Abstract

We predict the existence of discrete midband solitons in an array of coupled quadratically nonlinear cavities driven by an external optical field at the second harmonic. The tilted holding beam provides both subdiffractive propagation and maximum group velocity for the fundamental harmonic. This configuration allows for the existence of identical but counterpropagating midband solitons for the same system parameters. We study numerically the interaction dynamics of these solitons.

© 2008 Optical Society of America

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  1. T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
    [CrossRef] [PubMed]
  2. H. S.  Eisenberg, Y.  Silberberg, R.  Morandotti, and J. S.  Aitchison, "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
    [CrossRef] [PubMed]
  3. J.  Meier, G. I.  Stegeman, D. N.  Christodoulides, Y.  Silberberg, R.  Morandotti, H.  Yang, G.  Salamo, M.  Sorel, and J. S.  Aitchison, "Beam interactions with a blocker soliton in one-dimensional arrays," Opt. Lett. 30, 1027-1029 (2005).
    [CrossRef] [PubMed]
  4. T.  Pertsch, U.  Peschel, and F.  Lederer, "All-optical switching in quadratically nonlinear waveguide arrays," Opt. Lett. 28, 102-104 (2003).
    [CrossRef] [PubMed]
  5. K.  Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 053901 (2003).
    [CrossRef] [PubMed]
  6. O.  Egorov, F.  Lederer, and K.  Staliunas, "Subdiffractive discrete cavity solitons," Opt. Lett. 32, 2106-2108 (2007).
    [CrossRef] [PubMed]
  7. U.  Peschel, O.  Egorov, and F.  Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909-1911 (2004).
    [CrossRef] [PubMed]
  8. O.  Egorov, U.  Peschel, and F.  Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 056612 (2005).
    [CrossRef]
  9. O.  Egorov, U.  Peschel, and F.  Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 066603 (2005).
    [CrossRef]
  10. O. A.  Egorov, F.  Lederer, and Y. S.  Kivshar, "How does an inclined holding beam affect discrete modulational instability and solitons in nonlinear cavities?" Opt. Express 15, 4149-4158 (2007).
    [CrossRef] [PubMed]
  11. U.  Peschel, D.  Michaelis, and C. O.  Weiss, "Spatial Solitons in Optical Cavities," IEEE J. Quantum Electron. 39, 51-64 (2003).
    [CrossRef]
  12. N.  Akhmediev and A.  Ankiewicz, eds., Dissipative Solitons, Lecture Notes in Physics, (Springer, Berlin, 2005) pp. 450.
  13. S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
    [CrossRef]
  14. G. L.  Oppo, M.  Brambilla, and L. A.  Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028-2032 (1994).
    [CrossRef] [PubMed]
  15. S.  Longhi and A.  Geraci, "Swift-Hohenberg equation for optical parametric oscillators," Phys. Rev. A 54, 4581-4584 (1996).
    [CrossRef] [PubMed]
  16. M.  Peckus, K.  Staliunas, Z.  Nizauskaite, and V.  Sirutkaitis, "Stripe patterns in degenerate optical parametric oscillators," Opt. Lett. 32, 3014-3016 (2007).
    [CrossRef] [PubMed]
  17. K.  Staliunas and V. J.  Sanchez-Morcillo, "Localized structures in degenerate optical parametric oscillators," Opt. Communications 139, 306-312 (1997).
    [CrossRef]
  18. K.  Staliunas and V. J.  Sanchez-Morcillo, "Spatial-localized structures in degenerate optical parametric oscillators," Phys. Rev. A 57, 1454-1457 (1998).
    [CrossRef]
  19. C.  Etrich, D.  Michaelis, and F.  Lederer, "Bifurcations, stability, and multistability of cavity solitons in parametric downconversion," J. Opt. Soc. Am. B 19, 792-801 (2002).
    [CrossRef]
  20. D. V.  Skryabin and W. J.  Firth, "Intreraction of cavity solitons in degenerate optical parametric oscillators," Opt. Lett. 24, 1056-1058 (1999).
    [CrossRef]

2007

2005

O.  Egorov, U.  Peschel, and F.  Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O.  Egorov, U.  Peschel, and F.  Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 066603 (2005).
[CrossRef]

J.  Meier, G. I.  Stegeman, D. N.  Christodoulides, Y.  Silberberg, R.  Morandotti, H.  Yang, G.  Salamo, M.  Sorel, and J. S.  Aitchison, "Beam interactions with a blocker soliton in one-dimensional arrays," Opt. Lett. 30, 1027-1029 (2005).
[CrossRef] [PubMed]

2004

2003

T.  Pertsch, U.  Peschel, and F.  Lederer, "All-optical switching in quadratically nonlinear waveguide arrays," Opt. Lett. 28, 102-104 (2003).
[CrossRef] [PubMed]

U.  Peschel, D.  Michaelis, and C. O.  Weiss, "Spatial Solitons in Optical Cavities," IEEE J. Quantum Electron. 39, 51-64 (2003).
[CrossRef]

K.  Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

2002

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

C.  Etrich, D.  Michaelis, and F.  Lederer, "Bifurcations, stability, and multistability of cavity solitons in parametric downconversion," J. Opt. Soc. Am. B 19, 792-801 (2002).
[CrossRef]

2001

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

2000

H. S.  Eisenberg, Y.  Silberberg, R.  Morandotti, and J. S.  Aitchison, "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

1999

1998

K.  Staliunas and V. J.  Sanchez-Morcillo, "Spatial-localized structures in degenerate optical parametric oscillators," Phys. Rev. A 57, 1454-1457 (1998).
[CrossRef]

1997

K.  Staliunas and V. J.  Sanchez-Morcillo, "Localized structures in degenerate optical parametric oscillators," Opt. Communications 139, 306-312 (1997).
[CrossRef]

1996

S.  Longhi and A.  Geraci, "Swift-Hohenberg equation for optical parametric oscillators," Phys. Rev. A 54, 4581-4584 (1996).
[CrossRef] [PubMed]

1994

G. L.  Oppo, M.  Brambilla, and L. A.  Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028-2032 (1994).
[CrossRef] [PubMed]

Aitchison, J. S.

Brambilla, M.

G. L.  Oppo, M.  Brambilla, and L. A.  Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028-2032 (1994).
[CrossRef] [PubMed]

Bräuer, A.

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Christodoulides, D. N.

Egorov, O.

O.  Egorov, F.  Lederer, and K.  Staliunas, "Subdiffractive discrete cavity solitons," Opt. Lett. 32, 2106-2108 (2007).
[CrossRef] [PubMed]

O.  Egorov, U.  Peschel, and F.  Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 066603 (2005).
[CrossRef]

O.  Egorov, U.  Peschel, and F.  Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 056612 (2005).
[CrossRef]

U.  Peschel, O.  Egorov, and F.  Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909-1911 (2004).
[CrossRef] [PubMed]

Egorov, O. A.

Eisenberg, H. S.

H. S.  Eisenberg, Y.  Silberberg, R.  Morandotti, and J. S.  Aitchison, "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

Etrich, C.

C.  Etrich, D.  Michaelis, and F.  Lederer, "Bifurcations, stability, and multistability of cavity solitons in parametric downconversion," J. Opt. Soc. Am. B 19, 792-801 (2002).
[CrossRef]

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Fedorov, S.

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Firth, W. J.

Geraci, A.

S.  Longhi and A.  Geraci, "Swift-Hohenberg equation for optical parametric oscillators," Phys. Rev. A 54, 4581-4584 (1996).
[CrossRef] [PubMed]

Kivshar, Y. S.

Lederer, F.

O. A.  Egorov, F.  Lederer, and Y. S.  Kivshar, "How does an inclined holding beam affect discrete modulational instability and solitons in nonlinear cavities?" Opt. Express 15, 4149-4158 (2007).
[CrossRef] [PubMed]

O.  Egorov, F.  Lederer, and K.  Staliunas, "Subdiffractive discrete cavity solitons," Opt. Lett. 32, 2106-2108 (2007).
[CrossRef] [PubMed]

O.  Egorov, U.  Peschel, and F.  Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 066603 (2005).
[CrossRef]

O.  Egorov, U.  Peschel, and F.  Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 056612 (2005).
[CrossRef]

U.  Peschel, O.  Egorov, and F.  Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909-1911 (2004).
[CrossRef] [PubMed]

T.  Pertsch, U.  Peschel, and F.  Lederer, "All-optical switching in quadratically nonlinear waveguide arrays," Opt. Lett. 28, 102-104 (2003).
[CrossRef] [PubMed]

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

C.  Etrich, D.  Michaelis, and F.  Lederer, "Bifurcations, stability, and multistability of cavity solitons in parametric downconversion," J. Opt. Soc. Am. B 19, 792-801 (2002).
[CrossRef]

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Longhi, S.

S.  Longhi and A.  Geraci, "Swift-Hohenberg equation for optical parametric oscillators," Phys. Rev. A 54, 4581-4584 (1996).
[CrossRef] [PubMed]

Lugiato, L. A.

G. L.  Oppo, M.  Brambilla, and L. A.  Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028-2032 (1994).
[CrossRef] [PubMed]

Meier, J.

Michaelis, D.

U.  Peschel, D.  Michaelis, and C. O.  Weiss, "Spatial Solitons in Optical Cavities," IEEE J. Quantum Electron. 39, 51-64 (2003).
[CrossRef]

C.  Etrich, D.  Michaelis, and F.  Lederer, "Bifurcations, stability, and multistability of cavity solitons in parametric downconversion," J. Opt. Soc. Am. B 19, 792-801 (2002).
[CrossRef]

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Morandotti, R.

Nizauskaite, Z.

Oppo, G. L.

G. L.  Oppo, M.  Brambilla, and L. A.  Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028-2032 (1994).
[CrossRef] [PubMed]

Peckus, M.

Pertsch, T.

T.  Pertsch, U.  Peschel, and F.  Lederer, "All-optical switching in quadratically nonlinear waveguide arrays," Opt. Lett. 28, 102-104 (2003).
[CrossRef] [PubMed]

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

Peschel, U.

O.  Egorov, U.  Peschel, and F.  Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O.  Egorov, U.  Peschel, and F.  Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 066603 (2005).
[CrossRef]

U.  Peschel, O.  Egorov, and F.  Lederer, "Discrete cavity solitons," Opt. Lett. 29, 1909-1911 (2004).
[CrossRef] [PubMed]

U.  Peschel, D.  Michaelis, and C. O.  Weiss, "Spatial Solitons in Optical Cavities," IEEE J. Quantum Electron. 39, 51-64 (2003).
[CrossRef]

T.  Pertsch, U.  Peschel, and F.  Lederer, "All-optical switching in quadratically nonlinear waveguide arrays," Opt. Lett. 28, 102-104 (2003).
[CrossRef] [PubMed]

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Rosanov, N.

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Salamo, G.

Sanchez-Morcillo, V. J.

K.  Staliunas and V. J.  Sanchez-Morcillo, "Spatial-localized structures in degenerate optical parametric oscillators," Phys. Rev. A 57, 1454-1457 (1998).
[CrossRef]

K.  Staliunas and V. J.  Sanchez-Morcillo, "Localized structures in degenerate optical parametric oscillators," Opt. Communications 139, 306-312 (1997).
[CrossRef]

Silberberg, Y.

Sirutkaitis, V.

Skryabin, D. V.

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

D. V.  Skryabin and W. J.  Firth, "Intreraction of cavity solitons in degenerate optical parametric oscillators," Opt. Lett. 24, 1056-1058 (1999).
[CrossRef]

Sorel, M.

Staliunas, K.

O.  Egorov, F.  Lederer, and K.  Staliunas, "Subdiffractive discrete cavity solitons," Opt. Lett. 32, 2106-2108 (2007).
[CrossRef] [PubMed]

M.  Peckus, K.  Staliunas, Z.  Nizauskaite, and V.  Sirutkaitis, "Stripe patterns in degenerate optical parametric oscillators," Opt. Lett. 32, 3014-3016 (2007).
[CrossRef] [PubMed]

K.  Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

K.  Staliunas and V. J.  Sanchez-Morcillo, "Spatial-localized structures in degenerate optical parametric oscillators," Phys. Rev. A 57, 1454-1457 (1998).
[CrossRef]

K.  Staliunas and V. J.  Sanchez-Morcillo, "Localized structures in degenerate optical parametric oscillators," Opt. Communications 139, 306-312 (1997).
[CrossRef]

Stegeman, G. I.

Weiss, C. O.

U.  Peschel, D.  Michaelis, and C. O.  Weiss, "Spatial Solitons in Optical Cavities," IEEE J. Quantum Electron. 39, 51-64 (2003).
[CrossRef]

Yang, H.

Zentgraf, T.

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

U.  Peschel, D.  Michaelis, and C. O.  Weiss, "Spatial Solitons in Optical Cavities," IEEE J. Quantum Electron. 39, 51-64 (2003).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Communications

K.  Staliunas and V. J.  Sanchez-Morcillo, "Localized structures in degenerate optical parametric oscillators," Opt. Communications 139, 306-312 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. A

K.  Staliunas and V. J.  Sanchez-Morcillo, "Spatial-localized structures in degenerate optical parametric oscillators," Phys. Rev. A 57, 1454-1457 (1998).
[CrossRef]

G. L.  Oppo, M.  Brambilla, and L. A.  Lugiato, "Formation and evolution of roll patterns in optical parametric oscillators," Phys. Rev. A 49, 2028-2032 (1994).
[CrossRef] [PubMed]

S.  Longhi and A.  Geraci, "Swift-Hohenberg equation for optical parametric oscillators," Phys. Rev. A 54, 4581-4584 (1996).
[CrossRef] [PubMed]

Phys. Rev. E

O.  Egorov, U.  Peschel, and F.  Lederer, "Discrete quadratic cavity solitons," Phys. Rev. E 71, 056612 (2005).
[CrossRef]

O.  Egorov, U.  Peschel, and F.  Lederer, "Mobility of discrete cavity solitons," Phys. Rev. E 72, 066603 (2005).
[CrossRef]

S.  Fedorov, D.  Michaelis, U.  Peschel, C.  Etrich, D. V.  Skryabin, N.  Rosanov, and F.  Lederer, "Effects of spatial inhomogeneities on the dynamics of cavity solitons in quadratically nonlinear media," Phys. Rev. E 64, 036610 (2001).
[CrossRef]

Phys. Rev. Lett.

T.  Pertsch, T.  Zentgraf, U.  Peschel, A.  Bräuer, and F.  Lederer, "Anomalous Refraction and Diffraction in Discrete Optical Systems," Phys. Rev. Lett. 88, 093901 (2002).
[CrossRef] [PubMed]

H. S.  Eisenberg, Y.  Silberberg, R.  Morandotti, and J. S.  Aitchison, "Diffraction Management," Phys. Rev. Lett. 85, 1863-1866 (2000).
[CrossRef] [PubMed]

K.  Staliunas, "Midband Dissipative Spatial Solitons," Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

Other

N.  Akhmediev and A.  Ankiewicz, eds., Dissipative Solitons, Lecture Notes in Physics, (Springer, Berlin, 2005) pp. 450.

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Figures (4)

Fig. 1.
Fig. 1.

(Color online) An array of coupled cavities endowed with quadratic nonlinearity and driven by the holding beam at SH.

Fig. 2.
Fig. 2.

(Color online) Excitation of the counterpropagating MDCSs by means of a narrow additional pulse with the width equal to one (a) or two (b) cavities. (c) Amplitude profile of the MDCS. The diamonds depict the results of discrete model (1) whereas the full line corresponds to the results of quasi-continuous approximation (3). Parameters: E 0=7.25, Δ′1=-2, Δ2=-3, δ=2, C 1=0.7.

Fig. 3.
Fig. 3.

(Color online) Maximal amplitude of the bright MDCS depending on the coupling constant (a) and on the holding beam amplitude (b). The solid parts of these branches correspond to stable and the dashed ones to unstable solutions. The parameters are as in Fig. 2.

Fig. 4.
Fig. 4.

(Color online) (a) Moving DMCS. (b) Interaction dynamics of the counterpropagating MDCSs (b). The parameters are as in Fig. 2.

Equations (5)

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

i u n T + C 1 ( u n 1 + u n + 1 2 u n ) + ( i + 1 ) u n + u n * v n = 0 ,
i v n T + C 2 ( v n 1 + v n 1 2 v n ) + ( i δ + Δ 2 ) v n + u n 2 = E 0 exp ( iqn ) ,
u n ± = b 1 exp ( ± i π n 2 ) , v n = b 2 exp ( i π n ) ,
i u T + i D ( 1 ) u x + D ( 2 ) 2 u x 2 + i D ( 3 ) 3 u x 3 + ( i + 1 ) u + u * v = 0 ,
i v T + ( i δ + 2 ) v + u 2 = E 0 ,

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