design and additive manufacturing of broadband beamforming
TRANSCRIPT
DESIGN AND ADDITIVE MANUFACTURING OF BROADBAND
BEAMFORMING LENSED ANTENNAS AND LOAD BEARING
CONFORMAL ANTENNAS
by
Soumitra Biswas
A dissertation submitted to the Faculty of the University of Delaware in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Electrical and Computer Engineering
Summer 2019
© 2019 Soumitra Biswas All Rights Reserved
DESIGN AND ADDITIVE MANUFACTURING OF BROADBAND
BEAMFORMING LENSED ANTENNAS AND LOAD BEARING
CONFORMAL ANTENNAS
by
Soumitra Biswas
Approved: __________________________________________________________ Mark S. Mirotznik, Ph.D. Professor in charge of dissertation on behalf of the Advisory Committee Approved: __________________________________________________________ Kenneth E. Barner, Ph.D. Chair of the Department of Electrical and Computer Engineering Approved: __________________________________________________________ Levi T. Thompson, Ph.D. Dean of the College of Engineering Approved: __________________________________________________________ Douglas J. Doren, Ph.D. Interim Vice Provost for Graduate & Professional Education and Dean of
the Graduate College
I certify that I have read this dissertation and that in my opinion it meets the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Mark S. Mirotznik, Ph.D. Professor in charge of dissertation I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Keith Goossen, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ Shouyuan Shi, Ph.D. Member of dissertation committee I certify that I have read this dissertation and that in my opinion it meets
the academic and professional standard required by the University as a dissertation for the degree of Doctor of Philosophy.
Signed: __________________________________________________________ John Suarez, Ph.D. External Member of dissertation committee
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First and foremost I would like to thank my advisor Dr. Mark Mirotznik for his
support over the years. I would also like to thank my committee members for agreeing
to serve in my dissertation committee and offering invaluable suggestions and review
for this dissertation. I take this opportunity to thank Dr. Daniel Weile and Mr. David
Hopkins for their advices and insightful discussions we’ve had over the years.
Finally, I would like to dedicate this dissertation to my parents. Without their
support and love, it would not have been possible to finish this journey.
ACKNOWLEDGMENTS
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LIST OF FIGURES ..................................................................................................... viii ABSTRACT ................................................................................................................ xiii Chapter
1 INTRODUCTION .............................................................................................. 1
1.1 Motivation ................................................................................................. 5 1.2 Contributions ............................................................................................. 6 1.3 Dissertation Outline ................................................................................... 8
2 TRANSFORMATION OPTICS OVERVIEW ................................................ 11
2.1 Background .............................................................................................. 12 2.2 Coordinate Transformation ..................................................................... 13
2.2.1 Coordinate Mapping and Index Notation .................................... 13 2.2.2 Vector and Tensor Mapping ........................................................ 14
2.3 Conformal Mapping ................................................................................ 15 2.4 Transformation Optics (TO) .................................................................... 18 2.5 Quasi-Conformal Transformation Optics (QCTO) ................................. 19
3 DESIGN AND ADDITIVE MANUFACTURING OF MODIFIED GRIN LENS ................................................................................................................ 23
3.1 GRIN Lens Background .......................................................................... 24
3.1.1 Luneburg Lens ............................................................................. 25 3.1.2 Maxwell Fish-Eye Lens ............................................................... 27
3.2 Feed integration problems with GRIN lenses ......................................... 28 3.3 Modified Luneburg lens design using QCTO technique ......................... 30
3.3.1 3D QCTO Approximations ......................................................... 35 3.3.2 3D Full Wave Electromagnetic Simulation ................................. 37
TABLE OF CONTENTS
vi
3.4 Modified half Maxwell FISH-EYE lens design using QCTO technique 45
3.4.1 3D Full Wave electromagnetic simulation of modified fish-eye lens ............................................................................................... 48
3.5 Additive Manufacturing of Spatially Varying Permittivity Distributions ............................................................................................ 53
3.5.1 Space-Filling Curve for realizing graded permittivities .............. 55 3.5.2 Modified Luneburg lens fabrication ............................................ 58
3.6 Results ..................................................................................................... 60
3.6.1 Measurement Setup ..................................................................... 60 3.6.2 Experimental Data ....................................................................... 62
3.7 Conclusion ............................................................................................... 66
4 BROADBAND IMPEDANCE MATCHING OF QCTO TECHNIQUE ........ 68
4.1 Introduction ............................................................................................. 70 4.2 Reflections in QCTO-enabled designs .................................................... 75 4.3 Broadband Anti-Reflective (AR) Layer Design Methodology ............... 76
4.3.1 Klopfenstein Profile ..................................................................... 77 4.3.2 Exponential Profile ...................................................................... 79 4.3.3 Gaussian Profile ........................................................................... 80
4.4 Anti-Reflective layer design with QCTO-enabled modified GRIN lens 80 4.5 3D Full Wave Electromagnetic Simulation ............................................. 84 4.6 Anti-Reflective Layer Thickness and Device Performance .................... 99 4.7 Choice of graded profiles as anti-reflective (AR) layer parameter ....... 106 4.8 QCTO-inspired Generalized Vs Classical Luneburg lens ..................... 110 4.9 Conclusion ............................................................................................. 115
5 ULTA-WIDE BEAMSCANNING ANGLE LUNEBURG LENS ANTENNA DESIGN USING HIGH DIELECTRIC MATERIAL ............... 117
5.1 Introduction ........................................................................................... 117 5.2 High permittivity wide beamscanning angle lens antenna design ......... 119 5.3 3D Full-Wave Electromagnetic Simulation .......................................... 121 5.4 Multi-section anti-reflective layer ......................................................... 126 5.5 Conclusion ............................................................................................. 132
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6 ADDITIVELY MANUFACTURED CONFORMAL LOAD BEARING ANTENNA STRUCTURE (CLAS) ............................................................... 133
6.1 Introduction ........................................................................................... 134 6.2 Material selection and mechanical processing ...................................... 136 6.3 CLAS antenna design ............................................................................ 137 6.4 Additive Manufacturing for CLAS antenna fabrication ........................ 139
6.4.1 After curing the structural composite ........................................ 141 6.4.2 Before curing the structural composite ...................................... 143
6.5 Results ................................................................................................... 144 6.6 Conclusion ............................................................................................. 147
7 CONCLUSION .............................................................................................. 149
7.1 FUTURE WORK .................................................................................. 151
REFERENCES ........................................................................................................... 153
Appendix
PERMISSIONS .............................................................................................. 162
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Figure 2.1: Graphical representation of Coordinate transformation .......................... 14
Figure 2.2: Conformal Mapping Example .................................................................. 16
Figure 3.1: Luneburg Lens: (a) lens’s beamforming nature [82]; (b) Dielectric permittivity distribution ........................................................................... 26
Figure 3.2: Maxwell Fish-eye lens: (a) lens’s beamforming nature [76] ; (b) Dielectric permittivity profile .................................................................. 28
Figure 3.3: GRIN Lens with feed elements and beamswitching elements ................. 29
Figure 3.4: Illustration of the Luneburg lens: (a) virtual and (b) physical space used for QCTO mapping ; (c) coordinate grid of the original Luneburg lens obtained from QCTO mapping; (d) mapped coordinate grid of the modified Luneburg lens obtained from inverse coordinate transformation in physical space. ..................................................................................... 32
Figure 3.5: Permittivity profile for (a) cross sectional view of 2D original Luneburg lens, (b) cross sectional view of 2D modified Luneburg lens, (c) 3D representation of modified Luneburg lens permittivity distribution ....... 35
Figure 3.6: (a) 3D finite element setup of the modified Luneburg antenna modeled in COMSOLTM; (b) 3D Finite element meshing of the modified Luneburg lens modeled in COMSOLTM; (c) illustrations showing the positions of the waveguide feed sources used for the simulations along the planar surface of the modified Luneburg lens .................................................... 39
Figure 3.7: Simulated 3D radiation patterns (dBi) of the modified Luneburg lens at 30 GHz for source location at (a) pos -2, (b) pos -1, (c) pos 0, (d) pos 1, (e) pos 2, (f) pos 3, (g) pos 4, (h) pos 5, (i) pos 6, (j) pos 7, (k) pos 8 as shown in figure 3.6(c) .............................................................................. 45
Figure 3.8: Permittivity profile: (a) cross sectional view of 2D original Half Maxwell Fish-eye lens, (b) cross sectional view of 2D modified fish-eye lens, (c) 3D representation of modified fish-eye lens permittivity distribution .... 48
LIST OF FIGURES
ix
Figure 3.9: (a) Finite element mesh of the modified half Maxwell fish-eye lens modeled in COMSOLTM, (b) illustration showing the five positions of the waveguide source feed used for the simulations ..................................... 49
Figure 3.10: Simulated 3D radiation patterns (dBi) of modified half Maxwell fish-eye lens at 30 GHz for source location at (a) pos -2; (b) pos -1; (c) pos 0; (d) pos 1; (e) pos 2 ........................................................................................ 52
Figure 3.11: Simulated gain patterns of QCTO-enabled modified half Maxwell fish-eye lens as a function of azimuth angle and feed locations at 30 GHz ... 53
Figure 3.12: Voxelated permittivity values .................................................................. 54
Figure 3.13: The space-filling geometry used for generating spatially-varying Permittivities. By varying the number of turns, N, the local volume fraction of printed material, and thus its effective permittivity, is controlled. ................................................................................................ 56
Figure 3.14: The predicted and measured relative permittivity of the space-filling curve geometry as a function of volume fraction .................................... 57
Figure 3.15: Additive manufacturing system used to print modified Luneburg lens (nScrypt 3Dn-300) ................................................................................... 59
Figure 3.16: (a) FDM printing of the modified Luneburg lens with space filling curves using the nScrypt printer extruding polycarbonate filaments. (b) Fabricated lens antenna ........................................................................... 60
Figure 3.17: Measurement setup to characterize the modified Luneburg lens antenna gain as a function of azimuthal angle and frequency. The electric field was linearly polarized along the vertical axis .......................................... 61
Figure 3.18: Measured reflection coefficient (S11) of the WR28 open-ended waveguide with and without the presence of the modified Luneburg lens at three waveguide positions at the planar surface ............................................... 62
Figure 3.19: (a) Measured and simulated radiation patterns and beam steering performance of a Ka-band modified Luneburg lens antenna at 30 GHz; (b) measured realized gain at center excitation over the entire Ka-band; and (c) aperture efficiency at three feed locations as a function of frequency ................................................................................................. 65
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Figure 4.1: (a) Anti-Reflective layer with QCTO-enabled designs (in-general); (b) Anti-Reflective layer with QCTO-based modified Luneburg lens (in particular) ................................................................................................ 73
Figure 4.2: (a) 3D-approximate QCTO enabled modified Luneburg lens permittivity profile; (b) Reflection problems on lens’s radiation performances ......... 76
Figure 4.3: Klopfenstein Profile ................................................................................. 77
Figure 4.4: Anti-reflective design methodology: (a) 2D permittivity distribution of QCTO design; (b) AR layer design flowchart ......................................... 82
Figure 4.5: Designed Klopfenstein tapered anti-reflective layer: (a) Graphical representation of tapered permittivity distribution along the AR layer thickness; (b) 2D surface permittivity profile; (c) Axisymetrically rotated 3D permittivity profile ............................................................................. 83
Figure 4.6: 3D QCTO-approximate modified Luneburg lens with half-wavelength anti-reflective layer at the bottom surface ............................................... 84
Figure 4.7: (a) 3D finite element setup of the modified Luneburg antenna modeled in COMSOLTM, (b) illustration showing the positions of the waveguide feed sources used for the simulations .............................................................. 85
Figure 4.8: Simulated 3D radiation patterns (dBi) of anti-reflective layer enabled QCTO modified Luneburg lens antenna at 30 GHz for feed location at (a) pos -2 (-55˚) ; (b) pos -1 (-22˚); (c) pos 0(0˚); (d) pos 1 (22˚); (e) pos 2 (55˚) ......................................................................................................... 88
Figure 4.9: Simulated gain patterns of modified Luneburg lens antenna at 30 GHz for feed locations at pos -2, pos -1, and pos 0 with and without an anti-reflective layer ......................................................................................... 89
Figure 4.10: Fabricated QCTO-enabled modified Luneburg lens antenna with λ/2 thickness anti-reflective layer .................................................................. 90
Figure 4.11: Simulated and measured gain patterns of the modified Luneburg lens antenna at 30 GHz for feed locations at pos -2, pos -1, pos 0 with and without the presence of an anti-reflective layer ...................................... 91
Figure 4.12: Measured return loss (S11) at pos 0 with and without the presence of an AR layer ................................................................................................... 92
Figure 4.13: Realized gain comparison at Pos 0 (with and without AR Layer) ........... 93
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Figure 4.14: Aperture efficiency increase using an anti-reflective layer at the center excitation location of the modified lens antenna (pos 0) ......................... 94
Figure 4.15: Simulated 3D radiation patterns of QCTO-enabled modified fish-eye lens at 30 GHz for source locations: (a) pos -2; (b) pos -1; (c) pos 0; (d) pos 1; (e) pos 2 with full wavelength anti-reflective layer at the top surface and half wavelength anti-reflective layer at the bottom surface of the fish-eye lens ........................................................................................................... 98
Figure 4.16: Modified fish-eye lens’s simulated gain patterns as a function of azimuth angle at 30 GHz with and without the presence of AR layers ................. 99
Figure 4.17: Anti-reflective layer effects on device’s performance: (a) focal point at normal incidence, (b) focal point at 35˚ incidence, (c) focal point at -55˚ incidence and reflections due to phase aberration ................................. 102
Figure 4.18: Higher thickness anti-reflective layer effect on lens’s beamsteering performance and gain pattern. Example modified Luneburg lens with an anti-reflective layer thickness of (a) Half lambda, (b) full lambda, and (c) 1.5 * lambda; (d) Gain value increase with higher thickness anti-reflective layer as the impedance mismatch mitigates with the increasing thickness;(e) Beam steering angle reduction and lower gain value with gradual increase in anti-reflective layer thickness ................................. 106
Figure 4.19: Simulated realized far field gain patterns as a function of azimuth angle at five feed locations at 30 GHz for AR layer with Klopfenstein, Exponential and Gaussian permittivity profile ...................................... 108
Figure 4.20: Realized far field gain pattern as a function of frequency for Klopfenstein, exponential and gaussian permittivity profile AR layer; (a) at Edge excitation (pos -2), (b) at center excitation (pos 0) .................. 110
Figure 4.21: Classical vs Generalized Luneburg lens with half-wavelength anti-reflective layer: (a) focal length representation; (b) graphical representation of permittivity distribution; (c) 3D permittivity profile of modified Classical Luneburg lens and Generalized Luneburg lens with 6mm anti-reflective layer ; (d) Beamsteering performances at three excitation positions (Pos -2, Pos 0, Pos 2). [solid lines represent generalized Luneburg lens’s performance and dashed lines represent classical Luneburg lens’s performance] ................................................ 114
Figure 5.1: 180˚ beamscanning angle beamforming lens antenna with smart electronics feed networks and beamswitching networks ....................... 118
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Figure 5.2: (a) 2D representation of high dielectrics modified Luneburg lens (b) 3D representation of QCTO-enabled modified Luneburg lens’s permittivity distribution; (c) 3D permittivity distribution of QCTO-enabled modified Luneburg lens with broadband anti-reflective layer .............................. 121
Figure 5.3: (a) finite element mesh of the modified Luneburg antenna modeled in COMSOLTM numerical solver, (b) illustration showing the five positions of the waveguide source feed using for the simulations ........................ 122
Figure 5.4: Designed Luneburg lens’s beamscanning performance at 30 GHz ...... 126
Figure 5.5: Modified Luneburg lens with multi-section broadband anti-reflective layer: (a) 3D permittivity distribution of the lens; (b) excitation position of the lens structure ............................................................................... 128
Figure 5.6: Simulated 3D radiation pattern of multi-section anti-reflective layer enabled Luneburg lens antenna at: (a) Pos -3, (b) Pos -2, (c) Pos -1, (d) Pos 0, (e) Pos 1, (f) Pos 2, (g) Pos 3 ...................................................... 131
Figure 6.1: Curved surface CLAS structure mechanical process ............................. 137
Figure 6.2: CLAS antenna structure ......................................................................... 138
Figure 6.3: Antenna configuration ........................................................................... 139
Figure 6.4: Additive manufacturing system (nScrypt 3Dn-300) .............................. 140
Figure 6.5: 3D printing of CLAS antenna elements on curved surface ................... 142
Figure 6.6: Fabricated Antenna example .................................................................. 143
Figure 6.7: Antenna patterning on Uncured prepreg ................................................ 144
Figure 6.8: Fabricated example antenna ................................................................... 144
Figure 6.9: Comparison of measured and simulated impedance matching for the fabricated antenna after curing the composites ..................................... 145
Figure 6.10: Comparison of measured and simulated impedance matching for the fabricated antenna before curing the composites .................................. 146
Figure 6.11: Measured and simulated E-plane radiation pattern of the fabricated antenna after curing the composites ...................................................... 147
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Graded-Index (GRIN) spherical dielectric lens antennas such as Luneburg lens or
Maxwell fish-eye lens are an attractive choice for use as low-cost, wide angle and
wideband beamforming and beamscanning elements in a number of military and
commercial applications for satellite communication, remote sensing, and radar
imaging. When implementing these designs, however, there are many practical
challenges involved with the GRIN lens technology. First, the lens's spherical shape
complicates the integration of an antenna feed networks such as waveguide, antenna
arrays, detectors, and other associated external electronics. Second, practical
implementation of such a continuously graded permittivity profile is a challenge and
requires a robust fabrication approach to realize graded-index lens structures in a
minimum fabrication time with the ability of mass production. To solve the design
problem, a modified GRIN lens antenna, where portion of the lens’s spherical surface
will be modified into a flat surface, can be integrated with the feed networks in a
compatible way. However, this approach requires the optimization and redistribution of
permittivity profile inside the lens structures to ensure intended beamsteering and
electromagnetic performances. In this thesis, I will describe the detail design
methodology of quasi-conformal transformation optics (QCTO) based modified GRIN
lens structure design. Electromagnetic structures designed with QCTO technique
ABSTRACT
xiv
usually suffer from reflection problems at the planar excitation boundary due to the
absence of material’s magnetic response and result in degraded device performance. In
the following work, I will be addressing the reflection problems associated with the
QCTO approximations and design a novel anti-reflective layer along with the QCTO-
enabled modified GRIN lens antennas to mitigate the impedance mismatch problems
across the entire planar excitation surface. To solve the graded dielectrics realization
problem, I will be using fused deposition modeling (FDM) based additive
manufacturing technique to realize continuously graded dielectric lens antennas. In
addition, I will demonstrate the use of additive manufacturing to embed the antenna
elements within a curved surface load-bearing structure.
1
INTRODUCTION
Modern satellite and radar communication systems require wide angle and agile
beamscanning elements that combine high gain, high angular resolution, multiband
operability and, if possible, low fabrication costs. These properties are normally the
exclusive domain of either electronically steerable phased array antenna system or
mechanically rotating reflector antenna systems. Mechanical beamsteering systems are
often constrained by the speed of the mechanically rotating device to scan for targets in
the azimuth directions and does not meet the requirement of agile beamscanning.
Additionally, the presence of an antenna feed at the front of the reflector often results in
signal blockage. On the other hand, electronically scanning phased array antenna
technology is very sophisticated and agile but is often limited in spectral bandwidth.
Moreover, active phased array technology requires complex, and often very expensive,
hardware and external circuitries complicating the mechanical assembly of the elements
and increasing high fabrication costs [77]. All alternative to both mechanically steerable
antennas and active phased array technology is the use of Gradient-index (GRIN) lenses
for passive beamforming. GRIN lenses are an attractive choice for use as low-cost, wide
angle and multiband beamforming and may serve as an alternative to the expensive
phased array antenna systems and mechanically scanning reflector antennas. Because
Chapter 1
2
of these interesting features, GRIN lenses have been attractive research area in academia
and industry for numerous military and commercial applications. Among the variety of
GRIN lenses that have been explored the Luneburg lens and Maxwell fish-eye lens are
the most widely used for its high gain, wide angle, multiband nature and low cost.
Luneburg lens and half Maxwell fish-eye lens are typically spherical structures in which
every point on the surface acts as a focal point for a plane wave coming from the
opposite surface of the structures. Beamscanning is achieved by changing the antenna
feeds placed along the lens’s spherical surface. While interesting, this technology is
currently not mature enough to replace either phased array technology or reflector based
systems. There are numerous design and manufacturing challenges associated with the
practical implementation of GRIN lenses that need to be addressed and require new
solutions. First, the lens’s spherical shape is incompatible with the antenna feed
networks such as waveguide, antenna arrays, detectors, and other external associated
electronics. Second, the lenses have a continuously graded dielectric profile that varies
in three dimensions. Realization of such a spatially varying graded permittivity
distribution is a non-trivial fabrication challenge. In this dissertation, I explored methods
to design, optimize, and fabricate customized shaped GRIN lens antennas to be used as
agile beamscanning elements for radiofrequency communication and radar systems.
The principal contributions of this work are as follows;
(1) To address the geometry problem associated with integrating feed networks onto
GRIN lenses, I developed new analytical and computational methods to realize GRIN
3
lenses that contain flat surfaces. To this end, I developed new design and optimize
methods that resulted in an optimized GRIN lens permittivity distribution using quasi-
conformal transformation optics (QCTO), a subset of Transformation optics (TO).
Transformation optics is a powerful mathematical tool used to control the propagation
of electromagnetic waves in and around an electromagnetic structure. However, the
material parameters derived from this algorithm are normally complex with both
anisotropic and magnetic properties. As a result, fabrication of devices designed using
TO usually require the use of metamaterials which, unfortunately, functionally limits
the bandwidth. To eliminate the implementation problems, most investigators use quasi-
conformal transformation optics, an approximation of transformation optics, to optimize
the material parameters of spatially distorted electromagnetic structure. However,
QCTO is an approximated technique that often degrades device performance due to
impedance mismatches.
(2) My second major contribution was the development of novel design techniques to
mitigate the impedance mismatch problems associated with quasi-conformal
transformation optics. This new design methodology was implemented in the context of
QCTO-enabled modified Luneburg lens antenna and Maxwell fish-eye lens; however,
it can be extended to all other electromagnetic applications which utilize QCTO design
technique.
(3) Another limitation encountered by other investigators using GRIN beamsteering
lenses is the limited range of scan angles. My third contribution from this thesis was
4
demonstrating that very wide beamsteering angles (e.g. ±85 degrees) can be achieved
by employing QCTO technique with high dielectric constant materials.
To address the manufacturing challenge of complex electromagnetic structures,
including GRIN lens antennas, we have leveraged advances in additive manufacturing
(AM) to fabricate spatially varying effective permittivities. The realization of
continuously graded dielectric permittivity is a challenge and complicated. A lot of
fabrication difficulties are involved in manufacturing gradient dielectrics in a cost-
effective and scalable fashion. The first is the choice of material. As the device works
over wide spectral band, the material needs to be non-dispersive over the frequency
ranges of interests. A lot of investigators utilized subtractive method to realize
continuously graded dielectric structure [ 78-79]. In the subtractive method, parts of the
materials such as solid blocks, bars, rods of plastics or other materials were removed by
machine control through cutting, drilling, and grinding; and has the presence of air void
in the lattice structure [78-79]. The devices manufactured with the subtractive methods
lack the mechanical robustness. Additive manufacturing has recently come into the
focus among research communities to implement complex electromagnetic structure
[25,64,65,80]. Unlike subtractive method, additive manufacturing fabricates the
electromagnetic structure layer upon layer, and the end products are more mechanically
robust. It provides flexibility and precise control over the fabrication process. In this
dissertation, we will be utilizing fused deposition modeling (FDM) based additive
manufacturing technique to realize graded dielectrics for GRIN lens antennas. Space
5
filling curve based FDM approach, mentioned in [25,64,29,65] are adopted for this
purpose. Additive manufacturing also offers great flexibility in embedding the
electromagnetic functionality within a complex shaped load bearing structure. In this
dissertation, we will be exploring additive manufacturing technique to realize the
electromagnetic structure such as graded-index beamforming elements and conformal
load bearing antennas structure (CLAS).
1.1 Motivation
There is always strong motivation to build inexpensive, multiband and wide
beamscanning angle antenna and radar systems for the military and commercial
platforms in satellite communication, remote sensing, RF imaging and radar reflector.
Many applications operate simultaneously over multiple spectral bands including
satellite communications to internal communications. Having multiple devices for
different frequency bands communication is expensive, SWaP inefficient and
impractical. GRIN lens antennas are an attractive to alternative to solve these problems.
Classical GRIN lenses are incompatible to integrate with modern feed networks and
external RF circuitries; however, a geometrically modified GRIN lens structure with
novel design methodologies can solve this integration problem.
GRIN lenses have a continuously graded dielectric profile and realization of
such a continuously graded dielectric profile a robust, cost-effective and scalable
fabrication methods with the ability of mass production in minimum time. Additive
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manufacturing can also be used to embed the electromagnetic functionalities within the
structural composites for aerospace and defense applications.
This dissertation aims at the design, optimization, development and additive
manufacturing of GRIN lens based beamforming antenna system which can operate
over broader spectral bands with an agile and wide beamscanning angle capability, and
conformal load bearing antenna structure (CLAS) to embed the antenna functionalities
within the load bearing structure.
1.2 Contributions
A number of significant contributions resulted from this thesis to the field of
transformation optics and RF communications. Specifically,
A new way of realizing modified Luneburg lens antenna with a wide
beamscanning angle coverage from -55˚ to +55˚ using transformation
optics and additive manufacturing.
A novel design of a broadband anti-reflective layer for quasi-conformal
transformation optics (QCTO) enabled devices to mitigate the
impedance mismatch problems.
Realizing high dielectric material for designing wide beamscanning
angle (180˚ FOV) broadband modified luneburg lens design
First instance of embedding conformal antennas via additive
manufacturing technique on a complex shaped load bearing structure.
These new findings resulted in the following patent application and publications:
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Patent:
Soumitra Biswas, Mark Mirotznik. Modified Gradient Index Luneburg Lens
Antenna with Broadband Anti-Reflective Layer. US patent application
number: 62832934, Application filed: April 12, 2018.
Video Article:
Soumitra Biswas, Mark Mirotznik, Zachary Larimore, Paul Parsons.
Additively Manufactured Modified Gradient Index Luneburg Lens Antenna
with Broadband Anti-Reflective Layer. JOVE video Journal, in progress.
List of Journal Articles:
Biswas S, Lu A, Larimore Z, et al. Realization of modified Luneburg lens
antenna using quasi-conformal transformation optics and additive
manufacturing. Microwave Opt Technology Lett. 2019; 61:1022–1029.
Soumitra Biswas, Mark Mirotznik. Broadband Impedance Matching
Strategies for QCTO enabled designs. Nature Communications. In progress.
Soumitra Biswas, Zachary Larimore, Paul Parsons, Mark Mirotznik. High
Dielectric Additive Manufacturing for wide Beamscanning angle modified
Luneburg lens design. In progress.
8
List of Conference Papers:
Soumitra Biswas, Zachary Larimore, Mark Mirotznik. Additively
Manufactured Luneburg Lens based Conformal Beamformer. 2018 IEEE
International Symposium on Antennas and Propagation and USNC-URSI
Radio Science Meeting.
Soumitra Biswas, Mark Mirotznik. Customized shaped Luneburg Lens
Antenna Design by Additive Fabrication. 2018 - 18th International
Symposium on Antenna Technology and Applied Electromagnetics August 19
- 22, 2018, University of Waterloo, Waterloo, ON, Canada.
Soumitra Biswas, Mark Mirotznik. 3D Modeling of Transformation Optics
based Flattened Luneburg Lens using COMSOL Multiphysics® Modeling
Software. COMSOL Conference 2018.
Biswas et al. QCTO-enabled modified Luneburg lens antenna with broadband
anti-reflective layer. In progress.
1.3 Dissertation Outline
This dissertation is organized as follows:
Chapter 2 presents the overview of transformation optics algorithm; the
fundamental nature of electromagnetic material parameters under coordinate
transformation is discussed. The concept of coordinate transformation and conformal
mapping, and the trick to implement the conformal mapping numerically for designing
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transformation optics enabled devices are briefly introduced. A simplification of
transformation optics, known as quasi-conformal transformation optics, is discussed.
Chapter 3 presents the numerical modeling, fabrication approach, and
measurement results of QCTO-enabled modified GRIN lens. Finite element based
commercially available COMSOL Multiphysics numerical solver was utilized for the
2D and 3D design. The design was then realized using fused deposition modeling
(FDM) based additive manufacturing technique. Space-filling curve based additive
manufacturing approach was adopted as a fabrication method to realize the graded
dielectric GRIN structure.
Chapter 4 addresses the impedance mismatch problems present in the quasi
conformal transformation optics technique and to mitigate the mismatch losses, presents
a broadband anti-reflective layer along with the modified structure. The modeling of the
new devices along with anti-reflective layer and performance studies are shown.
In Chapter 5, based on the work from Chapters 3 and 4, we develop high
dielectric materials for designing ultra-wide beamscanning angle modified Luneburg
lens antenna using quasi-conformal transformation optics and additive manufacturing.
To achieve an ultra-wide beamscanning angle, the optimized material permittivity
profile becomes higher and to realize a higher permittivity value, we developed the high
dielectric materials.
Chapter 6 is more general in additive manufacturing scope in that we attempt to
embed conformal antennas on complex shaped load bearing structure for integrating the
electromagnetic functionalities within the structural composites. The ultimate goal is to
10
use the curved surface conformal antennas as a feed source for arbitrary shaped
beamforming GRIN lens antennas discussed in the previous chapters. Using
transformation optics, the luneburg lens can be designed in arbitrary shape including
cylindrical shape and in many applications, the conformal patch antennas can be used
as a feed source as waveguides or detectors are incompatible with non-planar shape
structure. By integrating the antenna functionality within the curved load bearing
structure, and the ability to additively fabricate the beamforming lenses on top of the
curved structure will improve the promise of realizing communication system on any
complex shaped structure. In this chapter, we are addressing the antenna integration
problem within the loadbearing structural composites using microdispensing printing.
But, the ultimate goal will be to use both the microdispensing and FDM simultaneously
on the curved structure in a single attempt. Future attempts will focus on the
multimaterial fabrication approaches of integrating the antenna and beamforming
elements on the curved surface load bearing structure in a single attempt to show the
robustness of additive manufacturing approach electromagnetic structure realization.
11
TRANSFORMATION OPTICS OVERVIEW
Transformation optics is a mathematical method to manipulate electromagnetic
waves to design novel electromagnetic structure [1]. This optimization scheme requires
the evaluation of physical laws in two different coordinate systems, and deals with the
coordinate transformations and mapping of physical quantities under coordinate
transformation. In this chapter, we review a brief overview of the transformation optics
theory which is used to design novel electromagnetic structures. Before going to the
example device design, a good understanding of the physics and mathematical
development of transformation optics, the concept of coordinate transformation and
index notations; mapping methods of the physical quantities such as field vectors and
tensors; and conformal mapping concept is necessary. To describe, the mathematical
reformulations of the physical laws specially Maxwell’s equations and change in
constitutive parameters under coordinate transformation are briefly discussed. This
chapter concentrates on the algebra and index notations of coordinate transformation
between two different sets of Cartesian coordinate systems, and the mapping of vector
field quantities and tensors; and the mathematical reformulations of the Maxwell’s
Chapter 2
12
equations under coordinate transformation has been shown to clarify the background of
transformation optics theory.
The implementation problems with the transformation optics scheme is
discussed and a simplified scheme of transformation optics scheme, known as quasi-
conformal transformation optics (QCTO), which is easier to implement is introduced
here.
2.1 Background
Transformation Optics (TO) is a methodology to control electromagnetic
waves in a prescribed fashion [1,3,5,6] and describes the reformulated physical laws
and quantities under coordinate transformation. This mathematical technique evolved
with form invariance principle of Maxwell’s equations, i.e., Maxwell’s curl equations
remain form invariant under coordinate transformation [1,3,6,8,10,15,81,83]. This
technique originally emerged for cloaking applications [1]; however, a lot of novel
electromagnetic devices have been designed using this technique including GRIN lens,
EM rotator, EM concentrator, polarization divider [5-60].
In transformation optics method, the material parameters change in tensorial
form and derived from the coordinate transformation. To explain the mathematical
development and derivation of material parameters under coordinate transformation, a
brief introduction of tensor algebra and coordinate mapping is worth noting. In this
chapter, we try to clarify the mathematical relationships and reformulations of physical
laws and quantities under coordinate transformation.
13
2.2 Coordinate Transformation
2.2.1 Coordinate Mapping and Index Notation
Coordinate mapping is the transformation of coordinates from one coordinate system to
another [2-4]. For example, a three-dimensional Cartesian coordinate set denoted by the
index notation 𝑥𝑥𝑘𝑘 = (𝑥𝑥1, 𝑥𝑥2, 𝑥𝑥3) = (x , y , z) is used to locate any quantity in that three-
dimensional space (Figure 2.1). We want to map the quantity of this coordinate set to a
different three-dimensional Cartesian coordinate set with index notation 𝑥𝑥𝑘𝑘′ = (𝑥𝑥1′, 𝑥𝑥2′ ,
𝑥𝑥3′) = (x' , y' , z'). The transformation of each point in 𝑥𝑥𝑘𝑘 coordinate set to a
corresponding point in 𝑥𝑥𝑘𝑘′coordinate set 𝑥𝑥𝑘𝑘 → 𝑥𝑥𝑘𝑘′ can be expressed as an arbitrary
function of change of variables [3,4]:
𝑥𝑥𝑘𝑘′ = 𝑥𝑥𝑘𝑘′ (𝑥𝑥𝑘𝑘) (2.1)
Equivalently, the inverse of the coordinate transformation (𝑥𝑥𝑘𝑘′ → 𝑥𝑥𝑘𝑘) can be expressed
as
𝑥𝑥𝑘𝑘 = 𝑥𝑥𝑘𝑘𝑥𝑥𝑘𝑘′ (2.2)
This coordinate mapping is immensely helpful to transform any physical quantity
between different coordinate systems and that mapping of physical quantities is
accomplished with the aid of a coordinate transformation matrix, known as jacobian
transformation matrix, which is basically a scaling factor of the unit basis vectors in the
new coordinate system. The jacobian transformation matrix can be mathematically
expressed as [1-6]:
14
𝜦𝜦𝑘𝑘𝑘𝑘′ ≜
𝜕𝜕𝑥𝑥𝑘𝑘′
𝜕𝜕𝑥𝑥𝑘𝑘 (2.3)
Figure 2.1: Graphical representation of Coordinate transformation
2.2.2 Vector and Tensor Mapping
All the electromagnetic structures’ working principle is based on the interaction of
electric and magnetic field vectors produced by the electric charges and currents
respectively; and to mimic the original electromagnetic functionalities of one
Coordinate set to a new Coordinate set requires the exact mapping of the field vectors
and corresponding media inside and around the structure [2]. If the field vector
quantities in the original space are E, H and in the distorted space are E', H' (figure
2.1); then the vector mappings can be accomplished with the aid of jacobian
transformation matrix using the following mathematical relation [7-8]:
𝑬𝑬 = [𝜦𝜦]𝑇𝑇𝑬𝑬′ ; 𝑯𝑯 = [𝜦𝜦]𝑇𝑇𝑯𝑯′ (2.4)
And the exact field mapping in the transformed space requires the redistribution of the
transformation media which are in tensorial form. If ε and μ are the metric tensors of
15
constitutive parameters in the original Coordinate space, then under coordinate
transformation the metric tensor changes as following [5,8]:
𝜀𝜀𝑖𝑖′𝑗𝑗′ = 𝜦𝜦𝑖𝑖𝑖𝑖′𝜀𝜀𝑖𝑖𝑗𝑗 𝜦𝜦𝑗𝑗
𝑗𝑗′ ; 𝜇𝜇𝑖𝑖′𝑗𝑗′ = 𝜦𝜦𝑖𝑖𝑖𝑖′𝜇𝜇𝑖𝑖𝑗𝑗 𝜦𝜦𝑗𝑗
𝑗𝑗′ (2.5)
Where εij and μij are the tensor components of the permittivity and permeability values
in the virtual space, and primes are the equivalent parameters in the transformed space.
In differential geometry, tensor transformations are usually performed according to their
tensor density, i.e. weighted by a power ‘w’ of the determinant of jacobian
transformation matrix and in the above expression the tensor transformation has a
density of weight ‘0’ [2]. A tensor density basically transforms a tensor quantity from
one coordinate set to another [2]. Using the tensor density weight ‘w’ in the above
expression, the generalized tensor mapping can be expressed as [2,8].
𝜀𝜀𝑖𝑖′𝑗𝑗′ =𝜦𝜦𝑖𝑖𝑖𝑖
′ 𝜦𝜦𝑗𝑗
𝑗𝑗′ 𝜀𝜀𝑖𝑖𝑗𝑗 |𝜦𝜦|𝑤𝑤 ; 𝜇𝜇𝑖𝑖′𝑗𝑗′ =
𝜦𝜦𝑖𝑖𝑖𝑖′ 𝜦𝜦𝑗𝑗
𝑗𝑗′𝜇𝜇𝑖𝑖𝑗𝑗 |𝜦𝜦|𝑤𝑤 (2.6)
2.3 Conformal Mapping
Transformation optics uses the concept of conformal mapping in complex analysis to
implement the coordinate transformation. Conformal mappings are used to change the
complicated domains of physical problems into a simpler one, and transform the
corresponding solutions of the original problems into new domain [11,3]. This mapping
preserves a special property in microscale, known as “angle-preserving” and this
property enables the transformation of a solution that was formulated for the original
16
complicated domain into a related problem for the newer domain [11,3]. In conformal
mapping, the angles between the intersecting grid lines are preserved locally and this
grid property preserves the magnitude of the physical problems in every point in space
enabling the transformation of solution in the new coordinate system. In this section, we
overview the concept of conformal mapping, its relation with analytic function and how
to implement conformal mapping for transformation optics applications.
Figure 2.2: Conformal Mapping Example
Conformal mapping appears in complex plane and deals with the complex
functions in two-dimensional space [11,3]. For example we have a two-dimensional
complex plane denoted by z=𝑥𝑥1 + j𝑥𝑥2 and a complex valued analytic function f(z)
(Figure 2.2) is working in this space. If we want to map this function in a different
complex coordinate set w-plane as a function of the z-plane where w=𝑥𝑥1′ (𝑥𝑥1, 𝑥𝑥2) +
j𝑥𝑥2′ (𝑥𝑥1, 𝑥𝑥2); then a conformal mapping or angle preservation exists in this coordinate
17
transformation if it satisfies the well-known Cauchy-Riemann equations in complex
analysis [11]
𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1=
𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥2 ; 𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥2= −
𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1 (2.7)
where 𝑥𝑥1′ and 𝑥𝑥2′ are both real valued variables of complex-valued function. These
Cauchy-Riemann equations have the following properties [11,8,3]:
• It must be a single valued function in its domain. Single valuedness is essential
for one-to-one mapping.
• The partial derivatives must be continuous and differentiable in every single
point in space. Differentiability preserves the angle during coordinate
transformation.
By differentiating the above two Cauchy-Riemann equations and commuting the mixed
partial derivatives, the equations can be simplified to the following Laplace’s equation
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥2)2 = 0 ;
𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥2)2 = 0 (2.8)
∆𝑥𝑥1′(𝑥𝑥1, 𝑥𝑥2) = 0 ; ∆𝑥𝑥2′(𝑥𝑥1, 𝑥𝑥2) = 0 (2.9)
where Δ is the Laplacian operator. In the above equation, the superscripts on the
parentheses in the denominator are an exponent, whereas the superscripts within the
parenthesis are an index. Likewise, the inverse transformation functions for 𝑥𝑥1 (𝑥𝑥1′, 𝑥𝑥2′)
and 𝑥𝑥2 (𝑥𝑥1′, 𝑥𝑥2′) satisfy the following Laplace’s equation
∆𝑥𝑥1 (𝑥𝑥1′ , 𝑥𝑥2′) = 0 ; ∆𝑥𝑥2 (𝑥𝑥1′ , 𝑥𝑥2′) = 0 (2.10)
18
Hence, a coordinate transformation that is angle preserving or conformally mapped will
satisfy the Laplace’s equations in its domain. Equivalently, any two-dimensional
coordinate transformation that satisfy the Laplace’s equations everywhere in its domain
will be conformal or locally orthogonal; and is the main trick to numerically implement
conformal coordinate mapping using Laplace’s equations under two dimensional
coordinate transformation.
2.4 Transformation Optics (TO)
Transformation optics (TO) scheme is the mathematical design methodology of
manipulating electromagnetic waves in novel way in a spatially distorted
electromagnetic structure and this technique is based on the form invariance principle
of Maxwell’s equations under coordinate transformation, i.e., Maxwell’s equations in a
distorted coordinate system can be expressed as a function of inhomogeneous and
complex permittivity and permeability tensor of original coordinate system [1,2,13,8,3].
Suppose, we have the time harmonic electromagnetic field vectors E and H in a
coordinate set which goes under coordinate transformation, and the mapped time
harmonic field vectors in the new coordinate set are E' and H'; then Maxwell equations
in the new coordinate set will remain form invariant [1,3,6,8]
∇′ × 𝑬𝑬′ = −𝑗𝑗𝜔𝜔𝜇𝜇′𝑯𝑯′ (2.11)
∇′ × 𝑯𝑯′ = 𝑗𝑗𝜔𝜔𝜇𝜇′𝑬𝑬′ (2.12)
And the electromagnetic field vectors in the new coordinate system can be expressed as
a function of original field vectors with the aid of jacobian transformation matrix [7-8]
19
𝑬𝑬′ = 𝜦𝜦𝑘𝑘𝑘𝑘′𝑇𝑇−1𝑬𝑬 ; 𝑯𝑯′ = 𝜦𝜦𝑘𝑘𝑘𝑘
′𝑇𝑇−1𝑯𝑯 (2.13)
The material tensors in transformation optics changes with a tensor density of weight
+1 in using the following relations [1-3,8]:
𝜀𝜀𝑖𝑖′𝑗𝑗′ =𝜦𝜦𝑖𝑖𝑖𝑖
′ 𝜦𝜦𝑗𝑗
𝑗𝑗′ 𝜀𝜀𝑖𝑖𝑗𝑗 𝛿𝛿𝑖𝑖𝑗𝑗
|𝜦𝜦| ; 𝜇𝜇𝑖𝑖′𝑗𝑗′ =𝜦𝜦𝑖𝑖𝑖𝑖
′ 𝜦𝜦𝑗𝑗
𝑗𝑗′𝜇𝜇𝑖𝑖𝑗𝑗 𝛿𝛿𝑖𝑖𝑗𝑗
|𝜦𝜦| (2.14)
where 𝛿𝛿𝑖𝑖𝑗𝑗 = 1 if i = j 0 if i ≠ j and 𝜦𝜦𝑖𝑖𝑖𝑖
′≜ 𝜕𝜕𝑥𝑥𝑖𝑖
′
𝜕𝜕𝑥𝑥𝑖𝑖 (2.15)
Here 𝜦𝜦𝑖𝑖𝑖𝑖′is the jacobian transformation matrix; εij and μij are the original permittivity
and permeability tensors of the electromagnetic structure, and εi'j' and μi'j' are the new
parameters of the distorted structure. These tensorial parameters are generally
anisotropic.
2.5 Quasi-Conformal Transformation Optics (QCTO)
In the transformation optics scheme, the derived material parameters are
complex anisotropic; magneto-dielectric and requires artificial metamaterial to realize.
These complex anisotropic materials that arise during coordinate transformation are
very difficult to realize. To avoid this implementation problem and eliminate the
anisotropy, a simplified scheme, known as quasi-conformal transformation optics, is
used [13]. In quasi-conformal transformation optics (QCTO) technique, a two-
dimensional transformation is performed using Laplace’s equations to calculate the grid
distortions during conformal coordinate transformation [13]. This simplification from
three dimension to two dimension makes the design implementation much easier. The
20
three-dimensional realization of the 2D QCTO-enabled design is obtained by rotating
or extending the 2D design using the symmetric or polarization specific nature of the
actual device.
In the two-dimensional QCTO approximation, the three-dimensional jacobian
transformation matrix are reduces to [13]
𝚲𝚲 =
⎣⎢⎢⎢⎡
𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥20
𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥20
0 0 1
⎦⎥⎥⎥⎤
(2.16)
Once Λ is known, the material parameters of the modified medium can be calculated
easily. The material permittivity in the transformation optics scheme described in
section 2.4 can be rewritten in a simplified way:
𝜀𝜀′ =𝜦𝜦 𝜀𝜀𝑟𝑟 𝜦𝜦𝑇𝑇
|𝜦𝜦| (2.17)
The quantity ΛΛT can be evaluated as
𝜦𝜦 𝜦𝜦𝑇𝑇 =
⎣⎢⎢⎢⎢⎡
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥2)2𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1+𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥2.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥20
𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1+𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥2.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥2𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥2)20
0 0 1
⎦⎥⎥⎥⎥⎤
(2.18)
The above expression can be simplified with the concept of conformal mapping in
which a complex analytic function relates two transformative coordinate space by
satisfying the Cauchy-Riemann equations as described in section 2.3. The Cauchy-
Riemann equations are
21
𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1= 𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥2 ; 𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥2= −
𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1 (2.19)
which implies
𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1+𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥2.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥2 =
𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥1.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥1+𝜕𝜕𝑥𝑥1′
𝜕𝜕𝑥𝑥2.𝜕𝜕𝑥𝑥2′
𝜕𝜕𝑥𝑥2 = 0 (2.20)
and the determinant of jacobian matrix in two-dimensional space
|𝜦𝜦| = 𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥2)2=
𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥2)2 (2.21)
This leads to the permittivity tensors as
𝜀𝜀′ = ε𝑟𝑟|𝜦𝜦|
⎣⎢⎢⎢⎢⎡
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥1′
(𝜕𝜕𝑥𝑥2)20 0
0𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥1)2 +
𝜕𝜕2𝑥𝑥2′
(𝜕𝜕𝑥𝑥2)20
0 0 1
⎦⎥⎥⎥⎥⎤
(2.21)
𝜀𝜀′ = 𝜀𝜀𝑟𝑟 1 0 00 1 00 0 1/|𝜦𝜦|
(2.22)
Where ε' and μ' are the new material parameters and in the example device this
technique will be used to design. The QCTO design method begins with a two-
dimensional coordinate mapping of the original electromagnetic device with a solution
to Laplace’s equations using a set of dirichlet and neumann boundary conditions [3,6].
Inverse Laplace’s equation based quasi-conformal transformation can be used for quasi-
isotropic transformation material design [10]. In the transverse polarized devices where
electric field is polarized in the z-direction, the material response becomes non-magnetic
22
and the modified material parameters for inverse coordinate transformation can be
simplified as [56]:
𝜀𝜀′ =𝜀𝜀𝑟𝑟
|𝜦𝜦−1| ; 𝜇𝜇′ = 1 (2.23)
23
DESIGN AND ADDITIVE MANUFACTURING OF MODIFIED GRIN LENS
In this chapter, I applied the quasiconformal transformation optics (QCTO)
technique discussed in previous chapter in a particular electromagnetic application
known as GRIN lens structure.
My specific contributions to this chapter are as follow:
I examined and applied the quasi-conformal mapping in the context of GRIN
lens structure with proper boundary conditions. The QCTO-enabled lenses
were designed in two dimensional space and extended to three dimensional
space using the rotational symmetry of the structure. Both the two-dimensional
and three-dimensional EM modeling were conducted using commercially
available finite element based COMSOLTM numerical solver.
I designed and modeled the three-dimensional modified Luneburg lens and
Maxwell fish-eye lens exploiting the rotational symmetry of the GRIN lenses.
The three dimensional designs assumed approximations in revolving the two
dimensional coordinate mapping and I verified through 3D EM modeling that
this approximation does not affect device beam steering performance in
Chapter 3
24
microwave frequency range. The side lobe levels of the modified Luneburg
lens antenna are considerably low. The designed modified Luneburg lens
provides a good beam steering performance from -55˚ to +55˚ over the entire
Ka-band (26 GHz - 40 GHz) frequency. I applied same approach to design a
modified Maxwell half-fish eye lens capable of beamsteering from -45˚ to
+45˚.
The three-dimensional modified Luneburg lens antenna designed with QCTO
technique was fabricated using FDM (Fused Deposition Modeling) based
additive manufacturing technique and experimentally characterized. Space
filling curve based additive manufacturing technique was utilized to realize
the graded dielectric structure of GRIN lens. The fabricated lens showed
similar beamsteering performance (-55˚ to +55˚) as the numerical predictions.
3.1 GRIN Lens Background
High directivity and agile beamscanning capability over broad spectral band is
a much needed antenna feature in RF and microwave communication system for a
reliable wireless communication system design. Conventionally these were achieved by
using phased array antenna system in high performance radar imaging and satellite
communications. Phased array antenna provides a sophisticated system advantages with
high gain, minimum sidelobes, and agile beam control. However, this technology is
limited by high expense, limited bandwidth and narrow beamscanning angle. To address
these problems, passive gradient-index (GRIN) lenses is an attractive alternative to the
25
expensive and bandwidth limited phased array systems for wide beamscanning angle
satellite communication and radar imaging. A suitable combination of GRIN lenses
along with external feed networks and beamswitching networks can lead to a cheap and
complete solution of communication system and to design such a system, every
component of the system including the beamforming lens antenna needs to be designed
precisely. A lot of GRIN lenses are available to be used as the beamforming antennas;
however, Luneburg lens and Maxwell fish-eye lens are the most widely used for their
wide beamscanning range; high gain and lower fabrication cost.
3.1.1 Luneburg Lens
The Luneburg lens is a spherical shaped gradient-dielectric structure in which
every point on the surface acts as a focal point for a plane wave incident from the
opposite surface of the lens (figure 3.1(a). The Luneburg lens geometry is the most
widely for its wide angle and multiband beamforming and beamscanning capability, and
low fabrication cost. Beamscanning is achieved by simply switching between focal
points along the lens’s spherical surface. The lens has a three-dimensionally varying
dielectric permittivity profile (figure 3.1(b)) mathematically expressed as [71]:
𝜀𝜀𝑟𝑟 = 2 − 𝑟𝑟 𝑅𝑅 2
where r is the radial distance from the center of the sphere, and R is the radius of the
sphere.
26
(a)
(b) Figure 3.1: Luneburg Lens: (a) lens’s beamforming nature [82]; (b) Dielectric
permittivity distribution
27
3.1.2 Maxwell Fish-Eye Lens
Maxwell fish-eye lens is another dielectric lens of GRIN lens family. Maxwell
fish-eye lens is a spherical shaped gradient-dielectric structure in which every point on
the lens surface acts as a focal point for signal excited at the point on the opposite surface
of the lens. However, half of the fish-eye lens acts similar to a focusing lens where lens’s
spherical surface acts as a focal point for plane wave coming from opposite surface of
the half lens (figure 3.2(a)) and the spatially varying three-dimensional dielectric
permittivity profile of the fish-eye lens can be expressed as [74, 76]:
𝜀𝜀𝑟𝑟 =4
1 + 𝑟𝑟 𝑅𝑅 2
where r is the radial distance from the center of the sphere, and R is the radius of the
sphere.
(a)
28
(b) Figure 3.2: Maxwell Fish-eye lens: (a) lens’s beamforming nature [76] ; (b)
Dielectric permittivity profile
3.2 Feed integration problems with GRIN lenses
GRIN lenses are passive in nature, and requires external feed sources to radiate
RF or photonic signals. When implementing these beamforming lens elements with the
external feed networks such as waveguides, antenna arrays, or detectors, there is
practical challenges [25]. The spherical nature of the lens’s surface complicates the
integration of feed networks and other associated external electronics as shown in figure
3.3. To address this compatibility issue of feed networks integration, many investigators
provide a flat surface by modifying the lens into a planar surface or encapsulating flat
background material [6,17,21,23,25,26,34,36]. The presence of extra surface reduces
29
the lens’s beam steering ability and modifying the lens’s spherical surface into a planar
surface with the intended beam steering angle requires optimization and redistribution
of material permittivity inside the dielectric structure. Among the variety of
optimization scheme that have been explored to optimize electromagnetic structure’s
material parameters under spatial distortion, transformation optics is the most widely
used to optimize material parameters of spatially distorted electromagnetic structure.
Figure 3.3: GRIN Lens with feed elements and beamswitching elements
In chapter 2, we overviewed the theory of transformation optics and in this section, we
will be exploring the quasi-conformal transformation optics (QCTO) technique to
design the modified Luneburg lens and Maxwell fish-eye lens and optimize the material
parameters.
30
3.3 Modified Luneburg lens design using QCTO technique
To modify the portion of the spherical surface of the Luneburg lens into a planar
surface, a two-dimensional quasi-conformal mapping was carried out by solving
Laplace’s equations in the physical space with a set of boundary conditions.
Figure 3.4(a) shows the original two dimensional Luneburg lens, known as the
virtual space, and bottom portion of the lens is transformed into a flat geometry, known
as the physical space as shown in figure 3.4(b). In the virtual space, the circle has a
permittivity distribution of two-dimensional Luneburg lens, surrounded by free space
in rectangular shape. In our design we considered the mapping of a portion of 2D
spherical lens (sector CDE in virtual space in figure 3.4(a)) into a planar surface (C'D'E'
in physical space in figure 3.4(b). When carrying out the coordinate mapping, we
pursued to find out the corresponding coordinates in the virtual space given that the
coordinates are at the physical space. In this way every single point of the boundary
CDE in virtual space conformally maps to C'D'E' in the physical space. The specific
geometries used for this design are shown in figure 3.4. For implementing the two
dimensional coordinate mapping we assumed a two dimensional spherical Luneburg
lens with a radius of 30 mm.
The coordinate transformation from virtual space to physical space were
determined by solving the Laplace’s equations in physical space using the following
Dirichlet-Neumann boundary conditions for coordinate mapping:
𝐴𝐴′𝐵𝐵′|𝑥𝑥′ = 𝐹𝐹′𝐺𝐺′|𝑥𝑥′ = 𝑥𝑥 ; 𝐶𝐶′𝐷𝐷′𝐸𝐸′|𝑥𝑥′ = 𝜃𝜃(𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟𝑟) ∙ 𝑥𝑥 2 ;
𝑟𝑟 ∙ ∇𝑥𝑥|𝐴𝐴′𝐺𝐺′,𝐵𝐵′𝐶𝐶′,𝐸𝐸′𝐹𝐹′ = 0
31
𝐴𝐴′𝐺𝐺′|𝑦𝑦′ = 𝐵𝐵′𝐶𝐶′|𝑦𝑦′ = 𝐸𝐸′𝐹𝐹′|𝑦𝑦′ = 𝑦𝑦 ; 𝐶𝐶′𝐷𝐷′𝐸𝐸′|𝑥𝑥′ = −𝑅𝑅2 − 𝑥𝑥2;
𝑟𝑟 ∙ ∇𝑦𝑦|𝐴𝐴′𝐵𝐵′,𝐹𝐹′𝐺𝐺′ = 0
(a)
(b)
32
(c)
(d)
Figure 3.4: Illustration of the Luneburg lens: (a) virtual and (b) physical space used
for QCTO mapping ; (c) coordinate grid of the original Luneburg lens obtained
from QCTO mapping; (d) mapped coordinate grid of the modified Luneburg lens
obtained from inverse coordinate transformation in physical space.
33
where n is the outward normal vector to the surface boundaries, x and y denote the
two dimensional coordinates in the virtual space and θ is the internal angle of the circular
sector of original Luneburg lens to be modified into planar surface as shown in figure
3.4(b). Figure 3.4 (c), (d) shows the coordinate grid of the virtual space and mapped
coordinate grid physical space after coordinate transformation. The new material
parameters of the modified Luneburg lens were then calculated as:
𝜀𝜀′ =𝜀𝜀𝑟𝑟
|𝜦𝜦−1| ; 𝜇𝜇′ = 1
The maximum permittivity distribution resulting from the QCTO technique and
the beamsteering angle of the lens can be controlled by varying the angle θ. For this
particular design the maximum permittivity value was limited within 2.9, which resulted
in an angle of θ=111o (figure 3.4(a)). The quasi-conformal mapping of the modified lens
was implemented using commercially available finite element based numerical solver
COMSOLTM Multiphysics. COMSOLTM simulation package offers a built-in capability
to solve Laplace equations with dirichlet and neumann boundary conditions, and
calculate the partial derivatives. The permittivity distribution of the two-dimensional
original and modified Luneburg lens, and the three-dimensional modified Luneburg lens
is shown in figure 3.5. The three dimensional implementation of the QCTO-enabled
modified Luneburg lens was achieved by revolving the two dimensional permittivity
distribution (figure 3.5(b)) along its center axis (z-axis) using COMSOLTM Multiphysics
numerical solver as shown in figure 3.5(c).
34
(a)
(b)
35
(c) Figure 3.5: Permittivity profile for (a) cross sectional view of 2D original Luneburg
lens, (b) cross sectional view of 2D modified Luneburg lens, (c) 3D representation
of modified Luneburg lens permittivity distribution
3.3.1 3D QCTO Approximations
The three-dimensional modified Luneburg lens antenna had a spatially varying
permittivity profile as shown in figure 3.5(c) and the magnetic response of the device
was ignored in the 3D QCTO-enabled device. In the original transformation optics
devices, materials have both electric and magnetic response. Hence, the non-magnetic
material parameters derived via rotational symmetry is an approximated profile and
such a non-magnetic, quasi-isotropic material approximations are not the exact material
36
transformation for rotationally symmetric three-dimensional electromagnetic structure.
The proper transformation optics design requires the presence of magnetic response
equal to the electric response [15]. This requirement of the presence of anisotropy will
be clear from the dispersion relation. The dispersion relation in the cylindrical
coordinate are given by [15]:
𝑘𝑘𝜙𝜙𝛽𝛽2
+𝑘𝑘𝜌𝜌
2
𝛼𝛼+𝑘𝑘𝑧𝑧
2
𝛼𝛼= 1
Where k = 𝑘𝑘𝜌𝜌𝜌𝜌 + 𝑘𝑘ϕϕ + 𝑘𝑘𝑧𝑧𝑧 is the normalized wavenumber relative to the air
in cylindrical coordinate. If ερ, εϕ, and εz are the permittivity components in cylindrical
coordinate, then, α and β in the above dispersion relation are expressed as follows (the
detail derivation is discussed in [15]):
α = εϕ = |𝜦𝜦|−1 ; β2 = ερ = εz
If 𝑘𝑘𝜙𝜙 is eliminated in the above dispersion relation, then the relation will be
simplified as an isotropic medium dispersion relation and that isotropic dielectric will
be ε = α = |𝛬𝛬|−1 [15-16] and we assumed this approximation in our three-dimensional
design of QCTO-enabled modified Luneburg lens.
As, 3D-approximate QCTO-enabled Luneburg lens required only dielectric material
without having any magnetic response, the design suffered from impedance mismatches
[15], and reflections were present at different excitation positions along the planar
surface of the modified lens. The reflection problems of QCTO designs can be seen
from 3D full wave electromagnetic simulation and is presented in section 3.3.2.
37
3.3.2 3D Full Wave Electromagnetic Simulation
To verify the beam steering performance of the modified Luneburg lens, 3D full
wave electromagnetic simulations were carried out using COMSOLTM numerical
solver. The 3D modified lens had the spatially varying permittivity distribution as
shown in figure 3.5(c) achieved using quasi-conformal transformation optics technique.
The lens was surrounded by air and perfectly match layer (PML) was applied around
the simulation domain to eliminate any spurious reflections from truncated
computational region (figure 3.6(a)). Figure 3.6(b) shows the finite element meshing
used in COMSOLTM. To show the beam steering functionality of the three-dimensional
modified Luneburg lens, the planar surface of the lens was excited with a waveguide
port at different feed positions. Figure 3.6(c) shows the different feed locations used in
the simulation. The waveguide excitation at each location should result in beamsteering
in the azimuth and elevation plane. At each feed location, 3D full wave electromagnetic
simulations were carried out at Ka-band and the 3D radiation patterns were computed
using COMSOLTM numerical solver. The computed 3D radiation patterns for each feed
location at 30 GHz are shown in figure 3.7. As expected the antenna’s main beam
steered over a fairly large range of angles (i.e. -55o to 55o) as the waveguide position
was moved at different locations along the planar surface of the modified lens.
38
(a)
(b)
39
(c)
Figure 3.6: (a) 3D finite element setup of the modified Luneburg antenna modeled
in COMSOLTM; (b) 3D Finite element meshing of the modified Luneburg lens
modeled in COMSOLTM; (c) illustrations showing the positions of the waveguide
feed sources used for the simulations along the planar surface of the modified
Luneburg lens
40
(a)
(b)
41
(c)
(d)
42
(e)
(f)
43
(g)
(h)
44
(i)
(j)
45
(k)
Figure 3.7: Simulated 3D radiation patterns (dBi) of the modified Luneburg lens at
30 GHz for source location at (a) pos -2, (b) pos -1, (c) pos 0, (d) pos 1, (e) pos 2,
(f) pos 3, (g) pos 4, (h) pos 5, (i) pos 6, (j) pos 7, (k) pos 8 as shown in figure 3.6(c)
3.4 Modified half Maxwell FISH-EYE lens design using QCTO technique
Similar to the Luneburg lens, half of the Maxwell fish-eye lens can also behave
as a beamsteering lens. In the half fish-eye lens, one side is planar and the other side has
a spherical surface. To modify the spherical surface of the fish-eye lens into planar to
make the device into a two-sided flat lens, we employed inverse coordinate
transformation based QCTO technique by solving Laplace’s equations in the physical
space with the following boundary conditions:
46
𝐴𝐴′𝐵𝐵′|𝑥𝑥′ = 𝐹𝐹′𝐺𝐺′|𝑥𝑥′ = 𝑥𝑥 ; 𝑟𝑟 ∙ ∇𝑥𝑥|𝐴𝐴′𝐺𝐺′,𝐵𝐵′𝐶𝐶′,𝐶𝐶′𝐷𝐷′𝐸𝐸′,𝐸𝐸′𝐹𝐹′ = 0
𝐴𝐴′𝐺𝐺′|𝑦𝑦′ = 𝐵𝐵′𝐶𝐶′|𝑦𝑦′ = 𝐸𝐸′𝐹𝐹′|𝑦𝑦′ = 𝑦𝑦 ; 𝐶𝐶′𝐷𝐷′𝐸𝐸′|𝑥𝑥′ = −𝑅𝑅2 − 𝑥𝑥2;
𝑟𝑟 ∙ ∇𝑦𝑦|𝐴𝐴′𝐵𝐵′,𝐹𝐹′𝐺𝐺′ = 0
The specific geometries used for our mapping are shown in figure 3.8(a), (b).
The new material parameters of the modified fish-eye lens were calculated as:
𝜀𝜀′ =𝜀𝜀𝑟𝑟
|𝜦𝜦−1| ; 𝜇𝜇′ = 1
The 2D permittivity distribution for the original and QCTO-enabled modified
fish-eye lens is shown in figure 3.8(a) and figure 3.8(b). For implementing the 2D
mapping we assumed a two dimensional semi-spherical fish-eye lens (figure 3.8(a))
with a radius of 30 mm surrounded by air. The three dimensional implementation of the
modified fish-eye lens was achieved by revolving the two dimensional permittivity
distribution (figure 3.8(b)) along its center axis (z-axis) using COMSOLTM Multiphysics
solver as shown in figure 3.8(c). The realization of the 3D permittivity profile assumes
the material parameters as non-magnetic and all-dielectric by exploiting the 3D QCTO-
approximations as described in section 3.3.1. As, in the QCTO approximation, the
magnetic response was eliminated by assuming the materials as all dielectric, the design
will suffer from impedance mismatch at different excitation positions. The effects of the
impedance mismatch will be clear from the 3D radiation pattern described in section
3.4.1.
47
(a)
(b)
48
(c)
Figure 3.8: Permittivity profile: (a) cross sectional view of 2D original Half Maxwell
Fish-eye lens, (b) cross sectional view of 2D modified fish-eye lens, (c) 3D
representation of modified fish-eye lens permittivity distribution
3.4.1 3D Full Wave electromagnetic simulation of modified fish-eye lens
To verify the beam steering functionality of the modified half fish-eye lens, 3D
full wave simulations were carried out using COMSOLTM solver. The lens has the
spatially varying permittivity distribution shown in figure 3.8(c). Figure 3.9(a) shows
the finite element meshing of the modified fish-eye lens using COMSOLTM solver. To
show the beamsteering performance and excite the lens, a waveguide port was placed
along the centerline of the bottom planar portion of the lens (figure 3.9(b)). The position
of the port along that line should result in beam steering. We chose five source locations
49
as waveguide excitation in this design. At each excitation a 3D full wave
electromagnetic simulation was conducted and the 3D radiation patterns were
calculated. For brevity, the simulated results for the five excitation positions at 30 GHz
frequency are shown in figure 3.10. The rotational symmetry of the lens allows similar
behavior in all other excitation locations as well:
(a)
(b)
Figure 3.9: (a) Finite element mesh of the modified half Maxwell fish-eye lens
modeled in COMSOLTM, (b) illustration showing the five positions of the
waveguide source feed used for the simulations
50
As expected the lens’s main beam steered over an azimuthal angle (i.e. -45o to
45o) as the source position was changed along the centerline. For brevity sake we only
present results for five feed locations resulting in beam steering in the azimuthal plane.
The rotational symmetry of the lens allows similar radiation patterns in the elevation
plane as well. Figure 3.11 shows the comparison of simulated gain patterns of the
modified half Maxwell fish-eye lens antenna at five different waveguide excitation
position (as shown in figure 3.9(b)) at 30 GHz frequency as a function of azimuth angle.
As expected, the lens’ gain value is much lower without anti-reflective layers due to the
fact that the impedance mismatch is higher at the top surface where the permittivity
profile is much higher than that of air resulting in lower gain value and higher sidelobes.
(a)
51
(b)
(c)
52
(d)
(e)
Figure 3.10: Simulated 3D radiation patterns (dBi) of modified half Maxwell fish-
eye lens at 30 GHz for source location at (a) pos -2; (b) pos -1; (c) pos 0; (d) pos 1;
(e) pos 2
53
Figure 3.11: Simulated gain patterns of QCTO-enabled modified half Maxwell fish-
eye lens as a function of azimuth angle and feed locations at 30 GHz
3.5 Additive Manufacturing of Spatially Varying Permittivity Distributions
Additive manufacturing (AM), also known as 3D printing, has emerged as a
wonderful manufacturing technique by realizing devices layer upon layer compared to
the traditional subtractive manufacturing method, which removes parts by parts from a
larger object and eventually realizing the final structure [78-79]. Additive
manufacturing has been widely used in a lot of electromagnetic applications to realized
54
RF and microwave structures. To fabricate our three-dimensionally varying permittivity
distribution of the modified Luneburg lens designed with QCTO technique, shown in
figure 3.5, we employed additive manufacturing method to realize the local permittivity
value of a large structure in small voxels from digitized data (figure 3.12). Recently,
space filling curve geometry based spatially varying graded dielectric structure has been
reported [64] and to fabricate the modified Luneburg lens, we employed fused
deposition modeling (FDM) based additive manufacturing method using custom
generated tool paths. In FDM, a thermoplastic filament is extruded from a heated nozzle
in layers and multiple layers were stacked vertically to build the solid object as shown
in figure 3.12 [64,25,65,29]. In developing an approach for realizing the modified
Luneburg lens structures using FDM, we employed the space-filling curve approach
described in [64].
Figure 3.12: Voxelated permittivity values
55
3.5.1 Space-Filling Curve for realizing graded permittivities
Recently, a method for applying space-filling curve geometries to realize
spatially varying graded dielectric structures has been shown by our group
[64,25,29,65]. As described in [64], the Peano-type space-filling curve was selected as
illustrated in figure 3.13 where a single unit cell is formed by printing the space filling
geometry. By changing the number of turns, N, in the space filling curve geometry the
local volume fraction of deposited material within a unit cell can be controlled.
Assuming the size of the unit cell, Λ, is much lower than the wavelength (i.e. Λ<<λ) an
effective local permittivity value results. Effective media theory can be used to quantify
the effective permittivity value as a function of the unit cell size, print parameters (e.g.
filament size and shape), bulk permittivity of the thermoplastic and order of the space
filling curve. To create a spatially varying permittivity distribution we then simply vary
the order of the space filling curve from unit cell to unit cell. Moreover, we can orient
the unit cells in such a way that the start and end points of the curves are at the same
corner (e.g. lower left and upper left). This allows the unit cells to be connected in rows
and columns such that each layer of a graded print is formed by a single continuous
curve (figure 3.13). Having a single continuous curve eliminates the necessity of
numerous beginnings and stops of the FDM print process and results in much higher
quality parts. To create spatially varying three-dimensional dielectric structure, the
design was discretized into two dimensional layers which were printed using space-
filling curves and successive layers were stacked together in the vertical direction.
56
Figure 3.13: The space-filling geometry used for generating spatially-varying
Permittivities. By varying the number of turns, N, the local volume fraction of printed
material, and thus its effective permittivity, is controlled.
To determine the effective permittivity value of the space-filling curve geometries
seven experimental calibration samples were fabricated and characterized. For these
samples, polycarbonate with higher permittivity value limited to 2.9 was used as the
printed material and air as the background material. The extruded filament diameter W,
layer height h, and unit-cell size Λ were fixed at 0.3 mm, 0.125 mm, and 3.0 mm,
respectively. The volume fraction for each sample was varied by changing the order of
the space-filling curve (i.e. number of turns). To create plates of 1 mm in thickness we
stacked ten 2D layers of the printed geometry shown in figure 3.13. In figure 3.15, the
57
measured effective permittivity value is presented as a function of volume fraction.
From figure 3.14, it is clear that this approach produced a small degree of anisotropy
indicated by the transverse components εx and εy, and through-thickness component εz
of the permittivity curves. This geometry was found to be isotropic in the xy-plane as
expected, but the through-thickness permittivity εz was found to be slightly lower
throughout the range of volume fractions. An effective media model for this geometry
was also developed with predicted values also shown in f. The details on the effective
media approach is described in [64,25]. For the design of the modified Luneburg lens
we used the effective permittivity value from the xy-plane (i.e. εxy).
Figure 3.14: The predicted and measured relative permittivity of the space-filling
curve geometry as a function of volume fraction
58
3.5.2 Modified Luneburg lens fabrication
The maximum permittivity value that we could realize was 2.9 and with this limited
permittivity value, we could fabricate the modified Luneburg lens for our design
methodology validation. The fish-eye lens is remained to be validated in the future
works. To fabricate the modified Luneburg lens, an additive manufacturing system,
nScryptTM 3Dn-300, as shown in figure 3.15 was utilized. The nScryptTM 3Dn-300
system is a quad deposition system with multiple print heads capable of depositing
custom and commercial inks and pastes via micro dispensing or extruding polymers via
FDM. For GRIN lens application, we only required the FDM print head. The printer is
also outfitted with a fiducial alignment camera, a 3D laser scanning system, and a 300
mm x 300 mm heated print bed. The printer has a resolution of printing linewidths as
narrow as 20 μm with stages that are capable of maintaining a positional accuracy within
of less than 1.0 μm.
For FDM printing our choice for the material was a polycarbonate obtained from
matterhackers.com that has excellent EM properties and mechanical strength. Prior to
printing, we performed EM characterization of this polycarbonate over a wide band of
frequencies (8-40 GHz), and found that the material was non-dispersive with a dielectric
constant of εr = 2.9 and a loss tangent of tanδ = 0.0005. Polymer extrusion was
performed at a nozzle temperature of 295˚C and a print bed temperature of 130˚C. The
nozzle used had a 200 μm inner diameter, and deposition occurred at 60 mm/s with layer
thicknesses of 100 μm.
59
Figure 3.15: Additive manufacturing system used to print modified Luneburg lens
(nScrypt 3Dn-300)
(a)
60
(b)
Figure 3.16: (a) FDM printing of the modified Luneburg lens with space filling curves
using the nScrypt printer extruding polycarbonate filaments. (b) Fabricated lens
antenna
3.6 Results
3.6.1 Measurement Setup
The additively manufactured modified Luneburg lens antenna performance was
characterized by measuring the return loss; gain pattern as a function of frequency, beam
steering angle, and location of the feed source. As an excitation source, we used an
open-ended rectangular waveguide (WR28) and placed along the flat surface of the
modified Luneburg lens antenna. The lens was designed to operate in the Ka-band from
26 GHz to 40 GHz. To measure the lens’s performance as function of azimuthal angle
61
and frequency the lens was centered and rotated around the center axis of the waveguide
feed under computer control using the setup shown in in figure 3.17. On the receive
side, a fixed vertically aligned standard gain horn antenna with a measured gain of 24
dBi was centered and aligned with the modified Luneburg lens. The orientation of the
lens with respect to the fixed standard gain horn antenna was varied automatically from
-90o to 90o in 1o increments. At each location the transmission coefficient (S21) was
recorded using an Agilent PNA E83684B vector network analyzer. To reduce the
unwanted reflections, radar absorbing material (RAM) was placed along all surfaces
surrounding the measurement setup.
Figure 3.17: Measurement setup to characterize the modified Luneburg lens antenna
gain as a function of azimuthal angle and frequency. The electric field was linearly
polarized along the vertical axis
62
3.6.2 Experimental Data
In figure 3.18 the measured reflection coefficients (S11) of the waveguide
feed with and without the presence of the modified Luneburg lens are shown as a
function of frequency for three waveguide feed positions. While S11 was reasonably
lower for all test conditions, due to the spatial gradient of the permittivity profile along
the planar surface of the modified lens (figure 3.6(c)), the reflection losses varied along
the center line of the planar surface as the waveguide feed was moved from the center
of the lens to the outer edge. Since the permittivity value of the design was maximum
at the center of the lens, the reflection losses were greatest at that location.
Figure 3.18: Measured reflection coefficient (S11) of the WR28 open-ended
waveguide with and without the presence of the modified Luneburg lens at three
waveguide positions at the planar surface
63
In figure 3.19(a), the measured gain pattern as a function of azimuth angle is
presented for each of the five feed locations at 30 GHz. Also shown in the figure are the
predicted beam patterns simulated using COMSOLTM solver. As expected when the
source was moved away from the center of the lens (pos 0), the main beam was steered
away from the center axis of the lens. This property was consistent across the entire Ka-
band frequency range. In fact, we were able to achieve a reasonably wide angle of beam
steering (i.e. -55˚ to +55˚) over the entire Ka-band. As shown in figure 3.19(a) the
maximum realized gain was highest at the edge positions (pos -2, pos 2) compared to in
the center (pos 0). The reason for this was the lower return losses of the waveguide feed
as the source was moved away from the center location. In section 3.5.2, we discussed
about the 3D QCTO approximations where the magnetic response of the QCTO-enabled
modified lens was ignored by assuming the design all-dielectric. As the magnetic
response was eliminated, the return loss was caused by the impedance mismatch
produced by the higher permittivity values at different excitation positions. At the center
location (pos 0), the lens has the highest permittivity value compared to the air and
creates maximum reflections, whereas at the edge (pos 2) the permittivity distribution
has a lower value nearer to that of free space. Due to the higher impedance mismatch at
the center, the lens was subjected to reflections and resulted in the lower gain value at
the center position compared to the edges.
Figure 3.19(b) shows the measured and simulated gain pattern of the Luneburg
lens as a function of frequency when the lens was excited at the center of the lens (pos
64
0). As expected, the lens’s measured gain pattern over the entire Ka-band complied well
with the predicted gain.
Figure 3.19(c) presents the measured aperture efficiency of the modified
Luneburg lens antenna as a function of frequency for three different feed positions. As
expected, the aperture efficiency is lowest when the feed location is placed at the center
of the lens and improves as the feed locations is moved towards the edge.
(a)
65
(b)
(c)
Figure 3.19: (a) Measured and simulated radiation patterns and beam steering
performance of a Ka-band modified Luneburg lens antenna at 30 GHz; (b) measured
66
realized gain at center excitation over the entire Ka-band; and (c) aperture efficiency
at three feed locations as a function of frequency
To mitigate the impedance mismatch problems arises in QCTO technique, in
general, and obtain a more uniform impedance matching across the entire planar surface
of the modified Luneburg lens antenna, in particular, which should result in better
aperture efficiency for all the excitation positions across the planar surface, we explored
new design methodologies which will be discussed in chapter 4.
3.7 Conclusion
This chapter describes the practically implementable three-dimensional
design and additive manufacturing based gradient dielectric structure fabrication
methodologies in the context of gradient index Luneburg lens and Maxwell fish-
eye lens. The design approach uses a different boundary conditions for QCTO
mapping and shows a better beamsteering performance of the modified luneburg
lens over other published QCTO based methods. The rotationally symmetric three-
dimensional designs of the QCTO based modified Luneburg lens and Maxwell fish-
eye lens were done using the numerical solver COMSOLTM which was also used
for 2D QCTO mapping, and the designed three-dimensional modified lenses were
verified via full-wave electromagnetic simulation with waveguide excitation. This
67
is the first instance of modeling 3D QCTO-enabled lens antenna designs using the
same numerical solver used for 2D QCTO mapping. The designed approach agrees
well with the analytical predictions and validated through experiments.
The designed lens was fabricated using space-filling curve based FDM
method described in [64]. This fabrication approach offers 1) the ability to fabricate
mechanically robust graded dielectric structures without need for a support
network, 2) the ability to print with very low electromagnetic loss polymers and 3)
the capability of being conducive to FDM printing, the most prevalent and cost
effective method available for additive manufacturing. We described and
experimentally validated our computational approach to predicting the effective
electromagnetic properties of these space-filling curves. We also demonstrated the
successful application of our design approach with two examples, a simple grid of
varying permittivity cells and a graded index lens that focused energy at microwave
frequencies. We are currently extending this new design and fabrication
methodology to a wide range of other applications, including graded index lens
antennas and passive beam forming structures.
Chapter 3 is based on and a reprint of the following paper: Biswas S, Lu A,
Larimore Z, et al. Realization of modified Luneburg lens antenna using quasi-conformal
transformation optics and additive manufacturing. Microwave Opt Technology Lett.
2019; 61:1022–1029. The dissertation author was the primary author of this paper.
68
BROADBAND IMPEDANCE MATCHING OF QCTO TECHNIQUE
In this chapter, we are exploring methodologies to mitigate the impedance mismatch
issues arises in the quasi-conformal transformation optics (QCTO) approximations
discussed in the chapter 2 and chapter 3, and propose a novel broadband anti-reflective
layer along with the modified surface of QCTO-enabled designs to achieve a uniform
impedance match across the entire QCTO-inspired modified surface. The anti-reflective
layer is made of all dielectric material and similar to the QCTO-enabled devices, the
anti-reflective layer has an inhomogeneous property.
My specific contributions to this chapter are as follow:
I proposed a novel broadband anti-reflective (AR) layer along with quasi-
conformal transformation optics (QCTO) inspired designs to improve the
performance.
I explored three different types of gradient dielectric profile which can be used
as an anti-reflective (AR) layer along with QCTO-enabled designs. I show that
the impedance mismatch issues exhibited in the QCTO-enabled design can be
mitigated by using the novel broadband anti-reflective layer. I introduce the
Chapter 4
69
detail design methodology of applying anti-reflective layer along with QCTO-
enabled designs.
I showed three different types of graded dielectric profile to apply as an anti-
reflective layer and present the relative study of device performance.
I addressed the effects of anti-reflective layer thickness on device
performance.
70
4.1 Introduction
Since the theory of transformation optics (TO) has been proposed [1] to design
electromagnetic cloaking with manipulated electromagnetic waves, it has seen an
unprecedented evolution as a powerful algorithm to design novel electromagnetic
structures in many applications [3,5-6,8,10,12-81]. This technique, based on the form-
invariance principle of Maxwell’s equations, is able to preserve the original
electromagnetic characteristics of the device which goes through some level of
geometry modifications. TO technique has emerged as a powerful tool for engineering
applications and has been used to design many novel electromagnetic devices including
gradient index (GRIN) structures [6,14,16,18-20,23-37]; illusion device and invisibility
cloaks [38-40]; waveguide benders [41-42], field concentrator [43-44], rotators [45],
reflectors [3,46-47], and in many other electromagnetic applications [48-
63,65,72,74,75]. Unfortunately, the material parameters derived from this technique are
complex anisotropic, magneto-dielectric, and requires artificial metamaterials [1] which
make the device performance bandwidth limited, and practical realization of such a
complex anisotropic permittivity and permeability tensor is non-trivial; and extremely
complicated and challenging.
To overcome the implementation problem of complex anisotropic magneto-
dielectric materials, a simplified scheme of TO technique known as quasi-conformal
transformation optics (QCTO) technique [15] is used which reduces the material
anisotropy by eliminating the magnetic response of the designs making the
implementation complexities much easier [15,16,81]. In the QCTO technique, a two-
71
dimensional coordinate mapping of the original electromagnetic structure is carried out
using a set of fixed and slipping boundary conditions and the new constitutive
parameters of the modified structure are calculated following the QCTO prescriptions
[15,16,22,23,25,29,48,50]. The three-dimensional realization of the modified structure
is then achieved by rotating, in case of rotationally symmetric devices, or extruding, in
case of transverse polarized devices, the two-dimensional index profile [32,25,16,56].
This approximated QCTO technique reduces the complexities in experimental
implementation; however, this technique is constrained in degraded device
performances as it introduces reflection problems on its transformed boundary which is
absent in TO technique. In chapter 3, we showed the impedance mismatch problems
presented in QCTO technique in the context of modified Luneburg lens antenna and
modified Maxwell fish-eye lens. These mismatches are due to the high permittivity
profile generated in QCTO mapping along the planar boundary of the modified devices
and this high permittivity value relative to that of free space introduces reflections at the
device’s modified surface. Even though, QCTO technique helps in implanting the
electromagnetic structure with dielectric materials; however, this degraded devices
performance becomes a limitation of QCTO-based designs as opposed to traditional
complex TO-based designs. In TO-based designs, the derived material parameters have
both magnetic and electric response, and the magnetic nature of the device ensures a
uniform impedance matching across the whole modified surface [1]. As in QCTO
approximations the magnetic response is ignored, the impedance mismatch is generated
by the permittivity value only. The higher the permittivity value of the QCTO-enabled
72
designs, the greater the reflections are; and these reflections cause performance
problems in the QCTO enabled designs which remained unaddressed. Quarter
wavelength based anti-reflective layer can mitigate the impedance mismatch for a
particular frequency [62,63]; however, this technique is limited by the narrow
bandwidth. For multiband operation, a broadband anti-reflective layer is a necessity
with QCTO-based designs which ensures a uniform impedance matching across the
entire modified surface.
In this work, we addressed these impedance mismatch issues presented in QCTO
technique by proposing a novel design methodology of a broadband anti-reflective (AR)
layer along with the QCTO-enabled designs (Figure 1(a)). The aim was to achieve a
uniform impedance matching across the entire planar surface while keeping the original
electromagnetic functions intact with improved device performance, and diversify the
conventional QCTO-enabled design technique for novel electromagnetic structure
design. To illustrate the use of an anti-reflective layer with QCTO-based devices, we
demonstrate and implement the design methodology of a broadband AR layer in the
context of modified Luneburg lens (Figure 4.1(b)) and modified Maxwell fish-eye lens
where the AR layer is attached along with the planar surface of the modified Luneburg
lens and two sides of the modified fish-eye lens. The proposed AR layer has an
inhomogeneous permittivity profile to ensure the impedance matching uniformly at
every single point of the modified surface. We expect that this methodology can be
extended to all other electromagnetic designs [43-63,65,72,74,75] which exploits
QCTO technique. We believe, this novel design approach will enhance the potential of
73
conventional QCTO-inspired designs by reducing the mismatch problems and will
become a powerful and practically implementable alternative to the original complex
TO-enabled designs.
(a)
(b)
Figure 4.1: (a) Anti-Reflective layer with QCTO-enabled designs (in-general); (b)
Anti-Reflective layer with QCTO-based modified Luneburg lens (in particular)
74
To show the design methodology of the proposed anti-reflective (AR) layer along
with QCTO-enabled modified GRIN lenses, we first investigated the QCTO design
technique for two different types of GRIN lenses: Luneburg lens antenna and half
Maxwell fish-eye lens which have been designed with QCTO technique in many
applications [6,14,16,18-20,23-37,74]. We studied the electromagnetic performance of
both the modified QCTO-enabled GRIN structures and showed the degraded gain
pattern due to the impedance mismatch associated with QCTO technique. To counter
the reflection problems resulted from QCTO approximations, we designed a spatially
varying all-dielectric anti-reflective(AR) layer along with the modified luneburg lens
and modified half Maxwell fish-eye lens to ensure a uniform impedance matching
across the entire boundary of the structure. This new design methodology has been
shown experimentally at Ka-band (26-40 GHz) frequency range for the luneburg lens
and the results are then compared with the conventional QCTO-based design to show
the performance improvement. From the measurement and predictions, we found that
our novel anti-reflective layer enabled QCTO design shows an unprecedented device
performance improvement compared to traditional only QCTO-enabled designs in both
the lenses. We anticipate that this novel anti-reflective layer enabled QCTO design
methodology will underscore the use of conventional QCTO technique in other
electromagnetic applications [43-63,65,72,74,75] as well with an improved
performance.
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4.2 Reflections in QCTO-enabled designs
In chapter 3, we showed the detail design methodology and results of the
modified Luneburg lens antenna and Maxwell fish-eye lens. In this section, we are
reusing those results to compare the device’s performances using an anti-reflective
layer. In QCTO-enabled modified Luneburg lens (figure 4.2(a)), the material response
was electric and non-magnetic; and the new material permittivity distribution of the
modified lens had a higher permittivity value at the center of the planar surface
compared to the edges. Due to the high permittivity value and non-magnetic nature, the
impedance mismatch was highest at the center and becomes gradually lower as the feed
location moved toward the edges. These impedance mismatches caused most reflections
at the center compared to the edges resulting in a lower gain value at the center of the
modified surface compared to the edges. The gain patterns of the modified Luneburg
lens antenna as a function of beamsteering angle are shown in in figure 4.2(b).
(a)
76
(b)
Figure 4.2: (a) 3D-approximate QCTO enabled modified Luneburg lens permittivity
profile; (b) Reflection problems on lens’s radiation performances
4.3 Broadband Anti-Reflective (AR) Layer Design Methodology
Ensuring a uniform impedance match across the entire planar surface of the
QCTO-inspired designs (figure 4.2(a), figure 4.1(a)), a spatially varying broadband anti-
reflective (AR) layer with an appropriate thickness needs to be designed which
gradually tapers the impedance of the modified surface to that of free space at every
single point. As the modified surface of the lens has an inhomogeneous permittivity
profile, the anti-reflective layer will be of graded-index nature. A thorough literature
review has been conducted to design such a continuously graded profile [66-67]. Several
design approaches are applicable in different contexts and we were exploring three main
77
approaches here described for Klopfenstein transmission line impedance taper [66],
exponential impedance taper [67] and gaussian profile in the context of QCTO-enabled
modified GRIN lenses. The detail design methodology and comparative study of
different graded permittivity profiles used as an anti-reflective layer along with QCTO-
enabled modified lenses are discussed:
4.3.1 Klopfenstein Profile
Klopfenstein taper describes the variation of characteristic impedance along a
transmission line as shown in figure 4.3 [66]. The relation between the logarithmic
change in characteristic impedance along the z-direction and the length of the
transmission line can be given as [66]:
ln𝑍𝑍 (𝑧𝑧) =12
ln𝑍𝑍0𝑍𝑍𝐿𝐿 +𝛤𝛤0
cosh𝐴𝐴 𝐴𝐴2𝛷𝛷 2
𝑧𝑧𝐿𝐿− 1,𝐴𝐴 ; 𝑓𝑓𝑓𝑓𝑟𝑟 0 ≤ 𝑧𝑧 ≤ 𝐿𝐿
Figure 4.3: Klopfenstein Profile
The corresponding index profile to achieve such a tapered characteristic impedance can
be expressed as [67]:
78
εAR(x, z) = εiεs(x, z) exp Γm A2 Φ2zL− 1, A ; for 0 ≤ z ≤ L
𝛤𝛤𝑚𝑚 =𝛤𝛤0
cosh𝐴𝐴 ; 𝛤𝛤0 =
12𝑙𝑙𝑟𝑟
εs√εi
In our design, εs (x, y) is the permittivity profile of modified GRIN lens and εi (x, y) has
the permittivity value of free-space. L represents the anti-reflective layer thickness
between the luneburg lens and air interface. Γm in the above expression signifies the
maximum ripple in the passband and the function Φ (x, A) is defined as
Φ (x, A) = −Φ (−x, A) = 𝐼𝐼1𝐴𝐴1 − 𝑦𝑦2
𝐴𝐴1 − 𝑦𝑦2𝑟𝑟𝑦𝑦 𝑓𝑓𝑓𝑓𝑟𝑟 |𝑥𝑥|
𝑥𝑥
0≤ 1
where I1 is the first kind modified Bessel function of order one. In order to apply
Klopfenstein profile as a permittivity profile of the anti-reflective layer, the integral
function Φ (x, A) in the above expression needs to be evaluated first. However, Φ (x,
A) is an inexpressible in closed form except for the following special values of the
parameters [66]. The special closed-form relationships are expressed as [66]:
Φ (0, A) = 0,
Φ (x, 0) = x/2,
Φ (1, A) =cosh(𝐴𝐴) − 1
𝐴𝐴2
Therefore, the function Φ (x, A) is computed through numerical analysis for simple and
rapid computation. The incomplete Bessel function is expanded in a power series [68-
69]
79
Φ (x, A) = 12
𝐴𝐴
2
4 𝑚𝑚
(1 − 𝑦𝑦2)𝑚𝑚
𝑚𝑚! (𝑚𝑚 + 1)!
∞
𝑚𝑚=0
𝑟𝑟𝑦𝑦 𝑥𝑥
0
The convergence of the series is uniform for −1 ≤ y ≤ 1. The term by term
integration of the above series are performed as [68]:
Φ (x, A) = 𝑟𝑟𝑚𝑚𝑏𝑏𝑚𝑚
∞
𝑚𝑚=0
where 𝑟𝑟𝑚𝑚 = 𝐴𝐴2𝑚𝑚 4𝑚𝑚 𝑚𝑚!(𝑚𝑚+1)!
and 𝑏𝑏𝑚𝑚 = 12
∫ (1 − 𝑦𝑦2)𝑚𝑚𝑟𝑟𝑦𝑦 𝑥𝑥0
The recursive functions are now computable and obtained through an integration
by parts:
𝑟𝑟0 = 1; 𝑟𝑟𝑚𝑚 =𝐴𝐴2
4𝑚𝑚(𝑚𝑚 + 1) 𝑟𝑟𝑚𝑚−1
𝑏𝑏0 =𝑥𝑥2
; 𝑏𝑏𝑚𝑚 =𝑥𝑥2 (1 − 𝑥𝑥2)𝑚𝑚 +2𝑚𝑚𝑏𝑏𝑚𝑚−1
(2𝑚𝑚 + 1)
the evaluation of the recursive function Φ (x, A) can be computed using custom
Matlab code.
4.3.2 Exponential Profile
The permittivity distribution of the anti-reflective layer for exponential taper can
be described according to the following formula [67]:
80
εAR(x, z) = εi exp zL
lnεs(x, z)√εi
for 0 ≤ z ≤ L
where εs (x, y) is the two-dimensional permittivity profile of the 2D modified
luneburg lens, εi is the permittivity of incident region which is free space. L is the
thickness of the anti-reflective layer.
4.3.3 Gaussian Profile
The permittivity profile for Gaussian taper can be expressed as [67]:
εAR(x, z) = εi exp 2 zL2
lnεs(𝑥𝑥,𝑦𝑦)
√εi ; for 0 ≤ z ≤
L2
εi exp 2 1 − 1 −zL2 ln
εs(𝑥𝑥,𝑦𝑦)√εi
; for L/2 ≤ z ≤ L
where εs (x, y) is the two-dimensional permittivity profile of the 2D modified
luneburg lens, εi is the permittivity of incident region which is free space. L is the
thickness of the anti-reflective layer.
4.4 Anti-Reflective layer design with QCTO-enabled modified GRIN lens
To show the design methodology of an anti-reflective layer with QCTO-enabled
GRIN lens antennas, we chose to use Klopfenstein profile as the permittivity profile of
the anti-reflective layer and the methodology began with the QCTO-enabled 2D
permittivity profile of the modified Luneburg lens (figure 4.4(a)). To compute the
81
proposed anti-reflective layer profile, the 2D COMSOLTM model was interfaced with
Matlab Livelink and the permittivity profile of the anti-reflective layer with appropriate
thickness was calculated using custom Matlab code as shown in the flowchart in figure
4.4(b). Figure 4.5(a) shows the graphical representation of the permittivity variation
along the entire AR layer thickness where, the permittivity value in each point at the
planar boundary of the modified Luneburg lens (figure 4.4(a)) tapers to that of free
space. In this particular design, the thickness of the anti-reflective layer was chosen as
λ/2 at 26 GHz. Figure 4.5(b) shows the surface permittivity profile of the proposed anti-
reflective layer having a thickness of λ/2. The 2D AR layer permittivity profile was then
axisymetrically rotated to realize the 3D anti-reflective layer (figure 4.5(c)) and finally
was coupled with the 3D QCTO-inspired modified Luneburg lens. The 3D permittivity
distribution of the modified Luneburg lens with anti-reflective layer is shown in figure
4.6.
(a)
82
(b)
Figure 4.4: Anti-reflective design methodology: (a) 2D permittivity distribution of
QCTO design; (b) AR layer design flowchart
(a)
83
(b)
(c)
Figure 4.5: Designed Klopfenstein tapered anti-reflective layer: (a) Graphical
representation of tapered permittivity distribution along the AR layer thickness; (b)
2D surface permittivity profile; (c) Axisymetrically rotated 3D permittivity profile
84
Figure 4.6: 3D QCTO-approximate modified Luneburg lens with half-wavelength anti-
reflective layer at the bottom surface
4.5 3D Full Wave Electromagnetic Simulation
To show the electromagnetic performances of the new QCTO-AR enabled
modified Luneburg lens antenna, a 3D full-wave electromagnetic simulations were
conducted using finite element based COMSOLTM multiphysics numerical solver. To
demonstrate the electromagnetic behavior of the device, we excited the planar surface
of the Luneburg lens antenna using the simulation setup as shown in figure 4.7(a) with
a waveguide port at five different locations (figure 4.7(b)).
85
(a)
(b)
Figure 4.7: (a) 3D finite element setup of the modified Luneburg antenna modeled in
COMSOLTM, (b) illustration showing the positions of the waveguide feed sources used
for the simulations
The 3D radiation patterns of the new lens with anti-reflective layer were computed at
each excitation positions as shown in figure 4.8 and the simulated gain pattern of the
86
lens antenna as a function of beamsteering angle is shown in figure 4.9. From figure 4.8
and figure 4.9, it is clear that the QCTO-enabled modified Luneburg lens with an anti-
reflective layer showed the equivalent beam steering performance (from -55˚ to +55˚)
similar to the lens without having an anti-reflective layer (figure 3.7). But using an anti-
reflective layer along with the QCTO-enabled design improved the gain value
significantly. The new lens antenna with anti-reflective layer had an almost flat gain
pattern (figure 4.9) at all excitation positions compared to the gradual drop in gain value
without using an anti-reflective layer. However, the use of an anti-reflective layer
reduced the peak gain value of the lens antenna at the edge excitation (pos -2 in figure
4.9) and this was due to the fact that the length of the anti-reflective layer shifted the
focal point of the modified lens by the distance of the AR layer thickness which
introduced scatterings at the edges.
87
(a)
(b)
(c)
88
(d)
(c) Figure 4.8: Simulated 3D radiation patterns (dBi) of anti-reflective layer enabled QCTO
modified Luneburg lens antenna at 30 GHz for feed location at (a) pos -2 (-55˚) ; (b) pos -
1 (-22˚); (c) pos 0(0˚); (d) pos 1 (22˚); (e) pos 2 (55˚)
89
Figure 4.9: Simulated gain patterns of modified Luneburg lens antenna at 30 GHz for
feed locations at pos -2, pos -1, and pos 0 with and without an anti-reflective layer
The modified Luneburg lens with an anti-reflective layer was fabricated using
space filling curve based additive fabrication method as described in chapter 3 and
figure 4.10 shows the fabricated Luneburg lens antenna. The lens was experimentally
characterized using the method and measurement setup as discussed in chapter 3 and
compared with the simulated predictions. Figure 4.11 compares the measured and
90
simulated gain patterns between the anti-reflective layer enabled modified Luneburg
lens antenna and the modified lens antenna without an anti-reflective layer. For brevity,
the results at three excitation locations are shown here. The rotational symmetry of the
lens should allow similar performances at other excitation positons along the entire
planar surface of the lens antenna. As expected, the measured gain of the modified lens
with an AR layer increased significantly compared to the lens without an AR layer and
had an almost flat gain pattern at all the excitation positions confirming the uniform
impedance matching across the entire planar surface of the modified lens structure.
Figure 4.10: Fabricated QCTO-enabled modified Luneburg lens antenna with λ/2
thickness anti-reflective layer
91
Figure 4.11: Simulated and measured gain patterns of the modified Luneburg lens
antenna at 30 GHz for feed locations at pos -2, pos -1, pos 0 with and without the
presence of an anti-reflective layer
92
Figure 4.12: Measured return loss (S11) at pos 0 with and without the presence of an
AR layer
Figure 4.12 shows the measured return loss of QCTO-enabled modified
Luneburg lens with an anti-reflective layer and QCTO-enabled lens without an anti-
reflective layer at the center excitation. The lenses were excited with a waveguide. We
are showing the return losses for only at the center position (pos 0 in figure 4.7(b)) as
the impedance mismatch was highest at the center of the QCTO-enabled modified lens
(pos 0 in figure 4.2(b)). From the figure, it is clear that the use of an anti-reflective layer
improves the impedance mismatch problem. Figure 4.13 shows the measured and
predicted gain pattern of the modified Luneburg lens with and without an anti-reflective
layer over the entire frequency band (Ka-band) when the waveguide excitation position
93
is at the center of the lens (pos 0 in figure 4.7(b), 4.2(b)). From the results, it is evident
that the measured results comply well with the numerical predictions calculated using
COMSOLTM numerical solver. Figure 4.14 compares the measured aperture efficiency
of the QCTO-enabled modified Luneburg lens antenna with and without an anti-
reflective layer when the excitation position is located at the center. We are here
showing the aperture efficiency only for the center position. It is clear that using an anti-
reflective layer improves the lens’s aperture efficiency in an unprecedented way at the
center from less than 20% to more than 70% making the anti-reflective layer based
QCTO design methodology a powerful alternative of the original TO-based designs.
Figure 4.13: Realized gain comparison at Pos 0 (with and without AR Layer)
94
Figure 4.14: Aperture efficiency increase using an anti-reflective layer at the center
excitation location of the modified lens antenna (pos 0)
In the modified Luneburg lens, anti-reflective layer was used on own side;
however, this technique can be used on multiple side as well. In the Maxwell half fish-
eye lens, the impedance mismatch problems arise on two sides: 1) at the bottom planar
excitation side; 2) at the top radiating side. To mitigate the impedance mismatch
problem, anti-reflective layers were designed on both sides of the fish-eye lens. The
gradually tapered anti-reflective layer profile was optimized using the Gaussian
distribution from the 2D QCTO-inspired fish-eye lens. Figure 4.14 shows the 3D
95
representation of permittivity profile of the modified half Maxwell fish-eye lens along
with anti-reflective layer at the top and bottom surface of the modified lens. In this
design, the thickness of the anti-reflective layer was chosen as λ/2 at the bottom surface
and at the top surface, the AR layer thickness was chosen as λ. Due to the limited
maximum permittivity dielectric material availability, the fish-eye lens was not
fabricated and here the device’s beamsteering performance was numerically calculated
using COMSOLTM solver. Figure 4.15 shows the simulated 3D radiation patterns of the
AR layer enabled modified fish-eye lens with a waveguide port excitation at five
different feed locations.
Figure 4.14 : Modified half Maxwell fish-eye Lens with anti-reflective layers: λ
thickness AR layer at top surface and λ/2 AR layer at the bottom surface
96
(a)
(b)
97
(c)
(d)
98
(e)
Figure 4.15: Simulated 3D radiation patterns of QCTO-enabled modified fish-eye
lens at 30 GHz for source locations: (a) pos -2; (b) pos -1; (c) pos 0; (d) pos 1; (e) pos
2 with full wavelength anti-reflective layer at the top surface and half wavelength
anti-reflective layer at the bottom surface of the fish-eye lens
99
Figure 4.16: Modified fish-eye lens’s simulated gain patterns as a function of azimuth
angle at 30 GHz with and without the presence of AR layers
4.6 Anti-Reflective Layer Thickness and Device Performance
Even though an anti-reflective layer can mitigate the impedance mismatch
problem significantly at most of the excitation positions along the entire planar surface
of the QCTO-enabled devices; however, a higher thickness anti-reflective layer does
reduce the device’s ability to beam steer at wide angles when the feed position is moved
at the extreme edges of the planar surface. This result from the fact that as the feed is
moved from the center location to the extreme edges, the presence of the anti-reflective
100
layer shifts the focal point of the structure farther from the surface of the structure
without an anti-reflective layer reducing the overall antenna gain at the extreme edges
compared to the other feed locations. The effect of the anti-reflective layer on device
antenna gain is explained in figure 4.17 in the context of modified Luneburg lens with
an anti-reflective layer:
1) the QCTO-enabled luneburg lens without an anti-reflective layer usually
works well at the extreme edges where the permittivity profile of modified lens is nearer
to that of free space and the impedance mismatch is minimal at the edge. However,
using an anti-reflective layer along with the modified surface forces the incoming wave
focal point to shift vertically by a distance of the anti-reflective layer thickness (figure
4.17(c)). This additional thickness creates a phase delay for the receiving signal and due
to this phase delays, scatterings are present at the edge resulting in lower peak gain and
higher side lobes at the edge compared to the other feed locations along the planar
surface. Figure 4.17 describes the phenomenon of the anti-reflective layer thickness
effect on the incoming electromagnetic wave at three excitation locations.
101
(a)
(b)
102
(c)
Figure 4.17: Anti-reflective layer effects on device’s performance: (a) focal point at
normal incidence, (b) focal point at 35˚ incidence, (c) focal point at -55˚ incidence
and reflections due to phase aberration
2) Several parametric studies were performed on the beamsteering performance
of the modified Luneburg lens with different thickness anti-reflective layers to
determine the optimal AR thickness which compensates the impedance mismatch at all
the feed locations along the planar surface. From these parametric studies, it was
observed that the higher the upper limit of the modified permittivity value the greater
the anti-reflective layer thickness needs to be designed to achieve a uniform impedance
matching and higher gain value. However, from the full-wave simulation it was
observed that if the length of the anti-reflective layer is greater than λ/2 (at lowest
frequency), then the anti-reflective layer will degrade the lens’ beam steering
103
performance. This is due to the fact that similar to the GRIN Luneburg lens the anti-
reflective layer also has a graded permittivity profile which inherently makes the anti-
reflective layer act like a GRIN lens structure. As a result, the incident wave within the
anti-reflective layer do not radiate along the desired direction and for higher thickness
layer, this deviation is significant. This thickness effect of the anti-reflective layer on
the lens’s beamsteering performance can be seen from a QCTO-enabled higher
permittivity modified example lens’s gain pattern as a function of beamsteering angle.
Figure 4.18 provides an example of a two-dimensional permittivity distribution with
three different thickness anti-reflective layer: (a) half lambda; (b) full lambda; (c) one
and half lambda. These lenses were excited with a waveguide port. Figure 4.18 (d) (e)
shows the gain pattern and beamsteering angle of the example lens at 30 GHz for three
different thickness AR layer at the center position (Pos 0) and edge (Pos -2). From the
figure 4.18(d), it is clear that the gain value of the modified lens increases with higher
AR layer thickness as the impedance matches more uniformly over the entire AR layer.
However, with an increasing AR layer thickness, the lens’s beam steering performance
starts reducing with degraded gain value as shown in figure 4.18 (e). This is due to the
fact that with increasing thickness the anti-reflective layer starts working like a perfect
GRIN lens and degrades the lens’s beamsteering angle. Also due to the higher shifts in
focal point (figure 4.17 (c), the gain value starts decreasing at the edge compared to
other feed locations.
104
(a)
(b)
105
(c)
(d)
106
(e)
Figure 4.18: Higher thickness anti-reflective layer effect on lens’s beamsteering
performance and gain pattern. Example modified Luneburg lens with an anti-
reflective layer thickness of (a) Half lambda, (b) full lambda, and (c) 1.5 * lambda;
(d) Gain value increase with higher thickness anti-reflective layer as the impedance
mismatch mitigates with the increasing thickness;(e) Beam steering angle reduction
and lower gain value with gradual increase in anti-reflective layer thickness
4.7 Choice of graded profiles as anti-reflective (AR) layer parameter
To minimize the impedance mismatch in quasi-conformal transformation optics
(QCTO) inspired devices requires the modified permittivity profile to be continuously
107
tapered to that of free space. To achieve this purpose, several graded-index profiles can
be adopted as an anti-reflective layer. In section 4.3, three different types of graded
dielectric permittivity profile (Klopfenstein, Exponential and Gaussian) were discussed
and in this section, a comparative study of using these three profiles as an anti-reflective
layer is presented in the context of QCTO-enabled modified Luneburg lens antenna.
Figure 4.19 shows the comparison of simulated gain pattern of the modified Luneburg
lens with a half wavelength anti-reflective layer (figure 4.5) using Klopfenstein,
exponential, and gaussian permittivity profile as an anti-reflective layer. The combined
lens-AR structure with these three dielectric profiles was excited with a waveguide port
at five different feed locations at the planar excitation side (figure 4.19) and the realized
far field gain as a function of azimuth angle was calculated at each location for these
three different permittivity profile. From the figure, it is evident that all the three
gradient-index profiles showed similar electromagnetic performance at each excitation
position. Figure 4.20 demonstrates the gain predictions for the Klopfenstein,
exponential, and gaussian permittivity profile over the entire Ka-band frequency range
at the center excitation and edge excitation. From the simulated results, it was observed
that Klopfenstein permittivity profile provided a slightly higher gain value compared to
that of exponential and gaussian profiles when the lens structure was excited at the edge
(figure 4.20(a)), and the exponential permittivity profile provided maximum gain value
when the excitation was at the center (figure 4.19(b)).
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Figure 4.19: Simulated realized far field gain patterns as a function of azimuth angle
at five feed locations at 30 GHz for AR layer with Klopfenstein, Exponential and
Gaussian permittivity profile
109
(a)
(b)
110
Figure 4.20: Realized far field gain pattern as a function of frequency for
Klopfenstein, exponential and gaussian permittivity profile AR layer; (a) at Edge
excitation (pos -2), (b) at center excitation (pos 0)
From the above numerical predictions, it is evident that all the three continuously
tapered permittivity profile can be used as a material parameter of the anti-reflective
layer to mitigate the impedance mismatch problems arise in QCTO-enabled devices.
4.8 QCTO-inspired Generalized Vs Classical Luneburg lens
As higher permittivity value deteriorates the beamsteering performance of
modified luneburg lens, a generalized luneburg lens might be helpful to keep the
dielectric permittivity value and the anti-reflective layer thickness lower. We
implemented the same thickness anti-reflective layer with the QCTO inspired modified
classical luneburg lens as well as generalized luneburg lens, and compared the
numerical results. From the previous discussions, it is clear that anti-reflective layer
needs to be of higher thickness with increased permittivity profile to compensate the
reflections. However, the higher thickness layer degraded the device performance from
beamsteering perspective. To achieve a lower permittivity profile, we used a generalized
luneburg lens whose permittivity profile is lower than classical luneburg lens with a
focal point away from the surface. In this section, we are conducting a comparative
study of modified classical luneburg lens and generalized luneburg lens with a same
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length anti-reflective layer and study the performance evaluation of the both lenses at
different excitation positons.
In the classical Luneburg lens, the focal point lies at the surface of the spherical
Luneburg lens for plane waves coming from the opposite side of the lens (figure 4.21(a))
[71]. However, in the generalized Luneburg lens, the focal point lies away from the
spherical surface by an arbitrary length for plane waves coming from opposite directions
[70]. The normalized permittivity profile of a generalized Luneburg lens is given by
[70]
𝜀𝜀𝑟𝑟 = 𝑒𝑒2𝜔𝜔 (𝜌𝜌 ,𝑠𝑠)
where 𝜔𝜔 (𝜌𝜌 , 𝑠𝑠) = 1𝑝𝑝𝑖𝑖 ∫
arcsin (𝑥𝑥𝑠𝑠)
(𝑥𝑥2− 𝜌𝜌2)1 21𝜌𝜌 𝑟𝑟𝑥𝑥
𝜌𝜌 = 𝑟𝑟𝑟𝑟
An analytical approximation for the above integral can be derived for discrete values of
s. For classical Luneburg lens (s=1); the focal distance lies at the surface of the Luneburg
lens and the permittivity distribution are expressed as [71]:
𝜀𝜀𝑟𝑟 = 2 − 𝑟𝑟𝑅𝑅2
For s > 1; the focal length lies away from the surface. With arbitrary value s=1.2; the
focal distance becomes 6 mm (figure 4.21 (a)) and we derived a general expression for
this Luneburg lens expressed by
𝜀𝜀𝑟𝑟 = 1.74 − 0.74 𝑟𝑟𝑅𝑅2
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Figure 4.21(b) shows the two-dimensional graphical permittivity distribution of
Classical Luneburg lens (s=1) and generalized Luneburg lens (s=1.2). Both the lenses
were modified using QCTO technique and implemented with an anti-reflective layer
having a thickness of 6mm. Figure 4.22(c) shows the 3D permittivity distribution of the
modified classical Luneburg lens and modified generalized Luneburg lens along with
the anti-reflective layer. The lenses were excited with an open-ended waveguide at three
excitation positions (pos -2, pos 0, pos 2 in figure 4.7 (b)) along the planar excitation
surface. Figure 4.21(c) shows the comparison of far-field radiation pattern for these
three excitation positions. From the numerical predictions, it is clear that both the QCTO
enabled modified classical Luneburg lens and generalized Luneburg lens with an equal
thickness anti-reflective layer showed similar performances.
(a)
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(b)
(c)
0 0.2 0.4 0.6 0.8 1 (r/R)
2
1
1.2
1.4
1.6
1.8
2
Perm
ittiv
ity D
istri
butio
n
Classical Luneburg Lens(s = 1) Generalized Luneburg Lens (s = 1.2)
114
(d)
Figure 4.21: Classical vs Generalized Luneburg lens with half-wavelength anti-
reflective layer: (a) focal length representation; (b) graphical representation of
permittivity distribution; (c) 3D permittivity profile of modified Classical Luneburg
lens and Generalized Luneburg lens with 6mm anti-reflective layer ; (d) Beamsteering
performances at three excitation positions (Pos -2, Pos 0, Pos 2). [solid lines represent
generalized Luneburg lens’s performance and dashed lines represent classical
Luneburg lens’s performance]
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4.9 Conclusion
In this chapter, the impedance mismatch problems arise in quasi-conformal
transformation optics (QCTO) approximation have been addressed by using a novel
broadband anti-reflective layer along with the QCTO-enabled modified electromagnetic
structure. The proposed anti-reflective layer has a graded dielectric profile which tapers
the permittivity value of the modified surface to that of free space. Any profile which
has a continuously tapered profile along the z-axis can be used as a material parameter
of the anti-reflective layer. We discussed three different types of profiles in this study
and showed the detail design methodology of the anti-reflective layer with QCTO-
enabled designs in the context of gradient-index luneburg lens and Maxwell fish-eye
lens antenna. A comparative study of different index profile and the effects of the anti-
reflective layer thickness on device performance has been shown in detail.
The design methodology has been validated via experiments at Ka-band (26-40
GHz) in the context of QCTO-enabled modified luneburg lens antenna. The measured
results comply well with the numerical predictions validating our design methodology.
We believe, this anti-reflective layer based design methodology can be extended to all
other applications which involve QCTO technique. An anti-reflective (AR) layer based
QCTO-enabled design will be of practical importance for making novel electromagnetic
structures where structural modifications of the device with good performances are of
particular interest.
Chapter 4 contributed to the following paper: Soumitra Biswas, Mark
Mirotznik. Broadband Impedance Matching Strategies of QCTO Enabled Designs. The
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dissertation author was the lead contributor of this idea and primary author of this
material.
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ULTA-WIDE BEAMSCANNING ANGLE LUNEBURG LENS ANTENNA DESIGN USING HIGH DIELECTRIC MATERIAL
In this chapter, I explore high permittivity dielectric material fabrication using
additive manufacturing technique to realize customized shaped beamforming lens
antennas. I designed a broadband 180˚ beamscanning angle modified luneburg lens
antenna for practical purposes using transformation optics. The design required high
permittivity dielectric material and the material was developed and characterized to
implement the high dielectric lens antennas.
5.1 Introduction
In automotive sensors and target tracking, an unlimited field-of-view is of high
demand for autonomous vehicle platforms and radar applications. Currently, this goal
is achieved by using smart electronics sensor systems to accurately estimate the
direction of arrival (DOA) and angular accuracy [84]. However, these systems are
highly limited in bandwidth and field-of-view. A modified Luneburg lens in conjunction
with the smart antennas, sensors and other electronics elements (figure 5.1) offers a high
field-of-view capability with high gain over multiple spectral bands and becomes a
promising choice as a low-cost solution for this purpose [85]. In chapter 3 and chapter
4, I discussed the methodology that enabled the design of a modified Luneburg lens
antenna to achieve a high field-of-view beamscanning angle using transformation optics
design scheme and anti-reflective layer. However, a wide beamscanning angle modified
Chapter 5
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Luneburg lens antenna requires a high dielectric permittivity value which is an
implementation challenge. Despite the huge promise of lens based automotive sensor
systems, a lot of fabrication challenges need to be addressed before the full use of lens
antenna system in radar applications and autonomous platform. The first and foremost
is the ability to fabricate high permittivity dielectric material. In chapter 3, we discussed
about the FDM based additive manufacturing technique to realize graded dielectrics.
However, we were limited by highest permittivity value of 2.9. In this chapter, we are
exploring to fabricate a higher permittivity dielectric lens antenna system.
Figure 5.1: 180˚ beamscanning angle beamforming lens antenna with smart
electronics feed networks and beamswitching networks
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5.2 High permittivity wide beamscanning angle lens antenna design
To design a modified Luneburg lens with wide field-of-view, I used quasi-
conformal transformation optics to modify the spherical lens into a planar one. Figure
5.2(a) shows the three-dimensional permittivity distribution of the modified Luneburg
lens antenna designed with QCTO technique. The theory and detail design methodology
of QCTO technique is described in chapter 2 and 3. As described in chapter 3, the lens
designed with QCTO technique suffers from reflection problems at its excitation
boundary. To counter the reflection problems and achieve a uniform impedance
matching across the entire planar surface, I implemented a broadband half-wavelength
anti-reflective layer based on Klopfenstein impedance taper. In chapter 4, I described
the detail design methodology of anti-reflective layer and readers are referred to chapter
4. Figure 5.2 (b) shows the three-dimensional permittivity distribution of the modified
Luneburg lens with half wavelength (at 26 GHz) anti-reflective layer. All the designs
were implemented using commercially available finite element based numerical solver
COMSOLTM Multiphysics simulation package. The two dimensional mapping of the
design was carried out using following boundary conditions:
𝐴𝐴′𝐵𝐵′|𝑥𝑥′ = 𝐹𝐹′𝐺𝐺′|𝑥𝑥′ = 𝑥𝑥 ; 𝑟𝑟 ∙ ∇𝑥𝑥|𝐴𝐴′𝐺𝐺′,𝐵𝐵′𝐶𝐶′,𝐶𝐶′𝐷𝐷′𝐸𝐸′,𝐸𝐸′𝐹𝐹′ = 0
𝐴𝐴′𝐺𝐺′|𝑦𝑦′ = 𝐵𝐵′𝐶𝐶′|𝑦𝑦′ = 𝐸𝐸′𝐹𝐹′|𝑦𝑦′ = 𝑦𝑦 ; 𝐶𝐶′𝐷𝐷′𝐸𝐸′|𝑥𝑥′ = −𝑅𝑅2 − 𝑥𝑥2;
𝑟𝑟 ∙ ∇𝑦𝑦|𝐴𝐴′𝐵𝐵′,𝐹𝐹′𝐺𝐺′ = 0
And the modified permittivity profile was calculated as
𝜀𝜀′ =𝜀𝜀𝑟𝑟
|𝜦𝜦−1|
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(a)
(b)
121
(c) Figure 5.2: (a) 2D representation of high dielectrics modified Luneburg lens (b) 3D
representation of QCTO-enabled modified Luneburg lens’s permittivity distribution;
(c) 3D permittivity distribution of QCTO-enabled modified Luneburg lens with
broadband anti-reflective layer
5.3 3D Full-Wave Electromagnetic Simulation
To verify the beamscanning angle of the modified lens antenna, 3D full wave
electromagnetic simulations were performed using COMSOLTM numerical solver. The
high permittivity graded dielectric lens has the spatially varying permittivity distribution
as shown in figure 5.2(b) and figure 5.3(a) shows the finite element meshing. To show
the beamsteering performance and excite the lens, an open-ended waveguide was used
along the centerline of the planar boundary of the modified lens antenna. The position
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of the port along the center line results in beam steering. For brevity, I am presenting
only five source locations as shown in figure 5.3(b). At each location full wave
electromagnetic simulation was conducted and the antenna radiation pattern was
calculated.
(a)
(b)
Figure 5.3: (a) finite element mesh of the modified Luneburg antenna modeled in
COMSOLTM numerical solver, (b) illustration showing the five positions of the
waveguide source feed using for the simulations
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Figure 5.4 shows the 3D radiation patterns of modified lens antenna computed
with COMSOLTM numerical solver at all the five source positions at 30 GHz. As
expected the antenna’s main beam steered over a fairly large range of angles (i.e. -85o
to 85o) as the source position was changed along the centerline. Here, the results for the
five feed locations as shown in figure 5.3 (b) is shown for the azimuthal plane. The
rotational symmetry of the lens structure makes the radiation patterns in the elevation
plane also same. Figure 5.5 shows the simulated gain patterns of the modified lens
antenna at 30 GHz frequency as a function of azimuth angle. From figure 5.5, it is
evident that, the lens has a beamsteering capability of -85o to 85o.
(a)
124
(b)
(c)
125
(d)
(e)
Figure 5.4: Simulated 3D radiation patterns (dBi) at 30 GHz for source location at
(a) pos -2; (b) pos -1; (c) pos 0; (d) pos 1; (e) pos 2
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Figure 5.4: Designed Luneburg lens’s beamscanning performance at 30 GHz
5.4 Multi-section anti-reflective layer
In the above design, the half-wavelength anti-reflective layer improves the impedance
mismatch with a wide beamscanning angle, however, there are still more rooms to
improve the gain value for higher resolution and more data rate. To increase the gain
value, the anti-reflective layer needs to be of higher thickness to more uniformly match
the impedance at all locations. However, using a thicker anti-reflective layer does reduce
the lens’s beamsteering angle and also increases the sidelobes. The details of the anti-
reflective layer thickness effects were explained in chapter 4. To achieve the wide
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beamscanning angle with increased gain value at most of the excitation positions, we
explored using a multi-section broadband anti-reflective layer as shown in figure 5.5. In
this design, a full-wavelength anti-reflective layer was used at most of portions of the
planar surface while a half-wavelength anti-reflective layer was used at the edges.
Figure 5.5 (a) shows the three-dimensional permittivity distribution of the modified lens
structure. To excite the lens structure and show the beamsteering performance and gain
pattern, the lens structure was excited with a waveguide port and the location of the
waveguide positions is shown in figure 5.5 (b). At each location, full-wave
electromagnetic simulations were performed and 3D radiation patterns were calculated.
Figure 5.6 shows the simulated 3D radiation patterns of the lens antenna at 30 GHz
frequency. Lens designed with multi-section anti-reflective layer shows a higher gain
pattern compared to gain pattern shown in figure 5.4.
(a)
128
(b)
Figure 5.5: Modified Luneburg lens with multi-section broadband anti-reflective
layer: (a) 3D permittivity distribution of the lens; (b) excitation position of the lens
structure
(a)
129
(b)
(c)
130
(d)
(e)
131
(f)
(g)
Figure 5.6: Simulated 3D radiation pattern of multi-section anti-reflective layer
enabled Luneburg lens antenna at: (a) Pos -3, (b) Pos -2, (c) Pos -1, (d) Pos 0, (e) Pos
1, (f) Pos 2, (g) Pos 3
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5.5 Conclusion
In this chapter, I designed a practically implementable high dielectric
permittivity modified Luneburg lens antenna with anti-reflective layer. The design
requires a higher value permittivity profile of 3.97 which is currently under fabrication
process using space-filling curve geometry. The modified lens antenna has a wide field-
of view (FOV) from -85˚ to +85˚. The designed lens antenna will be immensely useful
for target tracking in radar applications and autonomous platform. I am currently
exploring the use additive manufacturing technique to realize this design. Future
attempts will focus on implementing the lens antenna excited with smart electronics
which will be integrated within load-bearing platform. The smart electronics can be
integrated in electrically small PCB board or electrically large load-bearing platforms.
In the next chapter, we will explore the use of additive manufacturing to embed smart
electronics within load-carrying structure as part of the developing smart antenna fed
beamforming lens antennas on load-carrying platform such as aircrafts, ships and other
vehicles.
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ADDITIVELY MANUFACTURED CONFORMAL LOAD BEARING ANTENNA STRUCTURE (CLAS)
In this chapter, I explored the use of additive manufacturing technique to fabricate
curved surface conformal load-bearing antenna structure (CLAS). I applied additive
manufacturing technique to integrate the antenna functionalities within multifunctional
composite load-bearing structure. The objective was to demonstrate the ability and
scalability of additive manufacturing technique to embed electromagnetic
functionalities within the structural composites in a cost-effective and scalable way.
Chapter 6
134
6.1 Introduction
Modified Luneburg lens antenna excited with smart electronics and conformal
antennas is of particular interest in many defense and commercial platforms such as
aircrafts, naval ships, autonomous vehicle etc. for communication and navigation
purposes and to minimize the profile effects such as aerodynamic drag and weight.
Many of these platforms are increasingly moving towards multifunctional composite
structures antennas are integrated as part of the load-bearing structure [85-87]. This
integrated multi-functional composites reduces the aerodynamic drag and weight by
integrating the structural and electromagnetic functions together in a single load-
carrying structure. Conformal load-bearing antenna structure (CLAS), where smart
antennas and RF components are embedded as part of the load-carrying structure,
becomes a promising frontier in the development of integrated platform and these
incorporated smart electronics can be used as an excitation source for modified
Luneburg lens antenna for wide field-of-view communication and radar target tracking.
Conventional load-bearing antennas, where antennas and other feed networks
were protruded from the large structure, were very expensive and challenging to
fabricate, and often necessitated structural modifications for subsequent antenna
integrations on the load-bearing platform [87-88]. The emergence of CLAS antenna has
therefore become an attractive choice for its minimal aerodynamic drag and weight, and
increased electromagnetic performances. However, achieving this integrated smart
antennas and electronics system on load-carrying platform will require a perfect
135
combination of conventional materials and fabrication methods, such as multifunctional
composite, new materials and fabrication methodologies to integrate smart antennas as
part of load-carrying structure.
A specific problem in multifunctional structures often remain unaddressed is
the scalability of the manufacturing technique for the complete EM and composite
structure [86]. Traditional subtractive manufacturing of radiating elements using copper
films or Kapton sheets on large structure might take significant time, and requires the
use of different chemical etchants to realize the final antenna dimensions. Also
embedding the copper films on doubly curved surfaces is a challenge and can cause
draping issues. Weaving of carbon fibers can be incorporated quickly through fabric
looms, however the conductivity of carbon fibers is several order of magnitudes lower
than that of bulk copper. On the other hand, deposition of copper inks through additive
manufacturing technique presents many advantages over traditional subtractive
methods and the direct patterning of radiating elements on structural composite fabrics
remains largely unexplored.
In this dissertation, I explored the use of additive manufacturing to fabricate
conformal antennas as part of multifunctional load-carrying structure in a cost-effective
and scalable way. The embedded antennas will be used as a feed source for the
beamforming lens antennas described in chapter 3 ,4, 5.
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6.2 Material selection and mechanical processing
CLAS antennas are embedded on the load-bearing platform and the panel needs
to be manufactured from high stiffness materials. Multifunctional composite materials
are increasingly used for defense and commercial applications due to its high structural
performance as well as good electromagnetic functions. The materials not only satisfy
electromagnetic constraints (i.e. low material loss and wide range of dielectric
constants), but also possess attractive physical and mechanical properties which makes
it suitable for load-bearing platforms. Many of the reported CLAS demonstrators and
laboratory test specimens have taken the form of honeycomb stiffened sandwich panel
and current composite aircraft interior components often use glass fiber sandwich
construction [90]. We have chosen S-glass/Cyanate Ester fiber reinforced composite
prepreg (8 Harness Satin 6781, TenCate Inc.) as the base material for our design. The
composite prepreg is a lightweight fabric woven from S-glass fibers impregnated with
a thermoset cyanate ester resin. This prepreg is chosen for its reasonably low dielectric
constant (ε =4.2), low loss tangent (<0.01), and excellent structural and mechanical
properties [91]. A typical design approach of forming curved surface CLAS specimen
from unprocessed composite fabrics is shown in figure 6.1. The process starts with the
slicing and aligning of prepreg fabrics. The prepreg layers were then stacked together
into a composite laminate and went under autoclave processing to cure the prepreg
fabrics. To form the designed curvature of the structure, the laminates were cured with
the aid of a metallic support of prescribed curvature. The laminate was cured in
Autoclave under controlled temperatures and pressures as shown in figure subjects the
composite layup to controlled temperatures and pressures. After the mechanical
processing, the final product was a curved surface load-carrying specimen.
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Figure 6.1: Curved surface CLAS structure mechanical process
6.3 CLAS antenna design
To demonstrate the additive manufacturing based CLAS antenna fabrication
techniques, a dipole antenna operating at 6 GHz was designed using HFSS simulation
software following the approach discussed in [92]. Figure 6.2 shows the example CLAS
antenna design. To show the ability of additive manufacturing technique to embed
antennas on wide curvature and large surface areas, the inner radius of curvature of the
structure was chosen as 80 mm and the structure had a thickness of 1 mm.
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Figure 6.2: CLAS antenna structure
Figure 6.3 shows the dipole antenna configuration. The antenna consisted of 2 dipole
arms with each arm having 2 patches on each side. The design used a corporate feed
networks to match the impedance to a 50Ω microstrip line (ML) connected to a
subminiature version A (SMA) connector. The design used 3 impedance matching
transformer to keep the line dimensions with realizable resolution. The antenna had a
ground plane at the bottom of the curved surface structure.
139
Figure 6.3: Antenna configuration
All the design parameters were optimized to achieve the reflection coefficient (S11) less
than -10 dB with an antenna boresight gain of 8 dBi at 6 GHz resonant frequency. The
antenna dimensions are summarized as:
L 15.49 mm W 11.89 mm
D 23.4 mm λg 24.4 mm
T1 0.82 mm T2 3.741 mm
Wf 1.97mm T3 2.811 mm
6.4 Additive Manufacturing for CLAS antenna fabrication
To fabricate the designed antennas additively on the singly-curved surface, we
utilized a multi-material additive manufacturing system, nScryptTM 3Dn-300, as shown
140
in figure 6.4. The system used here is a quad deposition system with multiple print heads
capable of depositing custom and commercial inks and pastes through micro dispensing
or extruding polymers through FDM. The nScrypt is also outfitted with a fiducial
alignment camera, a 3D laser scanning system, and a 300 mm x 300 mm heated print
bed. The nScrypt system has the ability to print line elements as narrow as 25-500 µm
with a precise positional accuracy less than 1 µm.
Figure 6.4: Additive manufacturing system (nScrypt 3Dn-300)
For the curved surface CLAS antenna fabrication, we required only the micro
dispensing print head. Commercially available concentrated copper nanoparticle was
loaded into the micro dispensing nozzle and mounted onto a three-axis position with
141
nanoscale resolution. Pritor to printing the commercial copper inks, we measured the
electrical conductivity of the inks and we found that the elctrical conductivity of the
copper inks was only one-tenth of that of the bulk material.
The antenna integration at the top of the curved surface was conducted in two
ways: 1) after processing the composite; 2) before the cure of structural composite
fabrics.
6.4.1 After curing the structural composite
In the first method, the conductive inks were directly deposited on top of the
curved surface via microdispensing. Priting antenna features on the mechanically robust
curved surface was a fabrication challenge as the surface normal kept on changing
continuously and the deposited inks must wet the surface to facilitate the antenna
patterning [89]. In some instances, the prining head needed to bend upto angle 45˚ to
facilitate the antenna patterning on curved surface. Figure 6.5 shows the
microdispensing printing of the antenna on curved surface CLAS structure after
processing the composite laminate. The ground plane can also be printed in similar way.
However, to minimize the fabrication expense, we chose to use copper film as the
ground plane in this design.
142
Figure 6.5: 3D printing of CLAS antenna elements on curved surface
The printed antenna elements were extended to a excitation source via a SubMiniature
version A (SMA) connector for the electrical connection. The approximate printing time
depends upon the design size and printing speed, and for this design, it took about 5
minutes to complete printing. After printing the radiating elements on the exterior of the
curved surface and annealing the deposited inks in low temerature (350˚F) for an hour,
the final product is an example CLAS antenna prototype as shown in figure 6.6.
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Figure 6.6: Fabricated Antenna example
6.4.2 Before curing the structural composite
In thie first method, the antenna integration was successful, however, for more
complex shaped structures such as doubly curved surface, it is difficult to bend print
head beyond some extent. Also, printng antennas in the first approach does not
guarantee the environment protection of the printed antennas. To eliminate these
problems, in the second method, we deposited the antenna elements directly on uncured
prepreg composite fabric before the mechanical process. Figure 6.7 shows the patterning
of antenna elements on unprocessed composites. The printed prepreg layer along with
the subsequent composite layers were aligned and stacked into a laminate, cured in
autoclave following the approach discussed in figure 6.1. To form the desired curvature,
the laminate was cured using a mecahnical support. In this way, the patterned antennas
can even be embedded inside the composite laminate for external protection. Figure
6.8shows the prototype antenna fabricated before curing the composite.
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Figure 6.7: Antenna patterning on Uncured prepreg
Figure 6.8: Fabricated example antenna
6.5 Results
The fabricated CLAS antennas as shown in figure 6.6 and figure 6.8 were
experimentally characterized using a PNA Network Analyzer (E8364B, Agilent) and
145
anechoic chamber. Figure 6.9 presents the measured and simulated return loss (S11) of
the antenna fabricated after curing the composites (figure 6.6). It is clear that the
measured return loss complies well with the simulated one. The return loss is less than
-15 dB at 6 GHz.
Figure 6.9: Comparison of measured and simulated impedance matching for the
fabricated antenna after curing the composites
Figure 6.10 shows the return loss (S11) comparison of the antenna fabricated
before curing the composites. In this case, the resonant frequency of the fabricated
antenna shows a deviation of about 110 MHz from the resonant frequency and this is
due to the fact that during the fabrication process the dipole patch arrays became
misaligned resulting in a non-resonant nature of the dipole patch arrays.
146
Figure 6.10: Comparison of measured and simulated impedance matching for the
fabricated antenna before curing the composites
Due to the non-resonant nature of the CLAS antenna fabricated following the second
method, we only characterized the radiation pattern of the antenna fabricated following
the first method (antenna shown in figure 6.6) using a full anechoic chamber to compare
the radiation performance. Figure 6.11 shows the measured and simulated realized gain
pattern at 6GHz. From the figure, it is evident that the measured gain pattern matches
well with the simulated gain at boresight with a peak gain of about 8.2 dBi confirming
the constructive interference happening at far field.
147
Figure 6.11: Measured and simulated E-plane radiation pattern of the fabricated
antenna after curing the composites
6.6 Conclusion
In this chapter, I showed the development and fabrication technique of additive
manufacturing based conformal load-bearing antenna structure (CLAS) in a cost-
effective and scalable way. The antenna fabrication was explored in two different
approaches to embed the smart antennas within the load-carrying structure. The
fabricated antennas were measured, and the measured results comply well the simulated
predictions boosting the prospect of additive manufacturing to realize arbitrary shaped
CLAS antenna designs. Smart communication antennas or other electronics embedded
within the load-carrying structure can be beam steered using modified GRIN lens
148
antennas. Future work will investigate the integration of beamforming lens antennas
with the smart electronics embedded in load-carrying structure.
149
CONCLUSION
The main focus of this dissertation was to design, develop, and optimize three-
dimensional modified GRIN lens structure such as Luneburg lens antenna and Maxwell
fish-eye lens in a novel way, and explore the use of additive manufacturing techniques
to fabricate electromagnetic structure such as graded-index (GRIN) beamforming lens
antennas and load-bearing conformal antennas (CLAS). The design approach employed
the concept of transformation optics to optimize the material parameters of
geometrically modified Luneburg lens and Maxwell fish-eye lens antenna to achieve an
unchanged beam steering angle of original electromagnetic structure. The GRIN lenses
were designed with quasi-conformal transformation optics (QCTO) technique to
eliminate the anisotropy and magnetic response. However, electromagnetic structures
designed with QCTO technique usually suffer from reflection problems at the excitation
boundary due to the absence of device’s magnetic response. One of the novel aspects of
this thesis was that I addressed the fundamental design problems associated with QCTO
technique and invented a novel anti-reflective layer to counter the mismatch problems
introduced in QCTO optimization scheme. The nature of the proposed anti-reflective
layer’s permittivity profile and the detail design methodology were investigated and
Chapter 7
150
presented. I explored three-different types of profile as a material parameter in the
context of modified GRIN lens structure. However, I believe, any continuously graded
mathematical profile can be used as a potential permittivity profile of the anti-reflective
layer to minimize the impedance mismatch in QCTO-enabled designs.
The proposed anti-reflective layer along with the QCTO-enabled designs can
compensate the impedance mismatch problems significantly at all the excitation
position along the planar excitation surface, however, it was observed that a higher
thickness anti-reflective (AR) layer does actually degrade the device performance at the
extreme edges of the excitation surface. This is due to the fact that a higher thickness
(>λ/2) anti-reflective layer reduces the lens’s beam steering performance as similar to
the QCTO-enabled lens antennas, the anti-reflective layer also has a GRIN profile which
inherently makes the AR layer sort of GRIN structure. Also, the use of an anti-reflective
layer shifts the focal point of the modified lens by a length of AR layer thickness which
results in a lower gain value and higher side lobes at the edge excitation positions along
the entire planar boundary. However, the presence of anti-reflective layer improves the
device performance significantly at most of the excitation positions along the planar
surface. Using the QCTO technique and broadband AR layer, we designed a practically
implementable ultra-wide angle (-85˚ to +85˚) beamscanning modified Luneburg lens
antenna to effectively communicate and gather information from all-directions and this
design is particularly feasible for applications in 5G communications and autonomous
vehicle platform.
151
The modified GRIN Luneburg lens antennas designed with the QCTO technique
and broadband anti-reflective layer were implemented using fused deposition modeling
(FDM) based additive manufacturing (AM) technique. Space-filling curve geometry
base FDM method was utilized to realize the graded dielectric structures. We also
employed the additive manufacturing technique to implement conformal load-bearing
antenna structure (CLAS) in a cost-effective and scalable way. The CLAS antennas
were fabricated in two different ways to embed communication antennas and other
smart electronics on a curved surface load-carrying structure. CLAS antennas fabricated
using AM method ensures better conformability and integrability of the electromagnetic
functions on structural platforms.
7.1 FUTURE WORK
The wide angle (± 85˚) modified Luneburg lens antenna discussed in chapter 5
requires high permittivity (εr=4) dielectric material to implement and presents
fabrication challenges. Current materials are able to realize highest permittivity limit of
2.9. Future work will investigate the implementation of high-permittivity ultra-wide
beamscanning angle and high gain modified Luneburg lens antenna using additive
manufacturing technique. The candidate materials for achieving a higher permittivity
value are custom made polycarbonate filament and ceramics. Space filling curve
geometries discussed in chapter 3 can be used to realize the higher permittivity value
from custom made polycarbonate. On the other hand, in the case of using ceramics for
higher permittivity realization, Alumina has a high dielectric constant (εr=9) and is
152
available in large quantities. XJET inkjet printer can be used to mass-produce high
dielectrics ceramics.
Additionally, the fabricated lens antenna with wide beamscanning angle capability
can be integrated with RF electronics and radar sensors to track target in azimuth
direction for radar applications and RF imaging.
153
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PERMISSIONS
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