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 e¨ect 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 dichroism. In the subsequent sections, electro-optic e¨ects and light-induced e¨ects are described. Several types of electro-optic e¨ects 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 e¨ects, ``amorphous cholesterics,'' and the ¯exoelectric e¨ect. 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 elongated (nematogenic) molecules without mirror symmetry (chiral molecules) [1]±[3]. Typical representatives of these compounds are the derivatives of 159

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cholesterol. Thus, chiral nematic liquid crystals are generally called cholesteric liquid crystals (CLCs), although the name chiral nematic is more correct. 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 ˆ cos y, ny ˆ sin y, nz ˆ 0), where y ˆ q0 z ‡ constant, q0 is the wave number …q0 ˆ 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 inversion 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.

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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 ˆ 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 ˆ Pn. The spectral width of the selective re¯ection band is equal to Dl ˆ PDn, where Dn ˆ ne 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 j ˆ ‰2p d=P…ne2

no2 =ne2 ‡ no2 † 2 Š‰1=8…l 0 † 2 1=1

…l 0 † 2 Š:

…6:2†

(c) Close to the Bragg wavelength lB , the optical rotation becomes very large and changes its sign at l ˆ 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

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of Maxwell's equations was ®rst given in [13]. The exact solution of Maxwell'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. Experimental studies on a well planar oriented CLC with certain parameters con®rm the forward di¨raction e¨ect [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 :ˆ hP2 …cos y†i, 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 measuring 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 describes 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

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the pitch axis. If the local nematic refractive indices are given by ne0 and no0 , the average refractive indices with respect to the helix axis h can be written as neh ˆ no0 , noh ˆ …ne02 ‡ no02 † 1=2 . Precise measurements of the refractive indices of racemic (nematic) and the optically active (cholesteric) form of the same liquid crystal show that the refractive indices ful®ll this theoretical expectation within experimental error (G0:0005) [30], [31]. The optical anisotropy of the liquid crystal phase is determined by the anisotropy of the polarizability of the molecules and by the degree of their order which is described by the order parameter S. The obtained results indicate that the orientational order parameter S in a layer of a CLC is essentially the same as in the nematic phase occurring in the corresponding racemic mixture.

6.1.2

Induced Circular Dichroism and Suppression of the Absorption

If a small amount of dye, possessing linear dichroic absorption, is dissolved in a CLC, the helical arrangement transforms the linear dichroism into a circular one. The induced circular dichroism is given by D ˆ …Il Ir †= …Il ‡ Ir †, where Il and Ir are the light transmission coe½cients for leftand right-handed circularly polarized light, respectively. Induced circular dichroism was studied by several authors [32]±[37]. It was investigated even for a dye possessing, simultaneously, both positive and negative dichroism [37]. With the sign inversion of the linear dichroism, the sign of circular dichroism is changed, as well. The data given in [32]±[37] were obtained for structures where the pitch was greater than the absorption wavelength of the dye. However, the absorption is suppressed when the wavelengths of absorption and selective re¯ection coincide. Since the arrangement of the molecules is helical, the absorption of light with the di¨racted polarization undergoes an abrupt change near the region of selective re¯ection. On the short wavelength side of Bragg re¯ection, the electric vector is perpendicular to the long axis of the molecules, i.e., to the absorbing e¨ective oscillators in dyes with positive dichroism. As mentioned in [2], [38], this e¨ect is analogous in many respects to the anomalous absorption of X-rays (Borrmann e¨ect) that occurs as a result of di¨raction in ordinary crystals. Some measurements and the corresponding theoretical works [38]±[41] show a quantitative agreement between the theory and the experiment [41].

6.2

Electro-optics of Cholesteric Liquid Crystals

Several types of electro-optic e¨ects have been observed in CLCs, depending on the surface treatment (boundary conditions), the helical pitch P, the thickness-to-pitch ratio d=P, the dielectric anisotropy De, and the frequency of the applied ®eld. Some of these electro-optic e¨ects are caused by texture

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Figure 6.2. Textures in CLCs: (a) planar, (b) ®ngerprint texture, (c) focal conic, and (d) ``amorphous'' cholesteric texture.

changes. Figure 6.2 shows the the orientation of the liquid crystal for the typical cholesteric textures, namely the planar (Grandjean), focal conic, ``®ngerprint'' and ``amorphous'' cholesteric texture. In order to study electrooptic e¨ects, the liquid crystal is usually sandwiched between two parallel plates with transparent electrodes. Cholesteric liquid crystals with negative dielectric anisotropy De < 0 show a dynamic scattering for electric ®elds with low frequency [42]. A transparent ®eld-o¨ state can be obtained by preparing a planar texture. The electric ®eld induces hydrodynamic instabilities which cause a di¨use scattering appearance. Under appropriate conditions, the scattering state is maintained after removal of the ®eld, thereby providing an optical memory e¨ect (``storage mode''). The stored scattering state can be erased by the application of a high-frequency electric ®eld which reorganizes the planar structure. The hypothesis that this e¨ect is connected with the transition of a confocal texture to a planar texture was ®rst expressed in [43]. Experiments show that the erasure frequency is inversely proportional to the dielectric relaxation time [44]. In CLCs with positive dielectric anisotropy, an electric ®eld-induced cholesteric±nematic phase transition was theoretically predicted [45], [46] and experimentally observed [47], [48]. If the electric ®eld E is applied perpendicular to the helix axis h of a CLC, the helix unwinds like in a magnetic ®eld (Chapter 2). At su½ciently high ®eld strengths, the homeotropic nematic structure is stabilized (Figure 6.3). The critical ®eld strength E ˆ ECN depends on the pitch P, the dielectric anisotropy De, and the twist elastic constant K22 : ECN ˆ …p 2 =Po †…K22 =eo De† 1=2 :

…6:3†

The critical electric ®eld given by (6.3) in the analogous expression to the critical magnetic ®eld strength given by (2.22). If the electric ®eld is applied parallel to the helix, the situation is more complicated. For a short pitch, a shift of the re¯ection peak to shorter wavelengths (blue shift) can be observed [49]±[52]. It was assumed that above a threshold ®eld, a conical deformation of the planar texture leads to a contraction of the pitch, and thus lB is shifted to shorter wavelengths [52]. However, it could be shown that the blue shift of the selective re¯ection

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Figure 6.3. Schematic representation of the cholesteric±nematic phase transition.

results from a periodic distortion of the texture, rather than from a pitch contraction [46]. Using the continuum theory, a ®eld-induced pitch gradient in the CLC cell has been proposed [53]. The latter e¨ect would result in a shorter pitch, which provides an alternative explanation for the ®eld-induced wavelength shift. Experiments on dual-frequency addressable mixtures with a low-crossover frequency [54], [55] lead to new conclusions on the origin of the blue shift. It was shown that both a blue shift and a red shift occur in the same system. These two processes have di¨erent relaxation times. A reversible color change can be realized by switching between the low and high frequency, especially in the more stable systems with shorter pitch. The helix pitch is very sensitive to external in¯uences and, in particular, it may strongly depend on the temperature. The property of CLCs to change the re¯ected color with temperature (due to the temperature-dependence of the pitch) has been known for a long time, and CLCs are successfully used as thermochromic material [56]. The temperature-dependence of pitch is quite complex and so far not completely understood [57]. The temperaturedependence of pitch is an advantage for thermometric applications, but it can be a major problem for the use of CLCs as an electro-optic material. In the latter case, it is necessary to obtain mixtures with an operating voltage which is independent of the temperature. For this purpose, systems with a designed pitch-temperature dependence P…T† are needed. The critical ®eld strength for the cholesteric±nematic phase transition depends on the pitch, the twist elastic constant, and the dielectric anisotropy [(6.3)]. It was shown [58], [59] that a small negative temperature coe½cient dP=dT of the pitch is suitable to compensate the temperature-dependencies of the twist elastic constant and the dielectric anisotropy, so that the temperature-dependence of ECN becomes negligible. Five di¨erent ways to obtain a temperatureindependent ®eld strength ECN in systems with an induced spiral structure are described in [60].

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Figure 6.4. Transmittance hysteresis curve for the cholesteric±nematic phase transition.

6.2.1

Bistability and Color Change E¨ects in Cholesterics

For normal (homeotropic) surface orientation, a bistability e¨ect appears, i.e., the intensity-versus-voltage curve shows a large hysteresis [61] (Figure 6.4). As a matter of fact, the occurrence of a bistability is possible in all cases of surface treatment (including nontreated, nonrubbed cells), provided that the cell thickness d of the cholesteric sample is comparable to the helical pitch P [4], [62]. The appearance of certain textures depends on the boundary conditions, the value of the pitch, the thickness-to-pitch ratio, and the regime of the applied voltage. A deformed spiral superstructure, the so-called strain (scroll) texture, is organized under certain conditions [63], [62]. Various pitch±thickness ratios have been investigated [64], [65]. A CLC doped with dyes can even show a tristability e¨ect if the pitch is large [66]. The expression (6.3) for ECN was calculated for in®nitely thick ®lms without taking into account the boundary conditions. However, the cholesteric to nematic phase transition was investigated for di¨erent thickness [67], [68]. The in¯uence of the surface orientation was taken into account [68] by introducing a surface free energy per unit area F which leads to the following expression for VCN : VCN ˆ …8p 2 d 2 K22 =…Po2 eo De†

8Fd=…eo De†† 1=2 ;

…6:4†

where P0 is undisturbed pitch. The anchoring energy has a remarkable in¯uence on VCN [69]±[71]. The bistability behavior of the cholesteric±nematic transition was investigated for di¨erent treatments of the surface [72]. For this purpose, glass plates with transparent electrodes were coated by polyimid ®lms, and three kinds of cells were investigated with:

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(1) both sides unrubbed; (2) one side rubbed; and (3) both sides rubbed. Additionally, the rubbing strength was controlled. The largest value of the hysteresis width was obtained when both sides were unrubbed. The bistability discovered in [61] was also used for practical applications. The strain texture, which possesses the e¨ect of storage was applied in image converters [73]. A color projection display has been developed based on the bistability of the cholesteric-nematic phase transition [74], [75]. The cholesteric-nematic phase transition e¨ect is also successfully used in guest±host dichroic liquid crystal displays [76]. The phenomenon of aligning the dye molecules (or other doping molecules) by the liquid crystal matrix (Section 6.1.2) is called a guest±host interaction. Consequently, liquid crystal displays (LCDs) containing oriented dye molecules are commonly known as guest±host dichroic LCDs. The application of an electric ®eld changes the absorbing state into a nonabsorbing state or vice versa. The observed contrast ratio is a¨ected by the dichroic ratio, the concentration of the dye molecules, and the cell thickness. For a given thickness, a higher dye concentration leads to a higher contrast ratio, but the corresponding transmission is reduced. For a comprehensive review on this subject, see [77]. Another application of the cholesteric±nematic phase transition was discovered recently [78]. The spectral transmission characteristics were investigated for di¨erent incident and observing angles of a CLC with a pitch P V 0:8 mm. In this case, the wavelength of selective re¯ection is not in the visible range and the initial planar texture is transparent. At the electric ®eld strength ECN , the cholesteric±nematic phase transition takes place. After decreasing the applied electric ®eld strength below ECN , but not below a critical value EFC , one observes a state with uniform color. (The critical value EFC corresponds to the transition between the focal conic texture and the Grandjean texture.) For the Grandjean texture, the angular dependence of the wavelength lB of selective re¯ection (i.e., Bragg re¯ection) is characterized by dlB =dy 0 < 0, where the angle y 0 describes the direction of observation with respect to the surface normal. However, the ®eld-induced state occurring between ECN and EFC shows an angular dependence with dlB =dy 0 > 0, which is characteristic of a di¨ractive grating. The Bragg scattering from a polydomain structure for oblique incident light in re¯ected geometry was studied previously [79], [80]. In [81], the angular dependence of the wavelength of the di¨racted light was analyzed in greater detail. The geometry of observation of the scattering characteristics is shown in Figure 6.5. In the case of a polydomain structure, the angle j occurring in the Bragg condition [(6.1)] has the meaning of an e¨ective scattering angle. If the di¨raction pattern is studied in a transmission geometry (Figure 6.5(a)), the re¯ection condition with respect to the cholesteric planes, 0 g, and jin ˆ jout , is only ful®lled if the angle j is given by j ˆ 12 fyLC ‡ yLC

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(a)

(b)

Figure 6.5. (a) Schematic representation of the setup for investigating the scattering properties. (b) Theoretical angular dependence of lB . The values y 0 < 90 correspond to the transmission and y 0 > 90 to the re¯ection mode. Note that the two corresponding scales are very di¨erent.

thus [79]:

  lB ˆ pn sin 12farcsin…sin y=n† ‡ arcsin…sin y 0=n†g :

…6:5†

Equation (6.5) describes the angular dependence of the scattered wavelength l in the di¨raction limit, y 0 < 90 . For the case of observation in the re¯ection mode …y 0 > 90 †, the e¨ective angle j is given by j ˆ 12 fyLC 0 †g, and lB is given by [80]: …180 yLC   …6:6† lB ˆ pn cos 12farcsin…sin y=n† arcsin…sin y 0 =n†g : The calculated angular dependence is shown in Figure 6.5, as well. This ®gure describes all cases observed in the experiments very well. The color of the ®eld-induced state varies with the applied voltage in [78]. This electric ®eld-controlled color-change e¨ect was also observed in polymer-dispersed cholesteric ®lms [78] and gels [81], and is very promising for applications. Besides the voltage-controlled color e¨ect, the observed structure o¨ers another practical possibility: a laser beam passing through this structure can be de¯ected. For an He±Ne laser beam with oblique incidence (l ˆ 632 nm, j ˆ 45 ), a voltage-controlled variation of the de¯ection by about 30 was realized [78].

6.2.2

Electro-optics in Cholesterics with Medium Pitch and Amorphous Cholesteric Structure

Close to the Bragg wavelength lB , a CLC behaves ``pure'' optically active, i.e., it shows no linear birefringence. The rotation angle j of the polarization plane of normally incident light is given by (6.2). This case is denoted as a CLC with medium chirality (pitch) [82]±[86]. According to expression (6.3)

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for the cholesteric±nematic transition, a low control voltage can be obtained by a large spiral pitch. However, the Mauguin region is approached on increasing P. This is evident by an increase in the optical anisotropy (linear birefringence) of the structure. To avoid uncontrolled deformations of the pitch caused by the cell walls, called strain textures [62], [63], the condition P > d must be satis®ed. Estimating the in¯uence of linear birefringence near the Mauguin region, the general conditions for the optically active structure are l V DnP=2 and P > d [84]. Most frequently, a planar orientation of CLCs is obtained by the rubbing technique. However, in cells coated with polyamide (without rubbing), a new cholesteric state, called an ``amorphous'' cholesteric structure, is formed [82]±[85]. The surface, covered by polyamide, orients the molecules parallel to the surface, but without any preferable direction in the surface plane. Thus, a structure with random orientation of liquid crystal molecules, and a helical axis oriented normal to the surfaces, is obtained (Figure 6.2(e)). All parts of the amorphous cholesteric structure with medium chirality rotate the polarization plane by the same angle. The whole structure can be considered as optically active. In this case, the angle of rotation is independent of the direction of the polarization plane of the incident light. This is the basic property used for electro-optic application of the amorphous cholesteric structure with medium chirality. Characteristic features of this e¨ect are: (1) (2) (3) (4)

low demands on the surface conditions (nonrubbed cell); an ability to function in any position between crossed polarizers; wide and uniform viewing angle; and rise times less than 10 ms and decay times of 12±20 ms.

The electro-optic e¨ect was also studied in polymer dispersed liquid crystal (PDLC) ®lms [86]. With respect to nonchiral nematic PDLC ®lms, the transmission is lower, but the angular dependence is improved.

6.2.3

The Flexoelectric Electro-optic E¨ect in CLCs

It is well known that nematic liquid crystals are nonpolar. However, for a certain asymmetrical shape of the molecules, splay or bend deformations of the director ®eld lead to an electrical polarization [87]. This feature is known as the ¯exoelectric e¨ect. Theoretically, the in¯uence of an electric ®eld on CLCs for the case where the helical axis is oriented parallel to the plane of the sample was ®rst considered by Goossens [88]. Experimentally, the ¯exoelectric electro-optic e¨ect in CLCs can be observed in conventional sandwich cells with transparent electrodes when the helix axis of the CLC lies parallel to the glass surfaces [89]. In the absence of an electric ®eld, the CLC behaves as a uniaxial material with its optic axis perpendicular to the director and parallel to the helix axis. When an electric ®eld is applied normal to the pitch axis, the helix distorts, as shown in Figure 6.6. Thus, the optical axis is reoriented and the medium becomes biaxial. The deviation direction

170

G. Chilaya Figure 6.6. Flexoelectric e¨ect: The pattern of the director rotations, induced by an electric ®eld, applied perpendicular to the plane of the drawing.

changes with the polarity of the ®eld (Figure 6.6), and the corresponding angle is approximately proportional to the electric ®eld strength. The observed ¯exoelectric e¨ect can be explained by a linear coupling between the electric polarization and splay or bend deformations of liquid crystals. The following conditions are necessary to observe this e¨ect: (1) The homogenous alignment of the pitch axis is important in one direction, i.e., the formation of a uniform lying helix (ULH) texture. (2) The pitch of CLCs should be smaller than the wavelength of the incident light, since in this case one can consider the ULH texture as a uniaxial plate. Moreover, at a larger pitch, the di¨raction of light could suppress the ¯exoelectric e¨ect. (3) To achieve large deviation angles, it is necessary to minimize the contribution of the dielectric coupling (the unwinding of the helix) and thus the dielectric anisotropy of the materials should be small. The switching time is independent of the magnitude of the applied electric ®eld and the value is about 100 ms [90]. The static and dynamic properties of the ¯exoelectric e¨ect in CLCs are described in [91], [92]. Considerable e¨orts were made in order to optimize the respective parameters of the liquid crystals [93], [94]. The ¯exoelectric e¨ect was also studied in CLCs with a temperature-induced sign change of the dielectric anisotropy. In this way it was possible to perform measurements in the regions where De G 0. The dependence of the sign of the ®eld-induced deviation of the optic axis on the handedness of the helix was established, which supports once more a ¯exoelectric origin of observed e¨ect. In CLCs showing a temperature-induced twist inversion, it was possible to separate two linear electro-optic e¨ects in the vicinity of the inversion temperature, namely the electroclinic and the ¯exoelectric e¨ect. The reorientation of the optical axis due to the electro-

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clinic e¨ect is about 100 times smaller and 100 times faster than the ¯exoelectric e¨ect [95].

6.2.4

Instabilities and Domains

Several cases of dielectric, hydrodynamic, and ¯exoelectric instabilities and domain structures have been observed and extensively studied in CLCs. Their appearance depends on the initial orientation of molecules, the physical parameters of the material, and the applied electric ®eld. In CLCs with positive dielectric anisotropy De > 0, an electric ®eld applied along the helix axis of a planar (Grandjean) texture can induce a two-dimensional spatially periodic deformation which has the form of a square grid [96]. The period and threshold voltage of this ®eld-induced instability depend on the elastic constants, the dielectric anisotropy, and the sample thickness [97]. More frequently, the electrohydrodynamic instabilities were studied in CLCs with negative dielectric anisotropy, De < 0. The instability is caused by the torque induced by electrical conductivity acting against the elastic torque of the CLC. Similarly, instabilities are observed in the dielectric regime occurring at high frequencies. The threshold voltage increases with increasing frequency of the applied electric ®eld and the critical frequency is directly proportional to the conductivity [97], [98]. At higher voltages, turbulence sets in and a dynamic scattering e¨ect is observed. On switching o¨, the liquid crystal relaxes to the focal conic texture (storage mode) [42]. Theoretical and experimental results about the appearance of ¯exoelectric domains are given in [99]±[102]. The investigation of instabilities is often used for the determination of liquid crystal parameters, mostly the elastic constants. For more detailed discussions about the dielectric, hydrodynamic, and ¯exoelectric instabilities and domain structures in cholesterics, see [103], [104].

6.3

Light-Induced E¨ects in Cholesteric Liquid Crystals

Light-induced orientational nonlinearities and related e¨ects were extensively studied during the last two decades. The results have been reviewed in [105]±[108]. Several parameters of liquid crystals can change due to the in¯uence of the light ®eld. At low light intensity, the e¨ect could be due to the following processes: (1) (2) (3) (4)

change of the molecular conformation; change of the molecular interaction; local recrystalization; or photo-induced charge formation.

The nonlinear optical e¨ects, observed at high intensity of the light, are investigated for application aspects and are used in conventional nonlinear optics methods and schemes. Among them are: self-focusing, degenerate

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four-wave mixing, optical ®eld-induced birefringence, optical bistability, and switching. Light-induced changes of the complex birefringence, i.e., changes in the refractive index or absorption for a given polarization, can be due to Kerr-like nonlinearities. Due to the helical arrangement of the director, CLCs show a helically distributed feedback and thus provide numerous possibilities for the investigation of nonlinear optical e¨ects. Photo-induced e¨ects were observed:

. . . . . .

in absorbing or nonabsorbing materials; in strong or weak ®elds; in connection with a director reorientation or without director reorientation; in setups with or without mirror; under resonance or nonresonant conditions; and due to Kerr-like or non-Kerr-like e¨ects.

Some of these e¨ects are considered in the following sections.

6.3.1

Photo-Stimulated Shift of Pitch in Absorbing Cholesteric Liquid Crystals

Among absorbing liquid crystals, most of the studied systems consist of conformationally active molecules which are capable of trans±cis (E±Z) isomerization. Typically, in this case, an elongated rod-like molecule (trans isomer) transforms under the in¯uence of ultraviolet (UV) radiation into a bent or fractured form (cis isomer). In CLCs, this change of molecules can cause a change of pitch, and thus a shift of the wavelength of selective re¯ection. Consequently, a reversible color shift was observed in conformationally active dye-doped CLCs [109]. As a dye dopant both the cis and trans azobenzene were used. A trans±cis conversion can be accomplished by irradiation with the wavelength l ˆ 313 nm and cis±trans conversion with l ˆ 420 nm. Mixtures of cholesteryl chloride (CC) and cholesteryl nonanoate (CN) were used as a cholesteric solvent. The components do not absorb above 270 nm and photochemical decomposition can be ruled out. Two mixtures were studied: in experiments with pure trans azobenzene the weight ratio CC:CN was 25:75 (dP=dT < 0, according to [110]), and with pure cis azobenzene the weight ratio CC:CN was 35:65 …dP=dT > 0†. The concentration was chosen so that the change of P due to irradiation exhibited the opposite sign of dP=dT, in order to exclude the in¯uence of heating. The dependence of lB on the concentration of pure cis and trans azobenzene was studied. A rather small pitch p was observed for trans azobenzene, and a larger pitch for cis azobenzene. Consequently, the wavelength of selective re¯ection lB is shifted to larger values if a trans azobenzene-doped mixture is irradiated with light of 313 nm, whereas lB is shifted to smaller values if a cis azobenzene-doped mixture is irradiated with light of 420 nm. Experiments show that the shift can reach 60 nm. Similar

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values of the light-induced shift of lB were observed for cholesteric systems doped with another conformationally active dye [111]. A color shift can also be observed if the cholesteric compound itself is conformationally active [112], or if a photosensitive chiral compound is used as chiral dopant in an induced cholesteric mixture [113], [114], for example, a reversible red-shift of lB due to the photo-transformation of an optically active dopant was induced by an He±Cd laser beam (l ˆ 0:44 mm). The latter e¨ect can be attributed to a light-induced change of the helical twisting power [3] of the chiral dopant. The same e¨ect was observed in induced cholesteric polymers, too [115]. A photochemically induced cholesteric± nematic phase transition can occur in induced cholesteric systems [116] if the chiral dopant shows racemization under the in¯uence of UV radiation. For image formation due to UV irradiation, CLCs can be applied which show a transition to the isotropic phase [117] or an irreversible photo±chemical reaction connected with a change of pitch [118]. The high sensitivity of the pitch to light irradiation is very promising for optical data-storage applications.

6.3.2

Generation of Higher Harmonics in Cholesteric Liquid Crystals

The generation of higher harmonics is of particular interest in the ®eld of nonlinear optics. The nonlinear optical susceptibilities are due to the hyperpolarizabilities of the individual molecules and depend strongly on the molecular arrangement in the respective liquid crystalline phase. CLCs are not centrosymmetric. Thus, they can be expected to show to second harmonic generation. Experimental investigations of laser-induced second harmonic generation in CLCs are described in [119]. However, in [120] it was mentioned that this e¨ect might be connected to the presence of unmelted crystals in quasi-equilibrium with the cholesteric phase. There is no evidence of any discernible second-harmonic signal, suggesting that the molecular arrangement in these materials has an overall inversion symmetry. Third-harmonic generation in liquid crystals is clearly not forbidden by symmetry. In [121], independent con®rmation of the second-harmonic results of [120] and ®rst measurements of third-harmonic generation are reported. Large changes of the third-harmonic intensity occur at phase transitions. However, the attempt to achieve phase matching of third-harmonic generation in CLCs has not been successful in this ®rst study. The observation of third-harmonic generation was also realized in [122]. A mode-locked Nd:glass laser was used as the fundamental pump source. A typical pulse train lasted about 200 ns with the individual pulses separated by about 7 ns. The total energy in each train was about 0.03 J and the pulse width of the individual pulses was about 7 ps. In this work, it was shown that for light propagating along the helical axis of a CLC, third-harmonic generation is possible for 15 di¨erent phase-matching conditions. The following cases were considered:

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(1) both the fundamental and the phase-matched third-harmonic waves propagate in the same direction; or (2) the phase-matched third-harmonic wave is generated in a direction opposite to the fundamental wave; or (3) the third-harmonic generation occurs when the fundamental waves are propagating simultaneously in both the forward and backward directions. For a mixture of cholesteryl chloride and cholesteryl myristate (1.75 to 1 by weight), the satisfaction of the phase-matching condition 3k…o† ˆ k…3o† was predicted [123]. This mixture shows a helix inversion. Since the pitch of the sample can vary from left- to right-handedness, there should be two phase-matching peaks. Experimental results of phase-matched thirdharmonic generation are in good agreement with the theoretical calculations. The dependence of the nonlinear conversion e½ciency on the sample thickness D, and the absolute value of the third-order nonlinear susceptibility were investigated in [124]. The temperature-dependence of the pitch was used to achieve the phase-matching conditions for the third-harmonic generation. A drop in e½ciency of the third-harmonic generation was observed for cells with D > 10 mm, because of the inhomogenity of the pitch. Nonlinear-optical frequency conversion under conditions of selective re¯ection of the generated radiation was investigated theoretically in [125]. Third-harmonic generation was considered both for light propagating along the pitch axis and oblique to the pitch axis. For oblique incident light, the polarization characteristics of the generated waves are more complicated, owing to the more complicated linear optical properties of the CLC in this case. For both cases of harmonic generation, the e½ciency of nonlinear frequency conversion can increase strongly and reach a maximum at the boundary of the selective re¯ection region. According to [125], the phasematching conditions can be ful®lled for suitable CLCs and suitable pump conditions.

6.3.3

Theoretical Aspects of Pitch Dilation, Orientational Nonlinearities, and Optical Bistability in Cholesterics

The in¯uence of circularly polarized light with high intensity on the pitch of CLCs was theoretically studied in [126]. The wavelength of the light was assumed to be far from the Bragg condition. It was found that the pitch of the CLC increases under the in¯uence of such waves. Dilation of the pitch for light with a wavelength coinciding with the Bragg wavelength was considered in [127]. In this case, the light-induced change of the pitch leads to a mirrorless bistability. The physical principle of this bistability is similar to the predicted e¨ect for distributed-feedback structures and in the degenerate four-wave mixing process for an intensity-dependent refractive index medium [128], [129]. Like a static electric ®eld, the time-averaged optical ®eld couples

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to the local dielectric anisotropy and exerts torques within the CLC which may compete with the elastic torques determining its internal structure. The resulting increase of the pitch causes a change of the transmission at higher intensities. When the intensity is reduced, the appearance of the re¯ection occurs for smaller intensities than the disappearance for increasing power of light. Thus, optical bistability takes place. This bistability is a result of the light-induced pitch dilation and occurs even in the absence of external re¯ectors. The calculated critical intensity for the incident circularly polarized light is given by jEin j 2 ˆ …o=c† 2 …ea =eo e†K22 ;

…6:7†

where ea ˆ ee eo . Using typical values, ea =e ˆ 0:1, K22 ˆ 10 12 N, and an incident wavelength of 1 mm, a critical value of intensity of 1 MW/cm 2 was estimated. The in¯uence of the boundary conditions on the pitch change e¨ect in a CLC cell under the in¯uence of an intense optical ®eld was theoretically studied in [130]. Three cells with di¨erent anchoring conditions were considered: (1) Strong anchoring at the input side and weak anchoring at the output side. This case coincides with the situation considered in [127]. The pitch dilation is proportional to the local ®eld intensity in this case. (2) Strong anchoring conditions at both surfaces. A pitch dilation at the input side and a pitch contraction at the output side can be observed. As a result, the slope of the phase lag of the re¯ected ®eld from the CLC is slower than in a CLC in condition (1), and the re¯ectivity is nearly constant over a wide range of input intensities. (3) Weak anchoring at the input side and strong anchoring at the output side. These conditions create pitch contraction. Light-induced orientational nonlinearities in CLCs were theoretically studied in [131]. The most strong and rapid e¨ect is the director reorientation inside the unchanged periodic structure of the cholesteric. The mechanism of nonlinearity is connected with the spatially inhomogeneous reorientation of the director. The nonlinearity leads to self-focusing, self-rotation of the axis of elliptical polarization, and self-birefringence. The numerical estimations show that a light intensity of A 10 6 W/cm 2 is necessary to cause the predicted e¨ects. A qualitatively new approach, di¨erent from previous considerations [126], [131], was proposed in [132]. A nonlocal dependence of the helical pitch on the intensity of the incident light was assumed, and the re¯ection of ®elds from the cholesteric helix was taken into account. In [133], devoted to the orientational optical nonlinearity of liquid crystals, some additional e¨ects observed in cholesterics are considered: the giant optical bistabilty, grating orientational nonlinearity, pitch change due to the direct action of light; and to the thermal e¨ects, hysteresis, and optical bistability and nonlinearities connected with absorption. Here we cite some

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remarkable data from [133]. The giant optical nonlinearity in CLCs with at least one orienting surface should not occur. Due to heating problems in CLCs, an incident light power density of @ 10 3 W/cm 2 leads to a temperature increase of @ 2  C. An optical bistability device, utilizing the thermally induced change of the pitch, and other hysteresis e¨ects are theoretically discussed. Tabirian et al. [133] conclude that the investigation of orientational optical nonlinearities is only beginning. In this subsection, we considered some theoretical problems concerning light-induced nonlinearities in CLCs. In the next subsection we will consider experimental results observed due to this nonlinearity. Of course, some of the publications quoted in the following section include theoretical calculations, as well.

6.3.4

Generation of Tunable Radiation, Optical Bistability and Optical Switching, Pitch Dilation, and Other Photo-Optical Nonlinear E¨ects Realized in Cholesterics

The selective properties of CLCs and the ability to change the pitch make it possible to construct frequency tunable lasers. The idea of building a laser with distributed feedback using CLCs was expressed in [134], but without any calculations or estimations. The theoretical analysis for a distributedfeedback laser generation in a dye-doped CLC [135] leads to the conclusion that a pump power of A 10 5 W/cm 2 is required. A generation of light in dyedoped cholesteryl chloride was obtained for the ®rst time in the work described in [136], [137]. A three-component mixture (consisting of cholesteryl chloride, cholesteryl oleate, and cholesteryl pelargonate) with a positive temperature-dependence dP=dT > 0 was used. A benzanthrone derivative was used as a dye. Harmonic distortion of the induced cholesteric structure, detected by the distributed feedback laser, was also observed and studied [138]. Photo-induced periodic distortions of the director ®eld, due to the in¯uence of a laser beam on a CLC, were found and investigated, as well [139], [140]. A cw argon±krypton ion laser with the following wavelengths and maximum powers was used: 647 nm (A 160 mW), 515 nm (500 mW), and 488 nm (400 mW). Two types of CLC were investigated: mixtures of an absorbing nematic liquid crystal (azoxybenzene compounds), liquid crystals with cholesteryl caprate, and mixtures of a nonabsorbing nematic liquid crystal (5CB) with cholesteryl caprate. In both cases, photo-induced gratings were observed, similar to the electric ®eld-induced instabilities in CLCs [96], [97]. The appearance of the square-shaped periodic distortion of the director ®eld was found to be a power threshold phenomenon. The investigations indicate that the distortions are due to conformational transformations of the molecules in the case of the absorbing CLCs, and due to inhomogeneous heating in the case the nonabsorbing CLCs.

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In [141], a simple probe-pump laser technique was proposed to investigate the change of lB in CLCs under the in¯uence of intense laser radiation. An argon laser beam (l ˆ 514 nm), incident normal to the liquid crystal cell, was used as a pump beam, and an oblique incident laser beam from a laser diode (l ˆ 670 nm) as a probe beam. A nonabsorbing isothiocyanate mixture with lB ˆ 510 nm was investigated. An increase of the pitch at very low laser intensities (100±700 mW) was observed which depends on the polarization of the beam. An energy coupling between counterpropagating laser beams in CLCs, due to orientational nonlinearities, was theoretically considered and experimentally realized [142]. This interaction occurred in a case when the wavelength of the laser was close to the selective re¯ection wavelength of the CLCs. The pitch of an induced cholesteric mixture (consisting of the nematic liquid crystal 5CB and cholesteryl pelargonate) was varied by changing the concentration of the chiral dopant. With the irradiation of a ruby laser (800 ms duration) at intensities of 10±100 kW/cm 2 , an orientational backward stimulated scattering was observed. As predicted in [132], self-focusing and nonlinear optical activity of a CLC occurs only in the case of elliptic polarization. But the characteristics of the observed nonlinearities indicate a nonorientational mechanism of the observed e¨ects. A thermal mechanism is responsible for the nonlinearity in this case. It was concluded that the interaction of concurrent waves, and also self-focusing and nonlinear optical activity due to the orientational mechanism of the nonlinearity of CLCs, require high radiation intensities which exceed the damage threshold of the samples. Attempts to observe laser-induced nonlinear re¯ection in nonabsorbing CLCs were also made [143]. Cholesteric mixtures consisting of 2-methylbutyl-p-[( p-methoxy-benzilidine)amino] cinnamate and cholesteric oleyl carbonate with lB ˆ 532 nm, and mixtures of cholesteryl chloride and cholesteryl nonanoate with lB ˆ 591 nm were used. A pulsed, high-power Nd:YAG laser was applied. The series of experiments at power densities up to 4 MW/cm 2 (532 nm) and 8 MW/cm 2 (1064 nm) did not show any optically induced pitch dilation or distortion of the CLC cell. It was concluded that short laser pulses (15 ns) are not suitable to observe the light-®eldinduced pitch dilation predicted in [127]. Laser-induced gratings have been produced by mixing two laser beams at a small angle in dye-doped CLCs [144]. The sample (a mixture of n-( pmethoxy benzilidene)-p 0 -butylaniline and cholesteryl oleyl carbonate with lB ˆ 440 nm) was irradiated by two coherent argon ion laser beams. The beams were polarized in the direction parallel and perpendicular to the helix axis. A transient grating was observed at intensities of 50 mW per beam. The decay time of the transient grating was of the order of 1 second. At higher laser power, a ``persistent'' grating with a lifetime of several hours was formed. Unfortunately, the mechanism of formation and characteristic features of the gratings were not explained.

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The dynamics of the change of the pitch in dye-doped CLCs under the in¯uence of laser radiation was considered in [145]. Three di¨erent compositions of mixtures of cholesteryl chloride, cholesteryl pelargonate, and cholesteryl oleyl carbonate with lB ˆ 650, 570, and 486 nm were used (the third mixture also contained the nematic compound 4 0 -hexyl-4-cyanobiphenyl). The mixtures have di¨erent signs of dP=dT. A copper complex with an absorbing wavelength of 355 nm was used as a dye, and the third harmonic of an Nd:YAG laser (355 nm) with 11 ns pulses was used as a light source. The shift of the selective re¯ection wavelength was measured by means of a kinetic absorption spectrophotometer. An analyzing beam was incident normal to the cell and the laser pump beam was incident at an angle of A 20 with respect to the probe beam. Three di¨erent optical responses are observed. The fastest (U200 ns) process is an increase of the pitch …lB †. This nanosecond time phenomenon was explained by the formation of a transient grating caused by the modulation of the refractive index associated with the heat instantaneously (U10 ps) released by the dye molecules to their immediate neighborhood. The second process (30±600 ms) is a change of pitch according to its temperature dependence, i.e., a thermal process. The last (>1000 ms) process is the return of the pitch to its initial value after the heat is dissipated. Optically induced detuning of the selective re¯ectivity band in absorbing CLCs has been demonstrated for applications like mirrorless optical bistability and optical switching. Optical bistability in a dye-doped CLC with a distributed feedback was for the ®rst time observed by two groups independently and simultaneously [146], [147]. An important condition for achieving optical hysteresis is the presence of a feedback. In a periodic cholesteric structure, the feedback is due to the Bragg di¨raction of light. An optical hysteresis has been achieved in dye-doped CLCs at comparatively low-light intensities [146]. A pulsed nitrogen laser with a wavelength of 337.1 nm was used in [146] and optical hysteresis has been achieved in dye-doped CLCs at light intensities as low as A 10 kW/cm 2 with pulse lengths of G 8 ns. In [147], a circularly polarized He±Ne laser (633 nm) was used as a pump and a dyedoped mixture of the nematic liquid crystal 5CB with cholesteryl pelargonate and cholesteryl oleate was investigated. The hysteresis was observed at low intensities, A 10 W/cm 2 . Transient grating optical nonlinearities in dye-doped CLCs are investigated with laser-induced dynamic grating experiments using nanosecond and picosecond laser excitation pulses [148]. The chiral nematic liquid crystal CB15 was used, which was doped by 4% of a mixture of cetocyanide dyes. The Bragg wavelength of the CLC was lB ˆ 570 nm and the absorption band of the dye was at l ˆ 550 nm. In self-di¨raction experiments, laserinduced dynamic gratings have been excited by the second harmonic of Nd:YAG laser pulses at lexc ˆ 532 nm which were obtained in an asymmetric beam-splitting arrangement. Strong self-di¨raction has been observed for laser pulses of 15 ns duration and intensities of 1.1 GW/cm 2 . A further

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increase in the laser intensity results in permanent gratings. The CLC is transformed from a planar texture to the confocal texture. The latter state is stable so that an optical storage e¨ect occurs. The dynamic response of dye-doped CLCs with an absorption maximum near the selective re¯ection wavelength shows fast subnanosecond switching times. The dynamics of the grating formation was studied by using green excitation pulses of 80 ps duration, and the transient grating was probed with a continuous wave (cw) argon ion laser at a wavelength of lp ˆ 488 nm. Di¨erent response times were observed. A fast contribution to the dynamic gratings has been explained by resonant optical nonlinearity due to absorption saturation of the dye. A secondary process connected with thermal gratings has been observed as a result of radiationless recombination of the excited dye molecules. It was shown that the thermal grating gives rise to density modulations and elastic deformations of the liquid crystal host which change the e¨ective refractive index of the CLC and lead to the observed di¨raction e¨ects. The slower dynamics of the grating is explained by the relaxation of thermal gratings, density, and elastic deformations, which exhibit characteristic time constants from A 100 ns to hundreds of ms. According to a review on Kerr-like optical nonlinearities in liquid crystal [149], these experiments show that the photonic response may be much faster if short intense laser pulses are used instead of low-power cw laser radiation. This may open new aspects and perspectives in research and applications. In [150], the necessary conditions for the observation of resonance nonlinearities in dye-doped CLCs are summarized: (i) The medium should have a periodic helical structure and be thick enough to show a selective Bragg re¯ection that provides a distributed feedback. (ii) The dopant molecules should absorb light resonantly in order to provide a nonlinear response of the matter to the incident radiation. A threshold light intensity causes a change to the refractive index, which is su½cient to produce a certain phase delay and frustrate the Bragg condition. (iii) The resonance frequency of the dopant should be within the selective re¯ection band of the matrix. This enables both nonlinear response and feedback in the medium. Mirrorless optical nonlinear e¨ects (optical bistability, optical switching, and transient grating nonlinearity) as mentioned above, were observed only in dye-doped CLCs. In nonabsorbing CLCs, the realization of the lightinduced pitch dilation became possible when it was used in optical schemes as a mirror. In [151] it was shown theoretically that under intense optical radiation with a Gaussian pro®le, a retro-self-focusing e¨ect and a pinholing e¨ect occur in CLCs. These e¨ects have been observed in an experiment using a CLC dielectric resonator. The CLC dielectric resonator consists of an HR

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dielectric end mirror, a cw-operated Nd:YAG laser, a quarter-wave plate and a CLC end mirror. When the CLC element is used as a laser end mirror, it acts like a well-aligned concave mirror±pinhole combination. The retroself-focusing e¨ects are a result of pitch dilation due to nonlinear coupling of the intense optical ®eld with the CLC structure. More detailed experimental investigation of the unique properties of a CLC as a laser end mirror were done in [152]. It was shown that a CLC mirror can be used as a laser end mirror in a solid state laser oscillator to provide both TEM00 -mode operation for cw powers in excess of 1 W. No pinhole is required to obtain TEM00 -mode operation. CLCs have been investigated as a polarizing end mirror in an optical resonator for high-power pulsed solid state lasers [153]. In a resonator for an electro-optic Q-switched Nd:YAG laser (l ˆ 1064 nm), a CLC cell with a mixture nematic liquid crystal ZLI-2359 and the chiral compound CB15 was used instead of a dielectric mirror and a polarizer. In combination with an active Q-switched laser, pulses of 10 ns duration with peak intensities up to 1 GW/cm 2 have been realized. Compared with results obtained for a conventional resonator scheme, the pulses from the laser with a CLC mirror show a somewhat weaker modulation and an asymmetry, which can be explained by a decrease in re¯ection for strong optical ®elds during the pulse. Additional measurements show that the re¯ectivity of the CLC mirrors for circularly polarized light decreases from 90% at low intensities down to 25% for intensities of A 100 MW/cm 2 . A change of the re¯ectivity occurs in connection with the unwinding of the helix in high intense optical ®elds, as predicted in [127]. The pinholing e¨ect was also experimentally observed. A change in the re¯ectivity of nonabsorbing CLC mirrors under the in¯uence of pulsed laser irradiation was also observed in [154]. A mixture of the nematic liquid crystal E7 with the chiral compound CB15 and with lB ˆ 1064 nm was used. Cells were prepared with strong anchoring on one side and weak anchoring on the other side. CLC samples irradiated by a Nd:YAG laser operated in two regimes: (1) in a cw mode; and (2) in an acousto-optically Q-switched mode, emitting 500 ns pulses with a frequency of 4.5 kHz and intensities of 10 6 ±10 7 W/cm 2 . The re¯ectivity change was observed only under pulse irradiation and only under re¯ection of the incident light from the strong-anchoring side of the cell. The CLC mirror transmittance changed between 5% and 30±80%. The observed e¨ects were interpreted within the frame of the pitch-dilation model, predicted in [127]. The results presented in this subsection show that almost all predicted nonlinear e¨ects were observed and studied in CLCs. The results are very promising for applications in information techniques and photonics. However, many physical mechanisms, especially concerning processes at the molecular level, are still to be solved. Another challenge is the light-induced pitch unwinding in nonabsorbing mirrorless CLC structures which is not clari®ed, yet.

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In conclusion, CLCs are an important electro-optical and opto-optical material owing to the possibility of changing their unique optical properties easily in external ®elds. Due to their versatile behavior, many further interesting results can be expected in the future.

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