encoding of real-time spatial light modulators
Robert W. Cohn and Minhua Liang
We previously proposed a method of mapping full-complex spatial modulations into phase-only
modulations.The Fourier transform of the encoded modulations approximates that of the original
complex modulations.The amplitude of each pixel is encoded by the property that the amplitude of a
random-phasor sum is reduced corresponding to its standard deviation.
designed for phase-only spatial light modulators that produce 360° phase shifts.
are rare, experiments are performed with a 326° modulator composed of two In Focus model TVT6000
liquid-crystal displays.Qualitative agreement with theory is achieved despite several nonideal
properties of the modulator.
Optical information processing, spatial light modulators, liquid-crystal televisions,
phase-only filters, laser speckle, rough-surface scattering, statistical optics, binary and diffractive
optics, phased arrays, measurements of phase. r1996 Optical Society ofAmerica
Pseudorandom encoding is
Because such devices
Pseudorandom phase-only encoding1is a method for
designing phase-only filters and diffractive optical
elements that approximately produces the same
Fraunhofer diffraction pattern as would result from
a desired, but unrealizable, full-complex filter.
encoding procedure adds amplitude control to the
phase-only filter2through the addition of phase
offsets that have specified statistical properties.
The randomness of the phase offset at a given pixel,
as measured by the standard deviation of this ran-
dom variable, determines the effective attenuation
caused by the pixel insofar as it describes its effect on
the Fraunhofer pattern.
grammed in this way to have a specified value of
phase and effective amplitude.
resulting diffraction pattern approximates the de-
sired diffraction pattern from the full-complex modu-
lation is a result of the law of large numbers.
this situation the individual Huygens wave fronts
from all spatially separated pixels in the modulator
plane all coincide and coherently add together across
the Fraunhofer plane 1as illustrated in Fig. 12.
Each pixel can be pro-
The fact that the
the amount of randomness present in the input
pixels is not too large, then a coherent reconstruction
is observed that approximates the diffraction pat-
tern from the full-complex modulation.
domness is too great, only a speckle pattern is
observed. We also show that the quality of the
reconstruction, in terms of how well it approximates
the desired diffraction pattern, is closely related to
the average intensity transmittance of the desired
calculated directly from the full-complex modula-
tion, it can be used in advance of performing the
encoding to determine whether or not the phase-only
modulation provides adequate performance for the
The method is especially well suited for real-time
programming of spatial light modulators 1SLM’s2.
The mapping, because it requires one function calcu-
lation 1or table lookup2 per pixel, can be computed at
the frame rate by a serial electronic processor.
is in contrast to many diffractive-optic and filter-
design procedures that focus on optimal synthesis of
a desired diffraction pattern under the constraint of
are appropriate for the design of fixed-pattern filters;
however, as they are numerically intensive, they
usually cannot be performed in real time.
applications it may be adequate to precompute the
mapping off-line, but for other applications 1e.g.,
those in which the number of precomputed images
exceeds the amount of available memory or those in
If the ran-
Because this metric is
These latter approaches
R. W. Cohn is with the Department of Electrical Engineering,
University of Louisville, Louisville, Kentucky 40292.
with Neuristics Corporation, Baltimore, Maryland 21204-2316.
Received 9 June 1995; revised manuscript received 30 Novem-
r1996 Optical Society ofAmerica
M. Liang is
2488APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996
which the desired full-complex function is not neces-
sarily known in advance2, it may only be possible to
program the SLM by a real-time algorithm such as
Several devices can in principle produce a 2p
phase-modulation range, but far fewer are available
in practice.Parallel-aligned nematic liquid crys-
tals 1also referred to as birefringent liquid crystals2
have been known to produce pure phase modulation
for some time.3
However, such devices have become
available only recently.These include devices from
Hughes, Hamamatsu, and Meadowlark Optics.
Only the optically addressed SLM from Hamamatsu
has a 2p modulation range 1according to their techni-
ers have also demonstrated programmable kino-
forms using a birefringent liquid-crystal television
However, the device is a custom-
nematic display, and this device is not available
outside Epson. Using commercially available
twisted-nematic LCTV’s, several researchers have
These are achieved by the proper selection and
orientation of polarizers and@or wave plates.
interesting is the eigenpolarization mode in which
an elliptical polarization is transformed into an
identical polarization independent of the voltage
applied across the liquid crystal.6
195° and better than 64% uniformity of amplitude
were achieved for a panel from the In Focus model
TVT6000 video projector.
tified a configuration that produces 2.5p of modula-
tion for LCTV panels from an Epson E1020 video
However, Soutar and Lu later noted
tially across this device.8
able mirror device produces phase shifts through the
displacement of micromechanical pixels in a direc-
tion parallel to the optical axis.9
addressable devices with this type of pixel have not
yet been successfully demonstrated.
In spite of the limited availability of 2p modula-
tion it is still possible to experimentally demonstrate
the method of pseudorandom encoding on current
modulators and to produce diffraction patterns that
For the TVT6000 LCTV, several nonideal properties
Phase shifts of
Soutar and Monroe iden-
The 1phase-only2 deform-
must be recognized and controlled.
that, because the SLM does not produce full 2p
modulation, there is always a dc component of the
modulated light that is focused onto the optical axis.
In order to separate the dc peak from the designed
diffraction pattern, we add a phase ramp to the
desired modulation.As the SLM cannot produce all
values between 0 and 2p, multiple harmonics in
addition to the fundamental of the designed diffrac-
tion pattern are produced.
rier frequency in practical systems leads to an
undesirable reduction in the usable bandwidth of the
SLM, for our purpose of demonstration, this condi-
tion is acceptable.Before presenting the results of
the experimental demonstrations, we review the
method of pseudorandom encoding and describe how
the modulator is configured for the experiments.
Most critical is
Although use of a car-
The encoding procedure is designed for SLM’s com-
posed of one- or two-dimensional arrays of pixels.
However, the subsequent equations describing the
procedure 3in particular, Eqs. 142–1624 are presented
only as a function of one spatial coordinate.
done in part to simplify presentation, but more so to
emphasize that the performance of the procedure
depends directly on the total number of pixels.
generalization to two dimensions simply requires
the replacement of any spatial coordinate x or spatial
frequency component fxwith the two-dimensional
components 1x, y2 or 1fx, fy2.4
can be applied to any array of arbitrarily positioned
pixels, but in the experiments we considered only
regularly spaced arrays.
cally shaped apertures.
amplitude transmittance and can produce any value
of phase between 0 and 2p.
the value of the phase of the ith pixel located at
position xiis assumed to be its statistical average:
Review of the Method
The encoding procedure
The pixels have identi-
Each pixel has unity-
For purposes of design
where 7? ? ?8 represents the expected-value or en-
actually produced by the pixel is
The value of the phase
ci5 ci1 dci,
where dciis an unbiased random-phase offset.
tion. The distribution determines the design value
of the effective amplitude produced by the pixel.
The set of all uniform random distributions with
spreads between 0 and 2p is especially convenient
for this purpose. Amplitude control between unity
and 0 is provided according to
ai5 7exp1 jdci28 5 sinc1
where niis the spread of the uniform distribution.
Because the values of Eq. 132 are in the range
tion of a large number of wave fronts from independently phased,
equal-intensity point sources.
Fraunhofer diffraction geometry illustrating superposi-
10 May 1996 @ Vol. 35, No. 14 @ APPLIED OPTICS 2489
between zero and one, we always assume that the
full-complex modulation is normalized so that its
maximum amplitude is unity.
tude of less than one would reduce diffraction effi-
ciency and increase noise.2
To summarize the algorithm, we set the design or
effective complex modulation 1ai, ci2 equal to the
desired full-complex modulation.
inverted to solve for spread ni.
selected from a uniform unbiased distribution of
spread niby a random number generation routine.
The random-phase offset is added to the design value
of phase 3see Eq. 1224 to produce the actual modulation
The complex transmittance produced by a phase-
only SLM consisting of an array of N identical
subapertures centered at positions xiis
1A maximum ampli-
Equation 132 is
A value of dciis
exp1 jci2r1x 2 xi2,
where r1x2 is a function defining the clear area of the
subaperture. The effective complex transmittance
resulting from encoding each pixel is likewise writ-
aiexp1 jci2r1x 2 xi2.
The quality of the diffraction patterns produced by
pseudorandom encoding can be appreciated by con-
sideration of the expression for the expected inten-
sity of the diffraction pattern.
It has been shown to
I1fx2 5 0T1fx2021 N11 2 h2R21fx2,
where fxis the spatial frequency, T1fx2 and R1fx2 are
Fourier transforms of t1x2 and r1x2, respectively, and
the average intensity transmittance
can be viewed as a type of diffraction efficiency, as we
Equation 162, the expected value of the intensity, is
composed of two terms.
the specular component of the diffraction pattern
1caused by the coherent superposition of wave fronts
originating from the SLM2.
diffraction pattern of the desired full-complex modu-
lation, i.e., the square magnitude of Eq. 152.
second term represents the diffuse component 1or
speckle pattern2 that is due to the superposition of
randomly phased wave fronts.
tion is identical to the diffraction pattern from the
aperture of a single pixel.
terns of interest are much more directional than the
pattern of a diffuse scatter.
The first term represents
It is identical to the
Its spatial distribu-
Many diffraction pat-
This directionality gain
over the more nearly white background noise often
makes it possible to produce a close approximation to
the desired diffraction pattern.
The amount of noise, and thus the quality of the
reconstruction, is largely understood in terms of the
diffraction efficiency h.
indicates the fraction of energy that appears in the
coherent portion of the diffraction pattern.
phase-only modulation does not attenuate, the re-
maining fraction of the incident energy, 1 2 h, is
diffracted into the incoherent speckle background.
In Ref. 1 we interpreted hN as an effective number of
nonrandom phase-only pixels.
that increasing the diffraction efficiency 1or equiva-
lently, the number of nonrandom pixels2 reduces
diffuse scatter and would thus be expected to pro-
duce more accurate designs.
ing the relation between the desired modulation and
its pseudorandom encoding is that the more efficient
a full-complex modulation is, the greater is its
similarity to phase-only modulation.
An expression for the standard deviation of the
intensity of the diffraction pattern is also presented
in Ref. 1.It provides a precise statistical bound on
the accuracy of the diffraction pattern at every point
across fx. Although the expression is not duplicated
in this paper, it is used here to calculate error bars on
plots of the designed intensity patterns.
the effective number does not provide information as
detailed as the standard deviation, it can be calcu-
lated with much lower computational overhead.
Also it is well-enough correlated with the standard
deviation that it can be used to gauge the quality of
the diffraction pattern in real-time applications.
Standard deviation, h, and N can be used together to
define peak-to-background noise and the signal-to-
noise ratio as was done in Ref. 1.
The diffraction efficiency
Equation 162 shows
Another way of view-
The experiments are performed with two LCTV
video projectors.Oscilloscope measurements show
that, when black-and-white signals are applied to
the video input, the signals from the red, green, and
blue driver boards of the projector are essentially
identical.One LCTV is electrically reconnected to
the driver board for the red channel, and the other
LCTV is connected to the driver board for the blue
channel 1as illustrated in Fig. 22.
panels is connected through its own cable extender
that we fabricated from a 1-m, 23-conductor, flat-
laminated cable from Parlex Corporation and two
zero-insertion-force flexible-printed-circuit connect-
ers fromAmp Incorporated.
The green LCTV is left in the projector.
used to preview visually the image exactly as it is
displayed by the LCTV panels.
especially simple way to verify that the LCTV’s
respond correctly to the applied image.
when a RS-170 standard composite video is applied
to the video input of the projector, we observe that
Each of the two
This provides an
2490APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996
only 440 video lines and only ,94% of the viewable
portion of each line is displayed by the projector.
In terms of sending a 640 3 480 pixel digitized image
in video format from our frame grabber 1Dipix 360f
with optional video-display board2, only 600 3 440
pixels are displayed.Because the LCTV has 480 3
440 pixels, there is a 1:1 mapping from frame-
grabber lines to LCTV lines and a 5:4 mapping
between pixels on a line.
A second issue in the mapping of the digital pixels
is the modulation transfer function of the display.
Loudin noted that a single video line produces two
lines of video on the projected image from the
We also observed this.
that the projected lines are roughly half the bright-
ness possible.When two adjacent rows are turned
on, one fully bright row is displayed, and there is a
dim row to each side of the bright row.
the LCTV pixels cannot fully charge during the pixel
time slot, and thus the sampling time 1of the thin-
film transistors2 is extended to overlap with that of
the adjacent lines. With projector sharpness set at
minimum the effect is of roughly the same magni-
tude in both the horizontal and the vertical direc-
The optical apparatus used to produce and mea-
sure diffraction patterns is shown in Fig. 2.
collimated, nearly uniform-intensity laser beam of
632.8-nm wavelength and linear polarization illumi-
nates the entrance port of the optical system.
Standard optical components between the laser
source and the entrance port are used to filter and
expand the beam spatially, rotate the polarization,
and control the intensity of the beam.
consists of the components between the two quarter-
wave plates.Light from the output of the second
wave plate is focused with a 381-mm focal-length f@5
1Sorl F15@52 lens.The optical Fourier transform of
the SLM is observed with a video microscope that
consists of a Cohu 4915-2000 CCD video camera with
an active imaging area of 6.4 mm 3 4.8 mm, a
C-mount 120-mm extension tube–microscope barrel,
We further note
and a microscope objective.
observed on a video monitor or captured with the
frame grabber.The frame grabber is configured to
acquire images simultaneously while continuously
outputting a single image.
are viewed with a digital oscilloscope 1HP54503A2
that has built-in video triggering.
The camera output is
Individual video lines
The first LCTV 1LC12 is imaged onto the second one
1LC22 with a pair of 228.6-nm focal-length, firstname.lastname@example.org
imaging lenses 1Plummer lenses purchased from
MWK Industries2.The second LCTV is rotated by
180° to account for the image inversion produced by
gular aperture 1A12 at the Fourier plane between the
two lenses is set to pass only the signal information
surrounding the central diffraction order of the
LCTV grating–pixel pattern.
to describe the polarizations are chosen so that x, or
horizontal, is in the direction of the video scan line, y
is in the vertical direction of the video, and z is
opposite of the direction of laser propagation.
LCTV’s are oriented so that the laser light emerges
from the side of the panels on which the electrical
cable is visible. The first LCTV 1LC12 is oriented so
the cable is attached below the horizontal axis.
this coordinate system the first quarter-wave plate
1Q12 is illuminated from a red helium–neon laser
with a linear polarization of 70° from x and toward y.
The fast axis of the first wave plate is at 45°, and the
fast axis of the second quarter-wave plate 1Q22 is at
245°.The polarizer 1P2 is oriented to pass light at
70°, the same as the incident linear polarization.
The polarizer produces a substantial amount of
wave-front distortion across its full aperture.
illuminated area and consequently distortion are
reduced by placement of the polarizer near the focus
of the Fourier-transform lens.
ness and contrast controls are set to maximum, and
the sharpness is set to minimum.
The effect of maximum sharpness is clearly evi-
dent on broadly extended diffraction patterns.
particular, if the LCTV is driven by a digital image of
white noise, the envelope of the resulting speckle
pattern resembles the diffraction pattern of a single
pixel except that it is modulated in the horizontal
direction by six high-contrast stripes covering the
separation between adjacent diffraction orders of the
SLM.This scallop-shaped pattern is reduced for
the minimum sharpness setting.
duced the effect in our experiments by programming
multiple adjacent pixels with the same video level,
i.e., grouping pixels together into superpixels.
scalloped diffraction pattern is probably due to filter-
ing produced from the combination of sharpness
control, the LCTV pixel integration@charging time,
and the 5:4 mapping between the frame grabber and
the LCTV pixels.
Aperture A2 is chosen to be 15.2 mm 3 15.2 mm.
This limits the illuminated area of the SLM to 272
Alignment of the Two LCTV’s
The coordinates used
The projector bright-
We further re-
scope 1or optical powermeter where noted in text2; PC, personal
computer that contains a frame grabber; VP, video projector; LC1
and LC2, LCTV’s from the video projector; S, viewing screen; M,
video monitor; O, digital oscilloscope; P, polarizer; A1 and A2,
apertures; Q1 and Q2, quarter-wave plates.
Apparatus used for the experiments:VM, video micro-
10 May 1996 @ Vol. 35, No. 14 @ APPLIED OPTICS2491
pixels by 332 lines.
permits registration of the image of the first LCTV
onto the second LCTV to better than 0.5 pixel in both
x and y. We verified the alignment by modifying the
optics to make amplitude modulation evident.
did this by rotating the wave plates into an appropri-
ate orientation and placing the polarizer and the
Aperture A1 at the Fourier plane between the two
LCTV’s is also opened so that the edges of individual
pixel apertures from the first LCTV are observable
through the pixel apertures on the second LCTV.
Single-line and row-patterns are programmed onto
the LCTV’s, and micropositioners are used to align
the patterns.The microscope is mounted on trans-
lation stages, and the alignment is checked at sev-
eral positions across aperture A2.
errors only become noticeable close to the edges of
the aperture and appear to be caused by a slight
barrel distortion in the imaging optics.
alignment is set and@or tested in this way, aperture
A1 is once again set to pass only the central diffrac-
tion order, and the system is reconfigured according
to Fig. 2.
In addition to registration, aperture A2 is also
needed in order to improve optical flatness so as to
We determined this by measuring the point-spread
function of the SLM aperture with the video micro-
scope. For the modulated light to be separated
from the unmodulated light, the SLM, rather than
being programmed with a constant gray-scale level,
is programmed with a periodic ramp that displaces
the point-spread function from the optical axis.
1Details on selection of the ramp function and offset
are given in Section 4.2
spread function measured at the focal plane of the
transform lens with the diffraction pattern ideally
produced by a 15.2-mm aperture and a 381-mm
transform lens.The experimental intensity trace is
taken across the center of the intensity peak, and it
is aligned along a horizontal line of the camera in
order to improve resolution.
facturer-quoted dimensions for the CCD imager to-
For the available lenses this
Figure 3 compares the
The correct scale for
gether with the measured magnification of 9.76X
1nominally a 103 objective is used2 for the video
We also measured 1using a photodetector and a
powermeter in place of the video microscope2 the
variation in intensity of this central diffraction order
when the uniform gray-scale image from the frame
grabber is varied from full black to full white.
residual amplitude modulation is 63.7% of the aver-
age amplitude modulation.
mittance is at a gray scale of ,128, and there are two
local minima at ,40 and ,256.
We also measured energy utilization of the SLM
between the input face of Q1 and the output face of
Q2.We did this by measuring the intensity flux
across the 1effective2 input and output apertures 1A2
projected onto Q1 and Q2.2
maximum transmittance, only 0.325% of this inci-
dent energy is measured at Q2.
able feature of the eigenpolarization mode is its low
loss, we were initially concerned about the low
optical efficiency.We also configured the system as
a phase-only modulator by illuminating each LCTV
with a linear polarization and passing the linear
polarization that most closely approached phase-
only operation.The efficiency of this cascade is 1@6
that of the eigenpolarization arrangement.
has a similar phase-modulation range.2
The most significant loss factor is apparently
related to the shadow mask in each LCTV panel.
Measurements under a microscope indicate that the
pixel aperture is roughly 28 µm 3 28 µm on a pitch of
56 µm horizontally by 46 µm vertically, leading to a
fill factor of approximately 30%.
30% of the energy is transmitted through the first
shadow mask.Based on the Fourier-series analysis
of a square wave, only 30% of this energy remains in
the zero order at the filter plane between the two
lenses of the imaging system.
filtering of the nonzero orders, the second LCTV
ted flux. Overall, these losses would lead to a
transmittance of 0.86%.Thus in this configuration
even moderate fill factors can be the dominant
source of loss.
Amplitude-Modulating Properties of the SLM
The maximum trans-
With the SLM set for
Because one desir-
Owing to the spatial
We initially measured the phase shift by building a
Mach–Zehnder interferometer around the SLM ar-
rangement shown in Fig. 2.
was tilted to produce fringes vertical to the video
scan.The shift of the fringe between the row driven
with gray scale 0 and a row driven with another
value of gray scale was measured.
ments indicate that the phase increases monotoni-
cally from 0° to 326°.A rough curve that approxi-
mately describes the phase shift as a function of gray
scale is a curve that has a linear slope of 2° gray-
Phase-Modulating Properties of the SLM
The reference beam
spot perpendicular to the horizontal lines of the SLM.
Point-spread function of the SLM aperture as measured
The cross section is taken across the center of the
2492 APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996
scale level between gray-scale levels 0 and 140 and
0.4° per gray-scale level between 140 and 255.
This measurement procedure, however, appears to
be subject to several errors, including coherent noise,
curvature, low contrast, and vibration 1caused by air
turbulence2 of the fringes.
concerns about errors introduced by the additional
optical components of the interferometer.
measurement procedure was devised in order to
confirm the phase measurements.
also indicates the dependence of the phase modula-
tion on the spatial frequencies of the signal applied
to the SLM.To perform the measurement, one
loads a pseudorandom binary-level pattern into the
frame grabber.An aperture and a powermeter are
positioned at the focal plane of the transform lens
shown in Fig. 2 so as to intercept the central
diffraction order.Half the SLM pixels are ran-
domly selected and driven with a gray-scale value of
0.The other half are driven with a second gray-
scale value chosen between 0 and 255.
sity measured on the powermeter is recorded as a
function of gray scale.The measurement was re-
peated for various SLM pixel sizes, where an SLM
pixel is defined to be an n 3 m array of LCTV pixels
that are programmed with the same gray-scale
value.These results are plotted in Fig. 4.
Diffraction from a binary-level, phase-only modu-
lation can be understood by the analyses from Ref. 1
that were used originally to describe pseudorandom
encoding.In this case each pixel is programmed to
have an identical effective amplitude that produces
an expected intensity pattern consisting of a diffrac-
tion-limitd spot centered on the optical axis and a
broad background noise level.
from all SLM pixels mutually interferes at the
intersection of the optical axis and the focal plane
1i.e., fxequal to zero2, as illustrated in Fig. 1, the
intensity of the spot depends directly on c, the
difference between the two phases present on the
SLM.The phasing on axis can be evaluated by
deterministic analysis alone.
Furthermore, we had
Because the light
For an ideal phase-
only SLM in which each pixel can be controlled
independently of all others, the intensity of the
diffraction peak on the optical axis is
I102 5 N2R2102
1 1 cos c
where N is the number of SLM pixels and R1fx2
1introduced in Section 22 is the element factor corre-
sponding to the pixel-clear aperture.
lation of the spot for a phase difference of 180°.
practical measurements a finite detector size and
nearby speckles would introduce errors into the
measurement of power.
usually a large number, speckle noise is not a serious
source of error over most of the phase range.
Assuming that the SLM has adequate phase range,
it would also be possible to offset the lower gray-scale
value from zero to a larger value in order to more
clearly measure the 180° point.2
Based on the initial phase measurements with the
Mach–Zehnder interferometer, it is reasonable to
expect that there would be a simple and direct
relation between the phase difference and the non-
zero gray-scale value.Indeed, the curves in Fig. 4
show a qualitative agreement with this model in that
they resemble a cosine function for gray-scale values
of ,128.For gray-scale values of .128 the inten-
sity increases but much more slowly than for the
lower gray-scale values. This tracks with the inter-
ferometer measurements of phase in which the
phase sensitivity to gray scale decreased at large
values of gray scale.
The curves indicate that the phase-modulation
range increases with SLM pixel size.
mum range in these curves is for the 12 3 12 array.
The maximum value of I102 after the phase exceeds
180° is 0.57.Inverting Eq. 182 gives an estimated
modulation range of 278°.
there is barely a 180° range.
with the original interferometer measurements ap-
pears to be good for low spatial frequency patterns of
modulation; however, the actual phase modulations
needed for the encoding experiments have a range of
frequencies from high to low.
As would be
However, because N is
For the 4 3 1 SLM pixel
The cascade of two SLM’s was originally intended to
produce a phase modulator with a range of 360°.
The measured SLM does have a small degree of
residual amplitude modulation.
phase-modulation range that is close to 360° but only
for modulations with low spatial frequencies.
phase-modulating characteristics can be made inde-
pendent of spatial frequency by use of large SLM
pixels; however, this significantly reduces the num-
ber of available pixels for the encoding experiments.
1For the case of 12 3 12 SLM pixels and the available
clear aperture there are fewer than a thousand
Summary of SLM Properties
It also has a large
scale patterns and with superpixels of various sizes.
is normalized to the peak intensity with the gray scale equal to
On-axis diffraction intensity for random binary gray-
10 May 1996 @ Vol. 35, No. 14 @ APPLIED OPTICS 2493
Experimental Demonstrations of Pseudorandom
Because the available SLM does not achieve a 360°
range, there is usually a dc component in the spatial
modulation and a corresponding diffraction peak at
fxequal to zero.We chose to separate the dc compo-
nent by multiplying the modulation by a periodic
function.We accomplish this by adding together a
linear ramp function with the gray-scale values of
the encoded signal, then removing integer factors of
256 by using the module base-256 function, and
placing this image in the display memory of the
tion patterns are generated, of which the fundamen-
tal is usually the brightest.
pated from our presumed phase-only modulator.
The fundamental spatial frequency of the phase
ramp 1i.e., the carrier frequency2 also affects the
quality of the diffraction pattern.
evident in that the point-spread function decreases
with increasing phase ramp or carrier frequency.
This is illustrated in Fig. 5, in which diffraction
efficiency is plotted against the period of the carrier.
Efficiency for this measurement is defined as the
peak intensity normalized by the intensity of the
spread function when it is centered on the optical
axis 1specifically, when all pixels of the SLM are
programmed to gray-scale value 02.
sities are measured 1with the digital oscilloscope
from Fig. 22 across a single video line of the CCD
camera 1rotated 90° from horizontal lines of the
SLM2. Figure 5 can be viewed as a type of modula-
tion transfer function.The curve shows, for periods
between 20 and 70 SLM lines, that the diffraction
efficiency is close to flat.
rapidly at ,10 lines.
encoding demonstrations, presented below, a period
of 20 lines was found to be acceptable.
period increases to .20 lines, interference from
Adjustments Made for Demonstrations with the
It is this fundamental
This is most
The peak inten-
The efficiency falls off
For the pseudorandom-
overlap with the dc and the first-harmonic order
Fig. 42 indicates that individual pixels are not inde-
pendent of the modulation of nearby pixels.
surprising that there is still residual correlation
between pixels that are 10 and 20 lines apart.
Certainly, by any criteria, the resolution of the SLM
is far worse than one pixel.
The various issues relating to making the SLM
behave as an array of independently controllable
pixels cannot be well satisfied with the given con-
straints of maintaining a diffraction-limited aper-
ture, the low spatial resolution of the SLM, the lack
of 1:1 mapping between display board pixels and
SLM pixels along the scal line, and the need for a
large number of pixels. These constraints led us to
program the SLM as clusters of superpixels.
superpixel consists of 4 pixels along a single scan line
for 22,000 total pixels, which is roughly the same
number of pixels 116,3842 as used in the simulations
in Ref. 1.Using a 4 3 2 superpixel was not consid-
ered after noting, based on Fig. 4, that the total
phase-modulation range would increase to only 193°,
an increase of as little as 13°.
programmed as if it consists of 68 3 332 superpixels,
and the carrier has a period of 20 lines in the vertical
Thus the SLM is
larly shaped diffraction patterns centered on the
optical axis.The apodization is the product of a
sinc function and a Dolph window in the x, or
This window is chosen to
reduce the Gibbs ripple that can result from the
finite extent of the SLM truncating the sinc function.
The specific Dolph function selected is defined by the
sidelobe level of its Fourier transform d1fx2, which for
are known to further reduce the Gibbs ripple, but
they also increase transition bandwidth.
are more steeply tapered, which reduces the value of
diffraction efficiency h 3see Eq. 1724 and consequently
increases the speckle noise 3see Eq. 1624.
computer simulations of random-encoding-designed
diffraction patterns in Ref. 1, a 26-dB sidelobe level
was empirically found to provide a good trade-off
between reduction in the Gibbs ripple and back-
ground noise.The same apodization was applied
both in the horizontal and the vertical directions.
In the designs for the experiments presented here,
the apodizations vary in only the vertical direction.
The one-dimensional apodization is used with the
objective of further increasing the diffraction effi-
ciency and reducing the random noise inherent to
the encoding procedure.
sen to produce diffraction-intensity patterns of the
Design and Encoding of Full-Complex Modulations for
The apodizations are cho-
I1fx, fy2 ~ sinc21fx23d1fy2 p rect1fy@w242,
function of the carrier period.
intensity of the undeflected 1on-axis2 point-spread function.
unit for the carrier period is the number of horizontal SLM lines
Intensity of the deflected point-spread function as a
Intensities are normalized to the
2494 APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996
where sinc21fx2 is the point-spread function in the x
direction and d1fx2 is the Fourier transform of the
Dolph window.The amplitude in the y direction is
designed to approximate a rect function of width w
for values of 2, 3, and 4.The width is normalized in
terms of the unity width 1from peak to first null2 of
sinc21fy2, the point-spread function in the y direction.
From Fig. 3 it can be seen thatw 5 1 corresponds to a
physical distance of ,15 µm.
Each of the three apodizations are pseudoran-
domly encoded by Eqs. 112–132.
these modulations are then multiplied by a carrier
frequency with a period of 20 video lines before they
are loaded onto the frame grabber.
diffraction patterns of the apodizations are also
calculated. This is done with a 256 3 4096 array
fast Fourier transform on the 68 3 332 array de-
signed modulations 1zero padded to the size of the
y direction from the centers of the theoretical diffrac-
tion patterns are saved for comparison with the
measured diffraction patterns.
As described above,
The diffraction patterns are recorded with the CCD
camera in the video microscope rotated by 90° from
the x 1horizontal2 and the y coordinates of the SLM.
The resulting images are shown in Fig. 6.
the four images the CCD camera is saturated by
roughly the same factor with respect to the peak
intensity of the desired pattern in order to aid in
comparisons of the background levels.
the image of the point-spread function.
Summary of Experiments and Their Comparisons
In each of
Figure 61a2 is
section of this diffraction pattern is shown in Fig. 3.2
The cross sections corresponding to Figs. 61b2–61d2 are
shown in Figs. 7 and 81a2.
attenuator settings, the corresponding curves and
images were recorded under identical conditions and
within a few minutes of each other.
Figure 61a2 shows that a substantial amount of
energy is deflected into the fundamental order and
that there is much less energy at dc and the first
harmonic. The first diffraction order 1not shown2 to
the left of dc is much weaker than the fundamental
but stronger than dc and the first harmonic.
ures 61b2–61d2 show the successive widening of the
beam footprint. The first harmonic is still quite
faint for the shaped beams, but dc is now very bright.
As can be seen from the vertical axes in Figs. 7 and
81a2, the intensity of the fundamental is much less
than it is in Fig. 61a2. The sidelobe structure of the
saturated, closely resembles the point-spread func-
tion in Fig. 61a2.Each of these diffraction patterns
also shows, as expected, both a background speckle
pattern and sidelobes along the narrow direction of
the beam footprint.
Figures 7 and 81a2 provide more precise informa-
tion describing the measured diffraction patterns.
The three measured curves are normalized to the
tion in Fig. 132. The three theoretical curves were
Except for changes in
shown in Fig. 32.
designed to approximate a rect function in fyare of widths 1b2 w 5
2, 1c2 w 5 3, and 1d2 w 5 4.The images are oriented so that fyis
horizontal to the page.
Gray-scale images of the measured diffraction-pattern
1a2 the measured point-spread function 1cross section
The measured diffraction patterns that are
sured intensity curves are normalized with respect to the peak
intensity of the measured point-spread function in Fig. 3.
sections are taken from the centers of the diffraction patterns
shown in 1a2 Fig. 61b2 and 1b2 Fig. 61c2.
are scaled by 1.963 in 1a2 and by 1.533 in 1b2 with respect to the
peak intensity of the theoretical point-spread function from Fig.
3.Both measured curves are plotted with a 14-µm offset from
the center of the measured point-spread function.
Theoretical and measured diffraction patterns.Mea-
Theoretical intensity plots
10 May 1996 @ Vol. 35, No. 14 @ APPLIED OPTICS2495
initially normalized by the peak intensity of the
theoretical curve in Fig. 3, but the curves were found
to be somewhat lower than the experimental intensi-
ties. Multiplying the theoretical curves by vertical
scale factors 1between 1.53 and 1.96 as noted in the
corresponding figure captions2 helps to show just
how similar the curves are in shape.
measured data by 4–5 µm further improves the
correspondence, but to a much lesser degree.
these experiments the scale-factor difference be-
tween experiment and theory is the most significant
discrepancy.The offset in position is rather small
when compared with the diameter of the point-
spread function.These discrepancies are probably
due to one or more of the nonideal properties of the
actual SLM, none of which were factored into our
theoretical model.It is not our goal to account for
these differences quantitatively but rather to gain
insight into how the encoding method would behave
when 360° modulators become available.
reason we think that the degree of correspondence,
while qualitative, is nonetheless excellent, and we
are encouraged by this result.
Further evidence of the correspondence between
experiment and theory can be demonstrated by
repeated trials of the pseudorandom-encoding proce-
dure.An example of this is given in Fig. 8.
the experiment leading to the result in Fig. 81a2 is
repeated 120 times.The only difference between
each trial is that the random-number generator is
initiated with a different random seed for each
encoding.The average intensity of the 120 trials
1both experiment and theory2 is calculated, as are the
plus and minus one standard-deviation error bars of
the intensity pattern.The number of trials is ad-
equate to produce theoretical diffraction patterns
that, for purposes of plotting, are nearly identical to
the ensemble average that would be found for an
infinite number of trials 3i.e., found by Eq. 162 and the
equation for standard deviation in Ref. 14.
appreciate the small difference by noting that the
theoretical curve in Fig. 81b2 has almost equal-level
ripples, whereas the expected intensity for an infi-
nite number of trials has exactly equal-level ripples.
This degree of agreement between theory for a finite
and theory for an infinite number of trials was found
for the theoretical results in Fig. 81c2 as well.
The measured and the theoretical average curves
in Fig. 81b2 show more clearly the correspondence
than does Fig. 81a2.The discrepancies between the
two average curves indicate differences caused by
systematic errors in the experiment.12
bly, the measured curve exhibits more ripple and
higher sidelobes, or noise.
in Fig. 81c2 indicate that more random noise is being
introduced into the measurement than that for the
theory 1in which noise is due only to the random
statistics used in encoding2.
bounds are only ,60% wider than those for theory in
band.Nonetheless, we think that the degree of
agreement is quite good considering that the SLM is
so different from the assumed device, a 360° phase-
only SLM. Most important is that the diffraction
patterns produced do qualitatively agree with our
theory in that the beam shape is controllable and
there is a broad, near-uniform intensity speckle
background.As devices that do produce 360° be-
come available, it appears that it will be possible to
synthesize diffraction patterns with greatly im-
proved accuracies. Elimination of the need for a
carrier frequency will also make it possible to use the
entire bandwidth of the SLM for signal processing.
The larger error bounds
The measured error
5. Summary and Conclusions
A secondary goal of this paper has been to describe
the properties of the SLM and how they were mea-
sured.In most previous studies of electrically ad-
dressable SLM’s, the modulation transfer function
SLM Measurements and Characterization Procedures
pattern shown in Fig. 61d2; 1b2 the average of 120 cross sections with a different random seed for each experiment; 1c2 the error bars 1plus
and minus one standard deviation2 of the average intensity in 1b2. Curves are normalized the same as in Fig. 7 except that the measured
curves are plotted with a 15-µm offset, and the theoretical curves are scaled in intensity by a factor of 1.643.
Theoretical 1dashed curves2 and measured 1solid curves2 diffraction patterns:
1a2 cross section of the center of the diffraction
2496APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996
has been ignored.
applied different carrier frequencies, indicate that
spatial frequencies of ,5 SLM lines are substan-
tially filtered.There is even a nontrivial amount of
filtering at 10–15 SLM lines.
the phase-modulation range and causes the random-
phase components at nearby pixels to become statis-
tically correlated.This point may have been over-
looked in some previous interferometric studies of
SLM phase-modulating properties in which large
blocks of pixels were programmed with the same
Recently, two groups have focused on common-
path interferometric approaches in order to elimi-
One procedure is as follows:
112 the beam is split with a Ronchi ruling; 122 one
beam is passed through a reference portion of the
SLM 1programmed to gray scale 02, and the second
beam is passed through a portion of the SLM that is
modulated with a second gray-scale value; 132 the
beams are recombined and detected by camera; and
142 the phase shift of the sinusoidal interference
pattern is measured.13
The second procedure is as
112 a 50% duty-cycle square wave 1Ronchi
ruling2 is programmed onto the SLM with half the
gray-scale values set to 0 and the other half set to a
second level; 122 the diffraction pattern of the SLM is
detected by camera; and 132 the phase is decoded
based on the relative intensities of the first and the
third diffraction orders.14
path procedure is described here in which a pseudo-
random binary-phase-modulation pattern is applied
to the device. The method is useful in that it is an
in situ measurement procedure that requires no
extra beam splitters or combiners, and various-sized
superpixels can be evaluated.
intensity of the on-axis diffraction pattern is needed,
the measurement can be performed with a single
photodetector. Also, over most of the phase range,
the diffraction-limited spot is much brighter than
the background speckle noise, which can lead to
repeatable measurements of phase.
have fairly uniform characteristics across the sur-
face of the device and for which the amplitude
variation is also known, it will also be possible to
measure the phase with high precision.
Our measurements, in which we
The filtering reduces
An alternative common-
Because only the
For SLM’s that
ing two TVT6000 SLM’s.
326° was measured initially.
modulation was also found to depend on the spatial
frequencies present on the SLM.
Although it was found that increasing the size of
superpixels also increases the phase-modulating
range, the superpixels could not be made large
enough to simultaneously eliminate the sensitivity
of the SLM to spatial frequencies and provide a
reasonably large number of pixels for the encoding
Experimental Demonstration of Pseudorandom
A modulation range of
However, the phase
For high spatial
rier frequency and the apodization are slowly vary-
ing functions.However, the random-phasor portion
3see Eq. 1224 of the modulation pattern 1that controls
amplitude2 contains the highest spatial frequencies.
The portions of the apodization that are designed to
produce the smallest amplitudes will have the larg-
est random spread 3see Eq. 1324 and thus the most
rapid phase transitions between adjacent pixels.
Thus the low-amplitude portions are expected to be
the most in error owing to the modulation-transfer-
function limitations of the SLM.
nonlinear mapping between gray scale and phase
was not factored into the theoretical diffraction
that different portions of the apodization pattern are
phase encoded by different ranges of the nonlinear
mapping curve.Despite these properties, which
are significantly different from an ideal array of
independent 360° phase-only pixels, qualitatively
similar results were produced.
diffraction patterns very similar in shape to ideal,
but a broad background of speckle was observed.
The statistically averaged experimental results were
also quite similar to theory, and the error bounds
were only somewhat noisier than theory.
random encoding is a robust procedure in that the
theoretical predictions will reasonably match the
experimental results without use of a perfectly ideal
It is comforting to note that the car-
In addition, the
Not only were the
This research is sponsored by Advanced Research
Projects Agency through Rome Laboratory contract
F19628-92-K0021, U.S. Army Research Office con-
tract DAAH04-93-G-0467, and NationalAeronautics
and Space Administration cooperative agreement
NCCW-60 through Western Kentucky University.
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