Form birefringence in helical liquid crystals

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Form birefringence in helical liquid crystals

P. Allia, P. Galatola, C. Oldano, M. Rajteri, L. Trossi

To cite this version:

P. Allia, P. Galatola, C. Oldano, M. Rajteri, L. Trossi. Form birefringence in helical liquid crystals.

Journal de Physique II, EDP Sciences, 1994, 4 (2), pp.333-347. �10.1051/jp2:1994132�. �jpa-00247965�

Classification Physics Abstracts

61.30 42.10 78.20E

Form hire&ingence in helical liquid crystals

P. Allia, ^P.

Galatola,

C.

Oldano,

M.

Rajteri

and L. Trossi

Dipartimento di Fisica, Politecnico di Torino, Corso Duca degli Abruzzi, 24-10129 Torino, Italy

and Consorzio INFM, Section of Torino Politecnico, Italy

(Received

23 July 1993, accepted in final form 8 November

1993)

Abstract The optical properties of cholesteric and chiral smectic liquid crystals _are studied,

in the limit of pitch smaller than the wavelength of visible light, by _means of _a formalism

borrowing techniques developed to explain form birefringence effects in particulate media. The behaviour of actual stratified anisotropic substances is shown to be accurately approximated,

in this limit, by _an effective homogeneous uniaxial crystal, whose dielectric tensor elements _are

easily obtained from the _exact values corresponding to the stratified medium. In two notable cases, I-e-, for normal incidence of light and in smectic~c liquid crystals with _a critical value of the tilt angle, the optical properties of the stratified material may only be accounted for by

introducing _an effective medium suitably characterized by optical activity. In all the examined cases, the periodic media appear ^to be completely equivalent to homogeneous crystals displaying optical activity, _up to pitch values corresponding to _one half of the first Bragg diffraction band.

The model's results having higher relevance both in fundamental studies and applications _are described and discussed.

1 Introduction.

The

optical properties

^of

liquid crystals

have been

thoroughly explored

since the

discovery

of this class of

anisotropic

stratified

media, owing

to their intrinsic relevance both in fundamental

studies and in

applications.

The

problem

of

electromagnetic-wave propagation

in

anisotropic

stratified media is

canonically approached by making

_use of _a 4 _x 4 matrix

method, formerly

in- troduced

by

Berreman and

subsequently developed

into _a

variety

of alternative treatments

[I].

As is well-known, Berreman's equations, obtained after _some re-arrangement ^of the Maxwell equations, ^{involve the}

electromagnetic

field components that do not

display

_any

tangential discontinuity

at the interfaces between the

liquid crystal

and the

glass

substrates.

Although

_a number of aspects ^of

light propagation

in _many types of

liquid crystals (including

cholesteric and chiral smectic-C

phases)

have been

actively explored

in the last century, _some

interesting properties

of this class of

optically anisotropic

media have not

yet

been

highlighted.

In

particular,

the

development

of chiral

liquid crystals having

_a

pitch

_p

comparable

to, or

less than the

wavelength

I of visible

light

_[2], has made available

optical

media characterized

by

novel

features,

such _as

typical

form

birefringence

effects. When _{p <}

I,

these

materials,

inherently displaying

_a

spatial periodicity along

_one direction

(the z-axis),

behave in

fact,

in many respects, as

homogeneous

uniaxial

crystals

characterized

by

_an effective

z-independent

dielectric tensor.

The aim of this _paper is to

exploit

the standard 4 _x 4 matrix

approach

to describe the

optical

behaviour of cholesteric and chiral smectic-C

liquid crystals

in the above

limit, comparing

^their

properties

to those of _a suitable

anisotropic homogeneous

^medium. In sections 2 and 3 it is shown that the

analytical

methods

developed

to

investigate

the so-called form

birefringence

of

particulate

media _[3]

straightforwardly give

the real part ^{of the dielectric} tensor of the effective

homogeneous

^{medium. The}

imaginary

part, which is

responsible

for the

optical activity,

is evaluated in section 4. The results obtained in section 5 show

that,

under the

assumption

_p <

I,

these

periodic

media appear ^to be

completely equivalent

to

homogeneous crystals displaying optical activity.

In

particular,

the model will be shown to maintain its

validity

until the ratio

p/I

remains below the value

satisfying

the

Bragg

condition for

light diffraction,

of the order of _one for

cholesterics,

and of one-half for chiral smectics.

The

picture

^{of chiral}

liquid crystals developed

in this _paper leads to describe

specific optical

properties ^of anisotropic stratified media from _a

peculiar standpoint,

therefore

contributing

to a

deeper understanding

of _some

fascinating

aspects ^of these materials.

2. Form

birefringence

in stratified media:

general

considerations.

The concept of form

birefringence applies

to systems ^{where the}

optical anisotropy

is related _to the _presence of

particles having

sizes much

larger

than the molecules

constituting

^the

material,

but still much smaller than the

wavelength

of incident

light. Strictly speaking,

the form

birefringence

is related to the anisotropic

shape

of the

particles,

composed ^of an

optically isotropic

material. The

simplest

_case is

provided by

_an

assembly

of thin

alternating layers

of two

isotropic media,

with dielectric _{constants ei} and _e2,

respectively, giving

rise to _a

periodic

structure with

pitch

_p. In the limit p < I, this arrangement ^{behaves like} a

homogeneous

uniaxial

crystal

with

optical

axis

perpendicular

to the

layers

and with

diilectric

_constants

ijj and ii

(the subscripts

refer to the directions

parallel

and

perpendicular

to the

optical axis)

_[3, 4].

Let d~

(I

= 1,

2)

^{be the thickness of each}

layer (di

+ d2

"

P). By imposing

the condition of

continuity

of the

tangential

components of the electric field E and of the normal components ^of the dielectric

displacement

D at the

interfaces,

the effective dielectric constants of the uniaxial

crystal

_are found to be _[3]

fi$ a ii

"

~

_f~e~

= Z fi$ e ijj ⁼

~ _f~e/~

= e-I

,

(1)

~

~

~ ~

where the bars indicate _mean

values,

f~ ₌

d~/p

and the

subscripts

_o, e refer _to the

ordinary

and

extraordinary

_rays,

respectively.

It is

easily

checked that the difference fi~ fi~ is less than

zero,

indicating

that the

assembly

of

layers

behaves _{as a}

negative

uniaxial

crystal.

Although

^{these results} may be

easily

obtained

by using straightforward

arguments [3], ^this

example

^of "stratified" medium will _now be treated

by making

_use of _a 4 _x 4 matrix

method,

which is _{a more}

powerful approach

to

study

this class of

optical

media.

The Maxwell

equations

for _a

plane

_wave incident _{on a} stratified

medium,

whose

optical properties depend

_on a

single

_space ^coordinate _{z, can} ^be cast in the form

)

=

ikoa(z)~(z)

,

(2)

where ko

=

uJ/c,

_~b(z) is _a column _vector of

complex

_components

(E~, Hy,Ey, -Hz)

and

A(z)

is the 4 _x 4 Berreman's matrix, ^whose

general

expression ^is

given

elsewhere

[I].

The evolution operator

U(z), having

the

properties

ik(z)

=

U(z)1fi(0) U10)

=1

,

(3)

where I is the

identity

matrix, ^{satisfies the}

equation

)

=

ikoA(z)U(z) (4)

z

In the present case, A is _{a constant} matrix for each of the two

layers, having

the

simple

structure [1]

0

1-171~/f~

0 0

A~ ₌

j

,

(5)

0 0 _e~ m~

~

where I

= 1, 2, m = sin

9inc,

9inc

being

the incidence

angle

^of

light [the

incidence

plane

is

(x, z)].

As _a consequence of the

continuity

^{of the}

~b-vector, equation (4)

_can be

immediately

inte-

grated

_over the

length

of _one

pitch

_{p =} di +

d2, resulting

in

U(p)

=

exp(ikoA2d2) exp(ikoAidi) (6)

When _{p <}

I,

the kod~ terms _are much smaller then

unity,

and therefore

u(p)

a i +

ikoip

,

(7) where1= £~ f~A~. By using equation (I),

_one finds

0

1-m~lie

0 0

1

=

j

,

(8)

0 0 lo m~ ₀

where i~ ₌ z, ie =

(e-I)

^{~ This}

^equation

^{shows that the considered}

assembly

^of

layers

behaves _{as a}

single

uniaxial

crystal

with

optical

axis

along

_z, whose

extraordinary

and

ordinary

indices _are

given by equation (I),

_as

expected,

and which will be referred in the

following

_as

the effective

anisotropic crystal

_{or as} the model of the actual

periodic

_structure.

The present calculations _may be

easily

extended _not

only

to the _case of _an

arbitrary

number of

periodically

assembled

layers,

but also to

periodic

media whose

optical properties

_vary with

continuity along

_one

spatial

direction. The

study

of either

intrinsically anisotropic layers,

or continuous media

displaying

a local

anisotropy

would _{seem an}

obvious, straightforward

extension of the formalism. However, ^{in these} _cases, ^novel attractive

properties

_emerge, which will be described and discussed in the

following

sections.

3. Form

birefringence

in chiral

liquid crystals.

Chiral smectic-C

liquid crystals (S(

_can be considered from the

optical point

of view _as

locally

biaxial media with _a

spatially

^uniform rotation of the dielectric tensor around _an axis normal to the smectic

layers (referred

to _as the z-axis in this

paper).

Since the

biaxiality

of most

Sl's

is very

small, only locally

uniaxial media will be considered in the

following.

The _case of cholesteric

liquid crystals

_can be considered _as the

limiting

case

of

Sl's,

where the

angle

9 between the

optical

axis and _z

(tilt angle)

is

7r/2.

Let _us consider

a chiral

liquid crystal

sandwiched between

parallel glass substrates,

_so that the smectic

layers

are

parallel

to the

plates.

Berreman's matrix for

light propagation

in the

sample

reads

M

cos(qz)

I m~

le33

M

sin(qz)

0

~ ^{ao ai}

cos(2qz)

M

cos(qz)

_-ai

sin(2qz)

0

~~ 0 0 0 1 ^' j~~

-al

sin(2qz)

M

sin(qz)

_{ao + al}

cos(2qz)

m~ 0

where q =

27r/p,

_{e33 = El}

^sin~

9 ₊

e3cos~ 9,

_El and _e3

being

the

principal

components ^{of the} dielectric _tensor.

Finally,

_{m = ng sin fig}

(ng

^{is the refractive} index of the

glass substrates,

and fig the incidence

angle

in the

glass),

and M

= m cos 9 sin

9(el e3)le33,

_ao

" El

(1

+

e31e33)/2,

al " El(1

e31e33)/2.

The matrix of the

equivalent

uniaxial

crystal

is

obtained,

in the limit _{p <}

I, by taking

the average value of equation

(9)

_over the

length

of _one

pitch.

All terms of

A(z) containing

harmonic functions of _{qz are}

averaged

out, and the matrix _can be cast in the form

previously

obtained for the

assembly

of

layers [see Eq. (8)].

For

cholesterics,

the parameters i~ and ie are defined _as

i~ =

~~ ~~

i~ _{= ei}

(io)

Note that i~

corresponds

to the _average value of the

extraordinary (e3)

^and

ordinary (El)

indices of the

cholesteric,

while ie is coincident with the

ordinary

index of the

liquid crystal.

For chiral smectic-C

phases,

the

corresponding

parameters take the form i~ = ao ie

= e33 " El

sin~

_{9 + e3} cos~ ₉

(ll)

In both _cases, therefore,

1corresponds

_to Berreman's matrix of _a uniaxial

crystal

with optical axis

along

z and

principal

refraction indices fi~ ₌

@

and he

=

@, respectively.

It should be

explicitly

noted that, while cholesterics with positive dielectric anisotropy

always

_{appear as}

negative

uniaxial

crystals (tie

fi~ <

0),

chiral smectic-C

phases

behave either _as

positive

_or

negative

uniaxial

crystals, depending

_on the value of 9. The

angle

9rev where fi~ fi~

changes

from

positive

to

negative

is

given by

_[5]

cos~ 9rev

= +

-(3

+

6)

₊

[(3

+ 6)~

8(6

₊ 6~)]^~~~ =

~ 6 ₊

O(6~

,

(12)

where 6 ₌ (e3 El

)/(e3

+ _El

).

Generally speaking,

the

eigenvalues

of Berreman's matrix

A(z)

for

homogeneous layers give

the components

along

_z of the _wave vectors of the four

independent eigenmodes propagating

within the medium

(two

forward and two

backward-propagating

plane

waves) [I].

In the present case, the

diagonalization of1determines

the ratio

[

=

[z/ko,

where the components

[z

_are

1.62

1.605 60

j fl

1 58

1.604 56

l.54

0 20 40 60 50 0 20 40 60 80

INCIDENCE ANGLE / DEG INCIDENCE ~~NGLE / DEG

Fig. 1. Refractive indices

vs. the external incidence angle in the glass _(mg

=

1.75)

for _a right- handed cholesteric L-C- with the parameters

@

₌ 1.5,

@

₌ 1-I, _{p =}

~m/2.

Here and in the

following figures the full lines _are the exact solutions of the actual periodic structure, while the dashed lines refer to the homogeneous model with io and ie given by equation

(lo).

associated to the _waves

travelling

within the

equivalent

uniaxial

crystal.

The four

eigenvalues

turn out to be

la

= +

(i~

m~)~~~

lb

~~~~

=

+I(/~ (l _m~lie)~~~

In order to

verify

the limits of

validity

of this

model,

the

approximate

solutions _{are com-}

pared

to the exact values obtained

by integrating numerically

Berreman's

equations.

Let _us recall that the

optical properties

of the

periodic

medium _are

fully

described

by

the transfer

matrix

U(p, 9inc),

where the

dependence

on the incidence

angle

has been

explicitly

introduced.

A check of the

adequacy

of the model is

given by figure

I, drawn for _a

typical

cholesteric with

@

₌ 1.5 and

@

₌ 1.7 and for

p/lm

₌ 0.5,

being lm

₌

I/nm

with

n$

= (El +

e3)/2.

There,

the dashed lines show the variation with 9;nc of the

ordinary (left)

and

extraordinary (right)

indices of the model

crystal

^{obtained from}

equation (13).

The full lines

correspond

to

the exact

quantities

~/~0~

~^~ ~za(b) na(b) =

,

(14)

ko

where kza(b~ ⁼

(ip)-I _In(ua(b) ),

_{ua, ub}

being

the

eigenvalues

of the transfer matrix

U(p,

9inc) cor-

responding

to the two

forward-propagating eigenmodes. Strictly speaking,

the

quantities

_kz~(b)

are defined modulus

27r/p.

This fact is

intimately

relAted to the character of the

eigenmodes

of the

electromagnetic

field within the stratified medium: these _are Bloch _waves,

composed

of

an

infinity

of

plane

_waves. However, for _{p <} I,

only

_one Bloch component is

actually

domi- nant, so that the parameters n~(b)

correctly play

the role of effective indices for the dominant

components ^{of the} two

forward-propagating eigenmodes. Although

the deviations between the exact and the

approximate

indices _are

quantitatively

rather

small,

_a notable

qualitative

difference exists: in

fact,

the exact index

corresponding

to the

ordinary

_ray is not _a constant.

It may be concluded that the

optical

properties of the

sample

_are not well described

by

the effective

homogeneous

medium defined

by equations (10, 11).

Since a

possible biaxiality

of the model

crystal

is to be discarded _on the basis of obvious symmetry considerations, _a better

approximation giving

a non-constant _na value _can

only

be achieved

by introducing

_an ^effective

homogeneous anisotropic

medium

displaying optical activity.

In fact, the above mentioned difficulties _are

intimately

related to the so-called

pseudo-

rotatory power of chiral

liquid crystals.

It is worth

noting

that for normal incidence of

light,

the

eigenmodes propagating along

the z-axis _are characterized

by

_a

nearly

circular

polarization

and _a small difference between the refractive indices [6]. Such _a condition _ensures the

optical activity

of this

family

of stratified media.

The refractive indices _{nl, n2}

along

^the

optical

axis may be calculated

exactly

for both cholesterics and chiral smectics, and cast in the form [7]

ni =

((k$

^{+ q~}

^(4k$q~

⁺

^{~~k$) ~~)}

~~~

j

ko

~~~~

~~

o ~~~

~ ~~ ~

~~~~~~

~

~~~~~~~)~~~ ~

'

where km ₌

ko/G (reducing

to km ₌ ko (El +

e3)/2

for

cholesterics),

_{~ = al}

lao.

The real and

imaginary

parts ^of _nl ^and n2 are

plotted

in

figure

2 _as functions of the ratio

p/lm

for

a cholesteric with

right-handed

helix and

@

₌ 1.5,

/G

" 1.7. The

imaginary

part ^of the

refractive index _nl is _{non zero} when _p ££ Am,

indicating Bragg

reflection of the

eigenmode

with

nearly right

circular

polarization.

Near

p/lm

= I,

therefore,

the

approximate

model of

an

equivalent

uniaxial

crystal

is

intrinsically inadequate

to describe the

optical properties

of the stratified medium.

REFRACTIVE INDICES FOR CHOLESTERIC LC

f li i~

~

#

_~.~~

~'~fl 0.2 0.4 0.6 0.8 1.2

0.2 0.4 0.6 0.8 1.2

PITCH / WAVELENGTH

Fig. 2. Real and imaginary parts of the refractive indices _vs.

p/~m

for _a cholesteric L-C- with the

same values _as in figure _1, at normal incidence.

We may therefore _argue that the

optical properties

of the considered classes of chiral liquid

crystals

_may be described

by

the

approximate model, only

for _{p <} Am. However, the rota-

tory power of both cholesterics and chiral smectics cannot be accounted for without

explicitly

considering

the

optical activity

of the effective uniaxial

crystal.

4. Effective uniaxial

crystal

with

optical activity.

As it is well-known [8], ^the

optical activity

of _a

crystal

_may ^be described

by introducing

_a

pseudo

tensor g

represented by

_a real 3 _x 3 matrix. The components ^of _{g are} used _to

obtain, through

_a

straightforward procedure,

the corrections to the dielectric

teisor

e

accounting

for the rotatory _power.

Although

the actual values of the _{non zero} components ^of g are

clearly dependent

_on the

properties

^{of the} molecules

constituting

the

material,

the

strucfiure

_{of the 3}

x 3 matrix may be determined

purely

_on the basis of symmetry considerations. In

particular,

_a

diagonal

matrix of the type

gi 0 0

g = 0 gi 0

(16)

0 0 gjj

is

expected

for the considered classes of chiral

liquid crystals.

_{The gjj} component ^is

responsible

for the

optical activity

of the medium

along

its

optical

axis. Its value is

given by

gi =

~'~~

~'~~l'~~ ^~'~~~

,

ii?)

where _{ni, n2 are} the refractive indices

along

the

optical

axis

given by equations (15).

The components _gi are

responsible

for the

optical activity along

^directions

perpendicular

to the

optical

axis.

Straightforward

calculations show that the dielectric tensor of the effective uniaxial

crystal

with

optical activity

takes the

simple

form

i~ o 0

10 ^-ig3

⁰

(

₌ 0 i~ 0 + ig3 ⁰

igi

,

(18)

0 0 ie 0

-igi

0

where _g3

= gjj

cos9;nt,

_gi

" gi sin

9jnt,

9jnt

being

the

angle

between the _wave vector of the

ray

propagating

within the medium and the

optical

axis. For low incidence

angles,

b;nt is well

approximated by exploiting

the condition fi~ sin 9;nt _{" ng sin fig} e m. It should be

explicitly

noted

that,

while the values of i~, ie, _gjj are determined

by

equations

(10), (11)

and

(17),

the

quantity

_gi cannot be derived from the

theory

and must be

regarded

as the

only

free parameter of the present ^model.

5 Results.

5. I CHOLESTERICS AND ORDINARY CHIRAL SMECTICS. The results obtained

by studying

the effective medium described

by

the

complex

dielectric tensor

given

in

equation (18)

are

summarized

by figures

3-5, where the transmittance and reflectance of _a

typical

cholesteric

(with

_gl

=

0)

_are

plotted together

with those of the

corresponding

model

crystal

_as functions of the

pitch,

of the

sample thickness,

and of the incidence

angle, respectively,

all the other

parameters

being kept

fixed. The rotatory _power of both media is shown in

figure

6, where the

properties

of the transmitted

light

_are reported _{as a} function of the

pitch

and of the incidence

angle,

for _a

linearly polarized (TM)

incident beam. Similar _{curves are} obtained with _a TE incident beam. Note that in the model without

optical activity only

the

extraordinary

_ray is excited

by

incident

light having

this

polarization. Generally speaking,

the output beam turns out to be

elliptically polarized.

The rotation

angle

of the

major

axis of the

ellipse

is

given

in the

RC-RC xl 0^~ LC~RC

~j

~

j~

l

0.2 fl.4 0.6 0.8 0.2 fl.4 0.6 0.8

xl0.3 RC-LC LC-LC

~

^{~ ~~~}

~

j~

~

0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

PITCH / WAVELENGTH _PITCH / WAVELENGTH

a)

RC~RC

~ ^xl0~

LC~fC

d ----~~.~--~m=~~~jf ^~~

i ~

d

~

#

~

0.2 fl.4 fl.6 fl.8 ~) _{0.2 1).4 fl.6}

fl.8

~

RC- LC _{xl 0}3 LC-LC

~

~~

~j _fl.5

~

~

".

~~j

~

~ ,

'

j

0.2 fl.4 0.6 0.8 0 0.2 0.4 0.6 0.8

PITCH / WAVELENGTH PITCH / WAVELENGTH

b)

Fig. 3. Transmittance

(a)

and reflectance

(b)

_at normal incidence _vs.

p/~m

for _a cholesteric L-C- with the _same values _as in figure I, and thickness d

= 2~m. The incident and transmitted beams _are right

(RC)

and left

(LC)

circularly polarized.

upper

plots

^of

figure 6,

while the

ellipticity

is

provided by

the lower _curves. The results shown in

figures

3-6 _are

representative

of _a number of similar plots obtained with different parameter

RC-RC x10-~ LC-RC

l).5 1.5 2 fl.5 1.5 2

xl 0.3 RC-LC LC-LC

(

uJ

j~

~

_'

I

~

~

0.5 1.5 2 0.5 1.5 2

THICKNESS / WAVELENGTH THICKNESS / WAVELENGTH

a)

~xl 0-~ RC~RC ~ LC~RC

~~

~~ _{~ j}

~~<,~

j

2

~j

j

^~

~

o~ ~~~~

fl 0.5 1.5 2 0.5 1.5 2

xl03 RC~LC xl 0-4 LC-LC

f

^~

(

ul

d

0.5 1 1.5 2 0.5 1 1.5 2

THICKNESS / WAVELENGTH THICKNESS / WAVELENGTH

b)

Fig. 4. As figure 3 but _{as a} function of

d/~m

_{for p}

= 0.5~m.

values. It is therefore

possible

to conclude that the most

prominent optical properties

of cholesterics _are well described

by

the model for _p values not

exceeding

0.slur. For

higher

p

values,

the deviation between the exact and the

approximate

solutions

rapidly

increases, and

can reach _a value of _some percent at p =

0.81m (see

for instance the RC~RC _curves of

Fig. 3).

TM-TM x10-4 TE~TM

(

ul

(

/

~

20 4fl 60 80 21) 4fl 60 80

x10-4 TM~TE TE-TE

f

~

~ ~/

l

~

--,,_

~

20 40 60 80 20 40 60 80

INCIDENCE ANGLE MEG INCIDENCE ANGLE MEG

a)

TM-TM x10~ TE~TM

(

uJ

~j 0.5

ij b

20 40 60 80 21) 4fl 60 8fl

x10-3 TM-TE

~

TE-TE

f

~j

ij

#

/~~

20 40 60 80 20 40 60 80

INCIDENCE ANGLE ©EG INCIDENCE ANGLE MEG

b)

Fig. 5. As figure 3 but _{as a} function of the incident angle in the glass _{for p =} 0.5~m and incident and transmitted beams TE and TM linearly polarized.

Therefore,

the

approximate

model loses its

validity

_near the first

Bragg

diffraction

band,

_as indeed

expected.

Similar results _are obtained also in the _case of smectic-C

phases

for

virtually

all tilt

angles

~

£

u1 60

40

g

Z

~

i

O 00 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

~0 _{0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8}

PITCH / WAVELENGTH

a)

tJ

I

(

Z

5 10 15 20 25 30

~

O 0.2

~

~j _~~

~0 _{5 10 15 20} 25 30

INCIDENCE ANGLE / DEG

b)

Fig. 6. Polarization properties of the light transmitted by _a cholesteric slab for TM incident polar- ized beam _vs.

p/~m

at normal incidence

(a)

and _vs. the incidence angle in the glass _{for p}

= 0.5~m

(b).

The polarization is generally elliptical and the rotation angle refers to the major axis of the ellipse

[same

values _as in figure 1 with d =100~m in

(b),

d

= 50~m and _mg

= 1.6 in

(a)].

(see

for instance

Fig. 7),

with _one notable

exception, deserving

^further attention and

adequately

discussed in the

following paragraph.

TM~TM x10.3 TE~TM

uJ 8

~~

(

~

^~

~

_{l' ~~}

~

~~

~

~

/

o

~~

20 40 60 80 0 20 40 60 80

TM-TE TE~TE

fj

~

/

^f~/

~

I ~~

j~

~

/~

l

20 40 60 80 20 40 60 80

INCIDENCE ANGLE / DEG INCIDENCE ANGLE / DEG

a)

TM~TM TE-TM

~

°.°~

~~

^~~~

~

"°~~

/ _j ~

j

0.01

~

# _/~

^,

~) ~fj

o~

'

20 40 60 80 0 20 40 60 80

TM~TE TE~TE

~j

~

i

₎ ^~

d

~/~

~

/

20 40 60 80 20 40 60 80

INCIDENCE ANGLE MEG INCIDENCE ANGLE MEG

b)

Fig. 7. Transmittance

(a)

and reflectance

(b)

_vs. the incidence angle in the glass for _a chiral smectic- C with tilt angle 6 ₌

30°,

_p

= 0.3~m, d ₌ 2~m and the _same indices of refraction _as in figure 1.

5.2 CRITICAL _TILT ANGLE FOR CHIRAL SMECTICS. As it is well-known, ^the tensor g de- fined in

equation (16) plays

an essential role in

determining

the

optical properties

of cholesterics

(see Fig. 6) only

_near ^the

optical

axis. There, the refractive indices obtained

by neglecting

the

presence of _{g are} coincident and the

corresponding eigenmodes

_are

degenerate.

A similar situation _occurs in

S( liquid crystals

for the

particular

value 9r~v of the tilt

angle

⁹

corresponding

to the condition fi~ = fi~, therefore

marking

the

boundary

between the two

regions where the effective medium behaves

respectively

_{as a} positive and _a

negative

uniaxial

crystal.

The values of 9r~v obtained

by solving

equation

(12)

_range from 54.7° in the

limiting

case of

vanishing anisotropy

parameter 6, to 56.6° in the

highly anisotropic

_case d

= 0.2.

g

~i

fj

~

~

z O

( i

_'~~~0

10 20 30 40 50 ₆₀

10 20 30 40 50 60

INCIDENCE ANGLE / DEG

Fig. 8. Polarization properties of the light transmitted by

a chiral smectic-C slab for TM incident

polarized beam

us. incidence angle in the glass, with tilt angle 6

= 6rev, _p

# 0.2~m, d

= 50~m and

the _same indices of refraction

as in figure 1.

When 9

= 9r~v, the

eigenfunctions

_are

degenerate

^for _any direction of the internal _wave vector, so that the medium behaves _{as a} pure rotator, as shown in

figure

8. In this _case, the gjj component of the tensor g is still defined

by equation (17),

while the value of gi ^must be determined

through

_a

^belt-fit procedure.

We stress

again

that this is _a

theory

with _a

single

free parameter: the choice of _gi which makes the

approximate

solution coincident with the exact result at _one

specific angle

of incidence

automatically

_guarantees the

nearly perfect

superposition

of the entire curves.

The

interesting

behaviour of the

S(

as a pure anisotropic rotator

actually

_occurs

only

in _a

narrow interval of 9 values around 9r~v,

completely disappearing

for A9 of the order of few

tenths of _a

degree.

The critical

properties

of the

angle

9r~v,

already investigated

in [5] ⁱⁿ connection with the

optical properties

of chiral _{smectics near} the

Bragg

diffraction

bands,

_are

fully

confirmed

by

the present results.

The

dependence

of the terms gjj and _{gi on} the pitch _p turns out to be _a

particularly

interesting

feature of the considered model. The gjj term,

given by

equation

(17),

is well

approximated

for _{p «} I

by

~~~

~ll ^~

~4q~(~~/4~]

^~~~~

It is

easily

checked that in this limit gjj ^oc

q-3

oc

p3.

On the other

hand,

_an

essentially

linear

dependence

of gi on p has been

numerically

obtained

by

_means of _a best-fit evaluation

procedure.

The _{term gi turns out to} be

already

much

larger

than _{gjj even} at

moderately

small values of _p. For

instance,

when _p

=

0.21m,

_{one gets}

gjj ⁼ 3A _x

10~~,

_gi

= 3.6 _x 10-3 As _a

consequence, when _{p «}

I,

the

only surviving

components ^{of the} tensor g are

obviously

_gii

=

g22 = gi This

interesting

result is in

good

agreement with the models of

anisotropic

media based _upon the

analysis

of _an

assembly

of oriented

superconducting

circuits

having

_a helicoidal

symmetry _[9]

It should be

finally

noted that the terms gjj, gi

Play

_a

truly significant

role

only

in

particular

cases,

corresponding

to a

degeneracy

^{of the}

eigenfunctions

for

light propagation

within the

S(.

These conditions _occur either for 9inc ₌ 0 in all types ^{of chiral}

phases,

_or for

arbitrary

incidence in

Sl's

^{with critical} tilt

angle

9r~v. In both _{cases, a} small variation of either the incident

angle

9;nc _or the tilt

angle

9 from these values

brings

about _a strong increase in the role

played by

the anisotropy ^of the real part of the dielectric tensor, ^which

gives

rise to

linearly polarized eigenmodes,

and becomes the

dominating

contribution when

(AI(

e (tie fi~( » gjj, gi In this case, the effect of both gjj and _gi is

really marginal.

Note that the condition hi E£ gi ^turns out to be

already

satisfied when the tilt

angle

9 differs from 9rev

by

less than _one

degree.

As

a consequence, the contribution of _{gi may} be

completely neglected

without

losing

in accuracy for

practically

all 9 values but

9rev,

_as shown for instance

by figure

7, where the agreement between the

approximate

and the exact curves is indeed _very

good although

_gi has been _set to _zero.

6. Conclusions.

The

optical properties

of chiral smectic

liquid crystals (including

the

particularly important

class of

cholesterics)

have been shown _to be

accurately described,

in the limit of

pitches

small with respect to the

light wavelength, by

_a

specific

formalism

borrowing techniques formerly developed

to

study

the form

birefringence

of

layered isotropic

media. In

fact,

the standard

Berreman's 4 _x 4 matrix,

describing light propagation

within the stratified

medium,

_can be substituted

by

_a suitable average matrix

corresponding

to a

homogeneous

uniaxial

crystal.

The solutions obtained

by substituting

the effective matrix with the exact _one appear ^to be _{a very}

good

approximation of the exact results not

only

in the limit _{p <}

I,

but also _up to values of the ratio

p/I

_as

large

_{as one} half of the value

corresponding

to the first

Bragg

diffraction band.

Generally speaking,

therefore, the effective uniaxial

crystal correctly

simulates the

properties

of the actual stratified medium. However, under

particular

conditions such _a

simple approximation

is _no

longer

^valid. This

happens

when the exact functions

corresponding

to the

eigenvectors

of the effective medium _are

intrinsically degenerate, I.e.,

either at normal incidence _or at any 9;nc ^for smectics characterized

by

_a critical value 9rev of the tilt

angle.

In

correspondence

of the

degeneration points,

^the

optical

behaviour of the

layered

structures is well

approximated by explicitly introducing

in the model of uniaxial

homogeneous crystal

_new

terms

accounting

^for the

optical activity.

In this _case, the model contains four

independent

parameters, ^{I-e- the} lo, de components of the real part ^{of the dielectric tensor and two} non zero

components gjj, gi of the

pseudo-tensor describing

the

optical

activity. When _p becomes _very small with respect to

I,

_gjj is

predicted

to decrease _as

p3, rapidly becoming negligible,

^while _gi

is observed _to decrease

linearly

with _p. When _p becomes

comparable

to the intermolecular

distances,

_gi attains values

typical

of

crystals displaying optical activity.

In _our

opinion,

these results _are

particularly interesting

from _many

viewpoints.

Let _us underline _some aspects ^of

particular

relevance both in fundamental studies and in

applications.

As is

well-known,

helicoidal molecular arrangements ^have been invoked

long

_ago

by

Pas- teur [10] ^followed

by

Lindman

ill]

and _more

recently by laggard

et al. [12] ⁱⁿ order to

explain

the

optical activity

^of some types ^of

crystalline

structures. The present results indicate that the model

developed

in this work is

adequate

to

explain

the rotatory _power of

crystals having

a uniaxial symmetry. In the limit p <

I,

these media _appear to be

completely equivalent

_to

homogeneous crystals displaying optical activity.

Chiral smectic-C

liquid crystals

with _a tilt

angle

coincident with the critical value 9rev behave

as uniaxial media

simultaneously

characterized

by

_{a rotatory power} and _a

complete isotropy

of the real components ^of the dielectric tensor. These conditions

correspond

to _a pure

anisotropic

rotator with uniaxial symmetry. To _our

knowledge,

this is the

only reported

_case of materials

having

this notable property.

Cholesteric

liquid crystals

with

pitches

much smaller than the

wavelength

of visible

light

have been

recently proposed

_as

electro-optical

modulators [2]. ^{In that} case, the

samples

_are

produced

with the helix axis

parallel

to the

glass plates.

Exact calculations of the

optical

properties are therefore

particularly difficult,

because the formalism of Berreman's matrix is

no

longer adequate

to treat this

particular

_geometry, and _{a more}

complex approach

is needed.

In contrast, ^the approximate calculations

performed according

to _our model _{are very}

simple

and

provide

^{excellent results for all the}

practical

_purposes.

References

[ii

Berreman D.W., J. Opt. Soc. Am. 62

(1972)

502;

Berreman D.W., Mol. Cryst. Liq. Cryst. 22

(1973)175.

[2] Patel J-S- and Meyer R-B-, Phys. Rev. Lent. 58

(1987)

1538;

Komitov L.,

Lagerwall

S-T-, Stebler B. and Strigazzi A., submitted to J. Appl. Phys.

[3] ^{Born M. and Wolf} E., Principles of Optics (Pergamon Press, OXford, 1980) 705;

Yariv A. and Yeh P., Optical Waves in Crystals

(John

Wiley and Sons, New York,

1954).

[4] ^{Van der Ziel} J-P-, Illegems M. and Mikulyak R-M-, Appl. Phys. Lent. 28

(1976)

735.

[5] Oldano C., Phys. Rev. Lent. 53

(1984)

2413.

[6] ^de Gennes P-G-, The Physics of Liquid Crystals

(Oxford

University Press, London, 1975) 219.

[7] ^Allia P., Oldano C. and Trossi L., J. Opt. ^Soc. Am. B 3

(1986)

424.

[8] ^{Landau L. and Lifchitz} E., Electrodynamique des milieux continus

(Ed.

Mir, Moscou 1969) 438.

[9] Lindeu I-V- and Viitanen A-J-, Electronics Lett. 29

(1993)

150.

[10] ^Pasteur L., Ann. Chim. Phys. 24

(1848)

442.

[iii

Lindman K-F-, Ann. Phys. 63

(1920)

621; Ann. Phys. 69

(1922)

270.

[12] Jaggard D.L., Mickelson A-R- and Papas C-H-, Appl. Phys. 18

(1979)

211.

[13] Weiglhofer W-S-, Electronics Lett. 29

(1993)

844.

JOURN~L DE PHYSIQUE II -T 4 N'2 FEBRUARY 1994