## Impact of tensorial nature of the electro-optic effect on vortex beam propagation in photorefractive media

Optics Express, Vol. 16, Issue 10, pp. 7134-7141 (2008)

http://dx.doi.org/10.1364/OE.16.007134

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### Abstract

Influence of the anisotropic tensorial electro-optic effect of *LiNbO*_{3}:Fe photorefractive defocusing medium on propagation of a vortex beam is numerically and experimentally investigated. Characteristic behaviors are depicted by varying light polarization, sign of vortex angular momentum and propagation directions.

© 2008 Optical Society of America

## 1. Introduction

2. Z. Jaroszewicz and A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. **102**, 391–396 (1993). [CrossRef]

3. H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. **75**, 826–829 (1995). [CrossRef] [PubMed]

4. H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. **42**, 217–223 (1995). [CrossRef]

5. S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex-beam splitting,” Opt. Lett. **26**, 1004–1006 (2001). [CrossRef]

6. A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E **60**, 6111–6117 (1999). [CrossRef]

7. Z. Chen, M. Feng Shih, M. Segev, D. W. Wilson, R. Muller, and P. D. Maker, “Steady-state vortex-screening solitons formed in biased photorefractive media,” Opt. Lett. **22**, 1751–1753 (1997). [CrossRef]

8. M. Segev, B. Crosignani, A. Yariv, and B. Fischer, “Spatial solitons in photorefractive media,” Phys. Rev. Lett. **68**, 923–926 (1992). [CrossRef] [PubMed]

9. 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]

10. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Time-dependent evolution of an optical vortex in photorefractive media,” Phys. Rev. A **56**, 1713–1716 (1997). [CrossRef]

11. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. **77**, 4544–4547 (1996). [CrossRef] [PubMed]

*LiNbO*

_{3}:Fe crystals. Influence of sign of angular momentum is also investigated. Experimental results are confirmed by a time-dependent three dimensional numerical model which primarily relies on the calculation of the charge-distribution in the photorefractive medium to deduce both transverse components of the space-charge field. The tensorial character of the electro-optic effect is then taken into account for computation of the refractive index perturbation. Numerical simulations and experiments reveal the contribution of the anisotropic properties of the material with respect to the vortex angular momentum and polarization of the light.

## 2. Theoretical background and numerical model

*mπ*around the central singularity,

*m*represents the topological charge of the vortex. Electric field amplitude of a vortex beam is given by [12

12. I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. **119**, 604–612 (1995). [CrossRef]

*r⃗*

_{⊥}represents the coordinate in the transverse plane and z the position along the propagation axis,

*m*is the topological charge of the vortex,

*θ*is the azimuth angle and

*E⃗*induced by the light intensity

*I*, we solve the well known photorefractive system of equations [8, 13]:

*I*is the equivalent dark intensity,

_{d}*N*,

_{A}*N*,

_{D}*N*

^{+}

_{D}and

*N*are respectively the densities of shallow acceptors, deep donors, deep ionized donors and free electrons.

_{e}*s*is the photoexcitation coefficient,

*γ*is the recombination constant, k

_{B}is the Boltzmann constant,

*µ*is the electron mobility and

*T*is the temperature. Photovoltaic current directed along

*LiNbO*

_{3}c-axis (i.e Z axis) is considered throughout

*β*which is a polarization dependent component of the photogalvanic tensor. [

_{ph}*ε*] is the static dielectric tensor,

*ρ*is the charge density and

*J⃗*is the current density. From this set of equations usual assumptions are made. Acceptors density is assumed to be greater than the free electrons density (

*N*≫

_{A}*N*). Moreover generation time of conducting electrons is neglected with respect to the characteristic evolution time of space charge,

_{e}*Ñ*,

_{e}*Ñ*and

_{D}*Ñ*

^{+}

_{D}are free electrons, donors and ionized donors densities normalized to

*N*,

_{A}*Ñ*

^{+}

_{D}=1+

*where*ρ ˜

*is the space charge density normalized to e*ρ ˜

*N*,

_{A}*ρ*(

*r⃗*)d

*V*in the medium volume

*V*(d

*V*being an elementary volume) which is easier to solved than Eq.(3).

*LiNbO*

_{3}axis while in the second case propagation along Y axis is analyzed. In both case propagation is perpendicular to the c-axis to benefit from a large photovoltaic effect but the refractive index perturbation Δ

*n*induced by the space charge field differs. Furthermore, the tensorial nature of the electro-optic effect gives rise to an anisotropic space charge field. For 3

*m*group, like

*LiNbO*

_{3}, the refractive index modulation for ordinary (X or Y-polarized) and extraordinary (Z-polarized) beams are given by [14]. For propagation along X axis:

*n*for extraordinary polarization is identical to Eq.(11) while for X-polarized beam it yields to:

_{Z}*r*

_{22},

*r*

_{13}and

*r*

_{33}are the electro-optical coefficients and n

_{o}and n

_{e}are respectively the ordinary and extraordinary indices. E

_{Y}and E

_{Z}are the components of the space charge field

*E⃗*(

*r⃗*). Note that E

_{X}does not play a significative role for the chosen configurations. Vortex propagation can now be calculated for the different situations of interest by classical split-step Fourier method taking into account the time dependent photorefractive effect. For the numerical study we consider a CW 200

*µ*W vortex beam at 473 nm focused to a 26

*µ*m FWHM at the entrance of a 9 mm long

*LiNbO*

_{3}:Fe crystal. The input and output beams are presented on Fig. 1(a) and Fig. 1(b) in linear regime. The profiles are identical whatever the sign of angular momentum and the directions of propagation and polarization. The maximum intensity to dark irradiance ratio is set to

*E*=-7,7.10

_{ph}^{4}

*V*.cm

^{-1},

*N*=5.10

_{D}^{16}cm

^{-3}, and

15. M. Simon, S. Wevering, K. Buse, and E. Krätzig,“The bulk photovoltaic effect of photorefractive *LiNbO*_{3}:Fecrystals at high light intensities,”J. Phys. D **30**, 144–149 (1997). [CrossRef]

*m*=1 and

*m*=-1 for both polarizations are considered.

10. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Time-dependent evolution of an optical vortex in photorefractive media,” Phys. Rev. A **56**, 1713–1716 (1997). [CrossRef]

*m*=+1 (Figs. 2(a,b)) and for a topological charge

*m*=-1 (Figs. 2(c,d)). We specify that the phase of a positively charged vortex rotates anti-clockwise as the beam travels. Because of the intrinsic anisotropy of the photorefractive effect the beam spreads along the c-axis thanks to the deeper index modulation present in the upper and lower part of the beam as depicted in Figs. 2(b,d). Elliptical shapes of vortex and dark core are observed for both topological charges. However the sign of the topological charge impacts the light distribution. Indeed two bright areas whose location are dependent on vortex charge are present on each side of the beam. Symmetrical light distribution relative to the c-axis is observed for opposite topological charge (Fig. 2(c)).

*r*

_{13}and consequently a smaller photorefractive index change. But most importantly the two bright spots arrangement totally differs. While they are located on a quasi horizontal diameter for a positively charged vortex (Fig. 3(a)) they instead are titled approximatively at 30 degrees for a negatively charged vortex (Fig. 3(c)). Consequently, the intensity distributions for both charges are no more symmetric relative to the c-axis. The reason of this revealed behavior holds in the index distribution presented in Figs. 3(b,d). Indeed index modulation now depends on both transverse components

*E*and

_{Y}*E*of the space charge field (see Eq.(10)).

_{Z}*E*component has a symmetric distribution relative to the Y-axis while the

_{Z}*E*component is anti-symmetric relative to both Y and Z-axis. Moreover since electro-optical coefficients

_{Y}*r*

_{13}and

*r*

_{22}have similar values it yields that conjunction of both transverse space charge field components induces a complex refractive index distribution. To highlight the major role of the space charge field transverse component perpendicular to c-axis, a Y-propagating ordinary polarized vortex is also analyzed. Note that Δ

*n*only depends on

_{X}*E*(Eq.(12)). Calculated vortex intensity distribution is represented in Fig. 4 with corresponding index modulation. Comparison of light distributions for

_{Z}*m*=+1 (Fig. 4(a)) and

*m*=-1 (Fig. 4(c)) shows that they are symmetric with respect to the c-axis like for extraordinary polarized vortices depicted in Fig. 2.

## 3. Experimental results

*µ*m (FWHM) spot at the input face of a

*LiNbO*

_{3}:Fe sample (Fig. 6(a)). The 9 mm square sample is cut from a 1 mm thick c-oriented iron doped

*LiNbO*

_{3}wafer allowing vortex propagation either along the X or Y axis. Exit face of the crystal is imaged onto a CCD camera. Background illumination of the sample with a white light source is used to adjust the intensity to a dark irradiance ratio

*µ*W.

*µ*m FWHM vortex beam (Fig. 6(b)) is present at the exit face of the crystal. To start with, extraordinary polarized vortices are used for both sign of the topological charge for X-propagation. Results are presented in Fig. 6(c) and 6(d) for respectively

*m*=+1 and

*m*=-1 for an induction time of 1000 seconds sufficient to observe beam reshaping. Vortex stretching in the c-axis direction, peripheral brighter areas and strong ellipticity of the dark core are observed in accordance with numerical prediction. In addition, results for positive topological charge can be retrieved from negative topological charge by the symmetry relative to the c-axis of light distribution and conversely. Measurements for Y-propagating shows identical behavior.

*m*=+1 and

*m*=-1 charged vortex distributions. Results depicted in Fig. 7 clearly reveal that each pair angular momentum-propagation direction gives rise to a different intensity distribution.

## 4. Conclusion

*LiNbO*

_{3}:Fe photorefractive crystal. The influence of tensorial nature of electro-optic effect gives rise to a beam pattern characteristic of the polarization-momentum pair. Propagation along

*LiNbO*

_{3}X-crystallographic axis is found to be adequate for the demonstration. A full 3D photorefractive numerical model is used to successfully model the experimental observations. It shows that the observed phenomena is due to both the anisotropy of the space charge field and the rotating phase carried by the vortex. Influence of the two transverse components of the space charge field have to be taken into account to understand the vortex perturbation.

## Acknowledgments

## References and links

1. | A. S. Desyatnikov, Yu S. Kivshar, and L. Torner, “Optical vortices and Vortex Solitons” in |

2. | Z. Jaroszewicz and A. Kolodziejczyk, “Zone plates performing generalized Hankel transforms and their metrological applications,” Opt. Commun. |

3. | H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity,” Phys. Rev. Lett. |

4. | H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms,” J. Mod. Opt. |

5. | S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, “Soliton algebra by vortex-beam splitting,” Opt. Lett. |

6. | A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, “Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium,” Phys. Rev. E |

7. | Z. Chen, M. Feng Shih, M. Segev, D. W. Wilson, R. Muller, and P. D. Maker, “Steady-state vortex-screening solitons formed in biased photorefractive media,” Opt. Lett. |

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

9. | 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. |

10. | A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Time-dependent evolution of an optical vortex in photorefractive media,” Phys. Rev. A |

11. | A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Vortex evolution and bound pair formation in anisotropic nonlinear optical media,” Phys. Rev. Lett. |

12. | I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, “Optical wavefront dislocations and their properties,” Opt. Commun. |

13. | V. L. Vinetskii and N. V. Kukhtarev, “Wave front convolution in 4-wave interaction inmedia with nonlocal nonlinearity,” Sovi. Phys. JETP Letters |

14. | P. Yeh, |

15. | M. Simon, S. Wevering, K. Buse, and E. Krätzig,“The bulk photovoltaic effect of photorefractive |

**OCIS Codes**

(160.2100) Materials : Electro-optical materials

(160.5320) Materials : Photorefractive materials

(190.0190) Nonlinear optics : Nonlinear optics

(190.5330) Nonlinear optics : Photorefractive optics

(260.5950) Physical optics : Self-focusing

**ToC Category:**

Materials

**History**

Original Manuscript: January 15, 2008

Revised Manuscript: March 4, 2008

Manuscript Accepted: March 25, 2008

Published: May 2, 2008

**Citation**

Rémy Passier, Fabrice Devaux, and Mathieu Chauvet, "Impact of tensorial nature of the electro-optic effect on vortex beam propagation in photorefractive media," Opt. Express **16**, 7134-7141 (2008)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-10-7134

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### References

- A. S. Desyatnikov, YuS. Kivshar, and L. Torner, "Optical vortices and Vortex Solitons" in Progress in Optics, E. Wolf, ed., (Elsevier, Amsterdam, 2005) Vol. 47.
- Z. Jaroszewicz and A. Kolodziejczyk, "Zone plates performing generalized Hankel transforms and their metrological applications," Opt. Commun. 102, 391-396 (1993). [CrossRef]
- H. He, M. E. J. Friese, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Direct observation of transfer of angular momentum to absorptive particles from a laser beam with a phase singularity," Phys. Rev. Lett. 75, 826-829 (1995). [CrossRef] [PubMed]
- H. He, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical particle trapping with higher-order doughnut beams produced using high efficiency computer generated holograms," J. Mod. Opt. 42, 217-223 (1995). [CrossRef]
- S. Minardi, G. Molina-Terriza, P. Di Trapani, J. P. Torres, and L. Torner, "Soliton algebra by vortex-beam splitting," Opt. Lett. 26, 1004-1006 (2001). [CrossRef]
- A. Dreischuh, G. G. Paulus, F. Zacher, F. Grasbon, and H. Walther, "Generation of multiple-charged optical vortex solitons in a saturable nonlinear medium," Phys. Rev. E 60, 6111-6117 (1999). [CrossRef]
- Z. Chen, M. Feng Shih, M. Segev, D. W. Wilson, R. Muller, and P. D. Maker, "Steady-state vortex-screening solitons formed in biased photorefractive media," Opt. Lett. 22, 1751-1753 (1997). [CrossRef]
- M. Segev, B. Crosignani, A. Yariv, and B. Fischer, "Spatial solitons in photorefractive media," Phys. Rev. Lett. 68, 923-926 (1992). [CrossRef] [PubMed]
- 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]
- A. V. Mamaev, M. Saffman, and A. A. Zozulya, "Time-dependent evolution of an optical vortex in photorefractive media," Phys. Rev. A 56, 1713-1716 (1997). [CrossRef]
- A. V. Mamaev, M. Saffman, and A. A. Zozulya, "Vortex evolution and bound pair formation in anisotropic nonlinear optical media," Phys. Rev. Lett. 77, 4544-4547 (1996). [CrossRef] [PubMed]
- I. V. Basistiy, M. S. Soskin, and M. V. Vasnetsov, "Optical wavefront dislocations and their properties," Opt. Commun. 119, 604-612 (1995). [CrossRef]
- V. L. Vinetskii and N. V. Kukhtarev, "Wave front convolution in 4-wave interaction inmedia with nonlocal nonlinearity," Sovi. Phys. JETP Letters 30, 6 (1979).
- P. Yeh, Introduction to photorefractive nonlinear optics (Wiley-Interscience, New York, 1993).
- M. Simon, S. Wevering, K. Buse and E. Kratzig,"The bulk photovoltaic effect of photorefractive LiNbO3:Fe crystals at high light intensities," J. Phys. D 30, 144-149 (1997). [CrossRef]

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