ELECTROMAGNETIC MODELING OF QUASI-OPTICAL POWER COMBINERS

Transcription

1 ELECTROMAGNETIC MODELING OF QUASI-OPTICAL POWER COMBINERS by TODD WILLIAM NUTESON A dissertation submitted to the Graduate Faculty of North Carolina State University in partial fulfillment of the requirements for the Degree of Doctor of Philosophy ELECTRICAL ENGINEERING Raleigh 1996 APPROVED BY: Chair of Advisory Committee

2 Abstract NUTESON, TODD WILLIAM. Electromagnetic Modeling of Quasi-Optical Power Combiners. (Under the direction of Michael B. Steer.) A full-wave electromagnetic simulator is developed for the analysis of finite antenna elements and grid arrays in quasi-optical systems. This electromagnetic simulator employs an efficient Galerkin moment method technique with sub-domain sinusoidal basis functions. What makes this moment method analysis unique is that it is formulated using a combination of spatial and spectral domains taking full advantage of the strengths of each method to ensure accurate and efficient evaluation of the moment matrix elements. Incorporated into the moment method simulator are quasi-optical dyadic Green s functions which are derived by separately considering paraxial fields (quasi-optical modes) which are largely responsible for distant interactions in the quasi-optical system and the corrected open space (nonmodal) interactions responsible for near neighbor coupling. Two types of quasi-optical Green s functions included are the open cavity resonator Green s function and the grid lens system Green s function which is derived here for the first time. The method presented here is for analysis of structures of finite dimensions. All other quasi-optical modeling that has been done only considers the unit cell of the array and hence does not include mutual coupling from other unit cells. Results are presented here showing the mutual coupling from finite grid arrays. Other results presented here include the driving point impedance of several antenna elements and grid arrays in a quasi-optical open cavity resonator and lens system. Field profiles of the lens system are also presented and compared with measurements. The moment method simulator is also used to find the multiport parameters of a grid array in order to describe what the active devices in the system see. All simulations agree favorably with measurements.

3 Biographical Summary Todd William Nuteson was born in Tacoma, Washington, on November 16, He received the B.S. and M.S. degrees from Wright State University, Dayton, Ohio, both in electrical engineering, in 1991 and 1993, respectively. From 1991 to 1993 he worked at Wright State University as a Research and Teaching Assistant. He received Graduate Research Fellowships from the Air Force Office of Scientific Research in 1992 and While working towards his Ph.D. degree in electrical engineering at North Carolina State University, he held a Research Assistantship with the Electronics Research Laboratory in the Department of Electrical and Computer Engineering. While pursuing the M.S. and Ph.D. degrees, he has published over 20 papers in journals and conference proceedings. In June of 1996 he won the Prestigious Bronze Medallion for Outstanding Scientific Achievement presented at the 20th Army Science Conference. This is the second highest award for research from throughout the Army presented at the conference. His research interests include numerical modeling of microwave and millimeter-wave circuits, quasi-optical power combining, antennas, and electromagnetics. He is a member of the Institute of Electrical and Electronic Engineers and the Microwave Theory and Techniques, Antennas and Propagation, and Electromagnetic Compatibility Societies.

4 Acknowledgments I would like to express my gratitude to my advisor Dr. Michael Steer for his support and guidance during my graduate studies. It was a privilege to be part of his quasi-optical research group. I would also like to express my sincere appreciation to Dr. James Mink, Dr. James Harvey, Dr. Frank Kauffman, Dr. William Stewart, and Dr. Griff Bilbro for showing an interest in my research and serving on my Ph.D. committee. Special appreciation of financial and moral support from the Army Research Office through grant DAAH under the direction of Dr. James Harvey and by a subcontract from Scientific Research Associated, Inc. under U.S. Army Missile Command contract DAAH01-95-C-R111 under the direction of Mr. John Kreskovsky. A special thanks to Dr. Krishna Naishadham for consulting with me on my research and convincing me to pursue the Ph.D. degree. A very big thanks go to my past and present graduate student colleagues. First to Dr. Gregory Monahan for showing me how to do microwave measurements and keeping me on the right track. To Mr. Steve Lipa for working with me on calibration techniques for the HP 8510C network analyzer. To Mr. Huan-sheng Hwang, Mr. Konstantin Kojucharow, and Mr. Chris Hicks for assisting me with laboratory measurements. To Ms. Jaee Patwardhan and Mr. Ahmed Khalil for working with me on my moment method simulator. And to everyone who sits in Daniels Hall Room 336 and 334. And last I would like to thank the Daniels Hall Movie Society Group which kept me sane during my graduate studies. And finally, I wish to thank my parents for their support and encouragement.

5 Contents List of Figures xi List of Symbols xvii 1 Introduction Motivation For and Objective of This Study Dissertation Overview Original Contributions Publications Literature Review Background Method of Moments Quasi-Optical Power Combining Arrays Oscillators Arrays Amplifier Arrays Numerical Modeling of Quasi-Optical Systems Quasi-Optical Dyadic Green s Functions Introduction vii

6 viii 3.2 Dyadic Green s Function of Open Cavity Resonator Derivation of Lens System Dyadic Green s Function Modal Component = GEm Nonmodal Component = GEn Final Expression Modal Reflection Coefficient Conductor Losses Diffraction Losses Resonant Frequencies Other Dyadic Green s Functions Free Space Half Space Microstrip Dielectric Slab Method of Moments General Formulation Expansion Functions Quasi-Optical Moment Matrix Elements Open Cavity Resonator Lens System Nonquasi-Optical Moment Matrix Elements Spatial Domain Spectral Domain Excitation Vector... 60

7 ix Delta-Gap Voltage Generator Coaxial Current Probe Numerical Considerations Spectral Domain Moment Matrix Elements Convergence Issues Condition Number Valid Frequency Ranges Multiport Analysis Nodal Admittance Parameters Computed and Experimental Results Introduction Open Cavity Resonator Inverted L Antenna Rectangular Patch Antenna IMPATT Diode Oscillator Lens System Electric Field Profiles Measurement and Simulation Techniques Extended Unit Cell Grid Array Grid Array Multiport Parameters Two-Port Structure Four-Port Structure

8 x Patch Antenna Array Grid Array Finite Grid Arrays Versus Unit Cell Approach Tapered Antenna Conclusions and Future Research Conclusions Future Research References 122 A Multiport Analysis Using Half Basis Elements 132 A.1 Introduction A.2 Development A.3 Multiport Analysis A.4 Spectral Domain A.5 Spatial Domain A.6 Summary B Mixed Potential Integral Equation Formulation 139 B.1 Introduction B.2 Mixed Potential Integral Equation B.3 Method of Moments B.3.1 Electric Field Integral Equation Formulation B.3.2 Mixed Potential Integral Equation Formulation B.3.3 Scalar Potential Singularity B.4 Summary

9 List of Figures 1.1 Power capacities of microwave and millimeter-wave devices: solid line, tube devices; dashed line, solid state devices. After Sleger et al A cascaded quasi-optical oscillator and amplifier power combiner A quasi-optical power combiner configuration for an open resonator A grid amplifier/oscillator on a dielectric slab A grid amplifier/oscillator on a dielectric slab with X and Y polarizers A two-dimensional quasi-optical power combining oscillator A two-dimensional quasi-optical power combining amplifier Unit cell configuration for a grid oscillator Cross-section of the open cavity resonator Quasi-optical lens system configuration with a centered amplifier/oscillator array Cross section of the lens system showing test field Cross section of the lens system showing source field Fields excited by a current element in free space A non-confocal resonator geometry Equivalent transmission lines for the immittance approach Cross-section configuration of an infinitesimal x-directed current element on a: (a) dielectric slab; (b) microstrip xi

10 xii 4.1 Moment method flowchart utilizing C and FORTRAN An x-directed sinusoidal basis function Locations of x and y directed currents on a rectangular grid A delta-gap voltage generator A coaxial current probe Surface current magnitude: (a) x-directed; (b) y-directed; on a rectangular patch antenna in half space Driving point reflection coefficient: (a) magnitude; (b) phase; of a dipole antenna divided into N cells Convergence rate of the driving point reflection coefficient: (a) magnitude; (b) phase; of a dipole antenna Condition number of the moment matrix: solid line, N = 10; dashed line, N = 20; dotted line, N = A four-port unit cell structure with metal in the gap region A coaxial fed inverted L antenna Method of moments cell subdivision for the inverted L antenna with a deltagap voltage generator Driving point impedance: (a) magnitude; (b) phase; for the TEM 0,0,35 mode of the inverted L antenna: solid line, simulation; dashed line, measurement Driving point impedance: (a) magnitude; (b) phase; for the TEM 0,1,35 and TEM 1,0,35 modes of the inverted L antenna: solid line, simulation; dashed line, measurement Driving point impedance: (a) magnitude; (b) phase; for the TEM 0,2,35,TEM 1,1,35, and TEM 2,0,35 modes of the inverted L antenna: solid line, simulation; dashed line, measurement A coaxial fed rectangular patch antenna Method of moments cell subdivision for the rectangular patch antenna with a delta-gap voltage generator

11 xiii 5.8 Impedance Smith chart showing the driving point impedance of the patch antenna without the reflector: solid line, simulation; dashed line, measurement Impedance Smith chart showing the simulated driving point impedance of the patch antenna in the open cavity resonator: solid line, TEM 0,0,34 mode; dashed line, TEM 0,0,23 mode Simulated driving point reflection coefficient magnitude of the patch antenna: solid line, open cavity resonator; dashed line, half space Locations for three patch antennas in the open cavity resonator: all dimensions are in cm Simulated driving point reflection coefficient magnitude of patch antenna 1 for the TEM 0,0,35 mode: solid line, alone; dotted line, with patch antennas 2 and IMPATT diode and patch antenna impedance: solid line, negative impedance measurement of the IMPATT diode; dashed line, simulated driving point impedance of the center fed patch antenna in half space Simulated driving point impedance of the patch antenna in the open cavity resonator for the TEM 0,1,35 and TEM 1,0,35 modes: frequency range from GHz to 8.69 GHz IMPATT diode oscillator: (a) cavity spacing D =61.25 cm; (b) cavity spacing D =61.41 cm: solid line, measured oscillation in half space; dashed line, measured oscillation in the open cavity resonator; dotted line, simulated scaled driving point reflection coefficient magnitude of the patch antenna Oscillation frequencies on an expanded impedance Smith chart: (a) cavity spacing D =61.25 cm; (b) cavity spacing D =61.41 cm: solid line, oscillator in the open cavity resonator; dashed line, oscillator in half space. Note: this is an enlarged view of Fig Configuration for measuring the electric field intensity in the quasi-optical lens system Field distribution at the beam waist (z = 0): solid line, simulation; points, measurement Field distribution away from the beam waist (z=55 cm): solid line, simulation; points, measurement

12 xiv 5.20 Field distribution along the z-axis: solid line, simulation; dashed line, measurement Driving point reflection coefficient: (a) measurement technique; (b) delta-gap model; (c) moment method simulation technique Reflection coefficient magnitude for the shorted measurement probe: solid line, with absorber around coaxial probe; dashed line, without absorber Driving point reflection coefficient: (a) magnitude; (b) phase; of the extended unit cell: solid line, simulation; dashed line, measurement An extended unit cell along with the cell subdivision used Driving point reflection coefficient: (a) magnitude; (b) phase; of the extended unit cell: solid line, simulation; dashed line, measurement A 3 3 grid array with the driving point impedance being measured in the middle gap: (a) other gaps opened; (b) other gaps shorted Driving point reflection coefficient: (a) magnitude; (b) phase; of the 3 3 opened grid: solid line, simulation; dashed line, measurement Driving point reflection coefficient: (a) magnitude; (b) phase; of the 3 3 shorted grid: solid line, simulation; dashed line, measurement Driving point reflection coefficient magnitude in the corner gap of the 3 3 shorted grid: solid line, simulation; dashed line, measurement A 3 3 quasi-optical grid along with the cell subdivision used Driving point reflection coefficient: (a) magnitude; (b) phase; of the 3 3 grid: solid line, simulation; dashed line, measurement A 5 5 quasi-optical grid along with the cell subdivision used Driving point reflection coefficient: (a) magnitude; (b) phase; of the 5 5 grid: solid line, simulation; dashed line, measurement A two-port structure: (a) physical layout; (b) MoM modeling scheme; (c) V2 shorted; (d) V1 shorted; (e) measurement Magnitude of S 11 and S 21 for the two-port structure: solid line, simulation; dashed line, measurement A four-port structure with metal in the gap regions

13 xv 5.37 Driving point reflection coefficient magnitude of the four-port structure with terminals 3 and 4 shorted: solid line, simulation; dashed line, measurement A nine-port 3 3 patch antenna array in the open cavity resonator Simulated scattering parameters of the 3 3 patch antenna array: solid line, S 11 ; dashed line, S Simulated scattering parameters of the 3 3 patch antenna array: solid line, S 12 ; dashed line, S Simulated scattering parameters of the 3 3 patch antenna array: solid line, S 15 ; dashed line, S A nine-port 3 3 grid array in the lens system Simulated scattering parameters of the 3 3 grid array: solid line, S 11 ;dashed line, S Simulated scattering parameters of the 3 3 grid array: solid line, S 12 ;dashed line, S 13 ; dotted line, S Simulated scattering parameters of the 3 3 grid array: solid line, S 15 ;dashed line, S 25 ; dotted line, S Numbering scheme for the grid array consisting of 100 unit cells Driving point reflection coefficient of the grid: solid line, S 1,1 ;dashed line, S 45,45 ; dotted line, unit cell Simulated scattering parameters of the grid: solid line, S 45,45 ;dashed line, S 23,23 ; dotted line, S 1, Simulated scattering parameters of the grid: solid line, S 45,46 ;dashed line, S 45,43 ; dotted line, S 45, Simulated scattering parameters of the grid: solid line, S 1,2 ;dashed line, S 1,4 ; dotted line, S 1, A tapered antenna along with the cell subdivision used Driving point reflection coefficient magnitude of the tapered antenna: solid line, simulation; dashed line, measurement A 40 GHz grid amplifier unit cell layout

14 xvi 6.2 Moment method unit cell layout with equal sized cells Moment method unit cell layout with unequal sized cells A.1 Port definition for the unit cell of a grid amplifier/oscillator A.2 The location of the delta-gap voltage source and the corresponding half basis element at the tip of a feed line A.3 Configuration of x-directed sinusoidal basis functions with the source basis in the left column and the test basis in the right column B.1 Unequal size basis function: (a) rooftop and (b) pulse doublet

15 xvii List of Symbols A j a mn a st â x â y â z b b mn b st c c mn c st d D d mn d st E EFIE E in E m E mn ± E S E inc t E scat t E st T f mn f m,n,q F ji F x F y = GE = GE = GE0 = GEc = GEf = GEh = GEl Coefficient for numerical integration. Coefficient for modal source field. Coefficient for modal test field. Unit vector along the x axis in Cartesian coordinates. Unit vector along the y axis in Cartesian coordinates. Unit vector along the z axis in Cartesian coordinates. Radius of curvature for spherical lens. Coefficient for modal source field. Coefficient for modal test field. Speed of light. Coefficient for modal source field. Coefficient for modal test field. Dielectric substrate thickness and spacing above ground plane. Reflector and lens spacing. Coefficient for modal source field. Coefficient for modal test field. Electric field. Electric Field Integral Equation. Incident electric field. Electric field in a conducting plane. Scalar electric field Hermite-Gaussian traveling wave-beam with propagation in the ±â z direction. Electric source field excited by a source current. Transverse incident electric field. Transverse scattered electric field. Electric modal test field. Coefficient for modal source field. Modal resonant frequency. Spectral domain integrand argument for MoM elements. Focal length with respect to the x axis. Focal length with respect to the y axis. Electric field dyadic Green s function. Spectral domain electric field dyadic Green s function. Free space component of dyadic Green s function. Open cavity resonator component of dyadic Green s function. Open space component of dyadic Green s function. Half space component of dyadic Green s function. Lens system component of dyadic Green s function.

16 xviii = GEm = GEn = GEp = GEqo G xx E G xx E G xy E G xy E G yy E G yy E G yx E G yx E g mn Modal component of dyadic Green s function. Nonmodal component of dyadic Green s function. Paraxial component of dyadic Green s function. Quasi-optical component of dyadic Green s function. â x â x component of spatial domain dyadic Green s function. â x â x component of spectral domain dyadic Green s function. â x â y component of spatial domain dyadic Green s function. â x â y component of spectral domain dyadic Green s function. â y â y component of spatial domain dyadic Green s function. â y â y component of spectral domain dyadic Green s function. â y â x component of spatial domain dyadic Green s function. â y â x component of spectral domain dyadic Green s function. Coefficient for modal source field. He n (x) Hermite polynomial of order n and argument x. HG Hermite Gaussian. H m Magnetic field in a conducting plane. H mn ± Scalar magnetic field Hermite-Gaussian traveling wave-beam with propagation in the ±â z direction. H S Magnetic source field excited by a source current. H st T Magnetic modal test field. I Current vector for MoM. I i Current element for MoM. = It Transverse unit dyad. J m (x) Bessel function of the first kind and order m. J S Electric current density. J x Electric current density in the â x direction. K Kernel for Huygen s integral in free space. k 0 Free space wavenumber. k 1 Dielectric wavenumber. k x Spectral domain variable. k y Spectral domain variable. k z0 Free space propagation constant. k z1 Dielectric propagation constant. LG Laguerre Gaussian. L t p (x) Generalized Laguerre polynomial of order p and argument x. MoM Method of Moments. N Total number of unknowns in MoM. N HG Hermite Gaussian normalizing coefficient. N LG Laguerre Gaussian normalizing coefficient. N x Total number of x unknowns in MoM. Total number of y unknowns in MoM. N y

17 xix P in P loss P out r r R 1,mn R 2,mn R mn S S1 S2 T T 1,mn T 2,mn TE TEM TM V Wi x W i x W y i W y i X Ȳ Z Z 0 Z 1 Z f,ji Z in Z m Z qo,ji Z TE Z TM ZT TE Z TM T Z TE B Z TM B ZBL TE ZBL TM Incident power to a conducting plane. Power lost in a conducting plane. Total power reflected from a conducting plane. Observation or test location with respect to the origin. Source location with respect to the origin. Modal reflection coefficient for first lens. Modal reflection coefficient for second lens. Reflection coefficient of the traveling wave-beam modes. Closed surface that bounds the quasi-optical system. Infinite plane before the first lens. Infinite plane after the second lens. Transmission coefficient. Modal transmission coefficient for first lens. Modal transmission coefficient for second lens. Transverse Electric. Transverse Electromagnetic. Transverse Magnetic. Voltage vector for MoM. x-directed sinusoidal basis function. Fourier transform of x-directed sinusoidal basis function. y-directed sinusoidal basis function. Fourier transform of y-directed sinusoidal basis function. Gaussian mode parameter along the x axis. Gaussian mode parameter along the y axis. Moment matrix. Free space impedance. Intrinsic impedance of a dielectric. Open space moment matrix element. Input impedance. Intrinsic impedance of a metal. Quasi-optical moment matrix element. Equivalent TE mode impedance for the immittance approach. Equivalent TM mode impedance for the immittance approach. Equivalent TE mode impedance looking into the top for the immittance approach. Equivalent TM mode impedance looking into the top for the immittance approach. Equivalent TE mode impedance looking into the bottom for the immittance approach. Equivalent TM mode impedance looking into the bottom for the immittance approach. TE mode impedance load at the bottom. TM mode impedance load at the bottom.

18 xx Z1 TE TE mode dielectric impedance. Z1 TM TM mode dielectric impedance. α Polar coordinate variable for the spectral domain. α d,mn Modal diffraction losses due to the spherical reflector. α p,mn Modal conductor losses due to the planar reflector. α s,mn Modal conductor losses due to the spherical reflector. β Polar coordinate variable for the spectral domain. Γ Reflection coefficient. δ s Skin depth. ɛ 0 Permittivity of free space. ɛ r Substrate dielectric constant. λ 0 Free space wavelength. µ 0 Permeability of free space. π Circle circumference divided by circle diameter. σ Conductivity of a metal. ψ 1,mn Phase coefficient for first lens. ψ 2,mn Phase coefficient for second lens. ψ mn Phase coefficient of the traveling wave-beam modes. ω Radian frequency. ω s Spot size for spherical lens. Ω Quasi-optical system volume.

19 Chapter 1 Introduction 1.1 Motivation For and Objective of This Study The need for high-powered, light-weight sources at millimeter-wave frequencies is becoming more demanding. Currently tube devices such as klystrons or traveling wave tube (TWT) devices are used for producing large amounts of power at millimeter-wave frequencies but are becoming less desirable because of their low life span and bulky size due to their high voltage DC power supplies. On the other hand solid state devices are generally more desirable in terms of small size, light weight, high reliability and excellent manufacturability but are limited to the level of power that can be generated at millimeter-wave frequencies. Fig. 1.1 shows a comparison of the power levels for various tube devices and solid state devices [1]. Note that this data is from 1990 and since then the power levels of both tube devices and solid state devices have increased. A promising solution to these problems is quasi-optical power combining where the power from numerous solid state devices is combined in free space over a distance of many wavelengths. With the aid of lenses the electromagnetic fields are focussed and power is channeled predominately into a single paraxial mode. As an example of a quasi-optical power combiner see Fig Here the first stage consists of an oscillator array in an open cavity resonator (Fabry Perot resonator) where the millimeterwave source is generated. The active devices in the oscillator array lock together with the aid of the resonator allowing for power combining to take place. The power is then partially transmitted through the first lens where it passes through an amplifier array. In this stage the power is amplified by combining the power from all the active devices in the amplifier array and is then passed through a second lens. If more power is needed, several more amplifier stages could be employed. One of the main advantages of quasi-optical power combining is that the energy is not guided through metal interconnects which become very lossy at millimeter-wave frequencies but instead the energy is guided through free space or dielectric slabs. Applications where light weight millimeter-wave power sources are needed include the 1

20 2 following: near vehicle detection radar (collision avoidance radar), millimeter-wave LANs (60 GHz), cellular radio base stations, active missile seekers (94 GHz), and millimeter-wave imaging (100+ GHz). OUTPUT POWER (W) Gridded Tubes Si BJT VFET Klystrons TWT s PHEMT MESFET Gyrotrons Free Electron Laser IMPATT Gunn FREQUENCY (GHz) Figure 1.1: Power capacities of microwave and millimeter-wave devices: solid line, tube devices; dashed line, solid state devices. After Sleger et al. The strategy is to develop, using numerical field analysis, a multiport impedance model of the linear part of the quasi-optical system. This can then be interfaced with commercial microwave circuit simulators. Efficiency requires that volumetric discretization must be avoided. In the finite element method (FEM) and the finite-difference time-domain (FDTD) method, the entire three-dimensional structure is analyzed by discretizing the whole volume. For electrically large systems such as in quasi-optics where interfacing surfaces are distributed over electrically large distances, the use of FEMs and FDTD methods are very inefficient. By using a moment method utilizing Green s functions appropriate to the physical structure, discretization can be limited to planar surfaces. A series of developments [2 5] culminated in a straight forward methodology for developing a novel Green s function of a quasi-optical system. The electric field dyadic Green s function of a quasi-optical system is derived in two parts: one part describing the effect of the quasi-optical paraxial fields and the other part describing the remaining fields. This form of the dyadic Green s function is particularly convenient for quasi-optical systems because of its relative ease of development. Progress toward large, high-powered, efficient arrays is hampered by the relatively

21 3 Oscillator Array Lens Amplifier Array Lens Input Polarizer Output Polarizer Figure 1.2: A cascaded quasi-optical oscillator and amplifier power combiner. crude state of design technology including the lack of suitable computer aided engineering tools. In particular, the many unit active circuits in a large array cannot be individually optimized for efficiency and stability. This is because no simulation process has been developed to model impedances and stability criteria for a finite array where most of the array elements see different circuit conditions. The essential component of quasi-optical system modeling is development of circuit-level models of quasi-optical structures. The modeling of quasi-optical systems has generally been based on the unit cell approach where the minimum three dimensional cell of an array, generally containing a single active device, is modeled using symmetry of the structure to establish electrical and magnetic side-walls for the cell. A moment method or finite element program is then used to electrically characterize the cell and obtain the impedance presented to a single device. The unit cell approach assumes an infinitely periodic structure with no mutual coupling from other unit cells in the array. In order to obtain accurate modeling, structures of finite extent must be considered along with mutual coupling. In this dissertation an advanced method of moments approach combining spatial domain and spectral domain techniques to model quasi-optical systems is presented. The moment method implementation is developed in such a way that any quasi-optical dyadic Green s function, derived using the technique discussed previously, can be implemented. Two types of quasi-optical systems are presented here: (1) the open cavity resonator which uses the Green s function developed by Heron et al. [3,5] and (2) the grid lens system in which the Green s function is derived in this dissertation for the first time. With this formulation the field solver can be conveniently used in the development of circuit-level models of the passive linear elements in the quasi-optical systems which can then be interfaced with transient analysis (SPICE) and steady state analysis (harmonic balance).

22 4 1.2 Dissertation Overview Chapter 2 presents a review of the current literature in the field of quasi-optical power combining focusing mainly on the numerical modeling of quasi-optical systems. Also presented is a history of moment method techniques. In Chapter 3 the derivation of a dyadic Green s function for a quasi-optical lens system is presented. Also included in this chapter is the open cavity resonator dyadic Green s function including techniques for computing conductor and diffraction losses. Chapter 3 also includes other dyadic Green s functions such as free space, half space, microstrip, and dielectric slab. Chapter 4 focuses on implementing the quasi-optical dyadic Green s functions from Chapter 3 using the method of moments technique. The complete formulation including a unique moment method algorithm that combines both the spatial and spectral domain is given. Numerical considerations such as convergence and condition numbers of the moment matrix are discussed. Also the analysis for computing the multiport parameters using the moment method formulation is presented. Chapter 5 validates the moment method program by comparing simulations to measurements for the open cavity resonator and lens system. Measurements are included for inverted L antennas, patch antennas, grid arrays, and taper antennas. Other simulations including finite grid arrays versus the unit cell approach and multiport parameters are presented. In Chapter 6 a summary of the dissertation is given along with conclusions and suggestions for future work in this topic. 1.3 Original Contributions The original contributions presented in this dissertation are: The derivation of a dyadic Green s function for a quasi-optical lens system. The approach used in the derivation follows that of [3 5] and can be applied to various quasioptical systems. The derivation is given in Section 3.3 of Chapter 3. Commercial electromagnetic simulators do not include quasi-optical components such as lenses and reflectors because they use standard multilayered microstrip Green s functions. By using quasi-optical Green s functions, quasi-optical elements can be incorporated into electromagnetic simulators.

23 5 The incorporation of the quasi-optical dyadic Green s functions into a method of moments formulation. This required an advanced moment method implementation where both spatial and spectral domain techniques were used for accurate and efficient analysis of quasi-optical systems. The moment method simulator has a fast convergence rate along with well-conditioned moment matrices. One very strong feature of the simulator is that it works from DC to any frequency. Often moment method simulators break down at low frequencies but results presented in this dissertation show convergence at DC. The complete formulation is presented in Chapter 4. Finite quasi-optical grids are modeled. It is very common in quasi-optics to model a single unit cell. In doing so coupling from other unit cells in the array and edge effects are ignored. All simulations presented in this dissertation are for finite arrays were coupling from all unit cells is considered along with the edge effects. The unit cell model is done by assuming that the grid is infinite and periodic with all active devices in phase. With this assumption, electric and magnetic walls are placed along the unit cell edges to force these conditions. Coupling can only be considered by these boundary conditions and not by direct radiation and quasi-optical resonant modes as is the case for finite arrays. Verification of finite array simulations with measurements are presented in Chapter 5. Comparisons of finite grid arrays versus the unit cell is presented in Section 5.5 of Chapter 5. Multiport analysis of quasi-optical grid arrays. The multiport parameters are represented as admittance parameters which represent what the active devices in the system see. The multiport parameters give the coupling coefficients of the array which are ignored in unit cell models. The admittance port parameters are then converted to nodal admittance parameters to be compatible with microwave circuit simulators that use nodal analysis. These admittance parameters are needed for transient and harmonic balance simulations. The formulation is given in Section 4.7 of Chapter 4 and results are presented in Section 5.4 of Chapter 5. Driving point reflection coefficient measurements for several grid arrays in a quasioptical lens system. This technique uses an unbalanced coaxial line that is transformed to a balanced line without the aid of a balun. The measurements were used for verification of moment method simulations over a wide frequency band. The measurement technique and results are presented in Section 5.3 of Chapter Publications The work associated with this dissertation resulted in the following publications:

24 6 T. W. Nuteson, H. Hwang, M. B. Steer, K. Naishadham, J. Harvey, and J. W. Mink, Analysis of finite grid structures with lenses in quasi-optical systems, submitted to IEEE Trans. Microwave Theory Tech. M.B.Steer,T.W.Nuteson,C.W.Hicks,J.Harvey,andJ.W.Mink, Strategiesfor handling complicated device-field interactions in microwave systems, Proc. PIERS 1996 Symp., July H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella, A 2-dimensional slab quasi-optical power combining system, URSI Symp. Dig., p. 354, July J.Harvey,M.B.Steer,J.W.Mink,H.-S.Hwang,T.W.Nuteson,andA.C.Paolella, Advances in quasi-optical power combiners provide path to radar and communications enhancements, 20th Army Science Conference, June T. W. Nuteson, M. B. Steer, K. Naishadham, J. W. Mink, and J. Harvey, Electromagnetic modeling of finite grid structures in quasi-optical systems, IEEE MTT-S Int. Microwave Symp. Dig., pp , June H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella, Two-dimensional quasi-optical power combining system performance and component design, IEEE MTT-S Int. Microwave Symp. Dig., pp , June T. W. Nuteson, G. P. Monahan, M. B. Steer, K. Naishadham, J. W. Mink, K. Kojucharow, and J. Harvey, Full-wave analysis of quasi-optical structures, IEEE Trans. Microwave Theory Tech. vol. 44, pp , May H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella, A Quasi-Optical Dielectric Slab Power Combiner, IEEE Microwave Guided Wave Lett., vol. 6, pp , Feb H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella, A slab-based quasi-optical power combining system, Twentieth International Conference on Infrared and Millimeter Waves Dig., pp , Dec H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella, Quasi-optical power combining techniques for dielectric substrates, Proc. International Semiconductor Device Research Symposium, pp , Dec H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella, Quasioptical power combining in a dielectric substrate, Proc. International Symposium on Signals, Systems and Electronics, pp , Oct

25 7 T. W. Nuteson, G. P. Monahan, M. B. Steer, K. Naishadham, J. W. Mink, and F. K. Schwering, Use of the moment method and dyadic Green s functions in the analysis of quasi-optical structures, IEEE MTT-S Int. Microwave Symp. Dig., pp , May Special Note: The Paper entitled Advances in quasi-optical power combiners provide path to radar and communications enhancements, presented at the 20th Army Science Conference in Norfolk, Virginia, received the 1996 Prestigious Bronze Medallion for Outstanding Scientific Achievement for research presented from throughout the Army.

26 Chapter 2 Literature Review 2.1 Background Quasi-optical power combining techniques [6 8] provide a means for combining power from numerous solid-state millimeter-wave sources attached to radiating elements such as antenna arrays or grids. The power from the radiating elements is combined in free-space over a distance of many wavelengths to channel power predominately into a single paraxial mode. Mink [2] studied an array of filamentary current sources radiating into a plano-concave open resonator. This was the first theoretical investigation of power combining using a source array in a quasi-optical resonator, as shown in Fig The grid oscillator [9] and grid amplifier [10], shown in Fig. 2.2, are other methods of power combining where active devices or monolithic microwave integrated circuits (MMIC) are placed in a metal grid supported on a dielectric slab. Polarizers are used to isolate the output from the input as shown in Fig. 2.3 along with a slab for tuning the system. The complex device field interactions render it difficult to optimize efficiencies and ensure stable operation. However, computer aided analysis techniques are evolving to aid in design. The strategy is to develop, using numerical field analysis, a multiport impedance model of the linear part of the quasi-optical system. This can then be interfaced with commercial microwave circuit simulators. Efficiency requires that volumetric discretization be avoided. By utilizing Green s functions appropriate to the physical background, discretization can be limited to surfaces. In References [2 5] a series of developments culminated in a straight forward methodology for developing the dyadic Green s function of a quasi-optical structure. The dyadic Green s function was derived by separately considering paraxial and non-paraxial modes. It is not feasible to derive a mixed, scalar and vector, potential Green s function, as required in conventional space domain moment method techniques. Alternatively, we have adapted an efficient moment method field solver [11 13] to use dyadic Green s functions for the analysis of quasi-optical systems. 8

27 9 PLANAR REFLECTOR SOURCE ARRAY a^ y a^x a^ z PARTIALLY TRANSPARENT SPHERICAL REFLECTOR d D Figure 2.1: A quasi-optical power combiner configuration for an open resonator. Active grid surface E E Input beam Output beam Figure 2.2: A grid amplifier/oscillator on a dielectric slab. ACTIVE GRID SURFACE OUTPUT POLARIZER E INPUT BEAM E OUTPUT BEAM INPUT POLARIZER TUNING SLAB Figure 2.3: A grid amplifier/oscillator on a dielectric slab with X and Y polarizers.

28 10 This literature review consists of three main sections. The first is on moment method techniques. It is important to know what has been done using moment method analysis in order to see how it can be used with quasi-optical modeling. The second review is on quasi-optical power combining arrays including oscillator arrays and amplifier arrays. Here reviews are presented on experimental aspects of quasi-optical system development. The last and main focus of this literature review is on numerical modeling of quasi-optical systems. Reviews on what has been done with numerical modeling of quasi-optical systems is presented. 2.2 Method of Moments The method of moments (MoM), first introduced by Harrington [14], is a numerical technique used to solve integral equations. In electromagnetics, many problems are formulated as integral equations in which analytical solutions do not exist and therefore must be solved numerically. The MoM offers an accurate numerical solution for such problems where the integral equation is transformed into a set of linear equations which is then solved for with a computer. To have a good understanding of what the MoM is and how it works, a simple mathematical outline [15] illustrating the MoM procedure is given next. or Consider the following inhomogeneous equation Lu = f (2.1) Lu f = 0 (2.2) where L is a linear operator, u is unknown, and f is known. In order to solve for u an approximate solution for (2.2) is found by the following procedure known as the MoM. Let u be approximated by a set of basis functions or expansion functions given by n u n = α k φ k, n =1, 2,... (2.3) k=1 where φ k is the expansion function, α k is its unknown amplitude, and n is the total number of expansion functions. Replacement of u by u n in (2.2) and taking the inner product with a set of weighting functions or testing functions w m, where the left side of (2.2) is orthogonal to the sequence {w m },resultsin Lu n f,w m =0, m =1, 2,...,n. (2.4) Substitution of (2.3) into (2.4) yields n α k Lφ k,w m = f,w m, m =1, 2,...,n (2.5) k=1

29 11 which is the final matrix equation of the MoM [14, 16]. In matrix form we can write Ax = b (2.6) with each matrix and vector defined by x =(α 1 α 2 α n ) T (2.7) b =( f,w 1 f,w 2 f,w n ) T (2.8) and A =[a mk ] (2.9) where T denotes the transpose and a mk are the individual matrix elements given by a mk = Lφ k,w m. (2.10) The accuracy is highly dependent upon the choice and number of expansion and weighting functions used. The best accuracy is usually achieved when the same functions are used for both expansion and weighting which is known as the Galerkin method [17]. Also it is important to note that the computation time and memory size increases significantly with the increase in the number of basis functions used. For integral equations, the linear operator L will include a Green s function. The Green s function can best be described as the impulse response of the system. For example, an electric field Green s function would describe all of the electric fields at any location due to a current filament at a fixed position. In many applications dyadic Green s functions are used. Tai in Reference [18] gives a comprehensive and rigorous analysis of dyadic Green s functions. Some of the first widely used applications of the MoM were in wire antenna simulation where the MoM was used to solve either Pocklington s integral equation or Hallen s integral equation. The MoM was used to solve for the current distribution on the wire antennas which then could be used to predict the input impedance of the antenna or the farfield radiation. In both cases the free space Green s function was used. Often the Galerkin method was used with sub-domain sinusoidal expansion and weighting functions. In References [19 21], a complete formulation for wire antennas using the MoM is given. Other articles, such as [22,23], give numerical aspects of the stability and convergence of the moment method solutions. The type of structures analyzed included Yagi-Uda array antennas [19], single and multiple log-periodic dipole antennas [20], and wire grid modeling of airplanes [19]. While the MoM was being used to model one-dimensional wire grid geometries, research was also conducted in modeling of two-dimensional surfaces with the MoM. References [24 28] present MoM techniques for modeling arbitrary surfaces in free space. The

30 12 MoM programs were used to find the surface currents on the two-dimensional structure excited by an incident plane wave. Radar cross sections (RCS) for thin rectangular plates in free space are presented in [24]. Other popular studies of radiation and scattering from arbitrary surfaces are presented in [25 28]. The biggest applications to date of moment method solutions have been in the modeling of microstrip geometries including multilayered dielectrics for open and closed structures. There are two main approaches for modeling microstrip structures using the MoM, the first is the spectral domain approach [29 38] and the second is the spatial domain approach [39 46]. Applications of microstrip modeling include dispersion characteristics of printed transmission lines [29, 30, 34], input impedance and mutual coupling of microstrip antennas [31, 39, 41 43], phased array antennas [33, 36, 37], and microstrip discontinuities [32, 40, 35, 38]. There are several advantages and disadvantages for each moment method. For example in the spectral domain the microstrip dyadic Green s functions must be derived in closed form. A technique known as the immittance approach was developed by Itoh [30] where the Green s function is easily derived for multilayer geometries using a transverse equivalent transmission line for a spectral wave along with a coordinate transformation. This method is illustrated in Section 3.6 of Chapter 3 where several Green functions are derived using the immittance approach. In the spatial domain the microstrip Green s function is derived in terms of vector and scalar potentials expressed in terms of Sommerfeld integrals. Evaluation of the Sommerfeld integrals requires numerical analysis with special treatment for handling the complex poles due to surface waves in the dielectric and the oscillatory nature of the Sommerfeld integrals described by Bessel functions. Mosig has done several comprehensive studies on the numerical evaluation of Sommerfeld integrals [39,47,48]. In [49] Chow et al. developed a closed form approximation for the spatial domain scalar and vector potential microstrip Green s functions. Further development of the closed form Green s functions for multilayered microstrip geometries is given in [50, 51]. The main advantages of the spatial domain MoM approach is that it can model arbitrarily shaped structures by gridding the structure into unequal size cells and is also conceptually easier than the spectral domain MoM approach. The spectral domain requires transformation from the spatial domain to the spectral domain through the use of Fourier transforms. The Fourier transforms are not a problem when the gridding is done with equal size cells but does pose a problem for unequal size cells. A disadvantage of a spatial domain implementation is that the self-terms contain a singularity which does not exist in the spectral domain. This singularity is handled in the spatial domain by doing singularity extraction. It is very important to evaluate the self-terms correctly because they dominate the moment matrix. The biggest disadvantage for a spectral domain analysis is that numerical integration is required over an infinite range due to taking the Fourier transforms. It then becomes very difficult to numerically evaluate elements separated by electrically large distances because of heavy oscillations. Techniques for efficient evaluation of spectral domain moment method

31 13 elements are presented in [52 54]. In the spatial domain this is not a problem because everything is integrated over physical finite regions. The main problems of both MoM approaches is the amount of CPU time and memory required for simulating electrically large structures. These problems occur when filling and inverting the moment matrix. Work on improving the matrix fill times has been done in [55] using frequency interpolation and in [11] using spatial interpolation of the moment matrix. The combination of these interpolation techniques result in very efficient matrix fills. Since the moment matrix is usually dense and often ill conditioned, the inversion of the moment matrix remains a problem for matrices of large order. An approach that is being researched today is the use of wavelets in the MoM [56 59]. In this approach scaling functions and wavelets are used in place of the traditional orthogonal basis functions. When wavelets are used the moment matrix becomes sparse which allows for a fast solution of the inverted moment matrix. In this dissertation both the spatial and spectral domain MoM approaches are used concurrently taking full advantage of the strengths of each method in order to accurately and efficiently solve quasi-optical systems. 2.3 Quasi-Optical Power Combining Arrays Several review papers have been written on the subject of quasi-optical systems. One of the first was by Goldsmith [6] which deals with quasi-optical techniques using Gaussian beams along with various quasi-optical components at millimeter and submillimeter wavelengths. Since then these quasi-optical techniques have been used with many power combining oscillator and amplifier arrays to combine power in free space. The IEEE Transactions on Microwave Theory and Techniques devoted an entire issue on quasi-optics in the 1993 October edition [8]. A recent review paper by York [7] provides some of the latest quasi-optical power combining techniques for oscillators and amplifiers. A proposal to set figures-of-merit for standard characterization of spatial and quasi-optical power combining arrays was presented in [60]. This section will focus on the experimental work that has been done for quasioptical power combining including open cavity resonator oscillators, grid oscillators, and grid amplifiers. Two-dimensional slab power combining will also be presented in this section Oscillators Arrays The open cavity resonator shown in Fig. 2.1 has been used to produce several types of quasi-optical oscillators [61 68]. In [61 63, 68] Gunn diode oscillators were demonstrated