Influence of the incident angle in the performance

Influence of the incident angle in the performance
Influence of the incident angle in the
performance of Liquid Crystal on Silicon
displays
A. Lizana1, N. Martin1, M. Estapé1, E. Fernández2, I. Moreno3, A. Márquez2, C. Iemmi4,
J. Campos1 and M. J. Yzuel1
1
Departamento de Física, Universidad Autónoma de Barcelona, 08193 Bellaterra, Spain
Dept. de Física, Ingeniería de Sistemas y Teoría de la Señal, Universidad de Alicante, Ap. 99, 03080 Alicante, Spain
3
Dept. de Ciencia de Materiales, Óptica y Tecnología Electrónica, Universidad Miguel Hernández, Elche, Spain
4
Dept. de Física, Fac. de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina
*Corresponding author: [email protected]
2
Abstract: In this paper we experimentally analyze the performance of a
twisted nematic liquid crystal on silicon (LCoS) display as a function of the
angle of incidence of the incoming beam. These are reflective displays that
can be configured to produce amplitude or phase modulation by properly
aligning external polarization elements. But we demonstrate that the
incident angle plays an important role in the selection of the polarization
configuration. We performed a Mueller matrix polarimetric analysis of the
display that demonstrates that the recently reported depolarization effect
observed in this type of displays is also dependant on the incident angle.
2009 Optical Society of America
OCIS codes: (120.2040) Displays; (120.5410) Polarimetry; (230.3720) Liquid-crystal devices;
(230.6120) Spatial light modulators.
References and links
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Received 3 Mar 2009; revised 3 Apr 2009; accepted 6 Apr 2009; published 5 May 2009
11 May 2009 / Vol. 17, No. 10 / OPTICS EXPRESS 8491
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1. Introduction
The capability of the liquid crystal displays (LCD) to work as spatial light modulators (SLM)
have caused a widespread use of these devices in optical applications such as diffractive
optics [1], holographic data storage [2], optical metrology [3], or in programmable adaptive
optics [4]. As a consequence, optimizing the LCDs response has become a challenge to many
authors, and ways to obtain a desired intensity and phase response have been thoroughly
studied [5,6]. A type of LCD used in numerous optical applications is the Liquid Crystal on
Silicon (LCoS) display. These devices are LCDs that work in reflection, giving high phase
modulation. However, a certain amount of unpolarized light has been detected at the LCoS
displays reflected beam [7-9]. The origin of this depolarization effect was investigated in [9],
showing that it is related to temporal fluctuations of the liquid crystal orientation caused by
the electrical signal addressed to the display. This depolarization effect can adversely affect
applications, as for instance in diffractive optics where it was demonstrated to reduce the
diffraction efficiency [10].
Because of this depolarization effect, the Mueller-Stokes (M-S) formalism has been
adopted for LCoS displays, and a protocol to optimize the intensity [7] and phase [11]
modulation responses has been developed. By extension, this protocol is valid to characterize
any polarizing or depolarizing optical element. In [12] we showed that the intensity, phase and
degree of polarization of the light beam modulated at the LCoS display have a strong
dependence with the wavelength. In this work we study the modulation performance as a
function of another parameter: the angle of incidence. For that purpose we have performed a
complete polarimetric characterization of the LCoS display for different angles of incidence.
We show how the angle of incidence plays an important role and strongly affects the
modulation response. Then, we show that optimized phase modulation can be obtained for the
different angles, but the polarization configuration must be readjusted. This study can be
especially relevant for LCoS displays applications involving high numerical apertures, where
a wide range of incident angles are used, as for instance in optical trappings set-ups [13].
The outline of the paper is as follows. In Section 2, the experimental method and the setup used to characterize the LCoS display are described. In Section 3, the results of the
polarimetric analysis of the display are presented. In particular, the degree of polarization,
diattenuation, polarizance and retardance parameters are thoroughly studied as a function of
the addressed gray level and as a function of the incident angle. In Section 4, the phase and
intensity modulation provided by the LCoS display is analyzed as a function of the incident
angle, and optimized configurations are demonstrated for the different angular positions. The
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results are compared with those obtained with normal incidence and employing a beam
splitter. Finally, the conclusions are presented in Section 5.
2. Experimental characterization based on a synchronous method
In this work we have characterized the Mueller matrix of an LCoS display as a function of the
addressed gray level for five different incident angles: α=2º, 12.5º, 23º, 34º and 45º. The
characterization of the LCoS display has been performed by using a modification of the
methodology described at [7]. The proposed procedure is based on the method of synchronous
detection [14] and it is valid to characterize the Mueller matrix of any polarizing or
depolarizing optical element, being the LCoS display a particular case.
Fig. 1. Set up used to obtain the experimental LCoS Mueller matrix.
The experimental set-up used to implement this method is shown in Fig. 1. We use a 633
nm He-Ne laser and the LCoS display under analysis is a Philips model X97c3A0, sold as the
kit LC-R2500 by Holoeye. The LC-R2500 is a 2.46 cm diagonal reflective LCoS display of
the 45º twisted nematic type, with XGA resolution (1024 x 768 pixels), with digital data input
and digitally controlled gray scales with 256 gray levels. The pixels are square with a center to
center separation of 19 µm and an excellent fill factor of 93%. The LCoS is placed on the top
of a rotating platform that enables choosing the incident angle with a precision of 1º. We have
set a polarization state generator (PSG) at the incident beam and a polarization state detector
(PSD) at the reflected beam. The PSG is composed by a polarizer and an achromatic quarter
wave plate and the PSD is composed by an achromatic quarter wave plate and an analyzer.
Both polarizer and analyzer are fixed at 0º, considered parallel to the laboratory vertical
direction, and both wave plates can be electronically rotated by 360º.
It is well known that the Mueller matrix (M) of an optical polarizing element relates the
input and output states of polarization (SoPs), described by the Stokes vectors Sinput and Soutput.
By generating different input SoPs and measuring its corresponding output SoPs (using
simply intensity measurements), it is possible to construct an independent equations system
from which the whole Mueller matrix can be derived, as it was done in [7]. In this work, we
alternatively measure the SoPs reflected from the LCoS display by using the method of
synchronous detection [14]. The analyzer LP2 is fixed at 0º. Then, the intensity behind the
PSD is function of the Stokes parameters of the reflected beam, and of the phase-shift (φ) and
orientation (θ2) of the waveplate WP2. As a particular case, when using a quarter wave plate
(φ=π/2), the intensity as a function of the angle θ2 can be written as follows:
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S
S
S

 1
π
I  , θ 2  =  S 0 + 1 + 1 cos (4θ 2 ) + 2 sin (4θ 2 ) − S 3 sin (2θ 2 ) ,
2
2
2
 2
2

(1)
where S0, S1, S2 and S3 are the Stokes parameters of the light reflected from the LCoS display.
The intensity in Eq. (1) is a periodical signal with respect to the angle θ2 since it contains
several sinusoidal functions whose arguments are entire multiples of θ2. The synchronous
detection consists on the evaluation of the coefficients of the Fourier series of this function.
By performing a summation of intensities corresponding to N different equidistant values of
θ2, completing a rotation of 360º, some terms of Eq. (1) vanish due to the orthogonal
properties of the sinusoidal sampled functions. In particular the following relations hold:
2
N
N
∑
r =1
 2πri   2πrj  2
sin 
=
·sin 
 N   N  N
N
∑ sin
r =1
N
∑
r =1
N
∑ cos
r =1
2πri 
 2πrj 
 = δij ,
·cos
N 
 N 
2πri 
 2πrj 
 = 0,
·cos
N 
 N 
 2πri 
sin 
=
 N 
N
∑ cos
r =1
2πri 
 = 0.
N 
(2a)
(2b)
(2c)
where N is the number of samples and δij the Kronecker delta. Therefore, using these relations
it is possible to describe the reflected SoP as a function of summations of intensity
measurements obtained for the different equidistant analyzer angles θ2 as:
N
 N  π


π

  2·
I  , θ 2 ,r  − 4
I  , θ 2,r ·cos(4θ 2,r ) 


2


2
r =1 

  r =1
N
 S0 
π




 
8
I  , θ 2,r ·cos(4θ 2,r )


2
 S1  1 

r =1 
,
S  = N 
N

π
2


 
8
I  , θ 2,r ·sin (4θ 2,r )


S 
2

 3


r =1 
N



π

−4

I  , θ 2,r ·sin (2θ 2,r )


2

r =1 


∑
∑
∑
∑
(3)
∑
where N is the number of selected angles θ2 and θ2,r=r2π/N. On the other hand, keeping the
PSG polarizer fixed at 0º, the SoPs impinging the LCoS display only depend on the WP1
rotation angle θ1. The incident SoP can be expressed as:
S input
1




2
 cos (2θ1 ) 
=1
.
 sin (4θ1 )
2

 sin(2θ1 ) 
(4)
Then, Stokes parameters of the corresponding reflected beam can be written as:
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S kr (θ1 )
= mk 0 +
output
mk 1 mk1
m
+
cos(4θ1 ) + k 2 sin (4θ1 ) + mk 3 sin(2θ1 ) ,
2
2
2
(5)
where k = 0, 1, 2, 3, and mk,j (j = 0,1,2,3 ) are the different elements of the Mueller matrix.
Next, taking into account the orthogonal properties of the sinusoidal sampled functions (Eqs.
(2)), and performing a summation for different output SoPs (corresponding to equidistant
angles θ1 from a complete rotation of WP1) in Eq. (5), it is possible to obtain all the LCoS
Mueller matrix coefficients as a function of output SoPs summations as:
 N

S 0r

 rN=1

S1r

1  r =1
M=
N N r

S2
 r =1
 N

S 3r

 r =1
N
∑
−2
∑
∑
−2
∑
−2
∑
−2
r =1
N
S 0r cos(4θ1,r ) 4
∑
∑S
1
∑S
2
∑S
3
r =1
N
r =1
N
r =1
N
r
r
r
cos(4θ1,r ) 4
r =1
N
S 0r cos(4θ1, r ) 4
∑
∑S
cos(4θ1,r ) 4
r =1
N
∑S
cos(4θ1,r ) 4
r =1
N
∑S
r =1
N
r
1
r
2
r
3
cos(4θ1, r ) 4
r =1
N
∑S
cos(4θ1, r ) 4
r =1
N
∑S
cos(4θ1, r ) 4
r =1
N
∑S
r =1


(
)
θ
sin
2
r
1
,
0

r =1

N

r
2 S 1 sin (2θ1, r )
r =1
 ,(6)
N

2 S 2r sin (2θ1, r )

r =1

N
2 S 3r sin (2θ1, r )

r =1

S 0r sin (4θ1,r ) 2
N
∑S
r
1
r
2
r
3
sin (4θ1,r )
sin (4θ1,r )
sin (4θ1,r )
r
∑
∑
∑
where N is the number of selected equidistant angles θ1. For every value θ1,r the
corresponding Stokes parameters Skr are measured according to Eq. (3).
The advantage of this characterization procedure with respect to that used in [7] is that
here some redundant information is employed, which result in a reduction of the experimental
measurement error. In this work, these polarimetric measurements were acquired with steps of
51.4º (on both angles θ1 and θ2), and the whole system was automated by means of rotation
motorized devices with a precision of 0.1º.
3. LCoS display as a function of the incident angle: polarimetric analysis
In [7] we presented a rigorous study of the polarimetric properties of a twisted nematic LCoS
display, illuminated with a 633 nm laser beam at quasi-normal incidence. Here, we extend
that study to different incident angles in order to analyze its influence on the LCoS display
performance. In particular, we have analyzed the degree of polarization, diattenuation,
polarizance and retardance dependence with respect to the incident angle.
3.1 Degree of polarization as a function of the incident angle
As mentioned previously, the LCoS display has been shown to produce a reduction in the
degree of polarization (DOP) that depends on the gray level and on the input SoP [7-9,12].
Figure 2 shows the measured DOP as a function of the gray level for various angles of
incidence (α=2º, 12.5º, 23º, 34º and 45º), calculated from the experimentally measured Stokes
parameters as DOP = S12 + S 22 + S 32 S 0 [14]. By definition, the DOP takes values from 0 to
1 but in Fig. 2, the y axis has been zoomed in order to show the results more clearly. The
results correspond to three input SoPs: linear polarized light at 0º, linear polarized light at
135º and left-handed circular polarized light. Figure 2 shows some relevant information about
the DOP dependence with the incident angle. Note that some DOP values are slightly higher
than 1, as a consequence of the instrumental error associated to the intensity measurements in
Eq. (3) and its corresponding error propagation. From Fig. 2, we see that for all the selected
incident angles, the DOP depends on the input SoP. For a fixed input SoP, there is a quite
relevant difference in the DOP evolution as a function of the gray level. For quasi-normal
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incidence (Fig. 2(a)), the reflected light remains fully polarized (DOP close to one) for gray
level ranges below 100 or above 240. However, important depolarization effects are detected
for gray levels in between 100 and 240, reaching depolarization values higher than 10%. This
depolarization effect is related to SoP time fluctuations originated from the electrical signal
addressing of the LCoS display [9]. For low gray levels, the DOP remains close to one (Fig.
2(a), gray levels below 100) because the liquid crystal molecules are oriented basically
parallel to the glass windows and their orientation is not so sensitive to fluctuations in the
electrical signal. On the contrary, for gray levels above 240 (Fig. 2(a)), the LC molecules are
almost completely tilted parallel to the electric field direction, despite the fluctuations in the
electrical signal, and the reflected beam also remains fully polarized. However, for gray levels
in between 100 and 240, the liquid crystal molecules tilt has an intermediate value, and the
optical modulation is very sensitive to the fluctuations of the electrical signal, resulting in the
highest depolarization effect for gray level 180.
Fig. 2. Degree of polarization as a function of the gray level and for an angle of incidence equal
to: a) α=2º, b) α=12.5º, c) α=23º, d) α=34º and e) α=45º.
When increasing the incident angle (Figs. 2(b)-2(e)), we detect unpolarized light in the
gray level range around gray level 180 (as in the quasi-normal incidence case shown in Fig.
2(a)), but also for higher or lower gray level ranges, where the depolarization increases as the
incident angle increases. For instance, for incident angles α=12.5º and α=23º (Figs. 2(b) and
2(c)), depolarization overpass 5%, while it is greater than 10% for incident angles of α=34º
and α=45º (Figs. 2(d) and 2(e)). For high incident angles and for some input SoPs,
depolarization reaches approximately a 10% along the whole gray level range (black triangles
at Fig 2(d) and Fig 2(e)).
We want to emphasize that part of the depolarized light detected around the 180 gray level
and at oblique incidence can be attributed to the signal fluctuations discussed before.
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However, there is another depolarization source when the LCoS display is used at high
oblique incidences. The amount of depolarization is higher when increasing the incident angle
and it is not only caused by the fluctuations phenomena. In order to prove this last statement,
we have measured the DOP corresponding to different input SoPs and with the LCoS display
switched off (no voltage addressed). This measurement has been performed with the incident
angles of to 2º and 45º. The results are shown in Fig. 3.
Fig. 3. DOP as a function of different incident SoPs and with an incident angle equal to: a) 2º;
b) 45º. The LCoS display is switched off.
When no voltage is addressed to the LCoS display, the light reflected by the device is
almost fully polarized, for an incident angle equal to 2º and for all the tested incident SoPs
(Fig. 3(a)). Unlikely, for an incident angle equal to 45º, the DOP strongly depends on the
incident SoP although no voltage is addressed to the LCoS display (Fig. 3(b)). In fact, using
linear polarized light at 0º of the lab vertical the DOP is almost one but when using an input
linear polarized light at 135º or left-handed circular light, we reach values close to 10% of
depolarization. Therefore, Fig. 3 proves that we identified a new depolarization source which
is not originated by the fluctuations in the electrical signal addressed to the LCoS display
(which are the cause of the effective depolarization effect previously reported [9]). Moreover,
this new depolarization source is not simply a constant offset equally added to the effective
depolarization effect. A constant offset would mean that the DOP should be, along the whole
gray level range, equal or lower than the DOP measured with the LCoS switched off. We see
for example in the case of left-handed circular light and α=45º incident angle (squares in Fig.
2(e)) that the DOP is bigger than the value 0.9, measured in the off-state (Fig. 3(b)), for most
of the gray level range. Therefore, we conclude that the new depolarization probably depends
on the optical director distribution in the LC layer (which changes with the addressed
voltage). It would be necessary further experiments to get a tighter grip on which is the origin
of the new depolarization source detected. A list of possible depolarization sources are
described in [8,15].
3.2. Diattenuation, polarizance and retardation dependence with the incident angle
Using the synchronous method described in Section 2, we have measured the experimental
Mueller matrix of the LCoS display for the incident angles of α=2º, 12.5º, 23º, 34º and 45º.
The experimentally measured Mueller matrices provide the polarimetric information of the
analyzed LCoS display. On one hand, the first row of the Mueller matrix is related to the
diattenuation vector, which gives the dependence of the transmittance with the incident SoP
[16]. On the other hand, the first column of the Mueller matrix is related to the polarizance
vector, which indicates the capability of the polarization element to polarize a fully
unpolarized beam [16]. Finally, the bottom right 3x3 submatrix gives the information about
the retardance and depolarization of the optical polarization element.
Figure 4 shows the first row (Fig. 4(a)) and the first column (Fig. 4(b)) coefficients of the
obtained experimental Mueller matrices, as a function of the gray level and for an incident
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angle α=2º. The corresponding equivalent results are plotted in Figs. 4(c) and 4(d) for an
incident angle α=45º.
Fig. 4. First row and column Mueller coefficients as a function of the gray level for an incident
angle of: a, b) α=2º; c, d) α=45º.
All these Mueller matrix coefficients have values very close to zero (except m00 which is
equal to one). Similar results are obtained for the other measured angles of incidence. As these
coefficients remain null as a function of the gray level, the LCoS display can be regarded as a
non-diattenuating and non-polarizing polarization device independently of the chosen incident
angle.
Next, we have analyzed the coefficients of the bottom right 3x3 submatrix, which provide
the retardance and depolarization information. As an example, Fig. 5 shows a comparison, for
the different incident angles, of the evolution with gray level of the m21, m22 and m23
coefficients of the experimental Mueller matrices. The large coefficients modulation shown in
Fig. 5 has been observed also for all the 3x3 submatrix coefficients. Note that the coefficient
evolution as a function of the gray level, shown at Fig. 5, varies gradually when increasing the
incident angle, finally leading to large variations between the results for quasi-normal
incidence (Fig. 5(a)) in comparison with the incidence at α=45º (Fig. 5(e)). This result
indicates that the retardance and depolarization effects will have a relevant dependence on the
operating incident angle.
In order to extract more information from the Mueller matrices, we have used the
combined method exposed in [11], where the Lu-Chipman polar decomposition [14] (based
on the polar decomposition theorem [17]) is applied to the LCoS Mueller matrix. In this sense,
the Mueller matrix of any depolarizing element can be expressed as the product of three
Mueller matrices: the depolarizer, the retarder and the diattenuator matrices. By taking into
account the results shown in Fig. 4, the diattenuation matrix can be approximated to the
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Fig. 5. Mueller matrix third row coefficients as a function of the gray level and an incident
angle equal to: a) α=2º; b) α=12.5º; c) α=23º; d) α=34º and e) α=45º.
identity in all cases. Therefore, it is possible to write the Mueller matrix of the LCoS display
just as the product of the depolarizer and the retarder matrices. Finally, from the retarder
matrix, we are able to find the Jones matrix of the equivalent retarder [11]. Next, we focus on
the analysis of the Jones matrix for the equivalent retarder.
Non-absorbing reciprocal polarization devices in reflection are theoretically equivalent to
a linear retarder [18]. Then, if we concentrate on the equivalent retarder Jones matrix for the
LCoS we may consider that under normal incidence (forward and backward path after
reflection are the same) the LCoS can be expected to act as linear retarder, whose neutral lines
orientation and retardance depend on the addressed voltage. However, when increasing the
incident angle, the LCoS may act as an elliptical retarder since the forward and backward
paths in the LC layer are no longer coincident. In order to evaluate this effect, we have
calculated the eigenvectors and eigenvalues of the equivalent retarder Jones matrix, and the
eigenvectors orientation and ellipticity is derived as a function of the gray level. Note that the
eigenvectors indicate the neutral lines in a linear retarder and the phase-shift between
eigenvalues gives the retardance. Figure 6 shows the retardance as a function of the gray level
for all the incident angles used along the experience. The minimum phase value corresponds
to the gray level 0 and an incident angle α=2º (rhombus spots), whereas the maximum phase
is obtained for the gray level 240 and α=45º (circular spots). These results show that small
incident angles (2º-12.5º) show a higher phase-shift dynamic range than high incident angles
(34º-45º). This fact should be taken into account when searching configurations of maximum
phase modulation, as we show in Section 4.
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Retardance
270
250
230
210
190
170
150
130
110
90
0
20 40 60 80 100 120 140 160 180 200 220 240
2º
12.5º
23º
34º
45º
Gray level
Fig. 6. Retardance as a function of the gray level and different incident angles.
Fig. 7. Equivalent retarder eigenvectors as a function of the gray level for the incident angle α=2º.
Next, the equivalent retarder eigenvectors are represented in Fig. 7, as a function of the
gray level, and for quasi-normal incidence (α=2º). They remain almost linearly polarized in
the whole range of gray levels, and their orientation rotates counter-clockwise as the gray
level increases. Therefore, we can consider the LCoS display at quasi-normal incidence as a
linear retarder whose retardation (Fig. 6) and neutral lines orientation (Fig. 7) change with
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gray level, in agreement with [7]. Moreover, as expected in retarders, the eigenvectors are
orthogonal to each other at every gray level, pointing out that the LCoS display is a
homogeneous element [19].
Finally, in Fig. 8, we represent the eigenvectors as a function of the gray level for the other
incident angles α=12.5º, 23º, 34º and 45º. Now their ellipticity increases as gray level
increases, being this effect stronger as the incident angle increases. Thus these result show
that, by increasing the incident angle and the gray level, the LCoS displays becomes an
elliptical retarder [20] or even a circular retarder for the particular case of gray level 240 and
incident angle α=45º. Let us note that transmissive twisted nematic liquid crystal displays
have, in general, two eigenvectors that are elliptically polarized [6], while reflective twisted
nematic displays operating at perfectly normal incidence act as an equivalent linear retarder,
thus having linear eigenvectors [21]. The result in Fig. 7 for quasi-normal incidence verifies
this situation, but the results in Fig. 8 evidence that the equivalent linear retarder behavior is
lost when the angle of incidence increases.
4. LCoS response optimization results: comparison between different incident angles
The performance of LCoS displays in optical applications require the adjustment of optimal
PSG and PSD configurations, providing specific intensity and phase modulation regime.
Typically the two desired modulation regimes are: maximum intensity contrast modulation
with constant phase, or constant amplitude modulation with maximum phase modulation.
These intensity or phase regimes can be achieved by using elliptically polarization PSG and
PSD configurations [22]. In our previous works we have demonstrated a useful methodology
suitable to optimize the intensity response [7] or the phase response [11] of a LCoS display. In
fact, in [11] a PSG and PSD configuration giving high phase modulation and also a constant
intensity response was obtained for a 633 nm wavelength, and for the incident angle of α=2º.
Here we study how the optimized modulation results are affected by increasing the
incident angle. We have measured the intensity and phase of the LCoS response when setting
the optimized configuration for phase modulation used in [11]. The experimental modulation
results are shown in Fig. 9, where an optimized configuration designed for phase-only
modulation regime at the incident angle α=2º is then tested at higher angles α=12.5º and
α=45º. The lines represent the theoretical simulations calculated using the combined
formalism described in [11], being the continuous line the simulated intensity (left axis) and
the dotted line the simulated phase (right axis). The symbols represent the experimental values
and they have been measured following the techniques described in [7]. The black circles
represent the experimental intensity and the squares the experimental phase.
Figure 9 shows, in all cases, a great agreement between simulated and experimental
values. Figure 9(a) evidence a phase-only modulation response of the LCoS display, where a
phase modulation up to almost 360º is accompanied with a constant intensity modulation.
Figure 9(b) shows that the phase-only modulation response is only slightly modified at low
incident angles (α=12.5º), but Fig. 9(c) shows that the phase-only performance is lost for
α=45º. Figure 9(c) shows significantly lower values of phase-shift and noticeable coupled
amplitude. This result indicates that the optimization performed for a given angle of incidence
works well within a reduced incident angle range. Out of this range the response is rather
different and good modulation results require applying the optimization technique to obtain
the specific optimal PSG and PSD configurations. As an example, Fig. 10 shows the
modulation results for two phase modulation configurations and for two incident angles
(α=12.5º and α=45º). The PSG and the PSD optimized values are P1=95º, WP1=85º, P2=61º
and WP2=64º for the α=12.5º incident angle and P1=91º, WP1=104º, P2=54º and WP2=53º for
the α=45º incident angle.
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Fig. 8. Equivalent retarder eigenvectors as a function of the gray level for the incident angles
α=12.5º, α=23º, α=34º and α=45º.
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Fig. 9. Theoretical (lines) and experimental (spots) intensity and phase values, when using an
incident angle equal to: a) 2º; b) 12.5º; c) 45º. The rotation angle values of polarizers and
waveplates used at the PSG and PSD systems are: P1=88º and WP1=7º; P2=90º and WP2=-15º.
On one hand, Fig. 10(a) shows a very constant intensity response (black line and circles)
as a function of the gray level. Moreover, we obtain almost 2π phase-shift (dotted line and
squares). Therefore, the modulation response optimization with α=12.5º provides similar
results to those obtained with α=2º (Fig. 9(a)). Thus, for a small range of incident angles
(around 10º) a single optimization is enough. On the other hand, Fig. 10(b) (α=45º) gives also
a constant intensity response but the phase-shift is significantly shorter (only slightly over
240º) than the obtained at quasi-normal incidence (Fig. 9(a)). Then, even by optimizing the
LCoS display phase modulation response at high incident angles, the results remain worse
than the obtained at low incident angles (Fig. 9(a) and Fig. 10(a)). This result is in agreement
with Fig. 6, where the phase-shift between the equivalent retarder eigenvectors is higher for
quasi-normal incidence than for oblique incidence.
Fig. 10. Phase modulation optimization when using an incident angle equal to: a) α=12.5º; b) α=45º.
As we have shown above, by using high incident angles the obtained phase modulation
response is remarkably lower than by using quasi-normal incidence. However, there are some
optical applications where a right angle between the incident and reflected beam is required,
and a high phase-shift is also desired to achieve good diffraction efficiency [13, 23]. Normal
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incidence may be achieved in this case using a beam splitter, in a set-up like shown in Fig.
11(a), where the PSG and the PSD are located before and behind the beam splitter. While the
beam splitter permits to build this compact setup, it presents the disadvantage of loosing light
power (half power is lost on every pass) and eventually may introduce additional polarization
effects that must be taken into account. By following the procedure previously discussed, the
whole system composed of the beam splitter and the LCoS display has been characterized as a
polarization device, and its phase modulation response has been optimized. The results are
shown at Fig. 11(b). It shows modulation results very similar to those obtained with quasinormal incidence (Fig. 9(a)), with a very small reduction in the phase-shift, caused by the
retardance introduced by the beam splitter. However, in order to work at 90º between the
incident and the reflected beams, while maintaining a high phase-shift response, the beam
splitter option is recommended.
Fig. 11. (a). Experimental set-up. (b). Optimized phase modulation response obtained when
using the beam splitter set-up. On one hand, the intensity values are represented in continuous
line (simulation) and black circles (experimental values). On the other hand, the phase values
are represented with a dotted line (simulation) and squares (experimental values). The rotation
angle values of polarizers and waveplates used at the PSG and PSD systems are: P1=105º and
WP1=94º; P2=105º and WP2=82º.
5. Conclusion
Summarizing, in this work we provide a study of the performance of an LCoS display as a
function of the incident angle. We analyzed how optimized phase modulation configurations
employing elliptically polarized light respond when changing the incidence. Here we
presented results on the polarimetric properties of an LCoS display, illuminated by 633 nm
wavelength laser. The experimental measurements presented evidence that a previously
reported effective depolarization effect, which shows dependence on the addressed gray level
and on the input SoP, also presents an important dependence on the incident angle. Moreover,
we detected an additional source of depolarization, not related to the fluctuations of the
electrical signal, and which is more significant at high incident angles. In addition, the
polarimetric study revealed that LCoS display acts as a non diattenuating and non polarizing
element for every tested incident angle. On the contrary, we have observed a strong relation
between the LCoS display retardation and the incident angle. Moreover, the experimental
measurements show that LCoS display becomes an elliptical retarder when increasing the
incident angle. The retardance dependence with the incident angle has an important effect at
the LCoS display phase modulation response. We proved that incident angle deviations less
than 10º do not modify substantially the modulation properties. However, we showed that a
fixed configuration of polarizers and waveplates giving very good phase response at normal
incidence shows a degraded phase-only modulation response (reduced phase-shift and couple
amplitude modulation) when increasing the incident angle.
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We have also proved that optimized phase-only modulation configurations can be
achieved for every incident angle, although the optimization procedure must be applied in
each case. However, we have obtained less phase modulation depth in the phase-only
modulation configurations as the incident angle increases. Finally, in order to retain a good
phase modulation depth in a setup with perpendicular incident and reflected beams, we
included a beam splitter. The system composed of the beam splitter and LCoS display has
been characterized as a single polarization modulator, and the optimization process led to a
phase-only configuration providing results almost equivalent to those obtained with quasinormal incidence, in spite of the retardance introduced by the beam splitter. Therefore, when
good phase modulation is required, simultaneously with perpendicular incident and reflected
beams, the use of a beam splitter is recommended.
All these results are relevant since LCoS displays are becoming a device useful for a
number of optical applications, and care must be taken when selecting the incident angle. In
addition, these effects may be relevant when employing the device illuminated with a wide
range of incident angles, as it is the case for instance in optical trapping systems.
Acknowledgments
We acknowledge financial support from Spanish Ministerio de Educación y Ciencia
(FIS2006-13037-C02-01 and 02) and Generalitat de Catalunya (2006PIV00011). C. Iemmi
acknowledges support from Univ. Buenos Aires and CONICET (Argentina).
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