LIGHT REFRACTION BY A CHOLESTERIC PRISM

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LIGHT REFRACTION BY A CHOLESTERIC PRISM

J. Martin, O. Parodi

To cite this version:

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Classification

Physics Abstracts

2.440

-

9.528

LIGHT REFRACTION

BY

A CHOLESTERIC PRISM

J. C. MARTIN

Laboratoire de Minkralogie-Cristallographie, Universitk des Sciences et Techniques du Languedoc, place Eug&ne-Bataillon, 34060 Montpellier Cedex, France

0. PARODI

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

RBsum6. - La propagation de la lumikre dans un milieu cholestkrique a kt6 CtudiQ par de Vries [ 2 ] . Ses principaux resultats sont que, pour une frkquence donnee, il existe deux vibrations propres dont la polarisation est quasi circulaire sauf dans deux cas : pour les hautes frequences, et au voisinage de la bande de reflexion pour la vibration qui subit la reflexion totale. Dans ces deux cas, la polarisation est quasi lineaire. En etudiant la lumikre refractke par un prisme cholestkrique, nous avons trouve a) qu'il y avait quatre rayons refractes dans le cas des frequences elevees et 6 ) que dans tous les cas la polarisation des faisceaux refractes etait circulaire. Nous etudions la refraction de la lumi6re par un dipotre oblique et nous montrons que pour toute frequence (a l'exception de la bande de reflexion), on obtient quatre rayons refractes polarisb circulairement (et aussi quatre rayons reflechis). Nous avons calcule les intensites relatives des faisceaux refract& et rkflechis et nous predisons une anomalie au voisinage de la bande de reflexion pour celle des vibrations qui ne subit pas la reflexion totale.

Abstract. - Light propagation in a cholesteric medium has been completely discussed by de Vries [2]. The main results are that, for a given frequency, there are two eigenvibrations with quasi-circular polarizations, except i) for high frequencies and ii) near the reflection band (for the vibration having total reflection), where the polarization is linear. Looking at light refracted by a cholesteric prism, we have found i) that there were four refracted beams for high frequencies and ii) that for any frequency the refracted beams had a circular polarization. Refraction of light by an oblique interface is discussed and we show that for any frequency - except in the reflection band

-

there are four circularly polarized refracted beams (and also four reflected beams). The relative intensities of refracted and reflected beams are given, and an anomaly is predicted near the reflection band for the vibration having not total reflection.

1. Introduction. - Propagation of light in a cholesteric medium is complex. The only simple situation is the one where light propagates along the helical axis. It has been first studied by Mauguin [I], who gave the fundamental equations and a discussion for the case of high pitches

(A

$ p), but the first complete discussion

was given by de Vries [2]. A new presentation of de Vries theory has been given by de Gennes [3] and, in this paper, we will use de Gennes approach and notations.

In de Vries theory the dielectric properties of the medium were taken into account by using two local refractive indices, no and n,. It is in an attempt to measure these local indices with a cholesteric prism that we have obtained the unexpected results which are presented here.

These results are very different from those that should be expected, at first sight, from de Vries theory. So it has seemed useful t o give, in section 2, the main

results of this theory. In section 3, we derive the refraction angles for an interface making a small angle with the cholesteric planes (defined as normal to the helical axis). In order to derive the intensities of the refracted beams, one also has to look at reflection. This is done in section 4. The complete calculation of reflected and refracted intensities is given in Appendices A, B and C but the most interesting results are discussed in section 5 and compared with experimental results for the special case of high pitches (p 9 A).

2. Main results of de Vries theory. - Take the z-axis parallel to the helical axis. The director v has components

v x

= cos go z ; v y = sin go z ; v, = 0

.

Note that qo > 0 defines a right-handed helix (dextro component). We will treat only this case. The laevo case will be easily deduced, changing go into - go and

(*) Laboratoire associa au C . N. R. S. right into left.

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C1-274 J. C . MARTIN AND 0. PARODI

The local dielectric tensor is axial and has compo- For fixed z, both eigenvibrations have an elliptic

2

nents = n, (along v) and E, = n:. The following polarization, one of the ellipse axes being parallel to the notations will be used : director. If one now moves along the helical axik each - eigenvibration has a constant eccentricity, but its Ell

-

EL

.

-

X = ---

,

_~ ₌-2

-

----

_n EII

+

E L .

,

_{K ~ = -}o n _{ellipse axes are rotating, following the director.}

2 8 2

q0 I , \ The eccentricity is governed by p, with

The eigenvibrations (de Vries waves) are given by the _{As shown by de Vries, one has}

_I

_p,

₁

_<

_{1 and p2}_&₁ superposition of two circular vibrations with different except for two special cases.

intensities and wave vectors and can be written as I D 1

r

a) For high pitches w % w, or

-

%

-

Aj = a j exp

(

i[(pj

+

1) qo z

-

wt]

}

A

n,

-

no

m j =

p j a j e x p { i [ ( p j - l ) q o z - c o t ] } , p, and p2 tend to 1.

where

Aj and

mj

respectively are the right-handed and left- handed circular components.

&

is positive for any value of o except for o, < o < coo, where p? is negative. This corresponds to the reflexion band where total reflexion occurs for the 1-vibration, as can be seen on the dispersion curve (Fig. 1). For a given frequency, one finds four values of p (f pj). Two of them (+ pj) correspond to waves propagating in the

+

z direction (dola1

>

0 ; full line on Fig. 1). The two other ones

(- pj) correspond to waves propagating in the - z direction.

b) Near the reflection band.

1

p,

1

-t 1 for

o + coo ( o > w0) or o -+ o , ( o < me).

In both cases, one should have quasi-linear eigenvibrations. For any other value of o , the polarization of both vibrations should be quasi-circular.

In his paper, de Vries introduced an effective mean refractive index for each vibration

-

nj = pj n, with n, = qo c / o . ( 5 ) We have tried to measure this effective index by using a cholesteric prism, with incident light normal to the first face of the prism. The following surprising results were obtained :

a) f i r highpitches, instead of two linearly polarized beams, four circularly polarized beams were observed, corresponding to refractive indices

b) Near the reflection band, the refracted 1-vibration remained circularly polarized up to the extinction [4].

These results are very different from the one obtained between parallel plates. They are obviously due to refraction and reflection occurring at the second (oblique) face of the prism.

3. Refraction angles at an oblique interface. - Let 8 be the prism angle and take the Y-axis parallel to the interface. Light propagates in the cholesteric along the helical axis (Z-axis). Refraction laws are obtained'by writing boundary conditions for fields and inductions at the interface. Consider now one eigenvibration. It gives at the interface a field that cannot be characterized by a single wave vector since each of its circular components has a different projected wave vector

k,

:

kzj = (pj

-

1) q0 sin8 (1. h. c. p.).

The boundary conditions for fields and inductions imply conservation of the projected wave vector. Let n'

RG. 1. -Dispersion curve as deduced from de Vries theory. be the refractive index in the outer medium, then light

TO a given frequency correspond two values of I5 = ~5 9 0 . The _{propagating at frequency}_o_{has wave vector} wave vectors of different components are shown. In the reflection.

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Let rf be the angles of refraction for r. h. (+) and 1. h. (-) components of the j-vibration : One gets (Fig. 2) :

k'

sin yj' = k;

which can be written as the Snell-Descartes law n' sin r; = n

f

sin 0 (6)

with

+

n - =

j ( ~

+

j 1) nq (7)

in complete agreement with the experimental results for high pitches.

Light refraction for an eigenwave

FIG. 2. -Light refraction for an eigenvibration. The two circular components, having different projected wave vector on the

interface, give rise to two refracted beams.

Eq. (7) leads to two unexpected results.

a) One easily sees from the dispersion curve (Fig. 1) that p1 >

-

1 ; p,

>

1. A detailed inspection shows that, for w -, 0, n: and ny tend to

Z.

But, n; can be negative. This must be related to the physical meaning of the refractive index. Here we define the refractive index as c/vp where up is the phase-velocity. For a de Vries wave, the two components have different phase velocities, which means that one cannot define a single refractive index for such a wave. The refractive index introduced by de Vries is just the mean value of the two refractive indices corresponding to the two components

but has, by itself, no physical meaning. The existence of one negative refractive index just means that a wave having a positive group-velocity can have a component with a negative phase-velocity ;

b) Eq. (7) leads to four refracted beams, with cir-

cular polarizations, not only for high frequencies (or high pitches) but for any frequency outside the reflection band, where two refracted waves are predicted. However, these four beams will generally have very unequal intensities since the amplitudes of the r. h. and 1. h. components in each de Vries wave are very different, except for the special cases treated in section 2. The next step is now to calculate the intensities of these refracted beams. It is easily seen that a crude appli- cation of Fresnel laws leads to aberrant results. For a correct calculation, it is necessary to take into account the reflected waves when writing the boundary conditions at the interface.

4. Reflection angles at an oblique interfake.- A beam propagating along the helical axis will give rise, after reflection, to oblique beams (non-parallel to the helical axis). It is shown in Appendix A that, at small angles, the eigenvibration can be written as :

where

(2)

is an oblique de Vries-wave defined in Appendix A, provided that

k,

"0 40

.

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We will take as an assumption that this condition is realized for small prism angles 0, and give an a poste- rioti justification.

Let us look now a t the reflection of one of the incident de Vries wave, say the 1-vibration. Its main component is r. h. polarized. After reflection, it will be 1. h. polarized. We have now to choose vibrations propagating in the

-

Z direction. The 2'-vibration (p = - p,) has a 1. h. main component (Fig. 3). But its amplitude ratio p-, is different from p,, since from eq- (4)

Hence the 1-vibration will be reflected not only in a 2'-vibration, but also in a 1'-vibration with a much smaller amplitude. The same is true for the reflection of the 2-vibration which gives rise to a strong 1'-vibration and a weak 2'-one.

The angular laws can be deduced from the geometrical construction shown in figure 4. Define k i j as the

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C1-276 J. C. MARTIN AND 0. PARODI

ReFlected light

Incident light

L R I R L R crk

I ; .

L I L & t h d I

.+

R 1 r , I I 0 Wave v e c t o r I

________,

FIG. 3. -a) Relative intensities and polarizations of the components of the eigen vibrations propagating in the (+ 2)

and (- 2 ) directions. The arrows indicate the light rotation for a n observer standing along the

+

Z axis. L and R refer to the usual polarization conventions. b) The 1-vibration is reflected mainly in the 2' one. c) The 2-vibration is reflected mainly in the

1 '-one.

FIG. 4. -Geometrical construction of the kzi wave vectors (along X-axis) for the reflected waves. - lz and (- li

+

k$!) have the same projection on the interface. The reflection of the diffe-

rent components is detailed for the 11 + - 11 case.

The projection of the wave vector on the interface must be conserved. Hence

(pi

+

1) go sin 8 = (- pj

k

1) go sin 8

+

kij cos 8

.

We must here point out that, for small 8, relation (8) is satisfied except for the weak components (i = j) at very low frequencies (Alp

2

8/no). For any other cases our assumption of small k,'s appears as a posteriori justi- fied.

5. Intensities of the refracted beams. - A detailed calculation of the amplitudes of the refracted and reflected beams is given in Appendix B. In section 2, we referred to two special cases of interest (high frequency and around the reflection band) where the de Vries waves did not have a quasi-circular polarization. These are also the cases where one gets interesting results for the amplitudes of the ~efracted waves. In any other cases, the dominant components are

A,

and 33,.

Neglecting terms in p,/p,, one can apply the Fresnel formulas with the mean refractive index Z used as the cholesteric index : for small 8, if ajT and bjT respectively are the r. h. and 1. h. refracted amplitudes corresponding to the j-vibration,

where bj = paj. The formulas for the weak components are a little more complex but present no singula- rities outside the reflection band. In the same approxi- mation one has

where nj = +(nT

+

n7) is the mean refractive index for the j-vibration.

Let us now look to the special cases :

5.1 HIGH FREQUENCIES (O & OM). - This is the case for high pitches : p/A > 2/(ne

-

no). One easily gets the following results

and, for the refractive indices

Each de Vries wave gives rise to a group of two closed lines, the two groups being well apart.

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Then, neglecting terms in ( l l a ~ , ) ~ , one finds (Ap- pendix C ) 1 2 n a , , = c2[+ - -

A]

n

+

n arc0 n'

+ no

b2T = C 2 1 2 no n

+

ne arc, n'

+ no

When the pitch grows, the intensity ratio of the two components of one group tends to 1, and the Fresnel formulas become valid.

As an experimental test, we have measured the refraction of an He-Ne laser beam (6 238

A)

by a cholesteric prism. The prism angle was 70. The cholesteric material was obtained by adding a variable amount of choleste- rol benzoate to the eutectic mixture of PHT and MHT (*) at room temperature. Refractive indices were deduced from the angles of refraction. We have plotted

_-

on figure 5 the quantities Anj = nf - nj versus 2 n, = 2 q, c / o . The theoretical curve has slope 1. It can be seen that the agreement with theory is

An, An, 2 qoc

-

I I W 0.025 0.05 0.075

*

FIG. 5. - An1 and An, are plotted versus 2 go c/w = 2 ng.

The theoretical curve (full line) has slope 1.

(*) PHT : CPropoxy 4'-Heptyl Tolane ; MPT : CMethoxy 4'-Pentyl Tolane.

generally good, with better results for small n,'s. This can be easily explained. Small n,'s correspond to high pitches and to pj's close to 1. In this case, the four beams have strong intensities and the four refraction angles can be measured with a good accuracy. On the contrary, for higher n,'s, i. e. smaller pitches, two of the beams have very weak intensities and the accuracy for measurement of the refraction angles gets poor.

5 . 2 NEAR THE REFLECTION BAND. - That is around rcO = 1. The frequency range of the reflection band is

Take

CI a " , = I + & with 181

- - < - .

2 2 Then eq. (3) leads to

As a solution for eq. (13) take

with n a p = - + i u 2 for E > - 2 (16) n a p = - - + i 2 u for E < - - 2 One easily gets :

The 1-de Vries wave is damped in the reflection band. Above the reflection band, one gets

and, for the refracted intensities (neglecting terms in au) as shown in Appendix C

a 2 G(n'

+

3 ti)

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C1-278 J. C . MARTIN AND 0. PARODI We get similar results under the reflection band : with

and for the intensities

In both cases, for the calculation of b,,, we have neglected terms in a, taking

n

--

n,

--

no. However, this gives a good order of magnitude for b,, and it must be emphasized that, in spite of the fact that

I

p ,

I

-, 1 near the reflection band, the 1. h. component remains much weaker than the r. h. one. As both intensities tend to zero when one approaches the reflection band, the observation of the 1. h. component should be very difficult (Fig. 6).

b,, (L)

FIG. 6.

-

Amplitudes of the refracted waves near the reflection band. We have taken, for the sake of simplicity, ne = no = 7i and n' < E. One can see the dropping of both 1-components close to the reflection band and the enhancement of the ampli- tude b 2 ~ . In the reflecdon band ~ Z T has a phase-shift of a,

but keeps a constant amplitude.

A much more interesting effect should occur for the 2-de Vries wave. One finds

(n' - h) (ii - n;)

a,, = - b2

(n'

+

n ) (n'

+

n:)

The amplitudes of both waves remain nearly constant in the reflection band, but the r. h. component has a phase-shift of n. The behaviour of this r. h. component is quite surprising. Above the reflection band ( o > o,), one has :

(n'

-

6) (ii +no)

-,

a,, = b2 e .

(n'

+

Z )

(n'

+

no)

The amplitude has little change in the reflection band, and, for o < o,,

(n'

-

n) (i

+.

n,)

,-,

azT = - b,

(n'

+

G )

(n'

+

n,)

This shows an enhancement of this component around the reflection band.

It can be interpreted in the following way. The 2-vibration is, inside the cliolesteric, mainly 1. h. polarized. At the interface, it is partially reflected, giving rise to a mainly r. h. polarized reflected wave propagating in the

-

z direction. But total reflection occurs for this wave, giving a r. h. polarized wave [2] in the

+

z direction which enhances the initial r. h. component. A detailed inspection shows that these two waves have exactly the same wave vector projection on the interface and thus give rise to the same refracted beam.

There is, up to now, no experimental evidence for such an enhancement. There are essentially two reasons for that :

i) The mean refractive index 7i of the cholesteric we have used and the refractive index n' of the glass enclosing the prism had very closed values. As a conse- quence, the factor (n'

-

Ti) in eq. (17) was very low ;

ii) The hollow glass prism was made of parallel plates. Then a part of the 1. h. polarized beam was reflected on the second face of the glass plate, giving a r. h. beam which in turn had a total reflection on the glass-cholesteric interface. One then gets a r. h. beam which has exactly the same direction than the r. h. refracted component of vibration 1. In fact, we have observed a beam corresponding to a refractive index n: but it was not possible to decide whether i t was due to the refraction of the 1. h. component of the 2-vibration or to this parasite light.

Both experimental obstacles will be easily removed by using i) different cholesteric and glass materials and ii) non parallel glass plates.

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Appendix A. - OBLIQUE PROPAGATION AT SMALL ANGLES.

-

Oblique propagation in cholesterics has been discussed by several authors [5-71 with the help of heavy numerical calculations. It will be shown here that for small angles, one gets the much simpler results used in section 3. The medium has a translational symmetry in the xy plane. Hence we can write the eigenvibrations as

E(r, t) = E(z) exp i(k, x

-

o t )

.

(A. 1) Using Maxwell equations, one gets

ik,

We will assume here, and give further an a posteriori justification, that k, E, is much smaller than aE,/dz

and thus can be neglected. Using a second Maxwell equation, one obtains

Neglecting again k: Ey that we assume to be much

smaller than d2EY/dz2, one gets :

c2 aE,

i-k

-

= E ~ E , .

m2

az

Eq. (A.4) is exactly the propagation equation for normal waves found by de Vries. Its eigenvalues are de Vries waves. It must be emphasized that for oblique de Vries waves the electric field has a z-component given by eq. (A. 5).

We will now justify our assumptions. Using

reads

The validity condition of our assumptions is therefore

kx K O 90

.

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The justification for our second assumption

is a little more subtle. For a-j-vibration, this can be written as

k:

I

e i ( ~ j + 1)40z - e i ( ~ j - 1 )qoz I4

Three cases can occur. (A.7) a) For high frequencies, p is of order 1, K, $- 1, and (pj

+

1) and (pj

-

1) are of order K,. Then the condi; tion (8) ensures that (A.7) is satisfied.

b) Near the reflection band, for the 1-vibration, p j is very small, 7c0 of order 1, and the condition (A. 7) holds if (8) does.

c) For any other value of o we can neglect the weak component and take into account only the main one. It must be recalled here that both main components have wave vectors closed to rc,. The condition (8) ensures again the validity of (A. 7).

Appendix B. - BOUNDARY CONDITIONS ON THE INTERFACE.

-

Take the c-axis in the incident plane along the interface and the 6-axis normal to the interface. The r. h. and 1. h. components have different wave vectors along the E-axis and the boundary conditions can be written independently for each of them. If their amplitudes respectively are bi and a,, one has

b . = p a . ₂

2 .

Let us now write the amplitudes of the fields and inductions associated with a de Vries wave. Eor the r. h. component, omitting the phase-factor

exp i[(pj

+

1) go z

+

k, x - mt]

,

one has (see Appendix A)

+ 2

/

Ex = uj Bx = inf uj Dx = nj a,

In the same way, one gets for the 1. h. component (omitting again the phase-factor)

Ex = bj B, =

-

i n j b j D, = nf a j

Ey = ib, By = n J bj D, = inJ2 bj

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C1-280 J. C . MARTIN AND 0. PARODI

For the refracted waves, inductions and fields are transverse. As the refraction angle r is small, one can take for the x- and y-components of E, B and D the expressions (B. I), changing nf in n'.

Let us write now the boundary conditions. We have to use the

(g,

y,

I )

frame of reference. As the incidence, reflection and refraction angles are small, we can set

E, = Ex ; B, = B, ; D, = Dx

.

For the incident wave (i-vibration)

Bi = - ion' a i (r. h.) or = ion; bi (I. h.)

D,

= - en" a i (r. h.) or = - en;' bi (I. h.)

.

For the refracted waves, using eq. (6), one gets

+

B ₅- _-

-

ini @aiT _{(r. h.)} _or ₌_{in, 8biT} _{(1. h.)} D c - -

-

n+ n' 8aiT ( r h ) or = - n' q' Obi, (I. h.)

.

And, for the j-reflected wave,

C

Bc =

-

i

-

k . . SJ a!. IJ

-

ion$+ a t (r. h.)

with

nj+ = (- p j

+

1) n,

or, using eq. (9)

B _i- - _- in' Baij (r. h.) or = in; ebb (I. h.) Dc = nJ n+ eaij (r. h.) or = nf nlr Obij (I. h.)

.

Here, aiT(biT), bij(aij) respectively are the r. h. (1. h.) amplitudes (Ex) of the refracted and reflected waves due to the i-vibration.

Writing now the boundary conditions, we find four (instead of eight) independent linear equations :

Using now

a i

+

a:,

+

a:, = a,,

- -

n+ a,

-

n, a:,

-

n, ai2 = n' a,, b,

+

bfl

+

biz = biT

+

n; bi

-

n, bfl

-

n: biz = n' biT

.

one easily solves (B. 5) :

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Appendix

c.

-

REFRACTED AMPLITUDE FOR SPECIAL Or

CASES.

-

a) High frequencies (w %

w,,

or ~ c , % 2/09. 1

-

Then, from eq. ( 3 ) nO(nf

+

n )

+

--(n' _a"0

+

no)

I

.

or, to first order in l/alc0,

One gets from ( C . 1 ) the refractive indices

n: = no

k

n,; n,' = n,

+

nq (10) and, with help of eq. (4)

where terms in l/rco have been neglected. One also has from (1) and (5)

From this, one easily gets eq. (11). The other compo- nents a,,, b,,, b,, are obtained in a similar way.

b) Near the reflection band. - Above and under the reflection band, we make the assumption u

<

1. Then one can neglect terms in au. We will also neglect terms in p,/p2 which are of the order of a/8 (-- 1/80).

One has from (13) and (14)

This gives for o

>

o0

We will keep n: = n; = no except in the (n:

+

n;) From (B .7) one gets term :

i

4 n:

+

n; = 2 Gp, = - a - n(e"

-

e-")

.

D = - 2(nf+n0)(n'+n,)-2n: +---nq(n,-n 2

~ K o O ) ) One also has

=

-

2((nr

+

no) (n'

+

n,)

+

n i )

.

n l = (3

+

E ) nq

=

%

+

2 no - As n;/(n1

+

no) (nl

+

n,) ni/n2

-

l / ( a ~ c ~ ) ~ . One n i = (1

+

& ) n q z n can neglect the second term in the bracket.

Then Then we get for D'

-

D = - 2(n'

+

no) (n'

+

n,)

.

(c.

4) D' =

-

a (n' - 8

+ 6 )

(n'

+ no)

.

Let us now calculate a,,. From ( B .7) one gets

D a t ,

_-

4 no(n'

+

n,

+

n,)

+

a1

And one gets from eq. (B .7)

-I'

+

- (n,

-

no) (n' - no

+

n,)

.

arc0 ( C . 5 )

The next step is to neglect terms in a% in the r. h. s. Using eq. (C .3) and neglecting again terms in n:, one or eq. ( C . 2), which would lead to terms in au. The gets justification is that, as this component appears to be

_-

very weak, we are interested only in its order of magnitude.

This leads to set no = Ti and one gets It is now convenient to express a,, as a function of

c, = ( 1

-

a/lc0) a,. One gets _D'_biT

-

= - 2G(n'

+

3 i ) ( e U - e-")

nt(ne

-

no) bi

-

T% =

(I

+

&)4 [no(nl

+

n.)

+

C t arc0 which at turn leads to the expression given in section 5.

References

[I] MAUGUIN, C., Bull. SOC. Fran~. Miner. 34 (1911) 6,71. [5] BEWEMANN, D. W., SCHEFFER, T. J., Mol. Cryst. Liqu. [2] DE VRIES, H., Acta Cryst. 4 (1951) 219. _Cryst.₁₁_{(1970) 395.}

[31 DE GENNES, P. G., he physics of Liquid Crystals (Oxford _{[6] BILLARD,}_J.,_{MOZ. c w t .}₃_{(1968) 227.} University Press) 1974.