PHYSIC AL
REVIE%'
A
VOLUME
10,
NUMBER 4 OCTOBER
1974
Study
of pretransitional
behavior
of
laser-field-induced
molecular alignment
in
isotrolnc
nematic substances
George
K. L.
Mong
and Y.
R.
Shen
Department
of
Physics,
University
of
California
and
Inorganic
Materials Research
Division,
Lawrence
Berkeley Laboratory,
Berkeley,
California
94720
(Received
15
April
1974)
%'e
have measured the
optical
Kerr effect
and
the
intensity-dependent ellipse
rotation
in order
to
study
the pretransitional
behavior
of
field-induced molecular
alignment
in
the isotropic
nematic
substances p-methoxy-benzylidene p-n-butylaniline
(MBBA)
and
p-ethoxy-benzylidene-p-butylaniline
(EBBA).
The
results
agree
well with
predictions
of the
Landau-de Gennes model. Both
the
order-
parameter
relaxation
time and
the steady-state
field-induced
birefringence show
critical
divergence
towards the
isotropic
nematic transition with
a
(T
—
Ta)
'
temperature dependence.
In
the case
of MBBA,
our results are
also
consistent with the results from
light
scattering,
but the
method
we
use
is
perhaps
more
straightforward
and accurate. The nonlinear refractive
indices and
other
relevant
parameters
of
the
materials are derived from the
experiment.
I. INTRODUCTION
%'e
have shown recently
by
measuring
the
optical
Kerr
effect and
the
intensity-dependent
ellipse
ro
tation
that the nematic
compound
P-methoxy-ben-
zylidene
P-rr-butylaniline
(MBBA)
in its isotropic
phase
has
a
large
nonlinear
refractive
index
and
a pronounced
pretransitional
behavior.
'
The
large
field-induced
refractive index causes
a
moderate-
ly
intense laser
pulse
to
self-focus
readily
in
such
a
medium2'
and
induces other nonlinear
optical
effects such
as
stimulated Raman and
Brillouin
scattering.
'
On
the
other
hand,
measurements of
the
optical-field-induced refractive index and
its
pretransitional
behavior
yield
directly
informa-
tion about
the molecular orientational
properties
of
MBBA'
and
provides
a
stringent
test on
the
Landau-de
Qennes
model.
'
The same
information
can be obtained from light-scattering
experiments,
'
but the measurements are more complicated
and
less accurate.
%'e
have
now
extended our
optical
Kerr and
eQipse
rotation measurements to
P-
ethoxy-benzylidene-P-butylaniline
(EBBA)
which
is
homologous
to
MBBA.
We
have
found
similar
results in
EBBA
as in
MBBA.
In
particular,
the
results
again agree
well
with
the
predictions
of
the
Landau-de
Gennes model. Here we would
like
to
give
a
detailed account of our work on both
MBBA
and
EBBA.
In
Sec.
6,
we review
briefly
the
theories behind
our
measurements.
In
Sec.
ID,
we
show our
ex-
perimental arrangements
and
compare
our results
with the predictions of
the
Landau-de Gennes
mod-
el.
We discuss
our
results
in Sec.
IV
and
compare
r
them
with
results
obtained from other
measure-
ments.
II.
THEORETICAL BACKGROUND
We
first
give
a
brief review
on
the
theories
of
the
optical
Kerr effect and the
ellipse-rotation
effect.
%'e
then discuss these
effects in
connec-
tion
with
the pretransitional
behavior
of
liquid
crys-
talline
materials
in
the
isotropic phase.
A. Optical-field-induced
nonlinear
refractive
indices of an
isotropic
medium
The
optical
susceptibility
of a
medium
is,
in
general,
a
function of
the
applied
optical
fields.
For
a
medium with inversion
symmetry,
the
fieM-
induced
optical
susceptibility
in
the
lowest order
can
be
written
as~
&Xrr
=~XrrrIr(rd
=
rrr+rrr
—
rd)Er(rrr
)Er
(rd
)
~
By
symmetry,
the
third-order nonlinear
suscepti-
bility
tensor
X;,
rr
(&rr
=~+err'
—
rd')
of
an isotropic
medium has the
following
nonvanishing
elementse
(i,
j
=x,
y,
z):
(s)
(s)
Xllll
X
j
j
j
j
(3)
(3) (3)
(3)
X1122
Xj
jjj
s
X1212
Xjj
jj
(s)
(3)
X1221
X
j
J8
(3)
(s)
(s)
Xl ill
X1212 X1221
X1122
If
=
4P',
then
X1122
Xlp12
The
corresponding
nonlinear
polarization
is~'9
&'"(~)
=
g
~lx'„".
.
(~
=
~+
~'-
~')E
(~)E
(~')Ep(~')+
x',
.
",
.
(~
=
~+
~'-
~')E;(&)«(~')E*(&')
+X~rm",
r
(~
=
&u+
~'-
&u')Er(&o)E,
(rd')EP(~')]
1278
QEQRQE
K. L.
WONG
AND
Y. R. SHEN
10
In
general,
the
field-induced
refractive index
contains an electronic
part
due
to
field-induced
deformation of the
electron cloud around
mole-
cules
and
a
nuclear
part
due to
molecular
reori-
entation
and redistribution
by
the field.
According
to
Owyoung
et al.
"
we have for
an
isotropic
medium
time of
the order
of
10 "sec,
while the nuclear
part
can have
a
much slower
response.
There-
fore,
for
ordinary
laser
pulses,
we can
regard
the electronic
response
as
instantaneous and
write
X'„",
,
(~=~+~'
~')=24(o+2P),
X1212+X1221
12
(
+@
(3)
(3)
(4)
=~[a(ur,
(d')&(t
—
t')+2p'((d
(d'
t
—
t']
(X1221+
X12»)
(u,
u';
t
—
t')
E,
((d',
t)
= i.
*
e
8(t)
exp(ik'
r
—
i(()'t),
where
8(t)
is
the
amplitude
function.
The
field-
induced
susceptibility
becomes
t
exe(te,
e/;
()
=
f
I;
((x(
(,
(~/;
e(
ee),
-
k,
1
x
I
(g
I'
(t')
}2«e
l«e
.
'
(6)
The
electronic
part
and the nuclear
part
of
X&&k&
should
of course
have different
functional
depen-
dence
on time.
The electronic
part
has
a
response
I
where
0
and
P
are
contributions from
the
electron-
ic
part
and the
nuclear
part,
respectively.
If
X»21)2
and
Xt1222),
by
measuring
both
the optical
Kerr
effect
and the
ellipse
rotation.
'
We have
assumed in the above
discussion
mono-
chromatic
fields.
In
practice,
the
strong optical
field
may
be a
pulse
represented
by
the
field
com-
ponents
=
~»[o(&(),
~')5(t
—
t')
+
P'(((),
e',
t
—
t)]
For molecular reorientation
and
redistribution
governed
by
a diffusion
equation,
we
expect
the
response
function
P
to
have the form
P'(~,
~',
t)
=
[P(~,
~')
/r
l
e
"',
where
P(~,
&u')
is the
response
function for
an
in-
finitely
long
pulse
and
&
is the relaxation
time.
We shall
show
Eq.
(8)
explicitly, later,
for
an
isotropic
liquid-crystalline medium.
B.
Optical
Kerr effect
In the
presence
of
a
strong
linearly polarized
optical
beam,
an
isotropic
medium shows linear
'
birefringence. This
induced linear
birefringence
is
given
by
en,
=6nII
—
an
where
~nil
1
=
(2x/n)5XII
1.
From
Eqs.
(6)
—
(8),
we
find
&n,
(ur,
~',
t)
=(211/n)
6(x,
'»,
+
x'„'„',
)
(~,
~';
t
—
t')
I
&
I'(t')
«'
(10a)
=(e/")
e(,
')I&I'(()e
J
()
(',
(
(')I((I'(('')«',
-
(10b)
t
=('/")
'(,
')I&l'(()e[))(,
')/elf
e
e
'e
I()l
(el«
(10c)
If
the
variation of
Ih
I'(t)
is
negligible
in
a
time
r,
then
Eq. (10)
reduces
to the usual
expression
6n,
(((),
&u',
t)
=
(w/n)(
+pa)(~,
(d')
III'(t) .
from
Eq.
(3)
for
(d
=(d'
(12)
C.
Ellipse
rotation
The effect
of the
field-induced
refractive index
on
the
propagation
of
an
elliptically polarized
beam is most
easily
understood
by
transforming
P
(~)
and
E;((d)
into
the circular
coordinates
e+
--(x+iy)/)t2
and
e
=
(x
—
iy)/W2.
One
finds
5n,
=(2w/n)(5X
—
5X,
)
.
From
Eqs.
(6)-(8),
and
(12),
we obtain
(13)
where
E,
=(E,+iE„)/v2
and
P, =(P,+iP)/v
2
The.
induced circular
birefringence
seen
by
the beam
is
10
STUDY OF
PRETRANSITIONAL
BEHAVIOR
OF. ..
1279
(14a)
o(~,
~)libel'(t)+
'
e
' '
"Ibl'(t')dt'
(I
e*'
&I'
I
e*
el')
(14b)
In
the quasi-steady-state,
it
reduces
to
6&.
=
(&/2&)(&+
2P)(~,
~)
Ih
I'(~)
de
—
=
(&u/2c)bn,
(16)
D.
Landau-de Gennes model
for the
pretransi'-
tional behavior of
liquid
crystalline
substances
de
Gennes'
has successfully
applied
Landau's
theory
of
second-order
phase
transition to
de-
scribe
the
isotropic
—
mesomorphic
pretransition-
al
behavior of
liquid
crystalliqe
materials.
We
briefly
review the
theory
here. We
shall
limit
our discussion
to nematic
substances
only.
Let
Q;&
be the macroscopic
tensor order
param-
eter
which
describes
the ordering
in
molecular
orientation. As pointed
out
by
de
Gennes,
'
any
tensorial
property
of the
medium
can
be used
to
define
Q;;.
For
example,
we
can
define
2
Xt)
=
Xty
&0
+
3
&X
Q;~
where
y =Q;
sy«and
by
is
the
anisotropy
in
y,
&
when all
molecules are
perfectly
aligned
in
one
direction. The
free
energy
per
unit volume
in
the
isotropic
phase
is
given
by
0+
p
Q;,
Q~
—
4X;,
Es*'E;,
4=a(T
—
T*)
where
a
and
T*
are
constants.
We have
ne-
glected
in
the
above expression
higher-order
terms of
Q,
z
and
the
spatial
dependence
of
Q;z.
The
corresponding dynamic
equation
for
Q;;
is
(18)
f;,
(t)
=~&x(&&*&&
—
s
I
&I'6
g)(f),
(19)
where
&
is
a
viscosity
coefficient.
The
solution
of
the above
equation
is
x
(I
e,
*
el'
—
I
e+
el') .
As
the beam
traverses the
medium,
this
induced
circular
birefringence
leads to a
rotation
8
of the
polarization ellipse
with
From
Eq.
(1V),
we find
that
the
linear
birefrin-
gence
induced
by
a strong
linearly
polarized
field
along
i
is
on,
=
(2n'/n)
3by(Q,
,
—
Q,
,
)
=
(2
w/n)EyQ„.
If
lb
I'(t) is a
pulse
shorter than or comparable
with
T,
then
at
sufficiently
large
time
t,
both
Q„.
and
~&&
will decrease exponentially
with
a
time
constant
7.
We have
considered
here
only
the
nuclear
contribution to
the induced
refractive
in-
dex.
Then,
comparing
Eq.
(22)
with
Eq.
(10c),
with @=0,
we
find
p(&u,
~') =
2(&y)
/9A.
=
2(by)2/9a(T
—
T*)
.
(23)
Thus,
by
deducing
7'
and
P(~,
~')
from
experi-
mental
results
as a
function
of
temperature,
we
can
determine
v/a
and
(by)'/a.
III.
EXPERIMENTS
AND
RESULTS
A.
Sample
preparation
We made
measurements
on
the two
homologous
nematic compounds MBBA
and
EBBA.
The
sam-
ples
were
purchased
from Eastman
Kodak
and
Vari-Light
Corporation. They
were
used
without
further purification.
The
sample
was
placed
in
a
glass
cell
of 4 cm
long
with
end
windows
free
of
strain
birefringence.
The
cell was
pumped
under
vacuum
for several
hours
and
then
sealed
under
1-atm
pressure
of
N,
gas.
The
transition
temper-
atures
of
the
samples
prepared
this
way
showed
no
change
over
a
period
of
months.
The
cell was
then
placed
in
a closely
fitted
copper
block
and
thermally
controlled
by
a
Yellow-Spring
thermo
control unit.
The
temperature along
the
cell
was
found
to be uniformly
stabilized
to
within
+0.
03
C.
The clearing
temperature
T~
of
our
samples
are
42.
5
C
and
78.
5
C
for
MBBA
and
EBBA,
respec-
tively.
B.
Measurements
of
orientational
relaxation
times
QU(t)
=
[f
(ti)/v]
e
(t
0
)/i'dtl
where
T
=
v/~
=
v/a(T
—
T*)
.
(20)
(21)
We
used
a
single-mode
ruby
laser
Q
switched
by
cryptocyanine
in
methanol.
The single
spatial
mode
of the
laser beam was
achieved
by
placing
a
0.
8-mm
pinhole
inside
the
cavity.
The
output
1280
GEORGE
K.
.
L.
WONG
AND Y.
R.
SHEN
10
Scope
/
QD-I
I
I
f
Ruby
BS
Loser
F-I
50
Vo Beam
Splitter
I
P
I
I
Sample
I
j
Grating
r']
/
e
Light~+
Ruby
Light
pike
Filter
at 6328 A
FIG.
1.
Experimental
ar-
rargement
for
observing
molecular
orientational
re-
laxation
times in nematic
liquid
crystals.
BS,
beam
splitter;
P-1, P-2,
P-3
linear
polarizers;
D-1,
ITT
F4018
fast
photodiode;
D-2,
RCA
photomultiplier
7102;
F-1,
neutral-den-
sity
stacks.
He-Ne
Loser
Scope
pulse
width
was
about 10
nsec
(full
width
at
half-
maximum)
and the maximum
peak
power
was
about 50
kW.
The
experimental
arrangement
for
measuring
relaxation
timey is shown in
Fig.
1. The finite
ordering was induced
by
the
linearly polarized
laser
beam. The
subsequent
time variation
of
the
ordering
parameter
was
probed
by
a
40-mW
cw
He-Ne
gas
laser. The
polarization
of the
He-Ne
laser beam
was at
45'
to that
of
the
ruby
beam. The
polarizer
P-3
was crossed
with the
polarizer
P-2
so that
signal
could
reach
the photo-
multiplier
only
when the
medium
was birefringent
resulting
from
induced
ordering
in the
sample.
Both the
ruby
and
the
He-Ne
laser beams
were
telescoped
down to
-0.
5 mm inside the
sample
cell. The
He-Ne
laser
power
going
into the
sam-
ple
was about
10
mW and the
peak
power
of the
ruby
laser
pulse
was
about
10
kW.
For this
arrangement,
the
signal
at
the photo-
multiplier
was
proportional to
sin'(X5n,
),
where
E
is
a
constant
and
«,
is the linear
birefringence
at
the
He-Ne
frequency.
In our
experiments,
K«,
«1
and
hence the
photomultiplier
signal
was
I
proportional to
(«&)'.
Thus,
if
«,
or the
order
parameter
decayed as
e
'
',
the
signal
would
de-
cay
as
e
~'
.
We
found that our
measured
signals
always
had
perfect exponential
tails from which
we then
calculated the
orientational
relaxation
times
&.
In
Figs.
2
and
3,
we
present
our experi-
mental
data
of
T
as a
function
of
temperature
for
MBBA
and
EBBA.
Both curves show
clear
diver-
gence
as
7.
'
approaches
the
transition
temperature
They agree
very
well with
the theoretical
curves
given
by
7'=
v/a(T
—
T*)
in
Eq. (21)
if we
assume
~=~04"~
with
W=2800'K
as
suggested
I—
4P. 44
O
OJ
7
til
O
0
i
4—
z
o
3
X
UJ
MBBA
I I I I I
46
48
50 52
54
TElVIP
E
RAT
URE
(
t)
I
56 58
C
LLI
X
'z
O
a
I-
x
a
LLj
K
R
O
~
o
o
78
I
80
E 88A
I I I
82 84
86 88
TEMPERATURE
(
C)
I
90 92
FIG. 2.
Relaxation time
~
of the
order
parameter
as a
function of
temperature
for
MBBA. The
solid
curve is
the
theoretical
curve described in
the text.
The dots
are
the
experimental data
points.
FIG.
3.
Relaxation time
&
of the order
parameter
as
a function
of
temperature
for
EBBA.
The solid
curve is
the
theoretical
curve described in the text. The dots are
the
experimental
points.
10
ST
UDY
0
F P
RE
TRANSITIONA
L B
E
HA VIOR
OF. ..
1281
TABLE
I.
Results
of
optical
Kerr,
ellipse-rotation,
and
orientational
relaxation-time measurements on MBBA and
EBBA.
Material
(esu)
P
(esu)
Xii22
(~
&
~)
Xi2i2
(~
&
)
Xi22
i
(
&
~)
(esu)
(esu)
B
(esu)
v/a
(sec
'K)
(&X)'
a
T*
(erg
i
cm3
'K)
(K')
MBBA
&0.
01P
2.
7x
10
8
EBBA &0.
01P
1.
5x
10
9
o 0
Xi»i(~.
)
&0.
01X
f22i
((d. ~)
2.
2x
10
1.
5x
10
-ii
2(I00/T
T
-T+
T
-T"
9.
1x
10
e
1.
3x
10
8.
6x
10
ii
2800/T
T
—
T*
T
-T*
-7.
0
x10
e
1.
1x
10
8
314.7
6.
4x
10
~
350.
6
by
Stinson
and
Litster.
'
The values
of
)do/a
and
T*
deduced
from the
fit for
MBBA
and EBBA
are
given
in Table I.
For
MBBA,
the
relaxation
time
varies
fro~
-40
nsec at
temperature
far above
the
phase
transition to
&800
nsec near
the
transition.
The
relaxation time
for
EBBA
is
considerably
shorter. It varies
from
about
13
nsec
to
-170
nsec.
The results
for
MBBA
are in
good
agree-
ment
with those obtained
from
light
scattering
by
Stinson and Litster.
However,
we believe
our
measurements
are
more straightforward
and
ac-
curate, especially
when
7
is
long.
C.
Measurements
of intensity-dependent ellipse
rotation'
In
Fig.
4,
we show our
experimental
arrange-
ment for
ellipse-rotation
measurements
which
was
similar
to
that
used
by
Qwyoung
et af.
"
The
Fresnel rhomb
R-1
was
used
to produce a
laser
beam
of
desired
ellipticity.
The
single-mode
beam
was
focused into the
sample
by
a
15-cm
lens
L-1
so that
the
focus was
at
the
center
of the
sample
cell. The beam was
then recollimated
by
lens,
L-2.
The second
Fresnel rhomb
R-2
and the
Gian
polarizers
were oriented in
such
a
way
that
in
the
absence of
ellipse
rotation,
the
output
beam
from the Fresnel rhomb was
linearly polarized,
a
maximum
"transmitted"
signal
was
directed into
D-3,
and
a
minimum
"nulled"
signal
into
D-2.
The
purpose
of
D-3
was to
monitor
any
nonlinear
loss or
change
in the
spatial
profile
of the laser
beam.
Neutral
density
filter stack
F-1
was
used
to
vary
the
input power.
Laser
power
less
than
1 kW was
used in the
experiment.
If the
focusing
of the
beam
is weak
enough
so
that
self-focusing
is
absent,
then
geometric
optics
is
a
good
approx-
imation to describe the focused beam. Under
such
condition,
one
can
show'
that
for
a
single-mode
beam
with
a
Gaussian
profile,
the
signal
S
at
D-2
with
respect
to the
input
laser
power
P(t)
is
given
by
S(t)/P(t)
=
C(sin2$)'8',
„(t),
(24)
where
C
is a constant,
tang=
~E,
/E
~,
and we
have
assumed
e,
„«1.
Note
that
the above
result
is independent
of the
focusing
geometry
and
sam-
ple
length.
In
our
experiments,
we
confined
ourselves to
low
enough
power
so
that
8«1.
We
also chose
Q
=22.
5'.
Since we can
approximate
our laser
pulse
well
by
a
Gaussian
pulse
P
=Poe
'
',
we
have
from
E(l. (24)
S(0)/P
=
C(v'(d2/nc')'(o+
2pg)2Po,
(25)
where
S(0)
is the
signal
at
the
peak
of the
input
pulse
and
g
=
(1/br}e"'""[1
—
erf(1/2br)]
(27)
Because
of the
fluctuations
of the
ordering
pa-
rameter,
a
nematic
liquid
crystal
in its isotropic
phase
has
a
non-negligible
scattering
loss
coef-
ficient"
y
which varies
with temperature
as'
y=c(/(T
—
T*).
Since
the
ellipse
rotation
occurred
essentially
within the
focal volume
which was
lo-
27T
(gp
5,
„=,
coe2(t
(oP(t)+(25/e)
t
x
J(
.
e
t'
'
"'P(P)dt'),
(25)
Scope
Scope
il
f
hp-)
I
I
I
)--/
——
~&
--Scope
BS
RUby
F-I
P-I
R-
I
L-I
Sappy
L-2
R-2
F'-2
P-2
P-2
Loser
FIG.
4. Experimental
ar-
rangement
for
observing
ellipse-rotation
effect.
P-1, P-2,
Gian polarizers;
R-1, R-2, fresnel
rhombs;
L-1,
L-2,
15-cm
lenses;
F-1, F-2, neutral-density
stacks;
D-1, D-2,
D-3
ITT
fast
photodiodes.
