## Counterpropagating optical vortices in photorefractive crystals

Optics Express, Vol. 13, Issue 12, pp. 4379-4389 (2005)

http://dx.doi.org/10.1364/OPEX.13.004379

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### Abstract

We present a comprehensive numerical study of (2+1)D counter-propagating incoherent vortices in photorefractive crystals, in both space and time. We consider a local isotropic dynamical model with Kerr-type saturable nonlinearity, and identify the corresponding conserved quantities. We show, both analytically and numerically, that stable beam structures conserve angular momentum, as long as their stability is preserved. As soon as the beams loose stability, owing to radiation or non-elastic collisions, their angular momentum becomes non-conserved. We discover novel types of rotating beam structures that have no counterparts in the copropagating geometry. We consider the counterpropagation of more complex beam arrangements, such as regular arrays of vortices. We follow the transition from a few beam propagation behavior to the transverse pattern formation dynamics.

© 2005 Optical Society of America

## 1. Introduction

3. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A **45**, 8185 (1992). [CrossRef] [PubMed]

*spinning solitons*, their actual spin is more properly connected to their polarization state. In nonlinear self-focusing settings they appear as ring-like beams, displaying a phase singularity at the beam’s center: the intensity is vanishing at the center (and at the transverse infinity), and the phase is increasing counter-clockwise from zero to 2

*π*. The nagging problem in the studies of propagation of such soliton-like structures has been their stability [6].

7. M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. **78**, 2551 (1997). [CrossRef]

8. M. Shih and M. Segev, “Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. **21**, 1538 (1996). [CrossRef] [PubMed]

9. A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-focusing media,” Phys. Rev. A **57**, 522 (1998). [CrossRef]

*z*, the propagation direction; no real time has been involved. Temporal development of copropagating solitons was considered in only a few publications and in one transverse dimension (1D), displaying approach to steady state [10

10. N. Fressengeas, J. Maufoy, and G. Kugel, “Temporal behavior of bidimensional photorefractive bright spatial solitons,” Phys. Rev. E **54**, 6866 (1996). [CrossRef]

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

15. C. Rotschild, O. Cohen, O. Mandela, T. Carmon, and M. Segev, “Interactions between spatial screening solitons propagating in opposite directions,” J. Opt. Soc. Am. B **21**, 1354 (2004). [CrossRef]

16. M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E **68**, 025601 (2003). [CrossRef]

19. M. Belić, Ph. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B: Quantum Semiclass. Opt. **6**, S190–S196 (2004). [CrossRef]

20. D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E **58**, 3916 (1998). [CrossRef]

25. D. Briedis, D. E. Petersen, D. Edmunson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express **13**, 435 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-435 [CrossRef] [PubMed]

*not*remain stable as they propagate. They break up into a number (two, three, four, five) of fragments, after a propagation of several diffraction lengths. Different mechanisms have been proposed for improving their stability, for example the inclusion of nonlocal interaction in the medium [25

25. D. Briedis, D. E. Petersen, D. Edmunson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express **13**, 435 (2005). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-435 [CrossRef] [PubMed]

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

19. M. Belić, Ph. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B: Quantum Semiclass. Opt. **6**, S190–S196 (2004). [CrossRef]

*z*and

*t*. Novel types of rotating beam structures are discovered that have no counterpart in the copropagating geometry. The propagation of more complex CP beam arrangements, such as regular arrays of vortices, is also considered, and the transition from a few beam propagation behavior to the transverse pattern formation dynamics is followed.

## 2. The model

16. M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E **68**, 025601 (2003). [CrossRef]

*F*and

*B*are the forward and the backward propagating beam envelopes,

*Δ*is the transverse Laplacian,

*Γ*is the dimensionless coupling strength, and

*E*the homogenous part of the space charge field. The relaxation time of the crystal

*τ*also depends on the total intensity,

*τ*=

*τ*

_{0}/(1+

*I*), where

*τ*

_{0}is the dielectric relaxation time under

*uniform*background illumination. The quantity

*I*=|

*F*|

^{2}+|

*B*|

^{2}is the laser light intensity, measured in units of the background intensity. A scaling

*x*/

*x*

_{0}→

*x*,

*y*/

*x*

_{0}→

*y*,

*z*/

*L*

_{D}→

*z*, is utilized in writing the propagation equations, where

*x*

_{0}is the typical FWHM beam waist and

*L*

_{D}is the diffraction length. In our simulations we choose

*x*

_{0}=10

*µm*, so that for the light from the Nd:YAG laser

*L*

_{D}≈5

*mm*. The assumption is that the counterpropagating components interact only through the intensity-dependent space charge field. To make matters simple, we did not account for the temperature (diffusion) effects, although they are found to influence the interaction of CP beams [14

14. O. Cohen, S. Lan, and T. Carmon, “Spatial vector solitons consisting of counterpropagating fields,” Opt. Lett. **27**, 2013 (2002). [CrossRef]

26. K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, Ph. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: Anisotropic theory,” Phy. Rev. E **71**, 016610 (2005). [CrossRef]

18. M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, 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). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-708 [CrossRef] [PubMed]

## 3. Conserved quantities

*z*, integrated over the transverse coordinates, and time is the scaled

*t*/

*τ*. Applying the standard methods of dynamical analysis, we identify the following constants of motion.

*I*

_{tot}. By noting that:

*I*

_{tot}=

*I*

^{F}+

*I*

^{B}is independent of the propagation distance and time, and it does not depend on the input beam profiles.

*z*:

*E*=

*E*(

*I*)=

*E*(

*I*(

*x*,

*y*,

*z*)) depends only on the steady-state intensity. In this case, the total momentum is determined by the integral of input beam profiles only, and as they do not change in time, the total momentum also does not change in time.

*z*axis is given by:

*x*=

*ρ*cos

*φ*,

*y*=

*ρ*sin

*φ*:

*E*≠

*E*(

*φ*), it follows

*∂z*=

*∂z*=0, and hence the total angular momentum is a conserved quantity. In our model

*E*depends on

*I*, and if

*I*≠

*I*(

*φ*) then

*∂z*=0. However, when the modulational instability (MI) sets in, there appear azimuthal instabilities, and the angular momentum becomes non-conserved. It acquires

*z*-dependence, as well as

*t*-dependence, as MI is not uniformly distributed.

*E*=-

*I*/(1+

*I*), we can write the Hamiltonian

*H*for our system:

*∂H*/

*∂z*=0, which means that in this case the Hamiltonian represents an integral of motion (as expected).

*z*, are also constant in time. This is normal expectation when the transverse integrals are over the

*total*transverse space. In numerics, however, the total integrated transverse space includes only the total computational space, and there is always some transverse leakage involved. There are situations where this leakage is minimal, as for the stable soliton propagation, but there are situations where the leakage is larger, as in the case when the system radiates. In our case, whenever we have a stable soliton-like propagation of vortices, there is practically no leakage, and whenever such propagation becomes unstable, due to MI or radiation losses, the leakage becomes noticeable, and the conservation

*in time*is lost. This is most visible in the case of angular momentum, where even a tiny amount of radiation carries huge amount of momentum to transverse infinity. As mentioned, the momentum starts to vary in time, and also becomes

*z*-dependent, as the radiation is not uniform along the

*z*-axis. On symmetry grounds, when MI sets in the cylindrical symmetry is lost, and the angular momentum starts to change in

*z*. If the new symmetry of beams becomes a well defined discrete symmetry (tripole, quadrupole) one can define

*new*conserved angular

*quasimomenta*.

## 4. Transverse instabilities and stable structures

*Γ*or propagation distance

*L*we observe stable CP vortices. Nevertheless, when they break, they form very different stable

*filamented*structures in propagating over finite distances, corresponding to typical photorefractive crystal thicknesses, which are of the order of few

*L*

_{D}. In addition, they can form different stable

*dynamical*structures, such as stable rotating dipoles. It should be noted that in CP geometries, the absolute stability of propagation over indefinite distance is of secondary importance; the influence of both input faces, at any distance, must be felt equally. Hence, stable steady or dynamical structures arising over finite distances are of considerable experimental interest.

*1*. For lower values of

*Γ*or

*L*we see stable vortex propagation over the distances of interest (a few

*L*

_{D}). One can notice in the figure a narrow

*threshold*region which separates the stable vortices from other structures. The shape of the threshold region follows the general

*ΓL*=

*const*. form we derived earlier for the 1D case, in our papers [17

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

18. M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, 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). http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-708 [CrossRef] [PubMed]

*same*topological charge tend to form standing waves, whereas the vortices with the

*opposite*charges tend to form rotating structures. For higher values of the parameters, we identify the following quasi-stable situations: the transformation of a quasi-stable quadrupole into a stable tripole, several transformations of quadrupoles into quadrupoles, and a stable rotating dipole. Above the quasi-stable region, CP vortices produce unstable structures, i.e. constantly changing structures of unrecognizable shape.

*t*as well as in

*z*, as soon as the propagation becomes unstable.

*t*/

*τ*=200. They do not change in time. One can clearly see, especially in the case of stable dipole, the

*spiraling*of beam arms along the

*z*axis, which has been described previously in a number of papers treating

*copropagating*vortices and pairs of solitons [6,7

7. M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. **78**, 2551 (1997). [CrossRef]

27. Alexander V. Buryak, Yuri S. Kivshar, Ming-feng Shih, and Mordchai Segev, “Induced Coherence and Stable Soliton Spiraling,” Phy. Rev. Lett **82**, 81 (1999). [CrossRef]

## 5. Stable rotating structures

*breakup*of vortices, whereas in this case we have no vortices, but two Gaussian beams that carry no charge. Each beam collapses to a displaced soliton-like beam, which then starts to rotate

*indefinitely*. What is very interesting here is that, after the displaced soliton-like beams are formed and start to rotate, their phases acquire topological defect charges ±2

*π*(so that the total momentum is still conserved). This feature is typical of vortices. The splitup behavior is similar to the typical behavior of CP head-on Gaussian beams, which are known [17

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

19. M. Belić, Ph. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B: Quantum Semiclass. Opt. **6**, S190–S196 (2004). [CrossRef]

*not*followed by uniform rotation.

## 6. More complex beam structures

*lattice-like*arrangements is many-fold, in that one can address each pixel (its phase and intensity), monitor the interaction of

*copropagating*vortices by adjusting the distance between them, and control the interaction of CP beam arrays by choosing the distribution of charges and the position of arrays relative to each other. An additional point of fundamental interest is to follow what goes on in the inverse space, as the geometry changes from few CP beams to many CP beams.

*in-phase*, or pair-wise diagonally

*out-of-phase*. It is easy to distinguish the two arrangements in Figs. 7 and 8: the in-phase fragments rotate in the same sense, the out-of-phase rotate in the opposite sense. We also choose the forward and backward fields to be of the opposite charges, and position individual vortices head-on. Such geometries allow for

*stable*rotating structures.

*F*+

*B*, respectively. As it can be seen, because the angular moments of the forward and backward beams are symmetrical, the sum of the moments is not equal to the momentum of the sum. Also, large excursions in the moments are visible. This is expected in the system where four beams, each carrying a unit of momentum, collide, exchange momentum, and strongly radiate. There can be no question of angular momentum conservation in this strongly interacting nonintegrable system.

*regular*grouping of spots in the direct space. Although in both cases we deal with 4-on-4 vortices, which disintegrate into 4 fragments each, in the out-of-phase case the fragments conspire so that at any moment only 6 approximately hexagonal spots are highly visible, whereas in the in-phase case all 16 spots are visible at all times. This has consequence on the appearance of the inverse space distributions: in the out-of-phase case one still sees 6 prominent spots (and a number of satellites) reminiscent of the direct space distribution, whereas in the in-phase case one sees a square arrangement of spots, 4 of them more prominent, corresponding to the square-lattice-like arrangement in the direct space.

## 7. Counterpropagating arrays of vortices

*collective*behavior of all beams.

*large*distances, each pair of CP vortices will behave independently of the others, in the manner described above. As the distance is reduced, the interaction of

*copropagating vortices*starts to affect the interaction of CP vortices. As a rule, this interaction precipitates the, already present, instabilities of the basic CP two-beam system. In CP arrays, the filamentation of vortices happens sooner, at smaller

*z*distances, and at lower values of the

*Γ*parameter. The vortices break into four-petal clusters, as in the basic CP system. In addition, the presence of arrays brings to the fore collective features of the behavior of the system. Its evolution starts to acquire the standard

*pattern formation*characteristics [28

28. C. Denz, M. Schwab, and C. Weilnau, *Transverse pattern formation in photorefractive optics* (Springer, Berlin, 2003). [CrossRef]

*k*modes, corresponding to the symmetry of the lattice, and an increasingly sharp distribution of these spots in the inverse space. One can also follow the spatial and temporal

*chaotization*of transverse patterns, as the driving parameters change. Some of these aspects are exemplified in Figs. 10 and 11.

*spatially*chaotic, evolving into a

*granular*structure, in which different regions seem to execute different quasi-periodic motion. The chaotization proceeds from the lattice corners inward, owing to strong edge effects. In the inverse space one still sees the same basic arrangement of spots, corresponding to the underlying square arrangement of the vortices, however now with more irregular distribution of the additional higher-order mode and satellite spots, and with the temporal periodicity less apparent.

## 8. Conclusions

*z*and

*t*. Novel types of rotating beam structures are discovered that have no counterpart in the copropagating geometry. The propagation of more complex CP beam arrangements, such as regular arrays of vortices, is also considered, and the transition from a few-beam propagation behavior to the transverse pattern formation dynamics is followed.

## Acknowledgments

## References and links

1. | M. Padgett, J. Courtial, and L. Allen, “Light’s Orbital Angular Momentum,” Phys. Today, May Issue, 35 (2004). |

2. | V. I. Kruglov and R. A. Vlasov, “Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,” Phys. Lett. A |

3. | L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,” Phys. Rev. A |

4. | S. Trillo and W. Torruellas, eds., |

5. | Special Issue on solitons, ed. M. Segev, Opt. Phot. News13, No. 2 (2002). |

6. | Y. S. Kivshar and G. P. Agrawal, |

7. | M. Shih, M. Segev, and G. Salamo, “Three-dimensional spiraling of interacting spatial solitons,” Phys. Rev. Lett. |

8. | M. Shih and M. Segev, “Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,” Opt. Lett. |

9. | A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, “Solitary attractors and low-order filamentation in anisotropic self-focusing media,” Phys. Rev. A |

10. | N. Fressengeas, J. Maufoy, and G. Kugel, “Temporal behavior of bidimensional photorefractive bright spatial solitons,” Phys. Rev. E |

11. | M. Haelterman, A. P. Sheppard, and A. W. Snyder, “Bimodal counterpropagating spatial solitary-waves,” Opt. Commun. |

12. | 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. |

13. | O. Cohen, T. Carmon, M. Segev, and S. Odoulov, “Holographic solitons,” Opt. Lett. |

14. | O. Cohen, S. Lan, and T. Carmon, “Spatial vector solitons consisting of counterpropagating fields,” Opt. Lett. |

15. | C. Rotschild, O. Cohen, O. Mandela, T. Carmon, and M. Segev, “Interactions between spatial screening solitons propagating in opposite directions,” J. Opt. Soc. Am. B |

16. | M. Belić, Ph. Jander, A. Strinić, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional waveguides in a saturable photorefractive medium,” Phys. Rev. E |

17. | K. Motzek, Ph. Jander, A. Desyatnikov, M. Belić, C. Denz, and F. Kaiser, “Dynamic counterpropagating vector solitons in saturable self-focusing media,” Phys. Rev. E |

18. | M. Belić, M. Petrović, D. Jović, A. Strinić, D. Arsenović, K. Motzek, F. Kaiser, Ph. Jander, C. Denz, M. Tlidi, and P. Mandel, “Transverse modulational instabilities of counterpropagating solitons in photorefractive crystals,” Opt. Express |

19. | M. Belić, Ph. Jander, K. Motzek, A. Desyatnikov, D. Jović, A. Strinić, M. Petrović, C. Denz, and F. Kaiser, “Counterpropagating self-trapped beams in photorefractive crystals,” J. Opt. B: Quantum Semiclass. Opt. |

20. | D. V. Skryabin and W. J. Firth, “Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,” Phys. Rev. E |

21. | A. S. Desyatnikov and Y. Kivshar, “Spatial optical solitons and soliton clusters carrying an angular momentum,” J. Opt. B: Quantum Semiclass. Opt. |

22. | M. Belić, D. Vujić, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascossa, “Isotropic vs. anisotropic modeling of photorefractive solitons,” Phys. Rev. E |

23. | A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,” J. Opt. B: Quantum Semiclass. Opt. |

24. | C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, “Partially incoherent optical vortices in self-focusing nonlinear media,” Phys. Rev. Lett. |

25. | D. Briedis, D. E. Petersen, D. Edmunson, W. Krolikowski, and O. Bang, “Ring vortex solitons in nonlocal nonlinear media,” Opt. Express |

26. | K. Motzek, M. Belić, T. Richter, C. Denz, A. Desyatnikov, Ph. Jander, and F. Kaiser, “Counterpropagating beams in biased photorefractive crystals: Anisotropic theory,” Phy. Rev. E |

27. | Alexander V. Buryak, Yuri S. Kivshar, Ming-feng Shih, and Mordchai Segev, “Induced Coherence and Stable Soliton Spiraling,” Phy. Rev. Lett |

28. | C. Denz, M. Schwab, and C. Weilnau, |

**OCIS Codes**

(190.5330) Nonlinear optics : Photorefractive optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Research Papers

**History**

Original Manuscript: April 18, 2005

Revised Manuscript: May 24, 2005

Published: June 13, 2005

**Citation**

D. Jovi�?, D. Arsenovi�?, A. Strini�?, M. Beli�?, and M. Petrovi�?, "Counterpropagating optical vortices in photorefractive crystals," Opt. Express **13**, 4379-4389 (2005)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-13-12-4379

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### References

- M. Padgett, J. Courtial, and L. Allen, �??Light�??s Orbital Angular Momentum,�?? Phys. Today, May Issue, 35 (2004).
- V. I. Kruglov, and R. A. Vlasov, �??Spiral self-trapping propagation of optical beams in media with cubic nonlinearity,�?? Phys. Lett. A 111, 401 (1985). [CrossRef]
- L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, �??Orbital angular momentum of light and the transformation of Laguerre-Gaussian laser modes,�?? Phys. Rev. A 45, 8185 (1992). [CrossRef] [PubMed]
- S. Trillo, and W. Torruellas, eds., Spatial Solitons (Springer, New York, 2001).
- Special Issue on solitons, ed. M. Segev, Opt. Phot. News 13, No. 2 (2002).
- Y. S. Kivshar, and G. P. Agrawal, Optical Solitons, Academic Press, London (2003).
- M. Shih, M. Segev, and G. Salamo, �??Three-dimensional spiraling of interacting spatial solitons,�?? Phys. Rev. Lett. 78, 2551 (1997). [CrossRef]
- M. Shih, and M. Segev, �??Incoherent collisions between two-dimensional bright steady-state photorefractive spatial screening solitons,�?? Opt. Lett. 21, 1538 (1996). [CrossRef] [PubMed]
- A. A. Zozulya, D. Z. Anderson, A. V. Mamaev, and M. Saffman, �??Solitary attractors and low-order filamentation in anisotropic self-focusing media,�?? Phys. Rev. A 57, 522 (1998). [CrossRef]
- N. Fressengeas, J. Maufoy, and G. Kugel, �??Temporal behavior of bidimensional photorefractive bright spatial solitons,�?? Phys. Rev. E 54, 6866 (1996). [CrossRef]
- M. Haelterman, A. P. Sheppard, and A. W. Snyder, �??Bimodal counterpropagating spatial solitary-waves,�?? Opt. Commun. 103, 145 (1993). [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]
- O. Cohen, T. Carmon, M. Segev, and S. Odoulov, �??Holographic solitons,�?? Opt. Lett. 27, 2031 (2002). [CrossRef]
- O. Cohen, S. Lan, and T. Carmon, �??Spatial vector solitons consisting of counterpropagating fields,�?? Opt. Lett. 27, 2013 (2002). [CrossRef]
- C. Rotschild, O. Cohen, O.Mandela, T. Carmon, and M. Segev, �??Interactions between spatial screening solitons propagating in opposite directions,�?? J. Opt. Soc. Am. B 21, 1354 (2004). [CrossRef]
- M. Beli�?, Ph. Jander, A. Strini�? 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. Beli�?, C. Denz, and F. Kaiser, �??Dynamic counterpropagating vector solitons in saturable self-focusing media,�?? Phys. Rev. E 68, 066611 (2003). [CrossRef]
- M. Beli�?, M. Petrovi�?, D. Jovi�?, A. Strini�?, D. Arsenovi�?, 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). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-708">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-4-708</a> [CrossRef] [PubMed]
- M. Beli�?, Ph. Jander, K. Motzek, A. Desyatnikov, D. Jovi�?, A. Strini�?, M. Petrovi�?, C. Denz, and F. Kaiser, �??Counterpropagating self-trapped beams in photorefractive crystals,�?? J. Opt. B: Quantum Semiclass. Opt. 6, S190�??S196 (2004). [CrossRef]
- D. V. Skryabin, and W. J. Firth, �??Dynamics of self-trapped beams with phase dislocation in saturable Kerr and quadratic nonlinear media,�?? Phys. Rev. E 58, 3916 (1998). [CrossRef]
- A. S. Desyatnikov and Y. Kivshar, �??Spatial optical solitons and soliton clusters carrying an angular momentum,�?? J. Opt. B: Quantum Semiclass. Opt. 4, S58 (2002). [CrossRef]
- M. Beli�?, D. Vuji�?, A. Stepken, F. Kaiser, G. F. Calvo, F. Agullo-Lopez, and M. Carrascossa, �??Isotropic vs. anisotropic modeling of photorefractive solitons,�?? Phys. Rev. E 65, 066610 (2002). [CrossRef]
- A. V. Mamaev, M. Saffman, and A. A. Zozulya, �??Propagation of a mutually incoherent optical vortex pair in anisotropic nonlinear media,�?? J. Opt. B: Quantum Semiclass. Opt. 6, S318�??S322 (2004). [CrossRef]
- C. C. Jeng, M. F. Shih, K. Motzek, and Y. Kivshar, �??Partially incoherent optical vortices in self-focusing nonlinear media,�?? Phys. Rev. Lett. 92, 043904 (2004). [CrossRef] [PubMed]
- D. Briedis, D. E. Petersen, D. Edmunson, W. Krolikowski, and O. Bang, �??Ring vortex solitons in nonlocal nonlinear media,�?? Opt. Express 13, 435 (2005). <a href="http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-435">http://www.opticsexpress.org/abstract.cfm?URI=OPEX-13-2-435</a> [CrossRef] [PubMed]
- K. Motzek, M. Beli�?, T. Richter, C. Denz, A. Desyatnikov, Ph. Jander, and F. Kaiser, �??Counterpropagating beams in biased photorefractive crystals: Anisotropic theory,�?? Phy. Rev. E 71, 016610 (2005). [CrossRef]
- Alexander V. Buryak, Yuri S. Kivshar, Ming-feng Shih, and Mordchai Segev, �??Induced Coherence and Stable Soliton Spiraling,�?? Phy. Rev. Lett 82, 81 (1999). [CrossRef]
- C. Denz, M. Schwab, and C. Weilnau, Transverse pattern formation in photorefractive optics (Springer, Berlin, 2003). [CrossRef]

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