We experimentally ascertain the role of non locality in the spectral evolution of multifilament patterns generated by modulational instability in nematic liquid crystals.
© 2005 Optical Society of America
Non locality plays an important role in several areas of physics, from quantum-mechanics  to plasmas  to quantum optics,  to mention just a few. In optics a non local response may originate from mechanisms such as heat  or atom diffusion,  charge drift [6–7] or elastic intermolecular forces [8–9]. In nonlinear optics, a few major implications of non locality have been addressed with reference to spatial solitons (SS) and modulational instability (MI), both in general terms [4, 10–12] and with specific emphasis on nematic liquid crystals (NLC), even in the presence of significant walk-off . The non local response of NLC stems from the elastic interaction between elongated molecules subject to thermal or reorientational effects, the latter in the presence of an external (optical or low frequency) electric field. As underlined with specific reference to the reorientational response of undoped NLC, a non local response prevents catastrophic collapse, mediates long-range attractive interactions between solitons and supports breather-like propagation of narrow-waist filaments . In addition, spatial non locality supports the generation of spatial solitons by partially incoherent (speckled) input beams, smoothing out the resulting transverse index profile and allowing the guided-wave propagation of a co-polarized (weak) signal in the corresponding eigenmode(s) . Recently, multifilament formation owing to transverse modulational instability in NLC was discussed as an approach to the generation of multiple spatial solitons or nematicons . When a wide beam is employed in a non local system subject to a self-focusing response, however, the latter “globally” affects the propagation of the whole filament pattern,  resembling many-body interactions in a system consisting of several individual solitons.
In this Paper we discuss the role of non locality by investigating the evolution of the transverse spectrum of a filament pattern induced in NLC by one-dimensional modulational instability. At variance with previous investigations, the periodic pattern consists of self-localized nematicons which mutually interact through the medium non local response.
2. Sample and model
The optical nonlinearity of undoped NLC stems from light induced reorientation of highly anisotropic molecules and their optic axis or director n̑.[8–9] Due to the intermolecular forces, however, the refractive index change extends well beyond the excitation region, with a characteristic non local range which entails spectral low-pass filtering in transverse space. Thereby, fast transverse perturbations (i.e., refractive features on a scale smaller than the non local range) are filtered out [17–18]. This has a striking effect on modulational instability (MI), which tends to break a plane wavefront into a quasi-periodic pattern irrespective of the origin of nonlinearity and the degree of non locality. In either Kerr, photorefractive, quadratic or reorientational media, in fact, through frequency domain coupling in Fourier space MI promotes a power exchange from the pump spectrum (centered in the origin) to symmetrically displaced side-bands [19–20]. Additional non locality tends to distorts these sidebands in propagation, feeding low-frequency components at the expense of higher harmonics.
To experimentally verify this insight, we adopted a high non local medium, i.e., nematic liquid crystal E7, using a sample and the geometry as sketched in Fig. 1: two parallel glass slides spaced by L=75μm hold the NLC, and a third slide seals the input interface to prevent undesired light depolarization. Proper treatments of the interfaces determine the molecular alignment in the plane yc zc at an angle θ0 to the normal ẑc to the input interface. An elliptic Gaussian beam of wavevector k∥zc, waists U=200μm along yc and V>20μm along xc, (extraordinary) polarization in the cell plane yczc, is injected into the sample using cylindrical lenses. In such configuration, the flux S propagates with walk-off δ relative to k =(ky, kz) in the yczc plane (see Fig. 1). Considering the rotated reference system (x,y,z) with x//xc, the nonlinear contribution Ψ to the reorientation is governed by: 
K being the elastic constant (scalar approximation) and Δε the birefringence. The coefficient A(θ0) relates to non locality and, in our geometry, it takes the form:
with ky the transverse wavevector component with respect to the Poynting vector S(//z). From the Lorentzian (3) we can calculate the cutoff wavenumber for the transverse intensity: . It is important to stress that such cutoff does not depend on the material (Frank coefficient K) but only on the geometry (θ0). For the actual parameters of the liquid crystal mixture E7, we obtain the graph in Fig. 2 with a cutoff ky_cut=0.040μm-1 for θ0=45°, the actual molecular orientation in our samples.
3. Experimental results and discussion
Based on (3) we expect that, owing to the non local (attractive) interaction, the transverse intensity of a multifilament pattern evolves in propagation while undergoing Lorentzian filtering. Figure 3 below displays some typical experimental results: the wide input beam propagates in the NLC with wavevector k̂ = ẑc and a walk-off of 7°, in perfect agreement with calculated values from birefringence in E7 and molecular alignment at θ0=45°. Modulational instability occurs at powers as low as 30mW, producing a periodic transverse undulation clearly visible for z≥2mm (Fig. 3(a)). At P=60mW (Fig. 3(b)) MI yields a multisoliton pattern, with soliton-soliton attraction and an overall beam (self-) focusing. As the excitation level is further increased (Fig. 3(c)), the filaments tend to coalesce and agglomerate in more intense beams.
Such “agglomerates” are seeded by the non-uniform intensity and are wider and more intense than individual filaments, hence the Fourier spectrum of the whole transverse pattern changes as light propagates forward in the medium. Figure 4 maps the measured gain spectra versus ky and propagation distance z, i.e., the time-averaged Fourier-transformed transverse intensity normalized to its input spectrum. At P=30mW modulational instability is clearly visible in Fig. 3(a). The nonlinear process results into a sideband between 0.05μm-1 and 0.2μm-1, in good agreement with the estimates of Ref.  even though the geometry is slightly different here. At P=60mW (Fig. 4(b)) MI results in a filament pattern with a specific mean-periodicity highlighted by equally-spaced peaks in the transverse spectrum for z>2mm.
Finally, for P=90mW (Fig. 4(c)) a strong non local interaction takes place between filaments: power gets progressively coupled from side-bands towards a specific narrow band around 0.042μm-1, consistently with the estimated filtering action owing to medium elasticity. Such spectral behavior is underlined in Fig. 5 by comparing transverse intensity spectra at the three powers (Fig. 4) after propagation over 3.5mm.
We observe that the non local interaction allows the transverse pattern to evolve towards a mean intensity spectrum which complies with the medium cutoff. Otherwise stated, the non local contribution to filament-to-filament interaction is cumulative in propagation. Noticeably, light self-localization tends to preserve the filament identities despite coalescence, impeding the power (spectral) conversion towards a plane wave with ky=0.
While a nonlinear response acts on a plane wave by frequency-domain coupling power towards higher harmonics, we could argue that –conversely- non locality acts on a pattern of parallel filaments by frequency-domain coupling power towards lower harmonics.
In conclusion, for the first time, we experimentally studied the spatial spectrum of mutually interacting filaments and its evolution versus propagation in a non local medium. The filaments were generated via modulational instability in spatially non local nematic liquid crystals and evolved through progressive power coupling from high to lower frequencies, consistently with an active filtering action and the existence of a transverse wavevector cutoff.
We are grateful to C. Umeton (LICRYL, Univ. of Calabria) for the sample.
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