The concept of optical beam self-trapping in pyroelectric photorefractive medium is presented. We show that the temperature controlled spontaneous polarisation of ferroelectric crystals produces an optical nonlinearity that can lead to formation of 2-D spatial soliton named pyroliton. Experimental demonstrations performed in lithium niobate crystals illustrate that efficient self-trapping occurs either for ordinary or extraordinary polarisation under moderate temperature increase. For instance, a 15µm diameter pyroliton can be formed with a 10 degree temperature raise.
©2009 Optical Society of America
Since the prediction of beam self-trapping more than 40 years ago  followed by the first observation of stable trapping by Kerr effect 20 years later , the quest for more versatile and efficient nonlinear focusing medium remains. Quadratic medium , liquid crystals  or photorefractive (PR) medium  are able to give rise to self-confined beams that keep a constant transverse profile upon propagation. Such beams called spatial solitons [6, 7] have also been named specifically depending on the configuration. For instance Simultons applies to two frequency components solitons and Nematicons are formed in nematic liquid crystals. Photorefractive spatial solitons have attracted much attention [8–11] because of their appealing features. For instance, they can be created at very low light power level and are stable in 2-D due to the saturating character of the nonlinearity. Nevertheless, application of a strong external applied electric field is, most often, a necessary condition to obtain the local PR nonlinearity required in light trapping experiments . This drawback is not present in the case of photovoltaic photorefractive crystals [13, 14] but control of the photorefractive nonlinearity sign and amplitude is challenging  since the photovoltaic field is intrinsic to the crystal and is primarily intensity independent. In this paper we instead propose to use the spontaneous polarisation of ferroelectric crystal to induce a nonlinear optical effect which can be controlled using the pyroelectric effect. The concept of pyroelectric photorefractive solitons that can be formed by simple adjustment of the PR crystal temperature is introduced. Experimental demonstrations performed in lithium niobate (LiNbO3) samples are presented.
2. Theoretical model
Pyroelectric crystals exhibit a variation in spontaneous polarisation Ps as a function of temperature change given by its pyroelectric coefficient p=∂Ps/∂T. In short-circuit crystal this spontaneous polarisation change is known to induce transient electric currents fruitfully utilized in wide bandwidth pyroelectric photodetectors. In open-circuit configuration very high electric field can be generated by pyroelectric effect. Such large electric field induced in the surrounding of pyroelectric crystal can be used to ionise and accelerate charged particles [16-17]. Recently, steady-state temperature gradient was shown to suppress photorefractive damage in LiNbO3 . In this letter internal electric field generated by a temperature change DT in an open-circuit crystal is considered. This electric field variation Epy due to the spontaneous polarisation change is given by:
To model beam self-trapping in pyroelectric photorefractive crystal we consider a time-dependent band-transport model  with a single deep trap. Electrons which are dominant free charges are displaced by photovoltaic effect and under drift current along the crystal c-axis while diffusion is neglected. An open-circuit crystal at homogeneous and steady temperature is considered.
When illuminated with a light intensity distribution I, evolution of the charge density r is given by:Eqs. (3), (4) and (5), respectively
The electric field evolution is obtained by solving iteratively the set of Eq. (2-5)  in conjunction with the wave propagation equation. Starting from initial conditions, the electron density is calculated from Eq. (3) and space charge variation after a time step ∆t is deduced from Eq. (2). Then the ionized donor density is obtained from Eqs. (4). The electric field is deduced from Eq. (5). The inferred 2-D refractive index perturbation ∆n due to linear electro-optic effect is finally used to calculate light propagation along y-axis in the perturbed medium by a classical split-step Fourier transform method according to the nonlinear propagation equation.
To illustrate the concept of pyroelectric beam self-trapping we selected the widely available ferroelectric LiNbO3 crystal. It possesses well-known PR properties along with a pyroelectric coefficient p close to −6.10−5 C m−2 K−1 at 25°C [21, 22]. The pyroelectric field Epy is oriented along the crystal c-axis (z-axis) and can be either positive or negative by increase or decrease of the crystal temperature, respectively. From Eq. (1) an amplitude of 47 kV/cm can be deduced for the internal pyroelectric electric field Epy under a temperature increase of 20°C. A linearly polarized beam propagating along the crystal y-axis can sense the light-induced refractive index distribution ∆n=−0.5n3reffEz. Where Ez is the electric field component along the crystal c-axis, n is average refractive index and reff is the effective electro-optic coefficient. Two configurations are analyzed, no=2.3 and r13=8pm/V for ordinary polarized light and ne=2.2 and r33=31pm/V for extraordinary polarized light. Undoped LiNbO3 crystals are used for the experimental demonstration and a single iron deep center is considered in the model. Corresponding parameters are taken from ref .
Figure 1 depicts the predicted behavior of an extraordinary polarized beam, with a 16µm (FWHM) width at the entrance face, propagating in a 20 mm long LiNbO3 crystal whose temperature has been raised 10°C above initial temperature. When light is switched on progressive self-focusing is observed (Fig. 1(c)) until best confinement is reached (Fig. 1(d)).
The z-component distribution of the electric field, at the heart of the self-confinement process, is depicted in Fig. 1(e). Away from the beam location the electric field is equal to Epy while in the central part of the beam lower amplitude is obtained because of the presence of the photorefractive space charge field. Note that the minimum field amplitude approaches the photovoltaic field amplitude Eph=βγNA/(eμs). Therefore Eph depends on light polarization  through photovoltaic coefficient βph and on crystal doping concentration. In the presented simulation Eph is set to 19kV/cm for extraordinary polarization. Since the index change is proportional to the opposite of the electric field it thus results that a waveguide structure is created in the crystal providing Epy is larger than Eph. Moreover anisotropy of the effect is characterized by the two local minima present along the c-axis which are typical of a two-dimensional charge distribution as earlier observed for screening solitons . This anisotropic index distribution usually gives a slightly elliptical beam with a stronger confinement along c-axis. Note that for chosen parameters a circular beam with a similar size than the launched beam (16µm FWHM) is present at the exit face of the crystal when best confinement (Fig. 1(a), 1(d)) is reached which corresponds to a solitonic propagation. For parameters corresponding to an ordinary polarized beam (not shown) confinement is also observed. However the weaker adverse photovoltaic field and the electrooptic coefficient which is about 4 times lower than for extraordinary light give only partial beam diffraction compensation.
For the experimental demonstration either undoped photonic grade congruent or stoichiometric LiNbO3 crystals are used. Samples cut from photonic grade z-cut wafers have dimensions 8×20×0.5 mm3 along x, y and z crystallographic axes, respectively. The sample under test is placed between an insulating plastic cover and a metallic plate whose temperature is accurately controlled by a Peltier element. Such an arrangement provides a homogeneous temperature of the crystal with stability better than 0.1°C. Input and output faces of the crystal are observed on a CCD camera via imaging lenses.
A first experiment is performed with an extraordinary polarised light in a stoichiometric LiNbO3 crystal. A 532nm CW beam is focused to an 11µm FWHM spot at the front face and propagates over 20mm along the LiNbO3 y-axis direction. When the crystal is at room temperature the beam diffracts along propagation, its width at the output face approaches the crystal 500 µm thickness (Fig. 2(b) ) as witnessed by the Lloyd interferences produced by reflection on each side of the crystal. Moreover its spatial distribution is not Gaussian since no spatial filter was used to clean the launched beam. Nevertheless when the crystal temperature start to rise from room temperature to 40°C the 80µW beam self-focuses progressively. As shown in Fig. 2(d) beam confinement and clean-up are already obvious after 1 min of exposure while crystal temperature is not yet stabilized. As predicted by the numerical model, trapping is more efficient along the z-axis than along x-axis which is clearly observed in Fig. 2(e). When maximum confinement (t=tF) is reached a smooth and efficiently focused spot is obtained (Fig. 2(f)) with a 7 µm FWHM along c-axis and a 8 µm FWHM along x-axis. The large initial beam diffraction has thus been overcompensated which illustrates the remarkable self-trapping efficiency triggered by the spontaneous polarisation of the crystal. Note that self-trapping can also be obtained when crystal temperature is stabilized for several minutes before beam is switched turned on. Since this procedure is more consistent with the assumptions of our theoretical model it is used as a standard procedure in the following experiments. Assessment of the focusing strength at different time after temperature is set leads to an internal pyroelectric field decay rate τ of about 8h in our lithium niobate crystals. This slow decay due to the compensation of Epy is consistent with the dielectric response time of LiNbO3 in the dark.
To illustrate the influence of temperature change on self-trapping strength, the output beam width as a function of temperature variation ∆T has been measured. Results are depicted in Fig. 3 . Starting from room temperature, the crystal temperature is gradually raised. The beam FWHM is measured both perpendicular and along c-axis for best focusing at t=tF. Focusing strength increases with temperature to initially form an elliptical spot that gradually transforms to a round spot at ΔT=12°C. Larger temperature change tends to distort the trapped beam.
Additional results obtained in congruent LiNbO3 samples show similar behaviour than in stoichiometric samples. As an illustration and, in the meantime, to show the influence of light polarisation, two sets of experiment performed in congruent LiNbO3 are presented in Fig. 4 . It comes into view that optimum confinement leads to tighter beam for extraordinary polarisation (Fig. 4(d)) than for ordinary polarisation (Fig. 4(c)) as expected from theory. Uncertainties on the value of crystal parameters prohibit quantitative prediction of the time evolution but the overall behaviour is correctly described by our numerical calculation. Note that the induction time to reach best focusing (t=tF) is dependent on light polarization. tF=3 min (Fig. 4(d)) or tF=9 min (Fig. 4(c)) for extraordinary or ordinary polarisation, respectively. This response time disparity can be attributed to the photorefractive field which has larger amplitude for ordinary light and thus takes a longer time to build-up . In addition the photoionization cross section can be polarisation dependent because of crystal anisotropy .
For a given temperature change DT, which is the experimental parameter that dictates the amplitude of the nonlinear effect, the beam profile to launch in order to obtain propagation with an invariant transverse profile can be determined. Such a solitonic propagation is depicted in the numerical results in Fig. 1. The corresponding experimental demonstration of a spatial soliton supported by crystal spontaneous polarisation is shown in Fig. 5 . An extraordinary polarized 15µm FWHM pyroliton is obtained when crystal temperature is raised 10°C above room temperature as witnessed by the similar beam profile at the input (Fig. 5(a)) and output (Fig. 5(b)) face of the crystal. Note that these solitons are obtained in quasi-steady-state regime  since longer exposure time shows beam broadening. A solution to form steady-state narrow solitons could be to artificially increase the dark irradiance with the help of a uniform background illumination as for screening solitons . Also note that once spatial soliton is formed, beam can be switched off and crystal can be reset to room temperature without altering the induced guiding structure. Due to the high resistivity of LiNbO3 long living waveguides are thus memorised in the crystal and can thus be practically used to guide low power beams at different wavelengths. Indeed photorefractive effect is negligible for low power visible beam or at near I. R. telecommunication wavelengths. In such a situation induced waveguides are memorized even if LiNbO3 sample temperature changes.
The concept of beam self-trapping using spontaneous polarisation of ferroelectric crystal has been presented. Amplitude and sign of the nonlinear focusing effect is simply triggered and controlled by crystal temperature via the pyrolectric effect. Experimental demonstrations have been performed in LiNbO3 crystals and spatial solitons, named pyroliton, have been obtained. Photonic grade undoped congruent or stoichiometric LiNbO3 samples reveal efficient beam self-confinement under moderate temperature change on the order of 10°C. This powerful and easy to control optical nonlinearity can be advantageously employed to induce structures such as gratings, complex lattices or 3-D integrated circuits even inside large size medium. It is important to note that the reported effect could potentially be obtained in other pyroelectric photorefractive crystals providing characteristics such as long pyroelectric field decay time and large enough pyroelectric field amplitude are present.
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