ELECTROMAGNETIC MODELING OF

QUASI-OPTICAL POWER COMBINERS

by

TODD WILLIAM NUTESON

A dissertation submitted to the Graduate Faculty ofNorth Carolina State University

in partial fulfillment of therequirements for the Degree of

Doctor of Philosophy

ELECTRICAL ENGINEERING

Raleigh

1996

APPROVED BY:

Chair of Advisory Committee

Abstract

NUTESON, TODD WILLIAM. Electromagnetic Modeling of Quasi-Optical PowerCombiners. (Under the direction of Michael B. Steer.)

A full-wave electromagnetic simulator is developed for the analysis of fi-nite antenna elements and grid arrays in quasi-optical systems. This electromagneticsimulator employs an efficient Galerkin moment method technique with sub-domainsinusoidal basis functions. What makes this moment method analysis unique is thatit is formulated using a combination of spatial and spectral domains taking full advan-tage of the strengths of each method to ensure accurate and efficient evaluation of themoment matrix elements. Incorporated into the moment method simulator are quasi-optical dyadic Green’s functions which are derived by separately considering paraxialfields (quasi-optical modes) which are largely responsible for distant interactions inthe quasi-optical system and the corrected open space (nonmodal) interactions re-sponsible for near neighbor coupling. Two types of quasi-optical Green’s functionsincluded are the open cavity resonator Green’s function and the grid lens systemGreen’s function which is derived here for the first time. The method presented hereis for analysis of structures of finite dimensions. All other quasi-optical modeling thathas been done only considers the unit cell of the array and hence does not includemutual coupling from other unit cells. Results are presented here showing the mutualcoupling from finite grid arrays. Other results presented here include the drivingpoint impedance of several antenna elements and grid arrays in a quasi-optical opencavity resonator and lens system. Field profiles of the lens system are also presentedand compared with measurements. The moment method simulator is also used tofind the multiport parameters of a grid array in order to describe what the activedevices in the system see. All simulations agree favorably with measurements.

Biographical Summary

Todd William Nuteson was born in Tacoma, Washington, on November16, 1967. 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 to1993 he worked at Wright State University as a Research and Teaching Assistant.He received Graduate Research Fellowships from the Air Force Office of ScientificResearch in 1992 and 1993. While working towards his Ph.D. degree in electricalengineering at North Carolina State University, he held a Research Assistantship withthe Electronics Research Laboratory in the Department of Electrical and ComputerEngineering. While pursuing the M.S. and Ph.D. degrees, he has published over 20papers in journals and conference proceedings. In June of 1996 he won the PrestigiousBronze Medallion for Outstanding Scientific Achievement presented at the 20th ArmyScience Conference. This is the second highest award for research from throughout theArmy presented at the conference. His research interests include numerical modelingof microwave and millimeter-wave circuits, quasi-optical power combining, antennas,and electromagnetics. He is a member of the Institute of Electrical and ElectronicEngineers and the Microwave Theory and Techniques, Antennas and Propagation,and Electromagnetic Compatibility Societies.

Acknowledgments

I would like to express my gratitude to my advisor Dr. Michael Steer forhis support and guidance during my graduate studies. It was a privilege to be partof 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 Bilbrofor showing an interest in my research and serving on my Ph.D. committee.

Special appreciation of financial and moral support from the Army Re-search Office through grant DAAH04-95-1-0536 under the direction of Dr. JamesHarvey 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 onmy 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 measurementsand keeping me on the right track. To Mr. Steve Lipa for working with me oncalibration techniques for the HP 8510C network analyzer. To Mr. Huan-shengHwang, Mr. Konstantin Kojucharow, and Mr. Chris Hicks for assisting me withlaboratory measurements. To Ms. Jaee Patwardhan and Mr. Ahmed Khalil forworking with me on my moment method simulator. And to everyone who sits inDaniels Hall Room 336 and 334. And last I would like to thank the Daniels HallMovie Society Group which kept me sane during my graduate studies.

And finally, I wish to thank my parents for their support and encourage-ment.

Contents

List of Figures xi

List of Symbols xvii

1 Introduction 1

1.1 Motivation For and Objective of This Study . . . . . . . . . . . . . . 1

1.2 Dissertation Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Original Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.4 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Literature Review 8

2.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Quasi-Optical Power Combining Arrays . . . . . . . . . . . . . . . . . 13

2.3.1 Oscillators Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.3.2 Amplifier Arrays . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Numerical Modeling of Quasi-Optical Systems . . . . . . . . . . . . . 17

3 Quasi-Optical Dyadic Green’s Functions 22

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

vii

viii

3.2 Dyadic Green’s Function of Open Cavity Resonator . . . . . . . . . . 23

3.3 Derivation of Lens System Dyadic Green’s Function . . . . . . . . . . 26

3.3.1 Modal Component —=

GEm . . . . . . . . . . . . . . . . . . . 27

3.3.2 Nonmodal Component —=

GEn . . . . . . . . . . . . . . . . . 33

3.3.3 Final Expression . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.4 Modal Reflection Coefficient . . . . . . . . . . . . . . . . . . . . . . . 34

3.4.1 Conductor Losses . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.4.2 Diffraction Losses . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.5 Resonant Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 Other Dyadic Green’s Functions . . . . . . . . . . . . . . . . . . . . . 40

3.6.1 Free Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.6.2 Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.6.3 Microstrip . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6.4 Dielectric Slab . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4 Method of Moments 50

4.1 General Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.2 Expansion Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Quasi-Optical Moment Matrix Elements . . . . . . . . . . . . . . . . 55

4.3.1 Open Cavity Resonator . . . . . . . . . . . . . . . . . . . . . . 55

4.3.2 Lens System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Nonquasi-Optical Moment Matrix Elements . . . . . . . . . . . . . . 57

4.4.1 Spatial Domain . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.4.2 Spectral Domain . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Excitation Vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

ix

4.5.1 Delta-Gap Voltage Generator . . . . . . . . . . . . . . . . . . 60

4.5.2 Coaxial Current Probe . . . . . . . . . . . . . . . . . . . . . . 61

4.6 Numerical Considerations . . . . . . . . . . . . . . . . . . . . . . . . 65

4.6.1 Spectral Domain Moment Matrix Elements . . . . . . . . . . . 65

4.6.2 Convergence Issues . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6.3 Condition Number . . . . . . . . . . . . . . . . . . . . . . . . 66

4.6.4 Valid Frequency Ranges . . . . . . . . . . . . . . . . . . . . . 69

4.7 Multiport Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.7.1 Nodal Admittance Parameters . . . . . . . . . . . . . . . . . . 71

5 Computed and Experimental Results 73

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2 Open Cavity Resonator . . . . . . . . . . . . . . . . . . . . . . . . . . 73

5.2.1 Inverted L Antenna . . . . . . . . . . . . . . . . . . . . . . . . 74

5.2.2 Rectangular Patch Antenna . . . . . . . . . . . . . . . . . . . 75

5.2.3 IMPATT Diode Oscillator . . . . . . . . . . . . . . . . . . . . 82

5.3 Lens System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.1 Electric Field Profiles . . . . . . . . . . . . . . . . . . . . . . . 85

5.3.2 Measurement and Simulation Techniques . . . . . . . . . . . . 85

5.3.3 Extended Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.4 3 × 3 Grid Array . . . . . . . . . . . . . . . . . . . . . . . . . 92

5.3.5 5 × 5 Grid Array . . . . . . . . . . . . . . . . . . . . . . . . . 101

5.4 Multiport Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4.1 Two-Port Structure . . . . . . . . . . . . . . . . . . . . . . . . 103

5.4.2 Four-Port Structure . . . . . . . . . . . . . . . . . . . . . . . . 105

x

5.4.3 3 × 3 Patch Antenna Array . . . . . . . . . . . . . . . . . . . 106

5.4.4 3 × 3 Grid Array . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.5 Finite Grid Arrays Versus Unit Cell Approach . . . . . . . . . . . . . 112

5.6 Tapered Antenna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 Conclusions and Future Research 117

6.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

6.2 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

References 122

A Multiport Analysis Using Half Basis Elements 132

A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A.2 Development . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

A.3 Multiport Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

A.4 Spectral Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135

A.5 Spatial Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

A.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

B Mixed Potential Integral Equation Formulation 139

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

B.2 Mixed Potential Integral Equation . . . . . . . . . . . . . . . . . . . . 140

B.3 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

B.3.1 Electric Field Integral Equation Formulation . . . . . . . . . . 142

B.3.2 Mixed Potential Integral Equation Formulation . . . . . . . . 142

B.3.3 Scalar Potential Singularity . . . . . . . . . . . . . . . . . . . 143

B.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

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. [1] . . 2

1.2 A cascaded quasi-optical oscillator and amplifier power combiner. . . 3

2.1 A quasi-optical power combiner configuration for an open resonator. 9

2.2 A grid amplifier/oscillator on a dielectric slab. . . . . . . . . . . . . 9

2.3 A grid amplifier/oscillator on a dielectric slab with X and Y polarizers. 10

2.4 A two-dimensional quasi-optical power combining oscillator. . . . . . 15

2.5 A two-dimensional quasi-optical power combining amplifier. . . . . . 18

2.6 Unit cell configuration for a grid oscillator. . . . . . . . . . . . . . . 20

3.1 Cross-section of the open cavity resonator. . . . . . . . . . . . . . . 23

3.2 Quasi-optical lens system configuration with a centered amplifier/oscillatorarray. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

3.3 Cross section of the lens system showing test field. . . . . . . . . . . . 27

3.4 Cross section of the lens system showing source field. . . . . . . . . . 29

3.5 Fields excited by a current element in free space. . . . . . . . . . . . 33

3.6 A non-confocal resonator geometry. . . . . . . . . . . . . . . . . . . 35

3.7 Equivalent transmission lines for the immittance approach. . . . . . . 41

3.8 Cross-section configuration of an infinitesimal x-directed current ele-ment on a: (a) dielectric slab; (b) microstrip. . . . . . . . . . . . . . . 43

xi

xii

4.1 Moment method flowchart utilizing C and FORTRAN. . . . . . . . . 54

4.2 An x-directed sinusoidal basis function. . . . . . . . . . . . . . . . . 54

4.3 Locations of x−and y−directed currents on a rectangular grid. . . . 55

4.4 A delta-gap voltage generator. . . . . . . . . . . . . . . . . . . . . . 61

4.5 A coaxial current probe. . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.6 Surface current magnitude: (a) x-directed; (b) y-directed; on a rectan-gular patch antenna in half space. . . . . . . . . . . . . . . . . . . . 64

4.7 Driving point reflection coefficient: (a) magnitude; (b) phase; of adipole antenna divided into N cells. . . . . . . . . . . . . . . . . . . 67

4.8 Convergence rate of the driving point reflection coefficient: (a) magni-tude; (b) phase; of a dipole antenna. . . . . . . . . . . . . . . . . . . 68

4.9 Condition number of the moment matrix: solid line, N = 10; dashedline, N = 20; dotted line, N = 30. . . . . . . . . . . . . . . . . . . . 69

4.10 A four-port unit cell structure with metal in the gap region. . . . . . 71

5.1 A coaxial fed inverted L antenna. . . . . . . . . . . . . . . . . . . . . 74

5.2 Method of moments cell subdivision for the inverted L antenna with adelta-gap voltage generator. . . . . . . . . . . . . . . . . . . . . . . . 74

5.3 Driving point impedance: (a) magnitude; (b) phase; for the TEM0,0,35

mode of the inverted L antenna: solid line, simulation; dashed line,measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.4 Driving point impedance: (a) magnitude; (b) phase; for the TEM0,1,35

and TEM1,0,35 modes of the inverted L antenna: solid line, simulation;dashed line, measurement. . . . . . . . . . . . . . . . . . . . . . . . 77

5.5 Driving point impedance: (a) magnitude; (b) phase; for the TEM0,2,35,TEM1,1,35, and TEM2,0,35 modes of the inverted L antenna: solid line,simulation; dashed line, measurement. . . . . . . . . . . . . . . . . . 78

5.6 A coaxial fed rectangular patch antenna. . . . . . . . . . . . . . . . 79

5.7 Method of moments cell subdivision for the rectangular patch antennawith a delta-gap voltage generator. . . . . . . . . . . . . . . . . . . . 79

xiii

5.8 Impedance Smith chart showing the driving point impedance of thepatch antenna without the reflector: solid line, simulation; dashed line,measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.9 Impedance Smith chart showing the simulated driving point impedanceof the patch antenna in the open cavity resonator: solid line, TEM0,0,34

mode; dashed line, TEM0,0,23 mode. . . . . . . . . . . . . . . . . . . 80

5.10 Simulated driving point reflection coefficient magnitude of the patchantenna: solid line, open cavity resonator; dashed line, half space. . . 81

5.11 Locations for three patch antennas in the open cavity resonator: alldimensions are in cm. . . . . . . . . . . . . . . . . . . . . . . . . . . 82

5.12 Simulated driving point reflection coefficient magnitude of patch an-tenna 1 for the TEM0,0,35 mode: solid line, alone; dotted line, withpatch antennas 2 and 3. . . . . . . . . . . . . . . . . . . . . . . . . . 83

5.13 IMPATT diode and patch antenna impedance: solid line, negativeimpedance measurement of the IMPATT diode; dashed line, simulateddriving point impedance of the center fed patch antenna in half space. 84

5.14 Simulated driving point impedance of the patch antenna in the opencavity resonator for the TEM0,1,35 and TEM1,0,35 modes: frequencyrange from 8.62175 GHz to 8.69 GHz. . . . . . . . . . . . . . . . . . 84

5.15 IMPATT diode oscillator: (a) cavity spacing D = 61.25 cm; (b) cavityspacing D = 61.41 cm: solid line, measured oscillation in half space;dashed line, measured oscillation in the open cavity resonator; dottedline, simulated scaled driving point reflection coefficient magnitude ofthe patch antenna. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

5.16 Oscillation frequencies on an expanded impedance Smith chart: (a)cavity spacing D = 61.25 cm; (b) cavity spacing D = 61.41 cm: solidline, oscillator in the open cavity resonator; dashed line, oscillator inhalf space. Note: this is an enlarged view of Fig. 5.14. . . . . . . . . 87

5.17 Configuration for measuring the electric field intensity in the quasi-optical lens system. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.18 Field distribution at the beam waist (z = 0): solid line, simulation;points, measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.19 Field distribution away from the beam waist (z=55 cm): solid line,simulation; points, measurement. . . . . . . . . . . . . . . . . . . . . 89

xiv

5.20 Field distribution along the z-axis: solid line, simulation; dashed line,measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

5.21 Driving point reflection coefficient: (a) measurement technique; (b)delta-gap model; (c) moment method simulation technique. . . . . . 90

5.22 Reflection coefficient magnitude for the shorted measurement probe:solid line, with absorber around coaxial probe; dashed line, withoutabsorber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

5.23 Driving point reflection coefficient: (a) magnitude; (b) phase; of theextended unit cell: solid line, simulation; dashed line, measurement. 93

5.24 An extended unit cell along with the cell subdivision used. . . . . . . 94

5.25 Driving point reflection coefficient: (a) magnitude; (b) phase; of theextended unit cell: solid line, simulation; dashed line, measurement. 95

5.26 A 3 × 3 grid array with the driving point impedance being measuredin the middle gap: (a) other gaps opened; (b) other gaps shorted. . . 96

5.27 Driving point reflection coefficient: (a) magnitude; (b) phase; of the3 × 3 opened grid: solid line, simulation; dashed line, measurement. . 97

5.28 Driving point reflection coefficient: (a) magnitude; (b) phase; of the3 × 3 shorted grid: solid line, simulation; dashed line, measurement. 98

5.29 Driving point reflection coefficient magnitude in the corner gap of the3 × 3 shorted grid: solid line, simulation; dashed line, measurement. 99

5.30 A 3 × 3 quasi-optical grid along with the cell subdivision used. . . . 99

5.31 Driving point reflection coefficient: (a) magnitude; (b) phase; of the3 × 3 grid: solid line, simulation; dashed line, measurement. . . . . . 100

5.32 A 5 × 5 quasi-optical grid along with the cell subdivision used. . . . 101

5.33 Driving point reflection coefficient: (a) magnitude; (b) phase; of the5 × 5 grid: solid line, simulation; dashed line, measurement. . . . . . 102

5.34 A two-port structure: (a) physical layout; (b) MoM modeling scheme;(c) V2 shorted; (d) V1 shorted; (e) measurement. . . . . . . . . . . . 103

5.35 Magnitude of S11 and S21 for the two-port structure: solid line, simu-lation; dashed line, measurement. . . . . . . . . . . . . . . . . . . . . 104

xv

5.36 A four-port structure with metal in the gap regions . . . . . . . . . . 105

5.37 Driving point reflection coefficient magnitude of the four-port structurewith terminals 3 and 4 shorted: solid line, simulation; dashed line,measurement. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.38 A nine-port 3 × 3 patch antenna array in the open cavity resonator. . 107

5.39 Simulated scattering parameters of the 3×3 patch antenna array: solidline, S11; dashed line, S55. . . . . . . . . . . . . . . . . . . . . . . . . 108

5.40 Simulated scattering parameters of the 3×3 patch antenna array: solidline, S12; dashed line, S14. . . . . . . . . . . . . . . . . . . . . . . . . 108

5.41 Simulated scattering parameters of the 3×3 patch antenna array: solidline, S15; dashed line, S19. . . . . . . . . . . . . . . . . . . . . . . . . 109

5.42 A nine-port 3 × 3 grid array in the lens system. . . . . . . . . . . . . 110

5.43 Simulated scattering parameters of the 3×3 grid array: solid line, S11;dashed line, S55. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.44 Simulated scattering parameters of the 3×3 grid array: solid line, S12;dashed line, S13; dotted line, S14. . . . . . . . . . . . . . . . . . . . . 111

5.45 Simulated scattering parameters of the 3×3 grid array: solid line, S15;dashed line, S25; dotted line, S45. . . . . . . . . . . . . . . . . . . . . 111

5.46 Numbering scheme for the 10× 10 grid array consisting of 100 unit cells.112

5.47 Driving point reflection coefficient of the 10 × 10 grid: solid line, S1,1;dashed line, S45,45; dotted line, unit cell. . . . . . . . . . . . . . . . . 113

5.48 Simulated scattering parameters of the 10 × 10 grid: solid line, S45,45;dashed line, S23,23; dotted line, S1,1. . . . . . . . . . . . . . . . . . . . 114

5.49 Simulated scattering parameters of the 10 × 10 grid: solid line, S45,46;dashed line, S45,43; dotted line, S45,41. . . . . . . . . . . . . . . . . . . 114

5.50 Simulated scattering parameters of the 10 × 10 grid: solid line, S1,2;dashed line, S1,4; dotted line, S1,6. . . . . . . . . . . . . . . . . . . . . 115

5.51 A tapered antenna along with the cell subdivision used. . . . . . . . 116

5.52 Driving point reflection coefficient magnitude of the tapered antenna:solid line, simulation; dashed line, measurement. . . . . . . . . . . . 116

xvi

6.1 A 40 GHz grid amplifier unit cell layout. . . . . . . . . . . . . . . . . 120

6.2 Moment method unit cell layout with equal sized cells. . . . . . . . . 120

6.3 Moment method unit cell layout with unequal sized cells. . . . . . . 120

A.1 Port definition for the unit cell of a grid amplifier/oscillator. . . . . . 133

A.2 The location of the delta-gap voltage source and the corresponding halfbasis element at the tip of a feed line. . . . . . . . . . . . . . . . . . 134

A.3 Configuration of x-directed sinusoidal basis functions with the sourcebasis in the left column and the test basis in the right column. . . . 136

B.1 Unequal size basis function: (a) rooftop and (b) pulse doublet. . . . . 142

xvii

List of Symbols

Aj – Coefficient for numerical integration.amn – Coefficient for modal source field.ast – Coefficient for modal test field.ax – Unit vector along the x axis in Cartesian coordinates.ay – Unit vector along the y axis in Cartesian coordinates.az – Unit vector along the z axis in Cartesian coordinates.b – Radius of curvature for spherical lens.bmn – Coefficient for modal source field.bst – Coefficient for modal test field.c – Speed of light.cmn – Coefficient for modal source field.cst – Coefficient for modal test field.d – Dielectric substrate thickness and spacing above ground plane.D – Reflector and lens spacing.dmn – Coefficient for modal source field.dst – Coefficient for modal test field.E – Electric field.EFIE – Electric Field Integral Equation.Ein – Incident electric field.Em – Electric field in a conducting plane.E±mn – Scalar electric field Hermite-Gaussian traveling wave-beam with

propagation in the ±az direction.ES – Electric source field excited by a source current.Einct – Transverse incident electric field.

Escatt – Transverse scattered electric field.

EstT – Electric modal test field.

fmn – Coefficient for modal source field.fm,n,q – Modal resonant frequency.Fji – Spectral domain integrand argument for MoM elements.Fx – Focal length with respect to the x axis.Fy – Focal length with respect to the y axis.=

GE – Electric field dyadic Green’s function.=

GE – Spectral domain electric field dyadic Green’s function.=

GE0 – Free space component of dyadic Green’s function.=

GEc – Open cavity resonator component of dyadic Green’s function.=

GEf – Open space component of dyadic Green’s function.=

GEh – Half space component of dyadic Green’s function.

xviii

=

GEl – Lens system component of dyadic Green’s function.=

GEm – Modal component of dyadic Green’s function.=

GEn – Nonmodal component of dyadic Green’s function.=

GEp – Paraxial component of dyadic Green’s function.=

GEqo – Quasi-optical component of dyadic Green’s function.GxxE – axax component of spatial domain dyadic Green’s function.

GxxE – axax component of spectral domain dyadic Green’s function.

GxyE – axay component of spatial domain dyadic Green’s function.

GxyE – axay component of spectral domain dyadic Green’s function.

GyyE – ayay component of spatial domain dyadic Green’s function.

GyyE – ayay component of spectral domain dyadic Green’s function.

GyxE – ayax component of spatial domain dyadic Green’s function.

GyxE – ayax component of spectral domain dyadic Green’s function.

gmn – Coefficient for modal source field.Hen(x) – Hermite polynomial of order n and argument x.HG – Hermite Gaussian.Hm – Magnetic field in a conducting plane.H±mn – Scalar magnetic field Hermite-Gaussian traveling wave-beam with

propagation in the ±az direction.HS – Magnetic source field excited by a source current.HstT – Magnetic modal test field.

I – Current vector for MoM.Ii – Current element for MoM.=

It – Transverse unit dyad.Jm (x) – Bessel function of the first kind and order m.JS – Electric current density.Jx – Electric current density in the ax direction.K – Kernel for Huygen’s integral in free space.k0 – Free space wavenumber.k1 – Dielectric wavenumber.kx – Spectral domain variable.ky – Spectral domain variable.kz0 – Free space propagation constant.kz1 – Dielectric propagation constant.LG – Laguerre Gaussian.Ltp(x) – Generalized Laguerre polynomial of order p and argument x.MoM – Method of Moments.N – Total number of unknowns in MoM.NHG – Hermite Gaussian normalizing coefficient.NLG – Laguerre Gaussian normalizing coefficient.

xix

Nx – Total number of x unknowns in MoM.Ny – Total number of y unknowns in MoM.Pin – Incident power to a conducting plane.Ploss – Power lost in a conducting plane.Pout – Total power reflected from a conducting plane.r – Observation or test location with respect to the origin.r′ – Source location with respect to the origin.R1,mn – Modal reflection coefficient for first lens.R2,mn – Modal reflection coefficient for second lens.Rmn – Reflection coefficient of the traveling wave-beam modes.S – Closed surface that bounds the quasi-optical system.S1 – Infinite plane before the first lens.S2 – Infinite plane after the second lens.T – Transmission coefficient.T1,mn – Modal transmission coefficient for first lens.T2,mn – Modal transmission coefficient for second lens.TE – Transverse Electric.TEM – Transverse Electromagnetic.TM – Transverse Magnetic.V – Voltage vector for MoM.W xi – x-directed sinusoidal basis function.

W xi – Fourier transform of x-directed sinusoidal basis function.

W yi – y-directed sinusoidal basis function.

W yi – Fourier transform of y-directed sinusoidal basis function.

X – Gaussian mode parameter along the x axis.Y – Gaussian mode parameter along the y axis.Z – Moment matrix.Z0 – Free space impedance.Z1 – Intrinsic impedance of a dielectric.Zf,ji – Open space moment matrix element.Zin – Input impedance.Zm – Intrinsic impedance of a metal.Zqo,ji – Quasi-optical moment matrix element.ZTE – Equivalent TE mode impedance for the immittance approach.ZTM – Equivalent TM mode impedance for the immittance approach.ZTET – Equivalent TE mode impedance looking into the top for the

immittance approach.ZTMT – Equivalent TM mode impedance looking into the top for the

immittance approach.ZTEB – Equivalent TE mode impedance looking into the bottom for the

immittance approach.ZTMB – Equivalent TM mode impedance looking into the bottom for the

xx

immittance approach.ZTEBL – TE mode impedance load at the bottom.

ZTMBL – TM mode impedance load at the bottom.

ZTE1 – TE mode dielectric impedance.

ZTM1 – TM mode dielectric impedance.

α – Polar coordinate variable for the spectral domain.αd,mn – Modal diffraction losses due to the spherical reflector.αp,mn – Modal conductor losses due to the planar reflector.αs,mn – Modal conductor losses due to the spherical reflector.β – Polar coordinate variable for the spectral domain.Γ – Reflection coefficient.δs – Skin depth.ε0 – Permittivity of free space.εr – Substrate dielectric constant.λ0 – Free space wavelength.μ0 – Permeability of free space.π – Circle circumference divided by circle diameter.σ – Conductivity of a metal.ψ1,mn – Phase coefficient for first lens.ψ2,mn – Phase coefficient for second lens.ψmn – Phase coefficient of the traveling wave-beam modes.ω – Radian frequency.ωs – Spot size for spherical lens.Ω – Quasi-optical system volume.

Chapter 1

Introduction

1.1 Motivation For and Objective of This Study

The need for high-powered, light-weight sources at millimeter-wave frequencies isbecoming more demanding. Currently tube devices such as klystrons or traveling wavetube (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 andbulky size due to their high voltage DC power supplies. On the other hand solidstate devices are generally more desirable in terms of small size, light weight, highreliability and excellent manufacturability but are limited to the level of power thatcan be generated at millimeter-wave frequencies. Fig. 1.1 shows a comparison of thepower levels for various tube devices and solid state devices [1]. Note that this datais from 1990 and since then the power levels of both tube devices and solid statedevices have increased. A promising solution to these problems is quasi-optical powercombining where the power from numerous solid state devices is combined in freespace over a distance of many wavelengths. With the aid of lenses the electromagneticfields are focussed and power is channeled predominately into a single paraxial mode.As an example of a quasi-optical power combiner see Fig. 1.2. Here the first stageconsists of an oscillator array in an open cavity resonator (Fabry Perot resonator)where the millimeter-wave source is generated. The active devices in the oscillatorarray lock together with the aid of the resonator allowing for power combining to takeplace. The power is then partially transmitted through the first lens where it passesthrough an amplifier array. In this stage the power is amplified by combining thepower from all the active devices in the amplifier array and is then passed through asecond 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 isnot guided through metal interconnects which become very lossy at millimeter-wave

1

2

frequencies but instead the energy is guided through free space or dielectric slabs.Applications where light weight millimeter-wave power sources are needed includethe 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).

0.1 1 10 100 100010

10

1

10

10

10

10

10

10

10

−2

−1

1

2

3

4

5

6

7

Klystrons

Gyrotrons

TWT’s

Free−ElectronLaser

VFET

Si BJT

MESFET

IMPATTPHEMT

Gunn

FREQUENCY (GHz)

GriddedTubes

OU

TP

UT

PO

WE

R (

W)

Figure 1.1: Power capacities of microwave and millimeter-wave devices: solid line,tube devices; dashed line, solid state devices. After Sleger et al. [1]

The strategy is to develop, using numerical field analysis, a multiportimpedance model of the linear part of the quasi-optical system. This can then beinterfaced with commercial microwave circuit simulators. Efficiency requires thatvolumetric discretization must be avoided. In the finite element method (FEM) andthe finite-difference time-domain (FDTD) method, the entire three-dimensional struc-ture is analyzed by discretizing the whole volume. For electrically large systems suchas in quasi-optics where interfacing surfaces are distributed over electrically large dis-tances, the use of FEMs and FDTD methods are very inefficient. By using a momentmethod utilizing Green’s functions appropriate to the physical structure, discretiza-tion can be limited to planar surfaces. A series of developments [2–5] culminated ina 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 isderived in two parts: one part describing the effect of the quasi-optical paraxial fields

3

PolarizerInput Output

Polarizer

ArrayLensArray

Oscillator AmplifierLens

Figure 1.2: A cascaded quasi-optical oscillator and amplifier power combiner.

and the other part describing the remaining fields. This form of the dyadic Green’sfunction is particularly convenient for quasi-optical systems because of its relativeease of development.

Progress toward large, high-powered, efficient arrays is hampered by therelatively crude state of design technology including the lack of suitable computeraided engineering tools. In particular, the many unit active circuits in a large arraycannot be individually optimized for efficiency and stability. This is because nosimulation process has been developed to model impedances and stability criteriafor 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 hasgenerally been based on the unit cell approach where the minimum three dimensionalcell of an array, generally containing a single active device, is modeled using symmetryof the structure to establish electrical and magnetic side-walls for the cell. A momentmethod or finite element program is then used to electrically characterize the cell andobtain the impedance presented to a single device. The unit cell approach assumes aninfinitely 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 consideredalong with mutual coupling.

In this dissertation an advanced method of moments approach combin-ing spatial domain and spectral domain techniques to model quasi-optical systemsis presented. The moment method implementation is developed in such a way thatany quasi-optical dyadic Green’s function, derived using the technique discussed pre-viously, 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 etal. [3, 5] and (2) the grid lens system in which the Green’s function is derived in thisdissertation for the first time. With this formulation the field solver can be conve-

4

niently used in the development of circuit-level models of the passive linear elementsin the quasi-optical systems which can then be interfaced with transient analysis(SPICE) and steady state analysis (harmonic balance).

1.2 Dissertation Overview

Chapter 2 presents a review of the current literature in the field of quasi-optical powercombining focusing mainly on the numerical modeling of quasi-optical systems. Alsopresented is a history of moment method techniques.

In Chapter 3 the derivation of a dyadic Green’s function for a quasi-opticallens system is presented. Also included in this chapter is the open cavity resonatordyadic Green’s function including techniques for computing conductor and diffractionlosses. Chapter 3 also includes other dyadic Green’s functions such as free space, halfspace, microstrip, and dielectric slab.

Chapter 4 focuses on implementing the quasi-optical dyadic Green’s func-tions from Chapter 3 using the method of moments technique. The complete formu-lation including a unique moment method algorithm that combines both the spatialand spectral domain is given. Numerical considerations such as convergence and con-dition numbers of the moment matrix are discussed. Also the analysis for computingthe multiport parameters using the moment method formulation is presented.

Chapter 5 validates the moment method program by comparing simula-tions to measurements for the open cavity resonator and lens system. Measurementsare included for inverted L antennas, patch antennas, grid arrays, and taper anten-nas. Other simulations including finite grid arrays versus the unit cell approach andmultiport parameters are presented.

In Chapter 6 a summary of the dissertation is given along with conclusionsand 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

5

various quasi-optical systems. The derivation is given in Section 3.3 of Chap-ter 3. Commercial electromagnetic simulators do not include quasi-optical com-ponents such as lenses and reflectors because they use standard multilayeredmicrostrip Green’s functions. By using quasi-optical Green’s functions, quasi-optical elements can be incorporated into electromagnetic simulators.

• The incorporation of the quasi-optical dyadic Green’s functions into a methodof moments formulation. This required an advanced moment method imple-mentation where both spatial and spectral domain techniques were used foraccurate and efficient analysis of quasi-optical systems. The moment methodsimulator has a fast convergence rate along with well-conditioned moment ma-trices. One very strong feature of the simulator is that it works from DC toany frequency. Often moment method simulators break down at low frequenciesbut results presented in this dissertation show convergence at DC. The completeformulation is presented in Chapter 4.

• Finite quasi-optical grids are modeled. It is very common in quasi-optics tomodel a single unit cell. In doing so coupling from other unit cells in the arrayand edge effects are ignored. All simulations presented in this dissertationare for finite arrays were coupling from all unit cells is considered along withthe edge effects. The unit cell model is done by assuming that the grid isinfinite and periodic with all active devices in phase. With this assumption,electric and magnetic walls are placed along the unit cell edges to force theseconditions. Coupling can only be considered by these boundary conditions andnot by direct radiation and quasi-optical resonant modes as is the case for finitearrays. Verification of finite array simulations with measurements are presentedin Chapter 5. Comparisons of finite grid arrays versus the unit cell is presentedin Section 5.5 of Chapter 5.

• Multiport analysis of quasi-optical grid arrays. The multiport parameters arerepresented as admittance parameters which represent what the active devicesin the system see. The multiport parameters give the coupling coefficientsof the array which are ignored in unit cell models. The admittance port pa-rameters are then converted to nodal admittance parameters to be compatiblewith microwave circuit simulators that use nodal analysis. These admittanceparameters are needed for transient and harmonic balance simulations. Theformulation is given in Section 4.7 of Chapter 4 and results are presented inSection 5.4 of Chapter 5.

• Driving point reflection coefficient measurements for several grid arrays in aquasi-optical lens system. This technique uses an unbalanced coaxial line thatis transformed to a balanced line without the aid of a balun. The measurementswere used for verification of moment method simulations over a wide frequency

6

band. The measurement technique and results are presented in Section 5.3 ofChapter 5.

1.4 Publications

The work associated with this dissertation resulted in the following publications:

• 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, and J. W. Mink, “Strategiesfor handling complicated device-field interactions in microwave systems,” Proc.PIERS 1996 Symp., July 1996.

• 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 1996.

• J. Harvey, M. B. Steer, J. W. Mink, H.-S. Hwang, T. W. Nuteson, and A. C.Paolella, “Advances in quasi-optical power combiners provide path to radar andcommunications enhancements,” 20th Army Science Conference, June 1996.

• 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. 1251-1254, June 1996.

• 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 com-ponent design,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 927-930, June1996.

• 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. 701-710, May 1996.

• 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 GuidedWave Lett., vol. 6, pp. 73-75, Feb. 1996.

• 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 InternationalConference on Infrared and Millimeter Waves Dig., pp. 157-158, Dec. 1995.

7

• 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. In-ternational Semiconductor Device Research Symposium, pp. 243-246, Dec. 1995.

• H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paolella,“Quasi-optical power combining in a dielectric substrate,” Proc. InternationalSymposium on Signals, Systems and Electronics, pp. 89-92, Oct. 1995.

• T. W. Nuteson, G. P. Monahan, M. B. Steer, K. Naishadham, J. W. Mink, andF. K. Schwering, “Use of the moment method and dyadic Green’s functions inthe analysis of quasi-optical structures,” IEEE MTT-S Int. Microwave Symp.Dig., pp. 913-916, May 1995.

Special Note: The Paper entitled “Advances in quasi-optical power combiners pro-vide path to radar and communications enhancements,” presented at the 20th ArmyScience Conference in Norfolk, Virginia, received the 1996 Prestigious Bronze Medal-lion for Outstanding Scientific Achievement for research presented from throughoutthe Army.

Chapter 2

Literature Review

2.1 Background

Quasi-optical power combining techniques [6–8] provide a means for combining powerfrom numerous solid-state millimeter-wave sources attached to radiating elementssuch as antenna arrays or grids. The power from the radiating elements is combinedin free-space over a distance of many wavelengths to channel power predominatelyinto a single paraxial mode. Mink [2] studied an array of filamentary current sourcesradiating into a plano-concave open resonator. This was the first theoretical investi-gation of power combining using a source array in a quasi-optical resonator, as shownin Fig. 2.1. The grid oscillator [9] and grid amplifier [10], shown in Fig. 2.2, areother methods of power combining where active devices or monolithic microwave in-tegrated 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 alongwith a slab for tuning the system. The complex device field interactions render it dif-ficult to optimize efficiencies and ensure stable operation. However, computer aidedanalysis techniques are evolving to aid in design. The strategy is to develop, us-ing numerical field analysis, a multiport impedance model of the linear part of thequasi-optical system. This can then be interfaced with commercial microwave cir-cuit simulators. Efficiency requires that volumetric discretization be avoided. Byutilizing Green’s functions appropriate to the physical background, discretization canbe limited to surfaces. In References [2–5] a series of developments culminated in astraight forward methodology for developing the dyadic Green’s function of a quasi-optical structure. The dyadic Green’s function was derived by separately consideringparaxial and non-paraxial modes. It is not feasible to derive a mixed, scalar andvector, potential Green’s function, as required in conventional space domain momentmethod techniques. Alternatively, we have adapted an efficient moment method field

8

9

solver [11–13] to use dyadic Green’s functions for the analysis of quasi-optical systems.

Dd

ax

ya

za

SOURCEARRAY

PLANARREFLECTOR

PARTIALLYTRANSPARENTSPHERICALREFLECTOR

Figure 2.1: A quasi-optical power combiner configuration for an open resonator.

E

Input beam

Active grid surface

E

Output beam

Figure 2.2: A grid amplifier/oscillator on a dielectric slab.

This literature review consists of three main sections. The first is on mo-ment method techniques. It is important to know what has been done using momentmethod analysis in order to see how it can be used with quasi-optical modeling. Thesecond review is on quasi-optical power combining arrays including oscillator arraysand amplifier arrays. Here reviews are presented on experimental aspects of quasi-optical system development. The last and main focus of this literature review is onnumerical modeling of quasi-optical systems. Reviews on what has been done withnumerical modeling of quasi-optical systems is presented.

2.2 Method of Moments

The method of moments (MoM), first introduced by Harrington [14], is a numericaltechnique used to solve integral equations. In electromagnetics, many problems are

10

ACTIVE GRID SURFACEOUTPUT POLARIZER

INPUT POLARIZER TUNING SLAB

E

E

INPUTBEAM

OUTPUTBEAM

Figure 2.3: A grid amplifier/oscillator on a dielectric slab with X and Y polarizers.

formulated as integral equations in which analytical solutions do not exist and there-fore must be solved numerically. The MoM offers an accurate numerical solution forsuch problems where the integral equation is transformed into a set of linear equationswhich is then solved for with a computer. To have a good understanding of what theMoM is and how it works, a simple mathematical outline [15] illustrating the MoMprocedure is given next.

Consider the following inhomogeneous equation

Lu = f (2.1)

or

Lu− f = 0 (2.2)

where L is a linear operator, u is unknown, and f is known. In order to solve for uan approximate solution for (2.2) is found by the following procedure known as theMoM. Let u be approximated by a set of basis functions or expansion functions givenby

un =n∑k=1

αkφk , n = 1, 2, . . . (2.3)

where φk is the expansion function, αk is its unknown amplitude, and n is the totalnumber of expansion functions. Replacement of u by un in (2.2) and taking the innerproduct with a set of weighting functions or testing functions wm, where the left sideof (2.2) is orthogonal to the sequence {wm}, results in

〈Lun − f, wm〉 = 0 , m = 1, 2, . . . , n . (2.4)

11

Substitution of (2.3) into (2.4) yields

n∑k=1

αk〈Lφk, wm〉 = 〈f, wm〉 , m = 1, 2, . . . , n (2.5)

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 = (α1 α2 · · · αn)T (2.7)

b = (〈f, w1〉 〈f, w2〉 · · · 〈f, wn〉)T (2.8)

and

A = [amk] (2.9)

where T denotes the transpose and amk are the individual matrix elements given by

amk = 〈Lφk, wm〉 . (2.10)

The accuracy is highly dependent upon the choice and number of expan-sion and weighting functions used. The best accuracy is usually achieved when thesame functions are used for both expansion and weighting which is known as theGalerkin method [17]. Also it is important to note that the computation time andmemory size increases significantly with the increase in the number of basis functionsused.

For integral equations, the linear operator L will include a Green’s func-tion. The Green’s function can best be described as the impulse response of thesystem. For example, an electric field Green’s function would describe all of theelectric fields at any location due to a current filament at a fixed position. In manyapplications dyadic Green’s functions are used. Tai in Reference [18] gives a compre-hensive and rigorous analysis of dyadic Green’s functions.

Some of the first widely used applications of the MoM were in wire antennasimulation where the MoM was used to solve either Pocklington’s integral equationor Hallen’s integral equation. The MoM was used to solve for the current distributionon the wire antennas which then could be used to predict the input impedance of theantenna or the far-field radiation. In both cases the free space Green’s function wasused. Often the Galerkin method was used with sub-domain sinusoidal expansion and

12

weighting functions. In References [19–21], a complete formulation for wire antennasusing the MoM is given. Other articles, such as [22,23], give numerical aspects of thestability and convergence of the moment method solutions. The type of structures an-alyzed included Yagi-Uda array antennas [19], single and multiple log-periodic dipoleantennas [20], and wire grid modeling of airplanes [19].

While the MoM was being used to model one-dimensional wire grid ge-ometries, research was also conducted in modeling of two-dimensional surfaces withthe MoM. References [24–28] present MoM techniques for modeling arbitrary surfacesin free space. The 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 studiesof radiation and scattering from arbitrary surfaces are presented in [25–28].

The biggest applications to date of moment method solutions have been inthe modeling of microstrip geometries including multilayered dielectrics for open andclosed structures. There are two main approaches for modeling microstrip structuresusing the MoM, the first is the spectral domain approach [29–38] and the second isthe spatial domain approach [39–46]. Applications of microstrip modeling includedispersion characteristics of printed transmission lines [29, 30, 34], input impedanceand mutual coupling of microstrip antennas [31, 39, 41–43], phased array antennas[33, 36, 37], and microstrip discontinuities [32, 35, 38, 40].

There are several advantages and disadvantages for each moment method.For example in the spectral domain the microstrip dyadic Green’s functions must bederived in closed form. A technique known as the immittance approach was devel-oped by Itoh [30] where the Green’s function is easily derived for multilayer geometriesusing a transverse equivalent transmission line for a spectral wave along with a coor-dinate transformation. This method is illustrated in Section 3.6 of Chapter 3 whereseveral Green functions are derived using the immittance approach. In the spatialdomain the microstrip Green’s function is derived in terms of vector and scalar po-tentials expressed in terms of Sommerfeld integrals. Evaluation of the Sommerfeldintegrals requires numerical analysis with special treatment for handling the complexpoles due to surface waves in the dielectric and the oscillatory nature of the Sommer-feld integrals described by Bessel functions. Mosig has done several comprehensivestudies on the numerical evaluation of Sommerfeld integrals [39,47,48]. In [49] Chowet al. developed a closed form approximation for the spatial domain scalar and vec-tor potential microstrip Green’s functions. Further development of the closed formGreen’s functions for multilayered microstrip geometries is given in [50, 51].

The main advantages of the spatial domain MoM approach is that it canmodel arbitrarily shaped structures by gridding the structure into unequal size cellsand is also conceptually easier than the spectral domain MoM approach. The spec-

13

tral domain requires transformation from the spatial domain to the spectral domainthrough the use of Fourier transforms. The Fourier transforms are not a problemwhen the gridding is done with equal size cells but does pose a problem for unequalsize cells. A disadvantage of a spatial domain implementation is that the self-termscontain a singularity which does not exist in the spectral domain. This singularity ishandled in the spatial domain by doing singularity extraction. It is very importantto evaluate the self-terms correctly because they dominate the moment matrix. Thebiggest disadvantage for a spectral domain analysis is that numerical integration isrequired over an infinite range due to taking the Fourier transforms. It then becomesvery difficult to numerically evaluate elements separated by electrically large distancesbecause of heavy oscillations. Techniques for efficient evaluation of spectral domainmoment method elements are presented in [52–54]. In the spatial domain this is nota problem because everything is integrated over physical finite regions.

The main problems of both MoM approaches is the amount of CPU timeand memory required for simulating electrically large structures. These problems oc-cur when filling and inverting the moment matrix. Work on improving the matrixfill times has been done in [55] using frequency interpolation and in [11] using spatialinterpolation of the moment matrix. The combination of these interpolation tech-niques result in very efficient matrix fills. Since the moment matrix is usually denseand often ill conditioned, the inversion of the moment matrix remains a problem formatrices of large order. An approach that is being researched today is the use ofwavelets in the MoM [56–59]. In this approach scaling functions and wavelets areused in place of the traditional orthogonal basis functions. When wavelets are usedthe moment matrix becomes sparse which allows for a fast solution of the invertedmoment matrix.

In this dissertation both the spatial and spectral domain MoM approachesare used concurrently taking full advantage of the strengths of each method in orderto 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. Oneof the first was by Goldsmith [6] which deals with quasi-optical techniques usingGaussian beams along with various quasi-optical components at millimeter and sub-millimeter wavelengths. Since then these quasi-optical techniques have been used withmany power combining oscillator and amplifier arrays to combine power in free space.The IEEE Transactions on Microwave Theory and Techniques devoted an entire issueon quasi-optics in the 1993 October edition [8]. A recent review paper by York [7]

14

provides some of the latest quasi-optical power combining techniques for oscillatorsand amplifiers. A proposal to set figures-of-merit for standard characterization ofspatial and quasi-optical power combining arrays was presented in [60].

This section will focus on the experimental work that has been done forquasi-optical power combining including open cavity resonator oscillators, grid os-cillators, and grid amplifiers. Two-dimensional slab power combining will also bepresented in this section.

2.3.1 Oscillators Arrays

The open cavity resonator shown in Fig. 2.1 has been used to produce several typesof quasi-optical oscillators [61–68]. In [61–63,68] Gunn diode oscillators were demon-strated in an open cavity resonator. Other oscillator configurations include highelectron mobility transistors (HEMT) [64, 65], impact avalanche transit time (IM-PATT) diodes [66], and resonant tunneling diodes (RTD) [67]. These experimentswere done at X-band [61–64, 66, 68], 40 GHz [65], U-band [63], W-band [63], and75 GHz [67]. The work done on oscillators in an open cavity resonator has proventhat locking is achievable through the resonator allowing for power combining. In [68]power combining was demonstrated with 24 oscillators with an efficiency of 90%.

The most popular way of producing quasi-optical power combining oscil-lators has been using grid arrays, see Figs. 2.2 and 2.3, to combine power in freespace [9, 69–74]. Popovic et al. demonstrated a quasi-optical power combining oscil-lator at 9.7 GHz using a 5 × 5 grid array with 25 metal semiconductor field effecttransistors (MESFET) [9]. They reported an effective radiated power (ERP) of 37 Wwith a DC to RF conversion efficiency of 15%. Since then they have demonstrateda 6 × 6 grid array with 36 MESFETs [69] and a 10 × 10 grid array with 100 MES-FETs [70] reporting an ERP of 3 W at 3 GHz and 21 W at 5 GHz with efficienciesof 22% and 20%, respectively. In [71] a MESFET grid oscillator with 16 transistorsproduced 335 mW of power at 11.6 GHz with an ERP of 15 W and an efficiencyof 20%. Also reported in [71] was a 36 element MESFET grid oscillator which pro-duced 235 mW of power at 17 GHz with an ERP of 3.3 W and an efficiency of 7%.The largest amount of power produced from a grid oscillator to date was reportedin [73] with a 100 element MESFET grid producing a total radiated power of 10.3 Wat 9.8 GHz with an ERP of 660 W and an efficiency of 23%. A grid oscillator usingpseudomorphic high electron mobility transistors (pHEMT) was demonstrated in [74]with a 25 element grid producing 100 mW of power at 4.94 GHz and 4 cascaded 25element grids producing 265 mW of power at 5.15 GHz with an ERP of 8 W. In [72]a voltage controlled oscillator (VCO) with a 10% tuning bandwidth with less than2 dB power change was demonstrated for a MESFET grid containing a varactor diode

15

tuning grid.

Other techniques for combining power in free space have used patch anten-nas [75–78] and slot antennas [79]. In [75] a 4× 4 array of microstrip patch antennaswas reported. Using 16 Gunn diodes produced an equivalent isotropic radiated power(EIRP) of 22 W at 9.6 GHz with an efficiency of 1%. Also reported was a 16 elementMESFET array with patch antennas producing an EIRP of 10 W at 8.2 GHz with anefficiency of 26 %. A 16 element quasi-optical FET oscillator power combining arrayusing external injection locking with an ERP of 28.2 W at 6 GHz was demonstratedin [76]. In [77] a periodic second harmonic spatial power combining oscillator wasdemonstrated with a 4 element Gunn diode with microstrip patch antennas reportingan ERP of 25.7 dBm at 9.36 GHz with an efficiency of 10.2%. A MESFET arraywith a total radiated power of 7.92 W and an efficiency of 15% was reported in [78].Slot antennas with 9 MESFETs were used in [79] producing an EIRP of 2.4 W at10.11 GHz.

A new type of quasi-optical oscillator using a dielectric slab resonator waspioneered here at North Carolina State University [80]. This slab resonator shownin Fig. 2.4 is a two-dimensional version of the open cavity resonator shown in Fig.2.1. Here the energy is guided in the dominant transverse electric (TE) mode of thedielectric slab [80, 81].

Z

Y

X

PLANAR REFLECTOR

CURVED REFLECTOR

DIELECTRIC SLAB

OSCILLATORARRAY

GROUND PLANE

Figure 2.4: A two-dimensional quasi-optical power combining oscillator.

16

2.3.2 Amplifier Arrays

Several demonstration of quasi-optical amplifier arrays have been reported over thepast six years. Power combining using grid arrays has been very popular. The firstdemonstration of a grid amplifier was reported by Kim et al. [10]. The grid consistedof 50 MESFETs and produced a gain of 11 dB at 3.3 GHz. Each unit cell of the gridhad a MESFET pair resulting in a 5 × 5 grid. In [82] a unit cell of a grid amplifierwas reported with a gain of 10.5 dB at 5.92 GHz. Kim et al. [83] demonstrated a 100heterojunction bipolar transistor (HBT) grid amplifier. The amplifier consisted of a10 × 10 grid with a chip in each unit cell made up of HBT differential-pairs. A peakgain of 10 dB at 10 GHz with a 3 dB bandwidth of 1 GHz was reported. Maximumoutput power was 450 mW and the minimum noise figure was 7 dB. Also demonstratedby Kim et al. [84] was a 6.5 GHz to 11.5 GHz source using a grid amplifier with atwist reflector. The peak ERP was 6.3 W at 9.9 GHz. More recent quasi-opticalgrid amplifiers include a 36 element monolithic grid amplifier using HBTs [85]. Thisgrid amplifier had a gain of 5 dB at 40 GHz with a 3 dB bandwidth of 1.8 GHz.The maximum saturated output power was 670 mW at 40 GHz with a peak poweradded efficiency (PAE) of 4%. This was the first reported power measurement fora monolithic quasi-optical amplifier. In [86] a 36 element monolithic grid amplifierusing pHEMTs was demonstrated over a frequency range of 44 GHz to 60 GHz. Apeak gain of 6.5 dB at 44 GHz and 2.5 dB at 60 GHz was reported.

Other types of quasi-optical power combining amplifiers include microstripantenna arrays. In [87] a 9 MESFET array using microstrip patch antennas wasdemonstrated with an ERP of 6.06 W at 7.33 GHz. Quasi-optical microstrip amplifiersbased on multilayer coupled structures were used in [88]. Here the amplifiers wereplaced back to back sharing a common ground plane in order to provide isolation forthe input and output. A single stage had a gain of 7.54 dB at 10.04 GHz and a two-stage had a gain of 18.0 dB at 9.95 GHz. A two-level power combining pHEMT patchantenna lens amplifier with 8 dB of absolute power gain at 9.7 GHz was reportedin [89]. The quasi-optical antenna array amplifier was used for beamforming andbeam-switching. Also presented in [89] was a saturated class A polarization preserving24 MESFET patch array producing 0.7 W at 10 GHz with a PAE of 21%.

The first demonstration of a quasi-optical millimeter-wave amplifier wasreported in [90]. The unit cell consisted of a MMIC driver amplifier followed by a two-stage high powered amplifier chip. This circuit was used in a 6 × 6 array producinga 6 dB gain at 29 GHz.

Slot antennas are also used for quasi-optical amplifiers. In [89] an X-band two-stage low noise coplanar waveguide (CPW) pHEMT amplifier cell usinganti-resonant slot antennas was demonstrated. An active gain of 21.7 dB with a

17

3.2 dB noise figure and a 6% 3 dB bandwidth was reported. Other slot antennaconfigurations included a quasi-optical amplifier array using a direct integration ofMMICs with 50 Ω multi-slot antennas [91]. A 4×4 array with a 10 dB gain at 11 GHzwith a 4% bandwidth was reported. A CPW multiple slot antenna for active arraysand integrated antennas was demonstrated in [92] with a reported gain of 8 dB usingan HBT gain block MMIC chip.

In [93] a combination of slot antennas and patch antennas were used. Amonolithic plane wave pHEMT amplifier array was made up of a 5 × 7 input slotantenna array and a 4×8 output patch antenna array with a 7×8 pHEMT amplifierarray placed in between. The amplifier array was placed in an oversized waveguidewith a reported gain of 3 dB at 43 GHz. A quasi-optical traveling wave amplifier(TWA) using Vivaldi-type slots and a hybrid microstrip MESFET TWA circuit wasdemonstrated in [94]. The amplifier had a 50% fractional bandwidth at 3.5 GHz.

The two-dimensional approach used for the quasi-optical oscillators hasalso been pioneered here at North Carolina State University for quasi-optical ampli-fiers [95–97]. The two-dimensional dielectric slab power combiner for the amplifierstage is shown in Fig. 2.5. Here the RF energy is guided through the dielectric slabfor the TE mode and focussed with lenses. The active antenna array is placed be-tween the lenses to achieve the maximum concentration of the RF energy. In [95–97]the unit cell consisted of a Vivaldi-type antenna with a single MESFET. Real powergain of the complete system was demonstrated in [96] using 4 MESFETs operatingat 7.4 GHz. Other groups are now showing interest in two-dimensional quasi-opticalpower combining. In [98] a 10 element active lens amplifier on a dielectric slab wasreported. Yagi-Uda slot arrays were used with GaAs MMIC amplifiers producing apeak system gain of 11 dB at 8.25 GHz with a 3 dB bandwidth of 0.65 GHz.

2.4 Numerical Modeling of Quasi-Optical Systems

Quasi-optical power combining has been experimentally demonstrated but the powerlevels obtained so far are much smaller than what is needed for envisioned applica-tions. It is not competitive with conventional power combiners. In order to obtainhigher power we need to have computer aided engineering (CAE) tools available forquasi-optical systems so optimization of the system can be done. Being able to opti-mize the system using CAE tools will greatly speed up the development of high pow-ered quasi-optical systems by eliminating all unnecessary experimental work. Thissection will review the work that has been done on numerical modeling of quasi-opticalsystems.

18

convex lenses

dielectric slab

ground plane

dielectric slab

ground planeconcave lenses

guided input waves guided output waves

guided output wavesguided input waves

FET amplifiers

Figure 2.5: A two-dimensional quasi-optical power combining amplifier.

Many of the approaches used today to model grid arrays are based onthe methods used to model frequency selective surfaces (FSS) several years ago. In[99] periodic arrays of cross dipoles and Jerusalem crosses were analyzed. Thesestructures were analyzed using the Galerkin method in the spectral domain. A setof special entire-domain basis functions were developed to correctly account for thediscontinuous nature of the induced current at the cross junctions. Computed resultswere presented for the reflection coefficient of the FSS structures for arbitrary incidentfields. Infinite periodic arrays consisting of arbitrarily shaped patches were presentedin [100]. Here the arrays were modeled using a fast Fourier transform (FFT) basediterative approach to compute the scattered fields of the infinite array. Sub-domainrooftop basis functions were employed. Techniques for accuracy and efficiency forFSS are discussed in [101].

An impedance model for a quasi-optical diode array was presented in [102].This method models an infinite array of strips or slots loaded with two terminal semi-conductor devices such as diodes. The problem was solved using a unit cell approachwhere the unit cell was modeled as a waveguide discontinuity problem and solvedwith the MoM. Computed results were presented for the inductance and capacitanceof the diode array at 99 GHz.

In [103] a measurement technique for determining the driving point impedanceof active grid arrays was presented. This technique assumes that all of the active el-ements in the array are locked and in phase. With this assumption the grid arrayproblem was reduced by symmetry where only the unit cell was considered. Thedriving point impedance measurement was done using the dielectric waveguide mea-

19

surement (DWM) method where the unit cell was placed between two parallel metalplates emulating electric walls. The driving point impedance measurements were pre-sented for a dipole array, bow-tie array, double-Vee array, and a slot array. Simpleequivalent circuit models using only transmission lines were developed for each array.Results show that the dipole array had the highest driving point impedance with theslot array having the smallest.

Modeling of quasi-optical grids with thin rectangular patch and apertureelements were presented in [104]. The method of moments was employed with appro-priate edge conditions incorporated into the basis functions to compute the scatteringfrom periodic arrays with and without substrates and superstrates. For the apertureelements Floquet modes with the appropriate boundary conditions were used to ex-press the fields. The patch elements followed in a similar manner setting up theintegral equation for the unknown surface current density on the patch. Transmis-sion and reflection characteristics of the arrays were presented for arbitrary incidentfields.

Popovic’s group from the University of Colorado at Boulder has beendoing numerical work on quasi-optical grids [105–110]. A full-wave analysis usingthe MoM was developed by Bundy et al. [105, 108]. This method assumed that thegrid array was an infinite periodic structure with all of the active devices locked andin phase. With this assumption the problem was modeled by only considering theunit cell with special boundary conditions along the unit cell edges as shown in Fig.2.6. The moment method was formulated by relating the electric field to the surfacecurrent density on the metallic conductors on the grid and was represented by Fourierseries. In [105, 108] the driving point impedance of several unit cell configurationsincluding dipoles, cross dipoles, and bow-ties were given. The driving point impedancecould be computed for unit cells having a dielectric with or without metal on bothsides. The computed driving point impedances were validated by comparison to theEMF method [111]. In addition to the driving point impedance computations, themoment method simulator also computed the transmission coefficient of a passiveperiodic grid structure due to a normal incident plane wave. In these computationsthe grid was analyzed in free space with a single metallic surface and the dielectricwas added back with transmission line theory. In [106–108] results were presented forthe transmission coefficient of cross dipoles and Jerusalem crosses and were validatedwith measurements. Further analysis of cascaded quasi-optical grids were presentedin [109, 110]. Here each quasi-optical component was characterized by a multiportnetwork allowing for cascading of all the components in the system. The componentsin the system included a grid oscillator, a planar reflector (mirror), and a FSS wherethe grid and FSS could be loaded with active and/or passive lumped devices.

Here at North Carolina State University, Steer’s group has been doing

20

S SD

G

MESFET

MA

GN

ET

IC W

AL

LMA

GN

ET

IC W

AL

L

ELECTRIC WALL

ELECTRIC WALL

Figure 2.6: Unit cell configuration for a grid oscillator.

research on quasi-optical modeling [3–5, 112–119]. In [3–5, 112, 113] Heron et al. de-veloped an electric field dyadic Green’s function for the open cavity resonator shown inFig. 2.1. This Green’s function was comprised of resonant and nonresonant terms cor-responding to coupling from modal and nonmodal fields in the open cavity resonator.The Green’s function was then derived by considering the paraxial and nonparax-ial components where the paraxial components described the quasi-optical modes.Losses due to diffraction from the finite spherical reflector and conductor losses ofthe reflectors were also considered. Results for 1 port and 2 port antennas werecompared with measurements taken in an X-band open cavity resonator. The workpresented in this dissertation is based on the open cavity Green’s function wherea full-wave moment method simulator was developed incorporating the open cavityGreen’s function [117, 118]. This Green’s function analysis was further extended todeveloping a dyadic Green’s function for a quasi-optical grid system with lenses [119]and is also presented in this dissertation. Work is continuing on incorporating thefull-wave analysis from this dissertation with transient analysis [114] and steady state(harmonic balance) analysis [115] in order to describe a complete quasi-optical systemwith active devices.

The finite-difference time-domain (FDTD) method was applied to quasi-optical arrays in [120]. The grid structure analyzed was assumed to be infinite andperiodic so only the unit cell was modeled. The unit cell microstrip geometry consistedof two square metallic loops on the front side with a square aperture on the back side.Input and output polarizers were also included in the system. The unit cell wasexcited with a linearly polarized plane wave with a normal incidence on the unit cell

21

surface. In the FDTD analysis Floquet boundary conditions were used along theunit cell transverse edges and Berenger’s perfectly matched layer (PML) absorbingboundary conditions were used to terminate the fields along the longitudinal beamaxis. The FDTD analysis was compared with measurements for the transmitted andreflected powers through the system.

A generalized scattering matrix approach for analyzing quasi-optical gridsconsisting of grid amplifiers and grid oscillators was presented in [121]. The grid arraywas made up of cross dipoles with differential pair HEMTs embedded in the gaps.Again the structure was assumed to be infinite and periodic so only the unit cell wasconsidered. The gates of the HEMTs were connected to the horizontal dipole leadsproviding the input and the drains were connected to the vertical dipole terminalsproviding the output. The MoM was applied with spectral domain Green’s functionsincluding substrate and superstrate layers supporting the metallic layer. Instead ofusing electric and magnetic walls around the edges of the unit cell, Floquet harmon-ics were used to model the unit cell dimensions and the propagation constant of theincident field. Galerkin’s method with sub-domain rooftop basis functions were usedto find the currents allowing for the computation of the scattered fields in termsof Floquet harmonics reflected and transmitted by the grid. The scattering matrixrepresentation of the periodic grid array was then combined with power wave repre-sentations at the device ports allowing for the scattering matrix of the grid array tobe combined with the active device scattering matrix. Results for 1 port and 2 portdevices were presented.

Analysis of an oscillator array embedded in a rectangular waveguide waspresented in [122]. A 7 element oscillator was developed using planar dipole antennaswith a pHEMT in the gap of each dipole. The MoM was used to determine theelectric and magnetic fields of the passive waveguide system loaded with a dipolearray. A time domain analysis was then used to predict the oscillation frequency,locked amplitudes, and output power from the circuit. The output power of theoscillator array was 220 mW at 12.4 GHz.

Chapter 3

Quasi-Optical Dyadic Green’sFunctions

3.1 Introduction

The approach used in [3–5] to develop a dyadic Green’s function for a quasi-opticalopen cavity resonator is used here to develop a Green’s function for the quasi-opticallens system (see Fig. 3.2). The special insight to this technique is the derivation of=

GE in two parts:

=

GE ==

GEn +=

GEm (3.1)

where=

GEn and=

GEm refer to the nonmodal and modal fields, respectively. Thus weseparately consider the paraxial fields (quasi-optical modes) which are largely respon-sible for distant interactions in the quasi-optical system and the corrected open space(nonmodal) interactions responsible for near neighbor interactions. The nonmodal

component is found by removing the paraxial components=

GEp from the open space(i.e. free space, half space, microstrip, or dielectric slab) dyadic Green’s function

=

GEn ==

GEf −=

GEp (3.2)

where=

GEf represents the open space dyadic Green’s function. The complete Green’sfunction is then represented by

=

GE ==

GEf −=

GEp +=

GEm . (3.3)

The final dyadic Green’s function is evaluated in two parts=

GE ==

GEf +=

GEqo (3.4)

22

23

where=

GEqo ==

GEm − =

GEp represents the contribution of the quasi-optical modes and=

GEf represents the open space (direct radiation) contribution. The reorganizationof the Green’s function from that in (3.1) is required to separate the spatial domain,=

GEqo, and spectral domain,=

GEf , formulations.

3.2 Dyadic Green’s Function of Open Cavity Res-

onator

A dyadic Green’s function for the quasi-optical open cavity resonator was recentlydeveloped by Heron et al. [3–5]. The cavity resonator, shown in Fig. 3.1, consists ofa planar reflector at z = 0 and a partially transmitting spherical reflector with itscenter located at z = D. The planar reflector is assumed to be perfectly conductingwith infinite dimensions in the transverse direction. The spherical reflector is of finitedimension with a focal length with respect to the x and y axis, Fx and Fy, respectively.The medium in the cavity is free space. The dyadic Green’s function for this system

z

z = d z = D

y

z = 0

ε , μ00

ε , μ00

J S

PLANARREFLECTOR PARTIALLY TRANSMITTING

SPHERICAL REFLECTOR

Figure 3.1: Cross-section of the open cavity resonator.

is given as [3–5]

=

GE ==

GEc +=

GEh (3.5)

24

where

=

GEc = −Nm∑m=0

Nn∑n=0

Rmnψmn2 (1 +Rmnψmn)

(E−mn − E+

mn

) (E−mn − E+

mn

) =

It (3.6)

and=

GEh is the half space Green’s function given in Section 3.6.2. Nm and Nn repre-sent the number of transverse modes and

=

It= axax + ayay (3.7)

is the transverse unit dyad. The terms, Rmn and ψmn, represent the reflection coeffi-cient and phase, respectively, of the traveling wave-beam modes. The scalar electricmodal field, Emn, is given by the Hermite Gaussian traveling wave-beam as [123]

E±mn (x, y, z) =

√Z0

πXY m!n!

(1 + u2

)−1/4 (1 + v2

)−1/4

·Hem(√

2x/xz)Hen

(√2y/yz

)· exp

{−1

2

[(x/xz)

2 + (y/yz)2]

∓j[k0z +

1

2

(u (x/xz)

2 + v (y/yz)2)

−(m+

1

2

)tan−1 (u) −

(n+

1

2

)tan−1 (v)

]}(3.8)

where

u =z

k0X2, v =

z

k0Y 2

x2z = X2

(1 + u2

), y2

z = Y 2(1 + v2

)with the Gaussian mode parameters defined as [123]

X2 =1

k0

√FxD

(2 − D

Fx

)(3.9)

Y 2 =1

k0

√√√√FyD(

2 − D

Fy

). (3.10)

The Gaussian mode parameters X and Y determine the rate at which the fieldstrength decays in the ax and ay directions respectively. In the above expressions, Z0

and k0 represent the free space impedance and wavenumber, respectively, given by

Z0 =

√μ0

ε0, k0 = ω

√μ0ε0.

25

The Hermite polynomials, defined by [124]

Hen (x) = (−1)n exp(x2/2

) dn

dxn

[exp

(−x2/2

)], (3.11)

are orthogonal functions. The E±mn fields represent the desired wave-beam modes

with the beam waist at z = 0. The positive sign refers to propagation in the positiveaz direction and the negative sign refers to propagation in the negative az direction.An assumption is made that the electric field has only transverse components andno az component (quasi TEM mode). This assumption is valid since the sphericalreflector has a radius of curvature much greater than the wavelength of operation.This approximation holds true especially near z = 0 where the phase front is flat andthe fields are purely transverse. The antenna elements and arrays that are analyzedin the open cavity resonator are placed at z = d where d � D and is close to z = 0where we can say that the fields are TEM.

The cavity losses, consisting of conductor and diffraction losses, for eachmode are represented by Rmn and will be discussed in Section 3.4. The phase term,ψmn, is the ratio of the intensity for each mode of the outgoing wave-beam to theincoming wave-beam evaluated at the spherical reflector surface, z = D, given as [5]

ψmn =E+mn (x, y,D)

E−mn (x, y,D)

. (3.12)

This term gives rise to the cavity modal resonant frequencies which will be discussedin Section 3.5. A good approximation for ψmn can be found by evaluating (3.12) atx = y = 0.

The components of the cavity dyadic Green’s function given in (3.6) canbe written in matrix form as

=

GEc (x, y ; x′, y′) =

[GxxEc (x, y ; x′, y′) axax 0

0 GyyEc (x, y ; x′, y′) ayay

](3.13)

where each component is a function of its spatial distribution. The primed coordinatesdenote the source field and the unprimed coordinates denote the test field. Note thatthe Green’s function is not a function of the distance between the source and testfields which is common in most Green’s functions. This property enables the cavityGreen’s function to be a separable function in terms of the source and test coordinateswhich improves efficiency in computation as we will see in Chapter 4.

26

3.3 Derivation of Lens System Dyadic Green’s Func-

tion

A dyadic Green’s function for the lens system shown in Fig. 3.2 is derived for the firsttime. The system consists of a source, such as an amplifier or oscillator array, locatedat z = 0 with two lenses equally spaced from the center where each lens has a focallength with respect to the x and y axis, Fx and Fy, respectively. The system is fed bya transmitting horn antenna and the output is captured by a receiving horn antenna.The distances separating the horns, lenses, and source are very large in terms of thewavelength. The two lenses are not restricted to be identical.

z=−D z=0 z=D

a x

ay

az

LensTransmitting

Horn

Lens

Amplifier Oscillator/Array

Receiving

Horn

Figure 3.2: Quasi-optical lens system configuration with a centered ampli-fier/oscillator array.

The dyadic Green’s function is derived by determining the electromagneticfields within the system which are excited by an electric current source. The sameprocedure used in [2–5] is used here to find the fields. The electric field is related tothe Green’s function as

E (r) =∫Ω′

=

GE (r|r′) · JS (r′) dv′ (3.14)

where=

GE is the electric field dyadic Green’s function and JS is the electric currentdensity source used to excite the fields. The primed notation r′ = {x′, y′, z′} de-notes the source coordinates and the unprimed notation r = {x, y, z} denotes theobservation coordinates.

27

3.3.1 Modal Component —=GEm

In the lens system, the modal electric fields are approximated as the superposition ofHermite Gaussian traveling wave-beams, E±

mn. The first step in deriving the modalcomponent of the Green’s function is to introduce a test field into the system. Thetest field is excited from an incident field outside of the system at z � D. Forsimplicity let the incident field be a single wave-beam mode with unit amplitude asshown in Fig. 3.3 and also assume the electric field is in the ax direction. From Fig.

z=−D z=0 z=DS 1 S2

a x

ay

az

− −

ET HT,

1a

b

c d

Figure 3.3: Cross section of the lens system showing test field.

3.3 the electric and magnetic test field can be written as follows:

ET,st =

⎧⎪⎪⎨⎪⎪⎩astE

−st ax, z < −D(

cstE+st + bstE

−st

)ax, −D < z < D(

dstE+st + E−

st

)ax, z > D

(3.15)

HT,st =1

Z0

⎧⎪⎪⎨⎪⎪⎩

−astE−st ay, z < −D(

cstE+st − bstE

−st

)ay, −D < z < D(

dstE+st − E−

st

)ay, z > D

(3.16)

where a TEM approximation is used in order to find the magnetic field

H±mn = ± 1

Z0az × E±

mn. (3.17)

In any transverse plane (z = constant), the normalized traveling wave-beam in (3.8)satisfies the orthogonality condition∫

SE±st · H∗±

mnds =1

Z0

∫ ∞

−∞

∫ ∞

−∞E±stE

∓mndxdy = δsmδtn. (3.18)

28

The coefficients in (3.15) can be determined by the boundary conditions ateach lens in terms of transmission and reflection coefficients. At the first lens, z = −D,let the transmission and reflection coefficients be T1,st and R1,st, respectively, whichyields

astE−st ax = T1,stbstE

−st ax (3.19)

cstE+st ax = R1,stbstE

−st ax. (3.20)

At the second lens, z = D, let the transmission and reflection coefficients be T2,st andR2,st, respectively, which yields

bstE−st ax = T2,stE

−st ax +R2,stcstE

+st ax (3.21)

dstE+st ax = T2,stcstE

+st ax +R2,stE

−st ax. (3.22)

Solving for the unknown coefficients in (3.19)–(3.22) gives:

ast =T1,stT2,st

1 − R1,stψ1,stR2,stψ2,st(3.23)

bst =T2,st

1 − R1,stψ1,stR2,stψ2,st(3.24)

cst =R1,stψ1,stT2,st

1 − R1,stψ1,stR2,stψ2,st(3.25)

dst =R1,stψ1,stT

22,st

1 − R1,stψ1,stR2,stψ2,st

+R2,st

ψ2,st

(3.26)

where

ψ1,st =E−st (x, y,−D)

E+st (x, y,−D)

(3.27)

and

ψ2,st =E+st (x, y,D)

E−st (x, y,D)

. (3.28)

Since the lenses are equally spaced from the origin at a distance of D,

ψst = ψ1,st = ψ2,st (3.29)

29

and also since the phase fronts at each lens for every mode corresponds approximatelyto the surface of each lens, (3.27) and (3.28) can be evaluated at x = y = 0.

The next step in the derivation is to introduce a source. The source fieldis excited by a current density source placed in the system. The current density isgiven as

JS = Jx ax (3.30)

and is placed at z = 0 as shown in Fig. 3.4. From Fig. 3.4 the electric and magnetic

z=−D z=0 z=DS 1 S2

a x

ay

az

− −

Js

Es Hs,

ab

c d

fg

Figure 3.4: Cross section of the lens system showing source field.

source field can be written as follows:

ES =∑mn

⎧⎪⎪⎪⎨⎪⎪⎪⎩amnE

−mn ax, z < −D

(cmnE+mn + bmnE

−mn) ax, −D < z < 0

(fmnE+mn + dmnE

−mn) ax, 0 < z < D

gmnE+mn ax, z > D

(3.31)

HS =1

Z0

∑mn

⎧⎪⎪⎪⎨⎪⎪⎪⎩

−amnE−mn ay, z < −D

(cmnE+mn − bmnE

−mn) ay, −D < z < 0

(fmnE+mn − dmnE

−mn) ay, 0 < z < D

gmnE+mn ay, z > D

(3.32)

with the electromagnetic fields being the superposition of all quasi-optical modes.

The coefficients in (3.31) can be determined by the boundary conditionsat each lens in terms of transmission and reflection coefficients as was done for thetest field. At z = −D

amnE−mn ax = T1,mnbmnE

−mn ax (3.33)

30

cmnE+mn ax = R1,mnbmnE

−mn ax (3.34)

and at z = D

gmnE+mn ax = T2,mnfmnE

+mn ax (3.35)

dmnE−mn ax = R2,mnfmnE

+mn ax. (3.36)

One other boundary condition can be applied at the current source JS, at z = 0,where the electric field must be continuous,

(cmnE

+mn + bmnE

−mn

)ax =

(fmnE

+mn + dmnE

−mn

)ax. (3.37)

At z = 0, the Hermite Gaussian traveling incoming and outgoing wave-beams areequal (E+

mn = E−mn) which simplifies (3.37) further. Since there are only five equations

and six unknowns, the unknown coefficients are written in terms of gmn:

amn =T1,mn

T2,mn

(1 +R2,mnψ2.mn

1 +R1,mnψ1.mn

)gmn (3.38)

bmn =1

T2,mn

(1 +R2,mnψ2,mn

1 +R1,mnψ1,mn

)gmn (3.39)

cmn =R1,mnψ1,mn

T2,mn

(1 +R2,mnψ2,mn

1 +R1,mnψ1,mn

)gmn (3.40)

dmn =

(R2,mnψ2,mn

T2,mn

)gmn (3.41)

fmn =

(1

T2,mn

)gmn. (3.42)

The solution for gmn is found using the Lorentz reciprocity theorem which is appliednext.

The Lorentz reciprocity theorem is stated as follows [125]:

∮S

(Eb × Ha − Ea × Hb) · n ds =∫Ω

(Ea · Jb −Ha · Mb − Eb · Ja + Hb ·Ma) dv

(3.43)

31

where the independent electric and magnetic source currents Ja,Ma and Jb,Mb pro-duce the electric and magnetic fields Ea,Ha and Eb,Hb in a volume Ω which isbounded by the closed surface S and n is the unit outward normal on S. For the lenssystem shown in Figs. 3.3 and 3.4, the surface is given by S = S1+S2 which boundsthe volume Ω where S1 and S2 are transverse planes located at some z < −D andz > D, respectively. The Lorentz reciprocity theorem given in (3.43) is applied to thelens system by assigning the following:

Ea = ES Eb = ET,st

Ha = HS Hb = HT,st

Ja = JS Jb = 0Ma = 0 Mb = 0

which results in∮S

(ET,st × HS −ES ×HT,st) · n ds = −∫ΩET,st · JS dv. (3.44)

When the test and source fields given in (3.15), (3.16), (3.31), and (3.32) are appliedto (3.44) the following results are obtained. On surface S1 (z < −D)

∫S1

(ET,st × HS − ES × HT,st) · n ds = 0 (3.45)

and on surface S2 (z > D)

∫S2

(ET,st ×HS − ES × HT,st) · n ds =1

Z0

∫ ∞

−∞

∫ ∞

−∞2gmnE

−stE

+mn dxdy

= 2gmn (3.46)

where the orthogonality condition in (3.18) was applied. The volume integration overΩ for (−D < z < D) gives

−∫ΩET,st · JS dv = −

∫Ω

(cstE

+st + bstE

−st

)Jx dv. (3.47)

Taking the results of (3.45)–(3.47) and plugging into (3.44) yields

gmn = −1

2

∫Ω

(cmnE

+mn + bmnE

−mn

)Jx dv (3.48)

where the subscripts s, t have been changed to m,n for the test field. Substitutingthe test field coefficients (3.24) and (3.25) into (3.48) gives

gmn =−T2,mn

2 (1 −R1,mnψ1,mnR2,mnψ2,mn)

∫Ω

(R1,mnψ1,mnE

+mn + E−

mn

)Jx dv. (3.49)

32

The modal field is found by substituting all of the coefficients given in(3.38)–(3.42) and (3.49) into (3.31) yielding

ES = −∑mn

∫Ω (R1,mnψ1,mnE

+mn + E−

mn)Jx dv

2 (1 − R1,mnψ1,mnR2,mnψ2,mn)

·

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

T1,mn

(1+R2,mnψ2,mn

1+R1,mnψ1,mn

)E−mn ax, z < −D(

1+R2,mnψ2,mn

1+R1,mnψ1,mn

)(R1,mnψ1,mnE

+mn + E−

mn) ax, −D < z < 0

(E+mn +R2,mnψ2,mnE

−mn) ax, 0 < z < D

T2,mnE+mn ax, z > D .

(3.50)

Using the relationship between the electric field and Green’s function given in (3.14),the modal component Green’ function is given as

GxxEm = −∑

mn

(R1,mnψ1,mnE

+mn + E−

mn

)2 (1 − R1,mnψ1,mnR2,mnψ2,mn)

·

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

T1,mn

(1+R2,mnψ2,mn

1+R1,mnψ1,mn

)E−mn, z < −D(

1+R2,mnψ2,mn

1+R1,mnψ1,mn

)(R1,mnψ1,mnE

+mn + E−

mn) , −D < z < 0

(E+mn +R2,mnψ2,mnE

−mn) , 0 < z < D

T2,mnE+mn, z > D

(3.51)

for the axax component. Note that the primed notation represents the source locationand the unprimed notation represents the observation location. Since the source fieldwas derived with the source located at z = 0, the following simplification can bemade:

Emn (x, y) = E+mn (x, y, 0) = E−

mn (x, y, 0)

=

√Z0

πXY m!n!Hem

(√2x

X

)Hen

(√2y

Y

)exp

{−1

2

[(x

X

)2

+(y

Y

)2]}

.

(3.52)

For this derivation only the axax component of the dyadic Green’s functionwas considered. The ayay component is found by the same procedure but with thecurrent source and electric fields in the ay direction. When this is done the same resultis obtained. The axay and ayax components are zero because of the orthogonalityrelationship and the az az, azax, axaz , az ay, and ayaz components are all zero becausethe fields are assumed to have no components in the az direction. Thus the dyadicmodal component Green’s function can be written as

=

GEm = −∑mn

Emn2 (1 −R1,mnψ1,mnR2,mnψ2,mn)

=

It

33

·

⎧⎪⎪⎪⎨⎪⎪⎪⎩T1,mn (1 +R2,mnψ2,mn)E

−mn, z < −D

(1 +R2,mnψ2,mn) (R1,mnψ1,mnE+mn + E−

mn) , −D < z < 0(1 +R1,mnψ1,mn) (E+

mn +R2,mnψ2,mnE−mn) , 0 < z < D

T2,mn (1 +R1,mnψ1,mn)E+mn, z > D .

(3.53)

3.3.2 Nonmodal Component —=GEn

The nonmodal component is given as

=

GEn ==

GE0 −=

GEp (3.54)

where=

GE0 is the free space dyadic Green’s function (see Section 3.6.1) and=

GEp is theparaxial component of the dyadic Green’s function. The nonmodal fields are shownin Fig. 3.5 which are the fields that exist with the lenses removed from the system.

z=0

a x

ay

az

JsTraveling Paraxial

Wave−Beam

G Ep

=GE0

=−

G Ep

=

Radiated FieldsNon−Paraxial

Radiated FieldsNon−Paraxial

Figure 3.5: Fields excited by a current element in free space.

The paraxial component of the dyadic Green’s function is found from themodal component given in (3.53) with the lenses removed:

R1,mn → 0

R2,mn → 0

T1,mn → 1

T2,mn → 1

34

resulting in

=

GEp = −1

2

∑mn

Emn=

It

{E−mn, z < 0

E+mn, z > 0

. (3.55)

3.3.3 Final Expression

The final dyadic Green’s function is evaluated in two parts:

=

GE ==

GEl +=

GE0 (3.56)

where=

GEl ==

GEm − =

GEp represents the contribution of the quasi-optical modes ofthe lens system,

=

GEl = −∑mn

Emn2 (1 − R1,mnψ1,mnR2,mnψ2,mn)

·[R1,mnψ1,mn (1 +R2,mnψ2,mn)E

+mn

+R2,mnψ2,mn (1 +R1,mnψ1,mn)E−mn

] =

It (3.57)

for −D < z < D.

Several simplifications can be made such as assuming both lenses areidentical,

Rmn = R1,mn = R2,mn (3.58)

which results in

=

GE ==

GE0 −∑mn

Rmnψmn2 (1 −Rmnψmn)

Emn(E+mn + E−

mn

) =

It, −D < z < D. (3.59)

When the dyadic Green’s function is used such as in the method of moments, allfield interactions are generally done in one transverse plane in which for this Green’sfunction would be the z = 0 plane simplifying the final result to

=

GE ==

GE0 −∑mn

Rmnψmn(1 −Rmnψmn)

EmnEmn=

It, z = 0. (3.60)

3.4 Modal Reflection Coefficient

For the open cavity resonator, two types of losses are considered. These losses areconductor and diffraction losses. The conductor losses are due to the finite conduc-tivity in both the spherical and planar reflectors. Whereas the diffraction losses are

35

a result of the finite aperture size of the spherical reflector. With the combination ofthese losses, the modal value for Rmn can be found as

Rmn = −αs,mnαp,mnαd,mn (3.61)

where αs,mn and αp,mn represent the conductor losses due to the spherical and planarreflectors, respectively, and αd,mn represents the diffraction losses from the sphericalreflector. For real physical reflectors |Rmn| < 1. In the following sections, these losseswill be defined.

3.4.1 Conductor Losses

For the resonator geometry shown in Fig. 3.6, the electric fields are described by theHuygen’s integral equation [126,127]

E =∫S′

jk0 (1 + cos θ)

4πRe−jkRE ′ dS ′ (3.62)

where E is the field at the reflector S and E ′ is the field at the reflector S ′. After

z

b’

origin

x,y or r

a or r’

2D

θ

n

R

Figure 3.6: A non-confocal resonator geometry.

making several approximations, including a near field polynomial approximation forR in the exponential term, a far field approximation for R in the denominator, andunity for cos θ, a closed form solution in circular coordinates is given by [128]

E(r, φ,±D)p,tE0

= NLG(p, t)

(r√

2

w′s

)tLtp

(2r2

w′ 2s

)exp

[−r2

w′ 2s

](cos(tφ)sin(tφ)

). (3.63)

36

Using the same approximations in Cartesian coordinates, the solution for a rectan-gular aperture is given by

E(x, y,±D)m,nE0

= NHG(m,n)Hm

(x√

2

w′s

)Hn

(y√

2

w′s

)exp

[− (x2 + y2)

w2s

]. (3.64)

The mode indices m, n, p and t correspond to the x, y, r and φ directions respectivelyand NLG(p, t) and NHG(m,n) are normalizing coefficients for the Laguerre Gaussian(LG) and Hermite Gaussian (HG) modes respectively. The term Ltp(R) is the gener-alized Laguerre polynomial and Hm(X) is the Hermite polynomial. The spot size atthe reflectors is [126]

w ′s =

√λ0b ′

π

(2D

2b ′ − 2D

)14

(3.65)

where λ0 is the wavelength, 2D is the distance between reflectors, and b′ is the radiusof curvature. In both approximations infinite apertures are assumed. The closedform approximations in (3.63) describes the fields as LG mode families consideringa circular aperture whereas the closed form approximations in (3.64) describes thefields as HG mode families considering a rectangular aperture. The LGpt solutionsrelate to the LGmn solutions as follows [116]

HG00 = LG00, HG10 = LG(cosine)01, HG01 = LG(sine)01 and HG11 = LG(sine)02

where higher order LGpt and LGmn modes are distinct and will be neglected.

The losses due to the spherical and planar reflectors are computed byfinding the power loss in each conducting plane. The total power reflected from aconducting plane is given by

Pout = Pin − Ploss (3.66)

where Pin is the incident power, Ploss is the power lost in the conducting plane, andPout is the reflected power. Assuming a normalized incident power, Pin = 1 Watt, wecan write

αc = 1 − Ploss (3.67)

where αc = Pout/Pin is the loss due to the conducting plane. The average powerlost in the conductor is found by integrating the complex Poynting vector over theconducting plane surface [129]

Ploss =1

2Re

∫SEm × H∗

m · az dS (3.68)

37

where Em and Hm are the TEM fields in the conducting plane. Equation (3.68) canbe written as

Ploss =1

2Re

{1

Zm

} ∫S|Em|2 dS (3.69)

with the intrinsic impedance of the metal given by

Zm =1 + j

σδs(3.70)

where σ and δs are the conductivity and skin depth, respectively, of the conductorplane. The electric field in the metal can be described by the incident electric field as

Em = TEin (3.71)

where Ein is the incident electric field and T is the transmission coefficient givenby [129]

T = 1 + Γ =2Zm

Zm + Z0(3.72)

with the reflection coefficient

Γ =Zm − Z0

Zm + Z0. (3.73)

We can now write the power loss in (3.69) in terms of the incident electric field as

Ploss =2σδs

(1 + σδsZ0)2 + 1

∫S|Ein|2 dS. (3.74)

With the general result for power loss in a conducting plane given by(3.74), we can solve for the losses in the spherical and planar reflectors as

αs or p =√

1 − Ploss. (3.75)

Given the incident fields in (3.63) and (3.64) for the LG and HG modal fields, we cancompute the losses from (3.75) as

αs,pt =

√√√√1 − 2σδs

(1 + σδsZ0)2 + 1

∫ 2π

0

∫ a

0

∣∣∣E (r, φ,±D)p,t

∣∣∣2 rdrdφ (3.76)

αp,mn =

√√√√1 − 2σδs

(1 + σδsZ0)2 + 1

∫ a

−a

∫ a

−a

∣∣∣E (x, y,±D)m,n

∣∣∣2 dxdy. (3.77)

Note that the square root terms in (3.76) and (3.77) have a normalizing factor of√Pin = 1 W

12 . For the quasi-optical lens system, the same approach is applied with

the replacement of the intrinsic impedance of metal for the intrinsic impedance ofeach lens.

38

3.4.2 Diffraction Losses

The field on one reflector reproduces itself on the opposite reflector with the propor-tionality relationship

E = αdE′ (3.78)

where αd is the diffraction loss. Using the approximation for E given by (3.63) or(3.64) does not give an accurate result since infinite aperture size is assumed for theclosed form solutions. After making some simplifications to (3.62) [128, 130], theintegral equation can be written in circular cylindrical coordinates as

E (r, φ) =e−j2k0D

4πD

∫ 2π

0

∫ a

0exp

{jk0

[(2D − b′)

4Db′(r2 + r′2

)+rr′

2Dcos (φ− φ′)

]}

·E ′ (r′, φ′) r′dr′dφ′. (3.79)

Assuming that (3.79) can be separated in r and φ as E(r, φ) = R(r)Φ(φ) and using(3.78) yields

RΦ = αd,rR′αd,φΦ′ (3.80)

where αd = αd,rαd,φ. Then (3.80) can then be written as

αd,rαd,φR (r) Φ (φ) =∫ a

0K1 (r, r′)R (r′)

·{

1

2π

∫ 2π

0exp

[jk0rr

′

2Dcos (φ− φ′)

]Φ (φ′) dφ′

}dr′ (3.81)

where

K1 (r, r′) =

(jk0e

−j2k0D

2Dr′)

exp

[jk0

(r2 + r′2

) 2D − b′

4Db′

]. (3.82)

Using the following Bessel function relation

ejn[π/2−θ1]Jn (X) =1

2π

∫ 2π

0ej[X cos(θ1−θ2)−nθ2]dθ2

and assigning Φ (φ′) = e−jnφ′and αd,φ = 1 results in the reduced integral equation

αdR (r) =∫ a

0K2 (r, r′)R (r′) dr′ (3.83)

where

K2 (r, r′) =

(j(n+1)k0e

−j2k0D

2D

)J0

(k0rr

′

2D

)r′ exp

[jk0

(r2 + r′2

)(2D − b′

4Db′

)](3.84)

39

and J0 is the 0th order Bessel function of the 1st kind. For a particular point ri onthe reflector S, the one dimensional integral equation in (3.83) can be numericallyapproximated by the sum of the contributions of the field at point r′j on the reflectorS ′ as [127]

αd,iR (ri) =∑j

AjK2

(ri, r

′j

)R(r′j)

(3.85)

where Aj are the integration weighting coefficients, e.g. Simpson’s rule. Equation(3.85) is an eigenvalue problem which can be solved for the eigenvalues given by thediffraction loss αd,i. This gives a good approximation for the αd,r0 modes but not forhigher order angular modes. This is because of the assumption made where αd,φ = 1meaning no φ variation. The t = 0 mode does not depend on φ but all of the higherorder angular modes do.

Once a good approximation is made for the p = t = 0 mode, we canswitch back to Cartesian coordinates to compute the higher order mode diffractionlosses. In Cartesian coordinates the reflector is assumed to be rectangular. After somesimplifications of (3.62) [127, 130], the integral equation can be written in Cartesiancoordinates as

E =e−j2k0D

4πD

∫ +a

−a

∫ +a

−aexp

{jk0

[(2D − b′)

4Db′(x2 + x′2 + y2 + y′2

)+xx′ + yy′

2D

]}

·E ′ dx′dy′. (3.86)

Equation (3.86) can be separated in x and y as

E (x, y) = F (x)G (y) (3.87)

which can then be written as

F (x)G (y) = αd,xF′ (x′)αd,yG′ (y′) (3.88)

where αd = αd,xαd,y is the diffraction loss. Equation (3.88) can be solved for usingthe eigenvalue approach used in the previous case for the circular reflector [127].

The strategy for calculating the diffraction losses is first to numericallycompute the p = t = m = n = 0 mode in circular coordinates where a circularreflector is considered. Next choose an effective length in Cartesian coordinates andcompute the m = n = 0 mode in Cartesian coordinates for a rectangular reflector.When an effective length is found that agrees with the result found in the circularcoordinates, use that effective length to compute the diffraction loss for the higherorder modes.

40

3.5 Resonant Frequencies

At a resonant frequency, the phase of the traveling wave-beam should remain un-changed after one complete pass through the resonator. The resonant frequency foreach cavity mode occurs when the product Rmnψmn in the cavity Green’s function(3.6) approaches −1. This product will never equal −1 due to losses but can be-come very close because Rmn ≈ −1 for lower order modes. Evaluating (3.12) at(x, y, z) = (0, 0, D) yields

ψmn = e−j2θ (3.89)

where

θ = k0D −(m+

1

2

)tan−1

√D

2Fx −D−(n+

1

2

)tan−1

√D

2Fy −D. (3.90)

In order for Rmnψmn to approach −1, the condition θ = qπ for some integer q mustbe satisfied. Applying this condition to (3.90) yields an expression for the cavityresonant frequencies [126]

fm,n,q =c

2D

[q +

1

π

(m+

1

2

)tan−1

√D

2Fx −D+

1

π

(n +

1

2

)tan−1

√D

2Fy −D

]

(3.91)

where c = 1/√μ0ε0 is the speed of light and q is the longitudinal mode number

corresponding to an integer number of longitudinal half waves. Equation (3.91) can beused to predict all of the transverse mode resonant frequencies for a particular q family.The q families are separated by a frequency span of Δfq = c/ (2D). Degenerate modesoccur for m + n = constant if and only if Fx = Fy. The resonant frequencies for thelens system are found using the same approach.

3.6 Other Dyadic Green’s Functions

The dyadic Green’s functions for the open space are presented in this section, forfree space, half space, microstrip, and dielectric slab. All of the dyadic Green’sfunctions are derived in the spectral domain using the immittance approach [30].The spectral domain dyadic Green’s function is related to the spatial domain dyadicGreen’s function as the inverse Fourier transform

=

GE (x|x′; y|y′) =1

4π2

∫ ∞

−∞

∫ ∞

−∞

=

GE (kx, ky) ejkx(x−x′)ejky(y−y′) dkx dky. (3.92)

41

Using the immittance approach [30] we write the geometry of the problemas equivalent transmission lines for the transverse electric (TE) and the transversemagnetic (TM) cases as shown in Fig. 3.7. The components of the dyadic Green’s

J

1ε 0μ

Z1

TE / TM

BZTE / TM

ZT

TE / TM

ZTE / TM

TL

ZTE / TM

BL

z=d

z=0

Figure 3.7: Equivalent transmission lines for the immittance approach.

functions in the spectral domain are then given as [30]

GxxE (kx, ky) =

−1

β2

(k2x Z

TM + k2y Z

TE)

(3.93)

GyyE (kx, ky) =

−1

β2

(k2x Z

TE + k2y Z

TM)

(3.94)

and

GxyE (kx, ky) = Gyx

E (kx, ky) =−kxkyβ2

(ZTM − ZTE

)(3.95)

where

ZTE = ZTET ‖ ZTE

B (3.96)

ZTM = ZTMT ‖ ZTM

B . (3.97)

42

The equivalent impedances looking into the top for the TE and TM cases are

ZTET = ZTE

TL =Z0k0

kz0(3.98)

ZTMT = ZTM

TL =Z0kz0k0

(3.99)

where

kz0 =√k2

0 − β2 , Im (kz0) < 0 (3.100)

and

β2 = k2x + k2

y. (3.101)

The equivalent impedances looking into the bottom for the TE and TM cases are

ZTEB = ZTE

1

[ZTEBL cos (kz1d) + jZTE

1 sin (kz1d)

ZTE1 cos (kz1d) + jZTE

BL sin (kz1d)

](3.102)

ZTMB = ZTM

1

[ZTMBL cos (kz1d) + jZTM

1 sin (kz1d)

ZTM1 cos (kz1d) + jZTM

BL sin (kz1d)

](3.103)

where

ZTE1 =

Z1k1

kz1(3.104)

ZTM1 =

Z1kz1k1

(3.105)

and

kz1 =√k2

1 − β2 , Im (kz1) < 0. (3.106)

The wave number and impedance in the dielectric layer are given as

Z1 =Z0√εr, k1 =

√εrk0.

With everything defined for the equivalent transmission line model shownin Fig. 3.7, the dyadic Green’s functions for free space, half space, microstrip, anddielectric slab are derived in the next sections.

43

3.6.1 Free Space

The free space geometry is a special case of the dielectric slab configuration shownin Fig. 3.8(a) where the permittivity in the dielectric slab is set to the free spacepermittivity (ε1 = ε0). Here the Green’s function describes the electric field at someobservation point (x, y) which is produced from an infinitesimal current source locatedat (x′, y′). An assumption that the source and observation point are in the same plane(z = constant) is made. For the free space case the dielectric material has an εr = 1and the loads at the bottom for the TE and TM cases are

d

z

x

0ε , μ 0

(x , y , d)(x , y , d)/ /

SOURCEε , μ

01

0ε , μ 0

OBSERVATION

(a)

d

z

x

(x , y , d)(x , y , d)/ /

SOURCE ε , μ

01

0ε , μ 0

OBSERVATION

(b)

Figure 3.8: Cross-section configuration of an infinitesimal x-directed current elementon a: (a) dielectric slab; (b) microstrip.

ZTEBL =

Z0k0

kz0(3.107)

ZTMBL =

Z0kk0k0

. (3.108)

44

The equivalent TE and TM impedances looking into the bottom are then

ZTEB =

Z0k0

kz0(3.109)

ZTMB =

Z0kz0k0

(3.110)

and the total equivalent impedances of the top and bottom are

ZTE =Z0k0

2kz0(3.111)

ZTM =Z0kz02k0

. (3.112)

Using (3.111) and (3.112) in expressions (3.93)–(3.95) yields the dyadic componentsof the free space Green’s function:

GxxE0 (kx, ky) =

Z0

2k0

(k2x − k2

0

kz0

)(3.113)

GyyE0 (kx, ky) =

Z0

2k0

(k2y − k2

0

kz0

)(3.114)

and

GxyE0 (kx, ky) = Gyx

E0 (kx, ky) =Z0

2k0

(kxkykz0

). (3.115)

In the spatial domain the free space dyadic Green’s function is given as [18]

=

GE0 (r| r′) = −jωμ0

(=

It −∇t∇t

k20

)G0 (r| r′) (3.116)

where

G0 (r| r′) =e−jk0|r−r′|

4π |r − r′| (3.117)

and

|r − r′| =√

(x− x′)2 + (y − y′)2. (3.118)

45

Expanding the dyadic components in (3.116) yields:

GxxE0 (r| r′) = −jZ0

k0

G0 (r| r′)

·[k2

0 −jk0

|r − r′| −1 + k2

0 (x− x′)2

|r − r′|2 +j3k0 (x− x′)2

|r − r′|3 +3 (x− x′)2

|r− r′|4]

(3.119)

GyyE0 (r| r′) = −jZ0

k0G0 (r| r′)

·[k2

0 −jk0

|r − r′| −1 + k2

0 (y − y′)2

|r − r′|2 +j3k0 (y − y′)2

|r − r′|3 +3 (y − y′)2

|r − r′|4]

(3.120)

and

GxyE0 (r| r′) = Gyx

E0 (r| r′) =jZ0

k0(x− x′) (y − y′) G0 (r| r′)

·[

k20

|r − r′|2 − j3k0

|r − r′|3 − 3

|r − r′|4]. (3.121)

Looking at expressions (3.119)–(3.121) we observe that there are significant singular-ities when the source and observation point are co-located (|r − r′| = 0). Because ofthe singularities we choose to work with the Green’s function in the spectral domainwhere these singularities do not occur.

3.6.2 Half Space

The half space geometry is a special case of the microstrip configuration shown inFig. 3.8(b) where the permittivity in the dielectric is set to the free space permittivity(ε1 = ε0). In this configuration the source and observation point are located in thez = d plane. The half space case is similar to free space (εr = 1) but now the loadson the bottom are

ZTEBL = 0 (3.122)

ZTMBL = 0 (3.123)

because of the ground plane located at the bottom. The equivalent TE and TMimpedances looking into the bottom are then

ZTEB = j

Z0k0

kz0tan (kz0d) (3.124)

46

ZTMB = j

Z0kz0k0

tan (kz0d) (3.125)

and the total equivalent impedances of the top and bottom are

ZTE =Z0k0

2kz0

(1 − e−j2dkz0

)(3.126)

ZTM =Z0kz0k0

(1 − e−j2dkz0

). (3.127)

Using (3.126) and (3.127) in expressions (3.93)–(3.95) yields the dyadic componentsof the half space Green’s function:

GxxEh (kx, ky) =

Z0

2k0

(k2x − k2

0

kz0

)(1 − e−j2dkz0

)(3.128)

GyyEh (kx, ky) =

Z0

2k0

(k2y − k2

0

kz0

)(1 − e−j2dkz0

)(3.129)

and

GxyEh (kx, ky) = Gyx

Eh (kx, ky) =Z0

2k0

(kxkykz0

)(1 − e−j2dkz0

). (3.130)

In the spatial domain the half space dyadic Green’s function is given as [18]

=

GEh (r| r′) = −jωμ0

(=

It −∇t∇t

k20

)[G0 (r| r′) −G0 (ri| r′i)] (3.131)

where

|ri − r′i| =√

(x− x′)2 + (y − y′)2 + 4d2 (3.132)

is due to the image from the ground plane. Expanding the dyadic components in(3.131) yield:

GxxEh (r| r′) = Gxx

E0 (r| r′) −GxxE0 (ri| r′i) (3.133)

GyyEh (r| r′) = Gyy

E0 (r| r′) −GyyE0 (ri| r′i) (3.134)

and

GxyEh (r| r′) = Gyx

Eh (r| r′) = GxyE0 (r| r′) −Gxy

E0 (ri| r′i) . (3.135)

As was the case for free space, evaluation in the spatial domain is difficult due tosingularities which are avoided in the spectral domain.

47

3.6.3 Microstrip

The geometry configuration for a single layer microstrip is shown in Fig. 3.8(b).Between the ground plane (z = 0) and the conductor traces (z = d) is a dielectric andabove the conductor traces (z > d) is free space. In the Green’s function derivation,the ground plane and dielectric slab are assumed to be infinite in the transversedirections. For a single layer microstrip geometry the TE and TM loads at thebottom are

ZTEBL = 0 (3.136)

ZTMBL = 0 (3.137)

again due to the ground plane located at the bottom. The equivalent impedanceslooking into the bottom for the TE and TM cases are

ZTEB = j

Z0k0

kz1tan (kz1d) (3.138)

ZTMB = j

Z0kz1k0

tan (kz1d) (3.139)

and the total equivalent impedances of the top and bottom are

ZTE =jZ0k0 sin (kz1d)

Te(3.140)

ZTM =jZ0kz0kz1 sin (kz1d)

k0Tm(3.141)

where

Te = kz1 cos (kz1d) + jkz0 sin (kz1d) (3.142)

and

Tm = εrkz0 cos (kz1d) + jkz1 sin (kz1d) . (3.143)

The Te and Tm terms represent the TE and TM surface wave poles, respectively. TheTM surface wave has a zero cutoff frequency which means there is always at leastone TM surface wave pole for εr �= 1. The surface wave poles are real for lossless

48

dielectrics and complex for lossy dielectrics. Using (3.140) and (3.141) in expressions(3.93)–(3.95) yields the dyadic components of the microstrip Green’s function:

GxxEm (kx, ky) =

jZ0

k0

sin (kz1d)

·[kz0 (k2

x − εrk20) cos (kz1d) + jkz1 (k2

x − k20) sin (kz1d)

TeTm

]

(3.144)

GyyEm (kx, ky) =

jZ0

k0sin (kz1d)

·⎡⎣kz0

(k2y − εrk

20

)cos (kz1d) + jkz1

(k2y − k2

0

)sin (kz1d)

TeTm

⎤⎦(3.145)

and

GxyEm (kx, ky) = Gyx

Em (kx, ky) =jZ0

k0kxky sin (kz1d)

·[kz0 cos (kz1d) + jkz1 sin (kz1d)

TeTm

]. (3.146)

Numerical evaluation of (3.144)–(3.146) needs special attention because of the surfacewave poles in (3.142) and (3.143). Techniques for handling these surface wave polesare discussed in [31, 52].

3.6.4 Dielectric Slab

The geometry configuration for a single dielectric slab is shown in Fig. 3.8(a). Belowthe dielectric (z < 0) and above the dielectric (z > d) is free space with the conductortraces located on the z = d plane. The dielectric slab is assumed to be infinite inthe transverse directions. The dielectric slab geometry is similar to the microstripgeometry but without the ground plane where the TE and TM loads at the bottomare

ZTEBL =

Z0k0

kz0(3.147)

ZTMBL =

Z0kz0k0

. (3.148)

49

The equivalent impedances looking into the bottom are then

ZTEB =

Z0k0

kz1

[Te

kz0 cos (kz1d) + jkz1 sin (kz1d)

](3.149)

ZTMB =

Z0kz1εrk0

[Tm

kz1 cos (kz1d) + jεrkz0 sin (kz1d)

](3.150)

resulting in the equivalent impedances of the top and bottom

ZTE =Z0k0Te

2kz0kz1 cos (kz1d) + j (k2z0 + k2

z1) sin (kz1d)(3.151)

ZTM =Z0kz0kz1Tm

k0 [(εrk2z0 + k2

z1) cos (kz1d) + jkz0kz1 (εr + 1) sin (kz1d)]. (3.152)

Using (3.151) and (3.152) in expressions (3.93)–(3.95) yields the dyadic componentsof the dielectric slab Green’s function. Once again numerical considerations need tobe considered for handling singularities resulting from surface waves that occur in thedielectric layers.

The immittance approach used here can easily be expanded to containmulti substrate and superstrate layers. Only one dielectric layer is considered herebecause the quasi-optical structures analyzed in this dissertation contain at most onesubstrate layer.

Chapter 4

Method of Moments

4.1 General Formulation

The boundary value problem for the current distribution on the planar radiating el-ements in the quasi-optical system is formulated as an electric field integral equation(EFIE). The EFIE is formulated, for the first time, using the quasi-optical dyadicGreen’s functions from Chapter 3. The method of moments (MoM) [14] is used tosolve the EFIE for the unknown currents on the surfaces of the radiating elements. Toimprove performance, the MoM is formulated using a combination of spatial and spec-tral domain techniques. A spatial domain moment method procedure [11–13,27,28,43]is used to compute the moment matrix elements for the quasi-optical contributionsand a spectral domain moment method procedure [31, 32, 36–38, 52] is used to com-pute the moment matrix elements for the nonquasi-optical contributions. The surfaceof the antenna elements and arrays are divided into a grid of rectangular cells. Thegrid need not be restricted to rectangular cells and could, for example, consist oftriangular cells [28]. Equal size cells of dimension a × b are used in order to takeadvantage of symmetries [12, 13] and other numerical efficiencies such as interpola-tion of the moment matrix [11]. A set of basis functions are used to approximate thecurrent density on the antenna surfaces where each basis function is spanned overtwo rectangular cells and can be in the x or y direction. For the formulation pre-sented here, sinusoidal basis functions are used for both current expansion and testing(Galerkin method) [31,38]. The formulation presented in this chapter can be appliedto arbitrary planar geometries which can be divided into equal size rectangular cells.The formulation could also be extended to handle more complicated geometries bydividing the grid into unequal size cells consisting of rectangular and triangular cells.

The formulation begins with the boundary condition stating that the total

50

51

tangential electric field on the antenna surface is zero:

−Escatt (x, y) = Einc

t (x, y) (4.1)

where subscript t denotes the transverse components of the electric fields. Einct is

the incident electric field and Escatt is the scattered electric field. The incident field

is the electric field produced by the source that is used to excite the antenna. Theincident field, Einc

t , produces a surface current density, JS, on the antenna surfacewhich then produces a scattered field, Escat

t , where some of the field is coupled intothe quasi-optical system and the rest of the field is radiated out of the system.

The scattered field can be written in terms of the dyadic electric fieldGreen’s function, given in Chapter 3, as

Escatt (x, y) =

∫y′

∫x′

=

GE ·JS (x′, y′) dx′dy′. (4.2)

In order to solve for Escatt in (4.2), an approximation for the unknown surface current

density is needed. The unknown surface current density can be expanded in a set ofN basis functions

JS (x′, y′) =N∑i=1

IiWi (x′, y′) (4.3)

where Wi is the ith basis function and Ii is its unknown complex amplitude. Thebasis functions Wi can represent currents in the x or y directions

Wi (x′, y′) = W x

i (x′) ax +W yi (y′) ay. (4.4)

Substituting (4.3) into (4.2) and testing (4.1) with the same set of basis functions,known as the Galerkin method, yields

−N∑i=1

Ii

∫y

∫x

∫y′

∫x′

Wj (x, y) · =

GE ·Wi (x′, y′) dx′dy′dx dy

=∫y

∫xWj (x, y) · Einc

t dx dy (4.5)

where Wj is the test basis function and Wi is the source basis function. Primedcoordinates denote the source location and unprimed coordinates denote the testlocation. This procedure produces a set of linear algebraic equations to be solved forthe unknown currents Ii:

[Z] [I] = [V] (4.6)

where

Zji = −∫y

∫x

∫y′

∫x′

Wj (x, y) · =

GE ·Wi (x′, y′) dx′dy′dx dy (4.7)

52

and

Vj =∫y

∫xWj (x, y) · Einc

t (x, y) dx dy. (4.8)

The Z elements in (4.7) are known as the impedance or moment matrix elementsand the V elements in (4.8) are the voltage or excitation vector elements. Since theGreen’s function is comprised of quasi-optical and nonquasi-optical contributions, itis best to work with the moment matrix elements in the same manner. The momentmatrix elements can be divided into

Zji = Zqo,ji + Zf,ji (4.9)

where Zqo and Zf represent the quasi-optical and nonquasi-optical (open space) con-tributions, respectively, which are given by

Zqo,ji = −∫y

∫x

∫y′

∫x′

Wj (x, y) · =

GEqo (x, y ; x′, y′) · Wi (x′, y′) dx′dy′dx dy (4.10)

and

Zf,ji = −∫y

∫x

∫y′

∫x′

Wj (x, y) · =

GEf (x|x′; y|y′) · Wi (x′, y′) dx′dy′dx dy. (4.11)

It is important to note that GEqo in (4.10) is not a function of the distance between thesource and test location whereas GEf in (4.11) is a function of the distance betweenthe source and test location. For this reason, it is best to evaluate the quasi-opticaland nonquasi-optical moment matrix elements separately and then sum together toget the complete moment matrix elements. The final set of linear equations whensolving for the x and y currents may be written as⎡

⎣[Zxxqo,ji + Zxx

f,ji

] [Zxyf,jk

][Zyxf,li

] [Zyyqo,lk + Zyy

f,lk

]⎤⎦⎡⎣[Ixi]

[Iyk]⎤⎦ =

⎡⎣[V xj

][V yl

]⎤⎦ (4.12)

where

j, i = 1, 2, . . . , Nx

l, k = Nx + 1, Nx + 2, . . . , N

N = Nx +Ny.

Nx and Ny are the total number of x-and y-directed basis functions, respectively. Thesubmatrix [Zvu

ts ] denotes the contribution of v-directed testing of the field producedby u-directed current basis elements and the subscripts t and s refer to the individual

53

test and source basis elements, respectively. The voltage vectors[V xj

]and

[V yl

], of

length Nx and Ny, respectively, correspond to x-and y-directed testing of the incidentfield. The elements of the voltage vector will be discussed in Section 4.5. Similarly,[Ixi]

and[Iyk]

refer to the current expansion coefficients associated with each source

basis function. The moment matrix [Z] is a square matrix of order N . The momentmatrix is also symmetrical, due to the Galerkin method used, and diagonally strong.

A flowchart for the complete quasi-optical electromagnetic simulator isshown in Fig. 4.1. The inputs to the program are entered in a graphical user interfacewhere they are then parsed and passed to a program written in C. This program isthe brain for the simulator and controls the FORTRAN program. The FORTRANprogram is what is developed in this dissertation where it computes all the numericalresults. The C program determines what distinct moment matrix elements are neededto fill the entire moment matrix. By doing so the performance of the simulator isgreatly enhanced. After filling the moment matrix the C program passes it to theFORTRAN program where it solves for the rest of the parameters. The filling of themoment matrix and solving for the parameters can be done in parallel for efficientanalysis of electrically large arrays.

4.2 Expansion Functions

Sinusoidal basis functions are used for the current expansion functions and testingfunctions. An x-directed sinusoidal basis function is shown in Fig. 4.2 for a cell sizeof a× b. An x-directed sinusoidal basis function centered at (xi, yi) is given by

W xi (x) =

⎧⎪⎨⎪⎩sin [k0 (a− |x− xi|)]

b sin (k0a),

|x− xi| ≤ a|y − yi| ≤ b/2

0 , otherwise(4.13)

and for a y-directed sinusoidal basis function

W yi (y) =

⎧⎪⎨⎪⎩sin [k0 (b− |y − yi|)]

a sin (k0b),

|y − yi| ≤ b|x− xi| ≤ a/2

0 , otherwise .(4.14)

A basis function is spanned over two rectangular cells and the current amplitudes Iiare computed at the peak of each basis function as shown in Fig. 4.3.

The accuracy of the currents computed by the MoM is improved by in-creasing the number of basis functions (e.g. more cells). As a rule the grid shouldbe divided up into at least 10 cells per wavelength. Therefore evaluating over a largefrequency span, the number of cells should be determined from the highest frequencypoint needed. Numerical problems may also occur if too many cells are chosen.

54

Graphical UserInterface

Compute [V]

[Z][I]=[V]

Solve for [I]

ComputeParameters

Lex & YaccParser

Find Symmetries

Compute Z Elements

Fill Moment Matrix [Z]

Determine DistinctZ Elements

Write Results toData Files

Next Frequency

FORTRAN

C

Finished

Figure 4.1: Moment method flowchart utilizing C and FORTRAN.

y

x

y + b/2i

y − b/2i

yi

x − ai

xi

x + ai

Figure 4.2: An x-directed sinusoidal basis function.

55

Figure 4.3: Locations of x−and y−directed currents on a rectangular grid.

4.3 Quasi-Optical Moment Matrix Elements

4.3.1 Open Cavity Resonator

For a grid divided into equal size rectangular cells of dimension a × b, the momentmatrix elements for the open cavity resonator contribution are found by substitutingthe Green’s function given in (3.6) into (4.10) yielding

Zxxqo,ji =

Nm∑m=0

Nn∑n=0

Rmnψmn2 (1 +Rmnψmn)

·∫ yj+

b2

yj− b2

∫ xj+a

xj−a

[E−mn (x, y, d) −E+

mn (x, y, d)]W xj (x) dx dy

·∫ yi+

b2

yi− b2

∫ xi+a

xi−a

[E−mn (x′, y′, d) −E+

mn (x′, y′, d)]W xi (x′) dx′dy′

(4.15)

for the axax dyadic elements and

Zyyqo,lk =

Nm∑m=0

Nn∑n=0

Rmnψmn2 (1 +Rmnψmn)

·∫ yl+b

yl−b

∫ xl+a2

xl− a2

[E−mn (x, y, d)− E+

mn (x, y, d)]W yl (y) dx dy

·∫ yk+b

yk−b

∫ xk+ a2

xk− a2

[E−mn (x′, y′, d) − E+

mn (x′, y′, d)]W yk (y′) dx′dy′

(4.16)

for the ayay dyadic elements. The elements are computed on the plane z = d andcontain no cross-terms (e.g. Zxy = Zyx = 0). Since the Green’s function given

56

in (3.6) is a function of the source and test location and not the distance betweenthe two, the four dimensional integration can be divided into two separate doubleintegrations over the source and test fields. The double integration can be computedvery efficiently and suffers no convergence problems. Since the elements are a sum ofall the modes being considered, it is best to compute the double integration for eachmode as efficiently as possible. For the open cavity resonator with a metallic reflector,no more than Nm = 10 and Nn = 10 modes should ever need to be considered. Formost cases Nm = 3 and Nn = 3 modes are sufficient.

4.3.2 Lens System

For a grid divided into equal size rectangular cells of dimension a × b, the momentmatrix elements for the lens system are found by substituting the Green’s functiongiven in (3.57) into (4.10) yielding

Zxxqo,ji =

Nm∑m=0

Nn∑n=0

Rmnψmn(1 −Rmnψmn)

·∫ yj+

b2

yj− b2

∫ xj+a

xj−aEmn (x, y, 0)W x

j (x) dx dy

·∫ yi+

b2

yi− b2

∫ xi+a

xi−aEmn (x′, y′, 0)W x

i (x′) dx′dy′

(4.17)

for the axax dyadic elements and

Zyyqo,lk =

Nm∑m=0

Nn∑n=0

Rmnψmn(1 − Rmnψmn)

·∫ yl+b

yl−b

∫ xl+a2

xl− a2

Emn (x, y, 0)W yl (y) dx dy

·∫ yk+b

yk−b

∫ xk+ a2

xk− a2

Emn (x′, y′, 0)W yk (y′) dx′dy′

(4.18)

for the ayay dyadic elements. The elements are computed on the plane z = 0. Onceagain there are no cross terms as was the case for the open cavity resonator. Sincein most cases two identical lenses are used, the moment matrix elements above areformulated by assuming that two identical lenses are used. This is not a requiredassumption but instead keeps the formulation in its simplest form. The numericalapproach in computing the lens system elements is identical to the approach usedfor the open cavity resonator elements. For the lens system with dielectric lenses,

57

Nm = 1 and Nn=1 modes are sufficient because of the low reflections from the air todielectric interfaces. Whereas in the open cavity resonator, the reflections were quitehigh due to the air to metallic interfaces.

4.4 Nonquasi-Optical Moment Matrix Elements

4.4.1 Spatial Domain

The moment matrix elements for the nonquasi-optical elements (open space) havingequal size cells of dimension a× b, are found from (4.11) as

Zxxf,ji = −

∫ yj+b2

yj− b2

∫ xj+a

xj−a

∫ yi+b2

yi− b2

∫ xi+a

xi−aGxxEf (x|x′; y|y′)W x

j (x)W xi (x′) dx′dy′dx dy

(4.19)

Zxyf,jk = −

∫ yj+b2

yj− b2

∫ xj+a

xj−a

∫ yk+b

yk−b

∫ xk+ a2

xk− a2

GxyEf (x|x′; y|y′)W x

j (x)W yk (y′) dx′dy′dx dy

(4.20)

Zyxf,li = −

∫ yl+b

yl−b

∫ xl+a2

xl− a2

∫ yi+b2

yi− b2

∫ xi+a

xi−aGyxEf (x|x′; y|y′)W y

l (y)W xi (x′) dx′dy′dx dy

(4.21)

Zyyf,lk = −

∫ yl+b

yl−b

∫ xl+a2

xl− a2

∫ yk+b

yk−b

∫ xk+ a2

xk− a2

GyyEf (x|x′; y|y′)W y

l (y)W yk (y′) dx′dy′dx dy.

(4.22)

As mentioned earlier, direct evaluation of (4.19) and (4.22) would be very difficultdue to the singularity that occurs when the source and test location are at the samepoint (self-term). The self-terms are the dominate terms in the moment matrix andinaccurate evaluation of these terms will result in unreliable solutions. For the crossterms (4.20) and (4.21) no singularity occurs because the source and test fields arenever at the same location, but direct evaluation is still difficult due to the fourintegrations required. With these problems it is best to work in the spectral domain.

58

4.4.2 Spectral Domain

By using the open space dyadic Green’s functions in the spectral domain given inSections 3.6.1–3.6.4, the moment matrix elements are found by inserting the openspace dyadic components of (3.92) into (4.19)–(4.22) yielding

Zxxf,ji =

−1

4π2

∫ ∞

−∞

∫ ∞

−∞GxxEf (kx, ky) W

xj (kx, ky)

∗ W xi (kx, ky) dkx dky (4.23)

Zxyf,jk =

−1

4π2

∫ ∞

−∞

∫ ∞

−∞GxyEf (kx, ky) W

xj (kx, ky)

∗ W yk (kx, ky) dkx dky (4.24)

Zyxf,li =

−1

4π2

∫ ∞

−∞

∫ ∞

−∞GyxEf (kx, ky) W

yl (kx, ky)

∗ W xi (kx, ky) dkx dky (4.25)

Zyyf,lk =

−1

4π2

∫ ∞

−∞

∫ ∞

−∞GyyEf (kx, ky) W

yl (kx, ky)

∗ W yk (kx, ky) dkx dky (4.26)

where W x and W y are the Fourier transforms of the x-and y-directed sinusoidalbasis functions, respectively, and the (∗) denotes the complex conjugate. One of theadvantages of using the spectral domain is that the Fourier transforms of the basisfunctions can be evaluated in closed form as follows:

W xi (kx, ky) =

∫ yi+b2

yi− b2

∫ xi+a

xi−aW xi (x) e−jkxxe−jkyy dx dy

=2k0

sin (k0a)

(cos (kxa) − cos (k0a)

k20 − k2

x

)

·[sin (kyb/2)

(kyb/2)

]e−jkxxie−jkyyi (4.27)

W yi (kx, ky) =

∫ yi+b

yi−b

∫ xi+a2

xi− a2

W yi (y) e−jkxxe−jkyy dx dy

=2k0

sin (k0b)

(cos (kyb) − cos (k0b)

k20 − k2

y

)

·[sin (kxa/2)

(kxa/2)

]e−jkxxie−jkyyi. (4.28)

The moment matrix elements given in (4.23)–(4.26) now only involve two integrationsbut are of infinite limits. By using the even and odd properties of the integrands in(4.23)–(4.26), further simplifications can be made as follows [31, 52]:

Zxxf,ji =

−1

π2

∫ ∞

0

∫ ∞

0GxxEf (kx, ky)F

xxji (kx, ky) dkx dky (4.29)

59

Zyyf,lk =

−1

π2

∫ ∞

0

∫ ∞

0GyyEf (kx, ky)F

yylk (kx, ky) dkx dky (4.30)

Zxyf,jk =

−1

π2

∫ ∞

0

∫ ∞

0GxyEf (kx, ky)F

xyjk (kx, ky) dkx dky (4.31)

Zyxf,li = Zxy

f,kj (4.32)

where

F xxji (kx, ky) =

(2k0

sin (k0a)

)2 (cos (kxa) − cos (k0a)

k20 − k2

x

)2 [sin (kyb/2)

(kyb/2)

]2

· cos [kx (xj − xi)] cos [ky (yj − yi)] (4.33)

F yylk (kx, ky) =

(2k0

sin (k0b)

)2 (cos (kyb) − cos (k0b)

k20 − k2

y

)2 [sin (kxa/2)

(kxa/2)

]2

· cos [kx (xl − xk)] cos [ky (yl − yk)] (4.34)

F xyjk (kx, ky) =

−4k20

sin (k0a) sin (k0b)

[sin (kxa/2)

(kxa/2)

] [sin (kyb/2)

(kyb/2)

]

·(

cos (kxa) − cos (k0a)

k20 − k2

x

)(cos (kyb) − cos (k0b)

k20 − k2

y

)

· sin [kx (xj − xk)] sin [ky (yj − yk)] . (4.35)

The products of (W x)(W x)∗, (W y)(W x)∗, (W x)(W y)∗, and (W y)(W y)∗ result in com-plex functions where the real parts are even functions and the imaginary parts are oddfunctions. Since Gxx

Ef and GyyEf are even functions about the origin of the kx, ky space,

only the real part of the products of (W x)(W x)∗ and (W y)(W y)∗ show up in F xx andF yy, respectively. The imaginary part of the products are odd functions which resultsin an odd integrand that integrates to zero over the kx, ky space. Similarly Gxy

Ef is

an odd function so only the imaginary part of the product of (W x)(W y)∗ shows upin F xy. Also the product of (W y)(W x)∗ is equal to (W x)(W y)∗.

The integration is done only over the first quadrant of the kx, ky space butstill contains two upper limits that go to infinity. Changing the integration in (4.29)–(4.32) to polar coordinates will eliminate one of the infinite upper limits [31,52]. Thefinal moment matrix elements that will be numerically evaluated are given in polarcoordinates as follows:

Zxxf,ji =

−1

π2

∫ π/2

0

∫ ∞

0GxxEf (kx, ky) F

xxji (kx, ky) β dβ dα (4.36)

60

Zyyf,lk =

−1

π2

∫ π/2

0

∫ ∞

0GyyEf (kx, ky) F

yylk (kx, ky) β dβ dα (4.37)

Zxyf,jk =

−1

π2

∫ π/2

0

∫ ∞

0GxyEf (kx, ky) F

xyjk (kx, ky) β dβ dα (4.38)

Zyxf,li = Zxy

f,kj (4.39)

where

kx = β cosα (4.40)

ky = β sinα. (4.41)

4.5 Excitation Vector

The excitation vector elements from (4.8) are given as

V xj =

∫ yj+b2

yj− b2

∫ xj+a

xj−aW xj (x)Einc

x (x, y) dx dy (4.42)

V yl =

∫ yl+b

yl−b

∫ xl+a2

xl− a2

W yl (y)Einc

y (x, y) dx dy (4.43)

for the x and y directions. Two types of excitation are considered here. The first willbe the delta-gap voltage generator and the second will be the coaxial current probe.

4.5.1 Delta-Gap Voltage Generator

The delta-gap voltage generator is a simplistic source which is commonly used inmoment method analysis of wire antennas [131]. Although such sources do not existin practice, they do give good results for dipole antennas and have been shown towork well with printed transmission lines with electrically small widths. Fig. 4.4shows the implementation of a delta-gap voltage generator where a voltage source isplaced between two cells, in a fictitious gap, giving rise to an impressed electric fieldEincx = V/Δ confined entirely to the gap. As Δ → 0 we have

V xj =

∫ yj+b2

yj− b2

∫ xj+a

xj−aW xj (x)V δ (xj) dx dy = V. (4.44)

61

Therefore the expression for the voltage vector using a 1 Volt delta-gap voltage gen-erator is given as

Vp =

{1 for p equal to feed point0 otherwise.

(4.45)

The input impedance at the location of the delta-gap voltage generator can be com-puted as [36]

Zin =VpIp

(4.46)

where Ip is the current at the delta-gap computed by the MoM.

V

Δ

Figure 4.4: A delta-gap voltage generator.

4.5.2 Coaxial Current Probe

The coaxial current probe offers a more realistic method to excite currents on a patchantenna [31]. The coaxial probe is connected through the ground plane with thecenter conductor embedded vertically and terminated on the patch surface, wherethe outer conductor of the coax is connected to the ground plane. Fig. 4.5 shows theattachment of the coaxial probe to the patch antenna surface. The incident electricfields in (4.42) and (4.43) for the coaxial current probe are given as

Einct (x, y) =

∫y′

∫x′

=

GEf (x|x′; y|y′) · JF (x′, y′) dx′dy′ (4.47)

with a current density of [31]

JF (x′, y′) = IF δ (x′ − xp) δ (y′ − yp) az (4.48)

flowing on the inner conductor of the coaxial probe. The current is assumed to beconstant with an amplitude of IF . Substituting (4.48) into (4.47) gives the transverseincident electric field as

Einct (x, y) = IF

=

GEf (x|xp; y|yp) · az (4.49)

62

IF

a

b p x py,( )

x

y

CENTER CONDUCTOR

GROUND PLANE

RECTANGULARPATCH

COAXIAL CABLE

Figure 4.5: A coaxial current probe.

with the following transverse components

Eincx = IFG

xzEf (x|xp; y|yp) (4.50)

Eincy = IFG

yzEf (x|xp; y|yp) . (4.51)

The spectral domain Green’s functions can be used in (4.50) and (4.51), respectively,by using the following inverse Fourier transforms

GxzEf (x|xp; y|yp) =

IF4π2

∫ ∞

−∞

∫ ∞

−∞GxzEf (kx, ky) e

jkx(x−xp)ejky(y−yp)dkxdky (4.52)

GyzEf (x|xp; y|yp) =

IF4π2

∫ ∞

−∞

∫ ∞

−∞GyzEf (kx, ky) e

jkx(x−xp)ejky(y−yp)dkxdky. (4.53)

With this transformation to the spectral domain, substituting (4.50) and (4.51) into(4.42) and (4.43), respectively, gives the excitation vector elements

V xj =

IF4π2

∫ ∞

−∞

∫ ∞

−∞GxzEf (kx, ky) W

xj (kx, ky)

∗ e−jkxxpe−jkyypdkxdky (4.54)

V xl =

IF4π2

∫ ∞

−∞

∫ ∞

−∞GyzEf (kx, ky) W

yl (kx, ky)

∗ e−jkxxpe−jkyypdkxdky. (4.55)

By using the even and odd properties of the integrands in (4.54) and (4.55), as wasdone for the open space moment matrix elements, further simplifications can be madeas follows:

V xj =

jIFπ2

∫ ∞

0

∫ ∞

0GxzEf (kx, ky)F

xpj (kx, ky) dkxdky (4.56)

63

V yl =

jIFπ2

∫ ∞

0

∫ ∞

0GyzEf (kx, ky)F

ypl (kx, ky) dkxdky (4.57)

where

F xpj (kx, ky) =

2k0

sin (k0a)

(cos (kxa) − cos (k0a)

k20 − k2

x

) [sin (kyb/2)

(kyb/2)

](4.58)

· sin [kx (xj − xp)] cos [ky (yj − yp)]

F ypl (kx, ky) =

2k0

sin (k0b)

(cos (kyb) − cos (k0b)

k20 − k2

y

) [sin (kxa/2)

(kxa/2)

](4.59)

· sin [ky (yl − yp)] cos [kx (xl − xp)] .

Changing the integration in (4.56) and (4.57) to polar coordinates gives

V xj =

jIFπ2

∫ π/2

0

∫ ∞

0GxzEf (kx, ky)F

xpj (kx, ky) βdβdα (4.60)

V yl =

jIFπ2

∫ π/2

0

∫ ∞

0GyzEf (kx, ky)F

ypl (kx, ky)βdβdα (4.61)

which are the final expressions for the excitation vector elements that will be numer-ically evaluated. The dyadic components of the Green’s functions with respect tothe z-direction are found in [36]. The input impedance at the location of the coaxialcurrent probe can be computed as [31, 43]

Zin =1

I2F

N∑n=1

VnIn (4.62)

where Vn are the elements of the excitation vector and In are the currents computedby the MoM. One very important note is that the input impedance in (4.62) doesnot contain contributions from the incident field. Near resonance the incident fieldcontribution has a second-order effect and can be neglected. For low frequency com-putations the incident field contribution must be considered to obtain the correctimpedance. In CAD design it is important to characterize the linear part of the cir-cuit down to DC. The input impedance for frequencies below the first resonance iscomputed as [41]

Zin = − 1

IF

∫ d

0

(Einc + Escat

)· az dz. (4.63)

The MoM was used to compute the surface currents on a rectangular patchantenna in half space located 1 mm above the ground plane. The patch antenna was

64

0

15 0

5

0

0.5

1

1.5

X (mm)

Y (mm)

X Magnitude (Amps)

(a)

0

15 0

5

00.050.1

0.150.2

0.250.3

0.35

X (mm)

Y (mm)

Y Magnitude (Amps)

(b)

Figure 4.6: Surface current magnitude: (a) x-directed; (b) y-directed; on a rectangularpatch antenna in half space.

65

15 mm × 5 mm divided into a 15 × 5 rectangular grid with a center fed coaxialcurrent probe. Fig. 4.6 shows the x-and y-directed current magnitude distributionsat the half-wave resonance (f = 18.2 GHz). For this resonant mode, the current isdominant in the longitudinal direction (x direction) with current build up along theedges.

The charge distribution is related to the current through the continuityequation

∇ · JS + jωρS = 0. (4.64)

For this example most of the charge is along the longitudinal edges. Near DC allthe charge distribution is at the excitation feed point. At low frequencies the currentdistribution is constant with the exception at the feed point where it has a largediscontinuity. For the grid array problems analyzed in this dissertation, the currentdistribution is dominant in the longitudinal direction with the charge distributionalong the edges, similar to this patch antenna example. The current distributionis dominant in the longitudinal direction because the widths of the grid lines areelectrically small compared to the grid lengths.

4.6 Numerical Considerations

The FORTRAN program developed in this dissertation uses a numerical Gaussianquadrature integration routine for the double and quadruple integrations requiredfor computing the moment matrix elements. The MoM solution is computed usingLU decomposition and double precision is used in all calculations. Techniques forcomputing the spectral domain moment matrix reactions and convergence issues arepresent next along with the moment matrix conditioning. The valid frequency rangesfor the moment method simulator are also discussed.

4.6.1 Spectral Domain Moment Matrix Elements

The numerical integration of the moment matrix elements given in (4.36)–(4.39) arebest handled by dividing the integral into two parts [31, 38, 52],

Zf =−1

π2

[∫ π/2

0

∫ k0

0GF β dβ dα +

∫ π/2

0

∫ ∞

k0GF β dβ dα

]. (4.65)

At β = k0 the integrand of (4.36)–(4.39) has a zero in the denominator. This is not asingularity because it can easily be shown that the integrand has a limit at β = k0. In

66

the first integral in (4.65) the β variable ranges from 0 to k0 and in the second integralin (4.65) the β variable ranges from k0 to ∞. By keeping β = k0 as an endpoint in bothintegrals, this point is never computed when using a Gaussian quadrature numericalintegration routine. Also in the range 0 < β < k0, the moment matrix elements haveboth real and imaginary terms whereas in the range k0 < β < ∞, they only haveimaginary terms. The infinite upper limit can be truncated at 50k0 for the self-termsand 100k0 for all the other terms [52].

4.6.2 Convergence Issues

Convergence is a big concern in moment method solutions. As a rule of thumb thereshould be at least 10 cell subdivisions per wavelength to obtain reliable solutions. Cellsubdivisions consisting of 20 cells are often used for the most accurate solutions. Fig.4.7 shows the driving point reflection coefficient of a dipole antenna in free space fordifferent cell subdivisions. The dipole has a length of 30 mm and a width of 3 mm.In Fig. 4.8 the convergence rate of the reflection coefficient at 10 GHz is presented.At this frequency the dipole has a length of one free space wavelength. Convergenceof the reflection coefficient magnitude occurs between 10 to 20 cells per wavelengthand the convergence of the reflection coefficient phase occurs closer to 20 cells perwavelength. Above 20 cells per wavelength the convergence starts to break down forboth magnitude and phase. This error is due to the spectral domain moment matrixelements. When the cell sizes become too small the infinite limits in the numericalintegration need to be readjusted. By keeping the number of cells below 20 cells perwavelength, convergence is obtained and spectral domain errors are avoided. Thishas been the case for all of the structures analyzed in this dissertation.

4.6.3 Condition Number

One other numerical concern is the condition number of the moment matrix. Thecondition number is computed as follows:

COND (Z) = ‖Z‖ ‖Y‖ (4.66)

where

‖Z‖ = maxj

∑i

|zij | (4.67)

and

‖Y‖ = maxj

∑i

|yij | (4.68)

67

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

N=12N=10

N=8N=6N=4

(a)

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

0 2 4 6 8 10 12 14 16 18 20

PH

AS

E S

11

FREQUENCY (GHz)

N=12N=10

N=8N=6N=4

(b)

Figure 4.7: Driving point reflection coefficient: (a) magnitude; (b) phase; of a dipoleantenna divided into N cells.

68

0.82

0.84

0.86

0.88

0.9

0.92

0.94

0.96

0.98

1

0 5 10 15 20 25 30

MA

GN

ITU

DE

S11

NUMBER OF CELLS PER WAVELENGTH

(a)

-30

-25

-20

-15

-10

-5

0

5

0 5 10 15 20 25 30

PH

AS

E S

11

NUMBER OF CELLS PER WAVELENGTH

(b)

Figure 4.8: Convergence rate of the driving point reflection coefficient: (a) magnitude;(b) phase; of a dipole antenna.

69

with Z being the moment matrix and Y being the inverted moment matrix. For awell-conditioned matrix the condition number ranges from 1 to 1000 with 1 being thebest. Above 1000 the matrix becomes ill-conditioned. Fig. 4.9 shows the conditionnumber of the moment matrix for several cell subdivisions of the dipole antenna.In all three cases the condition number is well below 1000. The peaks in Fig. 4.9correspond to half-wavelength multiples of the dipole antenna. If the moment matrixwere to become ill-conditioned, it would occur at a resonant frequency where thematrix becomes singular. All of the moment matrices of the structures simulated inthis dissertation are well-conditioned when using LU decomposition.

0

50

100

150

200

250

300

0 2 4 6 8 10 12 14 16 18 20

CO

ND

(Z)

FREQUENCY (GHz)

Figure 4.9: Condition number of the moment matrix: solid line, N = 10; dashed line,N = 20; dotted line, N = 30.

4.6.4 Valid Frequency Ranges

One of the attractive features of this moment method simulator is that it can computefrom DC to any frequency. It is very common for electromagnetic simulators not towork at low frequencies. The MoM cannot handle DC (f = 0) directly but can handleany positive real number greater than zero. For example f = 10−6 Hz works in theMoM simulator and from Fig. 4.7 it is shown that the solution converges at DC.The upper frequency has no limit but the MoM cell subdivision is determined fromthe highest frequency and should obey the 10 cells per wavelength requirement forconvergence.

70

4.7 Multiport Analysis

The multiport parameters are found in terms of admittance (Y) parameters from themoment method solution. As a simple example consider a two-port problem wherethe Y-parameters are defined as follows:

y11 =I1V1

∣∣∣∣V2=0

(4.69)

y12 =I1V2

∣∣∣∣V1=0

(4.70)

y21 =I2V1

∣∣∣∣V2=0

(4.71)

y22 =I2V2

∣∣∣∣V1=0

. (4.72)

In this example two delta-gap voltage generators (V1 and V2) are used in the momentmethod simulator. First the moment matrix is computed and then inverted. TheY-parameters follow directly from (4.69)-(4.72) by using the two delta-gap voltagegenerators at the desired port locations. V1 is applied with V2 shorted and then V2

with V1 shorted.

For networks consisting of more than two-ports, the same approach canbe applied by placing delta-gap voltage sources at each port. A single port is excitedwhile all the other ports are shorted. This procedure is continued until all of the portparameters are computed. Note that since all of the structures are passive, the portmatrix will be symmetrical. If all of the delta-gap voltage sources are equivalent,the Y-parameters can be found directly from the inverted moment matrix. In mostsituations identical delta-gap voltage generators are used and have a value of unity.This analysis is not straight forward if using coaxial current probes as the sources.After the Y-parameters are found they are converted to scattering (S) parameters forcomparison with measurements,

S = Y1/20 (Y0 + Y)−1 (Y0 − Y)Y

−1/20 (4.73)

where Y0 is the characteristic admittance matrix. Also S-parameters make it easierto examine the coupling between ports.

71

4.7.1 Nodal Admittance Parameters

As discussed previously, the computation of the port parameters is straight forwardwhen using delta-gap voltage sources. The port parameters give coupling coefficientsfor the structure but are not able to be directly incorporated into a microwave circuitsimulator. A circuit simulator requires nodal parameters that are all referenced toa common terminal such as ground and obey Kirchhoff’s current law. In the portparameter matrix the ports are not necessarily referenced to the same place plusno constraints are applied to the moment method formulation forcing Kirchhoff’scurrent law. A general solution for computing nodal admittance parameters directlyfrom the port admittance matrix is currently being addressed. In Section 5.4.2 a four-port example is given where the port parameters have been successfully converted tonodal parameters.

The purpose for computing the nodal admittance parameters is so thatthe passive antenna or grid array of a quasi-optical system can be incorporated into amicrowave circuit simulator in order to get a complete description of the quasi-opticalactive circuit. Consider the four-port unit cell shown in Fig. 4.10. The active device

V2

V3

V1

V4

- + --

+

+

-

+

Figure 4.10: A four-port unit cell structure with metal in the gap region.

that would be located in the gap region is replaced by a piece of metal. The momentmethod simulator places delta-gap voltage sources (V1, V2, V3, and V4) from each gridlead to the piece of metal in the center (see Fig. 4.10). By placing a piece of metal inthe gap region, current is able to flow through the gap. For example, the metal piecein the gap region could represent the ground plane of a MMIC. With the delta-gap

72

voltage sources in place, the multiport parameters are computed. The same geometryconfiguration would also apply to a grid array consisting of many unit cells. For suchcases each four-port unit cell considers coupling from all other unit cells. For exampleif a 10 × 10 grid array is analyzed, there would be a total of 400 ports.

Another method for computing the port parameters and nodal parametersis presented in Appendix A. In this method half basis functions are placed at theedge of the grid lines (see Fig. A.2) leaving the gap regions without metal. Delta-gapvoltage generators are then placed between each half basis element and ground planedefining each port. This method was demonstrated in [46] for a microstrip geometry.Applying this formulation to a quasi-optical grid array poses several problems. Thegrid array has no ground plane in the circuit which makes it difficult to define acommon reference terminal to attach the delta-gap sources. For geometries with aground plane such as microstrip, the formulation has physical meaning but when theground plane is removed the formulation losses its physical meaning. Structures withground planes have image currents and can be defined as transmission lines. For thegrid array on a dielectric slab without a ground plane, the grid lines cannot be definedas transmission lines. In order to get this formulation to work with the grid arrays, acommon reference terminal will need to be forced in the moment method formulation.

Chapter 5

Computed and ExperimentalResults

5.1 Introduction

In this chapter results are presented, simulated and measured, for antenna elementsand arrays in quasi-optical systems including the open cavity resonator and lens sys-tem. The simulated results were obtained using the moment method technique devel-oped in Chapter 4, incorporating the quasi-optical Green’s functions from Chapter 3,where current distributions on the antenna surfaces are computed. From the currentdistribution the driving point impedance (or driving point reflection coefficient) ofthe antenna is found. The simulated results were verified with measurements thatwere taken using a Hewlett Packard model 8510C network analyzer.

Results are also presented in this chapter for electric field profiles in thequasi-optical lens system, computed and measured. Here the field profiles were mea-sured using a Hewlett Packard model 8566A spectrum analyzer. Other results includ-ing finite grid arrays versus the unit cell model, multiport parameters, and a taperedantenna used in two-dimensional quasi-optical power combining are presented.

5.2 Open Cavity Resonator

Two types of antenna elements were analyzed in the open cavity resonator shownin Fig. 2.1. The first was an inverted L antenna and the second was a rectangularpatch antenna. Both antennas were simulated and measured for the same open cavity

73

74

resonator geometry. The focal lengths of the spherical reflector with respect to thex and y axis were Fx = 89.4308 cm and Fy = 95.3839 cm, respectively, with adiameter of 25.4 cm. Both spherical and planar reflectors were made of aluminum(σ = 3.816 × 107 S/m). Also included in this section is a design example for anIMPATT diode oscillator.

5.2.1 Inverted L Antenna

Comparisons of measured and simulated results were made for an electrically shortinverted L antenna, shown in Fig. 5.1, located in the open cavity resonator at x =−90.6 mm and y = 15 mm, with d = 1.9 mm and L = 2.6 mm. The cavity spacingwas D = 62.0494 cm as determined in [4]. The antenna was constructed by extendingthe inner conductor, with a diameter of 0.9 mm, of an RG-141/U semi-rigid cableand bending it 90 degrees. For the moment method simulations the L antenna wasdivided into 10 cells with a delta-gap voltage generator placed between the first andsecond cells, as shown in Fig. 5.2. Only the longitudinal currents were consideredsince the transverse currents are negligible. The currents on the vertical element ofthe antenna were also ignored.

MEASUREMENTREFERENCE PLANE

ALUMINUMRESONATORBACKPLANE

COPPERFERRULE

SIMULATIONREFERENCE PLANE

RG−141/USEMI−RIGID COAXIAL CABLE

SMACONNECTOR

L

d

Figure 5.1: A coaxial fed inverted L antenna.

L

W

(x , y)

Figure 5.2: Method of moments cell subdivision for the inverted L antenna with adelta-gap voltage generator.

75

Fig. 5.3 shows the magnitude and phase of the driving point impedancefor the TEM0,0,35 mode. The agreement between the simulated and measured resultswere excellent. The magnitude and phase of the driving point impedance for theTEM0,1,35 and TEM1,0,35 modes are shown in Fig. 5.4. The TEM0,1,35 mode appearsfirst in frequency and then the TEM1,0,35 mode. The coupling in the TEM1,0,35 modeis much stronger than the coupling in the TEM0,1,35 mode. This is because of thelocation and polarization of the inverted L antenna. The difference between thesimulated and measured results were less than 1%. Fig. 5.5 shows the magnitudeand phase of the driving point impedance for the TEM0,2,35, TEM1,1,35, and TEM2,0,35

modes. The TEM0,2,35 mode appears first in frequency, with the TEM1,1,35 mode nextin frequency and then the TEM2,0,35 mode. The difference between the simulated andmeasured results were less than 2%. From Figs. 5.3–5.5, it is obvious that theinverted L antenna has very poor coupling into the cavity modes. The L antenna,at this frequency range, is electrically short, which explains the poor coupling factor.The simulated results are virtually identical to those in Reference [4]. Note that theprevious work [4] is restricted to short wire antennas whereas here it can be anyarbitrary planar geometry.

5.2.2 Rectangular Patch Antenna

A measurement of a coaxial center fed patch antenna, shown in Fig. 5.6, was takenwithout the spherical reflector. The rectangular patch had a length of L = 15 mmand a width of W = 5 mm. The height of the antenna above the ground planewas d = 1 mm. The simulation was done over a frequency range of 5.5 GHz to 8.5GHz. The patch antenna was modeled by the method of moments using the gridlayout shown in Fig. 5.7 with 16 cells and a delta-gap voltage generator placed inthe center. At first, both transverse and longitudinal currents were considered in themoment method simulation. For this frequency range, the transverse currents werevery small compared to the longitudinal currents so they were neglected in futuresimulations.

The driving point impedance of the patch antenna is shown in Fig. 5.8where good agreement between the simulation and measurement is observed. Nextthe same patch antenna is placed in the open cavity resonator at a location of x =y = 3 cm with D = 62.0494 cm. A simulation for the TEM0,0,34 mode and theTEM0,0,23 mode is shown in Fig. 5.9. The Q, computed as f/Δf , was found to be11,200 and 15,020 for the TEM0,0,23 and TEM0,0,34 modes, respectively. The patchantenna couples into the cavity modes much better than the inverted L antenna. Thisis because the patch antenna is a good radiator for this frequency range.

The driving point reflection coefficient was simulated for an identical patch

76

100

101

102

103

104

8.5022 8.5027 8.5032

MA

GN

ITU

DE

(O

hm

s)

FREQUENCY (GHz)

(a)

-87.5

-87

-86.5

-86

-85.5

8.5022 8.5027 8.5032

PH

AS

E (

Deg

rees

)

FREQUENCY (GHz)

(b)

Figure 5.3: Driving point impedance: (a) magnitude; (b) phase; for the TEM0,0,35

mode of the inverted L antenna: solid line, simulation; dashed line, measurement.

77

98

99

100

101

102

103

104

105

106

8.548 8.5505 8.553

MA

GN

ITU

DE

(O

hm

s)

FREQUENCY (GHz)

(a)

-87.5

-86.5

-85.5

-84.5

-83.5

8.548 8.5505 8.553

PH

AS

E (

Deg

rees

)

FREQUENCY (GHz)

(b)

Figure 5.4: Driving point impedance: (a) magnitude; (b) phase; for the TEM0,1,35

and TEM1,0,35 modes of the inverted L antenna: solid line, simulation; dashed line,measurement.

78

99

100

101

102

103

8.593 8.598 8.603

MA

GN

ITU

DE

(O

hm

s)

FREQUENCY (GHz)

(a)

-87.5

-87

-86.5

-86

-85.5

8.593 8.598 8.603

PH

AS

E (

Deg

rees

)

FREQUENCY (GHz)

(b)

Figure 5.5: Driving point impedance: (a) magnitude; (b) phase; for the TEM0,2,35,TEM1,1,35, and TEM2,0,35 modes of the inverted L antenna: solid line, simulation;dashed line, measurement.

79

MEASUREMENTREFERENCE PLANE

ALUMINUMRESONATORBACKPLANE

COPPERFERRULE

SIMULATIONREFERENCE PLANE

TOP VIEW

CROSS SECTIONAL SIDE VIEW

RG−141/USEMI−RIGID COAXIAL CABLE

SMACONNECTOR

RECTANGULARCOPPERPATCH ANTENNA

W

L

d

Figure 5.6: A coaxial fed rectangular patch antenna.

L

W

(x , y)

Figure 5.7: Method of moments cell subdivision for the rectangular patch antennawith a delta-gap voltage generator.

80

1 20.5

j

-j

S/C O/C

0.5j

-2j

2j

-0.5j

5.5 GHz

7.0 GHz

8.5 GHz

Figure 5.8: Impedance Smith chart showing the driving point impedance of the patchantenna without the reflector: solid line, simulation; dashed line, measurement.

1 20.5

j

-j

S/C O/C

0.5j

-2j

2j

-0.5j

5.5988 GHz 5.60375 GHz5.6088 GHz

8.2561 GHz

8.2608 GHz

8.2661 GHz

Figure 5.9: Impedance Smith chart showing the simulated driving point impedanceof the patch antenna in the open cavity resonator: solid line, TEM0,0,34 mode; dashedline, TEM0,0,23 mode.

81

antenna as in the previous example but now located in the open cavity resonator atx = −5.4 cm and y = 1.2 cm. The cavity spacing was D = 61.25 cm. Fig. 5.10 showsthe simulated driving point reflection coefficient magnitude for the q = 34 and q = 35mode families. A total of m+n = 10 modes were considered in the simulation but them + n = 6 mode set was the highest set that could be detected. From Fig. 5.10 wesee that the driving point reflection coefficient over this frequency span for the patchantenna in half space is flat whereas this is not the case when the antenna is placed inthe open cavity resonator. We see that at off resonant cavity modes the driving pointreflection coefficient is higher and at the resonant cavity modes it is much lower. Itis very important to know this behavior of the open cavity resonator when designingthe oscillator circuit to be used in this quasi-optical system. If you design your circuitto match the impedance in half space and then place your circuit in the open cavityresonator and try to operate it near a resonant cavity mode, the circuit may not workdue to the large impedance change that occurs from the resonant cavity modes. Anexample of this problem occurring in an oscillator design using an IMPATT diode isgiven in the next section.

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

8.55 8.6 8.65 8.7

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

m+n=4, q=34 m+n=5, q=34

m+n=0, q=35

m+n=1, q=35

m+n=6, q=34

m+n=2, q=35

Figure 5.10: Simulated driving point reflection coefficient magnitude of the patchantenna: solid line, open cavity resonator; dashed line, half space.

Next three patch antennas were placed in the open cavity resonator asshown in Fig. 5.11 with the same cavity spacing (D = 61.25 cm). Fig. 5.12 showsthe simulated driving point reflection coefficient of patch antenna 1 by itself (solidline) and with patch antennas 2 and 3 (dashed line). It is obvious from Fig. 5.12 thatthe other patch antennas load the open cavity resonator significantly and also shiftthe resonant mode frequency. This presents the case that a unit cell model of a singlestructure in the open cavity resonator would not be valid for an array design. The

82

finite array would need to be modeled in order to account for loading and couplingin the open cavity resonator.

y

x

(−5.4,1.2) (−2.5,1.3) (1.7,1.3)

1 2 3

ELECTRICAL CENTEROF CAVITY

(0,0)

Figure 5.11: Locations for three patch antennas in the open cavity resonator: alldimensions are in cm.

For all of the simulations a delta-gap voltage generator was used whereonly discretization in one direction was required (e.g. currents in the transversedirection are neglected). Since the electrical width of the patch antenna is small forfrequencies below 10 GHz, the delta-gap is a valid excitation. For frequencies above10 GHz, the transverse currents can no longer be neglected meaning the delta-gapexcitation is no longer valid. For these higher frequencies, the coaxial current probe isused where it excites both longitudinal and transverse currents. The coaxial currentprobe excitation requires discretization in both the x and y directions, hence moreunknown currents and longer computation time. So whenever valid, it is best to usethe delta-gap voltage generator for efficiency sake.

5.2.3 IMPATT Diode Oscillator

In [66] an array of oscillators using IMPATT diodes were designed and tested forpower combining in the open cavity resonator. A single oscillator was produced byplacing an IMPATT diode underneath a patch antenna. Oscillation occurs when thenegative impedance of the IMPATT diode equals the impedance of the patch antennafeed. In Fig. 5.13 the impedances of both the IMPATT diode [66] and patch antennaare shown. From Fig. 5.13 it looks as though the impedance match occurs around8.65 GHz. For this simulation the patch antenna was divided into 15 cells along the

83

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

8.6 8.602 8.604 8.606 8.608 8.61

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

Figure 5.12: Simulated driving point reflection coefficient magnitude of patch antenna1 for the TEM0,0,35 mode: solid line, alone; dotted line, with patch antennas 2 and 3.

longitudinal direction and 5 cells in the transverse direction. For the excitation acoaxial current probe was used in order to cover this wide frequency span. When thepatch antenna is placed in the open cavity resonator (x = −5.4 cm, y = 1.2 cm, andD = 61.25 cm) the driving point impedance changes significantly around 8.65 GHzas shown in Fig. 5.14.

Fig. 5.15 shows the IMPATT diode oscillator [66] in half space (solid line)and in the open cavity resonator (dashed line). Also plotted in Fig. 5.15 is the scaleddriving point reflection coefficient of the patch antenna showing the mode locations.When a cavity mode lies on top of the oscillation it rejects oscillation and causesthe oscillator to oscillate at a lower and higher frequency away from the mode as isindicated in Fig. 5.15(a). This is due to the significant impedance change at the cavitymode. The oscillation is not occurring simultaneously at two different frequencies butinstead is switching back and fourth. When the cavity spacing is adjusted so thatthe cavity mode does not lie on the oscillation frequency, the oscillation is enhancedas shown in Fig. 5.15(b). In Fig. 5.16 the oscillation frequencies are shown on anexpanded impedance Smith chart. These plots are an expanded view of the curvesshown in Fig. 5.14. From Fig. 5.16(a) we observe that the oscillator in the opencavity resonator is oscillating at the beginning and end of the resonant mode whereasin Fig. 5.16(b) the oscillation is not near a resonant mode but instead has a better

84

1 20.5

j

-j

S/C O/C

0.5j

-2j

2j

-0.5j

4 GHz

10 GHz 50 MHz7.27 GHz

18.57 GHz26.05 GHz

Figure 5.13: IMPATT diode and patch antenna impedance: solid line, negativeimpedance measurement of the IMPATT diode; dashed line, simulated driving pointimpedance of the center fed patch antenna in half space.

1 20.5

j

-j

S/C O/C

0.5j

-2j

2j

-0.5j

Figure 5.14: Simulated driving point impedance of the patch antenna in the opencavity resonator for the TEM0,1,35 and TEM1,0,35 modes: frequency range from8.62175 GHz to 8.69 GHz.

85

impedance match in the open cavity resonator than in half space.

5.3 Lens System

In this section measurements and simulations were performed for grid arrays in freespace and in the lens system shown in Fig. 3.2. The lens system consisted of twoconvex lenses made of Rexolite 1422 (εr = 2.56) with a diameter of 45.72 cm, focallength of 58.74 cm, and were spaced at twice the focal length. Field profiles for thelens system are presented along with the driving point reflection coefficients of severalgrid arrays. The driving point reflection coefficients are found in the gaps of the gridarrays representing what the active devices see. The simulations for the grid arraysuse delta-gap voltage generators located in the gaps of interest with the other gapsleft opened or shorted. A description of how the measurements were performed forthe grid arrays is also presented.

5.3.1 Electric Field Profiles

Electric field profiles of the convex lens were measured using an X-band horn as thesource located 85 cm from the lens and a small probe on the other side of the lens forreceiving the y-directed electric field as shown in Fig. 5.17 [132]. The horn aperturedimensions were 19.5 cm along the x direction and 14.3 cm along the y direction withthe TE10 mode being excited. Figs. 5.18 and 5.19 show the normalized electric fieldalong the x-plane (y = 0) at z = 0 (beam waist) and at z = 55 cm, respectively, at afrequency of 10 GHz. The beam spot size is 11 cm and 17 cm at z = 0 and z = 55,respectively. The numerical results were obtained by using the modal component ofthe Green’s function (3.53) in expression (3.14) with the current density representedas a point source. Fig. 5.20 shows the field distribution along the z-axis at x = 0 andy = 0 at 10 GHz.

5.3.2 Measurement and Simulation Techniques

The driving point reflection coefficient measurements were done using the techniqueshown in Fig. 5.21(a). Here we have a semi-rigid coaxial cable with the center con-ductor extended and a piece of wire conductor soldered to the outer conductor. Thecenter conductor is soldered to one side of the gap and the outer wire conductor issoldered to the other side of the gap. It seems as though this transition from theunbalanced coaxial line to the balanced twin lead line would result in a large discon-

86

-80

-75

-70

-65

-60

-55

-50

-45

-40

8.60 8.62 8.64 8.66 8.68 8.70

dB

FREQUENCY (GHz)

(a)

-80

-75

-70

-65

-60

-55

-50

-45

-40

8.60 8.62 8.64 8.66 8.68 8.70

dB

FREQUENCY (GHz)(b)

Figure 5.15: IMPATT diode oscillator: (a) cavity spacing D = 61.25 cm; (b) cavityspacing D = 61.41 cm: solid line, measured oscillation in half space; dashed line,measured oscillation in the open cavity resonator; dotted line, simulated scaled drivingpoint reflection coefficient magnitude of the patch antenna.

87

8.648 GHz

8.665 GHz

8.6435 GHz

(a)

8.648 GHz

8.65125 GHz

(b)

Figure 5.16: Oscillation frequencies on an expanded impedance Smith chart: (a)cavity spacing D = 61.25 cm; (b) cavity spacing D = 61.41 cm: solid line, oscillatorin the open cavity resonator; dashed line, oscillator in half space. Note: this is anenlarged view of Fig. 5.14.

88

z

y

x

HORNAPERTURE

FIRSTLENS

SECONDLENS

INPUTPOWER

PROBEMEASUREMENT

Figure 5.17: Configuration for measuring the electric field intensity in the quasi-optical lens system.

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

NO

RM

AL

IZE

D |E

y|

X POSITION (cm)

Figure 5.18: Field distribution at the beam waist (z = 0): solid line, simulation;points, measurement.

89

0

0.2

0.4

0.6

0.8

1

-10 -5 0 5 10

NO

RM

AL

IZE

D |E

y|

X POSITION (cm)

Figure 5.19: Field distribution away from the beam waist (z=55 cm): solid line,simulation; points, measurement.

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 10 20 30 40 50 60

NO

RM

AL

IZE

D |E

y|

Z POSITION (cm)

Figure 5.20: Field distribution along the z-axis: solid line, simulation; dashed line,measurement.

90

tinuity but measurements show that this is not the case. The calibrated referenceplane is located some distance down the line from the transition (see Fig. 5.21(a))and the measurements were deembedded taking into account the length of the linefrom the reference plane to the gap and the attenuation loss of the line over thisdistance. We neglected the discontinuity from the transition.

The model of this measurement technique is shown in Fig. 5.21(b) wherea delta-gap voltage generator is placed between the gap similar to a dipole antenna.The driving point impedance is simulated by placing a delta-gap voltage generatoracross the gap of interest where the current is computed through the gap using themethod of moments as shown in Fig. 5.21(c). The driving point impedance is thenfound using the technique in Section 4.5.1 from Chapter 4.

− +Vg

Ig

S11

(a)

(b)

(c)

Figure 5.21: Driving point reflection coefficient: (a) measurement technique; (b)delta-gap model; (c) moment method simulation technique.

One other important fact about the measurement probe that was used(see Fig. 5.21(a)) needs to be discussed. Going from the unbalanced coaxial lineto the balanced twin lead produced an unwanted standing wave of current on theouter conductor of the coaxial line. In order to remove this standing wave, absorbingmaterial was placed around the outer conductor of the coaxial line. Results for themeasurement probe shorted at the twin lead side is shown in Fig. 5.22, with and

91

without absorbing material around the coaxial line. When using the absorber (solidline) the response is smooth but without the absorber (dashed line) a ripple in theresponse is observed due to the standing wave of current on the outer conductor. Thisripple effect is greatly enhanced when taking measurements of grid arrays withoutusing the absorber around the coaxial line. The probe was shorted by soldering thetwin lead lines together at the end of the coaxial line. The measurement data fromthe shorted probe also gives information needed for deembedding. First we can findthe electrical length of the coaxial line with the measured phase data and the loss ofthe coaxial line with the measured magnitude data.

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

Figure 5.22: Reflection coefficient magnitude for the shorted measurement probe:solid line, with absorber around coaxial probe; dashed line, without absorber.

This measurement technique works well when measuring symmetricalstructures such as the ones presented in the following sections. For symmetricalstructures it does not matter on the polarity of the probe when it is attached but fornon symmetrical structures this is not the case. For example if the structure beingmeasured is in the corner gap of a grid array (see Fig. 5.26(a)), the polarity of theprobe matters significantly. The inner conductor should be soldered to the longestline of the two and the outer conductor to the shortest. If the polarity is reversedthe measurements do not have any correlation with the moment method simulations.With the correct polarity the measurements agree well with simulations but not aswell as the symmetrical cases. The measurement errors for non symmetrical struc-tures are greater than for the symmetrical ones but are good enough to validate themoment method simulations.

92

5.3.3 Extended Unit Cell

Fig. 5.23 shows the driving point reflection coefficient magnitude and phase of theextended unit cell (see Fig. 5.26(a)) in free space. The extended unit cell was ofdimension 93.8 mm × 93.8 mm with a gap spacing of 9.8 mm and a line width of6.35 mm. The unit cell was extended in order to have the same resonant lengths asthe 3 × 3 and 5 × 5 grid arrays with open gaps in the following sections.

The next extended unit cell analyzed is shown in Fig. 5.24 along with thedimensions of the structure. This extended unit cell is placed in the lens system atz = 0 on a dielectric substrate with εr = 2.56 and thickness 9.5 mm. Fig. 5.25 showsthe driving point reflection coefficient magnitude and phase at the node x = 0 andy = 0. In both unit cell examples there is a good correlation between simulations andmeasurements.

In many quasi-optical designs today, a MMIC would be placed in the gapof the unit cell. Most MMICs require an input and output impedance of 50 Ω. In Fig.5.23 a 50 Ω impedance occurs around 1.4 GHz. This is were the reflection coefficientmagnitude is closest to zero and the reflection coefficient phase is zero. This frequencyis also where the maximum radiation occurs making it ideal for receiving the RF signalwith the horizontal leads (input to MMIC) and transmitting the RF signal with thevertical leads (amplified output of MMIC). Also for the unit cell there is no couplingbetween the input and output ports. This is true only for symmetrical unit cells. InFig. 5.25 the 50 Ω impedance occurs around 2.1 GHz.

5.3.4 3 × 3 Grid Array

Measurements and simulations were performed in free space for the 3× 3 grid arraysshown in Fig. 5.26. The grid arrays consist of 9 unit cells where each unit cellwas of dimension 51.8 mm × 51.8 mm with the metallic grid lines having a lengthof L = 42 mm and a width of W = 6.35 mm. The gap spacing where the activedevice would be was 9.8 mm. Figs. 5.27 and 5.28 show the driving point reflectioncoefficient magnitude and phase in the center gap for the 3×3 grid array with openedand shorted gaps, respectively. The resonant frequencies are quite different for theopened and shorted cases. This is expected because of the change in the length ofthe dipole being measured in the grid (see Fig. 5.26).

To illustrate edge effects of finite grids, simulations and measurements forthe grid shown in Fig. 5.26(b) were performed for the corner gap with all the othergaps in the grid shorted. The driving point reflection coefficient magnitude is shownin Fig. 5.29 for the corner gap. The measurements are less accurate for the edge and

93

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

(a)

-150

-100

-50

0

50

100

0 1 2 3 4 5 6

PH

AS

E S

11

FREQUENCY (GHz)

(b)

Figure 5.23: Driving point reflection coefficient: (a) magnitude; (b) phase; of theextended unit cell: solid line, simulation; dashed line, measurement.

94

-30

-20

-10

0

10

20

30

-30 -20 -10 0 10 20 30

Y (

mm

)

X (mm)

Figure 5.24: An extended unit cell along with the cell subdivision used.

corner gaps due to the symmetry lost in the structure since a balun is not used. Theresults indicate that the input impedance of edge and corner gaps differ from that ofthe middle gap due to the finite extent of the grid. This variation as well as the directcoupling between cells is not incorporated in unit cell-based modeling of quasi-opticalsystems [107]. The unit cell approach assumes the grid array to be infinitely periodic.With this assumption the grid structure is modeled by only considering the unit cellwith magnetic and electric wall boundary conditions applied to the outer edges of theunit cell to emulate an infinite grid array. Therefor the coupling that is considered isa result of these boundary conditions and does not account for direct coupling fromnearby unit cells.

The next 3 × 3 grid array analyzed is shown in Fig. 5.30 along with thedimensions of the structure. This 3×3 grid array is placed in the lens system at z = 0on a dielectric substrate with εr = 2.56 and thickness 9.5 mm. Fig. 5.31 shows thedriving point reflection coefficient magnitude and phase at the node x = 0 and y = 0.In both 3 × 3 grid arrays, good agreement between simulation and measurements

is observed. The simulations for the grids in free space have better agreement withmeasurements than the grids on the dielectrics placed in the lens system. There couldbe two reasons for this. First the dielectric used was much thicker than dielectricsthat are used for general printed circuit boards. Thick dielectrics are more difficult tosimulate than thin dielectrics. This probably accounts for the weaker coupling fromthe simulation that occurs around 3.9 GHz in Fig. 5.31(a). The second source of

95

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

(a)

-150

-100

-50

0

50

100

150

0 2 4 6 8 10

PH

AS

E S

11

FREQUENCY (GHz)

(b)

Figure 5.25: Driving point reflection coefficient: (a) magnitude; (b) phase; of theextended unit cell: solid line, simulation; dashed line, measurement.

96

EXTENDEDUNIT CELL

UNIT CELL

W

L

Z in

L / 2

(a)

W

L

Z in

L / 2

(b)

Figure 5.26: A 3 × 3 grid array with the driving point impedance being measured inthe middle gap: (a) other gaps opened; (b) other gaps shorted.

97

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

(a)

-150

-100

-50

0

50

100

0 1 2 3 4 5 6

PH

AS

E S

11

FREQUENCY (GHz)

(b)

Figure 5.27: Driving point reflection coefficient: (a) magnitude; (b) phase; of the 3×3opened grid: solid line, simulation; dashed line, measurement.

98

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

(a)

-200

-150

-100

-50

0

50

100

150

200

0 1 2 3 4 5 6

PH

AS

E S

11

FREQUENCY (GHz)

(b)

Figure 5.28: Driving point reflection coefficient: (a) magnitude; (b) phase; of the 3×3shorted grid: solid line, simulation; dashed line, measurement.

99

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5 6

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

Figure 5.29: Driving point reflection coefficient magnitude in the corner gap of the3 × 3 shorted grid: solid line, simulation; dashed line, measurement.

-50

-40

-30

-20

-10

0

10

20

30

40

50

-50 -40 -30 -20 -10 0 10 20 30 40 50

Y (

mm

)

X (mm)

Figure 5.30: A 3 × 3 quasi-optical grid along with the cell subdivision used.

100

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

(a)

-150

-100

-50

0

50

100

150

0 2 4 6 8 10

PH

AS

E S

11

FREQUENCY (GHz)

(b)

Figure 5.31: Driving point reflection coefficient: (a) magnitude; (b) phase; of the 3×3grid: solid line, simulation; dashed line, measurement.

101

error may be in the measurement. Since the measurements were taken in the lenssystem, it was much more difficult to reach the grid array. A long semi-rigid coaxialcable had to be used in order to get to the grid and in doing so may have perturbedsome of the fields in the system. One final note is that the mutual coupling from theother elements in the grid is noticed which is neglected in unit cell models.

5.3.5 5 × 5 Grid Array

Measurements and simulations were performed for the 5× 5 grid array shown in Fig.5.30 placed in the lens system at z = 0 on a dielectric substrate with εr = 2.56 andthickness 9.5 mm. The dimensions of the grid are shown in Fig. 5.30 and are thesame as the 3 × 3 grid array with the exception of more unit cells. Fig. 5.33 showsthe driving point reflection coefficient magnitude and phase at the node x = 0 andy = 0. Once again the correlation between simulation and measurement is good.This result is similar to the 3 × 3 grid array result in Fig. 5.31 but the 5 × 5 gridarray shows stronger mutual coupling due to the added unit cells. The same sourcesof errors that were presented for the 3 × 3 grid array should be considered here.The simulations were efficient requiring 50 seconds per frequency point and 5 Mb ofmemory for simulation of the 5 × 5 grid array.

-80

-60

-40

-20

0

20

40

60

80

-80 -60 -40 -20 0 20 40 60 80

Y (

mm

)

X (mm)

Figure 5.32: A 5 × 5 quasi-optical grid along with the cell subdivision used.

102

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

(a)

-150

-100

-50

0

50

100

150

0 2 4 6 8 10

PH

AS

E S

11

FREQUENCY (GHz)

(b)

Figure 5.33: Driving point reflection coefficient: (a) magnitude; (b) phase; of the 5×5grid: solid line, simulation; dashed line, measurement.

103

5.4 Multiport Parameters

In this section the multiport parameters of of several antenna arrays are presented.These port parameters are computed using the method presented in Section 4.7. Theport parameters represent the coupling between the ports in the array. In order for theport parameters to be used with a circuit simulator, the port admittance matrix mustbe transformed to a nodal admittance matrix. Verification of multiport networks withmeasurements is very difficult. Measurements for a network consisting of more thantwo-ports is beyond the scope of this dissertation. In the following sections two-port,four-port, and nine-port examples are presented.

5.4.1 Two-Port Structure

Consider the two-port dipole structure shown in Fig. 5.34(a). Here the referenceterminal is defined to be in the center of the gap as shown in Fig. 5.34(b). In theMoM simulation we consider two delta-gap voltage generators, V1 and V2, in series.The admittance (Y) parameters y11 and y21 are computed by shorting out port 2

+ -

V1

(b)

+ -

Reference Plane

(e)

(d)

V2- +

V1

(c)

V2+-

(a)

Port 2Port 1

Figure 5.34: A two-port structure: (a) physical layout; (b) MoM modeling scheme;(c) V2 shorted; (d) V1 shorted; (e) measurement.

104

(V2 = 0) as shown in Fig. 5.34(c). The other two parameters, y12 and y22, are donethe same way by shorting out port 1 (V1 = 0) as shown in Fig. 5.34(d). Note that theproblem does not have to be solved twice but instead only once. The moment matrixis the same for both cases and all that needs to be modified is the voltage vectordepending on which source is used. We can get the Y-parameters directly from theinverted moment matrix in which we then can convert to scattering (S) parametersto compare with measured S-parameters.

In order to verify the MoM simulation for this structure, a two-port mea-surement was made. Referring to Fig. 5.34(e), the outer conductors of two semi-rigidcoaxial lines were soldered together. The inner conductors were then extended andsoldered to each terminal of the structure. An HP 8510C network analyzer was usedfor the two-port measurement with the calibrated reference plane located just beforethe coaxial probe as shown in Fig. 5.34(e). The measurements were then deembeddedmoving the reference plane to the plane of the two-port structure. In Fig. 5.35 theMoM simulation versus the measurement data is compared for the two-port structureshown in Fig. 5.34(a). Note that this is a passive structure (S12 = S21) and also asymmetrical structure (S11 = S22).

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

MA

GN

ITU

DE

FREQUENCY (GHz)

S21

S11

Figure 5.35: Magnitude of S11 and S21 for the two-port structure: solid line, simula-tion; dashed line, measurement.

105

5.4.2 Four-Port Structure

The four-port parameters of the structure shown in Fig. 5.36 were computed andthen converted to nodal admittance parameters. Instead of leaving the gaps open,metal was placed in the gap regions which could for example represent the groundplane of MMIC device that would be placed in the gap. Simulations were performedfor the four-port structure with a width of 6 mm, gap spacing of 5 mm, and a totallength of 100 mm. The structure is symmetrical and a reciprocal four-port network.

2 3 41

+ V1 - + V2 - + V3 - + V4 -

Figure 5.36: A four-port structure with metal in the gap regions

The input admittance between port 1 and 2 with ports 3 and 4 shorted(V3 = V4 = 0) was computed from the nodal admittance parameters and comparedwith measurements. The results are presented in Fig. 5.37 for the driving pointreflection coefficient magnitude. Good agreement between simulation and measure-ments is observed with the exception around 4 GHz to 6 GHz. This disagreementmay be due to error in the measurements. As discussed in Section 5.3.2 the error inthe measurement is due to the structure not being symmetrical with respect to themeasurement location.

The port parameters have the form of

[Y ] [V ] = [I] (5.1)

which are then modified to get the nodal parameters

[Y ′] [V ′] = [I] (5.2)

which are represented by the primed notation. The nodal parameters from (5.2) canthen be directly interfaced with microwave circuit simulators. A common referencepoint located in the center of the four-port structure was assumed in the computationof the nodal parameters. From the nodal parameters, the input admittance wascomputed as

yin =I

V ′1 − V ′

2

(5.3)

where V ′1 = V1 + V2 and V ′

2 = V2. The current I is the current flowing through thecenter of the gap and is determined from I1 and I2 using the overlapping sinusoidalbasis functions. Note that for lager arrays, the common reference point would notnecessarily be at the same location as in this example.

106

0

0.2

0.4

0.6

0.8

1

1 2 3 4 5 6 7 8 9

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.37: Driving point reflection coefficient magnitude of the four-port structurewith terminals 3 and 4 shorted: solid line, simulation; dashed line, measurement.

5.4.3 3 × 3 Patch Antenna Array

The 3 × 3 patch antenna array shown in Fig. 5.38 is represented here as a nine-port network. The patch antenna array is centered in the quasi-optical open cavityresonator (see Fig. 2.1) having the same cavity dimensions from Section 5.2. Thenine unit cells in the array had a unit cell dimension of 20 mm × 20 mm with a15 mm × 5 mm patch antenna centered in the middle. The patch antenna arraywas placed 1 mm above the planar reflector with the spherical reflector spaced at612.7 mm. For the simulations the patch antennas were divided into 6 cells withdelta-gap voltage generators placed in the centers. Only the TEM0,0,35, TEM0,1,35,and TEM1,0,35 modes were considered.

For all of the results presented in this section, the order in which themodes appear in the plots is as follows: the TEM0,0,35 mode first, the TEM0,1,35 modesecond, and the TEM1,0,35 last. The driving point reflection coefficient magnitudesfor patch antennas 1 (S11) and 5 (S55) are shown in Fig. 5.39. From this plot we seethat patch antenna 1 couples into all three modes but patch antenna 5 only couplesinto the TEM0,0,35 mode. Patch antenna 5 is located in the center of the open cavityresonator so it should couple in the TEM0,0,35 mode but not the other two modesbecause the other modes have a null in the center of the cavity. In Fig. 5.40 thecoupling between patch antennas 1 and 2 (S12) and patch antennas 1 and 4 (S14) is

107

1 2 3

654

7 8 9

Figure 5.38: A nine-port 3 × 3 patch antenna array in the open cavity resonator.

shown. Here S12 couples into the TEM0,0,35 and TEM0,1,35 modes whereas S14 couplesinto the TEM0,0,35 and TEM1,0,35 modes. In Fig. 5.41 coupling from patch antennas1 and 5 (S15) and patch antennas 1 and 9 (S19) is shown. S15 only has coupling in theTEM0,0,35 mode for the same reasons discussed previously but S19 has coupling in allthree modes. Even though patch antennas 1 and 9 have the furthest separation, thecoupling (S19) level is equivalent to all nearby antennas. From this we can say that allof the coupling is through the resonant cavity modes and not from direct radiation.The direct radiation is negligible because the patch antennas are well below resonantlengths. Also a simulation was done without the open cavity resonator and thecoupling between the elements (direct radiation) was very week over this frequencyrange. Coupling due to direct radiation will not show up until the antenna lengthsare close to resonance. From these results we can say that patch antennas 1, 3, 7, and9 couple into all three modes, patch antennas 2 and 8 couple into the TEM0,0,35 andTEM0,1,35 modes, patch antennas 4 and 6 couple into the TEM0,0,35 and TEM1,0,35

modes, and patch antenna 1 couples into only the TEM0,0,35 mode.

5.4.4 3 × 3 Grid Array

The coupling from unit cells in a grid array were computed for the 3 × 3 grid arrayshown in Fig. 5.42. Here each unit cell had a dimension of 34 mm × 34 mm and thewidth of the grid lines were 2 mm and the gap spacing was 8 mm. The lens systemdimensions were the same as in Section 5.3. Fig. 5.43 shows S11 and S55 of the grid

108

0.7

0.75

0.8

0.85

0.9

0.95

1

8.6 8.61 8.62 8.63 8.64 8.65 8.66 8.67

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.39: Simulated scattering parameters of the 3× 3 patch antenna array: solidline, S11; dashed line, S55.

0

0.05

0.1

0.15

0.2

0.25

8.6 8.61 8.62 8.63 8.64 8.65 8.66 8.67

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.40: Simulated scattering parameters of the 3× 3 patch antenna array: solidline, S12; dashed line, S14.

109

0

0.05

0.1

0.15

0.2

0.25

8.6 8.61 8.62 8.63 8.64 8.65 8.66 8.67

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.41: Simulated scattering parameters of the 3× 3 patch antenna array: solidline, S15; dashed line, S19.

array. From this we observe that there is a significant difference between the centerand corner of the grid. These edge effects are not considered when using the unitcell approach. Also we observed that S44 was close to S11 and S22 was close to S55

because they both have similar edge effects. In Fig. 5.44 coupling from cell 1 withcells 2, 3 and 4 is presented. Cell 1 has stronger coupling with the cells in the samerow (2 and 3) than cell 4. Also the coupling gets weaker as the distance separatingthe cells gets larger. Note that this coupling is due to direct radiation unlike the opencavity resonator case where all the coupling was due to resonant cavity modes. InFig. 5.45 coupling from cell 5 with cells 1, 2 and 4 is shown. Coupling to cell 5 isstrongest when cell 5 resonates (see S55 in Fig. 5.43) and the same is true for cell 1at resonant frequencies.

For these simulations the unit cell was divided into 10 cells with 6 cellsalong the horizontal leads and 4 cells along the vertical leads. Delta-gap voltagegenerators were place in the gaps of the horizontal leads as shown in Fig. 5.42. Thecoupling coefficients presented here can be either for the input or output since the unitcell is symmetrical. The coupling between the input and output ports is neglectedsince the input and output electric fields are orthogonal to each other.

110

1 2 3

654

7 8 9

Figure 5.42: A nine-port 3 × 3 grid array in the lens system.

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 1 2 3 4 5 6 7 8 9 10

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.43: Simulated scattering parameters of the 3 × 3 grid array: solid line, S11;dashed line, S55.

111

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3 4 5 6 7 8 9 10

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.44: Simulated scattering parameters of the 3 × 3 grid array: solid line, S12;dashed line, S13; dotted line, S14.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 1 2 3 4 5 6 7 8 9 10

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.45: Simulated scattering parameters of the 3 × 3 grid array: solid line, S15;dashed line, S25; dotted line, S45.

112

5.5 Finite Grid Arrays Versus Unit Cell Approach

The purpose of this section is to illustrate the differences between the unit cell modeland a finite grid array. In previous sections where finite grid arrays were analyzedwe saw the effects of finite grids including edge effects and coupling by nearby unitcells. A 10 × 10 grid is analyzed here where each unit cell is treated as a one-port.Fig. 5.46 shows the port numbering scheme along with the unit cell geometry. Theunit cell had the following dimensions, L1 = L2 = 24 mm, W = 2 mm, and the gapspacing was 8 mm with a delta-gap generator placed between the horizontal leads.Each unit cell was divided into 10 cells with the horizontal leads having 6 cells andthe vertical leads having 4 cells. The 10 × 10 grid array is represented as reciprocal100-port network. Due to the large quantity of ports only a selected few are presentedeven though all of the 100 ports were considered in all simulations in this section.

L

L

1

2

UNIT CELL

W

10099

1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 16 17 18 19 20

21 22 23 24 25 26 27 28 29 30

31 32 33 34 35 36 37 38 39 40

41 42 43 44 45 46 47 48 49 50

51 52 53 54 55 56 57 58 59 60

61 62 63 64 65 66 67 68 69 70

71 72 73 74 75 76 77 78 79 80

81 82 83 84 85 86 87 88 89 90

91 92 93 94 95 96 97 98

10 x 10 GRID ARRAY

Figure 5.46: Numbering scheme for the 10×10 grid array consisting of 100 unit cells.

In Fig. 5.47 the difference between the unit cell and finite grid array ispresented for different locations in the grid. The unit cell was simulated by onlyconsidering the unit cell whereas the 10× 10 grid array was computed with all of theunit cells being driven with a delta-gap voltage generator simultaneously. The drivingpoint reflection coefficients for a middle unit cell in the grid (S45,45) and a corner unitcell (S1,1) are compared to the unit cell. Here they start to very significantly afterthe first resonance. Cells located in the center of the grid had very similar responses

113

and slowly changed while getting closer to the edge of the grid.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.47: Driving point reflection coefficient of the 10 × 10 grid: solid line, S1,1;dashed line, S45,45; dotted line, unit cell.

The coupling coefficients were also computed for the 10 × 10 grid array.The main difference in these simulations compared to the previous unit cell simula-tions is that all of the unit cells are not sourced simultaneously but instead only onesource is used at a time with the others shorted. This is the same method used inSection 5.4.1 to compute the Y parameters for the two-port structure. The effect ofmoving from the center of the grid to the corner edge of the grid is presented in Fig.5.48. Here we see that there is quite a significant difference from the middle of thegrid and the outer edges. Coupling coefficients for unit cells around the center areshown in Fig. 5.49 and coupling coefficients for unit cells along the edges are shownin Fig. 5.50. The coupling is due to direct radiation because the grid was simulatedin free space with no quasi-optical components.

The simulation for the 10 × 10 grid array was quite computer intensiverequiring approximately 33 Mb of memory to run. The moment matrix size was880 × 880 taking about 15 minutes to fill the matrix and also about 15 minutes forsolving the matrix using standard LU decomposition. Therefore the complete solutiontook about 30 minutes per frequency. For the simulations presented in this section,a 101 frequency points were computed taking around 50 hours. With the size of theproblem and the total number of points taken, data storage required 45 Mb of diskspace to store a 100 port data file with 101 frequency points.

114

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

0 1 2 3 4 5 6 7 8 9 10

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.48: Simulated scattering parameters of the 10 × 10 grid: solid line, S45,45;dashed line, S23,23; dotted line, S1,1.

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0 1 2 3 4 5 6 7 8 9 10

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.49: Simulated scattering parameters of the 10 × 10 grid: solid line, S45,46;dashed line, S45,43; dotted line, S45,41.

115

0

0.05

0.1

0.15

0.2

0.25

0.3

0 1 2 3 4 5 6 7 8 9 10

MA

GN

ITU

DE

FREQUENCY (GHz)

Figure 5.50: Simulated scattering parameters of the 10 × 10 grid: solid line, S1,2;dashed line, S1,4; dotted line, S1,6.

5.6 Tapered Antenna

In [80, 96] tapered antennas, also referred to as Vivaldi antennas, are used in two-dimensional quasi-optical power combining oscillators and amplifiers (see Figs. 2.4and 2.5). The tapered antenna is shown in Fig. 5.51. Here the actual tapered antennais represented by the dashed line. For the moment method simulation rectangularcells were used with a staircase approximation to best fit the actual tapered antennageometry as shown in Fig. 5.51. In the actual antenna geometry there is a verysmall gap separating the two tapered sections but for the moment method simula-tion this gap did not exist. Even though the metal from the two tapered sectionswere connected in the moment method simulation, no current was allowed to flow be-tween the two tapered sections with the exception of the delta-gap voltage generatorrepresented as a diamond in Fig. 5.51. The driving point reflection coefficient wasmeasured and simulated at the diamond point shown in Fig. 5.51. The simulatedand measured driving point reflection coefficient magnitude is shown in Fig. 5.52 fora tapered antenna on a dielectric substrate with εr = 10.5 and a substrate thicknessof 0.635 mm. The agreement between the simulation and measurement is excellentfor such a crude approximation for the actual tapered geometry.

116

0

5

10

-5

-10

0 5 10 15 20 25X (mm)

Y (

mm

)

Figure 5.51: A tapered antenna along with the cell subdivision used.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14

MA

GN

ITU

DE

S11

FREQUENCY (GHz)

Figure 5.52: Driving point reflection coefficient magnitude of the tapered antenna:solid line, simulation; dashed line, measurement.

Chapter 6

Conclusions and Future Research

6.1 Conclusions

A full-wave moment method simulator has been developed for the analysis of quasi-optical systems. This technique uses a unique dyadic Green’s function which is derivedby separately considering paraxial and nonparaxial fields and was used to develop aGreen’s function for the grid lens system. The paraxial fields (quasi-optical modes)are largely responsible for distant interactions in the quasi-optical system and the cor-rected open space (nonmodal) interactions are responsible for near neighbor coupling.This form of the dyadic Green’s function is particularly convenient for quasi-opticalsystems because of its relative ease of development. With the Green’s function sep-arated into two parts, it is most efficient to compute the moment matrix elementsusing a combination of spatial and spectral domain techniques. The spatial domainanalysis is used to compute the quasi-optical mode contributions whereas the openspace interactions are computed with the spectral domain analysis. With the spec-tral domain analysis, the singularities that occur in the open space components caneasily be handled. Two types of quasi-optical systems were considered: the open cav-ity resonator and the grid lens system, where the radiating elements in each systemwere of finite size making no unit cell approximations. Structures analyzed in thesequasi-optical systems included inverted L antennas, patch antennas, grid arrays, andtapered antennas where the driving point impedance and multiport parameters werecomputed. Other results presented include field profiles of the lens system. As averification of the moment method, simulated results have been shown to comparefavorably with measurements.

Convergence issues of the electromagnetic simulator were also investi-gated. The moment method showed convergence when the structures were divided

117

118

into 10 to 20 cells per wavelength. In most cases 10 cells per wavelength was sufficient.In addition to convergence, the condition number of the moment matrix was com-puted indicating that the matrix was well-conditioned for the simulations presentedin this dissertation.

The electromagnetic simulator developed in this dissertation has severalsignificances and limitations. First the significance of the modeling work are as fol-lows:

1. Finite sized arrays are considered. There is no need to make the simplifyingassumption of an infinite periodic grid of identical unit cells as is required in allother quasi-optical system modeling approaches. By considering finite arrays,the coupling from every unit cell in the array is considered.

2. Quasi-optical components (e.g. lenses and reflectors) are considered in the for-mulation. In standard electromagnetic simulators, quasi-optical componentsare not considered including the unit cell approaches. The quasi-optical compo-nents are incorporated into the formulation by the unique quasi-optical Green’sfunctions. Other quasi-optical elements can easily be included by developingnew Green’s functions which can be easily interchanged in the moment methodformulation.

3. The simulator is capable of computing results from DC to any frequency. Thisis an important requirement in computer aided design. All results presentedhere converge at DC.

4. The end result of the simulator is the computation of the multiport parametersof the quasi-optical array. The coupling coefficients are directly found fromthese port parameters. For integration of the multiport parameters in transientand harmonic balance analysis, the port parameters are converted to nodaladmittance parameters.

Listed next are the limitations of the electromagnetic simulator:

1. In this formulation the gridding of the antenna element must consist of equalsized cells. Note that this is only true for cells on the same antenna. Forantennas that do not touch one another, the cell sizes may be different foreach antenna. This equal size cell requirement is due to the spectral domainformulation. A solution to this problem is addressed in Appendix B.

2. The structures considered here can only have one dielectric layer with or withouta ground plane. To include multilayered dielectrics, the complete quasi-opticalGreen’s function would need to be rederived.

119

3. Losses in the antenna conductors and dielectric substrate are not considered.However losses in the reflector and lenses are considered. The incorporation oflosses in the dielectric layer could easily be incorporated by considering the losstangent. This results in a complex dielectric constant which would need to beadded to the formulation. The conductor losses are more difficult to include.As it stands the conductors of the antenna elements are assumed to have zerothickness. Conductor losses will be significant at millimeter wave frequencies.

The techniques presented in this dissertation will aid in the design andanalysis of quasi-optical systems by accurately predicting the driving point impedancesand port admittances of the passive elements in the quasi-optical system. From asystem development point of view the design of each element in the array can beindividually optimized to achieve an optimum global solution in terms of stability,output power and efficiency.

6.2 Future Research

Several new features could be added to the full-wave moment method simulator de-veloped in this dissertation. The first and most important addition to the simulatorwould be the incorporation of unequal sized gridding for the geometry structures. Asthe simulator stands, unequal sized cells are allowed as long as the cell sizes thatdiffer are located in separate conductors that do not intersect with one another. Forexample this occurs in cross dipole grids where the the horizontal leads do not have tobe the same size as the vertical leads. But what would be nice is to be able to changethe cell size in the same conductor allowing for more versatility when modeling com-plicated structures. As an example consider the unit cell shown in Fig. 6.1 [85]. Herewe can see that this is a much more complicated structure than the ones modeledin this dissertation. The moment method simulator could accurately model this unitcell using the gridding shown in Fig. 6.2 for equal sized cells. The only problem withthis is that the computation time would be very significant because there is over 3500unknowns. If unequal sized cells could be used, the gridding size could be reduced asshown in Fig. 6.3 where there are only 572 unknowns. Further reduction may alsobe possible by somehow simplifying the bias lines.

The formulation for unequal sized cells has already been formulated and isgiven in Appendix B. This is for the nonquasi-optical elements only. As it stands in thecurrent moment method simulator, the quasi-optical moment matrix elements can beof unequal sized cells. The only reason that the nonquasi-optical elements need to haveequal sized cells is because they are computed in the spectral domain. The formulationpresented in Appendix B uses a mixed potential integral equation formulation in the

120

1 2

3

4

1 23

4

Input Lead

Output Lead

Collector Bias Line

Emitter Bias Line

Figure 6.1: A 40 GHz grid amplifier unit cell layout.

Figure 6.2: Moment method unit cell layout with equal sized cells.

Figure 6.3: Moment method unit cell layout with unequal sized cells.

121

spatial domain. In this formulation the singularities of the self-terms can be handled.Adding this formulation to the current moment method simulator, moment matrixelements could be computed several ways depending on which method best fits forthat particular moment matrix element.

Other additions that would be attractive for the simulator are actually forthe input and output stages of the program which are currently being developed inC. The geometry of the structure would be drawn with Cadence producing a Caltechintegrated format (CIF) file that describes all of the geometry gridding includingmultiple layers. The CIF file will then be parsed and sent to the C code where it will beused to tell the FORTRAN moment method simulator what to compute. The outputresults would then be manipulated with the C code in order to get the data in the rightformat to pass on to the microwave circuit simulators such as the transient analysisor harmonic balance for analysis of the complete quasi-optical circuit including theactive devices.

To help improve the CPU time of the moment method simulator, a betterand faster technique for solving the moment matrix could be employed. CurrentlyLU decomposition is used to solve the system. Other methods such as the conjugategradient method or other projection methods would help in decreasing CPU timeallowing for more efficient analysis of electrically large structures.

One of the main issues that needs to be addressed is device field interac-tions. Currently the passive and active elements in the circuit are modeled individ-ually and then cascaded together using port networks or circuit nodal parameters.When dealing at millimeter-wave frequencies the fields produced from the solid statedevices are going to have some serious effects with fields from the passive structure.Other considerations that have not been considered are the thermal issues. In quasi-optics heat dissipation is a serious problem for such systems as the grid array in freespace where a ground plane does not exist so therefore other methods for dissipatingthe heat need to be considered. So ultimately in quasi-optical modeling we would liketo see complete system models that predict total system performance including thesystem gain, field profiles, and efficiency of the quasi-optical power combiner. Themodel would include device field interactions, thermal conditions, and all quasi-opticalelements such as polarizers, lenses, reflectors, and tuning slabs.

122

References

[1] K. J. Sleger, R. H. Abrams, Jr., and R. K. Parker, “Trends in solid-state microwaveand millimeter-wave technology,” IEEE MTT-S Newsletter, pp. 11-15, Fall 1990.

[2] J. W. Mink, “Quasi-optical power combining of solid-state millimeter-wave sources,”IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 273-279, Feb. 1986.

[3] P. L. Heron, F. K. Schwering, G. P. Monahan, J. W. Mink, and M. B. Steer, “A dyadicGreen’s function for the plano-concave quasi-optical resonator,” IEEE MicrowaveGuided Wave Lett., vol. 3, pp. 256-258, Aug. 1993.

[4] P. L. Heron, G. P. Monahan, J. W. Mink, F. K. Schwering, and M. B. Steer,“Impedance matrix of an antenna array in a quasi-optical resonator,” IEEE Trans.Microwave Theory Tech., vol. MTT-41, pp. 1816-1826, Oct. 1993.

[5] P. L. Heron, Modeling and Simulation of Coupling Structures for Quasi-Optical Sys-tems, Ph.D. Dissertation, North Carolina State University, 1993.

[6] P. F. Goldsmith, “Quasi-optical techniques at millimeter and sub-millimeter wave-lengths,” in Infrared and Millimeter Waves, K. J. Button (Ed.), New York: AcademicPress, vol. 6, pp. 277-343, 1982.

[7] R. A. York, “Quasi-optical power combining techniques,” SPIE Critical Reviews ofEmerging Technologies, 1994.

[8] Special Issue on Quasi-Optical Techniques, J. W. Mink and D. B. Rutledge (eds.),IEEE Trans. Microwave Theory Tech., vol. MTT-41, Oct. 1993.

[9] Z. B. Popovic, M. Kim, and D. B. Rutledge, “Grid oscillators,” Int. Journal Infraredand Millimeter Waves, vol. 9, pp. 1003-1010, Nov. 1988.

[10] M. Kim, J. J. Rosenberg, R. P. Smith, R. M. Weikle II, J. B. Hacker, M. P. De Lisio,and D. B. Rutledge, “A grid amplifier,” IEEE Microwave Guided Wave Lett., vol. 1,pp. 322-324, Nov. 1991.

[11] T. W. Nuteson, K. Naishadham, and R. Mittra, “Spatial interpolation of the mo-ment matrix for efficient analysis of microstrip circuits,” IEEE MTT-S InternationalSymposium Digest, pp. 971-974, June 1993.

[12] K. Naishadham, T. W. Nuteson, and H. W. Yao, “Efficient prediction of radiationfrom printed transmission-line discontinuities,” IEEE Trans. Electromagnetic Com-pat., vol. EMC-35, pp.159-169, July 1993.

[13] K. Naishadham and T. W. Nuteson, “Efficient analysis of passive microstrip ele-ments in MMICs,” Int. Journal Microwave and Millimeter-Wave Computer-AidedEng., vol. 4, pp. 219-229, July 1994.

123

[14] R. F. Harrington, Field Computation by Moment Methods, New York: Macmillan,1968.

[15] D. G. Dudley, Mathematical Foundations for Electromagnetic Theory, New York:IEEE Press, 1994.

[16] R. F. Harrington, “Matrix methods for field problems,” Proc. IEEE, vol. 55, pp. 136-149, Feb. 1967.

[17] C. A. J. Fletcher, Computational Galerkin Methods, New York: Springer-Verlag, 1984.

[18] C. Tai, Dyadic Green’s Functions in Electromagnetic Theory, Second Edition, NewYork: IEEE Press, 1994.

[19] G. A. Thiele, “Wire antennas,” in Computer Techniques for Electromagnetics, R.Mittra (ed.), Oxford: Pergamon Press Ltd., pp. 7-95, 1973.

[20] W. A. Imbriale, “Applications of the method of moments to thin-wire elements and ar-rays,” in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra (ed.),New York: Springer-Verlag, pp. 5-50, 1975.

[21] P. A. Ramsdale, “Wire antennas,” in Advances in Electronics and Electron Physics,vol. 47, New York: Academic Press, pp. 123-196, 1978.

[22] E. K. Miller and F. J. Deadrick, “Some computational aspects of thin-wire modeling,”in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra (ed.), NewYork: Springer-Verlag, pp. 89-127, 1975.

[23] R. Mittra and C. A. Klein, “Stability and convergence of moment method solutions,”in Numerical and Asymptotic Techniques in Electromagnetics, R. Mittra (ed.), NewYork: Springer-Verlag, pp. 129-163, 1975.

[24] Y. Rahmat-Samii and R. Mittra, “Integral equation solution and RCS computation ofa thin rectangular plate,” IEEE Trans. Antennas Propagat., vol. AP-22, pp. 608-610,July 1974.

[25] J. R. Mautz and R. F. Harrington, “H-field, E-field, and combined field solutions forconducting bodies of revolution,” AEU, vol. 32, pp. 157-164, Apr. 1978.

[26] E. H. Newman and D. M. Pozar, “Electromagnetic modeling of composite wire andsurface geometries,” IEEE Trans. Antennas Propagat., vol. AP-26, pp. 784-789, Nov.1978.

[27] A. W. Glisson and D. R. Wilton, “Simple and efficient numerical methods for problemsof electromagnetic radiation and scattering from surfaces,” IEEE Trans. AntennasPropagat., vol. AP-28, pp. 593-603, Sept. 1980.

[28] S. M. Rao, D. R. Wilton, and A. W. Glisson, “Electromagnetic scattering by surfacesof arbitrary shape,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 409-418, May1982.

124

[29] T. Itoh and R. Mittra, “Spectral domain approach for calculating the dispersioncharacteristics of microstrip line,” IEEE Trans. Microwave Theory Tech., vol. MTT-21, pp. 496-499, July 1973.

[30] T. Itoh, “Spectral domain immittance approach for dispersion characteristics of gener-alized printed transmission lines,” IEEE Trans. Microwave Theory Tech., vol. MTT-28, pp. 733-736, July 1980.

[31] D. M. Pozar, “Input impedance and mutual coupling of rectangular microstrip an-tennas,” IEEE Trans. Antennas Propagat., vol. AP-30, pp. 1191-1196, Nov. 1982.

[32] R. W. Jackson and D. M. Pozar, “Full-wave analysis of microstrip open-end and gapdiscontinuities,” IEEE Trans. Microwave Theory Tech., vol. MTT-33, pp. 1036-1042,Oct. 1985.

[33] D. M. Pozar, “Finite phased arrays of rectangular microstrip patches,” IEEE Trans.Antennas Propagat., vol. AP-34, pp. 658-665, Jan. 1987.

[34] N. K. Das and D. M. Pozar, “A generalized spectral-domain Green’s function for mul-tilayer dielectric substrates with application to multilayer transmission lines,” IEEETrans. Microwave Theory Tech., vol. MTT-35, pp. 326-335, Mar. 1987.

[35] R. W. Jackson, “Full-wave, finite element analysis of irregular microstrip discontinu-ities,” IEEE Trans. Microwave Theory Tech., vol. MTT-37, pp. 81-89, Jan. 1989.

[36] D. M. Pozar, “Analysis and design considerations for printed phased-array antennas,”in Handbook of Microstrip Antennas, J. R. James and P. S. Hall (eds.), London: PeterPeregrinus Ltd., pp. 693-753, 1989.

[37] A. K. Skrivervik and J. R. Mosig, “Analysis of finite phase arrays of microstrippatches,” IEEE Trans. Antennas Propagat., vol. AP-41, pp. 1105-1114, Aug. 1993.

[38] G. Pan, J. Tan, and J. D. Murphy, “Full-wave analysis of microstrip floating-linediscontinuities,” IEEE Trans. Electromagnetic Compat., vol. EMC-36, pp. 49-59, Feb.1994.

[39] J. R. Mosig and F. E. Gardiol, “A dynamic radiation model for microstrip structures,”in Advances in Electronics and Electron Physics, vol. 59, New York: Academic Press,pp. 139-237, 1982.

[40] P. B Katehi and N. G. Alexopoulos, “Frequency-dependent characteristics of mi-crostrip discontinuities in millimeter-wave integrated circuits,” IEEE Trans. Mi-crowave Theory Tech., vol. MTT-33, pp. 1029-1035, Oct. 1985.

[41] J. R. Mosig and F. E. Gardiol, “General integral equation formulation for microstripantennas and scatterers,” Proc. IEE, pt. H, vol. 132, pp. 424-432, Dec. 1985.

[42] J. R. Mosig, “Arbitrarily shaped microstrip structures and their analysis with a mixedpotential integral equation,” IEEE Trans. Microwave Theory Tech., vol. MTT-36,pp. 314-323, Feb. 1988.

125

[43] J. R. Mosig, “Integral equation technique,” in Numerical Techniques for Microwaveand Millimeter-Wave Passive Structures, T. Itoh (ed.), New York: John Wiley &Sons, pp. 133-213, 1989.

[44] K. A. Michalski and D. Zheng, “Analysis of microstrip resonators of arbitrary shape,”IEEE Trans. Microwave Theory Tech., vol. MTT-40, pp. 112-119, Jan. 1992.

[45] R. Gillard, J. Corre, M. Drissi, and J. Citerne, “A general treatment of matchedterminations using integral equations - modeling and applications,” IEEE Trans. Mi-crowave Theory Tech., vol. MTT-42, pp. 2545-2553, Dec. 1994.

[46] G. V. Eleftheriades and J. R. Mosig, “On the network characterization of planarpassive circuits using the method of moments,” IEEE Trans. Microwave Theory Tech.,vol. MTT-44, pp. 438-445, Mar. 1996.

[47] J. R. Mosig and F. E. Gardiol, “Analytical and numerical techniques in the Green’sfunction treatment of microstrip antennas and scatterers,” Proc. IEE, pt. H, vol. 130,pp. 175-182, Mar. 1983.

[48] J. R. Mosig and T. K. Sarkar, “Comparison of quasi-static and exact electromagneticfields from a horizontal electric dipole above a lossy dielectric backed by an imperfectground plane,” IEEE Trans. Microwave Theory Tech., vol. MTT-34, pp. 379-387,Apr. 1986.

[49] Y. L. Chow, J. J. Yang, D. G. Fang and G. E. Howard, “A closed-form spatial Green’sfunction for the thick microstrip substrate,” IEEE Trans. Microwave Theory Tech.,vol. MTT-39, pp. 588-592, Mar. 1991.

[50] M. I. Aksun and R. Mittra, “Derivation of closed-form Green’s functions for a generalmicrostrip geometry,” IEEE Trans. Microwave Theory Tech., vol. MTT-40, pp. 2055-2062, Nov. 1992.

[51] G. Dural and M. I. Aksun, “Closed-form Green’s functions for general sources andstratified media,” IEEE Trans. Microwave Theory Tech., vol. MTT-43, pp. 1545-1552,July 1995.

[52] D. M. Pozar, “Improved computational efficiency for the moment method solution ofprinted dipoles and patches,” Electromagnetics, vol. 3, pp. 299-309, 1983.

[53] H. Y. Yang, A. Nakatani, and J. A. Castaneda, “Efficient evaluation of spectral in-tegrals in the moment method solution of microstrip antennas and circuits,” IEEETrans. Antennas Propagat., vol. AP-38, pp. 1127-1130, July 1990.

[54] U. V. Gothelf and A. Østergaard, “Full-wave analysis of a two-slot microstrip fil-ter using a new algorithm for computation of the spectral integrals,” IEEE Trans.Microwave Theory Tech., vol. MTT-41, pp. 101-108, Jan. 1993.

[55] E. H. Newman, “Generation of wide-band data from the method of moments byinterpolating the impedance matrix,” IEEE Trans. Antennas Propagat., vol. AP-36,pp. 1820-1824, Dec. 1988.

126

[56] B. Z. Steinberg and Y. Leviatan, “On the use of wavelet expansions in the method ofmoments,” IEEE Trans. Antennas Propagat., vol. AP-41, pp. 610-619, May 1993.

[57] K. Sabetfakhri and L. B. Katehi, “Analysis of integrated millimeter-wave andsubmillimeter-wave waveguides using orthonormal wavelets expansions,” IEEE Trans.Microwave Theory Tech., vol. MTT-42, pp. 2412-2422, Dec. 1994.

[58] G. Wang and G. Pan, “Full wave analysis of microstrip floating line structures bywavelet expansion method,” IEEE Trans. Microwave Theory Tech., vol. MTT-43,pp. 131-142, Jan. 1995.

[59] K. Sabetfakhri and L. B. Katehi, “Fast wavelet analysis of 3-D dielectric structuresusing sparse matrix techniques,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 829-832, May 1995.

[60] M. Gouker, “Toward standard figures-of-merit for spatial and quasi-optical power-combined arrays,” IEEE Trans. Microwave Theory Tech., vol. MTT-43, pp. 1614-1617, July 1995.

[61] S. L. Young and K. D. Stephan, “Stabilization and power combining of planar mi-crowave oscillators with an open resonator,” IEEE MTT-S Int. Microwave Symp.Dig., pp. 185-188, June 1987.

[62] M. Nakayama, M. Hieda, T. Tanaka, and K. Mizuno, “Millimeter and submillime-ter wave quasi-optical oscillator with multi-elements,” IEEE MTT-S Int. MicrowaveSymp. Dig., pp. 1209-1212, June 1990.

[63] H. Kondo, M. Hieda, M. Nakayama, T. Tanaka, K. Osakabe, and K. Mizuno, “Mil-limeter and submillimeter wave quasi-optical oscillator with multi-elements,” IEEETrans. Microwave Theory Tech., vol. MTT-40, pp. 857-863, May 1992.

[64] M. Kiyokawa and T. Matsui, “A new quasi-optical oscillator with Gaussian outputbeam,” IEEE Microwave Guided Wave Lett., vol. 4, pp. 129-131, May 1994.

[65] M. Kiyokaw, T. Matsui, and N. Hirose, “Dielectric loaded Gaussian beam oscillatorin the 40 GHz band,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 75-78, May 1995.

[66] G. P. Monahan, Experimental Investigation of an Open Resonator Quasi-OpticalPower Combiner Using IMPATT Diodes, Ph.D. Dissertation, North Carolina StateUniversity, 1995.

[67] T. Fujii, H. Mazaki, F. Takei, J. Bae, M. Narihiro, T. Noda, H. Sakaki, and K.Mizuno, “Coherent power combining of millimeter wave resonant tunneling diodes ina quasi-optical resonator,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 919-922,June 1996.

[68] M. Sanagi, E. Yamamoto, and S. Nogi, “Axially symmetric Fabry-Perot power com-biners with active devices mounted on both the mirrors,” IEEE MTT-S Int. Mi-crowave Symp. Dig., pp. 1259-1262, June 1996.

127

[69] Z. B. Popovic, R. M. Weikle II, M. Kim, K. A. Potter, and D. B. Rutledge, “Bar-gridoscillators,” IEEE Trans. Microwave Theory Tech., vol. MTT-38, pp. 225-230, Mar.1990.

[70] Z. B. Popovic, R. M. Weikle II, M. Kim, and D. B. Rutledge, “A 100-MESFET planargrid oscillator,” IEEE Trans. Microwave Theory Tech., vol. MTT-39, pp. 193-200,Feb. 1991.

[71] R. M. Weikle II, M. Kim, J. B. Hacker, M. P. De Lisio, and D. B. Rutledge, “PlanarMESFET grid oscillators using gate feedback,” IEEE Trans. Microwave Theory Tech.,vol. MTT-40, pp. 1997-2003, Nov. 1992.

[72] T. B. Mader, S. C. Bundy, and Z. B. Popovic, “Quasi-optical array VCOs,” IEEETrans. Microwave Theory Tech., vol. MTT-41, pp. 1775-1781, Oct. 1993.

[73] J. B. Hacker, M. P. De Lisio, M. Kim, C.-M. Liu, S.-J. Li, S. W. Wedge, and D. B.Rutledge, “A 10-watt X-band grid oscillator,” IEEE MTT-S Int. Microwave Symp.Dig., pp. 823-826, May 1994.

[74] W. A. Shiroma, B. L. Shaw, and Z. B. Popovic, “A 100-transistor quadruple gridoscillator,” IEEE Microwave Guided Wave Lett., vol. 4, pp. 350-351, Oct. 1994.

[75] R. A. York and R. C. Compton, “Quasi-optical power combining using mutual syn-chronized oscillator arrays,” IEEE Trans. Microwave Theory Tech., vol. MTT-39,pp. 1000-1009, June 1991.

[76] J. Birkeland and T. Itoh, “A 16 element quasi-optical FET oscillator power com-bining array with external injection locking,” IEEE Trans. Microwave Theory Tech.,vol. MTT-40, pp. 475-481, Mar. 1992.

[77] A. Mortazawi, H. D. Folt, and T. Itoh, “A periodic second harmonic spatial powercombining oscillator,” IEEE Trans. Microwave Theory Tech., vol. MTT-40, pp. 851-856, May 1992.

[78] P. Liao and R. A. York, “A high power two-dimensional coupled-oscillator array atX-band,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 909-912, May 1995.

[79] M. Rahman, T. Ivanov, and A. Mortazawi, “An extended resonance spatial powercombining oscillator,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 1263-1266, June1996.

[80] S. Zeisberg, A. Schuenemann, G. P. Monahan, M. B. Steer, J. W. Mink, and F. K.Schwering, “Experimental investigation of a quasi-optical slab resonator,” IEEE Mi-crowave Guided Wave Lett., vol. 3, pp. 253-255, Aug. 1993.

[81] F. Poegel, S. Irrang, S. Zeisberg, A. Schuenemann, G. P. Monahan, H. Hwang, M. B.Steer, J. W. Mink, F. K. Schwering, A. Paollela, and J. Harvey, “Demonstration of anoscillating quasi-optical slab power combiner,” IEEE MTT-S Int. Microwave Symp.Dig., pp. 917-920, May 1995.

128

[82] C. Y. Chi and G. M. Rebeiz, “A quasi-optical amplifier, IEEE Microwave GuidedWave Lett., vol. 3, pp. 164-166, June 1993.

[83] M. Kim, E. A. Sovero, J. B. Hacker, M. P. De Lisio, J. C. Chiao, S. J. Li, D. R.Gagnon, J. J. Rosenberg, and D. B. Rutledge, “A 100-element HBT grid amplifier,”IEEE Trans. Microwave Theory Tech., vol. MTT-41, pp. 1762-1771, Oct. 1993.

[84] M. Kim, E. A. Sovero, J. B. Hacker, M. P. De Lisio, J. J. Rosenberg, and D. B.Rutledge, “A 6.5 GHz source using a grid amplifier with a twist reflector,” IEEETrans. Microwave Theory Tech., vol. MTT-41, pp. 1772-1774, Oct. 1993.

[85] C. Liu, E. A. Sovero, W. J. Ho, J. A. Higgins, M. P. De Lisio, and D. B. Rutledge,“Monolithic 40-GHz 670-mW HBT grid amplifiers,” IEEE MTT-S Int. MicrowaveSymp. Dig., pp. 1123-1126, June 1996.

[86] M. P. De Lisio, S. W. Duncan, D. Tu, S. Weinreb, C. Liu, and D. B. Rutledge, “A44-60 GHz monolithic pHEMT grid amplifier,” IEEE MTT-S Int. Microwave Symp.Dig., pp. 1127-1130, May 1995.

[87] A. Mortazawi and B. C. De Loach, “A nine-MESFET two-dimensional power-combining array employing an extended resonance technique,” IEEE MicrowaveGuided Wave Lett., vol. 3, pp. 214-216, July 1993.

[88] T. Ivanov and A. Mortazawi, “Two stage double layer microstrip spatial amplifiers,”IEEE MTT-S Int. Microwave Symp. Dig., pp. 589-592, May 1995.

[89] J. Schoenberg, T. Mader, B. Shaw, and Z. B. Popovic, “Quasi-optical antenna arrayamplifiers,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 605-608, May 1995.

[90] J. Hubert, J. Schoenberg, and Z. B. Popovic, “High-power hybrid quasi-optical Ka-band amplifier design,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 585-588, May1995.

[91] H. S. Tsai and R. A. York, “Quasi-optical amplifier array using direct integration ofMMICs and 50 Ω multi-slot antennas,” IEEE MTT-S Int. Microwave Symp. Dig.,pp. 593-596, May 1995.

[92] H. S. Tsai and R. A. York, “Multi-slot 50 − Ω antennas for quasi-optical circuits,”IEEE Microwave Guided Wave Lett., vol. 5, pp. 180-182, June 1995.

[93] E. A. Sovero, Y. Kwon, D. S. Deakin, A. L Sailer, and J. A. Higgins, “A PHEMTbased monolithic plane wave amplifier for 42 GHz,” IEEE MTT-S Int. MicrowaveSymp. Dig., pp. 1111-1114 May 1995.

[94] A. Alexanian, H. Tsai, and R. A. York, “Quasi-optical traveling wave amplifiers,”IEEE MTT-S Int. Microwave Symp. Dig., pp. 1115-1118, May 1995.

[95] H. Hwang, G. P. Monahan, M. B. Steer, J. W. Mink, J. Harvey, A. Paollela, andF. K. Schwering, “A dielectric slab waveguide with four planar power amplifiers,”IEEE MTT-S Int. Microwave Symp. Dig., pp. 921-924, May 1995.

129

[96] H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paollela,“A Quasi-Optical Dielectric Slab Power Combiner,” IEEE Microwave Guided WaveLett., vol. 6, pp. 73-75, Feb. 1996.

[97] H. Hwang, T. W. Nuteson, M. B. Steer, J. W. Mink, J. Harvey, and A. Paollela,“Two-dimensional quasi-optical power combining system performance and componentdesign,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 927-930, June 1996.

[98] A. R. Perkons and T. Itoh, “A 10-element active lens amplifier on a dielectric slab,”IEEE MTT-S Int. Microwave Symp. Dig., pp. 1119-1122, June 1996.

[99] C. H. Tsao and R. Mittra, “Spectral-domain analysis of frequency selective surfacescomprised of periodic arrays of crossed dipoles and Jerusalem crosses,” IEEE Trans.Antennas Propagat., vol. AP-32, pp. 478-486, May 1984.

[100] T. A. Cwik and R. Mittra, “Scattering for a periodic array of free-standing arbitraryshaped perfectly conducting or resistive patches,” IEEE Trans. Antennas Propagat.,vol. AP-35, pp. 1226-1234, Nov. 1987.

[101] C. H. Chan and R. Mittra, “On the analysis of frequency-selective surfaces usingsubdomain basis functions,” IEEE Trans. Antennas Propagat., vol. AP-38, pp. 40-50,Jan. 1990.

[102] L. B. Sjogren and N. C. Luhmann Jr., “An impedance model for the diode grid,”IEEE Microwave Guided Wave Lett., vol. 1, pp. 297-299, Oct. 1991.

[103] A. Pance and M. J. Wengler, “Microwave modeling of 2-D active grid arrays,” IEEETrans. Microwave Theory Tech., vol. MTT-41, pp. 20-28, Jan. 1993.

[104] T. Wu, “Quasi-optical grids with thin rectangular patch/aperture elements,” Int.Journal Infrared and Millimeter Waves, vol. 14, pp. 1017-1033, May 1993.

[105] S. C. Bundy, W. A. Shiroma, and Z. B. Popovic, “Design-oriented analysis of gridpower combiners,” Proc. Workshop on Millimeter-Wave Power Generation and BeamControl, pp. 197-208, Sept. 1993.

[106] S. C. Bundy and Z. B. Popovic, “Analysis of planar grid oscillators,” IEEE MTT-SInt. Microwave Symp. Dig., pp. 827-830, June 1994.

[107] S. C. Bundy and Z. B. Popovic, “A generalized analysis for grid oscillator design,”IEEE Trans. Microwave Theory Tech., vol. MTT-42, pp. 2486-2491, Dec. 1994.

[108] S. C. Bundy, Analysis and Design of Grid Oscillators, Ph.D. Dissertation, Universityof Colorado at Boulder, 1994.

[109] S. C. Bundy, W. A. Shiroma, and Z. B. Popovic, “Analysis of cascaded quasi-opticalgrids,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 601-604, May 1995.

130

[110] W. A. Shiroma, S. C. Bundy, S. Hollung, B. D. Bauernfeind, and Z. B. Popovic,“Cascaded active and passive quasi-optical grids,” IEEE Trans. Microwave TheoryTech., vol. MTT-43, pp. 2904-2909, Dec. 1995.

[111] R. M. Weikle II, Quasi-Optical Planar Grids for Microwave and Millimeter-WavePower Combining, Ph.D. Dissertation, California Institute of Technology, 1992.

[112] P. L. Heron, G. P. Monahan, J. E. Bird, M. B. Steer, F. K. Schwering, and J. W.Mink, “Circuit level modeling of quasioptical power combining open cavities,” IEEEMTT-S Int. Microwave Symp. Dig., pp. 433-436, June 1993.

[113] P. L. Heron, G. P. Monahan, F. K. Schwering, J. W. Mink, and M. B. Steer, “Mul-tiport circuit model of an antenna array in an open quasi-optical resonator,” Proc.URSI, p. 84, June 1993.

[114] M. B. Steer, “Simulation of microwave and millimeter-wave oscillators, present capa-bility and future directions,” Proc. Workshop on Integrated Nonlinear Microwave andMillimeter-wave Circuits, Oct. 1992.

[115] M. B. Steer and S. G. Skaggs, “CAD of GaAs microwave circuits: historical perspec-tive and future trends,” Korea-United States Design and Manufacturing Workshop,pp. 109-118, Nov. 1993.

[116] G. P. Monahan, P. L. Heron, M. B. Steer, J. W. Mink, and F. K. Schwering, “Modedegeneracy in quasi-optical resonators,” Microwave and Optical Technology Letters,vol 8, pp. 230-232, Apr. 1995.

[117] 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 analysisof quasi-optical structures,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 913-916,May 1995.

[118] T. W. Nuteson, G. P. Monahan, M. B. Steer, K. Naishadham, J. W. Mink, K. Ko-jucharow, and J. Harvey, “Full-wave analysis of quasi-optical structures,” IEEE Trans.Microwave Theory Tech. vol. 44, pp. 701-710, May 1996.

[119] T. W. Nuteson, M. B. Steer, K. Naishadham, J. W. Mink, and J. Harvey, “Electro-magnetic modeling of finite grid structures in quasi-optical systems,” IEEE MTT-SInt. Microwave Symp. Dig., pp. 1251-1254, June 1996.

[120] A. Alexanian, N. J. Kolias, R. C. Compton, and R. A. York, “Three-dimensionalFDTD analysis of quasi-optical arrays using Floquet boundary conditions andBerenger’s PML,” IEEE Microwave Guided Wave Lett., vol. 6, pp. 138-140, Mar.1996.

[121] L. W. Epp, R. P. Smith, “A generalized scattering matrix approach for analysis ofquasi-optical grids and de-embedding of device parameters,” IEEE Trans. MicrowaveTheory Tech. vol. 44, pp. 760-769, May 1996.

131

[122] A. R. Adams, R. D. Pollard, and C. M. Snowden, “Method of moments and timedomain analyses of waveguide-based hybrid multiple device oscillators,” IEEE MTT-S Int. Microwave Symp. Dig., pp. 1255-1258, June 1996.

[123] G. Goubau, “Beam waveguides,” Advances in Microwaves, vol. 3, New York: Aca-demic Press, 1968, pp. 67-126.

[124] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover 1972.

[125] R. F. Harrington, Time-Harmonic Electromagnetic Fields, New York: McGraw-Hill,1961.

[126] G. D. Boyd and J. P. Gordon, “Confocal multimode resonator theory for millimeterthrough optical wavelength masers,” Bell System Technical Journal, pp. 489-509, Mar.1961.

[127] R. F. Soohoo, “Nonconfocal multimode resonators for masers,” Proceedings of IEEE,pp. 70-75, Jan. 1963.

[128] G. D. Boyd and H. Kogelnik, “Generalized confocal resonator theory,” Bell SystemTechnical Journal, pp. 1347-1369, July 1962.

[129] R. E. Collin, Foundations for Microwave Engineering, 2nd Edition, McGraw Hill,1992, pp. 53-56.

[130] A. G. Fox and T. Li, “Resonant modes in a maser interferometer,” Bell System Tech-nical Journal, pp. 453-489, Mar. 1961.

[131] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, New York: JohnWiley, 1981.

[132] J. Tuovinen, T. M. Hirvonen, and A. V. Raisanen, “Near-field analysis of a thick lensand horn combination: theory and measurements,” IEEE Trans. Antennas Propagat.,vol. AP-40, pp. 613-619, June 1992.

Appendix A

Multiport Analysis Using HalfBasis Elements

A.1 Introduction

In this appendix the multiport formulation for grid arrays is presented. This for-mulation is based on Eleftheriades and Mosig’s [46] work on multiport parametersfor microstrip structures. In [46] a moment method spatial domain mixed potentialintegral equation formulation was used. A similar technique is applied here to gridarrays in free space or on an ungrounded dielectric slab using the method of moments(MoM) formulated as an electric field integral equation applying a combination of spa-tial and spectral domain techniques as was presented in Chapter 4. In this techniquea rigorous spectral domain formulation is required to handle the half basis elements.The main difference between this formulation and the one presented in [46] is that forgrid arrays a ground plane does not exist. With a ground plane the terminations onthe conductors are referenced to the ground plane but with the grid array a commonreference does not exist.

A.2 Development

Consider the unit cell of a grid shown in Fig. A.1. It is assumed that the ports canbe conveniently placed at the gap where the active device will be connected. Theobjective is to find the terminal impedances at each port of the unit cell from theport currents determined using a MoM implementation with a delta-gap voltage ap-plied to each port. The port impedance matrix of each cell would allow the coupling

132

133

coefficients between cells to be calculated prior to the insertion of the active device.The active device itself is characterized by a frequency-dependent impedance (or scat-tering) parameter matrix at the four ports shown in Fig. A.1. Since the active deviceand the grid are characterized separately, the device field interactions, which accountfor the mutual coupling between the device and the electromagnetic field generatedin the grid, are neglected. Knowing the terminal network parameter matrices of thegrid cell and the device, standard network theory can be invoked to determine thenetwork description of the complete grid structure of finite size, but consisting ofarbitrary number of unit cells.

PORT TERMINATIONS

4

3

2

1

Figure A.1: Port definition for the unit cell of a grid amplifier/oscillator.

In order to determine the terminal impedance matrix of the unit cell, areference terminal which is common to each port, must be identified. Usually, such aterminal is assumed to be at ground potential [46]. Therefore, the delta-gap voltageexcitation is applied at each port as shown in Fig. A.2. The placement of basiselements is also shown. Note that in some cases such as free space a ground planeis nonexistent so therefore the reference terminal is assumed to be some commonpotential.

A.3 Multiport Analysis

The basis function at each port is described by a half sinusoidal basis function asshown in Fig. A.2. Using a half basis element allows for the current to be computedat the port. With the addition of the port elements, the system of linear algebraic

134

Vmt

δ

PORT TERMINATION

Figure A.2: The location of the delta-gap voltage source and the corresponding halfbasis element at the tip of a feed line.

equations given in (4.6) from Chapter 4 is expanded to[Zcc Zct

Ztc Ztt

] [Ic

It

]=

[0Vt

](A.1)

where superscript t denotes terminal port quantities and superscript c denotes quan-tities pertinent to currents induced on the conductor surface. Note that [Zcc] in (A.1)is the same as [Z] in (4.6). The excitation vector [Vt] is due to the delta-gap voltagegenerators at each port and is given a value of 1 V.

Equation (A.1) defines the complete moment matrix, which includes theinteractions associated with the non-port currents. However, we are interested inthe port impedance or admittance matrix, from which one can infer the couplingcoefficients between unit cells on the grid. We will now find these network matricesfor the ports.

We obtain the currents from (A.1) as[Ic

It

]=

[Zcc Zct

Ztc Ztt

]−1 [0Vt

](A.2)

or [Ic

It

]=

[Ycc Yct

Ytc Ytt

] [0Vt

]. (A.3)

From (A.3), we identify the port admittance matrix [Yt] = [Ytt], since [It] = [Yt][Vt]from circuit theory. Likewise, circuit theory yields (A.1) for the port impedance ma-trix, characterized by [

Vt]

=[Zt] [

It]. (A.4)

We can determine [Zt] easily from (A.1) as[Zt]

=[Ztt

]−[Ztc

][Zcc]− 1

[Zct

]. (A.5)

135

The analysis presented above is for port terminations consisting of one delta-gapsource per port. This analysis can easily be extended to allow more cells in thetransverse dimension at the port in which case more than one delta-gap is requiredfor each port. This analysis is given in [46].

In the following sections the formulation for the moment matrix elementsin (A.1) are given for both the spectral and spatial domains.

A.4 Spectral Domain

Filling the moment matrix in (A.1) now requires interactions containing a mixture ofhalf and complete sinusoidal basis functions. In order to evaluate these new elementsin the spectral domain, we need to consider the Fourier transforms of the half basiselements.

The Fourier transform of the sinusoidal basis function in Fig. 4.2 fromChapter 4 is rewritten here as

W xiT

(kx, ky) = 2AxBye−jkxxie−jkyyi (A.6)

where

Ax =k0

sin (k0a)

(cos (kxa) − cos (k0a)

k20 − k2

x

)(A.7)

By =sin (kyb/2)

(kyb/2)(A.8)

Dx =1

sin (k0a)

(k0 sin (kxa) − kx sin (k0a)

k20 − k2

x

). (A.9)

Dividing the sinusoidal basis function up into two halves yield the following Fouriertransforms:

W xiL

(kx, ky) = (Ax + jDx)Bye−jkxxie−jkyyi (A.10)

for the left half of the basis element and

W xiR

(kx, ky) = (Ax − jDx)Bye−jkxxie−jkyyi (A.11)

for the right half side. The subscripts T , L, and R in equations (A.6), (A.10) and(A.11) denote a total basis, left half basis and right half basis, respectively. Summingboth halves in (A.10) and (A.11) yields the total basis in (A.6).

136

A)

B)

C)

D)

E)

F)

Figure A.3: Configuration of x-directed sinusoidal basis functions with the sourcebasis in the left column and the test basis in the right column.

The different configurations for the xx moment matrix elements are shownin Fig. A.3. The moment matrix elements are formulated using the even and oddproperties of the integrand and converting to polar coordinates as was done in Section4.4.2. Referring to Fig. A.3, the xx moment matrix elements are given as follows:

case A (xx components of Zcc)

Zxxji =

−1

π2

∫ π/2

0

∫ ∞

04Gxx

E A2xB

2yCxCy β dβ dα (A.12)

case B (xx components of Ztt)

Zxxji =

−1

π2

∫ π/2

0

∫ ∞

0GxxE

(A2x +D2

x

)B2yCxCy β dβ dα (A.13)

case C (xx components of Ztt)

Zxxji =

−1

π2

∫ π/2

0

∫ ∞

0GxxE

[(A2x −D2

x

)Cx − 2AxDxSx

]B2yCy β dβ dα (A.14)

case D (xx components of Ztt)

Zxxji =

−1

π2

∫ π/2

0

∫ ∞

0GxxE

[(A2x −D2

x

)Cx + 2AxDxSx

]B2yCy β dβ dα (A.15)

137

case E (xx components of Zct and Ztc)

Zxxji =

−1

π2

∫ π/2

0

∫ ∞

02Gxx

E

[A2xCx − 2AxDxSx

]B2yCy β dβ dα (A.16)

case F (xx components of Zct and Ztc)

Zxxji =

−1

π2

∫ π/2

0

∫ ∞

02Gxx

E

[A2xCx + 2AxDxSx

]B2yCy β dβ dα (A.17)

where

Cx = cos [kx (xj − xi)] (A.18)

Sx = sin [kx (xj − xi)] (A.19)

Cy = cos [ky (yj − yi)] . (A.20)

The same procedure is applied for finding the cross term elements Zyx.

A.5 Spatial Domain

The formulation for the spatial domain in very simple compared to the spectral do-main. All that is required is a change of integration limits for each particular case. Forexample, the xx moment matrix elements computed in the spatial domain referringto Fig. A.3 are

case A (xx components of Zcc)

Zxxji = −

∫ yj+b2

yj− b2

∫ xj+a

xj−a

∫ yi+b2

yi− b2

∫ xi+a

xi−aGxxE W x

j (x)W xi (x′) dx′dy′dx dy (A.21)

case B (xx components of Ztt)

Zxxji = −

∫ yj+b2

yj− b2

∫ xj

xj−a

∫ yi+b2

yi− b2

∫ xi

xi−aGxxE W x

j (x)W xi (x′) dx′dy′dx dy (A.22)

case C (xx components of Ztt)

Zxxji = −

∫ yj+b2

yj− b2

∫ xj+a

xj

∫ yi+b2

yi− b2

∫ xi

xi−aGxxE W x

j (x)W xi (x′) dx′dy′dx dy (A.23)

138

case D (xx components of Ztt)

Zxxji = −

∫ yj+b2

yj− b2

∫ xj

xj−a

∫ yi+b2

yi− b2

∫ xi+a

xi

GxxE W x

j (x)W xi (x′) dx′dy′dx dy (A.24)

case E (xx components of Zct and Ztc)

Zxxji = −

∫ yj+b2

yj− b2

∫ xj+a

xj−a

∫ yi+b2

yi− b2

∫ xi

xi−aGxxE W x

j (x)W xi (x′) dx′dy′dx dy (A.25)

case F (xx components of Zct and Ztc)

Zxxji = −

∫ yj+b2

yj− b2

∫ xj+a

xj−a

∫ yi+b2

yi− b2

∫ xi+a

xi

GxxE W x

j (x)W xi (x′) dx′dy′dx dy. (A.26)

The cross elements (Zyx) are also computed with the same procedure where only theintegrations limits need to be changed for each case.

A.6 Summary

The formulation presented in this appendix has several problems. First of all thegeometry does not have a ground plane and therefore the conductors shown in Fig.A.1 are not defined as transmission lines. In [46] this formulation was used with amicrostrip geometry which had a ground plane that produced an image current. Herethe current is being forced at the edge of the gap regions but does not have any placeto flow, unlike the case with the ground plane. No current in this situation is beingallowed to flow through the gap whereas in Section 4.7 the multiport parameterswere computed by allowing current to flow through the gap. Other problems withthis formulation will occur at low frequencies near DC and at high frequencies abovethe first resonance.

Appendix B

Mixed Potential Integral EquationFormulation

B.1 Introduction

In this dissertation an electric field integral equation (EFIE) formulation that includesan electric field dyadic Green’s function was used. The electric field dyadic Green’sfunction exhibits a strong singularity for the self-terms. This is a significant problemsince the self-terms are the dominant elements in the moment matrix. To handlethe singularity, the self-terms are computed in the spectral domain which avoids thesingularity entirely. So in general for the EFIE formulation, the spectral domainis used for the self-terms and neighboring interactions and the spatial domain forfar away interactions. As the interactions get farther apart, the computation in thespectral domain becomes difficult due to oscillatory conditions.

When analyzing a multiport problem, half basis elements are assigned ateach reference port which requires computation of new interactions containing halfbasis elements. Computing interactions containing half basis elements in the spectraldomain requires reformulation of the moment matrix elements. This has been donebut the computation has a slower convergence. For the spatial domain the halfbasis elements are handled simply by changing the limits of integration. But againthe spectral domain must be used for overlapping cells due to the spatial domainsingularities. Another disadvantage of using the spectral domain is the requirementof equal size cells. This highly limits the flexibility for analyzing arbitrarily shapedgeometries. The quasi-optical moment matrix elements do not contain singularitiesand are computed in the spatial domain for unequal size cells including half basiselements.

139

140

In this appendix the nonquasi-optical elements have been formulated as amixed (scalar and vector) potential integral equation (MPIE). This keeps all compu-tations in the spatial domain and allows for the use of unequal size cells. A singularitystill exists in the self-terms but can be handled by singularity extraction. Singularityextraction can be done because the singularity is on the order of 1/R whereas in theEFIE formulation it is on the order of 1/R5.

B.2 Mixed Potential Integral Equation

The formulation of the MPIE starts with the boundary condition stating that thetotal tangential electric field on the conductor surface is zero:

−Escatt = Einc

t (B.1)

where subscript t denotes the transverse components of the electric fields. Einct is

the incident electric field and Escatt is the scattered electric field. The incident field

is the electric field produced by the source that is used to excite the structure. Theincident field, Einc

t , produces a surface current density, JS, on the conductor surfacewhich then produces a scattered field, Escat

t , where some of the field is coupled intothe quasi-optical system and the rest of the field is radiated out of the system.

The scattered field can be written in terms of the dyadic electric fieldGreen’s function as

Escatt =

∫S

=

GE (r|r′) · JS (r′) dS ′. (B.2)

Because of the strong singularity in=

GE , it is best to avoid using the electric fielddyadic Green’s function.

An alternative is to use the vector potential and scalar potential Green’sfunctions which are related to the scattered field as

Escatt = −jωA−∇V (B.3)

defining the vector potential as

A (r) =∫S

=

GA (r|r′) · JS (r′) dS ′ (B.4)

and the scalar potential as

V (r) =∫SGV (r|r′) qS (r′) dS ′. (B.5)

141

The vector potential Green’s function is a dyadic but does not contain cross elements(Gxy

A = GyxA = 0),

=

GA (r|r′) = GxxA (r|r′) axax +Gyy

A (r|r′) ayay. (B.6)

The scalar potential is a scalar quantity and contains a surface charge density, qs,which is related to the surface current density, JS, through the continuity equation

∇ · JS = −jωqS. (B.7)

The two potentials are also related by the Lorentz gauge

∇ · A = −jωμεV. (B.8)

B.3 Method of Moments

The method of moments (MoM) is formulated by expansion and testing of the MPIEusing Galerkin’s method to form a linear system of equations producing the momentmatrix. The expansion and testing is done using rooftop basis functions shown inFig. B.1(a) and is given as follows:

T si (s) =

⎧⎪⎪⎨⎪⎪⎩

[(s− si) /L

−i + 1

]/Wi , si − L−

i < s < si[1 − (s− si) /L

+i

]/Wi , si < s < si + L+

i

0 , otherwise

(B.9)

where s = x or y. Note that this is for unequal size cells which is a very importantnew feature allowing simulations of more complicated grid arrays such as the oneshown in Fig. 6.3 from Chapter 6.

The surface charge density is found using the continuity equation resultingin pulse doublets (see Fig. B.1(b))

Πsi (s) =

⎧⎪⎪⎨⎪⎪⎩

−1/(L−i Wi

), si − L−

i < s < si

+1/(L+i Wi

), si < s < si + L+

i

0 , otherwise

(B.10)

where s = x or y.

In the following sections the moment matrix elements are given for theEFIE and MPIE formulations.

142

Li- Li+

Wi

(a)

Li+

Li-

Wi

(b)

Figure B.1: Unequal size basis function: (a) rooftop and (b) pulse doublet.

B.3.1 Electric Field Integral Equation Formulation

For the EFIE formulation the moment matrix elements are given as follows:

Zxxji = −

∫ yj+Wj2

yj−Wj2

∫ xj+L+j

xj−L−j

∫ yi+Wi2

yi−Wi2

∫ xi+L+i

xi−L−i

T xj (x)GxxE (x|x′; y|y′)T xi (x′) dx′dy′dx dy

(B.11)

and

Zyxji = −

∫ yj+L+j

yj−L−j

∫ xj+Wj2

xj−Wj2

∫ yi+Wi2

yi−Wi2

∫ xi+L+i

xi−L−i

T yj (y)GyxE (x|x′; y|y′)T xi (x′) dx′dy′dx dy.

(B.12)

The other elements Zxyji and Zyy

ji are found by interchanging x and y variables. Againthis cannot be evaluated when x = x′ and y = y′ due to the strong singularity.

B.3.2 Mixed Potential Integral Equation Formulation

For the MPIE formulation the moment matrix elements are given as follows:

Zxxji = jω

∫ yj+Wj2

yj−Wj2

∫ xj+L+j

xj−L−j

∫ yi+Wi2

yi−Wi2

∫ xi+L+i

xi−L−i

T xj (x)GxxA (x|x′; y|y′)T xi (x′) dx′dy′dx dy

143

+1

jω

∫ yj+Wj2

yj−Wj2

∫ xj+L+j

xj−L−j

∫ yi+Wi2

yi−Wi2

∫ xi+L+i

xi−L−i

Πxj (x)GV (x|x′; y|y′) Πx

i (x′) dx′dy′dx dy

(B.13)

and

Zyxji =

1

jω

∫ yj+L+j

yj−L−j

∫ xj+Wj2

xj−Wj2

∫ yi+Wi2

yi−Wi2

∫ xi+L+i

xi−L−i

Πyj (y)GV (x|x′; y|y′)Πx

i (x′) dx′dy′dx dy.

(B.14)

The other elements Zxyji and Zyy

ji are found by interchanging x and y variables. Thereis a singularity when x = x′ and y = y′ but can be handled by extracting thesingularity which will be discussed in the next section. It should be pointed out thatthe scalar potential contribution is much more significant than the vector potentialcontribution.

B.3.3 Scalar Potential Singularity

Since the scalar potential is the dominant contribution, the singularity extractionfor the vector potential is neglected. What the singularity extraction does is allowsfor a smooth function to be extracted which can be numerically integrated. Thisis achieved by subtracting 1/R from the scalar potential Green’s function and thenadding back the 1/R contribution evaluated in closed form, as follows:∫

Sj

∫Si

GV dS′ dS =

∫Sj

∫Si

(GV − 1

R

)dS ′ dS +

∫Sj

∫Si

1

RdS ′ dS (B.15)

where

R =√

(x− x′)2 + (y − y′)2. (B.16)

The closed form expression is evaluated as∫Sj

∫Si

1

RdS ′ dS = 4

[a3 + b3

6− a3

6

1

cos (α)− b3

6

1

sin (α)

+a2b

2ln tan

(π

4+α

2

)− ab2

2ln tan

(α

2

)](B.17)

where a = L−i or L+

i depending on what cell you are in and b = Wi. α = tan−1 (b/a).

Using this MPIE approach, self-term elements were computed having re-sults that agreed with the spectral domain approach. This means the spectral domaincan be avoided allowing for the use of unequal size cells in the spatial domain.

144

B.4 Summary

The significance of this appendix has been the incorporation of the MPIE formulation.The moment matrix interactions can be computed three different ways: (1) EFIEspatial domain, (2) EFIE spectral domain, and (3) MPIE spatial domain. Limitationsto geometries containing equal size cells is no longer required. For simple structuresthat contain equal size cells it is most efficient to use the spectral domain, but forcomplex structures unequal size cells are needed which can now be handled.