Abstract

We study interaction of counterpropagating beams in truncated two-dimensional photonic lattices induced optically in photorefractive crystals, and demonstrate the existence of counterpropagating surface solitons localized in the lattice corners and at the edges. We display intriguing dynamical properties of such composite optical beams and reveal that the lattice surface provides a strong stabilization effect on the beam propagation. We also observe dynamical instabilities for stronger coupling and longer propagation distances in the form of beam splitting. No such instabilities exist in the single beam surface propagation.

© 2009 Optical Society of America

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  1. S. Suntsov et al., "Observation of one- and two-dimensional discrete surface spatial solitons," J. Nonlinear Opt. Phys. Mater. 16, 401 (2007).
    [CrossRef]
  2. Yu. S. Kivshar, "Nonlinear Tamm states and surface effects in periodic photonic structures," Laser Phys. Lett. 5, 703 (2008).
    [CrossRef]
  3. X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
    [CrossRef] [PubMed]
  4. A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
    [CrossRef]
  5. A. Ferrando, M. Zacar’es, P. F. de C’ordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
    [CrossRef]
  6. Y. Silberberg and I. Bar Joseph, "Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541 (1982).
    [CrossRef]
  7. M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993).
    [CrossRef]
  8. O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
    [CrossRef] [PubMed]
  9. M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
    [CrossRef]
  10. See, e.g., a recent review: M. S. Petrovic, M.R. Belic, C. Denz, and Yu. S. Kivshar, "Counterpropagating optical beams and solitons," to appear in Laser & Photonics Reviews (2010). http://www.arXiv.org, Preprint [physics.optics] arXiv:0910.4700
  11. E. Smirnov,M. Stepic, C. E. Ruter, V. Shandarov, and D. Kip, "Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array," Opt. Lett. 32, 512 (2007).
    [CrossRef] [PubMed]
  12. S. Koke, D. Trager, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, "Stabilization of counterpropagating solitons by photonic lattices," Opt. Express 15, 6279 (2007).
    [CrossRef] [PubMed]
  13. K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
    [CrossRef]
  14. M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, "Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals," Opt. Express 12, 708 (2004).
    [CrossRef] [PubMed]

2008 (1)

Yu. S. Kivshar, "Nonlinear Tamm states and surface effects in periodic photonic structures," Laser Phys. Lett. 5, 703 (2008).
[CrossRef]

2007 (5)

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

S. Suntsov et al., "Observation of one- and two-dimensional discrete surface spatial solitons," J. Nonlinear Opt. Phys. Mater. 16, 401 (2007).
[CrossRef]

E. Smirnov,M. Stepic, C. E. Ruter, V. Shandarov, and D. Kip, "Interaction of counterpropagating discrete solitons in a nonlinear one-dimensional waveguide array," Opt. Lett. 32, 512 (2007).
[CrossRef] [PubMed]

S. Koke, D. Trager, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, "Stabilization of counterpropagating solitons by photonic lattices," Opt. Express 15, 6279 (2007).
[CrossRef] [PubMed]

2005 (1)

A. Ferrando, M. Zacar’es, P. F. de C’ordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
[CrossRef]

2004 (1)

2003 (2)

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

2002 (1)

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

1993 (1)

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993).
[CrossRef]

1982 (1)

Y. Silberberg and I. Bar Joseph, "Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Arsenovic, D.

Bar Joseph, I.

Y. Silberberg and I. Bar Joseph, "Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Belic, M.

M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, "Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals," Opt. Express 12, 708 (2004).
[CrossRef] [PubMed]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

Bezryadina, A.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Carmon, T.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Chen, M.

Chen, Z.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Christodoulides, D. N.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Cohen, O.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Denz, C.

S. Koke, D. Trager, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, "Stabilization of counterpropagating solitons by photonic lattices," Opt. Express 15, 6279 (2007).
[CrossRef] [PubMed]

M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, "Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals," Opt. Express 12, 708 (2004).
[CrossRef] [PubMed]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

Desyatnikov, A.

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

Dreisow, F.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Ferrando, A.

A. Ferrando, M. Zacar’es, P. F. de C’ordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Fleischer, J. W.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Haelterman, M.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993).
[CrossRef]

Jander, Ph.

S. Koke, D. Trager, Ph. Jander, M. Chen, D. N. Neshev, W. Krolikowski, Yu. S. Kivshar, and C. Denz, "Stabilization of counterpropagating solitons by photonic lattices," Opt. Express 15, 6279 (2007).
[CrossRef] [PubMed]

M. Belic, M. Petrovic, D. Jovic, A. Strinic, D. Arsenovic, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, "Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals," Opt. Express 12, 708 (2004).
[CrossRef] [PubMed]

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

Jovic, D.

Kaiser, F.

Kartashov, Y. V.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Kivshar, Yu. S.

Koke, S.

Krolikowski, W.

Makris, K. G.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Mandel, P.

Motzek, K.

Neshev, D. N.

Nolte, S.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Odulov, S.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Pertsch, T.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Petrovic, M.

Segev, M.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Sheppard, A. P.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993).
[CrossRef]

Silberberg, Y.

Y. Silberberg and I. Bar Joseph, "Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

Smirnov, E.

Snyder, A. W.

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993).
[CrossRef]

Stegeman, G.I.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Strinic, A.

Suntsov, S.

S. Suntsov et al., "Observation of one- and two-dimensional discrete surface spatial solitons," J. Nonlinear Opt. Phys. Mater. 16, 401 (2007).
[CrossRef]

Szameit, A.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

T¨unnermann, A.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Tlidi, M.

Torner, L.

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Trager, D.

Uzdin, R.

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

Wang, X.

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

Zacar’es, M.

A. Ferrando, M. Zacar’es, P. F. de C’ordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
[CrossRef]

J. Nonlinear Opt. Phys. Mater. (1)

S. Suntsov et al., "Observation of one- and two-dimensional discrete surface spatial solitons," J. Nonlinear Opt. Phys. Mater. 16, 401 (2007).
[CrossRef]

Laser. Phys. Lett. (1)

Yu. S. Kivshar, "Nonlinear Tamm states and surface effects in periodic photonic structures," Laser Phys. Lett. 5, 703 (2008).
[CrossRef]

Opt. Commun. (1)

M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary waves," Opt. Commun. 103, 145 (1993).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E (3)

M. Belic, Ph. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, 025601 (2003).
[CrossRef]

K. Motzek, Ph. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066611 (2003).
[CrossRef]

A. Ferrando, M. Zacar’es, P. F. de C’ordoba, D. Binosi, and A. Montero, "Forward-backward equations for nonlinear propagation in axially invariant optical systems," Phys. Rev. E 71, 016601 (2005).
[CrossRef]

Phys. Rev. Lett. (4)

Y. Silberberg and I. Bar Joseph, "Instabilities, self-oscillations, and chaos in a simple nonlinear optical interaction," Phys. Rev. Lett. 48, 1541 (1982).
[CrossRef]

O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odulov, "Collisions between optical spatial solitons propagating in opposite directions," Phys. Rev. Lett. 89, 133901 (2002).
[CrossRef] [PubMed]

X. Wang, A. Bezryadina, Z. Chen, K. G. Makris, D. N. Christodoulides, and G.I. Stegeman,"Observation of two-dimensional surface solitons," Phys. Rev. Lett. 98, 123903 (2007).
[CrossRef] [PubMed]

A. Szameit, Y. V. Kartashov, F. Dreisow, T. Pertsch, S. Nolte, A. T¨unnermann, and L. Torner, "Observation of two-dimensional surface solitons in asymmetric waveguide arrays," Phys. Rev. Lett. 98, 173903 (2007).
[CrossRef]

Other (1)

See, e.g., a recent review: M. S. Petrovic, M.R. Belic, C. Denz, and Yu. S. Kivshar, "Counterpropagating optical beams and solitons," to appear in Laser & Photonics Reviews (2010). http://www.arXiv.org, Preprint [physics.optics] arXiv:0910.4700

Supplementary Material (8)

» Media 1: AVI (291 KB)     
» Media 2: AVI (252 KB)     
» Media 3: AVI (253 KB)     
» Media 4: AVI (1273 KB)     
» Media 5: AVI (284 KB)     
» Media 6: AVI (259 KB)     
» Media 7: AVI (267 KB)     
» Media 8: AVI (266 KB)     

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Figures (5)

Fig. 1.
Fig. 1.

(a) Problem geometry with the forward (F) and the backward (B) beams. (b,c) Sketches of the surface modes localized in the lattice corner or at the edge, respectively.

Fig. 2.
Fig. 2.

Corner surface modes, for different coupling constants. Top row: Intensity distributions of the forward field at its output face in the direct space and in the inverse space (insets). The layout of the lattice beams is only shown in the last figure. Bottom row: The same intensity distributions in 3D. The propagation distance is L=2.5LD =10mm. FWHM=14 µm of the input Gaussian beams, lattice spacing d=23 µm, |F 0|2=|BL |2=1, the maximum lattice intensity I 0=3.

Fig. 3.
Fig. 3.

Typical behavior of corner surface solitons. Symbols in the inset present possible outcomes. Blue dots denote the stable corner solitons, red triangles represent the diffracting modes, and green lozenges stand for the unstable modes.

Fig. 4.
Fig. 4.

Edge surface modes, as the coupling constant is increased. The figure layout is as in Fig. 2. The parameters are also as in Fig. 2.

Fig. 5.
Fig. 5.

Dynamical behavior of surface modes in two-dimensional lattice. Upper row: corner beams. Lower row: edge beams. Movies of the intensity distributions of both forward and backward fields are shown along the propagation direction, in time: (a) (Media 1), (b) (Media 2), (c) (Media 3), (d) (Media 4), (e) (Media 5), (f) (Media 6), (g) (Media 7), (h)(Media 8), for different coupling constants Γ. The cases (a) and (e) are zoomed, to display more clearly the diffraction of beams. Other parameters are as in Fig. 2.

Equations (3)

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

i F z = Δ F + Γ EF , i B z = Δ B + Γ E B ,
τ E t + E = I ( 1 + I ) ,
τ E t + E = ( I + I g ) ( 1 + I + I g ) .

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