Reconfigurable Transmitarray Antennas
by
Jonathan Yun Lau
A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Edward S. Rogers Department of Electrical and Computer Engineering University of Toronto
Copyright c 2012 by Jonathan Yun Lau Abstract
Reconfigurable Transmitarray Antennas
Jonathan Yun Lau
Doctor of Philosophy
Graduate Department of Edward S. Rogers Department of Electrical and Computer
Engineering
University of Toronto
2012
Transmitarrays have been shown to be viable architectures for achieving high-directivity reconfigurable apertures. The existing work on reconfigurable transmitarrays is sparse, with only a few experimental demonstrations of reconfigurable implementations. Fur- thermore, of the designs that have been presented, different approaches have been pro- posed, but the advantages and drawbacks of these approaches have not been compared.
Therefore, in this thesis we present a systematic study of the different approaches to designing reconfigurable transmitarrays, and present designs following these approaches with experimental validation.
First, we investigate the distributed-scatterer approach, which is modeled with layers of identical scattering surfaces. We characterize the beamforming capabilities and then present a Method of Moments technique for analyzing and optimizing designs that follow this approach. Then, we present experimental results for a unit cell with varactor-loaded dipoles following this approach. From these results, we demonstrate that the structure thickness following this approach is problematic for beamforming applications.
Taking the coupled-resonator approach, we next present a slot-coupled patch design that is significantly thinner and easier to fabricate than designs that follow the first ap- proach. Implementing this design in a fully reconfigurable transmitarray, we demonstrate two-dimensional beamforming. An advantage of this design is that it can also operate as
ii a reflectarray. Next, following the guided-wave approach, we present a transmitarray design that uses a bridged-T phase shifter and proximity-coupled differentially-fed stacked patches.
Not only does this design not require vias, it is has a large fractional bandwidth of 10 per- cent, which is unprecedented in reconfigurable transmitarrays. Implementing this design in a full transmitarray, we experimentally demonstrate reconfigurable two-dimensional beamsteering, as well as shaped-beam synthesis. The main contributions of this thesis are two-fold. First, we thoroughly and system- atically compare the transmitarray approaches, which has not been previously done in literature. Secondly, we experimentally demonstrate a reconfigurable array design that achieves better bandwidth, scan angle range, and beam-shaping capability, than existing designs, with reduced fabrication complexity and physical profile.
iii To God be the glory Acknowledgements
It was only through the support of many individuals and organizations that this thesis was successful. First, I would like to sincerely thank my supervisor Prof. Sean Hum for his constant support, guidance, and mentorship throughout my studies. When I joined the electromagnetics group, my knowledge of fields and waves was very limited at best.
I am extremely grateful for his patience and effort in guiding and teaching me. Not only have I grown academically, but I have also been greatly enriched and blessed by his regard for my personal development.
I would also like to thank Prof. Sarris, Prof. Eleftheriades, and Prof. Poon from the University of Toronto, and Prof. McNamara from the University of Ottawa for being on my supervisory and examination committees. Through their feedback and questions, I have not only gained valuable insight into the technical challenges, but more importantly, the motivation of the research.
I am also very grateful for the many friendships that I have built with the students and staff in the electromagnetics group. In particular, I would like to thank Marco and Tse for all the time spent answering my many questions about theory, simulation, and fabrication, and also Krishna and Tony for the many insightful conversations and memorable experiences.
I would also like to acknowledge the generous financial support provided by the Nat- ural Sciences and Engineering Research Council of Canada, the Ontario Graduate Schol- arship in Science and Technology, and the Edward S. Rogers Sr. Scholarship, without which this work would not have been possible.
Finally, I would like to thank my parents Edmund and Angela, my parents-in-law
Leonard and Belinda, and my brother and sister-in-law Josiah and Fiona for their un- conditional support and encouragement. Most of all, I would like to thank my beloved wife Melodie for sharing every step of this journey with me. I could not have asked for a better companion through life’s adventures.
v Contents
1 Introduction 1
1.1 ApproachestoTransmitarrays ...... 5
1.2 Motivation...... 10
1.3 Objectives...... 11
1.4 Outline...... 13
2 Background 14
2.1 ArrayAntennas...... 14
2.1.1 PhasedArrays...... 15
2.1.2 Reflectarrays ...... 18
2.1.3 FixedTransmitarrays...... 20
2.1.4 ReconfigurableTransmitarrays...... 25
2.1.5 FrequencySelectiveSurfaces ...... 27
2.2 ScatteringBehaviorofTransmitarrays ...... 28
3 Relevant Background Theory 30
3.1 Preliminaries ...... 31
3.1.1 CoordinateSystems...... 31
3.1.2 VectorPotentialandWaveEquations ...... 33
3.2 Radiation of Finite Structures ...... 36
3.2.1 RadiationfromaPointSource...... 36
vi 3.2.2 Radiation from a Finite Aperture ...... 38
3.2.3 Radiation from a Finite Array ...... 39
3.2.4 Directivity, Beams, and Side-lobes ...... 41
3.2.5 Gain,Losses,andEfficiency ...... 43
3.3 Infinite Arrays and Floquet Analysis ...... 45
3.3.1 Modeling Free-Space Propagation with Waveguides ...... 50
3.3.2 Green’sFunctions...... 51
3.3.3 Periodic Waveguides ...... 52
3.3.4 RectangularWaveguide...... 53
3.3.5 Parallel-Plate Waveguide ...... 57
4 Distributed-Scatterer Approach 61
4.1 Layered Scattering Surfaces for Beamsteering ...... 61
4.1.1 UniformScatteringSurface...... 67
4.1.2 GraduatedScatteringSurface ...... 76
4.1.3 Simulation Results ...... 86
4.2 Analysis of Discrete Layered Designs ...... 91
4.3 Transmitarray Unit Cell of Loaded Dipoles ...... 98
4.3.1 Unit Cell Design ...... 99
4.3.2 SimulatedResults...... 105
4.3.3 ExperimentalResults...... 107
4.4 Conclusions ...... 112
5 Coupled-Resonator Approach 114
5.1 TransmitarraysasFilters...... 115
5.2 Unit Cell with Slot-Coupled Patches ...... 121
5.2.1 Equivalent Circuit Model ...... 122
5.2.2 ImplementationGeometries ...... 123
vii 5.2.3 Simulation Results ...... 124
5.2.4 ExperimentalResults...... 130
5.3 Transmitarray Design and Implementation ...... 132
5.3.1 EffectofMutualCoupling ...... 135
5.3.2 Biasing Circuitry ...... 137
5.3.3 ExperimentalSetup...... 139
5.3.4 ElementCharacterization ...... 142
5.3.5 ExperimentalResults...... 143
5.4 Reflectarray-ModeOperation ...... 151
5.5 Conclusions ...... 155
6 Guided-Wave Approach 158
6.1 UnitCellDesign ...... 160
6.1.1 ReconfigurablePhaseShifter...... 160
6.1.2 AntennaandBalancedFeed ...... 162
6.1.3 Unit Cell Implementation ...... 163
6.1.4 Simulated and Experimental Results ...... 167
6.1.5 Stacked-PatchDesign...... 170
6.1.6 Simulated and Experimental Results ...... 172
6.2 TransmitarrayImplementation...... 178
6.2.1 MutualCoupling ...... 178
6.3 One-DimensionalBeamforming ...... 179
6.3.1 ArrayFabrication...... 181
6.3.2 ElementCharacterization ...... 183
6.3.3 ArrayBeamforming...... 184
6.3.4 Cross-Polarization ...... 193
6.3.5 Shaped-BeamSynthesis ...... 194
6.4 Conclusions ...... 195
viii 7 Comparison of the Approaches 198
7.1 DesignandOptimization...... 198
7.2 PhaseRangeandInsertionLoss ...... 199
7.3 Beamsteering and Side-lobe Levels ...... 200
7.4 Shaped-beamSynthesis...... 201
7.5 Bandwidth...... 202
7.6 ElementControl...... 202
7.7 PhysicalSize ...... 204
7.8 Fabrication ...... 204
7.9 Scalability and Extensibility ...... 206
7.10Summary ...... 206
8 Conclusions 210
8.1 Contributions ...... 213
8.2 ClosingRemarks ...... 215
Appendices 216
A Method of Moments Analysis for Dipole Unit Cell 216
B Multi-mode S-parameters 224
B.1 Definition of Multi-mode S-parameters ...... 224
B.2 Cascading multi-mode S-parameterblocks ...... 229
C Loss Budget Calculations 232
C.1 ArrayLosses...... 232
C.2 SpilloverLoss ...... 233
C.3 TaperEfficiency...... 234
ix D Stacked Patch Antenna and Feed Design 237
D.1 MicrostripTransition ...... 238
D.2 FeedMitre...... 239
D.3 StackedPatches...... 239
E Projection Matrix Algorithm 248
E.1 Source Listing for calc ff mat.m ...... 249
E.2 Source Listing for beamshape.m ...... 250
Bibliography 253
x List of Symbols
A, Ax,Ay,Az Magnetic vector potential AF Array factor
Az Azimuthal angle (measurement coordinate system)
Am MoM constant a Element width
B Susceptance
B Magnetic flux density
Bmn MoM constant b Element height
Cv Varactor capacitance
MN Cmn MoM constant d Layer spacing
δ( ) Dirac delta function
δm Kronecker delta function D Electric flux density D Directivity
Maximum dimension of radiating structure D
E, Ex,Ey,Ez Electric field intensity ǫ Permittivity constant
ε Efficiency
xi El Elevation angle (measurement coordinate system) f0 Operating frequency f˜ Fourier transform of f G¯ Dyadic Green’s function e G¯ Electric dyadic Green’s function
G Gain g Dipole gap size
Γmn Propagation constant in waveguide in z-direction for mode mn H Magnetic field intensity HPBW Half-power beamwidth
I Scalar current
J, Jx,Jy,Jz Current density k Free-space propagation constant k Wave vector kxm, kym, kzmn Floquet wavenumbers
Lv Varactor parasitic inductance l Dipole length
λ Free-space wavelength
M¯ Vector base for TE waveguide modes m Spectral index in x
M¯ Tensor base for TE waveguide modes Permeability constant
N Number of scattering surface layers n Spectral index in y
N¯ Vector base for TM waveguide modes
N¯ Tensor base for TM waveguide modes p Spectral component index
xii P Order of MoM sinusoids used to approximate currents
Prad Radiated power φ Azimuthal angle (beamforming coordinate system)
φ0 Azimuthal beam angle Ψ sin( ) or cos( ) ψ Electric scalar potential
Qo, Qe Multi-mode S-parameter matrices
Rvs, Rvp Varactor parasitic resistance (series/shunt)
RM Reflection coefficient for mode mn Mmn
Ra, Rb Multi-mode S-parameter matrices r Radial distance (beamforming coordinate system) r Observation (field) position vector r′ Source position vector
S11 Reflection coefficient
S21 Transmission coefficient T Transmission matrix
Aperture surface area S SLL Side-lobe level s Dipole position offset
T Total structure thickness
T M Transmission coefficient for mode mn Mmn t Substrate thickness
θ Polar angle (beamforming coordinate system)
θ0 Polar beam angle U Radiation intensity v Window function
W Spatial period of periodic scattering surface
xiii Wn Coefficient of window function
Wrad, Wrad Radiated power density w Dipole width ω Angular frequency (in radians)
Y Admittance
Z Impedance
xiv List of Tables
4.1 Minimum structure thickness ((N 1)d)...... 73 − 4.2 Varactor diode capacitance and biasing voltage ...... 101
4.3 Dipole unit cell design parameters ...... 103
5.1 Specialconfigurations...... 135 5.2 Pencil beam directivity, peak gain, side-lobe sevel, and beam width . . . 146
5.3 Transmitarraylossbudgetforbroadsidebeam ...... 151
6.1 Pencilbeamangles ...... 190
6.2 Pencil beam directivity, peak gain, side-lobe sevel, and beam width . . . 191
6.3 Transmitarray Loss Budget for Broadside Beam at 5 GHz ...... 193
7.1 Summaryofdesigncomparisons ...... 207
7.2 Comparison with existing designs ...... 208
xv List of Figures
1.1 Reflectarrayantenna ...... 4
1.2 Transmittarrayantenna ...... 5
1.3 Thin lens from a refraction perspective ...... 6
1.4 Thinlensfromdelayperspective...... 7
1.5 Thin lens from a Huygen’s wavelet perspective ...... 7
1.6 The lens as a collection of scattering surfaces ...... 8
1.7 The lens as a collection of resonators ...... 8
1.8 The lens as a collection of antennas and phase shifters ...... 9
1.9 Transmitarray/lensdesign ...... 11
2.1 Reconfigurablephasedarray ...... 15
2.2 Phasegradientofaphasedarray ...... 16
2.3 Rotmanlens...... 17
2.4 Equivalent Norton circuit for a receiving antenna ...... 29
3.1 Arraycoordinatesystem ...... 31
3.2 Beamformingcoordinatesystem ...... 32
3.3 Measurementcoordinatesystem ...... 33
3.4 Free-spaceradiation...... 35
3.5 Pointsourceinfree-space...... 38
3.6 Radiatingapertureinfree-space ...... 40
xvi 3.7 Radiatingarrayinfree-space...... 41
3.8 Side-lobes ...... 43
3.9 Transmitarraylosses ...... 44
3.10 Angled plane-waves due to different phase and magnitude periodicities . 45
3.11Infiniteperiodicarray...... 46
3.12 Radiated plane waves from an infinite periodic aperture ...... 49
3.13 Identical field solutions for freespace and periodic waveguide ...... 50
3.14 Rectangular waveguide T E10 mode and equivalent array and plane waves 54 3.15 Waveguide ports forscattering parameters ...... 56
3.16 Parallel-plate waveguide T EM mode and equivalent array and plane waves 58
4.1 Lensingwithscatteringsurface(s) ...... 62
4.2 Surface inside a waveguide with periodic boundary conditions ...... 63
4.3 Arrangement of multiple surfaces and points xm ...... 64 4.4 Powernormalization ...... 67
λ 4.5 The effects of the number of layers for d = 4 ...... 69 4.6 Theeffectsoflayerspacing...... 70
4.7 The effects of layer spacing (continued) ...... 71
4.8 S21 remapping...... 72 4.9 Maximumaveragetransmissionpower ...... 73
4.10 Susceptance ranges (solid - max, dashed - min) ...... 74
λ 4.11 S-parameters for N = 4 and d = 4 ...... 75 4.12 Computed real-part (arbitrary units) of normalized electric fields for dif-
ferent susceptance values (and phase shifts) ...... 76
4.13Graduatedphaseshift ...... 78
4.14 Computedelectricfields ...... 79
4.15 Fields in each mode for W =2.4λ ...... 80
4.16 Fields in each mode for W =7λ ...... 81
xvii 4.17Lateralpropagation...... 82
4.18 Beam powers of different modes p for scan angles θ1 ...... 83 4.19 Beam powers for different N and d ...... 84
4.20 Beam powers for different N and d (continued)...... 85
4.21 SEMCAD scattering surface implementation with lumped inductors and
capacitors ...... 87
4.22 Infinite scattering surface modes at different beamsteering angles..... 88
4.23 Implementation of finite scattering surface structure in SEMCAD .... 88
4.24 Radiated fields from a finite layered scattering surface ...... 89
λ 4.25 Simulated far-field pattern for a 180 mm surface (N =4,d = 4 ) ..... 90 4.26 Discretelayeredstructure ...... 92
4.27 Point current source a waveguide with a dielectric slab ...... 93
4.28Waveguidesetup ...... 98
4.29 Equivalent circuit models for dipoles ...... 99
4.30Unitcellgeometry ...... 100
4.31 Varactor diode series model L =0.4 nH, R = 2 Ω,C =0.15 pF 2.0 pF 100 v vs v − 4.32 Best S-parametersfromMoMoptimization...... 102
4.33 Effectofcenterdipolelength...... 104
4.34 Effectofdipolespacing...... 104
4.35Effectoflayerspacing ...... 105
4.36 MoM and FDTD S21 ...... 106
4.37 MoM and FDTD S11 ...... 106 4.38Fabricatedunitcell ...... 107
4.39Fabricatedunitcell ...... 108
4.40 Measured unit cell S-parameters...... 109
4.41 Summary of simulated (FDTD), numerical (MOM), and experimental
(EXP) and results at 6.4GHz ...... 110
xviii 4.42 Simulated S21 withlossandbiasingstructure ...... 111
5.1 Poleangleanddistance...... 117
5.2 Exampletwo-polecircuit...... 118
5.3 Example two-resonator complex poles and response, with s1 and s2 loci as
C is varied, with L = Lc = 1 nH and Z0 =50Ω...... 118 5.4 Example three-resonator complex poles and response ...... 120
5.5 Reconfigurable slot-coupled patches transmitarray element structure . . . 121
5.6 Circuitmodel ...... 123
5.7 Ideal simulated S21 for different configurations (Cp,Cs) ...... 126 5.8 Ideal transmission response at 5.7GHz ...... 127
5.9 Theeffectofvaractordiodeloss ...... 128
5.10 Simulated S21 in rectangular waveguide, with varactor losses ...... 129 5.11 Transmission response at 5.5GHz,withvaractorlosses ...... 130
5.12 Experimental waveguide test harness ...... 130
5.13 Measured S21 (magnitude) ...... 131
5.14 Measured S21 (angle)...... 132 5.15 Measured S-parametersforoptimalconfigurations ...... 133
5.16 Transmitarrayperformance ...... 134
5.17 Mutual coupling simulation waveguides ...... 136
5.18 Simulated mutual coupling for pairs of configurations ...... 137
5.19 Array element and bias network design (vertically exaggerated) ..... 140
5.20 Fabricated 6 6array ...... 141 × 5.21 Transmitarray experimental setup in the near-field scanner ...... 141
5.22 Phase range histograms for fabricated elements ...... 143
5.23 Co- and cross-polarizations in the two principal planes ...... 144
5.24 Measured far-field pattern as the elevation angle is scanned (E-plane) . . 145
5.25 Measured far-field pattern as the azimuth angle is scanned (H-plane) . . 147
xix 5.26 Measured aperture fields (array bounds denoted by rectangle) ...... 148
5.27 Measuredfar-fieldpattern(E-plane)...... 150
5.28 Measuredfar-fieldpattern(H-plane) ...... 150
5.29 Measured S11 for 1 V slot varactor bias and varied patch capacitance . . 152
5.30 S11 of unit cell at 5.5GHz ...... 152
5.31Reflectarraysetup...... 153
5.32 Reflectarray measured far-field pattern (E-plane) ...... 154
5.33 Reflectarray measured far-field pattern (H-plane) ...... 155
6.1 Bridged-T circuit topology ...... 161
6.2 Return loss and transmission phase a single bridged-T filter with L1 = 0.75 nH, L =1.25 nH, and C =0.3 pF 1.2pF...... 162 2 − 6.3 Balanced bridged-T circuit topology ...... 164
6.4 Biasingscheme ...... 164
6.5 Dimensions of the bridged-T phase shifter in microstrip ...... 165
6.6 Overallunitcelldesign ...... 166
6.7 Varactor diode model (C = 0.15 pF 2.0 pF, R = 3 Ω, R = 1.5 kΩ, − vs vp
Lv =0.4nH) ...... 167
6.8 Simulated results for the single-patch element ...... 168
6.9 Fabricated single-patch element ...... 169
6.10 Experimental results for the single-patch design ...... 169
6.11 Experimental S21 at 4.87GHz...... 170
6.12 Designofthestacked-patchelement ...... 171
6.13 Fabricatedstacked-patchelement ...... 172
6.14 Simulated results for the stacked-patch design ...... 173
6.15 Experimental results for the stacked-patch design ...... 174
6.16 Experimental S21 at various frequencies (solid - magnitude, dashed - phase)175
xx 6.17 Summary of simulated and experimental results (*denotes the results from thesingle-patchdesign)...... 175
6.18 Simulated far-field radiation pattern of the stacked-patch antenna . . . . 177
6.19 Setup of the side-by-side (x) mutual coupling simulation ...... 179
6.20 Mutual coupling between adjacent array elements ...... 179
6.21 Radiated fields from a finite layered scattering surface ...... 180
6.22 2D simulated results for a six-element stacked-patch bridged-T transmitarray181
6.23 Internalviewofthefabricatedarray ...... 182
6.24 Assembled 6 6transmitarray...... 183 × 6.25 Element classification for the 6 6array ...... 184 × 6.26 Element characterization 6 6array ...... 185 × 6.27 Measured far-field beam patterns at 5.0GHz...... 187
6.28 Measured far-field beam patterns at 4.7GHz...... 188
6.29 Measured far-field beam patterns at 5.2GHz...... 189
6.30 Comparison of expected (AF) and measured (EXP) far-field patterns at 5GHz ...... 192
6.31 Measured cross-polarization for φ0 = 45◦ ...... 194 6.32 Shaped-beam synthesis measurements ...... 196
7.1 Comparison between 2D 180 mm bridged-T transmitarray and four-layer
scatteringsurfaceapertures ...... 201
7.2 Moving pass band and resulting bandwidth ...... 203
A.1 Dipolegeometries...... 216
B.1 Waveguide port definitions (only two modes shown) ...... 224
B.2 S-parameter characterization for a single layer, in a waveguide section of
length 2d ...... 225
B.3 S-parametercases...... 226
xxi B.4 Scatterercascading ...... 229
C.1 Transmitarraysetup ...... 233
D.1 Return loss for the microstrip transition ...... 238
D.2 Return loss for the microstrip transition ...... 239
D.3 Mitresimulationsetup ...... 240
D.4 Returnlossfordifferentmitres...... 240
D.5 Stackedpatchlayoutwithfeed...... 241
D.6 Qualitative comparison of input impedance trajectories for single patch (loop-less) and when a stacked patch is added (with loop), for different
lowersubstratethicknesses ...... 243
D.7 Theeffectoflowerpatchlength ...... 244
D.8 Theeffectofupperpatchlength...... 244
D.9 Theeffectoflowerpatchwidth ...... 245
D.10Theeffectofupperpatchwidth ...... 246 D.11Theeffectofthefeedgap ...... 246
D.12Theeffectofuppersubstrategap ...... 247
xxii Chapter 1
Introduction
The antenna is an intrinsic part of any wireless communication or sensing system, tran- sitioning electromagnetic energy from its guided form within a structure to radiation in free-space, and vice versa. Antennas have evolved into all types of shapes and sizes, embedding themselves ubiquitously into everyday applications, such as personal commu- nications, home electronics, and transportation.
One of the major trends in antenna design has been in making the antenna less intrusive and improving aesthetics, while at the same time pushing the performance and increasing the number of supported frequency bands. For many years, antennas pervaded our view in the form of long monopole antennas on automobiles and large backyard satellite dishes. However, while some antennas have remained overtly visible, most are disappearing from view. For example, very few modern cellular handsets, GPS navigation devices, or laptop computers have protruding antennas. Antennas on automobiles and aircraft for the most part are integrated into the bodies of the vehicles. Antennas are becoming more discrete, conforming themselves to the form factors of the devices on which they reside. Consequently, due to proximity, device components have increased interaction with the radiating characteristics of the antenna, often making it difficult to clearly define the boundaries of the antenna on a device. The notion of radiation from a
1 Chapter 1. Introduction 2 single dedicated structure is changing.
In many instances, such as satellite and point-to-point communications or radar, the radiated power needs to be concentrated in a specific direction to maximize the power transmitted to the receiver, or to reduce interference to other systems operating within the same frequency band. In such cases, antennas with high gain and directivity are required, and two general approaches have been used to design such antennas. Aperture antennas, the first, achieve high directivity by creating large illuminated apertures. The larger the aperture and the more uniform the field phases and amplitudes on the aperture, the higher the directivity will be. Examples include the horn antenna and the widely recognizable parabolic reflector antenna. The other approach is to use an array of small antennas, each excited with a specific phase, to effectively create a large aperture of uniform fields.
However, directivity and gain can be achieved without the driven radiator being highly directive. Passive scatterers can also be used to modify the fields from a low directivity feed antenna and achieve high directivity. For example, Yagi-Uda antennas, which are commonly seen on rooftops for television signal reception, consist of a dipole surrounded by parallel parasitic elements. As the feed antenna radiates, currents are excited on the parasitic elements such that the fields radiating from the parasitic currents add constructively in a specific direction. In fact, the parabolic reflector of a parabolic reflector antenna can be viewed as a large passive scatterer illuminated by a small source.
Nevertheless, regardless of how directivity is achieved, a large effective aperture with uniform fields needs to be synthesized, which necessitates a structure with a large physi- cal size. Large antennas are not only cumbersome and unsightly, but they are also costly to use in many cases. For example, rigid reflector antennas, which are commonly used for high-gain satellite communications, are bulky and heavy, making them extremely expen- sive to launch into orbit. For this reason, antenna arrays have been used to synthesize large apertures. Chapter 1. Introduction 3
Antenna reconfigurability is also crucial in many applications. For instance, in a radar system where a high-directivity aperture needs to be scanned, typically a large reflector or antenna array is mechanically rotated rapidly, requiring a large amount of space and mechanical hardware. Reconfigurability is also required in communications systems where the transmitter or receiver are in motion such as satellite tracking, or for adapting to changing environments where there are moving scatterers and multi-path.
The ability to electronically reconfigure an antenna without mechanical movement is very desirable, because much less physical space is required and the reconfiguration speed is many orders of magnitude faster. Moreover, mechanical systems require significantly more maintenance and are more prone to failure over time.
Phased arrays, which are simply arrays of antennas driven with specific phases, have been proposed as elegant solutions for achieving high-directivity and electronic reconfig- urability. Unlike large reflector antennas or horn antennas which must be mechanically moved to redirect the beam, the beams produced by a phased array can be electroni- cally scanned very rapidly. Since the elements of the array can take planar or conformal arrangements, phased arrays are also generally less bulky than aperture antennas. How- ever, one major challenge with phased arrays is scalability. Since each element in the array needs to be individually excited, the length of the transmission lines needed to drive each element in the array increases quadratically with the size of the array, leading to quadratically increasing transmission line losses and space requirements. The losses and complexity associated with the feed network are major disadvantages.
For this reason, we can turn to spatial techniques for exciting an array, using a single feed antenna and an array of passive scatterers to synthesize a large effective aperture. A prominent example of this is the reflectarray antenna [1], shown in Figure 1.1, where an array of passive elements is spatially excited by a feed antenna and resonates, radiating in such a way that an aperture with uniform phase is created. A reflectarray can be reconfigured by electronically modifying the resonances of the elements. Chapter 1. Introduction 4
Figure 1.1: Reflectarray antenna
The transmitarray, also known as an array lens1, is a discretized lens, and is the dual structure of the reflectarray, as shown in Figure 1.2. From an optics perspective, a transmitarray is to a lens, what a reflectarray is to a mirror. A transmitarray has several advantages over a reflectarray. While the scanning range of a reflectarray is fun- damentally limited to one side of a plane, or 180◦, conceptually a spherical or cylindrical transmitarray surrounding an isotropic feed antenna can achieve a 360◦ scanning range. Furthermore, while feed blockage is a challenge for reflectarrays because the feeding an- tenna is on the same side of the array as the radiated fields, such is not the case for a transmitarray. In fact, with the feed out of the way, transmitarrays can not only produce beams in the far-field, but they can also be used for focusing fields from the feed into a spot in the near-field, which is not possible with reflectors due to feed blockage. Trans- mitarrays and reflectarrays can also be combined into more elaborate structures, as has been done with lenses and mirrors in catadioptric telescopes2.
Compared to traditional reflectors and lenses, transmitarrays and reflectarrays have the major advantage of being planar. Not only are planar antennas easier to integrate into everyday structures such as walls and roofs, they are more portable because they can
1The terms transmitarray and array lens will be used interchangeably in this thesis 2A catadioptric system is one where both refraction and reflection are used Chapter 1. Introduction 5
Figure 1.2: Transmittarray antenna be rolled up or dismantled in panels and easily reassembled. Unlike parabolic reflectors whose shape needs to be rigidly supported, the shape of a large planar transmitarray can simply be supported using stretched cables or membranes. Moreover, because the properties of transmitarrays can be arbitrarily designed, they can be made to take any form factor. If made to be reconfigurable, distortions in the shape of a transmitarray can be electronically corrected, which is a significant advantage when the antenna is operating in a remote location.
Although a criticism of transmitarrays and reflectarrays is that the feed significantly increases the profile of the antenna, this is in fact not a problem in many applications. For example, for space applications, physical profile is only a constraint when the antenna is being sent into orbit. Therefore, the feed structure can easily be collapsed and moved into place when the antenna is actually deployed. Alternatively, other feeding mechanisms such as Cassegrain reflection can be used to reduce the physical profile.
1.1 Approaches to Transmitarrays
In this thesis, we will explore different approaches to understanding reconfigurable trans- mitarrays. We begin with an illustration using thin lenses from optics. The most basic Chapter 1. Introduction 6 way to understand a thin lens is to assume that the media through which the waves prop- agate is homogeneous. In this case, the boundary conditions, described by Snell’s law, relate the material properties with the angles of incidence and refraction. In a properly shaped lens, all refracted waves converge on a focal point, as shown in Figure 1.3.
Figure 1.3: Thin lens from a refraction perspective
Alternatively, we can look at the paths on which waves propagate. Because different media have different associated phase velocities, waves propagating through different distances in the media experience different phase shifts and time delays. The focus of a lens is the point at which all paths to that point incur the same phase shift or delay. As illustrated in Figure 1.4, the two paths have different lengths, but the delay τ experienced on both paths is the same.
It is also possible to look at the lens from the Huygen’s principle, which views each point in the lens as a scatterer that produces waves in every direction, as illustrated in
Figure 1.5. Each scatterer will be excited by the incident wave according to the properties of the lens medium. In this case, the focal point is a result of the interference pattern created by an infinite number of tiny scatterers in the lens.
We can glean from these simple examples three approaches with which we can design a lens. If we perceive the lens as a collection of passive scatterers distributed in space, Chapter 1. Introduction 7
Figure 1.4: Thin lens from delay perspective
Figure 1.5: Thin lens from a Huygen’s wavelet perspective Chapter 1. Introduction 8 then we will seek to manipulate the properties and positions of the scatterers to control the focusing abilities of the lens. When the scatterers are small, then the scatterers can be perceived as layers of closely-spaced scattering surfaces, as shown in Figure 1.6. We will call this approach the distributed-scatterer approach.
Figure 1.6: The lens as a collection of scattering surfaces
For a lens, we are most interested in controlling the phase through the structure.
Since phase of the scattered fields has the greatest variation when the scatterers are near resonance, we can also see the transmitarray as a collection of coupled resonators, as illustrated in Figure 1.7, and take the coupled-resonator approach.
Figure 1.7: The lens as a collection of resonators
Finally, if we look at the delay or phase shift incurred over different paths, then we may be inclined to discretize the lens and use time delaying or phase shifting circuits to control the properties of the lens. In order to use such circuits, the free-space wave must first be coupled into transmission lines with antennas, as shown in Figure 1.8. This Chapter 1. Introduction 9 approach will be called the guided-wave approach.
Figure 1.8: The lens as a collection of antennas and phase shifters
These approaches are classified and named in this thesis for the purpose of comparing them systematically, and we note that the terminology is not commonly used in the community. Also, these three approaches are not mutually exclusive, since a particular structure can belong to multiple classes depending on how it is perceived. For example, a guided-wave transmitarray design may use a coupled-resonator phase shifter, or the guiding mechanism may be so short that one could argue that the waves are not in fact guided. Alternatively, the surfaces of a distributed-scatterer design may be resonant and tightly coupled. Nevertheless, for the purpose of this thesis, we will focus on the following concepts for each approach. For the distributed-scatterer approach, we will focus on scattered fields and surface impedances and lateral interaction of adjacent array cells. Although equivalent circuit models can be used to model a single cell, they cannot model the interaction between adjacent cells, which is an important consideration as we will see later. For the coupled-resonator approach, we will focus on the presence of resonant structures and how they interact. Unlike the distributed-scatterer approach, we will focus on longitudinal coupling of the resonators. Finally, for the guided-wave approach, we will focus on the structure as an assembly of microwave circuits connected by transmission lines. Chapter 1. Introduction 10
1.2 Motivation
This thesis demonstrates how transmitarrays can be used to effectively address three major challenges that have emerged in antenna design: directivity and gain, reconfig- urability, and hardware cost (in terms of monetary cost, complexity, or physical space).
Many solutions for each of the above challenges have been proposed, but typically ad- vances in one goal have led to trade-offs in another. The difficulty is thus in addressing the three challenges simultaneously.
Firstly, antenna directivity and gain are crucial. As the demand for wireless band- width continues to increase, higher frequencies will be needed. However, with increasing frequency, atmospheric attenuation also increases, resulting in the need for high-gain an- tennas. That is, transmitters and receivers must direct their power towards each other, rather than radiating in every direction. Such spatial directivity can also help alleviate the effects of multipath and co-channel interference that reduce the available bandwidth.
Furthermore, directivity is crucial for achieving high-resolution imagery in radar or re- mote sensing applications.
Secondly, the radiation characteristics from highly directive antennas often need to be reconfigurable when terminals are in motion or when the channel conditions change.
As frequency increases, channel variations in time increase also, resulting in the need for adaptive communication systems, for which real-time reconfigurable antennas are important. In radar applications, electronic reconfigurability can significantly reduce the space and hardware costs associated with the mechanical rotation of antennas, and improve reliability since no moving parts are required.
Thirdly, as wireless devices are becoming ubiquitous, more and more constraints are being placed on antenna shape, size, complexity, and cost. While single antennas have low associated hardware costs, they are typically not able to achieve the desired directivity and reconfigurability discussed previously. On the other hand, while phased- array antennas are able to achieve greater directivity and reconfigurability, they require Chapter 1. Introduction 11 large bulky feeding networks that result in significant transmission line losses. This underscores the importance of finding practical low-cost techniques for implementing arrays.
1.3 Objectives
Given the issues outlined above, the goal of this thesis is to demonstrate how the trans- mitarray paradigm can be used to achieve high-directivity reconfigurable apertures, while at the same time minimizing hardware cost and complexity. To achieve this goal, we first need to understand the fundamental problems, issues, and challenges. We can succinctly define the transmitarray or lens design problem as follows, as illustrated in Figure 1.9:
For given incident fields on one side of the transmitarray, produce outgoing
fields with specific phase and maximal power on the opposite side.
Figure 1.9: Transmitarray/lens design
For instance, the incident waves may be spherical waves produced by a nearby source or quasi-plane waves from a distant transmitter. The desired outgoing waves may be quasi-plane waves in one or more directions, or waves that converge to a nearby receiver.
We want the outgoing waves to carry as much power as possible, meaning that loss in the transmitarray and the reflections on the incident side must be minimized. Chapter 1. Introduction 12
As we will see in Chapter 2, reconfigurable transmitarrays have not been extensively studied in literature. Different approaches have been proposed for transmitarray design, but their relative strengths and limitations are not well understood. So, to systematically investigate transmitarray approaches, we will take three perspectives: the distributed- scatterer approach, the coupled-resonator approach, and the guided-wave approach. For each approach, we will identify the advantages and the limitations.
Following, detailed design methods and experimental implementation are crucial for demonstrating viability and how practical issues such as element control and fabrication complexity can be addressed. Only a few experimental array implementations have been presented and actual reconfigurable beamforming results have been very limited. Thus, drawing from the insight gained from studying the approaches, we will design planar reconfigurable transmitarrays for beamforming. In order to assert the practicality of transmitarrays, the experimentally tested arrays must be fully reconfigurable in two- dimensions, performing beamscanning over both azimuth and elevation angles.
To summarize, the three objectives and contributions of this thesis are:
1. Systematic examination and comparison of the different approaches to transmi-
tarrays to identify the strengths and limitations of each approach
2. Detailed description of design methods for reconfigurable transmitarray antennas
3. Design and experimental demonstration of reconfigurable transmitarray beamform-
ing to show that transmitarrays can be used to produce practical reconfigurable
high-gain apertures Chapter 1. Introduction 13
1.4 Outline
The outline of this thesis is as follows. In Chapter 2, we first review existing work on transmitarrays and related work in literature. Following, in Chapter 3, we provide relevant definitions and theoretical background for understanding the thesis. In Chap- ters 4, 5, and 6, we present three approaches and designs following these approaches: the distributed-scatterer approach, the coupled-resonator approach, and the guided-wave ap- proach. We compare the different approaches in Chapter 7 and present our conclusions in Chapter 8. Chapter 2
Background
In this chapter, we will review results and designs that have been proposed in literature that are relevant to this thesis. Because a transmitarray is essentially a discretized lens, it belongs to a class of antennas called quasi-optical or spatially-fed antenna arrays. For conciseness, we will not review dielectric lenses or Fresnel zone plates that are often used for focusing, because they are neither reconfigurable nor discretized. Therefore, we will first review the history of antenna arrays, focusing on spatially-fed arrays, from their origins in reflectarray design to the state-of-the-art in reconfigurable transmitarrays. Next, we will discuss the concept of maximum receivable power for an antenna and how it relates to transmitarrays. Finally, we will review beamforming techniques that will be used later in this thesis.
2.1 Array Antennas
Fundamentally, the maximum directivity of an antenna is proportional to its size. Rather than using a single large antenna, large radiating apertures can also be synthesized using an array of smaller antennas.
14 Chapter 2. Background 15
2.1.1 Phased Arrays
Phased arrays, which are arrays of antennas where each array element is directly fed by a transmission line with a specific phase and amplitude, have existed for many years. Be- cause of their conceptual simplicity, they have been popular for achieving high-directivity
fixed and reconfigurable radiating apertures.
To briefly review the governing principle of a phased array illustrated in Figure 2.1, consider the linear array with element spacing d shown in Figure 2.2. Suppose the elements of the phased array are excited with phase-shifted signals, where the phase difference between adjacent elements is φ. Consider parallel rays traced at an angle of
θ0 from the array axis. The additional distance of each subsequent ray to an imaginary aperture plane denoted by the solid line is d cos θ0. If k is the free-space phase constant, or the rate at which the phase changes in space for a propagating wave, then the rays arriving at the imaginary aperture will have all have uniform phases if
φ = kd cos θ0. (2.1)
In this case, the radiation from each element will add constructively, resulting in a beam directed at an angle θ0. From this, we can see that a phase gradient in the array excitation produces a narrow coherent beam, or a pencil beam, in a particular direction. A similar principle applies when the array is used to receive a signal from a particular direction.
Figure 2.1: Reconfigurable phased array
To drive each element, traditional phased arrays split the input signal into multi- ple feed lines using power dividers, and then phase shifters are used to manipulate the Chapter 2. Background 16
Figure 2.2: Phase gradient of a phased array radiated phase of each array element. Non-reconfigurable phased arrays typically use delay line phase shifters to create pencil beams in a specific direction. The amplitude of each element can also be manipulated by tailoring the power-splitting ratio of the power dividers to gain additional control of the radiated waves.
The body of research on phased arrays is vast and so it is not possible to thoroughly review the literature on phased arrays. The theory and design techniques of phased arrays have been well studied and many comprehensive books are available [2, 3]. Therefore, in the following sections, we will only highlight examples showing techniques that have been used to implement reconfigurability.
Because phased arrays require a phase shifter for each element, it is challenging to make each phase shifter reconfigurable for larger arrays. As a result, Rotman lenses were proposed as a way to introduce reconfigurability into linear phased arrays [4] without reconfigurable phase shifters. A typical Rotman lens, illustrated in Figure 2.3, consists of a convex-shaped parallel-plate waveguide, with input and output transmission lines connected by cone-shaped impedance matchers. In transmitting mode, by exciting a given input port, the different delays caused by the path distance to the output feed in the lens create different phase gradients across the antenna array, which result in a beam in a particular direction. Superposition by exciting multiple ports allows for multi- beam synthesis. While Rotman lenses provide an easy way to achieve reconfigurable Chapter 2. Background 17 beamsteering, the beam directions are discrete and limited to the number of inputs. Moreover, they can only be used for linear arrays and electronic input switching with a large number of inputs is non-trivial, particularly at high frequencies.
Figure 2.3: Rotman lens
To achieve independent phase control, each element requires an independently con- trollable reconfigurable phase shifter. A great deal of work has been done on monolithic- microwave integrated-circuit (MMIC) phase shifters, but complex fabrication processes are involved and the insertion losses are significant (more than 4 dB at 5 6 GHz) [5]. − More recently, with advances in semiconductor technology, researchers have studied electronic tuning techniques by integrating diodes in phased arrays. Varactor diodes, when reverse biased, provide a continuously tunable capacitance in the depletion region of the diode. They have been used to implement the extended-resonance power-dividing method for reconfigurable phased arrays without the need for separate power dividers and phase shifters [6]. At 2 GHz, a four-element linear array achieved a 20◦ reconfigurable scanning range with only 1 dB of insertion loss.
Alternatively, PIN diodes can be used as switches for changing signal paths. An as- sembly of Rotman lenses and PIN diode switches has been used to create an electronically reconfigurable phased array [7]. Operating at 35 GHz, a 10-element linear array achieved a scanning range of 49◦ experimentally with a gain of 15.6 dB. The scanning range was covered by seven beams spaced 7◦ apart, since Rotman lenses can only produce beams Chapter 2. Background 18 in discrete directions.
As mentioned in the introduction, one of the challenges with phased arrays is that the complexity, size, and losses of the feed network increase substantially with array size.
For this reason, researchers have turned to spatial feeding techniques for antenna arrays.
2.1.2 Reflectarrays
Spatial feeding of antenna arrays originates from optics, where a feed antenna illumi- nates the array elements, which in turn re-radiate the power with specific phases. Since reflectarrays have been a prominent example of discretized optics for microwaves [1], we will briefly review milestones in their development.
The reflectarray is analogous to a parabolic reflector, where the guiding principle in its design is that the phase delay from the source to the reflector, to the aperture plane must be equal for every possible radiation path. Thus, the primary challenge in reflectarray de- sign is the control of individual element phases. The first reflectarray proposed consisted of an array of open-ended waveguides, where the depth of each waveguide controlled the phase delay of the element [8]. As microstrip technology increased in popularity due to its ease of analysis and fabrication, the microstrip patch became the most popular reflectarray element. Many aspects of microstrip reflectarrays were studied, including control of element phases using microstrip stubs attached to each patch [9, 10], control of element phases using varied patch length [11, 12], and reflectarray feeding configurations
[13].
Compared to the parabolic reflector, the main drawback of the reflectarray is that the bandwidth tends to be narrow. As a result, work has focused on increasing the bandwidth of reflectarrays using stacked patches, achieving a fractional bandwidth of
10% [14]. Another technique for achieving large bandwidth is the use of true-time-delay
(TTD) lines. Reflectarrays have also been able to achieve fractional bandwidths of 10% by aperture-coupling microstrip delay lines to patch elements [15]. Chapter 2. Background 19
A number of techniques have been presented for electronically tuning reflectarrays [16, 17]. Most techniques either involve tuning a capacitance by manipulating of the
DC voltage across a varactor diode [18, 19, 20, 21, 22], micro-electromechanical systems
(MEMS) capacitor [23, 24], or by shorting and connecting components using PIN diodes
[20] or MEMS switches [25]. Other techniques also include mechanically raising and lowering patches [26], or changing the dielectric properties of the substrate using liquid crystal substrates [27, 28]. Amplifiers to augment the reflected power have also been proposed in fixed [29] and reconfigurable reflectarrays [22].
Generally, two approaches for reconfigurable reflectarrays have emerged. One ap- proach is to have the antenna of each element couple power into a transmission line, and then to use a reflection-type microwave phase-shifter to manipulate the phase of the signal that is re-radiated. To tune the reflection phase of each element, PIN diodes have been used to discretely change the length of the transmission line [20, 30]. Alternatively, varactor diodes have been used to tune capacitances, allowing continuous tuning of the reflection phase [20, 21, 22]. The other approach is to integrate the tuning component directly into the antenna, so that the phase of the re-radiated waves is manipulated by changing the resonance of the antenna. Varactor diodes have been directly connected to microstrip patches [18, 19], and MEMS switches have been used to connect or dis- connect structures that reactively load patches [31, 25]. These two approaches parallel the guided-wave and distributed-scatterer approaches to transmitarrays introduced in
Chapter 1.
Each tuning technology has its respective advantages and disadvantages. PIN diodes are inexpensive and readily available, but because they act as switches, multiple control lines are required for each element to achieve higher-resolution phase control. Varactor diodes are also readily available and allow for continuous phase tuning with a single control voltage, but are non-linear when the input power is large. Because the capacitance produced by the varactor diode is controlled by a DC bias voltage, a large input power Chapter 2. Background 20 affects the bias voltage, modulating the capacitance and producing distortion. MEMS technologies are capable of high power handling with low losses (3.5 dB for varactor diode vs. 0.5 dB for MEMS at 5.8 GHz [32]) and with virtually no distortion [32], but they are more challenging to fabricate. In terms of operating frequency, discrete components such as PIN and varactor diodes cannot operate at higher frequencies because of the parasitic effects caused by the component size and packaging. While tunable liquid crystal substrates are more suitable at millimeter-wave frequencies, they suffer from significant losses.
Regardless, given the dual nature of transmitarrays and reflectarrays, tuning tech- nologies that have been shown to be effective for reflectarrays can generally be applied in similar manners to transmitarrays. The advantages and disadvantages such as distortion and parasitic effects are all very similar. In the following sections, we will first review fixed transmitarrays, followed by reconfigurable transmitarrays, which are tuned using techniques similar to those that we have just discussed.
2.1.3 Fixed Transmitarrays
The transmitarray is similar to the reflectarray in that the phase delay from source to transmitarray element to the aperture plane must be equal for every possible radiation path, in order to produce a pencil beam. However, the most crucial difference between a reflectarray and a transmitarray is that in a reflectarray, all of the power will be reflected regardless of the frequency and unit cell design. In the extreme case with frequencies well below the resonant frequency of the cell, a reflectarray will reflect with 180◦ phase-shift, or a coefficient of 1, because the elements will have little effect and the reflecting ground − plane will dominate the response. Similarly, in the other extreme case with frequencies well above resonance, a reflectarray will also reflect with 180◦ (or 180◦) phase-shift. For − frequencies in between, a reflectarray will reflect with a phase shift between 180◦ and − 180◦. However, with a transmitarray, if the structure is not properly matched to free- Chapter 2. Background 21 space, all of the power will be reflected from the input of the transmitarray, resulting in zero transmission through the structure. Thus, the magnitude of transmission is an additional design consideration for transmitarrays.
An interesting observation is that the bandwidth limitation of reflectarrays can in part be attributed to the presence of the ground plane. To increase the bandwidth, the elements need to be less resonant. However, if the resonance of the elements is reduced, then the magnitude of the fields radiated by the elements is also reduced, and the response of the reflectarray becomes dominated by reflection off the ground plane. Because of this, it may be easier to achieve larger bandwidths using the transmitarray paradigm.
Fixed transmitarrays have been studied for many years, with most of the work taking the guided-wave approach, connecting two antennas together with a phase shifter. Two early microstrip-based transmitarray designs based on coupled patch antennas were pro- posed in 2006 [33]. The usefulness of the first design, which used slot coupling to connect the antennas, was deemed to be limited due to the limited bandwidth. The other design, which consisted of patches slot-coupled to a common transmission line, had an improved bandwidth, but at the cost of more substrate and metallization layers.
The concept of transmitarrays became more popular in the early 2000’s as integrated active antennas emerged. Transmitarray designs based on transmission line-coupled patch antennas were extended to include active elements [34]. A series of studies were presented on active transmitarrays [35, 36, 37], where a 4 2 transmitarray was pro- × totyped and measured. The elements consisted of an amplifier, with aperture-coupled patch antennas connected to the input and output of the amplifier. This structure was also used as power amplifier [38].
Development of transmitarray principles also progressed in the design of spatial power amplifiers, as this last example shows. Using a large number of low-power elements, a spatial power amplifier uses the principles of spatial power combining to achieve large power handling at high frequencies, while avoiding transmission line loss problems. A Chapter 2. Background 22 spatial power amplifier is very similar to a fixed transmitarray in that an array of elements is used as a lens. Fixed array lenses with amplifiers have been demonstrated for use as spatial power combiners [34, 39, 40, 41, 42]. The major difference between spatial power amplifiers and transmitarrays is that the design goal of a spatial power amplifier is to maximize power, with the collimating element very close to the array. On the other hand, for transmitarrays, the design goal is beam synthesis for a receiver that is very far away.
A5 5 transmitarray design was first experimentally demonstrated at 12 GHz in 2007 × [43, 44, 45]. The array elements each consisted of a stripline delay line between transmit and receive patch antennas, where the patches were connected to the stripline using coaxial probes. The phase delay of each element was varied by controlling the length of the delay line. While simple in principle, the implementation of such a structure was difficult, since the design required five conducting microstrip layers (one for each patch, and three for the stripline), all connected through vias.
To avoid the use of vias, an asymmetric fixed transmitarray design involving one microstrip-fed patch connected directly to a slot-coupled patch was proposed at 9.6 GHz
[46, 47]. In this design, additional stacked patches were also added to increase the bandwidth of the structure. A circular array with 208 elements experimentally achieved a directivity of 28 dBi with side-lobe levels of about 20 dB and 1.5 dB of insertion loss − through the structure.
In the transmitarray designs discussed so far, the phase shift through each element is manipulated by adjusting the length of the transmission line connecting the receive and transmit antennas. In this way, element phase shifts can be selected from a continuous range of values. Phase-quantized transmitarrays, where elements can only have discrete phase shifts have also been proposed, with 1-bit (0◦/180◦) and 2-bit (0◦/90◦/180◦/270◦) designs [48, 49]. It was suggested that phase quantization of the elements only has a minor impact on the array directivity, with only a 1 dB difference between the continuous phase and 2-bit phase designs in the configurations studied. Chapter 2. Background 23
The proposed 2-bit design [48] used two patch antennas connected by co-planar waveg- uide (CPW) transmission lines, where the lengths of the transmission lines were quantized to four lengths. A 20 11 array achieved a peak gain of 14.3dB at 9.6 GHz with a 3 dB × fractional gain bandwidth of 10.4%. While the simplicity of this CPW implementation was an advantage, requiring only two layers of substrate and three layers of metallization with no vias, higher than expected side-lobe levels of 6.6 dB were observed. Multi- − ple beam operation was also proposed, by changing the angle of the transmitarray feed antenna, where at 30◦ offset the array produced a peak gain of 11.5 dBi.
In the 1-bit design [49], two square patch antennas were connected using a single via, with one of the two patches rotated by 90◦ or 90◦ to achieve the 0◦ and 180◦ phase shifts. − Consequently, the input and output fields have orthogonal polarizations. A 20 20 array × was experimentally demonstrated at 9.8 GHz, achieving a gain of 23.6 dBi and side-lobe levels of 15 dB with a very large 3 dB fractional gain bandwidth of 18.4%. − Similar designs with probe-connected square patches were presented at V-band [50] and X-band [51, 52]. The 1-bit linearly polarized design was extended to a 2-bit circularly polarized design by rotating the patches by 0◦, 90◦, 180◦, or 270◦. A20 20 array × achieved a gain of 24.3 dBi with 13.7 dB side-lobe levels, with a large 3 dB fractional − gain bandwidth of 13% at 9.8 GHz [51, 52]. A similar 20 20 array achieved a gain × of 22 dBi with 16 dB side-lobe levels, with a large 3 dB fractional gain bandwidth of − 13.3% at 60 GHz [53]. The insertion loss through the structure was 0.46 dB.
A circularly-polarized lens based on rotated patch elements has also been proposed
[54]. Two stacked patches on either side of a ground plane were coupled by a cross-shaped slot. The phase shift through the structure was manipulated by rotating the patches.
At 12.4 GHz, the 349-element array had side-lobe levels of about 25 dB. The array − achieved a bandwidth of 7% where the axial ratio was less than 2 dB, with an insertion loss through the structure of 0.6 dB.
A different approach, the distributed-scatterer approach, has also been used to design Chapter 2. Background 24 transmitarrays, where array elements consist of four layers of coupled rings [55]. Without a ground plane, the waves passed through the structure without being guided. The phase shift through the structure was manipulated by adjusting the width of the rings and the size of the gap between the rings. At 30 GHz, the 21 21 array achieved a gain of × 28 dBi and side-lobe levels of 16.5 dB, with a 3 dB fractional gain bandwidth of 7.5%. − With each ring layer having about 0.25 dB of loss and 0.3 dB of loss from the foam spacers between the layers, the total insertion loss through the structure amounts to about 2 dB. One of the advantages or this coupled-ring design is that it is not restricted to a single polarization, but a drawback is that the layers need to be spaced approximately a quarter-wavelength apart, which makes the structure thick at lower frequencies.
A thinner structure, also employing layers of metallization, has also been proposed
λ using three metallization layers but a structure thickness of only 10 . The structure es- sentially consists of three closely spaced metal layers with large slots. The slots run hor- izontally and are connected end-to-end, effectively creating a long horizontal slot across the entire structure. The phase shift through the structure is manipulated by changing the width of the slots at different points, where the width the slot in each of the three layers may be different. One-dimensional beamforming at broadside is demonstrated at
30 GHz. Although very simple in design and fabrication, the drawback of this design is that it may be difficult to extend to two-dimensional beamforming.
To briefly summarize, many studies have demonstrated high-directivity transmitarray design at many different frequencies, with insertion losses as small as 0.46 dB and frac- tional bandwidths as large as 18%. Most of the designs consist of coupled-patch antennas, using delay lines to achieve phase shifting and stacked patches to increase bandwidth. All of the arrays, with some exceptions [54], are designed to produce a pencil beam when illuminated by a horn at broadside (at prime focus). That is, these transmitarrays are designed to convert the spherical waves produced by the feed antenna to broadside quasi- plane waves. Because these transmitarrays designs are fixed, new arrays would need to Chapter 2. Background 25 be fabricated to produce pencil beams at non-broadside angles. It is for this reason that we rarely see any fixed transmitarray results producing non-broadside beams.
2.1.4 Reconfigurable Transmitarrays
Following the development of tunability in reflectarrays, reconfigurability was introduced to transmitarrays. As is the case with reflectarrays, one of the trade-offs of introducing reconfigurability to transmitarrays is that the insertion loss through the structure will inevitably be increased, due to the additional complexity and resistances of the tuning mechanisms. Moreover, fabrication and experimental verification with reconfigurable ar- rays is more challenging due to the cost and control hardware for the tuning mechanisms.
With reconfigurable transmitarrays, it is more feasible to experimentally explore beamforming capabilities of non-broadside beams. There are certain challenges asso- ciated with beamforming that have often been overlooked with fixed transmitarrays that only produce broadside beams, particularly for beams with larger deviations from broad- side. For example, because the required phase differences between adjacent array ele- ments are larger, the mutual coupling between elements can cause significant element phase error.
Preliminary results for an electronically reconfigurable transmitarray design were first presented in 2008 [56]. Probe-coupled patches were loaded with varactor diodes connected to the ground plane to manipulate the resonance of the patch antenna, but only 90◦ of phase agility was achieved.
Reconfigurable array lenses using MEMS switches to achieve 2-bit phase control, allowing four possible discrete phase tunings, were also proposed for the purpose of an- tenna beamforming [57, 58]. Slot antennas were coupled using short stripline transmis- sion lines, and MEMS switches were used to control which transmission lines were used.
One-dimensional beamforming was demonstrated experimentally witha22 22 array at × 34.8 GHz, producing a gain of 9.2 dB and side-lobe levels of 13 dB at broadside. The − Chapter 2. Background 26 average insertion loss of the structure was 8.0 dB. The reconfigurable array was able to scan a pencil beam in an 80◦ range, producing 3.8◦ of gain at 40◦, and 6.4◦ of gain at − +40◦ off-broadside in the E-plane. This work is one of the first results on reconfigurable lenses that experimentally demonstrated electronic beamscanning.
In 2010, a fully reconfigurable 6 6 transmitarray was presented for beamforming [59]. × Probe-fed stacked patches were connected by three reflection-type phase shifters in series that used varactor diodes to tune the phase shift. Because the phase shifters were larger than the element spacing, the elements were grouped together, with four patch antennas connected to each phase shifter. In this way, there were effectively 9 tunable cells in the array. The reconfigurable transmitarray demonstrated beamforming at 0◦ and 9◦ degrees off-broadside at 12 GHz, producing 16 dBi of gain and side-lobe levels of about 13 dB. − The average insertion loss through the structure was 3 dB. Some of drawbacks of the design, however, include its physical thickness and fabrication complexity, as microstrip layers were soldered together at perpendicular angles.
While the above reconfigurable transmitarray designs follow the typical guided-wave approach, an alternative transmitarray design following the distributed-scatterer ap- proach has also been proposed [60, 61]. In this design, four layers of ground planes with varactor-loaded slots were used, similar in principle to the fixed coupled-ring trans- mitarray discussed previously [55]. Simulation results for a unit cell were presented, demonstrating 400◦ of phase range where the insertion loss was less than 3 dB.
In contrast to the wealth of work that has been presented for reconfigurable re- flectarrays, there have not been many investigations into reconfigurable transmitarrays, especially for beamforming applications. This motivates the need for a systematic un- derstanding of the different design approaches, as well as more experimental results to demonstrate the viability of the reconfigurable transmitarray paradigm. Chapter 2. Background 27
2.1.5 Frequency Selective Surfaces
Transmitarray and reflectarray design is also closely related to the design of frequency selective surfaces (FSS), as many FSSs are designed to behave as lenses or as reflectors.
For example, a surface made of small electrically small patches [62] was shown to improve the gain bandwidth to over 20%. A varactor diode-tuned high-impedance surface has also been proposed as a reconfigurable reflectarray [63]. FSS theory and application has been well-studied [64].
The concept of coupling two radiating structures separated by a ground plane, fol- lowing the guided-wave approach, can be found in many FSS designs, such as the spatial bandpass filter using two hexagonal patches coupled by a CPW resonator [65]. Alterna- tively, FSSs designs have also been proposed following the distributed-scatterer approach, such as the spatial bandpass filter using layers of patches and grids [66]. Because most
FSS designs are uniform, they are more applicable for spatial filtering applications than for beamforming, where the elements need to be varied across the aperture to produce a phase gradient.
Reconfigurable FSSs have also been proposed for lensing applications, such as a tun- able liquid crystal frequency selective surface (FSS) [67], and a varactor diode-based tunable FSS [68]. While these surfaces are reconfigurable, it is the operating frequency of the entire lens that was tuned in these designs. That is, phase tunability was not a de- sign objective, and the elements of the arrays could not be individually tuned. Thus, the surfaces are not suitable for beamforming applications where phase control is required.
Moreover, the structures do not have sufficient order to achieve 360◦ of phase tunability with acceptable insertion loss.
While many of the concepts between reconfigurable transmitarrays and FSSs are similar, the primary difference is that while the array elements of transmitarrays and
λ reflectarrays are typically resonant with sizes on the order of 2 , the element sizes of FSSs λ are electrically small (less than 10 ). The two major consequences are mutual coupling Chapter 2. Background 28 between the elements and element controllability.
Firstly, because FSS elements are very close together, their fields and currents are tightly coupled. Because of this, array beamforming, which is well understood, cannot be directly applied because it assumes that array elements can be individually manipulated.
When elements are highly coupled, it is very difficult to determine the required element tuning to produce a desired far-field pattern. While it may be possible for some FSS structures to produce a pencil beam where a linear phase gradient is desired, accurate beamshaping, or the production an arbitrary far-field pattern is next to impossible with high mutual coupling.
Secondly, due to the high element densities in FSSs, individual element control is very difficult in practice because of the large number of elements, and because there is very little space for control circuitry between the elements. Furthermore, the ratio of achievable far-field tunability versus complexity is much lower than that of antenna arrays, making FSSs less practical for beamforming applications.
2.2 Scattering Behavior of Transmitarrays
At this point we will briefly clarify a notion that has arisen in literature on the maximum proportion of incident power that an antenna can receive. This is pertinent to trans- mitarrays because an ideal transmitarray must receive 100% of the power on one side and re-radiate all of the power on the other side. A maximum receive antenna efficiency would impose a fundamental limit on the efficiency of a transmitarray. Intuitively, one could argue that any transmitarray structure necessarily scatters some power because it is a physical structure in the presence of incident waves, and thus the efficiency cannot be 100%. Because transmitarray elements are antennas, and antennas are known to have a scattering cross-section, some of the power must be scattered.
Thevenin or Norton equivalent circuits have been used to model receiving antennas Chapter 2. Background 29
[69], as shown in Figure 2.4, where Ya represents the admittance of the antenna, YL is the load, and Ig is the current that is generated when the antenna is excited. By this argument, maximum power is delivered to the load when it is conjugately matched to the antenna impedance, YL = Ya∗, resulting in a maximum receivable power of 50%.
Figure 2.4: Equivalent Norton circuit for a receiving antenna
This misconception was rigorously treated [70] by showing that while electric dipoles have a maximum receive efficiency of 50%, the theoretical maximum receive efficiency is
100% for electric dipoles over a ground plane, or a combination of electric and magnetic dipoles. Essentially the misconception is due to the fact that the equivalent circuit cannot fully capture the behavior the receiving antenna in free-space, especially with antennas containing ground planes. Therefore, following the guided-wave approach, if appropriate ground-backed antennas are used for the receive and transmit antennas, then theoretically it is possible to achieve zero return loss and insertion loss through the structure, as has been demonstrated in many transmitarray designs.
We note that this result does not apply to the distributed-scatterer approach, when the structure does not behave as an antenna. For example, consider an array where each element consists of two passive dipoles of different lengths. If the dipole lengths and spacings are tuned such that the waves that they scatter have equal magnitudes but opposite phases, then there will be zero power reflected. In this way, the transmitarray can achieve 100% transmission efficiency using only electric dipoles. This will be discussed in more detail in Chapter 4. Chapter 3
Relevant Background Theory
In this chapter, we will present the theory used to analyze and design transmitarrays.
In the first section, we will define the coordinate systems that will be used, and then we will introduce the vector potential and wave equations.
In the second section, we will present theory that is used for analyzing a finite radiat- ing structure in free-space. Beginning with a point source and far-field approximations, we will develop the radiation characteristics for a finite aperture, followed by the defini- tion of metrics for evaluating antennas such as directivity, gain, and loss. This theory is relevant for the experimental results for arrays that will be presented in later chapters.
Following, the third section presents the theory for analyzing infinite periodic aper- tures, such as an infinite antenna array. Unlike the finite case, far-field approximations cannot be made for infinite structures. Rather, Floquet analysis and waveguide Green’s functions are used. The theory for infinite periodic structures is relevant for unit cell design and characterization.
While in practice, a unit cell is first characterized prior to experimentation with full arrays, we have opted to review the theory of general finite radiating structures first because the theory can be presented with a better logical flow.
30 Chapter 3. Relevant Background Theory 31
3.1 Preliminaries
3.1.1 Coordinate Systems
In this thesis, three coordinate systems will be used. Note that in this thesis, we will only be considering lineary ˆ-polarized array elements.
Array and Element Coordinate System
When discussing an array element or the array locally, the Cartesian coordinate system will be used. We will always assume that both the incident and outgoing waves are propagating in the +ˆz-direction, and that the electric field is always polarized in they ˆ- direction, as shown in Figure 3.1. Unless otherwise specified, the transmitarray is always in the xy-plane and centered at the origin.
Figure 3.1: Array coordinate system
Spherical Coordinate System
The spherical coordinate system is used for describing radiated waves in the far-field, when the distance between the observation point and the source is very large. Following standard notation, r, θ, and φ, will be used for spherical coordinates, as shown in Figure Chapter 3. Relevant Background Theory 32
3.2. The relationship with Cartesian coordinates is given by
z x cos θ = tan φ = . (3.1) x2 + y2 + z2 y
In the spherical coordinate system, the unit vectors are
rˆ =x ˆ sin θ cos φ +ˆy sin θ sin φ +ˆz cos θ (3.2a)
θˆ =x ˆ cos θ cos φ +ˆy cos θ sin φ zˆsin θ (3.2b) − φˆ = xˆ sin φ +ˆy cos φ. (3.2c) −
Figure 3.2: Beamforming coordinate system
Measurement Coordinate System
The horizontal coordinate system, which uses elevation and azimuth angles to specify a direction, is used for measured array beamforming results. This is in following with what is common in the antenna measurement community. While similar to the spherical coordinate system, the azimuth is not to be confused with the azimuthal angle φ, and the elevation is not to be confused with the polar angle θ. In the measurement coordinate system, the array is arranged such that broadside, or the z-axis of the array, aligns with
0◦ in azimuth and elevation, and the y-axis of the array is aligned with 90◦ elevation. As shown in Figure 3.3, the relationship between the azimuth (Az) and elevation (El) and Chapter 3. Relevant Background Theory 33 the two other coordinate systems is given by y sin θ sin φ tan El = = (3.3a) √x2 + z2 sin2 θ cos2 φ + cos2 θ x sin θ cos φ tan Az = = . (3.3b) z cos θ
Figure 3.3: Measurement coordinate system
3.1.2 Vector Potential and Wave Equations
A fundamental approach for determining the electric field due to a current density J is through the use of a magnetic vector potential A, defined as
B = A. (3.4) ∇ × To understand why it is advantageous to use the vector potential, we will develop the wave equation for the vector potential from Maxwell’s equations. Assuming simple media where the electric and magnetic flux densities are linearly related to their respective field intensities (D = ǫE and B = H), Maxwell’s equations are
H = J + jωǫE (3.5) ∇ × E = jω H. (3.6) ∇ × − Chapter 3. Relevant Background Theory 34
Taking the curl of both sides and substituting, we get two wave equations for the electric and magnetic fields
H k2H = J (3.7) ∇×∇× − ∇ × E k2E = jω J (3.8) ∇×∇× − − where k = ω√ ǫ. Because any field can be decomposed into the sum of a curl and a solenoidal (curl-free) field, we can express the electric field as
E = ψ jωA (3.9) −∇ − where ψ is the electric scalar potential, producing a solenoidal field ψ. A trivial substi- ∇ tution of (3.4) and (3.9) into (3.6) reveals that Maxwell-Faraday’s equation is satisfied.
Substituting (3.9) into Maxwell-Ampere’s equation (3.5) gives
H = J + jωǫE ∇ × 1 ( A) = J + jωǫ( ψ jωA) ∇ × ∇ × −∇ − J = A jω ǫ ψ + ω2 ǫA − −∇ × ∇ × − ∇ J = ( A) 2A jω ǫ ψ + k2A. (3.10) − − ∇ ∇ −∇ − ∇ Since (3.4) only defines the curl of A, the divergence of A can be arbitrarily assigned.
Defining the divergence using the Lorentz gauge, where A = jω ǫψ, we can simplify ∇ − the wave equation for the magnetic vector potential to
2A + k2A = J. (3.11) ∇ −
With the Lorentz gauge, the resulting expression for the electric field becomes
jω E = k2 + ( ) A. (3.12) − k2 ∇ ∇ In Cartesian coordinates, the vector wave equation (3.11) simplifies into three scalar Chapter 3. Relevant Background Theory 35 equations, one for each component
2A + k2A = J (3.13a) ∇ x x − x 2A + k2A = J (3.13b) ∇ y y − y 2A + k2A = J . (3.13c) ∇ z z − z The vector potential is particularly useful because in free-space and some other scenarios, the components of A correspond directly to the components of J [71] (p. 141)
Jx(r′) Ax(r)= dV ′ (3.14a) 4π ′ r r V | − ′| Jy(r′) Ay(r)= dV ′ (3.14b) 4π ′ r r V | − ′| Jz(r′) Az(r)= dV ′ (3.14c) 4π ′ r r V | − ′| where r = (x, y, z) is the observation (field) coordinate and r′ = (x′,y′, z′) is the source coordinate, as shown in Figure 3.4. This means that for a current density directed only in thex ˆ,y ˆ,orz ˆ-direction, (3.11) can be reduced to a single scalar equation.
Figure 3.4: Free-space radiation
Observe that in a region without free charges where E = 0, (3.6) can also be ∇ simplified to
2E + k2E = jω J. (3.15) ∇ However, it is not possible to simplify this vector equation into a scalar equation, since any current density, regardless of its orientation, produces electric fields in all three components. For this reason, it is advantageous to solve for the vector potential A, and then obtain the electric field using (3.12). Chapter 3. Relevant Background Theory 36
3.2 Radiation of Finite Structures
In this section, we will develop the theory for analyzing the radiation characteristics of a finite structure in free-space.
3.2.1 Radiation from a Point Source
Consider ay ˆ-directed point current source, or a Hertzian source, located at r′ =(x′,y′, z′) radiating in free-space and observed at r =(x, y, z), as shown in Figure 3.5. The vector potential associated with the point source is
e jk r r′ A = − | − | y.ˆ (3.16) 4π r r | − ′|
We can calculate the electric field using (3.12) as
1 ∂ ∂ E = jω 1+ A (3.17a) x − k2 ∂x ∂y y 1 ∂2 E = jω 1+ A (3.17b) y − k2 ∂y2 y 1 ∂ ∂ E = jω 1+ A . (3.17c) z − k2 ∂z ∂y y Chapter 3. Relevant Background Theory 37
1 1 In the far-field when r r′ is large, we have r r′ n r r′ n+1 , meaning that we only | − | | − | ≫ | − | need to consider terms with the smallest power of r r′ . | − |
′ 1 ∂ ∂ e jk r r E (r) = jωµ − | − | x,y 2 r r − k ∂x ∂y 4π ′ ′ | − | jωµe jk r r (x x )(y y ) − | − | − ′ − ′ ≃ − 4π r r − r r 2 | − ′| | − ′| jk r r′ jωµe− | − | 2 r r sin θ cos φ sin φ (3.18a) ≃ 4π ′ | − | ′ 1 ∂2 e jk r r E (r) = jωµ 1+ − | − | y,y 2 2 r r − k ∂y 4π ′ ′ | − | jωµe jk r r (y y )2 − | − | 1 − ′ ≃ − 4π r r − r r 2 | − ′| | − ′| jk r r′ jωµe− | − | 2 2 r r 1 sin θ sin φ (3.18b) ≃ − 4π ′ − | − | ′ 1 ∂ ∂ e jk r r E (r) = jωµ − | − | z,y − k2 ∂z ∂y 4π r r | − ′| jk r r′ jωµe− | − | (y y′)(z z′) r r −r r −2 ≃ − 4π ′ − ′ | − ′| | − | jωµe jk r r − | − | sin θ cos θ sin φ (3.18c) ≃ 4π r r | − ′|
Here, Ex,y denotes the x-component of the electric field produced by ay ˆ-directed current. Representing the components of the electric field using spherical unit vectors following
(3.2c), the far-field radiated electric fields for a point source are
E (r) 0 (3.19a) r,y ≃ jω e jk r r′ E (r) − | − | cos θ sin φ (3.19b) θ,y ≃ − 4π r r | − ′| jω e jk r r′ E (r) − | − | cos φ. (3.19c) φ,y ≃ − 4π r r | − ′| Chapter 3. Relevant Background Theory 38
Likewise, forx ˆ andz ˆ-directed currents, we have [71] (p. 661)
E (r) 0 (3.20a) r,x ≃ jω e jk r r′ E (r) − | − | cos θ cos φ (3.20b) θ,x ≃ − 4π r r | − ′| jω e jk r r′ E (r) − | − | sin φ (3.20c) φ,x ≃ 4π r r | − ′| E (r) 0 (3.20d) r,z ≃ jω e jk r r′ E (r) − | − | sin θ (3.20e) θ,z ≃ 4π r r | − ′| E (r) 0. (3.20f) φ,z ≃
Figure 3.5: Point source in free-space
3.2.2 Radiation from a Finite Aperture
Next suppose we have a rectangular aperture, region , with ay ˆ-directed current distri- S bution that is limited to the z-plane, Jy(x′,y′, 0), as shown in Figure 3.6. The current distribution on the aperture can be viewed as a sum of point sources, and we can calculate the electric field using a surface integral of (3.19b) and (3.19c) as
jk r r′ jω cos θ sin φ e− | − | Eθ(r) = Jy(x′,y′, 0) dx′dy′ (3.21a) − 4π r r′ S jk r | r′− | jω cos φ e− | − | Eφ(r) = Jy(x′,y′, 0) dx′dy′. (3.21b) − 4π r r′ S | − |
For a large observation distance r r′ , we can approximate the magnitude as | |≫| | 1 1 1 = . (3.22) r r ≃ r r | − ′| | | Chapter 3. Relevant Background Theory 39
A similar approximation cannot be used for the argument of the exponential function because the exponential function changes rapidly even for large r r′ . Instead, if we | − | represent the observation and source coordinates in spherical coordinates,
x = r sin θ cos φ y = r sin θ sin φ z = r cos θ (3.23) x′ = r′ cos φ′ y′ = r′ sin φ′ z′ =0 then we can use the first order Taylor approximation
2 2 2 r r′ = (x x ) +(y y ) + z | − | − ′ − ′ = (r sin θ cos φ r cos φ )2 +(r sin θ sin φ r sin φ )2 +(r cos θ)2 − ′ ′ − ′ ′ 2r r 2 = r 1 ′ sin θ cos(φ φ)+ ′ − r ′ − r r r′ sin θ cos(φ′ φ). (3.24) ≃ − − With these approximations, we can express the far-field electric fielddue tothey ˆ-directed current distribution, using Cartesian coordinates for the source coordinates, as
jkr jωµ cos θ sin φ e− jkr′ sin θ cos(φ′ φ) E (r) J (x′,y′, 0)e − dx′dy′ θ ≃ − 4πr y jkr S jωµ cos θ sin φ e− jk(sin θ cos φ)(r′ cos φ′)+jk(sin θ sin φ)(r′ sin φ′) J (x′,y′, 0)e dx′dy′ ≃ − 4πr y jkr S jωµ cos θ sin φ e− jkˆr r′ J (x′,y′, 0)e dx′dy′ (3.25a) ≃ − 4πr y jkr S jωµ cos φ e− jkˆr r′ E (r) J (x′,y′, 0)e dx′dy′ (3.25b) φ ≃ − 4πr y S where ˆr is a unit vector in the direction of r.
3.2.3 Radiation from a Finite Array
Suppose now that we have a number of identical radiating apertures arranged in a grid, where element i is positioned at ri′ (xi′ ,yi′, zi′) with current distribution Ji(r′) and aperture region , as shown in Figure 3.7. Then, from (3.25a) and (3.25b), the far-field radiated Si electric field is simply the summation of fields from each element
jkr arr jω cos θ sin φe− jkˆr r′ E (r) J (x′,y′, 0)e dx′dy′. (3.26) θ ≃ − 4πr y,i i i S Chapter 3. Relevant Background Theory 40
Figure 3.6: Radiating aperture in free-space
Since the elements are identical (with a scaling factor of Ii),
J (r′)= I J(r′ r′ ) (3.27) i i − i we can substitute and change the integration regions to get
jkr arr r jωµ cos θ sin φ e− jkˆr r′ Eθ ( ) IiJy(x′ xi′ ,y′ yi′, 0)e dx′dy′ ≃ − 4πr i − − i S jkr jωµ cos θ sin φ e− jkˆr (r′ r′ ) I J (x′,y′, 0)e − i dx′dy′ ≃ − 4πr i y i S jkˆr r′ = Eθ(r) Iie− i . (3.28) i
Consequently, the electric field of a single element Eθ(r) and Eφ(r) and can be factored out from the expression, and the remaining summation is commonly called the antenna array factor (AF)
jkˆr r′ AF = Iie− i . (3.29) i The radiated fields from the array can be computed with a simple multiplication of the element pattern and the AF
Earr = E(r) AF. (3.30) Chapter 3. Relevant Background Theory 41
Figure 3.7: Radiating array in free-space
3.2.4 Directivity, Beams, and Side-lobes
From the fields given by (3.63) and (3.64), the time-averaged Poynting vector gives the average radiated power density as 1 W (r)= Re E(r) H∗(r) . (3.31) rad 2 { × } For an observation distance sufficiently far away from the source, the radiated power is radially directed, so we define
W (r)= W (r) W (r) r.ˆ (3.32) rad | rad | ≈ rad Since radiated power and distance have an inverse-squared relationship, we can normalize the radiated power by the distance using the radiation intensity defined for a far-field direction ˆr =(θ,φ) as
2 U(θ,φ)= r Wrad(r). (3.33)
Integrating over a closed surface S enclosing the entire structure, we can calculate the total power radiated by 1 P = Re E(r) H∗(r) dˆr. (3.34) rad 2 { × } S The directivity D is the ratio of the radiation intensity in a particular direction to the average radiation intensity, 4πU(θ,φ) D(θ,φ)= . (3.35) Prad Chapter 3. Relevant Background Theory 42
The directivity also often refers to the peak directivity. In the far-field region, we define a beam as the fields radiated in the direction in which the directivity is maximized. When the radiating structure is made to direct all of the radiated power in a single direction, this main beam is also called a pencil beam, as shown in Figure 3.8. We will call the formation of pencil beams beamsteering.
In (3.25a) and (3.25b), if we express the current as magnitude and phase components
J and J , we have | y| y jkr ′ ′ e− jkˆr r +j Jy(r ) E(r) E˜(r) J (r′) e ds′. (3.36) ≃ r | y | S
Since E is essentially determined by r in the far-field (r′ only has a minor effect on its value), the integrand is essentially the product of two sinusoids, one of which is J(r′) | | with zero-frequency. Therefore, for a fixed r, by orthogonality of sinusoids, the far-field electric field is maximized when the argument of the exponential is zero. This means that to create a pencil beam with maximal power in the direction ˆr, the phases of the currents at each point r′ on the aperture must be
J(r′)= kˆr r′. (3.37) −
The maximum directivity that can be achieved by a rectangular aperture of dimen- sions a b is [71] (p. 672) × 4πab D = . (3.38) max λ2 The deviation in θ or φ from the main beam direction at which the power density is halved is called the half-power beamwidth (BW ). The minimum theoretical beamwidth in degrees achievable for a uniform aperture of width a is [71] (p. 672)
1 0.443λ BW = 114.6 sin− . (3.39) min a There may also be other local maxima, called side-lobes, which are not the global maxi- mum. The difference in the magnitude between the largest side-lobe and the main beam Chapter 3. Relevant Background Theory 43 is the side-lobe level. When the objective is to create a pencil beam, it is desirable to have side-lobe levels as small as possible. For a uniformly excited aperture, the best achievable side-lobe level is 13.3 dB [71] (p. 674). The side-lobe level can be improved − by using non-uniform aperture field or current magnitudes, but at the cost of an increased beamwidth and reduced directivity.
Figure 3.8: Side-lobes
It is also possible to create multiple beams directing power in multiple directions, or creating a directivity pattern with a specific shape for a particular application. We will call this process beamshaping. The term beamforming will be used to refer to both beamsteering and beamshaping.
3.2.5 Gain, Losses, and Efficiency
The gain of the transmitarray antenna configuration shown in Figure 3.9 in a particular direction (θ,φ) is the ratio between the radiation intensity achieved, and the intensity that would result if all of the power delivered to the feed antenna was radiated isotropically.
Commonly, the gain refers to the maximum gain which is achieved in the direction of the main beam. Since the main beam is also in the direction of maximum directivity, the maximum gain and directivity are related by an efficiency coefficient ε as
4πUmax 4πUmax Gmax = = εDmax = ε . (3.40) Pin Prad Chapter 3. Relevant Background Theory 44
Ideally, it is desirable to have an efficiency coefficient of unity but a number of losses, illustrated in Figure 3.9, contribute to its reduction.
Figure 3.9: Transmitarray losses
Firstly, there are dissipative losses, which includes ohmic losses due to the finite conductivity of conductors as well as dielectric losses due to the non-zero conductivity in dielectric materials. Dissipative losses also include the parasitic resistances found in lumped components such as inductors, capacitors, and varactor diodes that may exist in the antenna. There may also be a small amount of dissipative loss in the feed antenna.
Secondly, since the transmitarray is finite, inevitably some of the power radiated by the feed antenna will not be intercepted by the transmitarray, resulting in spillover loss.
Thirdly, some of the power from the feed antenna that illuminates the transmitarray may be reflected back towards the feed resulting in reflection loss. These three sources of loss reduce the total radiated power from the transmitarray, Prad.
There are also sources of inefficiency that reduce the gain, but not Prad. These losses affect the gain by reducing the maximum directivity, diverting power away from the main beam direction. As discussed earlier, (3.37) defines the phases of the fields or currents required on the aperture of the transmitarray for a pencil beam. Phase error inefficiency results when the waves produced by the aperture do not all add constructively Chapter 3. Relevant Background Theory 45 in the main beam directly. When areas of the transmitarray are illuminated with little or no power, then the effective size of the aperture is reduced. Therefore, since the maximum achievable directivity is related to the aperture size by (3.38), variations in the magnitude of the aperture fields or currents reduce the maximum gain with taper inefficiency. Finally, when the polarization of the transmitter and receiver are misaligned, power in the cross-polarization is called polarization inefficiency. Appendix C contains more details on how these values are calculated.
3.3 Infinite Arrays and Floquet Analysis
To this point, we have examined the radiation characteristics of a finite aperture with an arbitrary current distribution. For practical implementations, a large aperture is typically designed in small repeated cells, forming an array. Because a transmitarray consists of repeating unit cells, this periodicity can be leveraged for the design and analysis of the array.
To understand how a unit cell will behave in the array, it is common to assume an infinite array of identical cells. To make the results more generalizable, we will use Floquet analysis and assume that both the magnitudes and phases of the cells are periodic, but the periodicities of the magnitude and phase may not be the same. That is, the phase may be shifted by a constant value from one cell to the next. This is so that the analysis can accommodate phase-shifted cells to produce plane-waves in different directions, as shown in Figure 3.10
Figure 3.10: Angled plane-waves due to different phase and magnitude periodicities Chapter 3. Relevant Background Theory 46
Consider an infinite array ofy ˆ-directed current sources in the z-plane with spacing a b, as shown in Figure 3.11. In this thesis, we will assume a rectangular grid. For × non-rectangular grids, we refer the reader to [3]. The magnitudes of the current sources are identical, but each subsequent source in the array has a phase shift of kx0a and ky0b in the x and y directions for some constants kx0 and ky0. If the function defining the current source at the origin is f(x, y), then the total current density is
j(kx0ma+ky0nb) J (x, y)= f(x ma, y nb)e− . (3.41) y − − n m
Figure 3.11: Infinite periodic array
∞ For brevity, let denote . Taking the two-dimensional Fourier transform of n n= −∞ Jy(x, y),
∞ j(kx0ma+ky0nb) j(kxx+kyy) J˜ (k ,k )= e− f(x ma, y nb)e dx dy. (3.42) y x y − − m n −∞ The spatial shifts ma and nb result in an extra factor of ejkxmaejkynb in the Fourier transform, giving
∞ j(kx0ma+ky0nb) j(kxx+kyy) J˜ (k ,k ) = e− f(x ma, y nb)e dxdy y x y − − m n −∞ jkx0ma jky0nb jkxma jkynb = e− e− f˜(kx,ky)e e m n j(kx kx0)ma j(ky ky0)nb = f˜(kx,ky) e − e − . (3.43) m n Chapter 3. Relevant Background Theory 47
Using the identity [3] (p. 484)
2π 2mπ ejkxma = δ k (3.44) a x − a m m we have
4π2 2mπ 2nπ J˜ (k ,k )= f˜(k ,k ) δ k k δ k k . (3.45) y x y ab x y x − x0 − a y − y0 − b m n 2mπ 2nπ Defining the Floquet wavenumbers as kxm = kx0 + a and kyn = ky0 + b , the original current field can be expressed with the inverse Fourier transform as
1 ∞ j(kxx+kyy) J (x, y) = J˜ (k ,k )e− dk dk y 4π2 y x y x y −∞ 1 ∞ j(kxx+kyy) = f˜(k ,k )e− δ(k k ) δ(k k ) dk dk . ab x y x − xm y − yn x y −∞ m n (3.46)
The Dirac delta functions cause the integral to be non-zero only at discrete points, leaving
1 jkxmx jkyny J (x, y)= f˜(k ,k )e− − (3.47) y ab xm yn m,n which is the Floquet series expansion of the periodic current density [3]. Substituting the current into the vector potential wave equation (3.11), we have
2 2 δ(z) j(kxmx+kyny) A + k A = f˜(k ,k )e− . (3.48) ∇ y y − ab xm yn m,n Since the forcing function of the wave equation is periodic in x and y, we also conclude that Ay is periodic with the same periodicity by Floquet’s Theorem [72]. Therefore, we expect Ay to take the form
jkzmn z j(kxmx+kyny) Ay = cmne− | |e− (3.49) m n for some constant cmn. To satisfy radiation boundary conditions, kzmn must be positive so that the exponential term with z decays as z . Substituting this back into → ±∞ (3.48) and comparing the terms for each m and n, we get
2 ∂ jkzmn z 2 jkzmn z δ(z) c e− | | + k e− | | = f˜(k ,k ) (3.50) mn ∂z2 zmn − ab xm yn Chapter 3. Relevant Background Theory 48 where k2 = k2 k2 k2 . If we integrate both sides of the equation with respect to zmn − xm − yn z from ε to ε, then second term on the left-hand side vanishes as ε 0 because it is − → continuous, leaving
ε 2 z=ε ∂ jkzmn z ∂ jkzmn z − | | − | | ˜ cmn 2 e dz = cmn e = f(kxm,kyn). (3.51) ε ∂z ∂z z= ε −ab − − The discontinuity of 2jk in the z-derivative between z = ε and z = ε results in − zmn − j cmn = f˜(kxm,kyn). (3.52) −2abkzmn Substituting this back into (3.49), we have
1 jkzmn z j(kxmx+kyny) A (r)= j f˜(k ,k )e− | |e− . (3.53) y − 2abk xm yn m n zmn We can find the expressions for the electric field with (3.12) to be [3]
1 kxmkyn jkzmn z j(kxmx+kyny) E (r) = f˜(k ,k )e− | |e− (3.54a) x 2abωǫ k xm yn m n zmn k2 k2 1 yn jkzmn z j(kxmx+kyny) E (r) = − f˜(k ,k )e− | |e− (3.54b) y −2abωǫ k xm yn m n zmn 1 jkzmn z j(kxmx+kyny) E (r) = k f˜(k ,k )e− | |e− . (3.54c) z 2abωǫ yn xm yn m n It is easy to see that the electric field is simply a summation of plane waves. Each plane wave with a specific m and n propagates in the directionxk ˆ +yk ˆ zkˆ , xm yn ± zmn and represents a component in the spatial frequency domain. In literature on FSS and periodic structures, this expansion of the electric field into a superposition of plane waves is also called the spectral expansion [64].
Unlike a finite aperture where some power is radiated in every direction, an infinite periodic aperture only radiates plane waves in discrete directions for specific integers m and n. The direction of propagation for each plane wave can also be expressed in spherical coordinates with θmn and φmn as
kxm = k sin θmn cos φmn (3.55a)
kyn = k sin θmn sin φmn (3.55b)
kzmn = k cos θmn (3.55c) Chapter 3. Relevant Background Theory 49 where cos θ = kzmn and tan φ = kxm . When k2 + k2 >k2, then k is necessarily mn k mn kyn xm yn zmn imaginary. In this case, the component is evanescent, and does not propagate because the electric field decays exponentially in z. Figure 3.12 illustrates the directions of radiated plane waves for different m and n = 0.
Figure 3.12: Radiated plane waves from an infinite periodic aperture
The dominant Floquet mode is the case when m = n = 0, where the constants kx0 and ky0 defined earlier produce a beam in the direction (θ0,φ0). In this case, kz00 =
2 2 2 kz00 kx0 k k k , and cos θ0 = and tan φ0 = . When high directivity is a design − x0 − y0 k ky0 objective, as in the case of transmitarrays, the dominant mode corresponds to the main beam of the antenna. In this case, only the dominant mode should propagate, and all other components should be evanescent so that power is only radiated in a single direction. Propagating non-dominant spectral components are similar to grating lobes in the design of finite antenna arrays.
We can adjust the cell spacing a and b such that only the dominant mode propagates
2π 2π 2 by selecting a and b to be less than one wavelength. If a > λ = k, then kxm = 2π 2 2 2 kx0 + a is less than k for only one value of m, and likewise for kyn and n. In this way, only one specific m, n component of the electric field propagates.
The element currents defined by f(x, y) affects the magnitude and phase of the plane wave in each direction, but it does not affect the directions in which the plane waves propagate. For an array of point radiators where f(x, y)= δ(r r′), the expressions can − Chapter 3. Relevant Background Theory 50
˜ be simplified with f(kxm,kyn) = 1.
3.3.1 Modeling Free-Space Propagation with Waveguides
Recall that the purpose of using Floquet analysis was so that the behavior of an infinite array of identical elements can be characterized by examining a single element. From the solution of vector potential (3.53), we can see that
jkxma jkx0a Ay(x + a, y, z) = Ay(x, y, z)e− = Ay(x, y, z)e− (3.56a)
jkynb jky0b Ay(x, y + b, z) = Ay(x, y, z)e− = Ay(x, y, z)e− . (3.56b)
If we limit the domain of interest to 0 x a and 0 y b, then the field solution of ≤ ≤ ≤ ≤ this finite region (finite in x and y) is identical to that of the infinite array in free-space, provided that the fields are continuous and smooth at the boundaries, as shown in Figure 3.13. That is, the boundary conditions at x = 0 and x = a, and y = 0 and y = b are
∂nA(0,y,z) ∂nA(a, y, z) = ejkx0a (3.57a) din din ∂nA(x, 0, z) ∂nA(x, b, z) = ejky0b (3.57b) din din for all z and integers n, where i is x, y, or z. This finite region is a waveguide with periodic boundary conditions, and in the following sections we will present Green’s functions, which are very useful for the analysis of fields in waveguides.
Figure 3.13: Identical field solutions for freespace and periodic waveguide Chapter 3. Relevant Background Theory 51
3.3.2 Green’s Functions
Up to this point, we have presented theory that allows the vector potential A and electric
field E to be directly calculated from an arbitrary current distribution J. However, for linear time-invariant systems, it is possible to completely characterize the behavior of the system with its response to a spatial impulse excitation. Using this technique, the process of calculating A or E from J can be decomposed into two steps where the more complex step of characterizing the system only needs to be performed once.
First, A or E is solved using Maxwell’s equations for a spatial impulse excitation, taking into consideration all of the boundary conditions of the system. The relationship between the point excitation at position r′ and the fields observed at position r is cap- tured by a Green’s function, commonly denoted by G(r r′). To solve the wave equation | (3.11), the forcing function is J is set to the Dirac delta function δ(r r′), oriented in − a specific direction. While in simple environments such as free-space, a current Jx in thex ˆ-direction only produces a vector potential Ax in the same direction, it may pro- duce vector potentials in all three components of A in more complex environments when scatterers or dielectrics are present. Therefore, since J is in three dimensions, we need to solve for the three components of A for each dimension of excitation. Conveniently, we can represent this as a rank two tensor called the dyadic Green’s function. In matrix ¯ form, G¯ (r r′) can be written as | ¯ G¯ (r r′) = G (r r′)ˆxxˆ + G (r r′)ˆxyˆ + G (r r′)ˆxzˆ (3.58a) | xx | xy | xz |
+G (r r′)ˆyxˆ + G (r r′)ˆyyˆ + G (r r′)ˆyzˆ (3.58b) yx | yy | yz |
+G (r r′)ˆzxˆ + G (r r′)ˆzyˆ + G (r r′)ˆzzˆ (3.58c) zx | zy | zz | where each component G (r r′) gives the magnetic vector potential in direction i at ij | position r, produced by a point excitation in direction j at r′. The dyadic Green’s function is the solution to the tensor equation
2 ¯ 2 ¯ ¯ G¯ (r r′)+ k G¯ (r r′)= ¯Iδ(r r′) (3.59) ∇ | | − − Chapter 3. Relevant Background Theory 52 where ¯I is the unit dyadx ˆxˆ +ˆyyˆ +ˆzzˆ. The second step of determining A or E for an arbitrary current distribution J involves a straight-forward integral using the Green’s function. If we right-multiply both sides of (3.59) by J(r′) and integrate over the entire volume V containing currents, then we have a tensor version of the wave equation (3.11), and it is easy to see that
¯ A(r)= G¯ (r r′) J(r′)dv′. (3.60) | V Alternatively, we can also calculate the electric field using (3.12) as
jω 2 ¯ E(r)= k + ( ) G¯ (r r′) J(r′)dv′ (3.61) − k2 ∇ ∇ | V and define the electric Green’s function as
¯ e ¯ 1 ¯ G¯ (r r′)= ¯I + ( ) G¯ (r r′) (3.62) | k2 ∇ ∇ | where the electric and magnetic fields are given by
¯ e E(r)= jω G¯ (r r′) J(r′)dv′ (3.63) − | V 1 H(r)= E. (3.64) −jω ∇ × Green’s functions are particularly useful for analysis of fields and currents in waveg- uides because they only depend on the boundary conditions of the waveguide, and are independent of the excitation.
3.3.3 Periodic Waveguides
The dyadic Green’s function for a waveguide with periodic boundary conditions (or a periodic waveguide) is given as [73]
′ jkzmn(z z ) ′ ′ j e∓ − j(kxmx+kyny) j(kxmx +kyny ) > G (r r′) = e− e , z z′ (3.65a) xx | −2abk2 k < m n zmn G (r r′) = G (r r′)= G (r r′) (3.65b) yy | zz | xx |
G (r r′) = G (r r′)= G (r r′)= G (r r′)= G (r r′)= G (r r′) = 0(3.65c) xy | xz | yx | yz | zx | zy | Chapter 3. Relevant Background Theory 53 where mπ k = + k sin θ cos φ (3.66a) xm a 0 0 nπ k = + k sin θ sin φ . (3.66b) yn b 0 0 As expected, it is identical to the vector potential solution of the infinite periodic aperture given in (3.53) when multiplied by jω . − In the simplified two-dimensional case with ay ˆ-directed current source that has no dependence on y, and a dominant mode or main beam directed at broadside (θ0 = φ0 = 0), we have the simplified electric dyadic Green’s function
′ jkzm(z z ) e j e∓ − G (r r′)= (cos(k x) cos(k x′) + sin(k x) sin(k x′)) (3.67) yy | −2a k x x x x m zm where mπ k = (3.68) x a and
k = k2 k2. (3.69) zm − x Note that in this case, there is no n summation, and since the source only produces an electric field in they ˆ-direction, all other components of the Green’s function are zero.
Physically, it is not possible to experimentally realize a periodic waveguide, and peri- odic boundary conditions only have limited support in commercial simulation packages. However, if the element only radiates withy ˆ-directed currents and is symmetric in both x and y, then it is possible to simulate certain cases of the periodic waveguide with rectangular or parallel-plate waveguide. Therefore, in the following sections we present the Green’s functions for these two waveguide types, and explain the significance of the waveguide propagation modes in relation to the spectral components of the infinite array.
3.3.4 Rectangular Waveguide
Rectangular waveguides are of interest because they are readily available in many differ- ent sizes. Consider a rectangular waveguide with walls that are perfect electric conductors Chapter 3. Relevant Background Theory 54
(PEC), as shown in Figure 3.14. All waves inside a rectangular waveguide can be de- composed into transverse electric (T E) or transverse magnetic (T M) modes. T E and
T M modes contain no electric and magnetic field components in the transverse direction
vp (ˆz), respectively. At frequencies below the cutoff frequency, fc = 2a , where vp is the phase velocity in the medium filling the waveguide, waves will only propagate in the
T E10 mode and all higher-order modes are evanescent and will decay exponentially inz ˆ
[74]. It is possible to perceive the T E10 wave as a pair of plane-waves reflecting inside the ˆ waveguide, with each direction k making an angle θTE10 with the waveguide walls. Since the propagation constant in for the T E mode is β = k2 π 2, we can calculate the 10 − a angle using β and the free-space wavenumber k
β cos θ = . (3.70) TE10 k
Figure 3.14: Rectangular waveguide T E10 mode and equivalent array and plane waves
From image theory, we can remove the PEC walls, and add the virtual sources created by they ˆ-directed current, as shown in Figure 3.14. The virtual sources create an infinite Chapter 3. Relevant Background Theory 55 array of alternating current sources which will produce plane waves at an angle of θ . ± TE10 If the structure of the cell is symmetric inx ˆ andy ˆ, we can characterize the behavior of an infinite array using a single cell in a rectangular waveguide [75].
With respect to infinite array case, because adjacent cells in x have phase shifts of
180◦, the T E10 mode corresponds to the Floquet wave numbers kx0a = k sin θ0 = π and ky0 = 0. As one would expect, θ0, the angle that the main beam makes with the normal
of the array plane, is exactly equal to θTE10 .
In order to use waveguide characterization, we require that only the T E10 mode propagates in the waveguide, so an appropriately-sized waveguide must be used to prevent other modes from propagating. The propagating waves in the different waveguide modes correspond directly to plane waves produced in different directions in the equivalent infinite array. If the cell size a is such that f < vp (assume for simplicity that b a), a ≪ then additional modes such as T E20 may propagate. In the corresponding infinite array, additional plane waves in different directions will also be produced. To prevent the T E20 mode from propagating, we require that a<λ. This is the same requirement to make all non-dominant modes evanescent in the infinite array case.
We are interested using a single quantity to represent the transmission response through a transmitarray. For an infinite periodic structure, we can use a transmission coefficient, which is a ratio of the transmitted and incident fields. Likewise, we can use a reflection coefficient, which is the ratio between the reflected and incident fields, to characterize the reflection. When only the T E10 mode is propagating in the waveguide,
+ then for an incident wave E1 , we can define scattering parameters (S-parameters) with ratios of the electric field waves shown in Figure 3.15
E1− S11 = + (reflection) (3.71a) E1
E2− S21 = + (transmission). (3.71b) E1
These S-parameters give the transmission and reflection characteristics of an infinite Chapter 3. Relevant Background Theory 56
array of identical cells, with an angle of incidence of θTE10 .
Figure 3.15: Waveguide ports for scattering parameters
To obtain the electric field using a Green’s function, consider the point current source in a rectangular waveguide shown in Figure 3.14. There are two general approaches for developing the Green’s function for rectangular waveguide. One method is to solve for the fields directly from the boundary conditions of the waveguide [76]. The other method is to derive how an impulse excites specific T E and T M modes [77]. While both methods result in the essentially the same Green’s function, we include both methods because the two derivation approaches will allow us to extend the Green’s functions later.
The electric Green’s function given in [76] is
e ′ ¯ j 1 jkz(z z ) G (r r′) = e∓ − | −2abk2 k m n z 2 2 (k k ) cos(k x) sin(k y) cos(k x′) sin(k y′)ˆxxˆ − x x y x y k k sin(k x) cos(k y) cos(k x′) sin(k y′)ˆyxˆ − x y x y x y
jk k sin(k x) sin(k y) cos(k x′) sin(k y′)ˆzxˆ ∓ x z x y x y
k k cos(k x) sin(k y) sin(k x′) cos(k y′)ˆxyˆ − x y x y x y 2 2 (k k ) sin(k x) cos(k y) sin(k x′) cos(k y′)ˆyyˆ − y x y x y
jk k sin(k x) sin(k y) sin(k x′) sin(k y′)ˆzyˆ ∓ y z x y x y
jk k cos(k x) sin(k y) sin(k x′) sin(k y′)ˆxzˆ ± x z x y x y
jk k sin(k x) cos(k y) sin(k x′) sin(k y′)ˆyzˆ ± y z x y x y 2 2 (k k ) sin(k x) sin(k y) sin(k x′) sin(k y′)ˆzzˆ (3.72) − z x y x y Chapter 3. Relevant Background Theory 57 where k = mπ , k = nπ , k2 = k2 + k2, k = k2 k2 1. x a y b c x y z − c The electric Green’s function given in [77] is
¯ e δ(r r′) j ∞ 1 G¯ (r r′) = − zˆzˆ M¯ ( k )M¯ ′ ( k ) | − k2 − ab k2k emn ± z emn ∓ z m=1 n c z +N¯ ( k )N¯ ′ ( k ) (3.73) omn ± z omn ∓ z where
hz cos(kxx) cos(kyy)e− ¯ e Momn(h) = zˆ (3.74a) ∇ × hz sin(kxx) sin(kyy)e− 1 N¯e (h) = M¯ e (h) (3.74b) omn k ∇ × omn
M¯ e′ (h) = M¯ e (h) (3.74c) omn omn x=x′,y=y′ N¯e′ (h) = N¯e (h) . (3.74d) omn omn x=x′,y=y′ Note the that the notation j = i and has been substituted. Here, M¯ and N¯ are − emn omn vector basis functions for T E and T M modes.
The primary difference between (3.72) and (3.73) is an impulse function in thez ˆzˆ component of (3.73). This term only has an effect when the observation point coincides exactly with source currents. The approach resulting in (3.72) can easily be generalized for other boundary conditions. However, the decomposition into T E and T M modes in (3.73) is necessary to model the presence of dielectric slabs inside the waveguide, as we will discuss further in Chapter 4.
3.3.5 Parallel-Plate Waveguide
While the use of rectangular waveguide allows us to simulate the behavior of an array
using a single element, the characterization is at an angle of θTE10 . Practically, it may be more likely that a transmitarray will be used to beamform at an angle closer to broadside, and so it would be more useful to have a broadside characterization of the array.
1Note that the semi-infinite summation was replaced with an infinite summation, forgoing the need for Neumann numbers. Chapter 3. Relevant Background Theory 58
To achieve this, consider the parallel-plate waveguide illustrated in Figure 3.16, where the top and bottom walls are PEC and the two side walls are perfect magnetic conductors
(PMC). By image theory, a virtual array of identical elements is simulated. Similar to the rectangular waveguide case, the fundamental mode of the parallel-plate waveguide, which is the transverse electric/magnetic (T EM) mode, is equivalent to the broadside plane- waves in the virtual array. Here, in terms of the Floquet wavenumbers of the infinite array case are kx0 = ky0 = 0, and correspond to a dominant mode with θ0 = φ0 = 0.
We note that this definition of a parallel-plate waveguide may differ from another com- mon use of the term. Often, a parallel-plate waveguide refers to a quasi-two-dimensional structure where the width a is infinite, resulting in no x dependence in the fields. In our case, where PMC walls are used, the structure supports fields with an x dependence.
Figure 3.16: Parallel-plate waveguide T EM mode and equivalent array and plane waves
While this parallel-plate waveguide is easy to implement in simulation, PMC is not easy to implement experimentally. For a limited bandwidth, it is possible to implement the PMC using a high-impedance surface [78]. Chapter 3. Relevant Background Theory 59
To develop the Green’s function for this parallel-plate waveguide, we can general- ize the bases from [77] given in (3.74a)-(3.74d) for both rectangular and parallel-plate waveguide using Ψ( ), where sin( ), Rectangular waveguide Ψ( )= (3.75) cos( ), Parallel-plate waveguide giving
hz M¯ (h) = Ψ′(k x) cos(k y)e− emn ∇× x y hz hz = xkˆ Ψ′(k x) sin(k y)e− +yk ˆ Ψ(k x) cos(k y)e− (3.76a) − y x y x x y hz M¯ (h) = Ψ(k x) sin(k y)e− omn ∇× x y hz hz =xk ˆ Ψ(k x) cos(k y)e− ykˆ Ψ′(k x) sin(k y)e− (3.76b) y x y − x x y 1 N¯ (h) = M¯ (h) omn k ∇× omn kxh hz kyh hz = xˆ Ψ′(k x) sin(k y)e− yˆ Ψ(k x) cos(k y)e− − k x y − k x y 2 kc hz +ˆz Ψ(k x) sin(k y)e− . (3.76c) k x y Next, define the tensor bases
¯ M± = M¯ ( k )M¯ ′ ( k ) mn emn ± z emn ∓ z k2 k k 0 y − x y jkzz ¯ = e∓ k k k2 0 S (3.77a) x y x ∗ − 0 0 0 ¯ N± = N¯ ( k )N¯ ′ ( k ) mn omn ± z omn ∓ z 2 2 2 2 kxkz kxkykz jkc kxkz jkzz ± e∓ ¯ = k k k2 k2k2 jk2k k S (3.77b) k2 x y z y z c y z ∗ ± 2 2 4 jkc kxkz jkc kykz kc ∓ ∓ where denotes component-wise multiplication, and ∗ T Ψ′(kxx) sin(kyy) Ψ′(kxx′) sin(kyy′) ¯ S = Ψ(kxx) cos(kyy) Ψ(kxx′) cos(kyy′) . (3.78) Ψ(kxx) sin(kyy) Ψ(kxx′) sin(kyy′) Chapter 3. Relevant Background Theory 60
Because the T EM mode cannot be decomposed into T E and T M modes, we need to add a term to the electric Green’s function given in (3.73) for the T EM mode. If we proceed through the derivation of (3.72) with PMC side wall boundary conditions, then for the T EM mode when m = n = 0, we have the additional term
j 0πx 0πy 0πx′ 0πy′ jkz jΨ(0) jkz − cos cos cos cos e∓ yˆyˆ = − e∓ yˆy.ˆ (3.79) 2abk a b a b 2abk We can then rewrite the electric Green’s function for both rectangular or parallel-plate waveguide as
¯ e δ(r r′) jΨ(0) jkz G¯ (r r′) = − zˆzˆ e∓ yˆyˆ | − k2 − 2abk j 1 M¯ ± + N¯ ± , z>0 (3.80) −2ab k2k mn mn < m n c z where the argument of the summation is zero when m = n = 0. When a structure is symmetric in both x and y, then the cell inside a parallel-plate waveguide characterizes the behavior of an infinite array with broadside incidence.
To briefly summarize, in this section we have presented the Floquet analysis for determining the fields radiated by an infinite periodic aperture. Following, we explained how an infinite periodic array can be simulated by a waveguide with periodic boundary conditions, and introduced the use of Green’s functions for waveguide analysis. Finally, we presented the analysis of two types of waveguides, rectangular and parallel-plate waveguides, which simulate two specific cases of the infinite array. We will build on this theory later in our discussion of the distributed-scatterer approach to transmitarrays in Chapter 4. Chapter 4
Distributed-Scatterer Approach
In this chapter, we will explore the distributed-scatterer approach to transmitarrays. In the first section, we will investigate the use of generalized layered scattering surfaces as a transmitarray. We will consider uniform scattering surfaces, followed by graduated-phase surfaces for beamsteering. In the second section, we will describe a method for analyzing discretized layered structures that uses the Method of Moments (MoM). Finally, we will present a transmitarray unit cell design based on loaded dipoles, with numerical, simulated, and experimental results.
4.1 Layered Scattering Surfaces for Beamsteering
As mentioned in Chapter 1, a lens can be perceived as a collection of passive scatterers distributed in space, as illustrated in Figure 1.6. When the scatterers are small, we can view a plane of scatterers as a scattering surface, and the entire lens can be perceived as layers of closely-spaced scattering surfaces. Therefore, to analyze transmitarrays from the distributed-scatterer approach, we will examine structures consisting of layered scattering surfaces. In this discussion, we will assume that the layers are all identical and evenly spaced, although the layers can vary laterally over the aperture of the lens. We note that while many other degrees of freedom exist for distributed-scatterer designs, such as
61 Chapter 4. Distributed-Scatterer Approach 62 non-identical layer spacings and inter-layer connections, these significantly increase the complexity of the design and fabrication processes. For this reason, the scope of this chapter is limited to structures that are easy to implement.
Consider a single infinite scattering surface in the xy-plane where the effective impe- dance of the surface for an incident plane wave can be real, imaginary, or complex at the operating frequency. For simplicity, we will assume that the impedance only varies along x. We are interested in designing a lens for redirecting a plane wave beam, as shown in
Figure 4.1.
Figure 4.1: Lensing with scattering surface(s)
If we assume that the surface is illuminated by a plane wave, then since the structure has no dependence on y, it can be modeled as a two-dimensional structure with all quantities constant in y. Next, we define an admittance per unit area for the surface
Y (x). Since the surface is infinite, we also assume that Y (x) is periodic in x, with spatial period W . That is, Y (x+W )= Y (x). Following the discussion in Section 3.3 of Chapter
3, we can then model the infinite array as a finite surface inside a waveguide with periodic boundary conditions with dimensions W b where b is arbitrary, as shown in Figure 4.2. × Suppose the surface is illuminated by ay ˆ-polarized plane wave. Then all of the currents that are excited on the surface will also bey ˆ-directed. The two-dimensional Chapter 4. Distributed-Scatterer Approach 63
Figure 4.2: Surface inside a waveguide with periodic boundary conditions scalar Green’s function for a current element in this case is given by (3.67). If the scattering surface is positioned at z = z′, then the currents on the scattering surface can be approximated as a sum of P 1 sinusoids with fundamental spatial period W . − P 1 − 2pπx 2pπx J(x, z)= δ(z z′) A cos + B sin (4.1) − p W p W p=0 Combining the Green’s function and the currents, we can find the electric field anywhere in the waveguide using (3.63) as
e E(x, z) = jωµ G (x, z x′, z′)J(x′, z′)dx′ dz′ − yy | V W P 1 − ′ ωµ 1 jkzm z z 2pπx′ 2pπx′ = − e− | − | A cos + B sin 2W k p W p W 0 m zm p=0 2mπx 2mπx′ 2mπx 2mπx′ cos cos + sin sin dx′. W W W W (4.2)
By orthogonality of the sinusoids over the integration of x′, we can find the electric field produced by a single surface at z = z′, for currents defined by Ap and Bp as
P 1 ′ − e jkzp z z 2pπx 2pπx E(x, z)= ω − | − | A cos + B sin . (4.3) − 2k p W p W p=0 zp Suppose now we have N identical scattering surfaces positioned at z = 0, d, 2d, . . . , (N { − 1)d . While the impedances of the different layers are identical, the currents excited on } each may be different, and so we distinguish the currents on layer i with a superscript Chapter 4. Distributed-Scatterer Approach 64 as J i(x, z), Ai , and Bi . Next, we define 2P 1 field coordinates in x as x = mW , m = p p − m 2P 1, 2,..., 2P 1 , as shown in Figure 4.3. { − }
Figure 4.3: Arrangement of multiple surfaces and points xm
Using n to index the observation layer and i to index the summation of the source layers, we can calculate the total electric field due to the currents on all of the layers, at each point xm in layer n as
N 1 P 1 − − e jkzp n i d 2pπx 2pπx E = ω − | − | Ai cos m + Bi sin m . (4.4) mn − 2k p W p W i=0 p=0 zp The total number of coordinates N(2P 1) is chosen such that it matches the number − i i of unknowns Ap and Bp. Next, suppose that the incident field is ay ˆ-polarized broadside plane wave given by
jkz Einc = E0e− . (4.5)
Then, the total electric field at each point xm is the sum of the incident field and the fields produced by the currents, given by
Etot = Einc + Emn. (4.6)
W Now, if we divide the surface into vertical strips of current, each with width 2P , then the current in each strip at xm in layer n is
J n(x , z)W In(x , z)= m . (4.7) m 2P
We can also calculate a voltage across each strip from the electric field as V (xm, z) =
E(xm, z)b. Taking the admittance per unit area of the surface Y (x) (which is the same Chapter 4. Distributed-Scatterer Approach 65 for each layer regardless of z), we multiply by the width of the strip and divide by the length of the strip to get the effective strip admittance. This admittance is necessarily equal to the ratio of the current and voltage
n W 1 J (xm,z)W Y (x ) = 2P . (4.8) m 2P b E(x , z)b m
Combining (4.6) and (4.8), we have (2P 1) N equations (one for each point x on − × m each layer) indexed by mn and (2P 1) N unknowns An and Bn in the equation − × p p n (assuming B0 = 0) n jknd J (xm) E0e− + Emn = (4.9) Y (xm) which can be expanded to
N 1 P 1 jkzp n i d jknd − − e− | − | i 2pπxm i 2pπxm E0e− ωµ Ap cos + Bp sin − 2kzp W W i=0 p=0 P 1 1 − 2pπx 2pπx = An cos m + Bn sin m (4.10) Y (x ) p W p W m p=0 and rewritten in matrix form as
Ai mn mn p mn gip hip = [v ] (4.11) i Bp where
ω + 1 cos 2pπxm , n = i mn 2kzp Y (xm) W gip = (4.12a) −jkzp|n−i|d ω e 2pπxm cos , n = i 2kzp W ω + 1 sin 2pπxm , n = i mn 2kzp Y (xm) W hip = (4.12b) −jkzp|n−i|d ω e 2pπxm sin , n = i 2kzp W mn jknd v = E e− . (4.12c) 0
i i Solving the system of equations, we can find Ap and Bp that define the current Chapter 4. Distributed-Scatterer Approach 66 distributions. The waves propagating in the +ˆz- and zˆ-directions are given by − N 1 P 1 − − jkzp(z id) + jkz e− − i 2pπx i 2pπx E (x, z) = E e− ω A cos + B sin 0 − 2k p W p W i=0 p=0 zp (4.13a) N 1 P 1 − − jkzp(z id) e − i 2pπx i 2pπx E−(x, z) = ω A cos + B sin . (4.13b) − 2k p W p W i=0 p=0 zp We can rewrite (4.13a) and (4.13b) in the form of a complex Fourier series
P 1 − 2pπ j( W x kzpz) E±(x, z)= cp±e− − ± (4.14) p= (P 1) − − where N 1 ω − e jkzpid ± (Ai jBi ), p> 0 − 2 2k p − p i=0 zp N 1 − jkid e± i c± = E jω A , p =0 . (4.15) p 0 − 2jk 0 i=0 N 1 − jkzpid ω e± i i (A p + jB p), p< 0 − 2 2k − − i=0 zp In this form, it is easy to see that each component represents a plane wave propagating with wave vector 2pπ k = xˆ k zˆ (4.16) − W ± zp where the direction of propagation (for propagating waves) is
1 2pπ 1 2pπ θ = tan− = sin− (4.17) p −Wk kW zp with θp = 0 at broadside from the array, as shown in Figure 4.1. In this formulation, positive p corresponds to negative θp angles. We can calculate the power in each beam with
1 2 = c± cos θ . (4.18) Pp η | p | p
There is a normalization factor of cos θp so that power is conserved. Because we are dealing with an infinite array meaning that there is infinite input power, conservation of power must be carefully treated. To illustrate why the normalization factor is needed, Chapter 4. Distributed-Scatterer Approach 67
consider the outgoing wave with angle θp shown in Figure 4.4. The segment of the outgoing plane wave in 0 x W is actually produced by a section of incoming plane ≤ ≤ wave with width W . Therefore, in order to properly characterize the power of the cos θp outgoing wave, we need to normalize it by cos θp.
Figure 4.4: Power normalization
2π If W is sufficiently small such that W > k, then only kz0 is real, meaning that only the fundamental mode propagates. In this case, we can define scattering parameters. If we define ports at a large distance away from the surface, then all fields with p = 0 are evanescent, and we can calculate the de-embedded scattering parameters to be
N 1 ω − S = E ejkidAi (4.19a) 21 0 − 2k 0 i=0 N 1 − ω jkid i S = e− A . (4.19b) 11 − 2k 0 i=0 4.1.1 Uniform Scattering Surface
We first need to investigate how many layers are required, and what is the optimal spacing between the layers. By observing how thin lenses and arrays function, we can infer that we need to manipulate the phase shift through the structure in order to control Chapter 4. Distributed-Scatterer Approach 68 the direction of the outgoing plane wave. To produce a lossless structure, the impedance of the surface must be purely imaginary at the operating frequency, meaning that we are only manipulating the susceptance of the surface.
If the excitation is a plane wave and the surface is uniform, then the currents on
i i the surface will be uniform as well. That is, in Ap = Bp = 0 for all p > 0 in (4.10). In this case, we can obtain an exact solution for the currents on each layer because the currents are not approximated by a finite sum of sinusoids. Therefore, (4.10) simplifies to N equations with N unknowns N 1 n − A0 η0 jk n i d i jknd + e− | − | A = E e− . (4.20) Y 2 0 0 i=0 Sweeping the susceptance for different numbers of layers N, each separated by a quarter-wavelength, we obtain the S-parameter plots shown in Figure 4.5. The value
λ of d = 4 is chosen as a starting point following existing distributed-scatterer designs [55, 61]. The shading denotes regions where the insertion loss is better than 3 dB.
Examining the progression of the transmission magnitudes and phases in Figure 4.5, we can see that as the number of layers is increased, phase range also increases with the number of layers, as expected. In order to achieve 360◦ of phase range with insertion loss better than 3 dB, we can see that we require N 4. This observation that at least four ≥ layers is required is consistent with existing layered designs [55, 61].
Figure 4.7 shows the effects of layer spacing d for N =4, 6, 8. As the layer spacing is
λ swept from 0 to 2 , resonances, or peaks in the response, move from very large values of B (right-hand side) to very negative values of B (left-hand side), passing by B = 0 when
λ . Because the resonances produce the peaks in S , when the resonances are far apart, 4 | 21| the regions where the insertion loss is better than 3 dB become discontinuous. In order to have 360◦ of phase range, these regions need to be continuous. The configurations where 360◦ or more phase range is achieved have horizontal dotted lines denoting the phase range.
To gain a more complete understanding of how N and d affect the structure, we wish Chapter 4. Distributed-Scatterer Approach 69
0 100 0 150
100 −5 50 −5 50 | | (deg) (deg) 21 −10 0 21 −10 0 21 21 |S |S S S
∠ −50 ∠ −15 −50 −15 −100
−20 −100 −20 −150 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S)
(a) N =1 (b) N =2
0 −100 0 0
−200 −5 −5 −200
−300 | | (deg) (deg) 21 −10 21 −10 −400 21 21 |S |S
−400 S S ∠ ∠ −15 −15 −600 −500
−20 −600 −20 −800 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S)
(c) N =3 (d) N =4
0 200 0 200
0 0 −5 −5 −200 −200 | | (deg) (deg) 21 −10 21 −10 −400 21 21 |S |S
−400 S S
∠ −600 ∠ −15 −15 −600 −800
−20 −800 −20 −1000 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S)
(e) N =5 (f) N =6
λ Figure 4.5: The effects of the number of layers for d = 4 Chapter 4. Distributed-Scatterer Approach 70
0 200 0 200 0 0
100 −5 100 −5 −5 −200 0 | | | (deg) (deg) (deg) 21 −10 0 21 −10 −100 21 −10 −400 21 21 21 |S |S |S S S S
∠ −200 ∠ ∠ −15 −100 −15 −15 −600 −300
−20 −200 −20 −400 −20 −800 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S) Susceptance (S)
(a) N =4, d =0.05λ (b) N =6, d =0.05λ (c) N =8, d =0.05λ
0 0 0 0 0 500
−200 −5 −200 −5 −5 0 −400 | | | (deg) (deg) (deg) 21 −10 −400 21 −10 −600 21 −10 −500 21 21 21 |S |S |S S S S
∠ −800 ∠ ∠ −15 −600 −15 −15 −1000 −1000
−20 −800 −20 −1200 −20 −1500 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S) Susceptance (S)
(d) N =4, d =0.15λ (e) N =6, d =0.15λ (f) N =8, d =0.15λ
0 0 0 200 0 0
0 −5 −200 −5 −5 −200 −500 | | | (deg) (deg) (deg) 21 −10 −400 21 −10 −400 21 −10 21 21 21 |S |S |S S S S
∠ −600 ∠ −1000 ∠ −15 −600 −15 −15 −800
−20 −800 −20 −1000 −20 −1500 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S) Susceptance (S)
(g) N =4, d =0.25λ (h) N =6, d =0.25λ (i) N =8, d =0.25λ
Figure 4.6: The effects of layer spacing Chapter 4. Distributed-Scatterer Approach 71
0 200 0 0 0 500
−200 −5 0 −5 −5 0 −400 | | | (deg) (deg) (deg) 21 −10 −200 21 −10 −600 21 −10 −500 21 21 21 |S |S |S S S S
∠ −800 ∠ ∠ −15 −400 −15 −15 −1000 −1000
−20 −600 −20 −1200 −20 −1500 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S) Susceptance (S)
(a) N =4, d =0.35λ (b) N =6, d =0.35λ (c) N =8, d =0.35λ
0 −100 0 0 0 200
−100 −5 −200 −5 −5 0 −200 | | | (deg) (deg) (deg) 21 −10 −300 21 −10 −300 21 −10 −200 21 21 21 |S |S |S S S S
∠ −400 ∠ ∠ −15 −400 −15 −15 −400 −500
−20 −500 −20 −600 −20 −600 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 −0.01 −0.005 0 0.005 0.01 Susceptance (S) Susceptance (S) Susceptance (S)
(d) N =4, d =0.45λ (e) N =6, d =0.45λ (f) N =8, d =0.45λ
Figure 4.7: The effects of layer spacing (continued) to investigate what is the best achievable average insertion loss for a given number of layers N and layer spacing d. To accomplish this, we first calculate S21 for a range of susceptance values jB. Following, we unwrap the phase curve, producing a monotonically decreasing phase curve. Because the phase curve is monotonic, we can define a function that relates the magnitude of S21 to its phase as
S = f( S ), (4.21) | 21| − 21 which is illustrated in Figure 4.8.
Since we desire 360◦ of phase range through the structure, we are interested in calcu- lating the average insertion loss over a 360◦ window. Where φ0 is a particular insertion phase, we can calculate the average transmission power as
1 φ0+2π = f(φ) 2dφ. (4.22) Pav 2π | | φ0 Note that we remap the magnitude as a function of phase, as opposed to simply taking the average with respect to B, because B is not linear with respect to phase. For Chapter 4. Distributed-Scatterer Approach 72
Figure 4.8: S21 remapping beamsteering, we expect that all insertion phases will be used with equal probability, so we take the average insertion loss with respect to phase, and not B.
To minimize the average insertion loss, we sweep φ0 and use the 360◦ window that has the largest . Figure 4.9 shows a plot of the maximum transmission power for different Pav number of layers N and layer spacing d. Regardless of N, the maximum transmission power (minimum insertion loss) is achieved with a spacing of a quarter-wavelength. How- ever, with higher N, the range of layer spacings that result in reasonable transmission power becomes larger, with the 15-layer case supporting layer spacings between 0.05λ and 0.45λ.
Although increasing N allows d to be reduced (maintaining a similar insertion loss), the overall thickness of the structure also increases with greater N. For a given average insertion loss level, we can calculate the minimum d required to achieve that loss. From that minimum layer spacing, we can calculate the minimum structure thickness ((N − 1)d) for different numbers of layers N, shown in Table 4.1. The results suggest that in general, increasing the number of layers does not reduce the minimum structure thickness.
However, the minimum structure thickness is significantly reduced for lower average insertion loss levels.
In this case where lossless surfaces are used, the average transmission power increases with N. However, in reality when the non-ideal dissipative scattering surfaces are used, Chapter 4. Distributed-Scatterer Approach 73
0
−0.2
−0.4
−0.6
−0.8
−1 N=3 −1.2 N=4
Average Power (dB) N=6 −1.4 N=8 −1.6 N=10 N=15 −1.8
−2 0 0.1 0.2 0.3 0.4 0.5 Layer Spacing (Wavelengths)
Figure 4.9: Maximum average transmission power
Table 4.1: Minimum structure thickness ((N 1)d) − N 0.375 dB 0.4 dB 0.5 dB 0.6 dB 0.8 dB 1.0 dB − − − − − − 3 - - - - 0.37λ 0.23λ
4 0.75λ 0.68λ 0.55λ 0.45λ 0.31λ 0.21λ
6 0.60λ 0.58λ 0.50λ 0.42λ 0.29λ 0.21λ
8 0.60λ 0.58λ 0.50λ 0.42λ 0.29λ 0.21λ
10 0.60λ 0.58λ 0.50λ 0.42λ 0.30λ 0.23λ
15 0.61λ 0.59λ 0.51λ 0.43λ 0.31λ 0.24λ Chapter 4. Distributed-Scatterer Approach 74 the insertion loss increases with the number of layers used, and so practically it is desirable to minimize the number of layers used.
The range of susceptances that produced the optimal transmission powers are shown in Figure 4.10. With a layer spacing of 0.25λ, the required susceptances ranged from about 0.005 to 0.005. As the spacing is decreased the maximum susceptance required − is increased rapidly, and as the spacing is increase the minimum susceptance required is decreased rapidly. As the number of layers is increased, the range of susceptances required is reduced. Intuitively, this is expected because the more numerous the layers, the less effect each layer needs to have. It is important to consider the required susceptance range from a technological perspective because surfaces with arbitrarily large positive or negative susceptances may not be practically realizable.
0.02 N=3 N=4 0.015 N=6 N=8 0.01 N=10 N=15
0.005
0
Susceptance (S) −0.005
−0.01
−0.015
−0.02 0 0.1 0.2 0.3 0.4 0.5 Layer Spacing (Wavelengths)
Figure 4.10: Susceptance ranges (solid - max, dashed - min)
For a given N and d, we can create a mapping between the phase shift S21 and the optimal susceptances B required to produce the phase shifts. To describe this mapping, Chapter 4. Distributed-Scatterer Approach 75 we can fit an equation to the phase shift and surface susceptance, which will be used later for calculating desired scattering surface susceptances. Figure 4.11 shows the S-
λ parameters for N = 4,d = 4 . The red line in the plot of S21 shows the equation of fit
5 B =( 2.34 10− ) S (4.23) − × 21 which is used in later analysis. A higher-order polynomial can be used for a closer fit, particularly for cases with larger N. Note that in Figure 4.11 while the average insertion loss is 0.4 dB, the worst-case insertion and return losses are actually 1 dB and 7 dB. − When losses are introduced into the structure in a realistic scenario, the insertion loss will only degrade.
0
100 −10 | 11 11
S 0 |S ∠ −20 −100
−30 −0.01 0 0.01 −0.01 0 0.01 Susceptance (S) Susceptance (S)
0
100 −10 | 21 21
S 0 |S ∠ −20 −100
−30 −0.01 0 0.01 −0.01 0 0.01 Susceptance (S) Susceptance (S)
λ Figure 4.11: S-parameters for N = 4 and d = 4
In this case, to produce a 90◦ phase shift, the susceptance required is jB = j0.0021. − Using Equations (4.13a) and (4.13b), we can compute the fields for the entire waveguide.
Figure 4.12 shows the fields for the waveguide when the surfaces are set to various con- Chapter 4. Distributed-Scatterer Approach 76 stant susceptance values. The differences in plots of the fields for the region z < 0 are caused by reflected waves, or a non-zero return loss.
(a) jB =0.003 ( 135◦) (b) jB =0 (0◦) (c) jB = 0.0019 (90◦) − − Figure 4.12: Computed real-part (arbitrary units) of normalized electric fields for different susceptance values (and phase shifts)
4.1.2 Graduated Scattering Surface
With a uniform impedance on the surfaces, the direction of the outgoing plane wave is clearly at broadside. In order to perform beamsteering with the surfaces, the impedance Chapter 4. Distributed-Scatterer Approach 77 must be varied across the surface. From optics and array theory, if we can create a periodic graduated phase shift across the surface, then the outgoing plane wave will have a deviation from broadside. For the periodic formulation above, we select W to correspond to the spatial period of the phase shift, as shown in Figure 4.13. The spatial period W here is not to be confused with the periodicity of a structure used to implement the scattering surface. For example, if the scattering surface is implemented using a wire grid with graduated spacings to produce a graduated impedance across the transmitarray, W refers to the distance over which the spacing sizes repeat, as opposed to the actual spacing between the wire grids. For a discrete antenna array, W refers to the distance over which element phases wrap around 360◦, and not to the actual spacing between the elements.
We can select the susceptance for the surfaces required to create a phase shifts between
180◦ and 180◦, linearly varying across x, using the mapping (4.23) obtained from the − uniform scenario. The susceptance as a function of x is
5 360◦x Y (x)= j( 2.34 10− ) 180◦ . (4.24) − − × W − W In this case, we have chosen the points at x = 2 to have zero phase shift, although any other point could have been selected to have zero phase because the array is infinite. Figures 4.14(a) and 4.14(b) show the fields in the waveguide for W = 2.4λ and
W = 7λ, which correspond to beamsteering angles of 24.6◦ and 8.2◦ respectively. We can see that the structure performs beamsteering on the incident broadside plane wave by redirecting the outgoing plane wave. We can also plot the fields for specific spectral components p from (4.14) which corresponds to beams in different directions. Note that in this discussion, the terms spectral component, mode, and beam refer to the same phenomenon. Figures 4.15 and 4.16 decomposes the fields into the different spectral components, showing the relative magnitudes of the waves propagating in the different directions. In Figure 4.15, we can see that modes where p 3 are evanescent. From ≥ array theory, these modes correspond to outgoing beams that are not within the visible Chapter 4. Distributed-Scatterer Approach 78
Figure 4.13: Graduated phase shift region of the spatial spectrum. For each visible mode, we can calculate the angle of the associated plane wave from broadside as
1 2pπ θ = sin− . (4.25) p kW Larger values of W , corresponding to more gradual phase gradients, result in angles closer to broadside. Larger values of W also result in more propagating modes. For example in the case of W = 7λ with the modes shown in Figure 4.16, all of the modes shown ( 4 p 4) are propagating. From (4.25), we can determine the modes p that − ≤ ≤ propagate by calculating the values of p which result in
2pπ 1. (4.26) kW ≤ For beamsteering, the mode associated with the desired outgoing plane wave is p =1 with angle θ . That is, we want the power in all other modes (p = 1) to be as small as 1 possible. Clearly, there is power in modes other than the p = 1 mode, which is primarily due to lateral wave propagation in the layers, as illustrated in Figure 4.17. Within the layers, the lateral field interaction is a result of both evanescent and propagating Floquet Chapter 4. Distributed-Scatterer Approach 79
(a) W =2.4λ (b) W =7λ
Figure 4.14: Computed electric fields Chapter 4. Distributed-Scatterer Approach 80
Figure 4.15: Fields in each mode for W =2.4λ Chapter 4. Distributed-Scatterer Approach 81
Figure 4.16: Fields in each mode for W =7λ Chapter 4. Distributed-Scatterer Approach 82 modes. We note that the term lateral propagation used here refers to the interaction of fields within the transmitarray in the lateral direction, and is not to be confused with surface waves. This is analogous to mutual coupling in array theory when there are discrete elements. This lateral propagation is particularly problematic when there is a phase wrapping at the edges of the periodic waveguide, because the current distribution on the surfaces for 180◦ of phase shift is substantially different than that for 180◦. −
Figure 4.17: Lateral propagation
For a desired beam angle θ1, we can calculate the spatial period W that produces the beam as 2π W = . (4.27) k sin θ1 Figure 4.18 shows a comparison of the power levels of the various beams. The abscissa is the beam angle of the desired beam θ1. From the plot, we see that for small beam angles, there is very little power in the undesired beams. However, as the beamsteering angle increases, the main beam power gradually decreases, and the amount of power in other directions increases rapidly, reaching 10 dB at a scan angle of 20◦. An infinite aperture which can be implemented by an infinite antenna array can create a beam in any direction
90◦ <θ< 90◦ with zero power radiated in other directions. In comparison to that, the − power in these undesired beams is a significant drawback of this approach.
From the plot, we can also see that the curves are clearly not monotonic as one might expect. These ripples are associated for cases when a particular mode has kzp = Chapter 4. Distributed-Scatterer Approach 83
0
−5
−10
−15 Beam Power (dB)
p: −2 −20 p: −1 p: 0 p: 1 p: 2 −25 0 10 20 30 40 50 60 Scan Angle (deg)
Figure 4.18: Beam powers of different modes p for scan angles θ1
0. Physically, this represents an outgoing mode wave propagating at θp = 90◦, and accumulating an infinite magnitude. The spikes in the curves correspond exactly to such cases. For example, to create a beam (with mode p = 1 at θ1 = 30◦, we set the spatial periodicity W =2λ, but this causes kz2 = 0. To createa beam at θ1 = 19.47◦, we set W =
3λ, but this causes k = 0. Similarly, we have k = 0for θ = 14.48◦, 11.54◦, 9.59◦,... . z3 zp 1 { } From Figure 4.9, we saw that it is also possible to use different layer spacings when more layers are used. If the undesired beams are caused by lateral wave propagation, then reducing the overall thickness of the structure by reducing the layer spacing may have an effect. Figures 4.20 and 4.19 show the beam powers for different N and d, and total structure thickness T .
Overall, the amount of power in undesired beams is correlated with the overall thick- ness T of the structure. In the case of Figure 4.19(a), the power in undesired beams is generally kept below 10 dB. However, average insertion loss also suffers, averaging − around 2.5 dB. Since this is a lossless structure, this means that almost half of the power is being reflected on the input side. As the structure thickness is increased, then the power in the undesired beams increases as well. Comparing Figures 4.19(f) and 4.19(e), Chapter 4. Distributed-Scatterer Approach 84
0 0
−5 −5
−10 −10 p: −2 p: −1 −15 p: −2 −15 p: 0 p: −1 Beam Power (dB) Beam Power (dB) p: 1 p: 0 −20 −20 p: 2 p: 1 p: 2 −25 −25 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Scan Angle (deg) Scan Angle (deg)
(a) N =4, d =0.05λ, T =0.15λ (b) N =6, d =0.05λ, T =0.25λ
0 0
−5 −5
−10 −10
−15 p: −2 −15 p: −2 p: −1 p: −1 Beam Power (dB) Beam Power (dB) p: 0 p: 0 −20 −20 p: 1 p: 1 p: 2 p: 2 −25 −25 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Scan Angle (deg) Scan Angle (deg)
(c) N =8, d =0.05λ, T =0.35λ (d) N =4, d =0.15λ, T =0.45λ
0 0
−5 −5
−10 −10
−15 p: −2 −15 p: −2 p: −1 p: −1 Beam Power (dB) Beam Power (dB) p: 0 p: 0 −20 p: 1 −20 p: 1 p: 2 p: 2 −25 −25 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Scan Angle (deg) Scan Angle (deg)
(e) N =6, d =0.15λ, T =0.75λ (f) N =4, d =0.25λ, T =0.75λ
Figure 4.19: Beam powers for different N and d Chapter 4. Distributed-Scatterer Approach 85
0 0
−5 −5
−10 −10
−15 p: −2 −15 p: −2 p: −1 p: −1 Beam Power (dB) Beam Power (dB) p: 0 p: 0 −20 −20 p: 1 p: 1 p: 2 p: 2 −25 −25 0 10 20 30 40 50 60 0 10 20 30 40 50 60 Scan Angle (deg) Scan Angle (deg)
(a) N =8, d =0.15λ, T =1.05λ (b) N =6, d =0.25λ, T =1.25λ
0
−5
−10
−15 p: −2 p: −1 Beam Power (dB) p: 0 −20 p: 1 p: 2 −25 0 10 20 30 40 50 60 Scan Angle (deg)
(c) N =8, d =0.25λ, T =1.75λ
Figure 4.20: Beam powers for different N and d (continued) Chapter 4. Distributed-Scatterer Approach 86 and Figures 4.20(a) and 4.20(b) we see that the distributions of power in the different beams are quite similar. Even though the number of layers N is different, the total thick- ness of the structure T is similar, suggesting that the behavior of the undesired beams is mainly determined by the overall thickness of the structure. For larger beamsteering angles and structure thickness, such as in Figure 4.20(c), the currents across the struc- ture become so mixed by lateral propagation that the result is near-uniform fields on the aperture, producing a plane-wave in the broadside direction that has far more power than any of the other beams, for scanning angles above 40◦.
4.1.3 Simulation Results
To validate the MoM results in the previous section, full-wave finite-difference time- domain (FDTD) simulations are used. The FDTD simulations conducted in this thesis use the SEMCAD-X simulation platform by SPEAG [79]. They were executed on servers equipped with graphics processing units for hardware acceleration.
The infinite scattering surfaces for four layers (N = 4) and quarter-wavelength spacing
λ (d = 4 ) were implemented in SEMCAD-X using periodic boundary conditions (PBC) and lumped elements. At 5 GHz, alternating lumped capacitors and inductors spaced 2.5 mm apart were used to approximate the uniform scattering surface, as shown in Figure 4.21.
Both inductors and capacitors were needed in order to produce both positive and negative susceptances. With PBC walls, the height of the waveguide can be arbitrarily selected, and so a height of 1 mm was used to minimize the simulation size. The inductors were
fixed at 0.5 nH while the capacitors were varied between 0.20 pF and 0.74 pF to generate the impedances needed to achieve 360◦ of phase shift in a uniform four-layer structure. The capacitor values were then graduated to produce a constant phase gradient across the aperture, corresponding to specific beamsteering angles.
The normalized power in the different beams for different beamsteering angles are shown in Figure 4.22 for simulated (FDTD) and numerically calculated (MOM) results. Chapter 4. Distributed-Scatterer Approach 87
Figure 4.21: SEMCAD scattering surface implementation with lumped inductors and capacitors
The normalized power of a particular mode p is the power in that beam divided by the power in the desired beam (p = 1).
The simulated and numerically results generally agree in shape, although some curves are shifted by a few dB. Likely, this is a result of the discretization of the scattering surface into lumped inductor and capacitors. Because pairs of inductors and capacitors are resonant, the simulation results were very sensitive to the FDTD gridding between the lumped elements.
Next, we investigate the field patterns produced by a four-layer scattering surface structure for a finite width W of 180 mm, which is 3λ. The width was selected to be consistent with other arrays discussed later in this thesis. Absorbing (radiation) conditions are used for the x and z boundaries, while periodic boundary conditions are used for the y boundaries to simulate an infinite array in the y direction. In the x direction, PECs are used to extend the edges of the array to radiation boundaries, as shown in Figure 4.23. In this investigation, we are not interested in edge diffraction effects because actual implementations of such a transmitarray would be significantly Chapter 4. Distributed-Scatterer Approach 88
0
−5
−10
p: −2 (FDTD) −15 p: −2 (MOM) p: −1 (FDTD) Normalized power (dB) p: −1 (MOM) −20 p: 0 (FDTD) p: 0 (MOM) p: 2 (FDTD) p: 2 (MOM) −25 0 10 20 30 40 50 Beam angle (deg)
Figure 4.22: Infinite scattering surface modes at different beamsteering angles
larger. The radiated field patterns for different beam angles θ1 are shown in Figure 4.24.
Figure 4.23: Implementation of finite scattering surface structure in SEMCAD
For small beamsteering angles (θ < 20◦), the structure produces fairly good results. However, for larger angles, the beams become less well-defined. For an array of size 3λ, phase wrapping is required for beamsteering angles greater than 19.5◦. If we compare the fields in Figures 4.24(c) and 4.24(d), we can see that the phase wrapping creates an anomaly in the fields within the layers, which leads to nulls and significant beams in Chapter 4. Distributed-Scatterer Approach 89
(a) Scattering surfaces θ =0.0◦ (b) Scattering surfaces θ =9.0◦
(c) Scattering surfaces θ = 18.9◦ (d) Scattering surfaces θ = 28.7◦
(e) Scattering surfaces θ = 39.2◦ (f) Scattering surfaces θ = 48.6◦
Figure 4.24: Radiated fields from a finite layered scattering surface Chapter 4. Distributed-Scatterer Approach 90 undesired directions.
λ If we sample the fields near the aperture (at 2 away), we can perform a near-to-far- field transformation. The far-field radiation patterns are shown in Figure 4.25. From the plot, we can see that the pencil beams are well defined for angles below θ = 20◦. However, the performance rapidly degrades beyond that angle where phase wrapping of the aperture fields is required. The structure seems to be unable to create a pencil beam at around 25◦, with significant direction error in the 23◦ and 26◦ beams. For larger angles, the far-field main beam magnitude decreases rapidly to levels comparable to its side-lobes, demonstrating that this approach may not suitable for beamsteering for angles larger than 20◦ from broadside.
0 deg 0 9 deg 19 deg −2 23 deg 26 deg −4 29 deg −6 39 deg 49 deg −8
−10
−12
Far−field pattern (dB) −14
−16
−18
−20 −80 −60 −40 −20 0 20 40 60 80 Azimuth (deg)
λ Figure 4.25: Simulated far-field pattern for a 180 mm surface (N =4,d = 4 )
To summarize, in this section we have investigated the use of layered scattering sur- faces for beamsteering. For infinite generalized scattering surfaces, we have shown nu- merically that due to lateral propagation of the waves, undesired spectral components or Chapter 4. Distributed-Scatterer Approach 91 beams are excited when a phase gradient is produced across the structure. This could be problematic for beamsteering applications desiring output beams at larger angles from broadside, as significant side-lobes will be produced in undesired directions. Furthermore, given the significant amount of lateral propagation, it would be difficult to perform accu- rate beamshaping with such a design. While this investigation only considered H-plane beamsteering, we expect that a similar result would be observed for E-plane beams.
Because this theoretical transmitarray based on scattering surfaces does not assume any particular structural implementation or tuning technology, it is not possible to discuss issues such as bandwidth and loss, since those are highly dependent on the implementa- tion. Furthermore, the range of achievable impedances will also be highly dependent on the physical implementation of the lens layers. Therefore, to obtain realistic results, we need to look at specific designs. In the next section, we present a method for analyzing layered transmitarray structures.
4.2 Analysis of Discrete Layered Designs
While scattering surfaces are interesting from a theoretical perspective, practically they may be difficult to realize, especially if reconfigurability is desired. Scattering surfaces are also often called frequency selective surfaces (FSS), and are typically implemented using electrically small structures. As mentioned in Chapter 2, control of individual structures in FSSs may be challenging due to high element densities. For this reason, in this section we will present a technique for analyzing and optimizing transmitarrays based on layers of planar structures, where the size of each element is electrically significant.
Consider a structure consisting of multiple dielectric slabs with identical planar metal- lic structures on each layer, as shown in Figure 4.26. Assume that the lateral periodicity of the elements is less than a free-space wavelength.
To analyze an infinite array of cells, we will place the unit cell inside a waveguide, fol- Chapter 4. Distributed-Scatterer Approach 92
Figure 4.26: Discrete layered structure lowing the theory presented in Section 3.3 of Chapter 3. Since the presence of the dielec- tric substrate significantly complicates the analysis, we will only develop the scattering characteristics for rectangular and parallel-plate waveguides, which respectively represent an angled incidence case and the broadside case of the generalized periodic waveguide, when the elements are symmetric. Unlike free-space or a homogeneous waveguide, anx ˆ- directed current in the presence of a dielectric slab produces a magnetic vector potential in all three directions, and therefore, dyadic Green’s functions are required. It is impor- tant to consider the dielectric slab, because it significantly changes how the structure resonates.
The outline of the analysis is as follows. First, we will determine the scattering characteristics of a single layer of the unit cell. We will then represent that behavior using a multi-mode scattering matrix, and then cascade the matrices to model the behavior of multiple layers. While it is possible to simply perform a Method of Moments (MoM) analysis on the entire structure with multiple layers, characterization of a single layer is computationally much faster, and also allows for quick optimization of parameters such as layer spacing and number of layers, without repeating the MoM computation for each configuration.
When an incident wave Ei(r) impinges on a conductor, it excites a current on the conductor. These currents, in turn, radiate other waves, called the scattered waves Es(r). Chapter 4. Distributed-Scatterer Approach 93
The total electric field is the superposition of the incident and scattered fields, given by
E(r)= Ei(r)+ Es(r). (4.28)
To determine the scattered fields, we first need to obtain the dyadic Green’s function for the waveguide with the dielectric slab. While dyadic Green’s functions have been developed for semi-infinite rectangular waveguides [80] with dielectric slabs, the solution is recursive, and is not generalized for fully infinite waveguides. To develop the required Green’s function, we begin with the dyadic Green’s function for a homogeneous rectangu- lar or parallel-plate waveguide given by (3.80). We are interested in adapting that result for a point current source on the surface of a dielectric slab of thickness t inside a waveg- uide, as shown in Figure 4.27. Because of the complications that arise from derivatives and curls on the dielectric interface, the source is positioned at z = ∆, with ∆ 0. − → While ultimately, the structure will contain multiple dielectric slabs, we only require the Green’s function for a single slab.
Figure 4.27: Point current source a waveguide with a dielectric slab
Consider now a TEM, TEmn or TMmn mode wave propagating in the waveguide, where ZmnM is the wave impedance in Regions I and III, and Zd,mnM is the wave impedance in Region II. We will use to represent the mode-type T EM, T E, or T M. Because the M sin( ) and cos( ) of the field solutions cancel when taking the ratio of the transverse electric and magnetic fields, the wave impedances for rectangular and parallel-plate waveguides Chapter 4. Distributed-Scatterer Approach 94 are identical. From [74] (p. 105,113), we have
TEM TEM Z = η Zd = ηd (4.29a)
TE kη TE kdηd kzmn TE Zmn = Zd,mn = kd = kd Zmn (4.29b) kzmn zmn zmn k η d d ZTM = zmn ZTM = kzmnηd = kzmn ZTM (4.29c) mn k d,mn kd ǫd,rkzmn mn where η and ηd are the intrinsic impedances in vacuum and the dielectric, and
mπ 2 nπ 2 k = k2 (4.30a) zmn − a − b mπ 2 nπ 2 kd = k2ǫ (4.30b) zmn d,r − a − b are the phase constants for mode m, n in the waveguide filled with vacuum or dielectric.
For each mode m, n, we can model the section of the waveguide with the dielectric slab as a simple transmission line using an ABCD matrix. Then, we can calculate equivalent
S-parameters, giving the reflection coefficient at the dielectric interface (S11), and the transmission coefficient through the dielectric slab (S21) [74] (p. 187) as
d 2 2 j sin(k t) ZM ZM zmn d,mn − 0,mn RmnM = (4.31a) d d 2 2 2Zd,mnM Z0M,mn cos(kzmnt)+ j sin(kzmnt) Zd,mnM + Z0M,mn 2Zd,mnM Z0M,mn TmnM = . (4.31b) d d 2 2 2Zd,mnM Z0M,mn cos(kzmnt)+ j sin(kzmnt) Zd,mnM + Z0M,mn We can use these coefficients to account for the effects of the substrate slab. In Region I, in addition to the usual zˆ-propagating waves, part of the +ˆz-propagating waves is − reflected by the dielectric. Therefore, we add a term including the reflection coefficient in Region I. On the other hand, waves that propagate into Region III need to pass through the dielectric, and so the waves are multiplied by the transmission coefficient in Region III. Since the transmission coefficient represents the transmission through length t of waveguide, we need to remove length t of propagation by including the de-embedding
jkzmnt d factor e . Note that this correction uses kzmn as opposed to kzmn. Therefore, adding the coefficients to (3.80), we obtain the Green’s functions for waves propagating in the Chapter 4. Distributed-Scatterer Approach 95
zˆ-direction (Regions I, z 0) and the +ˆz-direction (Region III, z t), given by − ≤ ≥
¯ e, 1 TEM jΨ(0) +jkz G −(r r′) = zˆzδˆ (r r′) 1+ R e yˆyˆ | −k2 − − 2abk j 1 ¯ ¯ 1+ RTE M¯ − + 1+ RTM N¯ − , z 0 −2ab k2k mn mn mn mn ≤ m n c zmn (4.32a)
¯ e,+ jΨ(0) TEM jkt jkz G (r r′) = T e e− yˆyˆ | − 2abk j 1 ¯ + ¯ + T TEejktM¯ + T TM ejktN¯ , z t (4.32b) −2ab k2k mn mn mn mn ≥ m n c zmn
¯ ± ¯ ± where Mmn and Nmn are dyads defined by (3.77a) and (3.77b). Following (3.63), we can find the total scattered electric field using
¯ e E (r)= jω G¯ (r r′) J(r′) dS′. (4.33) s − | S
Consider now an incident wave propagating in mode in the +ˆz-direction. Be- MMN cause we know that the total tangential electric field on a perfect electric conductor must be zero, we can use a MoM approach to find the currents J(r′) such that the tangential scattered electric fields cancel the incident fields on conducting surfaces. The details of the MoM analysis for the transmitarray unit cell based on dipoles discussed later in this
MN chapter are provided in Appendix A. For numerically determined coefficients Cmn , the scattered electric field for excitation can be expressed as MMN
TEM MN, ωµ 1+ R MN +jkz ωµ 1 E −(r) = yˆ C e s − 2abk 00 − 2ab k2k m n c zmn TE TM MN +jkzmnz 1+ R Mˆ − + 1+ R Nˆ − C Ψ(k x) cos(k y)e (4.34a) mn mn mn mn mn x y jkzmnt TEM MN,+ ωµe T MN jkz ωµ 1 E (r) = yˆ C e− s − 2abk 00 − 2ab k2k m n c zmn jkzmnt TE + jkzmnt TM + MN jkzmnz e T Mˆ e T Nˆ C Ψ(k x) cos(k y)e− (4.34b) mn mn − mn mn mn x y
ˆ ˆ for z < 0 and z > t, respectively, where Mmn± and Nmn± are basis vectors for the T E and Chapter 4. Distributed-Scatterer Approach 96
T M modes, given by
jkzmnz Mˆ ± (r) = xkˆ k Ψ′(k x) sin(k y)e∓ mn − x y x y
2 jkzmnz +ˆykxΨ(kxx) cos(kyy)e∓ (4.35a) 2 kxkykzmn jkzmnz Nˆ ± (r) = xˆ− Ψ′(k x) sin(k y)e∓ mn − k2 x y k2k2 y zmn jkzmnz yˆ− Ψ(k x) cos(k y)e∓ − k2 x y 2 kc kyjkzmn jkzmnz zˆ Ψ(k x) sin(k y)e∓ . (4.35b) ∓ k2 x y
Regardless of the type of unit cell used (e.g. coupled rings, patches, grids, slots), the scattered fields will assume the same form. The only difference would be in the coefficients
MN Cmn .
By taking the ratio of the scattered field with the incident field, we can obtain multi- mode S-parameters, where different modes are represented by different ports. In this way, we can characterize a single layer of the unit cell using an S-parameter matrix. Appendix B describes this process in greater detail. Let with uppercase M and MMN N denote the mode of the excitation, and with lowercase m and n denote the Mmn scattered mode. Then, for ports on the two sides A and B of the structure, we have the parameters Chapter 4. Distributed-Scatterer Approach 97
MN,( ) MN jk d 1+ R E − + R EM e zmn MNM s, mn mnM i − M , MN = mn MN jkzMN d M M EiM e SAA = MN,( ) (4.36a) 1+ R E − e jkzmnd MNM s, mn − M , MN = mn MN jkzMN d M M EiM e jk t MN,( ) jk t MN jk d e zmn T E − + e zMN T EM e zmn mnM s, mn MNM i − M , MN = mn MN jkzMN d M M EiM e SAB = MN,( ) (4.36b) ejkzmntT E − e jkzmnd mnM s, mn − M , MN = mn MN jkzMN d M M EiM e MN,(+) jk t MN jk d 1+ R E + e zMN T EM e zmn MNM s, mn MNM i − M , MN = mn MN jkzMN d M M EiM e SBA = MN,(+) (4.36c) 1+ R E e jkzmnd MNM s, mn − M , MN = mn MN jkzMN d M M EiM e jk t MN,(+) 2jk d MN jk d e zMN T E + e zMN R EM e zmn MNM s, mn MNM i − M , MN = mn MN jkzMN d M M EiM e SBB = MN,(+) (4.36d) ejkzMN tT E e jkzmnd MNM s, mn − M , MN = mn MN EM ejkzMN d M M i where
ωµΨ(0) EMN,(+) = ejkzmntT TEM CMN (4.37a) s,TEM − 2abk 00 MN,( ) ωµΨ(0) TEM MN E − = 1+ R C (4.37b) s,TEM − 2abk 00 MN,(+) ωµ(2 δ m)(2 δn) E = − − ejkzmntT TE/TM CMN (4.37c) s,TE/TMmn 2 mn mn − 2abkc kzmn MN,( ) ωµ(2 δm)(2 δn) TE/TM MN E − = − − 1+ R C (4.37d) s,TE/TMmn 2 mn mn − 2abkc kzmn a2 ETEMN = (4.37e) i M 2π2 2 2 TMMN b k Ei = 2 2 2 (4.37f) N π kzMN (4.37g) and where 1, m =0 δm = . (4.38) 0, otherwise Even though only the fundamental mode may be propagating, we need to consider several modes because the spacing d may be small enough that the evanescent fields interact Chapter 4. Distributed-Scatterer Approach 98 between layers. As the mode m and n indexes increase, all modes eventually become evanescent. Because the rate at which evanescent modes decay, the imaginary part of kzmn, increases with m and n, only a finite number of modes need to be considered. The exact number of modes that need to be considered will depend on the proximity of the layers d.
Finally, using the result from Section B.2 in Appendix B, several multi-mode S- parameters can be cascaded to model a transmitarray unit cell with multiple layers, as illustrated in Figure 4.28. From the cascaded multi-mode S-parameters, if we look at the
S-parameters corresponding to the fundamental mode (T E10 in rectangular waveguide and T EM in parallel-plate waveguide), then we have the reflection and transmission response of an infinite transmitarray of identical cells.
Figure 4.28: Waveguide setup
This analysis technique allows for rapid optimization of parameters, and will be used for a transmitarray cell based on loaded dipoles in the following section.
4.3 Transmitarray Unit Cell of Loaded Dipoles
In this section, we present the design, analysis, and simulation and experimental results for a unit cell based on layers of loaded dipoles. A few layered designs have been proposed for transmitarrays using coupled-rings [55] and loaded resonant slots [61]. In both cases, Chapter 4. Distributed-Scatterer Approach 99 the structures are bandpass structures. While dipoles are bandstop in nature, two dipoles of different lengths can exhibit bandpass behavior if one is inductive and the other is capacitive, as shown in Figure 4.29. In other words, the dipoles are bandpass when one dipole resonates slightly above the operating frequency, and the other slightly below.
This motivates the transmitarray unit cell design presented in this section, which consists of layers of simple loaded dipoles on a dielectric substrate. Although our motivation here is the bandpass nature of the dipoles, we will not approach the transmitarray design from a filter perspective. The concept of a transmitarray as a filter will be explored in further detail in Chapter 5. In this section, we will approach the design from a scattering perspective.
Figure 4.29: Equivalent circuit models for dipoles
4.3.1 Unit Cell Design
Consider a a b transmitarray unit cell consisting of three vertically-centered thin flat × dipoles of width w =1.0 mm. While only two dipoles are required for bandpass behav- ior, three are used for symmetry. The dipoles are mounted on a dielectric substrate of thickness t, with a dielectric constant of ǫd,r. A gap of size g exists in the middle of the center dipole, and is loaded with an impedance ZL, as shown in Figure 4.30. Following the results from [55, 61] and the scattering surface discussion earlier in this chapter, four layers of these dipoles are used.
The dielectric constant and thickness of the substrate is ǫr =2.94 and t =1.52 mm.
The load ZL is implemented using MGV100-20 GaAs Hyperabrupt varactor diodes in Chapter 4. Distributed-Scatterer Approach 100
Figure 4.30: Unit cell geometry
E28X packaging (2.3 mm 1.1 mm 1.3 mm) manufactured by Aeroflex/Metelics. × ×
Estimates for the series parasitic inductance and resistance are Lv = 0.4 nH and Rvs = 2 Ω, respectively, with a tunable capacitance range of C =0.15 pF 2.0 pF. Although v − the simplified equivalent circuit model shown in Figure 4.31 is used to model the varactor diode, we note that the actual behavior of the diode is much more complex, and these parameters are only estimates. The total impedance produced by the varactor diode is
1 ZL = Rvs + jωLv + (4.39) jωCv
The approximate bias voltage to capacitance mappings are given in Table 4.2.
Figure 4.31: Varactor diode series model L =0.4 nH, R = 2 Ω,C =0.15 pF 2.0 pF v vs v −
MN To find the scattered fields using (4.34b), values for the coefficients Cmn were nu- merically determined using the MoM approach for rectangular waveguide described in
Appendix A. The design parameters, listed in Table 4.3 were systematically swept in millimeter increments to optimize the design. The MoM code running in MATLAB on a desktop computer only required a few minutes to execute each configuration. If Chapter 4. Distributed-Scatterer Approach 101
Table 4.2: Varactor diode capacitance and biasing voltage Capacitance (pF) Bias Voltage (V)
0.15 14
0.2 11
0.3 8
0.4 6
0.5 5
0.7 3
0.9 2.5
1.2 2
2 0
FDTD simulations were used, the simulations would have taken a few hours for each configuration, even with accelerated hardware.
Although only a simple brute force approach was used, the main objective here was not to produce the optimal design at a particular frequency, but more so to understand the effects of the different parameters. An optimization algorithm such as gradient search can easily be implemented to achieve more efficient optimization. Rather than fixing an operating frequency, the length of the side dipoles were fixed instead, and the optimal operating frequency selected afterwards. This was done to reduce the number of matrix calculations. We note that while the structure is operated above the cutoff frequency of the T E20 mode (but below the cutoff frequency of the T E30 mode), the T E20 mode is not excited due to the symmetry of the structure.
In this case, an objective function was not explicitly defined because an optimization algorithm was not used. The best configuration was selected based on the size of the phase tuning range (at least 360◦) and low insertion loss throughout the tuning range. The best values achieved are shown in Table 4.3. A plot of the S-parameters of the optimal case is shown in Figure 4.32. Chapter 4. Distributed-Scatterer Approach 102
0
−5
−10
−15 Magnitude (dB)
−20 5.5 6 6.5 7 Frequency (GHz)
0.15 pF 0.2 pF 100 0.3 pF 0.5 pF 0 0.7 pF 1 pF
Phase (Deg) −100 1.5 pF 2 pF
5.5 6 6.5 7 Frequency (GHz)
(a) S11
0
−5
−10
−15 Magnitude (dB)
−20 5.5 6 6.5 7 Frequency (GHz) 0.15 pF
0.2 pF 100 0.3 pF 0.5 pF 0 0.7 pF 1 pF
Phase (Deg) −100 1.5 pF 2 pF
5.5 6 6.5 7 Frequency (GHz)
(b) S11
Figure 4.32: Best S-parameters from MoM optimization Chapter 4. Distributed-Scatterer Approach 103
Table 4.3: Dipole unit cell design parameters Parameter Sweep Range Best Value
Operating frequency 4 GHz 8 GHz 6.25 GHz − Side dipole lengths 20 mm 20 mm
Center dipole length 10 mm 20 mm 15 mm − Side dipole offset (s) 2 mm 15 mm 7 mm − Layer spacing 6 mm 20 mm 14 mm −
Figure 4.32(a) shows that the structure is clearly bandpass in nature, and manipula- tion of the varactor diode load changes the shape and location of the pass band. Since the phase is approximately linear across the pass band, to achieve 360◦ of phase shift, we desire the left edge of the 0.15 pF pass band to coincide with the right edge of the 2.0 pF pass band, so that good insertion loss is maintained with maximum phase shift. We will now briefly discuss the effects of the design parameters on the pass band behavior.
Figure 4.33 shows S21 for two other center dipole lengths. Clearly, the dipole length affects frequency, as is expected. More importantly, the dipole length affects the width of the pass band. From filter theory, the width of the pass band in an LC resonator is related to the quality factor of the resonator, which is related to the ratio between the inductance and the capacitance. By varying the dipole length, the equivalent inductance and capacitance values of Figure 4.29 are changed, resulting in a change of pass band width.
Figure 4.34 shows the effect of dipole spacing. In this case, we can see that the width of the pass band is also affected by the dipole spacing. The simplistic circuit model in Figure 4.29 cannot capture the effects of dipole spacing. However, since the spacing affects the coupling between the two dipoles, it also affects the quality factor of the resonator, and thus changes the width of the pass band.
Figure 4.35 shows the effect of layer spacing on S21. As mentioned earlier in the chap- ter, the optimal spacing for layered scattering surfaces is a quarter-wavelength. While Chapter 4. Distributed-Scatterer Approach 104
0 0
−10 −10
0.15 pF 0.15 pF Magnitude (dB) −20 Magnitude (dB) −20 5.5 6 6.5 7 0.27.5 pF 5.5 6 6.5 7 0.27.5 pF Frequency (GHz) 0.3 pF Frequency (GHz) 0.3 pF 0.5 pF 0.5 pF 100 0.7 pF 100 0.7 pF 1 pF 1 pF 0 1.5 pF 0 1.5 pF −100 2 pF −100 2 pF Phase (Deg) Phase (Deg)
5.5 6 6.5 7 7.5 5.5 6 6.5 7 7.5 Frequency (GHz) Frequency (GHz)
(a) 14 mm (b) 16 mm
Figure 4.33: Effect of center dipole length
0 0
−10 −10
0.15 pF 0.15 pF Magnitude (dB) −20 Magnitude (dB) −20 5.5 6 6.5 7 0.27.5 pF 5.5 6 6.5 7 0.27.5 pF Frequency (GHz) 0.3 pF Frequency (GHz) 0.3 pF 0.5 pF 0.5 pF 0.7 pF 100 100 0.7 pF 1 pF 1 pF 0 0 1.5 pF 1.5 pF −100 2 pF −100 2 pF Phase (Deg) Phase (Deg)
5.5 6 6.5 7 7.5 5.5 6 6.5 7 7.5 Frequency (GHz) Frequency (GHz)
(a) 5 mm (b) 10 mm
Figure 4.34: Effect of dipole spacing Chapter 4. Distributed-Scatterer Approach 105 that value may change slightly with the presence of the dielectric slabs, the optimal spacing of 14 mm is 0.3λ, which is still close to a quarter-wavelength. One important ob- servation from the plots is that as the spacing is moved away from the optimum, ripples appear in the pass band, significantly degrading the insertion loss. This means that the spacing between the layers needs to be approximately a quarter-wavelength to maintain good insertion loss.
0 0
−10 −10
0.15 pF 0.15 pF Magnitude (dB) −20 Magnitude (dB) −20 5.5 6 6.5 7 0.27.5 pF 5.5 6 6.5 7 0.27.5 pF Frequency (GHz) 0.3 pF Frequency (GHz) 0.3 pF 0.5 pF 0.5 pF 100 0.7 pF 100 0.7 pF 1 pF 1 pF 0 0 1.5 pF 1.5 pF −100 2 pF −100 2 pF Phase (Deg) Phase (Deg)
5.5 6 6.5 7 7.5 5.5 6 6.5 7 7.5 Frequency (GHz) Frequency (GHz)
(a) 10 mm (b) 18 mm
Figure 4.35: Effect of layer spacing
4.3.2 Simulated Results
The optimal design parameters obtained from the MoM calculations were then used to simulate the structure in a rectangular waveguide in full-wave FDTD simulations using
SEMCAD-X. A comparison of the simulated (FDTD) and calculated (MOM) results are shown in Figures 4.36 and 4.37, showing excellent agreement.
From the simulated results at 6.4 GHz, 400◦ of phase range is achieved using varactor capacitance values of 0.15 pF to 0.9 pF, with an insertion loss range of 1 dB to 4 dB over the tuning range. Chapter 4. Distributed-Scatterer Approach 106
0
−5 0.15 pF (FDTD) 0.15 pF (MOM) −10 0.2 pF (FDTD) 0.2 pF (MOM) −15
Magnitude (dB) 0.3 pF (FDTD) 0.3 pF (MOM) −20 5.5 6 6.5 7 0.5 7.5pF (FDTD) Frequency (GHz) 0.5 pF (MOM) 0.7 pF (FDTD) 0.7 pF (MOM) 1 pF (FDTD) 100 1 pF (MOM) 1.5 pF (FDTD) 0 1.5 pF (MOM) 2 pF (FDTD) Phase (Deg) −100 2 pF (MOM)
5.5 6 6.5 7 7.5 Frequency (GHz)
Figure 4.36: MoM and FDTD S21
0
−5 0.15 pF (FDTD) 0.15 pF (MOM) −10 0.2 pF (FDTD) 0.2 pF (MOM) −15 Magnitude (dB) 0.3 pF (FDTD) 0.3 pF (MOM) −20 5.5 6 6.5 7 0.5 7.5pF (FDTD) Frequency (GHz) 0.5 pF (MOM) 0.7 pF (FDTD)
0.7 pF (MOM) 100 1 pF (FDTD) 1 pF (MOM) 1.5 pF (FDTD) 0 1.5 pF (MOM) 2 pF (FDTD) Phase (Deg) −100 2 pF (MOM)
5.5 6 6.5 7 7.5 Frequency (GHz)
Figure 4.37: MoM and FDTD S11 Chapter 4. Distributed-Scatterer Approach 107
4.3.3 Experimental Results
To validate the numerical results, an experimental unit cell was fabricated on low-loss
Rogers Duroid 6002, with a dielectric constant of ǫr = 2.94 and thickness of 1.52 mm. In the prototype, the four layers are supported with spacings of 14 mm using additional pieces of Duroid 6002 substrate, as shown in Figure 4.38. Horizontal 1 mm-wide bias lines are used to effect a reverse bias on the varactor diodes on the center dipoles. The bias lines are connected to the dipoles using 100 kΩ resistors so that RF power would not
flow from the dipoles to the bias lines. Because the leakage current of the varactor diodes in reverse bias is small (100 nA), only a small voltage drop is incurred across the bias resistor. Simulations have shown that horizontally oriented lines have minimal effect on vertically polarized fields in a waveguide. A test harness is used to mount the structure inside a WR-187 waveguide, with inner dimensions of 47.55 mm 22.15 mm. The test × harness has small holes on the side for the bias wires, which are also used to hold the structure to the test harness.
Figure 4.38: Fabricated unit cell
Figure 4.39 shows the unit cell test setup used throughout this thesis. The setup consists of a unit cell test harness connected to two rectangular waveguides, which are Chapter 4. Distributed-Scatterer Approach 108 connected to a network analyzer. The network analyzer is calibrated using TRL (thru- reflect-line) calibrations to the ends of the WR-187 waveguide that connect to the unit cell. A 32-channel USB voltage controller is used to sweep the bias voltages of the unit cell.
Figure 4.39: Fabricated unit cell
The measured results of the fabricated unit cell are shown in Figure 4.40. At 6.4 GHz,
309◦ of phase range is achieved with varactor bias voltages in the range of 3 V 23 V. − The average insertion loss is 3.5 dB with the insertion loss ranging from 2.5dBto5.6 dB.
A comparison of the simulated and experimental S is plotted in Figure 4.41, showing | 21| a difference of about 2 dB to 3 dB in the insertion loss.
If we compare S of the simulated and experimental results from Figures 4.37 and | 11| 4.40, we see that they are similar, meaning that this difference cannot be attributed to reflection loss due to mismatch. This suggests that power is being dissipated in the structure in the experimental unit cell. Also, comparing S from Figures 4.36 and 4.40, | 21| the insertion loss in the experimental cell is significantly higher for low bias voltages, which correspond to larger capacitances. Since larger capacitances are associated with Chapter 4. Distributed-Scatterer Approach 109
0
−5
(dB)| −10 21 |S −15
−20 5.5 6 6.5 7 Frequency (GHz)
0
−5
(dB)| −10 21 |S −15 0.00V 01.25V −20 02.50V 5.5 6 6.5 7 03.75V Frequency (GHz) 05.00V 06.25V 07.50V 100 08.75V 10.00V
(dB)| 0
21 12.50V
|S 15.00V −100 20.00V
5.5 6 6.5 7 Frequency (GHz)
Figure 4.40: Measured unit cell S-parameters Chapter 4. Distributed-Scatterer Approach 110
0
−2
−4 | (dB) 21
|S −6
EXP −8 FDTD MOM
−10 100 200 300 400 500 600 700 Phase Shift (deg)
Figure 4.41: Summary of simulated (FDTD), numerical (MOM), and experimental
(EXP) and results at 6.4 GHz higher currents through the diode and dipole, it could be the case that power is being dissipated by conductor losses. Perfect electric conductors were assumed in the MoM and simulation results. Alternatively, the biasing circuitry, which was not modeled in
MoM and simulation, may have contributed to the loss.
To investigate this further, simulations were conducted in SEMCAD that included
finite conductivity (copper: 5.8 107 S/m) and dielectric loss (tan δ = 0.0012). Also, × the biasing lines, additional dielectric support structure, and biasing resistors (100 kΩ) were included. The resulting S-parameters are shown in Figure 4.42. From the plot, we see that the behavior of increased insertion loss, particularly for lower biasing voltages, is reproduced.
With more detailed simulations isolating the different sources of loss, it was deter- mined that conductor losses were the most significant, accounting for up to 1.6 dB of the loss for the larger capacitance configurations. Moreover, about 0.4 dB of loss could be attributed to dielectric loss. It was also determined that the bias lines and biasing resistors had negligible effect. Chapter 4. Distributed-Scatterer Approach 111
0
−5
(dB)| −10 21 |S −15
−20 5.5 6 6.5 7 Frequency (GHz)
0
−5
(dB)| −10 21 |S −15
−20 5.5 6 6.5 7 0.15pF Frequency (GHz) 0.20pF 0.30pF 0.40pF 100 0.50pF 0.70pF
(dB)| 0
21 1.00pF
|S 1.50pF −100 2.00pF
5.5 6 6.5 7 Frequency (GHz)
Figure 4.42: Simulated S21 with loss and biasing structure Chapter 4. Distributed-Scatterer Approach 112
In this design, the conductor and dielectric losses are quite significant, possibly due to the highly resonant nature of the dipoles. This is problematic for this particular design because these sources of losses cannot be practically reduced, since copper is already a very good conductor and Duroid 6002 a low-loss dielectric. Nevertheless, while the insertion loss was underestimated, the general behavior of the unit cell was predicted using simulation and MoM calculations, with sufficient accuracy to produce a functional transmitarray cell.
4.4 Conclusions
In this chapter, we thoroughly investigated the use of easy-to-fabricate layered structures for transmitarrays, following the distributed-scatterer approach. First, we studied the beamforming capabilities of a transmitarray consisting of layers of generalized scattering surfaces. Through numerical results and simulation, we have demonstrated that while this approach can produce beams at angles close to broadside, layered scattering surfaces produce significant undesired beams, or spectral components, for larger beamsteering angles. This phenomenon, caused by the lateral propagation of waves across the structure, is correlated with the overall structure thickness, and not with the number of layers used.
It is not possible to reduce the structure thickness and maintain 360◦ of phase tunability, without degrading the insertion loss. Longitudinal conducting walls could be used to isolate the cells, but would increase the fabrication complexity substantially.
Following we presented a method for analyzing and optimizing transmitarrays made with layers of planar elements on dielectric substrates. This approach models the struc- ture using a unit cell on a single layer, allowing rapid calculation of the response of a multi-layer structure. Computationally, this technique is orders of magnitude faster than other full-wave simulation techniques.
Finally, we presented a transmitarray unit cell based on three dipoles, where the center Chapter 4. Distributed-Scatterer Approach 113 dipole was loaded with a varactor diode. We demonstrated that while the structure can be used as a transmitarray cell, there are issues with conductor and dielectric losses, likely due to the resonant nature of the dipoles. While other types of elements can be used, such as rings or slots, practically it remains to be determined how such structures could be properly controlled without bias lines interfering with their operation. For these reasons, we have chosen not to fabricate a full transmitarray based on this layered approach, and instead will introduce a slightly modified approach, discussed in the next chapter. Chapter 5
Coupled-Resonator Approach
In Chapter 4, we saw that the beamforming capabilities of transmitarrays based on iden- tical layers was fundamentally limited, due to the physical thickness of the structure. A natural question that arises, then, is whether or not the thickness of the structure can be reduced if the layers are not identical, or if larger insertion losses are acceptable. If one approaches the problem from a purely distributed-scatterer perspective using Green’s functions and field theory, then it is difficult to insightfully optimize the structure when each layer can be different. As the number of layers increases, the design process degen- erates to pure numerical optimization.
Some FSS designs have been proposed that use non-identical layered structures to create phase-shifting surfaces [81] and spatial bandpass filters [66] that are electrically thin. However, one of the trade-offs of allowing non-identical layers, is that each layer must be independently tunable. In [66] where the surface consists of alternating layers of grids and patches, each layer corresponds to a specific component in a ladder-type bandpass filter. In general, each layer must take a different geometry for the filter to function properly. To make this structure reconfigurable, independent sets of bias lines would be required for each cell in each layer, and would be extremely challenging since there is no space in the structure for bias lines. For a design to be practical, we need to
114 Chapter 5. Coupled-Resonator Approach 115 minimize the number of control signals required for each cell.
Therefore, the goal of this chapter is to reduce the number of layers required and the thickness of the transmitarray. In order to systematically study distributed-scatterer structures with non-identical layers, we turn to filter theory and draw insight from equiv- alent circuit models. Because this approach will model the structure using resonators, we will call this approach the coupled-resonator approach. Coupled resonators can be implemented in both distributed-scatterer and guided-wave structures. However, since coupled-resonator filters in guided-wave structures have been extensively studied, we will focus on distributed-scatterer implementations in this chapter. Guided-wave approaches will be investigated in Chapter 6.
In this chapter, we will first look at the transmitarray from a filter theory perspective to determine minimum number of resonators that are required to achieve the desired phase tuning range. We will then present a simple unit cell design based on varactor- tuned patches and slots. Finally, we will present experimental results for a transmitarray realized with this unit cell design.
5.1 Transmitarrays as Filters
The observation that motivated the design of the dipole unit cell in Chapter 4 was that a bandpass structure has a phase gradient across the pass band. Consequently, we can create a tunable phase shifter using a filter with a movable pass band. Recall from Chapter 2 that both of the layered transmitarray designs mentioned were bandpass in nature [55, 61]. In fact, the majority of the FSSs overviewed were also bandpass structures.
A reconfigurable transmitarray with many unique layers is significantly harder to realize than a fixed transmitarray for several reasons. Firstly, because each cell needs to be independently controlled, biasing circuitry is needed that does not interfere with Chapter 5. Coupled-Resonator Approach 116 intended cell operation. Secondly, components used to enable reconfigurability, such as varactor diodes, tend to have variability, whether intrinsically or due to manufacturing processes used in building the transmitarray. While an advantage of having tunable components is that the tuning can be used to compensate for some of the variability, the larger the number of tuning components, the more difficult it is to isolate and compensate for the error.
Therefore, we are interested in the simplest design possible that can achieve 360◦ of phase range. Specifically, we seek a design that minimizes the number of substrate and metallization layers required. To quantify the minimum order of complexity required, we consider a transmitarray based on a reconfigurable bandpass filter. We emphasize that the objective here is to design a reconfigurable phase shifter, and not a spatial bandpass
filter, as many such designs have already been proposed in literature.
Consider a passive linear time-invariant filter with a particular magnitude and phase response. From filter theory, let the response be characterized by a set of poles (pi) and zeros (zi) on the complex plane, such that the transfer function is ΠM (s z ) H(s)= i=0 − i , (5.1) ΠN (s p ) i=0 − i where s = σ + jω is a point on the complex plane. The steady-state response at a particular operating frequency is defined by H(jω0) on the imaginary axis of the complex plane, and is equivalent to the S21 of the structure in a waveguide. It is well known that the magnitude response at a particular operating frequency jω0 is given by the product of the distances to each zero on the complex plane, divided by the product of the distances to each pole. Likewise, the phase response is given by the sum of the angles to each zero, minus the sum of the angles to each pole. Now, for a single pole or zero in the left-half plane where σ < 0 for stability, the corresponding angle with any point on the imaginary axis is between 90◦ and +90◦. Therefore, each pole or zero, if arbitrarily − moved, can contribute a maximum of 180◦ of phase response tunability in H(s), as shown in Figure 5.1. To maintain a constant magnitude response, the pole locus should maintain Chapter 5. Coupled-Resonator Approach 117
a constant distance from the operating frequency point jω0.
Figure 5.1: Pole angle and distance
For example, consider the two-pole circuit shown in Figure 5.2, which can model two inductively-coupled patches. The S21 of the circuit can be shown to be
2 2L Z0s S21 = 2 2 (5.2) (LZ0Cs + Ls + Z0)(LcZ0CLs + LLcs +2LZ0 + LcZ0) and the two positive-frequency poles from the denominator given by
2 2 + L + j 4LZ0 C L s1 = − − , 2LZ0C 2 2 2 + LLc + j L Lc 4LLcZ0 (Lc +2L) s2 = − − (5.3) 2LLcZ0C where L and C are the inductance and capacitance of the shunt resonators, Z0 is the port impedance, and Lc is the coupling impedance. Suppose the phase response of this structure is tuned by changing the capacitance C (by using a varactor diode, for example), which in turn changes the resonant frequency. Then, the loci of the poles are elliptical with foci on the real axis, as shown in Figure 5.3(a). Consider two capacitances
C = 1 pF and C = 4 pF that result in the pairs of poles shown. The corresponding frequency response for those two capacitances is shown in Figure 5.3(b).
Ultimately the desired effect of the tuning is to change the phase response of the circuit without significantly changing its magnitude response. The maximum phase response Chapter 5. Coupled-Resonator Approach 118
Figure 5.2: Example two-pole circuit
10 x 10 j ω 6 0 C=1pF 4 C=4pF s locus 1 2 s locus 2 0
−2
−4 Imaginary/Frequency Axis
−6 −4 −2 0 2 10 Real Axis x 10
(a) Pole loci on the complex plane
0 −10 −20 Magnitude −30 0 2 4 6 8 Frequency 9 x 10
C=1pF 200 C=4pF 0
Phase −200
0 2 4 6 8 Frequency 9 x 10
(b) Transmission response S21
Figure 5.3: Example two-resonator complex poles and response, with s1 and s2 loci as C is varied, with L = Lc = 1 nH and Z0 = 50 Ω. Chapter 5. Coupled-Resonator Approach 119 from tuning is achieved when the poles can be moved as far as possible along the locus.
In the example shown, it is not possible to achieve 360◦ of phase tunability given the elliptical curvature of the loci. Due to the trajectory of the poles (and zeros) of the structure, the magnitude response varies considerably over the range where most of the phase shift is developed. Furthermore, very large changes in C would be required to produce the large phase shifts needed in beamforming applications, which is not practical in many tunable components.
For different values of L and Lc, the sizes of the ellipses change, but large variations in inductance are needed to significantly alter the loci. In the degenerate case where the locus is an infinitely-large ellipse encircling the left-half plane, even though 360◦ of phase shift may be theoretically achieved, the variation of locus point distances to jω0 would be large, making the magnitude response vary too greatly. Therefore, a minimum of three resonances are needed to achieve 360◦ of phase tunability with an acceptable amount of variation in insertion loss.
It is well known that flatter transmission magnitude curves can be achieved by increas- ing a filter’s order, or the number of resonators. However, with microwave structures, each additional resonator requires more physical space, and incurs more loss and fabrica- tion complexity. For this reason, our design aims to achieve maximum phase tunability and transmission constancy with a minimum order filter using only three resonators.
Consider a system with three poles which are moved as shown in Figure 5.4. As the poles are shifted, the corresponding magnitude and phase responses are also shifted in
Figure 5.4(b). With more poles, each pole needs to move less to effect a given amount of total phase change, so the transmission magnitude can be kept more constant. If we consider a single frequency f0 in Figure 5.4(b), we see it is possible to achieve 360◦ of phase tunability with only small variations in the magnitude using three poles. Therefore, the proposed unit cell presented in the next section will use three conjugate pole-pairs to provide the necessary phase agility. Chapter 5. Coupled-Resonator Approach 120
6 jω 0 Pole Cfg. 1 5 Pole Cfg. 2 4 Trajectories
3
2
1
Imaginary/Frequency Axis 0
−1 −1 −0.5 0 0.5 Real Axis
(a) Poles on the complex plane
0 −10 −20 Magnitude −30 0 2 4 6 Frequency
200 Pole Cfg. 1 Pole Cfg. 2 0
Phase −200
0 2 4 6 Frequency
(b) Transmission response S21
Figure 5.4: Example three-resonator complex poles and response Chapter 5. Coupled-Resonator Approach 121
5.2 Unit Cell with Slot-Coupled Patches
The requirement for three tunable poles or zeros implies that there are three resonance points (where a resonance is a point at which S21 is real). Since microstrip patches are essentially leaky resonators and are easy to fabricate, two microstrip patches are used in this unit cell design. To make these resonators tunable, the patches are split in half and the halves are connected with varactor diodes. This tuning arrangement for patch antennas has previously been proposed for reflectarrays in [82]. To couple power from one patch to the other, a slot is made in the ground plane that is common to both patches.
By adding a varactor diode across the slot, a third tunable resonator is formed. Together, these structures act as three coupled tunable resonators. The structure of this unit cell design, which we will call the slot-coupled patches design, is illustrated in Figure 5.5.
Figure 5.5: Reconfigurable slot-coupled patches transmitarray element structure Chapter 5. Coupled-Resonator Approach 122
This structure is similar to a previously proposed transmitarray design [33], but with tunable components. Although one could perceive this design as two aperture-coupled patch antennas, it can also be seen as three closely spaced scattering layers, where the outer two layers are patches, and the center layer is a wire grid (an array of slots becomes a grid as the slots become large), as in some FSS structures [66]. While this design has resemblances to both distributed-scatterer and guided-wave designs, we do not consider it a guided-wave design because there are no transmission lines in the structure.
The advantage of this structure is its remarkable simplicity. Requiring only three layers of metallization, this design can be fabricated on two pieces of microstrip lami- nate. Furthermore, layer alignment and bonding is not critical because there are no vias connecting the layers.
5.2.1 Equivalent Circuit Model
We can approximately model the transmitarray element using three inductively-coupled shunt LC resonators, as shown in Figure 5.6. Due to the complex three-dimensional na- ture of the transmitarray structure, an equivalent lumped element circuit with quantities explicitly relating physical geometries to parameters cannot be developed. Nevertheless, we can discuss how the circuit components manifest themselves in the actual design, and how their values affect the overall behavior. This unit cell design can achieve a pole trajectory similar to that shown in Figure 5.4 using only tunable capacitances, which is advantageous since tunable inductors are not easily realized.
In the circuit model, each patch is modeled as a shunt LC resonator with parameters
Cp and Lp [71]. A transformer is shown to represent the impedance transformation by the patch to the free-space ports with intrinsic impedance η. The capacitance Cp combines the patch and varactor diode capacitances to form a single tunable capacitance. To model the slot, a T-network of inductors Lc and Ls can be used, as is done waveguide discontinuities [83]. Because the values of these inductances depend on the waveguide Chapter 5. Coupled-Resonator Approach 123
Figure 5.6: Circuit model dimensions, we can expect that the unit cell size, or equivalently the array element spacing, will affect the response of the unit cell. Finally, the varactor diode placed across the slot can be modeled as a variable capacitor Cs. Together with the shunt Ls element in the slot model, Cs and Ls form a third resonator between the two patch resonators.
5.2.2 Implementation Geometries
Next, we describe the design parameters of the unit cell and their relationships with the components in the equivalent circuit model.
The size of the patches, 13.0 mm 13.0 mm, was selected such that they resonated × around 5.7 GHz when loaded with the varactor diodes, for an intended array spacing of
30 mm 30 mm. A square shape was selected so that the distance between adjacent × array patches was maximized, in order to minimize mutual coupling. We note that while the patch was square, it only resonated in they ˆ-direction due to the loading produced by the varactor diodes.
The size of the slot was 5.0 mm 2.0 mm. The length of the slot (5.0 mm) is associated × with the series coupling inductance Lc in the equivalent circuit model, where smaller slot lengths result in higher values of Lc. If we examine the equivalent circuit model in Figure
5.6, the inductance Lc is needed for the structure to have two distinct resonances. If L 0, the equivalent circuit model degenerates into a single LC resonator. Accordingly, c → if the slot is very long in the implementation, then the structure will only have one Chapter 5. Coupled-Resonator Approach 124 resonance, which is not desired.
The patches were patterned on substrate with dielectric constant ǫr = 2.94. The thickness of the substrate affects the resonance quality factor of the patches. When the substrate is thick, the patches are less resonant because power is more easily radiated. On the complex plane discussed earlier in Figure 5.4(a), the thicker the substrate, the further the poles are located from the jω-axis, resulting in the movement of the poles having less impact on the phase response. On the other hand, the thinner the substrate, the closer the poles are located to the jω-axis, and so the ripples in the magnitude response are more pronounced. In the fabricated unit cell, a substrate thickness of3.0 mm was selected because it provided a balance between phase tuning range and magnitude variation of
S21. As with the unit cell design presented in Chapter 4, Aeroflex/Metelics MGV100-20
GaAs Hyperabrupt varactor diodes were used for both the patches and the slot, with reverse bias voltages of 0 V 20 V producing approximate capacitances of 0.15 pF − − 2.0 pF, as summarized in Table 4.2.
While there are three tunable resonators in the structure, the bias voltages of the two patch varactor diodes are kept the same because power does not flow through the structure when the patch resonant frequencies are different. Conveniently, this also re- duces the degrees of freedom in the reconfigurability to two, meaning that only two bias voltages are required for each unit cell. We define a configuration, denoted by (Cp,Cs) to be a pair of patch and slot varactor diode bias voltages.
5.2.3 Simulation Results
To characterize the unit cell, it was first simulated inside a 30 mm 30 mm waveguide × with periodic boundary conditions to simulate an infinite array, following the discussion in
Section 3.3 of Chapter 3. The structure was simulated using full-wave FDTD SEMCAD-
X simulations, assuming the ideal case with no losses and broadside illumination. The Chapter 5. Coupled-Resonator Approach 125 resulting S-parameters are shown in Figure 5.7 for different configurations. Many trial and error simulations were required to determine the (Cp,Cs) configurations that would produce the desired response. From the plot, we see a characteristic third-order pass band that is shifted as the capacitances are tuned, similar to the curves in Figure 5.4(b).
If we sample the results from Figure 5.7 at 5.7 GHz, then we can plot the transmission response shown in Figure 5.8. At 5.7 GHz, this ideal structure achieves 356◦ of phase range with an average insertion loss of 2 dB, ranging from 3.5 dB to 0.6 dB. The ripples in the curve correspond to the ripples in the pass band. This demonstrates that in the ideal case it is possible to create a transmitarray element with only two layers of substrate and three layers of metallization. We note that in the results shown, a substrate thickness of 2.5 mm was used instead of 3.0 mm to manipulate the size of the ripples in the passband. If a 3.0 mm substrate is used instead, the insertion loss variations would be slightly reduced, but the phase range would also be slightly reduced.
Because a waveguide with periodic boundary conditions cannot be realized experi- mentally, and the PMC walls of a parallel-plate waveguide are difficult to implement, we next investigate the behavior of the unit cell in rectangular waveguide as losses are in- troduced. Using WR-187 rectangular waveguide with dimensions 47.55 mm 22.15 mm, × the pass bands of the structure are shifted down in frequency by about 200 MHz. Be- cause the spacing of the simulated infinite array is changed, component values in the equivalent circuit model also change, resulting in a shift in the operating frequency from
5.7 GHz to 5.5 GHz. Nevertheless, as long as the measured and simulated results agree for the rectangular waveguide, the simulation results are validated, and so the simulation results with periodic boundary conditions should be able to predict element behavior in an infinite array. The array implementation discussed later in this chapter uses a spacing of 30 mm 30 mm, and so will function at 5.7 GHz. × When losses are introduced into the structure, the phase range is reduced because the quality factors of the resonators are reduced. Effectively, the poles on the complex plane Chapter 5. Coupled-Resonator Approach 126
0
−5 | (dB)
21 −10 |S
−15 4.5 5 5.5 6 6.5 7 Frequency (GHz)
0
(Deg) −200
21 (0.28,0.13pF)
S (0.33,0.14pF) ∠ −400 (0.37,0.15pF) 4.5 5 5.5 6 6.5 (0.41,0.16pF)7 Frequency (GHz) (0.45,0.17pF) (0.49,0.18pF) 0 (0.53,0.19pF) (0.57,0.2pF) −5 (0.61,0.21pF)
| (dB) (0.65,0.22pF) 11 −10
|S (0.73,0.24pF) (0.78,0.25pF) −15 4.5 5 5.5 6 6.5 7 Frequency (GHz)
Figure 5.7: Ideal simulated S21 for different configurations (Cp,Cs) Chapter 5. Coupled-Resonator Approach 127
0
−1
−2 | (dB)
21 −3 |S −4
−5 0 100 200 300 400 ∠ S (Deg) 21
Figure 5.8: Ideal transmission response at 5.7 GHz
are shifted leftward on the complex plane, resulting in pole movements having less effect on the phase. Modeling the varactor diodes with the simple series model from Chapter 4 shown in Figure 4.31, Figure 5.9 shows the effect as the varactor diode parasitic resistance is increased up to 2 Ω. While this loss only has a small effect on the transmission phase, the tuning phase range is reduced because the width of the bandpass region is reduced.
As a result, the range of configurations over which S is usable is reduced. | 21|
Figure 5.10 shows the S-parameters of the unit cell simulated in rectangular waveg- uide for different configurations, including varactor losses and parasitics (Lv = 0.4 nH,
Rvs = 2 Ω). Compared with Figure 5.7, there is almost no rippling in the transmission magnitude, and the phase curve is more linear with a 45◦ phase bandwidth of about 120 MHz, or about 2%.
If we sample the results from Figure 5.10 at 5.5 GHz, then we can plot the transmission response shown in Figure 5.11. The unit cell achieves 260◦ of phase tunability where the insertion loss varies between 2 dB and 5 dB. Clearly, the phase range has been significantly reduced with the introduction of losses. Chapter 5. Coupled-Resonator Approach 128
0 0.0 ohm 0.5 ohm 100 1.0 ohm 1.5 ohm (Deg) | (dB) −5 0 2.0 ohm 21 21 S |S
∠ −100
−10 5 6 7 5 6 7 Frequency (GHz) Frequency (GHz)
(a) Configuration 0.66 0.24 pF − 0 0.0 ohm 0.5 ohm 100 1.0 ohm 1.5 ohm (Deg) | (dB) −5 0 2.0 ohm 21 21 S |S
∠ −100
−10 5 6 7 5 6 7 Frequency (GHz) Frequency (GHz)
(b) Configuration 0.59 0.22 pF − Figure 5.9: The effect of varactor diode loss Chapter 5. Coupled-Resonator Approach 129
0 −5
| (dB) −10 21
|S −15 −20 4.5 5 5.5 6 6.5 7 Frequency (GHz)
(1.0,0.29pF) 0 (0.9,0.28pF) (0.78,0.27pF) (Deg) −200
21 (0.74,0.26pF)
S (0.69,0.25pF) ∠ −400 (0.66,0.24pF)
4.5 5 5.5 6 6.5 (0.63,0.23pF)7 Frequency (GHz) (0.59,0.22pF) (0.54,0.21pF) 0 (0.5,0.2pF) −5 (0.46,0.19pF) (0.43,0.18pF)
| (dB) −10 (0.4,0.17pF) 11 (0.35,0.16pF) |S −15 (0.29,0.15pF) −20 4.5 5 5.5 6 6.5 7 Frequency (GHz)
Figure 5.10: Simulated S21 in rectangular waveguide, with varactor losses Chapter 5. Coupled-Resonator Approach 130
0
−5 | (dB) 21
|S −10
−15 0 100 200 300 400 ∠ S (Deg) 21
Figure 5.11: Transmission response at 5.5 GHz, with varactor losses
5.2.4 Experimental Results
The unit cell was fabricated on Rogers Duroid 6002 (ǫr = 2.94) and is shown in Figure 5.12. We defer the discussion of the biasing circuitry to Section 5.3.2. It was tested with the same WR-187 rectangular waveguide setup and programmable voltage controller as the unit cell in Chapter 4.
Figure 5.12: Experimental waveguide test harness
The patch and slot voltages were systematically swept to create two-dimensional the
S plots shown in Figures 5.13 and 5.14. Because there are two degrees of freedom in the | 21| Chapter 5. Coupled-Resonator Approach 131 control, many different configurations can yield the same phase shift, but with different magnitude responses. The black line in the figures denotes the locus of configurations that yield maximal S for different desired phase shifts. | 21|
Locus of Optimal Configurations
0
−20 | [dB]
21 −40 |S
−60 20 15 20 15 10 10 5 5
Slot Varactor Bias [V] 0 0 Patch Varactor Bias [V]
Figure 5.13: Measured S21 (magnitude)
The S-parameters of the structure for the optimal configurations (configurations on the optimal locus) are shown in Figure 5.15. Comparison with Figure 5.10 reveals that they are very similar. Because the mapping between varactor diode capacitance and bias voltage is approximate, the capacitances shown in the legend are estimates, and so the simulated and experimental curves should not be individually compared. Also, because the RF chokes used in the biasing of the patches were self-resonant at 5.35 GHz, we expect the experimental and simulated results to diverge slightly as we move away from that frequency.
A comparison of the magnitude and phase responses at 5.5 GHz of both experimental and simulated results are shown in Figure 5.16, and we can see that the achieved trans- Chapter 5. Coupled-Resonator Approach 132
Locus of Optimal Configurations
400
300
200 [Deg]
21 100 S ∠ 0
−100 20 20 15 10 10
5 Slot Varactor Bias [V] 0 0 Patch Varactor Bias [V]
Figure 5.14: Measured S21 (angle) mission and reflection agree well for both S and S , with a maximum S of 2.7 dB. 21 11 | 21| − If we require that S vary by less than 3 dB ( 5.7 dB S 2.7 dB), or that S | 21| − ≤| 21| ≤ − | 11| be less than 10 dB, then the fabricated element achieves about 245◦ of phase tunability. However, depending on the beamforming algorithm used, larger variations in S can | 21| be accommodated. If the magnitude variation requirement is relaxed to 10 dB, then the element provides 330◦ of phase agility.
5.3 Transmitarray Design and Implementation
Although this unit cell design with slot-coupled patches does not achieve a good insertion loss for the entire 360◦ phase range, it is still possible to implement an array with the cell, since beamsteering is controlled by element phase and not element magnitude, as mentioned in the discussion in Section 3.2.4. Therefore, we present the design and char- acterization of a 6 6 planar transmitarray operating at 5.7 GHz, with 30 mm 30 mm × × Chapter 5. Coupled-Resonator Approach 133
0 −5
| (dB) −10 21
|S −15 −20 4.5 5 5.5 6 6.5 Frequency (GHz) 400
200 (Deg)
21 (0.99,0.28pF)
S (0.84,0.29pF)
∠ 0 (0.75,0.28pF) 4.5 5 5.5 6 (0.68,0.27pF)6.5 Frequency (GHz) (0.63,0.26pF) (0.55,0.24pF) 0 (0.5,0.23pF) −5 (0.45,0.21pF) −10 (0.4,0.2pF)
| (dB) (0.32,0.2pF)
11 −15
|S (0.22,0.2pF) −20 (0.15,0.19pF) −25 4.5 5 5.5 6 6.5 Frequency (GHz)
Figure 5.15: Measured S-parameters for optimal configurations Chapter 5. Coupled-Resonator Approach 134
0 |S | (EXP) 21 −5 |S | (SIM) 21 |S | (EXP) −10 11 |S | (SIM) 11 Magnitude [dB] −15 0 100 200 300 Configuration Number
300
200 ∠ S (EXP) 21 100 ∠ S (SIM)
Phase [Deg] 21
0 0 100 200 300 Configuration Number
Figure 5.16: Transmitarray performance
(0.57λ) element spacing. The size of 6 6 is sufficiently small to be easily and cost- × effectively fabricated, but sufficiently large so that the directivity is high enough to be measured by a planar near-field antenna scanner. This fabricated array is only a proof- of-concept, and a practical implementation of this array design would be significantly larger to capitalize on the advantages of the architecture. To realize larger sizes, the transmitarray could be assembled in panels.
In this section, we first quantify the mutual coupling of the unit cells, followed by a discussion on biasing circuitry. Following this, we describe the experimental setup and the characterization of the array elements. Finally, we present and discuss results from beamsteering experiments. Chapter 5. Coupled-Resonator Approach 135
Table 5.1: Special configurations
Name Cp (pF) Cs (pF) S21 Phase
A 0.74 0.26 130◦
B 0.66 0.24 171◦ C 0.59 0.22 154 − ◦ D 0.50 0.20 117 − ◦ E 0.43 0.18 85 − ◦ F 0.35 0.16 28 − ◦
5.3.1 Effect of Mutual Coupling
When combining elements into an array for beamsteering, mutual coupling is an im- portant consideration. If there is high mutual coupling between the elements, then the tuning of one element will affect neighboring elements, resulting in errors in the element phases. Since element phases are critical to beamforming, it will be significantly impacted by phase errors.
As the interaction between two adjacent array elements varies depending on how each element is tuned, we need to examine the mutual coupling between elements using a range of configurations. Table 5.1 shows names for the special configurations that are used in this subsection.
The mutual coupling between two elements can be characterized using two 30 mm × 30 mm parallel-plate waveguides, each containing an element. Two arrangements are used to quantify the E-plane and H-plane coupling, as illustrated in Figure 5.17. In both cases, the waveguides to the left-hand side of the elements in the figures are two isolated parallel- plate waveguides, with perfect electric conductor (PEC) and perfect magnetic conductor
(PMC) boundaries, into which a plane wave with broadside incidence is launched. The purpose of the conductor separation is to ensure that only element 1 is excited. On the right-hand side, however, the wall separating the waveguides is removed so that power Chapter 5. Coupled-Resonator Approach 136 is allowed to couple from element 1 into element 2. On both sides, elements experience the simulated effect of being in an infinite array. As the input port is excited, the S21 of the system reveals the amount of coupling between adjacent elements. We note that an alternative method of characterizing mutual coupling in a transmitarray has been proposed [84] (p. 77), but this technique assumes that the transmission magnitude is flat across the tuning range, which is not true for this element design.
(a) E-plane coupling
(b) H-plane coupling
Figure 5.17: Mutual coupling simulation waveguides
The E-plane and H-plane mutual coupling between two adjacent elements, using com- binations of special configurations is shown in Figure 5.18. Both identically-configured
(e.g. Configuration A for both elements 1 and 2, labeled A-A) and differently-configured
(e.g. element 1 with Configuration D and element 2 with Configuration A, labeled D-A) combinations are shown. From the plots, we see that E-plane coupling does not exceed Chapter 5. Coupled-Resonator Approach 137
25 dB and the H-plane coupling does not exceed 30 dB for the configurations tested. − − Thus, we conclude that the mutual coupling between elements using this slot-coupled patch design is insignificant.
E−plane Coupling 0
−20
| (dB) A−A 21 −40 B−B |S C−C −60 D−A 4.5 5 5.5 6 6.5 7 D−B Frequency (GHz) H−plane Coupling D−C 0 D−D D−E D−F −20 E−E | (dB)
21 F−F −40 |S
−60 4.5 5 5.5 6 6.5 7 Frequency (GHz)
Figure 5.18: Simulated mutual coupling for pairs of configurations
5.3.2 Biasing Circuitry
An obvious challenge with loading a slot with a varactor diode is that the diode cannot be readily biased. A voltage needs to be developed across the slot in the ground plane, but the ground plane needs to be electrically continuous at high frequencies. Any discontinu- ities along the edge of the slot significantly alter the behavior of the slot. Inter-digitated capacitors patterned directly onto the ground plane for the purpose of creating RF short circuits are not possible because sufficiently large capacitances cannot be achieved in the available space. Furthermore, the size of surface mount capacitors with respect to the slot also significantly alters the slot’s behavior. Chapter 5. Coupled-Resonator Approach 138
To solve this problem, two conducting layers separated by a very thin dielectric layer are used to compose the ground plane. As shown in Figure 5.19, the 5.0 mm 2.0 mm × slot is placed in the lower ground plane, with the two sides of the slot electrically isolated at DC. A larger 5.0 mm 5.0 mm rectangular hole is placed in the upper ground plane, × which has the same width as the slot, so that the biasing grooves do not electrically extend the slot. The length of the hole is large enough such that a varactor diode could easily be soldered to the lower ground plane. A small cavity (3.0 mm 1.0 mm 1.0 mm) × × is milled into the upper substrate (not shown) creating room for the slot varactor diode to fit in between the substrates. The two substrate layers are separated by a 0.04 mm
Rogers 3001 dielectric bonding film. With the thin film in place, a large capacitance is created between the upper and lower ground planes, resulting in a biased slot that appears electrically continuous at high frequencies, but has sides that are isolated for
DC voltages. This biasing solution has the benefit that there are no extra conducting layers needed, since two microstrip layers have four conducting planes, and bonding film is required anyways to bind the two layers together.
To bias the varactor diodes on the patch antennas, bias lines are placed on the same metallization layer as the patches, as shown in Figure 5.19. Independent control of reverse bias voltages on every element in the transmitarray is challenging. Unlike reconfigurable reflectarrays, where the biasing network can potentially be placed conveniently behind a ground plane, biasing mechanisms are fully exposed to the incident waves from the feeding antenna. It was observed in simulation thatx ˆ-directed lines (perpendicular to the electric-field) do not perturb the fields if they are sufficiently far away from the patches. On the other hand, resonances are easily excited in they ˆ-directed lines (parallel to the electric field). Therefore, while thex ˆ-directed segments of the lines can be simply implemented using microstrip lines, additional measures are needed to ensure that the yˆ-directed segments do not resonate. Because the currents flowing through the diodes in reverse bias are very small (about 100 nA), resistance can be added to the lines to Chapter 5. Coupled-Resonator Approach 139 prevent RF propagation, without affecting the DC bias voltages carried by the lines. While the optimal solution would be to use resistive lines realized from lossy thin films
(especially for scaling the design to larger arrays), such lines can be approximated using resistive loading at discrete locations along they ˆ-directed lines [85], as shown in Figure
5.19. It was determined through simulations that power is best attenuated in the lines with 10 kΩ resistors. The bias lines in the fabricated array are 0.4 mm thick with 0.3 mm gaps in between. To accommodate larger array sizes, the width of the bias lines and gaps could be significantly reduced without any problems, since the varactor diodes are reverse biased and do not carry any significant current.
5.3.3 Experimental Setup
The geometry of the array elements is identical to that of the unit cell. The only difference between the unit cell and the array implementation is the varactor diode model used.
Aeroflex/Metelics GaAs MGV100-21 varactor diodes with E28X packaging are used for the array, and have the same parasitic values as the MGV100-20 diode used earlier. The only difference is a very slight increase in the capacitance tuning range.
The fabricated 6 6 planar array is shown in Figure 5.20. A pyramidal feed horn × with a directivity of 17.6 dBi was placed such that the aperture of the horn was 300 mm from the ground plane of the array, which corresponded to an f/D ratio of 1.67. The feed horn was positioned such that the array was prime-focus fed. The feed horn was
180 mm long, with an aperture size of 140 mm 130 mm. The entire setup was placed × on a planar 5′ 5′ Nearfield Systems Inc. (NSI) near-field antenna scanner, as shown in × Figure 5.21.
The patches and slot biases of each element were connected with 40-pin ribbon cables to a series of custom-designed programmable USB voltage controllers. As there were
36 elements, 72 independent voltage channels were required. Three 32-channel voltage controller boards were used to control the elements. Each channel had an 8-bit digital-to- Chapter 5. Coupled-Resonator Approach 140
Figure 5.19: Array element and bias network design (vertically exaggerated) Chapter 5. Coupled-Resonator Approach 141
Figure 5.20: Fabricated 6 6 array ×
Figure 5.21: Transmitarray experimental setup in the near-field scanner Chapter 5. Coupled-Resonator Approach 142 analog converter followed by an operational amplifier, and produced a voltage between 0 V and 20 V.
5.3.4 Element Characterization
Repeated waveguide tests showed that the elements are somewhat sensitive to fabrication error, and the voltage-capacitance characteristics of the varactor diodes vary from diode to diode. So, the behavior of each element in the fabricated array must be measured.
To characterize the array, for each element, the open-ended waveguide (OEWG) probe of the scanner was moved to the broadside position of that element. Measurements were conducted with the waveguide opening positioned 24 mm away from the element-under- test in the longitudinal direction. A two-dimensional sweep of the two element control voltages was performed, with the patch voltage varying between 1 V and 20 V, and the slot voltage varying between 4 V and 20 V.
As each element was characterized, the bias voltages of all the other elements were kept at zero volts, since zero volt slot biases make the elements non-transmissive. The magnitude and phase received by the probe at that single position was taken to be the transmission response of the element. In this method, the near-field scanner is used only to position the probe, and no near-field scanning is actually performed. We note that while the gain of the OEWG probe is included in the measurements, the absolute magnitude and phase of the characterization is not important. By placing the probe at broadside for each element, the OEWG probe pattern consistently adds a constant magnitude and phase to each measurement.
As was done with the unit cell, the two-dimensional voltage sweep of each element produced a set of S21 magnitudes and phases for each element. Optimal configurations were determined for each desired phase shift, for each element. Ideally, all elements have the same optimal configuration curves, but in reality they differ due to fabrication errors and varactor diode variances. An advantage of having two degrees of freedom of control Chapter 5. Coupled-Resonator Approach 143 for each element is that these variances can be accommodated, to a certain degree. These optimal configurations were stored in lookup tables for beamforming.
The average 3 dB-variation phase range of the fabricated elements is 195◦, ranging between 149◦ and 244◦. If the magnitude variation is relaxed to 10 dB, then the average phase range of the elements is 284◦, ranging between 205◦ and 360◦. Figure 5.22 shows histograms of the phase ranges for the fabricated elements. We note that the variations in phase range did not depend on element position, so the variations cannot be attributed to the feed illumination. Nevertheless, despite the variation in the tuning range between the elements, the characterization shows that all of the elements exhibit a useful tunable phase range for beamforming purposes.
6 10
8 4 6
4 2 2 Number of elements Number of elements
0 0 100 150 200 250 200 250 300 350 400 Phase range [deg] Phase range [deg]
(a) 3 dB magnitude variation (b) 10 dB magnitude variation
Figure 5.22: Phase range histograms for fabricated elements
5.3.5 Experimental Results
An NSI planar near-field antenna scanner was used to measure the near-field response at 5.7 GHz, from which the far-field response was computed. Broadside beamsteering, formed by creating uniform phases on the aperture, was tested first and produced a well-defined pencil beam with an isolation of 18 dB between the co-polarized and − cross-polarized patterns, as shown in Figure 5.23. Chapter 5. Coupled-Resonator Approach 144
To produce pencil beams in different directions, the element phases were calculated using (3.37). Figure 5.24 shows the far-field gain of E-plane pencil beams directed from
25◦ to 25◦ in elevation at zero azimuth angle. Similarly, Figure 5.25 shows the far-field − gain of H-plane pencil beams directed from 25◦ to 25◦ in azimuth at zero elevation − angle. The gain of the array was established by comparing the radiated power density of the array with that of a standard gain horn. We note that there is a slight asymme- try between the positive- and negative-angled beams, which is due to a combination of element variations and fabrication error. The directivities (D), peak gain (G), side-lobe levels (SLL), and half-power beam widths (HPBW ) of the pencil beams are summa- rized in Table 5.2. The 1-dB bandwidth of the broadside beam was 280 MHz, which is a fractional bandwidth of 4.9%.
E−plane (co−pol) 15 E−plane (cross−pol) H−plane (co−pol) H−plane (cross−pol) 10
5
0 Gain (dBi)
−5
−10
−15 −50 −40 −30 −20 −10 0 10 20 30 40 50 Angle (Deg)
Figure 5.23: Co- and cross-polarizations in the two principal planes
From the plots, we see that the magnitude response decreases as the beam scans from broadside. A small amount of the reduction can be attributed to the patch antenna Chapter 5. Coupled-Resonator Approach 145
pattern, since at 25◦ off-broadside, the patch’s antenna pattern has losses of 0.5 dB and − 0.9 dB in the E-plane and H-plane, respectively. However, this scan loss is primarily − due to the fact that as scanning angle is increased, the range of required phase responses in the elements is also increased. Since the elements produce large insertion losses for a range of phase shifts, some elements in the array can effectively become disabled.
Thus, as scanning angle is increased, there is a reduction in the overall amount of power transmitted through the array, and the gain is reduced.
−25° 15 −20° −15° −10° 10 −5° 0° 5° 5 10° 15° 20° 0 25° Gain (dBi)
−5
−10
−15 −50 −40 −30 −20 −10 0 10 20 30 40 50 Elevation (Deg)
Figure 5.24: Measured far-field pattern as the elevation angle is scanned (E-plane)
We see the effect of the non-constant insertion losses for different desired element phases in Figure 5.26, where the near-field measurements are holographically back- projected to yield the fields on the array aperture. For the broadside pencil beam, the phase in Figure 5.26(b) is very constant over the entire aperture, and the magnitude in Figure 5.26(a) shows the effect of the expected feed taper. When a pencil beam with
θ0 = 15◦ in the H-plane is formed, we see from Figure 5.26(d), and from (3.37), that a Chapter 5. Coupled-Resonator Approach 146
Table 5.2: Pencil beam directivity, peak gain, side-lobe sevel, and beam width E-plane H-plane
θ0 D G SLL HPBW D G SLL HPBW (Deg) (dBi) (dBi) (dB) (Deg) (dBi) (dBi) (dB) (Deg)
-25 18.1 6.8 -9.6 24.2 17.3 6.0 -6.6 20.4
-20 19.4 8.8 -12.4 20.9 17.9 6.9 -6.4 26.4
-15 19.9 10.2 -14.2 19.8 18.9 9.1 -10.6 21.5
-10 20.2 11.2 -14.8 19.3 20.1 11.0 -12.1 16.5
-5 20.7 12.0 -18.3 17.6 20.6 11.9 -10.4 14.3
0 20.8 12.3 -14.9 17.1 20.8 12.4 -12.3 15.4
5 20.7 11.7 -17.4 17.6 20.8 11.6 -13.4 14.9
10 20.0 10.3 -21.1 19.8 20.7 10.6 -12.5 15.4
15 19.6 9.5 -19.4 20.9 19.8 7.9 -10.6 18.2
20 19.0 8.3 -9.3 18.7 18.6 6.4 -8.6 18.2
25 19.3 7.5 -7.9 17.1 18.3 6.5 -8.2 21.5 Chapter 5. Coupled-Resonator Approach 147
−25° 15 −20° −15° −10° 10 −5° 0° 5° 5 10° 15° 20° 0 25° Gain (dBi)
−5
−10
−15 −50 −40 −30 −20 −10 0 10 20 30 40 50 Azimuth (Deg)
Figure 5.25: Measured far-field pattern as the azimuth angle is scanned (H-plane)
full 360◦ of phase tunability is required. The array manages to create the required phase gradient, but the magnitudes of different elements deteriorate. From Figure 5.26(c), we see that certain elements effectively cease to transmit, resulting in reduced gain and di- rectivity. An alternative approach could be to configure such elements to transmit with a certain amount of phase error but with smaller insertion loss. While the overall gain may be improved, phase errors would result in degradation of the directivity and side-lobe levels.
The directivity of a hypothetical uniform aperture with the same dimensions as the array (180 mm 180 mm), given by (3.38) is 4π ab = 21.7 dBi. So, the array achieves × λ2 a directivity at broadside of 20.8 dBi, which is only 0.9 dB less than that of an ideal aperture. This difference is caused by the feed taper efficiency of 0.8 dB. There is − a significant difference between the array’s measured directivity and gain, which is due primarily to low efficiency of the feeding system and element losses. Losses will be Chapter 5. Coupled-Resonator Approach 148
(a) Magnitude (Broadside beam) (b) Phase (Broadside beam)
(c) Magnitude (15◦ beam) (d) Phase (15◦ beam)
Figure 5.26: Measured aperture fields (array bounds denoted by rectangle) Chapter 5. Coupled-Resonator Approach 149 discussed in greater detail later. Nevertheless, the measured half-power beam widths in both principal planes are very close to the theoretical value, given by (3.39), of 14.9◦. The theoretical first side-lobe level of a uniform aperture, 13.3 dB, is also very close − to the side-lobe levels achieved by this array. In fact, the magnitude tapering due to the feed resulted in smaller E-plane side-lobes, but at the cost of slightly wider beam widths.
At broadside, the 6 6 transmitarray demonstrates beamforming performance that is × very close to optimal.
To calculate the expected result from array theory, we need to determine the antenna pattern for a patch antenna, the excitation of each element, and the array factor, which is given by (3.29). The element pattern on principal planes of an L W patch antenna × can be calculated as [86] (p. 213)
kL E-plane: cos sin θ (5.4) 2 sin kW sin θ H-plane: cos θ 2 (5.5) kW sin θ 2
The radiation of each element can be calculated using the feed illumination and the measured element S21 responses. Figures 5.27 and 5.28 show that the expected (AF) and measured (EXP) far-field gain patterns correspond very well, confirming that mutual coupling is not a significant problem. The observed ripples are likely a result of diffraction effects at the edge of the array.
To account for the loss, which is the difference between the directivity and the gain, we can break down the losses into the types of losses described in Section 3.2.5, as shown in Table 5.3. Details on how the different quantities are calculated are provided in Appendix C. The loss budget shows that the expected gain is in good agreement with the measured gain. For non-broadside beamforming, the reduction in gain is primarily due to increased reflective loss. Chapter 5. Coupled-Resonator Approach 150
−20° (EXP) 15 −20° (AF) −10° (EXP) 10 −10° (AF) 0° (EXP) 0° (AF) 5 10° (EXP) 10° (AF) 0 20° (EXP) 20° (AF) −5 Gain (dBi) −10
−15
−20
−25 −50 −40 −30 −20 −10 0 10 20 30 40 50 Elevation (Deg)
Figure 5.27: Measured far-field pattern (E-plane)
−20° (EXP) 15 −20° (AF) −10° (EXP) 10 −10° (AF) 0° (EXP) 0° (AF) 5 10° (EXP) 10° (AF) 0 20° (EXP) 20° (AF) −5 Gain (dBi) −10
−15
−20
−25 −50 −40 −30 −20 −10 0 10 20 30 40 50 Azimuth (Deg)
Figure 5.28: Measured far-field pattern (H-plane) Chapter 5. Coupled-Resonator Approach 151
Table 5.3: Transmitarray loss budget for broadside beam Maximum theoretical aperture gain 21.7 dBi
Feed taper efficiency 0.8 dB − Feed horn efficiency 1.1 dB − Spillover loss 2.8 dB − Array loss 4.8 dB − Expected gain 12.2 dBi
Measured gain 12.4 dBi
5.4 Reflectarray-Mode Operation
A useful result from this unit cell design is its capacity to also function as a reflectarray element, if desired. When the slot varactor diode is biased with a low voltage (< 1 V), the slot ceases to be resonant, and as a result no power flows through the slot. In this case, the array element operates as a single-pole resonator, and the patch varactor diodes can be used to tune the reflection phase of the element like a reflectarray element [82].
The ability of the transmitarray element to function as a reflectarray is useful in many scenarios. A planar array of such elements can beam scan not only in the forward direction as a transmitarray, but also in the reverse direction as a reflectarray, effectively doubling the scanning range. If used in a conformal array, some parts of the array can be made to transmit, while other parts of the array can be made to reflect, all using the same basic array element.
Operating in reflectarray-mode with the patch varactor diode fixed at 1 V, the re- sulting S-parameters for the unit cell in rectangular waveguide are shown in Figure 5.29.
If the curves are sampled at 5.5 GHz, then the achievable S11 magnitude and phase re- sponse curves are shown in Figure 5.30. As a reflectarray element, this design can achieve
300◦ of phase tunability, with less than 4 dB of magnitude variation. This insertion loss is consistent with typical varactor diode-tuned reflectarray elements [82]. Chapter 5. Coupled-Resonator Approach 152
0
| [dB] −5 11 |S
−10 4.5 5 5.5 6 6.5 Frequency [GHz] 0.47pF 200 0.41pF 0.33pF 100 0.14pF 0.97pF [deg] 0 11 0.73pF S 0.59pF ∠ −100 0.50pF −200 4.5 5 5.5 6 6.5 Frequency [GHz]
Figure 5.29: Measured S11 for 1 V slot varactor bias and varied patch capacitance
0
−2
−4 | (dB)
11 −6 |S −8
−10 −200 −100 0 100 200 ∠ S (Deg) 11
Figure 5.30: S11 of unit cell at 5.5 GHz Chapter 5. Coupled-Resonator Approach 153
The 6 6 array that was used in the previous section as a transmitarray is shown × in Figure 5.31 in a reflectarray setup. A smaller pyramidal horn with a directivity of
14.1 dBi, and an aperture of 91 mm 64 mm and a length of 163 mm was placed × such that the center of the horn aperture was 200 mm from the array aperture plane, and 150 mm below the center of the array. The feed horn was tilted at an angle of
30◦ from the prime-focus position in the E-plane. This smaller feed horn was used for reflectarray-mode to reduce feed blockage.
Figure 5.31: Reflectarray setup
Desired element phases for a pencil beam are computed in the same way as the transmitarray case, using (3.37). The measured far-field gain patterns for the E-plane and H-plane are shown in Figures 5.32 and 5.33. In the E-plane, beams for positive elevation angles have much larger magnitudes than those for negative elevation angles. This is due to the tilt of the feed horn, and the effect of feed blockage. In the H-plane, the array achieves high-directivity beams in azimuth angles from 30◦ to 30◦, with a − directivity of 19.4 dBi at broadside. However, as the beam-scanning angle increases, the side-lobe levels increase from 16.3 dB at broadside to 5.0 dB at 30◦. In comparison to − − Chapter 5. Coupled-Resonator Approach 154 the transmitarray beamforming patterns shown in Figures 5.24 and 5.25, the scan angle has a much smaller effect on gain. This is because at 300◦, the phase range of the array in reflectarray mode is much larger.
Given that the peak gains of both the transmitarray and reflectarray are about the same at broadside, we conclude that the efficiencies are similar in the two modes. Due to the similarity between this cell in reflectarray mode and a similar existing reflectarray cell [82], we refer the reader to that work for details on the loss budget.
−30° 15 −25° −20° −15° 10 −10° −5° 0° 5 5° 10° 15° 0 20° 25° Gain (dBi) 30° −5
−10
−15 −50 −40 −30 −20 −10 0 10 20 30 40 50 Elevation (Deg)
Figure 5.32: Reflectarray measured far-field pattern (E-plane)
An interesting observation is that in Figure 5.32 a substantial lobe at an elevation angle of 30◦ is observed at all scanning angles. These lobes can be attributed to specular reflection, indicating that the array elements have reduced ability to couple power well at higher angles of incidence. Chapter 5. Coupled-Resonator Approach 155
−30° 15 −25° −20° −15° 10 −10° −5° 0° 5 5° 10° 15° 0 20° 25° Gain (dBi) 30° −5
−10
−15 −50 −40 −30 −20 −10 0 10 20 30 40 50 Azimuth (Deg)
Figure 5.33: Reflectarray measured far-field pattern (H-plane)
5.5 Conclusions
In this chapter, we approached transmitarray design from a coupled-resonator approach.
Because this approach condenses structures into filters that can be modeled with equiv- alent circuits, we can gain better insight into how geometries and dimensions can be optimized. If a reconfigurable transmitarray made of independently tunable layers was designed from a purely distributed-scatterer approach using Green’s functions and field theory, there would be too many degrees of freedom that would need to be considered.
The approach in this chapter allowed us to systematically determine the required complexity in the system, and design a unit cell with an equivalent circuit model. Al- though the exact parameters for the equivalent circuit model could not be explicitly derived from the geometries of the unit cell presented, the qualitative relationship be- tween the cell geometries and the components in the circuit gave us insight into how the transmission response could be manipulated. Chapter 5. Coupled-Resonator Approach 156
In this chapter, we first demonstrated the performance of a unit cell inside a rectan- gular waveguide. Extending this unit cell into a 6 6 transmitarray, we demonstrated × two-dimensional beamforming results over a 50◦ 50◦ scanning range. Although the × loss was fairly high, with most of the loss attributed to dissipative losses in the varactor diodes, specular reflection, and back-scatter, the far-field beam patterns corresponded well to predicted results.
The slot-coupled patch design presented in this chapter is very simple, requiring no vias and only three layers of metallization (one additional layer for biasing). Electrically, the design is thin at only 0.11 wavelengths, which is substantially thinner than the distributed-scatterer designs discussed in Chapter 4.
In addition to transmitarray operation, this design is also able to function as a re-
flectarray with a scanning range of 60◦ 30◦. With the ability to function as both as a × lens and a reflector, this design can be used in more sophisticated configurations, such as a conformal array where some parts of the array are operating as reflectors and other parts as transmitters.
One observation that we can make regarding the designs studied so far is that they can be modeled with ladder-type circuits. The layered scattering surfaces from Chapter 4 can be modeled by shunt LC resonators separated by transmission line segments, and the slot-coupled patches from this chapter can be modeled by shunt LC resonators separated by inductors. For both designs, the phase range was fairly sensitive to losses because of the way the losses shift the poles on the complex plane. As demonstrated in this chapter, the pole movement of ladder-type circuits is restricted, making the effects of loss difficult to mitigate. While non-ladder-type circuits, such as bridged circuits, can be used to achieve different behavior, it is fundamentally very difficult to achieve this in a distributed structure, because two components in space cannot be coupled together without affecting components in between. Therefore, in the next chapter, we will investigate transmitarrays that use the guided-wave approach, which will allow us Chapter 5. Coupled-Resonator Approach 157 to implement non-latter-type topologies. Chapter 6
Guided-Wave Approach
The literature review in Chapter 2 revealed that the majority of transmitarray designs follow the guided-wave approach, where two antennas are connected using a microwave circuit phase shifter. This is not surprising, since there are many advantages to this ap- proach. Firstly, by separating the transmitarray element into antenna and phase shifter, the design process can be modularized. Well established techniques from antenna and phase shifter design can be directly applied. Unlike the distributed-scatterer approach from Chapter 4, the mutual coupling between adjacent elements is much weaker and so array theory can be used. Secondly, because waves are guided into transmission lines between the input and output, a ground plane can be used to separate the two sides of the transmitarray. As a result, the transmitarray can basically be decomposed into receiving and transmitting phased arrays, for which the theory is well understood.
Of course, the guided-wave approach also has its challenges. The most significant challenge is likely the fabrication complexity that usually results from combining two antennas with a phase shifter. Due to the nature of transmitarrays, the input and output antennas need to have opposite orientations. While microstrip transmission lines are attractive because of their ease of fabrication, they are not symmetric about the ground plane, and as a result many designs have used other transmission line implementations
158 Chapter 6. Guided-Wave Approach 159 such as stripline. Most designs also require many layers of substrate and metallization that need to be pressed together with careful alignment, especially when vias are used.
For the implementation of the phase shifter, fixed transmitarrays typically use de- lay lines which are easy to implement. However, to make reconfigurable phase shifters, tuning components such as varactor diodes or MEMS capacitors are needed. Typically, microstrip implementations of microwave circuits require that the components be con- nected to the ground plane using vias. The use of vias can be challenging because they introduce a parasitic inductance between the component and the ground plane, which may be problematic at higher frequencies. Moreover, the use of vias in multi-layer designs can also introduce alignment issues in the fabrication process.
Reconfigurability also introduces certain challenges. Firstly, the input impedances of reconfigurable phase shifters generally vary over the phase tuning range, creating mismatch with the antennas. While antennas and phase shifters are often designed for
10 dB return loss, the simple cascading of three structures with this level of return loss can result in a poor overall insertion loss. Secondly, the physical size of an array lens element is an important consideration for both practicality and function. With reflection- type phase shifters that have been used, the physical size of the phase shifter forces the array cell spacing to be large, resulting in grating lobes. Integrated circuit phase shifters have small physical sizes, but also have larger insertion losses.
In this chapter, we will focus on a unit cell that uses bridged-T phase shifters and differential proximity-coupled patches. First, we will discuss the motivation for this choice of phase shifter and antenna. Following this, we will present waveguide simulation and experimental results for a unit cell. To increase the bandwidth, we will then extend the unit cell design to use stacked patches. Finally, we will present a 6 6 reconfigurable × transmitarray based on the stacked-patch design with beamforming results. Chapter 6. Guided-Wave Approach 160
6.1 Unit Cell Design
Using the guided-wave approach, the transmitarray element is implemented using two antennas connected by a reconfigurable phase shifter. Therefore, the choice of topology for these two components is crucial to the success of the design.
6.1.1 Reconfigurable Phase Shifter
Phase shifters and all-pass filters have been studied for many decades and are generally very well understood. We note that there is a wealth of work on reconfigurable MMIC phase shifters as well, but most designs have demonstrated significant insertion losses and are costly to fabricate [5]. Therefore, to prove the concepts described in this chapter, we have opted to fabricate the phase shifter on the same dielectric substrate as the antenna, and discrete (lumped) components are used.
Our requirement for reconfigurability means that components such as varactor diodes or MEMS need to be used, introducing two major challenges. Firstly, the tuning com- ponents have significant parasitic resistances and inductances. Secondly, since we desire a single control voltage for each cell, all tunable components must use the same bias voltage, limiting the capacitance and inductance values that can be used. Most non- reconfigurable phase shifter and all-pass filter designs consist of many components that must all have uniquely selected values. To make these designs reconfigurable, the values of these components would need to be tuned independently, which is not practical since each cell would require many control signals.
Reflection-type phase shifters based on quadrature hybrid couplers have been used for transmitarrays [59], but they tend to be physically large, since the lines of the coupler need to be a quarter-wavelength long and multiple sections may need to be cascaded to achieve sufficient phase range. Ladder, lattice, and bridged-T networks are other common
filter section topologies [87] (pp. 55-57). For a reconfigurable ladder phase shifter where Chapter 6. Guided-Wave Approach 161 only shunt resonances are tunable, a flat insertion loss cannot generally be achieved due to the trajectory of pole movement in the complex planes, as seen in Chapter 5. Lattice networks are non-planar, and so cannot be easily implemented without many vias. The use of vias increases fabrication complexity and also introduces a significant amount of parasitic inductance. Therefore, we employ the bridged-T topology, shown in Figure 6.1, because it is planar and can be tuned by changing a single capacitance value.
Figure 6.1: Bridged-T circuit topology
To understand how the bridged-T phase shifter operates, we can qualitatively describe the behavior of the bridged-T circuit for different values of C. When the capacitance C is small, the capacitors behave like open circuits and so the inductors predominate. In this case, the circuit can be simplified to the two L1 inductors in series, which produce a small negative phase shift (or a phase shift slightly less than 360◦). On the other hand when C is large, then it behaves like a short circuit, and so the circuit can be simplified to a single C capacitor in series, which produces a small positive phase shift. When C is such that the vertical branch with C and L2 is resonant with zero impedance, then center of the T-junction is grounded, and the circuit can be simplified to a pi-network with two L1 inductors in shunt and a C capacitor in series. If the values of L1 and C are appropriately selected, then the phase shift through the pi-network can be made to be approximately 180◦. In this way, when C is tuned from a small value to a large value, the phase shift through the structure will change from a value slightly less than 360◦ to 180◦, and then to a small positive value. The exact phase range will depend on the Chapter 6. Guided-Wave Approach 162 tuning range of the capacitance C. Figure 6.2 shows the return loss and transmission phase for the lossless bridged-
T circuit shown in Figure 6.1, where both capacitances are tuned together between
0.3 pF 1.2 pF, and the inductor values are L = 0.75 nH and L = 1.25 nH. The − 1 2 insertion loss over the entire tuning range between 3 GHz and 7 GHz is better than 0.4 dB.
At 5 GHz, a phase range of 220◦ is achieved, which is very large for a reconfigurable phase shifter with only two tunable capacitances and only a single degree of freedom in terms of tunable components. Furthermore, the circuit operates over a frequency range of several GHz with good return loss. As losses due to parasitic resistance in the capacitors are introduced, the insertion loss degrades by about a dB but good return loss and phase range are maintained. A drawback with using the bridged-T topology is that
LC resonator branches are not clearly distinguishable, unlike ladder or lattice designs.
This makes the design process less intuitive and increases the reliance on numerical optimization. To achieve 360◦ of phase range, two bridged-T structures can be cascaded.
0 400 0.3pF 0.4pF 300 0.5pF −10 0.7pF (Deg) | (dB) 200 0.9pF 21 11 1.2pF S |S −20 ∠ 100
−30 0 3 4 5 6 7 3 4 5 6 7 Frequency (GHz) Frequency (GHz)
Figure 6.2: Return loss and transmission phase a single bridged-T filter with L1 = 0.75 nH, L =1.25 nH, and C =0.3 pF 1.2 pF 2 −
6.1.2 Antenna and Balanced Feed
Because the input impedance of the phase shifter changes as it is tuned, we require an antenna feed that maintains a good match over the entire tuning range. If we simply Chapter 6. Guided-Wave Approach 163 use a narrowband probe feed with a patch antenna, then the structure would likely only transmit power over a small tuning range. A proximity-coupled differential, or balanced, feeding technique for microstrip patches [88] has been used to successfully match a varactor-tuned frequency agile antenna over a wide tuning range [89]. The observation that the proximity-coupled balanced feed can couple to a tuned patch antenna suggests that it can also provide a good coupling between a fixed antenna and a tuned phase shifter with a changing input impedance.
The use of an antenna with a balanced feed means that the phase shifter will need to operate on balanced signals as well. In fact, this simplifies the implementation of the bridged-T phase shifter. If an ordinary bridged-T phase shifter is implemented in microstrip, it would require two layers of metallization: the patterned conductor and the ground plane. A via would be needed to connect the T-junction to the ground plane. By mirroring the bridged-T structure to create two differential paths, as shown in Figure
6.3(a), a virtual ground is created, eliminating the need for any vias. While an additional varactor diode is required, the parasitic inductances and fabrication complexity of the vias are avoided. The circuit can be further simplified by removing an inductor and capacitor from the vertical branch, as shown in Figure 6.3(b). The resulting capacitance of the branch is halved and the inductance is doubled, so the effective resonant frequency of the vertical branch remains unchanged.
6.1.3 Unit Cell Implementation
In the implementation of the bridged-T phase shifter on microstrip, we can further sim- plify the circuit by replacing the inductors with short segments of transmission line. As with the unit cells in Chapters 4 and 5, Aeroflex/Metelics GaAs MGV100-20 varactor diodes in E28X packaging are used to produce a variable capacitance, with a tuning range of C =0.15 pF 2.0 pF. Voltages can be developed across the varactor diodes by − making small gaps in the transmission lines, and connecting them with large capacitors Chapter 6. Guided-Wave Approach 164
(a) Mirrored bridged-T (b) Simplified bridged-T
Figure 6.3: Balanced bridged-T circuit topology
(150 pF) used as RF shorts, as shown in Figure 6.4. Because varactor diodes effectively carry no current when they are reverse biased, 10 kΩ resistors can be used as RF chokes to connect the structure to bias lines.
Figure 6.4: Biasing scheme
The dimensions for the phase shifter, patch antenna, and the feed were optimized with FDTD simulations in SEMCAD-X, yielding the dimensions shown in Figure 6.5.
Combining two balanced bridged-T phase shifters and two differential proximity- coupled patch antennas, we have the unit cell illustrated in Figure 6.6. When incident Chapter 6. Guided-Wave Approach 165
Figure 6.5: Dimensions of the bridged-T phase shifter in microstrip waves excite the element, waves are first coupled into two microstrip lines with balanced signals using the proximity-coupled differential feed. To connect the input side to the output side of the cell, the two bottom microstrip lines are aperture-coupled to two other microstrip lines on the other side of the ground plane. The balanced signals then pass through two cascaded balanced bridged-T phase shifters, and are then proximity-coupled to the output patch antenna, which radiates the outgoing waves. We note that since the structure is passive, it is reciprocal, meaning that the same transmission response would be produced if the input and output sides are swapped.
Since the geometries of the design cannot be modeled analytically, they were opti- mized using various simulators. Using Agilent Advanced Design System (ADS), it was found that the bridged-T circuit yields the largest phase range and best insertion loss when the characteristic impedance of the underlying microstrip lines is 30 Ω. Therefore,
30 Ω microstrip lines on Rogers Duroid 6006 (ǫr =6.15) were chosen as the basis for the design.
In this chapter, a slightly more complex equivalent circuit is used to model the varac- tor diode than in the earlier chapters. This model includes a parasitic parallel resistance Chapter 6. Guided-Wave Approach 166
Figure 6.6: Overall unit cell design Chapter 6. Guided-Wave Approach 167
Rvp added in parallel with the capacitance, as shown in Figure 6.7. The series and par- allel parasitic resistances, determined by curve fitting, are Rvs = 3 Ω and Rvp =1.5 kΩ, and the series parasitic inductance is Lv = 0.4 nH. The parallel parasitic resistance models the leakage currents through the semiconductor in the diode, and together with
Rvs approximately model the voltage-dependent loss in the diode. Table 4.2 from Chap- ter 4 provides the approximate mapping between the reverse bias voltages and diode capacitance. We note that while this model works well for the arrangement used in this chapter, it is not appropriate for the designs in previous chapters because the simulated and experimental results do not agree with this model. Likely, this is because here the
fields are confined in a transmission line, which was not the case in the previous two chapters.
Figure 6.7: Varactor diode model (C = 0.15 pF 2.0 pF, R = 3 Ω, R = 1.5 kΩ, − vs vp
Lv =0.4 nH)
6.1.4 Simulated and Experimental Results
The unit cell was simulated inside a WR-187 rectangular waveguide, with dimensions
47.55 mm 22.15 mm, to characterize an infinite array of identical elements, following the × discussion on infinite arrays and waveguides in Chapter 3. At 4.8 GHz, the fundamental
TE10 mode of the waveguide simulates an angle of incidence of 41◦. The S-parameters of the waveguide section containing the element characterizes the return and insertion losses of the element in an array. The results from simulations are shown in Figure 6.8. At
4.78 GHz, the unit cell yields 395◦ of phase range, and an insertion loss varying between Chapter 6. Guided-Wave Approach 168
2.2 dB 4.5 dB with an average of 3.5 dB. The return loss is better than 11 dB over the − entire tuning range.
0
| (dB) −10 11 |S
−20 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz) 0.15pF 0 0.20pF 0.30pF 0.40pF
| (dB) −10 0.50pF 21
|S 0.70pF 0.90pF −20 1.20pF 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz) 2.00pF 200
(deg) 0 21 S ∠
−200 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz)
Figure 6.8: Simulated results for the single-patch element
The fabricated unit cell is shown in Figure 6.9 in a WR-187 test harness. The overall thickness of the element is 4.3 mm, which is 0.07λ in free-space. The measured perfor- mance of the cell is shown in Figure 6.10. While the simulated cell performs best at
4.78 GHz, the fabricated cell performs best at 4.87 GHz, representing a frequency shift of about 90 MHz. This minor shift in frequency is likely due to a shifted patch reso- nant frequency caused by fabrication error. Figure 6.11 shows a plot of the measured Chapter 6. Guided-Wave Approach 169 magnitude and phase at 4.87 GHz as a function of bias voltage.
Figure 6.9: Fabricated single-patch element
0
−5
| (dB) −10
11 0.00V
|S −15 1.75V 3.50V −20 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 5.25V6 Frequency (GHz) 7.00V 8.75V 0 11.00V −5 14.50V 21.00V
| (dB) −10 21
|S −15
−20 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz)
Figure 6.10: Experimental results for the single-patch design
At 4.87 GHz, the experimental unit cell yields 426◦ of phase range, and an insertion loss varying between 2.1 dB 4.3 dB with an average of 3.4 dB. The worst-case return − loss over the tuning range is 7.9 dB for a bias voltage of 7.5 V, but is generally better than 10 dB over the rest of the voltage range. For an insertion loss variation of less than Chapter 6. Guided-Wave Approach 170
0 400
−5 200
−10 0
Magnitude Phase (deg)
Magnitude (dB) Phase −15 −200 0 5 10 15 20 Bias voltage (V)
Figure 6.11: Experimental S21 at 4.87 GHz
3 dB, the bandwidth of the fabricated element is about 100 MHz, which is 2% fractional bandwidth.
We note that a slightly larger phase range was achieved in the experiment than in simulations because the bias voltage in the experiment was increased until the varactor diode reached breakdown. Likely, this resulted in a capacitance range larger than that specified in the data sheet. The insertion losses between the experimental and simulated results are in good agreement.
6.1.5 Stacked-Patch Design
A major advantage of the differential bridged-T approach is that since the feed, the bridged-T phase shifter, and the aperture coupling are all broadband in nature, it is possible to replace the single-patch antenna with a broadband antenna to increase the bandwidth of the element. To maintain the low-profile planar nature of the lens, here we use wideband stacked patched for this purpose.
The design of probe-fed and aperture-coupled stacked-patch antennas has been well- studied [90, 91]. However, unlike probe or aperture-coupling feeds, the position of the differential proximity-coupling feed cannot be moved in this design. Instead, the feed gap width, which is the size of the gap between the two balanced microstrip lines in the feed as shown in Figure 6.5, can be adjusted. The details on the design of the stacked Chapter 6. Guided-Wave Approach 171 patch antenna with a proximity-coupled differential feed can be found in Appendix D.
Practically, the various design parameters differ in the degree of precision with which their values can be selected. For instance, substrates are only commercially available for specific dielectric constants. On the other hand, patch and feed dimensions can be arbitrarily selected with high precision. Therefore, to design the stacked patches, we first choose the dielectric constants for the upper and lower substrates to be 2.3 (Rogers
Duroid 5880) and 2.94 (Rogers Duroid 6002), respectively. The same Duroid 6006 base substrate is kept from the single patch design for the feed lines and phase shifter. The discussion in [90] suggests that the dielectric constant of the substrate on which the upper patch is patterned should be as small as possible. While foam substrates are often used for this purpose, here Duroid 5880 is used instead, and an air gap is inserted between the lower and upper substrates to create a low effective dielectric constant between the two stacked patches. The entire structure is held together using vinyl screws and the air gap is supported using vinyl spacers. The arrangement of the substrates and metallization layers are shown in Figure 6.12.
Figure 6.12: Design of the stacked-patch element Chapter 6. Guided-Wave Approach 172
6.1.6 Simulated and Experimental Results
As was done for the single-patch unit cell, the stacked-patch design was tested in a WR-
187 rectangular waveguide. The fabricated stacked-patch element is shown in Figure
6.13.
Figure 6.13: Fabricated stacked-patch element
Designing for low insertion loss variation and bandwidth, the upper and lower patch sizes, feed gap width, and air gap size are 19.0 mm 2.3 mm, 18.0 mm 1.0 mm, 1.0 mm, × × and 1.6 mm, respectively. We note that while it is possible to select different parameters that would result in a slightly higher bandwidth, here we use the same feed gap size as the single-patch design so that the experimental feed and phase shifting layer can be reused. With the feed gap size fixed at 1.0 mm, very narrow patches are needed to match the patches to the feed. The overall thickness of the element is 10 mm, which is 0.17λ in free-space.
The simulation results are shown in Figure 6.14. Between 4.6 GHz and 5.2 GHz, over
400◦ of phase range is achieved, and an insertion loss variation of 1.7 dB 5.2 dB with − an average of 3.7 dB. The insertion phase is extremely linear resulting in low dispersion, which is not achievable with low-order ladder phase-shifter designs. The return loss is the primary performance difference between this stacked-patch design and the single-patch design. Greater bandwidth is achieved at the cost of higher return loss. Otherwise, in terms of phase tuning range and insertion loss, the two designs are comparable. Chapter 6. Guided-Wave Approach 173
0
| (dB) −10 11 |S
−20 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz) 0.15pF 0 0.25pF 0.35pF 0.45pF
| (dB) −10
21 0.5pF
|S 0.7pF 0.9pF −20 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 1.2pF6 Frequency (GHz) 2.0pF 200
(deg) 0 21 S ∠
−200 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz)
Figure 6.14: Simulated results for the stacked-patch design Chapter 6. Guided-Wave Approach 174
Measured results are shown in Figure 6.15. Between 4.7 GHz and 5.2 GHz, the ele- ment achieves over 400◦ of phase range, and an insertion loss variation of 1.9 dB 5.3 dB − with an average of 3.6 dB. While the insertion loss variation exceeds 3 dB, it only does so for a few bias voltages at 4.9 GHz. In these instances, the reduced S is also associated | 21| with an increased S , meaning that there is mismatch between the antenna and phase | 11| shifter for those configurations and frequencies. Otherwise, the observed insertion loss is primarily due to the dissipated loss from the parasitic resistances in the varactor diodes, since dielectric and conductor losses are insignificant in comparison.
0
−5
| (dB) −10 0.00V 11
|S −15 1.75V 3.50V −20 5.25V 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 7.00V Frequency (GHz) 8.75V 0 11.00V 14.50V −5 21.00V
| (dB) −10 21
|S −15
−20 4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6 Frequency (GHz)
Figure 6.15: Experimental results for the stacked-patch design
A summary of the magnitude and phase of the experimental results at various fre- quencies is shown in Figure 6.16, and we can see that the performance is similar (to within a dB) for the different frequencies. The bandwidth of the fabricated element, for an insertion loss variation of 3.4 dB is 500 MHz, or 10% fractional bandwidth, which is
five times the bandwidth of the single-patch element. We note that typically the use of a Chapter 6. Guided-Wave Approach 175 stacked patch only doubles the bandwidth. However, the reason why we see such a large improvement is because the single-patch element was not designed for bandwidth, but to have an excellent match to 30 Ω.
0 600
400 −5
200 4.7 GHz −10 Phase (deg) Magnitude (dB) 4.8 GHz 0 5.0 GHz 5.2 GHz −15 −200 0 5 10 15 20 Bias voltage (V)
Figure 6.16: Experimental S21 at various frequencies (solid - magnitude, dashed - phase)
A comparison of the simulated and experimental insertion losses, including the results from the single-patch element, is shown in Figure 6.17. Since the relationship between capacitance and bias voltage given in Table 4.2 is only approximate, phase shift is used in the abscissa to give a more accurate comparison. For both designs, the simulated and experimental results are in good agreement.
0 4.8 GHz (SIM) −2 4.8 GHz (EXP) 5.0 GHz (SIM) −4 5.0 GHz (EXP) 4.78 GHz (SIM)* −6 4.87 GHz (EXP)* Insertion Loss (dB) −8 0 100 200 300 400 500 600 Phase Shift (deg)
Figure 6.17: Summary of simulated and experimental results (*denotes the results from the single-patch design) Chapter 6. Guided-Wave Approach 176
Far-Field Radiation Pattern
The far-field element pattern of the stacked-patch antenna was characterized in FDTD by simulating a single stacked-patch antenna over an infinite ground plane. The differential ports of the antenna were driven by two sources with opposite phases. The normalized far-field patterns in the E- and H-planes of the differentially driven stacked patches are shown in Figure 6.18 for different frequencies. The peak directivity of the stacked patches at broadside was 7.7 dBi at 5 GHz, which is within the expected range for patch antennas
[71] (p. 842). For comparison, the theoretical pattern for a single 20.5 mm 3 mm × microstrip patch antenna calculated from (5.4) and (5.5) is also shown in the figure. We can see that in the H-plane the patterns at all frequencies are all nearly identical to the single patch pattern. In the E-plane, however, there are small variations in the directivity over frequency, likely because the electrical distance between the radiating slots of the patches changes slightly with frequency, changing the pattern.
The unit cell presented in this section addresses all of the major design objectives for the transmitarray. The cell achieves a phase range of well over 360◦ with a single bias voltage. While many layers of substrate are required, the designs can all be pat- terned using standard printed circuit board etching, making fabrication relatively easy.
Furthermore, there are no vias or soldered connections between the layers, making layer alignment and assembly straightforward. Physically, the unit cell is small enough to fit into a half-wavelength cell spacing, which is commonly used in array implementations.
Also, the thickness of the stacked-patch structure is only 0.17λ, which is significantly thin- ner than many reconfigurable transmitarray designs that have been proposed. Finally, this design achieves a large 10% fractional bandwidth. Although the average insertion loss of 3.6 dB means that more than half of the power is dissipated in the array, this is a trade-off of implementing reconfigurable structures. Therefore, in the next section we present a transmitarray implemented with this unit cell design. Chapter 6. Guided-Wave Approach 177
E−plane 0
−5
−10 Directivity (dBi) −15
−20 0 −5 −10 −15 −20 −15 −10 −5 0 Directivity (dBi) 4.7 GHz (stacked FDTD) 5.0 GHz (stacked FDTD) H−plane 5.2 GHz (stacked FDTD) 0 5.0 GHz (single theory)
−5
−10 Directivity (dBi) −15
−20 0 −5 −10 −15 −20 −15 −10 −5 0 Directivity (dBi)
Figure 6.18: Simulated far-field radiation pattern of the stacked-patch antenna Chapter 6. Guided-Wave Approach 178
6.2 Transmitarray Implementation
To use the unit cell in an array, the dimensions of the unit cell that were used in the previous section for WR-187 rectangular waveguide were re-optimized for a 30 mm × 30 mm periodic waveguide. The upper and lower patch dimensions, feed gap, and air gap size for the array implementation are 20.5 mm 3.0 mm and 17.5 mm 1.0 mm, × × 2.5 mm, and 1.6 mm, respectively. Simulations showed that the re-optimized design has very similar behavior, with insertion losses ranging between 1.7 dB 4.3 dB with an − average of 3.2 dB, and 500 MHz of bandwidth.
Following in the process used for the transmitarray in Chapter 5, we will first deter- mine the mutual coupling of the unit cell in an array. Then, we will characterize the individual elements in the fabricated array to quantify the amount of variability across the cells. Finally, we will present beamforming results with the array.
6.2.1 Mutual Coupling
To quantify side-by-side (x) mutual coupling between the array elements, three elements are arranged with 30 mm spacing, as shown in Figure 6.19, and simulated inside a three- cell waveguide with periodic boundary conditions. The ports of all three antennas have matched loads of 30 Ω. The feeds of the first antenna (port A) are driven differentially, and the response at the feeds of the second antenna (port B) is measured. End-to-end
(y) mutual coupling is quantified similarly, but with a vertical arrangement of the cells.
Assuming only nearest-neighbor mutual coupling, at least three cells are required. If only two cells are used, the periodic boundary conditions would simulate two excited antennas next to the antenna being measured, resulting in an overestimate of the mutual coupling. The mutual coupling SBA, shown in Figure 6.20, is calculated from the ratio of the voltage wave leaving the center antenna and the voltage wave entering the excited antenna. The isolation achieved for the side-by-side and end-to-end cases are 20 dB and Chapter 6. Guided-Wave Approach 179
18 dB, respectively, across the entire frequency of operation. This high degree of isolation ensures that the phase characteristics of the elements can be independently tuned in an array.
Figure 6.19: Setup of the side-by-side (x) mutual coupling simulation
0 Side−by−side (x) End−to−end (y)
−20 | (dB) ij
|S −40
−60 4 4.5 5 5.5 6 Frequency (GHz)
Figure 6.20: Mutual coupling between adjacent array elements
6.3 One-Dimensional Beamforming
To compare this transmitarray design with the layered scattering surface discussed in
Section 4.1.3 of Chapter 4, a one-dimensional six-element stacked-patch bridged-T linear transmitarray was simulated in SEMCAD-X. The array was illuminated by a plane wave, Chapter 6. Guided-Wave Approach 180 and produced pencil beams at different outgoing angles. For the varactor diodes, a parasitic inductance of Lv = 0.4 nH was assumed but no parasitic resistance was used
(Rvs = Rvp = 0) so that a fair comparison can be made with the simulation results from the lossless scattering surface. Figure 6.21 shows the radiated electric field when pencil beams are produced in different directions. Radiation and periodic boundary conditions are used for the x and y-boundaries, to simulate a six-element array in thex ˆ-direction and an infinite array iny ˆ-direction, respectively. The elements are spaced 30 mm apart synthesizing a 180 mm aperture, which is identical to the setup in Section 4.1.3.
(a) Bridged-T transmitarray θ =0.0◦ (b) Bridged-T transmitarray θ =9.0◦
(c) Bridged-T transmitarray θ = 19.0◦ (d) Bridged-T transmitarray θ = 29.0◦
(e) Bridged-T transmitarray θ = 39.0◦ (f) Bridged-T transmitarray θ = 49.0◦
Figure 6.21: Radiated fields from a finite layered scattering surface
From the figure, we can qualitatively see that as the scan angle increases side-lobe Chapter 6. Guided-Wave Approach 181 beams are produced, clearly visible in Figure 6.21(f). However, compared to the main beam, the power in the side-lobes is significantly smaller. This can be contrasted to the
field plots for the layered scattering surface structure in Figure 4.24. Employing a near- to-far-field transformation, we can obtain the far-field radiation pattern, shown in Figure
6.22. The simulated six-element array shows very good one-dimensional beamforming performance. At a scanning angle of 49◦, the simulated array has a scan loss of about 2 dB and side-lobe levels of 10 dB showing that the array is capable of beamsteering − with well-formed pencil beams.
0 deg 0 9 deg 19 deg −2 29 deg −4 39 deg 49 deg −6
−8
−10
−12
Far−field pattern (dB) −14
−16
−18
−20 −80 −60 −40 −20 0 20 40 60 80 Azimuth (deg)
Figure 6.22: 2D simulated results for a six-element stacked-patch bridged-T transmitar- ray
6.3.1 Array Fabrication
A 6 6 transmitarray was fabricated using chemical etching and the components were × individually hand-soldered. As mentioned in Chapter 5, this array size was selected so Chapter 6. Guided-Wave Approach 182 that it is small enough to be cost-effective and easily fabricated, but also large enough to effectively demonstrate beamforming, with a directivity large enough to be measured by the planar near-field scanner. The internal circuitry of the array is shown in Figure
6.23, without the upper Duroid 5880 patch substrate. The layer with the phase shifters and the lower Duroid 6002 patch layer are clearly visible, with the vinyl screws used to hold the structure together. Like the array in Chapter 5, vertical segments of bias lines are loaded with 10 kΩ resistors to dissipate any power coupled to the bias lines. Copper tape on the left and right sides of the array are used to prevent any spurious radiation from the bias lines. All of the bias lines are routed to a 40-pin ribbon cable header.
Figure 6.23: Internal view of the fabricated array
The assembled array is shown in Figure 6.24. Two 2.4 mm plexiglas panels support the substrates. The total thickness of the transmitarray is about 10 mm. Chapter 6. Guided-Wave Approach 183
(a) Front view (b) Back view
Figure 6.24: Assembled 6 6 transmitarray × 6.3.2 Element Characterization
Before the array can be used for beamforming, each element must be characterized to ensure that all of the elements are functioning properly. To characterize each individual element, the probe of the near-field scanner was moved in front of each element at a distance of 10 mm away to sample the fields from a single element. For each element, the bias voltages were swept from 0 V to 20 V. Due to edge effects and different feed illumination, we expect differences between the elements. To make fair comparisons, the
36 elements of the array can be divided into 9 classes of elements, where there are four elements in each class, as shown in Figure 6.25. We expect elements of the same class to all behave similarly.
Figure 6.26 shows plots of the transmission magnitude versus the transmission phase.
We note that the curves are normalized so that the peak measured value is 0 dB, since the absolute value of the measurements is not meaningful for this measurement. Overall, we can see that elements from the same class have similar curves. The elements of Class
A, positioned at the center of the array, have the highest magnitude while the elements of
Class I, positioned at the corners of the array, have the lowest magnitude, in accordance with the expected feed illumination taper. We can also see that the elements of Classes C, Chapter 6. Guided-Wave Approach 184
Figure 6.25: Element classification for the 6 6 array ×
F, and I have larger magnitude variations, which is likely due to edge effects. Compared to elements along the top and bottom of the array in Classes H and G, the elements along the sides of the array have much more variability. Observing that the variation is greater for elements of quadrants 1 and 3, it would seem that the variation is caused by the absence of a phase shifter beside the elements on the left side of the array (see Figure
6.23).
Regardless, all of the elements are capable of producing 360◦ of phase range, meaning that the fabricated array is functioning adequately, and can be used for beamforming.
6.3.3 Array Beamforming
Using the relationship between the bias voltages for each array element and its phase, pencil beams can be formed by assigning phases to the elements following (3.37). Figure
6.27 shows plots of measured far-field pencil beams in the two principal planes at 5.0 GHz.
From the plots, we can see that well-shaped beams are formed for scanning angles of
50◦ to 50◦, in both azimuth and elevation, corresponding to a scanning window size of − 100◦ 100◦. In this design, the scanning range is fundamentally limited by the antenna × pattern of the patch antenna, and can potentially be increased by using a less directive antenna for the unit cell. Chapter 6. Guided-Wave Approach 185
Class I Class H Class G 0 0 0 −2 −2 −2 −4 −4 −4 −6 −6 −6
Magnitude (dB) −8 Magnitude (dB) −8 Magnitude (dB) −8 −10 −10 −10 −100 0 100 200 300 −100 0 100 200 300 −100 0 100 200 300 Phase (Deg) Phase (Deg) Phase (Deg) Class F Class E Class D 0 0 0
−2 −2 −2 Quadrant 1 −4 −4 −4 Quadrant 2 Quadrant 3 −6 −6 −6 Quadrant 4
Magnitude (dB) −8 Magnitude (dB) −8 Magnitude (dB) −8 −10 −10 −10 −100 0 100 200 300 −100 0 100 200 300 −100 0 100 200 300 Phase (Deg) Phase (Deg) Phase (Deg) Class C Class B Class A 0 0 0 −2 −2 −2 −4 −4 −4 −6 −6 −6
Magnitude (dB) −8 Magnitude (dB) −8 Magnitude (dB) −8 −10 −10 −10 −100 0 100 200 300 −100 0 100 200 300 −100 0 100 200 300 Phase (Deg) Phase (Deg) Phase (Deg)
Figure 6.26: Element characterization 6 6 array × Chapter 6. Guided-Wave Approach 186
The peak directivity and gain of the broadside beam are 20.5 dBi and 14.2 dBi, re- spectively. As was done in Chapter 5, the gain of the array was established by comparing radiated power density of the array with that of a standard gain horn. The magnitude tapering observed as the scanning angle from broadside increases is typical for microstrip patch arrays, and can be attributed to the element pattern of the patch antennas. Al- ternatively, one could perceive the gain reduction as a result of the decreased effective aperture size when the array is angled with respect to the beam direction.
Beamforming measurements at 4.7 GHz and 5.2 GHz are shown in Figures 6.28 and 6.29 and demonstrate that the array is capable of producing well-defined beams in a
500 MHz bandwidth. We note that the gain bandwidth of the array and the bandwidth for the unit cell are approximately the same, because the gain of the array drops abruptly for frequencies outside of the unit cell bandwidth.
Table 6.1 summarizes the angle of peak gain for each pencil beam, and we can see that the beam angles are accurate to within a few degrees, except for the 50◦ beams ± where the error approaches 10◦. We can observe that there is a tendency for the beam angles to be less than the desired beam angle. This is possibly due to mutual coupling of the elements, where a phase-averaging effect reduces the phase gradient, causing the beam to tend towards broadside. Minor effects of beam squinting can also be observed, where higher frequency beams have smaller beam angles. For this small array, this error is insignificant in comparison to the beam width produced. However, for a large array where the beam widths are very narrow, the beam squint will be more noticeable.
The directivity (D), peak gain (G), side-lobe levels (SLL), and half-power beam widths (HPBW ) at 5 GHz of the pencil beams in the principal planes are listed in Table
6.2. Very good side-lobe levels better than 16 dB are observed for all beams in the − E-plane, but the side-lobe levels deteriorate to 10 dB for larger scanning angles in the − H-plane.
If we take the simulated element pattern from unit cell discussed in the previous Chapter 6. Guided-Wave Approach 187
−50° 20 −40° −30° 15 −20° −10° 0° 10 10° 20° 5 30° 40° 0 50° Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Elevation (deg)
(a) E-plane
−50° 20 −40° −30° 15 −20° −10° 0° 10 10° 20° 5 30° 40° 0 50° Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Azimuth (deg)
(b) H-plane
Figure 6.27: Measured far-field beam patterns at 5.0 GHz Chapter 6. Guided-Wave Approach 188
−50° 20 −40° −30° 15 −20° −10° 0° 10 10° 20° 5 30° 40° 0 50° Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Elevation (deg)
(a) E-plane
−50° 20 −40° −30° 15 −20° −10° 0° 10 10° 20° 5 30° 40° 0 50° Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Azimuth (deg)
(b) H-plane
Figure 6.28: Measured far-field beam patterns at 4.7 GHz Chapter 6. Guided-Wave Approach 189
−50° 20 −40° −30° 15 −20° −10° 0° 10 10° 20° 5 30° 40° 0 50° Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Elevation (deg)
(a) E-plane
−50° 20 −40° −30° 15 −20° −10° 0° 10 10° 20° 5 30° 40° 0 50° Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Azimuth (deg)
(b) H-plane
Figure 6.29: Measured far-field beam patterns at 5.2 GHz Chapter 6. Guided-Wave Approach 190
Table 6.1: Pencil beam angles Actual angle (E-plane) Actual angle (H-plane)
Desired angle 4.7 GHz 5.0 GHz 5.2 GHz 4.7 GHz 5.0 GHz 5.2 GHz
50 43 42 42 49 45 43 − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 40 38 38 35 39 38 35 − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 30 31 27 27 31 27 28 − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 20 19 17 17 17 19 16 − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 10 9 11 8 9 9 9 − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ − ◦ 0◦ 0◦ 1◦ 0◦ 1◦ 1◦ 1◦
10◦ 11◦ 11◦ 9◦ 10◦ 11◦ 9◦
20◦ 23◦ 19◦ 17◦ 23◦ 18◦ 16◦
30◦ 30◦ 31◦ 29◦ 30◦ 30◦ 29◦
40◦ 41◦ 38◦ 35◦ 42◦ 37◦ 40◦
50◦ 52◦ 41◦ 41◦ 48◦ 45◦ 44◦ section and combine it with the array factor in (3.29), we can calculate the expected beam pattern. Figure 6.30 shows that the array factor calculation and the measured beam patterns are in good agreement.
Table 6.3 summarizes the loss budget for the transmitarray. The feed horn loss was measured by comparing the measured directivity and gain of the feed horn, referenced with a standard gain horn. In this table, we subtract the different losses from the maxi- mum theoretical gain to obtain the expected gain, and the table shows that the measured gain is in good agreement. For array loss, which includes reflection and dissipation in the array, an estimated value of 2.5 dB is used. While the average insertion loss from unit cell measurements is 3.6 dB, all of the cells are set to produce a phase of 0◦ for a broadside beam. Looking at Figure 6.26, Class A, B, D, and E elements, which radiate most of the power, have transmission magnitudes above the average for 0◦ of phase shift. For this reason, we estimate that the array loss to be less than the average array loss. Chapter 6. Guided-Wave Approach 191
Table 6.2: Pencil beam directivity, peak gain, side-lobe sevel, and beam width E-plane H-plane
θ0 D G SLL HPBW D G SLL HPBW (Deg) (dBi) (dBi) (dB) (Deg) (dBi) (dBi) (dB) (Deg)
-50 18.9 11.2 -16.6 21.3 18.3 10.2 -10.3 23.7
-40 19.4 12.5 -14.1 20.9 18.9 11.9 -10.3 20.4
-30 19.5 12.8 -15.1 21.7 19.7 12.9 -11.8 21.2
-20 20.2 13.6 -16.1 18.3 19.2 12.3 -23.9 19.9
-10 20.3 14.1 -16.1 18.7 20.0 13.7 -16.0 19.3
-0 20.5 14.2 -17.9 16.6 20.5 14.2 -16.6 17.1
10 20.2 14.0 -15.8 18.4 19.9 13.8 -15.3 19.6
20 19.6 13.2 -16.1 19.6 20.1 13.4 -16.5 19.0
30 19.5 12.9 -17.6 21.3 19.7 13.0 -14.5 20.7
40 19.4 12.5 -16.5 19.3 18.3 11.1 -13.11 22.3
50 18.4 10.5 -20.1 25.8 17.7 9.4 -10.2 25.1 Chapter 6. Guided-Wave Approach 192
−30° (AF) 20 −30° (EXP) 0° (AF) 15 0° (EXP) 30° (AF) 10 30° (EXP)
5
0 Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Elevation (deg)
(a) E-plane
−30° (SIM) 20 −30° (EXP) 0° (SIM) 15 0° (EXP) 30° (SIM) 10 30° (EXP)
5
0 Gain (dBi) −5
−10
−15
−20 −60 −40 −20 0 20 40 60 Azimuth (deg)
(b) H-plane
Figure 6.30: Comparison of expected (AF) and measured (EXP) far-field patterns at
5 GHz Chapter 6. Guided-Wave Approach 193
Table 6.3: Transmitarray Loss Budget for Broadside Beam at 5 GHz Maximum theoretical aperture gain 20.5 dBi
Feed horn loss 0.5 dB
Taper loss 0.5 dB
Spillover loss 2.8 dB
Array loss 2.5 dB
Expected gain 14.2 dBi
Measured gain 14.2 dBi
Given that the peak insertion loss measured for the unit cell in the previous section is
1.9 dB, we estimate the array loss to be about 2.5 dB.
Another observation that we note is that the measured directivity is equal to the maximum theoretical directivity given by (3.38), which is 20.5 dBi. However, given that the reduction in directivity due to non-uniform magnitude on the aperture (taper loss) is
0.5 dB, these quantities should not be equal. Likely, this is because the effective aperture size of the array is slightly larger than the 180 mm 180 mm used to compute the × theoretical directivity, increasing the measured directivity.
6.3.4 Cross-Polarization
The cross-polarized field pattern of the broadside beam is shown in Figure 6.31(a), show- ing very good isolation of about 35 dB. In general, the cross-polarization of beams in − the principal planes is similar to that at broadside, and so those results are not shown here. For linearly polarized apertures, the cross-polarization tends to be the worst for beams in diagonal directions (D-plane), with φ0 = 45◦. Figure 6.31 shows the radiation patterns for beamforming directions diagonal to the principal planes. Vertical cuts (V- cut) plot the patterns for constant azimuth, and horizontal cuts (H-cut) plot the patterns for constant elevation. As the angle θ0 increases, we can see that the cross-polarization Chapter 6. Guided-Wave Approach 194
levels rise, reaching about 25 dB for θ = 30◦. Also, as the angle increases, the beam − 0 shape deteriorates more rapidly than beams in the principal planes shown in Figure 6.27.
20 20
10 10
0 0
−10 −10
Gain (dBi) −20 Gain (dBi) −20
−30 −30
−40 −40 −80 −60 −40 −20 0 20 40 60 80 −70 −50 −30 −10 10 30 50 70 90 V−cut Elevation/H−cut Azimuth (Deg) V−cut Elevation/H−cut Azimuth (Deg)
◦ ◦ ◦ ◦ ◦ ◦ (a) θ0 =0 (El: 0 , Az: 0 ) (b) θ0 = 10 (El: 7.1 , Az: 7.1 )
20 20 V−cut (Co−pol) V−cut (X−pol) 10 10 H−cut (Co−pol) H−cut (X−pol) 0 0
−10 −10
Gain (dBi) −20 Gain (dBi) −20
−30 −30
−40 −40 −60 −40 −20 0 20 40 60 80 100 −50 −30 −10 10 30 50 70 90 110 V−cut Elevation/H−cut Azimuth (Deg) V−cut Elevation/H−cut Azimuth (Deg)
◦ ◦ ◦ ◦ ◦ ◦ (c) θ0 = 20 (El: 14 , Az: 14.4 ) (d) θ0 = 30 (El: 20.7 , Az: 22.2 )
Figure 6.31: Measured cross-polarization for φ0 = 45◦
6.3.5 Shaped-Beam Synthesis
In addition to pencil beams, the transmitarray can also be used to perform shaped-beam synthesis, or beamshaping. To calculate the desired aperture fields for an arbitrary far-
field pattern, it is possible to simply use a two-dimensional inverse Fourier transform.
However, there are two subtleties that complicate the problem. Firstly, many arrays have elements where the phases can be controlled, but the amplitudes are fixed [92], as Chapter 6. Guided-Wave Approach 195 is the case for this transmitarray. Secondly, since only the magnitude of the desired far- field pattern is defined, the far-field phase can be arbitrarily selected. Without defined real and imaginary values for the far-field pattern, the inverse Fourier transform is not possible. While a solution is possible assuming that all far-field phases are zero, generally it will not be the optimal solution. These two things make the beamforming problem a non-convex optimization problem, for which iterated algorithms are required.
Using the projection matrix algorithm [93], desired element phases (for a fixed-element magnitude) were calculated for different shaped-beam patterns, and the transmitarray was configured to produce those phases. The implementation of the algorithm is provided in Appendix E. Figure 6.32 shows measured far-field patterns for two pencil beams, three pencil beams, a rectangular flat top, and a donut shaped pattern. The span of the plots shown is 80◦ to 80◦ in both elevation and azimuth. The desired far-field pattern can − be distinctly recognized in the measured results showing that this transmitarray can effectively be used for synthesizing more complex patterns, which is remarkable since the array is only 6 6 in size. ×
6.4 Conclusions
In this chapter, we followed the guided-wave approach to reconfigurable transmitarray de- sign, combining two antennas with a reconfigurable phase shifter. We presented a design based on a balanced bridged-T phase shifter and differential proximity coupling. We first discussed the motivation for these topologies, followed by single-patch and stacked-patch unit cell implementations. Then, the design and fabrication of a fully reconfigurable transmitarray was described and measurement results were presented.
The stacked-patch unit cell design achieves a phase tuning range of over 400◦, over a 500 GHz, which is 10% fractional bandwidth. With a thickness of only 0.17λ, the struc- ture is electrically thin compared other reconfigurable designs which have been proposed, Chapter 6. Guided-Wave Approach 196
(a) Two beams (b) Three beams
(d) Rectangle (e) Donut
Figure 6.32: Shaped-beam synthesis measurements Chapter 6. Guided-Wave Approach 197 and is also relatively easy to fabricate. The design has a flat insertion loss over the entire tuning range, averaging 3.6 dB of loss.
The unit cell and array results presented in this chapter clearly demonstrate that the guided-wave approach is able to address all of the transmitarray design objectives. Chapter 7
Comparison of the Approaches
With the detailed investigations of the different approaches presented in the previous chapters, we can make comparisons to understand the relative advantages and disad- vantages of each approach. In this chapter, we will discuss the approaches and designs that we have investigated in terms of ease of design and optimization, phase range, in- sertion loss, beamforming ability, bandwidth, element control complexity, physical size, fabrication complexity, and scalability.
It is important to distinguish the approach (distributed-scatterer, coupled-resonator, or guided-wave) from the designs presented (loaded dipole, slot-coupled patch, or bridged-
T), since each design is only a possible implementation of each approach. A design experimentally demonstrates what is achievable using a particular approach, but the limitations of the design may not apply to the approach in general. On the other hand, a limitation of an approach fundamentally limits that which is achievable by any design of that type.
7.1 Design and Optimization
From a design perspective, designs based on identical layered scattering surfaces are attractive because they can be easily analyzed. As shown in Chapter 4, a MoM approach
198 Chapter 7. Comparison of the Approaches 199 can be used to optimize planar designs on a dielectric substrate. This is attractive because many parameters can be rapidly optimized using standard optimization algorithms such as gradient search.
On the other hand, while circuit models can be developed for the coupled-resonator and guided-wave phase shifter designs, exact values relating the geometries to component values in the circuit model cannot be derived. We note that even though lumped com- ponent phase shifters are well understood in literature, electrically short microstrip lines and bends can only be accurately modeled numerically. This makes the design process reliant on full-wave simulations, which are much more time-consuming.
7.2 Phase Range and Insertion Loss
Using four layers, the loaded-dipole design achieves about 360◦ of phase range with an average of 5.1 dB of loss. About 2 dB of that loss is due to conductor and dielectric loss and the remainder is due to dissipated loss in the varactor diodes. We suspect that the loaded-dipole design has large conductor and dielectric losses because it is highly resonant, resulting in large fields and currents. Layered designs employing structures that are less resonant may have smaller conductor losses. For the other designs studied, the slot-coupled patch design only achieves 260◦ of phase range with an insertion loss varying between 2 dB to 5 dB, but the bridged-T design achieves 360◦ of phase range with an average insertion loss of 3.6 dB.
We can see that generally, regardless of design, there is about 3 dB of dissipated loss due the MGV100-20 varactor diodes. In fact, this dissipative loss is by far the most significant source of loss across all of the designs studied. This appears to be true regardless of whether the structure is a highly resonant dipole, or a non-resonant bridged-
T phase shifter. Therefore, since the guided-wave approach allows one to choose from a broad range of phase shifter and antenna implementations, by choosing less resonant Chapter 7. Comparison of the Approaches 200 structures such as the bridged-T phase shifter, conductor and dielectric losses can be minimized, yielding the best insertion loss and a full 360◦ of phase range.
7.3 Beamsteering and Side-lobe Levels
For a properly-spaced antenna array with low mutual coupling, the side-lobe levels or power radiated in undesired directions can be made arbitrarily low, simply by increasing the size of the array. On the other hand, the results from Chapter 4 for the distributed- scatterer approach with identical layers showed that even for an infinite surface size, the amount of power in the undesired spectral components increases with scanning angle. The distributed-scatterer approach may be suitable for transforming spherical waves from a horn to plane waves for a broadside beam, since the gradient of the required phase shifts across the surface is very flat. However, it is not suitable for reconfigurable beamforming where a large phase gradient is required on the aperture because lateral wave propagation causes large errors in the phase gradient.
For comparison, Figure 7.1 reproduces the simulated results of one-dimensional az- imuthal beamsteering for the 180 mm layered scattering surface and bridged-T trans- mitarrays from Figures 7.1(b) and 7.1(a). From the plots, we can see that the beam accuracy, scan loss, and side-lobe levels of the bridged-T design is far better than that of the layered scattering surfaces, particularly above a 20◦ scan angle.
The bridged-T design shown here represents only a particular implementation of a guided-wave transmitarray. Therefore, we can expect that another design could poten- tially improve the performance with even less mutual coupling. On the other hand, the layered scattering surface, implemented with LC lumped components, models a gener- alized scattering surface at a single frequency, and so we do not expect that a different implementation would have significantly improved performance. We note that conduct- ing walls could be added to discretize the surfaces into isolated cells to mitigate the Chapter 7. Comparison of the Approaches 201
0 deg 0 deg 0 0 9 deg 9 deg 19 deg 19 deg −2 −2 29 deg 23 deg 26 deg −4 39 deg −4 49 deg 29 deg −6 −6 39 deg 49 deg −8 −8
−10 −10
−12 −12
Far−field pattern (dB) −14 Far−field pattern (dB) −14
−16 −16
−18 −18
−20 −20 −80 −60 −40 −20 0 20 40 60 80 −80 −60 −40 −20 0 20 40 60 80 Azimuth (deg) Azimuth (deg)
(a) Bridged-T (b) Scattering surfaces
Figure 7.1: Comparison between 2D 180 mm bridged-T transmitarray and four-layer scattering surface apertures lateral propagation that results in side-lobes. However, this would significantly increase the fabrication complexity and physical weight of the transmitarray, making it very im- practical.
The slot-coupled patch design which has also been studied has larger scanning losses as the beam is scanned, because as the scan angle increases some elements effectively become disabled due to the limited phase range of the elements. However, one advantage of this design is that the array can function in both transmitarray and reflectarray modes, which effectively doubles the scanning range by allowing both forward and backward beamsteering.
7.4 Shaped-beam Synthesis
Compared to the distributed-scatterer approach, the guided-wave approach can produce array elements with very low mutual coupling. This is crucial for achieving precise control of aperture phases for shaped-beam synthesis. While it may be possible to create shallow phase gradients with the distributed-scatterer approach, it is not clear how arbitrary phase distributions can be accurately synthesized due to the lateral propagation of waves Chapter 7. Comparison of the Approaches 202 within the structure. If the surface is discretized into cells, and each cell is assigned a particular phase shift, the field interaction between adjacent cells would result in major phase errors. While it is possible to characterize the behavior of each cell as the phase shifts of adjacent cells are varied, the optimization required to accurately synthesize the aperture phases would be extremely complex, computationally intensive, and impractical.
7.5 Bandwidth
Both the layered scattering surface and slot-coupled patch designs effect a phase shift by moving a pass band. The structure orders (fourth for the loaded-dipole design and third for the slot-coupled patches design) were selected such that ideally a 360◦ phase range in
S21 is achieved between the limits of the tuning range for the operating frequency f0, as illustrated in Figure 7.2(a). Because of the pass band roll-off in S , the resulting | 21| bandwidth is relatively small. In order to increase the bandwidth, a higher-order pass band is needed where the S21 phase curve covers a larger phase range. In this case, the pass bands do not need to be shifted all the way to the limits, increasing the bandwidth, as shown in Figure 7.2(b). However, additional layers or resonators are required to achieve higher-order behavior, which increases the structure size, fabrication complexity, and most importantly the insertion loss.
On the other hand, for guided-wave structures where more complex phase-shifting mechanisms can be implemented using microwave circuits, it is possible to achieve large bandwidths using relatively low-order structures, as was demonstrated by the bridged-T design from Chapter 6.
7.6 Element Control
Element control, or biasing, is a practical challenge that is often overlooked in transmi- tarray design. The main challenge of element control is the routing of bias lines. For Chapter 7. Comparison of the Approaches 203
(a) S-parameters of moving pass bands (b) S-parameters for increased bandwidth
Figure 7.2: Moving pass band and resulting bandwidth a continuous scattering surface where currents are distributed throughout the cell, such as layers of small wire grids and patches, bias lines need to be placed on another plane, increasing the complexity of the structure. Even then, the coupling between the bias lines and the surface needs to be carefully handled. This problem can be alleviated by concentrating the radiating parts of the unit cell into a smaller area, which is the case for the loaded dipole, slot-coupled patch, and bridged-T unit cell designs studied. In this case, it is possible to route bias lines around the radiating parts of the cell on the same plane without requiring extra layers.
The second challenge is the number of bias lines required per cell, since a large array would require a large bank of programmable DC voltage sources. Also, element charac- terization and testing is significantly more time-consuming if multiple control voltages need to be swept for each cell. Of the unit cell designs studied, the loaded dipole and bridged-T designs can be tuned using a single control voltage, but the slot-coupled patch design requires two control voltages per cell. In general, a coupled-resonator approach requires multiple control voltages because the resonators need to be independently tuned, making that approach less practical. For this reason, the loaded dipole and bridged-T Chapter 7. Comparison of the Approaches 204 designs are more attractive than the slot-coupled patch design.
7.7 Physical Size
In order to achieve low insertion loss, the distributed-scatterer approach with identical layers requires a layer spacing of about a quarter-wavelength. With the four layers that are required to achieve 360◦ of phase tunability, the thickness of the entire structure is at least three-quarter wavelengths thick, which is 45 mm at 5 GHz. At lower frequencies commonly used for terrestrial communication systems, the thickness makes this approach impractical. On the other hand, the thickness of the slot-coupled patch and bridged-T designs are 0.11 and 0.17 wavelengths, respectively. These thicknesses are much more practical, especially if the transmitarray is to be integrated into other structures such as walls.
7.8 Fabrication
In terms of fabrication complexity, the distributed-scatterer approach with identical lay- ers, exemplified by the loaded-dipole design, is the simplest, since all of the metallization layers are only on one side of the substrates, and there are no vias connecting the different layers. However, because the layers need to be spaced a quarter-wavelength apart, some additional support hardware is required. The slot-coupled patch design is also via-less and requires the fewest layers, but it requires multi-sided patterning and a cavity needs to be milled into the substrate to accommodate the varactor diode in the slot. On the other hand, the guided-wave stacked-patch bridged-T design is more involved, requir- ing multi-layered patterning, multiple types of substrates, air gaps, and many lumped components. Although the RF components do not require any vias, vias and air-bridges are used for the biasing circuitry in our implementation. Generally, distributed-scatterer designs are less complex than guided-wave designs. Chapter 7. Comparison of the Approaches 205
We can also look at the element characterization from the slot-coupled patch array and the bridged-T array to compare the sensitivity of the two approaches to fabrication errors and component variations. Although both arrays were fabricated in-house using similar processes with hand-soldered components, the bridged-T array had much smaller performance variation across the elements. The phase tuning ranges of the slot-coupled patch elements varied by about 100◦, likely due to fabrication error. On the other hand, all the elements in the bridged-T array achieved more than 360◦ of phase range, and the magnitude variations observed in the characterization can be attributed to feed illumi- nation and edge effects.
Likely, the tolerance of the bridged-T design is due to three factors. Firstly, a varactor diode in the slot-coupled patch design is placed in a high-field region across a resonant slot. In this case, small variations in the varactor packaging and soldering can have a significant impact on the fields. On the other hand, the varactors in the bridged-T design are placed across a narrow gap in microstrip lines. With the fields tightly confined underneath the microstrip line, the solder and packaging are in low-field regions and have minimal effect. This issue of the solder and packaging being exposed to high-field regions exists for any distributed-scatterer design, and therefore must be carefully managed.
Secondly, unlike coupled resonators, the bridged-T phase shifter is not highly resonant, meaning that the phase does not change abruptly over the tuning range. This results in variations of component values having less impact on element behavior. Thirdly, the bridged-T design with the stacked patches has a large bandwidth, meaning that minor errors that detune the structure do not create significant mismatch between the components, as is the case for narrowband designs. Overall, the bridged-T design was far less sensitive to fabrication errors than the other designs in terms of bandwidth, insertion loss, and operating frequency.
Therefore, we can conclude that although the fabrication process of the guided-wave bridged-T design is more involved, it is not unreasonably complex, and its resilience to Chapter 7. Comparison of the Approaches 206 fabrication error makes it more attractive for practical applications.
7.9 Scalability and Extensibility
There are a number of ways in which the transmitarray designs can be scaled or extended.
The most obvious dimension is scaling in size. As the number of elements increases, el- ement control becomes more challenging because the length and number of bias lines increase. The slot-coupled patch design presented has the most complex biasing system, since each element requires two control lines. Nevertheless, all three of the designs pre- sented can be scaled in size by assembling the arrays in panels, with very fine bias line buses connecting to the panels. Since the reverse bias current of the varactor diodes is very small, the bias lines can be made very thin.
The more challenging dimension of scalability is frequency scaling. As the oper- ating frequency increases, the effect of parasitics and fabrication errors becomes more significant. Therefore, we expect frequency scalability to be correlated with resilience to parasitics and fabrication error, making the bridged-T structure most amenable for scaling to higher frequencies. However, it is not possible to make this conclusion without experimental verification. We note that while high frequency FSSs have been proposed, implementing individual element tunability for beamforming is extremely challenging, and has not been demonstrated.
Finally, guided-wave approaches also have the added advantage that amplifiers can be easily added to increase the gain. In the bridged-T design, two amplifiers can be added to the balanced microstrip lines without any significant redesign.
7.10 Summary
Table 7.1 provides a summary of the comparisons between the performances of the designs studied from the different approaches. It is important to distinguish the design and Chapter 7. Comparison of the Approaches 207
Table 7.1: Summary of design comparisons Criterion Loaded dipole Slot-coupled patches Bridged-T
(Distributed-scatterer) (Coupled-resonator) (Guided-wave)
Design optimization MoM (fast) Simulation (slow) Simulation (slow)
Insertion loss 5.1 dB 4.8 dB 3.6 dB
Beamsteering range 20 20 50 ± ◦ ± ◦ ± ◦ Reflectarray mode No Yes No
Shaped-beam synthesis Difficult Not investigated Good
Bandwidth Narrow Narrow Wide
Element control Single Multiple Single
Thickness Thick Thin Thin
Fabrication complexity Low Low Moderate
Fabrication tolerance Not investigated Poor Good the approach, as the performances of these designs only reflect the baseline of what is achievable with each approach. New designs can potentially achieve better results. However, with the distributed-scatterer approach, the observation of the limited scanning range due to lateral wave propagation is a limitation on the approach in general. In this case, the results from Chapter 4 place an upper limit on the approach as a whole.
From the table, it is clear that the guided-wave approach can address all of the major design criteria, as demonstrated by the experimental bridged-T array. Ideally, we would also like to compare the results with existing designs from literature, but only one other continuous-phase reconfigurable transmitarray with experimental two-dimensional beamforming results has been presented [59], and the figures of merit such as bandwidth, loss, and scanning range, were not provided.
Table 7.2 provides a summary of the comparison between the performances of the designs presented in this thesis and existing reconfigurable transmitarray designs. We note that because the frequency ranges of the designs are significantly different, the Chapter 7. Comparison of the Approaches 208
Table 7.2: Comparison with existing designs Criterion Bridged-T Slot-coupled Hybrid coupler MEMS switches
patches [59] [58]
Frequency 5 GHz 5.7 GHz 12 GHz 34.8 GHz
Element phase 360◦ 240◦ 360◦ 0◦/90◦/180◦/270◦ range (Discrete)
Insertion loss 3.6 dB 4.8 dB 3 dB 4.2 - 9.2 dB
Bandwidth 10% 5% 6% 4%
Beam scan 100 100 50 50 9 (limited 90 (one- ◦ × ◦ ◦ × ◦ ◦ ◦ window results available) dimensional)
Thickness 0.17λ 0.11λ 1λ 0.12λ
Control lines 1 2 1 2
per cell
Fabrication Moderate Low High High
complexity
figures of merit should not be directly compared.
To further support the viability of the transmitarray paradigm, we can compare the performance of the bridged-T transmitarray with that of a phased array implemented with similar technology. In terms of loss, if a phased array is implemented using a reconfigurable phase shifter, the phase shifter alone would have losses comparable to those of this transmitarray. Reconfigurable semi-conductor-based hybrid coupler phase shifters [59] or MMIC phase shifters [5] capable of producing 360◦ of continuous phase range have insertion losses of at least 3 dB 4 dB. Assuming that the illumination − efficiency of a transmitarray can be made to approach that of an ideal reflector antenna of 80% (or 1 dB) [71] (p. 920), if the feed network of a phased array results in more − than 1 dB of loss, it would be better to use a transmitarray instead. Although active elements can be incorporated into a phased array to compensate for these losses, the Chapter 7. Comparison of the Approaches 209 power consumption would become substantially higher than the 100 nA consumed by each varactor diode of the phase shifter of this transmitarray. Furthermore, because the scan angle of this transmitarray is primarily limited by the antenna pattern of the patches, a phased array using similar patches would have an equally limited scanning range. Finally, although the transmitarray feed increases the total antenna profile, this is not a significant drawback in many applications such as satellite communications, and alternative feeding mechanisms such as Cassegrain reflection can be used to reduce the total profile. Nevertheless, there remain certain advantages to phased arrays, such as amplitude control and access to individual signals which may be beneficial in certain applications.
Therefore, in light of the evidence presented in this thesis, we recommend the use of the guided-wave approach for reconfigurable transmitarray design for far-field beamform- ing. Although in some specific applications it may be advantageous to use a distributed- scatterer approach, the guided-wave approach is superior in almost all aspects for general beamforming applications. Chapter 8
Conclusions
The goal of this thesis was to demonstrate how transmitarrays can be used to implement high-directivity reconfigurable apertures. To this end, since transmitarrays have not been studied extensively in literature, the first objective was to systematically examine the capabilities and limitations of the different transmitarray approaches. To support the potential and practicality of the transmitarray paradigm, the second objective was to provide design methods for reconfigurable transmitarray antennas, and the third objective was to design and experimentally demonstrate a fully reconfigurable transmitarray.
In this thesis, we have systematically examined the three transmitarray approaches.
In Chapter 4, we first considered a transmitarray from the distributed-scatterer approach consisting of identical layered scattering surfaces. Using a two-dimensional analysis of an infinite transmitarray, we showed that while this approach is capable of producing phase shifts for uniform surfaces, problems arise when the transmitarray is graduated to create beams at different angles. Due to the thickness of the structure, lateral propagation of waves cause beams to be produced in undesired directions, particularly for larger beamforming angles. For an infinite aperture that should be able to produce perfect pencil beams, this drawback is significant. While this problem can be mitigated by reducing the structure thickness, the structure also becomes reflective, resulting in reduced transmitted
210 Chapter 8. Conclusions 211 power. Next, we presented a Method of Moments technique to analyzing transmitarrays with discretized planar elements. Using this technique, we designed and optimized a transmitarray unit cell consisting of varactor-loaded dipoles.
In Chapter 5, we investigated transmitarrays from a coupled-resonator approach.
The objective was to reduce the structure thickness and the number of layers required by allowing non-identical layers. Using filter theory, we showed that at minimum a structure with third-order resonance is required. Implementing a ladder-type filter using loaded patches and slots, we produced a transmitarray design based on slot-coupled patches using only three layers of metallization. A full transmitarray was implemented using the unit cell design, and beamforming was successfully demonstrated. In addition, a reflectarray mode of operation was demonstrated for the array, effectively doubling the scanning range by allowing both forward and backward scanning.
In Chapter 6, following the guided-wave approach of connecting two antennas with a reconfigurable phase shifter, we investigated different topologies and concluded that the bridged-T phase shifter with differential proximity-coupled patches was the most suitable for reconfigurable transmitarray cells. Combining those topologies, we presented a unit cell design that achieved the required phase range with acceptable insertion loss. Stacked patches were incorporated into the design to increase the bandwidth. Implementing a full transmitarray with the unit cell, we produced a reconfigurable transmitarray capable of beamforming and shaped-beam synthesis, with acceptable insertion loss, small electrical thickness, and a large bandwidth.
Finally, in Chapter 7, we compared the different approaches and associated design implementations, highlighting their strengths and limitations in terms of ease of design and optimization, phase range, insertion loss, beamforming ability, bandwidth, element control complexity, physical size, fabrication complexity, and scalability. Chapter 8. Conclusions 212
Through these investigations, we have accomplished the original goals and objectives:
1. We have provided a theoretical framework for understanding layered transmitar-
rays that did not exist before. We have systematically examined the different
approaches to transmitarrays, and we have generated practical insight on the
viability of the approaches through element designs.
2. We have presented detailed design procedures for three reconfigurable transmitar-
ray designs that follow the three different approaches.
3. We have demonstrated that reconfigurable transmitarrays can be practically real-
ized by producing a wideband, electrically thin, fully reconfigurable transmitarray
capable of beamforming and shaped-beam synthesis in a large scanning window.
Therefore, from comparison of the three approaches, with an emphasis on practical considerations such as ease of fabrication and element biasing, it is the position of this thesis that the best approach to reconfigurable transmitarray design is the guided-wave approach. The stacked patch bridged-T design which was presented in this thesis ad- dresses most of the challenges of transmitarray design at 5 GHz, including 360◦ continuous element phase tunability, good beamforming with low side-lobe levels, shaped-beam syn- thesis, large bandwidth, and single voltage control for each element. The only drawbacks are moderate fabrication complexity, reliance on full-wave simulation for optimization, and an insertion loss of about 3 4 dB. While the insertion loss is large compared to − fixed transmitarray designs which can achieve losses of less than 1 dB, most of the loss is due to dissipation by the varactor diodes. To improve the insertion loss, a different tuning technology such as MEMS capacitors with less loss could be contemplated for integration with transmitarrays in the future. Chapter 8. Conclusions 213
8.1 Contributions
The existing work on reconfigurable transmitarrays in literature is very limited. The most significant contribution of this thesis is in the area of linearly polarized reconfigurable transmitarrays with continuous phase tuning. While 1-bit or 2-bit designs, allowing two or four discrete phases, have been proposed, continuous phase tuning allows for greater beamsteering accuracy, good side-lobe control, and even shaped-beam synthesis, with fewer array elements.
The work presented in this thesis has culminated in the publication of articles in journals and conference proceedings, as listed below. An analysis from a filter-theory ap- proach was used to determine the minimum structure order required for a reconfigurable transmitarray, and a unit cell design was proposed and experimentally demonstrated [94, 95].
Following, an array implementation was experimentally demonstrated, presenting the
first ever transmitarray with reflectarray-mode of operation [96, 97]. This work, along with a hybrid-coupler based transmitarray [59] were the first two-dimensionally reconfig- urable transmitarrays, with continuous phase tuning, to present experimental beamform- ing results. At 0.14λ, this design is substantially thinner than the other design, which has a thickness of over a wavelength. Furthermore, beamsteering results were presented at many angles, whereas the other work only presented beamsteering results at a 9◦ scanning angle.
Next, a wideband reconfigurable transmitarray element design was presented, demon- strating a measured fractional bandwidth of 10% over the entire tuning range of the element [98, 99]. To emphasize the significance of this result, measured bandwidth val- ues have not been published for the other reconfigurable transmitarrays with continuous phase tuning presented in literature.
Finally, a journal article with a wideband transmitarray producing a large recon-
figurable beamscanning range of 100◦ 100◦ has been submitted for review. While a × Chapter 8. Conclusions 214 one-dimensional 2-bit lens has demonstrated a beamscanning range with a similar size [58], no two-dimensional reconfigurable transmitarray with a scanning range of that size has been experimentally demonstrated in literature. The article includes beamshaping as well, which has not been demonstrated for reconfigurable transmitarrays.
Journal Publications
[97] J. Y. Lau and S. V. Hum, “A planar reconfigurable aperture with lens and reflectarray
modes of operation,” IEEE Transactions on Microwave Theory and Techniques, vol. 58, pp. 3547-3555, 2010.
[95] J. Y. Lau and S. V. Hum, “Analysis and characterization of a multipole reconfigurable
transmitarray element,” IEEE Transactions on Antennas and Propagation, vol.
59, pp. 70-79, 2011.
[99] J. Y. Lau and S. V. Hum, “A wideband reconfigurable array lens element,” IEEE Transactions on Antennas and Propagation, (in press).
[100] J. Y. Lau and S. V. Hum, “Reconfigurable transmitarray design approaches for
beamforming applications,” IEEE Transactions on Antennas and Propagation,
(under review).
Conference Proceedings
[94] J. Y. Lau and S. V. Hum, “A low-cost reconfigurable transmitarray element,” in Pro- ceedings of the IEEE Antennas and Propagation Society International Symposium
(APS), June. 2009.
[96] J. Y. Lau and S. V. Hum, “Design and characterization of a 6 x 6 planar recon- Chapter 8. Conclusions 215
figurable transmitarray,” in Proceedings of the European Conference on Antennas and Propagation (EUCAP), Apr. 2010.
[98] J. Y. Lau and S. V. Hum, “A balanced bridged-T reconfigurable array lens element,”
in Proceedings of the IEEE Antennas and Propagation Society International Sym-
posium (APS), June. 2009.
8.2 Closing Remarks
Clearly, research in the area of reconfigurable transmitarrays has only begun, and there remains many challenges to address. For instance, as time progresses and MEMS fabrica- tion technologies become more mature, new low-loss tuning mechanisms such as MEMS capacitors will replace semiconductor-based varactor diodes. Not only will the inser- tion losses be improved, but power handling capabilities will also be increased, enabling the reconfigurable transmitarray paradigm to be used in high-power radar and satellite applications.
Certainly, inter-operation of transmitarrays and reflectarrays should be explored. The combination of reflectarrays and transmitarrays, or structures capable of both reflectar- ray and transmitarray operation can have endless applications if these reconfigurable technologies can be integrated into everyday structures. Furthermore, combinations of reconfigurable and fixed reflectarrays and transmitarrays should be investigated to pro- duce large low-cost reconfigurable apertures.
Antennas are a part of all wireless communication and sensing systems, and so re- configurable antennas can be expected to play an important role as our society becomes more and more wireless. In time, we can expect technologies to become more integrated, applications more enabling, and collaboration more convenient, all the while connected by antennas. Appendix A
Method of Moments Analysis for
Dipole Unit Cell
In this appendix, we will use the Method of Moments to determine the scattered electric fields, when dipoles on a dielectric substrate are excited in a rectangular or parallel-plate waveguide. Consider L thin flat dipoles in a waveguide with gaps g that are loaded with different impedances, as shown in Figure A.1. The dipoles are centered vertically in the waveguide and the distances s(1),...,s(L) are measured from the center of the waveguide, and can take positive or negative values. The dipoles are mounted on a dielectric substrate of thickness t, with a dielectric constant of ǫd,r.
Figure A.1: Dipole geometries
216 Appendix A. Method of Moments Analysis for Dipole Unit Cell 217
Let J(1)(r),..., J(L)(r) be the surface current on each dipole. These are the currents excited by the incident waves that we need to solve for. Because the dipoles are thin, we can assume that all currents are oriented in they ˆ-direction, so that J(i)(r)=yJ ˆ (i)(r).
Let J(r) be the superposition of all of the currents
L J(r)=ˆy J (i)(r). (A.1) i=0 Next, let S(1),...,S(L) be the surfaces of the dipoles, and S be the union of all the surfaces. From (3.63) and (4.32b), the scattered electric field from the dipoles is given by ¯ e, E (r)= jω G¯ −(r r′) J(r′) dS′. (A.2) s − | S e, ¯ − Note that we use the zˆ-propagating G¯ (r r′) because z = 0. The boundary conditions − | on the conducting surfaces of the dipoles forces the total tangential electric field on the dipoles to be zero
¯ nˆ (E (r)+ E (r))=n ˆ E (r) jω G¯ −(r r′) J(r′) dS′ =0. (A.3) × i s × i − | S Consider dipole (i). Assuming that the gap g is small, the electric field inside the gap can be approximated by E = V , where V is the voltage across the gap [101]. This y,gap − g (i) voltage is also equal to the product of the current and the impedance ZL (i) (i) a (i) w 2 +s + 2 (i) V IZL ZL (i) Ey,gap(r)= − = = J (r′)dx′ r in gap. (A.4) g g g a +s(i) w 2 − 2 (i) Since the gap is small, the E-field and J (r′) are almost uniform alongy ˆ in the gap. So,
b g b+g an integration with respect to y from −2 to 2 is simply a scaling by g. Thus, we can also write the total electric field in the gap as
(i) (i) a+w (i) b+g 2 +s 2 (i) ZL (i) E (r)= J (r′)dy′dx′ r in gap. (A.5) y,gap 2 (i) g a−w +s(i) b−g 2 2 Consider the expansion of a window function for the gap in y with with width g, centered
b at y = 2 1 nπy vg(y)= W cos (A.6) b n b n Appendix A. Method of Moments Analysis for Dipole Unit Cell 218 where W = g cos nπ sin θn and θ = nπg . We can limit the domain of the integrand of n 2 θn n 2b g (A.5) using the window function v (y′) so that the integration limits can be extended to
b l b+l −2 and 2 without changing the gap fields (i) (i) ZL (i) g r r′ ′ ′ r Ey,gap( )= 2 J ( )v (y ) dS in gap. (A.7) g (i) S Because the tangential fields on the dipole are zero, the onlyy ˆ-directed E-field on the surface S(i) of dipole (i) are in the gap. Therefore, using the same window function again, we can write the totaly ˆ-directed E-field as
(i) (i) (i) g ZL (i) g g (i) r r r′ ′ ′ r Ey ( )= Ey,gap( )v (y)= 2 J ( )v (y )v (y) dS on S . (A.8) g (i) S We can further generalize the total fields expression using window functions for the dipole geometries in x a w (i) a+w (i) 1, − s x s (i) 2 − ≤ ≤ 2 − u (x)= (A.9) 0, otherwise to yield L 1 (i) (i) (i) g g E (r)= Z J (r′)u (x) v (y′)v (y) dS′ r anywhere. (A.10) y g2 L S i=1 We can combine the total fields from (A.10) into (A.3) for they ˆ-component to get
E (r) jω G (r r′)J(r′) dS′ = E (r). (A.11) y,i − yy | y S They ˆ-directed orientation of the dipole suggests that surface currents will all bey ˆ- directed. Also, since the dipole is thin, we can assume that the currents are uniform alongx ˆ on the dipole. Therefore, we can describe the currents on each dipole using a Fourier series of sine and cosine basis functions. Because of the symmetry in the geometry, we do not expect any non-zero coefficients for the odd/sine bases. So, for each dipole we can approximate J(r) in terms of P + 1 cosine basis functions
P pπ(2y + l(i) b) J (i)(x, y, 0) = I(i)J (i)(x, y), J (i)(x, y)= u(i)(x)v(i)(y) cos − p p p l(i) p=0 (A.12) Appendix A. Method of Moments Analysis for Dipole Unit Cell 219 where v(i) is the window function for the currents on each dipole in y
(i) (i) 1, b l y b + l (i) 2 2 2 2 v (y)= − ≤ ≤ . (A.13) 0, otherwise
Note that the spatial periods of the expansions are equal to the dipole lengths and not 2b, as it is for the Green’s function. We can rewrite (A.11) as
P L (i) ZL (i) g g (i) I jω G (r r′) u (x)v (y′)v (y) J (r′) dS′ = E (r). p − yy | − g2 p − y,i p=0 i=0 S (A.14) (j) If we multiply both sides of (A.14) by Jq (r) and integrate over all of S for q =0,...,P ,
(i) (j) then we have P + 1 equations. When u (x) is multiplied by Jq (r) for i = j, the result (i) (j) (i) (j) is zero because u (x) and u (x) do not overlap. Note that Jq (r) and Jq (r′) do not eliminate because the function arguments are different. This eliminates all but one of the impedance terms, giving
P L (i) (i) (j) I jωµG (r r′)J (r′)J (r) dS′ dS p yy | p q p=0 S S i=1 P (i) ZL (i) g g (i) r (j) r r (j) r + Ip 2 u (x)v (y′)v (y)Jp ( ′)Jq ( ) dS′ dS = Ey,i( )Jq ( ). S S g S(i) p=0 (A.15)
We can rewrite the L(P + 1) equations in matrix form as
G (1)(1) G (1)(L) (1) Z (1) 0 0 I(1) v(1) [ ] . . . [ ] ZL [ ] . . . . . . . .. . + 0 .. 0 . = . (L)(1) (L)(L) (L) (L) (L) (L) [G] . . . [G] 0 0 Z [Z] I v L (A.16) T T (i) (i) (i) (i) (i) (i) (i)(j) where I = I0 ,...,IP , v = v0 ,...,vP , and the entries of matrix [G] (i) and [Z ] are
(j) ′ i j i ω Ψ(0) ω Jp (r ) G( )( ) = J( )(r) 1+ RTEM + qp q 2 S S 2abk 2ab m n kc kzmn ZZ ZZ X X 2 2 TE 2 TM kykzmn ′ ′ ′ ′ 1+ Rmn kx + 1+ Rmn Ψ(kxx)Ψ(kxx ) cos(kyy) cos(kyy ) dy dx dy dx (A.17) k2 ! ! ′ 1 i mπy nπy i Z(i) = J( )(r) W W cos cos J( )(r′)dy′dx′ dy dx. (A.18) qp 2 2 q m n p b g S S m n b b ! ZZ ZZ X X Appendix A. Method of Moments Analysis for Dipole Unit Cell 220
(i)(j) (i) (i) Note that both [G] and [Z] do not have any dependence on the loads ZL , and as well the Z-matrices does not have any dependence on frequency. Thus, the Z-matrices only need to be computed once, and the G-matrices need to be computed only once per frequency. This reduces the need for recomputing the matrices for different loads. For rectangular waveguide (where Ψ( ) = sin( )), we define (i) a+w +s(i) 2 mπx 0 m =0 A(i) = sin dx = . m (i) (i) (i) a−w +s(i) a 2a mπ mπs mπw 2 sin + sin m> 0 mπ 2 a 2a (A.19) For parallel-plate waveguide (where Ψ( ) = cos( )), we define (i) a+w +s(i) (i) 2 mπx w m =0 A(i) = cos dx = . m (i) (i) (i) a−w +s(i) a 2a mπ mπs mπw 2 cos + sin m> 0 mπ 2 a 2a (A.20) For both rectangular and parallel-plate waveguides, we define
(i) b+l (i) 2 mπ(2y + l b) nπy B(i) = cos − cos dy mn i (i) b−l( ) l b 2 l(i) m = n =0
(i) 2b cos nπ sin nπl m =0, n =0 nπ 2 2b = (A.21). l(i) nπ(b l(i)) b nπ( b 3l(i)) nπ(l(i) b) (i) cos − sin − − sin − 2mb = nl 2 2b 4nπ 2b 2b − (i) (i) − (i) nπ nπl nπ nπl bl cos 2 sin 2b cos 2 sin 2b π 2mb nl(i) 2mb+nl(i) otherwise − − − (A.22) Then we can simplify the entries in the matrices to
(i)(j) ω Ψ(0) (i) (j) (i) (j) TEM Gqp = A A B B 1+ R 2abk 0 0 q0 p0 2 2 ω 1 kykzmn + A(i)A(j)B(i)B(j) 1+ RTE k2 + 1+ RTM (A.23) 2 m m qn pn mn x mn 2 2ab m n kc kzmn k ! X X 2 i w i i Z( ) = W W B( )B( ) . (A.24) qp 2 2 m n qn pm b g m n X X (i) Next, we can compute vq . Regardless of propagation mode (T EM, T E, or T EM), they ˆ-component of the E-field is of the form
y,MN + Mπx Nπy jkzMN z E (r)= E Ψ cos e− (A.25) i a b Appendix A. Method of Moments Analysis for Dipole Unit Cell 221 where E+ is a constant. Because we are ultimately interested in scattering parameters where the magnitude of the output is divided by the magnitude of the input, we can assume that E+ = 1 without loss of generality. Thus, we have
(i) (i) a+w +s(i) b+l (i) 2 Mπx 2 qπ(2y + l b) Nπy vMN,(i) = Ψ dx cos − cos dy q (i) (i) (i) a−w +s(i) a b−l l b 2 2 (i) (i) = AM BqN . (A.26)
For a specific geometry, and when the incident fields are in mode M, N, we can calculate
MN,(i) MN,(i) MN,(i) the currents I0 , I1 ,... ,IP for all i using (A.16)
MN 1 MN I = ([G] + [Z] )− v . (A.27)
Note that both G and Z are independent of the excitation mode M, N. G must only be computed for every frequency and dipole length, Z for each gap size, and vMN for every excitation mode and dipole length.
MN,(i) MN,(i) MN,(i) With the currents I0 ,I1 ,...,IP , we can now obtain the scattered fields using
L P ¯ ( ) MN,(i) (i) E (r)= jω G¯ − (r r′) I J (r′) dS′. (A.28) s − | p p i=0 S p=0
Any fields in rectangular or parallel-plate waveguide can be decomposed into T E and T M modes (and T EM for parallel-plate waveguide). Conveniently, the bases of the Green’s function correspond directly directly to the different modes. For z < 0, the side without the dielectric slab, the scattered electric fields associated with mode M, N incidence for the three modes are Appendix A. Method of Moments Analysis for Dipole Unit Cell 222
MN,(−) Es (r) ωµ L yˆΨ(0) 1 = − 1+ RT EM e+jkz + 2 2ab S k m n kc kzmn i =0 P ′ (i) − − pπ(2y + l − b) ′ 1+ RTE Mˆ ( ) + 1+ RT M Nˆ ( ) IMN,(i) cos dS mn mn mn mn p l(i) p =0 (i) (i) L T EM a+w (i) P b+l ′ (i) ωµ yˆ 1+ R +s ′ pπ(2y + l − b) ′ = − e+jkz 2 Ψ(0)dx 2 IMN cos cos(0)dy a−w(i) b−l(i) p (i) 2ab k +s(i) l i=0 2 p=0 2 L ωµ 1 − − − 1+ RTE Mˆ ( ) + 1+ RT M Nˆ ( ) e+jkzmnz 2 mn mn mn mn 2ab m n kc kzmn i =0 a+w(i) b+l(i) ′ +s(i) P (i) 2 ′ ′ 2 MN pπ(2y + l − b) ′ ′ Ψ(kxx )dx I cos cos(ky y )dy Ψ(kxx) cos(ky y) a−w(i) b−l(i) p (i) +s(i) l 2 p=0 2 ωµ 1+ RT EM L P (i) (i) ωµ 1 = −yˆ e+jkz A IMN,(i)B − 0 p p0 2 2abk p 2ab m n kc kzmn i =0 =0 L P TE (−) T M (−) +jkzmnz (i) MN,(i) (i) 1+ R Mˆ + 1+ R Nˆ e Ψ(kxx) cos(kyy) A I B (A.29) mn mn mn mn m p pn p i =0 =0
( ) ( ) ˆ ( ) ˆ ( ) ¯ ± ¯ ± where Mmn± and Nmn± are the components of Mmn and Nmn associated with ay ˆ-directed current source (i.e. thex ˆyˆ,y ˆyˆ, andz ˆyˆ components)
( ) jkzmnz Mˆ ± (r) = xkˆ k Ψ′(k x) sin(k y)e∓ mn − x y x y
2 jkzmnz +ˆykxΨ(kxx) cos(kyy)e∓ (A.30) 2 ( ) kxkykzmn jkzmnz Nˆ ± (r) = xˆ− Ψ′(k x) sin(k y)e∓ mn − k2 x y k2k2 y zmn jkzmnz yˆ− Ψ(k x) cos(k y)e∓ − k2 x y 2 kc kyjkzmn jkzmnz zˆ Ψ(k x) sin(k y)e∓ . (A.31) ∓ k2 x y
Substituting L P MN (i) MN,(i) (i) Cmn = Am Ip Bpn (A.32) i=0 p=0 we have the scattered electric fields for z < 0 propagating in the zˆ-direction −
TEM MN,( ) ωµ 1+ R MN +jkz ωµ 1 E − (r)= yˆ C e s − 2abk 00 − 2ab k2k m n c zmn TE TM MN +jkzmnz 1+ R Mˆ − + 1+ R Nˆ − C Ψ(k x) cos(k y)e . mn mn mn mn mn x y (A.33) Appendix A. Method of Moments Analysis for Dipole Unit Cell 223
Likewise, for z > t, the scattered electric fields propagating in the +ˆz-direction is
jkzmnt TEM MN,+ ωµe T MN jkz ωµ 1 E (r)= yˆ C e− s − 2abk 00 − 2ab k2k m n c zmn jkzmnt TE + jkzmnt TM + MN jkzmnz e T Mˆ e T Nˆ C Ψ(k x) cos(k y)e− . mn mn − mn mn mn x y (A.34) Appendix B
Multi-mode S-parameters
B.1 Definition of Multi-mode S-parameters
Suppose we define two planes in the waveguide at z = d on either sides of the structure. ± At each plane, we can define ports corresponding to all T EM, T E and T M modes, as shown in Figure B.1. We can define ports for modes regardless of whether or not the mode propagates in the waveguide. For the z < 0 side, we will denote the ports ’A’ (e.g.
’TE10,A’), and for the z > 0 side, we will denote the ports ’B’ (e.g. ’TM32,B’). With these ports, we can define S-parameters that describe the waves observed at a particular port when another port is excited, as illustrated in Figure B.2.
Figure B.1: Waveguide port definitions (only two modes shown)
224 Appendix B. Multi-mode S-parameters 225
Figure B.2: S-parameter characterization for a single layer, in a waveguide section of length 2d
Looking at the scattered fields described by (4.34b), we can see that the scattered
fields in a particular TEmn or TMmn are given by the coefficients of Mˆ mn and Nˆmn,
y respectively. So, for example, if side A is excited with a TM32 wave where Ei has unit magnitude, then the scattered fields in TE10 mode on side B can be calculated from (4.34b) to be ω ejkz10tT TE E32 (r)= mn C32Mˆ (+) (B.1) s,TE10 2 10 10 − abkc kz10 where kx and kc are for m = 1 and n = 0. We can obtain scalar representations for the output fields by simply taking the coefficients of the mode bases
ωµΨ(0) EMN,(+) = ejktT TEM CMN (B.2) s,TEM − 2abk 00 MN,( ) ωµΨ(0) TEM MN E − = 1+ R C (B.3) s,TEM − 2abk 00 MN,(+) ωµ(2 δ m)(2 δn) E = − − ejkzmntT TE/TM CMN (B.4) s,TE/TMmn 2 mn mn − 2abkc kzmn MN,( ) ωµ(2 δm)(2 δn) TE/TM MN E − = − − 1+ R C (B.5) s,TE/TMmn 2 mn mn − 2abkc kzmn where denotes the mode. The Kronecker delta M 1, m =0 δm = (B.6) 0, otherwise is used to combine the positive and negative modes in the infinite summation, since there is no distinction between positive and negative modes in rectangular and parallel-plate waveguides ∞ ∞ ( )= (2 δ )( ). (B.7) − m m= m=0 −∞ Appendix B. Multi-mode S-parameters 226
Note that the uppercase M and N denote incidence mode while the lowercase m and n denote observed modes. Likewise, we can express the input fields in terms of TE or TM ˆ ˆ modes with MMN± or NMN± Mπx Nπy a2 TE incidence: Ey,TEMN (r)=Ψ cos = Mˆ y (r) i a b M 2π2 MN z=0 2 TEMN a E = (B.8) ⇒ i M 2π2 Mπx Nπy k2b2 TM incidence: Ey,TMMN (r)=Ψ cos = Nˆ y (r) i a b −jk2 N 2π2 MN zMN z=0 2 2 TMMN k b Ei = 2 2 2 . (B.9) ⇒ −jkzMN N π Because the bases in (4.35a) and (4.35b) have non-unity coefficients, the scalar represen-
y,TEMN y,TMMN tations of the incident fields Ei and Ei need to be scaled appropriately. There are four types of S-parameters that need to be derived from the scattered and incident fields, shown in Figure B.3. Due to the asymmetry of the dielectric substrate, the S-parameters will not be symmetrical. SAB is defined as the ratio of the outgoing waves at port A over the incident waves at port B. Note that SAA, for example, denotes a class of S-parameter where both excitation and observation ports are on Side A. For example, if Port 1 is TE10 on side A and Port 3 is TE30 on side A, then S31 is an SAA-type parameter, since both ports are on the same side. The port mode is not necessarily the same.
Figure B.3: S-parameter cases
Up to this point, we have not considered the port de-embedding distance d. In
jkzmnd calculating the S-parameters, we add the de-embedding factor e− to the outgoing Appendix B. Multi-mode S-parameters 227 wave, and ejkzMN d to the incident wave. Let denote the incident TEM, TE, or TM MMN port mode and denote the outgoing port mode. Mmn Both S and S are calculated using the outgoing waves in the zˆ-direction. For AA AB −
SAA, because the reflected waves from the dielectric cause the effective incident fields at z = 0 to be scaled by 1+ RMNM (note that the M, N is for the incident mode), the scattered waves need to be scaled by that amount. Also, if the incident port mode is the same as the outgoing port mode, then we need to include the incident waves that are reflected off the dielectric as well. Thus, we have
MN,( ) MN jk d − M zmn 1+ RMNM Es, mn + RmnM Ei e− M , MN = mn MN jkzMN d M M S = EiM e . (B.10) AA MN,( ) 1+ R E − e jkzmnd MNM s, mn − M , MN = mn MN jkzMN d M M EiM e jkzMN d For SAB, the scattered waves need to be scaled by the transmission coefficient e TMNM because the incident waves pass through the dielectric slab prior to exciting the conductor.
If the incident port mode is the same as the outgoing port mode, then we need to include the incident waves that pass through the dielectric as well. Comparing SAA and SBB , if the incident and outgoing port modes are exactly the same (M = m, N = n), then the calculated S-parameters are also the same. This is expected, since the conductors and dielectric are passive, and so the system must be reciprocal. Thus, we have
jk t MN,( ) jk t MN jk d zmn − zMN M zmn e TmnM Es, mn + e TMNM Ei e− M , MN = mn MN jkzMN d M M S = EiM e . AB MN,( ) ejkzmntT E − e jkzmnd mnM s, mn − M , MN = mn MN EM ejkzMN d M M i (B.11) Both SBA and SBB are calculated using the outgoing waves in the +ˆz-direction. For
SBA, the reflected waves from the dielectric cause the effective incident fields at z =0 to change, and so a factor of 1+ RMNM needs to be included, like the case with SAA. If the incident port mode is the same as the outgoing port mode, then we need to include the Appendix B. Multi-mode S-parameters 228 incident waves that pass through the dielectric as well. Thus, we have
MN,(+) jk t MN jk d zMN M zmn 1+ RMNM Es, mn + e TMNM Ei e− M , MN = mn MN jkzMN d M M S = EiM e . BA MN,(+) 1+ R E e jkzmnd MNM s, mn − M , MN = mn MN jkzMN d M M EiM e (B.12) jkzMN d Finally, SBB needs to be scaled by the transmission coefficient e TMNM because the incident waves pass through the dielectric slab prior to exciting the conductors, like SAB. If the incident port mode is the same as the outgoing port mode, then we need to include the incident waves that are reflected off the dielectric. Unlike SAA, the surface of the dielectric is a distance of d t away from the port, so the reflection coefficient needs an − additional factor to account for the thickness of the dielectric. Thus, we have
jk t MN,(+) 2jk d MN jk d zMN zMN M zmn e TMNM Es, mn + e RMNM Ei e− M , MN = mn MN jkzMN d M M S = EiM e . BB MN,(+) ejkzMN tT E e jkzmnd MNM s, mn − M , = MN MN mn EM ejkzMN d M M i (B.13)
While the waves for generalized S-parameters are typically scaled by √Z, such a scaling is not necessary if ports of different impedances are not connected to each other, and if we are only interested in S-parameters involving ports with the same impedance.
For example, consider the following equation from a particular signal flow where Zi are the port impedances. The signal A+ is first injected into the first S-parameter block. The signal that leaves from port 4 of the first block is injected into port 4 (or another port with the same impedance) in the second block. Following, the signal that leaves from port 3 of the second block is injected into port 3 (or another port with the same impedance) of the third block. A− is the signal that emerges from port 2 of the third Appendix B. Multi-mode S-parameters 229 block. This gives
+ A2− = S23S34S41A1
E−√Z E−√Z A−√Z = 2 3 3 4 4 1 A+ A+√Z A+√Z A+√Z 1 3 2 4 3 1 4 A− A− A− √Z = 2 3 4 1 A+. (B.14) A+ A+ A+ √Z 1 3 4 1 2 Clearly, if the input and output ports of interest have the same impedances (in this case ports 1 and 2), then we can ignore the scaling by √Z. In fact, we can arbitrarily scale (4.35a) and (4.35b) without any effect on the resulting scattering parameters.
B.2 Cascading multi-mode S-parameter blocks
Consider now multiple layers of unit cells in a waveguide separated by a distance of 2d. Each single-layer unit cell can be modeled by an S-parameter block where the odd numbered ports represent field modes on the one side of the dipole, and the even numbered ports represent field modes on the other side. We are interested in cascading the blocks to obtain S-parameters for multiple blocks connected in series, as shown in
Figure B.4.
Figure B.4: Scatterer cascading
For two-port networks, one would transform the S-matrix into a transmission ABCD matrix using well-known formulas. A generalized scattering matrix is defined as
b1 S1,1 ... S1,K a1 . . . . . . = . .. . . (B.15) bK SK,1 ... SK,K aK Appendix B. Multi-mode S-parameters 230
+ where ai = Ei /√Zi are the incident waves and bi = Ei−/√Zi are the outgoing waves from port i with port impedance Zi [102] (p. 50). We can rewrite the system of equations as
b1 . 1 0 ... 0 −S1,1 −S1,2 ... −S1,K . . 0 1 ... 0 −S2,1 −S2,2 ... −S2,K bK = 0. (B.16) ...... ...... a1 ...... . 0 0 ... 1 −S −S ... −S . K,1 K,2 K,K . a K We can reorder the variables and columns, swapping variable positions such that all of the odd numbered ai and bi are on the left and all of the even numbered ai and bi are on the right
a1 b1 a 3 b 3 . −S 1 −S ... −S − 0 0 −S 0 ... 0 −S . 1,1 1,3 1,K 1 1,2 1,K . −S2,1 0 −S2,3 ... −S2,K−1 0 1 −S2,2 0 ... 0 −S2,K aK−1 −S3,1 0 −S3,3 ... −S3,K−1 0 0 −S3,2 0 ... 0 −S3,K bK−1 = 0. (B.17) ...... ...... b2 ...... a −S − 0 −S − ... −S − − 1 0 −S − 0 ... 0 −S − 2 K 1,1 K 1,3 K 1,K 1 K 1,2 K 1,K b −S 0 −S ... −S − 0 0 −S 0 ... 1 −S 4 K,1 K,3 K,K 1 K,2 K,K a4 . . . bK a K
Note that the order of ai and bi are reversed for the right half of the matrix. If we perform elementary row operations and manipulate the matrix to reduced row echelon form, then
a1 b1 a 3 b 3 . . . 1 0 ... 0 −T1,1 −T1,2 ... −T1,K aK−1 0 1 ... 0 −T2,1 −T2,2 ... −T2,K bK−1 = 0 (B.18) ...... ...... b2 ...... a 0 0 ... 1 −T −T ... −T 2 K,1 K,2 K,K b 4 a4 . . . bK a K Appendix B. Multi-mode S-parameters 231 and we can see that the right-half of the matrix is simply the transmission matrix T. − Observe, with the first two equations as an example, we now have the odd numbered waves as a function of the even numbered waves
a1 = T1,1b2 + T1,2a2 + T1,3b4 + T1,3a4 + + T1,K 1bK + T1,K 1aK (B.19) − −
b1 = T2,1b2 + T2,2a2 + T2,3b4 + T2,3a4 + + T2,K 1bK + T2,K 1aK. (B.20) − −
If we define Qo and Qe as the right and left halves of the matrix in (B.17),
Qo = [ s1 e1 s3 e3 s5 e5 ... sK 1 eK 1] (B.21) − − − − − − Q = [ e s e s e s ... e s ] (B.22) e 2 − 2 4 − 4 6 − 6 K − K where si are the columns of S and ei are unit vectors, then the elementary row operations are equivalent to calculating T as
1 T = Q− Q . (B.23) − o e To cascade networks I and II, we equate the incident and outgoing waves of the appro- priate ports I II I II a2 = b1 b2 = a1
I II I II a4 = b3 b4 = a3 . . (B.24) . .
I II I II aK = bK 1 bK = aK 1 − − which amounts to simply multiplying the T matrices. If we wish to transform the transmission T -parameters of cascaded networks back into S-parameters, then we can define
Rb = [ e2 t1 e4 t3 e5 t5 ... eK 1 tK 1] (B.25) − − − − − − R = [ e t e t e t ... t e ] (B.26) a 1 − 2 3 − 4 5 − 6 − K K where ti are the columns of T, and
1 S = R− R . (B.27) − b a Appendix C
Loss Budget Calculations
In this appendix, we will define how different losses are calculated from simulated or experimental data.
C.1 Array Losses
Array losses include resistive loss, specular reflection, and back-scatter. Let Smn be the
S-parameters of the cell m, n, which will vary depending on the tuning of the cell. Smn | 11 | represents the power reflected from the input of the cell as specular reflection and back- scatter. The fractional resistive power loss of each cell can be found by summing the reflected and transmitted power, as
resistive loss = Smn 2 + Smn 2. (C.1) | 11 | | 21 |
We note that the loss is not expressed as the amount of power dissipated, but rather the proportion of power that is not dissipated.
To calculate the overall array loss, we need to weigh the losses of the individual elements using the incident power from the feed horn. If an element is extremely loss, but is only illuminated with a small amount of power, it will only have a small impact on the overall array loss. Consider the transmitarray setup shown in Figure C.1. Let f(x, y)
232 Appendix C. Loss Budget Calculations 233 denote the magnitude and phase of the fields from the feed horn at the transmitarray plane (z = 0). Then, the total power Fmn incident on each array element m, n is
Fmn = f(x, y),dxdy (C.2) mn A where is the surface area of the cell. The radiated power from each cell is therefore Amn F Smn 2. Therefore, we can calculate array losses by taking the ratio of the radiated mn| 21 | power and the incident power, following
mn 2 Fmn S array loss = m,n | 21 | . (C.3) F m,n mn
The data for Fmn can be obtained from near-field feed horn measurements. Assuming that most elements are approximately identical, we can use the S-parameters obtained from unit cell measurements for different configurations, and infer Smn based on the | 21 | tuning used for each cell.
Figure C.1: Transmitarray setup
C.2 Spillover Loss
The spillover loss is the power radiated by the feed which does illuminate any element of the transmitarray, and is assumed to be lost, since the input and output sides of the transmitarray are assumed to be isolated by an infinite ground plane. If the ground plane Appendix C. Loss Budget Calculations 234 is finite, then some of the waves from the feed will be diffracted around the transmitarray and contribute to the far-field beam pattern. Therefore, the spillover loss can be calcu- lated by the ratio of the total power incident on the array, and the total power radiated by the feed. f(x, y) dxdy spillover loss = A (C.4) Pfeed where is the surface of the entire transmitarray. For a directive feed horn, if we assume A that backward radiation of the horn is insignificant, then the power radiated from the feed can be estimated as
∞ ∞ Pfeed = f(x, y) dxdy. (C.5) −∞ −∞ We note that the spillover loss is not calculated from the power that is lost, but from the power that is captured, since it is expressed as a ratio in dB. No spillover loss corresponds to a ratio of 1, which is 0 dB.
C.3 Taper Efficiency
Taper efficiency is the reduction in gain and directivity due to non-uniform magnitudes of currents or fields on the radiating aperture. The taper efficiency is calculated by comparing the directivity with that of an ideal aperture with uniform field magnitudes.
Here, taper efficiency refers specifically to the taper efficiency at broadside, for a broadside beam formed when aperture phases are constant.
Let gmn(x, y) be the fields on the radiating aperture of the transmitarray produced by element m, n. gmn(x, y) depends on both the illumination Fmn and the transmission
mn coefficient S21 of each element.
g (x, y) F Smn (C.6) mn ∝ mn| 21 | and the total fields on the radiating aperture is simply the sum g(x, y)= m,n gmn(x, y),
mn as shown in Figure C.1. Note that g(x, y) can be estimated from f(x, y) and S21 , or Appendix C. Loss Budget Calculations 235 alternatively it can be obtained from holographic back-projection of near-field scanner measurements. In this thesis, g(x, y) was obtained by holographic back-projection.
For a broadside beam with uniform phase, the broadside directivity is
2 g(x, y) dxdy 4π | | D = S . (C.7) λ2 g(x, y) 2 dxdy | | S The ideal directivity for an aperture with uniform magnitude and phase is given by (3.38).
Thus, we can thus define the taper efficiency as
2 g(x, y) dxdy D | | taper efficiency = = S . (C.8) D ideal dxdy g(x, y) 2 dxdy | | S S A subtle point to note is that we need to use the output aperture fields g(x, y) and not the illumination fields f(x, y). While in reflectarray design, since the reflection magnitude is fairly constant for all elements, the difference between g(x, y) is small f(x, y), and so the taper efficiency can be calculated from f(x, y) [32]. However, for transmitarrays where some elements can potentially have zero transmission, it is important to use the magnitudes of the radiated fields. To illustrate this point, consider a four-element linear transmitarray where elements have unit area, and where the feed illuminates the elements with field magnitudes f(x, y)
1 2 2 1. (C.9)
If each element has the same insertion loss S = 1 = 3 dB, then the radiating 21 √2 − magnitudes g(x, y) will be
1 1 √2 √2 . (C.10) √2 √2
The directivity will thus be
2 1 + √2+ √2+ 1 4π √2 √2 18 4π D = = . (C.11) λ2 1 +2+2+ 1 5 λ2 2 2 Appendix C. Loss Budget Calculations 236
4π Since there are four elements with unit area, the ideal directivity is 4 λ2 , result in a taper efficiency of 0.9.
Next, consider another four-element transmitarray with the same illumination f(x, y), but where the two center elements have S21 = 1 and the two side elements have S21 = 0. The average insertion loss is 3 dB, which is the same as the previous case, but the − radiating magnitudes g(x, y) will be
0 2 2 0. (C.12)
The directivity will thus be
4π (0+2+2+0)2 4π D = =2 . (C.13) λ2 0+4+4+0 λ2 With the same ideal directivity as before, the resulting taper efficiency is only 0.5. Al- though the illumination f(x, y) and the average insertion loss of both arrays are the same, their directivities are different. Therefore, since insertion loss alone cannot be used to quantify the loss due to magnitude variations, the taper efficiency must be calculated including the effects of insertion loss, using g(x, y). Appendix D
Stacked Patch Antenna and Feed
Design
This appendix contains the procedures used to design the stacked patch antenna of the bridged-T unit cell design in Chapter 6. This appendix contains three sections. In the
first section, we examine the transition between the uncovered microstrip line of the phase shifter and the microstrip line covered by the Duroid 6002 substrate of the lower patch, as shown in Figure 6.6. In the second section, we optimize the mitre of the 90◦ bend needed for the parallel balanced microstrip lines to feed the patches. Finally, in the third section we present how the stacked patches are designed.
In this discussion, the specific feeding transmission line has a characteristic impedance of 30 Ω, and is implemented on 0.635 mm Rogers 6006 (ǫr = 6.15) substrate. Further- more, the two balanced transmission lines feeding the patch are separated by 11.75 mm, as shown in Figure 6.5. These values were obtained from the design of the phase shifter in Chapter 6.
237 Appendix D. Stacked Patch Antenna and Feed Design 238
D.1 Microstrip Transition
Using Agilent Advanced Design System (ADS), it was determined that the microstrip width on 0.635 mm Rogers Duroid 6006 (ǫr = 6.15) at 5 GHz to produce a 30 Ω char- acteristic impedance is 2.1 mm when the substrate is in air. When a 1.524 mm Rogers
Duroid 6002 (ǫr =2.94) substrate is added on top of the base substrate, the effective di- electric constant is increased, and so the microstrip width to produce a 30 Ω characteristic impedance changes to 2.0 mm.
The transition interface is between the section of microstrip that is under the Duroid
6002, and the section of microstrip that is in air. To investigate the transition, it was simulated using FDTD in SEMCAD-X, as shown in Figure D.1. Figure D.2 shows the simulated results when ideal 30 Ω transmission lines are used in both the 2.1 mm and
2.0 mm regions. The figure also shows the case when a uniform 2.1 mm transmission line is used in both regions. From the plot, we can see that there is a difference of about
10 dB at 5 GHz. However, since the return loss of 35 dB of the mismatched case is − very small, we will use 2.1 mm transmission lines in general to simplify the design.
Figure D.1: Return loss for the microstrip transition Appendix D. Stacked Patch Antenna and Feed Design 239
−25 2.1 mm, 2.0 mm 2.1 mm, 2.1mm −30
| (dB) −35 11
−40
−45 Return Loss |S −50
−55 0 2 4 6 8 10 Frequency (GHz)
Figure D.2: Return loss for the microstrip transition
D.2 Feed Mitre
Microstrip bends have been well-studied, and an empirical formula exists for the design mitres for microstrip bends [103]
M = 52+65exp( 1.35W/H),W/H 0.25, ǫ 25 (D.1) − ≥ r ≤ where M is the mitre in percent, W is the width of the microstrip line and H is the thickness of the substrate. Using that formula, a mitre of 54% should be optimal. How- ever, using SEMCAD-X simulations, shown in Figure D.3, it was determined that a 47% mitre resulted in the best return loss, as shown in Figure D.4. This minor difference may be because the additional Duroid 6006 substrate on top of the base substrate changes the field patterns of the microstrip line.
D.3 Stacked Patches
While design guidelines for stacked patches have been proposed for common designs, such as coaxial probe-fed patches [90], a generalized closed-form solution for optimal stacked patch design does not exist in literature. One of the reasons is because there are many Appendix D. Stacked Patch Antenna and Feed Design 240
Figure D.3: Mitre simulation setup
−25
| (dB) −30 11
45% 46% −35 47%
Return Loss |S 48% 49%
−40 0 2 4 6 8 10 Frequency (GHz)
Figure D.4: Return loss for different mitres Appendix D. Stacked Patch Antenna and Feed Design 241 design parameters, such as patch size and substrate dielectric constant and thickness, that all have complex interactions. The use of multiple substrates with different dielectrics and the fact that typical feeds generate complex current distributions on the patches means that mathematical treatment of stacked patches is generally not analytically tractable.
As a result, the design of stacked patches relies primarily on numerical techniques such as MoM or FDTD.
In this section, we will present a general design procedure for optimizing proximity- coupled differentially-fed stacked patches using repeated FDTD simulations. The layout of the substrates, patches, and feed are shown in Figure D.5. We will design the stacked patch antenna to match an input impedance of 30 Ω by examining the return loss and the trajectory over frequency of the input impedance on a 30 Ω Smith Chart. Since the base substrate also needs to support the phase shifter, we will assume that the dielectric constant and thickness of the base substrate is fixed.
Figure D.5: Stacked patch layout with feed Appendix D. Stacked Patch Antenna and Feed Design 242
To illustrate the design process, Figure D.6 shows the change in the input impedance trajectory when a stacked patch is added on top of an existing patch with a proximity- coupled differential feed. The different curves are for different lower substrate thicknesses.
The loop-less trajectories are for poorly matched single patches. When a stacked patch is added, the size of the trajectory is reduced, and a loop is introduced. To achieve a large bandwidth and a good match, we want to position the loop in the center of the
Smith Chart.
The general design procedure of the stacked patches is as follows:
1. Begin by choosing the lower substrate with a dielectric constant less than the base substrate (e.g. ǫr = 3). Use any commonly available thickness (e.g. 1.5 mm) for now. Place a lower patch on this substrate with a thin width comparable to the feed lines
(e.g. 1 mm) and a length approximately equal to a half-wavelength in dielectric constant of the lower substrate. Choose a small feed gap size (e.g. 1 mm). Simulate the patch without an upper substrate and patch for different lower patch lengths, and record the input impedance at the differential port. Use a lower patch length that produces a dip in S aligned with the desired operating frequency. Example Smith Chart and return | 11| loss plots are shown in Figure D.7. Note that without the upper substrate and patch, the
Smith Chart trajectory should be a large circle, and the matching should be relatively poor. This is desirable because as illustrated in Figure D.6, the trajectory will shrink to the right when the upper patch is added. For now, we want to focus on selecting an appropriate lower patch length.
2. Next, choose an upper substrate with as small a dielectric constant as possible
(e.g. ǫr = 2). Place the upper substrate above the lower substrate with an air gap of about 1 mm or 2 mm. The air gap serves to lower the effective dielectric constant of the upper substrate. If foam is used for the upper substrate, then the air gap may not be required. Place an upper patch on the upper substrate with a length of half-wavelength in the dielectric constant of the upper substrate. Simulate the stacked patches for different Appendix D. Stacked Patch Antenna and Feed Design 243
0.4mm (single) 0.6mm (single)
0.8mm (single) 1.1mm (single) 0.4mm (stacked) 0.6mm (stacked) 0.8mm (stacked) 1.1mm (stacked)
Figure D.6: Qualitative comparison of input impedance trajectories for single patch
(loop-less) and when a stacked patch is added (with loop), for different lower substrate thicknesses Appendix D. Stacked Patch Antenna and Feed Design 244
0
−5 | (dB)
11 16.0mm −10 16.5mm 17.0mm 17.5mm 18.0mm
Return Loss |S −15 18.5mm 19.0mm −20 4 4.5 5 5.5 6 6.5 Frequency (GHz)
(a) Smith Chart (b) Return Loss
Figure D.7: The effect of lower patch length upper patch lengths. Different upper patch lengths will shift the null in S in frequency. | 11| Use the upper patch length with the largest null close to the operating frequency. Figure D.8 shows example trajectories and return losses. The lower patch length can also be adjusted to shift the frequency of the null. At this point, the depth of the null is not important.
0
−5 | (dB)
11 19.0mm −10 19.5mm 20.0mm 20.5mm 21.0mm
Return Loss |S −15 21.5mm 22.0mm
−20 4 4.5 5 5.5 6 6.5 Frequency (GHz)
(a) Smith Chart (b) Return Loss
Figure D.8: The effect of upper patch length
3. The next step is to move the trajectory on the Smith Chart such that the loop Appendix D. Stacked Patch Antenna and Feed Design 245 circles around the center of the Smith Chart. This can be done by tuning the width of the patches and the feed gap size. Generally, the patch widths should move the loop roughly along constant resistance circles, as shown in Figures D.9 and D.10. The feed gap moves the loop in a different direction, roughly along constant conductance circles, as shown in Figure D.11. If the widths of the patches become too thin, or the parameters cannot be adjusted to move the loop to the center of the Smith Chart, then the thickness of the lower substrate needs to be adjusted, and the design process restarted from Step 1. Increasing the lower substrate thickness reduces the size of the circular trajectory on the Smith Chart.
0
−5 | (dB) 11
−10 1.0mm 2.0mm 3.0mm 4.0mm
Return Loss |S −15 5.0mm 6.0mm −20 4 4.5 5 5.5 6 6.5 Frequency (GHz)
(a) Smith Chart (b) Return Loss
Figure D.9: The effect of lower patch width
4. Finally, adjust the size of the air gap to change the size of the loop on the Smith
Chart. The air gap size affects the effective dielectric constant under the upper patch.
A larger air gap results in a tighter loop, and a smaller air gap results in a larger loop, as shown in Figure D.12. A tighter loop can be used to achieve a deeper null in S but | 11| with a narrower bandwidth, and a larger loop can be used to achieve a larger bandwidth, but with a shallower null in S . | 11| 5. Because the tuning of one parameter such as patch width may detune the frequency Appendix D. Stacked Patch Antenna and Feed Design 246
0
−5 1.0mm
| (dB) 1.5mm 11 2.0mm −10 2.5mm 3.0mm 3.5mm 4.0mm
Return Loss |S −15 4.5mm 5.0mm −20 4 4.5 5 5.5 6 6.5 Frequency (GHz)
(a) Smith Chart (b) Return Loss
Figure D.10: The effect of upper patch width
0
−5 | (dB) 11
−10 1.0mm 2.0mm 3.0mm 4.0mm
Return Loss |S −15 5.0mm 6.0mm
−20 3.5 4 4.5 5 5.5 6 6.5 Frequency (GHz)
(a) Smith Chart (b) Return Loss
Figure D.11: The effect of the feed gap Appendix D. Stacked Patch Antenna and Feed Design 247
0
−5 | (dB) 11
−10 0.5mm 1.0mm 1.5mm 2.0mm
Return Loss |S −15 2.5mm 3.0mm −20 4 4.5 5 5.5 6 6.5 Frequency (GHz)
(a) Smith Chart (b) Return Loss
Figure D.12: The effect of upper substrate gap of the S null slightly, multiple iterations may be needed to achieve an optimal design. | 11| Repeat the process with reduced tuning steps to optimize the matching and bandwidth of the stacked patches.
Although this appendix does not provide a complete overview on stacked patch design, more detailed literature on probe-fed stacked patch design is available [90]. The effects of the feed gap in the differentially-fed case are similar to the placement of the probe in the probe-fed case. Also, as mentioned earlier, the tuning of the air gap size has similar effects as changing the upper substrate dielectric constant. Manipulation of the patch sizes is similar for both differentially-fed and probe-fed cases, although patch widths using the differential feed tend to be much narrower. Appendix E
Projection Matrix Algorithm
In order to effectively use reflectarray or transmitarray cells, array beamforming tech- niques are used to control the elements. Antenna arrays have been studied extensively for many years, and many beamforming techniques have been presented. Because the far-field radiation pattern is simply the Fourier transform of the aperture fields, it is possible to determine the required element magnitudes and phases using a simple inverse
Fourier transform of the desired far-field pattern. However, there are two subtleties that complicate the problem. Firstly, many arrays have elements where the phases can be controlled, but the amplitudes are fixed [92]. In the case of most reflectarrays or trans- mitarrays, element amplitudes are constrained by the radiation pattern of the feeding antenna. Secondly, in most cases, only the magnitude of the desired far-field pattern is specified, and the phase is irrelevant. Without defined real and imaginary values for the far-field pattern, the inverse Fourier transform is not possible. While a solution is possible assuming that all far-field phases are zero, generally it will not be the optimal solution. These two things make the beamforming problem a non-convex optimization problem, for which iterated algorithms are required.
This appendix provides the MATLAB source listing used for performing beamshaping using the projection matrix algorithm [93]. This algorithm solves for element excitations
248 Appendix E. Projection Matrix Algorithm 249 in an array using an iterative approach. The algorithm optimizes the element currents to produce a desired far-field pattern, while constraining the magnitude of the currents.
The following additional input files are required:
1. input/phasemags.csv - Definition of element magnitudes for a given element phase
shift
2. input/elPat 5deg.csv - Definition of the element far-field pattern, in 5-degree
increments
The formats of these files are described inline in the source code listed below. The function in calc ff mat.m computes the T matrix used by the script beamshape.m.
E.1 Source Listing for calc ff mat.m
function [T, obPos] = calc f f mat(freq , elPos) i f (numel(elPos(1 ,:)) ˜= 2) error ( ’element positions must be (x, z)’); end
% Load in the element pattern. The format is: % Theta(deg) Phi(deg) Etheta(linear) Ephi(linear) % e.g. % 5, 10, 7.98E−12, 6.42E−11 % Note that the angles should be in 5 degree increments data = csvread ( ’input/elPat 5deg .csv ’); obPos = data(:, 1:2) ∗ pi / 180; elPat = data(:, 3) ∗ 1e12 ; % Just use the Etheta component
% Only use the observation positions for which 0 <= phi <= 180 obPos2 = 0 ∗ obPos ; elPat2 = 0 ∗ elPat ; n = 0; fo r i = 1:numel(obPos(:, 1)) i f (obPos(i , 2) >= 0 && obPos(i, 2) <= pi ) n=n+ 1; obPos2(n, :) = obPos(i, :); elPat2(n, 1) = elPat(i, :); end end obPos = obPos2(1:n, :); elPat = elPat2(1:n, :); obsN = numel(obPos(: , 1)); Appendix E. Projection Matrix Algorithm 250
k0 = 2 ∗ pi ∗ freq / 3e8; % Propagation constant N = numel(elPos(: ,1)); % Number of elements T = zeros (obsN, N); % The output matrix fo r n = 1:N x = elPos(n, 1); z = elPos(n, 2);
fo r m = 1:obsN theta = obPos(m, 1); phi = obPos(m, 2);
% Calculate the phase difference w.r.t the origin u = sin (theta) ∗ cos (phi); v = cos (theta ); phase = exp ( j ∗ k0 ∗ ( x ∗ u + z ∗ v));
% Calculate the element of T T(m, n) = elPat(m) ∗ phase; end end
E.2 Source Listing for beamshape.m
freq = 5e9; % Operating frequency
% Generate the element locations N = 36; elPos = zeros (N, 2); fo r row = 0:5 fo r col = 0:5 index = row ∗ 6 + col; elPos(index + 1, 1) = −0.075 + col ∗ 0.03; % x elPos(index + 1, 2) = 0.075 − row ∗ 0.03; % z end end
% Generate the matrix [T, obPos] = calc f f mat(freq , elPos); % T is the matrix, and obPos are the observation points M = inv (T’ ∗ T) ∗ T’; % The projection matrix
% Calculate the desired far −field pattern obsN = numel(obPos(: , 1)); % The number of observation points Fd = zeros (obsN, 1); % The desired far − field pattern % Pencil beam (can also produce other patterns) theta0 = 90 ∗ pi /180; % Pencil beam direction phi0 = 90 ∗ pi /180; % Pencil beam direction beam width = 10 ∗ pi / 180; % Width of the pencil beam fo r m = 1:obsN i f ( sqrt ( abs (theta0 − obPos(m, 1))ˆ2 + abs (phi0 − obPos(m, 2))ˆ2) < beam width / 2) Fd(m, 1) = 1; % Set far − field direction to be 1 for pencil point end end Appendix E. Projection Matrix Algorithm 251
% Load the characterization of the phase−magnitude. The format of each row is: % Phase(deg) El 1 mag(dB) El 2 mag(dB) El 3 mag(dB) ... El 3 6 mag(dB) % e.g. % 1 , −37, −33, −31, ..., −36 % Note that a row is required for each phase phasemag = csvread ( ’input/phasemags . csv ’ );
A = ones(N, 1); %A are the currents that we are solving for C = 0.3; % Parameter to adjust the step size iters = 1000; % Maximum number of iterations Fstep = ones(2 ∗ obsN, 1); % The step for each iteration err = 1e10; % Initial error lastErr = 0; % The error of the last iteration while (iters < 10 || ( abs (err − lastErr) > 1 e−3 && iters < 1000)) % For each iteration , there are 10 sub−iterations fo r subiter = 1:10 % Calculate the current far −field pattern F = T ∗ A;
% Avoid divide by zero Fabs = abs (F); n = numel(F); fo r i = 1:n i f (Fabs( i ) < 1 e −3) Fabs(i) = 1; end end
% Calculate how to adjust Fd Fstep = C ∗ ( abs (Fd) − abs (F)). ∗ F ./ Fabs;
% Step the currents Astep = (M ∗ Fstep ); A = A + Astep; end
% Display the error lastErr = err; err = norm( abs (Fd) − abs (F))ˆ2; disp ( sprintf ( ’Error: %g’, err));
% Force the maximum current to have zero phase phaseShift = 0; fo r i = 1:N i f ( abs ( abs (A( i )) − max( abs (A))) < 1 e −6) phaseShift = angle (A(i )); break ; end end
% Because elements have fixed magnitudes for given phases, we need to % set the magnitude fo r i = 1:N % Force the magnitude of A to be 1 Appendix E. Projection Matrix Algorithm 252
A(i) = A(i) / abs (A(i )) ∗ exp(− j ∗ phaseShift );
% Get the phase of the the current , and lookup the magnitude phase = round ( angle (A(i )) ∗ 180 / pi ); i f (phase < 0) phase = phase + 360; end mag = 10ˆ(phasemag(phase + 1, i + 1) / 20); A(i) = A(i) ∗ mag ; end
iters = iters + 1; end
% Output the result Amags = 20 ∗ log10 ( abs ( reshape (A, 6, 6).’)); Aphases = angle ( reshape (A, 6, 6).’) ∗ 180/ pi ; fo r i = 1:numel(Aphases) i f (Aphases(i ) < 0) Aphases(i) = Aphases(i) + 360; end end csvwrite (’output/phases.csv’, Aphases) ; Bibliography
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