# Three dimensional time dependent model with two photoactive centres for photorefractive spatial solitons in LiNbO3

F. Devaux, M. Chauvet, J. Safioui, R. Passier

► To cite this version:

F. Devaux, M. Chauvet, J. Safioui, R. Passier. Three dimensional time dependent model with two photoactive centres for photorefractive spatial solitons in LiNbO3. 3rd EOS Topical Meeting on Optical Microsystems (OMS09), Sep 2009, Capri, Italy. hal-00476858

HAL Id: hal-00476858

https://hal.science/hal-00476858v1

Submitted on 12 Apr 2021

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_1.png" /></figure>

Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
# Three dimensional time dependent model with two photoactive centres for photorefractive spatial solitons in LiNbO$_3$

Fabrice Devaux, Mathieu Chauvet, Jassem Safioui and R. Passier

Département d'Optique P.M. Duffieux, Institut Femto-ST, Université de Franche-Comté
16, route de Gray, 25030 Besançon cedex, France

Email: Fabrice.devaux@univ-fcomte.fr

## Summary

We present a new 3D time dependent model for photorefractive beam self focusing in undoped biased LiNbO$_3$ where iron impurities and Nb$_{Li}$ polarons are considered as two photoactive centres. Focusing or defocusing regimes are exhibited with respect to beam intensity and bias voltage in good agreement with experiments.

## Introduction

Photorefractive (PR) spatial solitons in LiNbO$_3$:Fe are usually modelised with Fe$^{2+}$/3+ single photoactive centre either in nominally pure samples where iron is present as impurity at very low concentration. Nevertheless, as it was demonstrated few years ago [1], intrinsic Nb$_{Li}$ polarons play a significant role as a second photoactive centre in LiNbO$_3$:Fe for high light intensity. Then polarons contribution to PR effect can be no more neglected in undoped materials. This work deals with a new 3D time dependent numerical model of beam self focusing in biased undoped LiNbO$_3$ where both iron and polaron photoactive centres are considered. This model enlightens the competition, for a given biased voltage, between drift and photovoltaic (PV) currents with respect to the light intensity along the propagation axis. We demonstrate that for low light intensity drift current is greater than PV current which leads to self focusing of the beam since for high intensity, PV current dominates along the first diffraction lengths. It leads, in a first stage, to a defocusing regime of the beam. While beam is enlarged by diffraction and defocusing effect, drift current is still stronger than PV current in the last part of the crystal. Then in a second stage of the process, the beam splits in two self focusing beams. These numerical results are in good agreement with recent experiments performed in our group.

## Numerical model and results

Numerical model is based on our previous 3D time dependent model where only one carrier is involved [2]. Rate equations for both photoactive centres and all parameters of the material are taken from [1]. Contribution of polarons to PV current, thermal diffusion and absorption are considered. From initial conditions for light distribution and electric species densities, time evolution of all electric species and space charge densities are calculated in the whole crystal volume. Then, from the space charge distribution, electric space charge field is calculated by three-dimensional discrete Fourier transform [2]. Then index perturbation induced by the electro-optical effect is calculated in the whole crystal volume and propagation of light in the perturbed medium is performed by a classical split-step-Fourier transform method that gives the new light intensity distribution. The whole procedure is repeated until steady-state regime is reached.
We present here the most important results obtained with this model where an extraordinary polarised Gaussian beam at 532 nm with a waist of 17µm is injected in a
2cm long undoped LiNbO₃ crystal. A static electric field of 33kV/cm is applied along the c axis. First, for low light intensity (5.5kW/m² Fig. 1a, 1b and 1c), defocusing effect of the PV current is screened by the drift current (Fig. 1a). Then, self focusing of the beam is observed and gives rise to a 2D spatial solitons (Fig 1b and 1c). When input intensity is increased up to 100kW/m², PV current becomes stronger than drift current along the first diffraction lengths while drift current still dominates in the last part of the crystal because of intensity decreasing along propagation axis (Fig 1d). Then during the process we observe, in a first step, a defocusing regime of the beam which enlarges and splits in two beams (Fig 1e and 1f). In a second step, since intensity of these two beams is smaller, focusing regime still dominates in the last part of the crystal which should lead to formation of two solitons as it was observed experimentally (Fig. 2e to 2h).

<figure><img src="image_2.png" /><figcaption>**Figure 1** : Numerical results for input intensities respectively of 5.5 and 100 kW/m², (a) and (d) : total current densities along propagation and c axes, (b) and (e) : output beam intensity profiles along c axis versus time, (c) and (f) pictures of output beams.</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>**Figure 2** : Experimental results : Time evolution of the output gaussian beam at 532 nm with a waist of 10µm injected in a 2cm long biased congruent LiNbO₃ crystal. (a) to (d) : low intensity. (e) to (h) high intensity.</figcaption></figure>

## Conclusion

We present a new 3D time dependent numerical model for PR self focusing of beam in undoped biased LiNbO₃ crystal where iron and polarons are considered as two photoactive centres. This model, in good agreement with experimental results, enlightens the contribution of both photoactive centres in the competition between drift and PV currents with respect to the beam intensity. It leads to self focusing of light for low intensity and defocusing and splitting of the beam in two self focusing beams for high intensity.

## References

[1] F. Jermann and J. Otten, JOSA B **10** 2085-2092 (1995).

[2] F. Devaux and al., JOSA B **25** 1081-1087 (2008).