Abstract

We show that nonlinear optical cavities under periodic modulations can exhibit dissipative Bloch waves. This can be achieved by a proper choice of the incident field in the cavity that must both compensate for losses and match with the modulation period. Dissipative Bloch waves, unlike their conservative counterparts, spontaneously emerge in the cavity and hence correspond to attracting solutions. This makes it possible to experimentally visualize the band structure of the cavity medium. As an illustration of this phenomenon, we performed our analytical investigations on a degenerate optical parametric oscillator with a modulated transverse refractive index.

© 2009 Optical Society of America

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  1. Dissipative Solitons: from Optics to Biology and Medicine, Lecture Notes in Physics, N.Akhmediev and A.Ankiewicz, eds. (Springer, 2008), Vol. 751.
  2. L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, B.Bederson and H.Walters, eds. (Academic, 1998), Vol. 40.
  3. N. N. Rosanov, in Spatial Hysteresis and Optical Patterns (Springer, 2002).
  4. K. Staliunas and V. J. Sanchez-Morcillo, Transverse Patterns in Nonlinear Optical Resonators (Springer-Verlag, 2003).
  5. A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
    [CrossRef]
  6. M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
    [CrossRef]
  7. R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
    [CrossRef] [PubMed]
  8. T. Ackemann, S. Barland, J. R. Tredicce, M. Cara, S. Balle, R. Jäger, M. Grabherr, M. Miller, and K. J. Ebeling, Opt. Lett. 25, 814 (2000).
    [CrossRef]
  9. P. L. Ramazza, S. Ducci, and F. T. Arecchi, Phys. Rev. Lett. 81, 4128 (1998).
    [CrossRef]
  10. S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
    [CrossRef]
  11. F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, Phys. Rep. 318, 1 (1999).
    [CrossRef]
  12. V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997).
    [CrossRef]
  13. Yu. A. Logvin, B. Schäpers, and T. Ackemann, Phys. Rev. E 61, 4622 (2000).
    [CrossRef]
  14. S. Residori, T. Nagaya, and A. Petrossian, Europhys. Lett. 63, 531 (2003).
    [CrossRef]
  15. Focus Issue on Dissipative Localized Structures in Extended Systems, Chaos 17, M.Tlidi, T.Kolokolnikov, and M.Taki, eds. (2007).
  16. K. Staliunas, M. Peckus, and V. Sirutkaitis, Phys. Rev. A 76, 051803(R) (2007).
    [CrossRef]
  17. K. Staliunas, Phys. Rev. Lett. 91, 053901 (2003).
    [CrossRef] [PubMed]
  18. K. Staliunas, Phys. Rev. A 70, 016602 (2004).
    [CrossRef]
  19. V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
    [CrossRef] [PubMed]
  20. M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
    [CrossRef]
  21. H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
    [CrossRef]
  22. V. A. Brazhnyi, V. V. Konotop, and M. Taki, Phys. Rev. A 80, 043814 (2009).
    [CrossRef]
  23. G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
    [CrossRef] [PubMed]

2009 (1)

V. A. Brazhnyi, V. V. Konotop, and M. Taki, Phys. Rev. A 80, 043814 (2009).
[CrossRef]

2007 (2)

K. Staliunas, M. Peckus, and V. Sirutkaitis, Phys. Rev. A 76, 051803(R) (2007).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
[CrossRef] [PubMed]

2004 (1)

K. Staliunas, Phys. Rev. A 70, 016602 (2004).
[CrossRef]

2003 (2)

K. Staliunas, Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

S. Residori, T. Nagaya, and A. Petrossian, Europhys. Lett. 63, 531 (2003).
[CrossRef]

2000 (4)

Yu. A. Logvin, B. Schäpers, and T. Ackemann, Phys. Rev. E 61, 4622 (2000).
[CrossRef]

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

T. Ackemann, S. Barland, J. R. Tredicce, M. Cara, S. Balle, R. Jäger, M. Grabherr, M. Miller, and K. J. Ebeling, Opt. Lett. 25, 814 (2000).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
[CrossRef]

1999 (4)

A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
[CrossRef]

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

1998 (2)

P. L. Ramazza, S. Ducci, and F. T. Arecchi, Phys. Rev. Lett. 81, 4128 (1998).
[CrossRef]

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

1997 (1)

V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997).
[CrossRef]

1996 (1)

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Ackemann, T.

Arecchi, F. T.

S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
[CrossRef]

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

P. L. Ramazza, S. Ducci, and F. T. Arecchi, Phys. Rev. Lett. 81, 4128 (1998).
[CrossRef]

Balle, S.

Barland, S.

Barsella, A.

A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
[CrossRef]

Boccaletti, S.

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

Brambilla, M.

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, B.Bederson and H.Walters, eds. (Academic, 1998), Vol. 40.

Brazhnyi, V. A.

V. A. Brazhnyi, V. V. Konotop, and M. Taki, Phys. Rev. A 80, 043814 (2009).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
[CrossRef] [PubMed]

Cara, M.

Colet, P.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Coulibaly, S.

V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
[CrossRef] [PubMed]

de Valcárcel, G. J.

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Ducci, S.

S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
[CrossRef]

P. L. Ramazza, S. Ducci, and F. T. Arecchi, Phys. Rev. Lett. 81, 4128 (1998).
[CrossRef]

Ebeling, K. J.

Fabre, C.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Ganne, I.

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

Gatti, A.

L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, B.Bederson and H.Walters, eds. (Academic, 1998), Vol. 40.

Glorieux, P.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
[CrossRef]

A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
[CrossRef]

Gonzàlez-Viñas, W.

S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
[CrossRef]

Grabherr, M.

Jäger, R.

Konotop, V. V.

V. A. Brazhnyi, V. V. Konotop, and M. Taki, Phys. Rev. A 80, 043814 (2009).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
[CrossRef] [PubMed]

Kuszelewicz, R.

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

Lepers, C.

A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
[CrossRef]

Logvin, Yu. A.

Yu. A. Logvin, B. Schäpers, and T. Ackemann, Phys. Rev. E 61, 4622 (2000).
[CrossRef]

Lugiato, L. A.

L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, B.Bederson and H.Walters, eds. (Academic, 1998), Vol. 40.

Maître, A.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Miller, M.

Nagaya, T.

S. Residori, T. Nagaya, and A. Petrossian, Europhys. Lett. 63, 531 (2003).
[CrossRef]

Ouarzazi, M. N.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
[CrossRef]

Peckus, M.

K. Staliunas, M. Peckus, and V. Sirutkaitis, Phys. Rev. A 76, 051803(R) (2007).
[CrossRef]

Petrossian, A.

S. Residori, T. Nagaya, and A. Petrossian, Europhys. Lett. 63, 531 (2003).
[CrossRef]

Ramazza, P. L.

S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
[CrossRef]

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

P. L. Ramazza, S. Ducci, and F. T. Arecchi, Phys. Rev. Lett. 81, 4128 (1998).
[CrossRef]

Residori, S.

S. Residori, T. Nagaya, and A. Petrossian, Europhys. Lett. 63, 531 (2003).
[CrossRef]

Roldán, E.

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Rosanov, N. N.

N. N. Rosanov, in Spatial Hysteresis and Optical Patterns (Springer, 2002).

Sagnes, I.

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

San Miguel, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Sanchez-Morcillo, V. J.

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

Sánchez-Morcillo, V. J.

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

Santagiustina, M.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Schäpers, B.

Yu. A. Logvin, B. Schäpers, and T. Ackemann, Phys. Rev. E 61, 4622 (2000).
[CrossRef]

Sirutkaitis, V.

K. Staliunas, M. Peckus, and V. Sirutkaitis, Phys. Rev. A 76, 051803(R) (2007).
[CrossRef]

Slekys, G.

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

Staliunas, K.

K. Staliunas, M. Peckus, and V. Sirutkaitis, Phys. Rev. A 76, 051803(R) (2007).
[CrossRef]

K. Staliunas, Phys. Rev. A 70, 016602 (2004).
[CrossRef]

K. Staliunas, Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997).
[CrossRef]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

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

Taki, M.

V. A. Brazhnyi, V. V. Konotop, and M. Taki, Phys. Rev. A 80, 043814 (2009).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
[CrossRef] [PubMed]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
[CrossRef]

A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
[CrossRef]

Taranenko, V. B.

V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997).
[CrossRef]

Tredicce, J. R.

Vaupel, M.

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

Walgraef, D.

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

Ward, H.

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
[CrossRef]

Weiss, C. O.

V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997).
[CrossRef]

Chaos (1)

V. A. Brazhnyi, V. V. Konotop, S. Coulibaly, and M. Taki, Chaos 17, 037111 (2007).
[CrossRef] [PubMed]

Europhys. Lett. (1)

S. Residori, T. Nagaya, and A. Petrossian, Europhys. Lett. 63, 531 (2003).
[CrossRef]

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

A. Barsella, C. Lepers, M. Taki, and P. Glorieux, J. Opt. B: Quantum Semiclassical Opt. 1, 64 (1999).
[CrossRef]

Opt. Lett. (1)

Phys. Rep. (1)

F. T. Arecchi, S. Boccaletti, and P. L. Ramazza, Phys. Rep. 318, 1 (1999).
[CrossRef]

Phys. Rev. A (5)

V. B. Taranenko, K. Staliunas, and C. O. Weiss, Phys. Rev. A 56, 1582 (1997).
[CrossRef]

K. Staliunas, M. Peckus, and V. Sirutkaitis, Phys. Rev. A 76, 051803(R) (2007).
[CrossRef]

V. A. Brazhnyi, V. V. Konotop, and M. Taki, Phys. Rev. A 80, 043814 (2009).
[CrossRef]

G. J. de Valcárcel, K. Staliunas, E. Roldán, and V. J. Sánchez-Morcillo, Phys. Rev. A 54, 1609 (1996).
[CrossRef] [PubMed]

K. Staliunas, Phys. Rev. A 70, 016602 (2004).
[CrossRef]

Phys. Rev. E (3)

M. Santagiustina, P. Colet, M. San Miguel, and D. Walgraef, Phys. Rev. E 58, 3843 (1998).
[CrossRef]

H. Ward, M. N. Ouarzazi, M. Taki, and P. Glorieux, Phys. Rev. E 63, 016604 (2000).
[CrossRef]

Yu. A. Logvin, B. Schäpers, and T. Ackemann, Phys. Rev. E 61, 4622 (2000).
[CrossRef]

Phys. Rev. Lett. (5)

K. Staliunas, Phys. Rev. Lett. 91, 053901 (2003).
[CrossRef] [PubMed]

P. L. Ramazza, S. Ducci, and F. T. Arecchi, Phys. Rev. Lett. 81, 4128 (1998).
[CrossRef]

S. Ducci, P. L. Ramazza, W. Gonzàlez-Viñas, and F. T. Arecchi, Phys. Rev. Lett. 83, 5210 (1999).
[CrossRef]

M. Vaupel, A. Maître, and C. Fabre, Phys. Rev. Lett. 83, 5278 (1999).
[CrossRef]

R. Kuszelewicz, I. Ganne, I. Sagnes, G. Slekys, and M. Brambilla, Phys. Rev. Lett. 84, 6006 (2000).
[CrossRef] [PubMed]

Other (5)

Dissipative Solitons: from Optics to Biology and Medicine, Lecture Notes in Physics, N.Akhmediev and A.Ankiewicz, eds. (Springer, 2008), Vol. 751.

L. A. Lugiato, M. Brambilla, and A. Gatti, in Advances in Atomic, Molecular and Optical Physics, B.Bederson and H.Walters, eds. (Academic, 1998), Vol. 40.

N. N. Rosanov, in Spatial Hysteresis and Optical Patterns (Springer, 2002).

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

Focus Issue on Dissipative Localized Structures in Extended Systems, Chaos 17, M.Tlidi, T.Kolokolnikov, and M.Taki, eds. (2007).

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Figures (1)

Fig. 1
Fig. 1

In the upper panel we show q 0 as a function of ( Δ 1 ) in the presence of periodic modulations V 1 ( x ) = 8 cos ( 2 x ) (thick curve). The neutral curve obtained in the absence of modulations is also shown (thin curve) for reference. The stability windows corresponding to gaps are shown by thick dashed lines and by shadowed domains. In the lower panels we show the neutral stability curves obtained in the first (A), second (B), and third (C) lowest bands. The thick, dashed, and dashed-dotted curves correspond to the first, second, and third bands. The horizontal dotted lines indicate A 0 ( th ) = 1 2 .

Equations (5)

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

i τ A 0 = ( Δ 0 i γ ) A 0 + 1 2 L 0 A 0 i A 1 2 + i E ( x ) ,
i τ A 1 = ( Δ 1 i ) A 1 + L 1 A 1 + 2 i A 0 A ¯ 1 ,
μ = μ ν q ( ± ) = i ± ( E ν q ( 1 ) + Δ 1 ) 2 4 ( A 0 ( st ) ) 2 .
( E ν 0 q 0 ( 1 ) + Δ 1 ) | E ν 0 q ( 1 ) q | q = q 0 = 0 .
τ A 1 2 [ v ν 0 q 0 ( 1 ) ] 2 x 2 A 2 ε 2 A + χ A | A | 2 = 0 .

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