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Optimal cascade operationof optical phased-array beam deflectors

James A. Thomas and Yeshaiahu Fainman

An optimal strategy for cascading phased-array deflectors is presented that allows for high-resolutionrandom-access beam steering with continuous scan-angle control but requires a minimum number ofcontrol lines. The system is analyzed theoretically by use of a Fourier optics approach and then verifiedexperimentally. A pair of 32-channel optical phased arrays fabricated by use of surface electrodes onlanthanum-modified lead zirconate titanate ~PLZT! was sandwiched together to form a functional two-stage phased-array cascade. Experimental results from the PLZT-based two-stage deflector are pre-sented that confirm the performance enhancements of the optimized cascading technique. A phase-staggered discrete–offset-bias protocol for controlling the cascaded system is shown to be optimal in termsof maximum diffraction efficiency and minimum number of control lines, while still providing for fullanalog scan control. © 1998 Optical Society of America

OCIS codes: 050.0080, 070.0030, 230.0080, 120.0380, 110.0180.

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1. Introduction

A multitude of useful static optical functions can beachieved with diffractive optical elements ~DOE’s!,ncluding beam deflection, anamorphic lensing, arrayeneration, and aberration correction. These oper-tions are achieved by the imposition of particularhase profiles on an incident wave front. Perform-ng such DOE functions in a dynamic fashion is theoal of an emerging optical phased-array technology.1

Optical phased arrays are direct functional analogs ofmicrowave phased-array antennas2 that allow agile,inertialess steering of microwave beams. The opti-cal arrays mimic the behavior of passive, space-fedmicrowave phased-array systems that have beenscaled down for use at optical wavelengths and thusshare the same fundamental beam-forming conceptsas their microwave precursors.

Optical phased arrays provide an elegant meansfor the inertialess, high-resolution random-accessbeam steering that is required by numerous applica-tions, including laser radar, laser communication,and laser projection display. For such applications

The authors are with the Department of Electrical and Com-puter Engineering, University of California at San Diego, La Jolla,California 92093-0407.

Received 26 March 1998; revised manuscript received 24 June1998.

0003-6935y98y266196-17$15.00y0© 1998 Optical Society of America

6196 APPLIED OPTICS y Vol. 37, No. 26 y 10 September 1998

high-resolution scanners with large apertures areneeded to provide a precision pointing capability forlight beams with potentially high light power. Typ-ical specifications call for more than 104 addressablepoints, which in turn requires at least an equal num-ber of independently controlled phase modulators inthe array.3 Numerous line-management techniqueshave been explored to reduce the number of controllines required to regulate fully such a phased-arraydeflector while still providing continuous scan-anglecontrol. Cascading phased arrays in tandem offers agood solution to this problem. With careful design ofthe cascade architecture and proper choice of thephased-array programming strategy, one can achievefull scan control with optimal diffraction efficiency~DE! by using a minimum number of control lines.

An optical phased array is typically composed of amultichannel array of electro-optic ~E-O! phase mod-ulators ~see Fig. 1!. However, because of the general

eed for small far-field spot patterns scannable overlarge angular range ~i.e., a large device apertureith small modulator sizes!, optical phased arrays

an require an enormous number of individual phasehifters. Previous research ~Ref. 1! has describedome basic methods for reducing the interconnectionsequired for high-resolution phased-array deflectors.hese include the restriction to one-dimensional

1-D! arrays for separate X–Y beam deflection andhe implementation of a simple coarse–fine cascadingcheme. The present paper specifically analyzes theroblem of addressing large 1-D arrays of phase mod-

o

ulators used for beam deflection and describes analternative cascading solution with improved perfor-mance. Experiments with optical phased-array de-flectors fabricated by use of lanthanum-modified leadzirconate titanate ~PLZT! are described that confirmthe usefulness of our approach. The specific at-tributes and implementation of this new optimal cas-cading scheme are discussed in the following sections.

The outline of this paper is as follows. In Section2 we present a brief review of earlier optical phased-array systems and discuss the addressing techniquesused in these systems. The majority of this paperthen deals with presenting a Fourier based theoret-ical analysis of cascaded optical phased-array deflec-tors and associated experiments with PLZT-basedphased-array devices that corroborate the theory.Section 3 is devoted to a theoretical description of asingle-stage phased-array deflector by use of Fourieroptics analysis, and several key features of the basicdeflector are pointed out. Section 4 introduces thegeneral concepts behind a two-stage phased-array de-flector. In Section 5 the Fourier analysis of Section3 is extended to modeling an aligned two-stagephased-array cascade. Here the key attributes ofthe two stages are pointed out, and the optimal pro-gramming strategy for such a cascaded array deflec-tor is determined. In Section 6 we describe thedesign and implementation of two 32-channel optical

Fig. 1. Basic concept of a 1-D phased-array deflector: ~a! near-fief the 1-D array ~with unfolded phases!. Quantization errors are

10

phased arrays that use PLZT and show the experi-mentally observed outputs from the two-stage deflec-tor. The observed data support the theoreticaldescription of Section 5. The overall results are thensummarized in Section 6.

2. Background

Several forms of optical phased arrays have beendeveloped in the past by use of various designs in awide variety of E-O materials for phase modulation.As early as 1971, Meyer4 demonstrated a multichan-nel array of bulk lithium tantalate modulators foroptical beam deflection. The deflector consisted of46 phase shifters in a 1-D geometry, and independentcontrol lines were provided for each channel to obtaincontinuous scan control. Several fundamental con-cepts of optical phased-array deflection were verifiedby this simple single-stage device. Shortly thereaf-ter, Ninomiya5 demonstrated a phased array of bulk-prism deflecting elements made with lithium niobate.This device exploited an array concept primarily toenhance the deflector resolution beyond that of a sin-gle deflecting element. In further research to sim-plify the device design, Ninomiya6 introduced a noveltwo-stage arrangement that included offset phaseelectrodes set in front of each prismlike deflectingelement. This allowed the phased array to create aphase-staggered approximation to an arbitrary linear

out and far-field intensity pattern, and ~b! equivalent phase profilen as dark notches.

ld layshow

September 1998 y Vol. 37, No. 26 y APPLIED OPTICS 6197

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6

phase profile, hence providing continuous deflection-angle control while maintaining enhanced resolution.~Lee and Zook7 described a similar cascade arrange-ment for resolution enhancement in a less openlydisclosed patent issued in 1972.!

Later efforts led to the design of fast, high-performance 1-D phased-array beam deflectors thatuse integrated-optics AlGaAs channel waveguides.8–10

Vasey et al.10 reported a 50-element rib waveguidedevice that exploited an integrated two-stage configu-ration analogous to Ninomiya’s arrangement. In thisdevice a pair of sawtooth-shaped electrodes across thearray of waveguides creates a sequence of identicalprismlike deflecting elements that, by themselves,form a discrete-state scanner. Separate block elec-trodes were placed in front of the primary electrodes asin the Ninomiya design so that continuous beam steer-ing could be achieved. This layout greatly reducedthe number of electrical connections required by the50-channel array.

McManamon et al.1 recently developed compact,high-resolution optical phased arrays based on anematic liquid crystal. These arrays have a largeaperture ~4 cm 3 4 cm! and contain up to 43,000phase modulators. To alleviate the burden of ad-dressing such a large number of modulators whilemaintaining continuous scan control, McManamon etal. used a two-stage coarse–fine cascade arrangementfor the phased arrays. The coarse stage performsbroad, discrete deflection, and the fine stage producescontinuous angular deflection over a narrow range.The two stages are designed so that, when combined,they yield continuous scan-angle coverage. Thecoarse stage is formed from a large phased array byuse of a reduced-leadout addressing architecture inwhich the modulators in the large array are intercon-nected to form a sequence of identical subarrays ~seeRefs. 1 and 11!. In the prototypical coarse deflectorthat contained 43,008 modulators, only 256 controllines were required to address the full array because256 modulators were grouped into each subarray.The fine deflector stage contained 512 independentlycontrolled modulators. This coarse–fine cascadingstrategy requires far fewer control lines than does afully addressed single-stage phased array but pos-sesses the full flexibility of a single-stage deflector.

An optical solid-state phased array designed forpotentially high-speed operation with PLZT was re-cently demonstrated.12 This single-stage phased-array device is capable of fast, continuous scanningbut requires relatively high drive voltages. Cascad-ing was examined here as a means of reducing thecontrol-line counts, hence to reduce the number ofhigh-voltage drivers required. A coarse–fine cascad-ing structure was initially considered for the PLZTphased arrays, but it was found that the Ninomiyascheme could also be applied to such phased-arraydeflectors and would offer better performance at-tributes. This paper presents these new results.

It should finally be noted that other techniques canreduce the line counts of phased-array deflectorswithout the use of cascading. Talbot and Song13


suggested using fixed resistive networks betweenelectrodes in a single-stage phased-array deflector toachieve a simple two- or three-terminal device. Sim-ilar arrangements have been used in phased arraysdedicated as variable-focal-length lenses.14,15 Suchschemes necessarily require that the E-O mediumexhibit perfect linear or quadratic behavior over thefull range of voltages dropped between electrodes,and they cannot fully compensate for changes in E-Obehavior arising from aging or material saturationbecause the resistance values are fixed. Further-more, in these schemes the phase modulators mustbe capable of higher-order ~i.e., greater than 2p!phase shifts because the fixed resistive networks can-not provide for 2p phase resets in the array program-ming. This places severe constraints on themodulators. The line-count reduction techniquesaccomplished through cascading can indeed performactive E-O device compensation and, in the case ofpiston-type phase shifters, require no more than 2prad of phase shift from any single modulator.

3. Phased-Array Beam-Steering Theory

An optical phased array can steer an incident beamby an angle uS by approximation of an ideal thinprism with a continuous linear phase functionfprism~x! 5 ax, where a 5 ~2pyl!sin~uS!, l is thewavelength of the optical beam, and uS is the speci-fied deflection angle. The output of the deflector isfound in the far-field diffraction pattern of the array@see Fig. 1~a!#. Because the phased array can imple-

ent only piecewise constant phase shifts, the ap-roximation to the ideal linear phase slope exhibitsuantization errors @see Fig. 1~b!#. These phase er-ors cause higher-order grating lobes to be introducednto the output that lower the intensity of the centralobe and restrict the region over which it can becanned unambiguously. For nonredundant scan-ing in one direction, the zero-order lobe can movenly as far as the position of the undeflected first-rder lobe. This defines the angular width of theseful scan zone to be Duzone 5 ~lyd!, where d is the

basic modulator width. The useful scan-zone rangeis inversely proportional to the modulator size d,whereas the output spot size is inversely proportionalto the full phased-array aperture of D 5 Md ~where Mis the number of modulators in the array!. It can beshown ~see Ref. 4! that, under the Rayleigh resolutioncriterion, an M-channel phased array is capable ofaddressing M resolvable positions within the usefulscan zone. Continuous scan-angle control is guar-anteed if there is independent control of each modu-lator in the array ~i.e., M control lines!, but thisecomes impractical and expensive as the number ofodulators becomes large.For modeling purposes consider a 1-D phased arrayith M pistonlike phase modulators that has a linearhase slope of a 5 Dwyd programmed across the ar-ay @see Fig. 1~b!#. The array functions as an M-stepchelon grating with a constant phase increment Dwnd step width d. Its general transmission function

b

fwdTttoaTds@ftlbtbto

ctst

er

fi

i

T0~x! can be described by an apertured sequence ofincreasing phase steps as

T0~x! 5 W~x! p (m50

M21

d~x 2 md!exp~imDw!, (1)

where W~x! is a windowing function of finite extent@W~x! 5 0, for uxu . dy2#, d is the modulator width, Dwis the phase increment between adjacent modulators,mDf is the phase value of the mth modulator, M isthe number of modulators in the array, the asteriskdenotes a convolution operation, and d~x! is an ide-alized d function. The windowing function W~x! per-mits a generalized description of the phased arraythat can include partial fill-factor effects and otherperiodic aberrations. The far-field intensity patternfrom such a phased-array deflector is proportional tothe square of the Fourier transform of T0~x! and cane written in the form ~see Ref. 2!

IOUT~u#! 5 C EF~u#!AF~u# 2 u#S!, (2)

where C is a constant of proportionality and the ele-ment factor EF and the array factor AF are defined as

EF~u#! ; u($W~x!%u2, (3a)

AF~u# 2 u#S! 5 ( sinHMpSdlD@u# 2 u#S#J

M sinHpSdlD@u# 2 u#S#J)

2

, (3b)

respectively. Here ($. . .% represents a Fouriertransform operation, and the generalized angular pa-rameters u# and u#S are defined as

u# ; sin~u! 5 ln, (4a)

u# ; sin~uS! 5 SDw

2pDSl

dD . (4b)

The element factor EF ~also known as the formactor! is defined as the Fourier transform of theindowing function, and it dictates the number ofiffraction orders with relatively strong intensity.he array factor AF ~also known as the interference

erm! represents the unattenuated diffraction spec-ra computed as the normalized Fourier transformf M infinitely narrow slits spaced a distance dpart. Plots of these factors are given in Fig. 2.he generalized angular coordinates in Eqs. ~4! areesignated with overbars and are equivalent to theines of their respective angles. The parameter u#Eq. ~4a!# is directly proportional to the spatial-requency component n of the object spectra; theranslation parameter u#S @Eq. ~4b!# is directly re-ated to the programmed phase slope of a [ Dwydecause a 5 ~2pyl!sin~uS!. Under paraxial condi-ions the sine parameters in Eqs. ~4a! and ~4b! cane reduced to the simple angles u and uS, respec-ively. Note that the form of Eq. ~2! shows that theutput intensity of a basic phased-array deflector

10

onsists of a fixed element-factor EF envelope and aranslatable array-factor AF term. The spectra de-cribed by the AF term can be shifted arbitrarily byhe programmable value u#S.

An analytic form for the element factor EF in Eq.~3a! can be given if a simple windowing function ofthe form W~x! 5 rect@xyw# 5 1 for uxu # wy2 and uxu 5 0lsewhere is assumed. This windowing functionepresents a clear aperture of width w # d over each

modulator. The element factor corresponding tothis simple window is a sinc envelope of half-widthDu#ENV 5 ~lyw! and can be written as

EF~u#! 5 FSwdDsincS u#

Du#ENVDG2

, (5)

where sinc~m! [ sin~pm!y~pm! and ~wyd! representsthe fill factor of the modulators. @Note that approx-imately ~dyw! diffraction orders will pass into theoutput with relatively strong intensity.#

Many key features of a phased-array deflector canbe explained with theoretical descriptions, which aresummarized in Fig. 2. This figure plots the calcu-lated output intensity of Eq. ~2! by use of Eqs. ~3b!and ~5! and depicts three diffraction orders of thefar-field intensity pattern from a 32-channel phased-array deflector with a fill factor of wyd 5 1y5. Notethat multiple spectra occur in the phased-array out-put whenever the fill factor is significantly less than1. The spectra are shifted laterally by an amountu#S 5 ~Dwy2p!~lyd!, which can be arbitrarily set by achange in the phase-step increment Dw. With con-tinuous control of Dw over the phase range @2p, p!,the phased array can steer an incident monochro-matic beam over a continuous range of angles withinthe nonredundant scan zone uu#Su , 1⁄2~lyd!. Thisscan zone is determined by the region over which thecentral lobe can be steered unambiguously withoutencroaching on the 61st order lobe regions.

The output spectra have spot widths du#spot propor-

Fig. 2. Central lobes of the normalized far-field intensity patternIOUT~u! of a 1-D echelon grating deflector with M 5 32 channels, a

ll factor of wyd 5 1y5 @solid envelope curve ~EF!#, and a deflectionangle of u#S 5 ~1y8!~lyd!. The ideal fill-factor envelope ~wyd 5 1!s shown as a dashed curve.


hflb

tb

t

da

o

6

tional to ~lyD!. For an M-channel array there are Mresolvable positions ~by use of the Rayleigh criterion!that fall within the nonredundant scan zone. Be-cause the intensity of the central lobe falls off with anincreasing deflection angle ~delimited by the element-factor envelope!, one can further restrict the allowedscan zone to guarantee a specific minimum DE of h $hMIN for all scan points but at the cost of fewer thanM resolvable scan positions. This is the same asrestricting the amount of allowed phase quantizationDf. ~See Appendix B for further discussion of theDE of phased-array deflectors.!

4. Cascading Concepts

Cascading is a useful strategy for achieving highswitching complexity by use of a sequential array ofsimple switching elements. It can be applied to re-duce the number of connections required to control acomplex switching network. Beam deflectors fallinto the category of optical switching elements, andarranging deflectors in a multistage cascade cangreatly reduce the number of control lines needed tosteer an optical beam, while still providing continu-ous, high-resolution scan control. In principle, a lin-ear sequence of R deflector elements, each capable ofaddressing N spots, could address as many as RN

resolvable spots, if multiplicative resolution enhance-ment16 is assumed. Several hardware-based cas-cading techniques have been investigated as a meansof increasing the resolution of existing E-O deflec-tors.17,18 Cascading has also been employed with

ybrid deflector technologies to derive continuous de-ection control from an inherently discontinuouseam-steering device.19 Application of such cascad-

ing techniques to phased-array deflectors can achievehigh-resolution scanning systems with a reducednumber of control lines. However, as we show be-low, optimal performance is achieved by the choice ofthe proper cascading strategy and its implementa-tion.

The fundamental principle behind a two-stage cas-cade is to set two deflectors with resolutions N1 andN2 in tandem and achieve a total scanner resolutionof NR 5 N1N2 points.16 This is called a multiplica-ive cascade. We can realize such an arrangementy coupling a discrete-state deflector ~stage 1! that is

Fig. 3. Basic optical layout of a two-stage phased-array cascadeby M1 control lines and stage 2 by M2 control lines. By itself, stagn the output screen!.


capable of addressing N1 disjointed positions to acontinuous-state deflector ~stage 2! that is capable ofsteering to N2 positions between each of the discretestates. By proper design of the two-stage combina-tion, the discrete-state and the analog-state scannerscan yield continuous scan-angle coverage of N1N2resolvable output positions. Note that, for achievingthe same far-field spot size from the coupled stages,both stages must possess the same device aperture.Phased arrays can be arranged in a multiplicativecascade, as shown in Fig. 3.

A. Aligned Two-Stage Phased-Array Cascade

In our experiments we used a particular cascade ar-rangement for the two-stage phased-array deflectorin which both stages are aligned and a minimumnumber of control lines is required. ~Throughoutthis paper, we refer to this specific geometry as analigned two-stage phased-array cascade.! This con-figuration is shown schematically in Fig. 4 and lendsitself well to theoretical modeling, as is detailed inSubsection 4.B. The first stage consists of a re-peated sequence of small, identical phased-array de-flectors g~x!, each containing M1 modulators of widthd1. There are M2 such subarrays in this stage thatare all controlled by the same shared M1 control lines.By itself, stage 1 is a discrete-state scanner. Stage 2is a large phased array with M2 channels—one chan-nel per subarray in stage 1. ~This is a minimumcontrol-line count configuration.! Its broad modula-ors have a width of d2 5 M1d1, and they are aligned

so as to overlap each of the subarrays in stage 1.Stage 2 requires M2 independent control lines andfunctions as a continuous-state scanner. The twostages can be programmed independently with phaseincrements Dw1 and Dw2, respectively. Both arraysfill the aperture D, hence generating the samediffraction-limited angular spot size lyD. But stage1 can support deflection only to discrete angles ofu#1~n! 5 n~lyd2! ~n is an integer!, and only M1 suchangles fall within the useful scan zone of width lyd2.Stage 2 can address M2 resolvable points over a nar-row, continuous angular range of width lyd2. The

eflection angles supported by the individual stagesre summarized in Table 1.The combination of the two stages can address

a reduced number of external connections. Stage 1 is addressedn deflect to only discrete output angles ~shown as thick tick marks

withe 1 ca

z

cFaTmPt11

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2Tapfptusm

Table 1. Deflection Angles Supported by the Individual Stages of an

NR 5 M1M2 resolvable points over the useful scanone with just M1 1 M2 control lines. This repre-

sents a logarithmic reduction in the number of con-trol lines needed for full deflection control. Forexample, with two 32-channel phased-array stages~M1 5 M2 5 32!, one can build an NR 5 1024-pointdeflector that would require only 64 control lines in-stead of the 1024 lines that would be required by asingle-stage deflector. Reducing the number of con-trol lines becomes crucial at high array densities andwhen the electronic driver costs become prohibitive.

The concept of cascading optical phased-array de-flectors to reduce the number of external connectionswas already proposed by McManamon et al.1 Theironcept used an architecture similar to that shown inig. 3 but without restricting the two stages to perfectlignment or aperture matching ~i.e., d2 5 M1d1!.heir technique employed a cascade-programmingethodology that can be called a coarse–fine protocol.atents on the implementation and programming ofhe coarse phased-array deflector of the form of stageshown in Fig. 4 have been issued to authors of Ref.

.

B. Two Programming Strategies

The coarse–fine protocol is the most direct method ofcontrolling a two-stage deflector. However, as isshown theoretically in Section 5 and experimentallyin Section 6, there exists an alternative strategy forprogramming an aligned phased-array cascade that

Fig. 4. Schematic layout of the unfolded phase distribution of analigned two-stage phased-array cascade with a minimum control-line configuration.

Aligned Two-Stage Phased-Array Deflectora

Stage Deflection Angle Range Scan Type

1 u#1~n! 5 ~lyd2! n is the integer value Discontinuous2 u#2~r! 5 r~lyd2! r [ @21y2, 11y2! Continuous

aAs shown in Fig. 4.

10

can provide better optical performance with the samephysical hardware. We call this second program-ming strategy a discrete–offset-bias protocol. Bothprotocols can use the same physical hardware butemploy slightly different programming strategies.Before a more rigorous analysis, we now give a briefdescription of the two basic strategies that can beused to program an aligned two-stage phased-arraycascade.

1. Coarse–Fine Programming ConceptUnder the coarse–fine protocol successive deflec-tions of the incident beam are performed by the twostages to achieve pointing to the desired outputposition u#S. Stage 1 is a discrete-state scanner thatsupports deflection to only discrete angles u#1~n! 5n~lyd2! ~n is an integer!, shown as thick tick markson the output screen illustrated in Fig. 3. Theseangles represent cases for which stage 1 is perfectlyblazed, i.e., each subarray in stage 1 generates nlworth of optical path delay across its aperture. Thesecond stage is then set to perform a slight angularcorrection by u#2 5 uS 2 u#1, which redirects the beamto the desired scan angle u#S. In this way both stagessimulate thin prisms with deflection angles u#1 and u#2,respectively, and both stages can therefore exhibituantization errors @see Fig. 5~a!#. The multiplica-ive behavior of the phase errors from the two stagesan lead to undesired noise lobes falling within thecan zone in the output and to a reduced intensity ofhe central peak. This is the protocol adopted in Ref..

. Discrete–Offset-Bias Programming Concepthe discrete–offset-bias protocol uses the sameligned-cascade hardware but employs a modifiedrogramming strategy that optimizes the optical per-ormance of the two-stage deflector cascade. Thehilosophy here is to permit imperfect blazing fromhe first stage and correct any phase mismatches byse of constant phase shifts generated by the secondtage. This concept was developed independentlyore than 20 years ago by Ninomiya5,6 and Lee and

Zook7 as a scheme for resolution enhancement of E-Odeflectors. Vasey et al.10 employed it to reduce thenumber of control lines required in a channel-waveguide beam deflector. We employed this im-proved programming strategy to design a moreefficient multistage phased-array deflector.

The discrete–offset-bias scheme for programmingan aligned two-stage cascade is shown schematicallyby the deflected wave fronts depicted in Fig. 3. Un-der the discrete–offset-bias strategy, each of thesmall subarray deflectors in stage 1 is programmed topoint the incident wave front directly to the desiredscan direction u#S, regardless of whether the discrete-state scanner ~stage 1! can support that angle byitself. The subarrays in the first stage generate anarray of tilted wave-front segments that are all di-rected toward u#S, and these segments will interferedestructively if perfect blazing is not achieved ~i.e., ifu#S Þ n~lyd2!, where n is an integer!. Stage 2 is used


tcnTt

1hd

w

coa

w

Dt

6

to introduce constant phase offsets to the tilted wave-front segments passed by stage 1 so that, at the out-put of stage 2, all the tilted wave fronts areseamlessly joined. This guarantees maximum con-structive interference into the far-field output angleu#S. Note that stage 2 is also programmed for deflec-tion to u#S in this scheme but is physically used only asa set of phase offsets. Because stage 2 just imple-ments constant phase shifts, phase-quantization er-rors are found in only stage 1 @see Fig. 5~b!#. Withhis form of cascade programming the intensity of theentral output lobe is maximized, and no spuriousoise lobes will be generated within the scan zone.hese qualitative statements are justified in quanti-ative terms in Section 5.

5. Aligned Two-Stage Phased-Array Theory

The far-field output from an aligned two-stagephased array can be derived by use of Eqs. ~2! and ~3!

Fig. 5. Two basic strategies for programming an aligned two-stage phased-array deflector of the form given in Fig. 4. Equiva-lent phase profiles for the two stages are shown: ~a! Coarse–fineprogramming; quantization errors occur in both stages. ~b!

iscrete–offset-bias programming; only stage 1 exhibits quantiza-ion errors.


if we note that one can view the aligned two-stagecascade shown in Fig. 4 as a single-stage phased-array deflector ~stage 2! that has a specialized win-dowing function W2~x! 5 g~x! ~a subarray from stage!. This type of analysis can be extended further toigher-order multistage arrangements. The win-owing function W2~x! is itself a phased-array deflec-

tor with M1 modulators of width d1 and aprogrammed phase increment Df1. The windowingfunction W2~x! is applied to each modulator in stage2, and it can be described analogously to Eq. ~1! as

W2~x! ; g~x! 5 W1~x! p (m50

M121

d~x 2 md1!exp~imDw1!,

(6)

here W1~x! is the windowing function of the indi-vidual modulators in the first stage, the asterisk de-notes a convolution operation, and d~x! is an idealizedd function. According to Eq. ~3a!, the element factorfor stage 2 EF2 is equal to the square of the Fouriertransform of its windowing function W2~x! given byEq. ~6!. This calculation is identical to that used toonvert the general phased array of Eq. ~1! to itsutput given in Eq. ~2!. By applying Eqs. ~2!, ~3b!,nd ~5! to the phased array described by W2~x! in Eq.

~6!, we can write the stage 2 element factor EF2 as

EF2~u#! 5 C9EF1~u#!AF1~u# 2 u#1!, (7a)

where

EF1~u#! 5 Sw1

d1D2

sinc2FSw1

l Du#G , (7b)

AF1~u# 2 u#1! 5 (sinHM1pSd1

l D@u# 2 u#1#JM1 sinHpSd1

l D@u# 2 u#1#J)2

, (7c)

here C9 is a constant of proportionality and EF1 andAF1 are the element factor and the array factor, re-spectively, of the stage 1 subarray g~x!. In Eqs. ~7!,w1 and d1 are the clear aperture and the full width ofthe modulators in stage 1, respectively, M1 is thenumber of modulators per subarray in stage 1, and u#1is the programmed deflection angle of the first stage.As derived from Eq. ~4b!, this angle is defined as u#1 [sin~u1! 5 ~Dw1y2p!~lyd1!, where Dw1 is the phase in-crement programmed onto the subarrays of stage 1.Equations ~7! show that the stage 2 element factorEF2 represents a multispiked envelope term of theform plotted in Fig. 2 but with broadened spikes ofwidth ~lyd2! that are laterally translated by u#1.

The far-field intensity pattern I2OUT from the full

two-stage phased-array cascade is obtained by appli-cation of Eqs. ~2! and ~3! to the stage 2 phased arrayand by use of Eqs. ~7! as the element factor for thisarray. The result can be written in the form

I2OUT~u#! 5 C EF1~u#!AF1~u# 2 u#1!AF2~u# 2 u#2!, (8a)

~

h

l

d

u

where EF1 and AF1 are defined above in Eqs. ~7b! and7c!, respectively, and

AF2~u# 2 u#2! 5 (sinHM2pSd2

l D@u# 2 u#2#JM2 sinHpSd2

l D@u# 2 u#2#J)2

. (8b)

where AF2 is the array factor for stage 2 and d2, M2,and u#2 in Eq. ~8b! are the modulator width, the totalnumber of modulators, and the programmed deflectionangle for stage 2, respectively. Note that u#2 is definedby Eq. ~4b! as u#2 [ sin~u2! 5 ~Dw2y2p!~lyd2!, where Dw2is the phase increment programmed into stage 2. Thespectra defined by the stage 2 array factor in Eq. ~8b!

ave narrow spot widths proportional to ~1yM2!~lyd2!and are translated laterally by u#2.

Fig. 6. Theoretical far-field intensity pattern IOUT from an alconfiguration. The cascaded arrays are programmed for deflectio#2! and ~b! discrete–offset-bias programming ~u#S 5 u#1 5 u#2!. T

w1yd1 5 1y5.

10

From Eq. ~8a! we see that the far-field intensitypattern from an aligned two-stage phased-array de-flector consists of a global fixed envelope ~EF1!, atranslatable array-factor envelope ~AF1!, and trans-atable diffraction spectra defined by AF2. These

terms are plotted in Fig. 6. The useful scan zone forthe two-stage deflector is limited to a width ofDu#zone 5 ~lyd1!, defined as the useful scan zone of asingle subarray deflector g~x! in stage 1. The outputspectra have an angular width proportional to ~lyD!,where D 5 M2d2 5 M1M2d1 is the full device aper-ture. Note that the two array-factor terms, AF1 andAF2 @Eqs. ~7c! and ~8b!#, can be translated indepen-

ently by the angles u#1 and u#2, which are functions ofthe programmed phase increments Dw1 and Dw2, re-spectively. The two fundamental protocols for pro-gramming cascaded phased arrays differ in the waythey select the translation angles u#1 and u#2. Below

two-stage phased-array cascade with a minimum control-lineS 5 ~3y8!~lyd1! by use of ~a! coarse–fine programming ~u#S 5 u#1 1are M1 5 4 modulatorsysubarray in stage 1; the fill factor is

ignedn to u#

here


1

~

Tts

t

wmbu

6

Ob

tau

ait

s

sp

6

we clarify the differences between the two program-ming strategies by examining a specific deflectionexample.

A. Aligned Two-Stage Cascade Example

Consider an aligned two-stage phased-array deflectorof the form shown in Fig. 4 with M1 5 4 modulatorsper subarray in stage 1, a fill factor of w1yd1 5 1y5,and an arbitrarily large number of subarrays ~M2 ..!. ~Stage 2 modulators are assumed to be 100%

filled with one modulator per subarray in stage 1.!The deflector is illuminated by a monochromaticbeam of wavelength l. The nonredundant scan zoneof this deflector has an angular width of Du#zone 5lyd1!, where d1 is the width of a phase shifter in

stage 1, and for simplicity this zone is assumed to becentered about the optical axis. Figure 6 shows theideal theoretical output from such a two-stagephased-array deflector that has been programmed fordeflection to u#S 5 3⁄8~lyd1! by use of the coarse–fineprotocol @Fig. 6~a!# and the discrete–offset-bias proto-col @Fig. 6~b!#. Note that the output spectra definedby the stage 2 array factor AF2 are plotted as narrowd functions for simplicity. The actual spectra willhave finite angular spot widths proportional to ~lyD!,where D 5 ~M1M2!d1 is the full device aperture.

1. Coarse–Fine Programming for u#S 5 3⁄8~lyd1!he calculated output of the aligned two-stage deflec-or programmed with the coarse–fine protocol ishown in Fig. 6~a!. Here stage 1 is programmed to

deflect the incident beam to one of its allowed deflec-tion angles n~lyd2!, choosing the specific angle u#1~n!hat lies closest to u#S. For this example we choose

the n 5 1 direction, giving u#1 5 ~lyd2! 5 1⁄4~lyd1!,here d2 5 4d1 for the specified deflector arrange-ent. Stage 2 must provide an additional deflection

y u#2 5 1⁄8~lyd1! so that the total angular deflection is#S 5 u#1 1 u#2 5 3⁄8~lyd1!.The coarse–fine programming translates stage 1’s

array-factor envelope AF1 to the designated angle u#1.The output spectra described by AF2 ~dashed verticallines in Fig. 6! are translated by u#2 so that one of thediffraction lobes will lie at the desired deflection an-gle u#S. Because the two translatable terms AF1 andAF2 are not coaligned, the central output lobe at u#Shas low intensity, and several noise lobes appear withstrong intensity within the useful scan zone @see Fig.~a!#. Note that the deflection angle of u#S 5 3⁄8~lyd1!

represents a worst-case situation for the coarse–fineprotocol here, as it requires stage 2 to exhibit maxi-mum quantization noise. A higher overall spot DEcould be achieved if the number of phase shifters inthe fine deflector ~stage 2! were increased, but thiswould require greater system complexity than theminimum configuration modeled in Fig. 6~a!. ~Fur-ther details are given in Appendix B.!

2. Discrete–Offset-Bias Programming foru#S 5 3⁄8~lyd1!

ptimal use of the incident light energy is achievedy means of programming the deflector cascade with


the discrete–offset-bias protocol. Figure 6~b! showshe expected output for this case. Here both stagesre programmed for deflection directly to u#S, i.e., u#1 5

#2 5 u#S 5 3⁄8~lyd1!. In this way the peaks of both the

translatable terms—the envelope term AF1 and theoutput spectra AF2—are coaligned directly above thedesired deflection angle u#S. Only one strong diffrac-tion order appears within the nonredundant scanzone of the output because the other spectral orderswithin this zone fall near the zeroes of the stage 1array-factor envelope AF1 @see Fig. 6~b!#. The strongcentral lobe has a maximum intensity that is limitedsolely by the fixed stage 1 element-factor envelopeEF1. Note that, under the discrete–offset-bias pro-tocol, the combination of stages 1 and 2 of the phased-array cascade work in unison to emulate a largesingle-stage array deflector programmed for deflec-tion to u#S. And unlike with the coarse–fine protocol,no quantization errors occur in stage 2 because it isresponsible for only flat phase offsets.

B. Discussion of the Cascading Theory

Analysis of the theoretical output as a function of thedeflection angle uS shows fundamental differencesbetween the two cascade-programming techniques.Under the coarse–fine protocol the output exhibitsdiscontinuous jumps in the central lobe intensity as afunction of scan angle because stage 1 operates in adiscontinuous fashion as the scan angle is increased.This protocol also generates noise lobes in the outputthat become most severe when there is maximummisalignment between the array-factor terms AF1and AF2 @as shown in Fig. 6~a!#. These effects areartifacts of the large quantization errors occuring instage 2 when the chosen deflection angle uS cannot beddressed by stage 1 alone, and they can be alleviatedn part by an increase in the number of modulators inhe fine stage ~stage 2! to reduce its phase-

quantization errors.Alternatively, the discrete–offset-bias protocol

achieves an optimal DE that is a monotonically de-creasing function of the deflection angle bracketedsolely by the fixed element-factor envelope EF1. Un-der this second protocol all the residual phase errorsoccur in only stage 1, and the intensity of the centrallobe is guaranteed to be maximum for all angles.~Appendix B gives specific formulas for the DE’s ofthe two protocols and quantitatively demonstratesthe effects described above.! Note that this compar-ative analysis is strictly valid for only an alignedtwo-stage cascade of the form shown in Fig. 4.

If alignment between the stages cannot be guaran-teed because of physical or hardware constraints, thecoarse–fine protocol must be used. In this case bothstages function as independent single-stage phased-array deflectors. If we assume completely filledmodulators, the ideal DE of each stage can be derivedfrom Eqs. ~4b! and ~5! and written as hi~Dwi! 5inc2@Dwiy2p#, where hi and Dwi are the DE and the

phase increment of the ith stage, respectively. ~Alightly more general treatment can be found in Ap-endix B.! Low-noise lobes and a minimum total

lh

pcp

6qmttcmaa~

rpv

m

rwtm4awtsalepodts

sip

qttlrbua

DE hMIN can still be achieved by oversampling of theinear phase profiles of the two stages so that hTOT [

1h2 $ hMIN, is satisfied for all deflection angles.This is equivalent to increasing the number of mod-ulators in both stages ~i.e., increasing M1 and M2! andrestricting the maximum allowed phase incrementsof the two stages, uDw1uMAX and uDw2uMAX, to be lessthan p. This is the approach employed by thecoarse–fine phased-array deflector described in Ref.1. Such an approach inherently requires more thanM1 1 M2 control lines to achieve a deflector resolutionof NR 5 M1M2 points and therefore needs a larger

hased-array architecture than an aligned two-stageascade programmed with the discrete–offset-biasrotocol, which would require only M1 1 M2 control

lines.The theoretical considerations show that an

aligned-cascade arrangement programmed with thediscrete–offset-bias protocol is the preferred cascadearrangement and makes the most efficient use of theexisting modulators. It achieves the highest overallDE for the least number of control lines. We deviseda novel compact design for an aligned two-stagephased-array cascade in which both stages are inte-grated onto a single PLZT wafer.20 Similar compactimplementations can be made with other opticalphased-array technologies such as liquid-crystal-based phased arrays.

6. Experimental Results

A. Lanthanum-Modified Lead Zirconate Titanate–BasedOptical Phased Array

To demonstrate the operation of a cascaded phased-array deflector, we employed a pair of 32-channeloptical phased arrays fabricated by using PLZT.PLZT is a transparent ferroelectric ceramic exhibit-ing strong E-O effects.21 We chose to use PLZT 9.0y5y35 for phase modulation because of its largeuadratic E-O coefficient, broadband optical trans-ission range, fast switching response, and good

hermal stability. It can be used at wavelengths inhe visible through the mid-IR spectral ranges andan provide very fast switching rates ~faster than 1s!. Our group previously demonstrated phased-rray beam deflection with PLZT in a single-stagerrangement.12 We call this type of device PAGESphased-array grating electro-optic scanner!.

Figure 7 shows a fabricated 32-channel phased ar-ay based on PLZT 9.0y65y35 and mounted in a gold-lated package. This PLZT composition offers aery strong quadratic E-O coefficient ~3.8 3 10216

m2yV2! with minimal residual linear effects. Wefabricated the electrodes on one surface of a 350-mm-thick PLZT wafer by using vacuum-evaporatedchrome gold, standard photolithographic techniques,and wet etching. The raw device was then mountedinto a 0.75-in. ~1.91-cm! square flat-pack packagewith an open aperture cut through the base plate sothat it could be used in transmission mode. Theactive window of the device measures 12.8 mm 3 10.0

m. Each modulator has a dual-aperture configu-

10

ation consisting of a central source electrode 160 mmide surrounded by two 160-mm-wide ground elec-

rodes that provide electrical isolation between theodulators. The modulator grating period is d 5

00 mm, and the interelectrode gaps ~clear apertures!re w 5 40 mm, yielding an effective fill factor ofyd 5 1y10. Voltage supplied to a specific modula-

or induces fringing electric fields within the sub-trate that cause a symmetric index change in bothpertures of that modulator. The strongest modu-ation occurs for light polarized perpendicular to thelectrodes. This device is a single-stage 1-D opticalhased array with independent control lines for eachf the modulators. One can also build a two-imensional deflector by crossing a pair of theseransverse 1-D arrays with a ly2 plate oriented at 45°andwiched between the devices.The phase response of the modulators was mea-

ured by use of a modified Young’s double-slit exper-ment in which one of the two slits is active ~with aositive applied voltage! and the other slit is

grounded. The relative phase shift of the active slitis determined by measurement of the lateral shift ofthe fringes in the far-field interference pattern.22

The measured average phase-modulation response ofthe PLZT modulators illuminated by a collimatedHe–Ne beam ~l 5 633 nm! polarized perpendicular tothe electrodes is shown in Fig. 8. The modulationcurve follows the expected quadratic E-O behavior upto 150 V and then flattens out as a result of saturationeffects in the PLZT. The measured full-wave volt-age ~V2p! of these modulators was 318 V. As is re-uired for a fully programmable phased-array device,his modulator design provides continuous phase con-rol over the range @0, 2p! for the given He–Ne wave-ength. Note that lower full-wave voltages can beealized by use of alternative electrode structures likeuried electrodes, but simple surface electrodes weresed here for the basic demonstration of a phased-rray cascade.

Fig. 7. Packaged 32-channel optical phased array made withPLZT. The active window has dimensions of 12.8 mm 3 10.0 mm.


c

ti

sfl

bflaftscptd

t

6

B. Cascade Experiments

We implemented an aligned two-stage cascade bymounting a pair of 32-channel test devices back toback so that the clear gaps from the first array werealigned directly over the respective gaps in the secondarray. In the actual mounting the two PLZT wafersrepresenting the individual phased-array stageswere separated by 1.5 mm with their electrode sur-faces facing outwards.

The control lines of the two devices were externallyinterconnected to simulate the cascade arrangementof Fig. 4. Specifically, for the stage 1 device, thecontrol lines were interconnected to form a series ofM2 5 8 subarrays containing M1 5 4 modulatorseach. The modulators in corresponding positions ineach of the subarrays were then connected in paral-lel. In this way stage 1 becomes a discrete-statescanner controlled by just four control voltages ap-plied in distributed fashion to each of the subarrays.~This implementation of a coarse array from a largearray of independent phase modulators is similar tothat described by Dorschner and Resler11 for a liquid-crystal-based optical phased array.)

The control lines of the stage 2 device were con-nected together to form a series of eight subarrayswith four neighboring modulators grouped into eachsubarray. Here the control lines for each subarraywere electrically ganged together. Thus stage 2 wasaddressed with eight control voltages, one voltage foreach subarray. The two-stage phased-array deviceis fully addressed by just MTOT 5 M1 1 M2 5 12ontrol voltages and is capable of addressing NR 5 32

resolvable positions. Note that a single-stage de-vice with an equivalent resolution would require 32independent control voltages. A scanning experi-ment was performed with the two-stage device toexamine the deflector output at each of the 32 fun-damentally resolvable output positions. An array of

Fig. 8. Average measured phase-modulation response of thePLZT modulators in the 32-channel phased-array device shown inFig. 7.


12 potentiometer-controlled voltage dividers wasused to provide static programming for the twostages. For each of the output positions the cascadewas programmed by use of both the course–fine pro-tocol and the discrete–offset-bias protocol, and theoutputs were compared. An expanded and colli-mated He–Ne laser beam polarized perpendicular tothe electrodes was used to illuminate the active win-dow of the coupled phased arrays. A Fourier trans-form lens ~ f 5 375 mm! was placed behind the phasedarrays to produce output spectra at the back focalplane of the lens. These spectra were imaged onto aCCD array with a 63 microscope objective. Foreliminating any undesired additional cosinusoidalenvelope over the diffraction orders, which is due tothe dual-aperture transmittance of the modulators, ahorizontal masking arrangement was employed sothat only a single gap from each modulator receivedillumination. In the vertical direction a 2-mm-wideslit mask was used across the active window of thedevices to reduce aberrations from the nonplanarityof the PLZT substrates. The angular width of thenonredundant scan zone ~lyd1! was roughly 0.1° forhis test device, which guarantees that all the exper-mental outputs lie well within the paraxial regime.

Figure 9 shows the recorded output of the two-tage deflector for all rightward-shifted resolvable de-ection angles u#S 5 ~ jy32!~lyd1!, whereas j 5 0, . . . ,

16. @The sequence of leftward-deflected beams~where j 5 215, . . . , 21! have the same appearanceas those shown in Fig. 10 but are reflected symmet-rically about the optical axis.# The topmost image inoth columns shows the output spectra when the de-ectors are turned off. Several diffraction orders arepparent in the output because of the low effective fillactor ~1y10! of the modulators. Only the three cen-ral diffraction orders ~0th and 61st orders! arehown in Fig. 9. The optical axis is delineated by theentral-order spot in the topmost images, and therimary nonredundant scan zone is centered abouthis spot with a width equal to the spacing betweeniffraction orders, i.e., ~lyd1! 5 0.1°. For each image

the intensity of the laser beam was adjusted to allowbright imaging of the central order without influenc-ing the spatial distribution or the relative intensity ofthe output lobes. Figure 10 shows plots of the abso-lute intensity of the CCD data for the case of uS 5~4y32!~lyd1!.

The two stages were programmed by use of bothhe coarse–fine protocol @Fig. 9~a!# and the discrete–

offset-bias protocol @Fig. 9~b!# in the manner detailedin Appendix A. All phase shifts were reduced bymodulo 2p to the range @0, 2p! in the actual program-ming. The output images in Fig. 9 clearly show pro-gressive lateral deflection of the far-field spectraunder both programming strategies. The discrete–offset-bias protocol displays good contrast of theshifted central lobe at all deflection angles. Thecoarse–fine protocol shows strong noise lobes in itsoutput at several deflection angles. The positionsuS 5 ~ jy32!~lyd1!, with j 5 28, 0, 8, 16, correspond tothe discrete states addressable by stage 1 alone. At

arSucPtaDOhwu

snt

d

these positions both protocols generate the same pro-grammed phase values for the two stages; hence theyexhibit identical output spectra. Under the coarse–fine protocol noise lobes become quite prominentwithin the scan pattern at angles at which stage 2must perform strong deflection ~i.e., where j 5 212,24, 4, 12!.

The scan profiles shown in Fig. 10 show that thedditional lobes appearing in the coarse–fine outputeduce the absolute intensity of the primary scan peak.mall noise lobes also appear at large deflection anglesnder discrete–offset-bias programming, and thesean be attributed to partial amplitude losses in theLZT modulators at high voltages. Measurements of

he primary scan peak intensity versus the deflectionngle coincide well with theoretical predictions for theE’s of the two protocols ~as outlined in Appendix B!.verall the discrete–offset-bias protocol producedigher peak intensities with only minimal noise lobes,hich can be attributed to scattering losses in the mod-lators at high drive voltages.Full scan-zone coverage was achieved with the two-

tage cascade, as shown in Fig. 11. Traces of alter-ating Rayleigh beam positions are shown, which fillhe nonredundant scan zone delineated by uuSu # ~1y

Fig. 9. One-dimensional beam-steering output by use of two 32-channel PLZT phased arrays in an aligned two-stage cascade withM1 5 4 and M2 5 8. The cascaded arrays were programmed for

eflection to various angles uS by use of ~a! the coarse–fine protocoland ~b! the discrete–offset-bias protocol. Only half of the 32 re-solvable scan outputs are shown.

10

2!~lyd1!. The scan-zone width extended over 129 pix-els on the CCD camera in our measurements. Theaverage spot width ~full width at half-maximum! of theoutput points was found to be 5.1 pixels, roughly 25%larger than the expected diffraction-limited width of129y32 5 4.0 pixels. This spot broadening is primar-ily due to phase distortions among modulators andaberrations in the optical setup. In a final set of ex-periments, we programmed the two-stage device fordeflection to angles of uS 5 ~1y64!~lyd1!, ~1yp!~lyd1! todemonstrate continuous beam-position control. Ac-curate subspot size-positioning control was confirmed.The two-stage phased-array deflector with a reducednumber of control lines is capable of continuousdeflection-angle control.

7. Conclusions

An optimal arrangement for cascading 1-D phased-array deflectors has been presented. Cascading is

Fig. 10. Horizontal traces through the CCD data shown in Fig. 9for a deflection angle of uS 5 jy32!~lyd1!. Strong secondary noiselobes appear in the scan zone under coarse–fine programming.


r

rhtalc

w

p

~pz

6

required by high-resolution scanning systems to re-duce the number of control lines significantly and canbe achieved with phased arrays while still allowingfor continuous deflection-angle control, restrictedvoltage requirements ~Vapplied # V2p!, and compensa-tion for nonideal E-O behavior and aging effects.Both theoretical and experimental results haveshown that an aligned multistage phased-array cas-cade programmed by use of the discrete–offset-biasstrategy can offer superior optical performance com-pared with the same minimum configuration pro-grammed with the coarse–fine strategy. Thealigned-cascade arrangement provides multiplicativeresolution enhancement, and the number of controllines required for a deflector of resolution NR can beeduced to the order of 2=NR.

The theoretical analysis introduced here employeda Fourier based examination of the aligned two-stagephased-array deflector and exploited the fact that thesubarrays in the first stage could be considered aswindowing functions for the second stage. Bothstages generate array-factor terms that can beshifted independently, depending on the phase rampsprogrammed into the respective stages. The maxi-mum DE for the central peak is obtained when botharray-factor terms are lined up over the desired out-put angle uS ~as with discrete–offset-bias program-ming! and when all modulators are 100% filled. Theecursive Fourier analysis can be applied directly toigher-order, aligned, multistage arrangements withhe consistent result that optimal performance ischieved when all translatable array-factor terms areined up over the output angle. Higher-order cas-ades would require even fewer control lines than a


two-stage cascade for the same overall deflector res-olution NR.

Two 32-channel optical phased arrays based onPLZT were fabricated and used to demonstrate thecascading concept experimentally and to show theimproved performance of the discrete–offset-bias pro-gramming strategy. The experimental two-stagedevice was able to address all 32 resolvable beampositions with just 12 control lines. It was furthercapable of continuous beam steering to any anglebetween these beam positions. Proof-of-conceptdemonstrations were made with the current PLZTarrays, which have a low fill factor of 1y10 as a resultof the broad surface electrodes on the substrates.Alternative electrode designs and the inclusion of len-slet arrays on both sides of the present devices wouldenhance the effective fill factor and generate a higheroverall DE for the central peak. The present exper-iments, however, verify that the two-stage phased-array deflector can provide precise and continuousdeflection-angle control with a reduced number ofcontrol lines.

The multistaging technique is a practical means forsimplifying phased-array deflector control withoutcompromising its beam-steering flexibility. Thebest performance is achieved with the least numberof control lines by use of an aligned phased-arrayarrangement and by programming of the stages withthe discrete–offset-bias strategy. Such phased-array cascading techniques are very useful and couldbe applied to all forms of space-fed phased-array sys-tems, including ultrasonic and microwave-basedphased arrays.

Appendix A: Phased-Array Programming Formulas

1. Single-Stage Beam Deflection

To deflect an incident monochromatic beam by anangle uS, it is necessary to program a phased arraywith an appropriate fixed phase increment Dw be-tween each phase shifter in the array. By means ofEq. ~4b!, the required phase-step increment Dw ~inradians! is given by

Dw 5 2pSl

dDsin~uS!, (A1)

here l is the wavelength of the incident beam, d isthe modulator spacing, and uS is the desired deflec-tion angle. The phase-step increment is a funda-mental parameter and dictates how the entire arrayshould be programmed. After Dw is determined, thephase shifts of the modulators in an M-channel

hased array can be set as wm 5 mDw 1 wAVE for m 50, . . . , M 2 1, where wm is the absolute phase shift ofthe mth modulator and wAVE is an arbitrary constantphase value. For monochromatic cw illumination,as is assumed here, all phase shifts can be reduced bymodulo 2p without a functional change in the oper-ation of the phased array. This helps to limit theamount of physical phase shift that must be producedby any modulator to the range @0, 2p!. In all our

Fig. 11. Intensity profiles of the two-stage deflector output for theprimary deflection angles of uS 5 jy32~lyd1!, j 5 214, . . . , 1 16solid curves! and j 5 11 ~dashed curve! under discrete–offset-biasrogramming. The data cover the central nonredundant scanone in the output.

)

u~

acftm~aupw

t

m

r

E

app

u

t

t

experiments the actual phase shifts programmed intothe modulators were reduced phase shifts. For de-flection by an angle uS, the phased array is pro-grammed with reduced phase shifts wm, given by

wm ; @wm#mod 2p

5 @mDw 1 wAVE#mod 2p, m 5 0, . . . , M 2 1. (A2

Here wm is the reduced phase shift of the mth mod-lator, Dw is the phase increment determined by Eq.A1!, wAVE is an arbitrary constant ~typically set to 0!,

and M is the number of modulators in the phasedrray. The programming of the phased array isompleted by conversion of the reduced phase shiftsrom Eq. ~A2! to corresponding control voltages andhen application of these voltages to their respectiveodulators. The phase-modulation transfer curve

Fig. 8! defines the conversion between phase shiftnd control voltage for the PLZT phase modulatorssed in our experiments. Note that using reducedhase shifts guarantees that all the applied voltagesill be less than a full-wave voltage ~V2p! of the mod-

ulators.

2. Two-Stage Beam Deflection

In the two-stage implementation of phased-arraybeam steering the individual stages are programmedwith separate phase increments Dw1 and Dw2, whichare functions of their individual deflection angles u1and u2, respectively, and their modulator spacings d1and d2, respectively. The values for the individualdeflection angles are dictated by the overall deflectionangle uS, as outlined in Subsection 4.B, and they aredependent on the type of cascade and the choice ofprogramming protocol.

We now derive the operational programming for-mulas used to set the phases in an aligned two-stagephased-array deflector by following either the coarse–fine protocol or the discrete–offset-bias protocol.The aligned-cascade layout is a minimum control-lineconfiguration of the form shown in Fig. 4 in whichstage 1 is the coarse ~or discrete! stage and stage 2 isthe fine ~or offset-bias! stage.

Consider establishing beam deflection by an arbi-rary generalized angle u#S 5 Q~lyd2!, where u#S [

sin~uS!, l is the beam wavelength, d2 is the modulatorspacing in stage 2, and Q is an arbitrary real-valued

ultiplicative constant. Q must be restricted to therange uQu # M1y2 to keep the central output spotwithin the nonredundant scan zone. Here ~lyd2!represents the angular spacing between the discretestates ~i.e., blazed orders! that can be addressed di-ectly by stage 1 alone, and M1 is the number of

modulators per subarray in stage 1. The constant Qcan conveniently be decomposed into an integer partn and a fractional part r such that Q 5 n 1 r and thefull deflection angle can be redefined as

u#S 5 QS l

d2D 5 ~n 1 r!S l

d2D , (A3)

10

where n [ round@Q# selects the integer value closestto Q and r [ Q 2 n. The parameter r is real valuedand is guaranteed to lie in the range @20.5, 0.5! bythis definition. With this decomposition the deflec-tion angles required by each phased-array stage inthe cascaded deflector can readily be determined.

A. Coarse–Fine Programming of an AlignedCascadeUnder the coarse–fine protocol the first stage is pro-grammed to deflect the incident beam by an angleu#1 5 n~lyd2! and the second stage by an angleu#2 5 r~lyd2!. The angle u#1 represents the nth blazeddiffraction order for the coarse deflector ~stage 1!.

ach subarray in stage 1 functions as an M1-channelphased-array deflector with beam steering to u#1.Stage 2 acts as a single M2-channel phased-arraydeflector with beam steering to u#2. If we note thatd2 5 M1d1 ~for a minimum aligned cascade!, whereM1 is the number of modulators per subarray in stage1 and d1 is the modulator spacing, we can rewrite thestage 1 deflection angle as u#1 5 ~nyM1!~lyd1!. Thisreformulation defines the stage 1 deflection angle interms of its modulator spacing d1. Now Eqs. ~A1!nd ~A2! can be used to determine the correspondinghase increments ~Dw1 and Dw2! and the reducedhase shifts ~w1m and w2m! for all the modulators in

both stages. The results are summarized in Table 2.Note that the generalized angles given above are spe-cifically defined as u#1 [ sin~u1! and u#2 [ sin~u2!, where

1 and u2 are the true angles.

B. Discrete–Offset-Bias Programming of anAligned CascadeUnder the discrete–offset-bias protocol, both stagesare programmed to deflect the incident beam by thefull angle of u#S 5 Q~lyd2! so that u#1 5 u#2 5 u#S. Inerms of their respective modulator widths d1 and d2,

the deflection angles to be generated by the twostages can be written as u#1 5 ~QyM1!~lyd1! and u#2 5Q~lyd2!, and Eqs. ~A1! and ~A2! can be invoked todetermine the specific reduced phases of each modu-lator in both arrays. These results are also summa-rized in Table 2. Note that the reduced phase shiftof the mth modulator in the offset-bias stage ~stage2!, given by w2m 5 @2pQm#mod 2p, can be simplifiedto w2m 5 @2prm#mod 2p by use of the decompositionQ 5 n 1 r and elimination of the integer multipleerm ~2pnm! in the modulo operand. Hence the re-

duced phases for the offset-bias stage are functionallyidentical to those of the fine stage under the coarse–fine protocol for the specified aligned phased-arraycascade. This shows the surprising fact that stage 2has the same reduced-phase programming when ei-ther protocol for this cascade arrangement is used.

Appendix B: Diffraction Efficiency of an AlignedTwo-Stage Phased-Array Deflector

The theoretical computation of the DE versus thedeflection angle for an aligned two-stage phased-array cascade is determined below for both types ofcascade programming. Even though the coarse–fine


ebfap

ptt

tptD

s

ea

Table 2. Summary of the Phase Programming Formulas for an Aligned Two-Stage Phased-Array Deflector Using Either the Coarse–Fine Protocol or

6

cascade can be used with a more general cascadearchitecture, we restrict this comparative analysis toan aligned configuration of the form shown in Fig. 4because this is the arrangement required for thediscrete–offset-bias protocol. The results are repre-sented graphically in the plot in Fig. 12.

1. Diffraction Efficiency under Coarse–Fine Programming

Under coarse–fine programming the two stages inthe phased-array cascade behave as independent de-flectors approximating smooth prisms with steppedphase profiles. The quantization errors in bothstages cause a reduced DE, and the total DE of thecascaded deflector is equal to the DE’s of the individ-ual stages multiplied together. Each stage can bemodeled as a single phased-array deflector ~i.e., as anchelon grating! that has a diffraction efficiency giveny the element-factor term of Eq. ~5!. The elementactor is a function of both the modulator fill factornd the programmed deflection angle of the givenhased array.

Fig. 12. Comparison of the DE versus the scan angle of an alignedtwo-stage phased-array deflector cascade by use of the discrete–offset-bias protocol ~upper curve! and the coarse–fine protocol ~low-r curve!. Cascade configuration: M1 5 4, M2 5 8, 100% filledpertures; minimum control-line arrangement ~Fig. 4!.

the Discrete–Offset-Bias Protocol ~n 5 in

Parameter

Total deflection angle u#S

Stage 1Deflection angle u#1

Phase increment Dw1

Reduced phase of mth modulator w1m

Stage 2Deflection angle u#2

Phase increment Dw2

Reduced phase of mth modulatora w2m

aw2m is the same for both protocols because ~2pQm!mod 2p 5 ~


The individual stages of the aligned cascade arerogrammed as coarse and fine deflectors followinghe procedure outlined in Appendix A. For deflec-ion by a total angle of u#S 5 ~n 1 r!~lyd2!, where n is

an integer and r is real valued ~uru # 0.5!, the coarseand the fine stages are programmed with generalizeddeflection angles of u#1 5 n~lyd2! and u#2 5 r~lyd2!,respectively. The DE’s of the individual stages,hcourse and hfine, are found by the application of Eq. ~5!o each stage with the appropriate fill factor andhase increment. The total DE is equal to the DE ofhe individual stages multiplied together. The totalE hTOT

C–F of a two-stage phased-array deflectorprogrammed with the coarse–fine protocol can bewritten as

hTOTC–F 5 hcoarsehfine 5 Sw1

d1D2

sinc2FSw1

d1DSDw1

2p DG3 Sw2

d2D2

sinc2FSw2

d2DSDw2

2p DG , (B1)

where ~w1yd1! and ~w2yd2! represent the fill factorsand Dw1 and Dw2 the programmed phase incrementsfor stages 1 and 2, respectively, and sinc~m! [sin~pm!y~pm!. For a total deflection angle of u#S 5~n 1 r!~lyd2!, the corresponding phase increments fortages 1 and 2 in an aligned configuration are Dw1 5

2p~nyM1! and Dw2 5 2pr ~from Table 2!, respectively,where M1 is the number of modulators per subarrayin stage 1. The values for n and r are derived fromthe the angle u#S by Eq. ~A3!. The maximum DEoccurs when there is no deflection and has the value~w1yd1!2~w2yd2!2. The DE in Eq. ~B1! varies in adiscontinuous way with the deflection angle becausethe integer parameter n is a nonlinear, discontinuousfunction of the deflection angle u#S. As u#S increases,the integer value n ~hence Dw1! for the coarse stageremains constant, while the fine deflector scanssmoothly from r 5 20.5 – 10.5 ~i.e., Dw2 5 2p – p!,and then n jumps to its next value of n 1 1. Thediscontinuous and nonmonotonic intensity behaviorof the coarse–fine programming scheme is demon-strated in the example plotted in Fig. 12.

-valued, r 5 real-valued, and Q 5 n 1 r!

Protocol

Coarse–Fine Discrete–Off-Set Bias

~n1r!~lyd2! Q~lyd2!

Coarse Deflector Discrete Deflectorn~lyd2! Q~lyd2!

2p~nyM1! 2p~QyM1!@2p~nyM1!m#mod 2p @2p~QyM1!m#mod 2p

Fine Deflector Offset-Bias Phasesr~lyd2! Q~lyd2!

2pr 2pQ~2prm!mod 2p ~2pQm!mod 2p

!mod 2p.

teger

2prm

tsa

Q

rih

sca

femcusFtcl

dmafij

rt

2. Diffraction Efficiency under Discrete–Offset-BiasProgramming

Using the discrete–offset-bias protocol causes the DEof the aligned two-stage phased-array deflector to bedictated primarily by the first stage ~the discretestage! because it is the only stage to exhibit phase-quantization errors. Under this programmingstrategy the terms AF1 and AF2 in Eq. ~8a! are guar-anteed to be at their peak values ~AF1 5 AF2 5 1! atu#S and the deflector DE is described simply by theelement factor term EF1 in Eq. ~7b!. The alignedwo-stage phased-array system behaves like a phase-taggered version of a large, single-stage phased-rray deflector that has modulators of width d1, an

effective fill factor ~w1yd1!, and a functional equiva-lent of M1M2 modulators. It is assumed here thatall the light passing through the first stage ~the dis-crete scanner! subsequently passes through the sec-ond stage ~the offset-bias stage! without furtherattenuation, so the full two-stage device can be mod-eled with the simple fill factor ~w1yd1! determined bystage 1. ~This is a reasonable assumption for two-stage cascades implemented by use of a butt-coupledpair of identical phased arrays, as in our experi-ments.!

Under discrete–offset-bias programming both stagesare programmed for deflection directly to the selectedangle u#S, which can be written as u#S 5 Q~lyd2!, where

is a real-valued constant and l and d2 are as pre-viously defined. The DE of this cascade is deter-mined solely by the term EF1 @Eq. ~7b!#, which can beewritten in terms of the fill factor and the phasencrement for stage 1. The resulting total DETOT

D–OB of an aligned two-stage phased-array cas-cade programmed with the discrete–offset-bias pro-tocol can be written as

hTOTD–OB 5 Sw1

d2D2

sinc2FSw1

d1DSDw19

2p DG . (B2)

Here w1 is the clear aperture of the modulators instage 1, d1 is the width of these modulators, and stage1 has a phase increment of Dw19 5 2p ~QyM1! ~fromTable 2!, where M1 is the number of modulators persubarray in stage 1 and Q represents the unitlessscan-angle parameter. Note that the stage 1 phaseincrement Dw19 takes on a slightly different valuehere than it would under coarse–fine operation ~seeTable 2!. The maximum DE under discrete–offset-bias programming is ~w1yd1!2. Because Q ~henceDw19! varies smoothly with the deflection angle u#S, itfollows from Eq. ~B2! that the DE of the aligned two-tage deflector operating as a discrete–offset-bias cas-ade is a smooth, monotonic function of the deflectionngle. Equation ~B2! is identical to the element-

factor envelope EF1 of the first stage, which repre-sents the fixed upper-bound limit to the DE of thecascaded system. Hence the discrete–offset-biasmode of operation achieves the optimal DE for suchan aligned phased-array cascade.

10

3. Comparison of Deflector Efficiencies

Figure 12 shows comparative plots of the DE versusthe scan angle for an aligned two-stage phased-arraydeflector programmed by use of both protocols andunder the assumption of the ideal case of 100% filledapertures in both stages. The aligned-cascade ar-rangement requires a minimum number of controllines for a given scanner resolution. A system con-figuration with M1 5 4 channelsysubarray in stage 1and M2 5 8 channelsysubarray in stage 2 was usedor the plots to mimic the configuration used in thexperiments described in Section 6. Because theodeling assumes 100% filled modulators, the DE

urves shown in Fig. 12 represent the maximumpper-bound limits on DE that can be obtained byuch an aligned two-stage cascade configuration.or partially filled modulators, as in our experiments,he essential features remain the same, but theurves are broadened and vertically compressed, thusowering the maximum possible DE.

The DE of the cascaded phased-array deflector un-er discrete–offset-bias programming is a smooth,onotonic function of the deflection angle and

chieves an overall higher DE than does the coarse–ne programming, which exhibits discontinuous

umps in its DE versus u#S profile. The discontinuousjumps occur at angles falling half-way between thediscrete states that can be addressed by the coarsestage ~stage 1! alone, and they exhibit a low DE as aesult of the large quantization errors occurring inhe fine stage ~stage 2! at these angles. The exper-

imental data from our test arrays described in Sec-tion 6 exhibit these same intensity trends. Notethat the large dips in the DE curve for the coarse–fineprogramming scheme could be reduced by an in-crease in the number of modulators in the fine stage,as discussed in Section 5, but at the cost of greaterphased-array complexity. Because the DE profilefor the discrete–offset-bias programming schemematches the element factor of the smallest diffractingelement in the system, it uniquely achieves the high-est overall DE that can be obtained by an alignedtwo-stage phased-array deflector with a minimumcontrol-line configuration.

This study was supported in part by the NationalScience Foundation and the U.S. Office of Naval Re-search. The authors also thank P. Soltan and M.Lasher of the Naval Research and Development Lab-oratories ~NRaD! in San Diego, California, for theirprogram assistance and technical support with thisproject.

References1. P. F. McManamon, T. A. Dorschner, D. L. Corkum, L. J. Fried-

man, D. S. Hobbs, M. Holz, S. Liberman, H. Q. Nguyen, D. P.Resler, R. C. Sharp, and E. A. Watson, “Optical phased arraytechnology,” Proc. IEEE 84, 268–298 ~1996!.

2. T. C. Cheston and J. Frank, “Phased array radar antennas,” inRadar Handbook, 2nd ed., M. I. Skolnik, ed. ~McGraw-Hill,New York, 1990!, Chap. 7, pp. 7.1–7.82.

3. R. M. Matic, “Blazed phase liquid crystal steering,” in Laser


Beam Propagation and Control, H. Weichel and L. F. DeSan- 13. P. J. Talbot and Q. W. Song, “Design and simulation of PLZT-

6

dre, eds., Proc. SPIE 2120, 194–205 ~1994!.4. R. A. Meyer, “Optical beam steering using a multichannel

lithium tantalate crystal,” Appl. Opt. 11, 613–616 ~1972!.5. Y. Ninomiya, “Ultrahigh resolving electrooptic prism array

light deflectors,” IEEE J. Quantum Electron. QE-9~8!, 791–795 ~1973!.

6. Y. Ninomiya, “High SyN-ratio electrooptic prism-array lightdeflectors,” IEEE J. Quantum. Electron. QE-10, 358–362~1974!.

7. T.-C. Lee and J. D. Zook, “Parallel array light beam deflectorwith variable phase plate,” U.S. patent no. 3,650,602 ~issued 21March 1972!.

8. C. H. Bulmer, W. K. Burns, and T. G. Giallorenzi, “Perfor-mance criteria and limitations of electro-optic waveguide arraydeflectors,” Appl. Opt. 18, 3282–3295 ~1979!.

9. D. R. Wight, J. M. Heaton, B. T. Hughes, J. C. H. Birbeck, K. P.Hilton, and D. J. Taylor, “Novel phased array optical scanningdevice implemented using GaAsyAlGaAs technology,” Appl.Phys. Lett. 59, 899–901 ~1991!.

10. F. Vasey, F. K. Reinhart, R. Houdre, and J. M. Stauffer, “Spa-tial optical beam steering with an AlGaAs integrated phasedarray,” Appl. Opt. 32, 3220–3232 ~1993!.

11. T. A. Dorschner and D. P. Resler, “Optical beam steerer havingsubaperture addressing,” U.S. patent no. 5,093,740 ~issued 3March 1992!.

12. J. A. Thomas and Y. Fainman, “Programmable diffractive op-tical element using a multichannel lanthanum-modified leadzirconate titanate phase modulator,” Opt. Lett. 20, 1510–1512~1995!.


based electro-optic phased array scanners,” Opt. Memory Neu-ral Net. 3, 111–117 ~1994!.

14. T. Tatebayashi, T. Yamamoto, and H. Sato, “Electro-optic vari-able focal-length lens using PLZT ceramic,” Appl. Opt. 30,5049–5055 ~1991!.

15. N. A. Riza and M. C. DeJule, “Three-terminal adaptivenematic liquid-crystal lens device,” Opt. Lett. 19, 1013–1015~1994!.

16. J. G. Skinner, “Optimal electrooptic deflection scheme,” Appl.Opt. 7, 1239–1240 ~1968!.

17. H. J. Boll, “Cascaded light beam deflector system,” U.S. patentno. 3,544,200 ~issued 1 December 1970!.

18. T. C. Lee and J. D. Zook, “Cascade operation of light beamdeflectors by fly’s eye lenses,” Appl. Opt. 10, 1965–1966 ~1971!.

19. K. M. Flood, B. Cassarly, C. Sigg, and J. M. Finlan, “Contin-uous wide angle beam steering using translation of binarymicrolens arrays and a liquid crystal phased array,” in Com-puter and Optically Formed Holographic Optics, I. Cindrichand S. H. Lee, eds., Proc. SPIE 1211, 296–304 ~1990!.

20. J. A. Thomas, M. E. Lasher, Y. Fainman, and P. Soltan, “PLZT-based dynamic diffractive optical element for high-speedrandom-access beam steering,” in Optical Scanning Systems:Design and Applications, L. Beiser and S. F. Sagan, eds., Proc.SPIE 3131, 124–132 ~1997!.

21. G. H. Haertling, “PLZT electrooptic materials andapplications—a review,” Ferroelectrics 75, 25–55 ~1987!.

22. R. Fleischmann and A. Lohmann, “Die Bestimmung einer ab-soluten Lichtphase durch Intensitatsmessung in der Beu-gungsfigur eines Gitters,” Z. Physik 137, 362–375 ~1954!.

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