Abstract

We demonstrate two alternative techniques for controlling and stabilizing domain walls (DW) in phase-sensitive, nonlinear optical resonators. The first of them uses input pumps with spatially modulated phase and can be applied also to dark-ring cavity solitons. An optical memory based on the latter is demonstrated. Here the physical mechanism of control relies on the advection caused to any feature by the phase gradients. The second technique uses a plane wave input pump with holes of null intensity across its transverse plane, which are able to capture DWs. Here the physical mechanism of control is of topological nature. When distributed as a regular array, these holes delimit spatial optical bits which constitute an optical memory. These techniques are illustrated in a degenerate optical parametric oscillator model, but can be applied to any phase-sensitive nonlinear optical cavity.

©2004 Optical Society of America

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References

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  1. P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University Press, Cambridge, 1997).
    [Crossref]
  2. L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” Adv. At. Mol. Opt. Phys. 40, 229 (1999).
    [Crossref]
  3. S. Trillo and W. Torruellas, eds., Spatial Solitons (Springer-Verlag, Berlin, 2001).
  4. K. Staliunas and V. J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, Berlin, 2003).
  5. W. J. Firth and C. O. Weiss, “Cavity and Feedback Solitons,” Opt. Photon. News 13, 55 (2002).
    [Crossref]
  6. D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
    [Crossref] [PubMed]
  7. S. Trillo, M. Haelterman, and A. Sheppard, “Stable Topological Spatial Solitons in Optical Parametric Oscillators,” Opt. Lett. 22, 970 (1997).
    [Crossref] [PubMed]
  8. K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-Localized Structures in Degenerate Optical Parametric Oscillators,” Phys. Rev. A 57, 1454 (1998).
    [Crossref]
  9. V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Pattern Formation and Localized Structures in Degenerate Optical Parametric Mixing,” Phys. Rev. Lett. 81, 2236 (1998).
    [Crossref]
  10. G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Parametric Oscillators,” Phys. Rev. E 63, 066209 (2001).
    [Crossref]
  11. W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controlable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. 76, 1623 (1996).
    [Crossref] [PubMed]
  12. G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and Evolution of Roll Patterns in Optical Parametric Oscillators,” Phys. Rev. A 49, 2028 (1994).
    [Crossref] [PubMed]
  13. G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
    [Crossref] [PubMed]
  14. We checked that a similar effect is achieved by introducing a linearly varying phase in the y direction that physically corresponds to tilting vertically the input field.
  15. M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and Control of Optical Localized Structures,” Europhys. Lett. 34, 109 (1996).
    [Crossref]
  16. M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
    [Crossref]
  17. V. J. Sánchez-Morcillo, I. Pérez-Arjona, F. Silva, G. J. de Valcárcel, and E. Roldán, “Vectorial Kerr Cavity Solitons,” Opt. Lett. 25, 957 (2000).
    [Crossref]

2002 (1)

W. J. Firth and C. O. Weiss, “Cavity and Feedback Solitons,” Opt. Photon. News 13, 55 (2002).
[Crossref]

2001 (2)

D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
[Crossref] [PubMed]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Parametric Oscillators,” Phys. Rev. E 63, 066209 (2001).
[Crossref]

2000 (2)

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
[Crossref]

V. J. Sánchez-Morcillo, I. Pérez-Arjona, F. Silva, G. J. de Valcárcel, and E. Roldán, “Vectorial Kerr Cavity Solitons,” Opt. Lett. 25, 957 (2000).
[Crossref]

1999 (1)

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” Adv. At. Mol. Opt. Phys. 40, 229 (1999).
[Crossref]

1998 (2)

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-Localized Structures in Degenerate Optical Parametric Oscillators,” Phys. Rev. A 57, 1454 (1998).
[Crossref]

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Pattern Formation and Localized Structures in Degenerate Optical Parametric Mixing,” Phys. Rev. Lett. 81, 2236 (1998).
[Crossref]

1997 (1)

1996 (3)

W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controlable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. 76, 1623 (1996).
[Crossref] [PubMed]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
[Crossref] [PubMed]

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and Control of Optical Localized Structures,” Europhys. Lett. 34, 109 (1996).
[Crossref]

1994 (1)

G. L. Oppo, M. Brambilla, and L. A. Lugiato, “Formation and Evolution of Roll Patterns in Optical Parametric Oscillators,” Phys. Rev. A 49, 2028 (1994).
[Crossref] [PubMed]

Brambilla, M.

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” Adv. At. Mol. Opt. Phys. 40, 229 (1999).
[Crossref]

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and Control of Optical Localized Structures,” Europhys. Lett. 34, 109 (1996).
[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 (1994).
[Crossref] [PubMed]

Colet, P.

D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
[Crossref] [PubMed]

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
[Crossref]

de Valcárcel, G. J.

V. J. Sánchez-Morcillo, I. Pérez-Arjona, F. Silva, G. J. de Valcárcel, and E. Roldán, “Vectorial Kerr Cavity Solitons,” Opt. Lett. 25, 957 (2000).
[Crossref]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
[Crossref] [PubMed]

Firth, W. J.

W. J. Firth and C. O. Weiss, “Cavity and Feedback Solitons,” Opt. Photon. News 13, 55 (2002).
[Crossref]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Parametric Oscillators,” Phys. Rev. E 63, 066209 (2001).
[Crossref]

W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controlable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. 76, 1623 (1996).
[Crossref] [PubMed]

Gatti, A.

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” Adv. At. Mol. Opt. Phys. 40, 229 (1999).
[Crossref]

Gomila, D.

D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
[Crossref] [PubMed]

Haelterman, M.

Hoyuelos, M.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
[Crossref]

Lugiato, L. A.

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” Adv. At. Mol. Opt. Phys. 40, 229 (1999).
[Crossref]

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and Control of Optical Localized Structures,” Europhys. Lett. 34, 109 (1996).
[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 (1994).
[Crossref] [PubMed]

Mandel, P.

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University Press, Cambridge, 1997).
[Crossref]

Oppo, G. L.

D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
[Crossref] [PubMed]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Parametric Oscillators,” Phys. Rev. E 63, 066209 (2001).
[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 (1994).
[Crossref] [PubMed]

Pérez-Arjona, I.

Roldán, E.

V. J. Sánchez-Morcillo, I. Pérez-Arjona, F. Silva, G. J. de Valcárcel, and E. Roldán, “Vectorial Kerr Cavity Solitons,” Opt. Lett. 25, 957 (2000).
[Crossref]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
[Crossref] [PubMed]

San Miguel, M.

D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
[Crossref] [PubMed]

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
[Crossref]

Sánchez-Morcillo, V. J.

V. J. Sánchez-Morcillo, I. Pérez-Arjona, F. Silva, G. J. de Valcárcel, and E. Roldán, “Vectorial Kerr Cavity Solitons,” Opt. Lett. 25, 957 (2000).
[Crossref]

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-Localized Structures in Degenerate Optical Parametric Oscillators,” Phys. Rev. A 57, 1454 (1998).
[Crossref]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
[Crossref] [PubMed]

K. Staliunas and V. J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, Berlin, 2003).

Scroggie, A. J.

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Parametric Oscillators,” Phys. Rev. E 63, 066209 (2001).
[Crossref]

W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controlable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. 76, 1623 (1996).
[Crossref] [PubMed]

Sheppard, A.

Silva, F.

Staliunas, K.

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-Localized Structures in Degenerate Optical Parametric Oscillators,” Phys. Rev. A 57, 1454 (1998).
[Crossref]

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Pattern Formation and Localized Structures in Degenerate Optical Parametric Mixing,” Phys. Rev. Lett. 81, 2236 (1998).
[Crossref]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
[Crossref] [PubMed]

K. Staliunas and V. J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, Berlin, 2003).

Stefani, M.

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and Control of Optical Localized Structures,” Europhys. Lett. 34, 109 (1996).
[Crossref]

Taranenko, V. B.

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Pattern Formation and Localized Structures in Degenerate Optical Parametric Mixing,” Phys. Rev. Lett. 81, 2236 (1998).
[Crossref]

Trillo, S.

Walgraef, D.

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
[Crossref]

Weiss, C. O.

W. J. Firth and C. O. Weiss, “Cavity and Feedback Solitons,” Opt. Photon. News 13, 55 (2002).
[Crossref]

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Pattern Formation and Localized Structures in Degenerate Optical Parametric Mixing,” Phys. Rev. Lett. 81, 2236 (1998).
[Crossref]

Adv. At. Mol. Opt. Phys. (1)

L. A. Lugiato, M. Brambilla, and A. Gatti, “Optical Pattern Formation,” Adv. At. Mol. Opt. Phys. 40, 229 (1999).
[Crossref]

Europhys. Lett. (1)

M. Brambilla, L. A. Lugiato, and M. Stefani, “Interaction and Control of Optical Localized Structures,” Europhys. Lett. 34, 109 (1996).
[Crossref]

Opt. Lett. (2)

Opt. Photon. News (1)

W. J. Firth and C. O. Weiss, “Cavity and Feedback Solitons,” Opt. Photon. News 13, 55 (2002).
[Crossref]

Phys. Rev. A (3)

K. Staliunas and V. J. Sánchez-Morcillo, “Spatial-Localized Structures in Degenerate Optical Parametric Oscillators,” Phys. Rev. A 57, 1454 (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 (1994).
[Crossref] [PubMed]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, “Transverse Patterns in Degenerate Optical Parametric Oscillation and Degenerate Four-Wave Mixing,” Phys. Rev. A 54, 1609 (1996).
[Crossref] [PubMed]

Phys. Rev. E (2)

M. Hoyuelos, P. Colet, M. San Miguel, and D. Walgraef, “Self-Similar Domain Growth, Localized Structures, and Labyrinthine Patterns in Vectorial Kerr Resonators,” Phys. Rev. E 61, 2241 (2000)
[Crossref]

G. L. Oppo, A. J. Scroggie, and W. J. Firth, “Characterization, Dynamics and Stabilization of Diffractive Domain Walls and Dark Ring Cavity Solitons in Parametric Oscillators,” Phys. Rev. E 63, 066209 (2001).
[Crossref]

Phys. Rev. Lett. (3)

W. J. Firth and A. J. Scroggie, “Optical Bullet Holes: Robust Controlable Localized States of a Nonlinear Cavity,” Phys. Rev. Lett. 76, 1623 (1996).
[Crossref] [PubMed]

V. B. Taranenko, K. Staliunas, and C. O. Weiss, “Pattern Formation and Localized Structures in Degenerate Optical Parametric Mixing,” Phys. Rev. Lett. 81, 2236 (1998).
[Crossref]

D. Gomila, P. Colet, G. L. Oppo, and M. San Miguel, “Stable Droplets and Growth Laws Close to the Modulational Instability of a Domain Wall,” Phys. Rev. Lett. 87, 194101 (2001).
[Crossref] [PubMed]

Other (4)

S. Trillo and W. Torruellas, eds., Spatial Solitons (Springer-Verlag, Berlin, 2001).

K. Staliunas and V. J. Sánchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, Berlin, 2003).

We checked that a similar effect is achieved by introducing a linearly varying phase in the y direction that physically corresponds to tilting vertically the input field.

P. Mandel, Theoretical Problems in Cavity Nonlinear Optics (Cambridge University Press, Cambridge, 1997).
[Crossref]

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Figures (8)

Fig. 1.
Fig. 1. A DOPO DW is advected towards the pump beam centre. A cylindrical pump wavefront, convergent along the horizontal direction, x, is used. Parameters: γ= 1, Δ0 = Δ1 = 0, P = 3, r 0 = 45, L = 100. At t = 0 a straight DW is excited at x = 0.56r 0. Plots are given at t = 50 (left), 600 (middle), and 2000 (right). Here, and in the rest of figures, the intensity of the signal field is plotted, except when noted.
Fig. 2.
Fig. 2. Final state, for the same parameters as in Fig. 1, reached at t = 2000 in the presence of noise. The right panel shows the intensity trace of the signal field, ∣A 12 , along the horizontal diameter. The noise variance is 3×10-3 in the dimensionless units of Eqs. (1) and (2).
Fig. 3.
Fig. 3. Selection and control of a single DW starting from noise. Rest of conditions as in Fig. 1. From left to right, plots are given at t = 200, 400, 800 (top row) and t = 1250, 2750, 3350 (bottom row). In order to remove the two dark ring cavity solitons bond to the DW, at t = 2750 the pump phase was made divergent along the vertical direction (see text).
Fig. 4.
Fig. 4. Selection and control of two DWs starting from noise. Parameters as in Fig. 1, except for the pump wavefront wich has two local maxima at the vertical lines x = ±0.28r 0 and is divergent along the vertical direction (see text). From left to right, plots are given at t = 10, 50, 200, (top row) and t = 400, 800, 1000 (bottom row).
Fig. 5.
Fig. 5. Control of DRCSs. A signal seed in a quasi-ordered shape (left) is controlled by an egg-box phase profile consisting of 3×3 phase maxima. At t = 100 (middle panel) the marks have developed into DRCSs that are pinned by the phase maxima. The right panel shows the erasure of the central DRCS obtained 10 time units after a narrow pulse superimposed to it has been injected. Other parameters as in Fig. 1.
Fig. 6.
Fig. 6. Control of DRCSs in the presence of noise. The right panel shows the intensity trace of the signal field along the horizontal diameter at t = 100 for the same initial condition and parameters as in Fig. 5 (the noise variance is 3×10-3).
Fig. 7.
Fig. 7. Stabilization of a DW by a doughnut shape pump. Same parameters as in Fig. 1. The time is t = 5, 50, 1000 from left to right.
Fig. 8.
Fig. 8. Codification in a 3×3 matrix. White corresponds to a positive value, dark gray corresponds to a negative value, and light gray (like in the circular holes) indicates a null value of the real part of the signal field. The left panel shows the field without any codification. After injecting appropriate signals, the system exhibits the distribution shown in the central panel where cells (3,1), (2,2), (3,2) have been marked (the phase value has been flipped from negative to positive). These marks can be changed after injecting new signals: The right panel shows new marks in cells (1,3), (2,2), (3,1). Time difference between the different panels is t = 10. Other parameters as in Fig. 1, except Δ1 = Δ0/2 = 1.2.

Equations (5)

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t A 0 = γ [ E ( 1 + i Δ 0 ) A 0 A 1 2 ] + 1 2 i 2 A 0 ,
t A 1 = ( 1 + i Δ 1 ) A 1 + A 0 A 1 * + i 2 A 1 .
D t F 0 = γ [ E ( κ 0 + i δ 0 ) F 0 F 1 2 ] + 1 2 i 2 F 0 ,
D t F 1 = ( κ 1 + i δ 1 ) F 1 + F 0 F 1 + i 2 F 1 ,
D t = t + ( ψ ) ·

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