## Elliptical solitons in nonconventionally biased photorefractive crystals

Optics Express, Vol. 15, Issue 2, pp. 536-544 (2007)

http://dx.doi.org/10.1364/OE.15.000536

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### Abstract

We theoretically predict and experimentally observe the two-dimensional (2-D) bright solitons in a nonconventionally biased strontium barium niobate (SBN) crystal. A theory describing light propagating in an SBN crystal with a bias field along an arbitrary direction is formulated. Then the existence of 2-D bright solitons in such a crystal is numerically verified. By employing digital holography, the index changes induced by Gaussian beams in an SBN crystal under different biasing conditions are visualized. Finally, skewed elliptical solitons are experimentally demonstrated.

© 2007 Optical Society of America

## 1. Introduction

1. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett **68**,923 (1992). [CrossRef] [PubMed]

2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,𡀝 Science **286**,1518 (1999). [CrossRef] [PubMed]

2. G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,𡀝 Science **286**,1518 (1999). [CrossRef] [PubMed]

5. W. Królikowski and S. A. Holmstrom, “Fusion and birth of spatial solitons upon collision,” Opt. Lett **22**,369 (1997). [CrossRef] [PubMed]

6. 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 **422**,147 (2003). [CrossRef] [PubMed]

3. M. Shih, P. Leach, M. Segev, M. H. Garett, G. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett **21**,324 (1996). [CrossRef] [PubMed]

4. Z. Chen, M. Mitchell, M. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett **21**,629 (1996). [CrossRef] [PubMed]

*c*axis of the photorefractive crystal. Then some natural questions arise: What will happen when the bias field is along an arbitrary direction? Can such nonconventionally biased schemes support different types of solitons such as bright, dark, vector [7

7. J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, “Vector solitons in (2+1) dimensions,” Opt. Lett **25**,643 (2000). [CrossRef]

8. Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett **78**,2948 (1997). [CrossRef]

9. C. Crognale and L. Rosa, “Vector analysis of the space-charge field in nonconventionally biased photorefractive crystals,” J. Lightwave Technol **23**,2175 (2005). [CrossRef]

## 2. Theoretical analysis

*c*axis component of the electric field can deform the index ellipsoid remarkably, i.e., contributions introduced by other components can be neglected [10]. This coincides with the fact that light-induced index changes in such crystals are proportional to the

*c*axis components of the space charge fields. Moreover, it should be noticed that only the bias field component perpendicular to the beam propagation direction can result in effective separation of the photo-excited electrons [11

11. P. Zhang, Y. Ma, J. Zhao, D. X. Yang, and H. Xu, “One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal,” Appl. Opt **45**,2273 (2006). [CrossRef] [PubMed]

*c*axis components of the space charge field as well as different distributions of light-induced index changes.

*c*axis of the crystal and propagating perpendicular to the

*c*axis. Figure 1 depicts the schematic diagram for a light beam propagating in a nonconventionally biased SBN crystal. The light beam propagates along the

*z*axis, and the direction of the bias field is parallel to the

*y*axis. The angle between the

*c*axis of the crystal and the

*y*axis is denoted as α. According to this geometry, during the process of photorefraction, bias fields will cause photo-excited electrons to drift and separate along the

*y*axis inside the crystal, and this will produce a space-charge field that modulates the index of the material via the electro-optic effect. Corresponding to different α, the

*c*axis components of the fields, which are proportional to the light-induced index changes, will possess different distributions. Furthermore, these index changes will modulate the propagation of the light beam. If the characteristic spatial scales are larger than the photorefractive Debye length, and the diffusion field can be neglected, the equations determining the steady state propagation of a light beam in a nonconventionally biased SBN crystal can be deduced from the model of Zozulya and Anderson [12

12. A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A **51**,1520 (1995). [CrossRef] [PubMed]

13. 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]

## 3. Nonlinear index changes and beam propagation

*B*(

*r⇁*) = √2exp[-(

*x*

^{2}+

*y*

^{2})/16] and different values of α are shown in Fig. 2, where the normalized index changes are equal to

*z*=15, respectively. The electrostatic potential induced by the input beam is shown in Fig. 2(c), where the orientation of the

*c*axis and the color index are both indicated. It is obvious that the potential is symmetric to the

*y*axis but not to the

*c*axis. This is because the photo-excited electrons have been drifted parallel to the bias field. Figures 2(d)-(h) depict the simulation results for α=0, π/4, π/2, 3π/4, and π, respectively, where from top to bottom they display index changes induced by the same input beam [see Fig. 2(a)], the output beams after nonlinear propagations, and the corresponding nonlinear index changes at

*z*=15. Recall that, as α changing, the polarization directions of the beams are always parallel to the

*c*-axis. It is revealed that the maximum index change at α=0 (the largest) is about three times as large as that at α=π/2 (the smallest), which cannot be seen from the figures as the color is normalized. From Figs. 2(d)-(h) we can see that different biased fields will cause different index changes as well as various nonlinear beam propagations. With a increasing from 0 to π, the crystal is gradually converted from a self-focusing medium into a self-defocusing medium, and the critical point is at α=π/2. Similar results are obtained for the case with α<0 due to the symmetries of Eqs. (1). From Figs. 2(e3) and (f3), it is obvious that such index changes may support 2-D elliptical bright solitons. So in the nonconventional biased SBN crystal, where bright solitons exist, the corresponding α must lie in the interval [-π/2, π/2].

## 4. Soliton solution and its properties

*B*(

*x, y, z*)=

*b*(

*x, y*)exp(i

*βz*), where β is the propagation constant, and the real envelope

*b*(

*x, y*) satisfies the following equations

13. 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]

*I*

_{max}versus the propagation constant β for the cases of α=0, π/8, π/4, 3π/8, and π/2, respectively. The full width at half maximum (FWHM) of the soliton intensity profile W

_{1}and W

_{2}, which are corresponding to the two principal axes of the ellipse and are indicated in Figs. 3(a), and the ratio W

_{1}/W

_{2}versus

*I*

_{max}are shown in Fig. 3(f). From Fig. 3, we can see that for the nonconventionally biased case the soliton profiles are also elliptical. However, the orientations of the ellipses are skewed due to the tilted bias fields. When α>π/4, the ranges of the eigenvalues β for the existed solitons will reduce fast with increasing α. Moreover the convergence speed of the iterative procedure becomes slower. For this reason, the maximum eigenvalues β in our calculation for the cases of α=3π/8 and π/2 are 0.35 and 0.18, respectively. For the case α=π/2 as well as the external field is biased perpendicular to the

*c*axis of the crystal, to observe solitons in experiment, the ratio between the soliton and the background beam intensities, i.e.

*I*

_{max}, cannot be too large because of the dramatically increase of W

_{1}/W

_{2}after

*I*

_{max}becomes larger than 2.

## 5. Soliton excitation by Gaussian beams

*B*(

*r⇁*) =√1.6 exp[-(

*x*

^{2}+

*y*

^{2})/36] in an SBN crystal with a bias field perpendicular to the

*c*axis, where the display areas for (a)-(o) are 400×400. Figures 4(a)-(e) shows the linear diffracted beam profiles at propagation lengths

*z*=0, 150, 300, 450, and 600, respectively, while Figs. 4(f)-(j) and Figs. 4(k)-(o) describe the beam profiles and their induced index changes after nonlinear propagations with

*z*=50, 100, 300, 450, 600, respectively. Figure 4(p) shows the FWHMs of the beam profiles along the dashed lines indicated in Fig. 4(a) and (f) versus the propagation lengths

*z*. Here, we also use W

_{1}and W

_{2}to denote the FWHMs of the long and short axes of the elliptical beam profiles. From these results we can see that a circular Gaussian beam indeed can evolve into a soliton in a nonconventionally biased SBN crystal. And it is also revealed that only the Gaussian beams with proper beam diameters and

*I*

_{max}can evolve into solitons, otherwise the spatial evolution of the input Gaussian beam will experience oscillations similar to that shown in Ref. [13

13. 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]

## 6. Experimental demonstration

*c*)mm and 0.025% by weight chromium dopant is used. We first visualize the index changes induced by a Gaussian beam in a nonconventionally biased SBN crystal employing digital holography. The experimental setup is as same as that described earlier in Ref. [15

15. J. Zhao, P. Zhang, J. Zhou, D. X. Yang, D. S. Yang, and E. Li, “Visualizations of light-induced refractive index changes in photorefractive crystals employing digital holography,” Chin. Phys. Lett **20**,1748 (2003). [CrossRef]

^{2}. After the crystal sample is illuminated for 3s without any background beam, the measured index changes are depicted in Fig. 5, where (a)-(d) are the 2-D maps of the index changes for α=0, π, π/2 and -π/2, respectively, and (e) is the 3-D display of (d). For obtaining Figs. 5(a) and 5(b), the amplitudes of the bias voltages are all 1kV, and in this case the maximum index changes are all about 1.5×10

^{-4}. For obtaining Figs. 5(c) and 5(d), bias voltages of 2kV are applied. In this case the maximum index changes are all about 3×10

^{-5}. It is obvious that although a higher bias field is applied perpendicular to the

*c*axis, the maximum index changes are much smaller than that in conventionally biased case. This is due to the differences between the amplitudes of the two vertical components of the light-induced electric fields (see numerically simulated results in Fig. 2).

_{1}(

*f*=15cm), and the beam profile at the rear face is monitored by a CCD camera through imaging lens L

_{2}(

*f*=5cm). The bias field can be applied onto the crystal parallel or perpendicular to the

*c*axis. A cold white light source is used to provide an incoherent background illumination. The total intensity of the He-Ne laser impinging onto the crystal is reduced to about 5μW and the intensity ratio between the soliton beam and the background beam can be adjusted by the white light source. Figure 7 shows the experimental results, where (a) is the beam profile of the input Gaussian beam (FWHM is 22μm) for generating solitons, (b) is the beam profile (FWHM is 42μm) on the rear face of the crystal after linear diffraction. Figures 7(c) and (d) are the soliton profiles under 3.6kV bias voltage with α=π/2 (in this case W

_{1}=46μm and W

_{2}=14μm) and α=-π/2 (in this case W

_{1}=46μm and W

_{2}=13μm), respectively. Figures 7(e)-(h) display the beam profiles observed on the rear face of the crystal with bias voltages of 1kV, 2kV, 3kV, and 3.8kV, respectively. And the normalized intensities with respect to the background illuminations for Figs. 7(c)-(h) are always adjusted to be about 1. It can be seen that, with the increase of the voltage, the FWHM of the short principal axis of the elliptical beam profile (W

_{2}) is reduced from 31μm to 13μm, while the long one (W

_{1}) is increased from 40μm to 45μm. The ratios W

_{1}/W

_{2}for the elliptical soliton profiles shown in Figs. 7(c), (d), and (h) are about 3.29, 3.54, and 3.46, respectively, which are approximately the theoretically expected values. Figures 7(i)-(l) display the output beam profiles with bias voltage of 3.6kV, and with normalized intensities of 2, 1.3, 0.8, 0.6, respectively. Combining with Fig. 7(c), it is clear that the FWHM of the output beam has a minimum at normalized intensity of about 1, and with increase or decrease of the normalized intensity, the input Gaussian beam cannot convergences into a soliton, which coincide with the theoretical expectation.

## 7. Summary

*c*axis of the crystal are experimentally observed. The experiments are in agreement with the theoretically predications. It is clear that bias fields along different directions can support 2-D bright solitons. These solitons are just like those formed in conventionally biased crystals except that they are elliptical and they have different orientations. While it is not surprising that elliptical spatial solitons can be generated with anisotropic nonlinearity or anisotropic correlation functions as in elliptical incoherent solitons [16

16. E. D. Eugenieva, D. N. Christodoulides, and M. Segev, “Elliptic incoherent solitons in saturable nonlinear media,” Opt. Lett **25**,972 (2000). [CrossRef]

17. O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, “Observation of elliptic incoherent spatial solitons,” Opt. Lett **29**,1248 (2004). [CrossRef] [PubMed]

## Acknowledgments

## References and links

1. | M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett |

2. | G. I. Stegeman and M. Segev, “Optical spatial solitons and their interactions: universality and diversity,𡀝 Science |

3. | M. Shih, P. Leach, M. Segev, M. H. Garett, G. Salamo, and G. C. Valley, “Two-dimensional steady-state photorefractive screening solitons,” Opt. Lett |

4. | Z. Chen, M. Mitchell, M. Shih, M. Segev, M. H. Garrett, and G. C. Valley, “Steady-state dark photorefractive screening solitons,” Opt. Lett |

5. | W. Królikowski and S. A. Holmstrom, “Fusion and birth of spatial solitons upon collision,” Opt. Lett |

6. | 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 |

7. | J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, “Vector solitons in (2+1) dimensions,” Opt. Lett |

8. | Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, “Self-trapping of an optical vortex by use of the bulk photovoltaic effect,” Phys. Rev. Lett |

9. | C. Crognale and L. Rosa, “Vector analysis of the space-charge field in nonconventionally biased photorefractive crystals,” J. Lightwave Technol |

10. | A. Yariv and P. Yeh, |

11. | P. Zhang, Y. Ma, J. Zhao, D. X. Yang, and H. Xu, “One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal,” Appl. Opt |

12. | A. A. Zozulya and D. Z. Anderson, “Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field,” Phys. Rev. A |

13. | 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 |

14. | V. I. Petviashvili, “Equation of an extraordinary soliton,” Sov. J. PlasmaPhys |

15. | J. Zhao, P. Zhang, J. Zhou, D. X. Yang, D. S. Yang, and E. Li, “Visualizations of light-induced refractive index changes in photorefractive crystals employing digital holography,” Chin. Phys. Lett |

16. | E. D. Eugenieva, D. N. Christodoulides, and M. Segev, “Elliptic incoherent solitons in saturable nonlinear media,” Opt. Lett |

17. | O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, “Observation of elliptic incoherent spatial solitons,” Opt. Lett |

**OCIS Codes**

(190.5330) Nonlinear optics : Photorefractive optics

(190.5530) Nonlinear optics : Pulse propagation and temporal solitons

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: November 7, 2006

Revised Manuscript: December 18, 2006

Manuscript Accepted: December 18, 2006

Published: January 22, 2007

**Citation**

Peng Zhang, Jianlin Zhao, Cibo Lou, Xiaoyu Tan, Yuhan Gao, Qian Liu, Dexing Yang, Jingjun Xu, and Zhigang Chen, "Elliptical solitons in nonconventionally biased photorefractive crystals," Opt. Express **15**, 536-544 (2007)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-15-2-536

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### References

- M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923 (1992). [CrossRef] [PubMed]
- G. I. Stegeman and M. Segev, "Optical spatial solitons and their interactions: universality and diversity," Science 286, 1518 (1999). [CrossRef] [PubMed]
- M. Shih, P. Leach, M. Segev, M. H. Garett, G. Salamo, and G. C. Valley, "Two-dimensional steady-state photorefractive screening solitons," Opt. Lett. 21, 324 (1996). [CrossRef] [PubMed]
- Z. Chen, M. Mitchell, M. Shih, M. Segev, M. H. Garrett, and G. C. Valley, "Steady-state dark photorefractive screening solitons," Opt. Lett. 21, 629 (1996). [CrossRef] [PubMed]
- W. Królikowski and S. A. Holmstrom, "Fusion and birth of spatial solitons upon collision," Opt. Lett. 22, 369 (1997). [CrossRef] [PubMed]
- 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 422, 147 (2003). [CrossRef] [PubMed]
- J. N. Malmberg, A. H. Carlsson, D. Anderson, M. Lisak, E. A. Ostrovskaya, and Yu. S. Kivshar, "Vector solitons in (2+1) dimensions," Opt. Lett. 25, 643 (2000). [CrossRef]
- Z. Chen, M. Segev, D. W. Wilson, R. E. Muller, and P. D. Maker, "Self-trapping of an optical vortex by use of the bulk photovoltaic effect," Phys. Rev. Lett. 78, 2948 (1997). [CrossRef]
- C. Crognale and L. Rosa, "Vector analysis of the space-charge field in nonconventionally biased photorefractive crystals," J. Lightwave Technol. 23, 2175 (2005). [CrossRef]
- A. Yariv and P. Yeh, Optical waves in crystals (Wiley, New York, 1984), Chap. 7.
- P. Zhang, Y. Ma, J. Zhao, D. X. Yang, and H. Xu, "One-dimensional spatial dark soliton-induced channel waveguides in lithium niobate crystal," Appl. Opt. 45, 2273 (2006). [CrossRef] [PubMed]
- A. A. Zozulya and D. Z. Anderson, "Propagation of an optical beam in a photorefractive medium in the presence of a photogalvanic nonlinearity or an externally applied electric field," Phys. Rev. A 51, 1520 (1995). [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]
- V. I. Petviashvili, "Equation of an extraordinary soliton," Sov. J. Plasma Phys. 2, 257 (1976).
- J. Zhao, P. Zhang, J. Zhou, D. X. Yang, D. S. Yang, and E. Li, "Visualizations of light-induced refractive index changes in photorefractive crystals employing digital holography," Chin. Phys. Lett. 20, 1748 (2003). [CrossRef]
- E. D. Eugenieva, D. N. Christodoulides, and M. Segev, "Elliptic incoherent solitons in saturable nonlinear media," Opt. Lett. 25, 972 (2000). [CrossRef]
- O. Katz, T. Carmon, T. Schwartz, M. Segev, and D. N. Christodoulides, "Observation of elliptic incoherent spatial solitons," Opt. Lett. 29, 1248 (2004). [CrossRef] [PubMed]

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