The theorem of equivalence between the Kirchhoff aperture integration (AI) and the physical optics (PO) approach applied to the interior walls of a open-ended waveguide is generalized to the case of any open-ended structure (OES).Peer Reviewe

Michalski, Krzysztof A.

Rius Casals, Juan Manuel

IEEE Transactions on Antennas and Propagation

English

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IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 6, JUNE 2002 891Communications______________________________________________________________________Generalization of the Equivalence Between PhysicalOptics and Aperture Integration for Radiation FromOpen-Ended StructuresJuan M. Rius and Krzysztof A. MichalskiAbstract—The theorem of equivalence between the Kirchhoff apertureintegration (AI) and the physical-optics (PO) approach applied to the inte-rior walls of a open-ended waveguide [1] is generalized in this letter to thecase of any open-ended structure (OES).Index Terms—Apertures, physical optics (POs), waveguides, open-ended.I. INTRODUCTIONIn a recent letter [1] Maci et al. use equivalence principle con-cepts [2] to prove the equivalence between the physical optics (PO)and Kirchhoff–Kottler approximations for the radiated field of anopen-ended waveguide. The purpose of this letter is to present a moregeneral theorem, of which the theorem of Maci et al. is a specialcase. The new theorem applies to the aperture fields and interior wallcurrents corresponding to an arbitrary open-ended structure, for whichmodal fields cannot in general be defined. In addition, some insightis given on the physical meaning of the radiated fields computed byaperture integration (AI) or PO approaches.II. DEMONSTRATIONConsider an open-ended structure (OES) with perfectly electricallyconducting (PEC) walls, as illustrated in Fig. 1(a). The OES, whichmay be of finite or semi-infinite extent, is excited by impressed electricand magnetic currents, Ji and M i, respectively. The resulting electricand magnetic fields are denoted byE andH , respectively. For later ref-erence, the exterior and interior surfaces of the OES walls are denotedby Sex and Sin, respectively and the mathematical surface coveringthe OES aperture by Sa. Furthermore, the exterior and interior regionsare denoted by V ex and V in, respectively.According to the equivalence principle [2], the field (E, H) in V exmay be found from the problem shown in Fig. 1(b), where the surfacecurrents are given asJexs=n^H on Sex (1)Jas=n^H on Sa (2)Mas=  n^ E on Sa (3)in which n^ is a unit surface normal vector directed into V ex. If thefields radiated by Jexsand (Jas, Mas) are denoted by (Eex, Hex) and(Ea, Ha), respectively, then(Ea + Eex; Ha +Hex) =(E;H) in V ex(0; 0) in V in: (4)Manuscript received August 10, 1998; revised April 7, 2000.J. M. Rius is with the Department of Signal Theory and Communications,Universitat Politècnica de Catalunya, 08034 Barcelona, Spain.K. A. Michalski is with the Electromagnetics & Microwave Laboratory, De-partment of Electrical Engineering, Texas A&M University, College Station,TX 77843-3128 USA.Publisher Item Identifier S 0018-926X(02)05733-2.The field (E, H) in V in may be found from the equivalent problemshown in Fig. 1(c) [2], where Jasand Masare given by (2) and (3),respectively, andJins= n^H on Sin (5)in which n^ points into V in. If the fields radiated by Jinsand (Ji, M i)are denoted by (Ein, Hin) and (Ei, Hi), respectively, thenEi +Ein   Ea; Hi +Hin  Ha =(0; 0) in V ex(E;H) in V in (6)or, equivalently,Ei +Ein; Hi +Hin =(Ea; Ha) in V ex(E + Ea; H +Ha) in V in (7)which leads to the conclusion that the field in V ex due to the aperturecurrents (Jas,Mas) is equal to the field due to the currents in the interiorof the OES, namely the induced current Jinsand the impressed currents(Ji, M i).Note that the currents of the external and internal equivalent prob-lems radiate in the absence of the OES. Hence, the “impressed field”(Ei, Hi) should not be confused with the modal-incident field, whichsatisfies the boundary conditions at the OES walls.An important observation regarding (7) is that this relationship holdseven if there are material objects present outside the OES of Fig. 1(a).Consider, for example, the infinite waveguide of Fig. 1(d), which isobtained by extending the waveguide of Fig. 1(a). Let us denote Sv =S + S0 the new virtual surface of Fig. 1(d), which partially superim-poses with the actual surface S.Clearly, the internal equivalent problem of Fig. 1(c) is equally appli-cable to both the original structure of Fig. 1(a) and its extended versionof Fig. 1(d). Hence, the following theorem: the field due to the aper-ture currents (Jas, Mas) is, in V ex, equal to that radiated by the interiorwall current Jinsplus the impressed field irrespective of whether thefield (E, H) used to compute the equivalent currents is derived fromthe OES (i.e., open-ended waveguide) or the extended structure (i.e.,infinite waveguide). That is, the theorem is valid for any field distribu-tion (E, H) that satisfies the wave equation in V in and the boundaryconditions in Sin. There is an infinity of such field distributions, eachsatisfying different boundary conditions in V ex.When (E,H) is taken to be a modal field propagating in the extendedstructure, the equivalent currents are easily determined and the field inV ex may be approximately computed by integrating either Jinsor Jasand Masagainst the free-space Green’s functions. Maci et al. refer tothese two approaches as POs and Kirchhoff–Kottler aperture integra-tion, respectively. Since a modal field is source free, the impressed field(Ei, Hi) is absent in (7) and the above theorem specializes to that ofMaci et al., which states that the PO and Kirchhoff-Kottler aperture in-tegration are equivalent in V ex. As is evident from (4), both approachesneglect the contribution of the exterior wall current, Jexs.Although arbitrary, the unique virtual surface Sv of Fig. 1(d) whichhas a practical sense in most cases is that for which the field solution isknown in analytical form when Sv has the same boundary conditionsof the original surface on S, Fig. 1(a), and arbitrary boundary condi-tions on S0. The above closed-form solution is usually obtained whenthe field in Sv is expressed in terms of modes. In this case, the gener-alization presented in this letter is the same as that suggested in [3] and0018-926X/02$17.00 © 2002 IEEE892 IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 6, JUNE 2002(a)(b)(c)(d)Fig. 1. (a) OES. (b) External equivalent problem. (c) Internal equivalent problem. (d) Extended structure. The virtual surface S partially superimposes with theactual surface S.that can be simply expressed invoking the modal currents in place ofthe PO currents.However, the generalized theorem presented in this paper may bea useful tool for field estimation even if a closed-form solution is notavailable for the field inSv : If this is the case, one must resort to numer-ical methods in order to obtain a good approximation of the aperturefields (Ea, Ha) and the aperture equivalent currents (Jas, Mas).To account for the contribution of Jexs, one can use an alternativeformulation of the equivalence principle [2] and place a PEC wall atSex in the exterior equivalent problem of Fig. 1(b). Now, in the pres-ence of the PEC wall that corresponds to the original OES, Jexsdoesnot radiate and (Jas,Mas) radiate the original field (E,H). To approxi-mately compute the radiation of (Jas,Mas) in the presence of the OES,reflections at surfaces and diffraction at edges may be taken into ac-IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION, VOL. 50, NO. 6, JUNE 2002 893count by using high-frequency techniques [4], thus leading to a betterapproximation than (Ea,Ha)—the free-space field of (Jas,Mas). Now,the equivalence between the interior-wall currents and the aperture cur-rents becomes approximate; the accuracy of one approximation withrespect to the other cannot be stated a priori and must be checked caseby case.ACKNOWLEDGMENTThe authors would like to thank the reviewers for their suggestionsto improve the clarity of the presentation.REFERENCES[1] S. Maci, P. Ya. Ufimtsev, and R. Tiberio, “Equivalence between physicaloptics and aperture integration for radiation from open ended waveg-uides,” IEEE Trans. Antennas Propagat., vol. 45, pp. 183–185, Jan.1997.[2] R. F. Harrington, Time-Harmonic Electromagnetic Fields. New York:MacGraw-Hill, 1961, ch. 3, sec. 3–5.[3] S. Maci, P. Ya. Ufimtsev, and R. Tiberio, “Equivalence between physicaloptics and aperture integration for antennas and scattering problems,” inProc. URSI Symp. Montreal, July 13–18, 1997.[4] A. K. Battacharyya, High-Frequency Electromagnetic Tech-niques. New York: Wiley, 1995.Design and Scan Performance of Large, Probe-Fed StackedMicrostrip Patch ArraysR. B. WaterhouseAbstract—A strategy is presented on how to design large, direct-con-tact microstrip patch arrays with broad bandwidths and useful scanningranges. It will be shown that to maximize these characteristics the lowerlayer of a stacked patch configuration must have a relatively high dielectricconstant, greater than 10 and the upper laminate must have a low dielectricconstant value. Doing so yields bandwidths in excess of 25% over a scan-ning range of 45 in the principle planes. Such arrays may be suitablefor millimeter-wave systems such as collision avoidance radars and micro-cellular mobile communication base stations.Index Terms—Antenna array feeds, microstrip arrays, stacked mi-crostrip antennas.I. INTRODUCTIONLarge, unobtrusive, and scannable/multibeam arrays are needed formany applications, including smart antennas for mobile communica-tion base stations and space-borne communication satellites. For thesesystems narrow beamwidths are required over a reasonable frequencyband (typically 20% of the center frequency). Due to its low produc-tion cost, low profile and ease of obtaining good polarization purity,the microstrip patch would appear to be an ideal element for the abovementioned array applications. However, in its conventional form, thisantenna is typically narrow band and also has limited scanning poten-tial in large arrays due to the excitation of surface waves. These trappedManuscript received October 19, 1999; revised June 15, 2001.The author is with the School of Electrical and Computer Systems Engi-neering, RMIT University, Melbourne VIC 3001, Australia (e-mail: r.water-house@ieee.org).Publisher Item Identifier S 0018-926X(02)05740-X.waves cause potential scan blindnesses, limiting the coverage area ofthe scannable array [1].Several investigations have been conducted on the scanning perfor-mance of microstrip patch arrays and also means of yielding mod-erate impedance bandwidths over a useful scanning range (for example,[2]). However, relatively large bandwidth/reasonably scannable con-figurations have yet to be developed. Design strategies have recentlybeen presented for several variations of individual probe-fed microstrippatches resulting in broad bandwidths (greater than 25% [3]). Thesestacked patch configurations have the potential to satisfy the require-ments for the applications given earlier, however their performance inlarge arrays and their scanning abilities have yet to be investigated. Ithas also been shown that by simply parasitically coupling a microstrippatch fabricated on a low dielectric constant laminate to a driven patchetched on a high dielectric constant material can result in a broadband,highly efficient antenna [4]. Such a configuration would allow for thesimple integration of the antenna directly with photonic and microwavedevices and circuits, minimizing the number of layers or material tran-sitions and interconnects required for the transceiver units. Once again,the performance of such a microstrip antenna in a large array environ-ment has yet to be investigated.In this paper, a thorough, theoretical investigation of the impedancebehavior and scanning potential of probe-fed microstrip patch arraysconsisting of broad-band stacked elements will be presented. Inparticular design methodologies to maximize both the bandwidth andscanning range of these arrays will be given. It will be shown that broadimpedance bandwidths over useful scanning ranges can be achieveddepending on the choice of dielectric constant for the lower microwavelaminate. Importantly, the developed design strategy does not requirematching networks to give the resulting bandwidths and thereforeyields very simple antenna structures and avoids potential blindnessesassociated with these networks [2]. All the arrays examined wereanalyzed using a full-wave spectral domain infinite array analysis.II. THEORY AND CONFIGURATIONFig. 1 shows the unit cell of the probe-fed stacked patch array. Herea rectangular stacked patch element is fed by a coaxial pin locatedat (p; p) from the center of the bottom patch. The lower patch re-sides on a grounded dielectric layer with a dielectric constant of "r1,a loss tangent, tan 1 and a thickness d1. The parasitic coupled patchis mounted on a laminate with a dielectric constant "r2, a loss tangent,tan 2 and a thickness d2. The unit cell size is a in the x-directionand b in the y-direction for the array. To investigate the performance ofthe probe-fed stacked patch configuration, the full-wave analysis pre-sented in [5] was used (refer to [5] for details and experimental verifica-tion). It should be noted that although hi–lo configurations would typi-cally incorporate edge feeding of the driven element, it has been shownthat edge and probe feeding these structures give similar responses [4].Therefore, for the ease of analysis, only probe-fed configurations arepresented here.III. PROBE-FED STACKED PATCH ARRAYSAn infinite array of probe fed rectangular stacked patches at broad-side ( = 0) was designed. Here "r1 = 2:2; d1 = 0:02c and"r2 = 1:07 and d2 = 0:025c, where c is the center frequency ofthe band of interest. The 10 dB return loss bandwidth of the antennawas approximately 27% and the response resembled that for a singlestacked patch (approximately 30%) using this combination of dielec-tric materials [3]. The design procedure is in fact fairly similar to [3] interms of dielectric constant selection; starting points for the conductor0018-926X/02$17.00 © 2002 IEEE

Generalization of the equivalence between physical optics and aperture integration for radiation from open-ended structures

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Generalization of the equivalence between physical optics and aperture integration for radiation from open-ended structures

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