Novel Metamaterial Blueprints and Elements for Electromagnetic Applications
Dissertation
Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University
By
Hayrettin Odabasi, B.S., M.S.
Graduate Program in Electrical and Computer Engineering
The Ohio State University
2013
Dissertation Committee:
Fernando L. Teixeira, Advisor, Prabhakar Pathak Roberto Rojas-Teran c Copyright by
Hayrettin Odabasi
2013 Abstract
In the first part of this dissertation, we explore the metric invariance of Maxwell’s equations to design metamaterial blueprints for three novel electromagnetic devices.
The metric invariance of Maxwell’s equations here means that the effects of an
(hypothetical) distortion of the background spatial domain on the electromagnetic
fields can be mimicked by properly chosen material constitutive tensors. The exploitation of such feature of Maxwell’s equations to derive metamaterial devices has been denoted as ‘transformation optics’ (TO). The first device proposed here consists of metamaterial blueprints of waveguide claddings for (waveguide) miniaturization. These claddings provide a precise control of mode distribution and frequency cut-off. The proposed claddings are distinct from conventional dielectric loadings as the former do not support hybrid modes and are impedance-matched to free-space. We next derive a class of metamaterial blueprints designed for low-profile antenna applications, whereby a simple spatial transformation is used to yield uniaxial metamaterial substrate with electrical height higher than its physical height and surface waves are not supported, which is an advantage for patch antenna applications. We consider the radiation from horizontal wire and patch antennas in the presence of such substrates.
Fundamental characteristics such as return loss and radiation pattern of the antennas are investigated in detail. Finally, transformation optics is also applied to design cylindrical impedance-matched absorbers. In this case, we employ a complex-valued
ii transformation optics approach (in the Fourier domain) as opposed to the conventional real-valued approach. A connection of such structures with perfectly matched layers and recently proposed optical pseudo black-hole devices is made.
In the second part of this dissertation, we move from the derivation of metamaterial blueprints to the application of pre-defined unit-cell metamaterial structures for miniaturization purposes. We first employ electric-field-coupled (ELC) resonators and complementary electric-field-coupled (CELC) resonators to design a new class of electrically small antennas. Since electric-field coupled resonators were recently proposed in the literature to obtain negative permittivity response, we next propose ELC resonators as a new type of waveguide loadings to provide mode control and waveguide miniaturization.
iii Dedicated to my family.
iv Acknowledgments
First and foremost, I would like to express my sincere gratitude to my advisor
Prof. Fernando L. Teixeira for his continuous guidance, encouragement, patience and understanding throughout my PhD. He has been a great advisor with patience, calmness and guidance and I felt very fortunate to work with him. His perspective and approach on scientific problems was one of the many things that I learned from him that I deeply appreciate.
I would also like to thank Prof. Prabhakar Pathak and Prof. Roberto Rojas for participating in my doctoral committee. I would like to thank Prof. Kubilay Sertel for supporting me in my last semester at OSU, which I am very grateful.
I would like to extend my sincere appreciation to Prof. Durdu O. Guney for intro- ducing me the metamaterial concept and supporting me during my stay in Houghton,
MI.
During my studies here at OSU, I have had many good friends that I always felt very fortunate. I would like to thank Ahmed Fouda not only for his friendship but also fruitful discussion on many topics during our coffee breaks. I would like to thank Erdinc Irci, Mustafa Kuloglu and Ugur Olgun for their friendship and help on various aspects of my study. I would like to thank Bunyamin Koz, Yunus Zeytuncu,
Yusuf Danisman, Oguz Kurt, Huseyin Acan, Fatih Olmez, Fatih Akyol, Huseyin
Ayvaz, Mustafa Yesil, Erdem Ozbek, Mehmet N. Tomac for their friendship over
v past five years. I would also like to thank to Seyit A. Sis, Fahri Sarac, Kamil Ciftci,
Sinan Savas, Selman Sakar, Orhan Bulan, Kasim Cologlu and all my friends for their valuable friendship.
I would like to thank my dear wife Zehra. Without her continuous support and love this work would have not been completed. Thank you for your endless love and support for myself and our daughter Ahsen Neva.
Finally, I would like to thank my parents, Halime and Mehmet Kadir, my sister
Hale for their unconditional love and support throughout my life.
vi Vita
March 03, 1983 ...... Born - Canakkale, Turkey
2005 ...... B.S. Electronics Eng. Gebze Institute of Technology, Turkey 2008 ...... M.S. Electrical Eng. and Computer Sci. Syracuse University, USA 2005-2007 ...... Grad. Research & Teaching Assistant Electronics Eng., Gebze Institute of Technology, Turkey 2007-2008 ...... Graduate Research Associate Electrical Eng. and Computer Sci., Syracuse University, USA 2008-present ...... Graduate Research Associate ElectroScience Laboratory, Electrical and Computer Eng., The Ohio State University, USA
Publications
Research Publications
H. Odabasi, F. L. Teixeira and D. O. Guney, “Electrically small, complementary electric-field-coupled resonator antennas,” Journal of Applied Physics, 113, 084903 (2013), DOI:10.1063/1.4793090.
H. Odabasi and F. L. Teixeira, “Analysis of Canonical Low-Profile Radiators on Isoimpedance Metamaterial Substrates,” Radio Science, 47, RS1002, 2012.
vii H. Odabasi and F. L. Teixeira, “Impedance-Matched Absorbers and Pseudo Black Holes,” J. Opt. Soc. Am. B., vol. 28, no. 5, pp. 1317–1323, 2011.
F. L. Teixeira, H. Odabasi and K. F. Warnick, “Anisotropic Metamaterial Blueprints for Cladding Control of Waveguide Modes,” J. Opt. Soc. Am. B., vol. 27, no. 8, pp. 1603–1609, 2010.
Conference Publications
H. Odabasi and F. L. Teixeira, “Metamaterial Claddings for Waveguide Miniatur- ization,” IIEEE International Symposium on Antennas and Propagation and USNC- URSI Natinal Radio Science Meeting, Orlando, 2013.
H. Odabasi and F. L. Teixeira, “Complementary Electric-Field-Coupled (CELC) Based Resonator Antennas,” IEEE International Symposium on Antennas and Prop- agation and USNC-URSI Natinal Radio Science Meeting, Orlando, 2013.
H. Odabasi and F. L. Teixeira, “Low-profile Antennas with Anisotropic Dispersive Metamaterial Substrate,” IEEE International Symposium on Antennas and Propaga- tion and USNC-URSI Natinal Radio Science Meeting, Chicago, 2012.
H. Odabasi and F. L. Teixeira, “Analysis of Electromagnetic Pseudo Black-Hole Devices Using FDTD in Cylindrical Grids,” 28th Annual Review of Progress in Applied Computational Electromagnetics, Columbus, 2012.
H. Odabasi and F. L. Teixeira, “Isoimpedance Anisotropic Substrates for Planar Antenna Profile Reduction,” USNC-URSI National Radio Science Meeting, Boulder, 2012.
H. Odabasi and F. L. Teixeira, “Analysis of Cylindrically Conformal Patch Antennas on Isoimpedance Anisotropic Substrates,” URSI General Assembly and Scientific Symposium, Istanbul, 2011.
H. Odabasi, F. L. Teixeira and W. C. Chew, “Analysis of Metamaterial Absorber Blueprints for Optical ‘Black Holes’,” URSI General Assembly and Scientific Sympo- sium, Istanbul, 2011.
H. Odabasi and F. L. Teixeira, “Impedance Analysis of Extremely Low-Profile Anten- nas Using Metamaterial Substrates,” IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, Istanbul, 2010.
viii Fields of Study
Major Field: Electrical and Computer Engineering
Studies in: Transformation Optics Prof. F. L. Teixeira Metamaterials Prof. F. L. Teixeira, Prof. Durdu O. Guney Finite Difference Time Domain Prof. F. L. Teixeira Antenna Miniaturization Prof. F. L. Teixeira
ix Table of Contents
Page
Abstract...... ii
Dedication...... iv
Acknowledgments...... v
Vita ...... vii
ListofTables...... xiii
ListofFigures ...... xiv
1. Introduction...... 1
1.1 IntroductionandBackground ...... 1 1.2 Contributions and Organizations of the Dissertation ...... 4
2. A Brief on Metamaterials and Transformation Optics ...... 8
2.1 OverviewofMetamaterials ...... 8 2.1.1 Introduction ...... 8 2.1.2 EffectiveMediumTheory ...... 10 2.1.3 Retrieval of Effective Parameters ...... 11 2.2 Overview of Transformation Optics ...... 15
3. Anisotropic Metamaterial Blueprints for Cladding Control of Waveguide Modes...... 20
3.1 Introduction ...... 20 3.2 Metamaterial Claddings for Waveguide Mode Control ...... 21 3.2.1 Circular Waveguide ...... 21
x 3.2.2 Rectangular Waveguide ...... 25 3.3 Examples ...... 27 3.4 ConclusionsandFurtherRemarks...... 30
4. Analysis of Canonical Low-profile Radiators on Isoimpedance Metamate- rialSubstrates...... 35
4.1 Introduction ...... 35 4.2 IsoimpedanceSubstrates...... 38 4.3 NumericalResults...... 45 4.3.1 Horizontaldipole...... 45 4.3.2 Rectangularpatch ...... 47 4.4 Conclusion ...... 54
5. Impedance Matched Absorbers and Pseudo Black Holes ...... 55
5.1 Introduction ...... 55 5.2 Metamaterial blueprints for reflectionless absorbers ...... 56 5.2.1 Planargeometries ...... 56 5.2.2 Curvedgeometries ...... 58 5.3 Material tensor properties on convex and concave surfaces ..... 59 5.4 Metamaterial blueprints of near-reflectionless absorbers on convex surfaces ...... 62 5.4.1 Approximate ‘PML-derived’ metamaterial blueprints . . . . 62 5.4.2 Simulations and backscattering results ...... 62 5.4.3 Optical ‘black hole’ metamaterial blueprints with embedded absorption ...... 67 5.5 Conclusion ...... 70
6. Electrically Small Complementary Electric-Field-Coupled Resonator An- tennas...... 71
6.1 Introduction ...... 71 6.2 ELC/CELC-Based Metaresonator Antennas ...... 74 6.3 Conclusion ...... 82
7. Metamaterial Claddings for Waveguide Miniaturization ...... 83
7.1 Introduction ...... 83 7.2 Rectangular Waveguides Loaded with Anisotropic Media ...... 85 7.3 TE Case: Electric and Magnetic Resonator Loaded Waveguide . . . 89 7.4 Conclusion ...... 95
xi 8. ConclusionsandFurtherRemarks...... 96
8.1 Conclusions...... 96
Appendices 100
A. FDTD Method for Dispersive Media in Cartesian Coordinates ...... 100
A.1 PML-PLRC-FDTDFormulation ...... 100
B. FDTD Method for Dispersive Media in Cylindrical Coordinates . . . . . 108
B.1 PML-PLRC-FDTDFormulation ...... 108
Bibliography ...... 112
xii List of Tables
Table Page
3.1 Cutoff frequencies of the dominant circular (cTE11) and rectangular (rTE10) modes for different s and following waveguide parameters: R = 1cm, a = 1.4cm, b = 1cm, d = 0.8R, and t = 0.1a ...... 33
4.1 Trade-off between patch dimensions and frequency bandwidth. ... 50
6.1 Simulationresultsofthefiguresofmerit...... 74
6.2 Measurementresultsofthefiguresofmerit...... 75
xiii List of Figures
Figure Page
2.1 Classification of possible material choices in isotropic media (where both ǫ and µ are scalar). Materials in the first quadrant are denoted as right-handed media. The second and fourth quadrants correspond to electric or magnetic plasmas that are opaque to electromagnetic waves. The third quadrant is of interest due to recent developments in metamaterial. Third-quadrant materials are denoted as left-handed media. Note that this classification refers to the real part of ǫ and µ only...... 11
2.2 Effective parameters for SRR. (a) Effective permittivity (b) Effective permeability (c)Unit cell configuration ...... 13
2.3 Effective parameters for ELC. (a) Effective permittivity (b) Effective permeability (c)Unit cell configuration ...... 14
2.4 Overview of transformation optics. The solid blue line represents an op- tical ray path. In empty flat space, the ray travels along a straight path. However, when the space is hypothetically distorted, the light travels along different path according to the underlying distortion. Trans- formation optics shows that the same ray path (as well as diffractive phenomena) can be mimicked with equivalent materials defined based on the relevant metric distortion (a) Empty flat space, light travels along straight path. (b) Transformed space, light travels non-straight path. (c) Equivalent physical space, light travels non-straight path. . 16
3.1 Cross-section views of (a) circular and (b) rectangular waveguides with a metamaterial cladding shown in yellow...... 21
3.2 ...... 24
xiv 3.3 Power density E~ × H~∗ distribution of the dominant mode on a rectan- gular waveguide with transversal dimensions a = 1.4cm, b = 1cm. The cladding is placed on the two lateral walls with [ǫ]=ǫ0 [Λ] and [µ]=µ0 [Λ], where [Λ] is given by equation (3.18). Four different values for s are again considered: (a) s = 1, (b) s = 2, (c) s = 3, (d) and s = 5. Similarly to the circular waveguide case, an increase in s produces a more uniform distribution within the core and overall more power con- finement in the cladding. The plots in (f), (g), and (h) screen out the cladding region and have resulting power distribution in the core normalizedtothepeakvaluethere...... 30
∗ 3.4 Power density E~ × H~ distribution for the TE11 mode on a rectangular waveguide with cladding on all four walls. Due to the bivariate nature of the TE11 mode and the presence of the cladding on all four walls, a change in s affects the mode distribution in both x and y. Similarly as before, an increase in s produces a more uniform distribution within the core and overall more power confinement in the cladding (and par- ticularly at the four corners). The plots in (f), (g), and (h) screen out the cladding region and have resulting power distribution in the core normalizedtoitspeakvaluethere...... 31
3.5 FDTD results showing cut-views of the steady-state electric and mag- netic field distributions along a rectangular waveguide with transversal dimensions a = 1.4cm, b = 1cm, cladding thickness t = 0.14cm, and length l = 9cm. The cladding is inserted along the two lateral walls, having [ǫ] =ǫ0 [Λ] and [µ] =µ0 [Λ] (see main text) with four values of s considered: (a) s = 1, (b) s = 2, (c) s = 3, (d) and s = 5. The source excitation frequencies are chosen between the two lowest cutoff frequencies; so that only the dominant mode propagates (the corresponding dominant mode cutoff frequencies are listed in Table I). 32
4.1 (a) Equivalent and (b) actual geometries. The isoimpedance metama- terial substrate with [ǫ] =ǫ [Λ] and [µ] =µ [Λ] (see the expression for [Λ] in the main text) is designed via TO principles so that the electro- magnetic fields above the planar antennas are identical for these two geometries...... 39
xv 4.2 Input impedance of a horizontal dipole for three configurations: (i) air, h = λ/4, (ii) isoimpedance metamaterial with s = 5, h = λ/16 (d = λ/64, t = 3λ/64), and (iii) HIGP with ǫr = 0.2 and µr = 5, h = λ/16 (d = λ/64, t = 3λ/64), where the ground plane dimensions are 1.5λ0 × 1.5λ0. Note how the performance of configuration (ii) is verycloseto(i)...... 40
4.3 S11 (negative return loss) of a horizontal dipole for the configurations considered in Fig. 4.2. Original and isoimpedance metamaterial con- figurations (i) and (ii) yield almost identical S11. HIGP configuration (iii)radiateswithreducedbandwidth...... 41
4.4 Electric field magnitude distribution along the xz-plane (side-view) for a horizontal dipole on top of air-backed, isoimpedance metamaterial- backed, and dielectric-backed ground planes: (i) air, h = λ/4; (ii) metamaterial with s=5, h = λ/16; (iii) air, h = λ/16; and (iv) HIGP with ǫr = 0.2 and µr = 5, h = λ/16. The lateral dimensions of the substrate are assumed infinitely large. The horizontal solid line indicates the ground plane level and the dashed line indicates the top surfaceofthesubstrate...... 43
4.5 Radiation pattern of horizontal dipole for the configurations considered inFig.4.2. (a)E-plane,(b)H-plane...... 44
4.6 Patch antenna geometry with isoimpedance metamaterial substrate. . 47
4.7 S11 (negative return loss) of a rectangular patch in four substrate con- figurations: (i) isoimpedance metamaterial with s = 3, (ii) dielectric (miniaturized size) with ǫr = 2.2, (iii) dielectric (miniaturized size) with ǫr = 10.2, and (iv) dielectric (original size) with ǫr = 2.2. In all cases, h = 1.2mm...... 48
4.8 Radiation pattern of a rectangular patch for three substrate configu- rations: (i) isoimpedance metamaterial with s = 3, (ii) dielectric with ǫr = 2.2, and (iii) dielectric with ǫr = 10.2. In all cases, h = 1.2mm and L = λ...... 49
4.9 E and H-plane mutual coupling configurations...... 51
xvi 4.10 E-plane mutual coupling between two co-planar rectangular patches for three configurations: (i) isoimpedance metamaterial (original size) with s = 3, (ii) dielectric (miniaturized size) with ǫr = 2.2, and (iii) dielectric (miniaturized size) with ǫr = 10.2. In all cases, h = 1.2mm. 52
4.11 H-plane mutual coupling between two co-planar rectangular patches for the same three configurations as considered the E-plane case. . . . 53
5.1 Behavior of PML-derived metamaterial blueprints on concave versus convexsurfaces...... 61
5.2 Spatial distribution, for two instants of time, of the electric field pro- duced by a point source (with location indicated by the star symbol) nearby a circular PEC cylinder (in white). In the top sequence, the PEC cylinder is stand-alone, producing a sharp reflection (backscat- tering). The mid and bottom sequences, the PEC cylinder is coated by a metamaterial absorber shell (whose outer boundary is indicated by a dashed circle) having constitutive parameters [ǫ] =ǫ0 [Λ](ρ,φ) and [µ] =µ0 [Λ](ρ,φ), with [Λ](ρ,φ) given by (5.10) and where the parameter ∆ρ is enforced to be zero or positive, in an ad hoc fashion. These two choices are both very effective in suppressing reflections (backscattering). 63
5.3 Same scattering geometry as before, except that the metamaterial coat- ing used has [Λ](ρ,φ) as given by (5.10) with negative parameter ∆ρ as directly computed from (5.9). Three snapshots of the electric field distribution are presented. Even though the frontal reflection is still strongly suppressed, the spurious field growth is clearly visible at the t = 11.72 ns and at t = 14.66 ns snapshots, with a particularly strong onsetinthe‘shadowregion’...... 64
5.4 Reflection coefficients computed for different choices of ∆ρ as in Fig. 5.2, and for three values of the conductivity σ0 factor (a) σ0 = 0.045, (b) σ0 = 0.075, and (c) σ0 = 0.105. Lower σ0 values produce less reflection from the ‘front-end’ of the coating (at about 2 ns) but more reflec- tion from the PEC cylinder itself (at about 5 ns); conversely, larger σ0 values decrease the reflection originated from the PEC cylinder but produce more reflection from the ‘front-end’ of the coating. The re- flection from the PEC cylinder with no coating (‘no absorber’) is also plottedforcomparison...... 66
xvii 5.5 ’Reflection coefficient’ results for σ0 = 0.5, showing the spurious field growth (at about 10 ns) originating from the metamaterial coating with ∆ρ < 0...... 68
6.1 Schematic configurations of (a) SRR, (b) ELC, (c) CELC-1, and (d) CELC-2, CELC-3 resonator antennas. For the CELC-1,-2,-3 configu- rations, the respective orientation is indicated in (e)...... 73
6.2 Return loss of the resonator antennas depicted in Fig. 6.1...... 76
6.3 3-D far-field radiation pattern for each of the antennas shown in Fig.6.1. 77
6.4 xy- and yz-plane radiation patterns for each of the antennas shown in Fig.6.1...... 78
6.5 Surface current plots for each of the antennas shown in Fig.6.1. . . . 79
6.6 (a) Bent monopole-excited CELC antenna. (b) Measured return loss. (c)Radiationpattern...... 81
7.1 Schematic configuration of a rectangular waveguide filled with uniform uniaxialmedia...... 85
7.2 Schematic configuration of a rectangular waveguide loaded with ELC and SRR. ELC resonators are placed at the lateral side walls while the SRR resides at the center of the waveguide. Perspective view (a), Side viewforELC(b)andSideviewforSRR(c)...... 88
7.3 Unit cell configurations for ELC(a,c) and SRR(b,d). For simulations PEC and PMC boundary conditions are applied on the y and x direc- tions respectively. The parameters depicted in (b,d) for ELC and SRR aregiveninthetext...... 90
7.4 Effective ǫ and µ response for ELC and SRR, respectively...... 91
7.5 Transmission coefficients for ELC, SRR, and ELC + SRR loaded waveg- uides, respectively. Note that the transmission coefficient for ELC + SRR loaded waveguide matches closely with respective ELC and SRR loadedwaveguides...... 92
xviii 7.6 Effective parameters for ELC + SRR loaded waveguide. Note that the first transmission happens around 6.5 GHz where both ǫ and µ are negative and the second transmission is around 8.5 GHz where both ǫ and µ arepositive...... 93
7.7 Dispersion diagram for ELC loaded and SRR loaded waveguide. Note that at SRR resonance waves are backward waves whereas at ELC resonance waves are forward waves...... 94
xix Chapter 1: Introduction
1.1 Introduction and Background
Over the past decade, the emerging field of ‘metamaterials’ has attracted a great deal of attention in the scientific community [1]–[8]. Metamaterials are artificially designed structures where sub-wavelength inclusions, which provide added degrees of freedom to control the material properties, are placed periodically (or non-periodically) into a host medium to yield effective material properties with unusual characteristics not found in any natural bulk medium. Thanks to dramatic advances in fabrica- tion technology in recent years, a host of metamaterials can now be fabricated with tailored spatial distributions of effective permittivity and permeability, allowing, for example, unprecedented control over the behavior of electromagnetic fields in such structures [1]–[16].
An important class of metamaterials is media with negative permittivity and
permeability, dubbed as left-handed media (LHM), negative refractive index (NIM),
double negative materials (DNG), or backward-wave materials. In its basic form,
LHM were first studied theoretically by Vesalego, who delineated some of their basic
unusual properties [17]. However, his study remained just a scientific curiosity for
over 30 years since there were no such materials in nature. In 2000, a composite
1 medium with simultaneously negative permeability and permittivity was proposed
and studied numerically [18]. In such composite media, metallic wire elements [19]
and split-ring resonators (SRRs) [20] are used a building blocks and combined to
build the desired response. In the same year, Pendry proposed the idea of a ‘perfect
lens’ [21] relying directly on negative refractive index, which drawn further interest
into LHM structures. In 2001, negative refraction was experimentally verified by
Shelby et al [9]. Motivated by the above pioneering studies, a tremendous amount
of research has been devoted to LHM ever since.
Another cornerstone for metamaterials occurred in 2006 when Pendry et al. [22] and Leonhardt [23] simultaneously proposed a recipe of how to obtain an ‘invisibil- ity cloak’, at least theoretically. This recipe is based on the metric invariance or
‘form-invariance’ feature of Maxwell’s equations [22]–[33], and provides the underly- ing basis for the so-called ‘transformation optics’ (TO) techniques. TO is a powerful approach for the systematic design of ”blueprints” for metamaterials constitutive tensors [22]–[33] and subsequent control of electromagnetic waves in a myriad of possible applications [22]–[25], [27], [29], [30], [34]–[55]. In addition to electromag- netic cloaks [22], [23], [36]–[39], some of the new devices that have been designed via TO include wave splitters [40], polarization rotators [41], field concentrators [42], electromagnetic masks [43], [44], perfectly matched layers [25], [34], [35], reflectionless waveguide bends [45], [46], waveguide loadings [47], [48], high-performance omnidirec- tional absorbers (so-called optical ”black holes”) [27], [29], [30], [49]–[54], and many others [55].
It is of note that the metric invariance of Maxwell’s equation first discovered very early by Weyl and later studied by many others in different contexts [22],
2 [24]–[26], [30], [56]–[60] but was never truly well-known or exploited for practical
material/device applications. This is because, in the conventional vector calculus
language, the coefficients associated with the metric of space are intertwined with
spatial derivative operators (i.e., curl, div, and grad operators), which obfuscates
the underlying metric/constitutive duality of Maxwell’s equations. The language of
differential forms (exterior calculus) [25], [57]–[62] provides perhaps the most direct
route to unveil the metric invariance of Maxwell’s equations because the only spa-
tial operator then present in Maxwell’s equations is the exterior derivative operator,
which is independent of any metric coefficient. Indeed, in the differential forms frame-
work, all the information about the metric and constitutive medium properties are
conveniently lumped together in (a pair of) Hodge star operators [25], [62]–[64]. Un-
fortunately, differential forms are still much less widely used than vector calculus in
classical electromagnetism. It is also of note that, in work that predates papers on
cloaking and other TO applications in metamaterials, the metric/constitutive dual-
ity was successfully explored for the design of perfectly matched layers (PMLs) [25],
[34], [35], [62] and for the implementation of consistent finite-difference schemes in
irregular meshes [24], [25], [65], for example.
Despite the great potential that metamaterials in general and TO in particular provide, the design of some metamaterials devices can be very challenging. Due to their resonant characteristics, metamaterials tend to be highly lossy and dispersive.
Such loss and narrowband characteristic can be detrimental to the performance of some proposed devices such as superlenses and invisibility cloaks.
One main focus of this dissertation is to exploit TO for the design of novel meta- material blueprints, viz. metamaterial claddings for waveguide miniaturization and
3 mode control, metamaterial substrates for the design very-low profile antennas and metamaterial shells for providing (near-)reflectionless omnidirectional absorbers (op- tical pseudo black holes). As noted, one basic challenge for subsequent fabrication arising from such designs is that the required media is typically anisotropic in both the permittivity and permeability. However, simplifications can often be made in the material tensors with minimal impact on the performance if it is know that the excitation fields will be of a particular form (for example, of a particular polarization) as opposed to a generic excitation.
1.2 Contributions and Organizations of the Dissertation
The major contributions of this dissertation can be summarized as follows:
• Metamaterial claddings are designed to control waveguide modes and provide
miniaturization of waveguides. By using the proposed claddings, better mode
uniformity within the core region can be obtained, together with a precise control
of the corresponding frequency cutoffs.
• New blueprints for metamaterial substrates are designed for low-profile anten-
nas, viz. horizontal wire antenna and a patch antenna. The specially designed
metamaterial substrates yield electrically-higher profile thus resulting in better
radiation characteristics.
• The integration of (perfectly) absorptive properties into the recently proposed
optical pseudo black hole devices is investigated. It is shown that reflection-
less optical black holes are not possible, even theoretically. Instead, near-
reflectionless optical pseudo black holes are proposed and studied.
4 • Complementary electric-field-coupled (CELC) elementary resonators are used
for the first time to design electrically small antennas (ESAs). The radiation
characteristics of CELC-based antennas are investigated, and new low-profile
antennas are designed.
• Rectangular waveguides loaded with ELC resonators are studied for waveguide
miniaturization purposes. The propagation of electromagnetic waves in such
waveguides is studied in detail, and in particular the unique properties of their
dispersion diagram are shown.
This dissertation is organized as follows:
Chapter 2 provides more detailed background information on metamaterials and transformation optics. Split-ring resonators and electric-field-coupled resonators are considered to obtain particular effective permittivity and permeability material de- signs. Effective medium theory is revisited for this purpose. As an example, we design a composite structure combining both such resonator elements. The metric invariance of Maxwell’s equations is explained in connection with transformation optics.
In Chapter 3, transformation optics is employed to design blueprints of metamate- rial claddings for waveguide mode control and miniaturization. By using the proposed claddings, better mode uniformity within the core region can be obtained, together with a precise control of the corresponding frequency cutoffs. At the same time, modal power distribution can be increasingly confined within the cladding region. In contrast to conventional material loading of waveguides, the proposed metamaterial cladding does not produce hybrid modes. Each resulting mode is homotopic to a hollow waveguide mode, in the sense that the former can be produced from a contin- uous deformation of the latter within the waveguide core. Physically, this means that
5 the mode distribution within the waveguide core is equivalent to that of a waveguide
with different (possibly larger) dimensions. We illustrate the analysis for circular and
rectangular perfectly electrically conducting (PEC) waveguides.
In Chapter 4, we analyze some the properties of canonical low-profile electromag-
netic radiators in the presence of ‘isoimpedance’ substrates. The latter are configured
as anisotropic metamaterials having constitutive tensors that provide impedance-
matching (to free space) for all incidence angles and polarizations. The metamaterials
for isoimpedance substrates are further devised to increase the electrical (equivalent)
thickness of the substrate without producing surface waves. We study the effect of
such metamaterial substrates on basic properties such as input impedance, return
loss, and radiation pattern of some canonical radiators.
In Chapter 5, we investigate metamaterial blueprints for the design of reflectionless
absorbers over nonplanar surfaces in connection with optical pseudo black holes. The
presence of fundamental theoretical constraints on the existence of a reflectionless
absorber on nonplanar surfaces are discussed based on analytical properties of the
associated constitutive tensors on the complex ω (angular frequency) plane. In view of such constraints, new metamaterial blueprints of near-reflectionless absorbers are
suggested for integration into optical pseudo black hole devices.
In Chapter 6, we study the radiation properties of electrically-small resonant antennas (ka < 1) composed of electric-field-coupled (ELC) and complementary-
electric-field-coupled (CELC) resonators and a monopole antenna. We use such par-
asitic ELC and CELC ‘metaresonators’ to design various electrically small antennas.
In particular, monopole-excited and bent-monopole-excited CELC resonator anten-
nas are proposed that provide very low profiles on the order of λ0/20. We compare
6 the performance of the proposed ELC and CELC antennas against more conventional designs based upon split-ring resonators (SRR).
In Chapter 7, we investigate waveguide miniaturization by means of ELC and
SRR metamaterial loadings, for TE modes. Towards this purpose, ELCs are placed at the sidewalls of the waveguides, to serve as a cladding. In addition, we studied the case where both ELC and SRR are used together to obtain both backward and forward waves in such waveguides. The presence of backward and forward waves is demonstrated using the dispersion diagram.
In Chapter 8, we summarize our main conclusions.
7 Chapter 2: A Brief on Metamaterials and Transformation Optics
2.1 Overview of Metamaterials
2.1.1 Introduction
As mentioned in the previous chapter, metamaterials are man-made structures with sub-wavelength inclusions that are periodically or non-periodically placed into a host medium to produce effective material properties with unusual characteristics.
Unlike natural bulk materials, which derive their electromagnetic response mostly from their molecular structure, metamaterials derive their electromagnetic response from their geometrical structure, such as the size and shape of the inclusions, in ad- dition the intrinsic properties of host material(s). This gives the designer a large pa- rameter space (degrees of freedom) to engineer the desired electromagnetic response.
Metamaterials can be often be represented in terms of effective medium parameters, whereby their sub-wavelength inclusions are ‘averaged out’ (in some since to be made clear in what follows) at the macro scale. In the following, we discuss very briefly the basic underpinning of effective medium theory, and the relation between metamate- rials and natural materials [5], [66].
8 Maxwell’s equations are written in macroscopic form as
∇ · D~ = ρ (2.1a)
∇ · B~ = 0 (2.1b)
∂B~ ∇ × E~ = − (2.1c) ∂t ∂D~ ∇ × H~ = + J~ (2.1d) ∂t
When electromagnetic waves interact with a material, they produce electric and magnetic (induced) moments and polarization fields that collectively define the macro- scopic effective permittivity and permeability response of the material. For example, the response of a (linear, time-invariant, isotropic) material to electric field can be described by means of the relation
D~ = ǫ0E~a + P~ = ǫ0E~a + ǫ0χeE~a (2.2)
where P~ is the electric polarization vector and χe is the electric susceptibility (that measures the ‘polarizability’ of a material). From the above, one can also define the relative permittivity of such a material as
ǫr =1+ χe (2.3)
which is also called dielectric constant. In a similar way, the magnetic permeability
of the material is affected by induced magnetic dipoles. The analogous relation for
the magnetic case is
B~ = µ0 H~ a + M~ = µ0 H~ a + χmH~ a (2.4)
9 where M is the magnetization vector defined as the averaged response of a material to applied magnetic field and χm is called magnetic susceptibility. From above we define the relative permeability as
µr =1+ χm. (2.5)
The electric and magnetic induced dipoles at the atomic/molecular level collec- tively define the response of a material to electromagnetic waves. Metamaterials mimic the same phenomena using sub-wavelength structures that are much larger than the atomic scale but smaller than the operating frequency so that effective the- ory can be applied at the macro scale. For example, split-ring resonators [20] and electric field coupled resonators [67] are designed to induce electric and magnetic dipoles , respectively, and thus effect the desired permeability and permittivity re- sponses at the macro scale. Fig. 1 shows a summary of material classification for different signs of ǫ and µ in isotropic media. We again stress that not all choices for
ǫ and µ illustrated in this Figure are readily available in natural bulk media (such as left-handed media in the third quadrant, where both ǫ and µ are negative).
2.1.2 Effective Medium Theory
In order for metamaterials to admit an effective medium description, the size of their inclusions should be smaller than the minimum wavelength of interest. Although not sharply defined a criterion, generally it is assumed that if the electric size of the inclusions is λ/4 [3] or less, it can be approximated as an effective medium. Note in particular that this criterion establishes distinction between metamaterials that admit an effective medium description and those that, broadly defined, do not (such
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Figure 2.1: Classification of possible material choices in isotropic media (where both ǫ and µ are scalar). Materials in the first quadrant are denoted as right-handed media. The second and fourth quadrants correspond to electric or magnetic plasmas that are opaque to electromagnetic waves. The third quadrant is of interest due to recent developments in metamaterial. Third-quadrant materials are denoted as left-handed media. Note that this classification refers to the real part of ǫ and µ only.
as dispersion engineered materials in general and photonic crystals in particular, for example [68]–[72]).
2.1.3 Retrieval of Effective Parameters
In this section, we illustrate a standard retrieval procedure for effective param- eters of some simple metamaterial structures. To keep the discussion as simple as possible, we restrict ourselves to isotropic media here, although the approach can be extended to anisotropic media. Though there are different algorithms to obtain effec- tive parameters, a great number of them use the same basic approach, which is based on the scattering parameters [73]–[79]. We present here a technique that is described in [73]. The goal of the retrieval procedure is to characterize a metamaterial ‘sample’
11 by means effective permittivity and permeability using the scattering parameters. In
our case, the method uses the reflection and transmission coefficients for a wave nor-
mally incident on a metamaterial slab. Then, the refractive index n and impedance
z are inverted from the scattering parameters. Next, the effective permittivity and
permeability is calculated using ǫ = n/z and µ = nz. There are certain requirements
on the retrieval procedure based on causality.
The transmission coefficient t for a plane wave incident on a planar slab with a
thickness d is given by
i 1 t = cos (nkd) − z + sin (nkd) eikd (2.6) 2 z where k = ω/c is the wavenumber of the incident wave. The reflection coefficient r is related to transmission coefficient as follows
r 1 1 = i z − sin (nkd) (2.7) t 2 z where a normalized transmission coefficients is defined as t′ = teikd. Basically, the following procedure amounts to inverting the relations in Eq.2.7 to solve for n and z.
As a result, the following relations are obtained
1 2 ′2 1 1 ′ cos (nkd)= 1 − r − t = Re − (A1r + A2t ) (2.8) 2t′ t′ 2 |t′|2 and (1 + r)2 − t′2 z = ± (2.9) s(1 − r)2 − t′2 ′ ′ 2 2 ′ 2 where A1 = rt + r(t ) and A2 = 1 − |r| − |t | are real valued functions equal to zero
when there is no loss.
Although the above equations are simple, there are ambiguities when choosing the
correct roots for n and z. These ambiguities are resolved by enforcing the causality
12 10 25
20
15 5 Re(µ) 10 Im(µ) x y ε µ 5 0 0 ε Re( ) −5 Im(ε)
−5 −10 2 3 4 5 6 7 8 2 3 4 5 6 7 8 Frequency (GHz) Frequency (GHz) (a) (b)
(c)
Figure 2.2: Effective parameters for SRR. (a) Effective permittivity (b) Effective permeability (c)Unit cell configuration
13 constraint so that Re (z) > 0 and Im(n) > 0. These conditions are sufficient to define n and z uniquely for any given frequency ω. Fig. 2.2 and 2.3 show the retrieved effective parameters for infinite array of SSR and ELC resonators respectively. The unit cell configuration for each case is also depicted in Fig. 2(c) and 3(c), respectively.
150 3 Re(µ) 2 Im(µ) 100 Re(ε) Im(ε) 1 x y ε 50 µ 0
0 −1
−50 −2 2 3 4 5 6 7 8 2 3 4 5 6 7 8 Frequency (GHz) Frequency (GHz) (a) (b)
(c)
Figure 2.3: Effective parameters for ELC. (a) Effective permittivity (b) Effective permeability (c)Unit cell configuration
14 2.2 Overview of Transformation Optics
As noted before, transformation optics is a powerful technique to design novel
optical devices that directly exploits the added freedom that metamaterials provide
in material response. Transformation optics is based on the metric invariance of
Maxwell’s equations that is the fact that the basic form of Maxwell’s equations is
invariant under coordinate transformations. The change on the metric can be used
to exactly determine the sought after material properties. Fig. 2.4 describes the
basic idea of the transformation optics [22]–[33]. Such feature of Maxwell’s equations
was recently employed as way of controlling electromagnetic waves for the design
of novel electromagnetic devices [22]–[25], [27], [29], [30], [34]–[55]. A hypothetical
distortion of the underlying space changes the path of electromagnetic waves and this
effect (the distortion of space) can be mimicked by properly chosen metamaterials. In
the following we present the basic mathematical relations underlying transformation
optics.
Maxwell’s equations in a medium with ǫ and µ constitutive parameters are given as ∂ǫ¯E~ ∇ × H~ = + J~ (2.10a) ∂t ∂µ¯H~ ∇ × E~ = − (2.10b) ∂t ∇ · ǫ¯E~ = ρ (2.10c)
∇ · µ¯H~ = 0 (2.10d)
Suppose that a coordinate transformation (x, y, z) → (x′, y′, z′) is effected. In the
transformed coordinates, Maxwell’s equations (2.10) are written in the same basic
form, that is ∂ǫ¯′E~ ′ ∇′ × H~ ′ = + J~′ (2.11a) ∂t 15 Figure 2.4: Overview of transformation optics. The solid blue line represents an op- tical ray path. In empty flat space, the ray travels along a straight path. However, when the space is hypothetically distorted, the light travels along different path ac- cording to the underlying distortion. Transformation optics shows that the same ray path (as well as diffractive phenomena) can be mimicked with equivalent materials de- fined based on the relevant metric distortion (a) Empty flat space, light travels along straight path. (b) Transformed space, light travels non-straight path. (c) Equivalent physical space, light travels non-straight path.
16 ∂µ¯′H~ ′ ∇′ × E~ ′ = − (2.11b) ∂t
∇′ · ǫ¯′E~ ′ = ρ′ (2.11c)
∇′ · µ¯′H~ ′ = 0 (2.11d)
The relation between two systems is defined through the following relations
−1 E~ ′ = ST E~ (2.12a) −1 H~ ′ = ST H~ (2.12b)
SǫS¯ T ǫ¯′ = (2.12c) detS SµS¯ T µ¯′ = (2.12d) detS SJ~ J~′ = (2.12e) detS ρ ρ′ = (2.12f) detS where S is the Jacobian of the transformation, and the overbar is used to represent
second-order-rank constitutive tensors. The proof of the above is given, for example,
in [28] and reproduced next for convenience using covariant notation. Equations
(2.11) can be rewritten in covariant form as
∂E ǫijk∂ H = ǫij j + J i (2.13a) j k ∂t ∂H µijk∂ E = µij j (2.13b) j k ∂t
ij ∂iǫ ∂jEj = ρ (2.13c)
ij ∂iµ ∂jHj = 0 (2.13d)
17 where the indices i,j,k each run from 1 to 3 and repeated indices are summed over.
′ Consider that there is a transformation in the spatial coordinates, i.e., xi → xi, so
that the fields E and H will be transformed according to:
′ i Ei = Si′ Ei (2.14a)
′ i ′ Ei = Si Ei (2.14b)
i′ where the Jacobian Si is defined as
′ i ∂x S ′ = (2.15) i ∂x
′ ′ ′ i i i i i The partial derivatives are rewritten as ∂i = ∂/∂x = ∂x /∂x ∂/∂x = Si′ ∂i. Substituting Eq. 2.14 into Eq.2.13a yields
j′ ′ ′ ∂ Sj Ej ijk k ij i ǫ ∂ S H ′ = ǫ + J (2.16) j k k ∂t ′ j′ ijk k ′ ′ using the fact that the left-hand side of above can be written as ǫ Sk Sj ∂j Hk =
′ ′ ′ ′ ijk k ′ ijk ′ k ijk k ijk k ǫ Sk ∂jHk + ǫ Hk ∂jSk , and using ǫ ∂jSk = ǫ ∂j∂kx , the above then can be rewritten as
′ ′ ′ ijk k′ j ij j ∂ (Ej ) i ǫ S S ∂ ′ H ′ = ǫ S + J (2.17) k j j k j ∂t The determinant is defined as
′ i′j′k′ n′ i′ j k′ ijk ǫ Sn = Si Sj Sk ǫ (2.18)
Inserting this last equation into the previous ones, we obtain
′ i′ j ′ ′ i i′j′k′ Si Sj ij ∂Ej Si i ǫ ∂ ′ H ′ = − ǫ + J (2.19) j k |S| ∂t |S|
The same procedure can be used for the second equation
′ i′ j ′ i′j′k′ Si Sj ij ∂Hj ǫ ∂ ′ E ′ = − µ (2.20) j k |S| ∂t
18 and the remaining equations are likewise transformed as follows
ij ∂iǫ Ej = ρ (2.21) and
′ j′ i ′ ij ′ Si ∂i ǫ Sj Ej = ρ (2.22) so that ′ ′ i ijk j i′ Si ǫ Sj i′ ij Si ρ ∂ ′ E ′ − S ǫ E ′ ∂ ′ = (2.23) i |S| j i j i |S| |S|
We can rewrite the above as
′ i′ j ij Si Sj ǫ ρ ∂ ′ E ′ = (2.24) i |S| j |S|
and similarly for the other divergence equation
′ i′ j ij Si Sj µ ∂ ′ H ′ = 0 (2.25) i |S| j
19 Chapter 3: Anisotropic Metamaterial Blueprints for Cladding Control of Waveguide Modes
3.1 Introduction
In this chapter, we explore the duality between metric and constitutive relations in a different context: to design metamaterial claddings for waveguide mode con- trol and miniaturization. By using the proposed claddings, better mode uniformity within the core region can be obtained, together with a precise control of the corre- sponding frequency cutoffs. At the same time, the modal power distribution can be increasingly confined within the cladding region. In contrast to conventional material loading of waveguides, the proposed (meta)material cladding does not produce hybrid modes. Each resulting mode is homotopic to a hollow waveguide mode, in the sense that the former can be produced from a continuous deformation of the latter within the waveguide core. Physically, this means that the mode distribution within the waveguide core is equivalent to that of a waveguide with different dimensions. We illustrate the analysis for circular and rectangular perfectly electrically conducting
(PEC) waveguides.
20 (a) (b)
Figure 3.1: Cross-section views of (a) circular and (b) rectangular waveguides with a metamaterial cladding shown in yellow.
3.2 Metamaterial Claddings for Waveguide Mode Control
3.2.1 Circular Waveguide
Throughout this analysis, we work in the frequency domain with the e−iωt con-
vention. We first illustrate our approach with a circular waveguide. We assume a
waveguide with a PEC wall of radius r. The metamaterial cladding exists in the region
d ≤ ρ ≤ R for some positive d and where ρ represents the usual radial coordinate, as illustrated in Fig. 3.1, and t = R−d is hence the thickness of the cladding. The objec- tive of the cladding is to mimic a deformation on the metric of space (“stretching” or
“squeezing” of space) for d ≤ ρ ≤ R. The required deformation can be accomplished by the coordinate transformation below
′ ρ for 0 ≤ ρ ≤ d ρ = ρ (3.1) d + s(ρ)dρ for d ≤ ρ ≤ R d R
21 where s(ρ) is a real positive function. The choice s(ρ) ≥ 1 corresponds to a “stretch- ing” and the choice s(ρ) ≤ 1 to a “squeezing” of space in the region d ≤ ρ ≤ R.
For simplicity, we assume s(ρ) to be constant in what follows, although the analysis below can be readily adapted to variable s(ρ).
In terms of the transformed variable ρ′, the fields inside the waveguide are gov- erned by Maxwell’s equations with constitutive parameters ǫ0 and µ0. When the equations are rewritten in terms of the variable ρ, however, they become equivalent to Maxwell’s equations of some transformed fields in an anisotropic medium with con- stitutive parameters [ǫ] =ǫ0 [Λ] and [µ] =µ0 [Λ]. The derivation of the tensor [Λ] can be found (in a different context) in [25], [34] and is summarized below for convenience.
In terms of ρ′, the three components of Faraday’s law are given by
1 ∂Ec ∂Ec iωµ Hc = z − φ , (3.2a) 0 ρ ρ′ ∂φ ∂z
∂Ec ∂Ec iωµ Hc = ρ − z , (3.2b) 0 φ ∂z ∂ρ′ 1 ∂ 1 ∂Ec iωµ Hc = ρ′Ec − ρ . (3.2c) 0 z ρ′ ∂ρ′ φ ρ′ ∂φ Ampere’s equation can be expanded into components similarly. Since ∂/∂ρ′=(1/s) ∂/∂ρ, we can rewrite the above as
ρ 1 ∂Ec ∂Ec iωµ Hc = z − φ , (3.3a) 0 ρ ρ′ ρ ∂φ ∂z ∂Ec 1 ∂Ec iωµ Hc = ρ − z , (3.3b) 0 φ ∂z s ∂ρ ρ 1 ∂ ρ 1 ∂Ec iωµ Hc = ρ′Ec − ρ . (3.3c) 0 z sρ′ ρ ∂ρ φ ρ′ ρ ∂φ It is clear that due to the extra factors 1/s and ρ/ρ′, the system in (3.3) does not retain the form of Maxwell’s equations. This is indicated by using the superscript c for
22 the corresponding field solutions. However, after some simple algebraic manipulations
(3.3) can be rewritten as
′ ρ′ 1 1 ∂ ∂ ρ Ec iωµ sHc = (Ec) − φ , (3.4a) 0 ρ s ρ ρ ∂φ z ∂z ρ ′ ρ ρ Hc ∂ ∂ iωµ s φ = sEc − (Ec) , (3.4b) 0 ρ′ ρ ∂z ρ ∂ρ′ z ′ ρ′ 1 ∂ ρ Ec 1 ∂ iωµ s (Hc)= ρ φ − sEc , (3.4c) 0 ρ z ρ ∂ρ ρ ρ ∂φ ρ where we used the fact that s is a function of ρ, but not φ or z. From the above equa-
tions (and from a similarly transformed Ampere’s law), it is clear that the transformed
fields
a c a ′ c a c Eρ = sEρ, Eφ =(ρ /ρ) Eφ, Ez = Ez (3.5)
a a a (and similarly for Hρ , Hφ, Hz ) obey Maxwell’s equations in an anisotropic medium
with material tensors [ǫ] =ǫ0[Λ] and [µ] =µ0[Λ], where
ρ′ sρ sρ′ [Λ] = diag , , . (3.6) sρ ρ′ ρ The tensor [Λ] can alternatively be written as [25]
[I] for 0 ≤ ρ ≤ d [Λ] = − (3.7) (det[S]) 1[S][S]T for d ≤ ρ ≤ R with 1 ρ [S] = diag , , 1 . (3.8) s ρ′ The matrix [S] represents the Jacobian of the transformation (dρ′,ρ′dφ,dz) → (dρ,ρdφ,dz).
In terms of ρ′, the field distribution corresponds to that of a hollow waveguide (i.e.,
with constitutive parameters ǫ0 and µ0), which has modal solutions of the form:
iωµ m ′ cos (mφ) Ec = 0 J (k ρ ) eikzz, (3.9a) ρ k2 ρ′ m mp −sin (mφ) mp 23 Figure 3.2: Power density E~ × H~∗ distribution of the dominant mode on a circular waveguide with R = 1cm, backed by a cladding with material parameters [ǫ] =ǫ0 [Λ] and [µ] =µ0, where [Λ] is given by equation (3.6). Four different values for the “contrast parameter” s are considered: (a) s = 1 (no cladding), (b) s = 2, (c) s = 3, (d) and s = 5. An increase in s produces a more uniform distribution within the core and overall more power confinement in the cladding. The plots in (f), (g), and (h) screen out the cladding region and have resulting power distribution in the core normalized to the peak value there.
−iωµ ′ ′ sin (mφ) Ec = 0 J (k ρ ) eikzz, (3.9b) φ k m mp cos (mφ) mp c Ez = 0, (3.9c) where kmp = χmp/Re, Re = d + s(R − d) (note again that s is assumed uniform
in the cladding), and χmp is the p-th root of Jm(·), the Bessel function of order
a a a m. Expressions for the magnetic field follow similarly. Eρ , Eφ, Ez in the circular waveguide with cladding are found by straightforward combination of (3.5) and (3.9).
24 3.2.2 Rectangular Waveguide
Now consider a rectangular waveguide with PEC walls as depicted in Fig. 3.1.
The waveguide has cross-section dimensions a × b, and is backed by a metamaterial
cladding with thickness t, as indicated. Assuming the origin of the coordinate system
at the center of the waveguide, the required metric deformation in the first quadrant
is described by the following coordinate transformation (x, y) → (x′, y′),
a ′ x for 0 ≤ x ≤ 2 − t x = a x a a (3.10a) − t + a − sx(x)dx for − t ≤ x ≤ ( 2 2 t 2 2 R b ′ y for 0 ≤ y ≤ 2 − t y = b y b b (3.10b) − t + b − sy(y)dy for − t ≤ y ≤ . ( 2 2 t 2 2 R Similar expressions apply for the other three quadrants. The effect of sx(x) and sy(x)
is analogous to the effect of s(ρ) considered previously. For simplicity, we consider sx(x) and sy(y) uniform and equal to s in the cladding, although the analysis below can be readily adapted to variable sx(x) and sy(x). In the case of uniform sx(x) and sy(y) (i.e., sx = sy = s inside the cladding), the transformation above reduces to
a ′ x for 0 ≤ x ≤ − t x = 2 (3.11a) a − t + s x − a − t for a − t ≤ x ≤ a 2 2 2 2 b ′ y for 0 ≤ y ≤ − t y = 2 (3.11b) b − t + s y − b − t for b − t ≤ y ≤ b . 2 2 2 2 In terms of the coordinates (x′ ,y′,z), Faraday’s law becomes
∂Ec ∂Ec iωµ Hc = z − y , (3.12a) 0 x ∂y′ ∂z ∂Ec ∂Ec iωµ Hc = x − z , (3.12b) 0 y ∂z ∂x′ ∂Ec ∂Ec iωµ Hc = y − x . (3.12c) 0 z ∂x′ ∂y′
25 Ampere’s equation is similar. The fields above satisfy Maxwell’s equations in terms
of (x′,y′,z) but not in terms of (x,y,z). As in the circular waveguide case, this is indicated by the use of the superscript c.
′ Using the fact that sς depends on ς only, and that ∂/∂ς =(1/sς ) ∂/∂ς, where
ς = x, y, the above can be rewritten as
c c c 1 ∂Ez ∂Ey iωµHx = − , (3.13a) sy ∂y ∂z c c c ∂Ex 1 ∂Ez iωµHy = − , (3.13b) ∂z sx ∂x c c c 1 ∂Ey 1 ∂Ex iωµHz = − , (3.13c) sx ∂x sy ∂y and, after some simple algebraic manipulations, as
s ∂ ∂ iωµ y (s Hc)= Ec − s Ec , (3.14a) s x x ∂y z ∂z y y x s ∂ ∂ iωµ x s Hc = (s Ec) − Ec, (3.14b) s y y ∂z x x ∂x z y ∂ ∂ iωµ (s s )(Hc)= s Ec − (s Ec) , (3.14c) x y z ∂x y y ∂y x x and similarly for Ampere’s law. Note that the last set of equations retains the form
of Maxwell’s equations, with field solutions
a c a c a c Ex = sxEx, Ey = syEy, Ez = Ez (3.15)
a a a (and similarly for Hx , Hy , Hz ) in a medium with material tensors given by [ǫ] =ǫ0
[Λ] and [µ] =µ0 [Λ], where [Λ] is obtained again using (3.7) with [S] now given by
−1 −1 [S] = diag sx ,sy , 1 . (3.16) which evaluates to diag {s−1, 1, 1} at the lateral side walls, [S]= diag {1,s−1, 1} at the top and bottom walls, (3.17) −1 −1 diag {s ,s , 1} at the four corners. 26 This matrix correspond to the Jacobian of the transformation (dx′, dy′, dz) → (dx,dy,dz).
From the above, it is seen that three different set of constitutive parameters are
to be used in the rectangular waveguide cladding:
diag {s−1,s,s} at the lateral side walls, [Λ] = diag {s,s−1,s} at the top and bottom walls, (3.18) 2 diag {1, 1,s } at the four corners. In terms x′ and y′, the field distribution corresponds to that of a hollow waveguide.
Therefore, the TM mode solutions can be written as:
−iωµ k ′ ′ Ec = 0 y cos (k x ) sin (k y ) eikzz, (3.19a) x k2 − k2 x y z
iωµ k ′ ′ Ec = 0 x sin (k x ) cos (k y ) eikzz, (3.19b) y k2 − k2 x y z c Ez = 0, (3.19c)
with kx = mπ/ae , ky = nπ/be, ae = 2t (s − 1)+a and be = 2t (s − 1)+b, and where m
and n are the mode indices. Similar expressions can be written for TE modes and for
a a a the associated magnetic fields. Finally, the expressions for the fields Ex,Ey ,Ez in the
rectangular waveguide with cladding can be found by straightforward combination
of (3.15) and (3.19).
3.3 Examples
In this section, we show results for circular and rectangular PEC waveguides
backed by the metamaterial claddings discussed above. For the cylindrical waveguide,
we assume cladding thickness t = 0.2R. For the rectangular waveguide, we assume
aspect ratio a = 1.4b and cladding thickness t = 0.1a. All plots below show field
distributions normalized to their respective peaks.
27 Fig. 3.2 depicts the power density distribution E~ × H~ ∗ of the dominant mode in the circular waveguide with s=1 (no cladding), s = 2, s = 3, and s = 5. From
Fig. 3.2(f)−(h), it is seen that inside the core, the power density becomes progressively more uniform for higher s. Moreover, Figs. 3.2(a)−(d) show that the power density becomes gradually more confined inside the cladding.
An example of a rectangular waveguide with cladding inserted on the two lateral walls is shown in Fig. 3.3, where a plot of the TE10 (dominant) mode for various values of s is presented. The TE10 power distribution is uniform along y and only the distribution along x is affected in this case. An example of a rectangular waveguide with cladding inserted on all four side walls is shown next in Fig. 3.4, with a plot of the TE11 mode whose distribution changes in both x and y directions as s increases.
The general observations made before regarding the mode uniformity in the core and mode confinement in the cladding also apply to the rectangular case.
As noted, the decrease on the cutoff frequencies for higher s is related to the increase on the dual-hollow waveguide. If the cladding has uniform s, a “dual-hollow” circular waveguide with radius Re = d + s(R − d) = d + st and a “dual-hollow” rectangular waveguide with width ae = 2t (s − 1) + a and height be = 2t (s − 1) + b result. In particular, assuming a circular waveguide with R = 1cm, and a rectangular waveguide with b = 1cm, the cutoff frequencies for the modes shown in Fig. 3.2 and
Fig. 3.3 are given in Table 3.1, for different s.
We next present three-dimensional (3-D) numerical results based on the finite- difference time-domain (FDTD) method. A time-harmonic electric dipole source is placed inside the rectangular waveguide, and excited by a frequency such that only the dominant mode propagates and all higher other modes are evanescent. Since one of
28 the effects of a higher s is to decrease the cutoff frequency, this implies that for higher
s, progressively lower frequencies are required to produce the propagating mode. The
FDTD simulations employ a 3-D grid with Nx×Ny×Nz = 56×40×360 nodes, unit cell size ∆s = 0.025 cm, and time step ∆t = 0.385 ps. Assuming that the origin is at one corner of the waveguide, the dipole source is positioned at (x, y, z) = (28, 10, 180)∆s.
A perfectly matched layer (PML) is used to suppress reflections from the waveguide ends along the z-direction.
Fig. 3.5 shows snapshots of the steady-state electric and magnetic field distribu- tions along the waveguide, for the different values of the cladding contrast parameter s. The cladding is inserted on the two lateral walls only. The horizontal-cut views shown are located at the mid-point of the waveguide along the y-direction. As noted, the frequencies shown for each s are chosen above the cutoff frequencies indicated in
Table 3.1 and below the cutoff frequency of the next higher order mode. Again, the gradual “spreading” of the electric field distribution along the transversal direction for higher s, which makes the field distribution more uniform in the core, is clearly visible in this figure. The magnetic field of the dominant mode, on the other hand, becomes increasingly confined inside the lateral cladding for higher s.
29 Figure 3.3: Power density E~ ×H~∗ distribution of the dominant mode on a rectangular waveguide with transversal dimensions a = 1.4cm, b = 1cm. The cladding is placed on the two lateral walls with [ǫ] =ǫ0 [Λ] and [µ] =µ0 [Λ], where [Λ] is given by equation (3.18). Four different values for s are again considered: (a) s = 1, (b) s = 2, (c) s = 3, (d) and s = 5. Similarly to the circular waveguide case, an increase in s produces a more uniform distribution within the core and overall more power confinement in the cladding. The plots in (f), (g), and (h) screen out the cladding region and have resulting power distribution in the core normalized to the peak value there.
3.4 Conclusions and Further Remarks
We have explored the known duality between metric and constitutive parameters in Maxwell’s equations to derive metamaterial blueprints for waveguide claddings that provide homotopic control of waveguide modes. The resulting metamaterial- loaded waveguides exhibit no hybrid modes. The obtained metamaterial blueprints can be viewed as isoimpedance materials with intrinsic impedance matched to free- space, regardless of frequency or propagation angle. The constitutive parameters are specified by a contrast parameter “s” that provides control over the mode uniformity
30 ∗ Figure 3.4: Power density E~ × H~ distribution for the TE11 mode on a rectangular waveguide with cladding on all four walls. Due to the bivariate nature of the TE11 mode and the presence of the cladding on all four walls, a change in s affects the mode distribution in both x and y. Similarly as before, an increase in s produces a more uniform distribution within the core and overall more power confinement in the cladding (and particularly at the four corners). The plots in (f), (g), and (h) screen out the cladding region and have resulting power distribution in the core normalized to its peak value there.
31 Figure 3.5: FDTD results showing cut-views of the steady-state electric and magnetic field distributions along a rectangular waveguide with transversal dimensions a = 1.4cm, b = 1cm, cladding thickness t = 0.14cm, and length l = 9cm. The cladding is inserted along the two lateral walls, having [ǫ]=ǫ0 [Λ] and [µ]=µ0 [Λ] (see main text) with four values of s considered: (a) s = 1, (b) s = 2, (c) s = 3, (d) and s = 5. The source excitation frequencies are chosen between the two lowest cutoff frequencies; so that only the dominant mode propagates (the corresponding dominant mode cutoff frequencies are listed in Table I).
32 cTE11 rTE10
s = 1 (no cladding) 8.790 GHz 10.714 GHz
s = 2 7.325 GHz 8.928 GHz
s = 3 6.279 GHz 7.653 GHz
s = 5 4.883 GHz 5.952 GHz
Table 3.1: Cutoff frequencies of the dominant circular (cTE11) and rectangular (rTE10) modes for different s and following waveguide parameters: R = 1cm, a = 1.4cm, b = 1cm, d = 0.8R, and t = 0.1a
(in the core region), the degree of field confinement in the cladding, and the resulting
waveguide cutoff frequencies. The isoimpedance property is closely related to the
perfectly matched layer concept [62], [80] but devoid of the absorptive properties of
the latter.
It should be emphasized that the type of waveguide geometries considered here are
inherently narrowband. This has nothing to do with the proposed cladding per se, but
follows more fundamentally from the strong chromatic dispersion at frequencies near
the cut-off (note that if one tries instead to operate the waveguide far from cut-off, an
even more detrimental type of dispersion ensues: modal dispersion caused by higher-
order modes). Narrowband operation facilitates the design and eventual fabrication of
cladding structures that approximate the constitutive parameters derived here (as the
latter need to be realized only over a narrow range of frequencies). It is also important
to stress that even though the metamaterial blueprints derived imply real-valued
constitutive tensors, the practical realization of the proposed claddings inevitably
33 entail losses arising from non-zero imaginary parts, as predicted by Kramers-Kronig
relations (and assuming a passive media realization). Narrowband operation also
facilitates the minimization of such losses (in the frequency range of operation).
It is beyond the objectives of this paper to dwell further into fabrication issues or potential applications. Suffice it to say here that the actual fabrication of the proposed cladding metamaterials is in principle no more difficult than the fabrica- tion of electromagnetic “cloaks” or “masks” [12], [13], [64], as they exhibit a similar anisotropic structure.
34 Chapter 4: Analysis of Canonical Low-profile Radiators on Isoimpedance Metamaterial Substrates
4.1 Introduction
It is well known that ‘low-profile’ antennas do not radiate well when placed in close proximity to a perfect electrical conductor (PEC) or conducting ground plane.
Given a desired radiation performance, it becomes necessary to place the antenna at a minimum distance away from the ground plane or to employ a substrate of suffi- cient thickness. This makes the system bulky especially at lower frequencies. Due to increasing demand for antenna miniaturization, strategies for profile reduction are of great interest [81]–[91]. A number of approaches have been utilized for substrate height reduction in the past, including: high-impedance structures such as electro- magnetic band-gaps (EBGs) [81]–[85], artificial magnetic conductors (AMC) [86]–[89], reactive impedance substrates [90], and impedance-matched ferrite layers (IML) [91].
For patch antennas, a traditional approach for miniaturization and substrate height reduction is to employ high-dielectric substrates. However, this negatively impacts radiation efficiency and bandwidth, and may cause unacceptable pattern degrada- tion due to the presence of stronger surface waves along the dielectric surface. Ap- proaches aimed at reducing surface waves include EBG structures in general (and
35 substrate perforation in particular) [81], [82], [84], [92] and ‘magnetodielectric’ sub- strates where magnetic properties are exploited to decrease the impedance mismatch at the air/substrate interface and compensate for capacitive effects [90], [93]–[95].
Recently, there has been an increasing interest in using metamaterials or sin- gle metamaterial components (i.e., split ring resonators (SRRs)) for antenna perfor- mance enhancement. Alici et al. has shown that electrically small monopole antennas
(≈ λ/10) can efficiently radiate when combined with a SRR [96], [97]. Similarly Jin et al. designed electrically small antennas using electrically or magnetically coupled capacitively-loaded loop (CLL) elements [98]. Similar studies have also been done for microstrip patch antennas employing metamaterial substrates composed of SRRs [99],
[100]. Metamaterials with negative index have also been investigated for directivity and performance enhancement [101]–[103]. In addition to above approaches, trans- mission line type electromagnetic metamaterials are also heavily employed for antenna applications [3], [104]. The above studies suggest that, metamaterials will play an important role for antenna performance enhancement.
More recently, there has been great interest in designing metamaterial-inspired devices based on ‘transformation optics (TO)’ concepts [22], [23], [25]. These have included electromagnetic cloaks [22], [23], [36], [38], [39], ground plane cloaks [13],
[37], wave splitters [40], polarization rotators [41], field concentrators [42], electromag- netic masks [43], [44], perfectly matched layers [25], [34], [35], reflectionless waveguide bends [45], [46], waveguide loadings [47], [48], high-performance omnidirectional ab- sorbers (so-called optical ”black holes”) [27], [29], [30], [49]–[54]. Transformation op- tics provides a design route for obtaining new electromagnetic functionalities. Besides above applications, TO have also been used in antenna applications [102], [105]–[108].
36 In [108], a metamaterial substrate was designed to reduce the profile of a horizontal
dipole above a PEC. However, such study was restricted to the far field behavior
only. Other metamaterial blueprints have recently been proposed as substrates for
radiation control and optimization [109], [110]. These proposals rely on negative con-
stitutive parameters, including anisotropic responses with negative tensor elements.
In contrast, the isoimpedance blueprints considered here correspond to anisotropic
tensors with strictly positive elements. In this paper, we examine in more detail
the TO approach considered in [108] to design anisotropic metamaterial blueprints
of ‘isoimpedance’ substrates. These substrates are aimed at providing small physical
height but increased electrical height and an intrinsic impedance that is matched to
free space for all incidence angles. As a result, no surface waves are supported at
the substrate/air interface. We extend the study in [108] to include patch antennas
and to examine in detail not only far-field behavior but also input impedance, re-
turn loss, and bandwidth characteristics. Note that previous proposals for surface
waves mitigation consisted in seeking µr ≈ ǫr by means of impedance-matched ferrite layers [91]. Although somewhat effective, this approach yields impedance-matched properties only at (near) normal incidence.
The metamaterial blueprints considered here consist of anisotropic permittivity
[ǫ] and permeability [µ] tensors that effectively mimic the electromagnetic properties
of a thicker (isotropic) substrate. This is illustrated in Fig. 4.1. From TO principles,
these two configurations produce identical fields above the respective substrates in an
infinite geometry (no lateral truncation). In practice, there are two basic limitations
for achieving this equivalence. First, the effect of substrate (lateral) truncation is
differently in the two cases. Second, fabrication constraints pose limitations on the
37 desired anisotropic response of the tensors [ǫ] and [µ] in the ‘actual’ configuration, especially if wideband operation is sought.
In the following, we revisit the design of the desired tensor blueprints [ǫ] and [µ] for isoimpedance metamaterial substrates and examine the effects of the above-mentioned
first limitation on the performance of two widely used types of low-profile radiators atop such substrates: horizontal dipoles and rectangular patches. The results are obtained numerically via a custom 3D finite difference time domain (FDTD) code optimized for such problem. Throughout our discussion, we assume an ideal material response (tensors) for [ǫ] and [µ]. Effects from non-ideal material response will be considered elsewhere. Although a discussion on the fabrication of such isoimpedance tensors is beyond the objectives of this paper, we note that their basic structure is very similar to metamaterials for cloaking applications [12], [13]. However, the proposed metamaterial substrates here poses challenges on traditional metamaterial design based on periodic unit cells because of the extremely low profiles considered
(h ≈ λ/20 for wire antenna and h ≈ λ/50 for patch antenna). Our primary goal is to delineate the ultimate performance characteristics such as input impedance, return loss, and far-field pattern of horizontal dipole and patch antennas placed on top of such substrates.
4.2 Isoimpedance Substrates
The derivation of the desired isoimpedance metamaterial substrate can be asso- ciated to a particular mapping of the metric of space in the region between ground plane and low-profile radiator. In general, a mapping of the metric can be mimicked
38 Figure 4.1: (a) Equivalent and (b) actual geometries. The isoimpedance metamaterial substrate with [ǫ] =ǫ [Λ] and [µ] =µ [Λ] (see the expression for [Λ] in the main text) is designed via TO principles so that the electromagnetic fields above the planar antennas are identical for these two geometries.
by properly chosen constitutive parameters [ǫ] and [µ] [22], [23], [25]. Fig. 4.1 illus- trates this duality. Though metamaterial blueprints can be obtained for any type of surfaces (i.e, nonplanar) [111], we illustrate the planar case here. Using the coordi- nate system shown, the desired metric transformation is performed for 0 ≤ z ≤ t and is given by: x′ = x, y′ = y, and
z 0 s(z) dz for 0 ≤ z ≤ t z′ = (4.1) R′ t + z for t ≤ z where s(z) is a real positive parameter and
t t′ = s(z) dz (4.2) Z0 The choice s(z) ≥ 1 corresponds to a ‘stretching’ of space along the vertical direction z for 0 ≤ z ≤ t (leading to t′ >t). For simplicity, we shall consider s(z) as a constant in what follows, even though this is not strictly necessary. In terms of the
39 1500 air, h=λ/4 meta (s=5), h=λ/16 1000 HIGP, h=λ/16
500
0 Input Impedance (Z) −500
−1000 0.5 1 1.5 2 2.5 3 3.5 Frequency (GHz)
Figure 4.2: Input impedance of a horizontal dipole for three configurations: (i) air, h = λ/4, (ii) isoimpedance metamaterial with s = 5, h = λ/16 (d = λ/64, t = 3λ/64), and (iii) HIGP with ǫr = 0.2 and µr = 5, h = λ/16 (d = λ/64, t = 3λ/64), where the ground plane dimensions are 1.5λ0×1.5λ0. Note how the performance of configuration (ii) is very close to (i).
40 0
−5
−10
−15 S11 (dB) −20
air, h=λ/4 −25 meta (s=5), h=λ/16 HIGP, h=λ/16 −30 0.6 0.7 0.8 0.9 1 1.1 1.2 Frequency (GHz)
Figure 4.3: S11 (negative return loss) of a horizontal dipole for the configurations considered in Fig. 4.2. Original and isoimpedance metamaterial configurations (i) and (ii) yield almost identical S11. HIGP configuration (iii) radiates with reduced bandwidth.
41 ‘equivalent’ coordinates (x′, y′, z′), the electromagnetic field in the region 0 ≤ z′ ≤ t′ is
governed by Maxwell’s equations with (isotropic) constitutive parameters ǫ = ǫ0 and
µ = µ0. In terms of the ‘actual’ coordinates (x, y, z), the field in the region 0 ≤ z ≤ t is
governed by anisotropic constitutive parameters [ǫ]=ǫ0[Λ] and [µ]=µ0[Λ], where [25]
diag{s,s,s−1} for 0 ≤ z ≤ t [Λ] = (4.3) diag{1, 1, 1} for t ≤ z The fields everywhere in the actual geometry can be obtained from those of the equivalent geometry by a simple scaling [22], [25]. In particular, the fields for z>t in
the actual geometry are identical to the fields for z′ >t′ in the equivalent geometry.
This implies, in particular, that the far-field radiation patterns are identical in the
two geometries. The substrate height in the actual geometry can be progressively
reduced by increasing the ‘contrast factor’ s.
We stress that the background medium in the equivalent geometry has ǫ = ǫ0 and
µ = µ0, implying that there is no impedance mismatch in the surrounding environ-
ment and, as a result, no surface waves are supported. Accordingly, the metamaterial
substrate is also impedance matched to free space and does not support surface waves.
It is shown analytically in the Appendix that the top surface of the above metamate-
rial substrate is reflectionless for all incidence angles, hence ‘isoimpedance’. However,
the lateral walls do not share this property, as also explained in the Appendix. In
other words, the above equivalence is exact only if the lateral dimensions in the xy
plane are infinite.
42 (a) air-backed, h = λ/4 (b) metamaterial with s=5, h = λ/16
(c) air-backed, h = λ/16 (d) HIGP with ǫr = 0.2 and µr = 5 , h = λ/16
Figure 4.4: Electric field magnitude distribution along the xz-plane (side-view) for a horizontal dipole on top of air-backed, isoimpedance metamaterial-backed, and dielectric-backed ground planes: (i) air, h = λ/4; (ii) metamaterial with s=5, h = λ/16; (iii) air, h = λ/16; and (iv) HIGP with ǫr = 0.2 and µr = 5, h = λ/16. The lateral dimensions of the substrate are assumed infinitely large. The horizontal solid line indicates the ground plane level and the dashed line indicates the top surface of the substrate.
43 10
0
−10
−20 Directivity (dB)
−30 air, h=λ/4 meta (s=5), h=λ/16 HIGP, h=λ/16 −40 −150 −100 −50 0 50 100 150 θ (Degrees) (a) E-plane
10
0
−10
−20 Directivity (dB)
−30 air, h=λ/4 meta (s=5), h=λ/16 HIGP, h=λ/16 −40 −150 −100 −50 0 50 100 150 θ (Degrees) (b) H-plane
Figure 4.5: Radiation pattern of horizontal dipole for the configurations considered in Fig. 4.2. (a) E-plane, (b) H-plane.
44 4.3 Numerical Results
4.3.1 Horizontal dipole
Ideally, a horizontal electric dipole should be at a height λ/4 above a PEC ground plane in order to radiate well [83], [89]. Even though a (synthesized) PMC ground plane would allow for a lower height, it is not the best choice because of the strong mutual coupling to the image dipole [81], [83]. Detailed analyses of dipole antennas near PEC and PMC ground planes can be found in [81], [83].
Here, we investigate isoimpedance substrates for maintaining the horizontal dipole performance while reducing the antenna height. Our specific purpose is to mimic the antenna configuration where dipole is λ/4 away from a PEC ground plane by using an equivalent anisotropic metamaterial substrate of lower height. For comparison, a high impedance ground plane (HIGP) is also considered. For this purpose, ǫr = 0.2 and µr = 5 are used to emulate an HIGP surface [108].
In the following simulations, a 3D-FDTD code is used for analysis. We chose the operating wavelength as λ0 = 300mm. The length of the dipole is .5λ0 with a
radius-to-length ratio of 0.004. An infinitesimal gap model [112] is used to accurately
model the thin wire structure. The center-fed dipole resonates at f = 935 MHz in
free space.
Next, we show that a horizontal dipole on an isoimpedance anisotropic metama-
terial substrate can exhibit similar performance to that of a horizontal dipole placed
at λ/4 distance above a PEC, while providing a much reduced substrate height. The
horizontal dipole antenna configuration is shown in Fig. 4.1. The ground plane size is
1.5λ0 × 1.5λ0. The dipole is placed at a distance of h = t + d above the PEC, where
the region corresponding to a thickness of t immediately above the ground plane is
45 filled with a metamaterial substrate. Figure 4.2 shows the input impedance for three cases: (i) dipole above PEC with h = λ/4, (ii) dipole on isoimpedance metamaterial substrate with [ǫ] =ǫ0[Λ] and [µ] =µ0[Λ] where Λ is given in ( 4.3) with s = 5 and h = λ/16 (d = λ0/64,t = 3λ0/64), and (iii) dipole on HIGP substrate with ǫr = 0.2 and µr = 5 and h = λ/16 (d = λ0/64,t = 3λ0/64). Cases (ii) and (iii) are backed by a PEC ground plane. As seen from Fig. 4.2, the impedance values are almost identical for configurations (i) and (ii). As mentioned above, the residual discrepancy is due to the finite size of the ground plane. Fig. 4.3 shows the corresponding return losses. Configurations (i) and (ii) have nearly identical return losses. Configuration
(iii) shows a reduced bandwidth. Fig. 4.2 and 4.3 indicate that the ‘original’ antenna performance is maintained by using the isoimpedance substrate.
Fig. 4.4 shows the snapshot of electric field magnitude |E| on the xz-plane for a dipole excited at f0 = 890 MHz. The ground plane is assumed infinite in this case.
It is seen that the field behavior above the transformed region is almost identical for configurations (i) and (ii); that is, the metamaterial configuration (ii) does mimic well the electromagnetic properties of the ‘original’ configuration (i). For comparison, we also plot the fields of a dipole over PEC with h = λ/16 (iii) and HIGP configuration
(iv). All snapshots are shown at the same time instant.
Fig. 4.5 shows the E-plane and H-plane far-field patterns, respectively, for the configurations above. Configurations (ii) and (iii) are backed by a PEC ground plane.
Fig. 4.5 shows that the pattern is only weakly perturbed. There is an increase of about
0.9 dB in the directivity for metamaterial substrate (ii) configuration relative to the original configuration and an increase of about 0.6 dB in the HIGP (iii) configuration.
46 Figure 4.6: Patch antenna geometry with isoimpedance metamaterial substrate.
In addition, the radiation efficiency of the metamaterial substrate case (ii) is better
than the HIGP case (iii), consistent with the findings in [108].
4.3.2 Rectangular patch
The patch antenna geometry considered in our study is rectangular with lateral dimensions w and l, as illustrated in Fig. 4.6. At first, we set the thickness at 1.2 mm
and the length of substrate L as λ, based on the operating frequency of f = 5.375
GHz. Notice that this thickness is much smaller than the wavelength, h ∼= λ/46. We analyze and compare four different combinations of substrates and patch sizes: (i) isoimpedance metamaterial substrate with s = 3, w=24.78 mm, and l=26.88 mm;
(ii) dielectric substrate with ǫr = 2.2, w=24.78 mm, and l=26.88 mm; (iii) dielectric
substrate with ǫr = 2.2, w=18.06 mm, and l=18.9 mm; and (iv) dielectric substrate
with ǫr = 10.2, w=8.232 mm, and l=9.282 mm. Note that in cases (iii) and (iv) the
47 0
−5
−10
S11 (dB) −15 meta (s=3) − original diel (ε =2.2) − original r −20 diel (ε =2.2) − minuat. r diel (ε =10.2) − minuat. r −25 3 4 5 6 7 Frequency (GHz)
Figure 4.7: S11 (negative return loss) of a rectangular patch in four substrate config- urations: (i) isoimpedance metamaterial with s = 3, (ii) dielectric (miniaturized size) with ǫr = 2.2, (iii) dielectric (miniaturized size) with ǫr = 10.2, and (iv) dielectric (original size) with ǫr = 2.2. In all cases, h = 1.2 mm.
48 10
0
−10
−20 Directivity (dB) meta (s=3) − L=λ −30 diel (ε =2.2) − L=λ r diel (ε =10.2) − L=λ r −40 −150 −100 −50 0 50 100 150 θ (Degrees) (a) E-plane
10
0
−10
−20 Directivity (dB) meta (s=3) − L=λ −30 diel (ε =2.2) − L=λ r diel (ε =10.2) − L=λ r −40 −150 −100 −50 0 50 100 150 θ (Degrees) (b) H-plane
Figure 4.8: Radiation pattern of a rectangular patch for three substrate configura- tions: (i) isoimpedance metamaterial with s = 3, (ii) dielectric with ǫr = 2.2, and (iii) dielectric with ǫr = 10.2. In all cases, h = 1.2mm and L = λ.
49 Patch Size % BW %
metamaterial (i) 24.78 × 26.88 (100%) 6.4 (100%)
dielectric (iii) 18.06 × 18.9 (51.24%) 2.3 (35.94%)
dielectric (iv) 8.23 × 9.28 (12.15%) 1.1 (17.18%)
Table 4.1: Trade-off between patch dimensions and frequency bandwidth.
lateral dimensions of the patch are reduced to maintain the same operating frequency
as the metamaterial substrate case. We denote configurations (i) and (ii) as ‘original’
size and (iii) and (iv) as ‘miniaturized’ size.
Fig. 4.7 shows the S11 factor for each of the above configurations. Clearly, con-
figuration (i) exhibits the largest bandwidth (S11≤-10dB), of about 6.4%. When a dielectric substrate (ǫr = 2.2) is used with the original patch size, the bandwidth de- creases to about 1.4%. Since the patch size is kept the same, the operating frequency also decreases down to f = 3.95GHz. As mentioned previously, patch size miniatur- ization is the main advantage of using high dielectric substrates. However, it degrades both the bandwidth and the radiation efficiency. This is illustrated by noticing that configurations (iii) and (iv) operate at the same frequency of configuration (i), but the bandwidths for (iii) and (iv) are reduced to about 2.3% and 1.1%, respectively.
This trade-off between patch size and bandwidth is summarized in Table 4.1. Fig. 4.8 shows the E-plane and H-plane radiation patterns for configurations (i), (iii) and (iv).
Notice that metamaterial configuration (i) provides stronger radiation at the zenith
(boresight gain) than either configurations (iii) or (iv).
50 (a) E-plane coupling
(b) H-plane coupling
Figure 4.9: E and H-plane mutual coupling configurations.
Next, we briefly illustrate how the isoimpedance metamaterial substrate affects the mutual coupling between adjacent patches in an array configuration. It is well known that surface waves are a major factor producing mutual coupling between patch antenna elements [84]. Accordingly, mutual coupling is expected to decrease if surface waves are mitigated. Fig. 4.9 depicts two basic coupling configurations, denoted here as E- and H-plane (where the arrow indicates the dominant current direction [84]).
Since the isoimpedance metamaterial substrates do not support surface waves, we expect mutual coupling to be lower than in conventional dielectric substrates. This is clearly corroborated in Fig. 4.10 and Fig. 4.11.
51 0
−5
−10 S11 meta (s=3) −15 diel (ε =2.2) r −20 diel (ε =10.2) r S (dB) S21 −25
−30
−35
−40 4 4.5 5 5.5 6 6.5 7 Frequency (GHz)
Figure 4.10: E-plane mutual coupling between two co-planar rectangular patches for three configurations: (i) isoimpedance metamaterial (original size) with s = 3, (ii) dielectric (miniaturized size) with ǫr = 2.2, and (iii) dielectric (miniaturized size) with ǫr = 10.2. In all cases, h = 1.2mm.
52 0
−5
−10 S11 −15 meta (s=3) diel (ε =2.2) −20 r
S (dB) diel (ε =10.2) −25 r S21 −30
−35
−40 4 4.5 5 5.5 6 6.5 7 Frequency (GHz)
Figure 4.11: H-plane mutual coupling between two co-planar rectangular patches for the same three configurations as considered the E-plane case.
53 4.4 Conclusion
We have analyzed the performance of two canonical low-profile radiators, viz. horizontal dipole and rectangular patch, placed above isoimpedance metamaterial substrates. These substrates are doubly anisotropic media designed via transforma- tion optics techniques in such a fashion that they are impedance-matched to free space. It was verified that the isoimpedance feature of the substrate can enhance the antenna characteristics and radiation because no surface waves are supported. In par- ticular, significant increase in the usable bandwidth is observed versus conventional
(isotropic) dielectric substrates. In addition, mutual coupling in array geometries can be mitigated for similar reasons. Note that a general tradeoff between efficiency and wide-band performance is present in other substrate miniaturization alternatives as well.
Throughout this study, we have assumed an ideal metamaterial response that is no losses or frequency dispersion present in the bandwidth of operation. Even though these assumptions provide good insight into the ultimate capabilities of such sub- strates, it ignores performance degradation due to fabrication limitations (especially degradation on the radiation efficiency due to material losses). These effects will be considered elsewhere. The derived permittivity and permeability constitutive ten- sors are diagonally anisotropic and proportional to each other; as such, they present similar challenges for realization as general electromagnetic cloaks.
54 Chapter 5: Impedance Matched Absorbers and Pseudo Black Holes
5.1 Introduction
Optical (pseudo) ‘black hole’ devices have been recently proposed [27], [49]–[53]
and built [50] based on transformation optics concepts. Most of the proposed optical
‘black holes’ are devices (metamaterial shells) intended to slow-down and guide in-
coming waves towards an inner core (while producing minimal reflections) where, de-
pending on the intended application, electromagnetic energy can be harvested and/or
efficiently absorbed (ideally, under minimal reflections as well). More generally, the
term ‘optical black hole’ can be used to denote a metamaterial device from which
no impingent electromagnetic wave ‘returns’ (or reflects from, at least in an approx-
imate sense and for a given range of frequencies). Because no gravitational effects
are implied, the term ‘black hole’ is used here in a very loose sense, hence the quotes
(perhaps a more precise term is ‘omnidirectional electromagnetic absorber’ [50]). This
terminology is adopted to conform with prior references on this topic [27], [49]–[53].
Media derived from transformation optics typically have inhomogeneous and anisotropic permittivity [ǫ] and permeability [µ] tensors [22], [25], [44], [45], [47]. In contrast, most previously proposed media for optical ‘black holes’ have consisted of nonmagnetic
55 isotropic media [49]–[53]. A nonmagnetic isotropic response is of interest because of
ease of fabrication, especially at optical frequencies; however, this implies that such
materials are not truly impedance-matched (reflectionless) and hence a low permit-
tivity gradient (or, equivalently, an electrically-large shell thickness spanning many
vacuum wavelengths) is necessary to minimize reflections.
For optical ‘black hole’ applications where the main objective is not to guide inci-
dent waves to a inner core but rather to simply produce very low levels of backscat-
tering for any incident wave direction, it is obvious that the performance crucially
depends on the amount of reflection produced. With this later objective in mind, we
investigate here metamaterial blueprints of (near-) reflectionless absorbers for inte-
gration into ‘black hole’ devices where the electrical thickness of the shell can be dras-
tically reduced. We discuss how some of these blueprints are related to those of ‘per-
fectly matched layers’, previously developed for numerical simulation purposes [34],
[80], [113], [114]. We point out some fundamental limitations on the existence of truly
reflectionless absorbers on curved surfaces and suggest some alternatives for integrat-
ing near-reflectionless absorptive effects into metamaterial blueprints of optical ‘black
holes’ devices.
5.2 Metamaterial blueprints for reflectionless absorbers
5.2.1 Planar geometries
We derive metamaterial blueprints of reflectionless, or perfectly impedance-matched, absorbers by extending the conventional (real-valued) coordinate-transformation ap- proach of transformation optics to complex-valued coordinate transformations. Con- sider, for example, the Cartesian coordinate x: by transforming it via x → x˜ =
56 x + i∆x/ω, where ω is the angular frequency and ∆x > 0 is a continuous function of x, it follows that any propagating eigenfunction of the form exp(ikxx) is (con- tinuously) mapped to an exponentially decaying function of the form exp(ikxx˜) = exp(ikxx − kx∆x/ω). This is effected without any perturbation of the boundary con- ditions (and in particular, phase matching) on the normal and tangential components of the electromagnetic fields along the transverse y-z plane, thus without producing reflections.
It turns out that an instance of such reflectionless absorber already exists and is provided by the so-called ‘anisotropic-medium’ (or ‘uniaxial’) formulation [25], [113],
[114] of the perfectly matched layer (PML) [80], originally developed as an absorb- ing boundary condition for numerical simulations. In Cartesian coordinates, the
PML material tensors that produce reflectionless absorption along x are written as
[ǫ] =ǫ0 [Λ]x and [µ] =µ0 [Λ]x, where [25]
[Λ]x = diag {1/sx, sx, sx} (5.1)
is a diagonal tensor in Cartesian coordinates (x, y, z) and sx is the complex-valued
‘stretching parameter’ along the (normal) x-direction. Typically, sx(x) = ax(x)+ iσx(x)/ω, where the real part ax(x) ≥ 1 controls the phase velocity (or equivalently, the ‘slowness’) of the wave inside the material and the imaginary part σx(x) ≥ 0 controls the amount of absorption, with the choice σx(x) = 0 producing no absorption.
Other, more involved functional dependencies of sx in terms of ω are also possible [115] but we are not going to consider those here. Assuming the absorber is located in the half-space x > 0, this is equivalent—from a transformation optics viewpoint—to a
57 complexification of the spatial metric via [25]
2 2 2 2 2 2 2 2 2 ds = dx + dy + dz −→ ds˜ = sxdx + dy + dz (5.2)
which is also equivalent to a analytical continuation of the x coordinate, x → x˜ based
on dx˜ = sxdx or, equivalently,
x x˜(x)= sx(x)dx (5.3) Z0 5.2.2 Curved geometries
Eq. (5.1) corresponds to the desired material blueprint for a planar reflectionless
absorber. For applications in optical ‘black hole’ devices, it is desirable to extend
the reflectionless property to curved surfaces. In cylindrical coordinates (ρ,φ,z) for
example, a similar complexification on the spatial metric along the differential element
in the radial direction ρ would produce
2 2 2 2 2 2 2 2 2 2 2 ds = dρ + ρ dφ + dz −→ ds˜ = sρdρ + ρ dφ + dz (5.4)
where now sρ = aρ(ρ)+ iσρ(ρ)/ω represents the complex stretching parameter along
ρ. Note that the (real-valued) choice
− L 1 s2 = 1 − (5.5) ρ ρ −1/2 i.e. aρ(ρ)=(1 − L/ρ) and σρ(ρ) = 0, with L> 0, recovers the parameters of an
optical Schwarzschild ‘black hole’ device as considered in [52].
Using transformation optics techniques, it is easy to show [25] that the above is
equivalent to a change on the constitutive tensors of the background medium to [ǫ]
=ǫ0 [Λ]ρ and [µ] =µ0 [Λ]ρ, where
[Λ]ρ = diag {1/sρ, sρ, sρ} , (5.6)
58 in a cylindrical coordinate basis (ρ,φ,z) representation. However, unlike the planar
case (5.1), the resulting medium ceases to be reflectionless, with a reflection coefficient
that now depends on the radius of curvature [116]. A truly reflectionless media in a
cylindrical surface entails applying a transformation on the radial coordinate ρ (and
not simply on the radial metric element) [34]. In other words, assuming an interface
at ρ = ρ0, let ρ ρ˜(ρ)= ρ0 + sρ(ρ)dρ = bρ(ρ)+ i∆ρ(ρ)/ω (5.7) Zρ0 for a point ρ in the interior of the absorber medium, with
ρ bρ(ρ)= ρ0 + aρ(ρ)dρ (5.8) Zρ0 and ρ ∆ρ(ρ)= σρ(ρ)dρ (5.9) Zρ0
Note that because both aρ(ρ) and σρ(ρ) are bounded, bρ(ρ) and ∆ρ(ρ) are continuous functions of the variable ρ by construction. The corresponding material tensors, derived using transformation optics techniques [34], read as
[Λ](ρ,φ) = diag {ρ/˜ (sρρ), (sρρ)/ρ,˜ (sρρ˜)/ρ} , (5.10) in a cylindrical coordinate basis. Metamaterial blueprints of reflectionless absorbers for spherical or more general (doubly-curved) surfaces also exist [117]. Note that these blueprints are three-dimensional and equally valid for any polarization.
5.3 Material tensor properties on convex and concave sur- faces
One important distinction between the reflectionless absorber tensors discussed above for planar surfaces (5.1) and for curvilinear surfaces (5.10) is the presence
59 of the radial parameters ρ andρ ˜ in the latter. These radial parameters lead to a fundamental difference between the corresponding material properties on convex and concave surfaces. Before examining this difference, we should first recall that constitutive tensors of causal passive media are free from singularities on the upper half of the complex-ω plane [118]; in particular they can only exhibit poles on the real axis or on the lower half of the complex-ω plane. This property is a condition, for example, for the derivation of Kramers-Kronig relations, which relate the real and imaginary part of a (causal, passive) constitutive tensor [118].
Let us briefly examine the singularities of the constitutive tensors [Λ]x and [Λ](ρ,φ) on the complex-ω plane. For [Λ]x, it is clear that the only singularity is due to the pole at ωx = −iσx(x)/ax(x). Since ax(x) and σx(x) are both nonnegative, this pole is located either on the lower half-plane or on the real axis. On the other hand, [Λ](ρ,φ)
(1) (2) has two poles: one at ωρ = −iσρ(ρ)/aρ(ρ) and the other at ωρ = −i∆ρ(ρ)/bρ(ρ).
(1) Again, because aρ(ρ) and σρ(ρ) are nonnegative, the pole ωρ is located either on the
(2) lower half-plane or on the real axis. However, the location of the pole ωρ depends on the sign of ∆ρ(ρ). Taking the integral in (5.9) for ρ>ρ0 (i.e. outwardly in the radial direction, which means that the cylindrical interface of the PML medium is concave as seen from the ‘outside’ medium), leads to ∆ρ(ρ) > 0. In contrast, taking the integral in (5.9) for ρ<ρ0 (i.e. inwardly in the radial direction, which means that the cylindrical surface is convex as seen from the ‘outside’), leads to ∆ρ(ρ) < 0.
The condition ρ>ρ0 is the typical geometry that is present in numerical simulations where the reflectionless PML medium is inserted at the outer boundary of the domain.
Conversely, the condition ρ<ρ0 is the case of interest for scenarios in which such
60 media are to be used for coating of finite objects, such as for optical ‘black hole’ devices.
Figure 5.1: Behavior of PML-derived metamaterial blueprints on concave versus con- vex surfaces.
If ∆ρ(ρ) ≥ 0, the tensor [Λ](ρ,φ) is singularity-free on the upper half-plane. On
(2) the other hand, if ∆ρ(ρ) < 0, the pole ωρ is present on the upper half-plane. This presence of a pole in the upper half-plane implies that the medium is not an absorber anymore: by examining the impulse response [Λ](ρ,φ) (t) given as
−1 [Λ](ρ,φ) (t)= F [Λ](ρ,φ) (ω) (5.11) h i for ∆ρ(ρ) ≤ 0 and where F denotes a Fourier transform, it is easy to verify [111],
(2) [119] that [Λ](ρ,φ) (t) exhibits exponential growth in time due to ωρ . This behavior will be illustrated in the next section and characterizes an active media (as opposed to an absorber media, which would provide exponential decay on [Λ](ρ,φ) (t)). These two situations are summarized in Fig. 5.1.
61 5.4 Metamaterial blueprints of near-reflectionless absorbers on convex surfaces
Because of the above limitation, we discuss next some alternatives for obtaining
near-reflectionless anisotropic absorbers on convex surfaces (left side of Fig. 5.1).
5.4.1 Approximate ‘PML-derived’ metamaterial blueprints
A possible approach to obtain metamaterial blueprints of near-reflectionless ab-
sorbers on convex surfaces is to employ approximate PML models. For example, the
tensor model in (5.10) can still be formally employed on convex surfaces as long as, instead of using Eq. (5.9), a different choice for ∆ρ(ρ) is made where the condition
∆ρ(ρ) ≥ 0 is enforced in an ad hoc fashion (for example, by ‘inverting’ the sign in front of the integral in Eq. (5.9) or ‘setting’ that integral to be zero). The choice
∆ρ(ρ) = 0, for example, would correspond to (5.4). Such material blueprints are
‘near-reflectionless’ in the sense that the magnitude of the reflection coefficient is not exactly zero but can be made arbitrarily small by increasing the (electrical size of the) radius of curvature—in the limit of electrically large radius, the reflection goes to zero as (5.10) recovers (5.1).
5.4.2 Simulations and backscattering results
Fig. 5.2 illustrates the effect of such metamaterial absorbers in suppressing reflec- tions. It shows, for two time instants, the ‘snapshot’ in the xy plane of the electric
field produced by a z-directed point source (electric dipole) in the presence of a nearby perfect electric conductor (PEC) circular cylinder. The PEC cylinder is coated by a metamaterial ‘shell’ with [ǫ] =ǫ0 [Λ](ρ,φ) and [µ] =µ0 [Λ](ρ,φ) with [Λ](ρ,φ) given by
(5.10) and where ∆ρ is chosen in one of two ad hoc ways: (a) as given by minus the
62 Figure 5.2: Spatial distribution, for two instants of time, of the electric field produced by a point source (with location indicated by the star symbol) nearby a circular PEC cylinder (in white). In the top sequence, the PEC cylinder is stand-alone, producing a sharp reflection (backscattering). The mid and bottom sequences, the PEC cylinder is coated by a metamaterial absorber shell (whose outer boundary is indicated by a dashed circle) having constitutive parameters [ǫ] =ǫ0 [Λ](ρ,φ) and [µ] =µ0 [Λ](ρ,φ), with [Λ](ρ,φ) given by (5.10) and where the parameter ∆ρ is enforced to be zero or positive, in an ad hoc fashion. These two choices are both very effective in suppressing reflections (backscattering).
63 Figure 5.3: Same scattering geometry as before, except that the metamaterial coating used has [Λ](ρ,φ) as given by (5.10) with negative parameter ∆ρ as directly computed from (5.9). Three snapshots of the electric field distribution are presented. Even though the frontal reflection is still strongly suppressed, the spurious field growth is clearly visible at the t = 11.72 ns and at t = 14.66 ns snapshots, with a particularly strong onset in the ‘shadow region’.
integral in (5.9) (∆ρ > 0) or (b) set to zero, ∆ρ = 0. Also shown is field distribution
with no absorber present. The point source is located 12.5 cm away from the surface
of the coating and is excited by a short pulse with center frequency 300 MHz. The
PEC cylinder has 1 m radius and the coating has 0.5 m thickness. A PEC cylinder
is used as the ‘core’ as this provides the worst-case scenario for evaluating the effi-
cacy of the absorber shell in reducing reflection. Note that the shell thickness is half
of the (vacuum) wavelength, whereas the thicknesses of previously proposed (non-
absorptive) ‘black hole’ shells span many wavelengths. This result is obtained using
a three-dimensional finite-difference time-domain (FDTD) algorithm. To eliminate
staircasing error in the circular geometry, we employ a cylindrical grid [120] instead
of a conventional (Cartesian) FDTD grid.
64 Fig. 5.2 serves to illustrate, at a qualitative level, the effectiveness of the absorber shell on suppressing the reflected field, with both choices ∆ρ > 0 and ∆ρ = 0 yielding similar performance. A more detailed (quantitative) analysis of the reflection coeffi- cient is presented below. On the other hand, Fig. 5.3 shows the scattering from the same geometry except that the metamaterial shell has [Λ](ρ,φ) with negative ∆ρ as computed from (5.9). Note that even though the frontal reflection is still strongly suppressed at first, this Figure clearly shows the onset of the spurious field growth inside the shell due to the upper half-plane pole, as discussed in the previous section.
For realizations of such absorbers based upon annular strips (columns) of peri- odic inclusions, σρ should assume a stepwise distribution behavior along ρ [50]. The
amount of reflection in this case would depend on the conductivity contrast between
adjacent strips and would increase for larger contrasts (an analogous issue exists for aρ, as well as for PML models on discrete grids [121]). To minimize reflections, it is there-
p fore desirable to adopt a tapered profile for σρ(ρ) such as σρ(ρ)=(σ0/ǫ0)|(ρ−ρ0)/δ| ,
where σ0 is the maximum value of the conductivity (in mhos/m), ǫ is the vacuum permittivity (in F/m), δ is the overall absorber thickness, and p represents the degree
of the taper [116], [119].
To quantify the level of residual reflections produced, we plot the ‘reflection’ or
‘backscattering coefficient’ as a function of time in Fig. 5.4. This coefficient is com-
puted here in the following way: The z-component of the electric field is first sampled
at a point 3.125 cm away from the coating surface (along the line segment from the
origin to the source location) with and without the presence of the coated cylin-
der; the reflection coefficient is then computed as the (magnitude of the) difference
between these two field values normalized by the peak field value observed at this
65 σ =0.045 σ =0.075 0 0 −20 −20 ∆ρ < 0 ∆ρ < 0 ∆ρ > 0 ∆ρ > 0 −40 ∆ρ = 0 −40 ∆ρ = 0 no absorber no absorber
−60 −60
−80 −80 Reflection (dB) Reflection (dB)
−100 −100
−120 −120 0 2 4 6 8 10 0 2 4 6 8 10 time(ns) time(ns) (a) (b)
σ =0.105 0 −20 ∆ρ < 0 ∆ρ > 0 −40 ∆ρ = 0 no absorber
−60
−80 Reflection (dB)
−100
−120 0 2 4 6 8 10 time(ns) (c)
Figure 5.4: Reflection coefficients computed for different choices of ∆ρ as in Fig. 5.2, and for three values of the conductivity σ0 factor (a) σ0 = 0.045, (b) σ0 = 0.075, and (c) σ0 = 0.105. Lower σ0 values produce less reflection from the ‘front-end’ of the coating (at about 2 ns) but more reflection from the PEC cylinder itself (at about 5 ns); conversely, larger σ0 values decrease the reflection originated from the PEC cylinder but produce more reflection from the ‘front-end’ of the coating. The reflection from the PEC cylinder with no coating (‘no absorber’) is also plotted for comparison.
66 same sample point. Any contribution to this coefficient originates from the residual backscattering. The results shown in Fig. 5.4 assume 32 annular strips for the coat- ing (so that each strip has about 1.5 cm thickness) and a tapered profile for σρ with p = 4. For simplicity, we set aρ = 1. Three results are shown for different values of σ0. These results serve to illustrate a basic trade-off when attempting to mini- mize the reflection: larger σ0 values increase the early-time ‘front-end’ reflection due to the (larger) contrast present between successive strips, but decrease the late-time
‘back-end’ reflection from the PEC cylinder itself (residual after a ‘two-way travel’ within the absorber); conversely, lower σ0 values decrease the conductivity contrast between successive annular strips, and hence the front-end reflection, but increase the back-end reflection from the PEC cylinder itself as the field absorption becomes less pronounced during the two-way travel within the absorber.
Note that the reflection coefficient does not show any marked degradation by considering ∆ρ > 0or∆ρ = 0, as opposed to ∆ρ < 0. Even though the reflection levels presented in Fig. 5.4 are characteristic of a particular choice of conductivity profile and number of ‘inclusion strips’, they are good indicators of the relative practical performance among the three ∆ρ choices considered. The spurious field growth for
∆ρ < 0 is present but not visible within the time window shown in Fig. 5.4. Next,
Fig. 5.5 shows a set of results with larger σ0, where this spurious growth appears earlier and is visible within this time window.
5.4.3 Optical ‘black hole’ metamaterial blueprints with em- bedded absorption
The near-reflectionless metamaterial model considered above can be used either as a ‘stand-alone’ coating (as simulated) or as the absorber core inside a conventional
67 σ =0.5 0 −20 ∆ρ < 0 ∆ρ > 0 −40 ∆ρ = 0 no absorber −60
−80 Reflection (dB)
−100
−120 0 2 4 6 8 10 time(ns)
Figure 5.5: ’Reflection coefficient’ results for σ0 = 0.5, showing the spurious field growth (at about 10 ns) originating from the metamaterial coating with ∆ρ < 0.
(non-absorbing) ‘black hole’ shell. In the latter scenario, the constitutive tensors of the absorber core would read as [ǫ] =ǫI [Λ](ρ,φ) and [µ] =µI [Λ](ρ,φ), with [Λ](ρ,φ)
given by (5.10) under the condition ∆ρ(ρ) ≥ 0, and with ǫI and µI representing the
values of the permittivity and permeability, respectively, of the inner boundary of the
(non-absorbing) shell.
Alternatively, a metamaterial blueprint can be designed to effect absorption through-
out the ‘black hole’ medium. To show this, let us first write the spatial component
of the metric of an optical Schwarzschild (non-absorbing) ‘black hole’ device, given
by [52] 1 ds2 = dρ2 + ρ2dφ2 + dz2 (5.12) 1 − L/ρ
68 The above can also be written as
ds2 = dρ¯2 + ρ2dφ2 + dz2 (5.13) by using the change of variables ρ → ρ¯, where
1 2ρ 1+ 1 − L/ρ − L ρ¯(ρ)= ρ + ρ 1 − L/ρ − ρ 1 − L/ρ + L ln 0 0 0 2 2ρ0 1+ p1 − L/ρ0 − L p p p (5.14)
for ρ ≤ ρ0, andρ ¯ = ρ for ρ>ρ0. Here, ρ0 is the radius of the device and L is a
positive parameter (‘Schwarzschild radius’) such that L<ρ0. Note thatρ ¯ decreases
monotonically as ρ decreases in the interval L<ρ<ρ0 and can assume negative
values, withρ ¯ → −∞ for ρ → L. The thickness t of the optical ‘black hole’ device
is chosen so that t<ρ0 − L to avoid the ‘singularity’ at ρ = L. In particular, an
optical ‘black hole’ device with thickness t can be used to coat an object with radius
b = ρ0 − t with L The generalized metamaterial blueprint for an optical Schwarzschild ’black hole’ with an embedded near-reflectionless absorption mechanism reads 2 1 σ (ρ) ds2 = + i ρ dρ2 + ρ2dφ2 + dz2 (5.15) 1 − L/ρ ω ! which is in the same genericp form of (5.4) but now with 1 σρ(ρ) sρ(ρ)= + i (5.16) 1 − L/ρ ω p so that corresponding material tensors are written as [ǫ] =ǫ [Λ]ρ and [µ] =µ [Λ]ρ with [Λ]ρ given by (5.6) in a cylindrical coordinate basis and with sρ now given by (5.16). 69 5.5 Conclusion We have discussed the derivation of metamaterial blueprints for (near-)reflectionless absorbers and their possible integration into optical pseudo black hole devices. Fun- damental theoretical constraints on truly reflectionless absorption models at convex surfaces were pointed out and explained by studying the analyticity of the associated constitutive tensors in the complex ω plane. In view of such constraints, ‘tweaked’ near-reflectionless absorbers blueprints were proposed. Simulation results of such ab- sorber models have shown their effectiveness in suppressing backscattering. Some basic trade-offs in the absorber performance stemming from the use of a stepwise conductivity distribution along the radial direction were also pointed by examining the residual backscattering produced by different conductivity profiles. This chapter has focused the analysis at the ‘blueprint’ (constitutive tensor de- scription) level. It is beyond the objectives of this study to dwell into fabrication issues. Suffice it so say that the proposed absorbers exhibit a layered structure that can facilitate eventual fabrication. Integration of spatially varying conductivity can be done, at least in the microwave frequency range, by means of stacking resistive sheets (R-cards) in decreasing resistivity values from the outer to the inner shell (such as in some radar absorber designs) and using a variable density of lossy magnetic particles, for example. 70 Chapter 6: Electrically Small Complementary Electric-Field-Coupled Resonator Antennas 6.1 Introduction Metamaterials have been widely used in many antenna applications as a means to improve overall performance [2], [3], [81]–[83], [86], [90], [93], [94], [96]–[98], [122]– [137]. In particular, new electrically small antennas (ESAs) have been designed thanks to miniaturization capabilities enabled by metamaterials [96]–[98], [124], [125], [130]– [133]. As summarized in [132], one effective approach to design ESAs is to employ metaresonators, i.e., split-ring resonators (SRRs) or complementary split-ring res- onators (CSRRs), excited by monopole or patch antennas [96], [97], [124], [125], [130]. Such combined structure (‘metaresonator antenna’) operates near the resonant frequency of the metaresonator, thus allowing for the miniaturization [132]. Resonant antennas composed of single negative materials were proposed by Isaacs [138]. Being perhaps the most popular element providing negative permeability response, the SRR [20] was employed to design ESA by Alici et al. [96], [97]. They showed that ESAs can be designed by properly exciting the SRR via a monopole antenna. CSRRs that exhibit negative permittivity were studied by Falcone et al. [139], [140]. CSRRs 71 were also used for low-profile planar antenna designs [125], [130]. Moreover, by em- ploying various resonators in tandem, added functionalities such as dual polarization, circular polarization, or multi-band characteristics can be obtained [124], [130]. SRRs are also employed in the design of multi-band and multifunctional printed monopole antennas [126]–[129], particularly for wireless applications. SRR and CSRR can be considered as parasitic elements inducing coupled (equivalent) magnetic or electric dipole responses, respectively [132]. To enhance the bandwidth of resonant antennas, active devices can also be used [134]–[137]. In this chapter, we investigate the use of electric-field-coupled (ELC) [67] and com- plementary electric-field-coupled (CELC) [141] resonators to design ESAs. It turns out that ELC and CELC resonators can also induce coupled electric and magnetic dipole responses [141]. The advantage here is that otherwise using either SRR or CSRR elements allows for coupling only to the perpendicular (w.r.t. the resonator plane) magnetic or electric field respectively, from the active component [97], [130]. This limits the choice of configurations to which SRR and CSRR resonators are use- ful for one particular orientation only. In contrast, ELC and CELC resonators can couple to both parallel and perpendicular components of the electric or magnetic field, respectively [67], [141]. In the following, we exploit this property to design new ESA configurations as depicted in Fig.6.1.Configurations depicted in Fig 1(d) in particular are both very low-profile. The design of a new low-profile bent-monopole CELC-coupled antenna is also presented. 72 Figure 6.1: Schematic configurations of (a) SRR, (b) ELC, (c) CELC-1, and (d) CELC-2, CELC-3 resonator antennas. For the CELC-1,-2,-3 configurations, the re- spective orientation is indicated in (e). 73 Table 6.1: Simulation results of the figures of merit. f0 (GHz) ka FBW −10dB (BW) Qrad Qmin efficiency SRR 2.82 0.58 0.030 0.99 % 33.3 4.34 39.8 % ELC 4.13 0.84 0.040 1.05 % 25.0 2.01 14.2 % CELC-1 4.31 0.84 0.040 1.29 % 25.0 2.04 36.3 % CELC-2 7.11 0.83 0.040 1.39 % 25.0 2.09 34.3 % CELC-3 7.49 0.87 0.055 1.88 % 18.2 1.90 55.0 % Bent CELC 4.50 0.68 0.058 0.92 % 17.2 3.03 23.2 % 6.2 ELC/CELC-Based Metaresonator Antennas Fig. 6.1 shows the five different antenna configurations considered here, viz., (a) SRR, (b) ELC and (c,d) CELC-1,-2,-3 resonator antennas, respectively. Fig. 1(d) depicts both CELC-2 and CELC-3 resonator antennas with their respective coordinates shown in Fig. 1(e). All resonators are excited through a monopole antenna, also depicted. The physical dimensions of the resonators are the same in all configurations. Following the definitions in Fig. 6.1, antenna parameters are as follows: the monopole wire length is equal to l = 9 mm for (a) SRR, (b) ELC, and (c) CELC-1, and equal to l = 2 mm for (d) CELC-2, -3; the inner wire radius is rw = 0.46 mm; and the coaxial outer radius is rc = 1.47 mm with ǫr = 2. The side length of square ground plane is lgp = 100 mm; the side length of the substrate is a = 7.8 mm; the width of the strips is w = 0.5 mm; the gap between strips is g = 0.4 mm; the distance between substrate edge and strips is s = 0.3 mm; and the length of the strip inside ELC and CELC-1,2,3 is t = 4.8mm. The distance between the monopole and 74 Table 6.2: Measurement results of the figures of merit. f0 (GHz) ka FBW −10dB (BW) Qrad Qmin SRR 2.84 0.58 0.061 1.94 % 16.3 4.28 ELC 4.11 0.84 0.089 1.38 % 11.2 2.03 CELC-1 4.32 0.84 0.082 2.13 % 12.2 2.03 CELC-2 7.12 0.83 0.051 1.59 % 19.6 2.08 CELC-3 7.49 0.87 0.082 2.55 % 12.2 1.90 Bent CELC 4.43 0.67 0.063 1.12 % 15.9 1.13 resonators is d1 = 0.1 mm for all cases, except for (c) CELC-1, where it is d1 = 0.2 mm. The distance between the ground plane and resonators is d2 = 0.6 mm for (a) SRR, (b) ELC (b) and (c) CELC. The FR-4 substrate has ǫr = 3.85 and loss tangent tan δ = 0.02. The thickness of the substrate is h = 1.6 mm with a copper thickness of 0.030 mm. The antennas are simulated using the software CST Microwave StudioTM, based on the finite integration technique. Return losses are measured using Agilent N5230A network analyzer. Fig. 6.2 shows the simulated and measured S11 data. Good agree- ment is observed between the two sets of results. The small discrepancy can be attributed to measurement imperfections and expected variations of FR-4 dielectric constant among samples [124]. Even though all resonators have same physical di- mensions, it is seen that the electrical dimensions for each antenna differ due to the different resonant frequencies. Figures of merit for above antennas are presented in 75 Figure 6.2: Return loss of the resonator antennas depicted in Fig. 6.1. Table 6.1 and Table 6.2. Using Foster reactance theorem, the quality factor is com- puted as Qrad = 1/FBW , where FBW is the fractional bandwidth. In order to calculate the minimum radiation quality of the antenna we determine the radius a for the minimum sphere that encloses each antenna, which is presented in Table 6.1 and Table 6.2. For CELC-2 and CELC-3, the resonant frequency is relatively higher but the minimum radius is smaller. This is due to the low profile characteristic of these two designs. Note that the minimum radius is determined for each configura- tion considering the ground plane effect, where the image of the antenna is taken into account. The minimum Q presented in Table 6.1 and Table 6.2 is calculated as 76 Figure 6.3: 3-D far-field radiation pattern for each of the antennas shown in Fig.6.1. 77 90 90 10 10 120 60 120 60 0 −20 150 30 150 30 −10 −40 180 0 180 0 210 330 210 330 xy plane xy plane 240 300 yz plane 240 300 yz plane 270 270 90 90 10 120 10 60 120 60 0 0 −10 150 30 150 −10 30 −20 −20 −30 180 0 180 0 210 330 210 330 xy plane xy plane 240 300 yz plane 240 300 yz plane 270 270 90 10 120 60 0 150 −10 30 −20 180 0 210 330 xy plane 240 300 yz plane 270 Figure 6.4: xy- and yz-plane radiation patterns for each of the antennas shown in Fig.6.1. 78 Figure 6.5: Surface current plots for each of the antennas shown in Fig.6.1. 79 1 1 2 Q = + (6.1) min 2 k3a3 ka where k is the wavenumber at the (resonant) frequency of operation f0. Notice that all antennas studied here have low radiation efficiency and narrow bandwidth, which is expected due to their small electrical sizes. The simulated 3D radiation patterns and the E- and H-plane patterns are shown in Fig. 6.3 and 6.4. ELC and CELC-1 designs have similar monopole-like patterns, whereas the SRR, CELC-2 and CELC-3 designs have a (z-oriented) patch-like pat- tern [96], [97]. The surface current distributions along the metaresonators are also depicted in Fig. 6.5. The computed antenna directivities are 6.1 dBi (SRR), 3.7 dBi (ELC), 4.6 dBi (CELC-1), 6.8 dBi (CELC-2) and 7.1 dBi (CELC-3) with correspond- ing efficiencies 40%, 14%, 36%, 34% and 55%, and gains 2.6 dB, 0.6 dB, 1.9 dB, 2.6 dB and 4.1 dB, respectively. We also study the case where the CELC resonator is excited through a bent monopole antenna, as depicted in Fig. 4(a). As stated previously, the CELC resonator can couple to a magnetic field parallel to its surface. We exploit this feature to design another antenna with a very low-profile. The CELC resonator has the same parameters as considered before. The wire height is hw = 4.72 mm and the wire length is l = 11 mm. The distance between the vertical wire and the CELC is d1 = 0.5 mm and the distance between horizontal wire and CELC is d3 = 2.2 mm. The S11 (along with the CELC-1 case, for comparison) and radiation pattern in the xy and yz planes are shown in Fig. 4(b,c), respectively. The resonant frequency for this bent excited CELC is slightly higher than the CELC-1 case. In the simulations it is seen that the resonant frequency is strongly dependent on the d3 parameter. The minimum 80 (a) (b) 90 10 120 60 0 150 −10 30 −20 180 0 210 330 xy plane 240 300 yz plane 270 (c) Figure 6.6: (a) Bent monopole-excited CELC antenna. (b) Measured return loss. (c) Radiation pattern. 81 radius a of the antenna is now considerably smaller. Moreover, the radiated power has increased significantly at zenith. On the other hand, the radiation efficiency is slightly smaller compared to the CELC-1 case. The radiation characteristics of this antenna are also summarized in Table 6.1 and Table 6.2. We mention in passing that due to the planar nature of this later design, reactive impedance surfaces can be used to further enhance this antenna performance [130]. 6.3 Conclusion We have designed electrically small metaresonator antennas composed of ELC and CELC resonators and a monopole antenna. We exploited the ability of ELC and CELC to induce equivalent electric or magnetic dipoles with different polarizations to- wards designing various ESA configurations. We have designed monopole-excited and bent-monopole-excited CELC antennas with extremely low profile. We observe that, as a general rule, CELC resonators are more suited than ELC counterparts for ESA configurations because of superior performance and flexibility of the former for exci- tation in different orientations. The results above clearly show that CELC resonators are a very attractive component for the design of electrically small metaresonator- based antennas. 82 Chapter 7: Metamaterial Claddings for Waveguide Miniaturization 7.1 Introduction The study of waveguides loaded with metamaterials has attracted much interest in recent years [142]–[148]. In the paper of Marquez et al. [142], it was pointed out that a rectangular waveguide below cut-off behaves as a 1-dimensional electric plasma for TE modes [142], [143]. Since the rectangular waveguide naturally provides nega- tive permittivity below cut-off, when combined with negative permeability materials, such loaded waveguide mimics left-handed media in the frequency range of negative permeability [142], [143], [149]. Thus, it provides an alternative approach for the realization of left-handed media [142], [144], [150]. Other loadings such as ferrites and dielectric filled corrugations were also proposed for realization of left-handed media. Negative refraction was experimentally demon- strated in a parallel-plate waveguide loaded with dielectric resonators [151]. Besides realization of left handed media, another potential application of metamaterial-loaded waveguides is miniaturization thereof [144], [148]. Recently, the controllability of backward and forward waves in metamaterial-loaded waveguide was studied by ro- tating the metamaterial element of the waveguide [148]. This may find use in, for 83 example, leaky wave antenna applications. Radiation characteristics of metamaterial loaded waveguides were also studied by Hrabar et al. [152]. It is also interesting to investigate what occurs when the waveguide is loaded with negative permittivity media. In fact, the duality between TE and TM modes is such that waveguides behave as magnetic plasma for TM modes below cut-off [145], [153], [154]. Thus, when loaded with negative permittivity media, they mimic left handed media similarly to the TE case. For the dominant TE mode, however, it turns out that transmission happens only for very high values of permittivity. On the other hand, negative permittivity doesn’t effect the transmission of the waveguide. Recently dielectric resonators were used in a rectangular waveguide that supports backward and forward waves below cut off [155]. It was analytically and numerically shown that negative permeability leads to backward waves and high positive permittivity leads to forward waves in a rectangular waveguide for TE modes. This corroborates the fact that metamaterial-loaded waveguides support backward waves if the metamaterial exhibits negative permeability and support forward waves if the metamaterial exhibits negative permittivity. Note, however, that the anisotropy of the loading plays an crucial role on the propagation behavior of the individual modes [144], [148], [149], [156]. Waveguides loaded with negative permeability media such as split-ring resonators (SRR) have been studied extensively in the literature. Wire media were also used for the investigation of TM modes. In this study, we investigate waveguides that are loaded with electric-field-coupled (ELC) resonators these are alternative building blocks for negative permittivity media, acting on the side walls of the waveguides as metamaterial claddings for TE modes. ELC resonators were recently proposed to 84 obtain negative permittivity media as an alternative to wire media. They provide flexibility to control more easily the permittivity response. ELC resonators were used on the two lateral sides of the rectangular waveguide for TE modes. Forward waves are obtained for TE modes for high values of permittivity way below the cut- off frequency of the dominant mode. In addition, when ELC resonators are used together with SRR resonator at the center of the waveguide, one obtains backward and forward waves simultaneously. Figure 7.1: Schematic configuration of a rectangular waveguide filled with uniform uniaxial media. 7.2 Rectangular Waveguides Loaded with Anisotropic Media Consider a metallic rectangular waveguide with a cross section a×b filled uniformly by a material with uniaxial permittivityǫ ¯ and permeabilityµ ¯ tensors given as: 85 ǫt 0 0 ǫ¯ = 0 ǫ 0 (7.1a) t 0 0 ǫz µt 0 0 µ¯ = 0 µ 0 (7.1b) t 0 0 µz where z is chosen as the propagation direction as depicted in Fig.7.1. Maxwell’s equations in such anisotropic media is written as follows (using e−iωt time convention) ∇ × E~ = iωµ¯H~ (7.2a) ∇ × H~ = −iωǫ¯E.~ (7.2b) The above can be decomposed into transverse and longitudinal components ∂ zˆ × E~ + ∇ × zEˆ = iωµ¯ H~ (7.3a) ∂z s s z s s ∇s × E~s = −iωµzzHˆ z (7.3b) ∂ zˆ × H~ + ∇ × zHˆ = −iωǫ¯ E~ (7.3c) ∂z s s z s s ∇s × H~ s = −iωǫzzEˆ z (7.3d) By taking a cross product of Eq. 7.3a and Eq. 7.3c withz ˆ and making use of the identityz ˆ×(∇s × zEˆ z)= ∇sEz andz ˆ× zˆ × E~s = −E~s, the transverse components can be written in terms of Ez and Hz as [118] 1 ∂E E~ = ∇ z + iωµ ∇ × zHˆ (7.4a) s ω2ǫ µ − k2 s ∂z t s z t t z 1 ∂H H~ = ∇ z − iωǫ ∇ × zEˆ (7.4b) s ω2ǫ µ − k2 s ∂z t s z t t z 86 We can then obtain the Helmhotz equation for Hz and Ez by substituting Eq. 7.4a into Eq. 7.3b and Eq. 7.4c into Eq. 7.3d: µ ∇2 + z ω2ǫ µ − k2 H = 0 (7.5a) s µ t t z z t ǫ ∇2 + z ω2ǫ µ − k2 E = 0 (7.5b) s ǫ t t z z t Solving the above yield the dispersion relations for TE and TM waves, k2 + k2 k2 = ω2ǫ µ 1 − x y (7.6a) z t t ω2µ ǫ t z k2 + k2 k2 = ω2ǫ µ 1 − x y (7.6b) z t t ω2ǫ µ t z One can then define the effective medium parameters of the waveguide as k2 + k2 ǫTE = ǫ 1 − x y (7.7a) eff t ω2ǫ µ t z TE µeff = µt (7.7b) TM ǫeff = ǫt (7.7c) k2 + k2 µTM = µ 1 − x y (7.7d) eff t ω2ǫ µ z t It can be seen from above that the waveguide modes can be controlled by three different constitutive parameters for TE and TM modes, respectively. For TE modes below cut-off ǫeff < 0; thus, when loaded with µt < 0 material it yields left-handed 87 (a) (b) (c) Figure 7.2: Schematic configuration of a rectangular waveguide loaded with ELC and SRR. ELC resonators are placed at the lateral side walls while the SRR resides at the center of the waveguide. Perspective view (a), Side view for ELC (b) and Side view for SRR (c). 88 behavior and backward waves are generated. When µz < 0, then ǫeff > 0 at any fre- quency yielding a forward wave in the negative µz region. When there is no magnetic loading, the sign of ǫt does not change the sign of ǫeff . The only way to have pass-band below cut-off is to have high ǫt values. The case when µt < 0 has been extensively studied in the literature [142]–[144]. The controllability of modes was recently stud- ied by controlling µt and µz simultaneously [148]. Though the sign of ǫt parameter doesn’t change the sign of ǫeff , it can lead to a significant waveguide miniaturization if the value of ǫt reaches very high values. Though single magnetic or electric loadings have been considered in the literature, they have not been considered together to the author’s knowledge. As clearly seen from Eq. 7.7a and Eq. 7.7b, all three components ǫt, µt and µz play a role on the waveguide modes. Thus, here we first consider the electric loading alone and next we study the waveguide with both electric and mag- netic loadings. Recently, a dielectric-resonator-loaded waveguide was also proposed used to control the waveguide modes and miniaturize the waveguides [155]. The case ǫt < 0, yielding backward waves, was studied in [145], [153], [154]. In most studies wire media is used to obtain negative permittivity response. The issue of spatial dispersion in wire media was discussed in [145]. 7.3 TE Case: Electric and Magnetic Resonator Loaded Waveg- uide It was discussed in the previous section that waveguide modes can be manipulated by changing the response of metamaterial loadings. Following Eq. 7.7a and Eq. 7.7b, TE modes can be controlled by changing three components of the constitutive parameters: ǫt, µt and µz. The cases for positive and negative µt and µz loadings has been extensively studied in the literature. Here we focus on ǫt loadings. Clearly, from 89 (a) (b) (c) (d) Figure 7.3: Unit cell configurations for ELC(a,c) and SRR(b,d). For simulations PEC and PMC boundary conditions are applied on the y and x directions respectively. The parameters depicted in (b,d) for ELC and SRR are given in the text. 90 15 ε eff µ 10 eff 5 0 −5 4 6 8 10 12 Frequency (GHz) 10 5 0 ε eff µ eff −5 4 6 8 10 12 Frequency (GHz) Figure 7.4: Effective ǫ and µ response for ELC and SRR, respectively. 91 Eq. 7.7a we see that transmission happens only for very high values of ǫt whereas negative ǫt does not lead any pass-band below cut-off. Furthermore, by combining the electric and SRR resonators one can obtain forward and backward waves below cut- off. In this study we focus on ELC resonators to control the ǫt values. ELC resonators were recently proposed to obtain negative permittivity response as an alternative for wire media [67]. Compare to wire media, ELC resonators have significant advantages and the permittivity response can be more easily controlled. In addition, ELC can be excited under different polarizations thus providing an additional degree of freedom. ELC and SRR loaded waveguides are shown in Fig.7.2, where ELC resonators are placed at the side walls of the rectangular waveguide and SRR resonators are placed at the center of the waveguide. 0 ELC SRR −20 Both −40 S21(dB) −60 −80 4 5 6 7 8 9 10 11 Frequency (GHz) Figure 7.5: Transmission coefficients for ELC, SRR, and ELC + SRR loaded waveg- uides, respectively. Note that the transmission coefficient for ELC + SRR loaded waveguide matches closely with respective ELC and SRR loaded waveguides. 92 10 5 0 −5 −10 ε eff −15 µ eff −20 4 5 6 7 8 9 10 11 Frequency (GHz) Figure 7.6: Effective parameters for ELC + SRR loaded waveguide. Note that the first transmission happens around 6.5 GHz where both ǫ and µ are negative and the second transmission is around 8.5 GHz where both ǫ and µ are positive. The waveguide has 9 × 9 mm cross section and 80 mm length. The unit cells for ELC and SRR are shown in Fig. 7.3. Following the definition in Fig 7.3, ELC resonator dimensions are as follows: the side length of the ELC is l = 6 mm, the width of the strip is w = 0.9 mm, the gap between the inner strips is g = 0.45 mm and the length of inner strip is t = 2.4 mm. Similarly, the length of the SRR ring is l = 5.2 mm, the width of the ring is w = 1.2 mm and the gap between rings is g = 0.6 mm. The substrate has ǫr = 2.2 and loss tangent tanδ = 0.0009. The thickness of the substrate is 0.49 mm with a copper thickness of 0.030 mm. First, we calculate the effective parameters of ELC and SRR using a standard retrieval procedure [73], [75], [79]. We particularly used the method detailed in [79] to choose the correct branch for ǫ and µ. The effective parameters for ELC and 93 9 8.5 8 ELC SRR 7.5 Frequency (GHz) 7 6.5 0 50 100 150 Phase Shift (degree) Figure 7.7: Dispersion diagram for ELC loaded and SRR loaded waveguide. Note that at SRR resonance waves are backward waves whereas at ELC resonance waves are forward waves. SRR are shown in Fig. 7.4. Fig. 7.5 shows the transmission coefficient for ELC loaded, SRR loaded and ELC + SRR loaded waveguide. Note that the ELC + SRR loaded waveguide has both transmissions peaks as the ELC loaded and SRR loaded waveguides with a small frequency shift at the latter. The waves at ELC resonance are forward waves because both ǫeff and µeff are positive , while the waves at SRR resonance are backward waves because both ǫeff and µeff are negative in that region. As a result, by simultaneously loading the rectangular waveguide with ELC and SRR both forward and backward waves are generated below the cut-off. The effective parameters of the ELC and SRR loaded waveguides are also shown in Fig. 7.6. Note that both ǫeff and µeff are negative at 6.6 GHz and both positive at 8.4 GHz indicating that the first one is backward type and the second one is forward wave. 94 Fig. 7.7 show the dispersion diagram for ELC loaded and SRR loaded waveguides. Note that dispersion diagram verifies the backward and forward waves in ELC loaded and SRR loaded waveguides. 7.4 Conclusion Rectangular waveguides loaded with metamaterial loadings were considered in this Chapter. First, a theoretical discussion was performed to determine effective param- eters of rectangular waveguide with uniform anisotropic fillings. It was shown that effective parameters of the rectangular waveguides with anisotropic fillings depend on the three components of the constitutive parameters. Distinct scenarios with differ- ent signs for these parameters were investigated. Backward waves occur when both ǫeff and µeff are negative and forward waves occur when both ǫeff and µeff positive. We also note that for the dominant TE mode only 1-D response is required in the material parameters. Furthermore, for TE modes both ELC and SRR are loaded simultaneously to obtain backward and forward waves. The proposed structures with ELC claddings provide an useful alternative for waveguide miniaturization under TE mode operation. 95 Chapter 8: Conclusions and Further Remarks 8.1 Conclusions Metamaterials have been the focus of extensive research over the past decade. The unusual properties of metamaterials have been exploited for many applications. In particular in this dissertation we focused on the use of metamaterials for waveguide and antenna problems. More recently, transformation optics was also proposed as a technique to design novel electromagnetic devices. In this dissertation, we have also explored the physics of some novel electromagnetic devices based on transforma- tion optics, e.g., metamaterial claddings, isoimpedance metamaterial substrates, and impedance-matched absorbers. In chapter 3 we studied mode control of waveguides using anisotropic metama- terial claddings. Transformation optics was exploited to design such claddings. The constitutive parameters are specified by a contrast parameter “s” that provides con- trol over the mode uniformity (in the core region), the degree of field confinement in the cladding, and the resulting waveguide cutoff frequencies. The obtained metama- terial blueprints can be viewed as isoimpedance materials with intrinsic impedance matched to free-space, regardless of frequency or propagation angle. Thus, the re- sulting metamaterial-loaded waveguides exhibit no hybrid modes. The close relation 96 between the isoimpedance property such metamaterials and perfectly matched layers was pointed out [62], [80]. In chapter 4 we have analyzed the performance of two canonical low-profile radi- ators, viz. horizontal dipole and rectangular patch on top of isoimpedance metama- terial substrates. These substrates are doubly anisotropic media designed via trans- formation optics techniques in such a fashion that they are impedance-matched to free space. It was verified that the isoimpedance feature of the substrate can enhance the antenna characteristics and radiation properties because no surface waves are ex- cited. In particular, significant increase in the usable bandwidth is observed versus conventional (isotropic) dielectric substrates. In addition, mutual coupling in array geometries can be mitigated for similar reasons. Note that a general tradeoff between efficiency and wide-band performance is present in other substrate miniaturization alternatives as well. In chapter 5 we have discussed the derivation of metamaterial blueprints for (near- ) reflectionless absorbers and their possible integration into optical pseudo black hole devices. Fundamental theoretical constraints on truly reflectionless absorption models at convex surfaces were pointed out and explained by studying the analyticity of the associated constitutive tensors in the complex ω plane. In view of such constraints, near-reflectionless absorbers blueprints were proposed. Simulation results of such absorber models have demonstrated their effectiveness in suppressing backscattering. Some basic trade-offs in the absorber performance stemming from the use of a stepwise conductivity distribution along the radial direction were also pointed by examining the residual backscattering produced by different conductivity profiles. 97 In chapter 6 we have designed electrically small metaresonator antennas composed of ELC and CELC resonators and a monopole antenna. We exploited the ability of ELC and CELC to induce equivalent electric or magnetic dipoles with different polar- izations towards designing various ESA configurations. We have designed monopole- excited and bent-monopole-excited CELC antennas with extremely low profile. We have observed that, as a general rule, CELC resonators are more suited than ELC counterparts for ESA configurations because of superior performance and flexibility of the former for excitation in different orientations. The results above clearly show that CELC resonators are a very attractive component for the design of electrically small metaresonator-based antennas. In chapter 7, rectangular waveguides loaded with metamaterial loadings were con- sidered. First, a theoretical analysis was done to find out the effective parameters of rectangular waveguide with uniform anisotropic fillings. It was shown that the effective parameters of the rectangular waveguides with anisotropic fillings depend on the three components of the constitutive parameters. Different scenarios for different signs for these parameters were investigated. Backward waves occur when both ǫeff and µeff are negative, whereas forward waves occur when both ǫeff and µeff positive. For TE modes only a 1-D response is required for material parameters. Furthermore, for TE modes both ELC and SRR can be loaded to obtain backward and forward waves simultaneously. We believe that the proposed structures with ELC claddings provide an attractive alternative for TE waveguide miniaturization. In conclusion, we believe that metamaterials can find practical use for some waveg- uide and antenna applications. Particularly, the promising sub-wavelength nature of 98 metamaterials can be exploited for the design of miniaturized waveguides and an- tennas. Yet, there are still some challenges such as efficiency and narrow bandwidth of metamaterial based antennas. These issues are currently being investigated for efficient electrically small antenna designs. 99 Appendix A: FDTD Method for Dispersive Media in Cartesian Coordinates A.1 PML-PLRC-FDTD Formulation In this Appendix, we briefly summarize the finite difference time domain (FDTD) method that is used throughout the dissertation for the simulation of transformation media. In order to accurately model the dispersive characteristics of materials, a piecewise linear recursive convolution (PLRC) technique is implemented into FDTD. Furthermore, we use the complex coordinate stretching approach to implement per- fectly matched layers (PML) [157]. In what follows, we summarize the approach described in [157]. In the frequency domain, modified Maxwell’s equations are written as follows (e−iωt convention), ∇s × E = iωB + σmH (A.1a) ∇s × H = −iωD + σeE (A.1b) for a conductive medium, where 1 1 1 ∇s =x ˆ ∂x +y ˆ ∂y +z ˆ ∂z (A.2) sx sy sz 100 where sx, sy and sz are frequency dependent complex stretching variables. Then we split Eq. A.1 as: 1 iωBsx + σmHsx = ∂xx × E (A.3a) sx 1 −iωDsx + σeEsx = ∂xx × H (A.3b) sx where, sx = ax + i (Ωx/ω). As such, we obtain: Ω iωa B − Ω B + a σ H + i x σ H = ∂ x × E (A.4a) x sx x sx x m sx ω m sx x Ω −iωa D + Ω D + a σ E + i x σ E = ∂ x × H (A.4b) x sx x sx x e sx ω e sx x Transforming back to time domain we get t −ax∂tBsx − ΩxBsx + axσmHsx + Ωxσm Hsx (τ) dτ = ∂xx × E (A.5a) Z0 t ax∂tDsx + ΩxDsx + axσeEsx + Ωxσe Esx (τ) dτ = ∂xx × H (A.5b) Z0 In a doubly dispersive medium, the constitutive parameters are given by B (t)= µ (t) ∗ H (t) (A.6a) D (t)= ǫ (t) ∗ E (t) (A.6b) Next, we show the implementation of dispersive media into FDTD algorithm for E field update. The H update can be obtained along similar lines. For an N -species Lorentzian dispersive medium, where frequency-dependent relative permittivity is given by 101 P 2 Gpωp ǫ (ω)= ǫ [ǫ∞ + χ (ω)] = ǫ ǫ∞ + ǫ (ǫ − ǫ∞) (A.7) 0 0 0 s ω2 − i2ωα − ω2 p=1 p p X where χ (ω) is the medium susceptibility, ωp is the resonant frequency for the pth species, αp is the corresponding damping factor, and ǫ0 and ǫ∞ are the static and infi- nite frequency permittivity, respectively. In the time domain, a complex susceptibility function can be defined P P (−αp−iβp)t χˆ (t)= χˆp (t)= iγpe u (t) (A.8) p=1 p=1 X X where 2 2 βp = ωp − αp (A.9a) 2q ω Gp (ǫs − ǫ∞) γ = p (A.9b) p 2 2 ωp − αp and p P Gp = 1 (A.10) p=1 X so that the actual susceptibility function is χ (t) = ℜe (ˆχ (t)). The electric flux is related to electric field via D (t)= ǫ0ǫ∞E + ǫ0χ (t) ∗ E (t) (A.11) so that, using the dispersive model given in the above, we obtain P D (t)= ǫ0ǫ∞E + ǫ0 ℜe [ˆχp (t) ∗ E (t)] (A.12) p=1 X 102 where t χˆp (t) ∗ E = E (t − τ)χ ˆp (τ) dτ (A.13) Z0 In the recursive convolution approach it is assumed that E (t − τ) varies linearly over each time interval. This approach is called piecewise-linear recursive convolution (PLRC). Using this approach, we discretize electric flux, when t = l∆t, as P l l l D = ǫ0ǫ∞E + ǫ0 ℜe Q (A.14) p=1 X where t l∆t l Q =χ ˆp (t) ∗ E = E (t − τ)χ ˆp (τ) dτ = E (l∆t − τ)χ ˆp (τ) dτ (A.15) Z0 Z0 Using the PLRC approximation, El−m−1 − El−m E (t − τ)= El−m + (τ − m∆t) (A.16) ∆t and substituting E (l∆t − τ) in the integral, we get l∆t l−1 (m+1)∆t l−m E (l∆t − τ)χ ˆp (τ)dτ = E χˆp (τ) dτ 0 m=0 " m∆t Z X Z El−m−1 − El−m (m+1)∆t + (τ − m∆t)χ ˆ (τ) dτ ∆t p Zm∆t # (A.17) We simplify above as l∆t l−1 l−m m l−m−1 l−m m E (l∆t − τ)χ ˆp (τ)dτ = E χˆp + E − E ςˆp (A.18) Z0 m=0 X 103 where we have defined (m+1)∆t m m χˆp = χˆp (τ) dτ (A.19) Zm∆t 1 (m+1)∆t ςˆm = (τ − m∆t)χ ˆm (τ) dτ (A.20) p ∆t p Zm∆t Using previously given dispersion model, above expressions are written explicitly (m+1)∆t iγ − − χˆm = χˆm (τ) dτ = p 1 − e (αp+iβp)∆t e (αp+iβp)m∆t (A.21) p p α + iβ Zm∆t p p (m+1)∆t m m ςˆp = χˆp (τ)(τ − m∆t) dτ m∆t Z 2 (A.22) iγp 1 −(αp+iβp)∆t −(αp+iβp)m∆t = 1 − [(αp + iβp)∆t + 1] e e αp + iβp ∆t with the desired recursive property m+1 m −(αp+iβp)∆t m m χˆp =χ ˆp e ⇒ χp = ℜe χˆp (A.23a) m+1 m −(αp+iβp)∆t m m ςˆp =ς ˆp e ⇒ ςp = ℜe ςˆp (A.23b) We next define ∆t iγ − χˆ0 = χˆ (τ) dτ = p 1 − e (αp+iβp)∆t (A.24) p p α + iβ Z0 p p ∆t 0 iγp −(αp+iβp)∆t ςˆ = χˆp (τ) dτ = 1 − (1+∆t (αp + iβp)) e (A.25) p ∆t (α + iβ )2 Z0 p p Using the definitions 104 l∆t l−1 − − − − − l l m 0 l m 1 l m ˆ0 (αp+iβp)m∆t Qp = E (l∆t − τ)χ ˆp (τ)= E χˆp + E − E ςp e 0 Z m=0 h i X (A.26) and reorganizing above as l−1 l 0 0 l−m 0 l−m−1 −(αp+iβp)m∆t Qp = χˆp − ςˆp E +ς ˆp E e (A.27) m=0 X l l the same can be also written recursively. Next, substituting Qp into D , we obtain P P P l 0 0 l 0 l−1 l−1 −(αp+iβp)∆t D = ǫ0ǫ∞E+ǫ0 ℜ χˆp − ςˆp E + ℜ ςˆp E + ℜ Qp e p=1 p=1 p=1 ! X X X (A.28) P P l 0 0 l 0 l−1 D = ǫ0 ǫ∞ + ℜ χˆp − ςˆp E + ǫ0 ℜ ςˆp E " p=1 # " p=1 # X X (A.29) P l−1 −(αp+iβp)∆t + ǫ0 ℜ Qp e " p=1 # X We then simplify above as l l l−1 l−1 D = ǫ0 λ0E + λ1E + P (A.30) where P 0 0 λ0 = ǫ∞ + ℜ χˆp − ςˆp (A.31) p=1 X P 0 λ1 = ℜ ςˆp (A.32) p=1 X 105 P l−1 l−1 −iωˆp∆t P = ℜ Qp e (A.33) p=1 X and ωˆp = βp − iαp (A.34) Next, we are ready to derive the time update scheme for Maxwell’s equations. From Eq. 5, we have l+1/2 l−1/2 −ax Bsx − Bsx − Ω Bl+1/2 + a σ Hl+1/2 + σ Ω Fml−1/2 = ∂ x × El ∆t x sx x m sx m x sx x (A.35a) l+1 l ax D − D sx sx + Ω Dl+1 + a σ El+1 + σ Ω Fel = ∂ x × Hl+1/2 (A.35b) ∆t x sx x e sx e x sx x t t where Fe (t) = 0 E (t) dτ and Fm (t) = 0 H (t) dτ. Rearranging above for time stepping, we haveR R l+1/2 l+1 l l−1/2 l−1/2 − (ax + Ωx∆t) Bsx +axσm∆tHsx =∆t ∂x x × E −axBsx −σmΩx∆tFmsx (A.36a) l+1 l+1 l+1/2 l l (ax + Ωx∆t) Dsx + axσe∆tEsx =∆t ∂x x × H + axDsx − σeΩx∆tFesx (A.36b) l+1/2 l+1/2 l+1 l+1 Since Bsx and Hsx and similarly Dsx and Esx are collocated in time, we can rearrange above again for convenience, using Eq. A.30, as l+1/2 l l−1/2 l−1/2 [− (ax + Ωx∆t) λ0µ0 + axσm∆t] Hsx =∆t ∂x x × E − axBsx − σmΩx∆tFmsx l−1/2 l−1/2 +(ax + Ωx∆t) µ0 λ1Hsx + Pmsx (A.37a) l+1 l+1/2 l l [(ax + Ωx∆t) λ0ǫ0 + axσe∆t] Esx =∆t ∂x x × H + axDsx − σeΩx∆tFesx l l − (ax + Ωx∆t) ǫ0 λ1Esx + Pesx (A.37b) 106 where other pertinent quantities are updated as l−1/2 l−1/2 l−3/2 l−3/2 Bsx = µ0 λ0Hsx + λ1Hsx + Pmsx (A.38a) l l l−1 l−1 Dsx = ǫ0 λ0Esx + λ1Esx + Pesx (A.38b) − − 1 − − Fml 1/2 = Fml 3/2 + Hl 1/2 + Hl 3/2 ∆t (A.38c) sx sx 2 sx sx 1 Fel = Fel−1 + El + El−1 ∆t (A.38d) sx sx 2 sx sx l−1/2 0 0 l−1/2 0 l−3/2 l−3/2 −iωˆ∆t Qmp,sx = χˆp − ςˆp Hsx +ς ˆp Hsx + Qmp,sx e (A.38e) l 0 0 l 0 l−1 l−1 −iωˆ∆t Qep,sx = χˆp − ςˆp Esx +ς ˆp Esx + Qep,sxe (A.38f) P l−1/2 l−1/2 −iωˆ∆t Pmsx = ℜ Qmp,sx e (A.38g) p=1 X h i P l l −iωˆ∆t Pesx = ℜ Qep,sxe (A.38h) p=1 X The above equations complete the PML-PLRC-FDTD update scheme, where the same is repeated for the y and z components. Note that since in the above the fields are split as Bsx = Bsxyyˆ + Bsxzzˆ = µyHsxyyˆ + µzHsxzzˆ (A.39) Dsx = Dsxyyˆ + Dsxzzˆ = ǫyEsxyyˆ + ǫzEsxzzˆ (A.40) in order to fully control all three components of constitutive tensors, the quantities that define dispersive characteristics of the material should be defined separately for each field component that are related to the respective constitutive components. 107 Appendix B: FDTD Method for Dispersive Media in Cylindrical Coordinates B.1 PML-PLRC-FDTD Formulation In many practical applications, we are confronted with the need to solve Maxwell’s equations in geometries with some underlying cylindrical symmetry such as in circu- lar waveguides, fiber optics, borehole problems, etc. A FDTD discretization in a Cartesian grid is less suited for such problems because of staircasing errors. Here, we describe a FDTD algorithm for dispersive media in 3-D cylindrical grids. We again apply the PLRC technique discussed in Appendix A to incorporate dispersion effects. In addition, the PML is implemented through unsplit-field formulation. This implementation follows the approach described in [120]. In an unsplit PML formulation, Maxwell’s equations are written as (e−iwt), iωΛ¯B + σmΛ¯H = ∇ × E (B.1a) −iωΛ¯D + σeΛ¯E = ∇ × H (B.1b) where Λ¯ is given by, s s s s s s Λ¯ =ρ ˆρˆ φ z + φˆφˆ ρ z +z ˆzˆ ρ φ (B.2) sρ sφ sz 108 Eq. B.1 can be rewritten as, iωBa + σmHa = ∇ × E (B.3a) −iωDa + σeEa = ∇ × H (B.3b) using auxiliary fields Ea = Λ¯ · E, Ha = Λ¯ · H, Da = Λ¯ · D and Ba = Λ¯ · B. These auxiliary fields are introduced for computational convenience. Following the PLRC algorithm developed in Appendix A, we obtain m l+1/2 l−1/2 l m l−1/2 l−1/2 ([µ] λ0 − ∆tσm) Ha = Ba − ∆t ∇ × E − [µ] λ1 Ha + Pm,a (B.4a) e l+1 l l+1/2 e l l ([ǫ] λ0 +∆tσe) Ea = Da +∆t ∇ × H − [ǫ] λ1Ea + Pe,a (B.4b) Other pertinent auxiliary equations are updated as l e l e l−1 l Da = [ǫ] λ0Ea + λ1Ea + Pe,a (B.5a) − − − e l 0 0 l 0 l 1 l 1 iω˜p∆t Qe,a,p = χe,p − ξe,p Ea + ξe,p Ea + Qe,a,pe (B.5b) P − e l l iω˜p∆t Pe,a = ℜe Qe,a,pe (B.5c) p=1 X h i and similarly, l−1/2 m l−1/2 m l−3/2 l−1/2 Ba = [µ] λ0 Ha + λ1 Ha + Pm,a (B.6a) − − − − − m l 1/2 0 0 l 1/2 0 l 3/2 l 3/2 iω˜p ∆t Qm,a,p = χm,p − ξm,p Ha + ξm,p Ha + Qm,a,p e (B.6b) P − − − m l 1/2 l 1/2 iω˜p ∆t Pm,a = ℜe Qm,a,p e (B.6c) p=1 X h i where ∆t e iγ − e e 0 p (αp+iβp)∆t χˆe,p = χˆm,p (t) dt = e e 1 − e (B.7) 0 αp + iβp Z h i 109 ∆t e iγp − e e ˆ0 ˆ e e (αp+iβp)∆t ξe,p = ξm,p (t) tdt = 2 1 − αp + iβp ∆t + 1 e 0 ∆t αe + iβe Z p p h i (B.8) similar equations are obtained for magnetic update where the superscript e is replaced with m instead. The update equations, however, are not complete yet. We have to define actual field in terms of auxiliary fields, i.e., for E field for example, −1 E = Λ¯ Ea (B.9) or, in terms of components, sρ Eρ = Ea,ρ (B.10a) sφsz sφ Eφ = Ea,φ (B.10b) sρsz ss Es = Ea,z (B.10c) sρsφ For convenience we introduce another auxiliary field Ee, so that we replace above equations with (iωsz) Ee,ρ =(iωsρ) Ea,ρ (B.11a) (iωsφ) Eρ = iωEe,ρ (B.11b) (iωsρ) Ee,φ =(iωsφ) Ea,φ (B.12a) (iωsz) Eφ = iωEe,φ (B.12b) (iωsφ) Ee,z =(iωsz) Ea,z (B.13a) (iωsρ) Ez = iωEe,z (B.13b) In the time domain, (az∂t + Ωz) Ee,ρ =(aρ∂t + Ωρ) Ea,ρ (B.14a) 110 (bρ∂t +∆ρ) Eρ = ∂tEe,ρ (B.14b) Then the update equation for the above becomes l l−1 l l−1 (az + Ωz∆t) Ee,ρ = azEe,ρ (aρΩρ∆t) Ea,ρ − aρEe,ρ (B.15a) l l−1 l l−1 (bρ∆ρ∆t) Eρ = bρEρ + Ee,ρ − Ee,ρ (B.15b) and similarly for the other components. 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