## Stabilization of counterpropagating solitons by photonic lattices

Optics Express, Vol. 15, Issue 10, pp. 6279-6292 (2007)

http://dx.doi.org/10.1364/OE.15.006279

Acrobat PDF (2948 KB)

### Abstract

We report on the stabilization of inherently unstable counterpropagating photorefractive spatial solitons by the use of one- and two-dimensional photonic lattices. We numerically investigate the dependence of the instability dynamics on period and amplitude of the lattice and present experimental verification for the dynamic stabilization of the bi-directional soliton state.

© 2007 Optical Society of America

## 1. Introduction

19. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in
linear and nonlinear waveguide lattices,”
Nature **424**, 817–823
(2003). [CrossRef] [PubMed]

20. A. L. Jones, “Coupling of optical fibers and
scattering in fibers,” J. Opt. Soc. Am. **55**, 261–271
(1965). [CrossRef]

21. S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide
directional couplers,” Appl. Phys. Lett. **22**, 46–48
(1973). [CrossRef]

19. D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in
linear and nonlinear waveguide lattices,”
Nature **424**, 817–823
(2003). [CrossRef] [PubMed]

22. D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear
arrays of coupled waveguides,” Opt. Lett. **13**, 794–796
(1988). [CrossRef] [PubMed]

23. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive
optically induced photonic lattices,”
Phys. Rev. E **66**, 046,602-5 (2002). [CrossRef]

24. H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in
waveguide arrays,” Phys. Rev. Lett. **81**(16), 3383–3386
(1998). URL
http://link.aps.org/abstract/PRL/v81/p3383. [CrossRef]

25. J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in
optically induced real time waveguide arrays,”
Phys. Rev. Lett. **90**(2), 023,902-4 (2003). URL
http://link.aps.org/abstract/PRL/v90/p023902. [CrossRef]

28. D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide
arrays,” Phys. Rev. Lett. **92**(9), 093,904-4 (2004). URL
http://link.aps.org/abstract/PRL/v92/p093904. [CrossRef]

29. D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, “Controlled generation and steering
of spatial gap solitons,” Phys. Rev.
Lett. **93**(8), 083,905-4 (2004). URL
http://link.aps.org/abstract/PRL/v93/p083905. [CrossRef]

23. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive
optically induced photonic lattices,”
Phys. Rev. E **66**, 046,602-5 (2002). [CrossRef]

30. E. Smirnov, M. Stepic, C. Ruter, V. Shandarov, and D. Kip, “Interaction of counterpropagating
discrete solitons in a nonlinear one-dimensional waveguide
array,” Opt. Lett. **32**, 512–514
(2007). [CrossRef] [PubMed]

## 2. Isotropic one-dimensional model

31. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-State Spatial Screening
Solitons in Photorefractive Materials with External Applied
Field,” Phys. Rev. Lett. **73**, 3211–3214
(1994). [CrossRef] [PubMed]

*F*and

*B*(|

*∂*

^{2}

*•|≪|2*

_{z}*ik*

_{0}

*∂*•|,

_{z}*∂*• ≃ 0,

_{t}*∂*

^{2}

*•≃0 where • stands for*

_{t}*F*and

*B*, respectively) of the forward and backward propagating beams is modeled in paraxial approximation:

*E*is the space charge field created by the beams inside the photorefractive crystal and Γ =

^{sc}*k*

^{2}

*n*

^{2}

_{0}

*w*

^{2}

_{0}

*r*

_{eff}

*E*denotes the photorefractive coupling constant, with

_{e}*n*

_{0}being the bulk refractive index,

*r*

_{eff}the effective coefficient of the electro-optic tensor and

*E*the externally applied electric field necessary for the screening effect to occur. The transverse

_{e}*x*-coordinate is scaled to the beam waist

*w*

_{0}whereas the propagation

*z*-coordinate is scaled with the diffraction length

*L*= 2

_{D}*kw*

^{2}

_{0}, with

*k*= 2

*πn*

_{0}/

*λ*and

*λ*denoting the laser wavelength [10

10. M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, “Self-trapped bidirectional
waveguides in a saturable photorefractive
medium,” Phys. Rev. E **68**, R025,601-4 (2003). [CrossRef]

*I*= |

*F*|

^{2}+ |

*B*|

^{2}is the total intensity scaled to the so-called dark intensity

*I*and

_{d}*E*is scaled to

^{sc}*E*. The relaxation time

_{e}*τ*depends on the total intensity as

*τ*(

*I*) =

*τ*

_{0}/(1+

*I*) with

*τ*

_{0}representing the dark relaxation time constant.

33. M. Carvalho, S. Singh, and D. Christodoulides, “Self-deflection of steady-state
bright spatial solitons in biased photorefractive
crystals,” Opt. Commun. **120**(5), 311–315
(1995). [CrossRef]

36. W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial
Solitons in Photorefractive Media,” Phys.
Rev. Lett. **80**, 3240–3243
(1998). [CrossRef]

*I*

_{pot}(

*x*) =

*A*cos

^{2}(

*πx*/

*p*) in the expression for the total light intensity

*I*= |

*F*|

^{2}+ |

*B*|

^{2}+

*I*

_{pot}, where

*A*and

*p*determine the peak intensity and periodicity of the lattice. As the lattice-forming waves are ordinary polarized, they do not interact directly with solitons [23

23. N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive
optically induced photonic lattices,”
Phys. Rev. E **66**, 046,602-5 (2002). [CrossRef]

*E*is simulated by using a second order split-step Fourier beam propagation method [37,38

^{sc}38. J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive
nonlinear optical system,” Phys. Rev. A **50**, 3471–3485,
(1994). [CrossRef] [PubMed]

^{-10}s) and response time of the photorefractive effect (on the order of 10

^{-3}- 1s).

*I*=1 each. We simulate their head on collision by launching both beams at the same lateral position perpendicularly to the crystal face. The propagation along a crystal of length

*L*= 5 and transverse extension of 20 was calculated on a grid consisting of 500 × 512 grid points. The coupling constant was chosen to be negative (Γ = -13.2) to ensure the self-focussing regime necessary for the formation of bright solitons. For this set of parameter values the soliton solutions in bulk media already exhibited very irregular temporal oscillations [12]. The numerical simulations started with only the lattice beams present. After evolving the corresponding space charge field over time period 10

*τ*

_{0}(with the time step of 0. 1

*τ*

_{0}) we launched both soliton forming beams and simulated the evolution of the whole system over time of 5000

*τ*

_{0}. The calculations have been carried out on the Morfeus-GRID at the Westfälische Wilhelms-Universität Münster, with the use of Condor [39

39. M. Litzkow, M. Livny, and M. Mutka, “Condor - A Hunter of Idle
Workstations,” in *ICDCS*, pp.
104–111 (IEEE-CS
Press, 1988). URL
http://www.cs.wisc.edu/condor/.

*level of dynamics*[12] defined as:

*lod*(

^{N}*lod*=0 zero for stationary solutions).

^{N}*p*= 1

*d*, where

*d*= 2 is the beam diameter of the incident Gaussian beams. This can be explained as follows: for small lattice periodicity, the self-trapped beam covers many lattice sites. As a result, the effect of the lattice is weaker, and in the limit

*p*→ 0 the medium can be regarded as homogeneous with higher refractive index. If the periodicity is comparable with the beam diameter, the soliton experiences maximal guiding by the lattice-induced refractive index modulation. For larger periodicity, the region in which the refractive index change is negligible increases, so that in the limit

*p*→ ∞ one ends up again with a homogeneous medium. Consequently the influence of the lattice decreases again. Furthermore, our simulations show that the decrease of the dynamics is not as rapid for smaller periodicity as it is for larger one.

*p*= 1

*d*for three different values of the lattice intensity

*A*. For weaker lattice the output oscillates irregularly. These oscillations are similar to those occurring in a bulk continuous medium. The presence of the lattice leads to solitons residing more frequently at the lattice sites what is reflected in the appearance of faint horizontal lines in Fig. 3(a). For stronger lattice we still observe some transient dynamics with oscillation periods which are quite short [Fig. 3(b)]. During these oscillations, the output couples to the neighboring lattice sites. However, after the initial oscillations the output becomes stationary and resides at the lattice site closest to the input waveguide. For even larger lattice intensity the range of time transient dynamics shortens. Notice that in the stable state a small fraction of the soliton is trapped at the input lattice channel while the majority of soliton power is confined in the neighboring site.

*lod*for small period and large lattice strength. This particular case is depicted in Fig. 5(a). The increase of

^{N}*lod*is caused by regular spatio-temporal oscillations of soliton beams. During these oscillations the solitons focus onto each other in the middle of the medium, then start to deflected to one side and break up. After that the sequence [as depicted in Fig. 5(b)-(d)] repeats with the spatial deflection going to the other side.

^{N}*lod*is due to the onset of fast oscillations. This interpretation can be explained as follows: On the one hand, the beams experience a reduction of the transverse separation of the lattice which decreases the transverse instability and leads to an increased focusing of the beams [Fig. 7 (b)]. On the other hand, the transverse separation increases the interaction region of the beams which supports the instability. In the case of beams launched off-site on lattices with low peak intensities and suitable periodicities, the complex interplay between these two effects combined with the intrinsic attraction to adjacent waveguides leads to the short time scale oscillations observed in Fig. 7. Apart from this, the intensity distribution of the stabilized state inside the crystal is characterized by the combination of attraction to adjacent waveguides and avoidance of the counterpropagating beam.

^{N}## 3. Experimental stabilization of soliton dynamics

8. P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, “Dynamic instability of self-induced
bidirectional waveguides in photorefractive
media,” Opt. Lett. **30**, 750–752
(2005). [CrossRef] [PubMed]

*λ*= 532nm) is split up into two parts with a power of 1

*μ*W each. These beams are focussed onto the two opposite faces of the Cerium-doped Strontium Barium Niobate crystal (SBN:60) and are polarized parallel to the crystal’s

*c*-axis exploiting the large electro-optic tensor element

*r*

_{33}. They propagate through the medium slightly tilted against the crystal’s

*a*-axis (

*L*= 23mm) in order to compensate for beam self-bending. Both beams are made mutually incoherent by reflecting one of them from a piezo-mounted mirror oscillating with a period significantly shorter than the response time of the photorefractive nonlinearity. The SBN crystal is biased by an external DC electric field

*E*= 1.9kV/cm applied along the crystal’s

_{e}*c*-axis [31

31. M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-State Spatial Screening
Solitons in Photorefractive Materials with External Applied
Field,” Phys. Rev. Lett. **73**, 3211–3214
(1994). [CrossRef] [PubMed]

*z*) refractive index modulation. The lattice peak intensity is adjusted by varying the power of the two interfering waves. Figure 9 compares the intensity of the interfering beams with the modulation of a guided plane wave used as a tool for characterization of the refractive index modulation.

*A*= 2.5mW corresponds to

*A*= 0.05 which is significantly lower than 0.60 necessary to stabilize the solitons in numerical simulations.

## 4. Soliton stabilization in two-dimensional photonic lattices

40. R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional
solitons in photonic lattices,” Phys.
Rev. Lett. **96**(2), 023,905-4 (2006). URL
http://link.aps.org/abstract/PRL/v96/p023905. [CrossRef]

41. D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch modes in
two-dimensional photonic lattices,” Opt.
Express **14**(5), 1913–1923
(2006). URL
http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-5-1913. [CrossRef] [PubMed]

42. A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, “Nonlinear photonic lattices in
anisotropic nonlocal self-focusing media,”
Opt. Lett. **30**(8), 869–871
(2005). URL
http://www.opticsinfobase.org/abstract.cfm?URI=ol-30-8-869. [CrossRef] [PubMed]

*μ*m and 18

*μ*m, respectively [Fig. 11(b)]. In our 10 mm long crystal this corresponds to approximately 5 diffraction lengths of linear propagation.

*μ*m) the potential induced by the lattice was too weak to arrest the instability of the counterpropagating beams. With the increased lattice period (to 6, 9, or, 12

*μ*m) the instability was practically removed for a certain range of lattice strength. The large lattice periods, however, strongly reduce the mobility of the beams as each beam can be fully trapped in a single lattice site. Such trapping imposes a constrain on the formation of bi-directional waveguide which becomes sensitive on the initial alignment of the beams. Thus, beams propagating in different directions inside the crystal will not attract, as their intensity overlap will be reduced by the trapping on different lattice sites. Therefore, in the following experiments we concentrated on the intermediate case of a lattice of period 6

*μ*m (

*p*= 0.3

*d*).

*μ*W each beam forms a spatial soliton, where the soliton size is equal to the input beam size. In Fig. 11(c,d) we show the digitally combined input and output of each beam as if they would propagate without interaction inside the crystal. When both beams co-propagate they start to interact. After the initial attraction, the beams exhibit oscillatory dynamics. The oscillations are depicted in Fig. 11(g, solid line) showing the (

*x*,

*y*) position of the output beam center of mass as a function of time. In this experiment we first established the soliton formation of the backward beam

*B*and then we launched the forward propagating beam

*F*. The observed oscillatory dynamics is similar to the 1D case, but appears in both transverse directions. The amplitude of the oscillations is dependent on the initial alignment of the beams.

## 5. Conclusion

## Acknowledgements

## References and links

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Interactions: Universality and Diversity,”
Science |

2. | W. Krolikowski, B. Luther-Davies, and C. Denz, “Photorefractive
Solitons,” IEEE J. Sel. Top. Quantum
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3. | Y. S. Kivshar and G. P. Agrawal, |

4. | D. Träger, N. Sagemerten, and C. Denz, “Guiding of Dynamically Modulated
Signals in Arrays of Photorefractive Spatial
Solitons,” IEEE J. Sel. Top. Quantum
Electron. |

5. | C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, “Spatial optical
(2+1)-dimensional scalar-and vector-solitons in saturable
nonlinear media,” Ann. Phys. |

6. | O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, “Collisions between Optical Spatial
Solitons Propagating in Opposite Directions,”
Phys. Rev. Lett. |

7. | D. Kip, C. Herden, and M. Wesner, “All-Optical Signal Routing Using
Interaction of Mutually Incoherent Spatial
Solitons,” Ferroelectrics |

8. | P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, “Dynamic instability of self-induced
bidirectional waveguides in photorefractive
media,” Opt. Lett. |

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

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waveguides in a saturable photorefractive
medium,” Phys. Rev. E |

11. | M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, “Counterpropagating self-trapped
beams in photorefractive crystals,” J.
Opt. B |

12. | P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, “Dynamic instability of
counterpropagating self-trapped beams in photorefractive
media,” Proc. SPIE |

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

14. | Y. Silberberg and I. B. Joseph, “Instabilities, Self-Oscillation, and
Chaos in a Simple Nonlinear Optical Interaction,”
Phys. Rev. Lett. |

15. | W. J. Firth and C. Pare, “Transverse modulational
instabilities for counterpropagating beams in Kerr
media,” Opt. Lett. |

16. | W. J. Firth, A. Fitzgerald, and C. Pare “Transverse instabilities due to
counterpropagation in Kerr media,” J.
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17. | F. T. Arecchi, S. Boccaletti, and P. Ramazza, “Pattern formation and competition in
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of counter-propagating fields,” Opt.
Lett. |

19. | D. N. Christodoulides, F. Lederer, and Y. Silberberg, “Discretizing light behaviour in
linear and nonlinear waveguide lattices,”
Nature |

20. | A. L. Jones, “Coupling of optical fibers and
scattering in fibers,” J. Opt. Soc. Am. |

21. | S. Somekh, E. Gamire, A. Yariv, H. L. Garvin, and R. G. Hunsperger, “Channel optical waveguide
directional couplers,” Appl. Phys. Lett. |

22. | D. N. Christodoulides and R. I. Joseph, “Discrete self-focusing in nonlinear
arrays of coupled waveguides,” Opt. Lett. |

23. | N. K. Efremidis, S. Sears, D. N. Christodoulides, J. W. Fleischer, and M. Segev, “Discrete solitons in photorefractive
optically induced photonic lattices,”
Phys. Rev. E |

24. | H. S. Eisenberg, Y. Silberberg, R. Morandotti, A. R. Boyd, and J. S. Aitchison, “Discrete spatial optical solitons in
waveguide arrays,” Phys. Rev. Lett. |

25. | J. W. Fleischer, T. Carmon, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of discrete solitons in
optically induced real time waveguide arrays,”
Phys. Rev. Lett. |

26. | J. W. Fleischer, M. Segev, N. K. Efremidis, and D. N. Christodoulides, “Observation of two-dimensional
discrete solitons in optically induced nonlinear photonic
lattices,” Nature |

27. | D. Neshev, E. A. Ostrovskaya, Y. Kivshar, and W. Krolikowski, “Spatial solitons in optically
induced gratings,” Opt. Lett. |

28. | D. Mandelik, R. Morandotti, J. S. Aitchison, and Y. Silberberg, “Gap solitons in waveguide
arrays,” Phys. Rev. Lett. |

29. | D. Neshev, A. A. Sukhorukov, B. Hanna, W. Krolikowski, and Y. S. Kivshar, “Controlled generation and steering
of spatial gap solitons,” Phys. Rev.
Lett. |

30. | E. Smirnov, M. Stepic, C. Ruter, V. Shandarov, and D. Kip, “Interaction of counterpropagating
discrete solitons in a nonlinear one-dimensional waveguide
array,” Opt. Lett. |

31. | M. Segev, G. C. Valley, B. Crosignani, P. DiPorto, and A. Yariv, “Steady-State Spatial Screening
Solitons in Photorefractive Materials with External Applied
Field,” Phys. Rev. Lett. |

32. | L. Solymar, D. J. Webb, and A. Grunett-Jepsen, |

33. | M. Carvalho, S. Singh, and D. Christodoulides, “Self-deflection of steady-state
bright spatial solitons in biased photorefractive
crystals,” Opt. Commun. |

34. | M. Shih, M. Segev, G. Valley, G. Salomono, B. Crosignani, and P. DiPorto, “Observation of two-dimensional
steady-state photorefractive screening solitons,”
Electron. Lett. |

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

36. | W. Krolikowski, M. Saffman, B. Luther-Davies, and C. Denz, “Anomalous Interaction of Spatial
Solitons in Photorefractive Media,” Phys.
Rev. Lett. |

37. | G. P. Agrawal, |

38. | J. B. Geddes, R. A. Indik, J. V. Moloney, and W. J. Firth, “Hexagons and squares in a passive
nonlinear optical system,” Phys. Rev. A |

39. | M. Litzkow, M. Livny, and M. Mutka, “Condor - A Hunter of Idle
Workstations,” in |

40. | R. Fischer, D. Träger, D. N. Neshev, A. A. Sukhorukov, W. Krolikowski, C. Denz, and Y. S. Kivshar, “Reduced-symmetry two-dimensional
solitons in photonic lattices,” Phys.
Rev. Lett. |

41. | D. Träger, R. Fischer, D. N. Neshev, A. A. Sukhorukov, C. Denz, W. Krolikowski, and Y. S. Kivshar, “Nonlinear Bloch modes in
two-dimensional photonic lattices,” Opt.
Express |

42. | A. S. Desyatnikov, D. N. Neshev, Y. S. Kivshar, N. Sagemerten, D. Träger, J. Jägers, C. Denz, and Y. V. Kartashov, “Nonlinear photonic lattices in
anisotropic nonlocal self-focusing media,”
Opt. Lett. |

**OCIS Codes**

(190.4420) Nonlinear optics : Nonlinear optics, transverse effects in

(190.5330) Nonlinear optics : Photorefractive optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: March 1, 2007

Revised Manuscript: April 18, 2007

Manuscript Accepted: April 19, 2007

Published: May 7, 2007

**Citation**

Sebastian Koke, Denis Träger, Philip Jander, Michael Chen, Dragomir N. Neshev, Wieslaw Krolikowski, Yuri S. Kivshar, and Cornelia Denz, "Stabilization of counterpropagating solitons by photonic lattices," Opt. Express **15**, 6279-6292 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-10-6279

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### References

- G. I. Stegeman and M. Segev, "Optical Spatial Solitons and Their Interactions: Universality and Diversity," Science 286, 1518-1523 (1999). [CrossRef] [PubMed]
- W. Krolikowski, B. Luther-Davies, and C. Denz, "Photorefractive Solitons," IEEE J. Sel. Top. Quantum Electron. 39, 3-12 (2003). [CrossRef]
- Y. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, San Diego, 2003).
- D. Träger, N. Sagemerten, and C. Denz, "Guiding of DynamicallyModulated Signals in Arrays of Photorefractive Spatial Solitons," IEEE J. Sel. Top. Quantum Electron. 12, 383-387 (2006). [CrossRef]
- C. Weilnau, M. Ahles, J. Petter, D. Träger, J. Schröder, and C. Denz, "Spatial optical (2+1)-dimensional scalarand vector-solitons in saturable nonlinear media," Ann. Phys. 11, 573-629 (2002). [CrossRef]
- O. Cohen, R. Uzdin, T. Carmon, J. W. Fleischer, M. Segev, and S. Odoulov, "Collisions between Optical Spatial Solitons Propagating in Opposite Directions," Phys. Rev. Lett. 89, 133,901-4 (2002).
- D. Kip, C. Herden, and M. Wesner, "All-Optical Signal Routing Using Interaction ofMutually Incoherent Spatial Solitons," Ferroelectrics 274, 135-142 (2002).
- P. Jander, J. Schröder, C. Denz, M. Petrovic, and M. R. Belic, "Dynamic instability of self-induced bidirectional waveguides in photorefractive media," Opt. Lett. 30, 750-752 (2005). [CrossRef] [PubMed]
- K. Motzek, P. Jander, A. Desyatnikov, M. Belic, C. Denz, and F. Kaiser, "Dynamic counterpropagating vector solitons in saturable self-focusing media," Phys. Rev. E 68, 066,611-4 (2003). [CrossRef]
- M. R. Belic, P. Jander, A. Strinic, A. Desyatnikov, and C. Denz, "Self-trapped bidirectional waveguides in a saturable photorefractive medium," Phys. Rev. E 68, R025,601-4 (2003). [CrossRef]
- M. Belic, P. Jander, K. Motzek, A. Desyatnikov, D. Jovic, A. Strinic, M. Petrovic, C. Denz, and F. Kaiser, "Counterpropagating self-trapped beams in photorefractive crystals," J. Opt. B 6, S190-S196 (2004). [CrossRef]
- P. Jander, J. Schröder, T. Richter, K. Motzek, F. Kaiser, M. R. Belic, and C. Denz, "Dynamic instability of counterpropagating self-trapped beams in photorefractive media," Proc. SPIE 6255, 62,550A (2006).
- M. Haelterman, A. P. Sheppard, and A. W. Snyder, "Bimodal counterpropagating spatial solitary-waves," Opt. Commun. 103, 145-152 (1993). [CrossRef]
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