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Observation of optical spatial solitons in a highly nonlocal medium

We report on the observation and quantitative assessment of self-trapped pulsating beams in a highly non-local nonlinear regime. The experiments were conducted in nematic liquid crystals and allow a meaningful comparison with the prediction of a scalar theory in the perturbative limit, while addressing the need for beyond-paraxial analytical treatments.

Several remarkable nonlinear physical phenomena may be classified as solitons, or solitary waves. Among the most famous is "the Great [water] Wave of Translation", originally reported by J. S. Russell and considered the first documented observation of solitons. [1] A fundamental characteristic unifies soliton phenomena: the discreteness arising in the framework of continuous fields (such as in electro-and fluid-dynamics). From a mathematical viewpoint, solitons are solutions of specific differential equations which are nonlinear and integrable. In general, however, nonlinear waves which keep a distinctive and unchanged identity along their propagation may be considered solitons, or solitary waves (SW), even when they are described by non-integrable systems. [2] In the applied nonlinear optics community such a distinction is often set aside, [3] as it will in this work.
Several approaches have been available to model SW: from heavy computations to sophisticated analytic transforms. More recently, a simple theoretical model was advanced which describes complex soliton-like dynamics and has direct relevance to the novel beam propagation we present here. In particular, Snyder and Mitchell described the propagation of optical spatial solitary waves (i. e., non-diffracting light beams which self-trap owing to the nonlinearity) in a highly non-local nonlinear medium (HNNM), i. e. a medium (described by an integro-differential constitutive relationship in the spatial coordinates) in which the optical field modifies the index of refraction in a kernel region much larger than the beam waist [4]. The model in Ref. [4], however, was attributed a limited practical relevance, mainly because of the apparent lack of a physical medium (i. e., an HNNM) able to support such solitons. Nevertheless, nonlocal media with a nonlinear response have been experimentally investigated to outline their specific features in relation to light localization and optical solitons, and among them atomic gases, [5] photorefractive and liquid crystals (LC). [6][7][8][9][10][11][12] Photorefractives are highly anisotropic crystals with a finite degree of nonlocality due to carrier drift and diffusion [6][7][8][9], while LC are liquids with a significant degree of molecular order and birefringence under appropriate anchoring. [10][11][12] In recent years, spatial optical solitons have been observed in a variety of nonlinear media and have attracted considerable interest for their potential applications to signal processing and communications. [3,[13][14] Such observations were mostly carried out in materials effectively described in terms of a local response (where the relation between the field and the induced nonlinear polarization can be taken as punctual), i. e., cubic or quadratic or photorefractive [15].
In this Letter, after showing theoretically that a highly-nonlocal response is appropriate to describe light propagation in nematic LC, in such HNNM we experimentally investigate nonlocal optical solitons and carry out a quantitative comparison with the model. Their features strongly differ from those of self-trapped beams in local media, and we trust that the results presented hereby are relevant in all those fields where non-locality cannot be neglected and a local picture fails; among them, Bose-Einstein condensation [16] and plasmas [17] are worth mentioning. Not only understanding the interplay between nonlocality and nonlinearity is fundamentally important, but it also leads to intriguing applications of spatial solitons: for instance, the recently demonstrated non-trivial logic gates based on non-local interactions between optical SW [18].
Theory. With reference to the highly-nonlocal regime, while the theory developed in Ref.
[4] provides insight and general guidelines on the dynamics of spatial solitons in HNNM, a more specialized description is required to model the response of a particular medium such as a nematic LC.
The experimental geometry of our nematic liquid crystal cell for planar orientation is sketched in Fig. 1. With proper anchoring boundaries, the molecular director ( ) z , y , x θ distribution can be initially adjusted to a background value by an externally-applied lowfrequency voltage. [19] By setting with respect to the linear spatial polarization of an incoming light beam, in the paraxial approximation using standard expressions for LC [10] at the first-order, from for finite radial distances from the beam axis (in 0 r = ), it stems that the dipole-induced perturbation is much wider that the beam itself, with: ) optically-induced correction to the background value 0 θ of the director angle, r the cylindrical radial coordinate, 0 ε the vacuum permittivity and E the electric field amplitude in the beam. K is the LC elastic constant (conveniently taken equal for splay, bend and twist of the molecules) and which describes the propagation of a linearly x-polarized light wave in a perturbed director field ( ( ) the Foch-Leontovich equation for the beam-envelope in the limit of a thick cell reads ). Eq. (4) describes the reorientational nonlinearity of LC, and supports a Gaussian solution with a waist ) z ( W (in intensity) periodically varying along the propagation distance. While coupled equations (1) and (3) which, for a flat-phase gaussian input, can be reduced to the equation of a normalized nonlinear one-dimensional oscillator, with position in a parabolic-like potential: (dots denote derivatives with respect to a normalized propagation coordinate), with ( ) An alternative insightful (albeit less rigorous) way to solve Eq. (4) consists in looking for transverse-localized waves, in the hypothesis that the optical beam is nearly trapped, i. e., 2 ) z , 0 r ( E = is almost z -independent. In essence, the on-axis intensity dependence is reduced to a power dependent term (for a Gaussian solution this is self-consistent), as in Ref. [4]. This approximation to the full dynamics encompassed by coupled Eqs. (1) and (3) accounts for the specific material constants and provides beam evolution laws in analogy to those in Ref. [4]: Thus, the beam generally undergoes nonlinear oscillations, their entity determined by the ratio between the input and the soliton power. The latter fixes the initial position of the particle in the bell-shaped potential. When s P P = the particle is initially settled in the stationary point, hence it remains still with z , corresponding to an exact soliton excitation.
Our treatment above corresponds to assume 1 s ≅ , hence it holds for excitations around the soliton value ( S P P ≅ ) and enables to deal with explicit expressions.
Eq. (6), in qualitative agreement with Ref. [4], will be used to interpret the experiments.
The dimensionless constant γ (i. e., the harmonic content of the soliton) identifies various self-trapping regimes. For 0 > γ the beam diffracts at first before it starts to periodically pulsate, while for 0 < γ it initially self-focuses. Λ grows linearly with 0 W and inversely with the square root of the power, because self-trapping stems from a balance between diffraction and self-focusing: the natural tendency of a beam to spread, stronger when 0 W is smaller, is overcome by the power-dependent response. In the case 0 = γ (i. e., S P P = ) nonlinearity exactly counteracts diffraction and the beam travels un-modified along z . Therefore, (for P P s ) we can state that spatial solitons in highly non-local media are pulsating beams.
The previous result can also be interpreted by observing that, according to Eq. (4), the LC system behaves as a graded-index (GRIN) lens with pitch depending not on beam waist, but on beam peak-intensity (or power in the limit of Eq. (6)). This agrees with the definition of an HNNM: the field intensity in a point modifies the medium in a region much larger that the beam spot. When the spot size is very small, the beam does not sense the GRIN and widens, until its waist gets large enough for the lens to become effective. Then, it starts focusing, and the process repeats itself in a cyclic fashion. Noteworthy, despite the specific material setting (LC exhibiting a reorientational response which is far from saturation) and within the paraxial frame, the general SW trend in an HNNM qualitatively resembles the case of a (logarithmic) saturable nonlinearity. [20] The non-local nonlinearity is able to trap intense and narrow-waist solitary beams. By looking at Eq. (6) i.e., there is no lower (upper) bound to its minimum (maximum) 0 W . Although derived within the limit of validity of Eq. (6) . Due to the blur, an actual m 5µ waist results overestimated by about 50%, but the error is larger than 100% for waists < m 1µ . Conversely, if the waist is > m 10µ , the error is under 10% (see Fig.1, lower right). Henceforth, we carried out the quantitative comparison with the theory considering only the maximum waist (from Eq. (7)) and the oscillation period (from Eq. (6)), the latter taken as twice the separation between the first two extrema in the graph of waist versus z (Fig. 2). ). By minimizing the standard deviation, . To further the comparison, Fig. 3b shows a plot of ) W max( / 1 2 versus P . The overall linearity is satisfactory and the standard deviation from the model is 4%. Fig. 3c graphs period versus power, and the best fit from Eq. (6). Given the previously evaluated α , here the fitting parameter is the minimum waist, yielding W 0 ≈1µm. While the latter could not be determined a priori, because light enters the sample through the interface and an LC transition layer, such wavelength-size estimate stems from a paraxial model and should, therefore, be taken as the indication of its inherent limit. Finally, in Fig.   3d the quantity 2 / 1 Λ is plotted versus P . Once more, there is good agreement with the calculation, and the standard deviation is 7%. The overall comparison between data and model is marginally affected by the limited experimental accuracy; however, it emerges clearly that we observed self-trapped optical beams in a HNNM. Finally, it is well worth underlining that an additional feature of spatial solitons in HNNM is their collisional behavior, as predicted in Refs. [4] and [21]. Indeed, the recently demonstrated incoherentlike and attractive interactions between SW in the same LC provide an independent proof of their highly-nonlocal dynamics. [22] A corollary of our model Eq. (6) is that the minimum waist of the oscillating SW directly relates to its maximum waist and period. However, since such minimum is estimated comparable to the light wavelength in the medium ( m 68 . 0 n / µ ≅ λ ) and the experimental data are masked by scattering, based on the paraxial approach (Eqs. (4)- (5)) and the small oscillation approximation (Eq. (6)) above we can only state that 0 W is rather small. While a vectorial treatment (e.g. Ref. [23]) is needed when dealing with a non-paraxiality figure close to unity, to circumvent the limits of our l inearized analysis leading to Eq. (6) we resorted to a numerical study of light propagation while solving the basic eqs. (1) and (3) for the nonlinear response of our voltage-biased LC cell.
The study, carried out with a split-step beam propagator in the paraxial regime, confirms that the SW undergoes oscillations in waist and intensity (up to one order of magnitude), with minimum values of the order of m 1µ . The numerical results ratify the widely oscillating character of the SW as observed in LC, and prompt for a non-paraxial vectorial theory. The latter will be considered elsewhere.

Conclusions.
Our investigation of an HNNM, employing the nematic liquid crystal E7 in the near infrared, demonstrates that highly nonlocal solitons can be excited and propagate as pulsating beams. In spite of the inherent limits of a paraxial analysis in the smalloscillation approximation, the model and the comparison with data indicate that the beam dynamics is primarily determined by the ratio between input and soliton power, with propagation of narrow-waist SW. The inferred non-paraxiality C, in fact, is more than one order of magnitude larger than for self-trapped beams in photorefractive and frequencydoubling crystals. In conclusion, not only nematic LC can be regarded as HNNM and sustain highly non local solitary waves, but they also prompt for the development of nonparaxial nonlinear models in optical soliton propagation.