LIGHT PROPAGATION ALONG THE HELICAL AXIS IN CHIRAL SMECTICS C

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LIGHT PROPAGATION ALONG THE HELICAL AXIS IN CHIRAL SMECTICS C

O. Parodi

To cite this version:

O. Parodi. LIGHT PROPAGATION ALONG THE HELICAL AXIS IN CHIRAL SMECTICS C.

Journal de Physique Colloques, 1975, 36 (C1), pp.C1-325-C1-326. �10.1051/jphyscol:1975154�. �jpa-

00216233�

JOURNAL DE PHYSIQUE Colloque C1, suppl&meni au no 3, Tome 36, Mars 1975, page C1-325

Classification

Physics Abstracts 7.130

LIGHT PROPAGATION ALONG THE HELICAL AXIS IN CHZRAL SMECTICS C

0. PARODI

Laboratoire de Physique des Solides ("), Universiti Paris-Sud, Centre d'Orsay, 91405 Orsay, France

and

Laboratoire de Cristallographie-Miniralogie, UniversitC des Sciences et Techniques du Languedoc, place E.-Bataillon, 34060 Montpellier Cedex, France

RCsumk. - On etudie la propagation de la lumi&re normale aux couches pour des smectiques C et des smectiques C chiraux. On montre que la thCorie de de Vries pour les cholestCriques peut 6tre appliquee aux smectiques C chiraux, a condition d'utiliser des indices de refraction effectifs, qui peuvent &tre mesurh dans la configuration non chirale.

Abstract. - Light propagation normal to the layers is discussed for smectics C and chiral smectics C. It is shown that de Vries theory for cholesterics can be applied to chiral smectics C if one uses, as refractive indices, effective indices which can be measured in the non-chiral configuration.

1. Introduction. - Light propagation along the helical axis in a cholesteric liquid crystal has been discussed by Mauguin and several others [I-31, the most illuminating discussion being that of de Vries [2].

A chiral smectic C has, as a cholesteric, a helical symmetry and the question arises as to know if de Vries' theory can be applied.

The first answer to this question has been given by Berreman [4] who has derived the components of the dielectric tensor and shown that for normal incidence its fundamental periodicity was p / 2 wherep is the pitch.

From this he deduced the existence of a Bragg reflection quite similar to the one found in cholesterics. However he did not show that all of de Vries' analysis could be applied to this case.

A second question is the choice of the refractive indices. In a cholesteric, the dielectric tensor is taken into account through the refractive indices, no and n,.

In the case where the cholesteric is obtained by adding a small amount of chiral material to a nematic, one generally uses the experimental value of no and n, measured in the pure nematic phase. In the same way, if a chiral smectic C is obtained by adding a small amount of chiral material to a smectic C, the natural choice of the refractive indices no and n, would be taken as the vaIues measured for propagation normal to the layer in the pure smectic C material.

In this paper a theoretical justification for such a procedure is given. In part 2, propagation of light nor- mal to the layers is discussed for a non-chiral smectic C.

In part 3, a dispersion equation for a chiral smectic C is derived and it is shown that this dispersion equation is quite similar to the one obtained by de Vries for cholesterics.

2. Propagation of light normal to the layers in a smectic C. - A smectic C liquid crystal is a periodic medium. However, as its period is of the order of a molecular length and much smaller than the optical wavelength, it can be considered as a homogeneous medium. With the z-axis taken as normal to the layers the electric field is described by

~ ( z , t ) = E

. ₍₁₎

Using Maxwell equations, one obtains

A

c2

D = EE = -- k E,,

o2 (2)

where E,, is the transverse component of E and : ^the

dielectric tensor.

Smectics C are, in general, biaxial [5]. Let 1, 2, 3 be the principal axes of the tensor (E, < E, < 8,).

The unit vector z of the 3-axis is parallel to the y-axis (Fig. 1) and the unit vector p of the 3-axis (which can be distinct from the director as defined by theory of elasticity [ 6 ] ) makes an angle 8 with the z-axis. Then :

can be written

n

where ^{c, = E} - E ~ , ^{E: = E~} - ^c1 ; 1 is the unit-tensor ; ,* + s ,

,u : ,u and z : z are dyadics.

(*) Laboratoire associe au C. N. R. S .

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1975154

0. PARODI

Eq. (3) is still valid but )r and z do not have an uniform orientation :

i ^px

^{p, = = = sin 0 sin COS sin 0 cos 0 . qo go z z} (3')

i ^{rx 7, Zz = = = cos 0 . - sin go go z z}

From this one obtains

E = - ^E, sin 0 cos e(E, cos go z + ^{E, sin go z)}

FIG. 1. - This illustrates the choice of axes. The planes P(xy) + ^{E, COS'} e

and Q(zx) are respectively parallel to the layers and to the tilt

plane. The axes 1, 2 and 3 are the principal axes of the dielectric and a dispersion

tensor. Due to theanisotropy of the orientation order parameter,

E I

c

and both are smaller than

(&I -

9

- &I). c2 a 2 ~ t r

o2 6z2 - (gL + '&(MI) Etr (4')

From D, = 0, one obtains where

( M ) = tos2

^{cos go z sin go z}

E, ⁼ - ^E. cos 8 sin 0 cos go z sin go z sin2 go z

E l + ^Ea c0s2 e ^Ex .

After some algebra one obtains the dispersion relation E, = - &,I + ^C, sin o / ( E ~ sin 0 + ^E, cos 0)

c2 1 It must now be pointed out that :

(> ^{k2 - (2) Etr = 0 (4)} i) eq. (4') is exactly the dispersion equation found by de Vries for cholesterics, and

where the two-by-two matrix (G2) is ii) that

Here 2 ^and are effective indices :

3. Chiral smectics C. - Here a new periodicity appears along the z-direction, due to the helical arrangement. As the pitch and the optical wave length are of the same order of magnitude, this periodicity has now to be taken into account. Therefore, the eigenvibrations are no longer the plane waves defined by eq. (I), but Bloch waves and, instead of eq. (I), one must write

E(z, t ) = E(z) e-'"' . _(1')

In the same way one gets, instead of eq. (2),

where E, and n", have been defined in section 2 for a non-chiral smectic C.

4. Conclusion. - All of de Vrics' analysis for cholesterics can be applied to chiral smectics C, with effective local refractive indices n", and Eo. Moreover, if the chiral smectic C is obtained by adding a small amount of chiral molecules to a non-chiral smectic C, n, and no can be directly measured on the non-chiral smectic C , with light propagating normal to the layers.

Such an experiment has been performed by M. Bru- net [7] with a good agreement between experimental results and theoretical predictions deduced from de Vries theory.

A perhaps better experimental procedure would be to use known mixtures of 1. h. and r. h. enantio- meres giving perfect smectic C phases. The effective refractive indices no and n, could then be directly measured on the racemic mixture.

References

111 MAUGUIN, C., BUN. SOC. Franc. MinPr. 34 (1911) 6, 71. [51 TAYLOR.

T. R., FERGASON, J. L. and ARORA, S. L., Phys. Rev.

121 DE VRIES, H. P., Acts Cryst. 4 (1951) 219. t e r t . 24 (1970) 3591

[3] See also

GENNES, P. G., The physics of Liquid Crysta[s [61 LEF"VRE, M.,

^{G' and}

(Oxford University Press) 1974. sr8, M., C. R. Hebd. Skan. Acad. Sci. 273B (1971) 403.

[7] BRUNET, M., Communication to the Vth International Liquid 141 BERREMAN, D. W., MoI. Cryst. Liqrc. Cryst. 22 (1973) Crystal Conference, Stockholm (1974), J. Physique

175. Colloq. 36 (1975) C 1-321.