Radiation from a monopole antenna in front of a conducting cylinder

Transcription

1 RADIO SCIENCE, VOL. 4,, doi:1.129/24rs312, 25 Radiation from a monopoe antenna in front of a conducting cyinder Quanxin Wang, Zhongxiang Shen, and Yiong Lu Schoo of Eectrica and Eectronic Engineering, Nanyang Technoogica University, Singapore Received 13 Juy 24; revised 14 November 24; accepted 22 February 25; pubished 19 May 25. [1] A monopoe antenna in front of a conducting cyinder is anayzed by the modaexpansion method and scattering matrix method. An infinite perfecty conducting pate is introduced at the top of the monopoe antenna to faciitate the moda-expansion anaysis. Scattered fieds due to the refecting cyinder are obtained by the scattering matrix method with the aid of cyindrica function s addition theorem. Enforcement of continuity conditions of the tangentia fied components across the regiona surfaces resuts in the matrix equation for determining the expansion coefficients. Numerica resuts for the return oss and radiation pattern of a monopoe antenna in front of a conducting cyinder are presented and are in good agreement with experimenta ones. The effect of the cyindrica refector s radius and its distance from the monopoe on the antenna s radiation performance is aso examined. Citation: Wang, Q., Z. Shen, and Y. Lu (25), Radiation from a monopoe antenna in front of a conducting cyinder, Radio Sci., 4,, doi:1.129/24rs Introduction [2] The probem of a monopoe antenna fed through an infinite ground pane by a coaxia transmission ine or a gap source has attracted extensive attention in the iterature over the past decades because of its geometrica simpicity, ow construction cost, and omnidirectiona property in the horizonta pane [King, 1956]. Many anaysis methods have been proposed to characterize the monopoe antenna s radiation performance, such as the integra equation method for an eectricay thick monopoe [Chang, 1968; Otto, 1967], the singuar integra equation formuation [Morris, 198], and the conservation of compex power technique [Do-Nhat and MacPhie, 1991]. The moda-expansion method, originay presented by Schwering et a. [1982], Morgan et a. [199], and Morgan and Schwering [1994], is computationay very efficient and has been empoyed to anayze various monopoe probems by Shen and MacPhie [1996a, 1996b], Biakowski [2, 22], and Chen et a. [21]. This paper appies the moda-expansion method to anayze the radiation characteristics of a monopoe antenna in front of a conducting cyinder. [3] The probem of a monopoe in front of a conducting cyinder is theoreticay important and practicay Copyright 25 by the American Geophysica Union /5/24RS312 usefu. The motivation of our study is to understand the radiation characteristics of a monopoe antenna when it is paced in front of a cyindrica conductor. The probem of a dipoe antenna in front of a conducting cyinder is basicay equivaent to the one considered in this paper. The understanding of the radiation characteristics of a monopoe/dipoe in front of a refecting cyinder is essentia and very usefu when one is to design base station antennas or switched beam antenna arrays used in various communication systems, such as the goba system for mobie communications, radio frequency identification, and wireess positioning system. [4] In order to faciitate the moda-expansion anaysis, an infinite perfecty conducting boundary is introduced at the top of the monopoe antenna so that discrete moda-expansion functions can be used to represent eectromagnetic fieds radiated by the antenna. The introduced conducting pate has no significant effect on the anaysis of the monopoe because the main radiation from the antenna is in the horizonta direction and the radiation is nu in the vertica direction. Because of the presence of a refecting cyindrica conductor, the radiated fieds from the monopoe wi be scattered toward a the directions and significanty affect the monopoe s radiation performance. The scattered fieds in the outer region can be derived using the scattering matrix method [Ragheb and Hamid, 1987; Tayeb and Maystre, 1997; Yonekura et a., 1999]. After taking the scattered fieds into account, appication of the boundary conditions that 1of9

2 tangentia eectric and magnetic fied components must be continuous across regiona interfaces resuts in the fied expansion coefficients, from which one can cacuate the current distribution aong the monopoe surface, radiation pattern, input impedance and refection coefficient or return oss aong the feed ine. Compared to many other numerica methods, our method can be easiy impemented and is computationay efficient. Moreover, the method can be appicabe to monopoe antennas of any radius in front of a sma or arge conducting cyinder. [5] As shown by Biakowski [2], the coaxia feed ine for monopoe antennas can be equivaenty represented by a sma gap feed when the dimensiona parameters of the coaxia ine are eectricay sma. Therefore we use a sma gap source in this paper, rather than the practica coaxia feed in [Shen and MacPhie, 1996a], to simpify the mathematica formuation. As wi be seen ater, this approximate mode can provide resuts in good agreement with experimenta data. [6] This paper is organized as foows. Section 2 presents the detaied formuation for the radiation probem of a monopoe in front of a conducting cyinder using the moda-expansion method and scattering matrix method. A panar conducting boundary is introduced at the top of the monopoe to confine the domain of interest. Section 3 provides numerica resuts for the monopoe antenna s return oss, input impedance and radiation pattern. A comparison between our numerica resuts with experimenta ones is conducted and a good agreement between them is observed. The effect of different parameters of the refecting cyinder on the monopoe s radiation performance is aso examined. Some concuding remarks are given in section Formuation [7] Figure 1 shows the geometry of a monopoe antenna fed by a gap source in front of a conducting cyinder. The monopoe is of radius a and has a height of h D with D being the gap where a votage source is appied. The conducting cyinder is of radius R a and at a distance of D away from the monopoe center. It is assumed that the conducting cyinder is semi-infinitey ong sitting on the same ground pane as the monopoe. As shown in Figure 1, two cyindrica coordinate systems having the same x-y pane and different origins are empoyed to faciitate the anaysis of the antenna structure. The infinite ground pane ies in the x-y pane, whie the monopoe and cyindrica conducting refector are aong the z and z axis, respectivey. A votage source is appied as the excitation aong the finite gap between the monopoe and the ground pane. [8] It is obvious that the fieds radiated by the gap source between the monopoe and ground pane are 2of9 Figure 1. Anaysis mode of a monopoe antenna in front of a conducting cyinder. axisymmetric under the absence of the conducting cyinder, which impies no f variation in a the radiated fied components. When the conducting cyinder of radius R a is present, it wi scatter the radiated fieds from the monopoe and the scattered fieds can be cacuated using the scattering matrix method. [9] As shown in Figure 1, the whoe structure between the ground pane and the upper conducting boundary can be divided into three regions: I, II, and. The eectromagnetic fieds in Regions II and can be expressed as the summation of their moda functions weighted with expansion coefficients to be determined. [1] A uniform eectric votage V is appied across the gap. Thus the reevant eectric and magnetic fied components in Regions I, II and can be expressed as foows. [11] In Region I, (r a, z D), we know E I z ¼ V D : ð1þ [12] In Region II, (r a, z = h + d), we have E II z ¼ 1 jwe X N2 H II f ¼ XN 2 n¼ n¼ B n g 2 2n J ðg 2n rþcos B n g 2n J 1 ðg 2n rþcos np ð z h Þ d ð2aþ np z h d ; ð2bþ

3 where J and J 1 are the first kind Besse function of order and order 1, respectivey, B n is the expansion coefficient to be determined, and g 2 2n ¼ k2 ð np=d Þ2 with k 2 = w 2 m e. [13] It shoud be mentioned that because of the existence of the cyindrica conducting refector, there shoud be f variation for the eectromagnetic fieds in Region II. However, because the radius of the monopoe antenna is usuay sma compared to the free-space waveength and when the conducting cyinder is not very cose to the monopoe, the effect of the scattered fieds due to the conducting cyinder on the f variation of the fieds in Region II is sma. Therefore in this paper we negect the f variation for the fieds in Region II to reduce the compexity in derivation and computation. [14] In Region, (r a, z ), we obtain E z ¼ 1 jwe X N1 n¼ H f ¼ XN 1 n¼ A n g 2 n H ð2þ ðg n rþcos npz ð3aþ A n g n H ð2þ 1 ðg n rþcos npz ; ð3bþ where H and H 1 are the second kind Hanke function of order and order 1, respectivey, A n is the expansion coefficient to be determined, and where the coefficient f t can be determined by enforcing the boundary condition on the surface of the cyindrica conducting refector, Ez inc þ E z sca ¼ Substituting (4a) and (4b) into (5), we have ð r ¼ R a Þ: ð5þ f t ¼ J tðg n R a Þ H ð2þ ðg n R a Þ H ð 2 Þ t ðg n DÞe jtq r ; ð6þ t where (D, q r ) are the cyindrica coordinates of the refecting cyinder s center in the x-y pane with being the origin. [16] Using Maxwe s equations and (4b), one can obtain the scattered magnetic fied Hf 1 sca ¼ XN n¼ where A n cos npz X f t H ð2þ t ðg n r p r ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ D 2 2rD cosðf q r Þ tan f ¼ r sinðfþ D sin ð q rþ r cosðfþ D cosðq r Þ : Þe jtf i ; ð7þ ð8aþ ð8bþ Substituting (8) into (7) yieds the foowing expression g 2 n ¼ k2 ð np= Þ2 : [15] Because of the existence of the cyindrica conducting refector, there are scattered eectric and magnetic fieds in Region. The scattered fied can be derived using the scattering matrix method [Yonekura et a., 1999; Wang et a., 23]. Using the addition theorem for cyindrica functions [Abramowitz and Stegun, 1965], the eectric fied radiated from the monopoe in Region can be expressed in the coordinate system (r, f ) as the incident fied for the refecting cyinder: Ez inc ¼ 1 X N1 A n g 2 n jwe n¼ npz X 1 cos a t J t ðg n r Þe jtf ; ð4aþ where a t = H t (2) (g n D)e itq r. The corresponding scattered fied can then be expressed as Ez sca ¼ 1 X N1 A n g 2 n jwe n¼ npz X 1 cos f t H ð2þ t ðg n r Þe jtf ; ð4bþ 3of9 Hf 1 sca ¼ XN n¼ ( A n cos npz X 1 f t jtdh ð2þ t ðg n r Þe jtf sinðq r fþ r 2 þ D 2 2rD cosðq r fþ ) þ H ð2þ t ðg n r Þe jtf g n ½r D cosðf q r ÞŠ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : r 2 þ D 2 2rD cosðf q r Þ ð9þ The tota eectromagnetic fieds in Region can then be obtained by summing the incident and scattered fieds together as foows: tota ¼ inc E þ E E z Hf ¼ tota z þ inc z sca sca : ð1aþ ð1bþ [17] The eectromagnetic fieds in Region can be expressed in terms of (r, f) using (8). Appying the

4 with c k ¼ g k H ð2þ 1 ðg k a Þ X1 b k ¼ H ð2þ ðg k h f H ð2þ t ðg k r Þþ X1 Þe jtf " # f t H ð2þ ðg k DÞ t i r¼a;r ¼D;f ¼ ð13bþ ð13cþ F mn ¼ Z h cos mpz np z h cos ð Þ dz d ð13dþ Figure 2. Surface current on the monopoe surface when the conducting cyindrica refector is absent (h =.24, a =.1, and =.81 ). boundary conditions that tangentia eectric fied and magnetic fied components must be continuous at r = a Ez tota ¼ 8 >< >: v D ; z D; ; D z h; E II z ; h z Hf ¼ II tota ð11aþ ð11bþ and Q m ¼ jwe v D 2 sin mpd 3 6 4d m D þ ð1 d m Þ 7 5; ð14þ mp with d mk = 1 for m = k, d mk = for m 6¼ k; e m = 1 for m =, and e m = 2 for m >. [18] Once the coefficient A n is known, the current distribution aong the monopoe surface can be given as I ¼ ¼ r¼a adf 8 Z X 2p N1 Ang n H ð2þ 1 ðg n rþcos npz XN1 An cos npz 9 >< >= n¼ n¼ ( ) X1 jtdh ð2þ t ðg n r Þe jtf sinðqr fþ ft r 2 þ D 2 2rD cosðqr fþ þ H ð2þ t ðg n r Þe jtf g n ½r D cosðf qrþš >: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi >; r 2 þ D 2 2rD cosðf qrþ r r¼a;r ¼D;f ¼df: ð15þ we can determine the expansion coefficients. By carrying out a simpe inner product on equation (11) and using the orthogonaity of trigonometric functions, we can obtain the foowing inear equation system for the expansion coefficient A k. X N 1 k¼ P mk A k ¼ Q m ; ð12þ where P mk ¼ c k d X N 2 n¼ e n g 2n J ðg 2n aþf mn F kn J 1 ðg 2n aþ e m g 2 k d mkb k ð13aþ 4of9 Figure 3. Simuated and measured return oss resuts of a monopoe antenna in front of a conducting cyinder (h = 34 mm and D = 28 mm).

5 Tabe 2. Convergence of the Return Loss and Input Impedance Resut With Respect to N 2, the Truncation Number of the Moda Expansion in Region II (f = 2. GHz, D = 6 mm, and N 2 = 8) N 2 Return Loss, db R in, W X in, W Figure 4. Simuated and measured H-pane radiation pattern of the monopoe antenna in front of a conducting cyinder (h = 34 mm, f = 2. GHz, z =, and D = 28 mm). The input impedance of the monopoe antenna can be cacuated as Z in ¼ V I ¼ V r¼a : ð16þ adf The refection coefficient or return oss aong the feed ine can then be found from G ¼ Z in Z c Z in þ Z c ; ð17þ where Z c is the characteristic impedance of the feed ine. In this paper, Z c =5W is chosen as the characteristic impedance of the coaxia feed ine. Tabe 1. Convergence of the Return Loss and Input Impedance Resut With Respect to N 1, the Truncation Number of the Moda Expansion in Region (f = 2. GHz, D = 6 mm, and N 2 = 6) N 1 Return Loss, db R in, W X in, W [19] Once the eectric current distribution aong the monopoe surface in (15) and the current aong the refecting cyinder s surface are obtained, one can then cacuate the far-fied radiation pattern of the monopoe antenna in front of a refecting cyinder. Image theory is empoyed in the process of computing the refections of the infinite ground pane. In the spherica coordinate system (r, q, f r ) the far-zone eectric fied has ony q component, so the far-fied radiation pattern can be cacuated from the foowing formua ~E ¼ jw m Z Z 4p ¼ ^a q jwm 4p 8 >< >: s ~J e jkr a e jkr r þr a e jkr r R ds þ m 4p Z h h Z r¼a r ¼a Z Zs ~J e jkr R ds sin qe jk ½ a cos ð f f rþsin qþz cos q Š dfdz 9 >= sin qe jk ½ Ra cos ð f f r Þsin qþz cos q Š df dz >; ; ð18þ where r and r are the distances of the observation point to and, and R and R are the distances of the observation point to the points on the surfaces of monopoe and conducting cyinder, respectivey. It shoud be mentioned that the surfaces s and s in (18) incude the surfaces of the monopoe and the conducting cyinder s images. Substituting (3) and (7) into (18), performing integration at the surfaces of the monopoe, the conducting cyinder and their images and considering Tabe 3. Effect of the Distance From the Introduced Conducting Boundary to the Monopoe End on the Antenna s Input Impedance (z in = R in + jx in, f = 2. GHz, and D = 6 mm) d R in, W X in, W of9

6 Figure 5. Effect of the distance between the monopoe and the refecting cyinder on the antenna s return oss. that the radius of the monopoe antenna is usuay sma compared to the monopoe s distance to the conducting cyinder, we have 8 E q ¼ jw m sin q 4p where c n ðþ¼2 q e jkr Z h D Z c n ðþ¼2 q r >< >: a XN 1 n¼ A n c n ðþd q 1 ðf r ; qþ g n H ð2þ 1 ðg n a þr a e jkd sin q cos ð q r f r Þ XN 1 ( ) 9 Þ h i f H ð2þ t ðg n r Þe jtf r ¼D;f ¼ n¼ cos npz cosðkz cos qþdz ð2aþ cos npz cosðkz cos qþdz ð2bþ d 1 ðf r ; qþ ¼ e jka cos ð f f rþsin q df ð2cþ d 2 ðf r ; qþ ¼ cosðf Þe jkr a cosðf f r Þsin q df ð2dþ 3. Numerica Resuts [2] In order to verify the formuation derived and the program written, an experimenta mode is constructed and its return oss and radiation pattern are measured. The parameters of the constructed mode are given as foows: a = 1.5 mm, R a = mm. It shoud be mentioned that these parameters wi be used in our numerica studies throughout the paper. In the theoretica anaysis, the ground pane and the introduced upper conducting pate are assumed to be of infinite extent, whie a finite ground pane of size 6cm 6cm is empoyed and no top conducting pate is present in the experiment. The ength of the conducting cyinder used in our experiment is = 25 mm. The measurement of the monopoe antenna s return oss and radiation pattern is conducted in an anechoic chamber using a vector network anayzer 853A. [21] Figure 2 iustrates the current distribution aong the monopoe surface when the refecting cyinder is absent. Our cacuated resuts are compared with those in the iterature [Morgan et a., 199]. It is seen that they are in exceent agreement, which vaidates our formuation and the written program. A n g n c n ðþ q þ X1 H ð2þ 3 1 ðg n DÞd 2 ðf r ; qþ 7 f t H ð2þ ðg n R a Þd3 t ð f 5 r; qþ t >= ; [22] Figure 3 shows the simuated and measured return oss resuts of a monopoe antenna in front of a conducting cyinder. It can be seen that our cacuated resuts agree very we with measured ones, which verifies the vaidity of our described formuation and the deveoped program. A comparison of our cacuated radiation pattern with measured one is presented in Figure 4. Again, a good agreement between them is observed and this aso vaidates our proposed anaysis. [23] Tabes 1 and 2 examine the convergence behavior of the return oss and input impedance resuts with Tabe 4. Effect of the Refector s Distance to the Monopoe on the Antenna s Return Loss and Input Impedance at f = 2. GHz D, mm Return Loss, db R in, W X in, W >; (19) d3 t ð f r; qþ ¼ e jtf e jkr a cosðf f r Þsin q df : ð2eþ of9

7 Figure 6. Effect of the distance between the monopoe and the refector on the antenna s H-pane radiation pattern at f = 2. GHz. Figure 7. Effect of the cyindrica refector s radius on the antenna s return oss at D = 6 mm. respect to the truncation numbers N 1 and N 2 for the expansion expressions of the fied components in Region and Region II (Figure 1), respectivey. It is seen from Tabe 1 that the input impedance resut does not change much when N 1 increases from 8 to 1. Simiary, it is noticed from Tabe 2 that N 2 = 8 wi be good enough to ensure convergent resuts. In fact, the vaue of the truncation number N 1 is reated to the spacing between the top conducting pate and the ground pane. The bigger the spacing between these two panes is, the arger is the number of N 1 required. [24] Regarding the truncation number t of the expansion expression in (4), it is dependent upon the eectric radius of the conducting cyinder: the bigger the radius is; the arger is the number needed. We usuay truncate the series on the basis of experience and numerica test. For exampe, t is from 1 to 1 when the radius of the conducting cyinder is.1 or ess. [25] Tabe 3 examines the effect of the introduced upper conducting pate on the antenna s input impedance. It is seen that the distance from the upper conducting boundary to the monopoe s top end has itte effect on both the rea and imaginary parts of the antenna s impedance, which vaidates our assertion that the introduction of the conducting pate has itte effect on the anaysis of the monopoe antenna. [26] The effect of the distance between the monopoe antenna and the cyindrica conducting refector on the antenna s return oss is examined in Figure 5. Three cases corresponding to D = 28. mm, D = 42.8 mm and D = 6. mm are shown. It is seen that the distance has a noticeabe effect on the antenna s return oss because the scattered fieds by the conducting cyinder can radiate back to the monopoe, change the current distribution aong the monopoe surface and in turn infuence the antenna s impedance or return oss. Tabe 4 aso provides some information on the effect of this distance on the antenna s return oss, which shows that the return oss is a minimum around D = 42.8 mm. [27] Figure 6 shows the effect of the distance between the monopoe and the conducting refector on the antenna s radiation pattern. Tabe 5 provides the antenna s gain vaues at and the ratio of forward radiation to backward radiation. It indicates that the gain at and Tabe 5. Effect of the Refector s Distance to the Monopoe on the Antenna s H-Pane Radiation Pattern (f = 2. GHz and z =) D, mm Gain at, db Forward/Backward Radiation Ratio Tabe 6. Effect of the Cyindrica Refector s Radius to the Monopoe on the Antenna s Return Loss and Input Impedance (f = 2. GHz and D = 6 mm) R a, mm Return Loss, db R in, W X in, W of9

8 the forward to backward radiation ratio decrease with the increase of the distance from the refector to the monopoe. [28] Figure 7 presents the effect of the radius of the cyindrica conducting refector on the antenna s return oss. Three cases are shown corresponding to R a = 1 mm, R a = mm and R a = 3 mm. It is noticed that with the increase of the radius of the refecting cyinder, the return oss wi increase at the resonant frequency, though the change is indeed sma. Tabe 6 iustrates the effect of the refector s radius on the antenna s return oss. [29] Figure 8 shows the effect of the cyindrica refector s radius on the antenna s radiation pattern. Tabe 7 tabuates the variation of the monopoe s gain and forward/backward radiation ratio with respect to the refecting cyinder s radius. It is seen that the antenna s gain and forward to backward radiation ratio increase when the radius of the refecting cyinder increases. This is expected because when the refecting cyinder becomes bigger more radiation wi be concentrated in the forward direction. 4. Concusion [3] The radiation characteristics of a monopoe antenna in front of a conducting cyinder driven a by a gap source over a ground pane has been studied in the paper. The formay exact moda expansion method and scattering matrix method have been empoyed to anayze the antenna probem. Numerica resuts for the monopoe s return oss and radiation pattern are in good agreement Figure 8. Effect of the cyindrica refector s radius on the antenna s H-pane radiation pattern at f = 2. GHz and D = 6 mm. 8of9 Tabe 7. Effect of the Cyindrica Refector s Radius to the Monopoe on the Antenna s H-Pane Radiation Pattern (f = 2. GHz and D = 6 mm) R a, mm with measured data. The effects of the refecting cyinder s radius and its distance from the monopoe on the antenna s radiation performance have been examined. Simuation resuts show that both the refector s radius and its distance from the monopoe can significanty affect the antenna s gain and forward to backward radiation ratio. It aso suggests that a shaped radiation beam may be obtained by pacing a number of refecting cyinders of suitabe radii at appropriate distances, which wi the subject of a future paper. References Gain at, db Forward/Backward Radiation Ratio Abramowitz, M., and I. A. Stegun (1965), Handbook of Mathematica Functions, Dover, Mineoa, N. Y. Biakowski, M. E. (2), On the ink between top-hat monopoe antennas, disk-resonator diode mounts, and coaxia-towaveguide transitions, IEEE Trans. Antennas Propag., 48, Biakowski, M. E. (22), Anaysis of a circuar patch antenna radiating in a parae-pate radia guide, IEEE Trans. Antennas Propag., 5, Chang, D. C. (1968), On the eectricay thick monopoe: Part 1. Theoretica soutions, IEEE Trans. Antennas Propag., 16, Chen, Z. N., K. Hirasawa, and K. Wu (21), A nove topseeve monopoe in two parae pates, IEEE Trans. Antennas Propag., 49, Do-Nhat, T., and R. H. MacPhie (1991), The admittance of a monopoe antenna fed through a ground pane by a coaxia ine, IEEE Trans. Antennas Propag., 39, King, R. W. P. (1956), The Theory of Linear Antennas, Harvard Univ. Press, Cambridge, Mass. Morris, M. E. (198), Anaysis of a finite ength tubuar monopoe antenna driven from a coaxia ine, Tech. Rep. SAND , Sandia Nat. Lab., Abuquerque, N. M. Morgan, M. A., and F. K. Schwering (1994), Eigenmode anaysis of dieectric oaded top-hat monopoe antennas, IEEE Trans. Antennas Propag., 42, Morgan, M. A., R. C. Hurey, and F. K. Schwering (199), Computation of monopoe antenna currents using cyindrica harmonics, IEEE Trans. Antennas Propag., 38, Otto, D. V. (1967), The admittance of cyindrica antennas driven from a coaxia ine, Radio Sci., 2(9),

9 Ragheb, H. A., and M. Hamid (1987), Simuation of a cyindrica refector by conducting circuar cyinder, IEEE Trans. Antennas Propag., 35, Schwering, F., N. N. Puri, and A. Stavridis (1982), The moda expansion soution of a top oaded monopoe, IEEE Trans. Antennas Propag. Symp., Shen, Z., and R. H. MacPhie (1996a), Moda expansion anaysis of monopoe antennas driven from a coaxia ine, Radio Sci., 31(5), Shen, Z., and R. H. MacPhie (1996b), Rigorous evauation of the input impedance of a seeve monopoe by moda expansion method, IEEE Trans. Antennas Propag., 44, Tayeb, G., and D. Maystre (1997), Rigorous theoretica study of finite size two-dimensiona photonic crystas doped by microcavities, J. Opt. Soc. Am. A Opt. Image Sci., 14(12), Wang, Q. X., Y. J. Zhang, and E. P. Li (23), Anaysis of finite-sized 2D coated eectromagnetic bandgap structures by scattering matrix method, paper presented at 6th Internationa Symposium on Antennas, Propagation and Eectromagnetic Theory (ISAPE 23), Inst. of Eectr. and Eectron. Eng. Antennas and Propag. Soc., Beijing, Aug. Yonekura, J., M. Ikeda, and T. Baba (1999), Anaysis of finite 2-D photonic crystas of coumns and ightwave devices using the scattering matrix method, J. Lightwave Techno., 17(8), Y. Lu, Z. Shen, and Q. Wang, Schoo of Eectrica and Eectronic Engineering, Nanyang Technoogica University, Nanyang Avenue, Singapore (ezxshen@ntu.edu.sg) 9of9