Chapter 12: Twisted and Chiral Nematicons

Urszula A. Laudyn and Miroslaw A. Karpierz

Warsaw University of Technology, Warsaw, Poland

12.1 Introduction

The uniqueness of nematicons is connected with the reorientational nonlinearity, by which light can change the initial orientation of liquid crystals. Such orientation, being the result of long-range interactions between molecules, is introduced at the boundaries and can be modified by electric or magnetic fields [1–3]. Solitary waves in nematic liquid crystals (NLCs) were investigated in capillaries [4–6], but the most often used configurations are planar cells with two parallel glass plates. In such cells, NLC can be homogeneously oriented, for example, in the homeotropic texture (where molecules are perpendicular to the glass plates) [7–9] or in the homogenic (planar) texture (where molecules are parallel to the glass plates) [10, 11]. Configurations in which the mean direction of the molecules (molecular director ) varies across the cell thickness, as in planar cells biased by a low-frequency external electric field [12–14], seem to be promising. A nonhomogeneous initial orientation can also be introduced by the boundary conditions, as in hybridly aligned nematics (molecules parallel to one plate and perpendicular to the second plate) or in twisted nematic liquid crystals (TNLC, in which the molecules are parallel to both plates but twisted within the layer thickness). This twisted orientation also characterizes chiral nematic liquid crystals (ChNLC, cholesteric liquid crystals).

Nematicons in TNLC and ChNLC have properties similar to standard nematicons in other configurations (Chapter 1): they are observed over distances of a few millimeters, they exist in a wide range of optical powers above a few milliWatts, they can be redirected by external electric fields, their excitation depends on light polarization, they can guide other signals, and their mutual interactions are attractive. However, at variance with other orientations, a cell filled by ChNLC offers the possibility of propagating nematicons (without external fields) in several independent layers. By choosing the birefringence and the period of orientation, either the independent propagation of nematicons or interaction characteristic for systems with discrete diffraction can be observed.

12.2 Chiral and Twisted Nematics

ChNLC consist of molecules with twisted alignment across the cell [1–3, 15, 16]. This results in a helical structure of finite pitch p, the latter defined as the distance along the helical axis after which the molecular orientation has turned by an angle 2π (Fig. 12.1). The ChNLC periodicity is p/2 as directors and are equivalent (for nonpolar molecules). Chirality can be induced by mixing a nematic liquid crystal with a chiral dopant. The pitch p of such a mixture can be easily modified because it depends on the concentration η of the chiral dopant: p = 1/(Hη), where H is the helical twisting power determined by the chemical structure of the chiral material [2, 16]. In the investigated ChNLC cells, the molecular director is parallel to the surrounding glass plates (interfaces) and twisted within the layer thickness. A similar orientation can be obtained for TNLCs, where the NLC molecules are anchored parallel to the glass plates, and these plates are mutually rotated by some angle. However, the maximum rotation in TNLCs is π/2, that is, corresponding to p/4.

The expression for the free energy of deformed ChNLCs is quite similar to that in NLC, and its density takes the form [1–3, 15, 16]

12.1

where K_ii are the elastic (Frank) constants for the three basic spatial distortions of the molecules. In the term related to a twist deformation, the parameter G = 2π/p describes chirality (in nematics G = 0). Energy minimization in unperturbed ChNLCs leads to , which can be fulfilled for example, for a director with the orientation angle θ = θ₀ + Gx. Such an orientation corresponds to a helical structure with an axis along the x direction (as in Figure 12.1).

External fields and surface interactions can easily deform the initial helical configuration. The electrical permittivity of ChNLCs for fields parallel to the helical axis is different from the average permittivity in perpendicular directions. Therefore, an external electric field also causes the reorientation of the whole helical axes [17–19]. It should be noted that the deformation occurs in two stages: first, the molecules are tilted toward the electric field direction and then, untwisting of the helical structure occurs.

Optical spatial solitons or nematicons are investigated in configurations in which a light beam propagates parallel to the glass plates. We assume a beam propagating in the z-direction with a dominating E_y component of the electric field (as in Figure 12.2). For light polarized along y, the refractive index varies across the cell from the ordinary value n_o in those planes in which the molecules are parallel to the z-axis (i.e., where θ = π/2) to the extraordinary value n_e in those where the molecules are parallel y (θ = 0). This can be compared to light propagation in a graded index waveguide. Therefore, the light is guided in a thin layer where θ = 0. By acting on the anchoring conditions, guiding index layers along x can be formed within the cell thickness, with location depending on θ at the glass/NLC interface and on the pitch p. For anchoring conditions such that θ₀ = π/2, the largest refractive index is at the edges of the cell: if the refractive index of the glass is lower than the extraordinary index n_e, the beam is guided at the NLC/glass boundary, partially extending into the isotropic medium (glass) and partially into the twisted liquid crystals. The asymmetry in molecular twist causes the appearance of beam walk-off.

12.3 Theoretical Model

The problem of light propagation in TNLCs and ChNLCs is quite complex, and its quantitative description requires an exact treatment of electromagnetic fields in anisotropic media [20]. However, for a qualitative description, the behavior of a light beam propagating in ChNLCs can be simply described by using the following approximated model [21–23]. This model assumes that the beam profile in the x-direction is invariant during propagation, that is, it assumes that the beam is guided by the linear waveguide created by the pitch. The results are in good agreement with experiments, and the description is adequate for both chiral and twisted structures.

The optic axis changes across the ChNLC cell, generally implying that all components of the electromagnetic fields are nonzero, even for linearly polarized input beams. However, only E_y and E_z cause molecular reorientation in the yz-plane. Along x, when the beam width ensures a good overlap with the guided mode (i.e., for a proper ratio of pitch and beam waist), the beam does not diffract appreciably in that direction. In this case, it can be assumed that the reorientation of the molecules is only in the yz-plane and is mainly connected with the twist deformation. Consequently, the electric permittivity tensor has the components ε _xy = ε _yx = ε _xz = ε _zx = 0 and . In this case, Maxwell's equations for monochromatic electromagnetic waves lead to

12.2

where k₀ = ω/c is the vacuum wave number.

For the beam propagating in the z-direction, it is convenient to introduce the ansatz

12.3

where ϕ(x)exp(iωt − ik₀Nz) and ψ(x)exp(iωt − ik₀Nz) are components of the planar waveguide mode with an effective refractive index N. The modal envelopes ϕ(x) and ψ(x) fulfill the equations

12.4

12.5

where ε ⁽⁰⁾ is the electric permittivity corresponding to the initial orientation of the nematics. In the presence of such fields, the Euler–Lagrange equation for energy minimization has the form

12.6

12.7

where is the optical anisotropy of the electric permittivity. Reorientation changes the local value of the electric permittivity tensor, which depends on the orientation angle θ as

12.8 .

For the paraxial approximation electric field components of the light beam E_z E_y, the slowly varying complex amplitude A fulfills the equation obtained from integration of Equation 12.3 over the cross section

12.9

where the coefficients γ₁ and γ₂ depend on the orientation angle θ as

12.10

12.11

At low optical power (the linear case), the coefficients γ have initial values equal to γ⁽⁰⁾. As light intensity increases (the nonlinear case), the dominant E_y component of a linearly polarized beam induces reorientation in the yz-plane. The liquid crystal molecules are forced to reorient parallel to the electric field, which increases the γ₁ coefficient. It modifies the diffractive broadening of the beam in the y-direction, whereas is responsible for self-focusing of the beam and the creation of a spatial soliton. It should be noted that the coefficient γ₁ has a saturable form and can be calculated for a given liquid crystal layer, as shown in Figure 12.3a.

The second nonlinear coefficient γ₂ is responsible for beam walk-off. This coefficient is zero in the symmetric case, that is, when the light beam profile ψ is symmetrically located around the layer in which θ = 0. This occurs with beams propagating inside the cell. If the beam is guided at the cell boundary (partially in the glass plate), the coefficient γ₂ ≠ 0 defines the walk-off angle. In such a case, at low intensities, γ₂ is the largest, and an increase in intensity causes a decrease in γ₂ (as shown in Figure 12.3b). It means that for higher intensities, the beam walk-off is smaller, that is, the beam direction changes with light power (Chapter 11).

12.4 Experimental Results

12.4.1 Nematicons in a Single Layer

Self-focusing has been observed in a ChNLC cell filled by capillary effect with 6CHBT (4-trans-4′-n-hexyl-cyclohexyl-isothiocyanatobenzene) doped with a chiral material. 6CHBT possesses low absorption and high nonlinear response [24, 25], with refractive indices n_o = 1.51 and n_e = 1.67 in the near infrared. The sample consisted of two glass plates glued together, with a gap controlled by a spacer. An alignment layer was deposited on the top plate to control the alignment of the liquid crystal molecules. The beam propagation in the cell was observed with a microscope objective and a CCD camera. The thickness of the cell was 50 μm, and the pitch of the ChNLC was about 25 μm. The long molecular axes were approximately parallel to the plane of the layers constituting the helical structure. The input beam waist was estimated to be about 2 μm by measuring the divergence during linear propagation in the NLC. Typical experimental results obtained with a Ti:Sapphire laser (λ = 793 nm) are presented in Figure 12.4 [26, 27]. Increasing the input power led to self-focusing and, finally, for an average power P = 9.6 mW, a spatial soliton was formed. The solitary beam had a transverse intensity distribution that remained unchanged after a propagation distance of about 3 mm (i.e., over 80 times the Rayleigh length).

Owing to the finite thickness of each layer in the chiral structure, self-focusing can balance diffraction and give rise to self-trapped solitons only in a limited waist range [22]. Owing to structural anisotropy connected with chiral orientation, the beam walks-off from the initial direction of propagation while changing its polarization. Indeed, as shown in Figure 12.5, crossing from the TE-like to TM-like polarization causes the nematicon to propagate in different directions. However, for the TM-like polarization (for which the Freedericksz threshold prevents nematicon formation), the light beam propagates again along the z-axis.

12.4.2 Asymmetric Configuration

It should be noted that beam walk-off was also observed in asymmetrically twisted liquid crystals [28–30]. Typical pictures are presented in Figure 12.6a. First, light propagation in a homogeneous medium was investigated to verify that the beam was launched parallel to the y-axis. This was achieved by applying an external field, as an applied field with E_y polarization renders the medium homogeneous. Next, without an external electric field, the direction of beam propagation was changed, showing walk-off. In the linear case, when the light power was too low to induce reorientation, beam diffraction was observed. Increasing the power modified the twisting angle and increased the effective refractive index. Light started to change its direction of propagation and eventually created a solitary wave via self-focusing. The beam was guided close to the TNLC/glass boundary. As a consequence, the effective direction of the birefringence axis was not perpendicular to the initial beam direction, showing walk-off.

A comparison of size and shape of such a nematicon for various optical powers and at different distances are presented in Figure 12.6b. The intensity cross section of the scattered light was normalized to its maximum value. The size and shape of the obtained nematicons do not change significantly with increasing input power. However, an increase in power can modify the walk-off. The changes in beam direction versus power are weak but measurable. Effectively, at a distance of 1 mm, the beam can change position by approximately 20 μm as the power goes from 15 to 55 mW.

12.4.3 Multilayer Propagation

A light beam propagates in the region where the ChNLC molecules are parallel to the electric field. The twist is periodic versus the cell thickness, which implies that there are a few high-index regions where the beam can propagate. The number of such layers is determined by the cell thickness and the chiral pitch: it corresponds to 2d/p.

In this configuration, the width of the cell (d = 50 μm) and the ChNLC pitch (p = 25 μm) are such that there are four layers with molecules oriented in the same way (Fig. 12.7a). As a result, there are four layers with a thickness of about 12 μm in which nematicons can be excited independently [26, 27]. By changing the vertical position along the cell, it is possible to independently launch as many solitons as the number of layers in the structure. The vertical position in a ChNLC cell is controlled by means of a microscope slide with micrometric patterns fixed to the (x, y, z) stage and a second CCD camera mounted at the butt of the cell.

The vertical position of the ChNLC cell was changed for a defined TE-like polarization and input power high enough (about 10 mW) to form nematicons. Typical results are shown in Figure 12.7b for the input positions marked on the photos. The position Δx = 0 corresponds to a light beam propagating at the boundary between a glass plate and an NLC. A nematicon is formed in the first layer marked Δx = 10 μm and can be generated in each of the subsequent layers. Indeed, four nematicons were formed in distinct layers, about 10–12 μm away from each other.

12.4.4 Influence of an External Electric Field

In a separate set of experiments, the influence of an external electric field on nematicon propagation was also investigated [31, 32]. An electric field (voltage) along the x-axis, thus parallel to the helix axis, leads to a bend deformation and director reorientation in the xy-plane. When a nematicon was generated in one of the middle guiding layers (for an input power of 15.4 mW and λ = 514 nm), it induced a channel waveguide with confinement along y. Applying an electric field along x changed the director orientation, and consequently, the soliton started to propagate at some angle to the z-axis. Figure 12.8a shows such a case corresponding to P = 15.4 mW and measured over a distance of z = 600 μm. A further increase in the electric field at fixed optical power increased the relative contribution of the electrically induced reorientation and destroyed the helical structure. As a result, the beam did not follow the molecule orientation, resulting in linear diffraction. Figure 12.8b plots the soliton trajectory as the voltage varies from 0 to 4.5 V. The displacement changed by 20 μm at a distance z = 600 μm. The maximum steering was estimated to be about , which is comparable with the birefringent walk-off in this material (the maximum walk-off was calculated to be ).

Furthermore, when the soliton propagated in the guiding layers nearest to the boundaries, the voltage required for steering shifted to higher values. This is caused by the fact that the anchoring conditions strongly inhibit molecular reorientation. In the middle layers, the director orientation changes are much less affected by the boundary conditions.

The polarization dependence of an input beam on steering was also investigated [31]; it was found that a change in input polarization alters the soliton direction for given beam, layer, and electric field. Changing the polarization causes the nematicon to propagate in a different direction relative to the z-axis. For an input polarization along y and a voltage of approximately 3.5 V, the beam propagates at an angle to the z-axis as a consequence of reorientation of the molecules. Changing the input polarization but keeping the voltage constant makes the beam propagate at an angle smaller than the z-axis.

Usually, in these kinds of experiments, one can see a diffractive background. This occurs because the field-induced reorientation changes the direction of the extraordinary axis in the xy-plane by the tilt angle, that is, a y-polarized input beam contains a small component along the new ordinary axis. With no voltage and sufficient power to create a nematicon, the wave has only an E_y component. With an external field, the nematicon propagates at a tilt angle relative to the yz-plane, with fields polarized along the new director. Furthermore, the small ordinary component diffracts in propagation.

12.4.5 Guiding Light by Light

Similar to other nematicons, it was verified in TNLCs and ChNLCs that a solitary wave can confine a second low power probe, for example, from a He–Ne laser [23, 30, 32]. Nematicon propagation in a ChNLC or TNLC cell introduces changes in the refractive index distribution and leads to the formation of an optical waveguide (Chapter 1). This can be verified by injecting a second copolarized low-power probe, generally at a different wavelength. When the pump beam diffracts, the probe beam diffracts as well (Fig. 12.9a). When a high-power TE-polarized beam (from an argon laser, in this case) forms a spatial soliton, the copropagating low-power probe is also confined (Fig. 12.9b). Moreover, the probe follows exactly the direction of the soliton. The pump beam causes a focusing “lensing” effect, and the probe becomes increasingly confined, too.

12.4.6 Nematicon Interaction

The interaction between two identical nematicons was also verified in ChNLCs [27]. If two nematicons are launched close to each other in one layer, they can attract and merge into one single self-trapped beam. This happens because of the spatially nonlocal response of liquid crystals, analogous to former experiments in planarly aligned NLC cells, for example, References 33, 34 and Chapters 1–2.

Two beams with w₀ = 2 μm and separated by 25 μm were launched in one layer of ChNLC to form two identical nematicons. The refractive perturbation created by one beam diffuses and affects the other beam, causing two initially parallel solitons to attract one another. Depending on the initial geometry and the strength of nonlinearity, the two-beam interaction can have different outputs, that is, the solitons can either drag each other or pass through each other. The distance at which two solitons collide decreases with the input power. Solitons can not only interact with each other but also merge in a single self-confined beam. An example of time evolution of soliton–soliton interaction is also reported. Two beams of 10 mW propagate in the yz-plane with initial separation of 25 μm. Owing to the slow ChNLC time response, after illumination, no nonlinear behavior is visible and diffraction is observed with an overlap of the two waves as they propagate. After 1 s, two solitons are formed and propagate, maintaining their initial separation (Fig. 12.10a). In a few seconds, the solitons start to attract each other. Eventually, owing to the nonlocality of the NLC, after 35 s, they collapse into one beam (Fig. 12.10b) giving rise to a stable solution.

12.5 Discrete Diffraction

The results presented in the previous section were obtained in a single layer of the ChNLC cell. However, the interaction between solitons propagating in different layers is also possible [35]. Owing to the fact that the refractive index distribution changes periodically along the x-direction, that is, perpendicular to the glass plates, the analyzed cell can be treated as an array of planar waveguides. In such structure, it is possible to obtain the conditions for which the standard continuous diffraction is substituted by a discrete one by way of a coupling between waveguides side by side. Discrete diffraction has numerous interesting features (Chapter 10). In ChNLCs, a beam can diffract in two ways: continuously in one direction (in the yz-plane) and discretely in the other (in the xz-plane). The reorientational nonlinearity can suppress both types of diffraction and form nematicons.

To enable the experimental observation of discrete diffraction, the coupling length L_c defining the distance at which light switches from one waveguide to the adjacent one should be shorter than the sample length. The coupling length can be modified by changing the amplitude and period of the refractive index modulation, that is, by changing pitch and birefringence in ChNLCs. Low birefringence ChNLCs with a pitch p < 10 μm should be used, as presented in Figure 12.11a. Numerical simulations of beam diffraction in such configuration are shown in Figure 12.11c. Note that discrete diffraction does not depend on beam waist but on the refractive index modulation. To obtain comparable amounts of discrete (along x) and continuous (along y) diffractions, the initial beam size along y needs to be properly engineered (in the simulations, the input beam is much wider in y than in x ).

The creation of spatial solitons requires an increase of refractive index, which must be higher for shorter coupling lengths. In the presence of reorientational nonlinearity, the maximum changes in refractive index are limited by the birefringence. The ratio between the nonlinear index changes necessary to create solitons and the maximum possible changes versus pitch is graphed in Figure 12.11b. A lower ratio requires lower reorientation; in turn a lower input power is necessary to form nematicons. As a consequence, pitch and birefringence cannot be too small. In addition, for low birefringence ChNLCs, the elastic constants are larger than in 6CHBT. Keeping in mind the previous discussion, the powers necessary to excite discrete nematicons are much larger than in the configurations discussed earlier.

The nematicon shown in Figure 12.11d was generated by a beam diffracting with comparable divergences in both dimensions, that is, its size in y was chosen so that Rayleigh and coupling lengths were equal. However, if these two divergences were different, the creation of spatial solitons would be more difficult (as in the example in Figure 12.11e) and sometimes even impossible.

In summary, discrete nematicons seem to be very peculiar of ChNLCs but require larger powers and more accurately controlled excitation conditions than standard nematicons.

12.6 Conclusions

TNLC and ChNLC form convenient geometry for creation of spatial solitons. They do not need an external electric field, and optimization of nematicon parameters can be done by choosing the pitch and the birefringence of NLCs. The properties of observed solitons are typical for nematicons. They can also be easily steered by external fields, by other beams, as well as by changing the polarization or light power. The prospective properties of ChNLCs are connected with the existence of multiple-layered structures created by periodical orientation of molecules. For high birefringent ChNLCs, it allows to propagate independent nematicons in different positions across the cell. On the other hand, for low birefringent ChNLCs it can be an interesting structure for application of discrete diffraction. This, among others, can be the basis of three-dimensional elements for light beam switching and routing.

Acknowledgments

We are grateful to Edward Nowinowski-Kruszelnicki, Marek Sierakowski, Katarzyna Jaworowicz, Michal Kwasny, Katarzyna Rutkowska, and Filip Sala for extensive contributions to this work. This work was supported financially by the National Science Centre.

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