. Lett. 82, 2346–2349 (1999).
[CrossRef]

F. De Rossi, C. Conti, and S. Trillo, “Stability, multistability, and wobbling of optical gap solitons,” Phys. Rev. Lett. 81, 85–88 (1998).
[CrossRef]

W. Chen and D. L. Mills, “Gap solitons and the nonlinear optical response of superlattices,” Phys. Rev. Lett. 58, 160–163 (1987).
[CrossRef] [PubMed]

C. Conti, S. Trillo, and G. Assanto, “Doubly resonant Bragg simultons via second-harmonic generation,” Phys. Rev. Lett. 78, 2341–2344 (1997).
[CrossRef]

A. E. Kozhekin, G. Kurizki, and B. Malomed, “Standing and moving gap solitons in resonantly absorbing gratings,” Phys. Rev. Lett. 81, 3647–3650 (1998).
[CrossRef]

Sov. Phys. JETP (1)

B. I. Mantsyzov and R. N. Kuzmin, “Coherent interaction of light with a discrete periodic resonant medium,” Sov. Phys. JETP 64, 37–44 (1986).

Other (2)

C. M. Bowden, M. Bertolotti, and C. Sibilia, eds., Linear and Nonlinear Nanoscale Optics (American Institute of Physics, New York, 2001).

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (U.S. Government Printing Office, Washington, D.C., 1964), Chap. 16.

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Figures (3)

Fig. 1
Fig. 1

Phase plane of Eq. (11) for (a) the attractive potential U at f0=-0.1 and (b) the repulsive potential at f0=0.1.

Fig. 2
Fig. 2

Evolution of the initial gap 2π pulse (the gray scale is proportional to n). The initial conditions at t=0 are fixed by n=-cos θ, P=-sin θ, θ=4 tan-1 exp 2(-x+x0)/1-u02, and Ω±=Ω0± sech2(-x+x0)/1-u02, where u0 is the initial pulse velocity. (a) Ω0+=1.45 and Ω0-=-0.87 correspond to f0=-0.4 and u0=0.2; (b) Ω0+=2.34, Ω0-=-0.48; f0=-0.4, and u0=0.55; (c) f0=0.07 and u00. The contour lines correspond to the perturbation f(x, t) calculated from Eq. (3) for the initial condition f(x)=2 f0sech2(-x+x0)/1-u02. Inset, square of the frequency of pulse harmonic oscillations as a function of perturbation obtained by analytical calculation from Eq. (14) (solid curve) and by numerical integration of Eqs. (1) (dashed curve).

Fig. 3
Fig. 3

Evolution of the incident pulse in the structure (the gray scale is proportional to n). Pulse duration, τ0=0.84; amplitudes Ω0 are (a) 2.701, (b) 2.70063, and (c) 2.70062. The contour lines in (c) show the perturbation f(x, t) calculated from Eq. (3). Inset, dependence of time delay τD on the depth of pulse penetration X.

Equations (31)

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Ωt++Ωx+=P,
Ωt--Ωx-=P,
Pt=n(Ω++Ω-),
nt=-P(Ω++Ω-),
Ω˜x+Ωt=-2 sin θ,
Ωx+Ω˜t=0,
θt=Ω,
Ω˜(x, t)=-θx(x, t)+f(x).
θxx-θtt=2 sin θ+fx(x).
f(x)=Ω˜(x, 0)+θx(x, 0).
θηη-θττ=sin θ+fη(η).
L=½θτ2-½(θη-f)2-(1-cos θ),
H=½θτ2+½θη2-fθη+½ f2+(1-cos θ).
ddτ -dη12θτ2+12θη2+(1-cos θ)
=ddτ -dηfθη.
θ=4 tan-1exp-η+ξ(τ)[1-u2(τ)]1/2,
ξττ=-14 ξ - sech(η-ξ)f(η)dη.
ξττ=-Uξ,U=f02 ξsinh ξ.
Uin=14 - sech(η-ξ)f(η)dη.
0ξ dξ(α-f0ξ/sh ξ)1/2=τ,
ξ=ξ0 sin ωτ,ω2=-f0/6.
ξττ-f06ξ+7f0180ξ3=0.
Ω+=1/2(Ω+Ω˜)=1/2[θt-θx+f(x)],
Ω-=1/2(Ω-Ω˜)=1/2[θt+θx-f(x)],
Ω+=(ξt+2)sech[2x-ξ(t)]+f(x)/2,
Ω-=(ξt-2)sech[2x-ξ(t)]-f(x)/2,
n=-cos θ=-1+2 sech2[2x-ξ(t)],
P=-2 sech[2x-ξ(t)]tanh[2x-ξ(t)].
Ω+(x=0, t)=Ω0 sech[(t-t0)/τ0],
Ω-(x=l, t)=0,Ω±(x, t=0)=0,
n(x, t=0)=-1,P(x, t=0)=0,

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