Structure and optical behaviour of cholesteric soliton lattices

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Structure and optical behaviour of cholesteric soliton lattices

P.B Sunil Kumar, G. Ranganath

To cite this version:

P.B Sunil Kumar, G. Ranganath. Structure and optical behaviour of cholesteric soliton lattices.

Journal de Physique II, EDP Sciences, 1993, 3 (10), pp.1497-1510. �10.1051/jp2:1993216�. �jpa-

00247922�

J. Phvs. II France 3 (1993) 1497-lslo _OCTOBER 1993, PAGE 1497

Classification

Physics

Abstracts

61.30 61.30J 78.20W

Structure and optical behaviour of cholesteric soliton lattices

P. B. Sunil Kumar and G. S.

Ranganath

Raman Research Institute,

Bangalore

560 080, India

(Received 12

Februaiy

1993, revised18

May

1993, accepted 16

July

1993)

Abstract. We consider the elastic

instability

of _a ferrocholesteric induced by a field. In _a

magnetic

field acting perpendicular to twist axis, at low fields

we get, a 2 _gr soliton lattice. Above

a threshold field, this will become unstable leading to _{a gr} soliton lattice for positive

diamagnetic anisotropy

_~y~ ~01 and _a N-W soliton lattice for Xa <0. We have also studied the

optical

reflection in such soliton lattices. At fields close _to the nematic cholesteric transition point the higher order reflections _{are more} intense than the

primary

order and each Bragg ^band

splits

into three sub bands. In these regions the electromagnetic _{waves are very} different _as regards their

polarizations,

and angle between E and H. This is in contrast to the undistorted structure where these _{waves are}

locally linearly

polarized with E

parallel

H. In addition~

optical

diffraction pattern has been computed for

propagation perpendicular

to the twist axis. In the ferrocholesteric soliton lattice due to

grain migration

alone _we

predict

_{a new type} of diffraction. This diffraction pattem ^for N-W lattice is very different from that of _{a gr} soliton lattice.

1. Introduction.

It is well known that when _a

magnetic

field H is

applied perpendicular

_to the twist axis of cholesterics

they undergo

a transition _to the nematic state at a critical field

Jfc~_~ [l, 2].

It _was also

pointed

out

by

earlier workers that for fields

0~Jf~Jfc~_~

the twist

angle

& varies

non-linearly

with distance

along

the twist axis

[1, 2] leading

to a soliton lattice ofw twist walls. This transition is

purely

due to the local

diamagnetic anisitropy

_xa of the cholesterics. In 1970 Brochard and de Gennes

[3]

indicated the

possibilities

of

obtaining stable,

ferronematic

(FN)

and ferrocholesteric

liquid crystals (FCh)

wherein the

ferromag-

netic

grains

_are

aligned

^with their

magnetization _M~ along

the local director. These authors also worked _out the

magnetic

field effects

by ignoring

the intrinsic

diamagnetic anisotropy

of the host matrix.

They predicted

_a soliton lattice of 2 _ar twist walls. Ferronematic systems ^have now

been

experimentally

realised

[4]

and initial

experiments

in

magnetic

fields indicate that

diamagnetic anisotropy

of the host matrix _cannot be

ignored.

Thus distortion of FCh in the presence of _a

magnetic

field will be much _more

interesting

when _we consider the effect of both

diamagnetic anisotropy

_xa and the

magnetization M~

^{of the}

grains.

In

fact,

_we find very different behaviours for xa ~ 0 and xa > 0 structures. We also consider the effects of

grain

segregation

in such structures.

1498 JOURNAL DE PHYSIQUE II N° lo

An

optical study

of these soliton lattices is _one

simple

_way of

elucidating

their structural

details. With this in mind _we have worked _out

optical

reflection in the

Bragg

mode and

diffraction in the

phase grating

mode. In many ways the

optics

of these soliton lattices is very different from that of normal cholesterics. In addition _we have also undertaken _an

analysis

of the

polarization

features of the

electromagnetic

waves in the

Bragg

bands.

2. Elastic

instability

in ferrocholesterics.

As

pointed

out earlier _we

incorporate

in _our

analysis

the

diamagnetic anisotropy

of the host matrix. Then the

generalized

Brochard-de Gennes free energy

density

of the FCh in _a

magnetic

field

(For

the

geometry

^{shown in}

Fig. I)

is

given by

k~~ j& 2 j & x~

Jf~

~

fk~

T

~~

2 dz ~ ~°

dz ^~ 2 ^~~~

~

~~~

~ _~" ~ _~

V

~~

~

where,

k~~ = twist elastic constant,

xa ^~

diamagnetic anisotropy,

M~ =

magnetization

^of the

grains,

f

₌

f(z)

= local volume fraction of the

grains

in the

field,

V ₌ volume of the

grain,

P ₌

pitch

of the FCh

=

2

ar/q~,

T

=

temperature,

k~

= Boltzmann constant.

Under the influence of _a

magnetic

field not

only

the director _n

gets

^{distorted but also the}

magnetic grains

will get redistributed from their initial uniform distribution. The distribution of

grains

is

given by

the

quantity fli(z)

=

f/f,

where

f

is the uniform volume fraction of the

grains

at Jf

=

0.

From the minimization of

F~

with respect ^fo

f,

_we get,

fli ₌ fli~

exp(po

Jf _cos

&) (I)

where, _p~

=

(M~ V/k~ T)

and fli~ is the normalization constant so chosen that

iP

^{# dz = P.}

⁽²⁾

o

p

-

H

x Yn

z

Fig.

I.

Geometry representing

_a

magnetic

field

applied perpendicular

to the twist axis of _a ferrocholesteric.

N° lo STRUCTURE AND OPTICS OF CHOLESTERIC SOLITONS 1499

The Euler

equation

for & is

~ ~

=

~~~

~(°~

^~ ₊ ^{~ ~~~ ~}

(3)

z~ _fj

~

fj

~

where,

~~ x~~~

^~~

^~~ M~~f ~~

Integrating equation (3)

and

using (I)

we get

II

-

) _(a

_~

+

6iSin~

&>~'~

(4>

where _a is _a constant,

~~

^~~

~~~

^~~~

^~~

"

~~

~~

~

2

fj

iP

From the minimization of the total free energy

F~

dz _we get

[3],

o

2

w

2 arqo

6j

₌

(a

+

6jsin~

_{& #)~~~ da.}

₍₅₎

o

We _can determine

#o,

a and P

by solving equations (2, 4, 5) simultaneously.

These

equations

have been solved

numerically.

When _xa

=

0 _we have _two

regimes. a)

_q~ 6

j ~ l, here _a

complete unwinding

takes

place

at a

critical field and

b) q~6j

ml, where such _a transition is _not

possible.

However when

xa # 0 such

regimes

do not exist and _a

complete unwinding

takes

place

for all values of qo

6j.

It _so

happens

^{that in} our case q~

6j

_~ l.

3. Structure of the soliton lattice.

For

positive diamagnetic anisotropy

_{xa >} 0 at fields lower than _a threshold

Jf~

=

M/xa

_we

get,

the Brochard-de Gennes 2

ar soliton lattice _a

periodic

_array of 2 _ar twist walls. However above this threshold this become unstable with each 2 _ar twist wall

splitting

into _a double

ar twist wall. This has been

depicted

in

figure

2a

(a

similar behaviour _was

predicted by

Hudfik

[5]

in smectic C * in _an electric

field).

In this process

grains migrate

out of the twisted

regions

to the

weakly

twisted

regions

where the director is oriented

nearly along

the field.

Figure 2b,

shows the

grain

concentration

profile.

For

comparison

_we

give

the &

profile

and

grain

concentration

profile

for the Brochard-de Gennes _case

(I.e.,

_x~

=

0)

in

figure

2.

For

xa~0 materials,

the behavior is very different. At low fields _we get the usual 2 _ar soliton lattice. For fields Jf _>

Jf~

=

(M/xa)

^the

equilibrium

orientation of the director in

a nematic makes _an

angle &~=cos~~(M/xaJf)

_{with the field}

_[6,7].

Then each

2 _ar wall

splits

into _a N twist wall and _a W twist wall. In the N wall the director _tums from

&~

to +

&~

^{while in the W} wall the director rotates from ₊

&~

to 2 _ar

&~.

It should be

remarked that these _are very similar _to the N and W walls described

by Belyakov

and

Dimitrienkov for SmC*

[6]

in _an electric field. In

figure 3a,

is shown the &

profile

_{over one}

period

of the N-W soliton lattice. In this _case the

segregation

of

magnetic grains

has _a different

1500 JOURNAL DE

PHYSIQUE

II N° 10

6.0

~

~

R"0

, /

W /

c _/

@ _/

6 I _4.0 /

, '

$

, '

/ / / /

Z-o /

/ / / / / / / /

0.0

z(cm)x10~

a)

1.4

1.3 ;,_ _.,"'

l.2

..._ ,:.'

i-i i-o

~

~-

o-a

Ah i

o-d .,

/

o?

0.6 _.,

0.5 ":, _,-

0.4 "....__ __.;./

0.3

z(cm)x10~

b)

Fig.

2. The &

profile

(a) and ~

profile

(b) of _a double _gr lattice shown for _one

period

(continuous line for Jf

<

Jf~~

dashed line for Jf

= 0), for the parameters _{Xa =} 5 _x

10~~,

M 1.8 _x 10~^~ G, Jf

=

7 609 G,

Jf~

= M/Xa ₌ ³⁶⁰ G, and po =

10~~ The _same for Xa =

0 is

given

(dotted line) for

comparison.

profile

from that of 2 _{ar or} double

ar soliton lattice. This is

depicted

in

figure

3b. It should be noticed that unlike the

previous

case where the

grains migrate

to the

weakly

twisted

regions

here the

grains

_go from the W wall _to the N wall.

4.

Bragg

reflections at normal incidence.

The dielectric tensor _s of cholesterics and Fch's and the soliton lattices

depend only

on z. In the _case of _a

plane

monochromatic _wave incident _on the medium

through

the

boundary

at

z =

0

only

four out of the six components of the E, ^H are

coupled.

In the Berreman formalism

N° 10 STRUCTURE AND OPTICS OF CHOLESTERIC SOLITONS 1501

d-o

7.0 R=0

~'

q/

,'

_Wall

~ 6.0

(

_/ ,

l~

_5.0

, '

$ ^/

/

4.0 _/

/ /

3.0 / ~ali

/ /

Z-O /

i-o

z(cm)x10~

a)

z-o

R>R.

1.6

1,2

~

fig -~"°

o-d

~,

0.4

~

o-o

z(cm)x10~

b)

Fig.

3. The &

profile

(a) and ~

profile

(b) of N-W lattice shown for _one

period.

_x~ ^-10~~

M

= 1.8 _x 10~ ^~G~ Jf

=

22 065 G and po 10~~

[8]

thex and _y components ^of E

I-e-, (E,, E,,)

and H

I-e-, (H,, H,

are

independent

and the Maxwell

equations

take the form _:

~(~~~

~

ikD(z) v/(z) (6)

where,

k

= w/c~ D is _a 4 _x 4 matrix and 1l'

=

(E,, H,, E,, H, )'. Integrating equation (6)

_we get

v~(z)

= F

(o, z) v~(o) (7)

where F is _a 4 _x 4 transfer matrix. The

eigen

vectors of the matrix F

(0,

P

gives

the net

electromagnetic

field for the proper modes. In the reflection band these modes

correspond

to

the _waves formed

by

the

superposition

of the _waves

propagating along

the forward and

1502 JOURNAL DE

PHYSIQUE

II N° lo

backward directions. The

electromagnetic

field _vector components at any

arbitrary

location

(nP

₊ h

(for

_any

integer

_n and 0

~ h

~ P ) are determined

by multiplying

the

eigen

_vector of F

(0,

nP

by

F

(nP, h). Using

this method we

study

the

polarization

_structure of the

eigen

waves.

In _an

experimental

situation the

sample

is

usually

confined between two

homogeneous transparent isotropic

media. To

analyze

such _{cases we} express the Berreman _vector

1l'(z)

_as a

superposition

of the _waves in the

confining

media

through

_a transformation

[9]

1l'=B4~.

Here B is _a 4 _x 4 matrix with columns

representing

the proper modes in the

isotropic

media

(the eigenvectors

of

D)

and 4~

=

[t~t~r~r~]'

with t~, t~

representing

the

strengths

of

E~, E,,

^modes

propagating

in the _{+ z} direction and r~, r~ in the _z direction. Thus when

[ii, i~], [rjr~]

and

[tjt~]

_are the

strengths

of the

incident,

reflected and transmitted E~ ^and

E~

^modes

respectively,

from

equation (7)

_{we get}

ij

_~~ ^tj

=

G ^~~

(8)

rj 0

r~ 0

where G

=B~~F~~B.

_From

equation (8)

we obtain 2 x2 transmission matrix

F~

and

reflection matrix

F~

to be such that _:

~~j

=

F~

~)

~2

12~

and

lrjj ^{=F~ ii}

~2

12~

The

eigenvectors

r of the matrix G

give

the proper modes of the

multilayer

in _terms of the modes in the

bounding isotropic

media. This allows _{us to}

analyze

the attenuated _waves in the reflection band in _terms of the forward and backward

propagating

_waves. For

example

in the

case of normal cholesterics _{we see} that _a

standing

_wave is formed

by

the circular _waves

propagating

in the _{+ z} and _z directions.

Thus,

in this _{case we}

have, r(I)

=

jr(2)

and

r(3

=

jr (4).

It should also be mentioned that the _vector

1l'(P

is

linearly

related to

1l'(0) through

the Berreman matrix F. We also _see that

9/(P)'s9/(P)

=

9/(o)'s9/(o>

=

o

(9)

where

o

^{i o o}

~_ -i o o o

o o o i

o o -i

since

F

(0,

P )

1l'(0 )

=

1l'(P )

we

get

^{from 9}

9'(o)'

Ft sF v/

(o)

= v/

(o)'

sv/

(o

N° lo STRUCTURE AND OPTICS OF CHOLESTERIC SOLITONS 1503

or

F' SF

= S.

(10)

Equation (10)

^{shows that the matrix F}

(0, P)

is

symplectic [10].

^{Then from the}

properties

of

symplectic

matrices _we get ^{that if A is} an

eigen

value then _{so are}

A*,

I/A and I/A ^*

(I).

Where A ^* is the

complex conjugate

of A. This property ^{of the}

simplectic

matrices

results in the

coupling

of the

eigenvalues.

For

example

far away from the reflection band _we have all the A's

complex

with AA *

=

I. Here _we have

only

two

independent eigenvalues.

Let this be A

j and A

~ then the other _{two are} fixed _as A

f

=

I/A

j and

Af

= IA

~. In the

Bragg

band

we have _two real

eigenvalues

with _one the inverse of the other. The other _two

eigenvalues

_are

complex

with unit modulas and _are

complex conjugates

^{of each other. On the other hand in} the

non-Bragg

band _we have

complex eigenvalues

with the modulus not

equal

to one Thus here all the

eigenvalues

_are

coupled

and _are

given

_{as (A}

j, I/A

j, A

f,

I/A

j*). Therefore,

in the _non-

Bragg regions

there _cannot be any

propagating

modes since the medium is _non

absorbing.

5. Phase

grating

mode.

Raman and Nath

[RN]

in 1935

[11] developed

a

theory

of

isotropic gratings

with

phase

modulation. This

theory

in its

simplest

form has been

applied

to

anisotropic

dielectric

gratings

like that of twisted

liquid crystals [12, 13].

Here _we make the

following assumptions

_:

I)

the

sample

is thin

it)

the

birefringence

is small and

iii)

the

periodicity

^of the medium is

large compared

to the

wavelength

of

light.

In the RN

theory

the emergent wavefront is described

by

U(z

=

Ao

_exp 2 arn~ t/A

where _t is the thickness of the

sample, Ao

is the

amplitude

of the incident

plane

wavefront and

n- is the refractive index for the vibration

perpendicular

to the twist axis at any

point

z

given by

sin~

_&

cos~

&

j ⁼ _~ ⁺

~

n- n~ no

n~ and no

being

the

principle

refractive indices

parallel

and

perpendicular

to the

director, respectively.

The diffraction pattem ^{is then}

given by

the Fourier transform of

U(z)

z (K)

=

j~ ^{U(z) exp(- jKz}

^dz

Where

K=~"j~~~,

6

being

the

angle

^of diffraction.

The above

assumptions

are not

always

valid in usual

experimental

situations. In such _{cases a}

multiple

^beam

analysis [14,15]

is very useful. In this paper ^we follow the method of

(')

We _are thankful to J. Samuel for discussions _on the

symplectic

nature of the matrix

F.

1504 JOURNAL DE

PHYSIQUE

II N° lo

Rokushima and Yamakita

[RY] [15]

and compare the results with those obtained

through

Raman and Nath

theory.

In the RY

theory

_we Fourier

expand

the dielectric _tensor and

electromagnetic

field components as

n,

F~~

(z)

=

jj

e~~ i exp

jiqz

t m

E~ =

~j

e~~

(x )

_exp

(- j (so

+ mq

)

z

)

/ ^~i

where q =

~ " and s~ = n, sin

p

P

Here _n~ is the refractive index of the incident medium and

p

is the

angle

of incidence. Thus

we can express the Maxwell

equations

for the

tangential

field components ^{in the}

=.r

plane

in the

form,

~

~

JC

_7~

(11)

where

e, o i o o

~ =

~~

and c

=

~~<

~~

~"

~~.>. +

~~

^{° °} _~~<

~~

~

~z

qE[

F,> ° °

qE[

q + I

>.

o o _e- o

Here e~~ ^are

(2

_{m + x}

(2

_{m +} submatrices with elements

e~~,~i ⁼

s~~i_,,,

q =

6,,1qt

and qy =

iq

_{+ s~.}

Equation (11)

is solved

numerically

for _~. We consider the _case of normal incidence

only.

6. Results.

6. BRAGG REFLECTION MODE. It is known

[16]

that soliton lattices exhibit

multiple Bragg

reflections. In addition _we find that when Jf

~

Jfc~_~

_some of the

higher

orders _{are more}

i-o

0.8

o.6 fl~

0.4

'

o-Z

o-o

2P/A

Fig.

4. The first three orders in the reflection spectra ^of a choiesteric _gr soiiton lattice for normal incidence at Jf

= 22065 G, and Xj =

10~~ _Continuous line is the

(E, -E,)

reflection. Broken line represents ^the (E, E,) reflection. The

(E, E,)

and

(E, E~)

reflections _are represented by the doted line. _n~ 1.595, and no 1.505.

N° lo STRUCTURE AND OPTICS OF CHOLESTERIC SOLITONS 1505

intense than the first order. In

figure

4, we show the reflection spectra ^of a ar soliton lattice. It should be remarked that this kind of

multiple Bragg

reflections _are

possible

in the _case of discret system ^{of thick}

birefringent plates

also

[17]. However,

in the _case of soliton lattices _we also have each

Bragg

band

splitting

into three

peaks.

By computing

the

eigenvalues

of the transmission matrix F for _one

period

we can get ^the nature of the

dispersion

_curve for _an infinite

sample.

In

figure

5 _we

show,

this

dispersion

_curve

in the first reflection band of the _same

ar soliton lattice. It has three sub-bands. The

eigen

modes of the transmission matrix

F~

are linear in all the three

regions.

For

regions

I and III _we have _a

propagating

mode and _an attenuated mode. The attenuated mode

experiences

reflection.

As shown in

figure

4 the reflected

light

in these

regions

_are of the _same

polarization

state as the incident

light.

The net vibration at any

point

is obtained

by adding

the forward and backward

propagating

waves of the _same

polarization.

These vibrations have E

perpendicular

to H with _a relative

phase

of ar/2 between them. The

corresponding

modes in I and III _are

orthogonal

to

0.6

,h ;?~:,

0A _=" h I I

1 [ _/&= / _~,

o-z I I ^'. ,I ^~= III

_

II

~~ -0.0

11 )

'~ -0.2 _j

_ f "~~ (

-0A _"..

~

,i _~, i

-0.6

~

2P/X

Fig. 5. The

imaginary

part ^{of the} wavevector W, (the eigenvalues of the matrix F (0, Pi) in the first reflection band of the _gr soliton lattice.

o.6

r,

0.5 j ^,

~ ~

' -

l I l ~

P~0.4 _j ^{/ , I ,}

~f

^/ ,

j ,

/

01 j-__ ^{I j ,}

/

$Z o.3 ^{I "=}

/~~., _t ^/

j

^{( ~li :, ~,~.,__}

j

~ ^{i ' /}

f

i,

j ,f [ ^'

~- o-Z ,

(

j / "; _j j ..,

/__~,? _? ', l ~/

(

^'

~:

o-1 ^l

j

/~~

ⁱ

^~~,?

i j /

'

,

/ , j

0,0 ^,

, j

z(cm)x10~

Fig.

6. The variation of

(E

_(~ with ₂ of the attenuated modes in

regions corresponding

to I and III (broken line) and in

region

II (doted line). The _same in the reflection band of normal cholesteric

(continuous line) is

given

for

comparison.

1506 _{JOURNAL DE}

PHYSIQUE

II N° lo

each other. In

region

II _we have

non-Bragg

reflection where both

eigen

modes _are attenuated.

The reflected

light

here is of

orthogonal polarization

to that of the incident

light.

The _{net waves} here

(which

is _a

superposition

of the forward and backward

propagating

_waves of

orthogonal polarizations)

_are

elliptical

with there

ellipticities varying along

the twist axis.

On the other hand, in the reflection band of _a normal cholesteric the _net attenuated mode is

locally

linear and is

coupled

to the local director. The E and H vectors of this mode are

parallel

to each other with _a

phase

difference of ar/2 between them,

In

figure 6,

_{we compare} the variation of

(E

^~ _{as a} function _z of these attenuated modes in the soliton lattices with that of _a usual cholesteric.

The reflection and transmission spectra ^of cholesteric soliton lattices

get

^altered

considerably

when _we include linear dichroism. This is shown in

figure

7 for two incident linear

polarizations.

^It should be noticed that _on the shorter _wave

length side,

the

E~

^mode

(marked Yl'~

is found to be

strongly

reflected _as well _as transmitted

compared

to the

E,

mode

(marked XX) I,e.,

_{we get, an} anomalous transmission.

o-d

YY

bb 0A

o-z xx

o-o

a)

o-zo

o-is

%l o:io

o.05

o.oo

2P

b)

Fig.

7. The transmission coefficient _it (a) and reflection coefficient _£W (b) of _{a «} soliton lattice in the presence of linear dichroism. _n~

= 1.595 ₊ 0.025 j, _no

=

1.505 ₊ 0.001j, other parameters

being

the

same as in

figure

4.

N° lo STRUCTURE AND OPTICS OF CHOLESTERIC SOLITONS 1507

6.2 PHASE _{GRATING MODE.} In the

simple theory

of RN _as the

sample

thickness increases the number of diffraction orders increases. While in the RY

theory

the number of diffraction orders

increases up ^to a certain thickness above which it decreases. We compare in

figure

8 the RN and RY theories. The intensities of the central and second orders _are

given

_{as a} function of the

sample

thickness. We _see that for the parameters ^{used RN}

theory

is valid

only

for thickness less than 7 _~m.

When the incident beam is in the

plane perpendicular

to the twist axis

making

an

angle

D with respect to the field direction such that 0~D

~i

_the diffraction pattem ^is

2

asymmetric.

^{This is} shown in

figure

^9. This feature is

typical

of _a soliton lattice. When the

structure is

absorbing

_we have also linear dichroism. Then the

development

of the soliton

lattice _can be

easily recognized.

Here _we find for D

=

0 all the orders _to be weak for

positive

dichroism. This

implies

that the

weakly

twisted

regions

_are

nearly parallel

to the field.

1-o

o-d °

f

~ _0.8

01

$Z

$

© ~'~

o-z

z

o-o

d(cm)x10~

Fig. 8. A

comparison

of the intensities in the central and second order of the phase

grating

diffraction pattern computed using RN theory (contineous line) and RY theory (broken line) _{as a} function of the sample thickness _« d_» for _{a gr} soliton lattice at H 22 060 with other parameters same as in figure 4.

io

l$

~

~

C

j

~ ^~~

lo

-8 -7 -8 -5 -4 -3 -2 -1

Diffraction Orders

Fig. 9. The asymmetry ^{in the} diffraction pattem of the _gr soliton lattice _{same as} in

figure

8 with Q

= gr/16.

1508 JOURNAL DE

PHYSIQUE

II N° lo

lo

~~ ^-,

l$

~

~

~

© io

$

lo

io

Diffraction Order a)

l>l

) iio~'

4~

~

io

~ ~

Diffraction Order b)

Fig.

^10. Diffraction due to the

segregation

of

grains.

For N-W soliton lattice (a) and for

gr soliton lattice (b). P

= 19 _~m, and the

grain

refractive index _n

= 1.505 ₊ 0.02 j.

We shall _now consider the

peculiar

features associated with the ferrocholesteric soliton lattices. One

important

feature is the

migration

of the

magnetic grains.

These

grains

_can be made up of either transparent ^materials

(e.g., garnets)

_or

absorbing

materials

(e.g., ferrites).

When these

grains

_are of the later type, ^{then due} to

migration

of the

grains,

_we will have _a nonuniform

absorption along

_z. This

nonuniformity

alone results in diffraction when the incident

polarization

^is

parallel

to the twist axis. This diffraction is

peculiar

to these systems.

This is

depicted

in

figure

^lo- In

figure

10a _we show~ the diffraction pattem ^due to N-W soliton lattice. This is very different from that of the soliton lattice in _Xa

~ 0 systems ^whose diffraction pattern ^{is shown in}

figure

lob. The

peculiar

feature of the diffraction in N-W soliton lattice is that

intensity

fluctuates from order to order and _we find the _even orders to be less intense than the odd

orders,

where _as for the double _ar lattice the

intensity

decreases

monotonicaly.

This feature in the pattem ^{could be used} to

distinguish

N-W lattices from soliton lattices in xa > 0 materials.

6.3 TEST _{FOR TWIST INDUCED} BIAXIALITY. It is known that cholesteric

liquid crystals

are in

principle

biaxial

[16].

The inherent

chirality

and the hindered rotation of the molecule due to

N° lo STRUCTURE AND OPTICS OF CHOLESTERIC SOLITONS j509

l>l

f

_lo

~

C

j

C

lo

-5 -4 -3 -1

Diffraction Order

Fig.

I. The computed diffraction pattern ^due to the variation of twist induced

biaxiality

along the twist axis of

a gr soliton lattice.

the twist _are the _two mechanisms

proposed [18]

_to

explain

this

biaxiality.

^It has been

argued

that the induced

biaxiality

increases with increase in twist. To _our

knowledge,

no direct

optical

measurements have been

proposed

_to

verify

this. Due to twist induced

biaxiality

in the soliton lattice the

biaxiality

will vary as function of _z. As _a

result,

in cholesteric soliton lattice for the

incident

light

with

polarization parallel

to the twist axis, these variation will

give

_a diffraction pattern. ^In

figure11,

_we show _one such diffraction pattern

computed by assuming

the

biaxiality

to vary

linearly

with twist. The pattem appears even for An= as small _as 5 _x

10~~ Thus,

the method appears to be

quite

^sensitive to twist induced

biaxiality.

7. Conclusions.

The structure of ferrocholesteric in _a

magnetic

field is worked _out for _an infinite

sample taking

into _account the effect of the

diamagnetic anisotropy

of the host, the

grain magnetization

and

the

grain segregation.

We find the 2 _ar soliton _to be unstable above _a threshold field. The

optical properties

of these _{structures are} studied for both

Bragg

mode and the

phase grating

mode. The soliton lattices exhibits

multiple Bragg

reflection. The _{net waves} inside the

Bragg

band is shown to have

interesting polarization

features. When the

grains

are

absorbing

soliton lattices exhibit anomalous transmission.

In the

phase grating

mode we

give

a

comparison

of the results obtained

using

the RN

theory

and RY

theory.

The soliton lattice exhibits

asymmetric

diffraction when

angle

of incidence D with respect to the field direction is between 0 and ar/2. It is shown that the

segregation

of the

grains gives

rise _to different diffraction pattems ^{in the double} ar and N-W lattice. It is also

possible

to test the twist induced

biaxiality using

the

phase grating.

Acknowledgements.

Our thanks _are due _to R.

Nityananda,

J.

Samuel,

K. A. Suresh and Y. Sah for useful _comments and discussions _we had with them and also to the referees for

helpful suggestions.

15 lo JOURNAL DE

PHYSIQUE

II N° lo

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R. B.,

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