Spatial and temporal locking of fundamental and second harmonic pulses was realized by means of photorefractive nonlinearity and highly mismatched harmonic generation. Due to the presence of both phase-locked and unlocked second harmonic pulses, a twin simultonic state was observed. Simultonic filamentation occurring at high pumping rates allowed us to determine a relation between the simulton’s waist and its intensity.
© 2010 OSA
Solitonic states are of great interest from both the scientific and applicative points of view. Up to now optical spatial solitons have been observed in different kinds of material nonlinearities, like for example Kerr [1,2] cascading [3–7] and photorefractive nonlinearities [8–10].
Kerr and cascading nonlinearities are indeed very fast, almost coherent with excitation, but also very weak, requiring high local intensities. The high response speed allowed in the past the realisation of both spatial [1–10] and temporal solitons [11–13] In 1999 Liu et al  demonstrated spatio-temporal solitons in LiIO3 crystals because of the simultaneous action of Kerr and cascading nonlinearities. However, the weak nonlinearities requires very intense laser beams to reach such soliton regime, as high as 70 GW/cm2, a value very close to the damage threshold of the material (100 GW/cm2). In this paper we demonstrate that such simultons  that consists of the simultaneous trapping, either in time or in space, of multiple wavelengths can be obtained in space at lower intensity by using the photorefractive effect while a locking in time is obtained through the quadratic effect.
Such photorefractive nonlinearity has advantages and disadvantages at the same time: generally speaking it is slow, but can be active already at very low light intensities, thanks to charge storing process involved. Among all, the photorefractive media is suitable for spatial soliton formation. During the last years big interest is directed to lithium niobate, mainly due to the ultra slow photorefractive relaxation time by which the written solitonic channels can act as optical waveguides long time after the soliton beam has been turned off. These soliton waveguides have been already used both for realizing switching devices  and for enhancing the second harmonic conversion efficiency through a confinement of pumping pulses within solitonic channels written by the second harmonic ones [17–19]. Lithium niobate has a very high nonlinear coefficient (d33 = 41.7 pm/V ) that usually cannot be fully exploited in bulk because of difficulties to obtain phase-matching, Periodically poled LiNbO3 has brought an elegant solution to this problem. However, it was recently shown that working in high dispersion regime (Δk ∝ 104 cm−1), thus very far from phase matching, reveals intriguing behavior. A very weak SH generation is possible even using a polarization coupling of type 0 (ee-e) linked to the d33 coefficient but in such a regime the generated SH splits into two pulses. One pulse travels with a lower speed than the fundamental pulse, as expected from the mismatching, but the second pulse is perfectly locked together with the pump pulse (they experience exactly the same phase- and group- velocities [21–24]). Such phenomenon will be adopted here in conjunction with the recently discovered photorefractive effect induced by pyroelectric effect , to demonstrate the spatio-temporal locking of the pump and second-harmonic pulses. In fact we shall force the temporal-locked second harmonic pulses to modify the refractive index and to give rise to a solitonic channel inside which the pump beam will be confined as well.
We have used a congruent lithium niobate crystal, cut from a commercially available z-cut wafer. The samples have a prismatic shape as shown in Fig. 1 .
The input face was tilted of 20° from the normal incidence in order to spatially separate the inhomogeneous second-harmonic (400nm) phase-locked pulse from the homogeneous second-harmonic pulse. By injecting the fundamental frequency (800nm) pulse at 48.8° from air, the beam is refracted at 20° and then will propagate exactly along the crystallographicdirection. Consequently the two second-harmonic pulses generated at the interface propagate along different spatial directions: the phase-locked pulse follows the fundamental frequency pulse along thedirection (i.e. generated at 20°) while the homogeneous pulse refracts at 18.6° according to the Snell-Cartesio law for the extraordinary refractive index at 400nm.
The experimental set-up is sketched in Fig. 2 . A Ti-Sapphire master oscillator generated 65 fs pulses at 800 nm. Such pulses were amplified in a regenerative amplifier, from which 100 fs pulses are emitted at 1 kHz repetition rate, with energies of the order of tens of μJ. After passing through a variable attenuator and a polarizer, pulses were focused onto the tilted input face of the sample at 48.8° of incident angle. In such a way the fundamental wave was travelling the whole crystal along the crystallographic direction. The input spot size was set at 18 μm FWHM. The sample was mounted on a Peltier heater that was set in order to realise an internal pyroelectric field as high as 35 kV/cm, enough for realising an effective screening and to induce the selfocusing regime.
The output beam was collected by a spherical mirror, in order to avoid chromatic aberrations, and then refocused at a very small angle (to limit spherical and coma aberrations) over the CCD sensor of an RGB camera, with a magnifying power of the order of 14. Filtering of IR light was provided by using coloured glasses.
For soliton stabilisation a background was applied by illuminating the whole crystal from the top-side with a high-power LED (~500-600mW @ 455 ± 5 nm), reaching an average intensity of the order of 4-6 W/cm2.
Low intensity – By injecting a fundamental beam at 800nm with input powers lower than 160 μW, no self-confinement was observable when applying the pyroelectric field (Fig. 3 ).
The generated second harmonic signal was too low to induce self-focusing; however its absorption was enough to have efficient self-defocusing because of the photovoltaic effect. As a result the beam showed depleted regions in the centre, where the photovoltaic field was strongest.
Simulton intensity – As the input average power was increased up to 160 μW, the generated second-harmonic became more important and efficient self-focusing occurred after the pyroelectric field was applied (Fig. 4 –Media 1). As it may be observed in the movie, the photorefractive nonlinearity is indeed cumulative: it requires some time to accumulate charges in sufficient quantities to induce self-focusing. Instead, the photovoltaic effect is almost instantaneous with the free-charge generation. Consequently the beam initially experiences a defocusing because of the photovoltaic field. Because of such defocusing the central part of the beam becomes depleted, and the light accumulates along the external ring, more concentrated in the upper and lower areas of the ring  (Fig. 4—first row from the top). Only after some time the second-harmonic signal generated on-axis by the initial beam becomes more intense inside a small area exactly at the centre where the input intensity is higher (Fig. 4—second row from the top). The generated second harmonic signal “writes” a guiding channel inside which the fundamental will enter as well. Such recursive procedure evolves very rapidly and, as a consequence, the whole system collapses into a solitonic state given by the simultaneous propagation of the fundamental and its harmonic pulses (Fig. 4—third row from the top). It should be stressed here that for such process, due to the special geometry of the crystal, we are dealing only with second-harmonic pulses that are perfectly locked together with the fundamental pulses.
Consequently the simulton is formed by pulses that are perfectly locked together in space and time. The whole process occurs within 5 minutes.
Twin simultonic state – We observe that the unlocked pulses also modify the refractive index and generate a solitonic state as well. Initially this additional process is slower that the development of the simultonic state, because it is not assisted by the fundamental beam but it regards only the second harmonic signal.
The unlocked second harmonic pulse is generated and propagates at a different angle with respect to the fundamental and locked harmonic pulses: this signal is visible in Fig. 4 at the third row of images from the top, to the left of the central spot. The unlocked second harmonic light experiences photorefractive self-focusing as well, and writes a soliton channel of its own. But due to the ultra-slow dielectric relaxation of lithium niobate, the solitonic channel written by unlocked pulses will be sensitive also for the fundamental pump. Then, the pump light will enter inside both channels created by the locked and unlocked pulses making their writing increasingly efficient [Fig. 4(D)]. Moreover, the two soliton channels interact by attracting each other. The final result is a twin simultonic state, constituted by two parallel channels very close to each other, almost overlapped but not coincident. It should be noted that these self-trapped states do not collapse into just a single channel, but instead remain separated, creating the twin state.
The wave equations describing the evolution of the fundamental and second-harmonic beams are:
These two waves, fundamental and second harmonics, are coupled together by the nonlinear quadratic nonlinearity but only the second harmonic is able to modulate the dielectric tensor by means of the electro-optic effect. Such modulation is indeed induced by the local space-charge field whose expression can be calculated using the Kukhtarev equations :
The equations have been solved numerically  in (1 + 1)D geometry using a split-step beam propagation algorithm. An ee-e nonlinear coupling between fundamental and second-harmonic signals was considered , with a d33 coefficient as high as and a Δk as large as . In fact at such very large phase-mismatch between the fundamental and second harmonic waves, the type-0 coupling is the most favoured because of the large nonlinear coefficient.
The numerical results are shown in Fig. 5 for an input intensity of the fundamental pulse of the order of 5 GW/cm2, which corresponds to 160 μW of average beam power (third-order nonlinearities have been neglected in the calculation because they should have a very fast response while the observed phenomena are much slower, within 5 min since the second-harmonic process started, and the pyroelectric field turned on). In the beginning, when no modifications in the refraction have yet to occur, both the locked and unlocked second-harmonic waves diffract along different directions (first two simulations from the top) due to the Snell-Cartesio refraction at the input face. Later on the second harmonic light, which is absorbed now by the lithium niobate host material, modifies the refractive index writing a guiding channel. Such phenomenon occurs initially just for the locked pulses: in fact the self-focusing of locked pulses induces waveguiding for the fundamental locked pulses as well. Consequently, the locked pulses generate a simultonic state.
However, the unlocked pulses modify the refractive index as well, even if with a lower efficiency and consequently slower. As soon as the unlocked pulses experience self-focusing, they attract the locked channel, creating a twin simultonic state, as a singular state of two parallel bound simultons (Fig. 6 ).
It should be pointed out that such bound beams are both simultons. In fact the ultra-slow photorefractive relaxation of lithium niobate fixes the guiding channels for long time. Consequently the fundamental pulses propagate at the same time inside the whole twin simultonic waveguides, i.e. inside the channels originated by both the locked and unlocked pulses. One more very important and peculiar behaviour is the non-collapsing nature of channels. Usually parallel attracting solitons collapse together giving rise to a single soliton channel. Instead here we observe a twin simultonic state. We believe that this phenomenon comes as a result of the simultaneous action of the attractive potential, generated by the photorefractive modification of the refractive index, and off-axis (large angle) k-vectors, that drive the two beams away from each other. An equilibrium is thus reached, for which attraction and repulsion are perfectly balanced, propagating a two-bound-beams state.
Filamentation into simultonic channels
As previously mentioned, a single simulton channel was generated pumping at 800nm with about 5 GW/cm2, which corresponds to 160 μW of average power. Thus, we considered this value as the fundamental simulton intensity, demonstrated from both the experimental and numerical points of view.
For much higher pumping intensities, the beam will give rise to nonlinear filamentation into elementary solitonic channels. In fact the soliton family could be considered a (not ortho-normal) basis for the nonlinear regime. Thus the nonlinear filamentation is a very powerful tool to originate many fundamental solitons-simultons and consequently to characterize them by means of the ratio energy-size.
Such solitonic channels are initially originated by both the unlocked and locked second harmonic pulses. However, as previously described for the twin-simultonic state, the ultra-slow photorefractive relaxation in lithium niobate causes each channel to write a long-lived waveguide, which is active for both the fundamental and second harmonic frequencies. As a consequence, the fundamental beam could be trapped into all nonlinear channels, i.e. into those generated by locked as well as unlocked second harmonic pulses.
Fitting each filament with hyperbolic secant functions, the experimental relation between peak-intensities and hyperbolic waists was derived, as shown in Fig. 8 .
The experimental points are distributed along a negative exponential best-fit curve of equation:
From the fit we realize that the simulton minimum size can be as small as 1.8μm.
We have demonstrated for the first time the spatial and temporal locking of pump and harmonic pulses in lithium niobate crystals. Such phenomenon was achieved by the simultaneous action of a photorefractive nonlinearity and of second-harmonic generation with very large phase-mismatch. The first nonlinear contribution was responsible for the spatial locking while the latter was responsible for the temporal locking. Such simultonic state was generated with a 5 GW/cm2 pump at 800nm, generating a second harmonic signal at 400nm. The numerical simulations confirmed the experimental observations Solitonic filamentation allowed us to derive a general relationship between the hyperbolic waist and the relative intensity.
This work has been supported by the contracts from Sapienza Università di Roma A) ricerche universitarie 2007(processi ottici nonlineari di generazione di armonica in materiali massivi e nanostrutturati altamente dispersivi) and B) ricerche di ateneo federato 2008 (generazione di seconda armonica in sistemi dispersivi e nano strutturati). E.F. is grateful to the Université de Franche Comté for the visiting professorship under which part of this work has been performed. A.M.D.G.
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