AIR FORCE INSTITUTE OF TECHNOLOGY

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1 MATERIAL PERTURBATIONS TO ENHANCE PERFORMANCE OF THE THIELE HALF-WIDTH LEAKY MODE ANTENNA DISSERTATION Jason A. Giad, Majo, USAF AFIT/DEE/ENG/08-04 DEPARTMENT OF THE AIR FORCE AIR UNIVERSITY AIR FORCE INSTITUTE OF TECHNOLOGY Wight-Patteson Ai Foce Base, Ohio APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

2 The views expessed in this thesis ae those of the autho and do not eflect the official policy o position of the United States Ai Foce, Depatment of Defense, o the U.S. Govenment.

3 AFIT/DEE/ENG/08-04 MATERIAL PERTURBATIONS TO ENHANCE PERFORMANCE OF THE THIELE HALF-WIDTH LEAKY MODE ANTENNA DISSERTATION Pesented to the Faculty Gaduate School of Engineeing and Management Ai Foce Institute of Technology Ai Univesity Ai Education and Taining Command in Patial Fulfillment of the Requiements fo the Degee of Doctoate of Philosophy Jason A. Giad, BS, MS Majo, USAF June 2008 APPROVED FOR PUBLIC RELEASE; DISTRIBUTION UNLIMITED

4 AFIT/DEE/ENG/08-04 MATERIAL PERTURBATIONS TO ENHANCE PERFORMANCE OF THE THIELE HALF-WIDTH LEAKY MODE ANTENNA Appoved: Jason A. Giad, BS, MS Majo, USAF // signed // D. Michael J. Havilla (Chaiman) Date // signed // D. Richad Decko (Dean s Repesentative) Date // signed // D. Pete Collins (Membe) Date // signed // D. William Bake (Membe) Date // signed // D. Gay A. Thiele (Membe) Date Accepted: // signed // M. U. THOMAS Date Dean, Gaduate School of Engineeing and Management

5 AFIT/DEE/ENG/08-04 Abstact Micostip taveling-wave antennas, often efeed to as leaky-wave antennas, have been shown to adiate when the dominant o fundamental mode is suppessed and the fist highe-ode mode is excited. One such micostip vaiation is the Thiele Half- Width (THW) antenna, which opeates fom GHz fo this eseach. Inceasing the bandwidth ove which the THW antenna adiates is desied, as is a fundamental undestanding of the popagation chaacteistics ove this egion. This dissetation seeks to vay o petub the mateial and physical popeties of the THW antenna, including stip-width vaiations and modifications of the substate laye, to achieve these esults. Thee methods will be used to examine the effects of vaying the mateial and physical popeties of the THW antenna and extact the adiation and popagation chaacteistics. Fo all thee methods, the esultant pecentage bandwidth impovement and/o degadation will be computed. The fist method to be used is the tansvese esonance method which models the coss section of the micostip stuctue as a tansmission line system to accuately pedict the tansvese popagation behavio fo a micostip antenna. It will be shown how the analysis can be used to extact the desied bandwidth of the adiation egime. The second method to be used is the Finite Diffeence Time Domain (FDTD) method. The popagation chaacteistics fo each antenna configuation can be extacted fom the longitudinal field pofile unde the antenna and compaed to the baseline THW antenna. iv

6 The last method involves pefoming a fequency domain full-wave analysis to deive the bandwidth infomation. Hee, the stip and mateial petubation ae modeled as equivalent suface and volume cuents adiating in a gounded-slab backgound envionment. Using vecto potentials and the gounded-slab Geen s function, a coupled integal equation fomulation is developed and subsequently solved via the Method of Moments (MoM). Examination of the natual-mode solution of the MoM matix leads to the popagation constant infomation. It is fom the popagation constant infomation that the modes of the antenna, the necessay length of the antenna, and the bandwidth of the antenna can be detemined. Using the popagation constants deived fom these methods, a complex plane analysis will be pefomed to show the migation of pole and banch point singulaities fom one quadant in the complex plane to anothe. By pefoming this analysis, insight can be gained into what factos dive the popagation and adiation popeties of the stuctue. The main contibutions found within this wok come in the ability to easily and quickly modify a micostip stuctue s mateials fo design puposes, efficiently detemine the popagation constants that define its opeating egimes, and gain physical insight into how these mateials influence the adiation and popagation chaacteistics of the stuctue. Anothe impotant contibution fom this eseach is gaining an oveall bette undestanding of the diffeent opeating egimes of the micostip antenna. v

7 Lastly, a FDTD code was developed to analyze the modified THW antenna. The Tansvese Resonance Code was also modified in ode to analyze this same antenna, but it will be shown late in this document that this code does not popely pedict the popagation chaacteistics of the modified stuctue. vi

8 Acknowledgments I would like to expess my sincee appeciation to my faculty adviso, D. Michael Havilla, fo his guidance and suppot thoughout the couse of this thesis effot. The insight and expeience was cetainly appeciated. I would, also, like to thank my sponso, D. Gay Thiele, fom the Ai Foce Reseach Laboatoy fo both the suppot and latitude povided to me in this endeavo. Lastly, I would like to thank my wife and my daughtes fo thei love and suppot duing this vey igoous pocess. Jason A. Giad vii

9 Table of Contents Page Abstact iv Acknowledgements vii Table of Contents viii List of Figues List of Tables x xiii I. Mateial Petubations to Enhance the Pefomance of the THW Antenna Intoduction Poblem Statement Potential Impact The Leaky Wave Antenna Popagation Constants Micostip Antenna Antenna Length Modes of Opeation Liteatue Seach of Poposed Methods fo Inceasing BW Poposed Method fo Inceasing Bandwidth II. Tansvese Resonance and Finite Diffeence Time Domain Intoduction Tansvese Resonance Tansvese Resonance Method Tansvese Resonance Code Vaiations of Micostip Stuctue Using TRC TRC Conclusions Finite Diffeence Time Domain Backgound FDTD Set-Up Extaction of Popagation Constants fom FDTD FDTD Methodology and Results III. Geen s Function Fomulation fo Full-Wave Analysis Intoduction Full-Wave Solution Spectal Coefficients viii

10 Geen s Function IV. Electic Field Integal Equation Fomulation and MoM Solution Intoduction Electic Field Integal Equation Fomulation Eigenmode Cuent Insetion of Geen s Functions into EFIEs Method of Moments MoM Implementation Intepetation of MoM Matix Elements Basis Functions Integation Path Newton Root Seach V. Results Intoduction MoM Solutions fo the Baseline Micostip Antenna MoM Solutions fo the Modified Micostip Antenna Physical Insight fom Radiation Regime Results Complex Plane Analysis of Poles and Banch Points Constaint Equation Analysis Conclusions fom Complex Plane Analysis VI. Conclusions Intoduction Oveview of Reseach Effot Recommendations fo Futue Reseach Conclusions Appendix A. Tansvese Resonance Method Appendix B. Micostip Antenna Chaacteistics Using TRC Appendix C. MoM Matix Element Extaction fom EFIEs Appendix D. Matlab Code Implementation Bibliogaphy ix

11 List of Figues Figue Page. Menzel's leaky-wave antenna design Thiele Half-Width antenna design Electic field distibution of dominant mode, EH Electic field distibution of fist highe-ode mode, EH Micostip popagation egimes Two methods used to modify the oiginal step discontinuities Leaky-wave micostip antenna loaded by capacitos Micostip with vaied ε 3 and µ 3 unde top conducto Repesentative tansvese t-line system fo the baseline THW antenna 2 0. TRC appoximation of leaky-wave pop const fo baseline THW antenna 23. Repesentative tansvese t-line system fo the modified THW antenna Popagation constants fo ad egime using TRC fo ε =.33 and ε 3 = Popagation constants fo ad egime using TRC fo ε =.03 and ε 3 = Popagation constants fo ad egime using TRC fo ε =3.33 and ε 3 = Modified THW antenna with PML layes and souce location Modified THW antenna with PML layes and souce location Modified THW antenna with inceased slab with PML and souce location Modified TFW antenna with PML layes and souce location FDTD y-diected electic field data fo THW antenna taken longitudinally Natual log of FDTD y-diected electic field data fo THW antenna Natual log of FDTD y-diected electic field data and best-fit α and β cuve 39 x

12 22. Nomalized FDTD y-diected electic field data and best-fit α and β cuve TRC and FDTD popagation constants fo baseline THW antenna FDTD pediction of popagation constants fo modified THW antenna FDTD pediction of popagation constants fo micostip with ε 3 = FDTD y-diected electic field data nea cut-off fo adiation egime (6 GHz) Natual log of FDTD y-diected electic field data nea cut-off (6 GHz) Natual log of FDTD y-diected electic field data and best-fit α and β cuve Coss-section of micostip backgound envionment Souce (y ) and obsevation (y) points and paths of inteaction Coss-sectional view of the modified full-width leaky micostip antenna The fou egions of inteaction between souce and obseve locations FDTD y-diected electic fields within the coss-section of THW antenna Riemann sheets in the complex p 2 and p planes Typical pole and banch point locations fo Bound Regime (lossy case) Migation path fo the pole and banch points fo Bound Regime Typical pole and banch point locations fo Radiation Regime (lossless case) Popagation constants fo the Baseline THW Antenna ove Radiation Regime Popagation constants fo adiation egime using MoM ε =.33 and ε 3 = Popagation constants fo adiation egime using MoM ε =.03 and ε 3 = Popagation constants fo adiation egime using MoM ε =3.33 and ε 3 = Popagation constants fo adiation egime using MoM fo ε 3 = ξ plane poles and banch points fo baseline THW antenna ove fou egimes 29 xi

13 44. ξ plane poles and banch points fo baseline THW antenna ove two egimes ξ plane poles and banch points fo baseline THW antenna using ε 3 = ξ plane poles and banch points fo modified THW antenna using MoM Close-up view of ξ plane poles and banch points fo modified THW ΜοΜ Close-up view of ξ plane poles and banch points fo modified THW FDTD ξ plane poles and banch points fo modified THW antenna using TRC ξ plane poles and banch points fo modified THW antenna using MoM Close-up view of ξ plane poles and banch points fo modified THW ΜοΜ Close-up view of ξ plane poles and banch points fo modified THW FDTD ξ plane poles and banch points fo modified THW antenna using TRC Complex p 2 -plane values ove ξ integation path fo baseline THW antenna Complex ξ p, ξ b, ζ z, and p 2 values fo baseline THW antenna fo fou egimes Complex ξ p, ξ b, ζ z, and p 2 values fo baseline THW antenna fo fou egimes Complex ξ p, ξ b, ζ z, and p 2 values fo baseline THW antenna fo fou egimes Complex ξ p, ξ b, ζ z, and p 2 values fo baseline THW antenna fo fou egimes Complex ξ p, ξ b, ζ z, and p 2 values fo baseline THW antenna fo fou egimes Repesentative tansvese t-line system fo the baseline THW antenna TRC leaky-wave popagation constants fo baseline THW antenna Multiple leaky-wave micostip antennas Leaky-wave antennas with vaying conducto widths Matlab Method of Moments code implementation flowchat xii

14 List of Tables Table Page. Test Matix fo Vaied Pemittivity Unde Top Conducto fo TRC Pedictions TRC Pedicted Bandwidth fo Vaied Pemittivity Unde Top Conducto Test Matix fo Vaied Pemittivity Unde Top Conducto Using FDTD FDTD Pedicted Bandwidth fo Vaied Pemittivity Unde Top Conducto Basis Functions Used To Repesent Unknown Cuents And Electic Fields BW of Baseline THW Antenna Using TRC, FDTD, and Method of Moments 0 7. Test Matix fo Modified THW Antennas Using Method of Moments Radiation Regime Bandwidth of Modified THW Antennas Using Methods 7 9. Micostip Antenna Widths and Pemittivities xiii

15 Chapte MATERIAL PERTURBATIONS TO ENHANCE PERFORMANCE OF THE THIELE HALF-WIDTH LEAKY MODE ANTENNA. Intoduction Eve since Heinich Hetz's geneation and detection of the fist mete-wavelength adio waves in 888 and Guglielmo Maconi's sending adio waves acoss the Atlantic Ocean in 90, the antenna has been a citical component in the ability to efficiently and pedictably tansfe data fom one device to anothe. This is tue whethe it be lowfequency voice communication between adios o ulta high-fequency data tansfe fom a gound station to a satellite obiting the Eath. Moden aicaft ely heavily on being able to etieve data fom GPS satellites, communicate with ai taffic contolles on the gound, as well as detect nea-by aicaft though the use of antennas positioned on the aicaft's suface. Typically, these antennas ae bulky, heavy, and costly due to thei complexity. They must be able to opeate in the equied fequency egime, have a gain and antenna patten that meets thei equied use (omni-diectional, naowbeam, boadband, etc.), and efficiently adiate and eceive EM enegy. In addition to all of these, with the advent of stealth aicaft, the need fo these antennas to be low obsevable becomes inceasingly moe impotant. One such antenna that meets most of these equiements is the micostip antenna.

16 .2 Poblem Statement The ability of a micostip stuctue to adiate as a leaky wave antenna has been widely investigated eve since Emet fist documented the existence of a adiation egion nea the cutoff of the highe-ode modes back in the 970s []. Since this discovey, micostip antennas have evolved to the point whee they ae used fo many militay applications. Due to thei elatively small size, ability to confom to a suface, lowe ada coss section, and low cost; micostip antennas have many benefits fo being used on aicaft sufaces. Howeve, with these benefits comes one significant deficiency: naow bandwidth. The taveling wave micostip antenna has been extensively analyzed by the electomagnetic community ove the past 28 yeas by Oline et al. [27, 42, 84, 89], Menzel [3], Nyquist et al. [3, 4, 6-8, 20-22, 3-33, 53, 57, 60], and Jackson et al. [24, 26, 27, 37, 84, 89]. Its baseline pefomance is well known and documented. Ove this time, eseaches have tied numeous methods to tackle the challenge of impoving its pefomance (bandwidth, gain, VSWR, etc.) [6-90]. Some of these methods include using an aay of micostip antennas (both in-plane and stacked), tapeing the width of the top conducto longitudinally, and placing capacitos peiodically along the edges of the top conducto. Howeve, all of these ideas have been met with limited esults. Why is this? What is the dominating facto that pevents the micostip fom achieving geate bandwidth? What othe potential methods can be employed to impove bandwidth? This eseach effot looks to investigate and answe these questions. 2

17 The pupose of this chapte is to povide the backgound necessay to undestand the poblem at hand and to give insight into why cetain methods wee chosen to investigate this poblem. A shot histoy of leaky-wave antennas will be given, including the taveling wave theoy behind thei existence. Seveal vaiations to the baseline micostip antenna will be discussed, including thei advantages and disadvantages..2. Potential Impact Although technology advancements ove the last 50 yeas have allowed fo the development of lighte-weight mateials, guidance and navigation systems that can fit onto a micochip, and smalle, moe efficient engines; as the available space to place antennas o any othe device on moden aicaft has become moe and moe limited. It has theefoe become inceasingly moe impotant to look to educe the size and weight of all of the aicaft's components. If the micostip antenna pefomance can be impoved, a single micostip antenna may be able to eplace two olde antennas, thus feeing up space fo othe mission essential equipment. Additionally, fewe antennas on an aicaft can also lead to less impact on the ada coss section of that platfom. Finally, impoved bandwidth pefomance will lead to enhanced ada imaging and taget detection capability..3 The Leaky-Wave Antenna A micostip antenna does not inheently adiate as a taveling-wave antenna. The fields poduced by the dominant EH 0 mode do not decouple fom the stuctue. It is 3

18 only until the dominant mode is blocked o esticted within the antenna that it can opeate in a highe-ode mode. It is in these highe-ode modes that the fields ae able to decouple fom the suface and, thus, adiate fom the stuctue. When this happens, the antenna is said to be opeating as a leaky-wave antenna..3. Popagation Constants Befoe the leaky-wave antenna can be analyzed, an undestanding of the electomagnetic theoy behind the popagation chaacteistics of the antenna is equied. Fom the popagation chaacteistics, the specific modes of opeation fo the micostip taveling-wave antenna can be deived and subsequently identified. As electomagnetic waves popagate, they must satisfy Maxwell's cul equations fo time-vaying hamonic fields H = J + jωd E = m jωb (.) (.2) whee J is the electic cuent density and m is the fictitious/equivalent magnetic cuent density. The fist equation is commonly called Ampèe's law and the second Faaday's law. Fo simple media (linea, homogeneous, isotopic), the magnetic and electic fields satisfy the elations i c m= m + m B = µ H D= ε E c m = σ m H (.3) (.4) (.5) (.6) 4

19 c J = σ E i c J = J + J (.7) (.8) with the i and c supescipts meaning impessed and conduction, espectively. Substituting (.3), (.5), and (.6) into (.2) leads to = i E m σ mh jωµ H i σ E m j ( m = ωµ + ) H jω = i E m jωµ ch (.9) (.0) (.) whee µ c is the effective complex pemeability. Similaly, substitution of (.4), (.7), and (.8) into (.) gives = + + i H J σ E jωε E i σ H = J + jωε ( + ) E jω = + i H J jωε ce (.2) (.3) (.4) whee ε c is the effective complex pemittivity. Taking the divegence of equation (.) poduces = ( E m i jωµ ch) i 0 = m jωµ H c (.5) (.6) The divegence of the cul (on the left hand side) is mathematically zeo. Using the continuity elation = i i m jωq mv (.7) in (.6) leads to the following divegence elation fo the magnetic field H, namely 5

20 q H = µ i mv c (.8) Similaly, taking the divegence of (.4) and using the second continuity elation i J = jωq i ev (.9) poduces the following divegence elation fo the electic field E, that is q E = ε i ev c (.20) Next, taking the cul of equation (.) gives = H i E m jωµ c and substitution into equation (.4) with the aid of the vecto identity 2 E = ( E) E (.2) (.22) leads to 2 i i ( E) E = m jωµ c( J + jωε E c ) (.23) Using equation (.20) and defining 2 2 k ω µε c c = gives q i ev 2 i i ( ) E = m jωµ cj + k 2 E εc Reaanging and using equation (.9) poduces the wave equation fo E 2 2 ( i i J ) E+ k E = jωµ cj + m i jωε c (.24) (.25) Similaly, the wave equation fo the magnetic field, H, can be deived as 2 2 ( i i m) H + kh= jωεcm J jωµ c i (.26) 6

21 Fo both wave equations, the souce tems on the ight-hand-side ae vey involved and complicated. Because of this, one method to solve fo the fields is to use the Hetzian potential method, which will be descibed late in this dissetation. The natual esponse of a system is examined by setting the impessed cuents in (.25) and (.26) equal to zeo. It will be shown late in this dissetation how the natual esponse solution leads to the desied popagation constants of leaky wave micostip antennas. If thee ae no impessed cuents, this esults in i i 2 2 J, m = 0 E+ k E = 0 (.27) with simila esults fo the magnetic field H. In ode to gain a bette undestanding of what is happening to the fields as they popagate thoughout an antenna and the space suounding it, it is suitable to use sepaation of vaiables to wite the wavenumbe, k, in tems of the thee sepaate constants, k x, k y, and k z. Doing so will show the influence in the x-, y-, and z- diections on the oveall wavenumbe as it is applied to (.25) and (.26). The wavenumbe can then be found using the well-known constaint equation k = k + k + k 2 (.28) 2 2 x y z whee, in geneal, each constant takes the fom of a complex popagation facto k η = β η jα η η = x, y, z (.29) with β being the phase constant, and α being the attenuation constant along the η diection. How these wave numbes inteelate in ode to define and physically 7

22 undestand the fou egions of opeation of the micostip taveling-wave antenna will be descibed late in this dissetation..3.2 Micostip Antenna Typically, a naowband antenna will have cuents that popagate down the length of the antenna, hit the end of the stuctue, and eflect back in the opposite diection. These cuents then constuctively and destuctively add to poduce standing waves along the length of the antenna. The cuents at the ends of the antenna ae diven to zeo, thus limiting how many wavelengths of a cetain fequency can fit along the length of the antenna. This limiting facto dives these standing wave antennas to be naow in bandwidth. If the antenna could be infinite in length (in theoy), designed to shed enegy befoe eaching the end, o "matched" at the ends so as to eliminate any cuent fom being eflected and standing waves fom foming along thei length, this would incease the bandwidth of the antenna. This type of adiating stuctue is efeed to as a taveling wave antenna. The micostip antenna is a simple, cost-effective stuctue fo adiating enegy as a taveling wave [2]. Howeve, thee main poblems exist: () the antenna must be finite in length fo pactical applications; (2) the antenna will only adiate enegy when it is opeating in the fist highe-ode mode, called the EH mode, o highe and not in the dominant mode; and (3) though the bandwidth is geate than that fo a standing wave antenna, the bandwidth fo the micostip antenna is still vey limited [2]. The following subsections addess these limitations and how they can be mitigated. 8

23 .3.3 Antenna Length Fo obvious easons, an antenna that must be extemely long in length in ode to avoid standing waves fom foming is neithe ealistic no pactical. Given this, the antenna will be finite in length and, thus, some cuent will be eflected when it hits the physical end of the antenna. As stated befoe, any cuent that is eflected back down the length of the antenna will constuctively and destuctively combine with the fowad taveling cuent to cause a standing wave. If this is the case, how can the micostip antenna opeate as a taveling wave antenna? The answe lies in the attenuation constant, α z, which is a measue of how much enegy escapes at a cetain fequency as the wave popagates along the guiding axis (i.e., the z-axis). As the enegy popagates down the length (l) of the antenna, it is attenuated at a ate of z e α l due to enegy leaking fom the stuctue. As this enegy is being "shed-off", less and less is being popagated down the length, to the point whee vey little enegy actually eaches the end of the antenna. If geate than 90% of the enegy has been attenuated befoe it eaches the end of the antenna, the micostip will suppot a taveling wave [5]. Fo the baseline THW antenna used in this eseach, the length equied is 20cm. 9

24 Figue : Menzel's leaky-wave antenna design [4] y w= 7.5 mm l= 220 mm x ε d= mm ε z Figue 2: Thiele Half-Width antenna design.3.4 Modes of Opeation The idea of using a micostip antenna to adiate EM enegy has not been aound as long as one might think. It was not until the 970s that Emet documented the existence of a adiation egion nea the cutoff of the highe-ode modes. Late in the same decade, Menzel pesented a pape that looked into this phenomenon futhe by puposely suppessing the dominant mode in his antenna design by cutting seven tansvese slots in the top conducto (see Figue ) [3]. Recently, D Gay Thiele of Analytic Designs, Inc. poposed a new micostip antenna that effectively blocked the 0

25 EH 0 Dominant mode Figue 3: Electic field distibution of dominant mode, EH 0 [4] EH Fist highe-ode mode Figue 4: Electic field distibution of fist highe-ode mode, EH [4] dominant mode. Instead of using tansvese slots, the Thiele Half-Width Antenna (THW) uses a longitudinal wall down the cente of the top conducto (see Figue 2). By blocking the dominant mode, both antennas ae foced to opeate in the fist highe-ode leaky mode. Choosing fequencies popely, these antennas will adiate as taveling wave antennas. If the fequencies ae too low, all fields will evanesce at the input of the antenna. If they ae too high, the fields will be bound to the stuctue and not adiate. By

26 looking at the elationship of the phase constant, β z, to the attenuation constant, α z, ove these fequencies, the opeating egions of the antenna can be defined. The dominant mode of the micostip, EH 0, will not adiate due to the natue of the fields that it suppots. Since the electic field lines up in the same diection unde the top conducto (see Figue 3), these fields will not be allowed to de-couple fom the stuctue and, thus, do not adiate. The highe-ode modes of the micostip (specifically, the odd numbeed higheode modes) will not be bound to the stuctue as with the dominant mode and will actually decouple due to the phase evesal at the longitudinal cente of the top conducto (see Figue 4). Highe-ode modes will popagate in thee identifiable egions o egimes: adiation, suface, and bound egimes. Below the cutoff fequency of the adiation egime, the attenuation constant, α z, will dominate, thus the micostip will appea to be a eactive load at the end of the input line. This non-popagating egion is called the eactive egime. As stated ealie, the constaint equation (.28) dictates the popagation behavio along the x-, y-, and z-diections, espectively with each complex wave numbe being defined as (.29). Focusing on the diection of popagation (the z-diection), a plot of the phase constant, β z, and the attenuation constant, α z, vesus fequency will eveal the thee popagation egimes: adiation, suface, and bound. 2

27 α z /k o β z /k o Figue 5: Micostip popagation egimes [5] Figue 5 shows the plot of nomalized (with espect to k 0 ) β z and α z vesus fequency. As stated befoe, fom 0 GHz up to the cutoff fequency, f c, this egion is called the eactive egime. It is chaacteized by a vey lage α z component, causing the micostip to appea as a eactive load at the input of the micostip. At the cutoff fequency, the phase and attenuation constants ae equal (β z = α z ). This is the stat of the adiation egime. In this egime, popagation is occuing in all diections. The adiation egime continues until the phase constant gows lage than k 0. At this point, thee is no attenuation in the diection of popagation (α z = 0) and lage attenuation in the adiated diection (α y ). Because of this, fields will no longe adiate. Howeve, they will continue to popagate in the longitudinal and tansvese diections. This is called the suface egime. This egime continues until β z > k. At this point, thee 3

28 is a lage attenuation in the tansvese diection (α x ) causing all fields to be bound unde the top conducto and only popagate in the longitudinal diection. This is called the bound egime. It is thus ecognized that the computation of the vaious popagation constants is cucial to undestanding the opeational egimes of the micostip leaky-wave antenna. Consequently, a majo thust of this dissetation is the calculation of these popagation constants..3.5 Liteatue Seach of Poposed Methods fo Inceasing Bandwidth In ode to tackle the challenge of inceasing the bandwidth of the micostip antenna, eseaches have tied numeous methods ove the past 28 yeas [6 90]. These ideas have been met with limited esults and have dawbacks that would hinde thei use on stealthy aicaft and missiles (size and ada signatue.). This section will highlight a few of the ideas developed by the antenna community, as well as talk to each one s applicability to stealth vehicle design. Given that the leaky-wave micostip antenna opeates with limited bandwidth (typically unde 20%) [2,6], the fist inclination would be to align seveal of these antennas (each of vaied widths) in paallel with each othe in ode to achieve the desied fequency coveage. The dawbacks of this include the need fo numeous antennas vesus a single antenna, the inceased costs associated with having to puchase numeous antennas, and inceased space equied to place these additional antennas on the aicaft. 4

29 Nalbandian's and Lee's wok shows that micostip antennas placed in an aay of oughly the same top-down coss-section as the THW antenna have the potential to incease the bandwidth of the adiation egion [7]. Howeve, the dawback of this wok was that the thickness of the micostip antenna went up fom 0.787mm up to.25cm. This incease in thickness could cause the antenna to have a highe Rada Coss Section (RCS), and thus make it impactical fo use on stealth aicaft. Futhe analysis is needed on this potential aea of eseach to show the tue impact on signatue, as well as bandwidth impovement ove the adiation egime. Anothe appoach to inceasing the bandwidth of the leaky wave micostip antenna is to modify the shape of the top conducto. By beaking-up the top conducto into sections of deceasing widths (Figue 6(a)), each section can be "tuned" to opeate ove a specific fequency egion. Howeve, with the intoduction of multiple widths in the same top conducto comes the intoduction of impedance mismatch and discontinuities that educe the opeating bandwidth and ceate spuious sidelobes [6]. In ode to match these sections and educe the discontinuity effects, the sections ae sepaated by tapeed tansition sections (Figue 6(b)) o by tapeing the top conducto linealy (Figue 6(c)). Modest bandwidth impovement is claimed by the authos by using these techniques, although the esults ae not clealy shown. Futhe analysis of this method is equied in ode to show the impact ove the adiation egime. The last method that has potential to incease the bandwidth of the leaky wave antenna is to peiodically place capacitos down the length of the top conducto (Figue 7). This method was shown by Luxey and Laheute to act as eithe a capacitive o an 5

30 (a) Oiginal multi-section micostip leaky-wave antenna (b) Inset a tapeed tansition section (c) Tape the steps linealy Figue 6: Two methods used to modify the oiginal step discontinuities. (a) The oiginal multisectional micostip leaky-wave antenna. (b) Insets additional tapeed sections between each two adjacent oiginal sections. (c) Tapeed linealy and the oiginal sections ae shown in solid lines. [6] inductive load (depending on the sepaation of the capacitos) that will move the β/k 0 = point to the ight (incease f high ) [8]. Although bandwidth impovement was not specifically claimed in the pape, the phase constant, β, was shown to shift as the distance between capacitos was vaied. No plots wee given showing the affects on the attenuation constant, α, and, thus, no conclusions can be dawn fom the pape egading how much bandwidth impovement can be gained fom this technique. Howeve, the manipulation of the popagation constants is the key towads obtaining bandwidth impovement ove the adiation egime. Because of this, the use of capacitos along the 6

31 W capacitos L gound plane top conducto h Figue 7: Leaky-wave micostip antenna loaded by capacitos [8] length of the top conducto should be consideed as a means fo potential bandwidth impovement. Futue investigation is equied. All of these methods mentioned offe, o claim to offe, some impovements to the bandwidth of the leaky-wave antenna. Howeve, they ae eithe incomplete o unclea as to what level of bandwidth impovement they can achieve within the adiation egime. Additionally, they do not give any insight as to how these new stuctues pomote adiation o how the changes in the stuctue influence the popagation and attenuation chaacteistics..3.6 Poposed Method fo Inceasing Bandwidth This eseach looks at a novel appoach to bandwidth impovement of the THW antenna modification to the mateial undeneath the top conducting stip (see Figue 8). 7

32 ε, µ ε 3, µ 3 ε, µ Figue 8: Micostip with vaied ε 3 and µ 3 unde top conducto The thought pocess fo choosing this appoach stems fom the opeating egimes themselves. When the taveling-wave antenna tansitions fom a suface mode to a bound mode, some physical phenomenology is causing the fields to become tapped undeneath the stip conducto. Why is this? Why is this tansition point located in the mateial unde the stip conducto at its edge? When the antenna tansitions fom a suface mode to a adiation mode, why is it that enegy is now allowed to adiate nomal to the suface vesus being tapped in the dielectic sheet? This eseach effot seeks to answe these questions. Intuitively, the eason fo choosing the antenna design seen in Figue 8 stems fom the phenomenology causing enegy to be physically tapped unde the stip conducto in the bound egime. This is simila to enegy being eflected at the junction of two diffeent mediums. If the mateial unde the stip conducto wee vaied, it might allow fo moe enegy to leak out into the dielectic sheet, thus pomoting suface wave leakage. If moe enegy wee allowed to ente the suface egime, then the thought is that this would lead to moe enegy being able to popagate away fom the suface in the adiation egime. Moe enegy adiating would tanslate to geate bandwidth due to inceased attenuation. 8

33 As shown in Figue 8, the baseline THW antenna will be modified slightly. Only the substate that lies diectly unde the top conducto will be modified to have a diffeent pemittivity (ε 3 ) than the pemittivity of the est of the dielectic slab (ε ). Both pemittivity values (ε and ε 3 ) will be aised and loweed and compaed to the baseline THW pefomance. In all cases, the attenuation constant, α z, and the phase constant, β z, will be computed. All thee egions ae assumed to be non-magnetic (µ = µ 3 = µ 0 ). The following chaptes will analyze the baseline antenna, as well as the modified micostip stuctue of Figue 8, using both computational as well as analytical methods. Insight into what causes the tansitions between the opeating egimes, as well as the ability to incease the bandwidth using the novel appoach descibed peviously is desied and will be investigated in this dissetation. 9

34 Chapte 2 TRANSVERSE RESONANCE AND FINITE DIFFERENCE TIME DOMAIN 2. Intoduction The simplistic natue of the micostip stuctue (stip conducto on a gounded dielectic slab) lends itself to being able to be analyzed using multiple methods. Fo example, it can be epesented as a tansmission line to be studied analytically o easily modeled in numeous compute codes to be studied computationally. Each method has diffeent advantages fo analyzing the micostip antenna, as well as limitations. This chapte will investigate the taveling-wave micostip antenna analytically, as well as computationally and compae both sets of data to known esults. The antenna stuctue will also be modified as descibed in the pevious chapte and analyzed again using the same analytical and computational methods to detemine the ability to impove/degade the bandwidth of the adiation egime. 2.2 Tansvese Resonance 2.2. Tansvese Resonance Method The simplistic natue of the taveling-wave micostip stuctue lends itself to being analyzed analytically. The antenna can be investigated in the x-y plane to find the tansvese popagation chaacteistics using the Tansvese Resonance Method [9]. 20

35 y W width = 2 2 W width = 2 2 ε, µ ε, µ x Z xl Z k 0, x Figue 9: Repesentative tansvese tansmission line system fo the baseline THW antenna Fom these, the longitudinal popagation chaacteistics can be easily calculated using the constaint equation. Looking at the micostip stuctue in the x-y plane, the coss section of the stuctue can be epesented as a simple tansmission line. Figue 9 shows the epesentative tansmission line system. Fo the tansmission line epesentation, the dielectic sheet (of pemittivity ε and pemeability µ ) becomes a load of impedance Z xl at the left side of the tansmission line. This impedance is an appoximation deived by Chang and Kueste fo a paallel-plate waveguide in conjunction with a gounded dielectic sheet [0]. At the othe end of the tansmission line is a shot cicuit load that epesents the shunt in the THW antenna model. By using tansmission line theoy, the shot can be tansfomed to the load location Z xl though distance w/2. By equating these impedances, the following elationship can be deived Χ( ζ ) kw=± nπ n=,3,5,... (2.) x whee n epesents the mode numbe, W is the width of the stip, and X is an expession that defines the eflection coefficient at the edge of the stip [0]. See Appendix A fo moe details on how (2.) is deived. 2

36 By setting n =, (2.) epesents a tanscendental equation fo the wave numbe k x of the fist highe-ode mode of the stuctue. Numeical solution of this equation gives the coesponding tansvese popagation constant (k x ). Fom this, the desied longitudinal popagation constant fo the micostip stuctue ( k = ε k k 2 2 z 0 x ) can easily be computed, whee ε is the elative pemittivity of Region. Fo the tansmission line epesentation of the Tansvese Resonance Method, thee is assumed to be no contibution fom the y-diection, thus k y =0. Equation (2.) will be the basis fo the Tansvese Resonance Code (TRC) used to investigate the micostip stuctues in this effot Tansvese Resonance Code (TRC) A woking copy of the TRC was obtained fom its autho, D Gay Thiele. The TRC is able to compute the popagation chaacteistics of a micostip stuctue, given the width, pemittivity, and pemeability of the antenna. The fist step in applying the TRC to the modified micostip stuctue is to validate the code fo a known case. Fom thee, the code can be alteed to assess its ability to accuately compute the popagation chaacteistics of the modified micostip stuctue. In ode to validate the TR code, the baseline THW antenna in Figue 9 was used (d =.787mm, W 2 =7.5mm, ε = 2.33) and the esults wee compaed with the known fequency ange of the adiation egime (fom 5.95 to 8.2 GHz) [5]. Figue 0 shows the esults of checking the TRC fo validity. The TRC esults match the fequency ange of 22

37 ζ z Popagation Constant of Baseline THW Antenna Using TRC, d=0.787mm, W=5mm 0.9 β/k o Nomalized Popagation Constants ε =2.33 ε ε α/k o Fequency in GHz Figue 0: TRC appoximation of leaky-wave popagation constants fo baseline THW antenna. 23

38 y W width = 2 2 W width = 2 2 ε, µ ε 3, µ 3 x Z xl Z k 0, x Figue : Repesentative tansvese tansmission line system fo the modified THW antenna the adiation egime fo this paticula micostip stuctue (pemittivity, height, width). The values in Figue 0 ae identical to those found by Zelinski [5], who also compaed his values to those validated by Lee [9]. Appendix B futhe investigates the micostip taveling-wave antenna, pimaily ove the adiation egime, to show the impact of vaying the width of the stip conducto and vaying the pemittivity of the dielectic sheet Vaiations of Micostip Stuctue Using TRC The TRC has shown excellent ageement with the known opeating egimes of the baseline micostip antenna. The next step in using the TRC is to look at the poposed modified micostip stuctue (Figue ) and detemine how well TRC pedicts the popagation constants. The fist step in modifying the TRC to handle vaied mateials unde the stip conducto was to split the code into thee steps: teat the micostip as a baseline (nonmodified) antenna and compute the admittance (o impedance) of the dielectic sheet using the standad method; use the modified pemittivity ε 3 to compute the new 24

39 Table : Test Matix fo Vaied Pemittivity Unde Top Conducto fo TRC Pedictions. Top Conducto Width (in mm) ε ε ε ε 3 ε tansvese popagation constant (k x ) of the tansmission line; and combine the two esults into (2.) and dive this equation to zeo to give the new tansvese popagation constant (k x ). Fom this, the longitudinal popagation constant (k z ) can be computed using the elation k = ε k k x. 2 2 z 0 In ode to compae esults to the pevious data, the physical dimensions of the modified antenna wee the same as the baseline THW antenna (d =.787mm, W=7.5mm). Fo the pemittivities, the values listed in Table wee used. Stating fom the baseline pemittivity (ε = 2.33) a epesentative lowe pemittivity (ε =.33) and highe pemittivity (ε = 3.33) wee chosen to show the affect of these changes to the sheet. The last pemittivity selected (ε =.03) showed the affect of making the sheet essentially made of ai. As befoe, all pemeabilities wee assumed non-magnetic (µ 0 =µ =µ 2 =µ 3 ). It can be seen in Figues 2-4 that the changes to the micostip stuctue caused damatic changes to the adiation egime. Figue 2 shows the case whee the dielectic sheet has a pemittivity of ε =.33 and the volume unde the top conducto has a pemittivity of ε 3 = Fo this case, TRC pedicts the bandwidth of the adiation egime to incease fom 2.25 GHz (fo the baseline case) up to 6.55 GHz. Figue 3 25

40 Table 2: TRC Pedicted Bandwidth fo Vaied Pemittivity Unde Top Conducto. ε ε 3 BW in GHz BW% % % % % ε ε 3 ε shows the case whee ε =.03 and ε 3 = Fom this, the new bandwidth can be computed to be GHz. Convesely, Figue 4 shows a decease in the bandwidth down to.4 GHz fo the case whee ε = 3.33 and ε 3 = Physically, these esults make sense although it is unknown at this point as to the validity of the amount of incease o decease in bandwidth. This is due to the uncetainty of whethe the modified TRC is accuate. Fo the cases whee the bandwidth inceases, the pemittivity ε of the dielectic sheet deceased. With the decease in pemittivity of the sheet, the fields aound the antenna stuctue ae less tightly bound to the substate, thus moe leakage can occu. The opposite holds tue fo the case whee the bandwidth deceases (whee the pemittivity ε of the dielectic sheet inceased). With the incease in pemittivity of the sheet, the fields aound the antenna stuctue ae moe tightly bound to the substate, thus less leakage. All thee of these cases ae compaed with the baseline case (ε = 2.33 and ε 3 = 2.33) in Table 2. Hee, the pecentage bandwidth is also computed. In ode to "nomalize" the bandwidth measuements, the pecentage bandwidth is used instead of absolute bandwidth. Resultant bandwidth is given as: BW% = (f high - f low ) / f c, whee f high is the fequency at which β/k 0 =, f low is the fequency at which α = β and is also called 26

41 the cutoff fequency fo the leaky egime, and f c is the cente fequency of the leaky egime. The fist two cases show significant incease in BW% - with one doubling the baseline value and the othe quadupling it. The last case cuts the BW% by almost a thid. These esults will be compaed late in this dissetation with the esults deived fom two othe pediction methods in ode to assess thei validity TRC Conclusions Oveall, the TRC gives vey good insight into the opeating fequencies of the adiation and popagation egimes of a taveling-wave micostip antenna. These fequencies ae deived diectly fom the popagation constants computed by the TRC fo a given set of paametes. Howeve, TRC does not give much insight as to what is diving the phenomenology within the stuctue o solve fo the cuents/fields within the stuctue. When computing the popagation chaacteistics of the modified THW antenna, TRC gives esults that show significant levels of incease/eduction in bandwidth fo the adiation egime. Without any known data to compae to, futhe analysis must be pefomed to validate these esults. This will be done in the following section using the FDTD method, as well as the following chaptes using a fequency domain full-wave analysis. 27

42 ζ z Popagation Constants Using TRC, ε 3 =2.33, d=0.787mm, W=5mm Nomalized Popagation Constants ε =2.33 ε =.33 ε 3 =2.33 ε=2.33 β ε=2.33 α ε=.33 β ε=.33 α ε ε ε ε 3 ε Fequency in GHz Figue 2: Popagation constants fo adiation egime using TRC fo ε =.33 and ε 3 =

43 ζ z Popagation Constants Using TRC, ε 3 =2.33, d=0.787mm, W=5mm Nomalized Popagation Constants ε =2.33 ε =.03 ε 3 =2.33 ε=2.33 β ε=2.33 α ε=.03 β ε=.03 α ε ε ε ε 3 ε Fequency in GHz Figue 3: Popagation constants fo adiation egime using TRC fo ε =.03 and ε 3 =

44 ζ z Popagation Constants Using TRC, ε 3 =2.33, d=0.787mm, W=5mm Nomalized Popagation Constants ε =2.33 ε =3.33 ε 3 =2.33 ε=2.33 β ε=2.33 α ε=3.33 β ε=3.33 α ε ε ε ε 3 ε Fequency in GHz Figue 4: Popagation constants fo adiation egime using TRC fo ε =3.33 and ε 3 =

45 2.3 Finite Diffeence Time Domain 2.3. Backgound The TRC povides a quick and faily simple way to pedict the popagation constants fo a micostip stuctue. Howeve, it does not compute the fields and cuents in the stuctue, no does it accuately pedict the constants when thee ae changes in the pemittivity of egion 3. A method that computes the fields within the stuctue, accommodates fo changes in the mateials, and computes the desied popagation constants is the Finite Diffeence Time Domain (FDTD) technique [92]. FDTD is a computational method fo solving the time-domain diffeential fom of Maxwell's equations diectly and discetely using a space-time gid. By placing the micostip stuctue into a ectilinea gid-space, discetizing it into a set numbe of finite elements that account fo mateial and dielectic popeties, and defining the location of the souce o input; the electic and magnetic fields fo each element can be computed fo a single time step. The electic and magnetic fields ae actually spaced ½ of a cell apat fom each othe in ode to accommodate fo the inteactions between them. The FDTD code actually computes E at one time step, then computes H one half of a time step late. By altenating between computing E, then H, then E, etc.; the esultant fields "popagate" in time []. In ode to investigate the fields in the micostip stuctue studied hee a gidspace of infinite extent is equied. Given that this is not computationally feasible, the gid-space will be tuncated by using a Pefectly Matched Laye (PML) as the oute laye of the FDTD gid [93]. PMLs essentially make the gid-space look infinite in extent by 3

46 absobing any incident fields and not allowing fo any fields to be eflected back to the micostip stuctue. The PML is matched pefectly to the othe layes it comes in contact with (thus, Γ = 0), but has absobing popeties that cause the fields to be attenuated FDTD Set-Up A copy of an FDTD code peviously used to analyze the THW antenna was obtained. The code was loosely based on the 3-D Hagness-Willis FDTD code, but modified in ode to optimally handle the THW stuctue and to compute the popagation constants fom the electic field data [5]. Befoe the pocess by which the popagation constants ae extacted is explained, it is necessay to undestand how the FDTD simulation was set-up. In ode to save time, the same PML thicknesses, substate thickness, ai gap thickness, and location of the souce pefomed by Zelinski [5] in his effots to measue the baseline THW antenna wee used in this eseach. His effots optimized the paametes to measue the baseline antenna. Figue 5 shows the oientation of the THW antenna and the location of the PML layes, PECs, and ai gap (ε 0 ). The location of the souce is indicated with a ed dot in the figue. The PMLs vaied fom six to twelve cells thick (fequency dependent) in ode to adequately absob enegy incident onto them. Plus, only two cells of ε substate o ai wee equied befoe enteing eithe PML. This left a minimum of eight cells that wee 32

47 y PEC PML fo Ai Region z Ai ε ε 3 ε PML to ε PML to ε x Figue 5: Modified THW antenna with PML layes and souce location (in ed) equied to be aound the micostip stuctue in ode to not intoduce any atifacts into the simulation. The baseline THW antenna is 66.9mm long x 7.5mm wide x 0.787mm tall. This tanslates to 42 x 47 x 5 cells at 6.7GHz (the longitudinal diection is scaled by a facto of 3). Combined with the dimensions of the gaps and PMLs, this comes to 58,790 cells equied to popely epesent the micostip stuctue. Two othe stuctues wee also investigated in ode to gain confidence in the FDTD esults. The fist was to incease the amount of ε dielectic sheet on the othe side of the shunt (Figue 7). The second was to make the micostip a full-width antenna with vaied ε 3 substate unde the stip, but to still include the shunt (Figue 8). The shunt is equied in ode to block the dominant mode and foce the antenna to opeate as a leaky wave antenna. The esultant popagation constants fo each wee detemined and 33

48 y Ai PML fo Ai Region ε ε 3 ε PML to ε PML to ε PEC x Figue 6: Modified THW antenna with PML layes and souce location (in ed) y PML to ε Ai PML fo Ai Region ε ε 3 ε PML to ε PEC x Figue 7: Modified THW antenna with inceased slab with PML layes and souce location (in ed) y Ai PML fo Ai Region ε ε 3 ε PML to ε PML to ε PEC x Figue 8: Modified TFW antenna with PML layes and souce location (in ed) 34

49 found to match the esults of the oiginal set-up in Figue 6. As such, the oiginal set-up can be used with confidence to detemine the popagation chaacteistics Extaction of Popagation Constants Using FDTD The pocess by which the popagation constants ae extacted fo a paticula micostip stuctue is a faily staightfowad one, albeit vey time consuming. A y- diected sinusoidal souce E i is intoduced into the stuctue and the fields ae popagated ove thousands of time step iteations (equivalent to 30 peiods at a given fequency). At the end of these 30 peiods, all fields within the stuctue have popagated and attenuated to a steady state. It is at this point that the magnitude of the y-diected fields can be used to compute the attenuation and phase constants fo the stuctue. The location of the cells fom which these fields ae taken is just cell to the left (in x) of the ed dot (souce location) in Figue 6. This point stays constant in x and y, but uns the length of the stuctue in z. Typical FDTD output can be seen in Figues 9 and 20. Fo this example, 325 cells ae used to epesent the length of the stuctue. The fist 2 cells ae used to epesent the PML in the z-diection and the souce can be seen at cell numbe 7. The y- diected electic fields popagate along the length of the antenna fom the souce down to cell numbe 37. The last 2 cells epesent the PML in the z-diection at the end of the length of the stuctue. In ode to extact the popagation constants fom the y-diected electic field data, it is necessay to stat with the expession fo that field 35

50 (, ) jk z z 0 (,) Re{ jk z E z j t y z ω = E e Ey z t = E e e ω 0 } (2.2) if the sinusoidal excitation is in steady state. Equation (2.2) can be e-witten using the expession fo kz = β z jα z α z E ( zt, ) = Ee z cos( ωt β z) (2.3) y 0 z Taking the natual log of both sides gives Ey (,) z t ln = ln e t z E 0 α z z ( cos( ω βz ) ) α z z = lne + lncos( ωt β z) = α z+ ln cos( ωt β z) z z z (2.4) The popagation chaacteistics can be computed using (2.4) and the natual log of the nomalized y-diected electic field data. Figue 20 shows a typical plot of such data. By measuing the distance in metes between peaks (o between nulls), the peiod of the electic field is obtained (λ z in the figue). Using the fundamental elation: β z = 2π / λ z, the desied phase constant, β z, can be computed. As an example, Figue 2 eveals that λ z /2=0.035 m, esulting in β z /k 0 = Examination of the amount of attenuation between peaks povides the value fo α z. In this case, the peak-to-peak slope of the ln E cuve in Figue 20 gives the value fo α / k 0 = In ode to check the accuacy of α z and β z, an exponential cuve is poduced using these values and ovelaid onto the natual log plot of the aw data (see Figue 2). Figue 22 shows the same data as Figue 2, but without taking the natual log of the values. 36

51 .6 FDTD E y Data (Peiod #30) E y Cell Numbe (z-diection) Figue 9: FDTD y-diected electic field data (magnitude only) fo THW antenna taken longitudinally with x fixed at cell #8 and y fixed at cell #7. The souce can be seen at cell #7 in z-diection. 37

52 0-0.5 Natual log of FDTD E y Data (Peiod #30) α z ln E y (db) λ z Distance fom souce (m) Figue 20: Natual log of FDTD y-diected electic field data (magnitude only) fo THW antenna taken longitudinally with x fixed at cell #8 and y fixed at cell #7. 38

53 0 Natual log of FDTD E y Data and Best-Fit α and β Cuve ln E y (db) FDTD α/k 0 = β/k 0 = z-diection cell numbe Figue 2: Natual log of FDTD y-diected electic field data (magnitude only) and coesponding best-fit α and β cuve. 39

54 0.8 Nomalized FDTD E y Data and Best-Fit α and β Cuve FDTD α/k 0 = β/k 0 = Nomalized E y z-diection cell numbe Figue 22: Nomalized FDTD y-diected electic field data (magnitude only) and coesponding best-fit α and β cuve. 40

55 2.3.4 FDTD Methodology and Results The fist step pefomed was to validate the code by epoducing the esults obtained by Zelinski duing his eseach. The paametes fo the THW antenna wee enteed into the code and the esults wee compaed with the TRC. As stated in the pevious section, the TRC has been shown to accuately pedict the popagation constants fo a standad micostip stuctue, thus the TRC data will be used as tuth data fo the non-modified micostip only. Fom Figue 23, it can be seen that the FDTD code did a vey good job pedicting the popagation constants fo the baseline antenna. The values fo the phase and attenuation constants ae within to 3% of the values obtained fom using the TRC fo the majoity of the adiation egime. The values close to cutoff (whee the α and β cuves intesect) tend to vay geate than the est of the values. This is due to the natue of the FDTD esults nea cutoff and will be explained late in this chapte. Oveall, it can still be said that the bandwidth of the baseline antenna can now be pedicted using FDTD with confidence. The next step is to modify the stuctue and analyze the esults using the same method. Befoe the modified micostip configuation could be tested, some modifications of the FDTD code wee equied. These modifications allowed fo the mateial unde the stip conducto to vay while keeping the est of the stuctue the same. In ode to validate the changes to the code, ε and ε 3 wee both set to the baseline pemittivity (2.33). The esults wee exactly the same as FDTD esults fo the baseline THW antenna (as shown in Figue 23), thus the code could then be used with modified ε and ε 3 values with confidence. 4

56 The baseline THW antenna was modified by inseting a diffeent pemittivity ε 3 unde the top conducto than the emainde of the substate (ε ), simila to that which was done ealie in this chapte using the TRC. Table 3 shows the test matix used fo this micostip configuation. The values chosen in the test matix give a ange of ealistic pemittivity values that cente aound the baseline THW antenna pemittivity (2.33), giving some values that ae lowe than the baseline case and one value that is geate. By using these values, it is desied that tends and physical insight can be gained without having to pefom dozens of test cases. 42

57 ζ z Popagation Constants Using TRC and FDTD fo Baseline THW Antenna 0.9 Nomalized Popagation Constants ε =2.33 β/k o TRC α/k o TRC β/k o FDTD α/k o FDTD ε ε Fequency in GHz Figue 23: TRC and FDTD pedictions of popagation constants fo baseline THW antenna. 43

58 Table 3: Test Matix fo Vaied Pemittivity Unde Top Conducto Using FDTD. Top Conducto Width (in mm) ε ε ε ε 3 ε Nomalized Popagation Constants ζ z Popagation Constants Using FDTD, ε 3 =2.33, d=0.787mm, W=5mm ε =2.33 ε =2.33 ε =.03 ε =.33 ε =3.33 ε ε ε =.03 ε ε 3 ε ε 3 =2.33 ε =.33 ε ε 3 ε ε 3 =2.33 ε=3.33 ε ε 3 ε ε 3 = Fequency in GHz Figue 24: FDTD pediction of popagation constants fo modified THW antenna. 44

59 Nomalized Popagation Constants ζ z Popagation Constants Using FDTD, ε 3 =.33, d=0.787mm, W=5mm ε =.33 ε =.33 β ε =.33 α ε =.03 β ε =.03 α ε =2.33 β ε =2.33 α ε ε ε =.03 ε ε 3 ε ε 3 =.33 ε=2.33 ε ε 3 ε ε 3 = Fequency in GHz Figue 25: FDTD pediction of popagation constants fo micostip with ε 3 =

60 Figues 24 and 25 show the affect of changing the pemittivity of egion. The plots show cases whee the pemittivity of the substate unde the top conducto was kept constant and the pemittivity of the est of the substate sheet was vaied. As with the TRC pedictions, the bandwidth and pecentage bandwidth of the modified micostip antenna was detemined fom the plots. Table 4 summaizes these esults fo the data seen in Figues 24 and 25 and compaes them to the coesponding TRC data obtained in the pevious section. Table 4: FDTD Pedicted Bandwidth fo Vaied Pemittivity Unde Top Conducto. ε ε 3 BW in GHz BW% BW in GHz BW% FDTD FDTD TRC TRC GHz 36.24% GHz 33.8% GHz 33.48% 6.55 GHz 7.0% GHz 32.7% 2.25 GHz 32.6% GHz 30.90%.4 GHz 2.2% GHz 75.89% 34.3 GHz 38.0% GHz 7.76% 8.7 GHz 72.3% GHz 65.76% 2.8 GHz 3.% ε ε 3 ε Fom the plots, seveal tends can be obseved: When ε = ε 3, the popagation constants match the TRC pedictions vey closely, as expected. As the pemittivity of the sheet is loweed, the cutoff fequency of the adiation egime is inceased, simila to deceasing ε in a loaded ectangula waveguide. As the pemittivity of the sheet is inceased, the cutoff fequency of the adiation 46

61 egime is deceased, simila to inceasing ε in a loaded ectangula waveguide. The bandwidth of the adiation egime does incease as the pemittivity of the sheet is loweed and the pemittivity of the mateial unde the top conducto is held constant (8.43% impovement by loweing ε fom 2.33 to.03 in Figue 24). The fields ae less tightly bound to the substate in the lowe pemittivity dielectic, thus moe leakage. The bandwidth of the adiation egime does decease as the pemittivity of the sheet is inceased and the pemittivity of the mateial unde the top conducto is held constant. The fields ae moe tightly bound to the substate in the highe pemittivity dielectic, thus less leakage. The FDTD code has a vey had time accuately pedicting both α and β as the fequency appoached the cutoff fequency whee attenuation becomes sevee (whee α = β ). This same obsevation was seen by Zelinski in his eseach using the FDTD code [5]. This is due to the wavefom attenuating too apidly (α > 0.) to be able to locate definitive peaks o zeo cossings in the aw Ey data (Figues 26 though 28). Without these values, α and β can not be computed with accuacy. This is one of the biggest deficiencies of the FDTD code when used to calculate the popagation constants fo a micostip stuctue and, thus, why anothe method is sought to compute the constants of the stuctue (i.e., an integal equation fomulation). The time equied to compute one data point (i.e., one attenuation and popagation combination at one fequency) was significant. The time equied to find one data 47

62 point would ange fom 2 to 0 hous to compute using a PC (2.66 GHz Dual Intel Xeon X5355 with 8GB RAM). As the fequency inceased within the leaky egime, the time equied to compute a data point would decease, but still emain above an hou. Fo the modified stuctue, TRC does not compae well with FDTD. This is most likely due to the lack of inclusion of the y-diected fields and the coupling that takes place between those fields and the cuents on the suface of the stip conducto. Oveall, the esults of this modification to the micostip stuctue wee good, but not damatic. The esults show that vaiations in the impedance looking out into the substate do affect the amount of enegy leakage acoss the adiation egime. The FDTD analysis of the vaiations of the baseline THW antenna is by no means intended to be a definitive look at what happens when modifying the mateial chaacteistics of the antenna. It is meant solely to be a "fist-look" to (a) see that changing the mateial popeties of the THW antenna does cause vaiations in the bandwidth of the adiation egime and (b) give a sense fo how much impovement can be expected. One of the biggest dawbacks of using FDTD to obtain the popagation constants of the antenna stuctues was the time equied to obtain the esults. Typical data uns would take 2 to 0 hous to obtain one popagation constant at one fequency on a PC. 48

63 Anothe deficiency of FDTD is that all that is possible is to pefom a numbe of test cases to see what the esults ae (i.e., it is somewhat limited in poviding fundamental physical insight into the opeational chaacteistics of the modified antenna). These, combined with the inability of FDTD to compute the popagation chaacteistics fo lage values of alpha (>0.) lead to the need to find anothe method to compute these constants. An integal-equation based fequency domain full-wave analysis is one such method that can moe definitively povide physical insight into what changes ae equied to lead to moe adiation. This method will be descibed in the following chaptes. 49

64 .2 FDTD E y Data (Peiod #30) E y Cell Numbe (z-diection) Figue 26: FDTD y-diected electic field data (magnitude only) nea cut-off fequency fo adiation egime (6 GHz). 50

65 0 Natual log of FDTD E y Data (Peiod #30) -2-4 ln E y (db) Distance fom souce (m) Figue 27: Natual log of FDTD y-diected electic field data (magnitude only) nea cut-off fequency fo adiation egime (6 GHz). 5

66 0 Natual Log of FDTD E y Data and Best Fit α and β Cuve -5 ln E y (db) FDTD α/k 0 = β/k 0 = z-diection cell numbe Figue 28: Natual log of FDTD y-diected electic field data (magnitude only) and coesponding best-fit α and β cuve (6 GHz). 52

67 Chapte 3 GREEN S FUNCTION FORMULATION FOR FULL-WAVE ANALYSIS 3. Intoduction The TRC allowed fo a quick computation of the popagation constants of the micostip stuctue, but did not popely handle vaiations within the substate laye. FDTD allowed fo vaiations in the stuctue, but was a vey slow method to compute the popagation constants of the micostip. Some FDTD data uns would take upwads of 0 hous to compute one popagation constant at one fequency on a stand-alone PC. In addition, fo fequencies in which α became significant, FDTD failed to convege. FDTD also suffeed a dawback in that physical insight into design paametes and how mateial popeties effect bandwidth is somewhat obscue. Because of this, a moe physically insightful and computationally efficient method of deiving the popagation constants fo all fequencies is desied. One such method is to develop an integal equation fomulation based on a full-wave vecto potential analysis in which micostip mateial petubations ae teated using equivalent cuents. 3.2 Full-Wave Solution Full-wave solutions (i.e., based on Maxwell s equations) to poblems such as the micostip antenna povide a igoous undestanding of the full electomagnetic inteactions between vaious elements in the stuctue. The full-wave analysis is an 53

68 appoach that typically elies on the use of a Hetzian potential (π ) which satisfies the Hetzian potential wave equation [2] 2 2 J π + k π = jωε (3.) whee J is the cuent. Field ecovey, which is essential fo enfocing bounday conditions, is obtained though the use of the elations 2 E = k π + π H = jωε( π) (3.2) (3.3) It will be shown in this chapte that the solution to (3.) is given by t J ( ') π ( ) = G( ') dv ' jωε whee G is the Hetzian potential dyadic Geen s function fo the cuent V J (3.4) immesed in the substate laye of the gounded-slab backgound envionment (see Figue 29). Although the goal of this chapte is to develop the necessay Geen s function, the oveall pocess to igoously compute the desied popagation constants is as follows:. Decompose the wave equation (3.) into pincipal (paticula) and eflected (complementay) contibutions to moe easily find the geneal solution. 2. Apply appopiate bounday conditions on the backgound envionment to uniquely compute the unknown spectal coefficients of the wave equation solution. 3. Identify the Hetzian-Potential Dyadic Geen's function of the geneic volumetic cuent souce embedded in the micostip backgound envionment by compaing the solution in step 2 with equation (3.4). 54

69 4. Fomulate the coupled Electic-Field Integal Equations (EFIEs) by enfocing bounday conditions on the equivalent volume and suface cuents of the modified THW antenna. 5. Solve the EFIEs using the Method of Moments which leads to the computation of the popagation constants. The Method of Moments will establish the inteactions between the equivalent volume cuents in the egion of inteest and the suface cuents on the stip conducto. It will be shown that the eigenvalues of the MoM matix ae inheently the popagation constants fo the modified THW stuctue. Fom solving fo these popagation constants at vaious fequencies, the leaky egimes of the stuctue can be detemined with the aid of the constaint equation. Steps -3 of this pocess will be caied out in this chapte and steps 4-5 will be pefomed in the following chapte Spectal Coefficients The fist step in obtaining the Hetzian potential dyadic Geen s function fo a geneic 3-D cuent immesed in egion is to fomulate the wave equation in each of the egions of the micostip backgound envionment. This involves decomposing the wave equation into pincipal and/o eflected contibutions fo each egion, as equied. Figue 26 shows the coss-section of the micostip backgound envionment. This backgound envionment only vaies in the y-diection, with the height of the substate being d. Fom the figue, thee distinct egions ae defined: egion is the substate laye of pemittivity ε, egion 2 is the ai laye above the antenna, and the thid egion is the 55

70 Ai Scattee d Wave y Region 2 (ε 2, µ 0 ) y=d Scattee d Wave J Pincipal Wave Region (ε, µ 0 ) y=0 PEC z x Figue 29: Coss-section of micostip backgound envionment. The stuctue is assumed to be infinite in extent along the x- and z-diections. pefect electic conducto (PEC) laye below the substate. Both egions and 2 have a pemeability of µ 0 and, thus, ae assumed to be non-magnetic. Since the equivalent volume cuent (due to mateial changes unde the stip) and equivalent suface cuent (due to pesence of the stip) ae cuents immesed in egion, we ae inteested in the Dyadic Geen s function of a geneic 3-D cuent immesed in this egion. Fo the micostip antennas used in this eseach, the souce J will always be located in egion. This is epesented as the pincipal wave, with a component taveling in the +y and the -y diections in an unbounded egion of pemittivity ε. This is analogous to the paticula solution with no boundaies pesent. Due to the discontinuities at the PEC/substate bounday and the substate/ai bounday, a scatteed wave will be pesent in both egions and 2. This scatteed wave is analogous to the 56

71 homogeneous solution when boundaies ae pesent. Additionally, since thee is no cuent in egion 2, thee is no pincipal contibution. Thus, fo each egion, the total Hetzian potential can be witten as π = π + π p π 2 = π 2 (3.5) (3.6) whee π p, π, and π 2 satisfy the following wave equations 2 p 2 p J π kπ jωε + = π + π = 2 2 k 0 π + π = k2 2 0 (3.7) (3.8) (3.9) with 2 2 k ω µε 0 = and 2 = ω 2 µε. Each of these vecto wave equations can be k2 0 2 decomposed into the scala wave equations, assuming sepaable solutions, as 2 p 2 p Jα ( ) π α( ) + kπ α( ) = jωε π ( ) + k π ( ) = α α π ( ) + k π ( ) = α 2 2α (3.0) (3.) (3.2) with α = x, y, z. Given that the backgound envionment is infinite in extent along the x and z diections, this pompts the Fouie tansfomation on these vaiables to help solve these equations. Conside the geneic 2-D tansfom pai j f% λ ( λ, y) f ( ) e dxdz (3.3) = 57

72 whee λ = xˆ ξ + zˆ ζ f ( ) f(, y) e λ d j 2 = % λ λ 2 (2 π ) (3.4) ( λ = λ λ = ξ + ζ ), = xx ˆ + yy ˆ + zz ˆ and d 2 λ = dξdζ. Upon Fouie Tansfomation, equations (3.0) to (3.2) simplify to 2 π p ( λ %, y) 2 p J% α (, ) p 2 πα ( λ, y) λ α y % = (3.5) y jωε 2 % π α ( λ, y) 2 p % 2 πα ( λ, y) = 0 y (3.6) 2 % π2 α ( λ, y) 2 p % 2 2π2α ( λ, y) = 0 y (3.7) whee p = λ k, 2 2 p = λ k with the positive squae oot chosen so that Re{p } > 0 and Re{p 2 } > 0 (this will be discussed in-depth in the following chapte). p The pincipal wave π% α used in (3.5) is assumed to exist in an unbounded medium of pemittivity ε, thus we can tansfom in y using the Fouie diffeentiation theoem, leading to 2 p 2 p J% α ( λ, η) ηπ % α( λη, ) p % πα( λη, ) = (3.8) jωε whee η is the tansfom vaiable associated with y and % π ( λ, η) % π ( λ, ye ) p p α = α jη y dy (3.9) j y J% (, ) J (, y) e η α λη = % α λ dy (3.20) 58

73 Solving fo ( p % π, ) λη in (3.8) gives α % ( λ, η) ωε % ( λ, η) ωε % π ( λ, η) = p = jp jp ) (3.2) p Jα j Jα j α 2 2 ( η + ) ( η+ )( η Fom this it can be seen that the poles of this equation ae located at η =± jp. (3.2), giving p The pincipal wave % π (, ) λ y can be ecoveed by taking the invese tansfom of α % % π λ % π λ η η (3.22) (, ) (, ) (, ) J j p p j η y α λη ωε jηy α y = % α e d 2 2 ( jp)( jp) e d π = π η η+ η Since the souce cuent does not exist ( J% α ( λ, y') = 0) outside of the souce egion, its Fouie epesentation is slightly diffeent than in (3.3), namely, the limits of integation ae as follows J% J% y e dy J% y e ' dy d j ' (, ) (, ') η y jηy α λη = α λ ' = α( λ, ') ' 0 (3.23) whee y ' is used as a integation vaiable within the souce egion. Inseting (3.23) into (3.22) leaves the esult d p p J% α ( λ, y') % π ( λ, y) = G% ( λ; y y') dy' (3.24) jωε α 0 whee G% λ y y G% λ y y dη (3.25) 2 ( )( ) jη ( y y') p p e ( ; ') = ( ; ') = π η+ jp η jp is the pincipal wave Hetzian-potential Geen's function in the spectal domain. Using Cauchy's Integal Theoem [2], the solution to (3.25) is 59

74 p y y' p p e G% ( λ; y y') = G% ( λ; y y') = (3.26) 2 p Substitution of (3.26) into (3.24) leads to the pincipal wave contibution d p y y' p e J% α ( λ, y') % π α ( λ, y) = dy' (3.27) 2 p jωε 0 with y as the field point, constant in egion. y ' as the souce point, and p as the y-diected popagation The scatteed wave (o eflected contibution) solutions to equations (3.6) and (3.7) ae well-known and ae given by π λ λ λ py py % (, y) = W + ( ) e + W ( ) e (3.28) α α α % π ( λ, y) = W ( λ) e + W ( λ) e (3.29) + p2y p2y α α α whee W βα ± ae the α-component spectal coefficients fo the +y and -y diected eflected waves in egion β. Thus, the total potentials in each egion can be epesented as % π = % π + % π = V e + W e + W e p + py + py py α α α α α α...y' < y < d (3.30) p py + py py % % %...0 < y < y' (3.3) π = π + π = V e + W e + W e α α α α α α py 2 py 2 % = % π = W + e + W e... y > d (3.32) π 2α 2α 2α 2α with α = x, y, z and whee p = λ k, 2 2 p = λ k, and ( d ± p λ ) y e ' J% ± ± α ( λ, y') Vα = Vα ( λ) = dy' (3.33) 2 p ( λ) jωε 0 ae the +y and -y diected waves emanating fom the souce. 60

75 Fom equations (3.30) - (3.32), thee ae twelve spectal coefficients W βα ± that must be detemined [2]. This will be done by applying the bounday conditions fo the backgound envionment (Figue 29). Note, the bounday conditions on π come fom the bounday conditions on E and H, specifically that E tang at y = 0 must be zeo and that E tang and H ae continuous at y = d. Additionally, the magnitudes of E and H will tang emain finite (< ) as y. It is typically easie to satisfy the bounday conditions on π than on E and H. Fom Havilla [2], the bounday conditions on π that ensue E tang = 0 at y = 0, continuity of E tang and H tang at y = d and EH<, fo y ae % π (, y ) 0 2 λ = α = x, yz, (3.34) α ε % π ( λ, d) = ε % π ( λ, d) α = x, z (3.35) α 2 2α % π α( λ, d) % π2 α( λ, d) ε = ε2 y y α = x, z (3.36) ε % π ( λ, d) = ε % π ( λ, d) (3.37) y 2 2y % π ( λ, d) % π ( λ, d) ε = (, ) + ( y 2y 2 y y ε [ jξπ% λ d jζπ% λ, )] 2x 2z d (3.38) % π (, y 0) 0 λ = = α = x, z (3.39) α % π (, 0) y λ y = = 0 y (3.40) The fist bounday condition elation (3.34) exists since the ai laye (egion 2) extends off to infinity but E and H emain finite, thus the potential will eventually be attenuated to zeo. Continuity acoss the ai/dielectic inteface at y = d leads to (3.35) and (3.36) 6

76 fo tangential and (3.37) fo nomal components. The PEC inteface at y = 0 will ceate the bounday conditions in (3.39) and (3.40). That is, at the PEC bounday, the tangential potential fields ae diven to zeo, as is the deivative of the nomal field. Lastly, (3.38) epesents the mixed/coupled bounday condition (i.e., how x and z-diected cuents couple into y-diected potential). Using these bounday conditions, the spectal coefficients can now be detemined. Applying bounday condition (3.34) to (3.32) esults in % π (, y ) 0 2 λ = α = x, yz, (3.4) α + p2y p2y ( W e W e ) lim % π = lim + = 0 (3.42) y 2α 2α 2α y + p2 y ( W2 α W2 αe ) lim 0 + = 0 (3.43) y W2 α 0 α = x, yz, (3.44) + py 2 % W e d y π = 2α 2α < < ( α = x, yz, ) (3.45) thus, the esultant Hetzian potential in egion 2 has only a upwad popagating wave associated with it. This is expected since the souce is located in egion and egion 2 extends to infinity and thee will be no scatteed wave in the -y diection. The next step is to apply the tangential bounday condition (3.39) at the PEC bounday (y = 0) to (3.3) % π (, y 0) 0 α λ = = α = x, z (3.46) % π α = x, z (3.47) α = Vα e + W αe + W αe = 0 W = W V α = x, z (3.48) + α α α 62

77 The tangential bounday condition (3.35) and the esult fom (3.45) at the aisubstate bounday (y = d) leads to ε % π ( λ, d) = ε % π ( λ, d) α = x, z (3.49) α 2 2α ( ) 2 ε V e + W e + W e = ε W e α = x, z (3.50) + pd + pd pd + p d α α α 2 2α + ε p2d + pd + pd pd W2 α = e ( Vα e + W αe + W αe ) α = x, z (3.5) ε 2 ε π = + + ( + + α ) p2( y d) pd pd pd % e V e W e W e, 2α α α ε 2 α = x z (3.52) Applying the second tangential bounday condition (3.36) with the newπ% 2α (3.52) and the oiginal π% α (3.30) gives % π α( λ, d) % π2 α( λ, d) ε = ε2 y y α = x, z (3.53) ε p ε V e + W e W e = p W e V e + W e + W e + pd + pd pd + p2 ( ) ( d d ) + pd + pd pd ( α α α ε α α α α ) ε 2 (3.54) Solving fo W α leads to p p pd + ( ) ( ) 2 2pd W = e V + W = Re V + W α p+ p α α α α 2 α = x, z (3.55) whee R = p2 p p + p 2 (3.56) Substituting this esult into (3.48) gives ( ) 2 pd W V Re V W + = + α = x, z (3.57) α α α α 63

78 o, solving fo W + α, leads to W = 2 pd + Re V V + α α α 2 pd Re α = x, z (3.58) Substituting (3.58) into (3.55) poduces the desied esult W + = Re 2pd + 2pd Re V Re V α α 2 pd α α = x, z (3.59) + The spectal coefficient W2 α can easily be detemined by inseting (3.58) and (3.59) into (3.50). Applying the nomal bounday condition (3.37), using the equations fo π% y and π%, and solving fo W + and, thus, 2 y 2 y π% 2 y gives ε % π ( λ, d) = ε % π ( λ, d) (3.60) y 2 2y ε ( ) + p2d + pd + pd pd W = e V e + W e + W e (3.6) 2y y y y ε 2 ε 2( ) % π = e V e + W e + W e (3.62) ( + + y ) p y d p d p d p d 2y y y ε 2 The coupled/mixed bounday conditions can now be used to solve fo the emaining two coefficients, W + y and W y. Stating with equation (3.38) and using the equation deived above fo π% 2 y at y = d leads to % π ( λ, d) % π ( λ, d) ε = (, ) + ( y 2y 2 y y ε [ jξπ% λ d jζπ% λ, )] 2x 2z d (3.63) 64

79 ε + pd + pd pd + pd + pd pd ε p2 ( Vy e + Wye + Wye ) + p( Vy e + Wye Wye ) = A ε 2 ε 2 whee (3.64) A= jξa + jζ A (3.65) x z + + A V e W e W e = + + (3.66) pd pd pd x x x x + + A V e W e W e = + + (3.67) pd pd pd z z z z Solving fo W y leads to W Re V Re W ε e ε pd 2pd + 2pd + 2 y = y y + ε p2 + p ε 2 A (3.68) The bounday condition (3.40) at the PEC bounday (y = 0) leads to % π (, 0) y λ y = = 0 y (3.69) = (3.70) + Wy Wy V y Substituting equation (3.68) fo W y gives an expession fo W + y W 2 pd + Vy Re Vy + + y = 2 pd ( ε ε2 ) e ( p ε ε + p ) pd ( Re + ) 2 2 A (3.7) Similaly, using equation (3.70) and the above solution leads to an expession fo W y W Re V Re V + 2pd 2pd + y y y = 2 pd ( Re + ) pd ( ε ε2 ) e ( p ε ε + p ) 2 2 A (3.72) 65

80 The next step involves the A potions of these two equations. Since the definition of A (and subsequently A x and A z ) includes aspects of W ± x and W ± z, these must be defined in tems of the souce and not the spectal coefficients. Havilla deives the following elationships [2] Wy = Wyx + Wyy + W yz (3.73) W = W + W + W y yx yy yz (3.74) whee W 2 pd ( N ) e jξ Ax 2 + 2pd 2pd pn 2 p C yxvx e CyxVx e + + yx = = 2pd 2pd pd 2 ( + Re ) ( + Re )( Re ) (3.75) W 2 pd + Re Vy + V + y yy = 2 pd + Re (3.76) W 2 pd ( N ) e jζ Az 2 + 2pd 2pd pn 2 p C yzvz e CyzVz e + + yz = = 2pd 2pd pd 2 ( + Re ) ( + Re )( Re ) (3.77) C yx = 2 jξ ( N )( R) ( pn + p) 2 2 (3.78) C yz = 2 jζ ( N )( R) ( pn + p) 2 2 (3.79) and 2 N ε/ ε 2 =. Now that all of the spectal coefficients have been computed, the t Hetzian potential dyadic Geen's function G) ( ' can be identified. 66

81 3.2.2 Geen's Function Befoe E and H can be computed, the Hetzian Potential Geen's function must be identified. The Geen's function can be found fom the spectal coefficients deived in the pevious section. Stating with the tangential components of the Hetzian potential in egion, π% x and π% z, we have m % π = % π + % π = V e + W e + W e α = x, z (3.80) p ± py + py py α α α α α α % % π α = x, z (3.8) d p y y' e Jα ( λ, y') py py α = dy' + W + αe + W αe 2 p 0 jωε Substituting fo W + α and W α gives % π % d p y y' 2pd + 2pd + 2pd e Jα( λ, y') Re Vα Vα py Re V Re V α + py α = dy ' + e + e 2pd 2pd 2p 0 jωε Re Re α (3.82) α Using the elationship fo V ± α and J% ( λ, y') in equation (3.33), equation (3.82) can be ewitten as d p J% α ( λ, y') % π α = G% ( λ; y y') + G% αα( λ; y, y') dy' α = x, z (3.83) jωε 0 with p y y' p e G % ( λ; y y') = 2 p (3.84) p φ p2 φ pφ 3 pφ4 Re e Re Re G% + αα ( λ; y, y') = α = x, z (3.85) p Re 2 pd 2 ( ) φ = 2 d + y y' φ 2 = y+ y' (3.86), (3.87) φ 3 = 2d y+ y' φ 4 = 2 d y y' (3.88), (3.89) 67

82 d y φ = y y' φ 4 = 2 d y y' φ 3 = 2 d y+ y' y 0 φ = + φ = 2 d + y y' 2 y y' φ 5 = 2 d + y+ y' Pincipal Field No coupling off of PEC Additional Gyx, Gyz p y y' Gxx, Gyy, Gzz only p e G% coupling tem ( λ; y y') = 2 p Figue 30: Souce (y ) and obsevation (y) points and paths of inteaction within backgound envionment. An examination of these tems within the backgound laye shows multiple paths and contibutions (see Figue 30). This figue shows the specific case whee the obsevation point (y) is located above the souce point ( y ' ). The fist path is the diect p path ( y - y ' ) fo when y > y '. This path epesents the pincipal field contibution G %. The second path shown is only fo the G%, G%, and G% xx yy zz contibutions no coupling will take place off of the PEC laye due to the equiement that the tangential electic fields on the PEC ae zeo. The last path shown (2d + y + y ' ) is included as the fouth coupling tem G% and G% yx yz. An infinite numbe of eflections take place within the substate egion, howeve only fou tems ae necessay in the eflected fields G % as all emaining eflections ae accommodated within these pimay tems due to the pole-seies summation of these tems. This will be confimed late in the esults of this eseach. 68

83 π% y leads to An examination of the nomal component of the Hetzian potential in Region % π = % π + % π = V e + W e + W e (3.90) p ± m py + py py y y y y y y d J% y ( λ, y') p = G% ( λ; y y') + G% yy ( λ; y, y') dy' + jωε 0 % (, ') % (, ') G ( ; y, y') dy' G ( ; y, y') d ' d d % Jx λ y Jz λ y yx λ + % yz λ jωε 0 jωε 0 y (3.9) whee pφ pφ2 pφ 3 pφ4 Re e Re Re G% + yy ( λ; y, y') = (3.92) p + Re 2 pd 2 ( ) φ φ C e C e C e + C e G% α = x, z (3.93) ++ pφ + p 5 p 3 + pφ 4 yα yα yα yα yα ( λ; y, y') = 2 pd 2 pd 2 p ( + Re )( Re ) and φ 5 = 2 d + y+ y'. Placing the above elationships fo the spectal-domain Hetzian potential Geen's function in dyadic fom gives t t t p G% ( λ; y, y') = G% ( λ; y y') + G% ( λ; y, y') (3.94) t t p p p p G% ( λ; y y') = IG% = xg ˆ % xˆ+ yg ˆ % yˆ+ zg ˆ % p zˆ ( λ ;, ') = % xx + yx + yy + yz + zz (3.95) t G% y y xg ˆ % xˆ yg ˆ % xˆ yg ˆ % yˆ yg ˆ % zˆ zg ˆ zˆ (3.96) Note that no t t t t G, G, G, o G xz zx xy zy tems ae equied in the nomal o tangential components of the eflected Geen s functions. This is due to the popeties of the intefaces at y = 0 and y = d. Coupling will only occu when an x o z-diected cuent 69

84 inteacts with the dielectic-ai bounday, which is accommodated by the t G yx and t G yz components in the eflected dyadic Geen s function. Summaizing, the Geen s function is identified as G% αα e ( p p ) e [ e + e e ] ( p + p ) e ( λ; y, y') = + p y y' 2 pd p( y y') p( y y') p( y+ y') p( y+ y') pd 2p 2 p[ p2 + p ( p2 p) e ] α = x, z (3.97) G% yy e ( λ; y, y') = + p y y' 2 p( y+ y') 2 2 pd p( y y') p( y y') p ( y+ y') ( pn 2 + p) e ( pn 2 p) e [ e + e + e ] pd 2p 2 p[ p2n + p+ ( p2n p) e ] 2 pd p( y+ y') p( y y') p( y y') p( y+ y') 2 yx λ = ξ 2 2 2pd 2pd pn 2 + p+ pn 2 p e p2 + p p2 p e (3.98) e [ e e e e ] G% + ( ; y, y') j ( N ) (3.99) [ ( ) ][ ( ) ] e [ e e e e ] G% + ( ; y, y') j ( N ) (3.00) [ ( ) ][ ( ) ] 2 pd p( y+ y') p( y y') p( y y') p( y+ y') 2 yz λ = ζ 2 2 2pd 2pd pn 2 + p+ pn 2 p e p2 + p p2 p e Note, since % contains the absolute value y y ', integals o deivatives opeating on p G αα this tem must be handled caefully (it is this tem located in the volume of inteest that is cucial in the analysis of the modified leaky-wave antenna investigated in this eseach). This will be addessed in the following chapte. Now that the Geen s function fo the micostip backgound stuctue has been identified, it can be used to develop the coupled Electic Field Integal Equation fomulation fo the modified THW antenna stuctue. The EFIEs can then be solved using the Method of Moments to find the popagation constants fo the modified antenna stuctue, as discussed in the next chapte. 70

85 Chapte 4 ELECTRIC FIELD INTEGRAL EQUATION FORMULATION AND METHOD OF MOMENTS SOLUTION 4. Intoduction Fo the modified taveling-wave micostip antenna unde investigation (see Figue 3), two cuents will exist: one on the suface of the PEC stip conducto J S and one within the volume unde the stip J = jωε ( ε) E. Since both of these cuents V 3 ae immesed in egion of the gounded slab backgound envionment (see Chapte 3, Figue 29), the Geen s function deived in the pevious chapte can be used to epesent the scatteed fields maintained by these cuents. This chapte will demonstate how these scatteed fields ae used to develop a coupled Electic Field Integal Equation (EFIE) fomulation fo the unknown cuents J S and J V and subsequently solved via the Method of Moments (MoM). Fom the MoM solution, the desied popagation chaacteistics can be computed fo the baseline antenna (stip conducto only), as well as fo the modified micostip stuctue. It is impotant to note that although the THW antenna is unde investigation, the full-width antenna shown in Figue 3 will be analyzed in this and subsequent chaptes. This is due to the ease of modeling the full-width antenna and its associated bounday conditions as compaed to the elative difficulty equied to apply the bounday 7

86 conditions (specifically, the vetical shunt) to the half-width antenna. Zelinski showed that fo the odd-numbeed highe-ode modes of opeation, thee was no diffeence between the esults fo the half-width vesus the full-width antenna [5]. Given that this eseach is focused on finding the popagation chaacteistics of the fist highe-ode mode (i.e., the EH mode) fo the antenna stuctue, the esults fo the full-width antenna will be the same as the half-width antenna in this mode. Additionally, it will be shown late in this chapte how the basis functions used to epesent the stip cuents and electic fields ae chosen such that they will dive the solution to an odd-numbeed highe-ode mode. The desied EH mode will theefoe be the lowest-ode odd mode. Thus, due to the pope choice of basis functions, the shunt is not equied fo the fullwave analysis. Note, howeve, the physical implementation of the antenna equies the shunt to be pesent in ode to block the dominant mode, and subsequent even highe ode modes, so that the antenna can opeate as a leaky-wave antenna. The fact that the shunt maintains odd-mode puity is impotant in the integal equation development as it allows a natual-mode cuent solution to be sought (since we ae inteested hee in only the popagation chaacteistics, a foced solution is not necessay due to this mode puity). Finally, it is impotant to mention that the length of the modified antenna modeled in this chapte is assumed to be infinite in extent along the guiding axis in ode to simplify the analytical development. Although the actual THW antenna is finite in length, its leaky-wave behavio effectively suppots a fowad taveling wave only (with minimal teminal eflection), thus appeaing as though it wee infinite in length. 72

87 i s E( ) = E ( ) + E ( ) y Ai Region 2 (ε 2 =ε 0, µ 0 ) y=d x = -w/2 C x = w/2 y=0 CS PEC Region 3 (ε 3, µ 0 ) z Region (ε, µ 0 ) x Figue 3: Coss-sectional view of the modified full-width leaky-wave micostip antenna. 4.2 Electic Field Integal Equation Fomulation The Geen s function fo the backgound envionment was deived in the pevious chapte. This expession can be used to find the scatteed electic field within the specific egion of inteest which, when inseted into the coupled EFIEs, can be solved using the MoM technique and the specific popagation chaacteistics of the antenna can be found. Figue 3 shows the modified leaky-wave micostip antenna. infinitesimally-thin PEC stip conducto suppoting suface cuent J S The is assumed to be infinite in extent along the guiding axis (i.e., the z axis) and located at y = d (igoously, y ). The egion 3 mateial suppoting volume cuent J is also = d V assumed to be infinite in extent along the guiding axis and has coss-sectional dimensions W W 2 2,0 < x < < y< d. The total electic field at any location is compised of two components: the impessed field and the scatteed field, and can be witten as i s E( ) = E ( ) + E ( ) (4.) 73

88 o in the ζ domain (tansfom on z pompted by the infinite guiding length) as i s E% ( ρ, ζ) = E% ( ρ, ζ) + E% ( ρ, ζ). (4.2) whee ρ = xx ˆ + yy ˆ. This fundamental field elation is utilized in the coupled EFIE fomulation in the following manne descibed next. In ode to ensue uniqueness, appopiate bounday conditions fo the modified stuctue must be satisfied. The bounday conditions of the backgound envionment ae natually satisfied since they ae built into the Geen s function development of Chapte 3. The emaining bounday/field conditions that equie enfocement ae at the suface of the stip conducto (suface contou C) and within egion 3 (coss section CS), namely tˆ E% ( ρζ, ) = 0... foρ C; tˆ= xˆ, zˆ (4.3) i s E% ( ρζ, ) = E% ( ρζ, ) + E% ( ρζ, )... foρ CS. (4.4) The spectal-domain bounday condition in equation (4.3) states that the total tangential electic field must be zeo at the suface of the stip conducto. Equation (4.4) is the elation that must exist on the total electic field within egion 3. With the aid of equation (4.2), the above bounday condition elations can be witten as ˆ s ˆ i t E% ( ρζ, ) = t E% ( ρζ, )... foρ C; tˆ= xˆ, zˆ (4.5) s i E% ( ρζ, ) E% ( ρζ, ) = E% ( ρζ, )... foρ CS. (4.6) These two bounday condition elations fom the basis of the coupled EFIEs. To s complete the EFIE development, an expession fo E% ( ρ, ζ ) must be found and is discussed next. 74

89 The spatial-domain scatteed electic field in egion can be witten in tems of the Hetzian potential using equation (2.45) s E ( ρ, z) = k + π ( ρ, z). (4.7) 2 ( ) Upon Fouie tansfomation of equation (4.7), the scatteed field in the ζ-domain becomes s E% ( ρ, ζ) = + % % % π ( ρ, ζ) (4.8) 2 ( k ) whee = % xˆ + yˆ + zˆ jζ. Note, the tem jζ esults fom application of the Fouie x y diffeentiation theoem. The Hetzian potential π ( % ρζ, ) can be found fom % π ( ξ, y, ζ) deived in Chapte 3 using an invese tansfom in ξ, namely j x π ( ρζ, ) = % % π( ξ, y, ζ) e ξ dξ (4.9) 2π Fom the pevious chapte, it was shown that the Hetzian potential % π ( ξ, y, ζ) can be expessed as (using the Geen s functions deived in Chapte 3 and the total cuent) t d G % ( y y '; ξζ, ) % π ξ ζ = % ξ y ζ dy (4.0) (, y, ) J(, ', ) ' jωε 0 Recall, the cuent is decomposed into two components, the suface conduction cuent on the top conducto and the volume polaization cuent in egion 3, thus J% ( ξ, y', ζ) = J% ( ξ, y', ζ) + J% ( ξ, y', ζ ) (4.) C V 75

90 Since bounday conditions must be applied in the x, y domain, and the cuents exist on the stip conducto and within egion 3, it is necessay to epesent J % ( ξ, y', ζ ) in tems of J % ( x', y', ζ ) using the invese tansfom elation, that is W /2 j x' J % ξ ( ξ, y', ζ) = J% ( x', y', ζ) + J% ( x', y', ζ) e dx' W /2 ( C V ) (4.2) within Region 3. Substituting (4.2) into (4.0) and the esultant into (4.9) leads to t d W /2 G% ( y y'; ξζ, ) jξx' jξx % (, ) = ( J% C( x', y', ) + J% V( x', y', )) e dx' dy' e d 2π jωε 0 W /2 π ρζ ζ ζ ξ (4.3) Intechanging limits of integation gives t % π ( ρζ, ) e J ( x', y', ζ) J ( x', y', ζ) dx' dy' dξ ( C V ) d W /2 G ( y y'; ξζ, ) jξ ( x x') % = % + % 2π jωε 0 W /2 and upon substitution into (4.7) poduces the desied expession fo the scatteed field 0 W /2 (4.4) t d W /2 s G% ( ) ( ) 2 ( y y'; ξζ, ) j (, ) ( x x E% ξ ρ ζ = k ') + % % e J% C( x', y', ζ) + J% V( x', y', ζ) dx' dy' dξ 2π jωε (4.5) As a final step, the volume cuent density J % V is elated to the total electic field (using volume equivalence) by the elation J% = jωε ( 3 ε) E%. Thus, upon eaanging V this volume cuent elation, the electic field E % can be witten as 76

91 J% (, ) E% (, ) V ρ ζ ρζ = (4.6) jω( ε ε ) 3 povided ε3 ε. In addition, the stip conduction suface cuent density can be epesented using the elation J% ( x', y', ζ) = J% ( x', ζ) δ( y' d) (4.7) C S Theefoe, using the elations (4.5)-(4.7) in equations (4.5) and (4.6) leads to the desied coupled EFIEs fo the unknown spectal-domain cuent densities ( k + ) tˆ 2 %% W t /2 j ( x x') [ G% ξ ( y d; ξζ, ) e J% S ( x', ζ) dx' 2π jωε W /2 d W /2 t + G% e J% dxdy d = tˆ E% fo C tˆ= xˆ zˆ 0 W /2 ( k + ) ] jξ ( x x') i ( y y'; ξζ, ) V ( x', y', ζ) ' ' ξ ( ρ, ζ)... ρ ;, 2 W /2 jξ ( x x') [ y d ξζ 2π jωε W /2 0 W /2 %% t G% ( ;, ) e J% ( x', ζ) dx' S jξ ( x x') V i ( y y'; ξζ, ) V ( x', y', ζ) ' ' ξ ( ρ, ζ)... jωε ( 3 ε) J % S and J % V (4.8) d W /2 t J% ( ρζ, ) + G% e J% dxdy d = E% foρ CS ] (4.9) Note, the baseline antenna involving the stip conducto only can easily be analyzed based on equation (4.8) alone since the volume cuent density vanishes when ε3 = ε. As mentioned peviously, we ae only inteested in the popagation constant ζ in this study, thus we ae not inteested in how the fields ae ultimately elated to the souce stength. In addition, it is assumed that the excitation of the leaky-wave antenna leads to a vey pue modal esponse (i.e., a nealy pue EH mode esponse fo the THW 77

92 antenna). Consequently, it is theefoe sufficient to seek a natual-mode (i.e., eigenmode) cuent solution to the above coupled integal equations, which is developed next Eigenmode Cuent In geneal, the esponse to an abitay excitation will esult in a supeposition of natual modes (i.e., eigenmodes) whose complex amplitude coefficients ae elated to the stength of the impessed souce. Howeve, due to the THW shunt design, the mode puity is vey high. Thus, we anticipate a single mode (the EH mode in this eseach) to be a vey good model of the cuents excited on the THW antenna. Since we ae pimaily only inteested in the field distibution j ζ z e (i.e., ζ = β jα ) and not on how the field stength is elated to the impessed souce stength, we can fomulate the integal equation fo natual mode cuents; that is, we ae only inteested in solving homogeneous and unfoced coupled integal equations. The unfoced coupled EFIEs can be developed as a special case of equations (4.8) - (4.9) and is discussed next. If we let ζ p epesent the th p natual mode popagation constant, the anticipated spatial field distibution is e m j ζ z p fo a fowad/evese taveling wave, espectively. In the spectal domain (i.e., the ζ domain), this exponential function manifests itself as a pole singulaity of the fom /( ζ ± ζ p ). Thus, a spectal-domain cuent density behavio nea an eigenmode can be epesented as [2] J% p ( ρ) J% ( ρζ, ) (4.20) ζ ± ζ p 78

93 whee J % p is the eigenmode cuent associated with the th p discete natual mode. Substituting (4.20) into (4.8) and (4.9) and multiplying though by ζ ± ζ poduces p ( k + ) tˆ 2 %% W t /2 j ( x x') [ G% ξ ( y d; ξζ, ) e J% ps ( x', ζ) dx' 2π jωε W /2 d W /2 t + G% e J% dxdy dξ = ζ ± ζ tˆ E% ρ, ζ) 0 W /2 jξ ( x x') i ( y y'; ξζ, ) pv ( x', y', ζ) ' ' ( p ) ( ] (4.2) ( k + ) 2 W /2 jξ ( x x') [ y d ξζ 2π jωε W /2 0 W /2 %% t G% ( ;, ) e J% ( x', ζ) dx' ps d W /2 t J% ( ρζ, ) + G% e J% dxdy dξ = ζ ± ζ E% ρ, ζ) jξ ( x x') pv i ( y y'; ξζ, ) pv ( x', y', ζ) ' ' ( p ) ( jωε ( 3 ε) Now, the impessed field ] (4.22) i E % is not influenced by the stuctue it is inseted into, simila to an ideal battey being inseted into a cicuit. Thus, i E % must be analytic nea the guiding-stuctue poles ζ ζ = m p. Theefoe, in the limit as m p ζ ζ, (4.2) and (4.22) educe to the unfoced coupled EFIEs ( k + ) tˆ 2 %% W t /2 j ( x x') [ G% ξ ( y d; ξζ, ) e J% S ( x', ζ) dx' 2π jωε W /2 d W /2 t + G% e J% dxdy dξ = foρ C tˆ = xˆ zˆ 0 W /2 ( k + ) jξ ( x x') ( y y'; ξζ, ) V ( x', y', ζ) ' ' 0... ;, 2 W /2 jξ ( x x') [ y d ξζ 2π jωε W /2 0 W /2 %% t G% ( ;, ) e J% ( x', ζ) dx' d W /2 t J% ( ρζ, ) + G% e J% dxdy dξ = ρ CS jξ ( x x') V ( y y'; ξζ, ) V ( x', y', ζ) ' ' 0... jωε ( 3 ε) S ] ] (4.23) (4.24) 79

94 since i lim ( ζ ± ζ ) E % ( ρ, ζ) = 0. Note, the subscipt p has been dopped fom ζ and ζ mζ p p the eigenmode cuent tems in equations (4.23) and (4.24) fo notational convenience. Finally, due to the pesence of the ε3 ε in the denominato of (4.24), it is bette (fom a numeical stability viewpoint), to utilize the elation J% V = jωε ( 3 ε) E% in both equations (4.22) and (4.23), leading to the desied unfoced coupled EFIEs 2 tˆ /2 ( k + %% W t ) j ( x x') J% S ( x ', ) [ G% ξ ζ ( y d; ξζ, ) e dx' 2π jωε W /2 d W /2 t ( ε ε ) + G% e E% dxdy d = fo C tˆ 0 W /2 0 W /2 jξ ( x x') 3 ( y y'; ξζ, ) ( x', y', ζ) ' ' ξ 0... ρ ; =, ε jξ ( x x') 3 ( y y'; ξζ, ) ( ', ', ) ' ' ( ', ', ) 0... ε ] xˆ zˆ (4.23) ( k 2 + %% W t ) /2 j ( x x') J% S ( x ', ) [ G% ξ ζ ( y d; ξζ, ) e dx' 2π jωε W /2 d W /2 t ( ε ε ) + G% e E% x y ζ dxdy dξ E% x y ζ = ρ CS ] Note, since the bounday condition in (4.23) must be enfoced at ρ C (4.24), this integal equation must be evaluated in the limit as y d. Next, the components of the dyadic Geen s function, suface cuent density and electic field will be substituted into (4.23) and (4.24) to complete the unfoced coupled EFIEs fomulation. 80

95 4.2.2 Insetion of Geen s Function into EFIEs The scala components of the coupled EFIEs can be identified by inseting the expession fo the dyadic Geen s function, suface cuent density and electic field into (4.23) and (4.24). Fom (3.94) to (3.96), the total Geen s function can be witten as t p p p G% ( ;, ') ˆ λ y y = x( G% + G% ) xˆ yg ˆ + % xˆ yˆ + ( G% + G% ) yˆ+ yg ˆ % zˆ+ zˆ ( G% + G% ) zˆ xx yx yy yz zz (4.25) Combining the pincipal and the eflected tems into a total Geen s function tem gives t G% ˆ ( λ ; y, y ') = xg% ˆ ˆ ˆ ˆ ˆ ˆ ˆ ˆ % xxx + yg% yxx + yg% yyy + yg% yzz + zgzzzˆ (4.26) whee the G%, G%, and G% xx yy zz tems contain both the pincipal and eflected field contibutions of the Geen s function. The suface cuent on the top conducto J % S and the electic field volume of egion 3 can be witten as E % within the J % ( x', ζ) = xˆ J % ( x', ζ) + zˆ J % ( x', ζ) (4.27) S x z E % ( x', y', ζ) = xˆ E % ( x', y', ζ) + yˆ E % ( x', y', ζ) + zˆ E % ( x', y', ζ) (4.28) thus, giving the following elations x y z t G% ( λ; y d) J% ( x ', ) xg ˆ ζ = % J% + yg ˆ % J% + yg ˆ % J% + zg ˆ % J % S xx x yx x yz z t G% ( λ; y, y ') E% ( ', ', ) ˆ x y ζ = xg% ˆ ˆ ˆ ˆ % % xxe % x + yg% yxe % x + yg% yye % y + yg% yze % z + zgzzez (4.30) which allows (4.23) and (4.24) to be witten as zz z (4.29) 8

96 d W /2 0 W /2 ( k + %% ) % % % % % % % % ( xx x yx x yz z zz z ) 2 W /2 jξ ( x x') ˆ e lim t [ xg ˆ J yg ˆ J yg ˆ J zg ˆ J dx' y d 2π jωε W /2 % % % % % % % % % % ( xx x yx x yy y yz z zz z ) ( ε ε ) ε 3 jξ ( x x') xg ˆ E yg ˆ E yg ˆ E yg ˆ E zg ˆ E e dx ' dy ' d ] ξ = (4.3) ( k + %% ) % % ( xx x + % % yx x + % % yz z + % % zz z ) 2 W /2 jξ ( x x') 2π W /2 ( ˆ % % ˆ ˆ ˆ xx x + % % yx x + % % yy y + % % yz z + ˆ % % zz z ) d W /2 ( ε3 ε) jξ ( x x') + ' 0 W /2 [ e xg ˆ J yg ˆ J yg ˆ J zg ˆ J dx ' jωε xg E yg E yg E yg E zg E e dxdy dξ ε '] xe ˆ % ( xy,, ζ) ye ˆ % ( xy,, ζ) ze ˆ % ( xy,, ζ) = 0 x y z (4.32) Befoe the k + %% 2 opeato can be bought inside the integals, the pincipal field potions (3.84) of the Geen s function must be consideed. The absolute value tem y y' in (3.84) causes a discontinuity at y = y', thus any integation with espect to y ' must be split into two integals one integal fom 0 to y and the othe fom y to d. Thus, fo only the tems containing the pincipal field ( p p G%, G%, and G% p ) xx yy zz d p y y' y δ p( y y') d p( y y') p y y' e e e d e dy ' = lim dy ' + dy ' = PV dy ' + 2p 2p 2p 2p δ y+ δ 0 (4.33) whee the PV notation indicates that the integal is evaluated in a Cauchy Pinciple Value sense. Upon using Leibnitz s ule of diffeentiation, the x and z diffeentiation opeatos can feely pass though the integal as well as the y opeato. Howeve, the opeato 2 y 2 will not feely pass though due to the y dependent limits of integation in (4.33). Havilla [2] showed that applying the 2 y 2 opeato to (4.33) esults in 82

97 w w 2 2 d p y y' 2 d 2 p y y' e e dy ' dx ' = δ( ρ ρ') dy ' dx y 2p ' (4.34) y 2p 2 2 w 0 w Examination of (4.3) and (4.32) eveals that the 2 y 2 opeato will only be pesent fo the tem G %, with ρ = ρ ' yy leading to the exta contibution in (4.34). Additionally, the x diffeentiation will only apply to the exp( jξ ( x x')) tem esulting in the facto jξ, thus the opeato 2 k + %% can be epesented as ξ ˆ ˆ ˆ ˆ ˆ y ζ ξ y ζ ˆ (4.35) 2 k j x y j z j x y j z Applying (4.35) to (4.3) and (4.32) esults in W /2 2 2 lim ˆ { [ { ˆ ( ) y d W /2 ( ) ( ) ( ) t x k ξ G% xxj % x + jξ G% yxj % x + G% yz J% z ξζg% zz J% z y y y y 2 2 ˆ y ( k ) G% 2 yxj % x G% yzj % z jξ G% xxj % x jζ G% zzj % z 2 2 ˆ + z ( k ζ ) G% zz J% z + jζ G% yxj % x + G% yzj % % z ξζg xxj % x y ( ) jξ ( x x') e dx' 2π jωε ( ) d W /2 ( ε3 ε) 2 2 ˆ + { x ( k ξ ) G% xxe % x + jξ G% yxe % x + G% yye % y + G% yze % z ξζg% zze % z 2πε 0 W /2 y y y y 2 2 ˆ y ( k ) G% 2 yxe % x G% yye % y G% yze % z jξ G% xxe % x jζ G% zze % z ( ) 2 2 ( ') + ˆ jξ x x z ( k ζ ) G% zze % z + jζ G% yxe % x + G% yye % y + G% yze % z ξζg% xxe % x } e dx ' dy '] dξ y }= 0 } (4.36) 83

98 W /2 [ { W /2 ( ) ( ) 2 2 ˆ x ( k ξ ) G% xxj % x + jξ G% yxj % x + G% yzj % z ξζg% zz J% z y y y y 2 2 ˆ y ( k ) G% 2 yxj % x G% yzj % z jξ G% xxj % x jζ G% zzj % z ( ) 2 2 ˆ + z ( k ζ ) G% zzj % z + jζ G% yxj % x + G% yzj % z ξζg% xxj % x y ( ) jξ ( x x') e dx ' 2π jωε ( ) d W /2 ( ε3 ε) 2 2 ˆ + { x ( k ξ ) G% xxe % x + jξ G% yxe % x + G% yye % y + G% yze % z ξζg% zze % z 2πε 0 W /2 y y y 2 2 ˆ y ( k ) G% ( ') 2 yxe % x G% yye % y G% yze % z δ y y jξ G% xxe % x jζ G% zze % z ( ) 2 2 ( ') ˆ jξ x x + z ( k ζ ) G% zze % z + jζ G% yxe % x + G% yye % y + G% yze % z ξζg% xxe % x } e dx ' dy '] d y ξ xe ˆ% ( x, y, ζ) ye ˆ % ( x, y, ζ) ze ˆ % ( x, y, ζ) = 0 } x y z y (4.37) Note, this fom assumes that the cuents/fields ae sufficiently smooth to allow this intechange and that the PV notation has been dopped fo convenience. Again, it must be noted that the Geen s function tems in (4.36) and (4.37) that ae multiplied by the suface cuent ae valid when y' = d and the subscipt p has been dopped fom all ζ. The next section will discuss how the Method of Moments will be used to discetize (4.36)-(4.37) and subsequently be solved fo the th p unknown eigenmode popagation constant of the modified micostip stuctue in Figue Method of Moments The solutions of (4.36) and (4.37) ae discete-mode stip cuents and volumetic electic fields. As such, they lend themselves to the use of a Galekin s Method of 84

99 Moments (MoM) solution. The coupled equations can then be solved fo points at the discete pole singulaities ζ p. It is fom these singulaity values that the popagation chaacteistics of the taveling wave micostip stuctue can be diectly detemined. The use of the Method of Moments to solve fo the unknown stip cuents and volumetic electic fields esults in the homogeneous matix equation [ L][ u ] = 0 (4.38) whee [ L ] is the matix defined by the Geen s function tems in (4.36) and (4.37) and the unknown vecto [ u ] is populated with the coefficients that define the stip cuents and volumetic electic fields. Fo non-tivial solutions, a popagation constant ζ p exists that causes the deteminant of [ L ], witten as L, to go to zeo. A oot seach can then be used to find ζ p fo cetain fequency/pemittivity combinations. Fom these popagation constants; the adiation, suface, and bound egime chaacteistics of the antenna stuctue can be detemined. This section has given a bief oveview of the Method of Moments and how it will be used to solve fo the popagation chaacteistics of the micostip stuctue. The next step in finding the popagation chaacteistics of the micostip stuctue is to apply the MoM method to (4.36) and (4.37). It will be shown, with insight gained fom the FDTD analysis in Chapte 2, that an appopiate choice of basis functions that closely model the fields and cuents within the stuctue will significantly impove computational efficiency. 85

100 4.4 MoM Implementation The pevious section discussed how the unknown fields and cuents within the EFIEs can be solved using the Method of Moments. This section will now apply the method to the coupled equations deived in Section 4.2 to build the desied matix [ L ]. This matix can then be used to find the popagation chaacteistics of the micostip stuctue by numeically seaching fo ζ values that satisfy the elation L = 0 (igoously, L < tol, whee tol is the specified solution accuacy found that tol = 0 6 esulted in conveging values fo ζ ). As peviously mentioned, an impotant step in the MoM technique is to expand the unknown fields and cuents. The unknown electic field at a souce point in Region 3 can be witten as a summation of weighted basis functions Ex %( ', y', ζ) = xe ˆ % ( x', y', ζ) + ye ˆ % ( x', y', ζ) + ze ˆ % ( x', y', ζ) x y z N N N xˆ a e ( x', y', ζ ) + yˆ b e ( x', y', ζ) + zˆ c e ( x', y', ζ) n xn n yn n zn n= n= n= (4.39) whee e, e, and e ae known expansion functions and an, bn, and c n ae the xn yn zn unknown expansion coefficients. Note, the same expansion is used fo a field point in Region 3. Similaly, fo the suface cuents J % ( x', d, ζ) = xˆ J % ( x', d, ζ) + zˆ J % ( x', d, ζ) S x z N xˆ d j ( x', d, ζ ) + zˆ e j ( x', d, ζ ) n xn n zn n= n= N (4.40) whee j and j ae known expansion functions and d and e ae unknown expansion xn zn n n coefficients (note, the suface cuent component 86 j yn is not suppoted since the stip

101 conducto is assumed to be infinitesimally thin). Although theoetically an infinite numbe of expansion tems ae geneally equied, only N tems ae utilized fo pactical implementation of the MoM method ( N is chosen sufficiently lage to each convegence within a specified toleance). Inseting (4.39) and (4.40) into (4.36) and (4.37), the esultant EFIEs can be decomposed into xˆ, yˆ, and z ˆ components. Binging the summation outside the integals and dopping the x ', y, ' and ζ notation within the Geen s functions and fields and cuents fo notational convenience esults in fo ˆx : ( ) /2 ( ') 2 2 lim { N [ W j ξ x e x ( k ξ ) G% xxdn jxn + jξ G% yxdn jxn + G% yzen jzn ξζg% zzen jzn d ' y d n= y x 2π jωε W /2 ( ) d W /2 ( ε3 ε) ( k ξ ) G% ae + jξ G% ae + G% be + G% ce ξζg% ce 2πε W /2 y 0 xx n xn yx n xn yy n yn yz n zn zz n zn ( ) ( ) ] } jξ ( x x') e dx dy dξ = N /2 ( ') 2 2 [ W j ξ x e x ξ % xx n xn + ξ % yx n xn + % yz n zn ξζ % zz n zn n= y 2π jωε W /2 ' ' 0 ( k ) G d j j G d j G e j G e j dx' d W /2 ( ε3 ε) ( k ξ ) G % a e + jξ G % a e + G % b e + G % c e ξζg % c e 2πε 0 W /2 y xx n xn yx n xn yy n yn yz n zn zz n zn ] jξ ( x x') e dx dy dξ N anexnx y n= (4.4) ' ' (,, ζ) = 0 (4.42) 87

102 fo ŷ : ( ) ( k ) G d j G e j j G d j j G e j dx' N /2 2 ( ') 2 [ W j ξ x e x + % 2 yx n xn + % yz n zn + ξ % xx n xn + ζ % zz n zn n= y y y 2π jωε W /2 ( ) d W /2 2 ( ε3 ε) ( 2 + k + 2 ) G % ae + G % be + G % ce + jξ G % ae + jζ G % ce 2πε W / y y 0 2 y fo ẑ : yx n xn yy n yn yz n zn xx n xn zz n zn ( ) ] N jξ ( x x') e dx dy dξ bneyn x y n= ' ' (,, ζ) = 0 ( ) (4.43) /2 ( ') 2 2 lim { N [ W j ξ x e x ( k ζ ) G% zzen jzn + jζ G% yxdn jxn + G% yzen jzn ξζg% xxdn jxn d ' y d n= y x 2π jωε W /2 d W /2 ( ε3 ε) ( k ζ ) G% c e + jζ G% a e + G% b e + G% c e ξζg% a e 2πε W /2 y 0 zz n zn yx n xn yy n yn yz n zn xx n xn ( ) ( ) ] } jξ ( x x') e dx dy dξ = N /2 ( ') 2 2 [ W j ξ x e x ζ % zz n zn + ζ % yx n xn + % yz n zn ξζ % xx n xn n= y 2π jωε W /2 ' ' 0 ( k ) G e j j G d j G e j G d j dx' d W /2 ( ε3 ε) ( k ζ ) G % c e + jζ G % a e + G % b e + G % c e ξζg % a e 2πε 0 W /2 y zz n zn yx n xn yy n yn yz n zn xx n xn ] N jξ ( x x') e dx dy dξ cneznx y n= (4.44) ' ' (,, ζ) = 0 (4.45) Note that the total field appeaing outside the integal in (4.42), (4.43), and (4.45) is epesented as a summation of expansion functions in x and y (not x ' and y ' ). Applying the Galekin testing opeatos (discussed in the next section) to (4.4) though (4.45) will esult in a matix of the fom (see Appendix C fo an explanation as to how (4.47) though (4.7) ae extacted fom (4.4) though (4.45)) 88

103 Amn Bmn Cmn Dmn Emn an Fmn Gmn Hmn Imn Jmn bn Kmn Lmn MmnNmn O mn c n = 0 Pmn Qmn Rmn Smn Tmn dn Umn Vmn Wmn Xmn Y mn e n (4.46) whee d W /2 ( ε3 ε) ( 2 2 ) ( ', ') (, ) jξ ( x x') Amn = k ξ G% xx + jξ G% yx exn x y exm x y e dx ' dy ' dx dy dξ 2πε 0 W /2 y d W /2 0 W /2 e xn ( x, y) e ( x, y) dx dy xm (4.47) d W /2 ( ε3 ε) jξ ( x x') Bmn = jξ G% yyeyn ( x', y') exm( x, y) e dx' dy' dx dy dξ 2πε 0 W /2 y (4.48) d W /2 ( ε3 ε) jξ ( x x') Cmn = jξ G% yz ξζ G% zz ezn ( x', y') exm ( x, y) e dx' dy' dx dy dξ 2πε 0 W /2 y (4.49) W /2 d jξ ( x x') 2 2 e Dmn = ( k ξ ) G% xx + jξ G% yx jxn ( x ') exm ( x, y) dx ' dx dy dξ y (4.50) 2π jωε W /2 0 W /2 d jξ ( x x') e Emn = jξ G% yz ξζ G% zz jzn ( x') exm( x, y) dx' dx dy dξ y (4.5) 2π jωε W /2 0 d W /2 2 ( ε3 ε ) ( 2 ( 2 ) ( ', ') (, ) jξ x x') Fmn = k + G% yx + jξ G% xx exn x y eym x y e dx ' dy ' dx dy dξ 2πε 0 W /2 y y (4.52) d W/2 2 ( ε3 ε ) 2 jξ ( x x') Gmn = ( k + ) G% ( ') ( ', ') (, ) ' ' 2 yy δ eyn x y eym x y e dx dy dxdydξ 2πε 0 W /2 y d W/2 0 W /2 e ( x, y) e ( x, y) dx dy yn ym (4.53) 89

104 d W /2 2 ( ε3 ε ) ( 2 ( 2 ) ( ', ') (, ) jξ x x') H mn = k + G% yz + jζ G% zz ezn x y eym x y e dx ' dy ' dx dy dξ 2πε 0 W /2 y y (4.54) W /2 d 2 jξ ( x x') 2 e Imn = ( k + ) G% ( ') (, ) ' 2 yx + jξ G% xx jxn x eym x y dx dx dy dξ (4.55) y y 2π jωε W /2 0 W /2 d 2 jξ ( x x') 2 e Jmn = ( k + ) G% ( ') (, ) ' 2 yz + jζ G% zz jzn x eym x y dx dx dy dξ (4.56) y y 2π jωε W /2 0 d W /2 ( ε3 ε) jξ ( x x') Kmn = jζ G% yx ξζ G% xx exn( x', y') ezm( x, y) e dx' dy' dx dy dξ 2πε 0 W /2 y (4.57) d W /2 ( ε3 ε) jξ ( x x') Lmn = jζ G% yyeyn( x ', y ') ezm ( x, y) e dx ' dy ' dx dy dξ 2πε 0 W /2 y (4.58) d W /2 ( ε3 ε) ( 2 2 ) ( ', ') (, ) jξ ( x x') M mn = k ζ G% zz + jζ G% yz ezn x y ezm x y e dx ' dy ' dx dy dξ 2πε 0 W /2 y d W /2 0 W /2 e zn ( x, y) e ( x, y) dx dy zm (4.59) W /2 d jξ ( x x') e Nmn = jζ G% yx ξζ G% xx jxn ( x') ezm( x, y) dx' dx dy dξ y (4.60) 2π jωε W /2 0 W /2 d jξ ( x x') 2 2 e Omn = ( k ζ ) G% zz + jζ G% yz jzn ( x ') ezm( x, y) dx ' dx dy dξ y (4.6) 2π jωε W /2 0 d W /2 ˆ ( ε3 ε) jξ ( x x') Pmn = lim t j G% yx G% xx exn ( x', y') jzm( x) e dx' dy' dx d y d 2πε 0 W /2 y { ζ ξζ ξ} (4.62) 90

105 d W /2 ˆ ( ε3 ε) jξ ( x x') Qmn = lim t { jζ G% yyeyn( x ', y ') jzm( x) e dx ' dy ' dx dξ} y d 2πε 0 W /2 y (4.63) d W /2 ˆ ( ε3 ε) 2 2 jξ ( x x') Rmn = lim t ( k ) G% zz + j G% yz ezn( x ', y ') jzm ( x) e dx ' dy ' dx d y d 2πε 0 W /2 y (4.64) { ζ ζ ξ} /2 ( ') lim ˆ { W j ξ x e x Smn = t jζ G% yx ξζ G% xx jxn ( x') jzm( x) dx' dx dξ} y d y (4.65) 2π jωε W /2 /2 ( ') 2 2 lim ˆ { W j ξ x e x Tmn = t ( k ζ ) G% zz + jζ G% yz jzn ( x ') jzm( x) dx ' dx dξ} y d y (4.66) 2π jωε W /2 d W/2 ( 3 ) 2 2 ( ') lim ˆ ε ε jξ x x Umn = t ( k ) G % xx + j G % yx exn( x', y ') jxm( x) e dx' dy ' dx d y d 2πε 0 W /2 y (4.67) { ξ ξ ξ} d W /2 ˆ ( ε3 ε) jξ ( x x') Vmn = lim t { jξ G% yyeyn ( x ', y ') jxm( x) e dx ' dy ' dx dξ} y d 2πε 0 W /2 y (4.68) d W /2 ˆ ( ε3 ε) jξ ( x x') Wmn = lim t j G% yz G% zz ezn ( x', y') jxm( x) e dx' dy' dx d y d 2πε 0 W /2 y { ξ ξζ ξ} (4.69) /2 ( ') 2 2 lim ˆ { W j ξ x e x X mn = t ( k ξ ) G% xx + jξ G% yx jxn ( x') jxm ( x) dx' dx dξ} y d y (4.70) 2π jωε W /2 /2 ( ') lim ˆ { W j ξ x e x Ymn = t jξ G% yz ξζ G% zz jzn ( x') jxm( x) dx' dx dξ} y d y (4.7) 2π jωε W /2 Note that the expansion coefficients (a n to e n ), each of length N, in (4.4) to (4.45) fom the unknown vecto [ u] in (4.46). 9

106 volume obseve volume souce s E ( ) E( ) = 0 ˆ t E s ( ) = 0 E xm E ym E zm J zm J xm E xn E yn Ezn y =d volume obseve suface souce J xn J zn Amn Bmn Cmn Dmn Emn an 0 Fmn Gmn Hmn Imn J mn b n 0 Kmn Lmn Mmn NmnO mn c n = 0 Pmn Qmn Rmn Smn Tmn dn 0 U en 0 mn Vmn Wmn Xmn Y mn suface obseve volume souce y=d suface obseve suface souce y=y =d Figue 32: Equation (4.46) split into the fou egions of inteaction between souce and obseve locations Intepetation of MoM Matix Elements The matix (4.46) is made up of the Geen s function tems found in (4.47) though (4.7). Each of these has a physical intepetation as to what they epesent within the micostip stuctue. This section will descibe each of these functions and what physical inteactions they epesent and will povide the eade with a bette undestanding of what is diving the esults seen in the following chapte. In ode to gain a bette undestanding of what each element of (4.46) epesents, efe to Figue 32. The figue clealy depicts fou distinct egions. These epesent the fou types of inteactions that take place in the micostip stuctue: volume-volume, volume-suface, suface-volume, and suface-suface. Fo each of these desciptions, the 92

107 fist tem epesents the obseve (field o cuent) and uses m as its designato. The second tems epesents the souce (field o cuent) and uses n as its designato. Thus, the geen section, fo example, would descibe the tems that take into account an x, y, o z-diected electic field (n th volume souce tem) acting on an x, y, o z-diected electic field (m th volume obseve tem). As an example, the B mn tem would epesent the n th x- diected volume obseved electic field due to the m th y-diected volume souce electic field. The matix tems A mn, B mn, C mn, F mn, G mn, H mn, K mn, L mn, and M mn account fo all of the volume-volume inteactions in the antenna. The volume obseve suface souce tems ae highlighted in the yellow egion and include the tems D mn, E mn, I mn, J mn, N mn, and O mn. Fo all of these tems, y' = d since they involve having the souce tem located on the stip conducto. As an example, the I mn tem would epesent the m th y-diected volume obseved electic field due to the n th x-diected suface cuent souce. The suface obseve volume souce tems ae highlighted in the blue egion and include the P mn, Q mn, R mn, U mn, V mn, and W mn tems. Fo all of these tems, y = d since they involve having the obsevation tem located on the stip conducto. This will be accounted fo in the integals themselves, with any deivatives with espect to y being taken befoe y = d is applied. As an example, the Qmn tem would epesent the m th z- diected cuent obseved on the suface due to the n th y-diected electic field volume souce. Lastly, the suface obseve suface souce tems ae highlighted in the white egion and include the S mn, T mn, X mn, and Y mn sub-matix tems. Fo all of these tems, 93

108 y' = d and y = d since they involve having the souce and the obseve located on the stip conducto. Again, any deivatives with espect to y ae being taken befoe y = d is applied. As an example, the Y mn tem would epesent the m th z-diected obseve suface cuent due to the n th x-diected souce suface cuent. Now that the vaious tems epesenting the inteactions within the micostip stuctue have been identified in (4.46), it is necessay to epesent these cuents and fields popely using appopiate basis functions. The next section will go though the pocess by which these basis functions wee chosen Basis Functions The last step in developing the matix that epesents the coupled EFIEs of inteest is to popely choose the basis functions. The basis functions will seve as the expansion and test functions found in (4.47) - (4.7) that epesent the electic fields and suface cuents within the micostip stuctue. Because of this, they should be epesentative of how the eal physical fields and cuents behave. Fo example, if it is known that a cuent will be at a maximum in the cente of the conducting stip (in x ' ), then it is ill-advised to choose a basis function that is odd in natue to epesent this cuent. The oddness of the function will always cause the magnitude of the cuent to be zeo at the cente ( x ' = 0), which is non-physical fo cetain modes. Sine and cosine basis functions wee chosen fo this eseach effot due to thei even/odd natue; thei ability to closely mimic the electic fields and cuents within the micostip stuctue; and thei othogonal chaacteistics. 94

109 Ey 2 Cell # in y Cell # in x Figue 33: The FDTD y-diected electic fields within the coss-section of Thiele half-width antenna. The vetical shunt is at cell #68 and the edge of the top conducto is at cell #8. In ode to popely choose which sine and cosine basis functions should be used to model the x-, y-, and z-diected fields and the x- and z-diected suface cuents, insight was needed as to how these behave within the stuctue. Looking at the stuctue in Figue 2 and knowing that the antenna is designed to opeate in the fist highe ode mode (see Figue 4), the even and odd natue of the fields and cuents can be deduced. Opeating in the fist highe ode mode equies the y-diected electic fields within egion 3 to be odd about the cente point ( x ' = 0). Physically, this means that the y-diected fields must always equal zeo at this point. A sine function in x ' will popely model these fields, given its odd natue. Thus, a easonable choice fo eyn ( x ') is (2n+ ) π x' eyn ( x') = sin n= 0,,2,... (4.72) W In ode to validate the choice of basis functions, an intensity plot of the electic fields inside Region 3 was analyzed fom the FDTD simulations pefomed ealie in this eseach (see Figue 33). The plot is of the y-diected electic fields in the coss-section 95