180 RLE Progress Report Number 131 - dspace.mit.edu

Section 3 Electromagnetics

Chapter 1 Electromagnetic Wave Theory andApplications.

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180 RLE Progress Report Number 131

Chapter 1. Electromagnetic Wave Theory

Chapter 1. Electromagnetic Wave Theory andApplicationsAcademic and Research Staff

Professor Jin Au Kong, Dr. Sami M. Ali, Dr. Pini Einziger, Dr. Tarek M. Habashy, Dr. Soon Y.Poh, Dr. Robert T. Shin

Visiting Scientists

Qizheng Gu, Dr. Alain Priou

Graduate Students

Christopher J. Adams, David V. Arnold, Robert G. Atkins, Maurice Borgeaud, Ike Chang, JudyChen, Nelson C. Chu, Hsiu C. Han, Jean-Fu Kiang, Cheung-Wei Lam, Check-Fu Lee, Kevin Li,Harold H. Lim, Freeman C. Lin, Son V. Nghiem, David M. Sheen, Albert A. Swartz, Michael J.Tsuk, Ann N. Tulintseff, Jiging Xia, Ying-Ching E. Yang, Herng A. Yueh

Professor Jin Au Kong and his research group work in the areas of electromagnetic wavetheory and applications. They are studing applications to microelectronic integrated circuits,microstrip antennas, geophysical subsurface probing, SAR imaging and microwave remotesensing, and transient electromagnetic pulse propagation and coupling problems. The study ofelectromagnetic wave propagation in integrated circuits is motivated by the need to achievecondensed circuit packages, shorter rise times, and smaller pulse widths without causing largesignal distortion and crosstalk. The methods of characteristics and integral transform based onscattering parameters are used to study the transient response.

Realistic theoretical models that are applicable to the active and passive remote sensing ofplowed fields, atmospheric precipitation, vegetation, and snow fields have been developed.The development of these theoretical models has been strongly motivated by the need foraccurate data analysis and interpretation, and scene simulation characteristics. Under the con-sideration that sea ice is a tilted uniaxial medium, the observed strong cross-polarized return inthe bistatic scattering coefficients is successfully predicted from the theoretical model. Exten-sive work has been accomplished in the development of theoretical models and applications topolarimetric remote sensing and terrain classification.

1.1 Electromagnetic Waves inMultilayer Media

Sponsors

Joint Services Electronics Program(Contracts DAAL 03-86-K-0002and DAAL 03-89-C-0001)

U.S. Navy - Office of Naval Research(Contract N00014-86-K-0533)

National Science Foundation(Contract ECS 86-20029)

U.S. Army Research Office(Contract DAAL03 88-K-0057)

International Business Machine CorporationSchlumberger-Doll Research

Project StaffProfessor Jin Au Kong, Dr. Sami M. Ali,Robert G. Atkins, Ike Chang, Qizheng Gu,Dr. Tarek M. Habashy, Check-Fu Lee, KevinLi, Dr. Soon Y. Poh, Michael J. Tsuk, Ann N.Tulintseff, Ying-Ching E. Yang

To provide shorter interconnects betweenchips, modern multilayer integrated circuitpackages for high-performance mainframecomputers employ not only conventionalstriplines, but also vertical transmission linesor so-called vias. Because vias may have thesame length as the striplines, the study oftransient electromagnetic wave propagationon the vias is equally indispensible to the

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understanding of the speed at which themultilayer integrated circuits can operate.

Idealized circuit packages consisting of cir-cular vias going through circular holes in theground planes that separate different layersof equal thickness are being considered.When only one via is present, we show that,by treating the layer between two groundplanes as a radial waveguide driven by acoaxial feed and making use of symmetry, theequivalent network parameters for each layercan be obtained. The multiple layers consti-tute a periodic structure and presents littledifficulty in computing the total transmissionmatrix. We also studied an infinite array ofvias and holes with equal spacing along twoperpendicular directions. By virtue of theimage theory, the problem is reduced to acoaxial transmission line with square outerwall and periodic circular diaphragm discon-tinuities. Both variational approaches as wellas the integral equation method can beapplied to the calculation of the character-istic impedance, the junction impedance, andthe effective propagation constant. The tran-sient response is then obtained by per-forming the fast Fourier transform on thefrequency domain response or by directLaplace inversion when the dispersion ofjunction capacitances is negligible. Ournumerical results based on the dimensions ofan actual package show that the degree ofreflection and signal distortion becomesintolerable when the input rise time is shorterthan 500 ps.

Another problem of practical interest inmicroelectronic packaging is the study ofpropagation characteristics of a shieldedmicrostrip line in the presence of crossingstrips. If the microstrip line and the crossingstrips are placed at two different interfaces ofa three-layer medium bounded by two par-allel conducting plates, the crossing strips areassumed to be periodic and the three layersto be uniaxially anisotropic.

We apply the network analytical method ofelectromagnetic fields to the hybrid-modeanalysis of the frequency dependent charac-teristics of the structure. The transverse fieldsin each region are expressed by their Fouriertransform and a Floquet harmonic represen-tation. Using Maxwell's equations andapplying boundary conditions, we obtain a

set ofcurrentand the

coupled integral equations for thedistributions on the microstrip linecrossing metallic strips.

The determinable equation for the dispersionrelation can be derived by applying Galerkin'sprocedure to the derived set of coupled inte-gral equations. The Brillouin diagrams areobtained numerically by seeking the roots ofthe obtained eigenvalue equation. The stop-band properties are numerically presented asa function of the spacing, length, and widthof the crossing metallic strips. The effects ofthe material and geometrical parameters arealso investigated.

The dyadic Green's function approach isused to formulate the integral equations forthe signal line and crossing strip currents.Galerkin's procedure is applied to obtain thedeterminantal equation for the dispersionrelation. The effects of substrate material onthe propagation characteristics are beinginvestigated numerically. We find that weakanisotropy in the second layer does not affectthe position and the width of the stopbandmuch. By the proper choice of substratematerial, one can minimize the width of thefirst stopband. At high frequencies, weencounter higher order stopbands which aredue to the interaction with higher ordermodes of the signal line.

The transient fields of a current source on alayered medium are calculated by using thedouble deformation technique, in whichcomplex integrals are deformed in the trans-verse wavenumber and frequency planes.Singularities from these complex planes cor-respond to physical modes of the structure,such as guided and leaky waves, and the rel-ative importance of each to the overallresponse can be discovered. Unlike theCagniard-de Hoop method, double deforma-tion can be applied to dispersive anddissipative media. Also, the causality of theelectromagnetic signal can be shown analyt-ically.

A modification to the double deformationtechnique is developed, which consists ofsplitting the Fourier transform of the sourcecurrent into two halves, one for times beforethe arrival at the observation point, and oneafter. This modification greatly increases therange of sources to which the double defor-



mation technique can be applied. Anotheradvantage of the modification is the indi-vidual causality and continuity of each mode.

Results have been computed both for lineand strip currents on the surface of a coatedperfect conductor, for cases where thedielectric coating is both lossless anddissipative. In most cases, only a smallnumber of modes suffices to reproduce theimportant features of the response, includingthe arrivals of reflected and lateral rays. Theimportance of each type of arrival dependson certain features of the time function,especially the initial slope. The response dueto a strip current resembles that of a linecurrent, with some smoothing of the sharperfeatures.

A rigorous dyadic Green's function formu-lation for the periodic structure is derived tostudy the dispersion properties of single andcoupled signal lines periodically loaded withcrossing strips. The passband and stopbandcharacteristics are investigated when crossingstrips are of finite or infinite length.

For crossing strips of finite length, thestopband properties are mainly affected bythe period, the length of crossing strips, andthe separation between the signal andcrossing strips. Also, at higher frequencies,higher order stopbands occur. For crossingstrips of infinite length, attenuation along thesignal line exists over the whole frequencyrange due to the power guided by the travel-ling wave along crossing strips.

The complex resonant frequencies of theopen structure of a microstrip antenna con-sisting of two circular microstrip disks in astacked configuration are rigorously calcu-lated as a function of the layered parametersand the ratio of the radii of the two disks. Byusing a dyadic Green's function formulationfor horizontally stratified media and thevector Hankel transform, the mixed boundaryvalue problem is reduced to a set of coupledvector integral equations. By employingGalerkin's method in the spectral domain, thecomplex resonant frequencies are calculatedand convergence of the results is demon-strated. It is shown that for each mode, thestacked circular microstrip structure has dualresonant frequencies which are associatedwith the two coupled constitutive resonators

of the structure and a function of the mutualcoupling between the two disk resonators.This mutual coupling is a function of thegeometrical configuration of the stackedstructure, the layered parameters, permit-tivities, permeabilities, and heights, and theratio of the radii of the two disks. The dualfrequency behavior of the stacked microstripstructure may be used to broaden the band-width or to provide for dual frequency use ofthe antenna.

Solutions to electromagnetic plane wavescattering by a coated perfectly conductingcylinder are being investigated. The scat-tering of cylindrical structures serves as amodel for the study of radio wave diffractionover the earth and radar cross section predic-tion. To obtain a solution that convergesquickly in the high frequency limit, theWatson Transformation is used to converteigenfunction series solution into a contourintegral. This is done by mathematicallyexpressing a series summed at integer valuesas an integral with its contour path tightlyencircling the real axis. The contour integralcan be easily evaluated by deforming theintegration path to encompass the upper halfplane. With the new contour, the integral iscalculated by picking up the residues in theupper half plane. The resulting residueseries, representing the diffraction field, isquickly convergent at high frequencies. Inthe shadow region, because no incident orreflected field exists, all radiation in theshadow region can be calculated by theresidue series solution. In the context of rayoptics, all scattering due to diffraction origi-nates from rays that "creep" around the air-coating interface of the cylinder. Each termof the residue series represents an excitedcreeping wave mode. The expression foreach term of the residue series yields a modeamplitude and a propagation constant for thecorresponding mode. The total diffractedfield is the sum of the radiation due to all ofthe creeping wave modes.

We also calculate the radar cross-section of atwo-dimensional slot in an infinite groundplane with an incident TM wave. In formu-lating the integral equation, the problem isseparated into two regions. The regionbelow the ground plane is treated as a par-allel plate waveguide. In the region abovethe ground plane, the slot aperture is

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replaced with a magnetic surface currentsheet, and the ground plane is removed byadding the necessary image sources. In uti-lizing the method of moments, Galerkin'smethod is used, with the basis functionbeing a pulse function.

This two-dimensional slot can be used topredict the contribution to the total RCS ofsurface features in vehicles, such as doorgaps. This two-dimensional problem bestapproximates the three-dimensional problemwhen the slot width is small compared to thewavelength, the slot length is longer than afew wavelengths, and the incident planewave's magnetic field is parallel to the slot.In this case, the contribution from the endsof the slots is small, and the transverse elec-tric field does not contribute significantlybecause this component cannot propagateinto the slot. In addition to the case wherethe slot is empty and is short circuited, theslot can be filled with any material and canbe backed with some material other than aperfect conductor.

We have incorporated double reflection con-tribution in the computation of radar scat-tering from flat plate surfaces. An integralexpression is derived by applying the tangentplane approximation at each surface, andcorrectly accounting for the potentially near-field interactions between the two perfectlyconducting plates. The initial complexity ofthis expression necessitates numerical evalu-ation of the integrals and results in a largecomputational burden. The requirement ofnumerical integration is eliminated, however,by applying the method of Stationary Phaseand expanding quadratically the exact inter-plate phase term. The two surface inte-grations are replaced by line integrals usingStokes' Theorem, and for polygonal plates,the contour is subdivided into linear seg-ments on which the integration is performedanalytically. In the case where the con-ducting plates are loaded with one or morelayers of dielectric, the equivalent electricand magnetic surface currents are derived byusing the reflection coefficients at each plate.

The resulting field expressions are derivedwith sufficient generality to permit incorpo-ration in standard RCS prediction codesallowing arbitrary geometry and illumination.Comparisons with measurements of several

simple two-plate targets confirm the accu-racy of the predicted field and demonstrate asignificant improvement over existing algo-rithms that neglect multiple reflections orassume a far field interaction between thesurfaces.

The pseudo-differential operator approach isbeing employed to derive absorbingboundary conditions for both circular andelliptic outer boundaries. The pseudo-differential operator approach employed byEngquist and Majda is modified. This mod-ification results in better absorbing boundaryconditions. In the case of circular outerboundary, the modified pseudo-differentialoperator approach leads to a condition equi-valent to that of Bayliss and Turkel's secondorder operator. In the case of elliptic outerboundary, again, this modified pseudo-differential operator approach also yields asecond order operator. A close examinationof this operator finds that width informationof the scatterers enters into the boundaryoperator. Numerical results demonstrate theeffectiveness of these operators.

Finite difference time domain (FDTD) tech-niques show great promise in their ability tosolve three-dimensional problems with arbi-trary geometry. One advantage of thismethod is the ability to model spatially ortemporally varying media. Advantages aredue to the complete discretization of bothspace and time. Considering the volume ofinformation being calculated, these tech-niques are very efficient and are well suitedto calculation on future parallel processingcomputers.

Our work in this area includes developmentof the algorithms into a general purposecomputer code that may be used to solve forthe transient response of electromagneticproblems with an arbitrary geometry. Inaddition to the transient response, frequencydomain parameters may be obtained byFourier transform of the time domain results.Since the fields are calculated throughoutspace and time all other desired parametersmay be calculated from the field quantities.

A new FDTD grid model is also used to solvemicrostrip problems in aniso-tropic mediahaving tilted optical axes expressed bypermittivity or permeability tensor with off-



diagonal elements. This grid model is indeeda superposition of two conventional gridswith some displacement that depends on theoptical axes of anisotropy. Implementationsof different boundary conditions are dis-cussed. With this model, the frequency-dependent characteristics of microstrip linesare investigated. The microstrips are assumedto be placed on top of anisotropic substrateswith tilted optical axes. The case withsuperstrates is also investigated.

In the finite difference computation, theopen-end termination is simulated by usingthe open-circuit, short-circuit technique. Thesource plane is implemented by using a mag-netic wall with a Gaussian pulse excited onthe surface under the strip. Because of thesymmetry of the problem, the region underconsideration can be reduced by half with amagnetic-wall at the center plane.

The fields at different positions are first cal-culated. Then the Fourier transform is takento give the field spectra from which thevoltage and current can also be obtained. Byusing these data, the effective permittivityand the characteristic impedance can bedetermined.

A finite element time domain technique fortwo-dimensional time domain scattering ofelectromagnetic waves is studied. Thecontrol region approximation, which calls forDelaunay and Dirichlet tessellation, has beensuccessfully applied to the frequency domainproblems in the past. Two double integralterms are obtained by integrating theHelmholtz equation about the Delaunaytessellation. The term involving the Laplaceoperator can be converted to a closed loopintegral of normal derivatives, which caneasily be approximated in finite differencemanner by utilizing the orthogonal propertyof Delaunay and Dirichlet tessellation. Theremaining term can be approximated bymultiplying the field at the node with thearea. In the time domain problem, the sameapproximation is applied to the waveequation, except the term involving timederivatives is used in time marching scheme.Alternatively, as in Yee's algorithm, the firstorder Maxwell's equations are solved by spa-tially and temporally separating the electricand magnetic fields. In the case of electricpolarization, the electric fields are placed at

the nodes and the magnetic fields are placedat the centers of triangular edges. The curl Hequation is integrated by applying Stoke'stheorem to convert it to a closed loop inte-gral of tangential magnetic fields. Thisequation can be used to advance electricfields in time. To update the magnetic fields,the second curl equation is used. Thisequation is approximated in the finite differ-ence manner by utilizing the orthogonalityproperty of the tessellation. The orthogonalproperty of the tessellation and finite differ-ence approximations are utilized which doesnot require integration. The equations forthe magnetic polarization case can be derivedfollowing a similar procedure.

To limit the computation domain, thescatterers are enclosed with artificial outerboundaries. Continuous smooth outerboundaries, such as circles and ellipses, arechosen. The second-order time domainabsorbing boundary conditions derived fromthe pseudo-differential operator approach isimposed at the outer boundaries. Theseboundary conditions are implemented withthe control region approximation to deter-mine necessary field quantities at theboundary. The time domain control regionapproach has been successfully applied tosimple geometries, such as conducting andcoated cylinders.

Publications

Ali, S.M., T.M. Habashy, and J.A. Kong,"Resonance in Cylindrical-Rectangularand Wraparound Microstrip Resonators,"paper presented at the IEEE AP-S Interna-tional Symposium and URSI RadioScience Meeting, Syracuse, New York,June 6-10, 1988.

Ali, S.M., C.W. Lam, and J.A. Kong,"Dispersion Characteristics of ShieldedMicrostrip Lines Crossed by PeriodicMetallic Strips in Multilayered AnisotropicMedia," paper presented at the IEEE AP-SInternational Symposium and URSI RadioScience Meeting, Syracuse, New York,June 6-10, 1988.

Ali, S.M., T.M. Habashy, and J.A. Kong,"Input Impedance of a Circular MicrostripAntenna Fed by an Eccentric Probe,"

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paper to be presented at the Progress inElectromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

Atkins, R.G., and R.T. Shin, "Effect of Mul-tiple Scattering on the Radar Cross-Section of Polygonal Plate Structures,"paper presented at the IEEE AP-S Interna-tional Symposium and URSI RadioScience Meeting, Syracuse, New York,June 6-10, 1988.

Chang, I.Y., R.T. Shin, and J.A. Kong, "Elec-tromagnetic Plane Wave Scattering by aCoated Perfectly Conducting Cylinder,"paper to be presented at the Progress inElectromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

Gu, Q., Y.E. Yang, and J.A. Kong, "TransientAnalysis of Frequency-Dependent Trans-mission Line Systems Terminated withNonlinear Loads," submitted to J.Electromag. Waves Applic.

Gu, Q., Y.E. Yang, and J.A. Kong, "TransientAnalysis of Frequency-Dependent Trans-mission Lines with Nonlinear Termi-nations," paper to be presented at theProgress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Habashy, T.M., S.M. Ali, and J.A. Kong,"Impedance Parameters and RadiationPattern of Cylindrical-Rectangular andWraparound Microstrip Antennas," sub-mitted to IEEE Trans. Antennas Propag.

Kiang, J.F., S.M. Ali, and J.A. Kong, "Propa-gation Properties of Strip Lines Period-ically Loaded with Crossing Strips,"accepted for publication in IEEE Trans.Microwave Theory Tech.

Lam C.W., S.M. Ali, and J.A. Kong, "A NewFinite-Difference Time-Domain GridModel for Microstrip Problems in Aniso-tropic Media," paper to be presented atthe Progress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Lee, C.F., R.T. Shin, J.A. Kong, and B.J.McCartin, "Absorbing Boundary Condi-

tions on Circular and Elliptic Boundaries,"submitted to J. Electromag. Waves Applic.

Lee, C.F., R.T. Shin, J.A. Kong, and B.J.McCartin, "Finite Element Time DomainTechniques for Two-Dimensional IrregularTriangular Grids," paper to be presented atthe Progress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Li, K., R.T. Shin, and J.A. Kong, "RadarCross Section Prediction of Slots inGround Planes," paper to be presented atthe Progress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Poh, S.Y., M.J. Tsuk, and J.A. Kong, "Tran-sient Response of Sources over LayeredMedia Using the Double DeformationMethod," paper to be presented at theIEEE AP-S International Symposium andURSI Radio Science Meeting, San Jose,California, June 26-30, 1989.

Sheen D.M., S.M. Ali, M.D. Abouzahra, andJ.A. Kong, "Analysis of Multiport Rectan-gular Microstrip Structures Using a Three-Dimensional Finite Difference TimeDomain Technique," paper to be pre-sented at the Progress in ElectromagneticsResearch Symposium, Boston, Massachu-setts, July 25-26, 1989.

Tsuk M.J., and J.A. Kong, "The Frequency-Dependent Resistance of Conductors withArbitrary Cross-Sections," paper to bepresented at the Progress in Electromag-netics Research Symposium, Boston,Massachusetts, July 25-26, 1989.

Tsuk M.J., and J.A. Kong, "Response ofLayered Media to Current Sources withArbitrary Time Behavior," paper presentedat the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Tulintseff, A.N., S.M. Ali, and J.A. Kong,"Resonant Frequencies of Stacked Cir-cular Microstrip Antennas," submitted toIEEE Trans. Antennas Propag.



Tulintseff, A.N., S.M. Ali, and J.A. Kong,"Resonant Frequencies of Stacked Cir-cular Microstrip Antennas," paper to bepresented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, San Jose, California, June26-30, 1989.

Tulintseff, A.N., S.M. Ali, and J.A. Kong,"Input Impedance and Radiation Fields ofa Probe-Fed Stacked Circular MicrostripAntenna," paper to be presented at theProgress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Xia, J., S.M. Ali, and J.A. Kong, "Analysis ofMicrostrip Discontinuities on AnisotropicSubstrates," paper to be presented at theProgress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Yang, Y.E., Q. Gu, and J.A. Kong, "TransientElectromagnetic Wave Propagation onVias of Multilayer Integrated Circuit Pack-ages," paper presented at the IEEE AP-SInternational Symposium and URSI RadioScience Meeting, Syracuse, New York,June 6-10, 1988.

1.2 Remote Sensing of EarthTerrain

SponsorsNational Aeronautics and Space

Administration (Contract NAG 5-270)

Project Staff

Professor Jin Au Kong, David V. Arnold,Robert G. Atkins, Nelson C. Chu, Harold H.Lim, Freeman C. Lin, Son V. Nghiem, Dr.Robert T. Shin, Albert A. Swartz, Herng A.Yueh

We have developed the layered randommedium model for both active and passiveremote sensing of earth terrain. For activeremote sensing, the model is applied to con-vention and polarimetric radar scatteringcoefficients. For passive remote sensing, themodel is used to calculate emissivities andbrightness temperatures. Effects of rough

surfaces are studied with the ExtendedBoundary Condition and Small PerturbationMethod. EM bias due to rough oceansurface is studied with a two-scale modeland the physical optics approximation. Fur-thermore, intervening effects such as phasefluctuations due to forest foliage, Faradaypolarization fluctuations, and wave nonlinearinteraction in the ionosphere are investigated.

We derived a mathematically rigorous andfully polarimetric radar clutter model used toevaluate the radar backscatter from varioustypes of terrain clutter such as forested areas,vegetation canopies, snow covered terrains,or ice fields. With this model, we can calcu-late the radar backscattering coefficients (uo)for the multi-channel polarimetric radarreturns, in addition to the complex cross cor-relation coefficients between elements of thepolarimetric measurement vector. The com-plete polarization covariance matrix can becomputed and the scattering properties ofthe clutter environment characterized over abroad range of incident angles and frequen-cies.

The random medium model with three-layerconfiguration is developed to study fullypolarimetric scattering of electromagneticwaves from geophysical media. This modelcan account for the effects on wave scat-tering due to weather, diurnal and seasonalvariations, and atmospheric conditions suchas ice under snow, meadow under fog, andforest under mist. The top scattering layer ismodeled as an isotropic random mediumwhich is characterized by a scalar permit-tivity. The middle scattering layer is modeledas an anisotropic random medium with asymmetric permittivity tensor with an opticaxis that can be tilted due to the preferredalignment of the embedded scatterers. Thebottom layer is considered as a homoge-neous half-space. Volume scattering effectsof both random media are described bythree-dimensional correlation functions withvariances and correlation lengths corre-sponding to the strengths of the permittivityfluctuations and the physical sizes of theinhomogeneities, respectively. The strongfluctuation theory is used to derive the meanfields in the random media under the bilocalapproximation with singularities of thedyadic Green's functions properly taken intoaccount and effective permittivities of the

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random media are calculated with two-phasemixing formulas. The distorted Born approxi-mation is then applied to obtain thecovariance matrix that describes the fullypolarimetric scattering properties of theremotely sensed media.

The three-layer configuration is first reducedto two-layers to observe fully polarimetricscattering directly from geophysical mediasuch as snow, ice, and vegetation. Suchmedia exhibit reciprocity as experimentallymanifested in the close proximity of themeasured backscattering radar cross-sectionsv,h and ahv and theoretically established in

the random medium model with symmetricpermittivity tensors. The theory is used toinvestigate the signatures of isotropic andanisotropic random media on the complexcorrelation coefficient p between Uhh and ovvas a function of incident angle. For theisotropic random medium, p has the value ofapproximately 1.0. For the untilted aniso-tropic random medium, p has complex valueswith both the real and imaginary partsdecreased as the incident angle is increased.The correlation coefficient p is shown tocontain information about the tilt of the opticaxis in the anisotropic random medium. Asthe tilted angle becomes larger, the magni-tude of p is maximized at a larger incidentangle where the phase of p changes its sign.

The effects on polarimetric wave scatteringdue to the top layer are identified by com-paring the three-layer results with thoseobtained from the two-layer configuration.The theory is used to investigate the effectson polarimetric radar returns due to a low-loss and a lossy dry-snow layers covering asheet of thick first-year sea ice. For the low-loss snow cover, both Uhh and vv areenhanced compared to those observed frombare sea ice. Furthermore, the boundaryeffect is manifested in the form of the oscil-lation on Chh and a,. The oscillation can alsobe seen on the real and imaginary parts ofthe correlation coefficient p. The magnitudeof p, however, does not exhibit the oscil-lation while clearly retaining the same char-acteristics as observed directly from theuncovered sea ice. In contrast to the low-loss case, the lossy top layer can diminishboth hh and av and depress the boundary-effect oscillation. When the thickness of thelossy top layer increases, the behavior of the

correlation coefficient p becomes more andmore similar to the isotropic case signifyingthat the information from the lower aniso-tropic layer is masked. At appropriate fre-quency, the fully polarimetric volumescattering effects can reveal the informationattributed to the lower layer even if it iscovered under another scattering layer. Dueto the physical base, the random mediummodel renders the polarimetric scatteringinformation useful in the identification, clas-sification, and radar image simulation ofgeophysical media.

We have derived the dyadic Green's functionfor a two-layer anisotropic medium. TheBorn approximation is used to calculate thescattered fields. With a specified correlationfunction for the randomness of the dielectricconstant, the backscattering cross-sectionsare evaluated. The analytic expressions forbackscattering coefficients are shown toinclude depolarization effects in the single-scattering approximation. It is also shownthat the backscattering cross-section (perunit area) of horizontal polarization can begreater than that of vertical polarization evenin the case of half-space. The bistatic scat-tering coefficients are first calculated andthen integrated over the upper hemisphere tobe subtracted from unity to obtain theemissivity. The principle of reciprocity is theninvoked to calculate the brightness temper-atures. It is shown that both the absorptiveand randomly fluctuating properties of theanisotropic medium affect the behavior of theresulting brightness temperatures both intheory and in actual controlled field measure-ments. The active and passive results arefavorably matched with the experimental dataobtained from the first-year and the multiyearsea ice.

Electromagnetic wave propagation and scat-tering in an anisotropic random medium isbeing examined with Dyson equation for themean field which is solved by bilocal andnonlinear approximations and with Bethe-Salpeter equation for the correlation of fieldwas derived and solved by ladder approxi-mation. The effective propagation constantsare calculated for the four characteristicwaves associated with the coherent vectorfields propagating in an anisotropic randommedium layer, which are the ordinary andextraordinary waves with upward and down-



ward propagating vectors. The longitudinal-component of the effective propagationconstant of the upward propagating wave isdifferent from the negative of that of thedownward propagating wave, not only forthe extraordinary wave but also for the ordi-nary wave. This is due to the tilting of theoptic axis which destroys the azimuthal sym-metry.

To describe the effect of the random mediumon electromagnetic waves, the strong permit-tivity fluctuation theory, which accounts forthe losses due to both of the absorption andthe scattering, is used to compute the effec-tive permittivity of the medium. For a mixtureof two components, only the frequency, thecorrelation lengths, the fractional volume,and the permittivities of the two constituentsare needed to obtain the polarimetric back-scattering coefficients. Theoretical pred-ictions are illustrated by comparing theresults with experimental data for vegetationfields and sea ice.

The correlation function plays an importantrole in relating the electrical response ofgeophysical media to its physical properties.Volume scattering effects of electromagneticwaves from geophysical media such as vege-tation canopies and snow-ice fields havebeen studied by using the random mediummodels. The correlation functions used inthe random medium model have beenextracted from digitized photographs ofcross-sectional samples for snow and lake iceand artificially grown saline ice. It wasshown that the extracted correlation lengthscorresponded to the physical sizes of icegrains, air bubbles, and brine inclusions.Also the functional forms of the extractedcorrelation functions were shown to bedependent on the shape and orientation ofembedded inhomogeneities.

Extending Debye's analytical work, we havederived the correlation function and the cor-relation length for isotropic random mediumwith spherical inclusions. The correlationfunction study is also generalized to ran-domly distributed prolate spheroids with pre-ferred alignment in the vertical direction forthe anisotropic random medium. A scalingscheme is employed to transform the surfaceequation of prolate spheroids to that ofspheres so that the same approach in the

isotropic case can be utilized to derive thecorrelation function. Most of geophysicalmedia are complex materials such as wetsnow which is a mixture of air, ice grains,and water content and multi-year sea ice,which consists of pure ice, air bubbles, andbrine inclusions. Therefore, the correlationfunction study for three-phase mixtures isalso established. Two different kinds ofinclusions with spherical and spheroidalshapes are considered. It is found that thereis a close relationship between the form ofthe correlation function and the distribution,geometrical shape, and orientation of thescatterers. Also, the calculated correlationlengths are related to the fractional volumesand total common surface areas. Theseresults can be utilized to identify the featuresignature and characteristics through itsmicroscopic structure. For instance, dry orslush snow can be distinguished from grainsizes, water contents, and density via thecomparison of the variances and correlationlengths. The form of the correlation functionprovides the information about the physicalshape and alignment of brine inclusions inaddition to the concentration of brine inclu-sions versus air bubbles for tracing sea-icesignatures such as thick first-year sea ice andmulti year sea ice.

The scattering of electromagnetic waves froma randomly perturbed periodic terrain surfaceis formulated by the Extended BoundaryCondition (EBC) method and solved by thesmall perturbation method (SPM). The scat-tering from periodic surface is solved exactlyand this solution is used in the SPM to solvefor the surface currents and scattered fieldsup to the second order. The random pertur-bation is modeled as a Gaussian randomprocess. The theoretical results are illustratedby calculating the bistatic and backscatteringcoefficients. It is shown that as the corre-lation length of the random roughnessincreases, the bistatic scattering pattern ofthe scattered fields show several beams asso-ciated with each Bragg diffraction directionof the periodic surface. When the correlationlength becomes smaller, then the shape ofthe beams become broader. Kirchhoffapproximation results show a good agree-ment with EBC/SPM method results for thehh and vv polarized backscattering coeffi-cients for small angles of incidence.However, the results obtained from the

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EBC/SPM method give significant depolar-ized returns when the incident direction isnot perpendicular to the row direction of theperiodic surface which the Kirchhoff approxi-mation is unable to predict.

The EM bias in altimetry can be solved moreaccurately by modeling the ocean surfacewith a two-scale model and by using a phys-ical optics technique. For normal incidence,a Gaussian height probability density, and forsmall slopes of the large-scale ocean waves,the physical optics integral can be solved forthe backscattered power by using the saddlepoint method. The ocean can be dividedinto large- and small-scales by choosing aseparation wavenumber. The small-scaleheight variance can be measured from exper-imental data and the small-scale slope andcurvature variances can be estimated byusing a suitable wavenumber spectrum.Then, considering the wind speed and small-scale height variance, the EM Bias can beestimated by calculating the centroid of thebackscattered power. The EM bias has beencalculated by using this method and com-pared with experimental data of wind speedand wave height statistics.

We have calculated the phase degradationthat an electromagnetic wave undergoeswhen it propagates through a forest, scattersoff a target, and then propagates backthrough the forest to the radar (two-wayproblem). The problem is configured with atransmitter located above the forest and areceiver located within it. The forest ismodeled as a two-layer random medium(two boundaries separating three regions),with the first layer representing free space,the second layer representing the forest, andthe third layer representing the ground.Scattering is considered to be mainly fromthe leaves, a good approximation for fre-quencies of 1 GHz and higher. Hence, theforest is modeled as a layer of identicalscatterers (representing the leaves) randomlydistributed in an air background. Thescatterers are characterized with correlationfunctions, which are statistical descriptionsof the physical size of the scatterers. Theeffective permittivity of the medium is calcu-lated by using strong fluctuation theory, thataccounts for the fact that the permittivity ofleaves that of air may be quite different(permittivity fluctuation variance not much

less than one). Defining the permittivity ofthe medium as the sum of the effectivepermittivity and an assumed fluctuatingpermittivity accounts for the scattering prop-erties of the medium.

Phase fluctuations are calculated for the caseof a plane wave normally incident upon theforest. The wave undergoes scattering as itpropagates through the medium. The totalelectric field in the medium is thus approxi-mated by an unperturbed incident field and afirst-order scattered field. Maxwell'sequations are used with this total field toobtain the wave equation. The fluctuatingpart of the resulting expression, which comesfrom the assumed fluctuating permittivity, isisolated and treated as a source term. Thenormalized first-order scattered field, con-taining information about the phase fluctu-ations, is then calculated by using a Green'sfunction for an unbounded medium (therebyneglecting boundary effects). The varianceof the scattered field is computed, givinginformation about the magnitude of thephase fluctuations.

The physical characteristics of correlationlength and scatterer permittivity must beknown to properly model the medium. Thesecharacteristics are determined by matchingexperimental data on attenuation throughforests with results from a code that calcu-lates attenuation by using the strong permit-tivity fluctuation theory. The numericalresults of the phase fluctuation calculationswere compared to those of a model using theparaxial approximation. Both models con-verge to the same result in the paraxial limit.

As radio waves propagate through theionosphere, wave scattering can occur as aresult of ionospheric density irregularitieswhich give rise to fluctuations in Faradayrotation angles, known as Faraday Polariza-tion Fluctuations (FPF). FPF have beenobserved with low-orbit satellite beaconsignals transmitted at frequencies 20, 40, and54 MHz, and also with geostationary satellitesignals at 136 MHz. It has been shownexperimentally that when the linearlypolarized waves, decomposed into two char-acteristic wave modes, were measured sepa-rately after transionospheric propagation,there was a loss of correlation between thetwo characteristic wave modes. Diffractive



scattering of radio waves by ionosphericdensity irregularities is responsible for thisphenomenon. The density irregularities wereconsidered to be isotropic and modeled bycorrelation functions having the same corre-lation length in all directions. Ionosphericplasma, however, is magnetized by thegeomagnetic field and ionospheric densityirregularities tend to elongate along the mag-netic field. The elongation results in the for-mation of field-aligned rod-like irregularities.Sheet-like irregularities were also predictedtheoretically and measured recently. There-fore, the geomagnetized ionosphere withdensity irregularities is modeled as agyrotropic random medium and the effects ofboth rod-like and sheet-like random densityirregularities in causing FPF of VHF radiosignals are studied by means of three-dimensional correlation functions.

The model is used to explain the intense FPFobserved in polarimetric records of 136-MHzsatellite signals received at Ascension Islandin 1980 and 1981. The VHF signals weretransmitted from the geostationary satelliteSIRIO and propagated through theionosphere near the Appleton equatorialanomaly crest where the ambient plasmadensity was high especially during the 1980solar-maximum period. For rod-like irreg-ularities, the theoretical results predict thefield-aligned enhancement of FPF. Theenhancement is shown to be stronger forlonger rod-like irregularities. Furthermore,the results also demonstrate an inverserelation between the strength of FPF and thewave frequency. For sheet-like irregularities,the results also exhibit the field-alignedenhancement of FPF and the decreasing FPFstrength with increasing propagation angle.The difference, however, is that the FPFstrength due to the sheet-like irregularitieshave slower decreasing rates at small propa-gation angle and have larger values at largepropagation angle than the FPF due to therod-like irregularities. For VHF waves, theRMS FPF due to the rod-like and the sheet-like irregularities are quite distinctive. Thissuggests that the RMS FPF data with multi-frequencies and multi-propagation anglescan be used to infer the size and shape ofionospheric irregularities.

During the transionospheric propagation of ahigh power radio wave beam, the nonuni-

form electromagnetic field interacts with thebackground plasma and gives rise to aponderomotive force and a thermal pressureforce. Both of the two nonlinear forcesmainly act on the electrons, but eventuallywill have an effect on the ions through theambipolar diffusion process. The spatialredistribution of the plasma density causedby the actions of these forces will change thelocal plasma permittivity along the beam pathand consequently lead to the focusing of theradio wave beam.

We examine the self-focusing phenomena bytaking into account both the ponderomotiveforce and the thermal pressure force as thetwo primary mechanisms. The thresholdpower intensity is determined by balancingthe natural diffraction and the nonlinearfocusing effects of the wave beam. The focallength for the concerned process is then esti-mated by solving the nonlinear waveequation.

To illustrate the self-focusing process, wecarried out a series of numerical simulationfor high frequency beams with various initialbeam widths propagating in the ionosphericplasmas. The peak field intensity and theelectron temperature along the beam path arecalculated numerically. It is also shown that,the thermal pressure force is predominantover the ponderomotive force in large inci-dent beam width cases; however, theponderomotive force is more significant if thebeam width is small.

Publications

Arnold, D.W., J.A. Kong, W.K. Melville, andR.W. Stewart, "Theoretical Prediction ofEM Bias," paper to be presented at theProgress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Atkins, R.G., R.T. Shin, and J.A. Kong, "ANeural Net Classifier for High RangeResolution Target Signatures," paper to bepresented at the Progress in Electromag-netics Research Symposium, Boston,Massachusetts, July 25-26, 1989.

Chu, N.C., S.V. Nghiem, R.T. Shin, and J.A.Kong, "Phase Fluctuations of Waves

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Propagating Through a RandomMedium," paper to be presented at theProgress in Electromagnetics ResearchSymposium, Boston, Massachusetts, July25-26, 1989.

Han, H.C., and J.A. Kong, "Self-Focusing ofa Radio Wave Beam During the Trans-ionospheric Propagation," paper to bepresented at the International Union ofRadio Science Commission Meeting,Boulder, Colorado, January 4-7, 1989.

Han, H.C., and J.A. Kong, "Self-FocusingInduced by the Ponderomotive Effect andthe Thermal Effect," paper presented atthe IEEE AP-S International Symposiumand URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Han, H.C., J.A. Kong, and T.M. Habashy,"Far Field Pattern of a VLF Antenna Arrayin the Ionospheric Plasmas," paper to bepresented at the Progress in Electromag-netics Research Symposium, Boston,Massachusetts, July 25-26, 1989.

Lim, H.H., H.A. Yueh, J.A. Kong, R.T. Shin,and J.J. van Zyl, "Contrast and Classifica-tion Studies of Polarimetric SAR Imagesfor Remote Sensing of Earth Terrain,"paper to be presented at the Progress inElectromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

Lim, H.H., A.A. Swartz, H.A. Yueh, J.A.Kong, and J.J. van Zyl, "Classification ofEarth Terrain Using Polarimetric SyntheticAperture Radar Imagery," paper presentedat the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Lin, F.C., J.A. Kong, and R.T. Shin, "Applica-tion of Three-Layer Random MediumModel for Microwave Remote Sensing ofSnow-Covered Sea Ice," submitted to J.Geophys. Res.

Lin, F.C., H.A. Yueh, J.A. Kong, and R.T.Shin, "Correlation Function Study forRandom Media with MultiphaseMixtures," paper to be presented at theProgress in Electromagnetics Research

Symposium, Boston, Massachusetts, July25-26, 1989.

Nghiem, S.V., and J.A. Kong, "FaradayPolarization Fluctuations in Trans-ionospheric Polarimetric VHF Waves,"paper to be presented at the Progress inElectromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

Nghiem, S.V., F.C. Lin, J.A. Kong, R.T. Shin,and H.A. Yueh, "Three-Layer RandomMedium Model For Fully PolarimetricRemote Sensing of Geophysical Media,"paper to be presented at the Progress inElectromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

Swartz, A.A., L.M. Novak, R.T. Shin, H.A.Yueh, and J.A. Kong, "The OptimalPolarizations for Achieving MaximumContrast in Radar Polarimetry," paper pre-sented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, Syracuse, New York, June 6-10,1988.


Yueh, H.A., J.A. Kong, R.M. Barnes, and R.T.Shin, "Calibration of Polarimetric RadarsUsing In-Scene Reflectors," paper to bepresented at the Progress in Electromag-netics Research Symposium, Boston,Massachusetts, July 25-26, 1989.

Yueh, H.A., J.A. Kong, J.K. Jao, and R.T.Shin, "K-Distribution and PolarimetricTerrain Radar Clutter," paper to be pre-sented at the Progress in ElectromagneticsResearch Symposium, Boston, Massachu-setts, July 25-26, 1989.

Yueh, H.A., R.T. Shin, and J.A. Kong, "Scat-tering of Electromagnetic Waves from aPeriodic Surface with Random Rough-ness," J. Appl. Phys. 64:(4):1657-1670(1988).



Yueh, H.A., S.V. Nghiem, F.C. Lin, J.A.Kong, and R.T. Shin, "Three-LayerRandom Medium Model for PolarimetricRemote Sensing of Earth Terrain," paperpresented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, Syracuse, New York, June 6-10,1988.

Yueh, H.A., R.T. Shin, and J.A. Kong, "Scat-tering of Electromagnetic Waves from aPeriodic Surface with Random Rough-ness," SPIE Proceedings No. 927, Florida,April 6-8, 1988.

1.3 Remote Sensing of Sea Ice

SponsorU.S. Navy - Office of Naval Research

(Contract N00014-83-K-0258)

Project StaffProfessor Jin Au Kong, Maurice Borgeaud,Freeman C. Lin, Son V. Nghiem, Dr. RobertT. Shin

For active and passive microwave remotesensing, the correlation function used in therandom medium model provides a direct linkbetween electrical behaviors and physicalproperties of geophysical media. The dis-tribution, shape, size, and orientation ofembedded inhomogeneities, such as icegrains in snow and brine inclusions in seaice, can be characterized by the functionalform of the correlation function, the variance,and the correlation lengths. Based on theprobability theory, analytical expressions ofcorrelation functions for two-phase mixtureswith randomly distributed inclusions ofspherical and spheroidal shapes are derived.It is shown that the functional form of thecorrelation function is determined by theshape and orientation of inclusions whilecorrelation lengths are related to the frac-tional volume of the scatterers and the totalcommon surface area.

In active and passive microwave remotesensing of sea ice, a correlation function ofexponential form is extracted from the photo-graph of a horizontal thin section taken froma sample of artificially grown saline ice that

closely resembles Arctic congelation sea ice.It is found that the extracted correlationlengths are consistent with the publishedaverage size of brine pockets. With the appli-cation of strong fluctuation theory and thebilocal approximation, the effective permit-tivity tensor is derived in the low frequencylimit for an unbounded uniaxial randommedium with two-phase mixtures. By usingthe extracted correlation lengths, the effec-tive permittivity tensor is computed as afunction of fractional volume of brine inclu-sions and compared with in situ measure-ments at 4.8 and 9.5 GHz.

We have also derived a general mixingformula for discrete scatterers immersed in ahost medium. The inclusion particles areassumed to be ellipsoidal. The electric fieldinside the scatterers is determined by quasi-static analysis and assuming the diameter ofthe inclusion particles to be much smallerthan the wavelength. The results are appli-cable to general multiphase mixtures, and thescattering ellipsoids of the different phasescan have different sizes and arbitraryellipticity distribution and axis orientation,i.e., the mixture may be isotropic or aniso-tropic. The resulting mixing formula is non-linear and implicit for the effective complexdielectric constant because the approach incalculating the internal field of scatterers isself-consistent. Still, the form is especiallysuitable for iterative solution. The formulacontains a quantity called the apparentpermittivity. With different variations of thisquantity, the result leads to the generalizedLorentz - Lorenz formula, the generalizedPolder - van Santen formula, and the gener-alized coherent potential - quasi-crystallineapproximation formula. The results havebeen applied to calculating the complexeffective permittivity of snow and sea ice.

For active remote sensing, the volume scat-tering effects of snow-covered sea ice arestudied with a three-layer random mediummodel for microwave remote sensing. Thestrong fluctuation theory and the bilocalapproximation are applied to calculate theeffective permittivities for snow and sea ice.The wave scattering theory in conjunctionwith the distorted Born approximation is thenused to compute bistatic coefficients andbackscattering cross-sections. Theoreticalresults are illustrated by matching exper-

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imental data for dry snow-covered thick first-year sea ice at Point Barrow. The radarbackscattering cross-sections are seen toincrease with snow cover for snow-coveredsea ice, due to the increased scatteringeffects in the snow layer. The results derivedcan also be applied to passive remotesensing by calculating emissivity from thebistatic scattering coefficients.

For passive remote sensing, snow-coveredice field are studied with a three-layer model,an isotropic random medium layer overlyingan anisotropic random medium. We havecalculated the dyadic Green's functions ofthe three-layer medium and the scatteredelectromagnetic intensities with Bornapproximation. The backscattering cross-sec-tions are evaluated for active microwaveremote sensing. The theoretical approachcan be extended to derive the bistatic scat-tering coefficients. After integrating thebistatic scattering coefficients over the upperhemisphere and subtracting from unity, wecan also compute the radiometric brightnesstemperatures for passive microwave remotesensing by invoking the principle ofreciprocity.

Since both snow and ice exhibit volumescattering effects, we model the snow-covered ice fields with a three-layer randommedium model having an isotropic layer tosimulate snow, an anisotropic layer to simu-late ice, and a bottom one simulating groundor water. The snow and ice are characterizedby different dielectric constants and corre-lation functions. The theoretical results areillustrated for thick first-year sea ice coveredby dry snow at Point Barrow and for artificialthin first-year sea ice covered by wet snow atCRREL. The radar backscattering cross-sec-tions are seen to increase with snow coverfor snow-covered sea ice because snowgives more scattering than ice. The resultsare also used to interpret experimental dataobtained from field measurements.

We have used strong fluctuation theory toderive the backscattering cross-sections. Thestudy of the strong fluctuation theory for abounded layer of random discrete scatterersis further extended to include higher orderco-polarized and cross-polarized secondmoments. The backscattering cross-sectionsper unit area are calculated by including the

mutual coherence of the fields due to thecoincidental ray paths and that due to theopposite ray paths which are correspondingto the ladder and cross terms in the Feynmandiagrammatic representation. It is proved thatthe contributions from ladder and cross termsfor co-polarized backscattering cross-sec-tions are the same, while the contributionsfor the cross-polarized ones are of the sameorder. The bistatic scattering coefficients inthe second-order approximation for both theladder and cross terms are also obtained. Theenhancement in the backscattering directioncan be attributed to the contributions fromthe cross terms.

A two-layer anisotropic random mediummodel has been developed for active andpassive microwave remote sensing of icefields. First, dyadic Green's function for thistwo-layer anisotropic medium is derived.Then, with a specified correlation functionfor the randomness of the dielectric constant,the backscattering cross-sections are calcu-lated with the Born approximation. It isshown that the depolarization effects exist inthe single-scattering process. By treating seaice as a tilted uniaxial medium, the observedstrong cross-polarized return in the bistaticscattering coefficients is successfully pre-dicted from the theoretical model. It is alsoshown that the backscattering cross-sectionof horizontal polarization can be greater thanthat of vertical polarization even in the half-space case. The principle of reciprocity andthe principle of energy conservation areinvoked to calculate the brightness temper-atures. The bistatic scattering coefficients arefirst calculated and then integrated over theupper hemisphere to be subtracted fromunity, in order to obtain the emissivity for therandom medium layer. It is shown that boththe absorptive and randomly fluctuatingproperties of the anisotropic medium affectthe behavior of the resulting brightness tem-peratures both in theory and in actual con-trolled field measurements. The results fromactive and passive microwave remote sensingmatch well with the experimental dataobtained from the first-year and the multiyearsea ice with detailed ground-truth informa-tion.

We have derived the dyadic Green's functionfor a two-layer anisotropic medium. TheBorn approximation is used to calculate the



scattered fields. With a specified correlationfunction for the randomness of the dielectricconstant, the backscattering cross-sectionsare evaluated. The analytic expressions forbackscattering coefficients are shown toinclude depolarization effects in the single-scattering approximation. It is also shownthat the backscattering cross-section (perunit area) of horizontal polarization can begreater than that of vertical polarization evenin the case of half-space. Furthermore, boththe absorptive and randomly fluctuatingproperties of the anisotropic medium affectthe behavior of the resulting brightness tem-peratures both in theory and in actual con-trolled field measurements. The active andpassive results matched favorably with theexperimental data obtained from the first-yearand the multiyear sea ice.

Publications

Lin, F.C., J.A. Kong, R.T. Shin, A.J. Gow,and S.A. Arcone, "Correlation FunctionStudy for Sea Ice," J. Geophys. Res.93(C11):14055-14063 (1988).

Lin, F.C., J.A. Kong, and R.T. Shin, "Corre-lation Functions Study for Random Mediawith Two-Phase Mixtures," submitted toJ. Electromag. Waves Applic.

Lin, F.C., J.A. Kong, R.T. Shin, A.J. Gow,and D. Perovich, "Theoretical Model forSnow-Covered Sea Ice," paper presentedat the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Nghiem, S.V., F.C. Lin, J.A. Kong, R.T. Shin,and H.A. Yueh, "Three-Layer RandomMedium Model For Fully PolarimetricRemote Sensing Of Geophysical Media,"paper to be presented at the Progress inElectromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

Sihvola, A., and J.A. Kong, "Effective Permit-tivity of Dielectric Mixtures," IEEE Trans.Geosci. Remote Sensing, 26(4):420-429(1988).

1.4 SAR Image Interpretationand Simulation

Sponsor

National Aeronautics and SpaceAdministration (Contract NAG 5-769)

U.S. Army Corps of Engineers - WaterwaysExperimental Station(Contract DACA39-87-K-0022)

Simulation Technologies

Project Staff

Professor Jin Au Kong, Freeman C. Lin, Dr.Robert T. Shin, Albert A. Swartz, Herng A.Yueh

For SAR image interpretation, we use thelayered random medium model. Earth terrainmedia, such as vegetation, forest, snow, andice exhibit strong volume scattering effects.To study polarimetric radar backscatter ofearth terrains from the point of view of elec-tromagnetic wave theory, we use a layeredrandom medium model to characterize theterrain clutter. The random medium isdescribed by a background permittivity witha fluctuating component; the randomness ofthe fluctuation is characterized by a three-dimensional correlation function with a vari-ance and horizontal and vertical correlationlengths. The variance corresponds to thestrength of the fluctuation, while the corre-lation lengths coincide with the geometricalsize of the basic scattering elements. Thepolarimetric backscattering coefficients canbe obtained from the electromagnetic wavetheory by calculating the covariance matrix ofthe polarimetric measurement vector.

We developed a wave approach by applyingthe Born approximation. An integralequation is formulated for the electric field byusing the unperturbed Green's function for alayered random medium in the absence ofpermittivity fluctuations. The random fluctu-ations are treated as induced scatteringsources in the integral equation, which willthen be solved by iteration to obtain aNeumann series. The Born approximation iscarried out to the second order to include thedepolarization effects. Physically, the Bornfirst-order and second-order approximationsaccount, respectively, for a single and doublescattering process. The calculated

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covariance matrix for a layered isotropicrandom medium has four of its elementsequal to zero, indicating absence of corre-lation between the HV and HH terms and HVand VV terms. The theoretical results areshown to be consistent with measurementdata obtained from MIT Lincoln Laboratory.

We have studied the Mueller matrix andpolarization covariance matrix for polarimetricradar systems. The clutter is modelled by alayer of random permittivity, described by athree-dimensional correlation function, withvariance, and horizontal and vertical corre-lation lengths. This model is applied byusing wave theory with Born approximationscarried to the second order, to find the back-scattering elements of the polarimetricmatrices. It is found that eight out of sixteenelements of the Mueller matrix are identicallyzero, corresponding to a covariance matrixwith four zero elements. Theoretical pred-ictions are matched with experimental datafor fields of vegetation.

Among many non-Gaussian statistics, theK-distribution has proven to be useful incharacterizing the amplitude distribution ofelectromagnetic echoes from various objects,including diverse ground surfaces, seasurface, and wave propagation throughatmospheric turbulence. We have developeda multivariate K-distribution model for thestatistics of fully polarimetric data from earthterrain with polarizations HH, HV, VH, andVV. In this approach, correlated polarizationsof radar signals, as characterized by acovariance matrix, are treated as the sum ofN n-dimensional random vectors which obeythe negative binomial distribution law. Sub-sequently, an n-dimensional K-distribution,with either zero or non-zero mean, is derivedin the limit of infinite illuminated area. Theprobability density function (PDF) of theK-distributed vector normalized by itsEuclidean norm is independent of the param-eter a and is the same as that derived from azero-mean Gaussian-distributed randomvector. High-order normalized intensitymoments and cumulative density functions(CDF) of experimental data obtained fromMIT Lincoln Laboratory and the Jet Propul-sion Laboratory are compared with theore-tical values of the K-distribution. The abovemodel is well supported by experimental datain the form of polarimetric measurements.

The evaluation of clutter backscatter coeffi-cients for HH, HV and VV polarimetricreturns is useful in the design and analysis ofoptimal radar target detection, discriminationand classification schemes. Another impor-tant application is the generation of randomclutter returns for Monte-Carlo simulations.A systematic approach for the identificationof terrain media such as vegetative canopy,forest, and snow-covered fields is developedusing the optimum polarimetric classifier.The covariance matrices for the variousterrain cover are computed from theoreticalmodels of random medium by evaluating thefull polarimetric scattering matrix elements.The optimal classification scheme makes useof a quadratic distance measure and isapplied to classify a vegetative canopy con-sisting of both trees and grass. Exper-imentally measured data are used to validatethe classification scheme. Theoretical proba-bility of classification error using the fullpolarimetric matrix are compared with classi-fication based on single features includingthe phase difference between the VV and HHpolarization returns. It is shown that the fullpolarimetric results are optimal and providebetter classification performance than singlefeature measurements.

Supervised and unsupervised classificationprocedures are developed and applied tosynthetic aperture radar (SAR) polarimetricimages to identify its various earth terraincomponents. For the supervised classifica-tion processing, the Bayes technique is uti-lized to classify fully polarimetric andnormalized polarimetric SAR data. Simplerpolarimetric discriminates such as the unnor-malized and normalized magnitude responseof the individual receiver channel returns inaddition to the phase difference between thereceiver channels are also considered.Covariance matrices are computed for eachterrain class from selected portions within theimage where ground truth is available underthe assumption that the polarimetric data hasa multivariate Gaussian distribution. Thesematrices are used to train the optimalclassifier, which in turn is used to classify theentire image. In this case, classification isbased on determining the distances betweenthe training classes and the observed featurevector, then assigning the feature vector tobelong to that training class for which the



distance was minimum. Another processingalgorithm based on comparing general prop-erties of the Stokes parameters of the scat-tered wave to that of simple scatteringmodels is also discussed. This algorithm,which is an unsupervised technique, classi-fies terrain elements based on the relation-ship between the orientation angle andhandedness, or ellipticity of the transmittedand received polarization state. These classi-fication procedures have been applied to SanFrancisco Bay and Traverse City SAR images.It is shown that fully polarimetric classifica-tion yields the best overall performance.Also, in some selected areas where theobserved amplitudes of the returns are quitedifferent from those of the training data, clas-sification techniques which were not basedon the absolute amplitudes of the returns,e.g., the normalized polarimetric classifier,produced a more consistent result in relationto the ground truth data.

Polarimetric radar backscatter data have beenused extensively to classify terrain cover.Since it is difficult to calibrate out the effectsof amplitude and phase errors induced byatmospheric effects, path loss, etc., absoluteamplitude and phase of radar returns are notreliable features for terrain classification pur-poses. The use of normalized polarimetricdata is proposed so that only the relativemagnitudes and phases will be utilized todiscriminate different terrain elements. It isshown that the Bayes classification errordoes not depend on the form of the normal-ization function if the unknown radar systemcalibration factor is modeled as a multipli-cative term in the received signal. This holdstrue for arbitrary probabilistic distributedpolarizations. Assuming a multivariateGaussian distribution for the un-normalizedpolarimetric data, the probability densityfunction (PDF) of the normalized data andits corresponding Bayes classifier distancemeasure are derived. Furthermore, byassuming a specific form of the covariancematrix for the polarimetric data, exact PDFsare given for HH, HV, VV and span type nor-malization schemes. Corresponding classi-fication errors are evaluated to verify theirindependence from all normalization func-tions.

There is considerable interest in determiningthe optimal polarizations that maximize the

contrast between two scattering classes inpolarimetric radar images. We have devel-oped a systematic approach for obtaining theoptimal polarimetric matched filter, i.e., thefilter that produces maximum contrastbetween two scattering classes. Themaximization procedure involves solving aneigenvalue problem in which the eigenvectorcorresponding to the maximum contrast ratiois optimal polarimetric matched filter. Toexhibit its physical significance, this filter istransformed into its associated transmittingand receiving polarization states written interms of horizontal and vertical vector com-ponents. For the special case where thetransmitting polarization is fixed, thereceiving polarization which maximizes thecontrast ratio is also obtained. Polarimetricfiltering is then applied to synthetic apertureradar images. It is shown, both numericallyand through the use of radar imagery, thatmaximum image contrast can be realizedwhen data is processed with the optimalpolarimetric matched filter.

We have developed a data processing algo-rithm that produces maximum contrastbetween two scattering classes, each repres-ented by its respective covariance matrix.We will derive an optimal linear decisionvector or decision functional that maximizesthe contrast or expected power return ratiobetween the two scattering classes. Thesuboptimal case of a fixed transmitpolarization will also be considered. Themaximization procedure involves solving aneigenvalue problem in which the theeigenvector yielding this maxima will corre-spond to the decision functional we seek. Todemonstrate the physical significance of thelinear weighting decision vector, we trans-form the vector into its associated transmitand receive polarization state in terms of hor-izontal and vertical vector components. Thistechnique is then applied to radar imagery toenhance contrast between different classeswithin a given database.

A clutter model is used to simulate fullypolarimetric returns for a coherent, stepped-frequency radar. The objective is to createsite-dependent synthetic clutter signaturesthat can be utilized in a hardware-in-the-loop test system. The fully polarimetric,multi-frequency, and multi-incident angletwo-layer anisotropic random medium model

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is employed to compute the normalizedbackscatter coefficients of terrain clutter.Polarization covariance matrices are calcu-lated for each of N high resolution rangebins, at each of the M discrete frequenciesthat comprise the stepped-frequency band-width. The covariance matrices are decom-posed, multiplied by complex Gaussian noise,and weighted according to the randomproduct model to account for the spatial vari-ability of clutter. This generates the normal-ized electric field backscattered from each ofthe N range bins, at each of the M discretefrequencies. These fields are coherentlyadded, taking into account the effects ofterrain shadowing and overlay, to realize thesingle-frequency polarimetric return that aradar would measure from the specifiedterrain. The radar return for each of the otherdiscrete frequencies is computed in a similarmanner. In addition, the real-time imple-mentation of this algorithm is considered inthe context of synthetic wideband pro-cessing. The result of this procedure yieldsthe clutter's coherent phase-history profile.The clutter phase-history returns can then becoherently superimposed on the targetphase-history returns. The combined (orclutter only) returns can be reduced to obtaineither 1) the coherent high resolution rangeprofile (HRRP) or 2) the non-coherentautocorrelation range profile.

By using the random medium model, syn-thetic aperture radar (SAR) simulations cannow be generated based on ground truthdata from a given terrain site. We first matchthe various elements within the data to thephysical parameters previously discussed,which are compiled in a database generatedfrom correlation function studies. Based onthese physical terrain parameters, we nextuse the random medium model to predict theradar backscatter from the various terrain ele-ments in order to generate a range cross-range terrain profile. We have taken intoaccount the change of incident angle alongthe ground swath as well as the terrain localincident angle, when the terrain backscatteris computed. Finally the effects of fading(i.e., speckle) are incorporated into the simu-lated imagery. Utilizing this procedure, we

are able to simulate the radar measurementswhich would have been actually recordedhad this terrain been imaged by an airborneSAR. There simulations are fullypolarimetric. The advantage to terrain simu-lation using the random medium model isthat, in general, most airborne radars operateat either a single frequency, of over some rel-atively small bandwidth. However, therandom medium model allows us to generatesimulations of the same terrain for a varietyof operating frequencies. In light of this fact,we see that terrain simulation based on therandom medium model is an extremely usefultool.

Development of high resolution andautocorrelation range profile algorithms,which are a special case of the above men-tioned terrain simulation, is continuing. Theprimary modification altered the way inwhich the terrain was sectioned into highresolution range bins. Formerly these binswere evenly spaced on a flat terrain;however, spacing was not uniform whenterrain elevation information was added. Thenew methodology takes this effect intoaccount. Now the terrain scattering ele-ments, located in each range bin are evenlyseparated both in the case of a flat terrainand for regions in which ground elevationoccurs. Since most terrain regions are notuniformly flat, this new algorithm for parti-tioning the terrain represents an improvementover the previous model.

Polarimetric calibration algorithms are devel-oped to provide exact solutions for com-pletely general target choices in polarimetricradar calibration. The transmitting andreceiving ports of the polarimetric radar aremodeled by two unknown polarizationtransfer matrices. These two matrices arefound by using three in-scene reflectors withdifferent scattering matrices. We have ana-lyzed all possible combinations of calibrationtargets and obtained solutions to all cases.Thus, if three scatterers with known scat-tering matrices, not necessarily man-made,are known to exist within a radar image, thewhole image can be calibrated by using theobtained exact solutions.



Publications

Borgeaud, M., J.A. Kong, R.T. Shin, and S.V.Nghiem, "Theoretical Models for Polari-metric Microwave Remote Sensing ofEarth Terrain," Proceedings of the 1988NA TO Advanced Research Workshop.

Borgeaud, M., S.V. Nghiem, R.T. Shin, andJ.A. Kong, "Theoretical Models for Polari-metric Microwave Remote Sensing ofEarth Terrain," J. Electromag. WavesApplic.

Kong, J.A., F.C. Lin, M. Borgeaud, H.A.Yueh, A.A. Swartz, H.H. Lim, L.M. Novak,and R.T. Shin, "Polarimetric Clutter Mod-eling: Theory and Application," GACIACPR-88-03, paper presented at the Polari-metric Technology Workshop, Huntsville,Alabama, August 16-19, 1988.

Lim, H.H., A.A. Swartz, H.A. Yueh, J.A.Kong, and J.J. van Zyl, "Classification ofEarth Terrain Using Polarimetric SyntheticAperture Radar Imagery," paper presentedat the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Lim, H.H., A.A. Swartz, H.A. Yueh, J.A.Kong, R.T. Shin, and J.J. van Zyl, "Classi-fication of Earth Terrain Using SyntheticAperture Radar Images," submitted to J.Geophys. Res.

Lin, F.C., J.A. Kong, R.T. Shin, A.J. Gow,and D. Perovich, "Theoretical Model forSnow-Covered Sea Ice," paper presentedat the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Lin, F.C., J.A. Kong, and R.T. Shin, "Corre-lation Functions Study for Random Mediawith Two-Phase Mixtures," submitted toJ. Electromag. Waves Applic.

Nghiem, S.V., F.C. Lin, J.A. Kong, R.T. Shin,and H.A. Yueh, "Polarimetric RemoteSensing of Earth Terrain with Three-LayerRandom Medium Model," GACIACPR-88-03, paper presented at the Polari-metric Technology Workshop, Huntsville,Alabama, August 16-19, 1988.

Swartz, A.A., L.M. Novak, R.T. Shin, D.A.McPherson, A.V. Saylor, F.C. Lin, H.A.Yueh, and J.A. Kong, "Radar RangeProfile Simulation of Terrain Clutter Usingthe Random Medium Model," GACIACPR-88-03, paper presented at the Polari-metric Technology Workshop, Huntsville,Alabama, August 16-19, 1988.

Swartz, A.A., H.A. Yueh, J.A. Kong, L.M.Novak, and R.T. Shin, "The OptimalPolarizations for Achieving MaximumContrast in Radar Polarimetry," J.Geophys. Res. 93(B1 2):15252-15260(1988).


Yueh, H.A, J.A. Kong, J.K. Jao, R.T. Shin,and L.M. Novak, "K-Distribution andPolarimetric Terrain Radar Clutter," sub-mitted to J. Electromag. Waves Applic.

Yueh, H.A., J.A. Kong, R.M. Barnes, and R.T.Shin, "Calibration of Polarimetric RadarUsing In-scene Reflectors," submitted toJ. Electromag. Waves Applic.

Yueh, H.A., A.A. Swartz, J.A. Kong, R.T.Shin, and L.M. Novak, "Bayes Classifica-tion of Terrain Cover Using NormalizedPolarimetric Data," J. Geophys. Res.93(B12):15261-15267 (1988).

Yueh, H.A., S.V. Nghiem, F.C. Lin, J.A.Kong, and R.T. Shin, "Three-LayerRandom Medium Model for PolarimetricRemote Sensing of Earth Terrain," paperpresented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, Syracuse, New York, June 6-10,1988.

Yueh, H.A., J.A. Kong, J.K. Jao, and R.T.Shin, "K-Distribution and PolarimetricTerrain Radar Clutter," paper presented atthe IEEE AP-S International Symposium

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and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

1.5 Microwave and MillimeterWave Integrated Circuits

Sponsor

U.S. Air Force - Rome Air DevelopmentCenter(Contract F19628-88-K-0013)

U.S. Navy - Office of Naval Research(Contract N00014-89-J-1107)

Project Staff

Professor Jin Au Kong, Dr. Sami M. Ali, Dr.Tarek M. Habashy, Jean-Fu Kiang, Check-FuLee, Dr. Soon Y. Poh, Michael J. Tsuk, AnnN. Tulintseff, Ying-Ching E. Yang

The leakage phenomenon is important in thearea of millimeter-wave integrated circuitsand integrated optics. Theoretical analysesand experiments have been performed toinvestigate this phenomenon. The leakage isdue to the TE-TM coupling occurring at thegeometrical discontinuities, and the leakypower in the form of surface wave propa-gates in the background medium.

An integral equation formulation based ondyadic Green's function is derived to solvefor the dispersion relation of single andcoupled dielectric strip waveguides. We areable to predict the leakage properties of themfrom these dispersion relations. For singledielectric strip waveguide, it is observed thatthe leakage occurs when the effectiverefractive index is smaller than that of asurface wave mode in the backgroundmedium. It is also observed that if the lowestTE-like mode is leaky, the lowest TM-likemode is non-leaky. The reverse is also truewhen the lowest order mode leaks, thesurface wave mode of opposite polarizationis excited. When the higher order modeleaks, the surface wave modes of bothpolarizations can be excited.

For two symmetrical dielectric strip wave-guides, both the even and odd modes areinvestigated. For the leaky mode, the totalleakage is due to the leakage from each indi-vidual strip waveguide. At the separation

where the even mode has a maximumleakage, it implies that the surface wavemodes excited by each waveguide add inphase. For the odd mode at about the sameseparation, these surface wave modes addout of phase, hence a null in the leakage isobserved.

Conventional microstrip antennas consistingof a single perfectly conducting patch on agrounded dielectric slab, have received muchattention in recent years due to their manyadvantages including low profile and lightweight. To increase the bandwidth, anumber of microstrip patches can be stackedin multilayer configurations, introducingadditional resonances in the frequency rangeof interest and achieving wider bandwidths.We have developed a method of analyzingcoaxial probe-fed microstrip antenna con-sisting of two circular microstrip disks in astacked configuration.

A spectral domain dyadic Green's functionfor multilayered uniaxially anisotropic mediacontaining three-dimensional sources arederived. Tractable forms are shown to beeasily deduced from the physical picture ofthe waves radiated by the primary sourcesand the multiple reflections from the stratifiedmedium. The formulation decomposes thedyadic Green's function into TE and TMwaves. The dyadic Green's function in thesource region is properly represented byextracting the delta function singularity. Asimple procedure to obtain the fields in anyarbitrary layer is described. Recursionrelations for appropriately defined reflectionand upward and downward propagatingtransmission coefficients are presented.Forms suitable for transmission line applica-tions in multilayered media are derived.

By using the dyadic Green's function formu-lation, a rigorous analysis of the two stackedcircular microstrip disks in a layered mediumis performed. A set of coupled integralequations for the current distribution on thedisks is derived by using the vector Hankeltransform. This coupled set is then solvedusing Galerkin's method. The choice of thecurrent basis functions is based on the cur-rents of the magnetic wall cavity. Complexresonant frequencies are calculated as afunction of the layered substrate, permit-tivities and thicknesses, and the ratio of the



two disks radii. The resonant frequencies oftwo different stacked configurations arestudied as function of the coupling inter-action. Critical coupling between the tworesonators occurs at the point where the realpart of the resonance curves for the two iso-lated resonators intersect. The splitting of thecomplex resonance curves at this point is afunction of the strength of the couplingbetween the two resonators. The dual fre-quency or wide band operation is shown tobe achieved by changing the coupling coeffi-cient. The input impedance and radiationpatterns of the stacked microstrip antenna iscalculated as a function of the layeredsubstrate, permittivities and thicknesses, andthe ratio of the radii of the two disks.

We rigorously analyzed the radiation problemof a circular patch that is center fed by acoaxial-line driven probe over a ground planeand situated in an arbitrary layered medium.The current distribution on both the patchand the probe is rigorously formulated usinga planar stratified medium approach. A set ofthree coupled integral equations is derivedwhich governs the axial current distributionon the probe, the radial current distributionon the patch, and the azimuthal magneticcurrent sheet across the aperture of thedriving coaxial line. This set of equations isthen solved by using the method ofmoments. The resulting matrix equation isobtained in terms of Sommerfeld-type inte-grals that take into account the effect of thelayered medium. These integrals are effi-ciently computed by a simple deformation inthe complex wavenumber domain.

We also formulated the problem in terms of aWeber transform that allows the developmentof Green's function of the layered mediumwith the probe and the microstrip patch aspart of the medium. Using the Weber trans-form automatically enforces the boundaryconditions on the probe and patch. Thisallows one to cast the problem as the sol-ution of a set of two coupled integralequations governing the tangential compo-nent of the electric field across the apertureof the coaxial line feed and the interfacewhere the patch lies. This set is then solvedby using the method of moments. Thecurrent distribution on both the probe andthe patch is then computed from the compo-nent of the magnetic field tangential to their

surfaces. Furthermore, from the computedelectric field across the aperture of thecoaxial line feed, one obtains the reflectioncoefficient for the TEM mode which allowscomputation of the input impedance acrossthe terminals of the probe.

Cylindrical microstrip antennas find manyapplications in high speed aircrafts and spacevehicles because of their conformity with theaerodynamical structure of these vehicles.We address the realistic problem of radiationfrom a cylindrical microstrip antenna excitedby a probe. Both the cylindrical-rectangularand the wraparound elements are discussed.The current distribution on the patch is rigor-ously formulated using a cylindricallystratified medium approach. A set of vectorintegral equations that governs the currentdistribution on the patch are derived. Thisset of equations is then solved usingGalerkin's method in which the patch currentis expanded in terms of a complete set ofbasis functions that take into account theedge singularity condition. The inputimpedance together with the radiation havebeen computed both exactly and in the smallsubstrate thickness limit where a single modeapproximation is employed.

We discovered that for thick substrates,hybrid modes are excited. Only in the caseof axially symmetric modes are the TE modesdecoupled from the TM modes and modes ofdifferent parity do not couple. The presenceof the dielectric substrate widens the band-width and broadens the radiation pattern.The radiation pattern is insensitive to thesubstrate thickness, especially for highdielectrics. For wraparound antennae, allcurrent modes with no axial variation tend toweakly radiate and, consequently, havenarrow bandwidth. When the TE01 mode isexcited, the wraparound antenna works as agood antenna. The rectangular-cylindricalpatch is, on the other hand, generally, lessradiating than the wraparound.

A rigorous analysis of the resonant frequencyproblem of both the cylindrical-rectangularand the wraparound microstrip structures iscarried out using two different approaches:integral equation formulation and a perturba-tional approach. By using Galerkin's methodin solving the integral equations, the complexfrequencies are studied with sinusoidal basis

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functions. The effect of the edge singularityof the patch current on the convergence isinvestigated. Numerical results show that theHE10 mode for the wraparound patch and thecylindrical-rectangular patches have narrowbandwidth and, thus, are more appropriatefor resonator applications. The TE01 andHE01 modes of the wraparound andcylindrical-rectangular patches, respectively,have wide bandwidth and are efficient radi-ating modes.

The wave propagation along microstrip lineson uniaxially anisotropic substrates havingtheir width periodically modulated is investi-gated by dividing the structure into consec-utive uniform sections. A rigorousformulation for calculating the capacitancematrix of uniform lines is derived. Aquasi-TEM approximation is then used toobtain the circuit parameters for eachsection. The transfer matrix is applied toobtain the dispersion relation of the periodicstructures. The effects of the anisotropy ofsubstrate material and the strip line geometryon the stopband characteristics are exploited.The properties of two coupled width-modulated lines are also investigated.

A rigorous dyadic Green's function formu-lation in the spectral domain is used to studythe dispersion characteristics of signal striplines in the presence of metallic crossingstrips. A set of coupled vector integralequations for the current distribution on theconductors is derived. Galerkin's method isthen applied to derive the matrix eigenvalueequation for the propagation constant.

The effects of the structure dimensions onthe passband and stopband characteristicshave been studied. For crossing strips offinite length, the stopband is mainly affectedby the period, the crossing strip length, andthe separation between the signal and thecrossing strips. For crossing strips of infinitelength carrying travelling waves, attenuationalong the signal line exists over the wholefrequency range of operation.

The open end, gap, and step in width dis-continuities placed on anisotropic substratesare rigorously analyzed. Both uniaxial andtilted uniaxial anisotropy are considered. Thematerials are assumed to be lossless and themetal strips to be infinitely thin. A dyadic

Green's function for layered anisotropicmedia is used to formulate a set of vectorintegral equations for the current distribution.The fundamental hybrid mode is assumed tobe propagating on the input and output ofmicrostrip lines. In solving the set of vectorintegral equations, the method of moments isemployed. The basis functions for thecurrent on the metal strip consider the edgeeffect. Both longitudinal and transversal cur-rents are considered in the calculation. Thepropagation constant for the infinitely longuniform microstrip line is first calculated.Then the propagation constant of the funda-mental mode is used to formulate theexcitation of the discontinuity problem. Atthe discontinuity, local basis functions areused to simulate the local currents near thediscontinuity. The scattering matrix are thenobtained to facilitate the construction ofequivalent circuit models.

Publications

Ali, S.M., T.M. Habashy, and J.A. Kong,"Probe Excitation of a Center-Fed CircularMicrostrip Antenna Employing a StratifiedMedium Formulation," paper to be pre-sented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, San Jose, California, June26-30, 1989.


Ali, S.M., T.M. Habashy, and J.A. Kong,"Input Impedance of a Circular MicrostripAntenna Fed by an Eccentric Probe," tobe presented at the Progress in Electro-magnetics Research Symposium, Boston,Massachusetts, July 25-26, 1989.

Ali, S.M., T.M. Habashy, J.F. Kiang, and J.A.Kong, "Resonance in Cylindrical-Rectangular and Wraparound MicrostripStructures," submitted to IEEE Trans.Microwave Theory Tech.




Ali, S.M., T.A. Habashy, and J.A. Kong,"Dyadic Green's Functions for Multilay-ered Tilted Uniaxially Anisotropic Media,"submitted to IEEE Trans. MicrowaveTheory Tech.

Habashy, T.M, S.M. Ali, and J.A. Kong,"Probe Excitation of a Center-Fed CircularMicrostrip Antenna Employing the WeberTransform," paper to be presented at theIEEE AP-S International Symposium andURSI Radio Science Meeting, San Jose,California, June 26-30, 1989.

Habashy, T.M., S.M. Ali, and J.A. Kong,"Impedance Parameters and RadiationPattern of Cylindrical-Rectangular andWraparound Microstrip Antennas," paperpresented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, Syracuse, New York, June 6-10,1988.

Habashy, T.M., S.M. Ali, and J.A. Kong,"Impedance Parameters and RadiationPattern of Cylindrical-Rectangular andWraparound Microstrip Antennas," sub-mitted to IEEE Trans. Antennas Propag.

Kiang, J.F., S.M. Ali, and J.A. Kong, "Propa-gation Properties of Strip Lines Period-ically Loaded with Crossing Strips,"accepted for publication in IEEE Trans.Microwave Theory Tech.

Kiang, J.F., S.M. Ali, andysis of Dielectric StripIntegral Equation Formsented at the ProgressResearch Symposium,setts, July 25-26, 1989

Tsuk M.J.,LayeredArbitrary

J.A. Kong, "Anal-Waveguides Usingulation," to be pre-in ElectromagneticsBoston, Massachu-

and J.A. Kong, "Response ofMedia to Current Sources withTime Behavior," paper presented

at the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Tulintseff, A.N., S.M. Ali, and J.A. Kong,"Resonant Frequencies of Stacked Cir-cular Microstrip Antennas," submitted toIEEE Trans. Antennas Propag.

Tulintseff, A.N., S.M. Ali, and J.A. Kong,"Input Impedance and Radiation Fields ofa Probe-Fed Stacked Circular MicrostripAntenna," to be presented at the Progressin Electromagnetics Research Symposium,Boston, Massachusetts, July 25-26, 1989.

1.6 High-Speed IntegratedCircuit Interconnects

Sponsor

Digital Equipment Corporation

Project Staff

Professor Jin Au Kong, Dr. Sami M. Ali, Dr.Tarek M. Habashy, Jean-Fu Kiang, Check-FuLee, Dr. Soon Y. Poh, Michael J. Tsuk, AnnN. Tulintseff, Ying-Ching E. Yang

A new method for analyzing frequency-dependent transmission line systems withnonlinear terminations is developed by usingthe generalized scattering matrix formulationfor the time domain iteration scheme. Pre-vious works have employed either theadmittance matrix, which results in extendedimpulse responses, or introduced artificialmatching networks, which could render thesolution unstable. In contrast, the general-ized scattering matrix approach is mostclosely tied to the concept of waves. There-fore, this approach achieves a shorter impulseresponse that leads to a smaller computermemory requirement and faster computationtime.

We have carried out the detailed procedurefor solving this kind of nonlinear transientproblem for a microstrip transmission linewith linear source resistance. The diode-liketerminal characteristics has been considered,and the speed of convergence is quite satis-factory.

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Time-Domain Finite-Difference modeling hasbeen applied to obtain propagation charac-teristics of microstrip structures. Maxwellequations are expressed in finite-differenceform, and the substrates are considered to beanisotropic in general. Positioning the com-ponents of the electric and magnetic fields atdifferent positions and evaluating them atalternate time steps, we obtain the compo-nents of Maxwell's equations. Boundary con-dition implementation is an important issuein this method. Electric and magnetic wallscan be implemented simply by setting theappropriate field components zero. Open-endtermination is simulated by using the open-circuit, short-circuit technique, to cancel outthe reflected waves. We have used thismethod to illustrate graphically the fieldpropagation along an open microstrip line.Field components are plotted in both spaceand time domains. The dispersive behaviourof the wave propagation can be observed. Itshould be noted that this method can beconveniently applied to obtain frequency-dependent characteristics of various micro-strip structures such as effective permittivity,characteristic impedance, scattering matrixelements, and equivalent circuit components.An improved source plane implementationbased on magnetic wall is applied for bettermodeling of the matched source. Thismethod is very powerful in obtaining boththe time-domain and the frequency-domaincharacteristics of microstrip lines, discontinu-ities, and coupled lines.

We are developing general purpose finite dif-ference time domain (FDTD) algorithms tosolve electromagnetic wave propagation inintegrated circuit problems, in particular thediscontinuities such as bends and corners.The geometry is read from a file whichassigns dielectric constants, permeability, andconductivity to each discrete mesh point. Tolimit the domain of computation, anabsorbing boundary condition has beenimplemented that simulates an outward prop-agating wave on the faces of the mesh, thuseliminating reflection from these faces. Spe-cific problems simulated so far include two-dimensional propagation from a point sourceto demonstrate the effectiveness of theabsorbing boundary condition and three-dimensional propagation around a 90-degreemicrostrip corner.

A new perturbation series, coupled integralequation approach for calculating the fre-quency dependent circuit parameters forquasi-TEM transmission lines with lossy con-ductors has been developed. The methodconsiders the addition of loss and dispersionto be perturbations on the lossless TEM caseand; therefore, the difference between thepropagation constant and the wavenumber infree space is a small parameter. We obtainthe lowest order term of the perturbationseries by solving two quasi-static problems;the electrostatic problem to get the capaci-tance, and the magnetoquasistatic problem,with the distribution of current inside thewire considered, which gives the frequency-dependent inductance and resistance. Bothof these problems are solved by using one-dimensional integral equations for quantitieson the surface of the conductor; this repres-ents a significant improvement in efficiencyover previous methods. For most cases ofpractical interest, the lowest order term of theseries will suffice. If, however, the change inthe propagation constant from the losslesscase, due to the altered inductance and theaddition of resistance, is significant, addi-tional terms in the perturbation series can becalculated.

The method has been applied to the case ofone or more wires embedded in a uniformdielectric. In the original magnetoquasi-static problem, the current is entirely directedalong the axis of propagation, and satisfiesthe frequency-domain diffusion equation.Outside the wire, the magnetic vector poten-tial is in the same direction, and obeysLaplace's equation. The boundary conditionsare the continuity of tangential and normalmagnetic field at the interface, which can beexpressed in terms of the current density,vector potential, and their derivatives. Sincewe can express the ratio of the frequency-dependent resistance to the DC resistance interms of the values of the volume current andits normal derivative on the surface of thewire only, we can use a pair of coupled inte-gral equations to solve for these quantitiesalone, which we can solve by Galerkin'smethod or other finite element methods.

In microelectronic packaging, a problem ofpractical interest is the study of propagationcharacteristics of a shielded microstrip line inthe presence of crossing strips in a multilay-



ered structure. We have investigated thedispersion characteristics of strip linescrossed by metallic strips and embedded inthe same isotropic layer and bounded by twoconducting planes. A rigorous dyadicGreen's function formulation in the spectraldomain is used and a set of coupled vectorintegral equations for the current distributionon the conductors is derived. Galerkin'smethod is then applied to derive the matrixeigenvalue equation for the propagation con-stant. The dispersion properties of the signallines are studied for both cases of finite andinfinite length crossing strips.

The effects of the structure dimensions onthe passband and stopband characteristicsare analyzed. For crossing strips of finitelength, the stopband is mainly affected bythe period, the crossing strip length, and theseparation between the signal and thecrossing strips. For crossing strips of infinitelength carrying travelling waves, attenuationalong the signal line exists over the wholefrequency range of operation.

Also considered are microstrip lines togetherwith a set of the crossing strips to be placedat two different interfaces of a three-layermedium bounded by two parallel conductingplates. The crossing strips are assumed to beperiodic and the three layers to be uniaxiallyanisotropic. The network analytical methodof electromagnetic fields is applied to thehybrid-mode analysis of the frequencydependent characteristics of the structure.The transverse fields in each region areexpressed by their Fourier transform in thetransversal-direction and a Floquet harmonicrepresentation in the longitudinal-direction.Using Maxwell's equations and applyingboundary conditions, we obtain a set ofcoupled integral equations for the currentdistributions on the microstrip line and thecrossing metallic strips. The determinantalequation for the dispersion relation can bederived by applying Galerkin's procedure tothe derived set of coupled integral equations.The Brillouin diagrams are obtained numer-ically by seeking the roots of the obtainedeigenvalue equation. The stop-band proper-ties are numerically obtained as a function ofthe spacing, length, and width of thecrossing metallic strips. The effects of thematerial anisotropy and geometrical parame-ters are also studied.

Publications



Gu, Q., Y.E. Yang, and J.A. Kong, "TransientAnalysis of Frequency-Dependent Trans-mission Line Systems Terminated withNonlinear Loads," submitted to J. ofElectromag. Waves Applic.

Habashy, T.M., S.M. Ali, and J.A. Kong,"Impedance Parameters and RadiationPattern of Cylindrical-Rectangular andWraparound Microstrip Antennas," paperpresented at the IEEE AP-S InternationalSymposium and URSI Radio ScienceMeeting, Syracuse, New York, June 6-10,1988.

Kiang, J.F., S.M. Ali, and J.A. Kong, "Propa-gation Properties of Strip Lines Period-ically Loaded with Crossing Strips,"submitted to IEEE Trans. MicrowaveTheory Tech.

Tsuk, M.J., and J.A. Kong, "Response ofLayered Media to Current Sources withArbitrary Time Behavior," paper presentedat the IEEE AP-S International Sympo-sium and URSI Radio Science Meeting,Syracuse, New York, June 6-10, 1988.

Yang, Y.E., Q. Gu, and J.A. Kong, "TransientElectromagnetic Wave Propagation onVias of Multilayer Integrated Circuit Pack-ages," paper presented at the IEEE AP-SInternational Symposium and URSI RadioScience Meeting, Syracuse, New York,June 6-10, 1988.

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Documents

180 RLE Progress Report Number 131 - dspace.mit.edu