Guo-Xin Fan 1 and Qing Huo Liu 1. 1 Klipsch School of Electrical and Computer Engineering. Recei ed 21 April 1998

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1 Measured H-plane and E-plane normalized radiation patterns for G 7, t 0, f 2.33 GHz are plotted in Figure 3. Similar estimated radiation patterns have been reported in 2. This slot antenna has syetri upper spae and lower spae radiation patterns; heneforth, only the half-spae portion of eah far-field radiation plane is presented. The normalized radiation patterns for G 3, t 15, f GHz are shown in Figure 4. The opolarization beamwidth of the H-plane is wider Ž than that of the struture of Figure 3 Ž However, the large transverse eletri field omponent Ex on the CPW line inreases the ross polarization above the level of the struture of Figure 3 in the H-plane. It is not neessary to add a uarter-wavelength transformer between the feed line and antenna when a narrower slow width is used. It is just neessary to hange the loading tuning length to improve the VSWR value. Experimental results show the tuning length is still shorter than the uarter wavelength 19.5 of the CPW line. 3. COCLUSIOS In general, the slot width of a slot antenna affets the bandwidth, and the slot length determines the resonant freueny. To obtain a wider radiation beamwidth, a narrower slot width is neessary. A narrower slot width needs a longer tuning length to improve the VSWR value, and results in a lower resonant freueny and a narrower bandwidth. This tuning length is shorter than a uarter wavelength of the CPW-fed line. After hanging the slot width and loading the tuning length, the beamwidth beomes wider in the H-plane as expeted. REFERECES 1. B. K. Kormanyos, W. Harokopus, L. P. B. Kathei, and G. M. Rebeiz, CPW-Fed Ative Slot Antennas, IEEE Trans. Mirowae Theory Teh., Vol. 42, Apr. 1994, pp H. C. Liu, T. S. Horng, and. G. Alexopoulos, Radiation of Printed Antenna with a Coplanar Waveguide Feed, IEEE Trans. Antennas Propagat., Vol. 43, Ot. 1995, pp J. M. Laheurte, L. P. B. Katehi, and G. M. Rebeiz, CPW-Fed Slot Antennas on Multilayer Dieletri Substrates, IEEE Trans. Antennas Propagat., Vol. 44, Aug. 1996, pp A. Illipiboon, R. K. Mongia, Y. M. M. Antar, P. Bhartia, and M. Cuhai, Aperture Fed Retangular and Triangular Dieletri Resonators for Use in Magneti Dipole Antennas, Eletron. Lett., Vol. 29, ov. 11, 1993, pp John Wiley & Sons, In. CCC Figure 3 Measured E-plane and H-plane radiation patterns for the CPW-fed slot antenna with G 7, t 0, and resonant freueny f 2.32 GHz A PML FDTD ALGORITHM FOR SIMULATIG PLASMA- COVERED CAVITY-BACKED SLOT ATEAS Guo-Xin Fan 1 and Qing Huo Liu 1 1 Klipsh Shool of Eletrial and Computer Engineering ew Mexio State University Las Crues, ew Mexio Reeied 21 April 1998 ABSTRACT: A three-dimensional freueny-dependent finite-differene time-domain ( FDTD) algorithm with perfetly mathed layer ( PML) absorbing boundary ondition ( ABC) and reursie onolution approahes is deeloped to model plasma-oered open-ended waeguide or aity-baked slot antennas. The algorithm is alidated and then applied to simulate the radiation of an inhomogeneous plasma-oered aity-baked slot antenna John Wiley & Sons, In. Mirowave Opt Tehnol Lett 19: , Key words: plasma-oered aity-baked slot antenna; FDTD; disper- ( ) sie media; perfetly mathed layer PML Figure 4 Measured E-plane and H-plane radiation patterns for the CPW-fed slot antenna with G 3, t 15, and resonant freueny f GHz I. ITRODUCTIO It is well known that the performane of antennas during re-entry into the earth s atmosphere an be signifiantly 258

2 affeted by the plasma sheath that forms around the vehile. The effets of plasma on slot antenna harateristis have been investigated mainly by variational tehniues 13 and the moment method Ž MoM. 4. In the variational tehniues, an aperture field distribution is assumed as those of the waveguide-dominant mode. This approximation ignores the effet of the higher modes, and therefore redues the auray of the result. The MoM an aurately obtain the aperture field by solving the integral euation of the aperture field, but is limited to the homogeneous plasma layer and avities of simple shapes sine it involves the omputation of various dyadi Green s funtions. This work presents a three-dimensional freueny-dependent FDTD algorithm with the perfetly mathed layer Ž PML. and reursive onvolution Ž RC. 5 and pieewise linear reursive onvolution Ž PLRC. 6 approahes to model plasma-overed open-ended waveguide or avity-baked slot antennas. II. THEORY Consider a general isotropi, ondutive, inhomogeneous, linear permittivity dispersive medium with permeability and ondutivity. Using the oordinate-strething approah 7 and following a similar proedure as in 8, the modified Maxwell s url euations with the split fields Ž x, y, z. in the time domain an be written as H Ž. Ž. Ž. ˆ E a H M Ž 1. t D Ž. Ž. Ž. ˆ H a D a E t t H Ž. Ž. E dt J Ž 2. where J and M are the imposed eletri and magneti urrent densities, and a and orrespond to the real and imaginary parts of the oordinate-strething variables, respetively 8. In this dispersive medium, the eletri flux density is related to the eletri field intensity by t H DŽ t. EŽ t. EŽ. Ž t. d Ž where is the free-spae permittivity, is the relative 0 permittivity at, and is the eletri suseptibility. The freueny-domain suseptibility funtions, as the transfer funtion of a linear system, an be generally expressed as a ratio of two polynomials or in a frational form, i.e., M M M Ý Ý Ý Ž. s s, Ž M M s s Ž 4. where s i, and s and are the omplex poles and the orresponding residues. Then the orresponding time-domain suseptibility funtions an be written as Ý s t 1 t Re R e U t 5 where Ut Ž. is the unit step funtion. In Ž. 5, M, R when all s and are real. For unmagnetized plasma, we have the following transform pair: p 2 p 2 Ž t, t 1 e. UŽ t. Ž i. where R1R2p 2, s1 0, s2, p is the angular plasma freueny, and is the ollision freueny. For plasma, 1 in E. Ž. 3. To simplify Ž. 3, we first introdue a unified pieewise approximation to EŽ. t over the time interval t mt, Ž m 1. t as follows: EŽ m 1. EŽ m. EŽ t. EŽ m 1. K tž m1. t a. t Ž 7. It is noted that E. Ž. 7 orresponds to the RC 5 when K 0 and to the PLRC a 6 when K a 1. Using Ž. 5 and the unified approximation Ž. 7, the onvolution integral in Ž. 3 is then transformed into the disrete onvolution suation: where Ý 6 DŽ n. EŽ n. Re Ž n. Ž 8. n1 Ý Ž n. ˆ Ž 0. ˆ Ž 0. EŽ n m. m 0 s a ˆ s m t 0 E n m 1 e Ž 9. R t, for s 0 ˆ Ž 0. R Ž 10. Ž s t e 1., for s 0 K R t, for s 0 2 ˆ Ž 0. Ž 11. K R 2 ts a s t 1 1 s t e, for s 0. From the above formulas, we derived the reursive relations for Ž n 1. and DŽ n 1.. With the help of these reur- sive onvolution euations, we an proeed to solve Maxwell s euations by using Yee s algorithm to disretize the split euations Ž. 1 and Ž. 2. We obtain the eletri field timestepping euations as follows: Ž. Ž. E n Ž H n. Ž n. 2 ˆ ž / 1 Ž. Ž. Ž. Ž. 0 E n tei n J ž n 2 Ž / 259

3 where Ž. and Ž. are some known oeffiients, and 0 1 ž / a Ž. 0 Ý Re½ t 2 1 ž / a s t Ž. e Ž n. Ž 13. t Ž. Ž. Ž. Ž. E n E n 1 E n E Ž n 1.. Ž 14. I I 8 8 The magneti field time-stepping euation is unhanged for magnetially nondispersive media, as given in 8. III. UMERICAL RESULTS The PML euations are applied to both the interior region and the mathed layers. Ten ells of PMLs are used as the absorbing boundary ondition outside the region of interest. The soure is an eletri dipole, and its time funtion is the first derivative of the BlakmanHarris window funtion with the entral freueny f 8. To validate the algorithm, we ompare the FDTD numerial results with analytial solutions. First, the transient wave propagation along an infinite retangular waveguide filled with nondispersive lossy medium is onsidered. The size of the waveguide setion is , and the eletri parameters of the filling medium are 0, 0, and 0.05 Sm. A ˆy-eletri dipole with f 10 GHz is used as the exiting soure. Figure 1 shows the E -field y waveforms of both the FDTD and analytial results. Seond, the radiations from a horizontal and a vertial eletri dipole in a plasma halfspae on an infinite onduting plane are examined. The onduting plane is loated at the z 0 plane. The parameters of the plasma halfspae are p rads and Hz. The soure is an ˆx- and ˆz-eletri dipole, respetively, and f 25 GHz. Figure 2Ž. a and Ž. b shows a omparison of the FDTD results with the analytial solutions. It an be seen that in all of the testing examples above, exellent agreement between analytial and numerial results is ahieved. Furthermore, the alulation also shows that the RC and PLRC approahes have a similar auray. The avity-baked slot antenna under onsideration is an open-ended waveguide as shown in Figure 3. The inhomogeneous plasma overing is divided into 15 homogeneous layers, and plaed parallelly on the slot aperture. Figure 3Ž. a dis- Figure 1 Ey-field from a ˆy-eletri dipole in a standard X-band retangular waveguide filled with a lossy medium. f 10 GHz,, 0.05 Sm, r Ž 11.43, 5.08, 0., and r Ž , 5.080, Figure 2 ear fields due to a Ž. a horizontal and Ž. b vertial eletri dipole. f 25 GHz, p rads, Hz, r Ž 0, 0, 6.25., and r Ž 0, 0, plays the transient near-field distribution right above the plasma layers. For omparison, the results in the absene of a plasma layer are also given in Figure 3Ž b.. The fields are normalized with respet to the peak value of the waveform without plasma layers. It an be seen that plasma has a signifiant ontribution to the power dissipation. In addition, the sustained osillations indiate that plasma is a highly resonant medium within this freueny band. The freueny response an be obtained through the fast Fourier transform Ž FFT. of the time-domain waveforms. To eliminate the trunation effet in the FFT introdued by the inompleteness of the alulated time-domain waveforms, and to redue the omputational time, we employ Prony s algorithm to extrapolate the late-time response 9. The original waveforms ontain n 12,000 time steps, and the extrapolated waveforms from n 6001 to 25,000 are obtained by using the original waveforms from n 3000 to Figure 4 shows both waveforms at the enter of the slot aperture of the plasma-overed slot antenna. Exellent agreement between the Prony extrapolated waveforms and the FDTD alulated waveforms is observed at the overlap region Ž n 3000 to n 12,000.. Beause of the exitation of the limited freueny band, only a few pairs of poles are of importane in the Prony extrapolation. Figure 5Ž. and Ž. d shows the slot aperture eletri field distribution at freueny GHz for both with and without plasma overing. The effet of plasma on the aperture field is obvious. As is expeted, the transverse distribution of the Ey-omponent is similar to that of the dominant mode in the waveguide, while the longitudinal distribution has been 260

4 Figure 3 ear field from a ˆy-eletri dipole in an inhomogeneous plasma-overed avity-baked slot antenna Ž. a with and Ž. b without plasma layers. The size of the avity is , f 9 GHz, Hz, r Ž 0, 0, , and r Ž 0, 0, Figure 5 Aperture field distribution Ž. a with and Ž. b without plasma layers at freueny f GHz Figure 4 Original Ž n 012,000. and Prony extrapolated transient waveforms of the slot aperture Ey-field of plasma-overed avity- baked slot antenna altered at the two ends due to the edge singularity ondition. It is worth pointing out that, unlike the variational tehniue and MoM, the FDTD algorithm proposed here does not assume the distribution of the aperture field in advane, and is appliable to omplex geometry and inhomogeneous overing. ote that the key problem of slot antenna analysis is to solve for the slot aperture fields. One the aperture fields are obtained, it is easy to alulate slot admittane and other antenna harateristi parameters by using the variational expression 2. We also an obtain the far-zone radiation pattern by using the nearfar field transformation. IV. COCLUSIOS We present a 3-D freueny-dependent FDTD algorithm with the PML absorbing boundary ondition to model plasma-overed open-ended waveguide or avity-baked slot antennas. The dispersion of plasma is taken into aount by using the reursive onvolution approahes. The algorithm is verified by several testing examples, and is applied to an inhomogeneous plasma-overed avity-baked slot antenna. Due to its flexibility and generality, the algorithm and omputer program an also be used to model arbitrary avitybaked slot antennaswith or without dispersive or nondispersive overing media. Moreover, it an provide the slot antenna harateristis over a wide range of freueny by using pulse exitation. ACKOWLEDGMET This work was supported by the Environmental Protetion Ageny through PECASE Grant CR , by the ational Siene Foundation through CAREER Grant ECS , and by Sandia ational Laboratories under the SURP Program. 261

5 REFERECES 1. J. Galejs and M. H. Mentzoni, Waveguide Admittane for Radiation into Plasma LayerTheory and Experiment, IEEE Trans. Antennas Propagat., Vol. AP-15, 1967, pp K. E. Golden and G. E. Steward, Self and Mutual Admittanes of Retangular-Slot Antennas in the Presene of an Inhomogeneous Plasma Layer, IEEE Trans. Antennas Propagat., Vol. AP-17, 1969, pp R. L. Fante, Admittane of an Aperture Antenna Radiating into a Warm Plasma Half-Spae, IEEE Trans. Antennas Propagat., Vol. AP-19, 1971, pp J. M. Jarem, The Input Impedane and Antenna Charateristis of a Cavity Baked Plasma Covered Ground Plane Antenna, IEEE Trans. Antennas Propagat., Vol. AP-34, 1986, pp R. J. Luebbers, F. P. Hunsberger, and K. Kunz, A Freueny- Dependent Finite-Differene Time-Domain Formulation for Transient Propagation in Plasma, IEEE Trans. Antennas Propagat., Vol. 39, 1991, pp D. F. Kelley and R. J. Luebbers, Pieewise Linear Reursive Convolution for Dispersive Media Using FDTD, IEEE Trans. Antennas Propagat., Vol. 44, 1996, pp W. C. Chew and W. H. Weedon, A 3D Perfetly Mathed Medium from Modified Maxwell s Euation with Strethed Coordinates, Mirowae Opt. Tehnol. Lett., Vol. 7, 1994, pp Q. H. Liu, An FDTD Algorithm with Perfetly Mathed Layers for Condutive Media, Mirowae Opt. Tehnol. Lett., Vol. 14, 1997, pp M. L. Van Blarium and R. Mittra, A Tehniue for Extrating the Poles and Residues of a System Diretly from Its Transient Response, IEEE Trans. Antennas Propagat., Vol. AP-23, 1975, pp John Wiley & Sons, In. CCC a hollow metal retangular waveguide are shown in Figure 1 Ž. a Ž. d. The mirostrip-slot oupled line struture is more useful in many MIC iruit designs, e.g., diretional ouplers, filters, modulators, and phase shifters beause there is an additional degree of freedom in design due to the existene of the slot 1, 2. El-Sharawy, Ed-Badawy, and Jakson have presented phase-shifting properties of a similar struture on ferrite substrates 3. Itoh used the spetral-domain iittane approah to alulate the freueny-dependent dispersion harateristis of a strip mode for a similar struture 4. But it did not inlude the freueny-dependent behavior of harateristi impedane as well as slot mode harateristis. In this paper, the freueny-dependent properties of a mirostrip-slot oupled struture on isotropi substrate have been analyzed and ompared with published data available for the finline and mirostrip line as a speial ase as well as the mirostrip-slot oupled line. 2. AALYTICAL FORMULATIO A mirostrip-slot oupled struture on a dieletri substrate inside a hollow metal retangular waveguide is shown in Figure 2. Consider that the strip and slot are entrally loated. Consider that the eletromagneti field is propagating in the z-diretion, and having x, z and time dependene as e jk x x, e j z, and e j t, respetively. The wave euation in the isotropi medium an be derived from Maxwell s euations. Sine the oupled struture supports a hybrid field, both the TE-to-x and the TM-to-x modes will be present. The Ex- and FREQUECY-DEPEDET AALYSIS OF MICROSTRIP-SLOT COUPLED LIE FOR PHASE-SHIFTIG AD IMPEDACE-TRASFORMIG ETWORK APPLICATIOS S. Palanihamy 1 and Animesh Biswas 1 1 Department of Eletrial Engineering Indian Institute of Tehnology Kanpur, U.P , India Reeied 30 Marh 1998 ABSTRACT: The properties of a mirostrip-plot oupled line on isotropi substrate are analyzed using the full-wae modal analysis method. The omputational method presented in this paper is general in nature, and an be applied to unilateral finline strutures and suspended substrate lines by hanging the width of the mirostrip or slot. The numerial results for the freueny-dependent normalized guide waelength ( g ) and harateristi impedane ( Z ) 0 0 for the dominant strip mode and slot mode of the mirostrip-slot oupled line on isotropi substrates are presented for suitable phase-shifting and impedane-transforming network appliations John Wiley & Sons, In. Mirowave Opt Tehnol Lett 19: , Figure 1 Variants of E-plane transmission lines Key words: E-plane planar iruits; finline; mirostrip-slot oupling 1. ITRODUCTIO Reently, E-plane uasiplanar transmission lines have been used most extensively in the design of mirowave and millimeter-wave integrated iruits. Some of the variants of E-plane transmission strutures on isotropi substrate inside Figure 2 Mirostrip-slot oupled line on isotropi substrate 262