Spatial versus Temporal Deterministic Wave Breakup of ... - ICFO

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Spatial versus Temporal Deterministic Wave Breakup of Nonlinearly Coupled Light Waves D. Salerno,1,2,3 S. Minardi, 4 J. Trull,1 A. Varanavicius,5 G. Tamosauskas,5 G. Valiulis,5 A. Dubietis,5 D. Caironi,1 S. Trillo,6 A. Piskarskas,5 and P. Di Trapani1 1 INFM and Department of Physics, University of Insubria, Via Valleggio 11, 22100 Como, Italy INFM and Department of Electronics, University of Pavia, Via Abbiategrasso 209, 27100 Pavia, Italy 3 Faculty of Sciences, University of Milano, Via Celoria 16, 20132 Milano, Italy 4 ICFO-Institut de Cie`ncies Foto`niques c/Jordi Girona 29, Nexus II E-08034 Barcelona, Spain 5 Department of Quantum Electronics, Vilnius University, Sauletekio Avenue 9, 2040 Vilnius, Lithuania 6 INFM and Department of Engineering, University of Ferrara, Via Saragat 1, Ferrara, Italy (Received 3 July 2003; published 1 October 2003)

2

We investigate experimentally the competition between spatial and temporal breakup due to modulational instability in

2 nonlinear mixing. The modulation of the wave packets caused by the energy exchange between fundamental and second-harmonic components is found to be the prevailing trigger mechanism which, according to the relative weight of diffraction and dispersion, leads to the appearance of a multisoliton pattern in the low-dimensional spatial or temporal domain. DOI: 10.1103/PhysRevLett.91.143905

The spontaneous breakup of an intense wave packet (WP) followed by multipeak is among the most spectacular processes of nonlinear optics. It appears as a dramatic manifestation of the universal phenomenon of modulational instability (MI). Spontaneous filamentation of high-power beams in Kerr media has been one of the first evidences of nonlinear dynamics in optics [1], followed by the observation of its temporal counterpart of great relevance in fiber telecommunications [2]. More recently, the importance of MI was significantly widened by studies concerning its role in nonlinear polarization evolution [3], incoherent illumination [4], and

2 mixing, i.e., second-harmonic (SH) generation [5–11]. The latter case constitutes a paradigm for systems where MI builds up from nonlinearly coupled pump waves (that exchange energy as they propagate). This case is also relevant to assess the general role played by MI in the stability of spatiotemporal (ST) solitons [7] and in periodic systems (quasi-phase-matching) [10]. The bottom line of these studies is that they deal with MI separately in the 1D temporal case [2,3,6] or in the spatial case (often 1D as well, when using elongated beams [5], waveguides [8], or ST solitons [7]). In this Letter, we address the competition of these two effects, a regime left thus far widely unexplored, despite the fact that ST-MI must be naturally expected in any bulk dispersive media. The richness of this topic is witnessed by new physics recently unveiled in

2 mixing by the discovery of nonlinear X waves [12] and ’’red solitons’’ (solitons without an apparent high-frequency component) [13], which have been interpreted as a manifestation of colored conical emission [14] (the natural form of space-time MI in normally dispersive media). Here we investigate a different regime, showing that the 1  1  1D (referring to longitudinal, transverse, and temporal coordinates, respectively) dynamics of a pulsed quasi-plane-wave (in 1D, i.e., elongated) beam at funda143905-1

0031-9007=03=91(14)=143905(4)$20.00

PACS numbers: 42.65.–k, 05.45.–a

mental harmonic (FH) undergoing frequency doubling in a bulk sample does not necessarily lead to ST (1  1D) breakup. Rather such breakup and multisoliton formation can occur in a subdimensional (1D) coordinate, either in space or time depending on the relative weight of diffraction and dispersion. We consider low-noise input WPs (commonly accessible thanks to high-quality laser sources) in the absence of any external perturbation (seed). In this regime, we show that the MI-induced breakup is triggered and ruled by a deterministic (in contrast with the stochastic nature of noise) waveenvelope modulation (WEM) which is an intrinsic feature of parametrically coupled pump WPs [5,15]. Our approach allows us also to address the fundamental questions left open by previous experiments on spatial MI [5,8,9,11] about (i) the observed regularity and stability of the soliton pattern [8], in spite of the stochastic nature of the noise-seeded process; (ii) the disappearance of the spatial modulation at high pumping [9]; (iii) the fact that the wave-breakup phenomenology has been described successfully in the frame of spatial (CW) models, in spite of the genuine ST nature of the driving MI process. As we show below, the MI seeding by the WEM process can explain all these features without the need to resort to environmental causes such as crystal imperfections and beam inhomogeneities. To illustrate the ST dynamics, we integrate numerically (via fast Fourier transform, split step, and Runge-Kutta algorithms, with 10 fs 5
m grid and 40
m step) the standard

2 1  1  1D coupled-wave equations for FH and SH envelopes Ej
z; x; t (carrier j!0 ), j  1; 2 [7,11] L^
!0 E1 
E2 E1 exp
i kz  0; L^
2!0 E2  i V@t E2  i@x E2 
E21 exp
i kz  0; (1) where L^
!  i@z  2k
! 1 @2xx  k00
!=2 @2tt , k00 is  2003 The American Physical Society

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group-velocity dispersion, V  k0
2!0   k0
!0  weighs group-velocity mismatch (GVM),  is the birefringence angle, and k  2k
!0   k
2!0 . Figure 1 displays the ST dynamics for a 1 ps, 500
m input WP at FH (parameter values for our noncritical type-I configuration are in the caption). In the first few millimeters of propagation, a high-contrast modulation develops both in space and time, leading to a ring-shaped profile in ST intensity map at FH [see Fig. 1(a)]. The corresponding SH (not shown) exhibits a complementary profile. The origin of this modulation is the different rate of FH-SH energy exchange experienced by the different portions of the wave, related to the different value of intensity over the bell-shaped input profile [5,15]. This leads to both temporal and spatial modulation at any given z, and to the appearance of the ST-ring structure. The number of oscillations (the same in space and in time) does not depend on the input diameter and duration and increases with the input intensity or z. As the number of rings increases with z, both the diffraction and dispersion lengths, Ldiff and Ldisp , associated with the ST-ring thickness decrease until one of the two becomes comparable to the nonlinear length LNL . At this point, the MI starts to play a dominant role leading to FH-SH self-trapping in the (spatial or temporal) dimension whose characteristic length approaches LNL . For the chosen ratio of input beam and pulse width, diffraction overcomes dispersion so that a pattern of spatial solitons get excited [Fig. 1(b)]. Here the

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role of noise is marginal and MI is seeded by the strong self-built WEM. Finally, in the case of longer propagation, ST effects start to play their role leading to a complete and chaotic fragmentation of the WP [Fig. 1(c)]. Note that, for the opposite case of dominant dispersion (i.e., for much shorter pulses or larger beams), a train of temporal solitons should be expected in the transient, with the stripes in Fig. 1(b) vertically oriented. In order to evaluate what can be an experimental evidence of the phenomenology described in Figs. 1(a) –1(c) in the frame of time-integrated (CCD based) detection, we plot in Figs. 1(d) –1(f) the corresponding time-integrated fluence profiles, both for the FH and SH WPs. The results show that in the early stage of the interaction [Fig. 1(d)] the two waves exhibit a very-low contrast modulation, which, however, keeps a clear evidence of the occurrence of complementary profiles. When spatial solitons are formed [Fig. 1(e)], the contrast increases markedly, disappearing again in the chaotic stage [Fig. 1(f)]. For the investigated regime, the development of a clean soliton pattern is strictly limited to the case of fairly weak input noise. In fact, for noise intensities or ST bandwidths exceeding slightly those used for the calculation of Fig. 1 (see caption), a direct transition occurs from the low-contrast case of Fig. 1(a) to that of Fig. 1(c). The experiment aimed to demonstrate the WEM seeding of the

2 MI in the spatial domain was performed by launching (at 10 Hz of repetition rate) a spatially filtered

FIG. 1. (a) –(c) Calculated spatiotemporal intensity map for a FH-WP propagating in a lithium triborate (LBO) crystal in regime of type I, noncritical, SH generation, for three different propagation distances; (d) –(f) corresponding time-integrated fluence profiles at FH (solid lines) and SH (dashed lines). The results are obtained from Eqs. (1) with   0, k  5 cm1 ,
 7  105 W1=2 (deff  0:85 pm=V), V  0:0472 ps=m, k00
!0   0:016 ps2 =m, k00
2!0   0:089 ps2 =m. The input is a Gaussian at FH with 1 ps FWHM duration, 500
m FWHM beamwidth, and 150 GW=cm2 peak intensity embedded in a Gaussian noise with 2% intensity and a ST bandwidth 5 times larger than that of the input pulse.

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FIG. 2. (a) –(b) Experimental transverse fluence profiles of the FH (solid lines) and SH (dashed lines) beams at the exit of a 15 mm LBO crystal, tuned for SH generation close to phase matching. Input WP: 1055 nm, 1 ps, 300  50
m2 FWHM, 2 GW=cm2 (a) and 13 GW=cm2 (b). (c) Transverse profile of the FH, superimposed with its spatial reverse; input WP as in (b) but with 1 mm width. (d) Spatial profile of the FH after 15 mm (solid line) and 50 mm (dashed line); input WP as in (c).

1 ps, 1055 nm pulse of elliptical [
300  50
m2 FWHM] shape into a 15 mm type-I LBO crystal, tuned for noncritical SH generation close to phase matching. The level of residual spatial noise of impinging beam was consistent with that used for the numerical calculations (see caption of Fig. 1). In the experiment, we mimicked the effect of internal propagation by changing the energy of the input WP. The results are presented in Fig. 2. At low pumping, the expected low-contrast FH and SH complementary profiles occur [Fig. 2(a)], while a highly contrasted array of mutually trapped FH and SH beams (spatial solitons) is clearly formed at high pumping [Fig. 2(b)]. In order to investigate the robustness of the process, we repeated the experiment by taking a 1 mm  50
m (FWHM) elliptical beam, for which the relative importance of noise-seeded over WEM-seeded dynamics is expected to increase due to larger beam size. Also in this case, however, we still observed a genuine, WEMdominated regime. A spectacular demonstration of that is given in Fig. 2(c), where the measured FH fluence profile is plotted against its spatial reverse. Even if the pattern is not regular, in the sense that the distance between neighboring solitons is not constant across the beam, it appears to be not only stationary (from shot to shot) and independent from the position of the beam inside the crystal, but perfectly symmetric, consistent with what should be expected when the soliton position is fully determined by the evolution of an initially symmetric beam. By replacing the 15 mm crystal with a longer one (50 mm), we were 143905-3

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able to observe the impact of further dynamics. The quenching of the contrast in modulation [Fig. 2(d), dashed line] proves the instability of the soliton array, as predicted by the numerical results in Figs. 1(c) and 1(f). In order to demonstrate experimentally the WEM seeding of the

2 MI in the temporal domain, we had to find out operating conditions that guarantee dominant dispersion. To this end, we have adopted the same titled-pulse technique used in Ref. [16], which allows for a dramatic shortening of the (effective) Ldisp and also for the removal of the (effective) GVM. Figures 3(a) –3(c) describe the calculated ST dynamics for the FH-WP in case of a 700 fs, 500
m, 50 -tilted input WP, undergoing type-I SH generation in a -barium borate (BBO) crystal close to phase matching (see the caption for details). The calculations are performed by solving directly the ST model [Eqs. (1)] in the presence of walkoff (
0), GVM ( V
0), and tilted-pulse input conditions. The results outline the occurrence of a dynamics whose key features are similar to

FIG. 3. (a) –(c) Calculated ST intensity map for a FH tilted WP propagating in a type-I BBO crystal and underlying SH generation close to phase matching, for z  3:5 mm (a) and z  7 mm (b); insets: zoom of the central part of the WP (a.u.); (c) autocorrelation at the beam center, relative to data in (b). Input-pulse parameter: 50 tilt, 700 fs FWHM duration, 500
m beam FWHM, 20 GW=cm2 peak intensity, Gaussian profile; input noise: Gaussian, 2% in intensity, 5 times the input pulse in ST bandwidth. Crystal parameters: k  5 cm1 ,   52 mrad, V  0:589 ps=m, k00
!0   0:1382 ps2 =m, k00
2!0   0:4388 ps2 =m. (d) Experimental SH scanning autocorrelation of the FH-WP exiting from a 7 mm BBO crystal, tuned for SH generation close to phase matching, in the presence of two-photon absorption. Input-pulse parameter: 50 tilt, 200 fs FWHM duration (with shoulders), 2 mm beam FWHM, ’ 13 GW=cm2 peak intensity.

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those shown in Fig. 1: For small z [Fig. 3(a)], a highcontrast, ST elliptically shaped modulation appears, with complementary profiles in the FH and in the SH (not shown) waves; for larger z [Fig. 3(b)], the results of WEM-seeded MI are evident, with appearance of FH and SH (not shown) self-trapped tilted stripes that propagate without spreading, in solitonlike regime. The calculated autocorrelation at the beam center is given in Fig. 3(c). The dynamics keeps his genuine 1D feature only for a short transient regime. In fact, further propagation leads to the occurrence of a chaotic fragmentation analogous to that shown in Fig. 1(c) (data not shown). The experiment [17] was performed by launching a ’ 50 tilted, 200 fs, 527 nm pulse at ’ 13 GW=cm2 peak intensity into a 7 mm type-I BBO, tuned for SH generation close to phase matching. Figure 3(d) shows the trace of a scanning SH autocorrelation measured for the FH-WP, imaged from the exit face of the BBO crystal (the measuring setup is the same as that described in Ref. [16]). The number of well distinguishable peaks confirms the temporal breakup of the WP into a stable (from shot to shot) train of equally spaced (by ’ 100 fs) ultrashort (’ 40 fs width) subpulses. Note that the laser pulse energy fluctuations were below 2%. The qualitative agreement with the calculated results in Fig. 3(b) and 3(c), which give subpulse interval and duration of 90 and 40 fs, respectively, indicates that a short train of solitonlike pulses was formed in the experiment also. Note that in this case the 40 fs subpulses give rise to a very small (effective [16]) dispersion length jLdispeff j  0:3 mm. In the experiment, the two-photon absorption (TPA) of the generated SH, which sharply flattens the pulse profile, and the presence of a fairly large pedestal in the input-pulse (see Ref. [16]) have both contributed in making the effective FWHM pulse width as large as the trace in Fig. 3 shows. Such relevant broadening was mimicked by the choice of longer input pulses in the calculations of Fig. 3, for which Gaussian profile and pure

2 was assumed in order to allow easy comparison with the Fig. 1. results. Numerical calculations that account for pulse shape and TPA gave optimum agreement with the experiment also. In conclusion, we have demonstrated the existence of a relevant mechanism, related to the wave-envelope dynamic, which competes with the noise-seeded modulational instability in determining the beam and pulse breakup occurring via

2 interaction. While the noiseseeded MI naturally occurs in the spatiotemporal domain, the present wave-envelope effect might enhance either the spatial or the temporal instability, according to the relative impact of diffraction and dispersion as settled by input-wave size and duration. By using suitably tilted pulses for compensating the group-velocity mismatch and enhancing the group-velocity dispersion, we obtained the breakup of the pulses in a train of temporal solitons,

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leading to the first ever reported observation of

2 temporal modulational instability. We mention that observed phenomenon is inherently related to the

2 interaction, being a consequence of the energy exchange between the fundamental and second-harmonic fields. Work is in progress concerning the investigation of regular-pattern formation in Kerr media. The authors acknowledge support from MIUR (COFIN01-FIRB01), DGI BFM2002-04369-C04-03 (Spain), Ministero de Education Cultura y Deporte (Spain), and EC CEBIOLA (ICA1-CT-2000-70027) contracts.

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