Cholesteric Liquid Crystals: Optics, Electro-Optics, and Photo-Optics

6 Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics Guram Chilaya Cholesteric liquid crystals (CLCs) show very distinctly that molecular structure and external ®elds have a profound eëct on cooperative behavior and phase structure (see also Chapters 2 and 3). CLCs possess a supermolecular periodic helical structure due to the chirality of molecules. The spatial periodicity (helical pitch) of cholesterics can be of the same order of magnitude as the wavelength of visible light. If so, a visible Bragg re¯ection occurs. On the other hand, the helix pitch is very sensitive to the in¯uence of external conditions. A combination of these properties leads to the unique optical properties of cholesterics which are of both scienti®c and practical interest. The in¯uence of electric ®elds and light irradiation on the optical properties of cholesteric structures are reviewed in this chapter. In Section 6.1 we consider the general optical properties of CLCs, such as Bragg di¨raction due to the periodical structure, refractive indices, and induced circular di- chroism. In the subsequent sections, electro-optic eëcts and light-induced eëcts are described. Several types of electro-optic eëcts have been observed in cholesteric liquid crystals [1]±[3]. Section 6.2 of this chapter is focused on some of the most recent results, e.g., bistability, color change eëcts, `àmorphous cholesterics,'' and the ¯exoelectric eëct. In addition, dielectric, hydrodynamic, and ¯exoelectric instabilities, as well as domain structures in cholesterics, are discussed brie¯y. Light-induced molecular reorientations and optical nonlinearities in transparent (nonabsorbing) and absorbing cholesterics, photostimulated shift of the pitch, optical bistability, and optical switching are presented in Section 6.3. The generation of higher harmonics and laser generation are considered, too. 6.1 General Optical Properties of Cholesteric Liquid Crystals The cholesteric phase appears in organic compounds which consist of elon- gated (nematogenic) molecules without mirror symmetry (chiral molecules) [1]±[3]. Typical representatives of these compounds are the derivatives of 159 160 G. Chilaya cholesterol. Thus, chiral nematic liquid crystals are generally called cholesteric liquid crystals (CLCs), although the name chiral nematic is more cor- rect. The cholesteric structure occurs not only in pure chiral compounds, but also in mixtures of achiral nematics with optically active (chiral) mesogenic or nonmesogenic dopants (induced cholesteric systems) [4]±[9]. Locally, a cholesteric is very similar to a nematic material. However, the direction of the preferable orientation of the molecules n (director) varies periodically in space. If the helical axis is oriented along z, the director n is given by (nx cos y, ny sin y, nz 0), where y q0z constant, q0 is the wave number q0 2p=P, and P is the helix pitch. The spatial period is equal to one-half of the pitch (because of the unpolarity of the cholesteric structure, see Figure 6.1). The helix may be right- or left-handed, depending on the absolute con®guration of the molecules. In some mixtures a helix sign in- version is observed when either the temperature or the concentration of the components is changed [10]. Figure 6.1. The arrangement of (a) the molecules and (b) the optical indicatrix in the cholesteric phase. 6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics 161 In various cholesteric systems, the period of the supermolecular structure (helical pitch) varies by a wide range (from @ 0.1 mm to several hundred mm). For the case of long pitch (low chirality) P g l (where l is the wavelength of light), the light propagating parallel to the helical axis may be described by a superposition of two eigenwaves having electric ®eld vectors parallel and perpendicular to the director. The long pitch case was studied for the ®rst time by C. Mauguin [11]. This type of con®guration can be obtained by the mechanical twist of nematics and is used in conventional twisted nematic displays [12]. In this case the structure behaves as a polarizing waveguide: the plane of polarization of linearly polarized modes follows the twist. For short pitch (high chirality), when l and P are comparable, the eigenwaves become elliptical, and in the limiting case circular. In this limiting case, selective re¯ection occurs due to Bragg di¨raction at a wavelength lB with mlB Pn cos j: 6:1 Here, m is the di¨raction order, j is the angle of light incidence, and n is the refractive index of the medium. The di¨raction in CLCs is responsible for some remarkable optical properties. The following characteristic features occur for light propagating along the axis of the helix. (a) Only the ®rst-order Bragg re¯ection is possible in this case. (This is con®rmed by both experimental results and theoretical considerations.) According to (6.1), the maximum of selective re¯ection occurs at the wavelength lB Pn. The spectral width of the selective re¯ection band is equal to Dl PDn, where Dn ne À no is the birefringence of a nematic layer perpendicular to the helix axis. The re¯ected light is circularly polarized and the sign of rotation coincides with the sign of rotation of the cholesterics helix. (b) On each side of the selective re¯ection band there are regions with a strong rotation of the plane of polarization of light. The rotatory power amounts to more than hundreds of revolutions per mm. The rotation of the plane of polarization depends strongly on the wavelength of incident light, and an anomalous dispersion of the rotatory power is observed. According to [13] the rotation angle is given by 2 2 2 2 2 0 2 0 2 j 2p d=P ne À no =ne no 1=8 l 1=1 À l : 6:2 (c) Close to the Bragg wavelength lB, the optical rotation becomes very large and changes its sign at l lB. The theory of the propagation of light along the optic axis was considered in [11], [13]±[17]. The kinematical approach could explain many experimental results. However, for quantitative explanation of the experiments, a more detailed consideration is necessary. For this purpose, the dynamical theory should be used [15]. Precise measurements of the re¯ection spectrum from a monodomain CLC show good agreement with theory [18]. A solution 162 G. Chilaya of Maxwell's equations was ®rst given in [13]. The exact solution of Max- well's equations for the general case is given in [16]. For the case of oblique incidence, ®rst- and higher-order di¨ractions are permitted. The polarization becomes elliptical in this case. Precise measurements of the re¯ection spectrum for the oblique incidence of this light were given in [19]. First- and second-order re¯ection spectra were calculated and observed. The analytical solution of Maxwell's equations by the dynamical theory of di¨raction [20] is in good agreement with experimental data [19]. The exact solution of Maxwell's equations has not been yet developed, because the theory is very complicated. The propagation of light perpendicular to the optical axis was studied in [21]. For a certain polarization of incident light, the cholesteric phase can be considered as a medium with a periodic gradient of the refractive index. The refractive index changes between ne and no and the period is half of the pitch. The periodicity in the phase and amplitude causes a di¨raction of polarized light. This di¨raction was used for investigating the temperature-dependence of the pitch. When the pitch of the CLC is larger than the wavelength of the visible light and if the linear birefringence is also large, it is possible to observe the forward di¨raction [22]. For a cholesteric layer between crossed polaroids, the presence of forward scattering is manifested in the form of selective dependence of the transmission coe½cients on the wavelength of light. Ex- perimental studies on a well planar oriented CLC with certain parameters con®rm the forward di¨raction eëct [23]. 6.1.1 Orientational Order Parameter and Refractive Indices The cholesteric phase is thermodynamically equivalent to the nematic phase. Both phases can be characterized by an orientational order parameter S : hP2 cos yi, where P2 is the second Legendre polynomial, and y is the angle between the long molecular axis and the local director n [24]. The presence of twist in the cholesteric phase complicates the problem of mea- suring S. Many of the methods applied successfully to nematics are not suitable for CLCs. Nevertheless, some measurements using optical methods were done in order to estimate S in CLCs [25]±[29]. A nematic liquid crystal with a uniform alignment of the director n behaves like a uniaxial crystal with positive optical anisotropy ne > no (where ne 1 nkL is the refraction index for the extraordinary beam and no 1 n?L is the refraction index for the ordinary beam). We can consider the cholesteric structure as a special case of a nematic structure when the director n de- scribes a helix. As is shown in Figure 6.1, the optical anisotropy in CLCs is negative, i.e., noh > neh, where neh 1 nkh and noh 1 n?h are the refractive indices for the extraordinary and ordinary beams, respectively. The index h indicates that the macroscopic optical axis corresponds to the direction of 6. Cholesteric Liquid Crystals: Optics, Electro-optics, and Photo-optics 163 0 0 the pitch axis. If the local nematic refractive indices are given by ne and no, the average refractive indices with respect to the helix axis h can be written as 0 02 02 1=2 neh no, noh ne no .

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