Abstract

Numerical studies, together with asymptotic and spectral analysis, establish regimes in which soliton pairs in degenerate optical parametric oscillators fuse, repel, or form bound states. A novel bound state that is stabilized by coupled internal oscillations is predicted.

© 1999 Optical Society of America

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References

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  1. N. N. Rosanov, Prog. Opt. 35, 1 (1996); M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994); W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).For a recent review of CS’s see W. J. Firth and G. K. Harkness, Asian J. Phys. 7, 665 (1998).
    [CrossRef] [PubMed]
  2. M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
    [CrossRef]
  3. K. Staliunas and V. J. Sánchez-Morcillo, Opt. Commun. 139, 306 (1997); S. Longhi, Phys. Scr. 56, 611 (1997); S. Trillo and M. Haelterman, Opt. Lett. 23, 1514 (1998).
    [CrossRef]
  4. C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).
    [CrossRef]
  5. C. Richy, K. I. Petsas, E. Giacobino, C. Fabre, and L. Lugiato, J. Opt. Soc. Am. B 12, 456 (1995); A. G. White, J. Mlynek, and S. Schiller, Europhys. Lett. 35, 425 (1996).
    [CrossRef]
  6. Various versions of this method can be found, e.g., in K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981); and in S. Longhi, Phys. Rev. E 55, 1060 (1997).
    [CrossRef]
  7. B. A. Malomed, Phys. Rev. A 47, 2874 (1993).
  8. G.-L. Oppp, A. J. Scroggie, and W. J. Firth, J. Opt. B 1, 133 (1999).
    [CrossRef]

1999 (1)

G.-L. Oppp, A. J. Scroggie, and W. J. Firth, J. Opt. B 1, 133 (1999).
[CrossRef]

1997 (3)

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

K. Staliunas and V. J. Sánchez-Morcillo, Opt. Commun. 139, 306 (1997); S. Longhi, Phys. Scr. 56, 611 (1997); S. Trillo and M. Haelterman, Opt. Lett. 23, 1514 (1998).
[CrossRef]

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).
[CrossRef]

1996 (1)

N. N. Rosanov, Prog. Opt. 35, 1 (1996); M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994); W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).For a recent review of CS’s see W. J. Firth and G. K. Harkness, Asian J. Phys. 7, 665 (1998).
[CrossRef] [PubMed]

1995 (1)

1993 (1)

B. A. Malomed, Phys. Rev. A 47, 2874 (1993).

1981 (1)

Various versions of this method can be found, e.g., in K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981); and in S. Longhi, Phys. Rev. E 55, 1060 (1997).
[CrossRef]

Brambilla, M.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Etrich, C.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).
[CrossRef]

Fabre, C.

Firth, W. J.

G.-L. Oppp, A. J. Scroggie, and W. J. Firth, J. Opt. B 1, 133 (1999).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Giacobino, E.

Gorshkov, K. A.

Various versions of this method can be found, e.g., in K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981); and in S. Longhi, Phys. Rev. E 55, 1060 (1997).
[CrossRef]

Lederer, F.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).
[CrossRef]

Lugiato, L.

Lugiato, L. A.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Malomed, B. A.

B. A. Malomed, Phys. Rev. A 47, 2874 (1993).

Oppp, G.-L.

G.-L. Oppp, A. J. Scroggie, and W. J. Firth, J. Opt. B 1, 133 (1999).
[CrossRef]

Ostrovsky, L. A.

Various versions of this method can be found, e.g., in K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981); and in S. Longhi, Phys. Rev. E 55, 1060 (1997).
[CrossRef]

Peschel, U.

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).
[CrossRef]

Petsas, K. I.

Prati, F.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Richy, C.

Rosanov, N. N.

N. N. Rosanov, Prog. Opt. 35, 1 (1996); M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994); W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).For a recent review of CS’s see W. J. Firth and G. K. Harkness, Asian J. Phys. 7, 665 (1998).
[CrossRef] [PubMed]

Sánchez-Morcillo, V. J.

K. Staliunas and V. J. Sánchez-Morcillo, Opt. Commun. 139, 306 (1997); S. Longhi, Phys. Scr. 56, 611 (1997); S. Trillo and M. Haelterman, Opt. Lett. 23, 1514 (1998).
[CrossRef]

Scroggie, A. J.

G.-L. Oppp, A. J. Scroggie, and W. J. Firth, J. Opt. B 1, 133 (1999).
[CrossRef]

Spinelli, L.

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Staliunas, K.

K. Staliunas and V. J. Sánchez-Morcillo, Opt. Commun. 139, 306 (1997); S. Longhi, Phys. Scr. 56, 611 (1997); S. Trillo and M. Haelterman, Opt. Lett. 23, 1514 (1998).
[CrossRef]

J. Opt. B (1)

G.-L. Oppp, A. J. Scroggie, and W. J. Firth, J. Opt. B 1, 133 (1999).
[CrossRef]

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

Opt. Commun. (1)

K. Staliunas and V. J. Sánchez-Morcillo, Opt. Commun. 139, 306 (1997); S. Longhi, Phys. Scr. 56, 611 (1997); S. Trillo and M. Haelterman, Opt. Lett. 23, 1514 (1998).
[CrossRef]

Phys. Rev. A (1)

B. A. Malomed, Phys. Rev. A 47, 2874 (1993).

Phys. Rev. Lett. (2)

C. Etrich, U. Peschel, and F. Lederer, Phys. Rev. Lett. 79, 2454 (1997); S. Longhi, Opt. Lett. 23, 346 (1998).
[CrossRef]

M. Brambilla, L. A. Lugiato, F. Prati, L. Spinelli, and W. J. Firth, Phys. Rev. Lett. 79, 2042 (1997); D. Michaelis, U. Peschel, and F. Lederer, Phys. Rev. A 56, R3366 (1997).
[CrossRef]

Physica D (1)

Various versions of this method can be found, e.g., in K. A. Gorshkov and L. A. Ostrovsky, Physica D 3, 428 (1981); and in S. Longhi, Phys. Rev. E 55, 1060 (1997).
[CrossRef]

Prog. Opt. (1)

N. N. Rosanov, Prog. Opt. 35, 1 (1996); M. Tlidi, P. Mandel, and R. Lefever, Phys. Rev. Lett. 73, 640 (1994); W. J. Firth and A. J. Scroggie, Phys. Rev. Lett. 76, 1623 (1996).For a recent review of CS’s see W. J. Firth and G. K. Harkness, Asian J. Phys. 7, 665 (1998).
[CrossRef] [PubMed]

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Figures (3)

Fig. 1
Fig. 1

CS velocity function f versus d. The solid (dashed) curves correspond to in-phase (out-of-phase) solitons and the thin (thick) curves correspond to μ=1.6 (1.9). The other parameters are δ2=-4.0, δ1=-1.8, γ1=1, γ2=0.8.

Fig. 2
Fig. 2

Interaction dynamics of χ2 CS’s. Only the central third of the computational window in x is shown in (a)–(c) and (e). At different values of pump parameter μ, in-phase CS’s (a) merge, μ=1.6; (b) form an oscillatory bound state, μ=1.8; (c) form a stable stationary bound state, μ=1.9; (d) generate a pattern through a switching wave, μ=2. (e) Out-of-phase solitons repel each other, e.g., at μ=2. The other parameters are the same as Fig. 1.

Fig. 3
Fig. 3

Dynamic interaction of CS’s for μ=2, δ1=-3, δ2=-12, γ1=0.3, γ2=1, for which each CS has a mode with eigenvalue pair λ-0.03±i4.14. (a) Spatial structure of the eigenmode: solid curve, Reu1; dashed line, Reu2. (b) Temporal evolution of signal energy Q=dxE12 for a slightly perturbed single soliton, showing damped oscillation. (c) Spatiotemporal evolution of E1 [the time window is much later than in (b)], showing the dynamic bound state. Only the central third of the computational window in x is shown. (d) Temporal evolution of signal energies of the two CS’s in (c), in the same time window, showing rapid undamped oscillations and slow energy exchange between the two solitons.

Equations (4)

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

-itE1=α1x2+δ1+iγ1E1+E2+μE1*, -itE2=α2x2+δ2+iγ2E2+E12/2.
Emx,t=Amx-xA+Bmx-xB+amx-xA,xB,t+bmx-xB,xA,t+O2,
LˆA-ta=-τxAξA+I/.
td=fd.

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