Quasi-periodic arrays of bright soliton-like beams are obtained experimentally in the picosecond regime as a result of the transverse modulational instability of a noisy continuous background in a planar CS2 waveguide. For a given propagation length, the array is stable from a laser shot to another and for a wide range of input intensities. The experimental period corresponds to the maximum gain of modulational instability only for the intensity just sufficient for soliton formation. On the other hand the mean period increases with the propagation length. We show by a numerical simulation that the leading edge of the pulse governs the dynamical formation of the array owing to the finite relaxation time of the reorientational Kerr nonlinearity in CS2.
© 2002 Optical Society of America
Though spatial solitons have been extensively studied in recent years [1–3] few works demonstrated the experimental generation of arrays of spatial solitons in the traveling-wave regime. Such arrays were obtained in bulk materials from transverse modulational instability (MI) or beam shaping in photorefractive  or quadratic [5, 6] crystals, as well as in Kerr liquids [7, 8]. To obtain one-dimensional MI, the incoming beams were shaped into highly elliptical stripes. In the particular case of positive Kerr media, self-focusing was prevented by segmentation of the CS2 medium by interference fringes . Surprisingly, the most straightforward way that consists in using a planar waveguide to obtain 1-D MI and to prevent self-focusing was considered only recently [9–12].
We studied in Ref.  the formation of a soliton array in CS2 from a weakly contrasted sinusoidally modulated input field (induced MI). We consider in the present paper the picosecond excitation of the waveguide by a transversely homogeneous background with spatial noise. In the second part, we present our experimental results and compare them to the theory of modulational instability, which predicts that the period of the array corresponds, for a given input intensity, to that for which the MI gain is maximum and hence should continuously vary with the input mean intensity. Some previous experimental results seem to support this prediction, in a Kerr medium  as well as in a quadratic one . As for our results however, the correspondence holds only for the minimum intensity allowing soliton formation. For greater intensities, the solitons remain located at the same places, with the same period. On the other hand output images for two lengths of the waveguide exhibit a mean period increasing with length. In the third part of this paper, we compare these results to a numerical simulation involving a constant spatial input noise and the finite relaxation time of the Kerr response in CS2. The good agreement allows us to conclude that the leading edge of the laser pulse governs the soliton array formation.
2. Experimental results
The experiment is depicted in Figure 1. The waveguide is a 10 μm-thick nonlinear layer (liquid carbon disulfide) sandwiched between two SK5 linear glass blocks of lower linear refractive index (Δn= - 0.04). Windows placed on the four sides of the rectangular guide allow the use of two different propagation lengths (L = 3 or 7 cm) by a π/2 rotation. 38-ps (FWHM) Gaussian pulses at 532 nm are delivered by a 10-Hz, Q-switched, mode-locked, and frequency-doubled Nd:YAG laser. The incident energy is adjusted by means of a half-wave plate and a Glan polarizer at the entrance of the experimental setup. A thin focal line homogeneous on about 1.5-mm width along x is smoothed from the nearly TEM00 laser beam by a spatial frequency filter and shaped by sphero-cylindrical optics. A second half-wave plate ensures linear TE input polarization. Two synchronized single-shot CCD cameras acquire the time-integrated output-face image and its spatial Fourier spectrum. The energy at the waveguide input is measured with a calibrated energy-meter (losses through the windows of the CS2 waveguide are taken into account). Due to imprecise knowledge of the actual launching conditions and matching efficiency between the incident focal line and the TE0 mode, we estimate a ± 10% uncertainty on the injected mean intensity. In the following, the experimental mean intensity is defined at the peak of the Gaussian temporal pulse at the input.
Figure 2 illustrates the transformation of an intense, homogeneous, and noisy pump wave through MI. The red curve of Fig. 2.(a) displays a typical noisy beam profile at the waveguide output when the nonlinearity is negligible, i.e. for a low input intensity. Though this output profile does not give the input profile of the noise before diffraction (absorption is negligible), it can be assumed that its contrast remains similar. The characteristics of the spatial noise are then the followings: its intensity contrast is about 15% and its intensity repartition is reproducible from one shot to another (the spikes are located at the same place in all shots). This key point, experimentally verified at the output for low intensities, is assumed to remain true at the input for high intensities. This noise is due to small defects of the geometry of the monomode laser and it is well-known that the beam profile of a laser, though not perfect, does not vary from one shot to another. On the other hand, experiments in Refs.  and  use an optical parametric generator, i.e. an amplifier of quantum noise: with such a source, variations of the beam profile are expected from shot to shot, owing to the random feature of the generation process.
The blue curve of Fig. 2.(a) displays the beam profile at the waveguide output for a high input intensity. As explained below with the MI formalism, the waveguide confinement prevents the beam from randomly self-focusing, and leads to the formation of a soliton array. As a consequence of the periodicity of the generated array, the growth of spatial harmonics is clearly seen from the corresponding output Fourier spectrum (blue curve of Fig. 2.(b)).
Output images corresponding to three different intensities and the two waveguide lengths are shown on Figure 3. From these images and others obtained for intensities ranging from 10 to 650 MW/cm2 (Figure 4), it appears that:
- For a given length L, solitons are located at constant places, therefore with the same period, from shot to shot and whatever the intensity. Above a threshold intensity, greater for the shortest length, irregular spikes are formed between the first generation of solitons.
- The mean period is 84 μm for L = 3 cm and 125 μm for L = 7 cm. Because the guides for the two lengths have the same characteristics, it can be inferred that some solitons disappear or merge during the propagation.
For a given input intensity, each spatial frequency of the initial noisy distribution can be associated to a parametric gain by the theory of modulational instability . The spatial period corresponding to the maximum gain is plotted on Figure 4 versus the incident mean intensity and compared with the experimental values. It can be seen that the experimental period is constant for a given length for a wide range of intensities. Though the lowest intensity leading to soliton formation has no precise definition, the corresponding period is approximately given by the theoretical curve of MI. On the other hand, this period remains constant for higher intensities and a clear discrepancy with the MI curve appears (see for example the point (e)). For the highest intensities, the appearance of spikes between the solitons can be interpreted as frequency doubling (two last red points). In the next section, we explain this behavior by a spatiotemporal dynamics of the generated arrays due to the non-instantaneous Kerr nonlinearity in CS2.
3. Numerical simulation
The propagation of the optical field is modeled by a scalar nonlinear Schrödinger equation, modified in order to take into account, by a convolution, the 2-ps decay time of the Kerr nonlinearity in CS2, due to molecular relaxation (see Ref.  for details). The numerical simulation uses the standard split-step algorithm for which the refractive index change is not only intensity- but also time-dependent and reads as
where n 2 stands for the usual (instantaneous) Kerr coefficient, I(t 1) being the intensity at time t 1 before t, i.e. earlier in the pulse, and τ the relaxation time of the nonlinearity. The spatial noise at the input is additive, white and its amplitude obeys a Gaussian statistics, with a random phase. As observed experimentally however, it is assumed that this noise is constant from one shot to another, whatever the intensity, and purely spatial: for a given abscissa x, the pulse is temporally perfectly Gaussian.
Figure 5 shows the output time-integrated intensity profiles for two input intensities. In good agreement with the experiment, solitons are located at the same place, with spikes between solitons for the highest intensity. Figure 6 shows, for I 0 = 440 MW/cm2, the evolution of the time-integrated intensity with the propagation length. Solitons begin to form from about L = 2.0 cm, with a mean period of 80 μm, very close of that corresponding to the maximum MI gain. However, some solitons merge or disappear, leading to a mean period of 110 μm for L = 7 cm.
For the same intensity as in Figure 6, Figure 7 shows the spatiotemporal repartition of intensity from the beginning (L = 0 cm) to the end (L = 7 cm) of the guide (see the movie). For L = 3 cm, solitons are formed around the peak of the pulse, with a period corresponding to the MI gain for this peak intensity. When light propagates further, a smaller gain per unit length is sufficient to form solitons. Hence, solitons are formed closer to the leading edge of the pulse, with a greater period corresponding to the smaller intensity in this leading edge, and tend to impose their shape to the rear of the pulse, because of the relaxation time of the nonlinearity. Indeed, Eq. (1) means that the leading edge, corresponding to the early arrived light at a given propagation distance, modifies the index seen by the later arriving light. This modification becomes important as soon as a soliton array has been formed, because such an array involves high enough intensities to impose a nonlinear index profile to the future. Since the period increases during the propagation, some solitons formed at shorter distance have disappeared or merged. At the trailing edge however, a second generation of spikes begins to form, that corresponds to frequency doubling in the time-integrated experimental data of Fig. 4. We showed in the case of induced MI  that the relaxation time of CS2 prevents periodic recurrence  to occur and leads to the formation of this second generation.
We show in the following that the experimental data can be explained by considering that the total MI gain that leads to the formation of solitons is a constant for a given level of input noise, whatever the soliton formation begins at the peak of the pulse or at its leading edge. The MI gain per unit length δ for an incident mean intensity I 0 is given by 
for the spatial frequencies 0 < |Ω| < Ωc = 2β√n 2 I 0/n 0 , n 2 and n 0 being respectively the nonlinear and the linear index and β the effective wave vector. The maximum growth rate
I 0 is the mean intensity at the time of the pulse where solitons begin to form and Ωc the cutoff frequency above which corresponding Fourier modes are spatially stable.
The soliton formation begins for a constant value of the total gain G,
where K is a constant. This relation is in good agreement with the experimental periods shown in Figure 4:
Eqs. (3) and (4) also predict that I 0 is inversely proportional to the propagation length. This prediction cannot be directly compared to the experiment, where only time-integrated profiles were recorded. For a 38-ps (FWHM) pulse, I 0 decreases to 3/7 of its peak value for T = - 20 ps (20 ps before the peak value). Comparison between Figures 7.(a) and 7.(b) rather shows a shift of less than 15 ps for the soliton formation. Actually, this simple model does not take into account the Gaussian shape of the pulse: on Figure 7.(b), the formation takes place in an area where the intensity is strongly increasing. Hence, there is less influence of the convolution with past intensities. On the other hand, simulations with other mean intensities show that lengths where solitons begin to form are actually inversely proportional to these mean intensities at the peak of the pulse.
A difference remains between experiment and simulation: while the evolution of the mean period is in good quantitative agreement, its absolute value is not the same: for L = 7 cm p experiment = 125 μm and p simulation = 108 μm. It is clear that this period depends strongly on the exact characteristics of the input noise. For example, very low input noise needs a high gain G to form solitons, resulting in a smaller period for a given length. We believe that a precise knowledge of the characteristics of the input noise in amplitude as well as in phase (the experimental noise shown in Fig. 2(a) being an intensity profile, the assumption of a random phase is questionable) is necessary to obtain a quantitative concordance.
Our experiments showed a striking discrepancy between the actual period of the soliton arrays and the period that is predicted by the modulational instability formalism. We have proposed a mechanism involving the relaxing time of the Kerr medium to explain this difference. Numerical simulations support this explanation qualitatively, and largely quantitatively. Results of Refs.  and  are in agreement with MI predictions because the involved nonlinearities (non-resonant electronic Kerr response or second harmonic generation) can be considered as instantaneous. However, the correspondence between MI and the observed period was obtained by applying coefficients to take account of the pulsed character of the source. It is also clear that, in an instantaneous medium and in absence of time resolution in the detection (CCD camera), slices of input noise that are not temporally correlated would result in uncorrelated soliton arrays that would blur each other. Moreover, in these references the input beam is issued from an optical parametric generator. The spatial noise for such a source is not reproducible from one shot to another, leading to transverse jitter of the arrays, unlike in our laser. Hence, we believe that work remains to do to understand more generally the precise relation between the spatiotemporal characteristics of the input noise and the observed output patterns. We plan to use parametric fluorescence as a random spatiotemporal source to progress in this direction.
This work has been supported in part by the European Union project QUANTIM under Grant No. IST-2000-26019. http://sucima.dipscfm.uninsubria.it/quantim/
References and Links
1. S. Trillo and W. E. Torruellas, Spatial Solitons (Springer Verlag, Berlin, 2001).
2. Special Issue on Solitons, Opt. Photon. News13, 17–76 (2002).
4. A. V. Mamaev, M. Saffman, and A. A. Zozulya, “Break-up of two-dimensional bright spatial solitons due to transverse modulation instability,” Europhys. Lett. 35, 25–30 (1996). [CrossRef]
5. R. A. Fuerst, D-M. Baboiu, B. Lawrence, W. E. Torruellas, G. I. Stegeman, S. Trillo, and S. Wabnitz, “Spatial modulational instability and multisolitonlike generation in a quadratically nonlinear optical medium,” Phys. Rev. Lett. 78, 2756–2759 (1997). [CrossRef]
6. A. Bramati, W. Chinaglia, S. Minardi, and P. Di Trapani “Reconstruction of blurred images by controlled formation of spatial solitons,” Opt. Lett. 18, 1409–1411 (2001). [CrossRef]
7. P. V. Mamyshev, C. Bosshard, and G. I. Stegeman, “Generation of a periodic array of dark spatial solitons in the regime of effective amplification,” J. Opt. Soc. Am. B 11, 1254–1260 (1994). [CrossRef]
8. H. Maillotte and R. Grasser, “Generation and propagation of stable periodic arrays of soliton stripes in a bulk Kerr liquid,” in Nonlinear Guided Waves and their Applications, Vol. 5 of 1998 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1998), 167–169.
9. C. Cambournac, H. Maillotte, E. Lantz, M. Chauvet, and J. M. Dudley, “Réseaux périodiques de solitons spatiaux dans un guide plan,” in Journées Nationales d’Optique Guidée ‘99, Limoges (France), 6-8 december 1999, (Recueil des communications, 109–111).
10. R. Malendevich, L. Jankovic, G. I. Stegeman, and J. S. Aitchison, “Spatial modulation instability in a Kerr slab waveguide,” Opt. Lett. 26, 1879–1881 (2001). [CrossRef]
11. R. Schiek, H. Fang, R. Malendevich, and G. I. Stegeman, “Measurement of modulational instability gain of second-order nonlinear optical eigenmodes in a one-dimensional system,” Phys. Rev. Lett. 86, 4528–4531 (2001). [CrossRef] [PubMed]
12. C. Cambournac, H. Maillotte, E. Lantz, J. M. Dudley, and M. Chauvet, “Spatiotemporal behavior of periodic arrays of spatial solitons in a planar waveguide with relaxing Kerr nonlinearity,” J. Opt. Soc. Am. B 19, 574–585 (2002). [CrossRef]
13. A. Hasegawa and W. F. Brinkman, “Tunable coherent IR and FIR sources utilizing modulational instability,” IEEE J. Quantum Electron. QE-16, 694–697 (1980). [CrossRef]
14. G. Van Simaeys, Ph. Emplit, and M. Haelterman, “Experimental demonstration of the Fermi-Pasta-Ulam recurrence in a modulationally unstable optical wave,” Phys. Rev. Lett. 87, 033902 (2001). [CrossRef] [PubMed]