tions required in FDTD can limit the accuracy when dealingwith near-zero power reflectivities.

REFERENCES

1. T. Ikegami, ‘‘Reflectivity of Mode at Facet and Oscillation Modein DH Injection Lasers,’’ IEEE J. Quantum Electron., Vol. QE-8,1972, pp. 470]476.

2. J. Buus, ‘‘Analytic Approximation for the Reflectivity of DHLasers,’’ IEEE J. Quantum Electron., Vol. QE-17, 1981, pp.2256]2257.

3. C. Vassallo, ‘‘Theory and Practical Calculation of AntireflectionCoatings on Semiconductor Laser Diode Optical Amplifiers,’’IEE Proc. Pt. J, Vol. 137, 1990, pp. 133]137.

4. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M.J. Robertson, ‘‘Semiconductor Laser Facet Reflectivity UsingFree Space Radiation Modes,’’ IEE Proc. Pt. J., Vol. 140, 1993,pp. 49]55.

5. P. C. Kendall, D. A. Roberts, P. N. Robson, M. J. Adams, and M.J. Robertson, ‘‘New Formula for Semiconductor Laser FacetReflectivity,’’ IEEE Photon. Technol. Lett., Vol. 5, 1993, pp.148]151.

6. C. J. Smartt, T. M. Benson, and P. C. Kendall, ‘‘Exact Analysis ofWaveguide Discontinuities, Junctions and Laser Facets,’’ Elec-tron. Lett., Vol. 29, 1993, pp. 1352]1353.

7. M. Reed, P. D. Sewell, T. M. Benson, and P. C. Kendall,‘‘Antireflection Coated Angled Facet Design,’’ IEE Proc., to bepublished.

8. J. Yamauchi, M. Mita, S. Aoki, and H. Nakano, ‘‘Analysis ofAntireflection Coatings Using the FD-TD Method with the PMLAbsorbing Boundary Condition,’’ IEEE Photon. Technol. Lett.,Vol. 8, pp. 239]241.

9. J. Yamauchi, S. Aoki, and H. Nakano, ‘‘Reflectivity Analysis ofOptical Waveguides with Coated and Tilted Facets Using theFDTD Method with the PML Absorbing Boundary Condition,’’Opt. Soc. Am. Tech. Digest Series, Vol. 6, Integrated PhotonicsResearch, Boston, 1996, pp. 422]425.

10. F. Schmidt, ‘‘An Adaptive Approach to the Numerical Solutionof Fresnel’s Wave Equation,’’ J. Lightwa¨e Technol., Vol. 11,1993, pp. 1425]1434.

Q 1997 John Wiley & Sons, Inc.CCC 0895-2477r97

OPTICAL ADAPTIVE BEAM FORMINGWITH SWITCHED STEP SIZEAlex V. Petrov,1 Shizhuo Yin,1 and Francis T. S. Yu11 Department of Electrical EngineeringPennsylvania State UniversityUniversity Park, Pennsylvania 16802

Recei ed 21 No¨ember 1996

ABSTRACT: The performance of con¨entional optical adapti e beamformers for rf communication and radars is shown to require a compro-mise between the con¨ergence speed and interference cancellation ratio.To alle¨iate this trade-off, a no¨el nonlinear algorithm with switched stepsize is proposed. Results of direct simulation and numerical solution arepresented to illustrate the approach. Q 1997 John Wiley & Sons, Inc.Microwave Opt Technol Lett 15: 16]20, 1997.

Key words: adapti e antenna; widrow algorithm; optical signal process-ing

INTRODUCTION

Adaptive beam forming is a technique to suppress an unde-sired signal penetrating through antenna side lobes by means

of automatic null steering in the direction of undesiredw xsource 1 . An adaptive property is vital for many advanced

w xcommunications systems and radars 2, 3 . Numerous studieshave indicated feasibility of broadband interference cancella-

Ž . w xtion with optical adaptive processors OAPs 4]9 . With theirpotential yet to be fully realized, OAPs have already demon-

Ž .strated a higher cancellation ratio CR , that is, the degree ofinterference suppression, than their microwave counterpartsw x9 .

In this Letter, we will focus on OAPs based on thew xWidrow algorithm 10 . According to this algorithm, interfer-

ence component in the main channel is estimated by filteringŽ .a sample of interference x t available from the auxiliary

Ž .channel with a certain weight function w t :

TŽ . Ž . Ž . Ž .y t s x t y t w t dt . 1H0

The weight distribution is chosen to minimize the mean-Ž .square error MSE V at the processor output with respect to

Ž .the reference signal d t :

Ž . Ž . Ž . � 2 Ž .4 Ž .e t s d t y y t : V s E e t ª V , 2min

� 4where E ??? stands for expected value. The optimum weightfunction is found by continuously modifying current weight

Ž .distribution w t, t in the gradient direction with a constantstep size m:

Ž .­ w t , t ­ VŽ . Ž . Ž .s ym s y2me t x t y t . 3

­ t ­ w

Ž .With m ) 0, w t, t converges to the optimum distribution,Ž . Ž . w xso that e t H x t y t , ;t g 0, T .

In the steady state, CR is finite for two reasons. First,OAPs with imperfect finite-time intergators need an interfer-ence component in the output to keep weights from vanishingw x w x4 . With progress in optical integrators 11 , this issue willbecome less critical. The second performance limitation israndom output fluctuation due to the gradient estimationerror. Because of this error, the increase in the step-sizevalue does not result in proportional gain in CR. This limita-tion, inherent to the continuous Widrow algorithm, will bediscussed below.

SPEED ///// CANCELLATION RATIO TRADE-OFFIN CONVENTIONAL OAP

To study the OAP performance, we modify the analysisw xprocedure for discrete LMS algorithm 10 . Specifically, two

consecutive eigenfunction expansions are used. First, time-w xdomain expansion 5 relates the continuous waveform to the

eigenvalue set. The second expansion is accomplished overthe eigenvectors of eigenvalues’ covariance matrix. These twodecompositions decouple differential equations governingvariation of the residual interference power at the output.The use of eigenfunctions to study the interference powersuppression is justified by the Parceval’s equality.

� Ž .4 w xIn an orthonormal basis F t over 0, T , the delayedkŽ . Ž .replicas of input signal x t y t and weight function w t are

decomposed as follows:

Ž . Ž . Ž . Ž . Ž . Ž .x t y t s l t F t ; w t s à F t . 4Ý Ýk k k kk k

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 1, May 199716

Ž . Ž .From 1 and 4 , the interference estimate can be ex-pressed as a function of eigenvalues:

Ž . Ž . Ž .y t s à l t . 5Ý k kk

In order to express the MSE in terms of eigenvalues, firstlet us introduce the autocorrelation function of reference

Ž . � Ž . Ž .4signal R t s E d t d t y t and cross-correlation func-ddŽ . � Ž . Žtion between reference and input signals R t s E d t x td x

.4 � 4y t . The latter can be also decomposed in F as usual:k

TŽ . Ž . Ž . Ž . Ž .R t s d F t ; d s R t F t dt . 6HÝd x k k k d x0k

Ž . Ž . Ž .Substituting 4 ] 6 for 2 gives MSE in terms of theeigenvalues:

Ž . � 4 Ž .V s R 0 y 2 d à q à à E l l . 7Ý Ýdd k k k 1 k 1k k , 1

Ž .From 7 , MSE is minimum for optimum weights à ,optwhich satisfy

Ž .LÃ s d , 8opt

� 4where L s E l l is covariance matrix of the eigenvaluesi j i jof the input signal. In fact, the last equation is the well-known

w xWiener-Hopf equation 6 defined in the eigenspace.With às à , MSE is minimum and equalsopt

TŽ . Ž .V s R 0 y à Là . 9min dd opt opt

It is convenient to express MSE in terms of V and mis-minalignment n s à y à :opt

T Ž .V s V q n Ln . 10min

Ž .With little loss of generality, x t can be treated as awide-sense stationary stochastic process. Then L is a posi-tive-definite symmetric matrix and can be orthogonally trans-formed:

y1 Ž .L s QCQ , 11

Ž .where C s diag c , . . . , c is a diagonal matrix of eigenval-1 Nues of L; Q is a matrix composed of eigenvectors of L. If we

y1denote the transformed misalignment vector as q s Q n ,Ž .10 becomes

T Ž .V s V q q Cq . 12min

For real-time operation, a feasible OAP must use anestimate of the gradient instead of the true gradient value. In

Ž .other words, in 3 the gradient of the current square errorˆ 2 10Ž .V s e t has to be used:

­Ã dVk Ž .s ym q N , k s 1 ??? N , 13kž /­ t dÃk

ˆŽ .where N t s dVrdà y dVrdà is gradient estimationk k kŽ .error. Usually N t is modeled as a zero-mean additive1

w xGaussian noise 10 .

Ž . Ž .Similarly to 3 , Eq. 13 can be rewritten for q space:

Ž .dq tk Ž Ž . Ž .. Ž .s ym 2c q t q h t , k s 1 ??? N , 14k k kdt

y1 Ž .where h s Q N. Notice that differential equations in 14are now decoupled.

The transient is defined as the time interval when theoutput error is predominantly due to the initial misalignment;

Ž . Ž .that is, 2cq t 4 h t . Then, solution to k th differentialk kŽ .equation in 14 is

Ž . Ž . ytrt k Ž .q t s q 0 e , k s 1 ??? N , 15k k

Ž .where q 0 corresponds to the initial weight misalignment,kŽ .y1and t s 2mc is the kth time constant. So, weightsk k

converge in the mean to the optimum with speed propor-tional to step size. In the transient, the MSE varies according

Ž . Ž .to 12 and 15 :

Ž . 2 Ž . y4mc k t Ž .V t s V q q 0 c e . 16Ýmin k kk

In the steady state, the gradient approximation error pre-vails. Although à ª à , weights and MSE experience ran-optdom fluctuations due to the gradient noise. To find resulting

Ž .excess MSE, first let us rewrite 14 in another form:

Ž .dq tk Ž . Ž . Ž .t q q t s z t , k s 1, . . . , N , 17k k kdt

Ž . Ž .where z t s ymh t t . This equation describes propaga-k k kŽ .tion of noise z t through the first-order filter with timek

constant t . Our objective is to find output noise varianceks 2 .q k

The output variance can be found in a closed form for theŽ . Ž .representative case when e t and l t are white Gaussiank

stochastic processes. Then, with the use of the orthogonalityprinciple and properties of Gaussian stochastic processes, thecovariance matrix of h can be first evaluated:

T� Ž . Ž .4 Ž .E h t h t s 4CV . 18min

Ž . � Ž .4From 18 and knowing that E h t s 0, the input vari-ance can be found:

� 2 Ž .4E h t Vk min2 Ž .s s s 19zk 4c ck

Now the output variance can be determined:

s 2zk2 Ž .s s . 20q k D f 2tk

Finally, if we apply the power conservation property

� 4 � 4 � 4 � 4tr C s c s tr L tr C s tr L ,Ý kk

Ž .y1 Ž .recall that t s 2mc , and use 12 , the excess MSE atk kthe OAP output is found:

� 2 Ž .4V E l tÝmin k kk Ž .V y V s m . 21min D f

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 1, May 1997 17

Figure 1 Results of computer simulation of OAP dynamics

Ž .From 20 , the steady-state residual interference power isŽ .proportional to the step size. Comparing this to 15 , we

conclude that constant step-size value imposes a constrainton the product of convergence speed and CR. In other words,OAPs employing high-performance integrators must compro-mise dynamic and steady-state parameters.

To verify and visualize this result, direct simulation ofOAP dynamics was performed with an ensemble of 500realizations of the nonstationary stochastic process governed

Ž .by 14 . Gradient noise was simulated as white Gaussiannoise of zero mean and unit variance. The ensemble-aver-aged value of output interference power for two step-sizevalues, m s 0.01 and m s 0.02, is plotted in Figure 1. Numer-ical results agree with conclusion on interrelated convergencespeed and CR.

SWITCHED-STEP-SIZE MODIFICATIONOF WIDROW ALGORITHM

SpeedrCR trade-off can be alleviated by modifying the basicalgorithm. Two approaches proposed so far, that is, hard-

w x Ž . Ž . w xlimiting 6 x t y t in 3 and dynamic gain adjustment 12suffer from hardware complexity.

Ž .A novel approach is to use in 3 a step size that isthresholded by output:

ˆm , if V G V ,d 0 Ž .m s 22½ ˆm , if V - V ,s 0

where m is a dynamic step size for the transient stage; m isd s< Ž .a static step size for the steady state; V s V 1 q «msm0 min d

is the threshold to switch between two step-size values;<V is steady-state MSE for m s m ; « s 0.1 ??? 0.2msmmin dd

secures switching between the step-size values.The operating scenario is as follows. Dynamic step size is

in use initially. For faster convergence, it must be the maxi-mum available gain. Eventually, residual interference powerbecomes less than V , and step size is switched to the static0value. The latter is selected to meet the CR specification.Figure 1 illustrates the approach by showing processor outputfor the same conditions as in the example in Section 2 andV s 0.4.0

There has been found no closed-form solution to thenonstationary nonlinear stochastic problem of finding MSEfor signal-dependent step size. Nonlinearity couples modestogether and makes eigenvalue expansion impossible. Tosolve the nonlinear stochastic differential equation governing

Ž .output MSE, we used a Markov chain model Figure 2 . Theoutput stochastic process was sampled, and the probability

Ž .distribution function pdf for each sample was found frombivariate distribution of the previous sample and noise. Thenonstationary output pdf and the residual power for the sameconditions as for the computer simulations are shown inFigures 3 and 4, respectively. The discrepancy between nu-merical solution and computer simulation results did notexceed 3%.

Though more elaborate step-size schedules could be bor-w xrowed from the area of neural networks 11 , a two-level

step-size function gives a reasonable compromise betweenperformance enhancement and added complexity. Real-time

Ž .implementation of 21 is possible in the microwave domainŽ .with high-speed p-i-n diode switches , at the interface be-

Žtween optical and microwave domains with a phototransistor. Žthat has its gate output controlled , or in optical domain with

.a fiber amplifier that has its pump controlled by the output .The proposed algorithm is compatible with other perfor-mance enhancement techniques, such as the passive listening

w xmode and loop gain regulation 8 .Simulations have also shown that the algorithm is robust

to changes in V . In particular, V s 0.05 and V s 0.5 yielded0 0 0the same CR and a 10% difference in convergence speed.This insensitivity to the threshold drift may be important for

Žapplications with unsteady thermal environment e.g., in air-.borne antennas .

CONCLUSIONS

Analysis of the continuous Widrow algorithm, as applied to awide class of input signals, reveals the trade-off betweenOAP convergence speed and cancellation ratio. As this trade-off is due to the constant value of step size, it is proposedthat a two-level output-controlled step size be used to providefor both fast convergence and high cancellation ratio. Imple-mentation of such an algorithm is possible with rf switches orphotodetectors. Results of computer simulations and a nu-

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 1, May 199718

Figure 2 Markov-chain]based algorithm used to characterize OAP with switched step size

Figure 3 Probability distribution function of the output error

Ž .Figure 4 Results of numerical solution solid line and computerŽ .simulation dashed line of OAP dynamics

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 1, May 1997 19

merical solution, based on the Markov chain model, agreethat the modified algorithm gives improved performance.

REFERENCES

1. R. Compton, Adapti e Antennas, Prentice-Hall, Englewood Cliffs,NJ, 1988.

2. B. D. Woerner, J. H. Reed, and T. S. Rappaport, IEEE Commun.Mag., Vol. 32, No. 7, 1994, p. 43.

3. R. Nitzberg, Adapti e Signal Processing for Radar, Artech House,Dedham, MA, 1992.

4. J. F. Rhodes, Appl. Opt., Vol. 22, 1983, p. 283.5. J. D. Cohen, Appl. Opt., Vol. 24, 1985, p. 4247.6. P. Das and C. DeCusatis, Acousto-Optic Signal Processing: Funda-

mentals and Applications, Artech House, Dedham, MA, 1991.7. R. Montgomery and M. Lange, Appl. Opt., Vol. 30, 1991, p. 2845.8. D. Friedman, D. Dwyer, and R. Iodice, SPIE, Vol. 1958, 1993, p.

83.9. W. LeComte, S. Henion, and P. Schulz, SPIE, Vol. 2155, 1994, p.

256.10. B. Widrow and S. Stearns, Adapti e Signal Processing, Prentice-

Hall, Englewood Cliffs, NJ, 1985.11. F. T. S. Yu, S. Yin, O. Leonov, and A. Petrov, ‘‘All-Optic

Adaptive Radar Processor: Performance Limitations and En-Žhancement’’ to be presented at the OSA Annual Meeting, 1996,

.paper MZ1 .12. COLSA, Inc., ‘‘Hybrid Electro-Optic Processor,’’ Final Tech.

Report No. RL-TR-91-164, 1991.11. A. Garga, private communication.

Q 1997 John Wiley & Sons, Inc.CCC 0895-2477r97

A RAY-BY-RAY ALGORITHM FORSHAPING DUAL-OFFSET REFLECTORANTENNASJ. O. Rubinos-Lopez1 and A. Garcıa-Pino1˜ ´ ´1 Dept. Tecnologıas de las Comunicaciones´E.T.S.I. Telecomunicacion´Universidad de VigoCampus Universitario36200 Vigo, Spain

Recei ed 26 No¨ember 1996

ABSTRACT: A no¨el GO method for designing dual shaped offsetreflector antennas with a prescribed phase and amplitude aperture distri-bution is presented. This technique consists of tracing rays consecuti elyby sol ing nondifferential equations. The solutions, which simultaneouslysatisfy the amplitude and phase requirements, together with the totalderi ability condition of the surface expressed in a new discretized way,are easier to control than the other techniques in the literature. Thisalgorithm accounts for the feed phase errors. Two examples are selectedto show the good performance of the algorithm. Q 1997 John Wiley &Sons, Inc. Microwave Opt Technol Lett 15: 20]26, 1997.

Key words: reflector; antennas; shaping

1. INTRODUCTION

In communication and radar applications, antennas with highdirectivity, low side lobes, and low cross polarization are

Contract grant sponsor: Spanish CICYT Project; Contract grant number:T1C95-0884.

often required. To achieve them, the reflector surfaces mustbe shaped to produce an optimal aperture distribution with aknown feed pattern. This problem was first solved for circu-

w xlarly symmetric geometries under GO assumptions 1, 2 ofthe phase and amplitude conditions. Because of the effect ofsubreflector blockage, offset solutions have been investigatedw x3]6 . There has long been controversy about the existence ofexact solutions to the problem; but it has finally been ac-

w x wcepted that several exact solutions are possible 7]10 . In 7,x8 , a complex notation for the ray directions is applied to the

synthesis of offset dual reflector systems. In this way, theexact formulation of the problem leads to a nonlinear sec-ond-order partial differential equation of the Monge]Amperetype. Galindo-Israel, Imbriale, and Mittra collected and de-veloped the GO principles and requirements, including thetotal derivability condition, into a set of nonlinear first-order

w xpartial equations 9 . The problem was also treated by Kildalw x10, 11 , who provided solutions by solving systems of fourlinear nondifferential equations for each ray. The reflectorsurfaces are discretized in rings, and it is necessary to traceall rays corresponding to a ring to establish the equations forthe next one. This technique does not account for the phaseerrors of the feed.

This Letter presents a method for synthesizing dual shapedoffset reflector antennas with a plane of symmetry. Thistechnique, which takes into account the phase errors of thefeed, is based on tracing rays consecutively by solving only

w xone or two equations, unlike doing it ring by ring, as in 10 .The solutions simultaneously satisfy the amplitude and phaserequirements, together with the total derivability condition,

w xwhich is expressed in a similar way as 12, 13 but extended tothree dimensions. Our method yields algebraic equationswhose solutions are easier to find and control than those

w xgiven by all other methods 5, 7]10 .In Section 2 we describe the geometry of the problem and

introduce the notation used for the synthesis method. Thedesign requirements, including the novel formulation of thetotal derivability condition, are presented in Section 3. Theproblem, reduced to solve just two nondifferential equations

Ž .per ray only one for rays in the symmetry plane , can betreated with the algorithm described in Section 4. The perfor-mance of the method is tested in Section 5 with two exam-ples. In both cases, the analysis of the shaped reflectorantennas is made by the use of physical optics. The agree-ment between desired and obtained amplitude distributionswas found to be excellent.

2. GEOMETRY DESCRIPTION

The geometry of a dual shaped offset reflector antenna isshown in Figure 1. The overall coordinate system is chosen tocoincide with the coordinate system of the aperture. The XZplane is considered a plane of symmetry.

A set of successive optical rays is launched from the feed.Ž .These rays are grouped into M rings m s 1 ??? M with

m s M defining the rim of the reflectors. Every ring consistsŽ .of N rays n s 1 ??? N, N even . A ray emerging from the

Ž .feed is characterized by u , f , the spherical coordinates inm n² :a feed coordinate system, x , y , z , with origin at theˆ ˆ ˆF F F

phase center of the feed, F. After reflecting from bothreflector surfaces, the ray ends at a point A defined bym, n

Ž .the polar coordinates r , c of the aperture with originm, n m , nat the offset point A . The reflector surfaces are described by0

MICROWAVE AND OPTICAL TECHNOLOGY LETTERS / Vol. 15, No. 1, May 199720