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

Rémy Passier; Fabrice Devaux; Mathieu Chauvet

doi:10.1364/OE.16.007134

1. Introduction

Vortices are entities that can be found in various physical systems with linear and nonlinear properties. In the field of optics, vortices are associated with phase singularities carried by optical beams [1]. The past decade has seen a renewed interest on optical vortices and various applications have been suggested, such as microlithography [2], optical trapping of particles and transfer of orbital angular momentum [3, 4] or soliton algebra [5]. Especially, propagation of optical vortex in nonlinear medium can lead to soliton vortex formation when beam diffraction is exactly compensated by a self-defocusing nonlinearity. For example soliton vortices have been demonstrated in medium with isotropic saturable thermal nonlinearity [6] or anisotropic photorefractive nonlinearity [7]. Vortex propagation in photorefractive medium was studied in detail [8, 9] and it was demonstrated, in particular, that intensity distribution due to the anisotropic nonlinear effect depends on the traveling vortex topological charge (angular momentum)[10, 11]. However only extraordinarily polarized light was considered in the literature which overlooks the influence of the directions of polarization and propagation of light in anisotropic materials.

In this paper we demonstrate significative influence of these parameters when propagation of light is performed in 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

Vortex beam presents a ring-shaped intensity profile due to an helicoidal wavefront in the transverse plane that forms a singularity in the beam center. The optical phase varies from 0 to 2mπ around the central singularity, m represents the topological charge of the vortex. Electric field amplitude of a vortex beam is given by [12]:

\vec{U_{em}} (\vec{r}) = \vec{A} (r_{⊥}, z) \exp (- i (ω t - kz)) \exp (im θ)

where 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 $\vec{A} = \vec{A_{0}} {(\frac{r_{⊥}}{ω_{0}})}^{∣ m ∣} \exp (\frac{- r_{⊥}^{2}}{2 ω_{0}^{2}})$ is the optical field amplitude. To calculate the space charge field E⃗ induced by the light intensity I, we solve the well known photorefractive system of equations [8, 13]:

\frac{\partial N_{D}^{+}}{\partial t} = s (I + I_{d}) (N_{D} - N_{D}^{+}) - γ N_{e} N_{D}^{+}

\vec{\nabla} \cdot {[ε] \vec{E}} = ρ

ρ = e (N_{D}^{+} - N_{A} - N_{e})

\vec{J} = e μ N_{e} \vec{E} + μ k_{B} T \vec{\nabla} N_{e} + β_{ph} (N_{D} - N_{D}^{+}) I \vec{c}

\frac{\partial ρ}{\partial t} = - \vec{\nabla} \cdot \vec{J} .

where I_d is the equivalent dark intensity, N_A, N_D, N ⁺ _D and N_e are respectively the densities of shallow acceptors, deep donors, deep ionized donors and free electrons. 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_ph which is a polarization dependent component of the photogalvanic tensor. [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_e). Moreover generation time of conducting electrons is neglected with respect to the characteristic evolution time of space charge, $(\frac{\partial N_{D}^{+}}{\partial t} = 0)$ . Following these hypothesis and Eq.(2), Eq.(4) and Eq.(5) we obtain :

{\tilde{N}}_{e} = \frac{ξ (I + I_{d}) ({\tilde{N}}_{D} - {\tilde{N}}_{D}^{+})}{{\tilde{N}}_{D}^{+}}

\frac{\partial \tilde{ρ}}{\partial t} = - μ {\vec{\nabla} [{\tilde{N}}_{e}] \cdot \vec{E} + {\tilde{N}}_{e} \vec{\nabla} \cdot \vec{E} + \frac{k_{B} T}{e} Δ {\tilde{N}}_{e}

Ñ_e,Ñ_D and Ñ ⁺ _D are free electrons, donors and ionized donors densities normalized to N_A,Ñ ⁺ _D=1+ $\tilde{ρ}$ where $\tilde{ρ}$ is the space charge density normalized to eN_A, $ξ = \frac{s}{γ N_{A}}$ and $E_{ph} = \frac{β_{ph} γ N_{A}}{e μ s}$ is the photovoltaic field.

To deduce the space charge distribution evolution induced by the vortex intensity the above system of equations is solved using a numerical iterative method. In conjunction, space charge field distribution is deduced by integration of Eq.(4) that gives the electrical field produced by charges distribution ρ(r⃗)dV in the medium volume V (dV being an elementary volume) which is easier to solved than Eq.(3).

\vec{E} (\vec{r}) = \frac{1}{4 π [ε]} \int \int \int_{V} ρ (\vec{r}') \frac{\vec{r} - \vec{r}'}{{∣ \vec{r} - \vec{r}' ∣}^{3}} d V

In this model discrete Fourier Transform is used to resolve the space charge field in 3D. Two configurations are considered. In the first case vortex beam propagates along X-crystallographic 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 Δn 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 3m 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:

Δ n_{Y} ≃ - \frac{1}{2} n_{o}^{3} (r_{22} E_{Y} + r_{13} E_{Z})

Δ n_{Z} ≃ - \frac{1}{2} n_{e}^{3} r_{33} E_{Z}

For propagation along Y axis, Δn_Z for extraordinary polarization is identical to Eq.(11) while for X-polarized beam it yields to:

Δ n_{X} ≃ - \frac{1}{2} n_{o}^{3} r_{13} E_{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 200µW vortex beam at 473 nm focused to a 26µm 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 $\frac{I_{max}}{I_{d}} = 260$ . Photovoltaic field is set to E_ph=-7,7.10⁴ V.cm^-1, N_D=5.10¹⁶cm^-3, and $\frac{N_{D}}{N_{A}} = 1.1$ and other parameters are extracted from [15]. Vortex topological charges m=1 and m=-1 for both polarizations are considered.

Fig. 1. Numerical results : Vortex intensity distribution at the crystal input (a) and output (b) in linear regime.

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Fig. 2. Numerical results : Vortex intensity distribution at the crystal output for extraordinary polarization with topological charges m=+1 (a) and m=-1 (b) and corresponding refractive index modulations (c, d). Arrows indicate sense of rotation of light.

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The first studied case corresponds to the familiar configuration of an extraordinary polarized vortex [10]. Results, independent of propagation direction, are presented in Fig. 2 at a typical instant of the formation process. The intensity distribution of the vortex beam and the corresponding refractive index modulation at the output of the crystal are depicted for a topological charge 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)).

Fig. 3. Vortex intensity for ordinary polarization and propagation along X-axis: charge m=+1 (a) and m=-1 (c) and corresponding refractive index modulations (b, d).

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Fig. 4. Vortex intensity for ordinary polarization and propagation along Y-axis: charge m=+1 (a) and m=-1 (c) and corresponding refractive index modulations (b, d).

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We now consider the case of an ordinary polarized vortex. Figures 3(a,b) shows the key configuration of a beam propagating in the X-axis direction. The beam ellipticity is less pronounced compare to the case of an extraordinary polarized beam due to a weaker electro-optical coefficient 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_Y and E_Z of the space charge field (see Eq.(10)). E_Z component has a symmetric distribution relative to the Y-axis while the E_Y component is anti-symmetric relative to both Y and Z-axis. Moreover since electro-optical coefficients 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 Δn_X only depends on E_Z (Eq.(12)). Calculated vortex intensity distribution is represented in Fig. 4 with corresponding index modulation. Comparison of light distributions for 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

Figure 5 shows the experimental setup. A CWlaser beam at 473 nm is enlarged and encoded by reflection on a phase hologram. Vortex topological charge sign is chosen by spatially selecting the +1 or -1 diffraction order. Polarization and intensity of the vortex are controlled by means of a polarizer and a half wave plate. The vortex beam is focused to form a 26µ 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 $\frac{I}{I_{d}} ≃ 260$ like in numerical simulations. Beam power is set to 200µW.

Fig. 5. Experimental setup.

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In linear regime a circular 150µm 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.

Figure 7 depicts experimental results for ordinary polarized vortices. Anisotropic diffraction of the beam forming two bright lobes and slight elliptical shape of the dark core can be observed. More interestingly, for propagation along X-axis, light distribution for a positive topological charge (Fig. 7(b)) presents a vertical symmetry while for a negative topological charge light distribution is clearly rotated (Fig. 7(d)). Such an asymmetric dependence versus topological charge is predicted from our numerical model (see Figs. 3(a,c)) and along with results in Fig. 6 it demonstrates the influence of the electro-optic effect anisotropy on propagation of an ordinary polarized beam with respect to the sign of the angular momentum.

Fig. 6. Experimental results : Vortex intensity distribution at the input (a) and at the output (b) before nonlinearity occurs. The insert shows the interferogram of the vortex. Vortex intensity distribution for extraordinary polarization at the output of a 9mm long LiNbO ₃:Fe sample for topological charges m=+1 (c) and m=-1 (d) at t=1000 s for propagation along X axis.

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Finally, for propagation along Y-axis, influence of the topological charge on the beam distribution tilt is exhibited in Fig. 7(a,c). For such configuration vortex is slightly tilted throughout the nonlinear photorefractive effect. As predicted by theory and contrary to X-propagation, vertical symmetry is observed between 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

Impact of the electro-optic effect on propagation of vortex beam in photorefractive media is established. Demonstration is realized in self-defocusing 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.

Note that this fundamental interaction effect can be viewed as an analysis technique. Indeed for given crystal configuration, polarization and momentum of vortex can be deduced from observed intensity pattern or otherwise, for given vortex properties, crystal orientation could be assessed.

Fig. 7. Experimental vortex intensity for charges m=1 (a, b) and m=-1 (c, d) for ordinary polarization and for propagation along Y axis (a, c) and X axis (b, d).

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Acknowledgments

Authors would like to thanks Bruno Wacogne for preparing the holographic mask.

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

Abstract

1. Introduction

2. Theoretical background and numerical model

3. Experimental results

4. Conclusion

Acknowledgments

References and links

Cited By

Figures (7)

Equations (13)

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