Wideband Structural and Ballistic Radome Design Using Subwavelength Textured

THE CATHOLIC UNIVERSITY OF AMERICA

Surfaces

A DISSERTATION

Submitted to the Faculty of the

Department of Electrical Engineering and Computer Science

School of Engineering

Of The Catholic University of America

In Partial Fulfillment of the Requirements

For the Degree

Doctor of Philosophy

Paul Eugene Ransom, Jr.

WASHINGTON, DC

2016

Wideband Structural and Ballistic Radome Design Using Subwavelength Textured

Surfaces

Paul Eugene Ransom, PhD

Director: Ozlem Kilic, D.Sc.

This dissertation presents a methodology for designing and fabricating wideband structural and ballistic radomes using conventional composite and ballistic materials. The methodology employed centers on transforming the radome design into an impedance matching problem utilizing electrically compatible materials. Included in this dissertation is a thorough overview of both structural composite and ballistic materials, with the aim of identifying the compatible conventional materials by highlighting both advantageous and detrimental electrical properties.

Moreover, I describe the current state of the art in radome design and performance. As with all impedance matching problems there are standard techniques for developing impedance matching solutions, in this dissertation I describe the most common analytical methods for impedance matching. In addition to analytical methods for designing impedance matching structures, iterative methods are explored. The impedance matching solutions developed through the analytical and iterative methods are implemented using subwavelength textured surfaces. The efficacy of the textured surfaces is controlled by the accuracy of the numerical modelling, many of the common electromagnetic subwavelength modelling techniques like effective medium theory (EMT), are not sufficient to design textured surfaces because EMT breaks down. To address this short fall, the rigorous coupled wave analysis method was employed. Fabrication of properly modelled

textured surfaces was accomplished using both subtractive and additive manufacturing techniques.

The advantages and pitfalls of each manufacturing technique is explored and conclusions are provided. Finally, to validate this methodology I present experimental results of radomes designed and fabricated using this new methodology.

This dissertation by Paul Eugene Ransom Jr. fulfills the dissertation requirement for the doctoral degree in Electrical Engineering approved by Ozlem Kilic, Dr. Sc. as Director, and by

Nader Namazi, Ph.D., Mark Mirotznik, Ph.D., and Steven Russell, Ph.D., as Readers.

Dr. Ozlem Kilic, Director

Dr. Nader Namazi, Reader

Dr. Mark Mirotznik, Reader

Dr. Steven Russell, Reader

Dedication

As children, my sisters and I would spend our summers in Fort Pierce, Florida, with my mother’s family. These opportunities to connect with my aunts, uncles, cousins, and especially my grandmother were times I still cherish. My grandmother, Laura Idella Grier, was the unquestioned matriarch and head of our family. She always had the highest of expectations for me and would often refer to me as “Dr. Paul,” declaring that I would one day be a doctor. She was a steadfast servant of God, rock of her family, and friend to many. I dedicate this work, in loving memory, to my beloved grandmother, Laura Idella Grier.

iii

Dedication iii

List of Figures viii

List of Tables xiv

Acknowledgements xv

Chapter 1: Introduction 1

Contributions to Radome Design 4

Original Publications 5

Overview of Dissertation 6

Chapter 2: Structural Composites Background 8

Structural Composite Materials 8

Matrix Systems 15

Structural Core Materials 19

Electrical Properties of Structural Composite Materials for Radomes 22

Chapter 3: Composite Armor Background 26

Ballistic Armor Design Considerations 27

Chapter 4: Current State of Radome Design and Performance 30

Non-structural Radomes 30

Conventional Structural Radome 36

Sandwich Wall Materials 37

Ballistic Radomes 41

Chapter 5: Wideband Impedance Matching Methodologies 44

Wideband Impedance Matching by Dielectric Layers 44

Analytical Methods 45

Tapered Structures or networks 52

Iterative Optimization 55

Textured Surfaces 57

Chapter 6: Numerical Methods 63

Multilayered Dielectrics 64

Rigorous Coupled Wave Method 67

Iterative Design 82

Chapter 7: Wideband Structural Radome Design 90

Conventional Radome Design Methods 90

Antireflective Surface Radome Approach 92

Chapter 8: Ballistic Radome Wall Configuration Simulations 110

Ballistic Protection Radome Numerical Examples 113

Chapter 9: Antireflective Surface Fabrication Methods 118

Subtractive manufacturing - Computer Numerically Controlled (CNC) Machining

118

Additive Manufacturing Implementation 122

Chapter 10: Experimental Validation 128

Measurement System Background 129

Alternating Slope AR Ballistic 5-30GHz Radome FDM Iterative Design 132

Klopfenstein AR Surface Experimental Validation 134

Alternating Slope AR Structural Composite K-Band Radome 138

Chapter 11: Conclusion 141

References 144

Appendix A 150

Rigorous Couple Wave Analysis Enhanced Transmittance Matrix Approach 150

Analytical solution for rectangular and hexagonal permittivity distributions 153

vii

List of Figures

Figure 1.1 Radome wall configuration and associated frequency performance 1

Figure 1.2 Multilayer Dielectric Slab EM Configuration 2

Figure 1.3 Broadband Radome “Gold Standard” configuration and insertion loss. 3

Figure 2.1 (a). Illustration of structural composite layup. (b) Balsa Wood structural composite. 9

Figure 2.2 (a) Bundle of fibers. (b) 2D plain weave woven cloth. 9

Figure 2.3 Example of a single stack or multi-stack laminate 10

Figure 2.4 Plain Weave 14

Figure 2.5 Satin Weave 14

Figure 2.6 Honeycomb core. 20

Figure 2.7 Foam core. 21

Figure 3.1 Non-Armor Piercing Ballistic Protection Layers 26

Figure 3.2 Ballistic Protection Enhanced Design 28

Figure 4.1 Geodesic fabric radome 30

Figure 4.2 Complex Permittivity of E-glass 2D plain weave woven cloth and an E-glass plain weave vinyl

ester laminate. 32

Figure 4.3 Complex permittivity of S-glass 2D plain weave woven cloth and an S-glass plain weave epoxy

laminate. 33

Figure 4.4 Complex permittivity of Astroquartz 2D plain weave woven cloth and an Astroquartz plain weave

epoxy laminate. 34

Figure 4.5 Insertion loss for Astroquartz, S-glass and E-glass laminates as a function of normalized thickness

Figure 4.6 Sandwich Radome Configuration 37

Figure 4.7 Radome Wall Categories 37

Figure 4.8 Structural core material loss calculated at 40GHz 39

Figure 4.9 Sandwich composite insertion loss for Astroquartz, S-glass and E-glass face sheets. 40

Figure 4.10 Non-AP Ballistic radome insertion loss for Figure 3.2 configuration 42

viii

Figure 4.11 Amor piercing ballistic radome insertion loss for Figure 3.2 configuration. 43

Figure 5.1 Antireflective Conceptual Approach 44

Figure 5.2 Microwave engineering impedance matching model 45

Figure 5.3 Quarter-wavelength multilayered configurations 48

Figure 5.4 (a) Semi-infinite or half space medium configuration. (b) Dielectric slab configuration 48

Figure 5.5 Example of Klopfenstein, exponential and optimized multilayered tapered surfaces 54

Figure 5.6 General iterative optimization algorithm 56

Figure 5.7 – 1-D and 2-D Periodicity 58

Figure 5.8 – Common Grating Lattice Types and Fill Factors for CNC Implementation 59

Figure 6.1 Multilayered Dielectric 64

Figure 6.2 Two-dimensional periodic dielectric grating and problem geometry 67

Figure 6.3 Structural composite radome wall physical configuration and lay-up. 82

Figure 6.4 Direct Design Method Algorithm 83

Figure 6.5 Indirect Design Method Algorithm 84

Figure 6.6 Permittivity profile for Example 5. 86

Figure 6.7 Permittivity comparison between iterative design and Klopfenstein taper. 88

Figure 6.8 Transmission comparison between iterative design example 5 and Klopfenstein taper example 1

from normal incidence to 60° incidence. 88

Figure 7.1 Mode Matching Generalized Scattering Matrix 91

Figure 7.2 Slab transmission 94

Figure 7.3 Structural Composite Face sheet Permittivity and Loss Tangent Comparison 95

Figure 7.4 Permittivity and Loss Tangent of Structural Sandwich Composite Core Foams 95

Figure 7.5 Structural composite wall configuration without an AR surface and the associated transmission

loss exhibited by the wall configuration. 97

Figure 7.6 Structural composite radome wall physical configuration and lay up – Klopfenstein FDM

Polycarbonate Taper. Permittivity profile of two Klopfenstein AR surfaces implemented using FDM

additive manufacturing. 100

Figure 7.7 Structural Composite insertion loss without AR surfaces Structural radome insertion loss

simulation assuming Klopfenstein AR surfaces are layed up in accordance with Figure 7.6. 100

Figure 7.8 Structural composite radome wall physical configuration and lay up. Permittivity profile of two

AR surfaces designed using simulated annealing and pattern search optimization routines; and

implemented using FDM additive manufacturing. 101

Figure 7.9 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss

simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.8.

102

Figure 7.10 Structural composite radome wall physical configuration and lay up. Permittivity profile of two

AR surfaces designed using simulated annealing and pattern search optimization routines; and

implemented using FDM additive manufacturing. 103

Figure 7.11 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss

simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.10.

103

Figure 7.12 Radome configuration in a side view. Permittivity profile of two AR surfaces designed using

simulated annealing and pattern search optimization routines; and implemented using subtractive

manufacturing 105

Figure 7.13 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss

simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.12.

105

Figure 7.14 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss

simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.15.

107

Figure 7.15 Radome configuration in a side view. Permittivity profile of two AR surfaces designed using

simulated annealing and pattern search optimization routines; and implemented using subtractive

manufacturing 107

Figure 8.1 Ballistic Armor Material Permittivity and Loss Tangent 110

Figure 8.2 Ballistic Armor sandwich using S-glass epoxy backing material and a Spectra shield impact

material along with its associated transmission response. 111

Figure 8.3 Ballistic Armor sandwich using S-glass epoxy backing material and an Alumina ceramic are along

with its associated transmission response. 112

Figure 8.4 Ballistic armor sandwich configuration with cyanate ester backing material, ceramic impact

material, and Spectra shield backing layer, along with its associated transmission response. 112

Figure 8.5 Ballistic Armor S-glass Spectra Shield radome configuration and associated transmission loss

prediction 113

Figure 8.6 Ballistic Armor S-glass Alumina radome configuration and associated transmission loss prediction

114

Figure 8.7 Ballistic Armor S-glass Spectra Shield radome lay-up and configuration 114

Figure 8.8 Ballistic Armor S-glass Alumina radome lay-up and configuration 115

Figure 8.9 S-glass ceramic ballistic armor insertion loss without AR surface and no impedance matching

layer. 116

Figure 8.10 Ballistic Armor S-glass Alumina Spectra radome configuration and associated transmission loss

prediction 116

Figure 8.11 Ballistic Armor S-glass Alumina radome lay-up and configuration 117

Figure 9.1 Discrete AR Surface fabricated using CNC machining 119

Figure 9.2 Discrete Ka-band AR surface transmitted energy measurement and predicted performance

results. 120

Figure 9.3 Klopfenstein subwavelength grating 120

Figure 9.4 (a) The black curve illustrates Klopfenstein permittivity profile; the blue curve represents the

effective dielectric constant when the radius varies according to effective medium theory at the center of

the band; the red curve represents the effective dielectric constant when the radius varies according to

the RCWA and optimization at the center of the band. (b) Comparison of the normalized diameter

using RCWA and EMT to determine the radius. 121

Figure 9.5 Illustrates the single and doubly truncated Klopfenstein grating, implemented using subtractive

manufacturing. 122

Figure 9.6 Illustration of FDM printing process shows heated nozzle extruding the thermoplastic feedstock.

124

Figure 9.7 (a) Images of FDM printed test samples used to determine the relationship between local volume

fraction and effective dielectric constant. (b) Cross-hatching fill pattern used to fill the outline. The

effective dielectric constant is proportional to the local volume fraction of polymer to air. 125

Figure 9.8 Measured data and Maxwell-Garnett fit for the effective dielectric constant of these samples as a

function of volume fraction. 126

Figure 9.9 (a) Comparison of the measured and predicted transmission energy through an AR FDM

fabricated slab. (b) Compares the design AR surface permittivity profile (red curve) to the actual

fabricated permittivity profile (blue curve). 126

Figure 10.1 Transmission and reflection measurement set up. Transmit and receive horns are aligned and

attached to a vector network analyzer. 129

Figure 10.2 Illustration of the four states of EM energy for free space measurements. 130

Figure 10.3 Collimating Lens and Focused Beam Measurement System 131

Figure 10.4 (a) Permittivity profile of the iterative design alternating slope AR surface. (b) Ballistic radome

full system configuration. 132

Figure 10.5 (a) Insertion loss for ballistic armor configuration from 2-40 GHz over incidence angles 0-50°. (b)

Insertion loss for ballistic armor with iterative designed AR surface applied. 133

Figure 10.6 (a) Image of the cross-hatched iterative AR surface. (b) Measured vs. predicted insertion loss of

ballistic radome at 0° incidence angle. 133

Figure 10.7 (a) Image iterative AR surface bonded to ballistic armor. (b) Measured vs. predicted insertion

loss of ballistic radome at 0° incidence angle. 134

Figure 10.8 Klopfenstein AR surface permittivity profile for a ballistic armor core and the associated

transmission loss prediction for the total radome lay up. 135

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Figure 10.9 (a) Klopfenstein AR Surface ballistic radome. (b) Comparison of predicted and measured

Klopfenstein AR surface transmission loss. 136

Figure 10.10 (a) Comparison of measured and predicted Klopfenstein AR surface ballistic radome

transmission loss. (b) Comparison of measured and predicted Klopfenstein AR surface ballistic radome

return loss. 136

Figure 10.11 (a) Illustration of the insertions loss without the Klopfenstein AR surface. (b) Illustration of

insertion loss with Klopfenstein AR surface. 138

Figure 10.12 (a) Predicted insertion loss of the ballistic radome with Klopfenstein AR surface. (b) Measured

insertion loss of the ballistic radome with Klopfenstein AR surface. 138

Figure 10.13 (a) K-band iterative design permittivity profile. (b) Transmission of each FDM AR surface

compared to the predicted transmission 139

Figure 10.14 (a) Illustration of structural composite with K-band iterative design AR surface. (b) Simulated

and measured transmission loss results for structural composite with and without K-band iterative AR

surface. 140

Figure 10.15 (a) Astroquartz radome configuration. (b) Comparison of astroquartz radome insertion loss to

structural composite radome with K-band iterative design AR surfaces. 140

Figure 13.1 Antireflective surface structures for a rectangular packed hole array 153

Figure 13.2 Antireflective surface structures for a hexagonal packed hole array. 154

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List of Tables

Table 2-1 Properties of some commercially available high-strength fibers 11

Table 2-2 Relative characteristics of thermoset resin matrices 16

Table 2-3 Core material properties 22

Table 2-4 Electrical Properties of Structural Composite Materials 24

Table 3-1 Ballistic Armor Materials 29

Table 4-1 Buckling failure due to wind speed and panel thickness 31

Table 4-2 Derived Structural Properties for Example 1 39

Table 4-3 Ballistic radome physical configuration 42

Table 5-1 CNC dielectric constant dynamic range 60

Table 6-1 Computational Demand of Iterative Algorithm Using the Direct Design Method 85

Table 6-2 Computational Demand of Iterative Algorithm Using the Indirect Design Method 86

Table 7-1 AR Surface Designs 97

xiv

Acknowledgements

Since beginning this journey in earnest more than eight years ago, I’ve gotten married, had two boys, and took on several challenging projects at work – all of which at times left me feeling this work might become a “dream deferred.” As I am now on the precipice of completing this journey I am eager to acknowledge the many people that have helped and encouraged me to make my dream a reality.

I am grateful for the love and support of my wife, Mya Ransom, who often picks up the slack for me and keeps our beautiful children at bay during the many nights when I’m held up in my home office reading and writing. Her partnership and encouragement have spurred me on.

To my sons, Khyrie, Paul III, and Jaxon, you bring me such pride and inspire me to work hard if only to show you that hard work pays off if you see it through. I am thankful for my sisters

Shanee and Tafaya for their constant support and encouragement throughout my life. They uplift and inspire me to work harder, achieve greater, and be better. Their confidence in me motivates me to reach their expectations. Finally, I am ever indebted to my mother Rhonda Grier, who has been the steady example of grace, strength, and perseverance. A single mother who sacrificed much for myself and my two sisters, it has always been my goal to make my mother’s sacrifice worthwhile. Even back to my high school years, I worked hard academically and even played hard athletically simply to make her proud. This doctorate is another testament to her sacrifice and leadership. Ultimately, though, I don’t think I can ever make her as proud of me as I am of her.

A heartfelt thank you to my dissertation advisors: Dr. Mark Mirotznik and Dr. Ozlem

Kilic. I started this process with Dr. Mirotznik, who left Catholic University of America for The

University of Delaware during the second year of my candidacy. Nevertheless, Dr. Mirotznik xv

has remained a strong advocate, advisor, and friend throughout this journey. I truly appreciate his guidance and can unequivocally say that without his mentorship and persistence I would not have completed this milestone. To Dr. Kilic, who agreed to serve as my advisor following Dr.

Mirotznik’s move to Delaware, I am ever grateful for your patience, perseverance, and guidance as I plodded through this work. Many thanks also to Dr. Steve Russell and Dr. Nader Namazi for lending their time and assistance as members of my dissertation committee, and to Peggy

Bruce for helping me to resolve various enrollment challenges I created juggling my life-work- school responsibilities!

I would also like to thank Mr. Shaun Simmons, Dr. Brandon Good, Mr. Tony Wilson,

Mrs. Carrie Erickson, Ms. Janette Lewis, Mr. Zachary Larimore, Dr. Thomas Miller, Mr. Bruce

Crock and numerous colleagues at the Naval Surface Warfare Center Carderock for their encouragement and support throughout my candidacy. A special thanks to Mr. Simmons for the use of his 1-D dielectric recursive solver and Mr. Larimore for fabricating the FDM antireflective surfaces.

xvi

Chapter 1: Introduction

To improve the transmission and reflection response of structural and ballistic radomes researches have used various techniques. The most effective radome design techniques address several parameters namely insertion loss, weight, cost, complexity and environmental susceptibility. Radome is derived from the term radar dome, which refers to a cover placed over an antenna to protect it from the environment. Radomes are principally used to protect antennas and their associated electronics. The most advanced radome design is conducted by the military community. In response to both the environment in which radomes reside as well as the advanced antennas in which they are required to protect; military radomes must have significant capabilities. Some of the most common radome wall configurations are illustrated in Figure 1.1 along with their bandwidth capacities.

Today’s advanced antennas are large, integrated, multi-band components with a multitude of functions.

Figure 1.1 Radome wall configuration and associated frequency performance

Figure 1.2 Multilayer Dielectric Slab EM Configuration

푗푘 푙 −푗푘 푙 퐸 + 1 푒 𝑖 𝑖 휌 푒 𝑖 𝑖 퐸 [ 푖 ] = [ 푖 ] [ 푖+1,+] , 푖 = 푀, 푀 − 1, … 1 ( 1.1 ) − 푗푘𝑖푙𝑖 −푗푘𝑖푙𝑖 퐸푖 휏푖 휌푖푒 푒 퐸푖+1,−

They are integrated within structures of all forms; whether that structure is a land vehicle, aircraft, naval platform or unmanned aerial vehicle. Broadband structural radomes are typically designed using the multilayer wall configurations shown in Figure 1.1. Non-structural radomes

(i.e. environmental covers) are designed to handle wind and rain loads, and employ simple single layer laminates also shown in Figure 1.1 The most common approach to the design of structural wideband radomes implements the multilayer radome wall [1], which is modelled using the equivalent transmission line model [2] or the multilayer dielectric model [3] illustrated

Figure 1.2 and described by ( 1.1 ).

Currently, the “gold standard” for broadband radomes is the C-sandwich radome, with a honeycomb core sandwiched between three thin cyanate ester quartz laminate skins illustrated in

0.005” Quartz/Cyanate Ester Resin

Figure 1.3 Broadband Radome “Gold Standard” configuration and insertion loss.

Figure 1.3 [4]. While this design produces excellent broadband performance it is not a structural radome. In fact, its structural capabilities only extend to endure wind velocities up to 45 mph and a shock of 40G’s for 0.011 seconds.

Broadband ballistic radomes do not currently exist because most ballistic protection materials have poor electrical properties for RF transparency. Conventional ballistic protection materials include Kevlar, Spectra, Dyneema Alumina, and other ceramics. Alumina and other ceramics typically have large dielectric constants that are highly dispersive. These elements, make it challenging to design a broadband ballistic radome with acceptable insertion loss for most applications. In this dissertation, I employed new broadband antireflective surfaces and iterative design methods to realize wideband impedance matching networks suitable for structural and ballistic materials. These new methods enabled the design of wideband, broad incidence structural and ballistic radomes. Indeed, the robustness of this approach allowed the marriage of conventional structural composites and ballistic materials. The consequence of this

union resulted in the creation of multifunctional radomes that retain all of their structural and ballistic characteristics while adding attractive wideband RF transparency not previously available.

Contributions to Radome Design

In this dissertation I present several developments that have advanced radome design.

The concept of radome design by combining antireflective surface technology and conventional structural composites or ballistic materials represents a significant contribution and advancement in radome design.

1.1.1 Iterative Design

In Chapter 6 I present a new iterative design method for designing wideband antireflective surfaces. The development of the indirect design method represents an improvement in antireflective surfaces design because it is more efficient and enables a more comprehensive optimization result. Using this new method, I designed wideband structural composite and ballistic radomes.

1.1.2 Ballistic Radomes

The ballistic radome designs and examples presented in Chapter 9 represent a significant contribution to radome design technology. Current radome design technology has not produced ballistic radomes with the ballistic protection capabilities described in Chapter 3 and the bandwidth and performance demonstrated in Chapters 8 and 10.

1.1.3 Non-Monotonic Antireflective Surfaces

Using the indirect design method was an enabling concept that led to the development of non-monotonic antireflective surfaces described in Chapters 7 and 8. This is a new type of

antireflective surface that is only realizable using additive manufacturing techniques. To fabricate these new AR surfaces, I used Fused Deposition Modelling (FDM), which also represents a new approach to fabricating subwavelength surfaces. The description of this fabrication method is found in Chapter 9 and will be published in Electronic Letters “Fabrication of Wideband Antireflective Surfaces using Fused Deposition Modeling”.

1.1.4 Experimental Validation of Antireflective Wideband Structural Composite and

Ballistic Radomes

Chapter 10 presents the experimental validation of several ballistic radomes designed using the direct and indirect design methods. In all cases the experimental validation of radomes designed using the antireflective surface approach represents a major contribution to radome design. Moreover, the addition of non-monotonic FDM antireflective surfaces for radome design advances wideband radome technology and helps the community deliver more capable radomes.

Original Publications

What follows are papers and presentations that I have published resulted from this work.

1. P. Ransom, Z. Larimore, S. Jensen, M. Mirotznik, “Fabrication of Wideband Antireflective

Surfaces using Fused Deposition Modeling”, Electronic Letters

2. P. Ransom and M.S. Mirotznik, 'Broadband Antireflective Surfaces using Tapered

Subwavelength Surface Texturing', IEEE International Symposium on Antennas and

Propagation, Orlando FL, 2013

3. Good, B., Ransom, P., Simmons, S., Good, A. and Mirotznik, M. S. (2012), Design of graded index

flat lenses with integrated antireflective properties. Microwave Optical Technology Letters 54:

2774–2781

4. M.S. Mirotznik, B. Good, P. Ransom, D. Wikner and J.N. Mait, ‘Iterative Design of Moth-Eye

Antireflective Surfaces at Millimeter Wave Frequencies’, Microwave and Millimeter wave

Technology Letters, Vol. 52, No. 3, March 2010, pp. 561-568.

5. M.S. Mirotznik, B. Good, P. Ransom, D. Wikner and J.N. Mait, ‘Design of Inverse Moth-eye

Antireflective Surfaces’, IEEE Trans on Antennas and Propagation, Vol. 58, No. 9, September

2010, pp. 2969-2980.

6. P. Ransom, “Aperstructures: An Integrated Self-Collimating Photonic Crystal”, Ships and Ship

System Symposium Proceedings, 13-14 November 2006.

7. P. Ransom, “Aperstructures in LO Systems”, Have Forum Symposium Proceedings “23-25 April

2007”

8. P. Ransom, “Comparison of Theoretical and Experimentally Measured Propagation Loss in

Photonic Crystals”, Electromagnetic Code Consortium (EMCC), 8-10 May 2007

9. P. Ransom, “Advanced Composite Materials”, Tri-Service Metamaterials Conference 8-10

December 2009

Overview of Dissertation

The objective of this work was to design wideband structural and ballistic radomes using conventional structural and ballistic materials. I was able to accomplish this objective by employing a design methodology focused on addressing the fundamental challenge of

minimizing insertion loss. Insertion loss in radomes is controlled by two mechanisms: reflection and material loss. In this dissertation, Chapter 2 Structural Composite Materials describes, in detail the most commonly used structural materials. In addition, I detail the most popular ways these materials are configured to produce structural composites (i.e. sandwich configurations).

Chapter 3 describes common ballistic materials and their associated design configurations.

Chapter 4 provides an overview of the current state of radome design for broadband structural and non-structural radomes. Chapter 5 presents impedance matching methodologies using antireflective textured surfaces as well as the equivalent transmission line (i.e. multilayer) and generalized scattering model approaches to radome design. In addition to the introduction of antireflective textured surface radomes, Chapter 5 also provides the reader with designs and simulations illustrating the effectiveness of the textured surface design methodology. Chapter 6 describes the numerical methods used to design and predict the electromagnetic response of structural and ballistic radomes. In Chapters 7 and 8, the reader will find several wideband structural composite and ballistic designs, using both discrete and tapered antireflective surfaces.

This chapter is intended to give the reader a better sense of the effectiveness of this method as well as compare and contrast different antireflective surface designs. Chapter 9 describes the fabrication methods used to realize the designs presented in Chapters 7 and 8. Finally, Chapter

10 presents the experimental validation of the designs described in Chapter 9.

Chapter 2: Structural Composites Background

In this dissertation, I developed an EM design methodology that was flexible and robust enough to work with a wide variety of structural materials that have limited loss. To comprehensively describe my radome design approach it is first necessary to discuss in detail the key properties of structural composites and ballistic armor materials.

Structural Composite Materials

Structural composites are a combination of two or more individual components; 1) the reinforcement material providing the structural characteristics and 2) the matrix resin systems providing the binding agent for the composite. There are an abundance of reinforcement and matrix materials and the combination of the two is used to build structural composites. The choice of materials is dictated by a host of requirements such as strength, weight, cost, and now electromagnetic (EM) properties. Figure 2.1 (a) shows an example of a balsa wood sandwich composite. Figure 2.1 (b) presents the most common configuration of a standard structural composite.

a b

Figure 2.1 (a). Illustration of structural composite layup. (b) Balsa Wood structural composite.

a b

Figure 2.2 (a) Bundle of fibers. (b) 2D plain weave woven cloth.

Specifically, this example shows a sandwich composite including a lightweight structural core with two thin outer skins known as facing. The lightweight outer skins are typically comprised of fiber reinforcement materials, however in some instances particles or whiskers are also used. Particles are frequently used as fillers to reduce material cost. However, since they have no preferred orientation they provide minimal mechanical properties [5]. Whiskers, however are extremely strong but are difficult to disperse uniformly within a matrix, because they are single crystals. Fibers on the other hand have very long aspect (length/diameter) ratios,

due to their strength and stiffness advantages over the previous materials are the dominant reinforcement for composites [5]. Reinforcing the woven cloth, particles or whiskers with a matrix system results in the outer skin facing shown in Figure 2.1 (b), this structure is commonly known as a laminate and is illustrated in Figure 2.3. Several factors contribute to the strength of individual fibers. Table 2-1 illustrates that in addition to material type, a fiber’s diameter and surface flaws also influences its properties. Specifically, as the diameter decreases the fiber strength increases thereby reducing the surface flaws and subsequently reduces the variability in the fiber strength [5].

Figure 2.3 Example of a single stack or multi-stack laminate

Table 2-1 Properties of some commercially available high-strength fibers

Fiber Type Tensile Tensile Elongation Density, Coefficient of Fiber strength, modulus, at failure, % g/cm2 thermal Diameter, ksi msi expansion 10-6 °C µm Glass E-glass 500 10 4.7 2.58 4.9-6.0 5-20 S2-glass 650 12.6 5.6 2.48 2.9 5-10 Quartz 490 10 5.0 2.15 0.5 9

Organic Kevlar 29 525 12 4.0 1.44 -2.0 12 Kevlar 49 550 19 2.8 1.44 -2.0 12 Kevlar 149 500 27 2.0 1.47 -2.0 12 Spectra 1000 450 25 0.7 0.97 - 27

PAN-based Carbon Standard 500-700 32-35 1.5-2.2 1.80 -0.4 6-8 modulus Intermediate 600-900 40-43 1.3-2.0 1.80 -0.6 5-6 modulus High modulus 600-800 50-65 0.7-1.0 1.90 -0.75 5-8

Pitch Base Carbon Low modulus 200-450 25-35 0.9 1.90 - 11 High modulus 275-400 55-90 0.5 2.0 -0.9 11 Ultra-high 350 100-140 0.3 2.2 -1.6 10 modulus

Note: Table data referenced from [5].

2.1.1 Glass Fibers

Although, there are many more types of glass fibers than those mentioned in Table 2-1,

E-glass, S2-glass and Quartz are three of the most common glass fibers. They all have attractive properties such as their high tensile strength, high impact resistance, low cost, and good chemical resistance glass fibers have become a staple of the structural composite industry. Of the glass fibers E-glass is the most prevalent because it provides the best balance between cost and

structural performance (i.e. tensile strength 500 ksi and modulul 70 GPa). S2-glass is also very popular because it provides 40% stronger fibers and handles elevated temperatures better, with a minimal cost penalty. Quartz fibers are somewhat of a specialty fiber, in that they provide excellent electrical loss properties due to their ultrapure silica glass content, however the price to pay for this electrical property is steep. Quartz is typically used for applications that require substantial electrical performance like radomes, but can become cost prohibitive.

2.1.2 Organic Fibers

Another class of fibers are the Aramids. These are organic fibers that have stiffness and strength greater than glass fibers and less than carbon fibers. The most common type of aramid fiber is the commercial product made by Dupont® known as “Kevlar”. While aramid fibers are susceptible to compression loads thereby, limiting their use in high-strain, compressive or flexural loads applications, their extreme toughness make them well suited for ballistic protection. Aramid fibers have an ability to absorbs large amounts of energy during fracturing and undergo plastic deformation in compression making them an excellent backing material for ballistic armor. As illustrated in Table 2-1 their relatively low density suggests they are lighter weight than glass fibers, however they lack adhesion to matrix materials. The most popular

Kevlar fibers are listed in Table 2-1.

Ultra-High Molecular Weight Polyethylene (UHMWPE) fibers are an additional organic fiber produced from Gel-spun polyethylene. They are extremely strong high modulus fibers.

They are commercially produced under the names Dyneema and Spectra.

2.1.3 Carbon and Graphite Fibers

Perhaps the most prevalent fibers used in high-performance composite structures are carbon and graphite fibers. Interestingly, carbon and graphite fibers are used interchangeably however, for completeness graphite fibers are different in that they are subjected to heat treatment above 3000°F, have carbon content greater than 99% and have elastic moduli greater than 345 GPa. Conversely, carbon fibers have lower carbon content (i.e. 93-95%) and are heat treated at lower temperatures [5]. Both fibers exhibit superior tensile strength, high moduli, and compressive strength and have excellent fatigue properties. This superior structural performance does come at a cost, however when the application requires superior structural performance, carbon fibers are the reinforcement material of choice and there are a wide variety of carbon fiber products to choose from.

2.1.4 Woven Fabrics

To obtain the structural benefits of fibers they must be transformed or integrated such that they form a two dimensional layer. One of the most common ways this is done is by weaving the fiber yarn or yarn into a cloth using a loom, an example of a woven cloth is illustrated in

Figure 2.2. Woven cloths can have many different arrangements of weaves and materials and those arrangements are called hybrid weaves. Weaves are also classified according to their weave patterns. Two of the most prevalent weave patterns are the plain and satin weave shown in Figure 2.4 and Figure 2.5, respectively.

Figure 2.4 Plain Weave

Figure 2.5 Satin Weave Another common fiber transformation method is the prepreg reinforcement. Prepregs are formed using either unidirectional fibers or woven cloth impregnated with a controlled amount of resin. The resin is advanced to the point where it is semisolid or tacky. Prepregs enable superior control over the composite thickness and are the preferred laminate for high- performance composites. The woven cloth fibers used for prepregs can be glass, carbon, and in some instances aramid.

Matrix Systems

Matrix resin systems are the second material that makes up all structural composites. The matrix is the binder material for the fiber. The essential function of the matrix is to transfer the load from the structure to the fibers and to transfer load from fiber to fiber. It is usually the outer surface material and therefore provides abrasion resistance, toughness, impact resistance and any damage tolerance. Polymeric matrix systems are categorized as thermosets or thermoplastics.

Thermosets are low molecular weight, low viscosity monomers (≈2000 centipoise) that are converted during curing into three-dimensional crosslinked structures that are infusible and insoluble [5]. Thermoplastics were developed as a replacement for thermosets during the early

80’s and 1990’s, because of their potential for increased toughness and more damage tolerant, because they do not crosslink during cure. In addition, thermoplastic consolidate and thermoform in minutes or seconds while thermosets may require hours to cure [5].

Thermoplastic also exhibit low moisture absorption and thermoplastic prepregs do not require refrigeration during storage. Although thermoplastics have potential to replace thermosets to date only a handful of thermoplastic are used today.

Table 2-2 Relative characteristics of thermoset resin matrices

Polyesters Used extensively in commercial applications, relatively inexpensive Vinyl Esters Similar to polyester, tougher and better moisture resistance Epoxies High-performance matrix for primary continuous fiber composites. Appropriate for temperature ranges up to 250-275°F. Superior high temperature performance than polyesters and vinyl esters Bismaleimides High-temperature resin matrix appropriate for temperature ranges up to 275-350°F Cyanate Esters High-temperature resin matrix appropriate for temperature ranges up to 275-350°F, with epoxy like processing. Better suited for EM applications due to good electrical properties. Polymides Very-high temperature resin system appropriate for temperature ranges up to 550-600°F. Very difficult to process. Phenolics High temperature resin system good smoke and fire resistance, most common for aircraft interiors.

2.2.1 Thermosets

2.2.1.1 Polyester

Polyester matrix resin system is a lower cost resin system that has limited use in high- performance structural composites, due to its lower temperature stability, mechanical properties and inferior weathering resistance. Polyesters are not fabrication friendly due to their tendency to cure at room temperatures in addition to their relatively short pot life. To improve their curing properties both inhibitors and catalysts are added to this resin system. The combination of these disadvantages makes the polyester resin system one of the least appealing for structural composites.

2.2.1.2 Vinyl Ester

Vinyl esters are closely linked to polyester resins with significant differences, that make this resin system more appealing for structural composites. Vinyl esters have lower crosslink densities and are tougher than higher crosslinked polyesters. Moreover, they exhibit better resistance to water and moisture degradation.

2.2.1.3 Epoxy

Epoxy resin systems are the most common type of matrix resin systems, owing their popularity to a combination of excellent strength, adhesion, low shrinkage and processing versatility. Epoxy can be either a resin system or adhesive and usually consists of at least one major epoxy and a curing agent. Most epoxies have several minor epoxies and curing agents that make up the compound. These minor epoxies are usually incorporated to provide additional features like viscosity control, elevated temperature compliance and improve moisture absorption. Perhaps the reason epoxy resin systems are so dominant is because their properties are so well understood and many of their deficiencies can be addressed through the use additives and fillers.

2.2.1.4 Bismaleimides (BMIs)

Bismaleimides were developed to bridge the gap between epoxies and Polymides [5].

BMIs have excellent temperature properties, in fact they are commonly used in temperature ranges between 430 – 600°F; however, they also have a tendency to suffer from imide corrosion.

This form of hydrolysis requires greater care be taken when BMI resins are used with conductive fibers.

2.2.1.5 Cyanate Ester

Cyanate esters are low dielectric, low loss matrix resins that can be extremely useful for designing radomes and other EM applications. In addition to their low dielectric constant cyanate esters have low water absorption (0.6 – 2.5%), which results in better dimensional stability and low outgassing. Due to their moderate crosslinking densities cyanate esters are relatively tough as well, and can be toughened using some of the same mechanisms used for rubbers and thermoplastics. They have good temperature properties (375 – 550°F) and are inherently flame resistant, because of the limited market demand relative to other resins, cyanate esters tend to be expensive. Epoxies and BMI’s have better adhesion than cyanate esters.

2.2.1.6 Polyimide

Polyimides are both thermosets and thermoplastics and their major advantage is their high temperature properties (500-600°F). They are more difficult to process than BMIs and epoxies, because they are processed at temperatures up to 700°F. They tend to give off water which results in voids and porosity issues that impact their mechanical properties.

2.2.1.7 Phenolic

Phenolic matrix resins are high temperature, low flammability and low smoke resins. For this reason, they are typically used in aircraft interior structure or in applications where flame resistance is paramount. They are brittle and hard to process.

2.2.2 Thermoplastics

Thermoplastics are high molecular weight resins that are fully reacted prior to processing.

They do not crosslink they melt and flow instead. The lack of crosslinking prevents thermoplastics from being inherently brittle and as a result, they can be reprocessed.

Thermoplastic are typically tougher, have lower moisture absorption and shorter curing times.

Although there are definite advantages that thermoplastics have over thermosets, thermoplastics are not as prevalent in commercial and non-commercial communities. The reasons why thermosets have remained dominant are [5]:

1. High cost for processing due to the elevated temperature (500-800°F) required to process

thermoplastics as compared to thermosets

2. Difficulty handling thermoplastic prepreg due to the lack of tack of thermoplastic

prepregs.

3. The tendency of fibers to wrinkle and buckle with thermoplastics known as

thermoforming.

4. The improvement in toughness and damage tolerance of thermoset resin systems.

5. Solvent and fluid resistance of amorphous thermoplastics.

Structural Core Materials

Sandwich composites are composite materials that are lightweight and they have high stiffness and high strength-to-weight ratios. A sandwich composite requires the facesheets to carry the bending loads (tension and compression) while the core carries the shear loads. An excellent way to increase composite stiffness and have minimal effect on weight is increasing the core thickness. In fact, for honeycomb structures doubling the thickness increases the stiffness six times and quadrupling the thickness increases the composite stiffness 37 times. The face sheets that make up a sandwich composite are typically very thin (i.e. 0.010 – 0.125”) carbon, glass, aramid or aluminum fibers. Some of the more popular sandwich cores are described in sections 2.3.1, 2.3.2, and 2.3.3

2.3.1 Honeycomb Core

Honeycomb cores are periodic macro cellular structure that can be made of aluminum, glass, aramid paper, aramid fabric or carbon fabric [5]. Hexagonal, flexible, and over expanded cores are the three most popular cellular configuration used today [5]. The choice of cellular configuration is determined by the application, for instance a flexible core is most likely used in applications that requires molding the sandwich composite in the form of a shape. The bond of the face sheet to the core is an important part of the sandwich composite construction and the adhesives that are used to form this bond must be tailored to the core material and structure to insure optimal performance.

Figure 2.6 Honeycomb core.

2.3.2 Foam Cores

Foam cores are most popular in the boat building and light aircraft industries [5]. Foam cores are made by blowing and foaming agents that expands during fabrication to produce a porous cellular structure. In general, the higher the core density the greater the percentage of closed cells. Moreover, most structural foam cores are closed cell which means their cells are discrete. Open cell foams are weaker and also absorb water, although they are good for sound absorption.

Figure 2.7 Foam core.

Table 2-3 Core material properties

Name Foam Core Density, Maximum Characteristics pcf Temperature, °F Polystyrene 1.6-3.5 165 Low-density, low cost, closed cell capable of thermoforming Polyurethane 3-29 250-350 Low-high density closed cell foam capable of thermoforming, thermoplastic and thermoset, co- cured, secondary bonding available Polyvinyl chloride 1.8-26 150-275 Low – high density foam, can contain open cells, thermoplastic or thermoset, co-cured Polymethacrylimide 2-18.7 250-400 Expensive high-performance closed cell foam, can be thermoformed co- cured, secondary bonded

2.3.3 Syntactic/Solid Cores

Syntactic cores consist of a matrix such as epoxy or cyanate ester that is filled with hollow microspheres of glass or ceramic. Syntactic cores generally have higher densities than foam or honeycomb cores, they tend to be supplied as pastes as well. If the syntactic core is co- cored with the facesheets an additional adhesive is not required. Ceramic microspheres are added to syntactic cores to provide improved high temperature performance.

Electrical Properties of Structural Composite Materials for Radomes

In this section I will discuss the fundamental electrical properties of optimal structural composite radome materials. To achieve perfect impedance matching to an air interface the real

′ part permittivity (휀푟) must equal unity. Similarly, the imaginary permittivity which governs material loss must be driven to zero. The previous sections have provided an overview of the structural characteristics of the most frequently used composites in the academic and commercial communities.

휀′′ δ = 푡푎푛 휀′ ( 2.1 )

A review of those materials leads to a downselect process in which structural composites are evaluated not only on their structural characteristics but also on their electrical properties. The electrical properties can be simplified to an assessment of their reflection coefficient in the radome passband. The reflection coefficient (Γ) is governed by the impedance at the interface between the outer surface and the incident medium, which is usually air. Equation ( 2.2 ) describes the elementary reflection coefficient for dielectrics and is determined by the complex relative permittivity (휀푟).

휂1 − 휂0 휇0 Γ = , 휂1 = √ 휂1 + 휂0 휀0휀푟

( 2.2 ) 휇0 휂0 = √ 휀0

−7 Where 휇0 is the free space magnetic permeability 4휋 ∙ 10 henries/meter and 휀0 is the free

−12 space permittivity 8.854 ∙ 10 farads/meter, 휀푟 is the complex relative permittivity given by

′ ″ ′ ″ 휀푟 = 휀 − 푗휀 . Where 휀 and 휀 are the real and imaginary part of the complex relative permittivity 휀푟, respectively. To minimize the reflection coefficient, the impedance (η1) of the material should be equivalent to the impedance at the interface (η0). In general, the interface is air and the impedance of air is 휂0 = 377Ω or 휀푟 = 1. In addition to minimizing the reflection coefficient optimal radome materials must exhibit minimal material loss which is described by loss tangent given by equation ( 2.1 ). Table 2-4 presents the measured electrical properties of structural materials that are candidates for radome design.

Table 2-4 Electrical Properties of Structural Composite Materials

Material Name Fiber Resin Real Permittivity Loss Architecture 24-40 GHz Tangent

Glass E-glass 50 oz 3-D Epoxy 4.4-4.8 0.01-0.13 S2-glass 8 oz Plain Epoxy 3.83 0.016-0.03 Quartz 8 oz Plain Epoxy 3.10 0.016

Organic S2/Kevlar 50/50 24 oz Plain Phenolic 4.1 0.040 Kevlar UD 0/90 PP 3.51 0.017 Polypropylene Vectran Sentinel UD 0/90 Thermoplastic 3.15 0.002 Dyneema UD 0/90 Thermoplastic 2.43 0.006 Spectrashield UD 0/90 Thermoplastic 2.43 0.001 Aramid UD 0/90 3.67 0.063

This table provides the complex real permittivity and the loss tangent for frequencies between

24GHz and 40GHz. suggests that the S-glass, Astroquartz, Vectran, Dyneema and Spectra- shield are good radome candidate materials. Given their lower complex permittivity and loss tangent values. This table provides the electrical properties at millimeter wave frequencies (24

GHz – 40 GHz). Radomes also operate at microwave frequencies (300 MHz – 20 GHz).

Typically, it is wise to evaluate radome materials at the highest frequency of operation because material loss is typically greatest at high frequencies because more wavelengths can propagate, therefore measuring material loss at high frequencies represents a worst-case scenario.

Additionally, because these materials are non-dispersive (i.e. the complex permittivity does not

change greatly with frequency) the complex permittivity at lower frequencies is the same as the permittivity at higher frequencies.

Clearly, if the structural composite material is dispersive the material evaluation should be conducted within the passband.

Chapter 3: Composite Armor Background

Recent advancements in radome design have begun to address not only structural characteristics and environmental protection, but also consider creating radomes with ballistic protection capabilities [6]. Ballistic protection provides impact resistance designed to withstand the effect of high velocity projectiles. Light weight ballistic armor is typically comprised of a rigid solid ceramic layer bonded to glass (most likely polyethylene) or aramid fibers with an epoxy binder acting as a kinetic energy and projectile fragment catcher. Ceramic ballistic armor operates using a system of layers’ approach shown in Figure 3.1. The first layer is typically formed by ceramic materials that dampen the initial impact of the projectile by providing a sufficiently rigid barrier. The ceramic must also fracture the tip of the projectile, dissipating the kinetic energy of the projectile in order to distribute the impact to the second layer. The second layer is usually comprised of ductile material such as polyethylene or aramid fibers and is known as the backing. The backing is used to absorb the kinetic energy from the projectile fragments and the deformation of the ceramic [7].

Figure 3.1 Non-Armor Piercing Ballistic Protection Layers

Aluminum Oxide (AL2O3), boron carbide (B4C), and silicon carbide (SiC) are commonly used commercial ceramics in ceramic ballistic armor systems. Common backing materials are laminates with Kevlar™, Spectra™ or Dyneema™ [8] fibers and an epoxy matrix.

Ballistic Armor Design Considerations

Ballistic armor systems are designed to satisfy requirements for performance, weight, and application. Ceramic and backing thickness along with the arrangement of any additional separator layers are the predominant design considerations. If the ballistic armor system is required to be high performing (i.e. armor piercing (AP) projectiles), it typically requires the ceramic layer to be thicker (8-8.5 mm) [8]. The thickness of the backing is designed to compensate for all the energy that is distributed due to the fracturing ceramic and all fragments produced at impact.

Ballistic protection and weight savings improvement are accomplished by employing one of two design configurations. The first armor system configuration employs the use of confinement. This simply refers to bonding a layer of fiberglass prepreg to the ceramic layer.

This confinement approach creates a uniform compression condition thereby reducing its fragmentation upon impact [8]. The confinement improvement is manifested as a reduction in perforation at the backing layers. Confinement also reduces the shock wave of the projectile after impact. The second configuration seeks to dampen the projectile impact shock wave by adding a separator layer between the ceramic and backing layers. The separator is typically an epoxy matrix filled with boron carbide, silicon carbide, or alumina ceramic microspheres. The separator layer enables a thinner and consequently lighter ceramic and backing layer (i.e. 4-4.5 mm) due to the reduction in shock wave reflection amplitude.

Ballistic materials have electrical properties that vary from generally low loss dielectrics to extremely lossy dielectric. In addition to wide ranging loss components, ceramics tend to

have large real part dielectrics that are highly dispersive. Materials with large dielectrics result in large reflection coefficients. Moreover, material that share an interface with air or a low dielectric material typically result in impedance mismatches that produce large reflection coefficients. This difficulty was addressed by [6], by integrating graded permittivity layers between the ceramic and backing layers; then incorporating an antireflective surface at each air interface. The graded permittivity layers provide an impedance match between the ceramic layer and the backing layer, while the anti-reflective surface provides an impedance match between the fiberglass and free space layers. This system was demonstrated in [6], and shown to be highly effective.

Figure 3.2 Ballistic Protection Enhanced Design

Table 3-1 Ballistic Armor Materials

Monolithic Density Vickers Hardness Modulus Strength (MPa) Ceramics (g/cm3) (kg/mm2) (GPa) Alumina 3.95 Alumina-mullite 3.52-3.56 1130 237 350 whiskers Boron carbide 2.51 2790 440 155* Silicon carbide 3.21* 2800* 476* 324* Aluminum 3.25* 1170* 308* 428* nitride Backing Materials Kevlar 1.44 19 550 Spectra 0.97 25 450 Dyneema 0.97 25 450 *Information obtained from [9].

Table 3-1 presents the mechanical properties of common ballistic armor materials.

Included in this table are ceramic fracturing materials along with several common backing materials.

Chapter 4: Current State of Radome Design and Performance

Radome technology is deployed in commercial automobiles, aircraft and terrestrial towers. Automobiles are heavily equipped with integrated antenna arrays, patch antennas and traditional mast antennas. Commercial aircraft depend on numerous antennas for communication and navigation and they are protected from the environment using radomes.

However, much of the radome research is conducted for military application because the antennas that are protected by radomes are usually fundamental to mission success or mission failure. Moreover, the antennas operate in harsh environments, with critical weight constraints.

To design radomes that address these elements the military community has invested heavily in radome technology.

Figure 4.1 Geodesic fabric radome Non-structural Radomes

One of the most well-known types of radomes is the geodesic radome which is presented in Figure 4.1. Geodesic radomes are comprised of panels that are attached to a metal or dielectric frame.

Table 4-1 Buckling failure due to wind speed and panel thickness Safety Critical Maximum Worst Case Maximum Radome Factor at Buckling Panel Stress Frame Stress Frequency Core 150 mph Wind 150 mph 150 MPH of Thickness Wind 2 2 Speed Operation Speed (lb/in ) (lb/in ) 0.25 inch 180 mph 1.44 916.25 765.21-1005.71 9.0 GHz

0.5 inch 222 mph 2.19 814.76 572.12-981.41 5.0 GHz

1.0 inch 290 mph 3.73 780.18 526.3-956.91 3.5 GHz

The panel sections are usually spherical or flat facets that form an orange peel, triangular, hexagonal or pentagonal shape. The panels are comprised of either sandwich composites, solid laminates or thin membranes fabricated from various strong fabrics [10]. These geodesic panels are the classic example of a non-structural radome. In general, these panels are required to withstand environmental conditions, like rain, snow and wind loads up to 230 mph. Table 4-1 illustrates the typical structural requirements for geodesic radomes. The most common method for designing non-structural radomes like the panels used in geodesic radomes is to employ solid laminates or thin high-strength fabrics as environmental protection layers. The laminates and fabrics are selected based on their structural and electrical properties.

Figure 4.2, Figure 4.3, and Figure 4.4 shows the complex relative permittivity of E-glass,

S-glass and Astroquartz as a woven cloth and as a laminate. In many cases one or more of these materials form the basis of a non-structural radome. The inherent structural properties discussed in Section 2.1 in addition to the electrical properties illustrated in Figure 4.2 through Figure 4.4 make these materials ideal for radome design.

Figure 4.2 Complex Permittivity of E-glass 2D plain weave woven cloth and an E-glass plain weave vinyl ester laminate.

For non-structural radomes the inherent structural properties of the laminate

(compression, tension and shear properties) are sufficient to satisfy the stresses associated with the application. In those cases, the radome is designed such that the insertion loss associated with the laminate or fabric does not exceed the maximum allowable insertion loss for the application.

′ ′′ Figure 4.2 presents the complex relative permittivity (휀푟 = 휀푟 − 푗휀푟 ) for an 8-ounce E- glass woven fabric and an 8-ounce vinyl ester infused E-glass laminate. An optimal laminate or

′ ′′ woven cloth should have a real part near unity ( 휀푟 > 1.5) and a very small imaginary part (휀푟 >

−0.005). The real permittivity value for the woven cloth is approximately 3 and when the vinyl ester matrix is infused into the cloth through laminate processing the real part is increased to 4.5.

This E-glass material has a woven cloth and laminate imaginary permittivity of -0.05 and -0.1,

respectively. The increase in imaginary permittivity is important because it increases the insertion loss. My impedance matching method is most effective addressing impedance mismatches which are almost exclusive caused by the real part of the permittivity; the method does not effectively address insertion loss caused by material loss. Therefore, selecting composite materials with low loss is critically important to designing high performance radomes.

Figure 4.3 presents the complex relative permittivity for a 24-ounce S-glass woven cloth and a 24-ounce epoxy infused S-glass laminate. Similar to the E-glass complex relative permittivity this laminate has a greater real and imaginary part permittivity, however the value of the real part permittivity is only 3.5 instead of 4.5 as was the case for the E-glass laminate.

Moreover, the loss factor is approximately -0.1 which is comparable to the E-glass laminate.

The S-glass woven cloth and laminate are better candidates for radome design given its complex relative permittivity characteristics.

Figure 4.3 Complex permittivity of S-glass 2D plain weave woven cloth and an S-glass plain weave epoxy laminate.

Figure 4.4 Complex permittivity of Astroquartz 2D plain weave woven cloth and an Astroquartz plain weave epoxy laminate. Recall from Section 2.1.1 that quartz fibers have excellent electrical properties, but they are the most expensive fibers. Astroquartz is a commercial fiber that is constructed using quartz fibers. Figure 4.4 presents the complex relative permittivity for an 8-ounce Astroquartz woven cloth and an 8-ounce epoxy infused Astroquartz laminate. Of the three materials presented here,

′ it is clear from Figure 4.4 that Astroquartz has a laminate real permittivity (휀푟 = 3.0) closest to

′′ unity and it also has the smallest laminate imaginary permittivity (휀푟 = −0.05). These comprise the advantageous electrical properties discussed in Section 2.1.1. Figure 4.5 presents a comparison of insertion loss for the three laminates (E-glass, S-glass and Astroquartz) as a function of wavelength. These curves illustrate the impact of material properties and thickness on the insertion loss. The Astroquartz and S-glass laminates provide the best insertions loss performance, regardless of thickness. However, quartz fibers and specifically, Astroquartz is a high cost material (~$100/yd.). S-glass and E-glass are relatively low cost ($5-$10/yd)

alternatives to quartz. Clearly S-glass is the most cost effective high-strength fiber considering its insertion loss performance and cost.

Figure 4.5 Insertion loss for Astroquartz, S-glass and E-glass laminates as a function of normalized thickness

An analysis of the insertion loss illustrated in Figure 4.5 reveals the major penalty of using laminates for non-structural broadband radomes, which is the radomes have a minimum intrinsic insertion loss. Most radome applications can accept 0.5 dB of loss in the passband. However, to insure relatively low insertion loss for S-glass or Astroquartz, the laminate should be electrically

휆 thin (i.e. 푡 < 푚𝑖푛). 20

Conventional Structural Radome

The sandwich radome is the most common structural radome configuration. Sandwich radomes are more popular structural radomes than their monolithic radome counterpart because they offer greater flexibility in design parameters. Monolithic or single layer radomes

(laminates) are typically electrically thin and incorporate some form of fiber reinforcement within each layer. The fiber inclusion is used to improve the radome mechanical properties. In general, sandwich radomes have three categories A-sandwich, B-sandwich, and C-sandwich. A- sandwich configurations consist of a low density core sandwiched between two higher density thin structural skins, whereas B-sandwich radomes are the inverse, consisting of a high density core and lower density thin outer skins. The C-sandwich also known as the multilayer radome has a wall configuration consisting of 5 or more layers. Figure 4.7 provides an illustration of the four radome categories commonly employed. Sandwich radomes can be constructed using a variety of materials for the core and the outer skins; however, the number of suitable structural materials is limited.

Figure 4.7 Radome Wall Categories

Figure 4.6 Sandwich Radome Configuration

Sandwich Wall Materials

An example of a sandwich radome is illustrated in Figure 4.6. In general, sandwich radomes require low electrical loss materials for both the outer skins and core material. The outer skin components of a sandwich radome is typically a laminate. The core material is usually a low loss low density structural material such as honeycomb or polyurethane foam.

Radomes require this outer skin to have a real part permittivity (휀′) close to unity to minimize the impedance mismatch. Since the thickness of the outer skin is usually much less than the minimum passband wavelength of the radome passband the overall loss associated with the outer

skin has a negligible contribution to the reflection coefficient. Selection of an outer skin material that minimizes the impedance mismatch is more critical than the outer skin material loss. In contrast, the core material typically provides the structural stiffness for the radome and is required to be much thicker than the outer skins. In this case, the radome designer requires the core material to have a much smaller material loss component. Figure 4.8 presents the material loss for several core materials and an S-glass Cyanate Ester laminate calculated at 40 GHz. This plot illustrates the importance of selecting a low loss core material. The structural foam exhibits significant material loss (i.e. loss > 1dB) as the thickness extends pass 1”. Whereas the polypropylene and the S-glass laminate exhibit negligible loss up to 5”. Quartz honeycomb and structural foam are popular choices for sandwich composites because of their lightweight high stiffness characteristics, however, their use in structural radomes must be carefully weighed against their material loss properties. Polypropylene provides excellent material loss properties, but it is a significantly heavier material and must be used in applications where weight concerns are not a top priority.

Figure 4.8 Structural core material loss calculated at 40GHz

The structural composite geometry used to conduct the insertion loss predictions in

Figure 4.9 was chosen to match the Hexcel F161/7781 fiberglass epoxy laminates in [13]. With the assumption that the laminate structural properties

Table 4-2 Derived Structural Properties for Example 1

Radome Radome Core Derived Derived Derived Face Sheet Thickness Passband Thickness Compression Tension Flexure Material (inch) (GHz) (inch) (ksi) (ksi) (ksi) Astroquartz 0.08 2-18 1 73.2 92 94.1

S-glass 0.08 2-18 1 73.2 92 94.1

E-glass 0.08 2-18 1 73.2 92 94.1

Figure 4.9 Sandwich composite insertion loss for Astroquartz, S-glass and E-glass face sheets.

(i.e. Table 4-2) will closely resemble the properties given in Table A1.4 of [13]. The core material used in the insertion loss predictions was a 6.2 lbs/ft3 closed cell foam known as

Divinycell H. Figure 4.9 shows the insertion loss calculated for Astroquartz, S-glass and E-glass sandwich composites. The core thickness for Figure 4.9 (a) and (b) is 1” and Figure 4.90.5”, respectively. The best performing composites are the S-glass and Astroquartz variants. This result was also observed in Figure 4.5. Clearly S-glass is the best value face sheet material for radome application because it’s 10-15% stronger than E-glass and provides insertion loss comparable to Astroquartz.

Ballistic Radomes

Ballistic protection is typically categorized by its resistance to armor piercing and non- armor piercing projectiles. This work will focus on non-armor piercing ballistic protection

(NAPB). NAPB protection provides impact resistance designed to withstand the effect of non- armor piercing projectiles. Light weight ballistic armor is typically comprised of a composite consisting of a rigid solid ceramic and glass or aramid fibers with protective fabric. Ceramic ballistic armor operates using a system of layers approach, where the first layer is typically formed by ceramic materials. The function of the ceramic material is to dampen the initial impact of the projectile by providing a sufficiently rigid barrier. The ceramic must also fracture the tip of the projectile, dissipating the kinetic energy of the projectile in order distribute the impact to the second layer. The second layer is usually comprised of ductile material such as fiberglass or aramid fibers and is known as the backing. Whereas the backing is used to absorb the kinetic energy from the projectile fragments and the deformation of the ceramic [7].

Aluminum Oxide (AL2O3), boron carbide (B4C), and silicon carbide (SiC) are commonly used commercial ceramics in ceramic ballistic armor systems. These materials vary in terms of their suitability as a radome material due to their RF properties. For example, Aluminum Oxide, also known as Alumina has a relatively small lossy component (i.e. loss tangent = 0.001 at X-band

[12]); however, Alumina has a relatively large dielectric constant which makes RF transparency a challenge. Indeed, ballistic material properties have made the concept of ballistic radomes fantasy; however, I show in this work that with the proper choice of materials combined with my impedance matching methodology renders ballistic radomes achievable.

Figure 4.10 Non-AP Ballistic radome insertion loss for Figure 3.2 configuration

Table 4-3 Ballistic radome physical configuration

Polyethylene Epoxy Separator Ceramic Layer Backing Layer Thickness (in) Thickness (in) Thickness(in) Thickness (in) Polyethylene Layer 0.05 0 0.1772 0.1772 (Non-AP) Epoxy Layer (Non-AP) 0 0.05 0.1772 0.1772

Polyethylene Layer (AP) 0.05 0 0.5625 0.5

Epoxy Layer (AP) 0 0.05 0.5625 0.5

Given the ballistic armor configuration illustrated in Figure 3.2, Figure 4.10 presents the

insertion loss simulation for those two configurations. The physical configuration is presented in

Table 4-3. The ballistic radome configuration is dependent on the level of ballistic protection

required, for this example I assume non-armor piercing ballistic protection. Non-armor piercing

ballistic protection results in a less harsh reflection coefficient than does armor piercing configurations, thereby reducing the complexity of the radome design. Figure 4.11 presents the insertion loss for an armor piercing configuration. Certainly, the insertion loss reported in Figure

4.10 and Figure 4.11 illustrates that standard ballistic materials are not suited for use as radomes as currently constituted. Design approaches must be developed in order to address the insertion loss if the objective is to provide ballistic protection and RF transparency using the current suite of ballistic materials. Chapter 0 of this dissertation presents a design approach for ballistic radomes that uses standard ballistic materials.

Figure 4.11 Amor piercing ballistic radome insertion loss for Figure 3.2 configuration.

Chapter 5: Wideband Impedance Matching Methodologies

Advancements in antenna technology have increased the need for wideband broad incidence radomes. To address these advanced antennas radomes typically employ two primary design techniques: (1) Transmission-Line Method [3], illustrated in Figure 1.2, and (2) the

Generalized Scattering Method illustrated in Figure 7.1. The aim of this effort was develop a methodology for designing wideband structural and ballistic radomes, using conventional structural composite and ballistic protection materials. Chapters 2 and 3 present a comprehensive overview of both structural composites and ballistic materials. From this list of materials, I evaluated the electrical properties to identify suitable materials with compatible electromagnetic properties for my radome design methodology.

Figure 5.1 Antireflective Conceptual Approach

Wideband Impedance Matching by Dielectric Layers

A common method utilized by the optics community to increase transparency is to apply antireflective coatings to low loss substrates with the objective of suppressing the Fresnel reflections at the air substrate interface.

Figure 5.1 illustrates this approach, two mechanisms prevent structural composites from being RF transparent and subsequently effective radomes. Those two mechanisms are Fresnel reflections which are a consequence of an impedance mismatch between the radome materials and the incident media. The second mechanism is the inherent material loss of the structural composite materials. Material loss discussed in Section 2.4, is often described as loss tangent

휀′′ (δtan), where 훿 = . However, δtan is difficult to alter without changing the material’s 푡푎푛 휀′ chemical composition, which may impact its structural properties. The better approach to addressing material loss is to select structural composite materials with small loss tangents (i.e.

δtan < 0.005). Indeed, impedance mismatch and dispersion can be addressed by designing antireflective surfaces that are capable of suppressing Fresnel reflections over the passband.

Analytical Methods

Impedance matching techniques have been studied in a variety of areas, however much of the prevailing work originated in microwave engineering community (i.e. matching transmission line impedance to various load impedances).

Matching Load Z0 network

Figure 5.2 Microwave engineering impedance matching model

Figure 5.2 presents an illustration of the general impedance matching model used in microwave engineering. The characteristic (Z0) and load (ZL) impedance describe the fundamental characteristics of the problem while the matching layer is derived such that (Γ) the reflection coefficient is minimized.

푍 − 푍 Γ = 0 퐿 푍0 + 푍퐿 ( 5.1 )

Elementary reflection coefficient The objective of impedance matching is to determine the optimal matching network. The determination of what constitutes an optimal network is dependent on the problem statement. In general, shorter (fewer sections or layers) networks are better than longer networks. Matching networks are determined using a number of techniques including analytical and iterative methods. The most common analytical methods used to determine the optimal matching network are quarter wave transformation, multi-section or multilayer matching networks and tapered networks.

5.2.1 Quarter wave transformer matching network

A quarter-wave transformer network is the simplest form of a matching network used in transmission line theory, additionally it is the basis for more complex forms of matching networks. Moreover, the quarter-wave transformer can only be applied to load impedances that are strictly real, i.e. no reactive impedance. Lastly, the quarter-wave transformer is a narrowband solution because it operates on a single frequency. In electromagnetics, quarter- wave transformers are only valid for lossless or very low loss dielectrics and is also a narrowband technique. Given these parameters, the quarter-wavelength transformer is designed such that the thickness of the layer is electrically equivalent to ¼ wavelength within the materials and the intrinsic impedance (휂2) is given by ( 5.2 ) The intrinsic impedance equation is derived from the well-known theorem that maximum power transfer is achieved when the input and load impedance are equal. Therefore, the characteristic impedance of the transmission line or in

electromagnetics, the intrinsic impedance of the dielectric slab must be set such that the input reflection coefficient is forced to zero.

휂2 = √휂0휂푠푢푏

1 ( 5.2 ) 푡 = 휆 푙푎푦푒푟 4 휂2

Quarter-wavelength transformer

Where 휂0 and 휂푠푢푏 are the intrinsic impedance of the input half space and the substrate layer, respectively. Impedance matching techniques are applied to the general configurations illustrated in Figure 5.4. Figure 5.4 (a) represents an infinite medium substrate with intrinsic impedance (ηsub) while Figure 5.4 (b) represents a dielectric slab substrate also with intrinsic impedance (ηsub). In the case of the dielectric slab configuration you must apply the quarter- wavelength slab at each interface. This symmetric impedance matching concept is applied to the substrate medium at all interfaces. For most radome applications the passband is typically larger than a single frequency, therefore broader bandwidths are desired. To address broadband

휆 radome requirements the quarter-wavelength transformer can be extended by adding multiple 4 sections where the intrinsic impedances are designed to force the reflection coefficient to zero at specific frequencies within the passband. Figure 5.3 illustrates the concept of applying multiple quarter-wave sections to an infinite half space and dielectric slab.

Figure 5.4 (a) Semi-infinite or half space medium configuration. (b) Dielectric slab configuration

Figure 5.3 Quarter-wavelength multilayered configurations

5.2.2 Multilayered matching network or planar surfaces

To increase the bandwidth of the quarter-wavelength transformer, multiple sections or layers of quarter-wavelength transformers are used. Each layer thickness is matched to a corresponding frequency within the passband. The intrinsic impedances can be determined using a number of techniques, however I will discuss the binomial and Chebyshev polynomial techniques for determining intrinsic impedances. The binomial transformer yields a maximally flat passband because this technique requires the magnitude of the reflection coefficient (ρ) to equal the reflection coefficient (Γ) and the first N-1 first derivatives with respect to frequency

휋 vanish at frequency (f0) where 휃 = [14]. Whereas, the Chebyshev transformer allows the ρ to 2 vary between 0 and some maximum reflection (ρm) in an oscillatory manner. This is known as an equal ripple in the passband, which may be acceptable because the equal ripple Chebyshev transformer yields much greater bandwidth than that of the binomial technique.

5.2.3 Binomial Transformer Design

The binomial technique results in a reflection coefficient with a maximally flat passband, to realize this flat passband the intrinsic impedances are determined using ( 5.3 ) through ( 5.6 )

[14].

푁 휂푠푢푏 − 휂0 푁 (푓) = ∑ 훤푛 exp(−푗2푛휃) = exp (−푗푁휃) 푐표푠 (휃) 휂푠푢푏 + 휂0 푛=0

푁 ( 5.3 ) −푁 휂푠푢푏 − 휂0 푁 = 2 ∑ 퐶푛 exp (−푗2푛휃) 휂푠푢푏 + 휂0 푛=0

Binomial intrinsic impedances

푁 Where the binomial coefficients (퐶푛 ) is given by ( 5.4 )

푁! 퐶푁 = 푛 (푁 − 푛)! 푛! ( 5.4 )

The intrinsic impedances for each section can be calculated using ( 5.5 )

휂푛+1 −푁 푁 휂푠푢푏 ln = 2휌푛 = 2 퐶푛 ln 휂푛 휂0 ( 5.5 )

−1 2휌푚 1 휃푚 = cos | 휂 | 푠푢푏⁄ 푁 ln( 휂0) 휋푓 휃 = 2푓 0 ( 5.6 ) 훥푓 2(푓 − 푓 ) 0 푚 4 −1 2휌푚 1 = = 2 − ⁄휋 cos | 휂 | 푠푢푏⁄ 푁 푓0 푓0 ln( 휂0)

Bandwidth calculation for N-section binomial quarter-wavelength transformer

5.2.4 Chebyshev Transformer Design

The Chebyshev transformer results in a significantly wider bandwidth than the binomial transformer design, because the Chebyshev transformer can be designed such that each matching section forces the reflection coefficient (ρ) to zero at a specified frequency. The increased bandwidth does however, result in ripples within the passband. The total number of passband ripples are proportional to the total number of layers or sections that comprise the transformer.

The ripple characteristics exist because the reflection coefficient (ρ) is made to behave like

Chebyshev polynomials. What follows are the design equations for the Chebyshev transformer.

푇푛(푥) = 2푥푇푛−1 − 푇푛−2 ( 5.7 ) Chebyshev polynomial recurrence formulation

Replace x with cos 휃 yields

푇푛(cos 휃) = cos 푛휃

cos 휃 −1 cos 휃 ( 5.8 ) 푇푛 ( ) = 푇푛(sec 휃푚 cos 휃) = cos 푛 [cos ( )] cos 휃푚 cos 휃푚

Chebyshev equation to map lower and upper passband 휃푚 to 푥 = 1 and 휋 − 휃푚 to 푥 = −1.

In practice Chebyshev transformer sections are usually no more than four discrete sections therefore I have included the first four Chebyshev transformer equations in ( 5.9 ).

푇1(sec 휃푚 cos 휃) = sec 휃푚 cos 휃

2 푇2(sec 휃푚 cos 휃) = sec 휃푚 (1 + cos 2휃) −1

3 푇3(sec 휃푚 cos 휃) = sec 휃푚 (cos 3휃 + 3cos 휃) −3 sec 휃푚 cos 휃 ( 5.9 )

4 2 푇4(sec 휃푚 cos 휃) = sec 휃푚 (cos 4휃 + 4cos 2휃 + 3) −4 sec 휃푚 (cos 2휃 + 1)

First four Chebyshev polynomials mapped to 휃푚

−푗푁휃 훤(휃) = 2푒 [훤0 cos 푁휃 + 훤1 cos(푁 − 2)휃 + ⋯ + 훤푛 cos(푁 − 2푛)휃 + ⋯]

−푗푁휃 훤(휃) = A푒 푇푁(sec 휃푚 cos 휃)

휂푠푢푏 ( 5.10 ) ln ⁄휂 퐴 = 0 2푇푁(sec 휃푚)

Reflection coefficient of N section Chebyshev transformer

휂푠푢푏 1 ln( ⁄휂 ) sec 휃 = cosh [ cosh−1(| 0 |)] 푚 푁 2휌푚

훥푓 4휃 = 2 − 푚 푓 휋 0 ( 5.11 ) 휂 훤 ≃ 1 ln 푛+1 푛 2 휂푛

Computation of the bandwidth and characteristic impedances.

1 In ( 5.10 ) the last term in the series is ( ) Γ푁 for N even and Γ(푁−1)/2 cos 휃 for N odd. Equation 2 2

( 5.11 ) is used to compute the characteristic impedances of each section. Once the reflection coefficient at each section is determined using ( 5.9 ) and ( 5.10 ).

Tapered Structures or networks

A continuously varying impedance or taper can be used to design multilayered impedance networks as well. The continuous taper is very similar to the impedance matching networks discussed in Sections 5.2.3 and 5.2.4 except the impedance matching network is assumed to have an infinite number of sections. Obviously, in practice the impedance network sections will be truncated to some finite number of sections. Several common techniques for designing tapered impedance networks include the exponential [15], Gaussian, Klopfenstein

[16] , polynomial [17] tapers or calculating a tapered impedance using general optimization routines.

Given a non-dispersive, lossless medium, it has been determined by various researches

[18] and [19] that the Klopfenstein gradient index profiles is the optimum taper, in that given a

maximum reflection coefficient specification, the Klopfenstein profile yields the shortest taper length [19].

1 Γ 푧 ln[휂(푧)] = ln 휂 휂 + 0 [퐴2ϕ (2 − 1, A)] , for 0 ≤ x ≤ L 2 0 푠푢푏 cosh 퐴 퐿

푥 퐼 (퐴√1 − 푦2) 휙(푥, 퐴) = ∫ 1 푑푦, for|x| ≤ 1 0 퐴√1 − 푦2 ( 5.12 ) 1 휂 퐴 = 푐표푠ℎ−1 [ ln ( 푠푢푏)] 2Γ푚 휂0

Klopfenstein Taper

−푎푧 휂(푧) = 휂0푒 푓표푟0 < 푧 < 퐿 휂 ln 푠푢푏⁄휂 sin 훽퐿 훤(휃) = 0 푒−푗훽퐿 ( 5.13 ) 2 훽퐿

Exponential taper and resulting reflection coefficient

Figure 5.5 presents a visualization of three types of impedance tapers used in this dissertation. The first is the Klopfenstein taper calculated using ( 5.12 ) the exponential taper given in ( 5.13 ) and an impedance taper determined using an iterative design method.

Tapered impedance networks like the Klopfenstein distribution are designed to behave like high pass filters, however tapered subwavelength gratings cannot achieve high pass filter operation because at certain wavelength the grating begins to propagate non-zeroth order fields.

At these frequencies the taper breaks down. Therefore, subwavelength tapered gratings are designed such that the taper length is determined by 휆푚 the maximum wavelength within the

passband and the period is determined to insure zeroth order propagation is preserved. When both requirements for zeroth order propagation and minimum taper length are observed the resulting subwavelength grating no longer operates as a high pass filter it becomes a bandpass filter. To translate the 휂(푧) given in ( 5.12 ), ( 5.13 ) or any other method used to calculate impedance values to a geometric taper, effective medium theory (EMT) equations are typically employed [21]. Effective medium theory provides an initial radius starting point to realize a tapered subwavelength grating; however, to more closely reflect the actual dielectric profiles it may be useful to refine the taper geometry using more rigorous computational methods like the

RCWA method. Chapter 9 describes in further detail the fabrication approach I used to design subwavelength gratings.

Klopfenstein taper Exponential taper

Iterative Optimized taper

Figure 5.5 Example of Klopfenstein, exponential and optimized multilayered tapered surfaces

Iterative Optimization

An optimization algorithm is a common method to determine or refine an impedance matching network. In most cases the analytical methods described in Section 5.2 are the starting point for the solution and an iterative optimization algorithm is applied to the analytical solution to refine the solution. I implemented an iterative optimization algorithm as shown in Figure 5.6.

Here, the solution for the reflected energy of a multilayered structure was calculated as a function of frequency and angle of incidence. The optimization algorithm is then used to refine the characteristic impedances such that an objective function is minimized. The objective function is chosen to facilitate satisfying the design criteria. A number of iterative optimization algorithms could be used to refine a design. These include traditional derivative-based algorithms, genetic algorithms or direct pattern search algorithms.

An advantage of both genetic and pattern search algorithms is that they do not require derivatives, and they work well on non-differentiable, stochastic, and discontinuous objective functions. Both simple genetic algorithms and direct pattern search algorithms have been implemented and tested for determining optimized impedance networks. Although both methods produced comparable results, the pattern search algorithm was often computationally less expensive. Similar to genetic algorithms, a pattern search can be effective in finding a global minimum because of the nature of its search method.

5.4.1 Pattern Search

Pattern search is an optimization algorithm that is part of the Matlab™ optimization toolbox and works by searching or polling a set of points within a mesh or grid. This grid expands or contracts depending on polling success or finding a solution within the mesh that

satisfies the objective function. After a successful poll the previous point moves to the successful poll location and the mesh is expanded.

Figure 5.6 General iterative optimization algorithm

If a successful poll is not found, then the mesh is contracted and the current point is retained.

The search can be stopped using a number of different criteria, such as: reaching a minimum pattern size, or exceeding a maximum number iterations set by the user, or specifying and attaining a minimum distance between current points in successive iterations.

5.4.2 Genetic Algorithm

A genetic algorithm (GA) is a method for solving both constrained and unconstrained optimization problems based on a natural selection process that mimics biological evolution. The algorithm repeatedly modifies a population of individual solutions. At each step, the genetic algorithm randomly selects individuals from the current population and uses them as parents to produce the children for the next generation. Over successive generations, the population

"evolves" toward an optimal solution. [20]. I utilized the genetic algorithm optimization that is part of the Matlab™ optimization toolbox.

Textured Surfaces

Sections 5.2 and 5.3 focused on methods to determine effective impedance matching networks, i.e. determining optimal intrinsic impedance for impedance matching. This section discusses textured surfaces which is an approach to implementing or realizing the optimal impedance values calculated in Sections 5.2 and 5.3. Textured surface implementation is the procedure to take a dielectric constant distribution or profile and realize a physical structure.

Implementation at radio frequency (RF) is typically accomplished by fabricating subwavelength grating structures using computer numerically controlled (CNC) machining techniques. A subwavelength grating is defined as a dielectric material with either a 1-D or 2-D periodicity (Λx,

Λy), that over a broadband of frequencies exhibit dielectric properties of a homogeneous medium. In order for the subwavelength grating to exhibit effective dielectric properties over the operational band the grating periodicity must satisfy the requirement for zeroth order propagation given in ( 5.14 ).

휆0 Λ푥 < 1 2 2 2 ⁄2 [max(푛푠 , 푛푖 ) − (푛푖 sin 휃푖 cos 휙푖) ] + |푛푖 sin 휃푖 cos 휙푖|

휆0 Λ푦 < 1 ( 5.14 ) 2 2 2 ⁄2 [max(푛푠 , 푛푖 ) − (푛푖 sin 휃푖 cos 휙푖) ] + |푛푖 sin 휃푖 cos 휙푖|

Zeroth order diffraction x-dimension periodicity requirement, Zeroth order diffraction y- dimension periodicity requirement

Figure 5.7 – 1-D and 2-D Periodicity

Although, ( 5.14 ) does not specify any particular type of periodicity, the grating structures designed in this effort are all 2-D lattice structures. One of the key aspects of implementation is optimizing the effective dielectric constant dynamic range, which is the ability of the fabricated structure to reproduce the dielectric properties of the distribution from the largest dielectric constant value down to unity. Figure 5.7 presents an illustration of a 1D and

2D periodicity while Figure 5.8 provides an example of a unit cell with a rectangular and hexagonal lattice. The light gray circles represent the elemental material and the dark gray represents the background material. The elemental geometry can be increased or decreased, the relative area of the elements determines the overall fill within the unit cell. This fill factor criterion is used to control the effective dielectric constant of the structure. Figure 5.8 shows that the hexagonal lattice realizes a greater fill factor than the rectangular lattice. Given that filling factors govern the effective dielectric constant dynamic range for antireflective surface.

Increasing the effective dielectric constant dynamic range allows the design of a wider range of dielectric profiles. Therefore, to insure maximum flexibility in the design of antireflective

surfaces, all of the subwavelength gratings were designed using the hexagonal lattice. The maximum fill factor for a hexagonal lattice is 0.9069, however, I restricted the maximum fill factor for my designs to 0.866 to insure the structures could be easily fabricated.

Figure 5.8 – Common Grating Lattice Types and Fill Factors for CNC Implementation

Setting the maximum fill factor and lattice type allows the effective dielectric constant dynamic range to be set, thereby bounding the dielectric profile design space. To bound the dielectric profiles for CNC implementation I modelled the subwavelength grating structures using the

Rigorous Coupled Wave Analysis (RCWA). The RCWA calculates the transmission and reflection response of the subwavelength grating structure rigorously. After calculating the transmission response, I extracted the effective dielectric constant using ( 5.15 ).

−푗푘1푙1 휏1휏2푒 푇 = −2푗푘 푙 1 − 휌1휌2푒 1 1

휂1 − 휂푎푖푟 휂푎푖푟 − 휂1 ( 5.15 ) 휌1 = , 휌2 = , 휏1 = 1 + 휌2, 휏2 = 1 + 휌2 휂1 + 휂푎푖푟 휂1 + 휂푎푖푟

Transmission equation for single slab [3]

휋푎(푧)2 푣 = 훼 푓 휀 훬2

휀ℎ − 휀푏 훼휀 = 휀ℎ + 휀푏 ( 5.16 ) 휀푏푎푐푘(푣푓훼휀 + 1) 휀푒푓푓 = 1 − 푣푓훼휀 Effective Medium Theory

Table 5-1 CNC dielectric constant dynamic range

CNC Implementation Dielectric Constant Dynamic Range 0.866 Fill Factor

Slab Dielectric Minimum Effective AR Material Total Dynamic Range Constant Dielectric Constant Polycarbonate 2.9 1.53 1.53 – 2.9 ABS – 3.5 3.5 1.74 3.5 – 1.74

휋푎(푧)2 푣 = 훼 푓 휀 훬2

휀ℎ − 휀푏 훼휀 = 휀ℎ + 휀푏 ( 5.16 ) 휀푏푎푐푘(푣푓훼휀 + 1) 휀푒푓푓 = 1 − 푣푓훼휀 Effective Medium Theory

Table 5-1 provides an example of the dielectric constant range for two materials. A subwavelength grating will achieve a minimum dielectric constant of 1.53 when the background material is polycarbonate and the element material is air and the grating has a hexagonal lattice fill factor of 0.866. The subwavelength grating will realize a dielectric constant of 2.9 when the hexagonal lattice fill factor is zero.

5.5.1 Continuously varied textured surfaces

At these frequencies the taper breaks down. Therefore, subwavelength tapered gratings are designed such that the taper length is determined by 휆푚 the maximum wavelength within the passband and the period is determined to insure zeroth order propagation is preserved. When both requirements for zeroth order propagation and minimum taper length are observed the resulting subwavelength grating no longer operates as a high pass filter it becomes a bandpass

filter. To translate the 휂(푧) given in ( 5.12 ), ( 5.13 ) or any other method used to calculate impedance values to a geometric taper, effective medium theory (EMT) equations are typically employed [21]. Effective medium theory provides an initial radius starting point to realize a tapered subwavelength grating; however, to more closely reflect the actual dielectric profiles it may be useful to refine the taper geometry using more rigorous computational methods like the

RCWA method. Chapter 9 describes in further detail the fabrication approach I used to design subwavelength gratings.

Chapter 6: Numerical Methods

In this chapter I will describe the numerical methods used to design and analyze the antireflective structures presented in Chapters 7 and 0. In general, subwavelength gratings like those presented in Section 7.2 are simulated using effective medium theory [22] or rigorous electromagnetic models. Effective medium theory approaches are closed form expressions that provide an effective dielectric constant for subwavelength grating geometries that satisfy certain criteria. Namely, the normalized period of the subwavelength grating must produce zeroth order propagation, this is accomplished when ( 5.14 ) is satisfied. Closed form expressions for 2- dimensional subwavelength gratings are very difficult to determine [22]. Moreover, effective medium theory breaks down as the normalized period gets closer to unity, this is known as approaching the resonance region of the structure. Antireflective structures designed in this dissertation cover a wideband of frequencies; at the low frequency portion of the passband the normalized period is much smaller than one, and EMT expressions are valid. However, at the higher portion of the passband the structures begin to enter the resonance region and effective medium theory expressions begin to breakdown. To insure a robust design, I chose to simulate the antireflective structures using the RCWA model. The RCWA model is one of the most widely used methods for accurate analysis of diffracted electromagnetic waves by periodic structures. Due to the rigorous nature of this model, the solution is valid regardless of grating period and incident electromagnetic wavelength.

Multilayered Dielectrics

Textured surface and homogeneous dielectrics are the most common application of multilayered impedance matching networks used in electromagnetics. Efficient computation of their electromagnetic response is key to designing multilayered dielectrics. Calculating the electromagnetic response of a homogeneous multilayered dielectric is straightforward and is often accomplished using a recursive formulation. Given the structure illustrated in Figure 6.1 the recursive formula presented in [23] can be used to determine its electromagnetic response, the explicit formulation is given in equations ( 6.1 ), ( 6.2 ), ( 6.3 ) and ( 6.4 ).

Figure 6.1 Multilayered Dielectric

푟 퐸⊥ 퐵0 Γ⊥ = 푖 = 퐸⊥ 퐴0

푡 퐸⊥ 1 ( 6.1 ) 푇⊥ = 푖 = 퐸⊥ 퐴0

Perpendicular Polarization (Horizontal)

푟 퐸∥ 퐶0 Γ∥ = 푖 = 퐸∥ 퐷0

푡 퐸∥ 1 ( 6.2 ) 푇∥ = 푖 = 퐸∥ 퐷0

Perpendicular Polarization (Horizontal)

퐴푁+1 = 퐶푁+1 = 1

퐵푁+1 = 퐷푁+1 = 0

푒휓푖 퐴 = [퐴 (1 + 푌 ) + 퐵 (1 − 푌 )] 푗 2 푗+1 푗+1 푗+1 푗+1

푒−휓푖 퐵 = [퐴 (1 − 푌 ) + 퐵 (1 + 푌 )] 푗 2 푗+1 푗+1 푗+1 푗+1 ( 6.3 ) 푒휓푖 퐶 = [퐶 (1 + 푍 ) + 퐷 (1 − 푍 )] 푗 2 푗+1 푗+1 푗+1 푗+1

푒−휓푖 퐷 = [퐶 (1 − 푍 ) + 퐷 (1 + 푍 )] 푗 2 푗+1 푗+1 푗+1 푗+1

Formulation of the reflection and transmission coefficients for an N-layer stack of planar slabs having permittivity, or permeability.

cos 휃푗+1 휀푗+1(1 − 푗 tan 훿푗+1) 푌푗+1 = √ cos 휃푗 휀푗(1 − 푗 tan 훿푗)

cos 휃푗+1 휀푗(1 − 푗 tan 훿푗) 푍푗+1 = √ cos 휃푗 휀푗+1(1 − 푗 tan 훿푗+1)

휓푗 = 푑푗훾푗 cos 휃푗 ( 6.4 )

훾푗 = ±√푗휔휇푗(휎푗 + 푗휔휀푗)

휃푗 = complex angle of refraction in the jth layer

Oblique incidence N-layer slab refractive function

This formulation is a highly efficient way to compute the electromagnetic response of multilayered dielectrics like the one depicted in Figure 6.1. However, this efficient formulation can only be used for textured surfaces when they are implemented as subwavelength gratings whose individual layers behave like an effective dielectric constant. In those cases, effective medium theory (EMT) can be used to determine the effective dielectric constant of each layer.

Computing the electromagnetic response of subwavelength gratings when the EM wave enters the resonance region of the grating will lead to erroneous results. Therefore, the electromagnetic response of subwavelength gratings must be calculated using more rigorous methods. Section

6.2 presents the RCWA method.

Rigorous Coupled Wave Method

The RCWA algorithm was originally reported by Moharam and Gaylord in [24], and over the years since the publishing of the original paper several authors have improved the implementation. The RCWA is implemented by first separating the problem into three regions as illustrated in Figure 6.2. Figure 6.2 is a depiction of a 4-layer subwavelength grating structure with 2-dimensional cylindrical elements. The grating structure is assumed to be infinitely periodic in the lateral directions. RCWA can handle multilayered, infinitely periodic dielectric structure that are sandwiched between to semi-infinite half spaces. The objective of the RCWA is to obtain the exact solution of Maxwell’s equations for the electromagnetic diffraction by a grating structure.

1. Incident Region

2. Grating Region

3. Exit Region

Figure 6.2 Two-dimensional periodic dielectric grating and problem geometry

First, the permittivity, r, or the index of refraction, ns of the grating are given and the period of the gratings along the x and y axes, denoted by x and y respectively. Again, the

periodicity should be smaller than the material wavelength (i.e. o o ) x  ,  y  ns sin()cos() ns sin()sin() to avoid activating any diffractive orders, other than the zeroth order. The diameter of the hole and the depth of each layer are denoted by dn and hn.

(−j퐤0⋅퐫) 푬푰(풓) = 푢̂푒

퐤0 = 훼0풙̂ + 훽0풚̂ + 푟00풛̂ 2휋 ( 6.5 ) 훼 = 푛 푘 sin 휃 cos 휙, 훽 = 푛 푘 sin 휃 sin 휙, 푟 = 푛 푘 cos 휃 with 푘 = 0 1 0 1 00 1 휆 Incident electric field written in vector notation

Now we describe the fields outside the grating region beginning with the incident electric field in the incident region. The polarization vector 퐮̂ is given below:

퐮̂ = (cos 휓 cos 휃 cos 휙 − sin 휓 sin 휙)퐱̂ + (cos 휓 cos 휃 sin 휙 + sin 휓 cos 휙)퐲̂

− (cos 휓 sin 휃)퐳̂

The position vector r is given below: ( 6.6 ) 퐫 = 퐱̂푥 + 퐲̂푦 + 퐳̂푧

푖푛푐 [−푗(훼0푥+훽0푦+푟00푧)] 퐸푥 (푟) = (cos 휓 cos 휃 cos 휙 − sin 휓 sin 휙)exp 풙̂

푖푛푐 [−푗(훼0푥+훽0푦+푟00푧)] 퐸푦 (푟) = (cos 휓 cos 휃 sin 휙 + sin 휓 cos 휙)exp 풚̂ ( 6.7 ) 푖푛푐 [−푗(훼0푥+훽0푦+푟00푧)] 퐸푧 (푟) = − cos 휓 sin 휃 exp 풛̂

Incident electric field in the incident region.

Where 휃 is the angle between the z-axis and the 퐤0 vector,휙, is the angle between the x- axis and the plane of incidence (defined by 퐤0 and the z-axis) and 휓, represents the polarization angle defined as the angle between the polarization vector 퐮̂ and the plane of incidence.

Separating out the electric field components of ( 6.5 ) yields ( 6.7 ). After writing the incident field where 휃 is the angle between the z-axis and the 퐤0 vector,휙, is the angle

푬푅퐸퐹(풓) = ∑ 푹mnexp (−푗풌1mn ∙ 풓) m,n

푬푻푹푨푵(풓) = ∑ 푻mnexp (−푗풌2mn ∙ (풓 − 풉풛̂) m,n ( 6.8 ) 풌1mn = αm풙̂ + 훽n풚̂ − 푟mn풛̂

풌2mn = αm풙̂ + 훽n풚̂ + 푡mn풛̂

Diffracted electric fields in the incident and exit regions

between the x-axis and the plane of incidence (defined by 퐤0 and the z-axis) and 휓, represents the polarization angle defined as the angle between the polarization vector 퐮̂ and the plane of

incidence. The diffracted electric field in the incident region is comprised solely of the reflected fields and the diffracted electric field in the exit region is comprised solely of the transmitted electric field.

Considering this geometry includes two-dimensional elements the diffracted electric fields are expressed in ( 6.8 )

2πm 2πn 훼 = 훼 + , 훽 = 훽 + , and 푚 0 dx 푛 0 dy

2 2 2 2 2 2 √(푛1푘) − 훼푚 − 훽푛, 훼푚 + 훽푛 ≤ (푛1푘) 푟푚푛 = { 2 2 2 2 2 2 −푗√훼푚 + 훽푛 − (푛1푘) , 훼푚 + 훽푛 > (푛1푘) ( 6.9 ) 2 2 2 2 2 2 √(푛2푘) − 훼푚 − 훽푛, 훼푚 + 훽푛 ≤ (푛2푘) 푡푚푛 = { 2 2 2 2 2 2 −푗√훼푚 + 훽푛 − (푛2푘) , 훼푚 + 훽푛 > (푛2푘)

The reflected electric field components in the incident region are given in ( 6.11 ) while the transmitted electric field polarization components are given in ( 6.12 ). Now that we have written the electric fields in regions I and III we simply use Maxwell’s curl equation ( 6.10 ) to determine the incident and diffracted magnetic fields in those same regions.

풙̂ 풚̂ 풛̂

1 1 휕 휕 휕 훻 × 푬 = −jωμ푯 ≡ 퐻 = − 훻 × 푬 = − 푗휔휇 푗휔휇 휕x 휕y 휕z [Ex Ey Ez] ( 6.10 )

Faradays Law relating the magnetic field to the curl of the electric field in differential form.

푟 퐸푥 (푟) = ∑ 푹mnexp [−푗(α푚푥 + 훽푛푦 − 푟푚푛z)]퐱̂ m,n

푟 퐸푦(푟) = ∑ 푹mnexp [−푗(훼푚푥 + 훽푛푦 − 푟푚푛푧)]퐲̂ m,n ( 6.11 )

푟 퐸푧 (푟) = ∑ 푹푚푛 exp[−푗(훼푚푥 + 훽푛푦 − 푟푚푛)] 퐳̂ 푚,푛

Reflected electric field in the incident region

The incident magnetic field is given in ( 6.13 ), while the diffracted magnetic fields are also determined using ( 6.10 ) and are given below in ( 6.13 ) and ( 6.14 ).

푡 퐸푥(r) = ∑ 퐓mnexp [−푗 (훼푚푥 + 훽푛푦 + 푟푚푛(푧 − ℎ))퐱̂] m,n

푡 퐸푦(r푦) = ∑ 퐓mnexp [−푗 ((훼푚푥 + 훽푛푦 + 푟푚푛(푧 − ℎ)))퐲̂] m,n ( 6.12 )

푡 퐸푧(ℎ) = ∑ 퐓mnexp [−푗 ((훼푚푥 + 훽푛푦 + 푟푚푛(푧 − ℎ)))퐳̂ m,n

Transmitted electric field in the exit region

1 푖푛푐 [−푗(훼0푥+훽0푦+푟00푧)] 퐻푥 (푟) = (훽0푢푧 − 푟00푢푦)exp 풙̂ 휔휇0

1 푖푛푐 [−푗(훼0푥+훽0푦+푟00푧)] 퐻푦 (푟) = (푢푥푟00 − 푢푧훼0)훼0exp 풚̂ 휔휇0 ( 6.13 )

1 푖푛푐 [−푗(훼0푥+훽0푦+푟00푧)] 퐻푧 (푟) = (훼0푢푦 − 훽0푢푥)exp 풛̂ 휔휇0

Incident magnetic field components

푟 1 퐻푥 = (훽푛 + 푟푚푛) ∑ 푅푚푛exp ([−푗(훼푚푥 + 훽푛푦 − 푟푚푛푧)] 휔휇0 푚,푛

푟 1 퐻푦 = (푟푚푛 + 훼푚) − ∑ 푅푚푛exp ([−푗(훼푚푥 + 훽푛푦 − 푟푚푛푧)] 휔휇0 푚,푛 ( 6.14 )

푟 1 퐻푧 = (훼푚 − 훽푛) ∑ 푅푚푛exp ([−푗(훼푚푥 + 훽푛푦 − 푟푚푛푧)] 휔휇0 푚,푛

Diffracted magnetic fields in the incident region

푡 1 퐻푥 = (훽푛 − 푡푚푛) ∑ 푇푚푛exp ([−푗(훼푚푥 + 훽푛푦 + 푡푚푛(푧 − ℎ))] 휔휇0 푚,푛

푡 1 퐻푦 = (푡푚푛 − 훼푚) ∑ 푇푚푛exp ([−푗(훼푚푥 + 훽푛푦 + 푡푚푛(푧 − ℎ))] 휔휇0 푚,푛 ( 6.15 )

푡 1 퐻푧 = (훼푚 − 훽푛) ∑ 푇푚푛exp ([−푗(훼푚푥 + 훽푛푦 + 푡푚푛(푧 − ℎ))] 휔휇0 푚,푛

Diffracted magnetic fields in exit region

Now that the electric and magnetic fields in the incident and exit region have been determined we must now write the electric and magnetic fields in the grating region. To determine these field components, we begin with Maxwell’s coupled curl equations. Given the problem geometry illustrated in Figure 6.2 we must compute the fields within each grating layer

(퐸푝, 퐻푝) as well. The permittivity within each layer is a periodic function of x and y, but in the

z-direction the permittivity is assumed to be constant. The transverse components within each layer are then written as an expansion in terms of Floquet space.

1 퐸(푟) = 훻 × 퐻(푟) 푗휔휀 1 휕 휕 휕 휕 휕 휕 퐸(푟) = [( 퐻 − 퐻 ) + ( 퐻 − 퐻 ) + ( 퐻 − 퐻 )] 푗휔휀 휕푦 푧 휕푧 푦 휕푧 푥 휕푥 푧 휕푥 푦 휕푦 푥 1 휕 휕 퐸퐼퐼 = ( 퐻 − 퐻 ) 푥 푗휔휀 휕푦 푧 휕푧 푦 ( 6.16 ) 1 휕 휕 퐸퐼퐼 = ( 퐻 − 퐻 ) 푦 푗휔휀 휕푧 푥 휕푥 푧 1 휕 휕 퐸퐼퐼 = ( 퐻 − 퐻 ) 푧 푗휔휀 휕푥 푦 휕푦 푥 Electric Field in the grating region

1 퐻(푟) = − 훻 × 퐸(푟) 푗휔휇0 1 휕 휕 휕 휕 휕 휕 퐻(푟) = − [( 퐸푧 − 퐸푦) + ( 퐸푥 − 퐸푧) + ( 퐸푦 − 퐸푥)] 푗휔휇0 휕푦 휕푧 휕푧 휕푥 휕푥 휕푦 1 휕 휕 퐻퐼퐼 = − ( 퐸 − 퐸 ) 푥 푗휔휇 휕푦 푧 휕푧 푦 0 ( 6.17 ) 퐼퐼 1 휕 휕 퐻푦 = − ( 퐸푥 − 퐸푧) 푗휔휇0 휕푧 휕푥

퐼퐼 1 휕 휕 퐻푧 = − ( 퐸푦 − 퐸푥) 푗휔휇0 휕푥 휕푦 Magnetic field in the grating region.

푝 푝 퐸푥 = ∑ 푆푥푚푛(푧) exp (−푗[푘푥푚푥 + 푘푦푛푦) 푚,푛 ( 6.18 ) 푝 푝 퐸푦 = ∑ 푆푦푚푛(푧) exp (−푗[푘푥푚푥 + 푘푦푛푦) 푚,푛

푝 푝 퐸푧 = ∑ 푆푧푚푛(푧) exp (−푗[푘푥푚푥 + 푘푦푛푦) 푚,푛 Fourier Expansion of the electric field components in the grating region.

Equations ( 6.16 ) and ( 6.17 ) are Maxwell’s coupled electric and magnetic field equations in the grating region, to determine the field components in the grating region we Fourier expand the fields in each layer of the grating region according to ( 6.18 ).

푝 휀0 푝 퐻푥 = −푗√ ∑ 푈푥푚푛(푧) exp (−푗[푘푥푚푥 + 푘푦푛푦) 휇0 푚,푛

푝 휀0 푝 퐻푦 = −푗√ ∑ 푈푦푚푛(푧) exp (−푗[푘푥푚푥 + 푘푦푛푦) 휇0 푚,푛

푝 휀0 푝 퐻푧 = −푗√ ∑ 푈푧푚푛(푧) exp (−푗[푘푥푚푥 + 푘푦푛푦) ( 6.19 ) 휇0 푚,푛 2휋푚 푘푥푚 = 푘0 − 훬푥 2휋푛 푘푦푚 = 푘0 − 훬푦 Fourier expansion of the magnetic field components in the grating region.

We then write Maxwell’s equations in Fourier space by substituting ( 6.18 ), ( 6.19 ), and

( 6.20 ) into ( 6.16 ) and ( 6.17 ). For brevity, I leave the majority of the simplification and algebra to the reader. Several key points to recognize are that the permittivity and permeability distributions are multiplied by the Fourier transforms of the electric and magnetic field, resulting in the product of two infinite sums and it can be simplified using the Cauchy product rule. The second notable operation to recognize is that the partial derivatives of the complex space harmonic amplitudes become ordinary derivatives when because they are only a function of z.

푝 푝 2휋푚 2휋푛 휀푟 (푥, 푦) = ∑ 휀푚푛 exp (푗[ 푥 + 푦) 훬푥 훬푦 푚,푛

푝 푝 2휋푚 2휋푛 휇푟 (푥, 푦) = ∑ 휇푚푛 exp (푗[ 푥 + 푦) 훬푥 훬푦 푚,푛

푝 −1 푝 2휋푚 2휋푛 [휀푟 (푥, 푦)] = ∑ 휉푚푛 exp (−푗[ 푥 + 푦) 훬푥 훬푦 ( 6.20 ) 푚,푛

푝 −1 푝 2휋푚 2휋푛 [휇푟 (푥, 푦)] = ∑ 휒푚푛 exp (−푗[ 푥 + 푦) 훬푥 훬푦 푚,푛 Fourier transform of the material permittivity and permeability within the grating region in the x-y direction.

1 휕 휕 퐸퐼퐼 = ( 퐻 − 퐻 ) 푥 푗휔휀 휕푦 푧 휕푧 푦

휕 휕 1 휕 휕 푝 퐻푦 = [ ( 퐸푦 − 퐸푥)] − 푗휔휀0휀(푥, 푦)퐸푥 휕푧 휕푦 푗휔휇0 휕푥 휕푦

푝 휕 푝 휕푈푦푚푛 (푧) 휀0 퐻푦 = − √ ∑ exp (−푗[푘푥푚푥 + 푘푦푛푦] 휕푧 휕푧 휇0 푚,푛

푝 푝 2휋푚 −푗휔휀0휀(푥, 푦)퐸푥 = −j휔휀0 ∑ 휀푚푛exp (−푗 [ 푥 훬푥 푚,푛

2휋푛 푝 + 푦]) ∑ 푆푥푚푛(푧)exp(−푗[푘푥푚푥 + 푘푦푛푦]) 훬푦 푚,푛

푝 푝 푝 −푗휔휀0휀(푥, 푦)퐸푥 = −j휔휀0 ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) ∑ 휀푚−푟,푛−푞푆푥푟푞(푧) 푚,푛 푟,푞 휕 1 휕 휕 [ ( 퐸푦 − 퐸푥)] 휕푦 푗휔휇0 휕푥 휕푦

푘푦푛 1 푝 푝 = [ ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (푘푥푚푆푦푚푛(푧) − 푘푦푛푆푥푚푛(푧))] ( 6.21 ) 푗휔휇0 휇(푥, 푦) 푚,푛

푘푦푞 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)])] 푗휔휇0 푚,푛 푟,푞

푝 휕푈푦푚푛 (푧) 휀0 − √ ∑ exp (−푗[푘푥푚푥 + 푘푦푛푦] 휕푧 휇0 푚,푛

푝 푝 = −j휔휀0 ∑ exp (−푗[푘푥푚푥 + 푘푦푛푦]) ∑ 휀푚−푟,푛−푞푆푥푟푞 (푧) + 푚,푛 푟,푞

푘푦푞 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)])] 푗휔휇0 푚,푛 푟,푞 푝 휕푈푦푚푛(푧) 푝 푝 푘푦푞 푝 푝 푝 = 푘0 ∑ 휀푚−푟,푛−푞푆푥푟푞(푧) + ∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)] 휕푧 푘0 푟,푞 푟,푞 Derivation of the Fourier space magnetic field spatial harmonic in the y-direction.

1 휕 휕 퐸퐼퐼 = ( 퐻 − 퐻 ) 푦 푗휔휀 휕푧 푥 휕푥 푧

휕 푝 퐼퐼 휕 1 휕 휕 퐻푥 = 푗휔휀0휀(푥, 푦)퐸푥 − [ ( 퐸푦 − 퐸푥)] 휕푧 휕푥 푗휔휇0휇(푥, 푦) 휕푥 휕푦

푝 휕 푝 휀0 휕푈푥푚푛(푧) 퐻푥 = −√ ∑ exp (−푗[푘푥푚푥 + 푘푦푛푦] 휕푧 휇0 휕푧 푚,푛

푝 푝 2휋푚 2휋푛 푝 푗휔휀0휀(푥, 푦)퐸푦 = j휔휀0 ∑ 휀푚푛exp (−푗 [ 푥 + 푦]) ∑ 푆푦푚푛(푧)exp(−푗[푘푥푚푥 + 푘푦푛푦]) 훬푥 훬푦 푚,푛 푚,푛

퐼퐼 푝 푝 푗휔휀0휀(푥, 푦)퐸푦 = j휔휀0 ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) ∑ 휀푚−푟,푛−푞푆푦푟푞 (푧) 푚,푛 푟,푞 휕 1 휕 휕 [ ( 퐸 − 퐸 )] 휕푥 푗휔휇(푥, 푦) 휕푥 푦 휕푦 푥

푘푥푚 1 푝 푝 = [ ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (−푘푥푚푆푦푚푛(푧) + 푘푦푛푆푥푚푛(푧))] 푗휔휇0 휇(푥, 푦) 푚,푛 ( 6.22 )

푘푥푟 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휒푚−푟,푛−푞[−푘푥푟푆푦푟푞(푧) + 푘푦푞푆푥푟푞(푧)])] 푗휔휇0 푚,푛 푟,푞

푝 휕푈푥푚푛(푧) 휀0 − √ ∑ exp (−푗[푘푥푚푥 + 푘푦푛푦] 휕푧 휇0 푚,푛

푝 푝 = j휔휀0 ∑ exp (−푗[푘푥푚푥 + 푘푦푛푦]) ∑ 휀푚−푟,푛−푞푆푦푟푞 (푧) + 푚,푛 푟,푞

푘푥푟 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)])] 푗휔휇0 푚,푛 푟,푞 푝 휕푈푥푚푛(푧) 푝 푝 푘푥푟 푝 푝 푝 = −푘0 ∑ 휀푚−푟,푛−푞푆푦푟푞(푧) + ∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)] 휕푧 푘0 푟,푞 푟,푞 Derivation of the Fourier space magnetic field spatial harmonic in the x-direction.

1 휕 휕 퐻퐼퐼 = − ( 퐸 − 퐸 ) 푥 푗휔휇 휕푦 푧 휕푧 푦

휕 휕 1 휕 휕 푝 퐸푦 = [ ( 퐻푦 − 퐻푥)] + 푗휔휇0휇(푥, 푦)퐻푥 휕푧 휕푦 푗휔휀0휀(푥, 푦) 휕푥 휕푦 휕 휕푆푝 (푧) 퐸 = 푦푚푛 ∑ exp (−푗[푘 푥 + 푘 푦] 휕푧 푦 휕푧 푥푚 푦푛 푚,푛

푝 푝 2휋푚 푗휔휇0휇(푥, 푦)퐻푥 = j휔휇0 ∑ 휇푚푛exp (푗 [ 푥 훬푥 푚,푛

2휋푛 휀0 푝 + 푦]) (−푗√ ) ∑ 푈푥푚푛(푧)exp(−푗[푘푥푚푥 + 푘푦푛푦]) 훬푦 휇0 푚,푛

푝 푝 푝 푗휔휇0휇(푥, 푦)퐻푥 = 푘0 ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) ∑ 휇푚−푟,푛−푞푈푥푟푞(푧) 푚,푛 푟,푞 휕 1 휕 휕 [ ( 퐻푦 − 퐻푥)] 휕푦 푗휔휀0휀(푥, 푦) 휕푥 휕푦 ( 6.23 )

푘푦푛 1 휀0 푝 푝 = ( [ (푗√ ) ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (푧)(푘푥푚푈푦푚푛 − 푘푦푛푈푥푚푛)]) 푗휔휀0 휀(푥, 푦) 휇0 푚,푛

푘푦푞 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞(푧) − 푘푦푟푈푥푟푞(푧)])] 푘0 푚,푛 푟,푞 휕푆푝 (푧) 푦푚푛 ∑ exp (−푗[푘 푥 + 푘 푦] = 푘 ∑ exp(−푗[푘 푥 + 푘 푦]) ∑ 휇푝 푈푝 (푧) + 휕푧 푥푚 푦푛 0 푥푚 푦푛 푚−푟,푛−푞 푥푟푞 푚,푛 푚,푛 푟,푞

푘푦푞 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞(푧) − 푘푦푟푈푥푟푞(푧)])] 푘0 푚,푛 푟,푞 푝 휕푆푦푚푛(푧) 푝 푝 푘푦푞 푝 푝 푝 = 푘0 ∑ 휇푚−푟,푛−푞푈푥푟푞(푧) + ∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞(푧) − 푘푦푟푈푥푟푞(푧)] 휕푧 푘0 푟,푞 푟,푞 Derivation of the Fourier space electric field spatial harmonic in the y-direction.

1 휕 휕 퐻퐼퐼 = − ( 퐸 − 퐻 ) 푦 푗휔휇 휕푧 푥 휕푥 푧

휕 휕 1 휕 휕 푝 퐸푥 = − [ ( 퐸푦 − 퐸푥)] − 푗휔휇0휇(푥, 푦)퐻푦 휕푧 휕푥 푗휔휀0휀(푥, 푦) 휕푥 휕푦 휕 휕푆푝 (푧) 퐸 = 푥푚푛 ∑ exp (−푗[푘 푥 + 푘 푦] 휕푧 푥 휕푧 푥푚 푦푛 푚,푛

푝 푝 2휋푚 −푗휔휇0휇(푥, 푦)퐻푦 = −j휔휇0 ∑ 휇푚푛exp (푗 [ 푥 훬푥 푚,푛

2휋푛 휀0 푝 + 푦]) (−푗√ ) ∑ 푈푦푚푛 (푧)exp(−푗[푘푥푚푥 + 푘푦푛푦]) 훬푦 휇0 푚,푛

푝 푝 푝 −푗휔휇0휇(푥, 푦)퐻푦 = −푘0 ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) ∑ 휇푚−푟,푛−푞푈푦푟푞 (푧) 푚,푛 푟,푞 휕 1 휕 휕 [ ( 퐻푦 − 퐻푥)] 휕푥 푗휔휀0휀(푥, 푦) 휕푥 휕푦

푘푥푚 1 휀0 푝 = ( [ (−푗√ ) ∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (푘푥푚푈푦푚푛(푧) ( 6.24 ) 푘0 휀(푥, 푦) 휇0 푚,푛

푝 + 푘푦푛푈푥푚푛(푧))])

푘푥푟 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞(푧) + 푘푦푞푈푥푟푞(푧)])] 푘0 푚,푛 푟,푞 휕푆푝 (푧) 푥푚푛 ∑ exp (−푗[푘 푥 + 푘 푦] = −푘 ∑ exp(−푗[푘 푥 + 푘 푦]) ∑ 휇푝 푈푝 (푧) + 휕푧 푥푚 푦푛 0 푥푚 푦푛 푚−푟,푛−푞 푦푟푞 푚,푛 푚,푛 푟,푞

푘푥푟 푝 푝 푝 [∑ exp(−푗[푘푥푚푥 + 푘푦푛푦]) (∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞(푧) + 푘푦푞푈푥푟푞(푧)])] 푘0 푚,푛 푟,푞 푝 휕푆푥푚푛(푧) 푝 푝 푘푥푟 푝 푝 푝 = −푘0 ∑ 휇푚−푟,푛−푞 푈푦푟푞 (푧) + ∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞 − 푘푦푞푈푥푟푞] 휕푧 푘0 푟,푞 푟,푞 Derivation of the Fourier space electric field spatial harmonic in the x-direction.

휕푈푝 (푧) 푘 푦푚푛 푝 푝 푦푞 푝 푝 푝 = 푘0 ∑ 휀푚−푟,푛−푞푆푥푟푞(푧) + ∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)] 휕푧 푘0 푟,푞 푟,푞 휕푈푝 (푧) 푘 푥푚푛 푝 푝 푥푟 푝 푝 푝 = −푘0 ∑ 휀푚−푟,푛−푞푆푦푟푞(푧) + ∑ 휒푚−푟,푛−푞[푘푥푟푆푦푟푞(푧) − 푘푦푞푆푥푟푞(푧)] 휕푧 푘0 푟,푞 푟,푞 휕푆푝 (푧) 푘 푦푚푛 푝 푝 푦푞 푝 푝 푝 ( 6.25 ) = 푘0 ∑ 휇푚−푟,푛−푞푈푥푟푞(푧) + ∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞(푧) − 푘푦푟푈푥푟푞(푧)] 휕푧 푘0 푟,푞 푟,푞 휕푆푝 (푧) 푘 푥푚푛 푝 푝 푥푟 푝 푝 푝 = −푘0 ∑ 휇푚−푟,푛−푞 푈푦푟푞(푧) + ∑ 휉푚−푟,푛−푞[푘푥푟푈푦푟푞 − 푘푦푞푈푥푟푞] 휕푧 푘0 푟,푞 푟,푞 Maxwell’s Equations transformed in Fourier space

Maxwell’s equations in Fourier space are commonly expressed in block matrix form because they are more numerically convenient to solve.

푑푈푦푚푛(푧) 푑푧 0 0 퐊 훍−1퐊 훆 − 푲ퟐ훍−1 푑푈 (푧) 푦 푥 풚 푈푦푚푛 푥푚푛 0 0 −훆 + 푲ퟐ훍−1 −퐊 훍−1퐊 푈 푑푧 풙 푥 푦 푥푚푛 = −1 ퟐ −1 푑푆푦푚푛(푧) 퐊 훆 퐊 흁 − 푲 훆 0 0 푆푦푚푛 푦 푥 풚 푑푧 ퟐ −1 −1 [푆푥푚푛 ] [푲풙휺 − 흁 −퐊푥훆 퐊푦 0 0 ] ( 6.26 ) 푑푆푥푚푛(푧) 푑푧

Maxwell’s coupled equations in Fourier space written in block matrix form, for two- dimensional grating geometry.

Equation ( 6.26 ) is the block matrix equation for the Fourier expansion of the fields in the grating region. Most solutions of the RCWA assume non-magnetic materials however, I have provided the Fourier expansion for both non-magnetic and magnetic materials. Equation

( 6.26 ) is solved by recognizing that it is an eigenvalue problem and finding the eigenvectors and eigenvalues for the spatial harmonics U and S the magnetic and electric field spatial amplitudes, respectively. The solution for the spatial harmonics is exact; however, the accuracy

of the solution is dependent on the number of terms (i.e. m, n) retained in the expansion of the fields. As such the size of the problem has the potential to be very big, in fact ( 6.26 ) is actually a (4n x 4n) matrix [25]. Several authors have revised the RCWA to more efficiently solve the eigenvalue problem of ( 6.26 ). We solve the eigenvalue problem of RCWA using the enhanced transmittance approach presented by [26]. Moharam’s formulation addresses the numerical instability resulting from the inversion of an ill-conditioned matrix (i.e. the diagonal elements of the matrix are very small) by scaling the elements of the ill-conditioned matrix appropriately.

Now that the eigenvectors and eigenvalues are determined they are used along with the boundary conditions (in the case of non-metallic media, the tangential electric and magnetic fields at the boundary is continuous) to determine the transmittance and reflectance. The U and S space harmonics are now known and are substituted into Fourier expansions of the electric and magnetic fields given in ( 6.18 ) and ( 6.19 ). Using ( 6.14 ) and ( 6.15 ) and the boundary conditions we can solve for the transmittance and reflectance at the boundary (z=0, z=h) and each layer of the grating (zp). The complete numerical implementation of the algorithm is completed in the Appendix A.

Iterative Design

Sections 7.2 and 8.1 present simulations for antireflective (AR) surfaces matched to conventional structural composite and ballistic materials. I show that this method is very effective towards the aim of developing wideband structural composite and ballistic radomes.

To effectively design the antireflective surfaces presented in Sections 7.2 and 8.1 I used the direct and indirect design approach.

Figure 6.3 Structural composite radome wall physical configuration and lay-up.

6.3.1 AR Surface Direct Design Approach

In the direct design method, I determined the AR surface geometry by calculating the transmission and reflection response of the subwavelength structure directly. The RCWA described in Section 6.2. was used to calculate the response of the subwavelength structure (i.e.

hole size, hole depth and hole spacing. The subsequent AR surface geometry is refined to satisfy

design criteria for the AR surface. This iterative optimization process is illustrated in Figure 6.4.

Figure 6.4 Direct Design Method Algorithm

Using the direct design method is a straightforward method for designing and calculating

the response subwavelength antireflective surfaces, although it is not very efficient. Recall

Section 6.3.3 describes the computational cost of iterating an AR design using the both the

indirect and direct design methods. The inefficiency of the direct design methods reduces the

effectiveness of the AR surface optimization because it shrinks the number of AR surface

geometries that can be evaluated. This reduced number of evaluations minimizes the probability

of determining the optimal AR surface geometry.

6.3.2 AR Surface Indirect Design Approach

I developed the indirect design method to address the inefficiency associated with the

direct design method. The indirect design method is a three-step approach to determining the

AR surface geometry. The first step is to determine the optimal dielectric constant for the structure. As illustrated in Figure 6.5 the optimal dielectric can be determined using an iterative optimization or an analytical calculation. Once the dielectric is determined it is then translated into an AR subwavelength geometry using effective medium theory (EMT). The final step is to evaluate the accuracy of the EMT and correct the geometry if necessary. The correction is accomplished using RCWA and an optimization routine.

Figure 6.5 Indirect Design Method Algorithm

6.3.3 Computational Cost of RCWA for Iterative Design

To illustrate the computing cost of direct modelling and optimizing tapered AR surfaces I will present two examples. For each example the RCWA algorithm was run on a Dell Precision

M4600 computer with 8.0 GB RAM, Core I7 8 core processor running Windows 7. The algorithm was run in the Matlab™ programming environment. The first example considered is an AR surface designed using the direct design method from Section 7.2.3 . The radome configuration is illustrated in Figure 6.3; in this design I used four AR surfaces each of which is comprised of 40 layers. The overall modelling problem is considerably large and the

computational cost is reflected in the computer processing unit (CPU) solution time. The CPU solution time is the major bottleneck, considering the objective is to determine the optimal permittivity for each AR surface using an iterative method. To approximate the total solution time for the direct design I computed the product given the following parameters. First, I considered the computation time required by the RCWA to solve the geometry illustrated in

Figure 6.3 at a single frequency and incidence angle denoted as ΔTg. The second consideration was to determine the total number of discrete frequencies (nF), incidence angles (nQ) and objective function evaluations (Θ) required to satisfy the stopping criteria. The resulting solution computation time was given by:

퐶푃푈푡푖푚푒 = Δ푇푔 ∗ 푛퐹 ∗ 푛푄 ∗ Θ ( 6.27 )

Table 6-1 Computational Demand of Iterative Algorithm Using the Direct Design Method

ΔTg nF nQ Θ CPU Time Memory (sec) (hrs) Structural 28.1 51 1 27000 11128 766MB Composite Design 6

Table 6-1 illustrates the computational cost of iterative design using the RCWA. The computational cost could be reduced by running the RCWA on a parallel computing architecture.

The reduction using parallel computing would result in a linear improvement in solution time. In the second example I designed and modelled the AR surface using the indirect design method. I modeled the radome lay-up using the indirect design methodology. Calculating the transmission and reflection coefficients using this recursive formulation significantly reduced the

computational CPU solution time, helping to insure the optimal solution was attained during the optimization procedure. Table 6-2 presents the computational cost of modelling the radome lay- up using the 1-D multilayer dielectric formulation of Section 6.1.

Table 6-2 Computational Demand of Iterative Algorithm Using the Indirect Design Method

ΔTg (sec) nF nQ Θ CPU Time Memory (hrs) Structural Composite 0.009 51 1 27955 3.6 693MB Design 7

6.3.4 Iterative Structural Composite AR Surface Radome

After determining the AR surface geometry (i.e. layer thickness) and dielectric constant using the formulations described in Section 6.1 and the optimization routines. I translated the geometry to the grating structure illustrated in Figure 6.3 As an example, consider the permittivity profile presented in Figure 6.6; the curve with the red markers represents the profile for the air to face sheet interface. While the yellow markers show the profile for the face sheet to core interface. This permittivity profile was iteratively determined using the formulation in

Section 6.1 with a pattern search optimization routine. The formulation was used to calculate the

Figure 6.6 Permittivity profile for Example 5.

total transmitted energy of the radome lay-up (see Figure 6.3) from 4-18GHz at 0° incidence angle. An optimization algorithm was used to refine the thickness and dielectric constant of each of the 40 layers that comprised the tapered AR surface such that the objective function was minimized. The objective function I chose to minimize was the negative sum of transmission coefficients in decibels, as given by ( 6.28 ).

푀 퐹 = 푚푖푛 [∑ −20 ∗ log10푇(푓푘)] 푘=1 ( 6.28 )

Objective function of iterative design

Clearly, there are a number of effective optimization algorithms that could be used to refine the index profile. Some of the algorithms include traditional derivative based algorithms, genetic algorithms or direct pattern search algorithms. I settled on the pattern search algorithm because the pattern search method is computationally less expensive and provides an acceptable probability of finding the global solution [27]. While there are a number of good tapered AR coatings to employ, I chose to use the Klopfenstein refractive index taper ( 6.29 ) as a starting point for my iterative design.

푧 푛(푧) = √푛 푛 exp [Γ 퐴2ϕ (2 − 1, A)] , for 0 ≤ x ≤ L 푖푛푐 푠푢푏 푚 퐿 푥 퐼 (퐴√1 − 푦2) ( ) 1 휙 푥, 퐴 = ∫ 푑푦, for |x| ≤ 1 ( 6.29 ) 0 퐴√1 − 푦2 1 푛 퐴 = 푐표푠ℎ−1 [ ln ( 푠 )] 2Γ푚 푛푖푛푐

Klopfenstein index of refraction taper equation

Using the iterative method described in Section 6.3, Figure 6.7 and Figure 6.8 illustrate that I was able to beat the insertion loss performance, exhibited by the Klopfenstein taper at

Figure 6.7 Permittivity comparison between iterative design and Klopfenstein taper.

Figure 6.8 Transmission comparison between iterative design example 5 and Klopfenstein taper example 1 from normal incidence to 60° incidence.

normal incidence as well as across a broad range of incidence angles. Moreover, Figure 6.7 also shows that I was able to derive an index taper that was shorter than that offered by the

Klopfenstein taper. The structural radome was designed to have a passband from 4-18GHz and the AR surface was formed using a thermoplastic resin with a dielectric constant of 3.5. The

grating structure could be fabricated using fused deposition modelling (FDM) which is described thoroughly in Section 9.2.1. Figure 6.8 presents the predicted insertion loss results for the radome layup using the iterative AR surface design and the Klopfenstein AR surface permittivity.

Chapter 7: Wideband Structural Radome Design

To properly design the walls of a radome, the EM design must account for all of the first order EM effects that occur during the interaction of the EM waves with the radome wall. These interactions have a profound effect on the EM performance parameters such as transmission efficiency, reflection and insertion phase delay. Radome insertion loss which is a measure of the reduction of the strength of the EM signal while passing through the radome’s.

푃푡 is the power transmission coefficient.

퐼퐿(푑퐵) = −20 log10 푇, ( 7.1 ) Insertion loss

Insertion loss given by ( 7.1 ) is the sum of the losses due to reflection from the radome’s wall and absorption within the wall (which is governed by the electrical loss tangent of the radome wall materials) [1].

Conventional Radome Design Methods

Two general approaches are usually employed in the design of radomes: (1) Equivalent

Transmission-Line Method [2] and (2) the Mode-Matching Generalized Scattering Matrix (MM-

GSM). An overview of each design method is presented in Sections 7.1.1 and 7.1.2.

7.1.1 Equivalent Transmission Line Method

This method translates the layers of a radome into their equivalent impedances and thicknesses, then the reflection (Γ) of the system is calculated using the matrix formulation shown in ( 1.1 ). To reduce the insertion loss in this system, the impedances and thicknesses of each layer is optimized.

The optimization can be conducted using an optimization routine, or analytical methods like quarter-wavelength phase cancellation and Chebyshev techniques. The challenge with this method is that all the layers must be described using equivalent impedance (all layers must be dielectrics) and the resulting impedances may not be easily realized.

7.1.2 Mode Matching Generalized Scattering Method

To address radome wall configurations that include Frequency Selective Surfaces (FSS) or other metallic periodic layers the Mode-Matching Generalized Scattering Matrix Method

(MM-GSM) is typically used. This method can handle both periodic a FSS and a homogeneous dielectric at normal and oblique incidence angles.

The MM-GSM is a method that computes the composite S-Parameters of multiple cascaded screens. The MM-GSM computes the modes within each FSS layer and outside of the

FSS. The field within the FSS layer is computed in terms of their waveguide modes, whereas the fields outside the FSS (i.e. dielectric layers) are computed in terms of their Floquet modes.

Figure 7.1 Mode Matching Generalized Scattering Matrix

The waveguide modes are represented in a scattering matrix of forward and backward traveling modes that describes all self and mutual interactions of scattering characteristics, including

contributions from both propagating and evanescent modes. Finally, to compute the system transmission and reflection the individual scattering matrices for each layer are cascaded together to obtain the generalized scattering matrix (GSM) [28], [29]. From the GSM, the EM performance of the radome wall is determined. This approach has limitations in that the waveguide modes that are necessary to compute the fields of the internal layers are only available for limited number of shapes (i.e. rectangular, circular, crosses, etc.) This limits the flexibility of this technique. In order to address these shortfalls several hybrid techniques and finite element modeling based techniques are used.

Antireflective Surface Radome Approach

The radome design approach described in this dissertation implements an elegant EM design methodology utilizing antireflective (AR) surfaces as the key component to minimizing insertion loss. Antireflective surfaces can be implemented as subwavelength coatings or appliques, or they can be implemented as subwavelength periodic grating structures.

Subwavelength coatings are usually implemented by chemically altering the properties of the coating to produce the desired permittivity. The major advantage of AR coatings is that they don’t suffer from upper bandwidth limits, because they are inherently zeroth order structures.

However, because it is more difficult to chemically “dial-in” permittivity for a wide range of materials this approach is costly. Although, AR subwavelength gratings have an upper bandwidth limit their implementation is less complex and provides a level of flexibility that makes this approach far more attractive. Moreover, the upper bandwidth limit is a function of the grating implementation and therefore can be address in using simple design rules.

Antireflective surfaces are a subset of textured surfaces described in Section 5.5, they provide a wideband impedance matching layer to the radome wall with minimal impact on the radome wall structural, ballistic, or environmental characteristics. Moreover, antireflective surfaces can be implemented using both discrete and continuously tapered subwavelength gratings enabling both wideband capability and large incidence angularity acceptance. Using this design methodology enables radome wall configurations to remain flexible as long as the constituent layers don’t exhibit excessive loss and allow the use of well-known structural materials. While I do not conduct extensive mechanical testing as a part of this research; the radome materials presented in this work have a well-established history of use as structural elements.

The key metric in evaluating the efficacy of a radome design is the insertion loss, which primarily depends on the EM material properties of the constituent layers of the radome wall.

The principal EM material properties are the complex relative permittivity, electrical loss tangent and thickness of the constituent layers. Additional factors that affect the insertion loss are the dimensions and periodicities of the unit cell elements of the embedded structures (if any), the operating bandwidth, polarization of the antenna signal, and the range of incidence angles impinging on the radome wall. The interaction of these effects which are the reflection (Γ) and transmission coefficients (Τ) illustrated in Figure 7.2. One key component of radome design is the selection of suitable materials. Indeed, material selection is paramount in the design of effective radomes. The perpendicular and parallel polarization Fresnel reflection and transmission equations for non-magnetic materials are given in ( 7.2 ), ( 7.3 ), ( 7.4 ) and ( 7.5 ).

Figure 7.2 Slab transmission

휀2 휀1 2 cos 휃푖 − √ √1 − sin 휃푖 휃 휀1 휀2 Γ⊥ = 휀2 휀1 2 cos 휃푖 + √ √1 − sin 휃푖 휃 ( 7.2 ) 휀1 휀2

Perpendicular Polarization Fresnel Reflection

휀1 휀1 2 − cos 휃푖 + √ √1 − sin 휃푖 휀2 휀2 Γ∥ = 휀1 휀1 2 cos 휃푖 + √ √1 − sin 휃푖 ( 7.3 ) 휀2 휀2

Parallel Polarization Fresnel Reflection

2 cos 휃푖 T⊥ = 휀1 휀1 2 cos 휃푖 + √ √1 − sin 휃푖 휀2 휀2 ( 7.4 )

Perpendicular Polarization Fresnel Transmission

휀1 2√ cos 휃푖 휀2 T∥ = 휀1 휀1 2 ( 7.5 ) cos 휃푖 + √ √1 − sin 휃푖 휀2 휀2 Parallel Polarization Fresnel Transmission

Figure 7.3 Structural Composite Face sheet Permittivity and Loss Tangent Comparison

Figure 7.4 Permittivity and Loss Tangent of Structural Sandwich Composite Core Foams

7.2.1 Design Method for Wideband Structural Radomes

Combining antireflective surfaces with the structural composites described in Section 2.1 illustrates the utility of this design approach and highlights the importance of selecting structural composites with advantageous electrical properties. To best amplify this point I have included simulations of sandwich radomes with an S-glass epoxy face sheet and H100 foam core described in Figure 7.3 and Figure 7.4, respectively. I designed antireflective surfaces to transform these conventional structural composites into highly effective radomes from 4-18 GHz and over incidence angles from 0-60°. Figure 7.5 presents the wall configuration and associated insertion loss across frequency and incidence angle for the structural composite without an antireflective structure. Clearly, this wall configuration cannot act as a radome because of the significant insertion loss exhibited across frequency and incidence angle. The majority of the insertion loss exhibited is attributable to the Fresnel reflections due to the impedance mismatch at the two material interfaces. To minimize the impedance mismatches at each material interface

I designed antireflective surfaces two antireflective surface materials. The AR surface designs are presented in Table 7-1. By designing antireflective surfaces that transition the impedance at the face sheet to free space interface and at the face sheet to structural core interface I was able to significantly minimize the Fresnel reflection and subsequently improve insertion loss. The antireflective surfaces can be constructed using a non-dispersive polycarbonate or an ABS thermoplastic with a dielectric constant of 2.9 -0.0066j and 3.5-0.005j, respectively.

Figure 7.5 Structural composite wall configuration without an AR surface and the associated transmission loss exhibited by the wall configuration.

Table 7-1 AR Surface Designs

Antireflective Surface Design Parameters

AR Implementation Multilayered Example AR Material Approach Approach 1 FDM Klopfenstein Polycarbonate

2 FDM Iterative Optimized Polycarbonate Tapered 3 FDM Iterative Optimized ABS-3.5 Tapered 4 Grating Iterative Optimized Polycarbonate Discrete 5 Grating Iterative Optimized Polycarbonate Tapered

To determine the optimal dielectric constant values and layer thickness for the polycarbonate sheets I used an iterative design methodology illustrated in Figure 5.6 and discussed in Section

5.4. Adhering to those bounds I was able to compute optimized dielectric profiles for both discrete and tapered antireflective surfaces. The dielectric profiles for each design are provided along with the taper grating geometry and the insertion loss for the complete wall configuration.

7.2.2 AR Surface Bandwidth to Thickness Ratio

In general, antireflective surfaces are evaluated based on their ability to minimize the single or multilayer slab reflectance over the required passband using the shortest or thinnest possible AR surface. Consequently, one of the criteria that must be determined is the maximum thickness of the AR surface. To insure I have developed the optimal antireflective geometry for our passband I use the approach described in [30] to determine the minimum possible thickness for our AR surface as a function of the required bandwidth. A commonly used thickness to bandwidth rule is given by ( 7.6 ). This equation relates the reflectance (휌0) of a slab with a perfectly conducting backing to the thickness and bandwidth of the slab.

∆휆 2휌 = 0 휆0 (휋푑⁄ ) ∗ |휀 − 휇| 휆0 ( 7.6 )

Single Layer Bandwidth to Thickness Rule for Absorbers

Our system does not have a perfectly conducting layer and is multilayered as well. To relate the reflectance (휌0) given a single path through the slab to the bandwidth and thickness, I simply used ( 7.7 ), insuring the reflectance does not include the two-way path effect.

2 |ln 휌0|(휆푚푎푥 − 휆푚푖푛) < 2휋 ∑ 휇푠,푖푑푖 푖 ( 7.7 ) K. Rozanov Thickness to bandwidth ratio for perfectly conducting multilayer slabs

Clearly, ( 7.7 ) only includes the bandwidth, reflectance, thickness and static permeability, it says nothing about the particular approach used to achieve the thickness to bandwidth ratios set forth by ( 7.7 ). Using ( 7.7 ), I set my passband to 4-18GHz which is a 4.5:1 bandwidth and my maximum reflectance to -15dB. Using these parameters, the minimum possible thickness for my slab is 10.218 mm or 0.4048”. In the simulations to follow, each antireflective surface approach will set the maximum AR surface thickness using ( 7.7 ).

7.2.3 AR Structural Composite Numerical Examples

In this section I have provided five numerical examples of antireflective surfaces designed using the methodologies described in Section 7.2. In all cases the AR permittivity profile is provided for each AR surface. Moreover, the insertion loss through the structural composite with and without the AR surfaces is also presented. Lastly, the radome configuration, which includes the geometry of each AR surface as well as their location within the radome is provided. All of the examples provide insertion loss performance from 4-18GHz over incidence angles 0-60°. The insertion loss objective is to experience no greater than 1dB of insertion loss over the 4-18GHz bandwidth out to 40° incidence angle.

Example 1 Polycarbonate Klopfenstein AR implemented using FDM

In this example I designed a Klopfenstein AR taper using equation ( 5.12 ) and a given taper length determined using Rozanov’s formulation ( 7.7 ). In this example the Klopfenstein

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taper is implemented using the FDM approach which enables the minimum dielectric constant to approach unity as is illustrated in Figure 7.6. The taper is applied at the air-face sheet interface and the foam core – face sheet interface. Both AR surface structures are in excess of 0.14λ thick, which is approximately 0.4”.

Figure 7.6 Structural composite radome wall physical configuration and lay up – Klopfenstein FDM Polycarbonate Taper. Permittivity profile of two Klopfenstein AR surfaces implemented using FDM additive manufacturing.

Figure 7.7 Structural Composite insertion loss without AR surfaces Structural radome insertion loss simulation assuming Klopfenstein AR surfaces are layed up in accordance with Figure 7.6. The insertion loss is improved; however, the desired bandwidth is not achieved. The FDM implemented Klopfenstein AR surfaces attain 2.83:1 bandwidth not 4.5:1. Moreover, this performance costs the radome designer a total of 1.6” in additional thickness.

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Example 2 Polycarbonate AR surfaces iteratively optimized and implemented using FDM

In this example I designed an AR taper using the iterative approach described in Section

6.3 and setting a maximum thickness using Rozanov’s formulation ( 7.7 ). In this example the optimization produced a taper that alternates between positive and negative slopes. Moreover, the profile contains discrete jumps and constant slopes. I have provided the permittivity profile in Figure 7.8. The bandwidth using the iteratively designed AR surfaces is 3.33:1 which is an improvement over the Klopfenstein AR surfaces. Similar to the Klopfenstein example the cost penalty is significant. The AR surface adds an additional 1.6” of thickness to the structural composite. It is implemented using the FDM approach which enables the minimum dielectric constant to approach unity as is illustrated in Figure 7.6.

Figure 7.8 Structural composite radome wall physical configuration and lay up. Permittivity profile of two AR surfaces designed using simulated annealing and pattern search optimization routines; and implemented using FDM additive manufacturing.

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Figure 7.9 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.8.

Example 3 ABS-3.5 AR surfaces iteratively optimized and implemented using FDM

In the two previous designs I showed that the addition of an AR surface reduces the

Fresnel reflections and subsequently improves the insertion loss of a sandwich composite. I also showed that the optimized AR taper that has an alternating slope with discontinuities produces a broader band radome. However, in both cases there was a thickness penalty because the design required four 0.14λ AR surfaces (one for each impedance mismatch interface). In this example I present an optimized AR taper that has alternating slopes with discontinuities, however these AR surfaces are not fabricated using polycarbonate which has a dielectric constant of 2.9. Instead I designed these AR surfaces using an ABS material with a dielectric constant of 3.5.

The increased dielectric constant enables a shrinking of the individual AR surface thickness. These thinner AR surface structures are illustrated in the permittivity profile in Figure

7.10. The normalized taper length of ABS tapers is 0.1λ for the face sheet to structural core taper and 0.11λ for the air- face sheet taper. A review of the insertion loss prediction shown in Figure

7.11 illustrates the improvement is consistent with both polycarbonate and ABS AR surfaces.

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The bandwidth for the ABS AR surface radome is 2.82:1. The overall thickness contribution from the AR surfaces is 1.24” while the polycarbonate AR surfaces add 1.6” of thickness to the structural composite. Certainly, there is a trade space between bandwidth and thickness, and the material properties of the AR surface can be exploited to reduce thickness while not sacrificing performance.

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Figure 7.10 Structural composite radome wall physical configuration and lay up. Permittivity profile of two AR surfaces designed using simulated annealing and pattern search optimization routines; and implemented using FDM additive manufacturing.

Figure 7.11 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.10.

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Example 4 Polycarbonate iterative optimized tapered AR Grating

The preceding AR tapers have all been FDM implemented tapers which have a greater dynamic permittivity profile range. FDM implemented tapers can also have alternating slopes, in many cases this extra degree of freedom helps to improve the overall impedance matching performance of the taper. In the next two examples I present AR grating tapers that can be implemented using a more conventional subtractive manufacturing approach like CNC machining. The permittivity profile for these AR gratings were calculated using the methodologies described in Section 6.3.1 while adhering to the rules presented in Section 5.5.

These gratings were implemented using a hexagonal periodicity (shown in Figure 5.8). The permittivity profile illustrated in Figure 7.12 is monotonic and does not approach unity. I constrain the dielectric constant because in my optimization approach I do not simulate the grating geometry, instead I simulate the effective dielectric constant for each layer and translate the permittivity to appropriate grating geometry. This approach is computationally more efficient and allows for a more comprehensive optimization as was described in Section 6.3.1.

Figure 7.12 also provides a side view of the radome configuration. The AR surfaces are represented by the blue and white rectangles, the yellow rectangle represents the structural face sheets and the brown rectangle is the structural core. It is clear from this image that the AR surfaces become a significant percentage of the entire structure.

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Figure 7.12 Radome configuration in a side view. Permittivity profile of two AR surfaces designed using simulated annealing and pattern search optimization routines; and implemented using subtractive manufacturing

Figure 7.(13 Structural Composite insertion loss without AR surfaces( (b) Structural radome insertion loss simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.12. a) b) The bandwidth of this AR surface configuration is 3.33:1 which is comparable to the insertion loss performance of the FDM designs. Although the FDM designs provide a more flexible permittivity profile the insertion loss performance for the grating is comparable. The

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value of FDM tapers is that the high frequency limit that gratings experience due to the zeroth order requirement ( 5.14 ) happens at much higher frequencies for FDM tapers. For example, given a grating with a periodicity (Λ) of 4.5 mm, the zeroth order requirement limits the high frequency operation at 40 GHz, while using an FDM taper the periodicity can be as small as 1.35 mm which results in a high frequency operation up to 100 GHz.

Example 5 ABS iterative optimized discrete tapered AR Grating

In this final example I designed a 3-layered discrete AR taper in a polycarbonate material. Similar to example four the permittivity profile for this AR grating was calculated using the methodologies described in Section 5.4 while adhering to the rules presented in Section

5.5. These discrete gratings were also implemented using a hexagonal periodicity (shown in

Figure 13.2). The permittivity profile illustrated in Figure 7.15 is monotonic and does not approach unity.

Figure 7.15 also provides a side view of the radome configuration. The AR surfaces are represented by the blue and white rectangles; the yellow rectangle represents the structural face sheets and the brown rectangle is the structural core. The bandwidth of this AR surface is 3.27:1 which is comparable to the insertion loss performance of the previous grating design. Similar to example 4, the AR surfaces become a significant percentage of the entire structure.

( (

a) b)

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Figure 7.14 Structural Composite insertion loss without AR surfaces (b) Structural radome insertion loss simulation assuming iteratively optimized AR surfaces that are layed up in accordance with Figure 7.15.

Figure 7.15 Radome configuration in a side view. Permittivity profile of two AR surfaces designed using simulated annealing and pattern search optimization routines; and implemented using subtractive manufacturing

7.2.3.1 Structural Composite Simulations Observations

The previous five radome design simulations illustrate the effectiveness of my design approach. Designing anti-reflective surfaces to reduce Fresnel reflections at impedance interfaces makes it possible to use conventional structural composite materials as wideband

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radome materials. The approach is most effective when the structural composite materials exhibit acceptable levels of material loss within the passband. A review of Section 2.1 shows that there exists a substantial set of structural composite reinforcement materials (i.e. glass fibers) and binding agents (i.e. epoxy and cyanate ester) that possess these electrical properties.

A review of the simulations presents a number of interesting observations; I will highlight several of these observations. First, comparing the Klopfenstein taper transmission performance to the transmission performance of the antireflective surfaces designed using general optimization routines suggests that the optimization tapers provides better bandwidth and angular performance.

Moreover, example 3 the FDM ABS-3.5 is uniquely impressive because the bandwidth is 2.8:1 and the structure thickness was reduced by 25%. This was principally accomplished by using an

ABS which is a thermoplastic material with a dielectric constant 3.5 instead of polycarbonate which has a dielectric constant of 2.9. Moreover, each iterative optimized design outperformed the Klopfenstein taper which produced a bandwidth of 2.8:1.

The grating designs implemented using subtractive manufacturing techniques can only produce monotonically increasing or decreasing permittivity profiles, which appears to not have a significant impact on bandwidth performance at these frequencies. However, for ultra- wideband performance (<8:1) FDM implementation is the better choice because it does not have the upper bandwidth limit that CNC subwavelength gratings experience. It is also interesting to review the transmission performance of the continuously tapered design of example 1. In general, the discrete transmission performance is as good as or better than the continuously

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tapered designs. This is also reported by [31] in their comparison of discrete to continuously tapered gratings. Although, the findings in [31] dealt solely with gratings and my FDM designs do not hold the same period to wavelength ratios (Λ/λ). I too found that discrete gratings outperform continuously tapered gratings.

The transmission performance using the subwavelength grating implementation began to degrade at the band extrema and oblique angles. In the case of the antireflective surfaces produced using the FDM implementation, they too degraded at the band extrema, however the drop-off in performance was not as severe. While not a significant difference, it is interesting that there is a consistent performance drop-off.

Lastly, I included the transmission results for a Klopfenstein impedance taper as a reference to compare the antireflective surfaces designed using an optimization routine to the transmission performance of antireflective surfaces designed using the Klopfenstein impedance taper. In most communities the Klopfenstein taper is considered the optimum taper profile [19], however, the iteratively designed AR surfaces outperformed the Klopfenstein transmission performance.

Chapter 8: Ballistic Radome Wall Configuration Simulations

Combining antireflective surfaces with the ballistic armor configurations described in

Section 7.2 again illustrates the utility of this design approach and highlights the importance of selecting materials with advantageous electrical properties. However, ballistic armor consists of fewer electromagnetically compatible materials than structural composites. As a consequence, the radome wall configurations tend to be more complex. In general, ballistic armor materials consist of ceramic materials and glass fiber backing materials (i.e. Spectra, Dyneema and

Kevlar).

Figure 8.1 Ballistic Armor Material Permittivity and Loss Tangent

Figure 8.1 presents the real permittivity and loss tangent for the ballistic armor materials I used to design the ballistic armor radomes in the succeeding sections. The simulations from

Section 5.2 illustrates the utility of this design approach for structural radomes, and also provides insight into the best ways to implement our antireflective design approach. 110

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Figure 8.2 Ballistic Armor sandwich using S-glass epoxy backing material and a Spectra shield impact material along with its associated transmission response.

To show the value of this approach for ballistic armor configurations we will apply antireflective surfaces to three specific ballistic armor configurations. The first ballistic armor configuration will be a symmetric sandwich design using S-glass backing materials with a Spectra-shield core.

Figure 8.2 illustrates the physical configuration along with the associated transmission response assuming an S-glass epoxy face sheet. The second ballistic armor configuration is a sandwich design using cyanate ester face sheets with a ceramic core. Figure 8.3 illustrates the physical configuration along with its associated transmission response. The final ballistic armor configuration is an asymmetric configuration consisting of a ceramic core with a spectra shield and an S-glass backing layer. Figure 8.4 shows the final ballistic armor configuration along with the transmission response for that configuration.

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Figure 8.3 Ballistic Armor sandwich using S-glass epoxy backing material and an Alumina ceramic are along with its associated transmission response.

Figure 8.4 Ballistic armor sandwich configuration with cyanate ester backing material, ceramic impact material, and Spectra shield backing layer, along with its associated transmission response.

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Ballistic Protection Radome Numerical Examples

Example 1 – Ballistic Armor Spectra Shield and S-glass Epoxy Radome Simulation

In this example a sandwich ballistic armor configuration of S-glass face sheets with a

Spectra-shield core was simulated. Figure 8.5 (a) presents the insertion loss of the original sandwich configuration while Figure 8.5 (b) shows the insertion loss with four discrete AR surfaces applied as described in Figure 8.7 (a).

The discrete AR surfaces were designed using the methodology defined in Section 7.2.

Figure 8.7 (b) provides the permittivity profile for each AR surface. In this example each AR surface is 0.14λ thick and monotonically increasing or decreasing. The radome bandwidth is

3.72:1 which nearly satisfies my requirement of 4.5:1 however, the designed required a total AR thickness of 1.6” to achieve this bandwidth.

a b Figure 8.5 Ballistic Armor S-glass Spectra Shield radome configuration and associated transmission loss prediction

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a b

Figure 8.7 Ballistic Armor S-glass Spectra Shield radome lay-up and configuration

a b Figure 8.6 Ballistic Armor S-glass Alumina radome configuration and associated transmission loss prediction

Example 2 - Ballistic Armor Ceramic and S-glass Cyanate Ester Radome Simulation

Example 2 is a sandwich ballistic armor configuration with S-glass face sheets and a 0.5”

Alumina core. Alumina is a ceramic material with a complex relative permittivity of 9.0- j0.072i. In order to transition the impedance from the face sheet to ceramic core a 0.092” slab with a dielectric constant of 6 was applied between the alumina core and S-glass face sheet.

Figure 8.6 (a)illustrates the insertion loss of the sandwich ballistic armor without the AR surface

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and impedance matching layer. Clearly, the addition of the AR surface and impedance layer are effective in transforming the ballistic armor into a ballistic radome. Due to the application of the impedance layer only two AR surfaces were required for this design. The total AR surface thickness was only 0.797”, whereas example 1 required four AR surfaces and resulted in a total

AR thickness of 1.6”. Example 2 represents a 63% reduction in AR surface thickness. The effectiveness of the impedance sheet can also be seen by observing the insertion loss of the design without the impedance matching layer illustrated in Figure 8.9.

a b

Figure 8.8 Ballistic Armor S-glass Alumina radome lay-up and configuration

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Figure 8.9 S-glass ceramic ballistic armor insertion loss without AR surface and no impedance matching layer.

a b

Figure 8.10 Ballistic Armor S-glass Alumina Spectra radome configuration and associated transmission loss prediction

Without the impedance layer the bandwidth is decreased from 4.1:1 down to 2.23:1. Figure 8.8

(a) and (b) present the ballistic radome configuration and AR surface discrete permittivity profiles, respectively.

Example 3 - Ballistic Armor Ceramic and Spectra Shield Radome Simulation

Example 3 is a ballistic armor configuration that combines the armor elements from examples 1 and 2. The configuration includes the Alumina core for projectile fragmentation as well as the

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Spectra-shield and S-glass backings to catch the ceramic and projectile fragments. This configuration provides an upgrade in ballistic protection from the previous configurations. A review of Figure 8.10 (a) demonstrates that the added ballistic protection destroys and transmission performance. In fact, the insertion loss is greater than 2dB throughout the passband and incidence angles. By applying the AR surfaces and impedance matching layers as shown in

Figure 8.11(a) I was able to transform the highly reflective ballistic armor into a ballistic radome with a 4.5:1 bandwidth. To achieve this bandwidth, I designed AR surfaces with permittivity’s illustrated in Figure 8.11(b) and the AR design system only adds 1.134” of thickness.

Figure 8.11 Ballistic Armor S-glass Alumina radome lay-up and configuration

Chapter 9: Antireflective Surface Fabrication Methods

Fabricating a continuously varying dielectric profile can be quite challenging. A common way to fabricate continuously varying dielectrics is to build subwavelength gratings.

Here, a periodic subwavelength textured surface is used to create effective dielectric properties.

When the cross sectional area of the structure varies with depth (e.g. tapered hole) a continuously varying dielectric constant can be effectively constructed. At microwave frequencies these structures are commonly fabricated using standard machining techniques (i.e. computer numerically controlled milling), however for broadband and high frequency applications this method breaks down [22] and more precise fabrication techniques are required. Moreover, it is difficult to fabricate the subwavelength features using CNC machining if the index of refraction varies non-monotonically or the AR surface needs to conform to a non-planar surface. Using additive manufacturing is an alternative fabrication approach that can be used to realize subwavelength gratings. Specifically, I used an additive manufacturing technique called Fused

Deposition Modelling (FDM) to fabricate non-monotonic graded subwavelength structures.

Subtractive manufacturing - Computer Numerically Controlled (CNC) Machining

Textured surfaces designed to operate in the microwave band are often fabricated using CNC machining. CNC machining is straightforward and provides very good results provided the structure adheres to the subwavelength requirement described in Section 5.5. CNC machining was used to fabricate the sample shown in Figure 9.1 and illustrates utility of this method. The physical dimensions of the structure illustrate the fidelity that we are able to achieve with CNC machining and demonstrates the limit of this approach.

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Λ= 2.8 mm h =1.33 mm 1 h =2.26 mm 12” 2 h =6.0 mm 3 d =2.54 mm 1 12” d =1.27 mm 2 e =9.0-0.02j r Fabricated using CNC milling

Figure 9.1 Discrete AR Surface fabricated using CNC machining

This antireflective surface was designed to operate in the Ka-band (30-40 GHz) which is in the millimeter wave regime, consequently the periodicity (Λ) of this textured surface is just 2.8 millimeters. In general, most CNC machines will assert a repeatable and precision down to

0.001” or 25µm. My design requires precision near the limit of the advertised accuracy of the fabrication technique. Furthermore, given the geometry described in Figure 9.1, a 25µm misalignment could yield a period as small as 2.55mm or as large as 3.05. Uncertainties of that magnitude would have an impact on the AR surface performance. Figure 9.2 presents the transmission results of this AR surface by comparing the transmitted energy of the AR surface to the transmitted energy of the sample without the AR surface. Clearly, the addition of the AR surface significantly improves the transmission by reducing the reflections at the interface.

9.1.1 Continuously Tapered Textured Surfaces CNC fabrication

Figure 9.3 presents an example of a tapered subwavelength grating implemented using the CNC subtractive manufacturing approach.

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Figure 9.2 Discrete Ka-band AR surface transmitted energy measurement and predicted performance results.

Figure 9.3 Klopfenstein subwavelength grating

CNC machining of tapered surfaces requires the hole diameter to decrease continuously with depth. The rate at which the diameter decreases is determined by the permittivity profile for the design. For example, the Klopfenstein impedance profile given in Figure 9.4, can be implemented using CNC machining. Figure 9.4 (a) presents the Klopfenstein permittivity profile derived using RCWA and EMT implemented on a hexagonal lattice with a period of 0.1969”.

Figure 9.4 (b) presents the Klopfenstein normalized hole diameters as a function of normalized

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taper length using RCWA and an optimization routine to derive the red curve and EMT to derive the blue curve.

a b

Figure 9.4 (a) The black curve illustrates Klopfenstein permittivity profile; the blue curve represents the effective dielectric constant when the radius varies according to effective medium theory at the center of the band; the red curve represents the effective dielectric constant when the radius varies according to the RCWA and optimization at the center of the band. (b) Comparison of the normalized diameter using RCWA and EMT to determine the radius.

The red curve which is the permittivity produced by the RCWA optimized hole diameters shown in Figure 9.4 (b), while the blue curve is the permittivity profile derived using the EMT hole diameters shown in Figure 9.4 (b). Equation ( 9.1 ) presents the volume fraction formula used to derive the normalized hole diameters for the EMT shown in Figure 9.4 (b). The black curve represents the calculated permittivity profile for a Klopfenstein taper. Clearly, the red curve does a better job reproducing the Klopfenstein taper; however, at a taper length of 0.1λ the curves diverge because the physical geometry (fill factor cannot be fabricated) does not allow the permittivity to approach unity. The effective dielectric constant is truncated at approximately

1.53 when the substrate material has a permittivity of 2.9, recall Section 5.5 and Table 5-1.

The consequence of this truncation is that CNC machining does not allow the permittivity to approach unity and the Klopfenstein taper must be modified to account for this subtractive

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manufacturing limitation. Subtractive manufacturing also experiences taper profile truncation, caused by end mills with short cutting lengths.

Figure 9.5 Illustrates the single and doubly truncated Klopfenstein grating, implemented using subtractive manufacturing.

Because the end mill cutting length is determined by the end mill flute length and must satisfy the cutting length ratio CL = 1.5*D [32], certain small deep holes cannot be fabricated.

Consequently, it is difficult to exactly reproduce the Klopfenstein geometry truncation using

CNC machining. Instead the Klopfenstein fabrication truncation is produced from Figure 9.5.

Additive Manufacturing Implementation

Additive manufacturing techniques have been developed that can address some of the limitations that I discussed in Section 9.1. To achieve near unity permittivity and eliminate profile truncation due to small hole deep depth limits set by end mill cutting length ratios I have used additive manufacturing to implement some of my designs. Specifically, I employ a technique known as Fused Deposition Modelling.

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9.2.1 Fused Deposition Modelling

Fused deposition modelling (FDM) is a fabrication technique that extrudes a thermoplastic feed stock using a heated nozzle. The extruded filament is then stacked layer upon layer to build a solid outline. When the extruded filament is deposited onto a previously deposited layer, the hot extrudate partially melts the previous layer creating a bond and then rapidly cools to lock in the desired shape. Rigid structures are fabricated by filling the interior of the outline with a raster pattern of polymer, such as a simple cross-hatching pattern Figure 9.7. Figure 9.6 illustrates the

FDM printing process. I fabricated several continuously tapered AR surfaces using FDM and will present the fabricated samples in the sections to follow. The design methodology for fabricating AR subwavelength gratings using FDM shows that FDM provides an improvement over conventional subtractive fabrication methods. Specifically, I show that fused deposition modelling (FDM) is a flexible and effective method for fabricating nearly continuously varying or discrete AR surface coatings. Moreover, this method can produce nearly arbitrary dielectric profiles even on curved surfaces.

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Figure 9.6 Illustration of FDM printing process shows heated nozzle extruding the thermoplastic feedstock.

FDM AR surfaces are reproduced by first establishing the permittivity profiles or effective dielectric constant as a function of depth. The effective dielectric constant will be a function of the local volume fraction of polymer to background material (normally air). Since the diameter of the extruded plastic fibers can be varied between 50 µm to 300 µm, and the fibers can be separated by a distance as small at 1µm, thereby creating a wide range of effective dielectric constants even at relatively high frequencies (e.g., <100 GHz). To determine the precise relationship between the FDM fill volume and the effective dielectric constant, we used a

3DN-300 printer, sold by nScrypt Inc., to print several rectangular test samples (200 mm x 200 mm x 6.3 mm). We deposited polycarbonate feed stock at various fill volumes (15%, 30% and

50%; Figure 9.7) and measured the dielectric constant over the K-band using a free space

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focused beam system [33]. The experimental data was analyzed and found to fit to a standard

Maxwell-Garnett mixing formula given by ( 9.1 ) where PC = 2.9 and air =1.0 are the dielectric constants of polycarbonate and air, respectively, and PC is the volume fraction of the polycarbonate. Figure 9.8 presents the measured data and the Maxwell Garnett fit equation that was used to determine the volume fill for any effective dielectric the Maxwell-Garnett fit can be used.

Figure 9.7 (a) Images of FDM printed test samples used to determine the relationship between local volume fraction and effective dielectric constant. (b) Cross-hatching fill pattern used to fill the outline. The effective dielectric constant is proportional to the local volume fraction of polymer to air.

2훿 (휀 − 휀 ) + 휀 + 2휀 휀 = 휀 푃퐶 푃퐶 푎푖푟 푃퐶 푎푖푟 푒푓푓 푎푖푟 2휀 + 휀 + 훿 (휀 − 휀 ) 푎푖푟 푃퐶 푃퐶 푎푖푟 푃퐶 ( 9.1 ) Maxwell Garnett effective dielectric equation

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Figure 9.8 Measured data and Maxwell-Garnett fit for the effective dielectric constant of these samples as a function of volume fraction.

a b

Figure 9.9 (a) Comparison of the measured and predicted transmission energy through an AR FDM fabricated slab. (b) Compares the design AR surface permittivity profile (red curve) to the actual fabricated permittivity profile (blue curve).

Figure 9.9 presents two graphs that provide an illustration of an FDM printed AR surface with an alternating slope permittivity profile. Figure 9.9 (a) compares the predicted transmitted energy through the AR surface (red curve) to the measured transmitted energy through the AR surface

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(black curve). The predicted transmitted energy was calculated using the fabricated permittivity profile given in Figure 9.9 (b). The fabricated permittivity profile was determined by measuring the transmission across 18-40 GHz, then using this data to extract the effective dielectric constant of the AR surface across the 2-40 GHz band. Figure 9.9 demonstrates that FDM printing is an accurate method for implementing complex permittivity profiles that cannot be realized using conventional fabrication methods like CNC machining.

This chapter described two manufacturing techniques for fabricating textured surfaces and implementing permittivity profiles. In Chapters 5 and 6 I described the method for calculating the permittivity profiles and textured AR surface geometry. Sections 9.1 and 9.2 discussed the fabrication methods for translating the profiles and textured surfaces to physical structures. Of the two methods, additive manufacturing is able to realize a wider range of permittivity profiles and textured surfaces. Moreover, FDM printing can provide better impedance matching at high frequencies than more traditional fabrication techniques.

Chapter 10: Experimental Validation

In this work I have presented a novel method for the design and fabrication of wideband radomes using textured surfaces for impedance matching. The method hinges on modelling and fabrication of textured surfaces that provide the broadband impedance match for conventional structural composites and ballistic protection materials. This section substantiates this methodology by presenting experimental results. For each example I first present the AR surface model and experimental transmission measurements to illustrate the efficacy of the AR design approach, second I provide the radome prediction and compare it with the experimental results.

Finally, I compare the radome performance using this method to the radome performance in the absence of the AR surface, and compare this performance to the conventional radomes of comparable geometry. In Section 7.2 I described the radome design methodology and discussed the types of AR surfaces that can be designed. Indeed, AR textured surfaces can have dielectric profiles that are continuously varying with depth or discretely varying with depth. In this chapter

I present both cases for comparison and discussion. Moreover, the dielectric profile can also be non-monotonic. Section 10.2 presents a 5-35 GHz ballistic radome with an AR surface designed using an unconstrained iterative method, resulting in a permittivity profile that is non-monotonic.

This permittivity profile cannot be implemented using conventional subwavelength grating techniques and was therefore implemented using additive manufacturing, specifically fused deposition modelling (FDM). Section 10.3 presents a 4-18GHz ballistic radome that uses a continuously tapered Klopfenstein subwavelength AR grating fabricated using subtractive manufacturing. Section 10.4 presents a K-band structural radome with an AR surface designed using the unconstrained iterative method and was fabricated using FDM.

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Figure 10.1 Transmission and reflection measurement set up. Transmit and receive horns are aligned and attached to a vector network analyzer.

Measurement System Background

The measurement data for each of the examples described in the ensuing sections was acquired using the set up illustrated in Figure 10.1. This free space measurement set up is very effective for capturing the insertion loss and return loss of planar material systems. Figure 10.1 presents an anechoic chamber free space configuration where the transmit and receive are aligned and separated such that they are in the far field of the antennas. The antennas are connected to a broadband source and receiver; in most cases this is a vector network analyzer.

The vector network analyzer allows the user to capture both the magnitude and phase of the transmitted and received EM wave. The sample under test is located in the center of the chamber.

This measurement technique is most often used to acquire the transmission and reflection response of the material under test. Transmission measurements are set up to determine the amount of EM energy that is transmitted through a material, while the complement of this

130

measurement is the reflection measurement, which are conducted to determine the amount of EM energy that is reflected by the sample. Free space measurements allow EM energy to exist in four states: EM energy can be transmitted through the sample, reflected by the sample, absorbed by the sample or diffracted off the sample. Figure 10.2 provides an illustration of the four states of EM energy in a free space measurement environment. To accurately characterize a material’s transmission and reflection response the EM wave must exist in only the first three states. The fourth energy state is a major source of error in free space measurements. EM diffraction is typically caused by illumination of the test sample edges. A review of Figure 10.2 illustrates this phenomenon. To minimize diffraction effects the illumination spot is designed to be smaller than the sample by employing large samples (i.e. illumination does not reach the edge of the sample). Secondly, ensuring the EM wave behaves like a plane wave is typically accomplished by separating the antennas in accordance with ( 10.1 ) the far field equation or designing the system with a collimating lens.

R sample T Receive Transmit Antenna Antenna absorbed

  

diffracted Figure 10.2 Illustration of the four states of EM energy for free space measurements.

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Lens

Source Plane Wave

Focal distance 2퐷2 푑 = 휆 Figure 10.3 Collimating Lens and Focused Beam Measurement System ( 10.1 ) Far Field Equation

The collimating lens transforms the EM wave from a spherical near field wave into a quasi-plane wave that behave like a plane wave. Figure 10.3 provides an illustration of the function of the collimating lens. Finally, Figure 10.3 also presents an illustration of the free space measurement system used to acquire the data to follow. This system is equipped with collimating lens and a vector network analyzer for accurate measurement results.

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Alternating Slope AR Ballistic 5-30GHz Radome FDM Iterative Design

In this example an AR surface was designed using the iterative design process discussed in Sections 6.3 and 7.2. The AR surface permittivity profile was unconstrained such that the optimal permittivity was allowed to have both positive and negative slopes, discontinuous slopes and constant slopes. The permittivity profile is shown in Figure 10.4 (a) and the full ballistic radome configuration is provided in Figure 10.4 (b). Moreover, the AR surface was fabricated using the FDM printing process which constructed 40 – 0.004” layers AR surface, resulting in an overall thickness of 0.157”.

Figure 10.6 (a) presents a picture of the AR surface one can see the cross-hatched structure described in Section 9.2.1, also presented in Figure 10.6 (b) is a comparison between the measured and predicted insertion loss of the iteratively designed AR surface at 0° incidence angle. The prediction shows excellent agreement. Figure 10.5 (b) presents the measured insertion loss for the ballistic armor from 2-40 GHz across incidence angles 0-50°.

a b

Figure 10.4 (a) Permittivity profile of the iterative design alternating slope AR surface. (b) Ballistic radome full system configuration.

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a b

Figure 10.6 (a) Image of the cross-hatched iterative AR surface. (b) Measured vs. predicted insertion loss of ballistic radome at 0° incidence angle.

Figure 10.5 (a) Insertion loss for ballistic armor configuration from 2-40 GHz over incidence angles 0- 50°. (b) Insertion loss for ballistic armor with iterative designed AR surface applied.

In Figure 10.5 (b) the reader will find the measured insertion loss of the ballistic armor with the iterative AR surface applied. The addition of just the AR surface transforms the highly reflective ballistic armor into a wideband ballistic radome. The insertion loss is less than 1dB from 6.5-35GHz delivering an impressive 5.4:1 bandwidth. Moreover, this is all accomplished by adding only 0.314” of thickness to the original ballistic armor configuration. Figure 10.7 is a comparison between the measured and predicted insertion loss of the full ballistic radome at 0°

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incidence angle. The prediction shows very good agreement from 2-25GHz. Figure 10.7 (a) presents a picture of the ballistic radome, which is simply the graded ballistic armor core with the two iterative designed AR surfaces applied to the outer surface. Figure 10.7 (b) presents the measured (black curve) and predicted (red curve) insertion loss for the ballistic radome. The predicted insertion loss shows good agreement with the measured insertion loss from 2-25 GHz, however, above 25 GHz the measured and predicted insertion loss diverge. This is likely due to measurement error due to instability of the calibration at higher frequencies. Also provided for reference is the insertion loss of the ballistic armor at 0° incident angle.

a b

Figure 10.7 (a) Image iterative AR surface bonded to ballistic armor. (b) Measured vs. predicted insertion loss of ballistic radome at 0° incidence angle.

Klopfenstein AR Surface Experimental Validation

In this example an AR surface was designed using the Klopfenstein taper ( 5.12 ). The

AR surface was designed to provide an impedance match for a ballistic armor core. The AR surface was a subwavelength grating fabricated using CNC machining.

135

b a

Figure 10.8 Klopfenstein AR surface permittivity profile for a ballistic armor core and the associated transmission loss prediction for the total radome lay up.

The AR surface was 0.5” thick and consisted of 25 0.015” layers where the hole varied in accordance with Figure 9.4 (b). The final layer was 0.111” thick and the subwavelength grating was design using a hexagonal lattice with a periodicity of 0.1969”. An image of the subwavelength grating is presented in Figure 10.9 (a) and the full ballistic radome lay-up can be found in Figure 10.8 (a). Figure 10.8 (b) presents the permittivity profile for the Klopfenstein AR surface considering the fabrication truncations associated with CNC machining. As consequence of this limitation I machined the Klopfenstein fabricated truncation profile given in Figure 9.5.

Figure 10.9 (a) presents a picture of the Klopfenstein AR surface while Figure 10.9 (b) presents a comparison between the predicted and measured transmission coefficient for the Klopfenstein subwavelength grating. The transmission prediction for the Klopfenstein AR surface was performed using the RCWA and shows good agreement between the predicted and measured transmission Figure 10.9(b). The good agreement between predicted and measurement also suggests that the CNC machining was precise.

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b a

Figure 10.9 (a) Klopfenstein AR Surface ballistic radome. (b) Comparison of predicted and measured Klopfenstein AR surface transmission loss.

a b

Figure 10.10 (a) Comparison of measured and predicted Klopfenstein AR surface ballistic radome transmission loss. (b) Comparison of measured and predicted Klopfenstein AR surface ballistic radome return loss.

Figure 10.10 (a) presents the insertion loss of the ballistic armor with and without the AR surface at 0° incidence angle from 2-18 GHz and Figure 10.10 (b) compares the reflected energy with and without the AR surface. The AR surface is able to reduce the insertion loss to less than

1.5dB across the entire 2-18GHz band; without the AR surface, the radome reflects over -5dB of the energy. Additionally, the modelling technique accurately predicts the insertion loss and return loss for the radome structure. Figure 10.11

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Figure 10.12

138

a b Figure 10.11 (a) Illustration of the insertions loss without the Klopfenstein AR surface. (b) Illustration of insertion loss with Klopfenstein AR surface.

a b

Figure 10.12 (a) Predicted insertion loss of the ballistic radome with Klopfenstein AR surface. (b) Measured insertion loss of the ballistic radome with Klopfenstein AR surface.

Alternating Slope AR Structural Composite K-Band Radome

In this last example an AR surface was designed using the unconstrained iterative design methodology from Section 7.2. The goal was to design an AR surface at K-band with a thickness of 0.15”. Recognizing this thickness is approximately 4 times greater than the minimal thickness achievable according to the ( 7.7 ) of 0.9 mm. The novelty of this AR surface design is that the permittivity profile exhibits an alternating slope distribution, illustrated in Figure

139

10.13(a). This permittivity profile cannot be implemented using subtractive manufacturing and instead was fabricated using fused deposition modelling. The transmission response of the two fabricated AR surfaces was measured to estimate the accuracy of the fabrication. The results of the transmission measurements are found in Figure 10.13(b). The measured and predicted transmission response show good agreement which suggests I was able reproduce a close approximation of the design permittivity profile. The measured and predicted results shown in

Figure 10.14 provide an excellent example of the effectiveness of the AR surface and the radome design methodology. Clearly, the addition of the AR surface reduces the insertion loss of the structural composite. Moreover, the results illustrate the effectiveness of the alternating permittivity profile shown in Figure 10.13. Figure 10.15

Figure 10.15

a b

Figure 10.13 (a) K-band iterative design permittivity profile. (b) Transmission of each FDM AR surface compared to the predicted transmission

140

a b b

Figure 10.14 (a) Illustration of structural composite with K-band iterative design AR surface. (b) Simulated and measured transmission loss results for structural composite with and without K-band iterative AR surface.

a b

Figure 10.15 (a) Astroquartz radome configuration. (b) Comparison of astroquartz radome insertion loss to structural composite radome with K-band iterative design AR surfaces.

Chapter 11: Conclusion

This dissertation presented a methodology for designing and fabricating wideband structural and ballistic radomes using conventional composite and ballistic materials. The methodology employed centered on transforming the radome design into an impedance matching problem utilizing electrically compatible materials. Chapters 2 and 3 provided a thorough overview of both structural composite and ballistic materials and identified the compatible conventional materials by highlighting both advantageous and detrimental electrical properties.

Chapter 4 provided a description of the state of the art in radome design and performance. While

Chapter 5 presented the standard techniques for developing impedance matching solutions, including the most common analytical methods for impedance matching. In addition to analytical methods for designing impedance matching structures, iterative methods were explored. The impedance matching solutions developed through analytical and iterative methods were implemented using subwavelength textured surfaces. Because the efficacy of the textured surfaces depends on the accuracy of the numerical modelling techniques employed, I modeled the antireflective surfaces using the rigorous coupled wave analysis method. Chapter 6 provided a detailed description of this method and the iterative methods used for the design of antireflective surfaces.

Chapter 7 provided several numerical examples of my radome design approach. The examples illustrated the wideband and broad incidence performance that radomes could achieve by employing this approach. By applying antireflective surfaces to conventional structural composite or ballistic materials, I was able to transform conventional structural materials into wideband radomes.

141

142

Specifically, Section 7.2.3 demonstrated the approach with several examples. In addition to demonstrating the effectiveness of the antireflective surfaces I also showed that improving the antireflective surfaces impedance matching performance also improves the performance of the radome. For example, the antireflective surfaces designed using the pattern search iterative method described in Section 5.4 produced tapers with better bandwidth and angular performance than the Klopfenstein taper illustrated Section 7.2.3. Moreover, the iteratively design non- monotonic taper in example 3 produced comparable bandwidth and angular performance as the generally accepted optimal Klopfenstein taper although it was 25% thinner.

Chapter 8 presented several radome simulation for ballistic protection materials and many of the conclusions found in Chapter 7 were similar to those observed in Chapter 8.

Chapter 8 also introduced impedance matching systems that not only included AR surfaces but also employed impedance matching layers that were simple homogenous slabs. The impedance matching layers were designed to reduce the impedance mismatch between the high dielectric core and the lower dielectric outer skins. In several cases the impedance matching systems were thinner than the less complex AR systems presented in Chapter 7. Chapter 9 focused on the fabrication of properly modeled textured surfaces. I introduced subtractive and additive manufacturing techniques. The advantages and pitfalls of each manufacturing technique was explored and conclusions were provided.

Finally, Chapter 10 provided measurements of several AR surfaces, structural composites and ballistic radomes to validate this methodology. The experimental validation results presented in Sections 10.2 to 10.4 illustrate the accuracy of the modeling approach as the antireflective measured and simulated insertion loss showed good agreement. Figure 9.9, Figure 10.5, Figure

143

10.7, Figure 10.9, Figure 10.10, Figure 10.12 and Figure 10.14 all provide good examples of the effectiveness of the design approach, given the accuracy of the modelling and the improvement of the insertion loss. Moreover, using the antireflective surface method is the best method for transforming ballistic armor into ballistic radomes, and this method is a practical, easy to implement, high performing alternative to conventional radome design methods.

This dissertation presented a methodology to design broadband antireflective surfaces that create a wideband impedance matching system for structural and ballistic materials that transform conventional structural composites and ballistic armor into wideband, broad incidence radomes. Indeed, the robustness of this approach allowed the marriage of conventional structural and ballistic materials with novel antireflective surfaces. The radomes created retained all of their structural and ballistic characteristics while adding an attractive wideband RF transparency not previously available.

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Appendix A

Rigorous Couple Wave Analysis Enhanced Transmittance Matrix Approach

푑푈푦푚푛(푧) 푑푧 −1 ퟐ −1 0 0 퐊푦훍 퐊푥 훆 − 퐊풚훍 푑푈푥푚푛(푧) 푈푦푚푛 0 0 −훆 + 퐊ퟐ훍−1 −퐊 훍−1퐊 푑푧 퐱 푥 푦 푈푥푚푛 훀 = = −1 ퟐ −1 푑푆 (푧) 퐊 훆 퐊 흁 − 퐊 훆 0 0 푆푦푚푛 푦푚푛 푦 푥 풚 ퟐ −1 −1 [푆 ] ( 13.1 ) 푑푧 [퐊퐱휺 − 흁 −퐊푥훆 퐊푦 0 0 ] 푥푚푛 푑푆푥푚푛(푧) 푑푧

Faradays Law relating the magnetic field to the curl of the electric field in differential form. Solve Maxwell’s Fourier transformed coupled equations using enhanced transmittance matrix approach. Compute eigenvalues of ( 13.1 ).

푝 퐪푚푛 = eigenvalues(훀) ( 13.2 ) Elements of the eigenvalues of Ω

푝 퐰푚푛 = eigenvectors(훀) ( 13.3 ) Elements of the eigenvectors of Ω.

풑 풑 풑 풑 퐰풎풏 ⋯ 퐰 퐰ퟏퟏ ⋯ 퐰ퟏ풏 +ퟏ ퟏ풏 ퟐ 퐖풑 = [ ⋮ ⋱ ⋮ ] 퐖풑 = [ ] ퟏ 풑 풑 ퟐ ⋮ ⋱ ⋮ 풑 풑 퐰풎ퟏ ⋯ 퐰풎풏/ퟐ 퐰풎ퟏ ⋯ 퐰풎풏 ( 13.4 )

푝 Eigenvector matrix of Ω, 퐰푚푛 eigenvectors. Where m and n are spatial harmonic and p represents each grating layer.

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풑 풑 풒풎풏 ⋯ ퟎ √풒 ⋯ ퟎ √ +ퟏ ퟏퟏ ퟐ 풑 풑 퐐ퟏ = ⋮ ⋱ ⋮ 퐐ퟐ = ⋮ ⋱ ⋮ 풑 풑 ퟎ ⋯ √풒 ퟎ ⋯ √풒 [ 풎풏/ퟐ] [ 풎풏] ( 13.5 )

푝 Diagonal matrix of eigenvalues (푞푚푛) that are positive and square rooted. Where m and n are the spatial harmonics and p represents each grating layer.

푝 푒−푘0푞11푑 ⋯ 0 퐗푝 = [ ⋮ ⋱ ⋮ ] 푝 0 ⋯ 푒−푘0푞푚푛푑 ( 13.6 )

Diagonal matrix of elements, where p represents each grating layer.

퐕풑 = 퐖풑퐐풑 ( 13.7 ) Product matrix of eigenvalue elements and eigenvector matrix.

1 2 퐕푠푠퐗 퐕푠푝퐗 퐕푠푠 퐕푠푝 + 푐1 퐈 ퟎ 1 2 − 퐖푠푠퐗 퐖푠푝퐗 −퐖푠푠 −퐖푠푝 푐1 풋퐘퐈퐈 ퟎ 퐓푠 1 2 [ +] = [ ] [ ] 퐖푝푠퐗 퐖푝푝퐗 −퐖푝푠 −퐖푝푝 푐2 0 퐈 퐓푝 − 1 2 푐 0 −푗퐙퐈퐈 ( 13.8 ) [ 퐕푝푠퐗 퐕푝푝퐗 퐕푝푠 퐕푝푝 ] 2

Diffracted reflected amplitudes at incident region boundary TE and TM polarizations.

1 2 퐕푠푠 퐕푠푝 퐕푠푠퐗 퐕푠푝퐗 + sin 휓 훿푖0 퐈 ퟎ 푐1 1 2 − 푗 sin 휓 푛퐼훿푖0 cos 휃 −풋퐘퐈 ퟎ 퐑푠 퐖푠푠 퐖푠푝 −퐖푠푠퐗 −퐖푠푝퐗 푐1 [ ] + [ ] [ ] = 1 2 [ +] −푗 cos 휓 푛퐼훿푖0 0 퐈 퐑푝 퐖푝푠 퐖푝푝 −퐖푝푠퐗 −퐖푝푝퐗 푐2 − cos 휓 cos 휃 훿 0 −푗퐙퐈 1 2 푐 ( 13.9 ) 푖0 [ 퐕푝푠 퐕푝푝 퐕푝푠퐗 퐕푝푝퐗 ] 2

Diffracted reflected amplitudes at incident region boundary TE and TM polarizations.

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푝−1 푝−1 푝−1 + 푝 푝 푝 + 퐖 퐗 퐖 푐푝−1 퐖 퐗 퐖 푐푝 [ 푝−1 푝−1 푝−1] [ − ] = [ 푝 푝 푝] [ −] 퐕 퐗 −퐕 푐푝−1 퐕 퐗 −퐕 푐푝 ( 13.10 ) Diffracted amplitudes within the grating region

퐕푠푠 = 퐅푐퐕11

퐖푠푠 = 퐅푐퐖1 + 퐅푠퐕21

퐕푠푝 = 퐅푐퐕12 − 퐅푐퐖2

퐖푠푝 = 퐅푠퐕22

퐖푝푝 = 퐅푐퐕22 ( 13.11 )

퐕푝푝 = 퐅푐퐖2 + 퐅푠퐕12

퐖푝푠 = 퐅푐퐕21 − 퐅푠퐖1

퐕푝푠 = 퐅푠퐕11

Fc and Fs are diagonal matrices with elements exp (−푘0푞1푚푛푑) and exp (−푘0푞2푚푛푑).

퐖푝 퐖푝퐗푝 푃 푃 푝 −1 푓푝+1 [ ] [퐖 퐗 퐖 ] [ ] 퐓 퐕푝 −퐕푝퐗푝 퐕푃퐗푃 −퐕푝 푔푝+1

푾푝 푾푝푿푝 푃 −1 푃 푝 −1 푓푝+1 =[ ] [푿 0] [퐖 퐖 ] [ ] 퐓 푽푝 −푽푝푿푝 0 푰 퐕푃 −퐕푝 푔푝+1

−1 ( 13.12 ) 퐖푝 퐖푝퐗푝 퐗푃 0 퐚푝 =[ ] [ ] [ ] 퐓 퐕푝 −퐕푝퐗푝 0 퐈 퐛푝

퐚푝 푃 푝 −1 푓푝+1 [ ] = [퐖 퐖 ] [ ] 퐛푝 퐕푃 −퐕푝 푔푝+1

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퐖푝(퐈 + 퐗푝퐛푝(퐚푝)−1퐗푝) = [ ] 퐓푝 퐕푝(퐈 − 퐗푝퐛푝(퐚푝)−1퐗푝)

−1 푓푝 퐖푝 퐖푝퐗푝 퐈 [ ] 퐓푝 = [ ] [ ] 퐓푝 푔푝 퐕푝 −퐕푝퐗푝 퐛푝(퐚푝)−1퐗푝

퐓 = (퐚푝)−1퐗푝퐓푝

All reflected and transmitted amplitudes within the grating region. Numerically stable computation and simplification. Method is stable because inversion is performed on matrix that is no longer ill conditioned, due to the substation of T = (a푝)−1X푝T푝 .

Analytical solution for rectangular and hexagonal permittivity distributions

Rectangular distribution of cylinders within a medium.

Figure 13.1 Antireflective surface structures for a rectangular packed hole array

2 푚 2 푛 푝 푝 퐽 (휋푑 √( ) + ( ) ) 1 푝 훬푥 훬푦 푝 푑푝(휀푠 − 휀ℎ ) 휀푚푛 = ( 13.13 ) 2훬푥훬푦 2 2 √(푚 ) + ( 푛 ) 훬푥 훬푦

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2 1 1 푚 2 푛 푑푝 ( 푝 − 푝) 퐽1 (휋푑푝√( ) + ( ) ) 휀 훬푥 훬푦 푝 푠 휀ℎ 휉푚푛 = 2훬푥훬푦 2 2 √(푚 ) + ( 푛 ) 훬푥 훬푦

st Rectangular permittivity distribution, where J1 denotes the Bessel function of the 1 kind order 1. Hexagonal distribution of cylinders within a medium

Figure 13.2 Antireflective surface structures for a hexagonal packed hole array.

휋푑푝 푛2 √ +푚2 푝 푝 퐽1 ( 훬 3 ) 푝 푑푝(휀푠 − 휀ℎ ) 휀푚푛 = (1 + cos(휋(푚 + 푛))) 2√3훬 2 √푛 2 3 + 푚

휋푑푝 푛2 1 1 √ +푚2 푑푝 ( 푝 − 푝) 퐽1 ( 훬 3 ) ( 13.14 ) 푝 휀푠 휀ℎ 휉푚푛 = (1 + cos(휋(푚 + 푛))) 2√3훬 2 √푛 2 3 + 푚

st Hexagonal permittivity distribution, where J1 denotes the Bessel function of the 1 kind order 1.

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