We introduce the generation of dense trains of light-bullets in nonlocal nonlinear dielectrics. We exploit stable spatio-temporal self-trapped optical packets stemming from the interplay between local electronic and nonlocal reorientational nonlinearities, considering a seeded temporal modulation instability by specifically referring to nematic liquid crystals.
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Solitons have been recognized and investigated in several areas of physics, and can be considered general phenomena in nonlinear science [1,2]. Optical solitons (both in time and in space) exhibit particle-like interactions and are good candidates for the realization of all-optical signal processing devices [3,4]. As the dimensionality of light localization phenomena increases when including both temporal and spatial self-actions, the stability of such nonlinear solutions is a non-trivial issue. The very existence of media and nonlinear regimes where multi-dimensional solitons can be observed is a subject of current investigation. Spatial solitons confined in both transverse dimensions and supported by a purely Kerr response are known to be unstable (thus undergoing catastrophic collapse) , whereas when other mechanisms are present, such as the saturation of the nonlinear response [6–8] or an ongoing nonlinear absorption [9,10], stable (2 + 1)D solitons and their interactions have been predicted and observed . In pioneering theoretical work, Snyder and Mitchell revealed that a nonlinear nonlocal material response can prevent collapse while relaxing the excitation conditions required to generate spatial solitons . As foreseen by Shen , nematic liquid crystals (NLC) have been recognized as nonlinear media where spatial optical solitons are accessible . Spatial solitons and their main features were reported in NLC at mW powers, where they are possible due to the reorientational nonlinearity of these materials, which are non instantaneous in time and highly nonlocal in space [15–17]. Spatio-temporal (3 + 1)D solitons (STS) or light bullets, i. e. self-localized light wave-packets which conserve both their spatial and temporal profiles in propagation, were first postulated by Silberberg in the early nineties  and have since attracted a lot of interest. Nevertheless, the experimental conditions for the excitation and propagation of STS were found to be challenging to achieve , or have been associated to nonlinearities tailored ad hoc but weakly connected to either realistic responses or excitation regimes .
We recently presented an original approach towards the observation of STS in highly nonlocal nonlinear systems, taking advantage of the cooperative action of two nonlinearities acting on different time scales . By exploiting a nonlocal response (characteristic of several media, including thermo-optic media, photorefractive ferroelectrics and nematic liquid crystals) and the (always present) instantaneous electronic Kerr contribution, we have predicted optical bullets (self-trapped wave-packets in time and space) and anti-bullets (i. e. temporal dark dips in a bright self-trapped background) in realistic optical systems such as reorientational molecular media, excited by pulsed laser beams. A similar approach was independently undertaken by Gurgov and Cohen with reference to a generic Gaussian nonlocality . In this Paper we discuss the generation of temporally dense bullet bursts by way of temporal modulation instability (MI) in spatially self-trapped filaments. Such trains of bullets could be obtained via the interplay of an electronic Kerr response and the reorientational nonlocal nonlinearity in NLC.
2. Field-matter model
Modulational instability stems from the inherent instability of homogeneous solutions in self-focusing media. In both time and the space domains, MI can be described in terms of a spectrally selective nonlinear gain which, seeded by the background noise, results in a periodic modulation of the wave envelope (Fig. 1 ). Since this perturbation can evolve into a comb of solitons, MI is usually considered as a soliton precursor [3,23–26]. We consider a light beam from a periodically pulsed laser source producing a train of pulses of duration and repetition time σ shorter than the response time (typically >1ms) of the nonlocal nonlinear medium. Under this assumption, in a slow/nonlocal self-focusing dielectric, the spatial dynamics is insensitive to the pulsed nature of the excitation and each wave-packet propagates in an invariant confining potential [21,22]. In the self-confined regime, we can express the excitation in the form , where Z and t are the space and time propagation coordinates, ω0 and β are the optical frequency and propagation constant, respectively, and F represents a normalized transverse profile ().
The evolution of the temporal envelope A is governed by the 1D nonlinear Schrödinger (integrable) model (NLS):15,16]. for a planar cell of thickness d (Fig. 2a ), the light-matter interaction in space is ruled by:Eq. (2) the time average operator < >t indicates a time-mobile mean due to the slow reorientational response. Equation (2) supports (2 + 1)D bright solitary waves [14–17]. An example of an experimentally observed nonlocal soliton is presented in Fig. 2(b). Equation (1) admits the existence of temporal bright solitons for β2<0 and of dark solitons for β2>0. In the former case, since the pulses are localized within a stable (2 + 1)D spatial soliton, the overall field distribution corresponds to stable STSs and the laser repetition rate 1/σ can be tuned to achieve the required combination of peak and average powers in .We stress here that, since the nonlocal response tends to be slow in time, the required repetition rate could also be adjusted by externally modulating (e. g. chopping) a source with a much higher repetition rate. When long pulses are launched in the soliton waveguide, temporal MI can break up the envelope A and eventually gives rise to a sequence of interacting self-localized wave-packets [23,24] or temporal solitons. An intrinsic problem with this approach to periodic pulse generation is the nature of noise, which triggers the process and results in poor mutual coherence and stability among the pulses, i.e. in significant differences between consecutive realizations. In addition, the propagation length is rather critical, because a well defined pattern of spaced solitons occurs only at a specific propagation section: for extended propagation the solitons eventually interact and deteriorate the desired spectral features of the train. The introduction of a continuous-wave (CW) seed spectrally centered in one of the MI gain peaks can control the periodic modulation of the pulse envelope, driving its break up and allowing the generation of equi-spaced and regularized narrow pulses [27,28].
3. Optical bullet trains via temporal modulation instability
The approach we adopt here consists in exploiting the substantial independence of the nonlocal nonlinearity from the wavelength, such that we can sustain the spatial solitons with a CW source slightly detuned from the pulse peak wavelength. An intrinsic advantage of this approach is the dependence of the effective temporal nonlinearity from the soliton average power. As a consequence, the optimum propagation distance can be tuned to the actual length of the nonlinear medium. We computed the nonlinear eigenmodes of the system (2) (lifting the assumption of a Gaussian spatial profile of Ref .) and obtained the soliton GVD by determining the mode dependence in function of the wavelength. We referred to the material parameters of the commercial liquid crystal E7, accounting also for its weak dye-doping in order to alter bot h the sign and the size of β2 (usually positive) [21,28].
Figure 3 shows the waveguide and eigenmode profiles for a power of P = 2.35mW at λ = 850nm, as obtained from Eq. (2) using a relaxation algorithm. The calculation was carried out for an NLC cell of thickness L = 100μm, boundary conditions θ0 = θ(X = 0) = θ(X = L) = π/6, a dye-dopant with resonance at λres = 930nm and a corresponding strength Γres = 10−9 . The soliton waveguide effective index, area, nonlinearity and GVD were found to depend on the average power via the index perturbation induced by the reorientational nonlinearity (Fig. 4(a-d) ), with β2 = −1.54ps2/m at P = 2.3mW. Equation (1) was solved using a finite-difference approach and a flat-top pulse was employed to generate pulse trains via MI.
In the example shown in Fig. 5 , we superimposed wide-band noise (Δλ = 255nm) of average energy spectral density ESDnoise = −9.3dBm/THz2 and power Pnoise = 2.3mW on a pulse of central wavelength λ = 850nm, duration τ = 4.4ps (FWHM) and peak power P = 200W propagating in the self-induced waveguide generated by the spatial soliton of Fig. 3. The output section in Z = 33mm (Fig. 5(c)) corresponds to the maximum energy transfer to the MI side bands.
Figure 5(c) shows that the generated output train possesses a periodic structure, but the modulationally unstable process leads to poor coherence between the wavepackets, with an outcome strongly dependent on each realization of the seeding noise (Media 1). The average noise of 2.3mW in this example is comparable to the soliton average power. However, we like to underline that the noise power considered here must not be included in the average contribution sustaining the nonlocal soliton, as it is usually associated with the pulse train emitted by the pulsating (mode-locked, Q-switched, cavity dumped, etc.) laser. Thereby, the noise power is to be intended as an average over the temporal window of the pulse. Figure 6 displays a train realization from a pulse peaking at λ = 850nm when the average soliton excitation is carried almost entirely by a CW seed centered at λ = 843.3nm, i. e. at the MI gain peak. (Media 2) In this case the process is more efficient and the optimum propagation distance is 26mm. For Pseed = Pnoise (ESDnoise = - 9.3dBm/THz2), MI gives rise to a regular train with period T corresponding to the frequency detuning, i. e. .
In order to evaluate pulse-to-pulse stability, we define the cross-correlation Hn of the nth realization with the mean output profile:
We can neglect the phase change of the CW seed with respect to the various output realizations An, as it only affects the delay between trains. The maximum of |Hn(t)| goes to 0 when An it is completely uncorrelated to the mean train profile, whereas it tends to the energy in the case of equal realizations, i. e. A1 = A2 = ... = An. Therefore, the average max[|Hn(t)|/En] on a series of realizations qualifies the pulse-to-pulse regularity. Figure 7(a) plots Ψ for various noise powers, i. e. different noise seeds.
In the presence of a seed and for noise powers of comparable power, the pulse train maintains a regular structure. The stabilizing effect of the seed is clear, as highlighted by the superposition of 50 realizations for ESDnoise = −20dBm/THz2 in Fig. 7(b) (7dBm/THz2 in Fig. 7(c)) corresponding to an average noise power Pnoise = 0.22mW (11mW). We emphasize that, even though the CW seed introduces a non localized background in time, it only contributes to shape the proper excitation; hence, the modulation could in principle be directly superimposed to the pulse envelope, the required mean power for the spatial problem being provided by the pulsed source with a suitable repetition rate. Hence the train of output temporal solitons is effectively a dense light-bullet train as it is an eigensolution of the (3 + 1)D propagation problem.
In conclusion, the synergetic approach between the (2 + 1)D nonlocal self-trapped propagation ruled by Eq. (2) and the temporal evolution governed by Eq. (1) can effectively deliver high rep-rate trains of temporal solitons in a self-confined beam, i. e. dense light-bullets trains. Nonlinear nonlocal materials such as nematic liquid crystals -with the required doping to tune the GVD- appear as the ideal setting to demonstrate spatio-temporal light localization and bullet interactions.
We thank A. Pasquazi for enlightening discussions. This work was funded by an NSERC Discovery grant. IBB acknowledges support from NSERC-USRA and NSERC PGS fellowships. MP acknowledges Marie Curie People support through the project TOBIAS PIOF-GA-2008-221262.
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