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Singular Optics of Liquid Crystal Defects

Etienne BRASSELET

Laboratoire Ondes et Matière d'Aquitaine (LOMA), Talence, France

1.1. Prelude from carrots

In the late 19th century, a carrot entered into the history of material sciences: as Reinitzer scrutinized the temperature behavior of cholesterol derivatives extracted from a carrot, he noticed "two melting points" (Reinitzer 1989). He thought that his discovery was worthy for consideration by physicists and contacted Lehmann in 1888 for further investigation. These were the early days of thermotropic liquid crystals, for which the temperature is the state parameter. Singularly, defects and light, with all of its colors, are ubiquitous to the observation of liquid crystals (Dierking 2003). This chapter focuses on the singular optical aspects of liquid crystal defects, which are discussed in section 1.6 after having reviewed the regular counterparts in section 1.5. For the sake of consistency, the chapter starts by providing the reader with the necessary, but non-exhaustive, information about liquid crystal structures, optics and behavior under external fields. A jump start is given by Figure 1.1, where the carrot's orientation rotates by 2p per full-turn around the normal direction to the table, providing an artistic wink at liquid crystals, light and defects.

1.2. Liquid crystals, optics and defects: a long-standing trilogy

The pioneering experimental investigations by Lehmann were made possible by an instrument he developed, a temperature-controlled polarizing optical microscope (see Figure 1.2(a), in which Lehmann is standing nearby his apparatus). Two of his drawings representing the optical observations of the materials he received from Reinitzer, are presented in Figures 1.2(b) and (c). These sketches simply and beautifully celebrate the links between liquid crystals, optics and defects, which still stand vividly today, as this chapter will illustrate. In fact, these pictures demonstrate how irregular structures spontaneously occur in liquid crystals, which, once tamed, pave the way for several modern optics applications worth exploring.

Figure 1.1. Colorful arrangement of carrots recalling an orientational defect with unit charge. Picture taken from https://app.emaze.com/user/lisaadjeroud

On one hand, the black crosses in Figure 1.2(b) are related to local axisymmetric supramolecular structuring of the anisotropic optical properties of liquid crystals. As we shall see, this is one of the situations enabling the interplay between the topology of liquid crystals and that of light. On the other hand, the threads running randomly through a uniform background in Figure 1.2(c) correspond to complex clusters of defects of various kinds, called "oily streaks" by Reitnizer in his first letter to Lehmann in 1888 (Kelker 1973). This illustrates how liquid crystal structures can either act as optical scatterers or be considered as tangible information recorded in the fluid, which will be also discussed.

It took a few decades of abundant debates, experiments and conceptual developments (Kelker 1973, 1988; Lagerwall 2013) before Friedel articulated that liquid crystals represent mesomorphic states of matter, with the potential to express a rich polymorphism (Friedel 1922). In the latter work, Friedel also introduced the terms smectic, nematic and cholesteric. Such terminology refers to the different kinds of spatial organization of the constituting anisotropic building blocks of liquid crystals, and is still in use today. Nowadays, the classification is much richer and we refer to Oswald and Pieranski (2005) for a comprehensive overview.

Figure 1.2. (a) Lehmann posing close to his crystallization microscope. (b and c) Sketches by Lehmann of some of his observations, dated from April 4, 1888 (Kelker and Knoll 1989) and August 30, 1889 (Kelker 1988), respectively, which emphasize the emergence of spontaneous irregular structures in liquid crystals

1.3. Polarization optics of liquid crystals: basic ingredients

1.3.1. The few liquid crystal phases at play in this chapter

In this chapter, three types of thermotropic liquid crystals will enter the discussion: smectic A, nematic and cholesteric. All three types are made up of elongated molecules represented as cylinders in Figure 1.3, where n is a unit vector called director that refers to the local average orientation of the molecules and satisfies the equivalence n = -n. From a structural point of view, the smectic A phase combines a positional order along one spatial dimension, illustrated as layers in Figure 1.3(a), with an orientational order associated with a director pointing along the normal to the layers. The nematic phase, illustrated in Figure 1.3(b), is only characterized by an orientational order. Finally, the cholesteric phase merely corresponds to a nematic phase endowed with supramolecular chirality, where the director field adopts a one dimensional helical order, in the absence of external constraints (see Figure 1.3(c)). A cholesteric liquid crystal is thus characterized by (i) its helix pitch and (ii) its right or left handedness. The helix pitch p is the distance over which the director rotates by 2p, although the space inversion invariance of the director implies that the physical period is p/2.

1.3.2. Liquid crystals anisotropy and its main optical consequence

As suggested by the axisymmetric representation of the liquid crystal building blocks in Figure 1.3, here, we will deal with uniaxial smectic A and nematic phases. In other words, any of their physical property (electrical, magnetic, optical, thermal, etc.) is characterized by two scalar quantities: one associated with the direction along and the other associated with a plane perpendicular to it . Regarding the cholesteric phase, it is usually prepared by adding a small fraction of a chiral dopant into a nematic phase. This has the advantage of preserving the physical properties of the host phase, while the helix pitch can be tuned at will, typically from infinity to a few hundred nanometers, by adjusting the nature and concentration of the dopant.

Figure 1.3. Illustration of the three liquid crystal phases appearing in this chapter. Note that in panels (a) and (b) the orientation of the molecules fluctuates with time. In panel (c), the depicted regular helix made up of one molecule is only for the sake of illustration, recalling that there is no positional order and that there are microscopic orientational fluctuations

Discussing the anisotropic nature of liquid crystals leads to the introduction of one of its most salient features when dealing with direct observations by optical means: their birefringence. This refers to the fact that the electronic response of the material irradiated by light depends on its polarization state, which informs on how the real electric field vector oscillates. Considering a fully polarized monochromatic paraxial light beam1, the trajectory of the tip of the electric field vector at a given position lies in a plane transverse to the beam propagation direction and is elliptical. The latter ellipse is called the polarization ellipse. Disregarding the field intensity, the latter ellipse is specified by two parameters: its orientation angle 0 = ? = p and its ellipticity angle -p/4 = ? = p/4 (see Figure 1.4). Note that the angles ? and ? allow an arbitrary polarization state to be represented unequivocally by a point on the unit sphere, called the Poincaré sphere of polarization (Born and Wolf 2013), whose longitude and latitude are respectively given by the angles 2? and 2?. The birefringence is defined as the difference between the refractive indices experienced by linearly polarized light oriented along a direction parallel and perpendicular to n, namely, . The birefringence explains the extent to which liquid crystals can modify the polarization state of light.

In order to quantitatively grasp how the polarization state of light can be modified as light propagates through liquid crystals, as a consequence of their optical anisotropy, we introduce a few simple, yet relevant, polarization optics facts that enable us to handle the concepts presented in this chapter. Of course, the optics of liquid crystals are by no means restricted to such a simple view. We can refer to Yeh and Gu (2009); Khoo and Wu (1993) for further reading about applied and fundamental aspects.

Figure 1.4. Polarization ellipses and their characteristic angles (?, ?). Linear, elliptical and circular polarization states refer to ? = 0 (a), 0 < |?| < p/4 (b) and |?| = p/4 (c), respectively. We define the polarization handedness as that of the helix formed by the electric field vector at a given time. Namely, for a wave propagating toward z > 0, sketches with ? < 0 (blue color) referring to right handedness, while those with ? > 0 (red color) refer to left handedness

1.3.3. Polarization state representation in the paraxial regime

Let us consider, from now on, a fully...