1. Introduction

Photorefractive nonlinear medium provides an intensity-dependent, saturable nonlinear response which can be conveniently exploited for the purpose of spatial-soliton generation. The existence of photorefractive (PR) spatial soliton, first predicted in Ref. [1] has stimulated substantial research interest because of an attractive property of these spatial solitons in photorefractive materials, that is the very low laser power, in the range of mW and lower, necessary for their generation and manipulation.

The investigation of PR spatial solitons is thus accessible with conventional continuous-wave laser sources and allows to foresee a number of realist applications in waveguiding, switching and optical information processing systems, owing to the compactness and relatively low cost of the experimental devices. PR spatial solitons are based on an optical nonlinearity which results from charge excitation, transport and trapping and gives rise to a space-charge filed that in turn modulates the refractive index by means of the electro-optic (i.e. Pockels) effect.

Although the first observations [2, 3] and advanced experiments [4, 5] were realized in strontium barium niobate (SBN) crystal, it was recently demonstrated that lithium niobate (LiNbO₃) can be efficiently employed for forming narrow bright screening spatial solitons [6]. At first planar and channel waveguides in LiNbO₃ has been produced by dark solitons, that leave the material modified for a long time [7]. Dark photovoltaic (PV) solitons have been observed in LiNbO₃, due to the defocusing nature of the photovoltaic effect [8]. However has been observed that bright solitons are much more stable than dark solitons and offer the simplest way to photo-induce slab and channel waveguides.

Lithium niobate is available in good optical quality and it is a widely used material for optoelectronic applications, with large electro-optic, nonlinear and acusto-optic coefficients. The possibility of generating and employing spatial solitons in LiNbO₃ for writing optimum single-mode waveguide memorized for long time, represents a convenient low-power light technique [6], advantageous compared to other light-induced techniques such as direct UV writing [9], micromachining [10] or ion and proton exchange [11].

An important issue for complete solitonic waveguide characterization in photorefractive media, such as lithium niobate, and a prerequisite for understanding the physical mechanisms underlying the writing process is an accurate investigation of the dynamics of the soliton formation leading to the refractive index change across the waveguide.

Complete characterization of the dynamics of the soliton development requires measuring not only the output light intensity distribution but also the wavefield phase at the exit plane of the photorefractive crystal. To the best of our knowledge, a technique for the simultaneous measurement of the transient dynamics of both the intensity and phase of the output beam propagating in the photorefractive medium has not been presented before in the literature. We show here how reconstruction of amplitude and phase of the wavefront is possible as it emerges from the crystal exit face while the writing process of the soliton formation takes place. We were able to follow the dynamics of the process of the wavefront at the output of an externally biased photorefractive LN crystal by means of an interferometric method based on digital holography (DH) technique [12]. Recently, DH has been employed as a full field, high spatial resolution technique for measuring stationary refractive index profiles of femtosecond laser written waveguides [13] or for in-situ characterization of electric poling of LN z-cut crystals or for measuring internal fields [14,15]. For studying the temporal behaviour of photorefractive spatial solitons sequences of digital holograms have been recorded and numerically reconstructed to obtain quantitatively amplitude and phase maps evolution during the process of formation. Comparison of the experimental results with numerical simulations within the framework of a time-dependent one dimensional band transport model is also presented and discussed.

2. Experimental setup

The experimental set-up to generate and to study the dynamics of the soliton formation is shown in Fig. 1. It is based on a classical Mach-Zehnder configuration in which a laser source emitting at a wavelength of λ = 514 nm from a Ar+ laser is split into two beams, the reference and the object beam respectively. The object beam is directly focused with beam-waist of 29 μm onto the input face of a congruent undoped LiNbO₃ crystal with dimensions 10mm × 7mm × 0.5(c) mm (Altechna Co. Ltd.). The object beam is linearly polarized along the crystallographic Z-axis and its intensity on the entrance face of the crystal is I_max = 2.13 W/cm². The crystal is externally biased along the Z direction by a static, high voltage field of 3.5 kV/mm and illuminated by a background blue laser emitting at a wavelength of 473nm, along the Y crystallographic direction. The background beam provides a uniform illumination of the sample with an intensity I_b =1.2 mW/cm². The object beam propagates trough the crystal length along the X-axis. The wave front emerging at the exit face of the crystal, orthogonal to the X axis, is imaged by a 10 × microscope objective (MO) in a plane at a distance d in front of the CCD array plane which records the interference pattern between the plane wave reference beam and the object beam.

 

Fig. 1. Scheme of set-up used for soliton formation

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We made use of DH in microscope configuration to reconstruct the complex wave field U_L(Y,Z)= |U_L(Y,Z,t)|exp[i φ_L(Y,Z,t)] at the exit plane X=L=7mm of the crystal as function of time t. Whole wave field reconstruction during the formation process of the photorefractive bright soliton means that both the time dependent intensity I_L(Y,Z,t)=|U_L(Y,Z,t)|² and phase φ_L(Y,Z,t) distributions are numerically reconstructed from a sequence of digital holograms recorded at different instant of time, after the externally applied voltage is suddenly switched on. The numerical reconstruction process of the digital holograms is based on the Fresnel transformation method [12].

DH is an imaging method in which the hologram resulting from the interference between the reference and the object complex fields, w₁(x, y) and w₂(x, y) respectively, is recorded with a CCD camera and reconstructed numerically. The hologram is multiplied by the reference wavefield in the hologram plane, namely the CCD plane, to calculate the diffraction pattern in the image plane (see Fig.1). The reconstructed field _Γ(v,μ) in the image plane, at a distance d from the CCD plane along the beam path, is obtained by using the Fresnel approximation of the Rayleigh-Sommerfield diffraction formula

where r(ξ, η) ≡ w ₁(ξ, η) is the reference wave which in the case of a plane wave is simply given by a constant value, h(ξ, η) = |w ₁(ξ, η)+w ₂(ξ, η)|² is the hologram function, λ is the laser source wavelength and d is the reconstruction distance, namely the distance measured between the object and the CCD plane along the beam path. The coordinates (v,μ) are related to the image plane coordinates (x’, y’) by v = x́/λd and μ = ý/λd. It is clear that the object field Γ(v,μ) is the Fourier transform of the hologram h(ξ,η) multiplied by the reference wave r(ξ,η) and the chirp function exp[(iπ/λd)(ξ ² +η ²)]. The pixel size (Δx́, Δý) in the image plane is related to that (Δξ, Δη) of the CCD array through the equations Δx¹ =λd/NΔξ and Δy¹ =λd/NΔη where N is the pixel number of the CCD array. The great advantage of this technique is that it allows to reconstruct numerically the complex field of the object beam. The two-dimensional amplitude A(x’, y’) and phase ϕ(x’, y’) distributions of the object wavefield can be re-imaged by using one hologram acquisition and performing simple calculations on the object wavefield Γ(v,μ) reconstructed from the numerical solution of the diffraction problem in the computer:

3. Experimental results

Figures 2–4 show the temporal evolution of the field intensity and phase. The spatial resolution of the reconstructed wave front is determined by the size λd / NΔξ of the reconstruction pixel, where the Δξ = 6.7 μm is the pixel size of the CCD detector and N × N = 512 × 512 is the number of pixels composing the digital holograms. For a reconstruction distance d = 100 mm, the corresponding size of the reconstruction pixel is 18.50 μm, that scaled for the actual magnification M = 24.5 gives a final value of 0.75 μm for the spatial resolution of the reconstructed maps. It important to note that by means of DH it is possible to extract information both in amplitude and phase of the reconstructed optical field under investigation.

In Fig. 2(a) is displayed the Gaussian beam intensity map after propagation inside the crystal before the external voltage is applied. Figures 2(b)–2(c) shows intermediate steps at t=84 s and t=172s of the soliton formation during the application of the bias field. The voltage was switched off at a time t=194s. In order to show that the written waveguide lasted longer than the writing time, we characterized the modal profile of the solitonic waveguide at about 20 minutes after the external filed was turned off. Figure 2(d) shows the intensity distribution form retrieved from numerically reconstructed hologram that was recorded at t =1482 s.

 

Fig. 2. Maps of beam intensity during the soliton formation

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Figure 3 shows a comparison between the initial and final amplitude profile, along z axis. In fact, the beam profile on the crystal exit face for t=0 s (blue line) and t =1482 s (black line) is shown. Experimental results are fitted with expected theoretical curve (red lines). The experimental curve for t=0 is fitted with a Gaussian curve shape giving the expected information that the actual profile is Gaussian before the soliton formation process takes place. On contrary at t =1482 the fitting curve is in good approximation a squared-hyperbolic secant confirming that the soliton regime has been achieved [6].

 

Fig. 3 Intensity profile of light beam after and before soliton formation

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Figures 4(a)–4(d) display the corresponding time evolution for the phase distribution at the exit face of the crystal. The retrieved phase maps are shown at the same time as in Fig. (3). In Fig.4(c) it is noticeable a central circular area in the phase map. The optical field is clearly confined where the soliton waveguide is written in respect to the surround area. Moreover, it is important to note that the phase map remains stable even after the voltage is switched off [see Fig 4(d)].

 

Fig. 4. Phase maps of laser beam during soliton formation

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However we have not already investigated completely how much time it remains stable and if changes occurs during the expected decay process. Nevertheless that topic will be the subject of further studies. Figures 5(a) and 5(b) report the corresponding movies displaying the evolution of the soliton formation respectively in amplitude and phase.

 

Fig. 5. (780 KB- 225KB ) Movie of intensity (a) and phase [Media 1] (b) maps during soliton formation [Media 2]

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4. Numerical simulations and comparison with the experimental data

Numerical simulations of the dynamics of the soliton formation, with the beam propagation method, were based on the non-stationary 1+1D ( one propagation direction and one diffraction direction ) band transport model developed in ref. [16]. Compared to other approaches, such as the general treatment of Zozulya and Anderson [17, 18] that requires solving globally an elliptic-type equation for an anisotropic electrostatic potential in the photorefractive medium, this simplified model leads to an explicit expression for the change of the induced refractive index Δn_b(t) = -(1/2) n_b ³r₃₃E_sc(t) in terms of the time dependent local space-charge field E_sc(t),namely

where n_b =2.2 is the extraordinary refractive index, r₃₃ =32pm/V the effective electro-optic coefficient of LiNbO₃ crystal for the geometry under consideration, E_ph is the photovoltaic field, assumed to be directed along the crystal Z axis, E_o is the amplitude of the external electric field, k_b the Bolzmann constant, T the crystal temperature, e the electron charge, T_d the dielectric response time of the crystal medium in the dark, I_d the equivalent dark irradiance, with I_d ≪ I_b and the quantity I_n =I+I_b+I_d is the generalized light intensity, taking into account both the beam local intensity I, thermal and photo-excitation.

In the paraxial approximation the slowly varying amplitude A(X,Y,Z,t) of the optical field is related to beam optical intensity by I= |A|² and it is determined by the standard propagation equation

where k=2πn_b/λ is the wavevector in the medium and the time dependence of A(X,Y,Z,t) accounts explicitly for the non-stationary evolution of the induced refractive index Δn_b(t). By inserting Eq. (3) in Eq. (4) the envelope evolution equation can be derived in normalized form, namely

In Eq. (5) we have introduced the non linear function

and we have defined the complex amplitude U=A/√I_d normalized with respect to the dark irradiance, the dimensionless transverse coordinates Ỹ=Y/w₀, Z̃ = Z/w ₀ normalized with respect to the beam-waist w₀, the longitudinal coordinate x̃ = x/kw ² ₀ and the following dimensionless quantities N²=k²n_b ²r₃₃w₀ ²(E₀+E_ph)/2, N_ph ²=k²n_b ²r₃₃w₀ ²E_ph/2 related to the drift and photovoltaic mechanism of charge transport and D= k²n_br₃₃w₀k_bT/2e which is characteristic of the diffusion mechanism of transport. The time dependence of f(|U|²,t) in Eq. (5a) is governed by the dielectric response time in the dark T_d = ∑/I_d.

To compare with the experimental results we have numerically solved the beam envelope propagation Eq. (5) using an iterative beam propagation method, with the value of the experimental parameters and an circularly symmetric input Gaussian field, polarized parallel to the Z crystallographic axis

Movies of the numerically simulated amplitude I_L(Y,Z,t)=|U_L(Y,Z,t)|² and phase distribution in the ỹ-z̃ plane, at the exit plane of the crystal, are shown in Figs. 6(a) and 6(b), respectively. The movie of the evolutions of output intensity of the nonlinear propagation after the externally applied voltage is suddenly switched on at t = 0 shows clearly the shrinking of the input beam diameter along the z direction. The output beam profile converges rapidly within to an elliptically shaped soliton solution elongated in the y direction, in agreement with the experimental observations which show that the circular input Gaussian beam changes into an elliptical shape on a time scale of the order of few tenths of seconds after a dc bias of 3.5 kV/mm.

The theoretical results were obtained with two fitting parameters: E_ph -24.5kV/cm for optimum beam width fitting and ∑ = T_dI_d =600 W∙s cm^-2 for best temporal fitting. The fitted value of the photovoltaic field E_ph is in good agreement with that reported by Fazio et al [6], where screening-photovoltaic solitons were observed in congruent lithium niobate crystal electrically biased along the c-axis, while the estimated value of ∑ is slightly lower. The parameter ∑ is characteristics of the relaxation of the self focusing process and this discrepancy can be attributed to the slight different input and background beam intensities of our experimental conditions compared to those of Ref [6]. However, we point out that the numerical simulations could not reproduce the observed time evolution of the output beam width at larger times. A stable elliptically-shaped steady-sate is obtained, whereas the beam width is observed to perform several oscillations with a width decreasing over both the y and z directions, until a quasi steady-state soliton channel is written.

 

Fig. 6. (250KB-580KB) Movie of simulated intensity (a) and phase [Media 3] (b) maps during soliton formation [Media 4]

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The evolution of the phase distribution gives a further impressive signature of the soliton formation. The movie in Fig. 6(b) shows the non linear phase build up ϕ_L(Ỹ,Z̃,t) = Arg⌊U_L(Ỹ,Z̃,t)⌋ at the exit face x=L of the crystal after the dc bias. The phase changes gradually from the parabolic-like shape distribution of the Gaussian input beam to an almost uniform phase profile across a strip elongated in the y direction over the spatial extent of the output beam, when the soliton channel is formed at longer times.

The numerical simulations agree quite well with the experimentally measured phase distribution at shorter times, whereas the phase distribution at longer times is found to be uniform over a larger spatial extent compared with the experimental results.

This again is attributed to the dimension discrepancy between the theoretical model and experiment. Indeed, the nonstationary space charge field given by Eq. (3) was actually derived in the framework of a 1+1D model and its accounts quite reasonably of the anisotropic nature of the photorefractive medium and of the externally applied electric field, at least at shorter time scale and for low value of the intensity, but a full nonstationary three-dimensional model is expected to account explicitly of the refractive index distribution induced by the optical beam, at larger times when the charge carriers have time to redistribute themselves.

Furthermore, the recently reported observations [19] of large self-deflection of solitons beam in undoped LiNbO₃, attributed to the low level of impurity acceptors N_A present in the samples, seems to suggest that for low N_A in undoped LiNbO₃ samples the expression of the space charge field given by Eq. (3) should be modified, even within the framework of a one dimensional model, through the inclusion of higher order terms depending on the spatial gradient of the field.

5. Conclusion

We have reported and discussed results about experimental investigation and numerical simulation on temporal behaviour of photorefractive spatial bright-soliton in LiNbO₃.

To the best of our knowledge, it is the first time that a full characterization of the soliton radiation, either in amplitude and phase is provided during the formation process of the bright soliton. The phase map allow to visualize and evaluate in quantitative manner and in full field mode the wavefront as it emerges at the exit face of the crystal. We believe that the possibility to have and handle the quantitative information will allow to verify theoretical model and corresponding numerical simulation of the dynamics of the soliton development while it is formed inside the crystal.