Helical wave vector

We may also define the helical wave vector k-lTzIp, which then appears in the twist term of the free energy density equation for an N phase, i.e.,... 

Figure 37. An electric field E applied perpendicular to the helix axis of a cholesteric will turn the director an angle ) and thereby the optic axis by the same amount. The director tilt is coupled to the periodic splay—bend director pattern shown below, which is generated in all cnts perpendicular to the new optic axis. In this inverse flexoelectric effect, splay and bend will cooperate if and have the same sign. The relation between E and 0 is shown for a positive helical wave vector k (right-handed helix) and a positive average flexoelectric coefficient e= (Cs+ b)-When the sign ofE is reversed, the optic axis tilts in the opposite direction (0—>-0).

We start by constructing an orthonormal basis a, b, c (where c is a unit vector along the wavevector k, with a and b in the two-dimensional vector space orthogonal to k). The significance of this ansatz is that any vector function F(x,y,z) is divergence free if and only if its Fourier coefficients F(k) are orthogonal to k, that is if k -F(k) =0. Thus, F(k) is a linear combination of a(k) and b(k). Lesieur defines the complex helical waves as... 

Using an LG03 doughnut the theoretical prediction is that the frequencies should scale as 2 3 4 when going fi-om opposite helicity Poynting vector and spin to linearly polarised helical wave fi ont through to the same helicities. This was verified experimentally (see Figure 5). 

Fig. 11. Fourier transform exchange J q) for the c-axis direction in Dy in both ferromagnetic (78 K) and helical (98 K) temperature regions. The peak near q = 0.2 corresponds to the peak in the generalized susceptibility x(9) and is a signature of the incommensurate helical 4f spin ordering occurring with the same wave vector. Note that the peak remains, even below consistent with a transition driven not by the exchange but by the magnetostriction. (After Nicklow 1971.)...

Here 0 denotes the angle between the wave vector and the helical axis n = [uo + nc)/2. (Note that the reflection wavelength corresponds to a periodicity of p/2 rather than p this is because a rotation of the director by tt does... 

Note that the waves can propagate along the helical axis in either direction, and, correspondingly, the wave vectors k can be positive or negative. The generated third-harmonic intensity is given by Eq. (7), which is of course independent of the coordinate system we choose. If the beam has a finite cross section, then the total third-harmonic power is ... 

The admixture of the chiral compound to a conventional nematic induces a macroscopic helical structure whose wave vector, that is, inversed pitch qo = 27t/Po depends on concentration (c), in general, nonmonotonically. Only in the case of the ideal mixture, without any short-range intermolec-ular interactions, we have the linear relationship... 

The incorporation of a chiral dopant into an achiral smectic C matrix results in the helical twisting of the structure of the mixture. In the simplest case, the wave vector of the helix is proportional to the concentration of a dopant. 

Taking only the bilinear P-0 coupling into accoimt, i.e., g = ge + Qpi + 9q + gpf, one obtains for the helical pitch p (resp. the corresponding wave vector... 

In Section 9.2.1 of this Chapter we discussed field-induced changes in the microstructure of liquid crystals. However, field-induced unwinding of the cholesteric (macroscopic) helix (see Section 9.3.2.3 of this Chapter) shows that the transition from a twisted to a uniform nematic may also be considered as a phase transition. In the latter case the field energy term competes with a rather small elastic energy proportional to nematic-like elastic moduli and the squared wave vector of the helical superstructure Kq As the pitch of the helix p = lKlq is large, the field threshold for the transition is very low. On the other hand, between the two extreme cases (a microstructure with a molecular characteristic dimension and a... 

We have, so far, considered the non-helical C case. With the helix present we would have found the transition C A pushed even slightly further upwards in temperature because now, in addition, we have to bring up energy to unwind the helix at the transition. We would then have found another contribution to AT being proportional to q, where q is the value of the helical smectic C wave vector at the transition. In any case, AT still turns out to be quite small (like all chiral perturbations), in practice often less than a degree. 

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