COMPLEX FIELD DISPLAYS USING A COMERCIAL TNLCD IN A
SIMPLIFIED MOSTLY-PHASE CONFIGURATION
A. Serrano-Herediaa, V. Arrizón-Peñab, R. Ponce-Díaza, J. Ramos-Oliveiraa
a
Centro de Óptica, ITESM, Campus Monterrey. Av. Eugenio Garza Sada #2501, Col.
Tecnológico, C.P. 64849, Monterrey N.L. serrano@itesm.mx; rponce@itesm.mx;
jaro@itesm.mx.
b
Instituto Nacional de Astrofísica, Óptica y Electrónica. Apartado Postal 51 y 216 , Puebla,
Pue. 72000, México. arrizon@inaoep.mx
ABSTRACT
In this work we propose to encode a complex field in a commercial twisted-nematic liquid-crystal display
(TNLCD), using a simplification to the phase-mostly configuration set up; which consists in a generator of
elliptic polarization at the input, and a detector of elliptic polarization at the output. Both the generator and
the detector are formed by a quarter wave plate and a linear polarizer, appropriately arranged. In our case, a
quarter-wave plate is removed from the detector. The remaining components, two polarizers and a wave plate
are conveniently arranged and oriented. This modification reduces complexity and cost of the mostly-phase
setup arranged with the TNLCD, and aloud to encode a complex field. Experimental result is shown.
1. INTRODUCTION
A twisted nematic liquid crystal display (TNLCD) is a device that changes the state of polarization of light. In
a properly controlled environment, consisting in a set of suitably configured optical elements and the TNLCD,
the change in the state of polarization of light can be used to modulate the phase and/or amplitude of an input
light signal. Although this device is manufactured with the purpose of generating amplitude modulation, it
has been modified in order to obtain the so called mostly-phase modulation. In this configuration, the TNLCD
modulates the phase of a light beam in a significant range, while the amplitude remains almost constant.
The simplest mostly-phase setup requires two external polarizers, whose orientations must be properly
oriented1. Unfortunately, the phase modulation obtained with this setup is coupled with an amplitude
modulation whose standard deviation is relatively high. The standard deviation of the amplitude keeps an
inverse relation with the operating wavelength λ.
A modified mostly-phase configuration3-4, that substantially reduces the variance of the coupled amplitude,
employs a generator of elliptic polarization, at the input, and a detector of elliptic polarization, at the output.
Both the generator and the detector of elliptic polarization are formed by a quarter wave plate and a linear
polarizer, appropriately arranged. This modified setup, that here will be referred to as optimum mostly-phase
configuration, gives a large phase range and low intensity variation.
For our proposed mostly-phase setup we maintain the use of an elliptic polarization at the input of the LCD
device, but employ only a linear polarizer at the output. It is shown that the performance of this simplified
mostly-phase setup is quite similar to that of the optimum setup. In section 2 we present a brief summary of
the theory of light propagation through a TNLCD. Our proposal and experimental verifications are
respectively presented in sections 3 and 4. Section 5 presents examples of reconstruction of complex fields.
2. THEORY OF LIGHT TRANSMITION ON A TNLCD
In a cell of a TNLCD, the liquid crystal molecules are parallel to the glass plates containing the liquid crystal 5
because the molecules are anchored to rubbing groves at the inner faces of the glass plates. These groves
appear oriented with the x-axis; at the input plane, and with the y-axis, at the output plane forcing the
molecules to form a helix with the z-direction as the axis. If an electric field is applied between the glass
plates, the molecules tend to obtain the orientation of the field.
A simplified physical model of the TNLCD cell assumes that the twist of the molecules is a linear function of
z. The Jones matrix of the cell, can be expressed as
sin 
sin  

cos   i
 T

 ,
(1)


M (  )  e i R(  T )

sin

sin

 T

cos   i

 

where β is the birefringence of the liquid crystal, given by,

d
(ne  no ) ,

(2)
d is the thickness of the liquid crystal layer in the cell, no and ne are the ordinary and extraordinary refraction
indices for light propagated along the z-axis, and γ is defined by
  T2  2 .
(3)
In addition, T is the total twist angle of the molecules in the cell, and R(T) denotes the rotation matrix
cos T sin T 
R(T )  
.
 sin T cos T 
(4)
To complement the above simplified physical model of a TNLCD cell, it is assumed that when a voltage is
applied to the cell, along the z-axis, all the molecules are tilted by the same angle
electric field. For this simple model the tilted cell can also be described by the Jones matrix in Eq. (1) by the
substitution of the extraordinary refractive index ne by the new index ne(τ), given by
1
sin 2 
cos 2 


n e2 ()
n o2
n e2
(5)
For each value of the tilt angle, the above parameters  and  adopt different values, providing different
modulation properties to the LCD cell. The complete analysis of light propagation across a TNLCD arranged
in the mostly-phase setup requires the Jones matrices of linear polarizers and wave plates. The Jones matrix of
a linear polarizer, whose axis forms an angle  with the x-axis of the reference system (see Fig. 1), is
 cos 2 
P
sin  cos 
sin  cos  
,
cos 2  
(6)
and the Jones matrix of a wave plate, whose slow axis is aligned with the x-axis, is expressed as
W0  e

and φ is given by

i
e i

 0
0

e i 
 z '
( ne  no' ) ,

 z '
( ne  no' ) .

(7)
(8)
(9)
n'o and n'e are the fast and slow retarder refraction indices, and Δz is the plate thickness. In addition, the Jones
matrix of a plate wave whose optical axis forms an angle  with the x-axis of the reference system, is obtained
in terms of the matrix W0 and the rotation matrix [defined in Eq. (4)], as
2
W  R( ) W R( ) .
(10)
Although these assumptions are not exactly true6-7, we have found that the above simple model is quite useful
to obtain rough predictions of the behavior of the TNLCD in different situations.
3. THE SIMPLIFIED OPTIMUM MOSTLY-PHASE SETUP
The configuration of the optimal setup for phase-only modulation with a TNLCD, proposed by Davis et. al 2,
is shown in Fig. 1. the input retarder is illuminated by a linearly polarized beam transmitted by the input
polarizer, it then generates an elliptically polarized beam that illuminates the LCD device.
Each pixel of the LCD device is electrically excited by discrete voltage levels, numbered with the “gray level”
index g=0:255 that correspond to different values of birefringence (). For the discrete domain of  the liquid
crystal cell generates a collection of elliptic polarization states. The output couple retarder-polarizer must be
designed with the aim of collecting all the elliptic polarization states generated by the LCD, and to transmit
light which is modulated in a significant phase range, with almost constant amplitude. For example, a phase
modulation range of 350 with a coupled amplitude standard deviation below 2% of the average amplitude,
using λ=458 nm, has been reported by Davis et al4.
Figure 1. Set up of optimized phase-mostly
configuration, elliptic polarization generator and
detector.
Figure 2. Simplified set up for a phase-mostly
configuration.
From numerical simulations based on eqs. (1-4), it is found that when the LCD cell is illuminated by an
appropriate elliptically polarized beam, the generated polarization states are all intermediate between linear
and circular polarization states. Thus, it can be expected that these elliptic states, illuminating an appropriately
oriented linear polarizer, may transmit a modulated field with highly uniform amplitude. Based on this idea,
we studied the performance of the simplified mostly-phase setup shown in Fig. 2.
4. CHARACTERIZATION OF PHASE MODULATION
To obtain the appropriate configuration of the setup in Fig. 2, we employed the basic physical model for light
propagation through the LCD cell. Our specific strategy was to find numerically the appropriate elliptic
polarization of the light beam (orientation angle α and ellipticity ε) to be used at the input of the LCD cell,
together with the orientation angle (ΨP2) of the linear polarizer, at the output of the cell. The theoretical model
requires the range of β that according to the plot in Fig. 3 is related with the cell rotation power. Since we
required to optimize the setup for a wavelength =457nm, we measured the total rotation power provided by
the employed TNLCD (the SLM LC2000 of HoloEye Photonics AG) under such illumination. The maximum
measured rotation angle (100 o) corresponds to the modulation level g=255. Considering this angle in Fig. 3,
it is found that the range of birefringence provided by the TNLCD is β=0:3.26 rad.
3
Figure 3. Rotation θ of generated elliptic polarization versus β.
The numerical optimization searched for the orientation (α) and ellipticity (ε) of the input elliptical
polarization state, and the orientation of the output linear polarizer (ΨP2). The ellipticity is defined as
  Em E ,
(11)
M
where Em and EM are the components of the electric field along the minor and the major axes of the elliptic
polarization state. For a given set of parameters (α, ε, ΨP2 ) we compute the Jones vector J1(α, ε), for the input
elliptic polarization, and the Jones matrix JP2(ΨP2), for the output linear polarizer. The output polarization
state is computed by the relation
(12)
J 2 (  )  J P 2 ( P 2 ) M (  ) J1 (  , ) ,
The search method started by setting initial trial parameters α, ε, ΨP2. Then, a local optimum is obtained
around these initial parameters with a steepest descent method. We then established the cost function equal to
the standard deviation of the computed amplitude modulation: the objective was to find parameters α, ε, Ψ P2
for which the standard deviation has values below a limit s0. A set of solutions fulfilling such requirements are
obtained and the one for which the phase modulation range is closest to 2 is chosen. One of the results
provided by the optimization process, corresponds to the parameters α=8 o, ε=0.29 and ΨP2=127 o.
4.1. EXPERIMENTAL VERIFICATION
For the generation of the input elliptic polarization obtained in the numerical simulation, we employed a half
wave plate designed for 633nm. We experimentally set the angle  between the input polarizer and the slow
axis of the available wave plate, such that the polarization state had the required ellipticity ε=0.29. The angle
was 16 o. The major axis of the generated elliptic state was 2 o respect to the retarder’s slow axis.
In order to orient the generated elliptic state at the required angle α= 8 o, the configuration of the components
were: ΨP1= 8o Ψ = 8o
ΨP2= 127o. The setup performance evaluated under these conditions showed a
phase modulation close to 2, but an amplitude modulation with a standard deviation around 12%. Imposing
changes to the configuration we reduced the amplitude variation. The final obtained angles, ΨP1= 21 o Ψ =
8o
ΨP2= 106o, provided a satisfactory modulation, both in amplitude and phase. This modulation was
experimentally obtained by displaying a binary grating in the TNLCD and measuring the relative intensities
of the grating diffraction orders. The result is shown in Fig. 4, the phase modulation range is about 1.98 rad.
and the standard deviation of the amplitude modulation is less than 2% of the average amplitude.
Figure 4. Modulation curves of simplified phase-mostly configuration.
4
5. RECONSTRUCTION OF COMPLEX FIELDS
We designed a Computer Generated Hologram to generate first-order Bessel beams function with finite
support, in order to have a complex field:
(13)
B( r , )  A J 1 ( 20 r ) exp( i ) circ( r / R )
where ρ0 is the radial spatial frequency of the beam, R is the radius of the circular beam support, and A is
normalization constant. The continuous function should be sampled in order to obtain the modulation element
cnm, required to establish the pixelated complex function. The sampling resolution (δx=33μm) corresponds to
the pixel pitch of the LCD. The radius of the support for the modulation is R=37 δx=1.22mm, this restricts to
one bright ring of the first-order Bessel beam.
The sampled complex function should be encoded, because our phase-mostly modulation curves have a small
amplitude variation, then real values should be encode with phase values. We implemented double-pixel
coding Arrizon et. al2. The encoded CGH signal field was separated from the noise using a 4-f spatial filtering
setup, whose transforming lenses had a focal length f = 40cm. At the Fourier plane of this setup, the signal
band (centered at the zero frequency) is transmitted by a low pass spatial filter. Images of the generated firstorder Bessel beams, with one ring are shown in Fig 5 (a) y (b). These patterns, experimentally recorded at
z=5cm and z=90cm from the second transforming lens of the 4-f setup, show the high invariance of the
synthesized beam.
(a)
(b)
Figure 5. Experimental intensity distribution of the complex field displayed in the TNLCD, (a)
at 5 cm and (b) at 90 cm from the second transforming lens of the 4-f spatial filtering set up.
CONCLUSIONS AND REMARKS
We proposed a simple technique to use a commercial TNLCD as a phase modulator and display a complex
field. We implemented a numerical simulation in order to optimize a mostly-phase setup, couple of linear
polarizers plus a wave retarder (elliptic generator), and a polarizer (linear detector). With this simulation we
optimized the elliptic polarization input state and the orientation of the output polarizer in order to minimize
the variance of the setup amplitude modulation, maintaining a phase modulation range close to 2 rad. The
orientation of setup components, suggested by the numerical optimization process, only required slight
experimental adjustments, in order to obtain a satisfactory mostly-phase modulation. The experimentally
obtained phase range was approximately 2 rad. and the standard deviation of the amplitude was less than 2%
of the average amplitude. This modulation curves aloud to display a complex field, using a encoding
algorithms, results show a good behavior of the TNLCD as a phase-mostly modulator.
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