Chapter 1: Nematicons

Gaetano Assanto, Alessandro Alberucci and Armando Piccardi

Nonlinear Optics and OptoElectronics Lab, University ROMA TRE, Rome, Italy

1.1 Introduction

The term nematicon was coined to denote the material, nematic liquid crystals (NLC), supporting the existence of optical spatial solitons via a molecular response to light, a reorientational nonlinearity. Nematicons was first used in the title of Reference 1, after three years since the first publication on reorientational spatial optical solitons in NLC [2]. Since then, a large number of results, including experimental, theoretical, and numerical, have been presented in papers and conferences and formed a body of literature on the subject. In this chapter we attempt to summarize the most important among them, leaving the details to the specific articles but trying to provide a feeling of the amount of work carried out in slightly more than a decade.

1.1.1 Nematic Liquid Crystals

Liquid crystals are organic mesophases featuring various degrees of spatial order while retaining the basic properties of a fluid. In the absence of absorbing dopants, they are excellent dielectrics, transparent from the ultraviolet to the mid-infrared, with highly damaged thresholds, relatively low electronic susceptibilities, and significant birefringence at the molecular level and in the nematic phase. In the latter phase, their elongated molecules have the same average angular orientation, although their individual location is randomly distributed as they are free to move (Fig. 1.1a). NLC exhibit a molecular nonlinearity; when an electric field is present, the electrons in the molecular orbitals tend to oscillate with it and give rise to dipoles which, in turn, react to and tend to align with the field in order to minimize the resulting Coulombian torque [3–5] (Fig. 1.1b–c). This torque is counteracted by the elastic forces stemming from intermolecular links: equilibrium is established when the free energy of the system is minimized, as modeled by a set of Euler–Lagrange equations. Because the polarizability of the molecules is higher along their major axes, their reorientation toward the field will increase the optical density, both at the microscopic and macroscopic levels. It is noteworthy that an initial orthogonality between the field and the induced molecular dipoles corresponds to a threshold effect known as Freedericksz transition [3]. For static or low frequency fields, reorientation leads to a large electro-optic response with a positive refractive index variation for light polarized in the same plane of the field lines and the long molecular axes [3]. For fields at optical frequencies, the average angular orientation or molecular director in the nematic phase corresponds to the optic axis of the equivalent uniaxial crystal; hence, the refractive index for extraordinarily polarized electric fields (i.e., with field vector coplanar with both optic axis and wave-vector) will increase with the orientation angle θ (Fig. 1.1c–d for wave-vectors along z).

Figure 1.1 (a) Sketch of molecular distribution in the nematic phase and definition of director ; the ellipses represent NLC molecules. (b) Director orientation in the absence of electric field: the angle θ0 is determined by anchoring at the boundaries. (c) In a positive uniaxialNLC, a linearly polarized electric field can induce dipoles and rotate the molecular director towards its vector; the resulting stationary angle θ is determined by the equilibrium between the electric torque and the elastic intemolecular links. (d) Extraordinary refractive index versus angle between wave vector and director for a positive uniaxial NLC with n|| = 1.7 and n⊥ = 1.5.

The reorientational mechanism described above is neither instantaneous nor fast (see Chapter 13), but can be very large, with effective Kerr coefficients n2 of about 10−4 cm/W2 [6], that is, eight to twelve orders of magnitude larger than that in CS2 and in electronic media, respectively [7]. Therefore, nonlinear effects can be observed in NLC even with continuous wave lasers, at variance with many other nonlinear dielectrics often requiring pulsed excitations.

Nevertheless, the reorientational response is not the only available response in NLC. Owing to their fluidic nature, a high electric field can change the portion of molecules aligned to the director, that is, can affect the order parameter [8], particularly in the presence of dye dopants [9]. Doped NLC also features an enhanced reorientational nonlinearity because of the Janossy effect [10]. As a result of thermo-optic effect, a nonlinear response also stems from temperature changes, modifying the refractive indices mainly via the order parameter in phase transitions [6] (see Chapter 9). Moreover, NLC can show the photorefractive effect [4] and fast electronic nonlinearities (see Chapter 14).

1.1.2 Nonlinear Optics and Solitons

In nonlinear optics, the basic example of an intensity-dependent refractive index is the Kerr response n(I) = n0 + n2I. When n2 is positive, the index increases with the light intensity and, in the case of a finite beam, it gives rise to a lens-like refractive distribution, which is capable of self-focusing the excitation. Such a mechanism can actually compensate for the natural diffraction of the beam, resulting (in the simplest case) in a size/profile-invariant spatial soliton. Otherwise stated, the excitation beam deforms the refractive index distribution of the nonlinear (initially uniform) dielectric, generating a transverse graded-index profile that acts as a waveguide, that is, confines the field into a guided mode. The fundamental soliton in space is the lowest order mode guided by the self-induced dielectric waveguide. Spatial solitons of a Kerr nonlinearity, the so-called Townes solitons [11], tend to be unstable in two transverse dimensions because the exact balance of diffraction and self-focusing is achieved at a critical power [12, 13]. They are stable in one dimension (e.g., in planar waveguides [14]) or in the presence of higher order effects as compared to the Kerr law, such as saturation of the nonlinear change in index [15, 16], multiphoton absorption [17], discreteness [18, 19], and nonlocality [20]. In most cases they are observable in actual media although, being no longer exact solutions of an integrable differential system, they should be rigorously referred to as spatial solitary waves [21]. The terms soliton and solitary wave are interchangeably used throughout this chapter.

1.1.3 Initial Results on Light Self-Focusing in Liquid Crystals

As discussed in Section 1.1.1, several terms can contribute to the nonlinear response of NLC. Experiments conducted in the early 1980s demonstrated that, in undoped NLC, the dominant contribution is the reorientational nonlinearity [6, 22, 23]. An equivalent Kerr response was measured with light beams passing through the thickness of a planar cell, the latter behaving as a lens, the focus of which is dependent on the input power. For Rayleigh distances much smaller than the NLC layer thickness, rings could be observed in the diffraction pattern [24].

An experiment on self-focusing in the bulk of a dye-doped NLC layer was carried out in 1993 by Braun et al. [25], who imaged the scattered light from a beam propagating in a cylindrical geometry with NLC subject to Freedericksz threshold. Various phenomena were observed, including undulation, filamentation, and nonstationary evolution along the capillary; they were interpreted and modeled with joint reorientational and nonlinear Schrödinger equations [26, 27]. After such a pioneering work, self-localization of light as a consequence of thermo-optic effects in capillaries was reported by Derrien et al. [28]; the interplay between thermal and reorientational responses was addressed by Warenghem et al. [29] (see Chapter 9). The use of suitably built planar cells with the director tilted by an external bias to avoid the Freedericksz threshold allowed Peccianti et al. to observe the profile-invariant spatial solitons at a few milliWatts [2]. Unbiased planar cells with pretilt determined by rubbing permitted the detailed study of walk-off [30] (see Chapter 6). Figure 1.2 sketches the basic mechanism of nematicon formation via a purely reorientational response.

Figure 1.2 Basic physics of nematicons. An extraordinarily polarized bell-shaped beam with wave-vector along is launched in an NLC layer with director lying in the plane yz. The major axes of the molecules are at an angle with the wave-vector, thanks to a pretilt (the arrows indicate the molecular director). (a) In the linear regime light does not affect the angular distribution of the director: the beam diffracts as in homogeneous media. (b) Conversely, at high powers the director is perturbed and reorientates toward , increasing θ and thus the refractive index (Fig. 1.1d). The perturbation is stronger where the intensity I is higher; hence, an index well is created by the light beam itself, leading to the formation of a waveguide and a self-trapped nematicon. Noticeably, the perturbation extends far beyond the beam profile owing to the elastic links between molecules. For the sake of simplicity, in this illustration the role of walk-off is ignored (Section 1.2.1).

Finally, nematicons were also reported in slab waveguides with homeotropically aligned NLC [31], in one-dimensional arrays of coupled waveguides [18, 32] (see Chapter 10) and in twisted/chiral NLC [33,...