Multistability and non linear dynamics of the optical Fréedericksz transition in homeotropically aligned nematics

HAL Id: jpa-00247538

https://hal.archives-ouvertes.fr/jpa-00247538

Submitted on 1 Jan 1991

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Multistability and non linear dynamics of the optical Fréedericksz transition in homeotropically aligned

nematics

G. Abbate, P. Maddalena, L. Marrucci, L. Saetta, E. Santamato

To cite this version:

G. Abbate, P. Maddalena, L. Marrucci, L. Saetta, E. Santamato. Multistability and non linear dy-

namics of the optical Fréedericksz transition in homeotropically aligned nematics. Journal de Physique

II, EDP Sciences, 1991, 1 (5), pp.543-557. �10.1051/jp2:1991188�. �jpa-00247538�

J

Phys

II1

(1991)

543-557 _MAI 1991, PAGE 543

Classification

Physics

Abstracts

61 30G 42.65 64.70M

Multistability and

_non

linear dynamics of the optical

Frkedericksz transition in homeotropically aligned nematics

G.

Abbate,

P.

Maddalena,

L

Marrucci,

L. Saetta and E. Santamato

Dipartimento

di Scienze Fisiche, Pad 20, Mostra d'oltremare, 80125

Napoh, Italy

(Received

28 November 1990,

accepted

7

February 1991)

Rksi~mk. On

prbsente

_un moddle

simple

et nouveau pour dkcnre la

dynamique

de la

rbonentation optique obtenue par un rayon laser dans _un cnstal

hquide nbmatique alignb hombotropiquement

Contrairement _aux

prkcbdents

moddles sur ce sujet, on a rendu compte de

l'kchange

du moment

angulaire

entre la lurnidre et le milieu On _a trouvb _une

dynamique

nche et

assez mattendue, mime dans le cas d'une

polansation

linbaire de la lurradre incidente

Abstract. A _new

simple

model _is

presented

_to describe the

dynamics

of the

optical

reorientation induced by _a laser beam _{into a}

homeotropically

aligned nematic liquid

crystal.

Unlike previous ^models on the

subject,

_we accounted for transfer of

angular

momentum from

light

to the

sample

A nch and somewhat

unexpected dynamics

_is found also _m the _case of linear

polar~zation

of the incident

light

1. Introduction.

The

Optical

Frkedencksz Transition

(OFT)

_was first observed _m Nematic

Liquid Crystals (NLC) by

_{using a}

linearly polanzed

laser beam at normal incidence

[I].

The effect consists m a

Strong

optically

induced reorientation of the _mean direction _n of the molecules _m the

Sample,

when the laser

intensity

exceeds _a charactenst~c threshold The _{unit vector n iS} called the molecular director.

It _iS

commonly accepted

that th~S effect _iS very Sim~lar _to the well known Frkedencksz transition induced _{in nematics}

by

extemal Static fields

[2].

^It iS

argued,

_m

fact, that,

_{m mew} of the linear

polanzation

of the

light

inside the

Sample,

the

liquid crystal

molecules feel _an

average

optical

field which iS constant both _m intensity and _in direction.

Consequently,

all models

proposed

_in the literature to

explain

the

phenomenon

retain th~S picture ^{and lead} to results that _are very Sim~lar _to the _ones obtained _m the Static-field _case

[3]

Nevertheless,

_Some recent expenments

[4]

^have Shown that the interaction between nematics and

light

iS

essentially

different from the interact~on between _nematics and Static fields because

liquid crystals directly couple

with the

angular

_momentum camed

by

^the

optical

field. In

Simple

geometnes, the transfer of

angular

_momentum from the radiation to the

liquid

crystal

_may ^lead to Self-Induced Stimulated

Light Scattenng [5].

544 JOURNAL DE

PHYSIQUE

II lK° 5

This suggests ^{that also} m the

simplest

_case,

namely

_m the

optical

Frkedencksz transition with

linearly polanzed light

at normal

incidence,

the

coupling

between the

angular

momentum of the

light

and the LC medium cannot be

ignored

_so that

completely

new

features should appear

having

no

analog

in the static-field _case

In th~s work _a

simple

model _is

presented

^for the OFT w~th

linearly polarized light

at normal incidence where

angular

momentum balance is taken _into account. It will be shown that

unforeseen effects _as

multistability, hysteresis

and

oscillating

relaxation toward

steady-state

appear

naturally

_{as a} consequence of

angular

momentum transfer All these

effects,

that may

occur also _{very near to} the threshold for the

OFT,

cannot be denved from the known models

and their observation _{was never}

reported

_m the

literature, although they

should be

certainly

observable _{m accurate} measurements

The _main difference between our model and all others

reported

_m the literature _is that _we allow for _a

precessional

motion of the director _n around the direction of propagation ^{of the} laser beam _even if the laser beam is

linearly polanzed

In all previous

works,

_m

fact,

_{n was} bound to stay m the

plane

formed

by

the

polanzation

direction and the propagation direction of the beam

Releasing

these constraints leads _to the _occurrence of _new

phenomena,

related to

angular

momentum

transfer,

that will be the _main

object

of

study

of this work.

Since _our theoret~cal model _seems to be

interesting

_m its _own

nght

and differs

sensibly

from all other models existing m the

literature,

it w~ll be the

object

of the present paper The

expenments

^{about the} observation of the _new effects

expected

^from the model will be

presented

_m

_separate

_paper.

Th~s work _is

organized

_as follows _m section 2 the relevant geometry is

presented

and the

general _equations

_govemmg the motion of the molecular director _as well _as the

change

_m the

polarization

of the laser beam _are denved from _a

general LagrangJan formalism, presented

elsewhere

[6]

In _section 3 the equations are

specialized

to the _case under

study

and _some

approximations

are used to put them m a

simpler

form The

validity

of the

approximations

^involved is also discussed

Finally,

m section 4 the _new effects

expected

on the basis of the model _are

presented

and their eventual

expenmental

observation _is discussed

2. The model.

The results

presented

_m this section have been

already reported

_m

_[6]

We consider _a th~n film of NLC of thickness L between

glass plane

walls coated w~th surfactant for

homeotropic alignment.

The

anchonng

at the wall _is

supposed

to be

strong.

^The

local

alignment

of the molecules in the film is accounted for

by

the unit vector

n(r).

Due to the

homeotropic anchoring,

_in the absence of extemal field _{n is}

uniformly

directed

along

the z-axis normal to the

sample

walls At _some instant _a monochromat~c laser beam with _an

arbitrary polanzation

_is focused at normal incidence onto the film When the

intensity

I of the beam at the

sample

reaches the characteristic threshold I~~ for ^the

OFT,

the molecular dJrector _is reoriented and

changes

_m time until _a final

steady

state _is reached. As shown

by

the

expenment reported

_m

[4],

the final

steady

state may also be time

dependent,

corresponding

to a continuous precession of _n around the _z-axis- In the _case of l~near

polanzation

of the incident

beam, however,

both

expenment

^and

theory

lead to a time

independent

final _state.

The basic equations govemmg the interaction between the director _n and the laser

optical

field have been denved m

[6]

_m the limit of

plane

wave

(all

fields

depend

on z

only)

and of slow

envelope.

Both these

approximations

are

largely exploited

_m the literature and

usually

met m

ordinary experiments.

In

particular,

the consistency of the slow

envelope approxi-

lK° 5 MULTISTABILITY OF OPTICAL FREEDERICKSZ TRANSITION 545

mation

(often

referred _{to as} the Geometric

Optics Approximat~on (GOA))

_requires that the

matenal

birefringence

is

low,

_viz. An

= n~ n~ ^«

I,

where _n~ and n~ are the

extraordinary

and

ordinary

indexes of the matenal. This condition is

usually

satisfied _m

ordinary

nematic

liquid crystals. Moreover, light absorption

is

neglected,

_so that the

only

effect of the

sample

on the laser beam _{is a}

change

of its

polanzation.

In the

following,

the director _n w~ll be described

by

_means of _its

polar

and azimuthal

angles

0 and

4 [n

=

(sin

0 _cos

4,

sin 0 _sin

4,

_cos 0

)].

Because nematics _are not

polar,

_n and

n are

physically indiscemible,

_so that _we have one-to-one

correspondence

with distinct

physical

states

by taking

0 in the range

[0, l12 ar]

and

4

m the range

[-

_{ar, ar}

The

polanzation

of the laser beam _may be described

by

_means of _its

ellipticity

e and

ellipse

onentation

angle #r. (The

absolute value of _e is the ratio between the minor and

major

_axis of the beam

polarization ellipse

and its sign is positive ^for left-handed

polanzation

and

negative

for

nght-handed polanzation.

The

angle

_{#r is} the

angle

formed

by

the

ellipse

_major axis w~th respect to an x-axis fixed _m the

laboratory.)

It turns out to be useful

introducing

also the

quantity I=

₌

(Iiw)

_e, where _w

=

2

ariA

_is the

frequency

of the laser beam and I _its intensity

(average Poyntmg

vector

along z).

It _is ev~dent that

I=

is the component of the average

angular

momentum carded

by

the beam

along

its propagation ^direction

The

time-independent

equation ^of motion for the azimuthal

angle 4

^{and the}

polar angle

0 of the director _can be obtained from the m~nlrnization of the total

(elastic

+

optical)

free-

energy

j6j

F =

Fo

₊

w, (i)

where

Fo

_is the elastic free energy

density

of the

NLC,

_g~ven

by

Fo

= (k~~

sm~

_{0 + k~~}

cos~

₀

sin~

_if

(d4idz)~

₊

2

+

(kjj ^sm~

0 ₊ k~~

cos~

₀

(do idz)~ (2)

2

and W _is the energy

density

of the

optical field,

that _can be wntten out m terms of _e and

pas

W

=

(In~/c)

₊

(1/2 c)

An

(if

I ₊

fi

cos

[2

_#r

4 )] (3)

In equations

(2)

and

(3) k,,

denote the NLC elastic constants and _c the

speed

^of

light (cgs

units are

used). Finally,

the

birefnngence

An

(if)

_{as seen}

by

the

optical

_wave is

given by

~ ~ n~ n

~

~ ~~~

~/n~ ^cos~

~ ₊

nj Sm~

_~ ^~

As stressed in

[6],

^{the total}

free-energy

F must be considered _{as a} Hamiltoman H for the whole system,

provided

we introduce the

quantities

p_~ = (k~~

sin~

_{if +}

_k~~ cos~

_if

sm~

_O

(d4/dz)

p~ =

(kj, sm~

if ₊

k~~ cos~

_if

)(dir/dz)

p~ =

I=

= +

(I/w)

_e

(5)

as moments conjugate to the

generalized

coordinates

4,

Wand #r,

respectively,

and _express F

as F

= H ~,

o,

p, p _~, p _~, p _~

)

Immediate consequences of this Hamiltoman formulation _is

that,

_m

time-independent

states, H itself

(I

e the total

free-energy F)

_is constant

along

the

sample (F

= const

)

and _so

546 JOURNAL DE

PHYSIQUE

II lK° 5

is the quantity

p~ + p _~ =

(k~~ sm~

₀

+

k~~ cos~

₀

) sm~

0

(d4 /dz)

₊

(I/w )

e ₌ const.

(6)

The conservation law

(6)

_expresses the conservation of the total

(elastic plus optical)

average

angular

momentunl flux

along

the beam

propagation

^direction.

The distortion

qpgles 4 (z)

and if

(z)

_as well _as the

polarization

state #r

(z), e(z) along

the

sample

_are obtained from Hamilton's

equations

_:

d4/dz

= ~

~~

~ ~

(7a) (k~

_sin 0 + k

~~ cos 0 sin if

dir/dz

= ~

~°

~

(7b) kji

_sin if +

k~~

cos if

dp~/dz

=

(I/c)

An

(if ) $7~

sin

[2(#r 4 )] (7c)

dp~/dz

=

(1/2 c) An'(if (I

₊

ficos _[2(#r

4 )])

₊

+ sin if _cos 0 (k~~

kj

j

(dir/dz)~

_[k~~ 2(k~~

k22) sm~ ill (d4 /dz)~) (7d)

~~"~~

"

~~°'~

~~ ^~~

~~ fi

_~°~

i~~~' ~ ~i

_~~~~

de/dz

=

(w/c)

An

(~ /fi

sin

j2(#i ~ )j (70

Equations (7a-o

must be

supplemented

with

boundary

conditions at the

sample

walls For

homeotropic alignment

these _are

0

(0)

= 0

(L)

= 0

d4/dz (0)

=

d4 /dz(L)

= 0

(8)

Moreover,

_we must impose

e(0)

= eo and

#r(0)

= #ro ^at the

input plane

z ₌

0,

where eo and #ro are the

ellipticity

and

ellipse angle

of the incident laser beam.

These results hold for

time-independent

states

only.

In the _not

stationary

case we must add

phenomenologJcal

v~scous terms _on the left of

equations (7c)

and

(7d) having

the form y

sm~

₀

a41at

and _y

aolat, respectively,

where _y is _an

appropnate viscosity coefficient,

and

change

all

total

_{denvatives into}

_pamal

_denvat~ves

_(d/dz

-

alaz) [7].

In th~s _case, the

angular

momentunl balance equation assumes the fornl

,L

y

a4 fat ^sin~

0 dz

=

(I/w)

_he,

(9)

o

where he

=

e(L) e(0)

is the

change

of the

polanzation ellipticity

suffered

by

the beam _m traversing ^the

sample.

Finally,

_we note that

equations (7e)

and

(7f~

can be rewritten _m terms of Stokes' reduced

parameters

s,

(i

=

1, 2,

3 _m the

simpler

form

ds/dz

=

fly

x s

(10)

where _s

=

(sj,

_s~,

s~)

with

si =

ficos

2 #r

s~ =

fi

sin 2 #r

s~ = e

(I1)

lK° 5 MULTISTABILITY OF OPTICAL FREEDERICKSZ TRANSITION 547

and fl

=

fli(0, 4 )

_{is gJven}

by nj

=

(w/c)

An

(o (cos (2 ~ ),

_sin

(2 ~ ),

o

(12)

The

boundary

condit~ons _on equat~on

(12)

_are

ev~dently s(0)

=

(sjo,s~o,s~o),

where sic, s~o and _{s~o are} the Stokes

parameters

^{of the incident beam.}

3. The

simplified

model.

Equat~ons (7a-fl

describe the laser induced molecular reonentation _m the NLC and the

change

m

polarization

of the beam in the

plane-wave

and slow

envelope

approximations ^It is

evident that these equations are very

awkward,

_{so we} need

simplification.

In view of the

boundary

conditions

(8),

wh~ch must hold at any ^time t, we may

expand 0(z, t)

_as

0

(z, t)

=

~0 £ 0~(t)

_sin

(narz/L ) (13)

For laser

intensity

_very close to the threshold to induce the

OFT,

0 _is small and _we may retain

only

the first term in the _sum

(13),

_so that

O

(z,

t

)

= ~

j

(t)

sin

(arz/L) (14)

Moreover,

since if _is

small,

_we retain

only

terms up ^to

if~

m equations

(7b)

and

(7c)

This

yields

the

following

approximate equation for 0 _:

~~

=

(l

k0 _~)

~~~

^{k0 ~~ + » ~}

i

₀ ^~ _Ail

)

l ₊ sjo cos 2

4

,

(15)

at az @z 3

where _we

posed

~33

T =

yj L~

~

~~

n~

8

=

n~

~

9-5&~

8&~

~

^I

~Ith

~

car ~ ~

k~~ ~~

~~

n~

L~(&

^~ l

I~~ is the threshold intensity ^{for the OFT w~th}

linearly polanzed light.

The _same expression of I~~ was obtained with _a different

approach

in

[2]

Expenments

show that the azimuthal

4 angle

_may rotate _over several _{ar, so we} cannot consider it small _in

general.

Nevertheless _{we assume}

that,

_{in mew} of the

boundary

conditions

(8),

which _{are «} free for

4,

we may

neglect

the _twist of

4

_in the intenor of the cell. In other

548 JOURNAL DE

PHYSIQUE

II lK° 5

words,

_{we assume}

4 (z, t) large,

but

a41az

small

(a4iaz

_is

exactly

_zero at the wallq At the zero-order

approximation

_we have

a4 laz

_m

0,

so that

4

₌

4 (t) only. Inserting

then equation

(14)

into equation

(15), multiplying by

sin

( arz/L)

and

integrating

from _z

= 0 to _z

= L

yields

_a

single equation

for

ifj(r)

_:

ifj(r)

=

ar~ifj (1- _kill)

+

ar~iifj(I -Aif))(1+ _{sjocos2 4 ), (17)}

where

ifj

=

dill/dr

and the reduced time

r is given

by

the first of

equations (16).

The evolution equation for

4 (r )

is obtained

directly

from the

angular

momentum balance equation

(9). Inserting expression (14)

for

if(z, r)

ml and assuming

4 independent

of z, the

integration

_in

equation (9)

_can be

performed yielding

~2

~²

I

~ (T

"

iS3(L,

_{T S301}

,

(18)

2(1

₊ L

where _we introduced the dimensionless thickness

I

=

(w/c) n~(&

I L

Within these

approximations,

the equation

(10)

_goveming the

change

of the beam

polanzation

_can be solved

analytically.

It _is convenient passing ^to the _new

independent

vanable

u(z,

_T defined

by aulaz

=

(w/c)

An

(if(z,

_r

)).

Then _we obtain

aslau

=

hi _(r

x s, with

hi _(r

=

(cos

2

4 (r ),

_sin 2

4 (r ), 0),

whose solution _is

S(U(z,

_T

))

"

fli(fli'So)

₊

[(fli

_x

_So)

_x

fli)

_Cos

(u)

₊

(fly

_x

so)

_Sin

(U) (19)

Moreover,

_using equations

(4)

and

(14)

and _since if _«

I,

_we find

.L

U(Z,

_T

"

(W/C)

An

(i~ (z,

_r

))

dz

" "

(T)[ (~/L) _{(1/2 ir)}

_Sin

_{(2 irz/L ))}

,

(20)

where _we introduced the

birefringence phase angle

_a

(r)

_g~ven

by

«(T)

=

U(L, T)

=

°~()~~~ _i~l(T)

=

Pi~l(T) (21)

The

angle

_a is the

phase

difference accumulated

by

the

optical

_{wave in}

traversing

the film with

respect

to _an

ordinary

_wave

The

polanzation

state

s(r

=

s(I,

r of the

light

_emerging from the cell _is obtained from equation

(19)

_as a function of

4(r)

and

ifj(r), by substituting

for _u its

end-point

value

a(r).

In

particular

_{we may} evaluate

s~(I,

r and _{insert it into} equation

(18), obtaining

Finally, using

_{again equation}

(21),

_{we may} rewrite

equation (17)

in terms of

a

(r)

_:

a(r)

=

(nr2(a ^{(k12 p)} a2-jiaii _(Alp) aj(1+sjocos2 ~)j ⁽²³⁾

Equations (22)

and

(23)

_are the _main result of this _section.

They

form _a set of _two first-

order

ordinary

differential

equations

in the unknown functions

4(t)

and

a(t)

Once _a

M 5 MULTISTABILITY OF OPTICAL FREEDERICKSZ TRANSITION 549

solution

(4 (r ),

_a

(r ))

of equations

(22)

and

(23)

_is

found,

the

polar angle

if

(z,

_r

)

_{is g~ven}

by

equation

(14)

and the

light polar~zation

state at each

point

_in the

sample

_{is g~ven}

by

_equation

(19)

w~th _u derived from equation

(20)

We notice that

although ifj

_is

supposed

_very

small,

_we cannot consider the

birefnngence

phase angle

_{a as a} small

quantity,

^{because the} coefficient

p

_in

equation (21)

_may be very

large

in

practical

_cases. For

example,

_in the _case of the _nematic 5 CB

(n~

₌

17,

_n~

=

1.5)

and

assuming a film th~ckness L of

75~m

and _a

wavelength =0.515~m,

we have

p

=

~~~ ~~

= 70 In

particular,

_we cannot

expand

sin a and _{cos a in}

equation (23)

_{as a}

4

power senes of _a, unless

ifj

«

p

^~~

=

0.01,

_m

typical

conditions.

4. The _case of linear

polarization.

Although

_equations

(22)

and

(23)

hold _in

general

for any

polanzation

state of the incident

beam,

in the

following

we assume the

polarization

to be

linear,

_so ^that we have sjo =

I,

_s~o

= 0 and s~o = 0. In this case, we see that _a

= 0 and

4

= 0 _{or ± ar are} trivial

steady

solutions of equations

(22)

and

(23). Obv~ously, 4

= ± ar represent the _same

physical

state

Stnctly speaking,

when _a

= 0 the value of

4

_is undetermined. It _is not _{so in}

equations (22)

and

(23)

To understand the real

physical

_mean~ng of these equations we must

study

their behavior _in the hm~t of _very small

a. The solution of

equations (22)

and

(23)

for small a, in

fact,

_represents the initial

dynamics

of the system ^{when it} is moving from the undistorted

homeotropic al~gnment. Retaining only

terms _m the first order in _a, equations

(22)

and

(23)

can be solved

yielding 4(r)

=

tan~~ (tan 40e~~"~~~)

Ii

^{~ i l + tan}

~

4

o e

~ _"^~~~

~ ~ ~° ~~~

2 ^~ ~~ _~ ~

4 ~~

l ₊ tan ^~

4

o

'

~~~~

where ao « I and _ar

SW

o 5 _{ar are} the initial values of _a and

4, respectively.

If _no extemal field

except

the

optical

field _is

applied

to the

system,

^{the induced} reorientation starts from noise and ao and

40

must be considered _as random initial data Notice that

40

is not bounded to be small.

Equations (24)

show that whenever ao and

40

may

be, 4 (r

tends

asymptotically

to 0 _{or ± ar as r}

- cc, and _a

(r

tends

asymptotically

to _zero for

I

=

I/I~~

_< I and to infinite for

I= I/I~~

_» I.

Then,

_m order _to induce the reorientation starting ^{from noise} m the

undistorted _state

a =

0,

_we must have

I» _I,

ie. I must be

larger

than the threshold

I~~ ^{for the OFT The second of} equations

(24) shows, however, that,

for

I»I

_and

40

#

0,

_a

(r

decreases first and then _increases again

passing through

_a min~mum value.

Only

for

40

= 0 _we have _an

exponential

_grow of _a

(r ),

as claimed _m all _previous works. Th~s not

monoton~c behavior of the

birefnngence

_a m time could be

observed,

for

example, by

prepanng the system m a

slightly

rotated state with _n

lying

_{m a}

plane form~ng

_an

angle do

# 0 with the

polanzation

of the laser beam.

As the time goes on, a increases until saturation is reached. Saturation _is

govemed by

higher-order

terms in a m equation

(23).

An _exact solution of equations

(22)

and

(23)

for linear

polanzation

_is

4

= 0 and _a

(r

_g~ven

by

ao a1

a

(r )

₌

,

(I

_»

1) (25)

w~(I-

I) _T

"0+

("i~"o)e

^~

JOURNAL DE PHYSIQUE Ii T V 5 MA< (W( 28

550 JOURNAL DE

PHYSIQUE

II M 5

where ao » 0 _is the initial value of _a and

2p(I- _I)

ai

(1)

=

(26)

2Ai-k

The solution

(25)

_was

already

studied in

Ong's

_paper

_[8]

This solution shows that if aj > 0 then _a

(r )

tends

asymptotically

to the _saturation value _a~

= a j. If _aj

<

0, instead,

a

(r)

_grows

asymptotically

to

infinite,

_so that saturation is

governed eventually by

terms of the order of _a ^~ _or

h~gher,

which _we have

neglected

_{in equation}

(23)

Since

4

m 0

along

this

solution,

the

polanzation

of the beam stays linear for _any value of _a

(r)

The _{case a}

j ^< 0 _was discussed _{in some} detail

by Ong [8],

who showed

that,

_in this _{case, a}

first

order

(I

e.

discontinue)

OFT _{is to} be

expected

The cond~tion

a j ~ 0 _is

always

satisfied if the matenal constants _{are so} that k

» 2 A. In _mew of

equations (16),

th~s _is

precisely

the _same condition obtained

by Ong [8].

It is worth noting that PAR

~p-azoxyanisole)

and _a few mixtures of nematic

hquJd crystals

_can have k _» 2 A

[9]. But,

to _our

knowledge,

_{any attempt}

made _to observe _a first-order OFT _in these matenals failed. For this _{reason in} this work _we

assume aj

positive

We pass now to consider _{in some} detail the

steady

states of the system

a)

The

steady-states.

The

steady-states

of the system ^{for linear}

light polarization

_are

obtained

by

_{setting sjo}

"

I,

_s~o

=

0 and

4

= if

= 0 _in

equations (22)

and

(23)

The

resulting time-independent

states

($, b)

_{are g~ven}

(implicitly) by

sin 2

$

_{sin &}

= 0

I(

_{I + cos 2}

_{4 )}

₂

~

~ (27)

Ai(I

+ cos 2

4 )

k

Since _{a is} positive semidefinite _we must retain

only

solutions of

equations (27)

with

& ? 0. Below the threshold

(I

< I

),

_no distortion _can be

optically

induced in the

sample

and the

only

stable state _is the _state

a =

0. We shall

consider, therefore, only

states at laser intensity

I

_{greater than the threshold. At all}

_steady

states the

polarization

of the beam

emerging from the

sample

_is still

linearly polar~zed

_as the input

light,

_as

required by

the

angular

momentum conservation If this _were not the _{case, m}

fact, angular

momentum would be

deposited

into the

sample, producing

rotation of the molecular d~rector _n around the beam

axis.

A first

steady

state is

$

= 0

(or

_±

ar)

and &

= a j

(I),

_as

_{given by}

_equation

₍₂₆₎

_{This is the}

state that the system should reach at saturation,

provided

the

4 angle

_is bound to stay on the

polanzat~on plane

of the incident

light.

As

prev~ously mentioned,

this state _was the

only steady

state considered _m the literature In th~s state, the

polarization

of the beam remains

unchanged (linear)

at any point m the

sample.

We w~ll refer to th~s state as the unrotated state of the

system

Besides this state,

however,

_we have _{two more sets} of

time-independent

states,

correspond-

ing ^to n out of the

light polanzation plane

We will refer to these bets of states as the rotated

states of the system The two sets of states are g~ven

by

&

= nor and

$

= ±

$~(i)

(n

=

0,

1, with _n odd and _{n even,}

respectively

For each _n,

$~ (i)

is found

implicitly

from

equation (27)

_as

tan~ $~

=

i(

^~

^~

^{~ ~~"} _l

(28)

2

p

knot

lK° 5 MULTISTABILITY OF OPTICAL FREEDERICKSZ TRANSITION 551

For the set of

steady

_states with odd _n, the

optical

retardation

phase

_{a is} locked _at

ar and

sample

behaves _{as a}

-plate

rotated at _an

angle $~(i)

In these states, the

polanzation

of 2

the beam _emergJng from the

sample

_is still

linearly polarized,

but _is rotated with

respect

to the incident

polanzation

direction of _an

angle

2

$~(i), _depending

on the incident power This fixed-retardation behavior may be useful for

applications.

A standard

Lyapounov's stability analysis

shows that these states are stables foci

(unless $~

is very

small,

m which _case

they

_are

stable

nodes)

It _is

expected, therefore,

that the director _n

approaches

these states with

damped

oscillations

In the set of

steady

states with _{even n,} the

sample

behaves _{as a}

A-plate

In these states, the beam emerges with the _same

polanzation

_as it enters,

although

the

polanzation

state may vary inside the film. These states are found to be unstable saddle points ^and should be not

observed

A

plot

of

4~(I)

_{for both sets of states}

is shown _m

figure I,

the dashed _curves

refernng

to the unstable states

6

11 12 13

1/l~~

Fig

I

Steady-state

director

4-angle

_{as a} function of intensity

I,

Solid curves refer to a = nw with odd _n and _are stable states Dashed refer to _a

= nw with _{even n} and _are unstable states In the numerical calculations _we used

typical

values for nematic 5 CB _as

reported

_m Ong's _paper [3] ^and film thickness L ₌ 75 ~Lm and

wavelength

A

= 0 _{514 ~Lm}

The rotated _states coalesce w~th the unrotated state

4

₌

0,

_a

=

aj(I)

_at charactenstic

branching

_points

I

=

i~ (n

=

0,

1, gJven

by

_a

j(i~)

= nor, viz

i~= ^~~~~~" =l+I(2A-k) _{(n=0,1, ). (29)}

2p-2Anar p

In the last _{term we} used the fact that

p

_is

large.

We _notice that because _m all

practical

_cases

p

is

large

and A and k _are of the _same order of

magnitude,

the difference

hi

=

i~

~

i~

_is

usually

_very

small,

I-e the

branching

_{points on} the _curve

&(I)

are very dense In accordance with the

alternating stability

of the rotated states, it turns out that the unrotated state

changes

its

stability

at the

branching

points

(29).

Unlike claimed _m all previous

work,

the present

analysis shows, therefore,

that the unrotated state is sometime destabilized

by

the

4 degree

of

freedom. This _is made evident

by linearizing

_equations

(22)

and

(23)

around the unrotated

552 JOURNAL DE

PHYSIQUE

II lK° 5

state

4

=

0,

_a

= a

j(I).

_The fluctuations &a and

SW

around th~s state _are found to be

~ _{" ~}

~ ~

" ° ~~~ "

~~~

~ ~ ~

&~ (~ )

_{_}

&~~

exp

(- ^{(~r2 I)}

^{~~~ "} r -

~ ~°~ _{~~~ "}

" ~

(30)

a j cc for sin _a

j < 0

We _see that the fluctuations of the

angle

_a

(or if)

_are

damped

out, ^{but the} fluctuations of the

4 angle

_{may grow m} time if _{sin aj}

< 0 The motion of the director _n is henceforth

asymptotically

^stable if sin a1 » 0 and

asymptotically

unstable if sin _{al <} 0.

Therefore,

if the laser intensity

I

_is

so that

i~

_<

I

<

i~

~ j with odd _n

,

(31)

the molecular director will deviate out of the

plane

of

polanzation

of the

light ~plane

4 =0)

and w~ll reach

finally

some

equilibrium

state which _is rotated at an

angle

4

# 0. The saturated undistorted state

4

=

0,

_a

= a

j(I)

is unstable and cannot be reached

by

the system.

It should be

pointed

out,

however,

that _{even m} the _cases where the unrotated state

4

= 0 is

stable,

it may

happen

that the

dynamics

of the system may dnve it to some other rotated

(4

#

0)

stable state,

preventing

it from

reaching

the state

4

=

0,

_{a = a}

j(I).

b)

The

dynamics

The system is charactenzed

by

a

large

number of

steady

states, so its

dynamics

_is

complex

The differential

equations (22)

and

(23)

have been

numencally integrated

for the _case of linear

polanzation

The results _are

reported

_m

figure 2,

where the

6

1/l~(O)=

1045 X " COO

4

z

/

0

2

4

-6

-6 -4 -2 0 2 4 6

ny

Fig.

2.

Trajectories

of the director _{n m} the

(nm n~)-plane

for linear

polarization

and

I

= 045. For

this intensity no stable state ex~sts at 4 ₌ 0

lK° 5 MULTISTABILITY OF OPTICAL FREEDERICKSZ TRANSITION ₅₅₃

trajectories followed

by

the director _{n m} the

xy-plane

_are drawn. The

figure

refers to _{a case} where

I

is m the interval

(31),

_so that _no stable state exists w~th

4

=

0. The

figure

shows that if the initial state _is close

enough

to the undistorted state _a

=

0,

the system will evolve towards _a rotated final state,

execut~ng damped

oscillations _m both _n~

= sin if _cos

4

and

n,

= sin if sin

4

Th~s

oscillating approach

to

equilibrium

should be

experimentally

observ- able. Similar results _are obtained _m the _case where

I

_is

not in the interval

(31).

The unrotated state _{can now} be

reached,

since it is

stable,

but many

trajectories

still _exist that fall

spiraling

around _some other rotated state

Moreover _we found that the

dynam~cs

of _n is very sensitive _to the

ellipticity

parameter x = sm~

~(s~o)

^{of the}

input

beam A value of _{x as} small _as 0

01,

for

example

dnves the

2 '

director _{to a} quJte different final

steady-state,

_{as it is} shown in

figure

3 We notice that the symmetry m the

change 4

_-

4

_is

strongly

broken for _x #

0,

_so

that,

for instance, the _state C which _is the

syrnrnetric

of the stable state

B,

becomes

actually

unstable

6

' 1050 x = 001

4

z

/ 0

2

4

-6

-6 -4 -2 2 4 6

ny

Fig

3

Trajectones

of the director _{n m} the

(n~n~)-plane

for almost linear

polanzation (X

=

0 01) and

I

= 105

Trajectones starting

from small _a end onto the state B instead of A, as it

happens

for exactly linear

polanzation.

Unlike state B, the state C is unstable

5. The _case of circular

polarization.

Putting

_sjo

=

0,

_s~o

= ± I _m

equations (22)

and

(23) they

_can be solved m the form ao a~

a(r)=

_{-a~ as r-cc}

w~(I-2)T

ao +

(a~ ao)

e ~

~ (r )

=

~

o

(» 21) ii ^~~j~j~~~

ds

-

~

_{o +}

fir

as r _- cc

,

(32)

554 JOURNAL DE

PHYSIQUE

II lK° 5

with

~~~21~~

fi

=

(ar~i)

^~

a~

p (I- ₂₎

a~ =

(33)

Al

_k

Wee _see

that, provided I

» 2

(which

_is the threshold to induce reorientation with

circularly polarized light)

after _a transient the

system

is put into un~form rotation with constant

asymptotic angular velocity fi _[10]

_The

asymptotic

motion of _{n is}

actually

_a uniform precession around the beam

propagation

direction with fixed tilt

angle 0~

=

11

_and

angular velocity fi.

_This

precessional

motion _was

effectively

observed _m

[3].

^A

plot

of

fl

_and

if~

_as functions of the reduced

intensity I

_{is shown}

m

figure

4 The _zeroes of

fi _correspond

_{to the}

_vanish~ng

_{of the}

angular

momentum transfer

[see Eq (18)]

8

8

6

~ -

_ z

~ '

i~

^~ 4

-

~

~n -

A ^C

z z

0 0

Z 2 2 2 2 3 2 2 Z 2 2.3

1/lt~(0) 1/lt~(0)

Fig

4 Polar

angle

~ and

angular velocity

^fl as functions of

I

_for circular

polarization

of the input beam

lK° 5 MULTISTABILITY OF OPTICAL FREEDERICKSZ TRANSITION 555

The

stability analysis

of the solution

(32)

is made in the usual way

by considenng

small fluctuations

SW (r)

and

&a(r)

We find

-~w~(I-2)T 8a(r)

=

Bare

^~

~4(T)

"

81~(r)

=

~2

^{CDS a~- a~~~~ ~~}

j

~ "2

~"°~

^{~~ ~~ ~~~}

(34~

Both the fluctuations of the

angle

_a

(or if)

and of the

angular velocity fi

_{tend to be}

damped

out _m time above the

threshold,

so the

precession

motion is

orbitally

stable We notice

however,

that the

hysteretic

behavior of tins reg~me as

reported

in

[3]

_is not found _m the hm~t of the

present

^{model ThJs} may be due to some amount of twist inside the

sample

which tends to overstabilize the precession motion,

leading

to

hysteresis

A

generalization

of the present model accounting also for twist should be then

adv~sable,

_in order to

explain

the

expenmental

observations.

6. The _case of

elliptical polarization.

In the _case of

elliptical polarization,

the director

dynamics

_can be studied

solv~ng equations (22)

and

(23) numencally. Typical

results _are shown _in

figures

5 and 6.

The threshold to induce reonentation for

general

_{x is given}

by

ith(x)

=

~

~

(35)

For

I

»

i~~(x),

_two

possible asymptotic

reg~mes were found _:

I) Stationary steady

states

2)

Limit

cycles.

Limit

cycles, however,

have _never been found for _x

< I

=

0 4 As shown _m

figures

4 and

5,

for _x

» I, the limit

cycle

_is reached when the laser

intensity I

is increased to destabil~ze the two foci at the center of the

figures

The hm~t

cycle

corresponds actually

to a

non-un~fornlprecession-nutation

of _n around the laser beam axis.

The observation of this kind of regimes was

actually reported

in

[4]

^The

persistent

oscillation

reg~mes observed _m the second of

[4]

_was not

found, however,

from _our model As _we

pointed

out at the end of section 5 for the

hysteresis

of the rotation

regime,

the occurrence of

persistent

oscillations could be

essentially

related to the presence of twist _in the

sample,

wh~ch

is

completely neglected

_in the

present

^model.

7. Final remarks.

Although

the model

presented

_in th~s paper is _in many

respects oversimplified,

it shows

clearly

that the transfer of

angular

momentum from _a

l~ght

beam to _a

l~quid crystalline

medium

leads,

in

general,

to

peculiar dynamical

etTects

having

_no

analog

_in the d-c--field _case Some of these effects have been observed

experimentally

_using circular _or

ell~ptical

polanzation [4]

In all

reported

_{expenments on} the

Optical

F?kedencksz Transition with linear

polanzation

at normal

incidence, instead,

neither

multistability

_was found _nor

oscillating

relaxation to

equilibrium

_was

reported [I].

We believe that this is due _to the fact that _m usual

experiments

the

phase

retardation

a was measured

by simply looking

at the diffraction

rings

_in the far field

pattem beyond

the

sample (yielding

to _an accuracy on a of

roughly

_±

ar)

and that the power of the incident beam _was varied _m

relatively large steps.

^In

556 JOURNAL DE

PHYSIQUE

II lK° 5

1/I~(0)=

_{2.030 x}

= 070

5

/

5

-1

-1 5 0 5

ny

Fig.

5 Trajectones of the director _{n m} the (nm

n~)-plane

^for

elliptical polarization (X

= 0 7 and

I=

_{19 All} trajectories end _{m a} stable state

1/I~(O)=

1.900 _{X "} 070

5

5

-1

-1 5 0 5

ny

Fig

6 Trajectones of the director _n m the

(nm n~)-plane

^for

^elliptical polanzation

(X _" 0 7) and

I

=

2.03 All trajectories ^end m a limit

cycle.