Bragg reflection, cholesteric

Cholesterics are characterised by a "single twist", which is characterised by a relative rotation of flat layers of molecules. (These layers are illusory, and useful for illustrative purposes only, since there is no evidence of lamellar ordering perpendicular to the layers.) The optical novelties of these cholesteric phases are due to the pseudo-Bragg reflections from the helical... 

The natural pitch P of cholesteric liquid crystals is in general in the order of 102 nm, comparable with the visible band of light. The Bragg reflection from cholesteric liquid crystal occurs at the wavelength... 

In the helical structure, the optical ellipsoid of the smectic C phase rotates together with the tilt plane. Like in cholesterics, we can imagine that helical turns form a stuck of equidistant quasi-layers that results in optical Bragg reflections in the visible range. Therefore, like cholesterics, smectic C liquid crystals are onedimensional photonic crystals. However, in the case of SmC, the distance between the reflecting layers is equal to the full pitch Pq and not to the half-pitch as in cholesterics, because at each half-pitch the molecules in the SmC are tilted in opposite directions. Hence, we have a situation physically different from that in cholesterics. 

A cholesteric forms a helical structure and its optical properties are characterised by the tensor of dielectric permittivity rotating in space. We are already familiar with the form of the cholesteric tensor (see Section 4.7). It was Oseen [1] who suggested the first quantitative model of the helical cholesteric phase as a periodic medium with local anisotropy and very specific optical properties. First we shall discuss more carefully the Bragg reflection from the so-called cholesteric planes . 

Fig. 12.5 Comparison of the non-polarized light transmission by a stack of dielectric layers and a cholesteric liquid crystal (CLC). The two materials have the same Bragg reflection frequency (numerical calculations, for parameters see the text), (a) Transmission spectra on the frequency scale showing the absence of high harmonics in the case of CLC (b) blown transmission spectra at the wavelength scale showing the flat form of the CLC Bragg band and oscillations of transmission at the edges of the band...

Figure 12.14 shows the calculated transmission spectra of a cholesteric mixture in zero field and at E = 5.7 V/pm. In this case, the pitch is 0.4 pm and the cell thickness d=10 pm. The incident light is circularly polarised. Upon application of the field, a strong second Bragg reflection band emerges. The transmission is almost... 

D. W. Berreman and T. J. Scheffer, Bragg reflection of light from single-domain cholesteric liquid crystal films, Phys. Rev. Lett., 25, 577 (1970). 

The above solutions describe the characteristics of linear wave propagation in a cholesteric liquid crystal along the helical axis. In particular, if X - 11 < la I, then m. is purely imaginary, and the corresponding wave should be totally reflected. We realize that x 1 or X = 2 nc/(j)t =p (la 1 1 usually) is just the condition for Bragg reflection from the helical structure. If we transform Eq. (5) back into the lab frame, we have... 

Thus, we are now interested in the case of Bragg reflection, or more accurately, we want to establish why only one diffraction maximum of selective reflection (Fig. 6.4(b)) is observed, in experiments with the normal incidence of light, on the planar texture of the cholesteric liquid crystal,... 

Selective Bragg Reflection from a Cholesteric Liquid Crystal... 

Bragg reflections of light from these cholesteric samples are obviously due to the periodic structure of the cholesteric phase, and the wavelength of the reflected light is determined by the pitch p of the helix. Using circular... 

Figure 2.16. Samples for demonstration of the Bragg reflection from cholesteric liquid crystal.

In this expression, a is a factor proportional to the scattered light intensity, Sij q) is the amplitude of the Fourier component with wavevector q of the dielectric tensor fluctuation, while vectors f and i are polarizations of, respectively, reflected and incident light waves. In the context of Bragg reflection from the cholesteric helix, we know already from the expression (2.25) that there is just one Fourier component with wavevector 2q. Its amplitude is complex because the second term in the expression (2.25) can be written as... 

Cholesteric liquid crystals (CLCs) show very distinctly that molecular structure and external fields have a profound effect 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 reflection occurs. On the other hand, the helix pitch is very sensitive to the influence of external conditions. A combination of these properties leads to the unique optical properties of cholesterics which are of both scientific and practical interest. 

This case, analyzed by Fergason [119] and others [116], gives rise to irridescent colors due to Bragg reflection. We also see the reflection of this band in the case of an absorbing mode cholesteric (Fig. 15). However, one component of the polarized light always gets absorbed. 

Cholesteric liquid crystals, e.g., those of cholesteroylnonaoate (see Sec. 3.2), produce a Bragg-type scattering, which depends on temperature and angles of incidence and observation. Either total reflection or total transmission of circular polarized light is observed, which effect provides the basis of the dark-bright liquid crystal display in the Schadt-Helfrich cell (Fig. 3.5.3) as well as color reflection. 

Now we would like to understand why only one diffraction maximum is observed in the normal reflection from the cholesteric helix and why the reflected light is circularly polarized. Therefore, at first, we write the Bragg condition on account of possible higher diffraction orders ... 

Fig. 12.3 The geometry for discussion of the Bragg diffraction in a cholesteric (a) and illustration of the wavevector conservation law (b). k,- and are wavevectors of the Incident and reflected beams, qo is the helix wavevector...

The reflection from cholesteric and blue phases is Bragg-type scattering, similar to the diffraction of X-rays by crystals. The wave vector of the incident light Ko, the wave vector of the scattered light Kg, and the wave vector of the dielectric constant component q must satisfy the Bragg condition ... 

Fig. 1. Schematic representation of (a) nematic, (b) smectic and (c) cholesteric (or chiral nematic) liquid crystalline phases. In the nematic phase only orientational correlations are present with a mean alignment in the direction of the director n. In the smectic phase there are additional layer-like correlations between the molecules in planes perpendicular to the director. The planes, drawn as broken lines, are in reality due to density variations in the direction of the director. The interplane separation then corresponds to the period of these density waves. In the cholesteric phase the molecules lie in planes (defined by broken lines) twisted with respect to each other. Since the molecules in one plane exhibit nematic-like order with a mean alignment defined by the director n, the director traces out a right- or left-handed helix on translation through the cholesteric medium in a direction perpendicular to the planes. When the period of this helix is of the order of the wavelength of light, the cholesteric phase exhibits bright Bragg-like reflections. In these illustrations the space between the molecules (drawn as ellipsoids for simplicity) will be filled with the alkyl chains, etc., to give a fairly high packing...

---