Complex amplitude reflectance of the liquid crystal light
valve
Kanghua Lu and Bahaa E. A. Saleh
The complex amplitude reflectance of the liquid crystal light valve (LCLV) is determined as a function of the
writing intensity and applied voltage using an approximate model. The input and output polarizers are
assumed to have arbitrary directions. The theoretical results based on this model match our experimental
measurements. This theory allows us to optimize the operation of the LCLV as an intensity or phase-only
spatial light modulator. When the polarizers are orthogonal and the input polarizer is at -34° with the front
liquid crystal director, the intensity reflectance reaches 100% (compared to 81% for the conventional
configuration). Phase-only modulation is realizable by use of appropriate applied voltage bias and configuration of polarizers. Key words: Spatial light modulator, liquid crystal displays, optical processing.
1. Introduction
The liquid crystal light valve (LCLV) was developed
in the 19 7 0s14 and has been extensively used as a
spatial light modulator (SLM). 5,6 However, no explicit analytical expressions are available for its reflectance as a function of writing intensity and applied
voltage. This paper aims to fill in this gap.
Using a model for light propagation through the
inhomogeneous medium of the twisted nematic liquid
crystal, we use Jones calculus to derive the Jones matrix of the device. Under some approximation we
determine an expression for the dependence of the
complex amplitude reflectance on the writing light and
the applied voltage. Input and output polarizers are
assumed to have arbitrary directions. This theory is
verified by experimental measurements of intensity
reflectance and phase shift in different conditions.
Using our theory we undertake a systematic study of
the polarization configurations and the device parameters that optimize the operation of the LCLV as an
intensity or phase-only spatial modulator. We find
that, when the polarizers are orthogonal and the input
polarizer is oriented at -34° with the front liquid
crystal director, the efficiency of intensity modulation
The authors are with University of Wisconsin, Department of
Electrical & Computer Engineering, Madison, Wisconsin 537061691.
Received 20 October 1989.
0003-6935/91/172354-09$05.00/0.
©1991 Optical Society of America.
2354
APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991
reaches 100% for monochromatic reading light (compared to 81% for the conventional configuration).
Phase-only modulation is also found to be realizable by
use of appropriate applied voltage bias and a configuration in which both polarizers are parallel to the front
liquid crystal director. The intensity distortion can be
reduced below 10%.
II. Complex Amplitude Modulation of the LCLV
A liquid crystal material is an anisotropic medium
which can be treated locally as a uniaxial crystal whose
optical axis is parallel to the director of the molecules.
In a twisted nematic liquid crystal cell (LC),7 - 21 the
molecules rotate gradually in a helical fashion, and the
material is therefore inhomogeneous. It is locally a
uniaxial crystal whose optical axis rotates helically in
the direction of twist.
To examine the propagation of polarized light along
the twist axis of an LC, the Jones calculus may be
used.9"12"19 For an LC of thickness d, when no electric
field is applied, there is no molecular tilt, i.e., the tilt
angle = 0. Twist angle ois linearly distributed along
the twist axis z with a maximum twist angle a at z = d,
so that o = azid. The material can be divided into
incremental slices orthogonal to the z-axis. Each slice
acts as a homogeneous uniaxial crystal8"19 whose Jones
matrix can be written as a function of its ordinary and
extraordinary indices of refraction, n and ne, and its
direction. The overall Jones matrix is the product of
the individual matrices:' 9
JT=
f
N
j=1
J,
Neinatic Liquid Crystal
where Jj is the incremental Jones matrix of a layer
between zj and zj+l and N is the number of layers. In
the limit N
JT =
Diclectric MilTor
X,19,21
-
exp(-i)R(a)
)[
-
cos(y)
-
i
sin(y)
-sin(-y)
-sin(y)
cosl(y) + isin(y)
CdS-CdTe
Heterojunction
,
4
(1)
.4-
Writing
Light
_
where R(a) is the rotation matrix:
R(a) = [cos(a)
[-sin(a)
sin(a)]
(2)
cos(a)j'
^ = V.E2+ gl2
1= A In,
-
(3)
no]
=rd
= n,+nj
Function
Generator
Configuration of the LCLV modulator with normal incident
light and arbitrary input and output polarizations.
Fig. 1.
When an electric field is applied in the direction of
the twist axis, the equilibrium tilt and twist angles are
altered in accordance with the Oseen-Frank equation;13 in general, the tilt and twist angles vary nonlinearly along the z-direction, i.e., 0 = f(z,V), and so =
g(z,V), where V is the applied voltage. Each incremental Jones matrix Jj is a function of the applied
voltage.
Because of the nonlinear distributions of the tilt and
twist angles, it has not been possible to determine an
explicit expression for JT similar to Eq. (1). The problem can only be addressed numerically. To circumvent this difficulty we consider the following approximation:
(1) Since the tilt angle 0 = f(z, V) is a smoothly
varying function with even symmetry around the midpoint (z = d/2),113 , 7 it is approximated by its mean
value, so that it is treated approximately as a homogeneous tile. The dependence of this constant value of 0
on V has been expressed in the form15 21
=
A
no
(5)
=
d [ne(O) + n0 ] = 0° + /3,
(6)
[ne(O)
where ne(O) is the equivalent extraordinary index of
refraction given by
cos 2(O)+ sin 2 (0)
1
n2
n2
n,1(O)
(7)
'
and 4O is a constant phase shift, which is ignored thereafter.
For a given LC, 3 is defined in a region (,flnmaxl,
where
=
/max
7rd
(n
-
n.);
represents the maximum value of ,-which occurs
in the absence of the applied voltage. Equations (1)(7) are applicable to a transmissive twisted nematic
liquid crystal cell.
Our experimental measurements have demonstrated that the accuracy of this simplified model is quite
satisfactory. 2 ' Furthermore, the predictions of this
model are consistent with experimental results obtained by others.2 2 - 2 4
An LCLV is a reflective twisted nematic liquid crystal cell (see Fig. 1) that can be equivalently unfolded in
the form of a cascade of two identical transmissive cells
with mirror-symmetric structures., By use of Eq. (1),
the Jones matrix of the LCLV is
l#max
0,
a=
2
f[e(-
V-Vc)]
V< Vc
where V is the rms voltage applied on the LC, V, is the
threshold voltage, and V0 is the excess voltage.
(2) Since the twist angle so = g(z,V) is a smoothly
varying function with odd symmetry around the midpoint (z = d/2)," ' 3 17 it is approximated by a linear
function of z, so = azid.
When these two assumptions are used, an explicit
expression for JT can be obtained. Using the approximate model, Eqs. (1)-(3) continue to apply but with
-
(J = exp(-i
2
)
cos(2y)
-
i - sin(2y)
+
i-; [1
-
cos(2y)]
i
[1
-
cos(2y)]
1
// 3 2 cos(2'y) + i -/3sin(2Y)J
(a2+
10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS
I ' (8)
2355
90
LQ
Lu,
z
GLu
0
N~
0
-90
a
90
180
270
360
BETA
(b)
(a)
Fig. 2. (a) Dependence of intensity reflectance R on : and input polarization angle 6. The input and output polarizers are orthogonal. (b) A
topographic view of the relation in (a); R is a periodic function of with a period of 900. The first optical threshold occurs when A = 174.3°.
where a, Al,and y are as before. Considering the effects of a polarizer and an analyzer at arbitrary angles
61 and 02 as shown in Fig. 1, the complex amplitude
reflectance of the entire device is
R = Vit exp(-ib),
where
+ (i)
R = f(-)
+{
cos(2y)] cos(V
t2)}
sin(2,y) coS(it
1 + 02)}
(9)
is the intensity reflectance, and
-
is~
phas
~
shift.
~ ~~~[(O)
a
[1 -cos(2y)]
2 + nr,
n
cos(2p +
sin(2y) sin(2l)2
142.
In this case, the dependence of R on and A when a =
450, is shown in Fig. 2. We find that when
2
2
= n - (n7r) -a , n = 1,2,3,....
R = 0 regardless of A/; therefore, O3n acts as the nth
optical threshold. Furthermore, R is a periodic function of ipwith a period of 90°. The phase has a linear
dependence on with jumps of iroccurring whenR = 0.
2356
1
th osP -I
is the phase shift.
If a polarizing beam splitter is used instead of the
common beam splitter with a pair of polarizers (Fig. 1),
the condition A -2 = 90° must be satisfied. When
^61 = , Eqs. (9) and (10) become
R =-l
where S(I) is a monotonically increasing function of I,
and s() = S(I)/S(o) is called the switching ratio, which
may be determined from an electrical model of the
circuit.' However, it is convenient to measure the
switching ratio experimentally, as explained in the
[1 - cos(2y)] sin(\61 + 1k2) - sin(2y) cos(P + 'I2)
(a) + ()
cos(2)] cos(tj(10)
tan'
(11)
V = S() Vext,
-
[1 - cos(2y)] sin(ik1 + Vt2 ) -
=2
Equations (9) and (10), together with Eqs. (3)-(7),
describe the reading side of the LCLV modulator. At
the writing side, for the LCLV using a CdS-CdTe
heterojunction photosensor, voltage V across the LC
layer is related to the intensity of writing light I and the
rms value of the externally applied voltage Vext by
APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991
next section. Therefore, the normalized voltage in Eq.
(4) can be written in the form
v
~S(O)
V,,
V~ V S)Vet
s~) Vet
V.
C
V.
S(O)
In summary, the LCLV is characterized by the internal parameters a, /3ma., VIS(), VIS(0), and s(I).
The operating point is characterized by the external
variables Vext, I, and the setting of polarizers 4(1 and 02.
The dependence of the complex amplitude reflectance
A? = R exp(-ib) on these variables is established as
follows:
(1)R and 5are expressed as functions of the auxiliary
variable is and parameters ihl, 2, and a, given by Eqs.
(9), (10), and (3). For example, for an LCLV with a =
I
:I
Zr
LCLV
Fig. 3. Dependence of phase shift on j3and input polarization
angle 46.The input and output polarizers are parallel.
1.0
<0.8
LI
o
X
0.6
LL\
N
0.4
0
0.2
0.0
0
1
2
3
4
NORMALIZED VOLTAGE
Fig. 4. Dependence of the normalized parameter /#max on the
normalized voltage (V-V,)/V6, where n0 = 1.5 and ne = 1.7. When
ne is changed by +0.1, this curve is changed only very slightly.
450 when 11 - 12 = 900, the R-3 dependence for
different q1 is plotted in Fig. 2. When 11 = 412, the b-j
dependence for different 461 is plotted in Fig. 3.
(2) The ratio W//max is itself related to the normalized voltage (V - V,)/V,, by Eqs. (4), (5), and (7), as
illustrated in Fig. 4 for several values of ne. This
relationship is monotonically decreasing and is approximately linear when f/lmax is within the region
from 0.3 to 0.9, with an average slope of -0.75. The ne
has a small effect on this relationship.
(3) Finally, the normalized voltage (V - V,)IV0 is
determined by the product s(I) Vext. The features of
s(I) are discussed in the next section.
Ill.
Experimental Verification
A. Verification of Theory
To verify the theory and to estimate the parameters
of the device, we conducted some experiments in which
CRT
Fig. 5. Experimental system to measure the dependence of the
intensity reflectance R on external voltage Vext and writing intensity
I.
the dependences of R and on I and Vext were measured for several polarization angles i61 and P2 We use
RP1,42 and 414'2 to denote the setting of the polarizers.
The LCLV used in our experiments is model H-4060
made by Hughes Aircraft Co. The total twist angle a
= 450 is given by the manufacturer.
The system used to measure the dependence of the
intensity reflectance R on Vext and I is shown in Fig. 5.
A uniform pattern displayed on the screen of a CRT
monitor is imaged on the writing side of the LCLV.
The writing intensity I is controlled by varying the
Brightness. An external voltage Vext is applied using a
function generator. A polarizing beam splitter is used
and the LCLV is rotated to change ^i6. This limits us
to the case of V/1 - '12 = 900. However, i1p= 02 can be
inferred by using the relation Rp = 1 -Rpo+0.
We have measured Rp,9,o+ vs Vext with fixed I for
several values of A. These results are compared with
the theoretical relationship between Rp,9 o+, and : displayed in Fig. 2. For example, in the case of / = -34°,
the experimental relationship of R-3 4,5 6 to Vext is
shown in Fig. 6(a). The corresponding theoretical
relationship between R- 34 56 and A is copied from Fig.
2(a) into Fig. 6(b) for convenience. We find that these
two functions have similar structures with the same
number of lobes and the same values of maxima and
However, the
minima within the region A e (#max].
order in which these maxima and minima appear is
inverted. This is an indication of a decreasing 3 as a
function of Vext, as is also expected from the theory (see
Fig. 4). Similar comparisons are held for other values
of A,as shown in Fig. 6(e) where the T-axis is reversed
for convenience. In all cases, the number and values of
the maxima and minima were identical to those in the
theoretical plots. Our experiment has demonstrated
that the same relationship between Rp,9 o+p and f is
repeated for each 90° period of , as predicted by
theory. Another theoretical prediction, the optical
10
June 1991 / Vol. 30, No. 17 / APPLIED OPTICS
2357
1.0
LUi
z
0.8
LU
-i 0.6
LILUi
0.4
zLU
1- 0.2
0.0
Vext
ca)
(b)
1.0
Ld
LU
()0.8
so Illlilllll
LU
LI
* 0.6
1111111111111111111111111111
a m 11X111
slzllllllilllllllllllllllll .
X
H
t
111111llllT
tE
t
: ::: :: : i
eLlllllllll
II II I .
.,.
,
LL
r
.
I- 0.4
W m
h a ._.__...___.._..... .J
i 1 1111
I I g g , . .... .. .......
En
l l llllllll II : _ s _ _ : : . _ _ _ . . _ : n : : . . _
1111111111 H * ._ ..v .: .___ ; _ 4
4
I I I I 1 1TTTT
= II : _ : i S . _ _ . : _ : 7 . . . . n
I 11111111
X n 1n| N
Xr s
H
=
* ._. * .. .x_
_ 4
= : _: = : : __ ' ; _: =
n 1 1 1 1 1 1 1 F I
,
z- 0.2
_ 4
_r
I z _ = _ = = 7 . . . n
0.0
, .,:,,
I,,,,
= = : : : : . r : -r n . . . -r
_, I I I 1_1_'_'_'_'
l l ,, . _. l l : .. X
J; J
_ 7
WRITING INTENSITY
I
(d)
(c)
=O
0
/
tY
vex
= -22.50 0.0.
0A
27U
Vex
A
180
90
0
P
I =-450
vex:
C0.5
270
18U
90
0
1
XA
A
vexe
2358
= -67.5'
04 :5
0
270
Ce)
A
180
90
0
13
APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991
Fig. 6. (a) Measured intensity reflectance R-34,56 VS Vext at a fixed
writing intensity I = 500 W cm-2 . At V8,t = 0, corresponds to
3max = 2480 when X = 633 nm. (b) Theoretical dependence of R-34,56
on 13. The peak value of the second lobe from the left is 100%. (c)
Measured intensity reflectance R-34,56 vs I when the external voltage
bias is V.xt = 12.2 V. (d) Theoreticalresultcorrespondingto (c). (e)
Experimental dependence (left) of R on Vext is compared with the
theoretical dependence (right) of R on 13for different input polarization angles ip. Since 13is a monotonically decreasing function of Vext,
it is plotted in the reverse direction so that experiment and theory
can be conveniently compared.
Film
1.4
Lj 1.2
1.0
0
r)0.8/
N/
0.6
y 0.4-
0
0.2
0.0
0
4
.
8
.,.
.
12
.
16
EXPERIMENTAL VOLTAGE
Fig. 7. Correspondence between the experimental Vext and the
theoretical normalized voltage (V - V,)/V 0 when s(I) = 2.1.
threshold, was also observed. When setting the value
of Vext so that fi = fli, R = 0 for all iA. Thus, rotating the
LCLV does not alter the intensity reflectance, as is
evident in Fig. 6(e). We, therefore, conclude that the
theoretical model, although approximate, leads to device characteristics that are generally consistent with
experimental observations.
By comparing the ordinates in Figs. 6(a) and (b), a
one-to-one correspondence between points of the abscissas can be established. Ambiguities due to the
nonmonotonic nature of the functions are resolved by
using the order in which the maxima and minima appear. The device parameter flmax = 2480 can be determined by the experimental R when Vext = 0. Once the
f3-Vext relation is known, the theoretical relationship
between 3 and the normalized voltage (V -V,)IV
(shown in Fig. 4) can be used to determine the correspondence between the theoretical normalized voltage
(V - V,)/V 0 and the experimental voltage Vext. This
relationship is plotted in Fig. 7. Its linearity is reasonably good. A best-fit straight line gives an estimation
of the parameters V0 IS(O) = 28 V and VcIS(O) = 2.9 V.
The experimental setup used to observe the phase
modulation is shown in Fig. 8(a). The LCLV replaces
the mirror in one arm of a Michelson interferometer.
A pattern with a grey value varying in one direction as a
sawtooth function is addressed on the CRT. The angle it' = t'2 can be changed by rotating the LCLV. The
displacements of the interference fringes are observed.
Although no attempts were made to measure a accurately, the qualitative results are consistent with the
theoretical plot in Fig. 3. For example, 50, has the
maximum modulation sensitivity and 690,9o (which is
identical to -9-o,-9o) undergoes little change when
varying I or Vext Similar experimental results have
been reported in Ref. 24.
B. Measurement of Switching Ratio
We now turn to the estimation of switching ratio
S(1). Here we rely on the experimental measurements
of function R vs Vext for fixed I [see Fig. 6(a), for
example] and function R vs I for fixed Vext [see Fig.
6(c), for example]. Since R is determined completely
(a)
(b)
Fig. 8. (a) Michelson interferometer with an LCLV phase-only
modulator replacing the mirror in one arm. A sawtooth grey level
pattern is addressed on the CRT. (b) Interference pattern demonstrating phase modulation.
by the products(I) Vext. We identify matching pairs of
points (1, Vextj) and (I2, Vext2) on the two functions that
have the same R; ambiguities due to the nonmonotonic
nature of the functions are resolved by using the order
in which the maxima and minima appear. Therefore,
we equate
S(Ii)Vextl = S2)Ye.12,
where Vext2 and I, are of fixed values. When I2 = 0, the
tj. When I2 is equal to an
corresponding Vext = V%_
arbitrary nonzero value I, the corresponding Vexti =
V tj, then the switching ratio can be determined by
ex
SU) = VIL
V=exti
This method was applied to the data in Figs. 6(a) and
(c). The switching ratio s(I) obtained is shown in Fig.
10 June 1991 / Vol. 30, No. 17 / APPLIED OPTICS
2359
2.2
2.0
0
<1.8
Z 1.6
I-1.4
1.2
1.0
260
400
60
WRITING INTENSITY
Fig. 9. Dependence of switching ratio s(I) on writing intensity I.
Experimental data are based on Figs. 6(a) and (c). Its best match
with Eq. (12) gives the device parameters Sat = 2.4 and k = 1.94%
cm2 .'W1
9. Since this relationship exhibits initial linearity for
small I and saturation for large I, we fit it to a twoparameter function:
= kI
1
+
(12)
/
Ssat
which has an initial value s(0) = 1, initial slope k, and a
saturation value Ssat for large IL The best fit is accomplished for sat = 2.4 and k = 1.94% cm 2 _ jLW-1 The
quality of fit is excellent as is evident from Fig. 9.
Equation (12) is consistent with an equivalent electrical circuit in which the main contribution of the writing light is to change the capacitance of the CdS-CdTe
heterojunction.
Now that we have estimated all the parameters of
the device, we provide one more verification of the
theory by comparing experimental results for R- 34 .56 as
a function of I at a fixed V,a = 12.2 V, with the
corresponding theoretical relationship. These results
are consistent as Figs. 6(c) and (d) show.
IV. Operation of the LCLV as a Spatial Intensity or
Phase-Only Modulator
As shown earlier, the dependence of the complex
amplitude reflectance of the LCLV on writing intensity I can be changed by selection of polarizer angles Al
and V'2 (or only one angle p when a polarizing beam
splitter is used) and voltage bias Vt. This offers
great flexibility in the operation of the device for different applications.
A.
LCLV as a Spatial Intensity Modulator
If an LCLV is used as an optical intensity modulator,
high efficiency, high modulating sensitivity, and good
linearity are usually expected. Because of the high
efficiency of the polarizing beam splitter in comparison with other beam splitters, we limit our discussion
to this case.
Figure 2(b) shows the dependence of the intensity
reflectance R on V/and Al,where the control variables
Vext and I are embedded in . The design involves
2360
APPLIED OPTICS / Vol. 30, No. 17 / 10 June 1991
selection of the angle and the interval over which ,B
varies so that R is changed from 0 to 1. Within the
range of a single period, A'e [-90°,0] and 3 (,flma],
where &m. = 2480 in our LCLV, there are three points
(A,B,C) at whichR = 1. There are also lines at whichR
= 0, one at /3I = 174.30 and an oblique line extending
from the points ( = 0,# = 0) and ( = -90', =
173.40). There are several possible choices for the
operating lines, each corresponds to a horizontal line
passing through points A,B, or C and extending to the
right or left to the first zero. Lines extending to the
right correspond to positive modulations, those extending to the left correspond to negative modulations.
The lines associated with point A are not suitable since
small values of require high voltages and involve
nonlinearity. Point C involves only negative modulation. Operation at point B appears to be the most
suitable one. This choice requires / = - 3 4 ° and a
voltage bias Vext = 12.2 V, corresponding to iB1 for
positive modulation. The experimental and theoretical R-I dependences of this choice are shown in Figs.
6(c) and (d).
We now compare the option of s =-34°
to the
conventional option of A/= 00. In the former case, the
maximum value of R is 1; whereas in the latter it is only
0.81. Furthermore, when ,6=-34°, the slope of R vs f
at R = 0.5 is -5.5hr. For , = 0, the slope at R = 0.41 is
-2.9/7r. One may be interested in the phase distortion
caused by the intensity modulator. As discussed in
Sec. II, when using the polarizing beam splitter, the
phase distortion is at a rate of 2f regardless of Af. The
maximum phase distortion for the operating value of i
is 90° in the VI = -34° case, and 180° in the t = 0
case. Thus, the proposed operating angle A = -34°
offers a higher sensitivity, greater efficiency and dynamic range, and smaller phase distortion. These
conclusions, however, are applicable only for monochromatic reading light because ,3depends on wavelength X. An investigation of the nonmonochromatic
case has been presented elsewhere.2 5
B.
LCLV as a Spatial Phase-Only Modulator
When the LCLV is used as a spatial phase modulator, it is expected to have a dynamic range of 2r, good
linearity, and high sensitivity. For phase-only operation, the intensity reflectance should be kept constant
and large.
As is clear from Fig. 2, which is applicable to the
polarizing beam splitter, there is no extended region
for which R is flat and high. We must therefore examine other polarizer configurations not allowed by the
polarizing beam splitter. We found that the configuration A1 = V'2 = 0 satisfies the above requirements
most closely. This configuration is realized by use of a
single polarizer. oo and Ro,0 are plotted in Figs. 10(a)
and (b). When A is large, the -: dependence is sufficiently linear with a slope of 4, and R varies between 0.9
and 1 for > 127°.
The requirement of a 27r dynamic range necessitates
that the operating range of extends between two
values 03, and A1 (as the writing intensity varies be-
720
I
U-
/
Co
360
0~
0
180
90
BETA
(a)
270
(DEG.)
1.0
U0.8-\t
LU
()
zJ
10.6
LU
U- 0.4-
z 0.4
Z
\\
0.0-
0
90
180
270
BETA (DEG.)
(b)
Fig. 10. (a) Theoretical dependence of bo,o on /. When: > 90°, the
relation is approximately linear with a slope of 4. (b) Theoretical
dependence of R0 ,0 on /3. The efficiency is high and undergoes
relatively small variation when :3is large.
find the corresponding Imax. These points are represented by the solid curve in Fig. 11. Thus, for each
Vext, the operating line is a vertical line extending from
I = 0 to I = Imax on the solid curve. The dotted curve
represents values of I and Vext for which ,3= 13c. Clearly, we must operate below this line. Since the solid
curve crosses the dotted line at Vext = 9.2 V, the operating range must be limited to the shaded region. Within this region any value of Vext satisfies all -requirements. However, to minimize Imax we must operate
near the critical line Vext = 9.2 V.
Although the relationship between phase shift 6 and
f3 in Fig. 10 is reasonably linear for large A3,
the relationship between fl and I is not linear, mainly because of
the nonlinearity of the switching ratio (see Fig. 9).
This nonlinear effect may be alleviated by precompensation of the CRT used to produce the writing light.
The previous analysis was limited to an LCLV with
twist angle a = 450 (the only commercially available
choice). This is not the optimal choice for phase-only
operation. In fact, a device with zero twist angle a = 0
and Alj = VP2 = 0, yields R = 1 and = 4, as may be
readily seen from Eqs. (9) and (10), with the curves in
Figs. 4 and 9 remaining unchanged. In this case, phase
modulation is achieved by tilting the liquid crystal
molecules and thus changing ne(O). The device then
operates as in the well-known tunable birefringence
effect. 2 6 4 In this case, a liquid crystal material of large
birefringence should be selected so as to obtain a large
value of /3max, and therefore to increase the dynamic
range and reduce the effect of nonlinearity of the
switching ratio.
References
V/)
z
z
z
81
12
14
16
APPLIED VOLTAGE (VOLTS)
Fig. 11. Operation lines within the shaded area have a dynamic
range for phase modulation >3600 and associated intensity distortion <10%.
tween 0 and Imax), such that ,B-,/I = 90°. The
requirement of high reflectance R is satisfied if / > ti,
where &3is some critical value [see Fig. 10(b)]. For
example, if R must be greater than 0.9, /3 = 1270. The
distortion of R is then <10%.
For each value of Vxt we determine the value of /3
for I = 0 and determine /3I = AU- 900, from which we
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