Page 1

Pseudorandom phase-only

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.

Key words:

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

1.

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.

Introduction

The

Each pixel can be pro-

The fact that the

In

If

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

full-complex modulation.

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

given application.

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

phase-only modulation.

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

This

These latter approaches

For some

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-

ber 1995.

0003-6935@96@142488-11$10.00@0

r1996 Optical Society ofAmerica

M. Liang is

2488APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996

Page 2

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

pseudorandom encoding.

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-

caldatasheetdatedOctober19932.

ers have also demonstrated programmable kino-

forms using a birefringent liquid-crystal television

1LCTV2 panel.4

However, the device is a custom-

modifiedversionofacommerciallyavailabletwisted-

nematic display, and this device is not available

outside Epson. Using commercially available

twisted-nematic LCTV’s, several researchers have

identifiedphase-mostlymodulationcharacteristics.5–7

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

projector.7

However, Soutar and Lu later noted

thatthemodulationcharacteristicscanvarysubstan-

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

arequitesimilartothosepredictedfor2pmodulators.

For the TVT6000 LCTV, several nonideal properties

Epsonresearch-

Most

Phase shifts of

Soutar and Monroe iden-

The 1phase-only2 deform-

However, fully

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-

2.

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

This is

3The

The encoding procedure

The pixels have identi-

Each pixel has unity-

For purposes of design

ci5 7ci8,

112

where 7? ? ?8 represents the expected-value or en-

semble-average operator.

actually produced by the pixel is

The value of the phase

ci5 ci1 dci,

122

where dciis an unbiased random-phase offset.

offsetisselectedfromaprespecifiedrandomdistribu-

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

The

ai5 7exp1 jdci28 5 sinc1

ni

2p2,

132

where niis the spread of the uniform distribution.

Because the values of Eq. 132 are in the range

Fig. 1.

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

Page 3

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

11, ci2.

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

t1x2 5o

i51

N

exp1 jci2r1x 2 xi2,

142

where r1x2 is a function defining the clear area of the

subaperture. The effective complex transmittance

resulting from encoding each pixel is likewise writ-

ten

t1x2 5o

i51

N

aiexp1 jci2r1x 2 xi2.

152

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.

be1

It has been shown to

I1fx2 5 0T1fx2021 N11 2 h2R21fx2,

162

where fxis the spatial frequency, T1fx2 and R1fx2 are

Fourier transforms of t1x2 and r1x2, respectively, and

the average intensity transmittance

h ;

1

No

i51

N

ai

2

172

can be viewed as a type of diffraction efficiency, as we

describe below.

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

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

Since

Equation 162 shows

Another way of view-

Although

3.

The experiments are performed with two LCTV

panelsfromtheredchanneloftwodifferentTVT6000

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

Experimental Configuration

Each of the two

This is

This provides an

In fact,

2490APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996

Page 4

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

LCTV.10

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-

tions.

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

Apparently,

A

The SLM

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

A.

The first LCTV 1LC12 is imaged onto the second one

1LC22 with a pair of 228.6-nm focal-length, f@4.5

imaging lenses 1Plummer lenses purchased from

MWK Industries2.The second LCTV is rotated by

180° to account for the image inversion produced by

the4fimagingsystem. AnEalingadjustablerectan-

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

Both

In

The

The projector bright-

In

We further re-

The

Fig. 2.

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

Page 5

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

videomicroscopeimmediatelyafterthesecondLCTV.

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

achievenear-diffraction-limitedFraunhoferpatterns.

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.

thespatialcoordinateswasdeterminedbythemanu-

facturer-quoted dimensions for the CCD imager to-

For the available lenses this

We

The alignment

After the

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

microscope.

B.

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

shouldproduceanidenticalreductioninthetransmit-

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

The maximum trans-

With the SLM set for

Because one desir-

1It also

Therefore only

Owing to the spatial

C.

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

These measure-

Fig. 3.

and predicted.

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

Page 6

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

A second

The procedure

The inten-

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

2

,

182

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.

expected,Eq.182indicatesthatthereisperfectcancel-

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

1In

However, because N is

The maxi-

For the 4 3 1 SLM pixel

The correspondence

D.

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

invidiaul pixels.2

Summary of SLM Properties

It also has a large

The

Fig. 4.

scale patterns and with superpixels of various sizes.

is normalized to the peak intensity with the gray scale equal to

zero.

On-axis diffraction intensity for random binary gray-

Each curve

10 May 1996 @ Vol. 35, No. 14 @ APPLIED OPTICS 2493

Page 7

4.

Encoding

Experimental Demonstrations of Pseudorandom

A.

Available SLM

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

framegrabber.Severalharmonicallyrelateddiffrac-

tion patterns are generated, of which the fundamen-

tal is usually the brightest.

diffractionpatternthatwecomparewiththoseantici-

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-

As the

overlap with the dc and the first-harmonic order

becomesincreasinglysignificant.

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

direction.

Figure51andalso

It is

Each

Thus the SLM is

B.

Beam Shaping

Threeapodizationsaredesignedtoproducerectangu-

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

horizontal, direction.11

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

thesedesignsis26dB.Higher-sidelobe-levelDolphs

are known to further reduce the Gibbs ripple, but

they also increase transition bandwidth.

pseudorandomencoding,higher-sidelobe-levelDolphs

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

form

Design and Encoding of Full-Complex Modulations for

As for

In the

The apodizations are cho-

I1fx, fy2 ~ sinc21fx23d1fy2 p rect1fy@w242,

192

Fig. 5.

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

per period.

Intensity of the deflected point-spread function as a

Intensities are normalized to the

The

2494 APPLIED OPTICS @ Vol. 35, No. 14 @ 10 May 1996

Page 8

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

fast-Fourier-transformarray2.

y direction from the centers of the theoretical diffrac-

tion patterns are saved for comparison with the

measured diffraction patterns.

As described above,

Theoretical

Crosssectionsinthe

C.

with Theory

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

1The cross

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

dccomponentinFigs.61b2–61d2,althoughmoreheavily

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

peakintensityoftheexperimentalpoint-spreadfunc-

tion in Fig. 132. The three theoretical curves were

Except for changes in

Fig-

Fig. 6.

intensity:

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

Fig. 7.

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-

Cross

Theoretical intensity plots

10 May 1996 @ Vol. 35, No. 14 @ APPLIED OPTICS2495

Page 9

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.

Offsetting the

In

For this

Here,

One can

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.

Most nota-

The larger error bounds

The measured error

5. Summary and Conclusions

A.

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

Fig. 8.

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

Page 10

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

gray-scale value.

Recently, two groups have focused on common-

path interferometric approaches in order to elimi-

nate vibrations.13,14

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

follows:

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

B.

Encoding

Weattemptedtodevelopa360°modulatorbycascad-

ing two TVT6000 SLM’s.

326° was measured initially.

modulation was also found to depend on the spatial

frequencies present on the SLM.

frequenciesthemodulationrangedecreasesto,180°.

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

experiments.

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

pattern. Theuseofthecarrierfrequencyalsomeans

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.

experimentalresultsleadustoconcludethatpseudo-

random encoding is a robust procedure in that the

theoretical predictions will reasonably match the

experimental results without use of a perfectly ideal

SLM.

It is comforting to note that the car-

In addition, the

Not only were the

These

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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