Terahertz Dielectric Resonator Antennas for High Speed Communication and Sensing: From theory to design and implementation 1839533552, 9781839533556

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IET TELECOMMUNICATIONS SERIES 103

Terahertz Dielectric Resonator Antennas for High Speed Communication and Sensing

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Terahertz Dielectric Resonator Antennas for High Speed Communication and Sensing From theory to design and implementation Rajveer S. Yaduvanshi

The Institution of Engineering and Technology

Published by The Institution of Engineering and Technology, London, United Kingdom The Institution of Engineering and Technology is registered as a Charity in England & Wales (no. 211014) and Scotland (no. SC038698). † The Institution of Engineering and Technology 2022 First published 2021 This publication is copyright under the Berne Convention and the universal Copyright Convention. All rights reserved. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may be reproduced, stored or transmitted, in any form or by any means, only with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries concerning reproduction outside those terms should be sent to the publisher at the undermentioned address: The Institution of Engineering and Technology Michael Faraday House Six Hills Way, Stevenage Herts, SG1 2AY, United Kingdom www.theiet.org While the authors and publisher believe that the information and guidance given in this work are correct, all parties must rely upon their own skill and judgement when making use of them. Neither the author nor publisher assumes any liability to anyone for any loss or damage caused by any error or omission in the work, whether such an error or omission is the result of negligence or any other cause. Any and all such liability is disclaimed. The moral rights of the author to be identified as author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing in Publication Data A catalogue record for this product is available from the British Library

ISBN 978-1-83953-355-6 (hardback) ISBN 978-1-83953-356-3 (PDF)

Typeset in India by MPS Limited Printed in the UK by CPI Group (UK) Ltd, Croydon

Contents

About the author Preface

xi xiii

1 Dielectric resonator antennas (DRAs) and its synthesis 1.1 Introduction 1.2 CDRA (cylindrical DRA): design and modeling using silicon-radiating element 1.3 Terahertz or quantum devices characteristics 1.3.1 Theory of TDRA 1.3.2 Terahertz DRA or quantum DRA near fields/far fields 1.3.3 Radiation parameters 1.3.4 Drude’s model theory 1.4 Terahertz MIMO DRA parameters 1.4.1 Microwave DRAs vs optical DRA parameters 1.4.2 Optical DRAs 1.4.3 Radiated fields 1.5 Main functions of terahertz DRA 1.5.1 Some important parameters of microwave and terahertz DRA 1.6 THz DRA model design parameters 1.7 Rectangular nano-DRA design parameters 1.7.1 Design steps 1.8 Conclusion References

1 2

15 16 17 17 18 19

2 Dielectric resonator antennas—a comprehensive review 2.1 Introduction 2.2 Propagation of light 2.3 Design of a terahertz dielectric resonator antenna 2.4 Fabrication and testing 2.5 Terahertz antenna far-field radiations: flowchart 2.6 Mathematical analysis of terahertz RDRA 2.7 Approximate analysis of a rectangular quantum antenna 2.8 Terahertz DRA simulation results 2.9 Conclusion References

25 25 29 32 33 33 34 38 40 40 40

4 6 6 10 11 11 12 13 13 14 14

viii 3

4

Terahertz dielectric resonator antennas Light–matter interaction in terahertz dielectric resonator antennas (DRA) 3.1 Introduction 3.2 Light–matter interaction theory in a quantum antenna 3.3 Theory of quantum entanglement 3.4 Conclusion Reference

43 43 44 47 51 51

Terahertz dielectric resonator antennas design and modeling 4.1 Introduction to terahertz DRA 4.2 Mathematical formulations used to describe working of quantum DRA 4.3 Cylindrical terahertz DRA 4.4 Conical terahertz DRA 4.5 Conclusion References

58 60 62 62 63

5

Surface plasmon polytrons (SPP) into terahertz DRA 5.1 Introduction 5.2 Working principle of TDRA 5.3 Terahertz CDRA design and simulations 5.4 Terahertz DRA main features 5.5 Mathematical formulations used in TDRA 5.6 Terahertz DRA applications 5.7 Conclusion References

65 65 66 68 69 74 77 77 77

6

Terahertz conical dielectric resonator antenna—design, simulation and implementations 6.1 Introduction 6.2 Design structure of conical THz DRAs 6.3 Model-1 multiband conical TDRA 6.4 Mathematical modeling of terahertz conical DRA 6.5 Equivalent electrical circuit of conical terahertz DRA 6.6 Conclusion References

81 81 83 84 90 96 98 98

7

Cylindrical terahertz and optical DRA—design and analysis 7.1 Introduction 7.2 Model 2 TCDRA at 10-THz resonant frequency 7.2.1 Design computations 7.3 Terahertz antennas detailed description 7.4 Theory of terahertz cylindrical DRA and mathematical formulations

53 53

101 101 102 102 104 106

Contents 7.5 Optical CDRA description 7.6 Conclusion References 8 Spherical terahertz and optical DRA—design and implementations 8.1 Introduction 8.2 Design of terahertz spherical DRA at 511 THz 8.3 Mathematical formulations of terahertz spherical DRA 8.4 Results and discussions 8.4.1 Super directivity in spherical DRA 8.5 MIMO (multi-input–multi-output) spherical DRA 8.6 Conclusion References 9 Rectangular terahertz DRA—design, simulation and implementations 9.1 Introduction 9.2 Propagation of light 9.3 Design and simulation of terahertz dielectric resonator antenna 9.4 Synthesis of a terahertz rectangular DRA at optical frequency and its radiation theory 9.5 Mathematical analysis of resonant modes excited into a terahertz rectangular DRA 9.6 Terahertz optical RDRA at 484 THz 9.6.1 Approximate analysis of a rectangular terahertz DRA and its controlled electromagnetic fields 9.7 Conclusion References 10 Equivalent circuit analysis on terahertz and optical dielectric resonator antennas (DRAs) 10.1 Introduction 10.2 Quantum DRA-equivalent circuit mathematical analysis for mixed circuits 10.2.1 Impedance (Zin) 10.2.2 The frequency-dependent resistance is also called dynamic resistance of the circuit 10.2.3 Two resonant modes, i.e. fundamental and higher order 10.2.4 Second resonant mode 10.3 Higher order resonant modes 10.4 Bandwidth (BW) of terahertz DRA 10.5 Simulated results based on MATLAB 10.6 Design development and evaluation of NDRA 10.6.1 Resonant frequency of TRDRA formulations

ix 114 132 133 135 135 143 143 148 148 153 156 158

161 161 166 170 170 171 176 176 188 188

191 191 194 194 196 197 198 200 202 202 202 202

x

Terahertz dielectric resonator antennas 10.7 Synthesis of NDRA radiation theory 10.8 Drude’s model 10.9 MATLAB program 10.10 Conclusion References

202 210 210 211 211

11 Optical DRA for retinal applications—next generation DRAs 11.1 Introduction 11.2 Optical antenna arrays basic requirements 11.3 Optical antenna design 11.4 Entanglement 11.5 Modeling of optical antennas 11.6 Light–matter interaction 11.7 Theory of coupled resonant modes 11.8 Designs of terahertz DRAs simulation results for various shapes 11.9 Conclusion and applications References

215 215 219 223 224 224 225 227 229 229 229

12 Conclusion and futuristic vision 12.1 Introduction 12.2 Patient-centric healthcare system outline 12.3 Thumb DRA sensors integrated with patient-centric healthcare system 12.4 Thumb DRA design and implementations 12.5 Conclusion

231 232 233

Appendix A: Case studies Appendix B: Terahertz absorbers Appendix C: Antenna measured values in anechoic chamber Appendix D: Dielectric materials and resources Appendix E: Dual-band graphene antenna design and implementation Appendix F: Miniaturization design techniques Appendix G: Gaussian beam feed process Appendix H: Silicon dielectric resonator antenna at 5-THz frequency Appendix I: DRA designing process Appendix J: DRA design case study Appendix K: Vector network analyzer process for calibration Glossary Index

241 247 251 341 347 353 359 373 377 381 389 391 393

233 234 239

About the author

Rajveer S. Yaduvanshi is a professor in the Department of Electronics & Communication Engineering at Netaji Subhas University of Technology, Sec-3, Dwarka, New Delhi, India. He has more than 35 years of experience in the fields of teaching and research in radio frequency and microwave engineering. He has written 3 books on the topic of antennas and published more than 107 research papers. He has organized the international conference ICPR-2017. His research interests involve quantum antennas, terahertz dielectric resonator antennas, nano dielectric resonator antenna design, optical antennas, radio frequency sensors, microwave devices, millimeters wave (mm-wave) antennas, vehicular antennas and retinal quantum antennas. He has written three books on the topic of antennas and organized the international conference ICPR-2017 in Delhi. He holds a PhD from Delhi University in Electronics and Communication Engineering. He has served as Dean R&D at East Campus. He was selected as Director EQDC (Electronics & Quality Development Centre) by Govt. of Gujarat. Prof. Rajveer Yaduvanshi received Academic Excellence awards and Excellence in Management awards by Delhi Government, India. He has supervised seven PhD theses and thirty-three PG theses. Seven more PhD scholars are currently registered under him. His current research interests are nano DRA, sensors, retinal photoreceptors and photonics.

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Preface

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This book covers microwave and terahertz dielectric resonator antenna (DRA) technologies and their applications to high-speed communications and sensing. Microwave, terahertz and optical DRAs use ceramic/silicon/TiO2/teflon and graphene materials. These DRAs are efficient and used for high-speed communication due to their wide bandwidth and less lossy property. They can perfectly work at microwave, terahertz and optical spectrum regime. These are compact in size that makes them very well suited for advanced applications in sensing, scanning, imaging and biomedical applications. They have futuristic capability to design optical DRAs in visible frequencies, so that DRAs can be used as implant antennas for retinal photoreceptors. The authors introduced new DRA geometries such as conical optical DRAs, cylindrical optical DRAs and rectangular optical DRAs, which can provide ultrafast data-transfer rate due to large bandwidth and multimode operations. Optical DRAs are useful for devices such as sensors (LiDAR) for scanning and imaging. Spherical optical DRAs have features of super directivity that can be used in quantum radars. Cylindrical optical DRAs with photodiodes can be used for wireless energy harvesting, i.e. rectennas. Conical optical DRAs can be used at basic R, G, B color wavelengths. Optical DRAs of rectangular shape, conical shape, cylindrical shape and spherical shape and structures if embedded with photodiodes can be used for eye retinal prosthesis applications.

The authors explained thoroughly the differences among microwave DRAs, terahertz DRAs and other optical DRAs. ●

●

A mathematical study of rectangular, cylindrical, conical and spherical microwave, terahertz and optical DRAs has been developed. The concepts on Maxwell’s and Schro¨dinger equations and Dirac equations have been worked out for far-field equations. The resonant frequency and plasmon frequency, light–matter interaction solutions have been presented with computational capabilities. A theoretical study of directivity (super directivity), quality factor, radiated quantum fields, radiation pattern and quantum entanglement features of quantum DRAs has also been included.

xiv ●

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Terahertz dielectric resonator antennas The miniaturization techniques of optical DRAs using higher order modes and metal or graphene crafting are included. A mathematical description of optical DRAs-equivalent circuits and quality factor and impedance equations has been well defined by using dynamic impedance terms. Design and simulations of RF absorbers are to be used for renewable energy harvesting and absorption of unwanted RF signals in chip-to-chip high-speed communication and sensing applications. Absorbers have special applications in MIMO designs to improve isolation. The mechanisms of quantum eye gage optics for estimating exact focal point of the images have been derived. This book will help one to solve the following problems:

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Design and implementation of microwave/terahertz/optical DRAs useful for biomedical applications. Special geometries DRAs to improve directivity and other radiation parameters tuning. The high-speed communication compact RF devices design techniques along with bandwidth and polarization tuning. Terahertz designs of DRAs will have large bandwidth for 5G and beyond applications to fetch high data rates. LiDAR, radar, satellite communication and wireless personal communication antennas using various dielectric, semiconductor, graphene PDMS and polymer materials. Design and implementation of terahertz/optical/microwave DRAs for selfdriving autonomous cars, sensor and scanners for noninvasive biomedical applications. Design and implementations of small size terahertz/optical DRAs for chip-tochip communications in SOC (system on chip). DRAs for use as photodetectors. Analysis of equivalent circuits at microwave, terahertz and quantum devices.

Case studies with prototype models have been included at microwave and mmwave frequencies. This book covers new geometry DRAs at a terahertz and optical spectrum. This book will be very useful to Industries, academia, research scholars, engineers, scientists, PG and UG students working in the fields of electronics, microwave, communication, sensing, radars, satellites and remote sensing, navigation sensing, healthcare sensors, smart building sensing and renewable energy harvesting, etc. Prof. Rajveer S. Yaduvanshi, PhD NSUT, Dwarka, New Delhi 110078

Chapter 1

Dielectric resonator antennas (DRAs) and its synthesis

Abstract Dielectric resonator antenna (DRA) has high efficiency and low losses at high frequencies. Terahertz (THz) and optical DRA are new trend topics of research in millimeter wave (mm wave), optics and photonics. They are simple in design, highly efficient, built with high dielectric constant ceramics and polymers materials (er ¼ 10–1,600), have design flexibility, easy to fabricate and stable operations at high temperatures. DRAs have excellent features of working throughout the frequency spectrum, right from microwave to the optical regime. Light (photons) are bosons and metal (electron–positrons) at terahertz frequency, metal acts as gaseous (plasma) at terahertz frequencies; hence it is termed fermions. The plasmon frequency gets coupled to terahertz DRA (TDRA) in proximity feed thus creates radiations in TDRA at its resonant frequency. Bosons and fermions scattering and transportation mechanism can be solved by giving quantum-mechanical treatment to these TDRAs. Dirac and Maxwell’s equations have used to get desired solution using creation and annihilation operators theory. Electromagnetic (e.m.) far fields are solved as desired outcome. The radiation pattern is based on input to feed, current density fluctuations and retardation potentials. The fluctuations in quantum antennas must be controlled to minimum threshold value. The radiated field is a state of jointly coherent for the bosons (i.e. photons) and for the fermions (positrons–electrons). These are basic concept of radiations into DRAs at terahertz and optical frequencies. Hence, interaction between these bosonic and fermionic fields causes the e.m. field to change and induce surface currents that radiate out in space. This change in e.m. fields must be optimized to a threshold value. Light-imaging detection and ranging operates at 200-THz frequency. Polymer, ceramics and polymer composites can be used at mm wave or terahertz and optical frequencies. TDRA can be built with low-quality factor materials, and absorbers are designed using materials having high-quality factor materials. In TDRAs, input is LASER, it interacts with noble metals such as gold or silver, light–matter interaction takes place resulting into surface plasmon resonance. The silver nano waveguides have been used to provide feed mechanism to TDRAs of different

2

Terahertz dielectric resonator antennas

shapes and geometries, i.e. cylindrical, rectangular, spherical and conical shape DRAs. LASER input interacts with silver metal and dielectric SiO2 substrate. Hence, in TDRA, surface plasmon polytron waves are generated due to light– matter interaction, these will give rise to plasmon frequency generation. This plasmon frequency is always kept lesser than LASER input frequency, so as to enable it to propagate in forward direction. Here, we propose a TDRA fully integrated with a photonic crystal waveguide for broadside radiation pattern.

1.1 Introduction Microwave dielectric resonator antennas (DRAs) have been in use due to its highperformance parameters, i-phone 12 mobile phone device is built by millimeterwave (mm-wave) technology for high-speed communication. DRAs are efficient and stable against temperature variations. DRA has design flexibility for choosing different aspect ratios. Hence, DRAs have capability to excite applicationspecific resonant modes. These modes can be operated in a controlled manner such as merged modes, multimodes (harmonics) and single operating mode as dominant mode. Terahertz devices will be bridging the gap between electronics and photonics. The noninvasive THz imaging has become an effective tool of application in many industrial and research sectors. Gas sensors are new devices at optical frequencies for monitoring environmental parameters such as sulfur, carbon mono oxide, oxygen and nitrogen contents in the environment. These parameters can be monitored on real-time basis. Microfluidic and biosensing are new applications for terahertz DRAs (TDRAs). Wireless sensors and Internet of Things (IoT) are two main areas of applications for TDRAs. Input is given by LASER to nano waveguide that becomes proximity feed to DRA [1–9]. The surface plasmon polytron (SPP) phenomenon takes place at metal dielectric interface, when laser is connected at the edge of nano waveguide. In terahertz antennas, we have an expression for the are interaction Hamiltonian between the electromagnetic (e.m.) fields of photons and the Dirac fields of electrons and positrons comparing it with bosonic and fermionic operators and these canonical communication rules (CCR) of the bosonic operators and the canonical anticommutation relations (CAR) of the fermionic operators can be used to calculate all the matrix elements of interaction Hamiltonian and all the quantum-statistical properties of the field formed. This interaction Hamiltonian may also be used to calculate radiation pattern correction to the Dirac wave function. Thus, the Dirac current field in turn manifests itself as correction to the far-field e.m. field radiation pattern, thus, correction to the mean-square fluctuation-radiated quantum e.m. fields in any state [10–19]. It defines a trilinear form in the electron– positron–photon creation–annihilation operators based on the theory of the Dyson series. Feynman diagrams are applied to compute or estimate radiated quantum e. m. fields. The light is in fact a quantum e.m. field, when it falls on material cavity, it starts immediately interacting with the matter within the cavity [19–37]. A coarse

Dielectric resonator antennas (DRAs) and its synthesis

3

approximation of this matter is to regard as a sea of electrons and positrons and then apply the formalism of quantum electrodynamics to describe its interaction with the incident photons, both before the light hits the cavity matter directed based on boundary conditions. Bosonic creation and annihilation operators satisfy the CCR [38–42]. If the field is fermionic, they will satisfy CARs. Finally, we shall be able to compute the mean and covariance of the e.m. field in a state. Hence, due to Planck’s constant in quantum antennas, radiation energy comes in quanta. The mean-square fluctuations are at least of the order of magnitude of Planck’s constant. The frequency bands in the range of 275–3,000 GHz, which are known as terahertz communication bands, are useful for highspeed communication [43–47]. Hence, it can be used for wireless communication in 5G and beyond or IoT for high data rate transmission. Nano-photonics is a branch of science that deals with transmission and reception of signals at terahertz, optical frequencies and travel in nanometer dimensions devices such as silver nano waveguide. DRAs have been developed at nanotechnology due to high efficiency, stability and less loss. A nano antenna has two important features, such as high radiation efficiency and super directivity [48–59]. Basic working principle of a quantum or TDRAs (cavity resonator antenna) of different shapes enclosing a terahertz or quantum e.m. field comprising photons interacting with electrons and positrons using the canonical quantization method of Feynman, Schwinger, Tomonaga and Dyson. The quantum e.m. field is regarded as an ensemble of quantum harmonic oscillators, whose Hamiltonian is treated as a quadratic form in the creation and annihilation operator fields and the matter field within the cavity in which it interacts is regarded as an ensemble of electrons and positrons modeled using the second quantized Dirac Hamiltonian. The interaction between these bosonic and fermionic fields causes the e.m. field to change and induce surface currents which radiate out in space. The statistics of this radiated field, namely that of quantum fluctuations in this radiated field in a given by state of the electron–positron–photon field is analyzed [58,59]. This quantum field theoretical analyses of the cone, rectangular, spherical and cylindrical cavities have been solved in this book. An example of TDRA design and modeling is given in Figure 1.1. TDRA dimension and volume is selected based on operating frequencies. Basic physics of working of DRA differs in microwave and optical region as metal acts as gaseous state in optical or quantum frequencies. The feeding mechanism of the TDRA consists of the silver (Ag) nanostrip feed in the place of microstrip feed. The Gaussian pulse excitation is given to nano-feed line for proper phase matching in the place of wave port signal. As the nano waveguide is inserted into substrate in place surface, an interface is created by metal and dielectric substrate. This interface acts as channel to carry SPPs waves. This is plasmon frequency, which depends on dielectric properties of materials for forward propagations. Hence, its configuration is wavelength dependent. Dispersion characteristics are given by Drude’s theory [1–7].

4

Terahertz dielectric resonator antennas

h Feed line

h2 d s

DR

Substrate (a)

Feed line

Path-2 x z

Path-1 Excitation O/4

h3 h1

(b)

(c)

Figure 1.1 CDRA design with input excitation of Gaussian laser beam to silver nano waveguide of TDRA: (a) DR element with feed and substrate— top view, (b) DR with inserted nano wave—side view guide, (c) nano waveguide fields—inner view

1.2 CDRA (cylindrical DRA): design and modeling using silicon-radiating element The TDRA, cylindrical DRA (CDRA) of the silicon (Si) with relative permittivity
r ¼ 11:56 is placed at the top of the substrate, dimensions diameter (dÞ ¼ 0:510 mm; height ðhÞ ¼ 0:325 mm. The proposed CDRA is excited using a rectangular nano waveguide of width w with the stub of lengths. The value of the stub length is s ¼ 0:375 mm for impedance match, 3-dB AR bandwidth is fine here. The dispersive properties h of silver
nano waveguide i can be described based on Drude’s model as
Ag ¼
o
1  fp2 =ff ðf þ igÞg

. Here,
1 ¼ 5; fp ¼ 2;175 THz and

g=2p ¼ 4:35 THz, f is the operating frequency,
Ag is the permittivity of silver, fp is the plasma frequency, g is the collision frequency and it is inverse of the relaxation time,
1 is the offset of real part of the dielectric constant and
o ¼ 8:85  1012 F=m. These E-fields tell us about excited states of TDRA, i.e. resonant modes. The numbers of half-wave variation in x, y, z directions tell us that fields are propagating in particular direction. These are state of short magnetic dipoles or short electric dipoles. The E-fields and H-fields provide physical insight of TDRA. These are shown in Figures 1.2–1.4. Nanotechnology is multidisciplinary area of science, merges physics, chemistry, material, biology, medicine and engineering branches for innovations on current trends of optical DRAs at nano regime. The physics of optical antennas involves a dominance behavior of skin effects, SPPs and light–matter interaction (creation and annihilation). It is well known that light can penetrate through metallic walls at optical frequencies. Light–matter interaction at optical domain is studied to design quantum devices [11,55–57].

Dielectric resonator antennas (DRAs) and its synthesis

Figure 1.2 TDRA feed to generate SPP waves and E-fields

Figure 1.3 Surface currents in TDRA

5

6

Terahertz dielectric resonator antennas

Figure 1.4 Radiated fields in TDRA

1.3 Terahertz or quantum devices characteristics 1.3.1

Theory of TDRA

In TDRA, there can be zero, one or more than one particle having a specified momentum and helicity. In the case of fermionic fields like the second quantized Dirac field, we cannot have more than one particle having a definitive value of momentum and spin. Complete frequency spectrum is shown in frequency in Figure 1.5. If, aðk Þ ; aðk Þ; k ¼ 1; 2; . . . are canonical bosonic creation and annihilation operators, they satisfy the CCR ½aðmÞ ; aðk Þ ¼ d½k  m; ½aðk Þ; aðmÞ ¼ ½aðk Þ ; aðmÞ  ¼ 0, while if cðk Þ ; cðk Þ; k ¼ 1; 2; . . . are canonical fermionic

Dielectric resonator antennas (DRAs) and its synthesis Photonics

Electronics 10 cm

Radio waves 106

108

10 Pm

1 mm Microwave 1010

THz Gap

Infrared

1012 Terahertz Region

100 nm Visible light

Wavelength 10 m

7

1014

10 mm

Ultraviolet

X-Ray

1016

1018

10 fm Gamma Rays 1020 Frequency

Figure 1.5 Terahertz spectrum is vacant for high-speed communications creation and annihilation operators, they satisfy the CAR ½cðk Þ; cðmÞ þ ¼ d½k  m; ½cðk Þ; cðmÞþ ¼ ½cðk Þ ; cðmÞ þ ¼ 0: Here, ½A; B ¼ AB  BA; ½A; Bþ ¼ AB þ BA. It is clear therefore that cðk Þ2 ¼ 0 ¼ cðk Þ2 : Here, k represents a definite value of momentumpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi and spin/helicity. If j0 > is a vacuum boson state, then ffi aðk1 Þn1    aðkr Þnr = n1 !    nr !j0  jðk1 ; n1 Þ; . . . ; ðkr ; nr Þ > represents a normalized state of the bosonic field in which there are nj bosons having momentum and helicity index kj for each j ¼ 1,2, . . . , r. It is clear then from the CCR that     pffiffiffiffi a kj ðk1 ; n1 Þ; . . . ; ðkr ; nr Þ  nj ðk1 ; n1 Þ; . . . ; jj ; nj  1 ; . . . ; ðkr ; nr Þ >; 1jr

(1.1)

While if k 2 = fk1 ; . . . ; kr g; then aðk Þjðk1 ; n1 Þ; . . . ; ðkr ; nr Þ  0 In short, a(k) annihilation, a boson, having momentum–helicity k and if there is no such boson with momentum–helicity k in the state, then a(k) annihilates the entire states giving zero. Likewise, a(k)* creates a boson having momentum– helicity k:      pffiffiffiffi a kj ðk1 ; n1 Þ; . . . ; ðkr ; nr Þ  nj ðk1 ; n1 Þ; . . . ; kj ; nj þ 1 ; . . . ; ðkr ; nr Þ >; 1  k  r; = fk1 ; . . . ; kr g aðk Þ jðk1 ; n1 Þ; . . . ; ðkr ; nr Þ jðk1 ; n1 Þ; . . . ; ðkr ; nr Þ; . . . ; ðk; 1Þ >; k 2 It can be verified that these two rules are in agreement with the CCR. From these CCR, it is evident that N ðk Þ ¼ aðk Þaðk Þ is the number operator, i.e. when acting on a state, it gives the number of bosons in that state:  ! r   X    d k  kj nj ðk1 ; n1 Þ; . . . ; ðkr ; nr Þ > N ðk Þðk1 ; n1 Þ; . . . ; ðkr ; nr Þ    j¼1 The state described before are called the occupation number states or simply the number sates. They are Eigen states of the number operators but not of the

8

Terahertz dielectric resonator antennas

creation and annihilation operators. The second quantized bosonic field is a superposition of the annihilation and creation operators: X ~ ðx Þ fðxÞ ¼ ½aðk Þck ðxÞ þ aðxÞ c k

where satisfy the classical bosonic field equation Lck ðxÞ ¼ 0; x ¼ ðt; rÞ with, for example

 1 @m @ jmu þ m2 L¼ 2 In the Klein–Gordon case, or in e.m. field case where the bosons are photon having zero mass, L is the wave operator:

 1 @m @ jmu L¼ 2 (Klein–Gordon particles have zero spin while the photon has spin 1 with, helicities 1; the zero helicity not being allowed. This corresponds to the fact that any state of photon polarization can be expressed as a superposition of left and right circularly polarized states.) It follows that to get a state definite field amplitude for positive frequencies, the state should be an Eigen state of the annihilation operators a(k) and such a state called a coherent state can be obtained (in the discrete momentum–helicity setting) as

 

PQ 1 P uðk Þnk aðk Þnk ∅ðuÞ ¼ exp  j0 > fnk ; k 2 I gjuðk Þj2 k2I nk ! 2 !

X XY pffiffiffiffiffiffi  1 ¼ exp  uðk Þnk = nk ! nj ; j 2 I > juðk Þj2 k 2 k
I k The second quantized Dirac fermionic field can be expressed on the other hand in terms of fermionic creation and annihilation operators. In the photon case, the antiparticle of a photon is again a photon and so its creation operator at a given momentum–helicity is the adjoint of the corresponding annihilation operator. On the other hand, the antiparticle of the electron is another particle, a positron, and hence the Dirac field should be expressed as a superposition of electron annihilation operators and positron creation operators. The creation of a positron of the positive energy is according to Dirac equivalent to annihilation an electron of negative energy. Thus, the Dirac field is expressed as the superposition X ½bðk Þck ðxÞ þ cðxÞ hk ðxÞ y ðx Þ ¼ k

Dielectric resonator antennas (DRAs) and its synthesis

9

where b(k) annihilation, an electron, with momentum–spin k and c(k) annihilations, a positron, with momentum–spin k. Equivalently, cðxÞ creates a positron with momentum–spin k. ck ðxÞ and hk ðxÞ are solutions to the free Dirac equation:  m  igm @m  mck ðxÞ ¼ 0; ig @m  m hk ðxÞ ¼ 0 Given a quantum matter field hðxÞ, either bosonic or fermionic, it satisfies a classical wave equation of the Klein–Gordon type, or the 3D wave type or the Dirac type with certain boundary conditions. This second quantized field can therefore be expressed as superposition of Eigen functions corresponding to the boundary condition with coefficients being particle creation and annihilation operators. If the field is bosonic, these creation and annihilation operators will satisfy CCRs while if the field is fermionic, they will satisfy CARs. If H corresponds to the Hamiltonian of the first quantized theory, then we can write the second quantized Humiliation of field as ð hðxÞ HhðxÞd 3 x This is a function of an infinite number of particle operators acting in a bosonic or fermionic Fock space. In optical DRA or TDRA, SPP phenomenon occurs due to strong light–matter interactions. A simplified analysis would be to model the matter as just the second quantized electron–positron field using Dirac’s relativistic wave equation. In this process, wave function is an operator field, and then to include the photon interaction term in the usual way. In this way, the total Hamiltonian of the photon and matter field splits into three terms as follows: 1. 2.

3.

One, the Hamiltonian of the e.m. field described as a quadratic form in the photon creation and annihilation operators within the cavity. Two, the Hamiltonian of the Dirac field with cylindrical cavity (in CDRA) boundary conditions on the wave function described as a quadratic form in the electron–positron creation and annihilation operators. Three, the Hamiltonian of the interaction between the photon and electron– positron field described as a quadratic form in the electron–positron creation– annihilation operators multiplied with a linear form in the photon creation– annihilation operators.

This interaction Hamiltonian can be used to compute the amplitudes for scattering, but since we are primarily interested in the statistics of the e.m. wave field pattern, we shall describe this interaction using the Dirac current density expressed as a quadratic form in the electron–positron creation–annihilation operator fields that drive the photon field using the wave equation for the e.m. field in the presence of a current density. The previous quantum theory can be applied to understand the concept of quantum physics as stated later: the cavity can be modeled as cylindrical cavity or conical shape cavity or spherical cavity.

10

Terahertz dielectric resonator antennas

Since light is in fact a quantum e.m. field, when it falls on cavity, it starts immediately interacting with the matter within the cavity. A coarse approximation of this matter is to regard as a sea of electrons and positrons and then apply the formalism of quantum electrodynamics to describe its interaction with the incident photons, both before the light hits the cavity matter directed based on boundary conditions. The field in the cavity is actually a planar quantum e.m. field in certain state and we wish to calculate the probability that such a field will be formed.

1.3.2

Terahertz DRA or quantum DRA near fields/far fields

To perform cavity quantum e.m. field computation, we must solve the joint Maxwell–Dirac equations within the cavity with the appropriate boundary conditions corresponding to what shape of the cavity we are using. The cavity shape can be cone, cylindrical or spherical. This means that we expand the Maxwell field in terms of basis functions that vanish on the boundary and then the coefficients in this expansion will be the photon creation and annihilation operator fields. Likewise, we must expand the Dirac field within the cavity using the same orthogonal basis functions whose coefficients will now be the electron–positron creation and annihilation operator fields. The resulting Hamiltonian of these fields will then comprise three components: one, the Hamiltonian of the free photon field expressed as a quadratic form in the photon creation and annihilation operators, two the Hamiltonian of the free electron–positron field expressed as a quadratic form in the electron–positron creation and annihilation operators and finally three, the interaction Hamiltonian between the electron–positron field and the photon field. It is this last interaction term that is of importance in scattering theory of light and the formation of fields. Using the interaction picture of Dyson, we can expand the solution to the Schro¨dinger evolution operator for this quantum system as a Dyson series in powers of the interaction Hamiltonian. Then we can, using the technology of Feynman diagrams, calculate various terms of the scattering matrix to determine the probability amplitude of obtaining a certain image field on the retinal screen. The basic physics behind this computation does not depend on what shape we choose for the cavity. No matter what shape we choose, we can in principle determine a complete orthonormal basis of test functions that satisfy the appropriate boundary conditions for the given cavity and then expand the Maxwell and the Dirac field in terms of these test functions. Before the interaction between light and matter, the system was in an initial state and after this interaction, including the interaction of light with the matter, the state of the whole system changes with time and this change can be determined by the action of the unitary evolution operator on the initial state with the unitary evolution operator constructed from the total Hamiltonian of the interacting electrons, positrons and photons using the Dyson series. The final state of this system is then equivalently determined once we know the matrix elements of the evolution operator between any given initial state and a final state of the particles. From this final state, the probability of getting a given field is computed from the basic rules for quantum probability.

Dielectric resonator antennas (DRAs) and its synthesis

11

Thus, we shall by perturbation theory be able to calculate the change in the e. m. fields pattern caused by the Dirac current in term of operators and then by assuming a definitive state of the photon–electron–positron field, we shall be able to compute the mean and covariance of the e.m. field in this state. Their operating wavelengths are very small, mostly in the nanometer size. Optical antennas work on Drude’s theory and have nature of nonlinearity. Optical antenna have far field with finite values of Planck’s constant.

1.3.3 Radiation parameters The signal-to-noise ratio (SNR), mean-square fluctuation in S11, S12, S21 and S22 in a given state, fluctuation variance from state to state, second quantized current density, mean of current density dependence of state of system, status of photons, electrons and positron builds current density in system at any state, average current with fluctuations develop field pattern, i.e. radiation pattern is defined as theta vs mean square of e.m. fields, as well as theta vs mean value of current, frequency of minimum fluctuations value will be considered important, and state will change if large fluctuation occurs. Hence, optimization by minimizing the mean-square value of fluctuation is a Lagrangian problem. The mean-square fluctuation of power along with SNR is an important term for power in radiated fields. Thus, SNR is an average value of the field square divided by mean-square value of the fluctuating component in terms of Planck’s constant. Quantum field radiations occur due to light–matter interactions. In fact, as Feynman had observed in his path integral approach to quantum mechanics, the quantum amplitude for a particle to go from one space–time point to another is given by a sum over paths of the phase function that is action integral over the given path. The probability of going from the first to the second is then the modulus square of this sum and it contains interference terms between any two paths. On the other hand, in the limit when Planck’s constant h goes to zero, the rapid phase variations between neighboring paths will cancel out except in the vicinity of the classical path where the action integral has zero variation. Thus, in the limit as Planck’s constant goes to zero, the particle will move only along the classical path. Letting Planck’s constant converge to zero is equivalent to making S/h very large, which can equivalently be accomplished by increasing the dimensions of our system so that S becomes large. This means that for large sized objects, only classical mechanics will matter.

1.3.4 Drude’s model theory "

eAg

fp2 ¼ e0 ea þ f ðf þ igÞ

#

where g is the collision frequency, fp is the plasmon frequency, e is the electron charge, T is the temperature, w is the radian frequency, mc is the chemical potential, Г is the scattering rate.

12

Terahertz dielectric resonator antennas

In order of Planck’s constant in quantum antennas, radiation energy comes in quanta. Thus, discrete particle distributions of electrons, positrons and photons do matter and this discreetness is also responsible for producing quantum fluctuation in the radiation field pattern, which are emerging technology. " # fp2 eAg ¼ e0 ea þ f ðf þ igÞ F e0 ¼ 8:85  1012 m ea ¼ 5 Drude’s model for dispersion in dielectrics, i.e. plasma frequency: sffiffiffiffiffiffiffiffi ne2 ep ¼ e0 m

1.4 Terahertz MIMO DRA parameters Any deviation in the excitation position at the input port can disturb the field components resulting in linear polarization. A wideband circular polarization (CP) response can be obtained due to the merging of the fundamental and higher order orthogonal degenerate mode pairs. The antenna can provide high peak gain in the operating impedance pass band of the antenna. The MIMO diversity performance parameters, including the envelope correlation coefficient (ECC), diversity gain (DG), channel capacity loss (CCL), mean effective gain (MEG) and total active reflection coefficient (TARC), have to be within the acceptable limits. The MIMO measurements using far-field parameters are usually calculated as given in the following:   2 ÐÐ
    þ Efi  Efj dW Eqi  Eqj    
 rei j ¼ ÐÐ ÐÐ
ÐÐ   þ E  E  dW  þ E  E  dW  E  E Eqi  Eqi fi qj fj fi fj qj The DG of the antenna is calculated using mathematical calculations as explained in the following equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Gd ¼ 10ep where ep ¼ ð1  j0:99 re j2 Þ Similarly, the CCL of the proposed antenna remains as follows: R Þ CCL ¼

 log2 detðy r r e11 e12 yR ¼ re21 re22

Dielectric resonator antennas (DRAs) and its synthesis

13

The MEG details are given as follows: " # M X 2 MEGi ¼ 0:5hi;rad ¼ 0:5 1  jSij j j¼1

where M is the number of ports in MIMO antenna and is the radiation efficiency of the antenna. Furthermore, the TARC is the ratio of square root of total reflected power to the square root of total incident power. TARC based on S-parameters is defined as follows: sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jS11 þ S12 ejq j2 þ jS21 þ S22 ejq j2 Gta ¼ 2 Theta is phase of input signal. Here are few parameters listed in the following, which differentiates classical antennas from optical.

1.4.1 Microwave DRAs vs optical DRA parameters The basic difference between a classical and a quantum antenna is the size. Classical antennas have surface currents. The former has length dimensions of order ranging from a few centimeters to several meters, while the latter has length dimension of the order nanometer, which is the typical order of magnitude of the radius of an atom. The physical quantities measured by a classical antenna are actually quantum averages w.r.t. a given pure or mixed state of electrons, positrons and photons. The classical case, the SNR is infinite, which is equivalent to setting Planck’s constant to zero after doing a quantum-mechanical analysis.

1.4.2 Optical DRAs The optical antennas shall have SPP or surface charge density due to the involvement of Planck’s constant [1–4]. The physical quantities measured by a quantum antenna are not only quantum averages of the field amplitude but also quantum averages of the higher moments of the e.m. field in a given state of the electrons, positrons and photons. Design of spherical optical DRA is given in Figure 1.6.

Titanium dioxide (resonating element) Silver nanostrip Teflon (substrate) Silver

Figure 1.6 Spherical optical DRA

14

Terahertz dielectric resonator antennas

These higher order moments are taken about its mean value. In particular, apart from the average field amplitude, the mean-square fluctuations of the e.m. field around its mean value become significant in a quantum antenna. The quantum fluctuations of the field around a given mean value is to be interpreted as quantum noise in the system and the ratio of the square of the mean field amplitude to the mean-square amplitude fluctuation is called the SNR of the antenna. The mean-square fluctuations in the field arise due to the Heisenberg uncertainty principle or equivalently due to the noncommutative of the operator field. These mean-square fluctuations are at least of the order or magnitude of Planck’s constant and hence the SNR of a quantum antenna is usually bounded above by Planck’s constant.

1.4.3

Radiated fields

A typical example of how the analysis of the radiation field is carried out in quantum field theory is as follows: 1. 2. 3.

We interpret the e.m. four potential as a canonical position vector and its time derivative as a canonical momentum vector. In quantum mechanics, we have the Heisenberg commutation relations. The mean field energy in the Gibb’s state is defined by quantum-mechanical process as given by the following:

Define the annihilation and creation operators, derive Heisenberg commutation relations and express the field energy as radiated far fields. The reason for this discrepancy lies in the fact first observed by Max Planck when he proposed his quantum-mechanical law of black-body radiations as an improvement over the earlier Wien’s displacement law: radiation energy comes in quanta and not as a continuum as the Maxwell wave theory of radiation predicts. Thus, in an optical or quantum DRA, the discrete particle distribution of electrons, positrons and photons do matter and this discreteness is also responsible for producing quantum fluctuations in the radiation field pattern, which are of the order of Planck’s constant.

1.5 Main functions of terahertz DRA 1. 2. 3. 4. 5.

Light in light out principle of working. Input excitation by semiconductor LASER. Plasmonic resonance takes place into substrate. Electromagnetic coupling (proximity feed) takes place to quantum DRA through plasmonic resonance. Capacitive coupling takes place due to proximity feed.

Dielectric resonator antennas (DRAs) and its synthesis 6. 7. 8. 9. 10. 11. 12.

15

Screening of electron–hole takes place near band gap due to accelerated charge carriers. Initially very small band gap energy was observed. LASER provided higher photon energy then initial band gap energy. LASER accelerated photons due to excitation. LASER created clouds of charge carrier (screening of fields). Quick change in radiated field takes place (due to fluctuations). Quantum states.

1.5.1 Some important parameters of microwave and terahertz DRA 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32.

Career multiplication ratio. Signal-to-power ratio. Receiving gain. Transmitting gain. Dispersion. Spectral response. Beam forming. Bandwidth. Directivity. Localized energy. Coupled energy. Light emitting. Spectroscopy. Electromagnetic states. Scattering for microwave imaging. Creation and annihilation. Absorption. Coherent control. Field enhancement for scanning. Wave fronts. Top down fabrication. Control and utilize material at nanoscale. Electron–hole pair combines to emit a photon. Plasmons—electron oscillations. Electromagnetic response. Device that efficiently converts localized energy into propagating optical radiations, and vice versa. Diffraction limit. Metals do not behave as perfect conductors at optical frequency. Capacitive driven by proximity feed. Surface plasmon resonances are material and shape dependent. Quantum absorber or emitter. Control light–matter interaction.

16

Terahertz dielectric resonator antennas

33. 34. 35. 36. 37. 38. 39. 40.

Local density of electromagnetic states. Energy dissipation by dipoles. Quantum-mechanical treatment. Power dissipated. Antenna efficiency. MIMO radar for scanning. Graphene absorber due to strong light–matter interaction. The field enhancement, field localization, absorption, high-resolution microscopy and spectroscopy, photovoltaic (rectenna), light emission and coherent control. Optical antennas are still infancy stage study. Classical antenna downsizing does not work due to radiation penetrates into metals, which results into plasma oscillations. Optical antenna increases interaction with absorber or emitter radiation, enhances light–matter interaction. Bottom up designs by colloidal solution. Resonant light scattering. Fundamental and higher order modes. Electron energy loss spectroscopy. Near-field microscopy for scanning. Field intensity distribution. Impedance matches for maximum energy transfer from source to antenna.

41. 42. 43. 44. 45. 46. 47. 48. 49. 50.

1.6 THz DRA model design parameters The dimensions of the proposed rectangular DRA (RDRA) and feeding mechanism parameters are determined by considering the following dimension guidelines: the DRA length and width is selected as and optimized height is taken in order to maintain the aspect ratio equivalent to while splitting the DRA into two equal halves using the single-port configuration. This high value of aspect ratio allows the excitation of the fundamental and higher order design dimensions, let a ¼ 2b, aspect ratio a/h ¼ 2. The isolation between the ports is maintained by gap, which is optimized using the parametric analysis. It is interesting to note that the good isolation is maintained at gap, which separates feed line by the distance equivalent to l0/2. For CP generation, the excitation is given at the corner of the feed lines which allows the traveling wave field to travel with quarter-wavelength path difference along the edges of the respective feed lines. In this motif, the width of the feed lines is taken equivalent to l0/4, where l0 is the operating wavelength of the DRA. This path difference in the traveling field at the edges of the feed line provides the 90 phase difference. The MIMO diversity performance parameters include ECC, DG, CCL, MEG and TARC. The specific feeding technique excites the orthogonal degenerate mode pairs inside the DR resulting in CP.

Dielectric resonator antennas (DRAs) and its synthesis

17

1.7 Rectangular nano-DRA design parameters The design step of nano-DRA is given in Figure 1.7. The formulations used to compute resonant frequency for terahertz RDRA is

 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kz d ¼ ð
r  1Þk02  kz2 kz tan 2

1.7.1 Design steps To design nano-RDRA, for ground plane, assign the material silver (Ag) of minimum height. The substrate of same size is placed on ground plane. Assign this substrate material as silicon dioxide (SiO2). Use of low loss tangent materials for substrate can provide high gain, greater efficiency and lower dielectric losses. The nanostrip waveguide of silver material (Ag), having length half the length of ground plane, desired width and desired height based on frequency of excitation is used. The impedance matching of feed is obtained for 50 W. For input excitation, we have to define the 50-W lumped port. The port height is equal to the height of substrate and width greater than width of nanostrip. The nano-RDRA of silicon of required dimensions, based on resonant frequency, is placed on SiO2 (permittivity ¼ 3.9) substrate with proximity feed arrangements. The materials are Si, SiO2 from material library. The Gaussian pulse excitation is given through the nanostrip feed line of TDRA. The excitation position is chosen at the edge of the feed line having width equivalent to the quarter wavelength. SPP generates due to edge feeding into silver waveguide. The RDRA allows the generation of specific resonant mode based on aspect ratio. The substrate used for TDRA is SiO2 having relative permittivity, (3.9) and equal to dimensions of the ground plane. Nano rectangular RDRA design

z

HR WS LS LP

H sub

LSUB /L G

y

WP 0

15 W /W sub G

Figure 1.7 Nano-RDRA (rectangular DRA)

30 (μm)

18

Terahertz dielectric resonator antennas

Table 1.1 Design dimensions Parameters

Material

Dimensions (mr, m)

LSUB WSUB HSUB LG WG HG LN1 WN1 LN2 WN2 WS LR1

Silicon dioxide (er ¼ 4, mr ¼ 1 and mass density ¼ 2,220)

45 45 5 45 45 5 21 3 21 3 0.2 6

Silver (er ¼ 1, mr ¼ 0.99998 and mass density ¼ 10,500) Silver (er ¼ 1, mr ¼ 0.99998 and mass density ¼ 10,500)

Silicon (er ¼ 11.9, mr ¼ 1 loss tangent ¼ 0.001 and mass density ¼ 2,330)

BR1 HR1 LR2 BR2 HR2

6 6 6 6 6

specifications are given in Table 1.1. The nanostrip feed line is made of silver of dimensions and stub length, substrate layers of height h.

1.8 Conclusion Microwave DRAs have been in use in mobile communication such as Wi-Fi, Wimax, WLAN and for mm-wave applications. TDRA has excellent applications for sensing and high-speed communication applications. These TDRAs can be developed as light-imaging detection and ranging (LiDAR) for self-driving cars. The terahertz antennas radiation pattern controls on the wave function operator of the Dirac field of electrons and positrons. The quantum cavity photon and electron– positron fields will then be expressible in terms of the free quantum fields plus additional perturbation terms involving the classical current and field sources. Once this is done, we can in principle calculate the far-field antenna pattern produced by the cavity surface currents induced by the tangential components of the quantum magnetic field operators as well as that produced by the Dirac field of electrons and positrons and then design these classical control fields so that the far-field quantum Poynting radiation pattern has a mean value and correlations in a given quantum coherent state of the photons and electrons–positrons within the cavity as close as possible to specified values. The applications of terahertz frequency in communication are intra-chip> 1) Abs Component Output Gain Frequency 192.68 THz Rad. effic. –3.298 dB Tot. effic. –3.529 dB Gain 4.297 dB

z y x

Figure 3.3 Light–matter interaction and radiation pattern of THz DRA at frequency 192.68 THz

quantum state of the wave field like a coherent state, or a number states, and then compute the statistics of the wave operator field. On the retinal surface, the action of creation and annihilation operators on coherent states or on photon number states is used. After we do such an analysis, we can ask a more interesting question: how does light within the cylindrical cavity resonator interact with the matter field? Specifically, when light propagates within the cylindrical eye cavity, it interacts with electrons, atoms and molecules of the fluid medium of which the eye cavity is composed and its propagation therefore gets affected by this medium. In classical wave field theory, we would model this fluid medium as a plasma composed having a definitive permittivity, permeability and conductivity and would analyze the propagation of the e.m. field in this plasma using the Vlasov equations that are in fact the coupled Boltzmann equation for the particle distribution function of the plasma and the Maxwell e.m. field. The result of this analysis using linearized perturbation theory would be dispersion relation between the oscillation frequencies of the plasma and the e.m. field and the wave vector. However, at the quantum level the description of the interaction between light and matter is more subtle. A simplified analysis would be to model the matter as just the second quantized electron–positron field using Dirac’s relativistic wave equation with the wave function being an operator field and then to include the photon interaction term in the usual way. In this way, the total Hamiltonian of the photon and matter field splits into three terms: one, the Hamiltonian of the e.m. field described as a quadratic form in the photon creation and annihilation operators within the cavity; two, the Hamiltonian of the Dirac field with cylindrical cavity boundary conditions on the wave function described as a quadratic form in the electron–positron creation and annihilation operators; and three, the Hamiltonian of the interaction between the photon and electron–positron field described as a quadratic form in the electron– positron creation–annihilation operators multiplied with a linear form in the photon

Light–matter interaction in terahertz antennas

47

x

Optical DRA

Fields due to light matter interaction

Laser

Figure 3.4 Light–matter interaction in optical DRA resulting into SPP and e.m. field radiations into space

creation–annihilation operators. This interaction Hamiltonian can be used to compute the amplitudes for scattering but since we are primarily interested in the statistics of the e.m. wave field pattern on the retinal surface, we shall describe this interaction using the Dirac current density expressed as a quadratic form in the electron–positron creation–annihilation operator fields that drives the photon field using the wave equation for the e.m. field in the presence of a current density. Thus, we shall by perturbation theory be able to calculate the change in the e.m. fields pattern on the retinal surface caused by the Dirac current in terms of operators and then by assuming a definitive state of the photon–electron–positron field, we shall be able to compute the mean and covariance of the e.m. field on the retinal screen surface in this state. Figure 3.4 is an example of light–matter interaction. Quantum field radiations in optical antennas or terahertz antennas occur due to light–matter interactions.

3.3 Theory of quantum entanglement Entanglement is the process of generation of two coherent states of photons. These states are found to be synchronized. They are useful in secure quantum communication. Radiated far fields are defined as single state, by making use of operator fields. The theory of entanglement has been discussed as follows: X ½bðk Þck ðxÞ þ cðk Þ hk ðxÞ (3.1) yð0Þ ðxÞ ¼ k

48

Terahertz dielectric resonator antennas The free cavity constrained e.m. four potential is defined as X A m ðx Þ ¼ ½aðk Þqk ðxÞ þ aðk Þ qk ðxÞ 

(3.2)

k

where aðk Þaðk Þ are the photon annihilation and creation operators, while bðk Þ bðk Þ are the electron annihilation and creation operators and cðk Þcðk Þ are the positron annihilation and creation operators. We get, for an approximate value of the Dirac current operator,     J m ¼ yð0Þ þ yð1Þ am yð0Þ þ yð1Þ ¼ J mð0Þ ðxÞ þ dJ m (3.3) where J mð0Þ ¼ yð0Þ am yð0Þ ; dJ m ¼ yð0Þ am yð1Þ þ yð1Þ am yð0Þ Now, J mð0Þ ¼ ¼

X k;m X



ðbðk Þcm ðxÞ þ cðk Þ hk ðxÞÞ am ðbðmÞcm ðxÞ þ cðmÞ hm ðxÞÞ ½bðk Þ bðk Þck ðxÞ am cm ðxÞ þ cðmÞ cðk Þam hm ðxÞhk ðxÞ

k;m

þ bðk Þ cðmÞ am hm ðxÞck ðxÞ þ cðmÞ cðk Þam hm ðxÞ (3.4) Which is the free Dirac current, i.e., in the absence of the interactions with the photon field? We have already indicated how to compute the far-field radiation pattern produced by this field and how to evaluate the moments of this field. Specifically, if Gðx  yÞ denotes the causal Green’s function for the wave operator, then the electromagnetic four potentials produced by the Dirac current are given by ð A m ðx Þ ¼ Gðx  yÞJ m ðyÞd 4 y ð ð (3.5) ¼ Gðx  yÞJ mð0Þ ðyÞd 4 y þ Gðx  yÞdJ m ðyÞd 4 y ¼

Aðm0Þ ðxÞ þ dAm ðxÞ

Remark: We can consider a fermionic coherent state rather than a fermionic number state. Such a state is parameterized by a Grassmannian vector variable g ¼ ðgb ðk Þ; gc ðk ÞÞ and is denoted by j∅ðgÞ>, and the action of electron and positron annihilation operators on the state is bðk Þj∅ðgÞ> ¼ gb ðk Þj∅ðgÞ>; cðk Þj∅ðgÞ> ¼ gc ðk Þj∅ðgÞ> In order that the canonical anticommutation relations (CAR) ½bðk Þ; bðmÞþ ¼ ½c½k ; cðmÞþ ¼ ½bðk Þ; cðmÞþ ¼ 0

Light–matter interaction in terahertz antennas

49

holds good, we require that the Grassmannian parameters satisfy the following anticommutation rules: gb ðk Þgb ðmÞ þ gb ðmÞgb ðk Þ ¼ 0; gc ðk Þgc ðmÞ þ gc ðmÞgc ðk Þ ¼ 0; gb ðk Þgc ðmÞ þ gc ðmÞgb ðk Þ ¼ 0; Further, by our analogy with bosonic coherent states, we impose the requirement that   @ ; g ðmÞ ¼ dðr; sÞdðk; mÞ; r; s ¼ b; c @gr ðk Þ s þ And that  @ bðk Þ j∅ðgÞ> ¼ j∅ðgÞ>;  @gb ðk Þ @ cðk Þ j∅ðgÞ> ¼ j∅ðgÞ> @gc ðk Þ 



We then get ¼ ∅ðgÞ ¼ ¼ ¼ gb ðk Þ gb ðmÞ On the one hand, while on the other h∅ðgÞjbðmÞbðk Þ j∅ðgÞi   ¼ h∅ðgÞ

   @ bðmÞ∅ðgÞi @gb ðk Þ

  ¼ h∅ðgÞ

   @ bðmÞ∅ðgÞi @gb ðk Þ

  ¼ h∅ðgÞ

   @ gb ðmÞ∅ðgÞi @gb ðk Þ

     @  ∅ðgÞi ¼ dðk; mÞ  h∅ðgÞgb ðmÞ @g ðk Þ  b

¼ dðk; mÞ  gb ðmÞh∅ðgÞjbðk Þ j∅ðgÞi ¼ dðk; mÞ  gb ðmÞhbðgÞ∅ðgÞj∅ðgÞ  dðk; mÞ  gb ðmÞgb ðk Þ

(3.6)

50

Terahertz dielectric resonator antennas This is in agreement with the CAR ½bðmÞ; bðk Þ þ ¼ dðm; k Þ Provided that we assume that CAR ½gb ðmÞ; gb ðk Þ þ ¼ dðm; k Þ

and likewise, ½gc ðmÞ; gc ðk Þ þ ¼ dðm; k Þ By imposing such restrictions, we can calculate easily the momentum of the current density field and hence of the radiated field is a state that is jointly coherent for the bosons (i.e. photons) and for the Fermions. We observe that the perturbation of the current density of the Dirac field caused by the interactions between the electron–positron field and the photon field is given up to first order in the photon field and second order in the Fermion field by an expression of the form dJ m ¼ yð0Þ am yð1Þ þðyð1Þ am yð0Þ

¼ eyð0Þ ðxÞam Se ðx  yÞgm Am ðyÞyð0Þ ðyÞd 4 y þ h  c:

(3.7)

This expression is manifestly trilinear in the operators. Specifically, it is quadratic in the electron–positron field and linear in the photon field, totally yielding a trilinear term. It can be expressed as X m

dJ m ðxÞ ¼ F1 ðxÞjk; m; q bðk Þ bðmÞ þ F2m ðxjk  m  qÞbðk Þ cðmÞ k;m;q

þF3m ðxjk  m  qÞcðk ÞbðmÞ þ F4m ðxjk  m  qÞcðk ÞcðmÞ aðqÞ þ h  cÞ

(3.8)

It should be noted that the photon operators aðqÞ; aðqÞ commute with all the electron–positron operators bðqÞ; bðqÞ ; cðqÞ; cðqÞ : From this expression, it is clear that the photon operators tend to couple the other modes of the electron–positron field and hence produce additional terms in the far-field radiation pattern. If we have a state jy > of the electron–positron field in which there are ne ðk Þ ¼ 0; 1 electrons with momentum-spin index k, np ðk Þ ¼ 0; 1 positron with momentum-spin index k and nph ðk Þ ¼ 0; 1; 2; . . . photon with momentum– helicity index k for k ¼ 1,2, . . . , then we can calculate easily the moments of the currents fluctuation field dJ m ðxÞ in this state by simply applying the following rules: bðk Þjne ðk Þ ¼ 0 > ¼ 0; bðk Þjne ðk Þ ¼ 1 > ¼ jne ðk Þ ¼ 0 >; cðk Þnp ðk Þ ¼ 0 > ¼ 0; cðk Þnp ðk Þ ¼ 1 > ¼ np ðk Þ ¼ 0 >;  bðk Þjne ðk Þ ¼ 0 > ¼jn  e ðk Þ ¼ 1 >; bðk Þ jne ðk Þ ¼ 1 > ¼ 0;   cðk Þ np ðk Þ ¼ 0 > ¼ np ðk Þ ¼ 1 >; cðk Þ np ðk Þ ¼ 1 > ¼ 0;

Light–matter interaction in terahertz antennas

51

in view of the Pauli-exclusion principle and likewise, for the photon numbers qffiffiffiffiffiffiffiffiffiffiffiffiffiffi   aðk Þnph ðk Þ > ¼ nph ðk Þ > nph ðk Þ  1 >; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi   aðk Þ nph ðk Þ > ¼ nph ðk Þ þ 1 > nph ðk Þ þ 1 > more precisely, we can evaluate the moments < yjdJ m1 ðx1 Þ      dJ mm ðxm Þjy > By noting that the quantity dJ m1 ðx1 Þ      dJ mm ðxm Þ

(3.9)

Is a homogeneous polynomial of degree 3 in the electron–positron–photon operators with the photon operator appearing with a total degree of m and the electron–positron operators appearing with the total degree of 2. We can also evaluate the previous moment in joint coherent state ∅ep ðgÞ∅ph ðuÞ > :

3.4 Conclusion Quantum computing and quantum communications are two different fields using quantum entanglement. Entanglement can have wide applications in secure communication. Coherent states are secured and utmost important in quantum communications. Quantum mechanics can provide analytical solution for quantum communications. Light–matter interaction can increase confinement. This confinement is very useful at terahertz frequencies for healthcare and security purpose. Terahertz scanners and spectroscopy are useful as biosensors to the mankind. Protein, skin and fat testing sensors can be developed based on light confinements technologies at terahertz frequencies.

Reference [1] S. Lepeshov and A. Gorodetsky, “All-dielectric optically tunable metasurface for terahertz phase and amplitude modulation,” J. Phys.: Conf. Ser. vol. 1461, p. 012203, 2020 IOP Publishing. doi:10.1088/1742-6596/1461/1/ 012203.

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Chapter 4

Terahertz dielectric resonator antennas design and modeling

Abstract Theory of operation of a terahertz (THz) dielectric resonator antenna (DRA) is explained and mathematical expressions for rectangular DRA, spherical DRA, cone DRA and cylindrical DRA are broadly given in this chapter. THz emission is produced solely by the photo-induced current in the silicon (Si) DRAs, also telecom wavelength of 1,550-nm DRA is presented. Our finding is that THz waveforms are lower than those for pumping at 1,100 nm, due to the nonuniform absorption of different parts of the excitation spectrum that is also called plasmon frequency. DRAs exhibit an infinite number of resonant modes. In this chapter, the development of high-efficiency DRAs at optical frequencies and THz frequencies has comprehensively explained using mathematically equations for optical currents densities and quantum fields.

4.1 Introduction to terahertz DRA Terahertz (THz) cavity that can be modeled as given in Figures 4.1–4.3 is spherical, cylindrical and conical shape, respectively. Since light is in fact a quantum electromagnetic (e.m.) field, when it falls on cavity, it starts immediately interacting with the matter within the cavity. A coarse approximation of this matter is to regard as a sea of electrons and positrons and then to apply the formalism of quantum electrodynamics to describe its interaction with the incident photons, both before the light hits the cavity matter directed based on boundary conditions. The field in the spherical, cylindrical and conical cavity is actually a quantum e.m. field in certain state and we wish to calculate the probability that such field will be formed. The penetration of light wave into silver nano waveguide takes place to generate surface plasmon polytrons (SPP). The e.m. response is an ensemble of collective electron oscillations (plasmons) coupled plasma or gaseous form of metals. Optical antennas are also not typically driven with galvanic transmission lines— localized oscillators are instead brought close to the feed point of the antennas, and electronic oscillations are driven capacitive coupling, their properties may be strongly shaped and materials dependent owing to surface plasmon resonances

54

Terahertz dielectric resonator antennas Silicon (resonating element)

Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Lumped port

Figure 4.1 Spherical terahertz DRA Silicon (resonating element)

Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Lumped port

Figure 4.2 Cylindrical terahertz DRA [1–6]. The optical dielectric resonator antennas (DRAs) are fed by localized light emitters, not by real currents, i.e., local density of e.m. states, which can be expressed in terms of Green’s function tensor G. The cellular communication in the 5G and beyond can be in the sub-THz band, i.e., 100–350 GHz, because the data rate of 20–100 Gbps is needed. Here, atmospheric path loss in THz frequency can be more than 100 dB/km. There is a need to operate in frequency windows below 1 THz, where the atmospheric attenuation is relatively low, and the high-gain antennas can be good solution for this. Hence, solid-state communication sources with a power of over 10 mW are required [1]. The gain required for such antennas is minimum of 15.27–20.71 dB for the frequency of 100–350 GHz for a data rate of 20 Gbps. There is need to maintain effective cellular wireless communication at a 0.1–10 THz band to develop powerful sources, and efficient detectors for cellular wireless communications systems, for expanding the transmission range more than 100 m. The hybrid antenna array for beam forming massive multiple-input and multiple-output can significantly improve the spectrum efficiency [2]. Here, the study of optical antennas is still in

Terahertz dielectric resonator antennas design and modeling

55

z

y

Theta

Phi

x

Figure 4.3 Super directive far field radiations in terahertz CDRA

its infancy and some properties directly derive from classical and microwave antenna theory, the direct downscaling of antenna designs into the optical regime is not possible because radiation penetrates into metals and gives rise to plasma oscillations. An optical antenna is designed to increase the interaction area of a local absorber or emitter with free radiation, thereby making the light–matter interaction more efficient. In the excitation of certain resonant modes, the DRAs may be used as resonant cavities or efficient radiators. With knowledge of its resonant modes, one can qualitatively predict the antenna behavior and can estimate the produced far field during design stage itself [3]. The generation of short vertical electric dipole and horizontal magnetic dipole is the origin of radiation. Their design resonance frequencies depend on the dimensions, geometry and permittivity of ceramics used. At optical, the nanoscale version of the microstrip line is known as nanostrip waveguide. We have numerically investigated the basic elements of photonics (DRAs) and plasmonic (nanostrip waveguide). By observing the fundamental antenna parameters, the performance of DRAs at optical frequencies have been explained. The Tx mode of DRA by means of the propagation vector orientation is also explained. The magnetic field vectors of the fundamental mode of the nanostrip waveguide show their compatibility for modal coupling to DRA. Nano waveguide dispersive properties have been described by the Drude model. The coupling of the magnetic field coming from the nanostrip’s fundamental mode to DRA’s takes place. The optical communication bands are 185–205 THz: L-band (1.625
1.565 mm ¼ 0.06 mm), C-band (1.565
1.530 mm ¼ 0.35 mm) and

56

Terahertz dielectric resonator antennas

S-band (1.530
1.460 mm ¼ 0.70 mm). The cylindrical dielectric resonator antenna (CDRA) is made from silicon materials. The structure is multilayer, i.e., ground plane is made from silver, then the substrate is SiO2 materials and DRA is made from silicon materials. The whole DRA structure is excited via a nano waveguide made from a noble silver metal (Ag) whose conductive properties are calculated using the Drude model. DRA dielectric is made of Si, with er ¼ 11.9 and estimated loss tangent, tan d ¼ 0.0025. It is excited by microlasers such as microdisks and photonic crystal lasers. The length of the transmission lines is characterized to the wavelengths (l) of incoming and outgoing radiations [4]. However, while working at the optical frequencies, most of the incident light is transparent through the metals. This gives rise to plasmonic free electrons; thus the feed line is analyzed considering shorter effective wavelength (leff), which depends on the material properties. The properties of the noble metals are explained as per Drude’s model [5]. The radiating power flows perpendicular to the feed waveguide’s plane. However, depending on the excited radiation resonating mode, nano dielectric resonator antennas (NDRAs) can radiate either end-fire or broadside. To perform cavity quantum e.m. field computation, we must solve the joint Maxwell–Dirac equations within the cavity, and with the appropriate boundary conditions, these are corresponding to what shape of the cavity we are using. The cavity shape can be cone, cylindrical or spherical. This means that we expand the Maxwell field in terms of basis functions, which vanish on the boundary. Then the coefficients in this expansion will be the photon creation and annihilation operator fields. Likewise, we must expand the Dirac field within the cavity using the same orthogonal basis functions, whose coefficients will now be the electron–positron creation and annihilation operator fields [6]. The resulting Hamiltonian of these fields will then comprise three components: one, the Hamiltonian of the free photon field expressed as a quadratic form in the photon creation and annihilation operators; two, the Hamiltonian of the free electron–positron field expressed as a quadratic form in the electron–positron creation and annihilation operators; and finally three, the interaction Hamiltonian between the electron–positron field and the photon field. It is this last interaction term that is of importance in scattering theory of light and the formation of fields. Using the interaction picture of Dyson, we can expand the solution to the Schro¨dinger evolution operator for this quantum system as a Dyson series in powers of the interaction Hamiltonian. Then, by using the technology of Feynman diagrams, we can calculate various terms of the scattering matrix to determine the probability amplitudes. The basic physics behind this computation does not depend on what shape we choose for the cavity. No matter what shape we choose, we can in principle determine a complete orthonormal basis of test functions that satisfy the appropriate boundary conditions for the given cavity and then expand the Maxwell and the Dirac field in terms of these test functions. Before the interaction between light and matter, the system was in an initial state and after this interaction, including the interaction of light with the matter, the state of the whole system changes with time and this change can be determined by

Terahertz dielectric resonator antennas design and modeling

57

the action of the unitary evolution operator on the initial state with the unitary evolution operator constructed from the total Hamiltonian of the interacting electrons, positrons and photons using the Dyson series. The final state of this system is then equivalently determined once we know the matrix elements of the evolution operator between any given initial state and a final state of the particles. From this final state, the probability of getting a given field is computed from the basic rules for quantum probability. Thus, we shall by perturbation theory be able to calculate the change in the e.m. fields pattern caused by the Dirac current in terms of operators and then by assuming a definitive state of the photon–electron–positron field, we shall be able to compute the mean and covariance of the e.m. field in this state. Their operating wavelengths are very small, mostly in the nanometer size. Optical antennas that work on Drude’s theory have the nature of nonlinearity. Optical antennas have far field with finite values of Planck’s constant. The physics in optical antennas involves the dominance behavior of skin effects, SPP and light–matter interaction (creation and annihilation). It is well known that light can penetrate through metallic walls at optical frequencies. Light–matter interaction at optical domain is studied to design quantum devices. In optical DRA, SPP phenomenon occurs due to strong light–matter interactions. A simplified analysis would be to model the matter as just the second quantized electron–positron field using Dirac’s relativistic wave equation. In this process, wave function is an operator field, and then the process includes the photon interaction term in the usual way. In this way, the total Hamiltonian of the photon and matter field splits into three terms: 1. 2.

3.

One, the Hamiltonian of the e.m. field is described as a quadratic form in the photon creation and annihilation operators within the cavity. Two, the Hamiltonian of the Dirac field with cylindrical cavity (in cylindrical DRA) boundary conditions on the wave function is described as a quadratic form in the electron–positron creation and annihilation operators. And three, the Hamiltonian of the interaction between the photon and electron– positron field described as a quadratic form in the electron–positron creation– annihilation operators multiplied with a linear form in the photon creation– annihilation operators.

This interaction Hamiltonian can be used to compute the amplitudes for scattering, but since we are primarily interested in the statistics of the e.m. wave field pattern, we shall describe this interaction using the Dirac current density expressed as a quadratic form in the electron–positron creation–annihilation operator fields that drive the photon field using the wave equation for the e.m. field in the presence of a current density. The previous quantum theory can be applied to understand the concept of quantum physics as stated in the following. Antennas have characteristic dimensions of the order of a wavelength of light, demanding fabrication accuracies better than 10 nm. The advent of nano-science and nanotechnology provides access to this length scale with the use of novel topdown nanofabrication tools (e.g. focused ion beam milling and electron–beam

58

Terahertz dielectric resonator antennas

lithography) and bottom-up self-assembly schemes. The fabrication of optical antenna structures is an emerging opportunity for novel optoelectronic devices. In light-emitting devices, an electron and a hole pair combines to emit a photon. The antenna serves to make both the excitation and the emission more efficient. It can be expressed in terms of the system’s dyadic Green’s function G. This reflects the fact that the decay from the excited state happens in response to the emitter’s own field. Thus, the excited state lifetime of the quantum emitter is determined by the Green function G of the system in which the emitter is embedded. Rigorously treating the metal as a strongly coupled plasma is required, which leads to a reduced effective wavelength. This effective wavelength is related to the external (incident) wavelength. The plasma wavelengths of the metal n1 and n2 are constants that depend on the geometry and dielectric parameters of the antenna. The effective wavelength is roughly a factor of 2–6 shorter than the free-space wavelength. Laser Gaussian input pulse for nonlinear phase matching (coherent states) of metals at optical frequency. The absorption by infrared-active optical phonons takes place beyond 7 THz. The mode-locked fiber lasers operate at wavelengths of 1.1 and 1.55 mm with pulse repetition rates of 10 and 20 MHz at 10-mW power. The compact THz photonic devices operate up to multi-THz frequencies that are compatible with Si CMOS technology. The spectrum of potential applications depends on the available spectral bandwidth, signal-to-noise ratio and data acquisition speed. In general, the techniques for THz generation and detection exploit either photoconductivity or optical nonlinearity. Photoconductive techniques for THz emission and detection are widely used due to their simplicity, compactness and possibility of direct coupling to fiber optics. THz emission from photoconductivity was first demonstrated using Si. These DRAs are investigated at optical frequencies (C-band). Here, another potential application of optical nanoantennas in wireless broadcasting and wireless optical links is used between two points at the nano- and microscale. Here, we develop the idea of wireless links using optical nanoantennas, inspired by the concept of microwave links at radio frequencies.

4.2 Mathematical formulations used to describe working of quantum DRA In order to describe the propagation of e.m. waves a spherical cavity, we must express the Maxwell equation in the frequency domain using spherical polar coordinates. An elegant formulation of this is based on the theory of multipole moments, where we account one component of the E field that is E lm ðm; r Þ ¼ fl ðrÞLglm ð^r Þ !

!

where L ¼
i r  ris the vector angular momentum operator when used in quantum mechanics, and glm ð^r Þ are the standard spherical harmonics.

Terahertz dielectric resonator antennas design and modeling

59

Likewise, we assume that one component of the magnetic field is H lm ðu; rÞ ¼ glðrÞLglm ð^r Þ ^r :E lm ¼ 0; ^r  H lm ¼ 0; these are longitudinal components. We select fl at gl to that E lm and H lm satisfy the Helmholtz equation (i.e. the ware equation in the frequency domain):  
2  E lm 2 r þk ¼0 H lm This result in fl and gl satisfies modified D Bessel differential equations that have far fields behavior in the form of spherical waves propagating radially outward. These components in addition satisfy Maxwell equations: divE lm ¼ 0; divH lm ¼ 0 Hence, we can build up all lollops for spherical wave propagation using E lm and H lm . The coefficients that we choose for expanding the e.m. field are that of E lm r  H lm ; H lm ; r  E lm that determine the Hamiltonian (field energy of the e.m. field) within the cavity as a quadrature form in these coefficients and hence these coefficients upon second quantization become the creation and annihilation operators of the e.m. field. The surface current density in the spherical surface generated by the boundary conditions of these field operators generates a quantum e.m. field in space, where mean-square fluctuations may be determined in the coherent state of the field defined by the joint Eigen states of the annihilation operators. Finally in order to discuss the interaction of the matter within the spherical DRA cavity with the quantum e.m. field, we solve the following stationary Dirac equation: ðða
irÞ þ bmÞy ¼ Ey For the four-component wave function y by expanding, y as a linear combination of the basis function obtained earlier and denote the solved By þEn ; 0 yn ; cn ; we care to express the general time dependent on wave function as X
 yðt; r Þ ¼ aðnÞe
iEnt yn ðr Þ þ b nÞþ eiEnt cn ðr Þ n


where faðnÞg are the electron annihilation operators and b nÞþ are the positron creation operators. The interaction energy operator between electron–positron and the photon field is then ð    !  ! ! ! ! ! HlmðtÞ ¼ yþ t; r aA; eA t; r y t; r d 3 r

60

Terahertz dielectric resonator antennas

Therefore, in a polynomial of third-order electron–position–photon operators and their interdependence on perturbation theory or the Dyson series, we can use Hlm ðtÞ; the effect of the e:m: field on matter and the effect of matter on fields formation. For e.m. fields and changed matter within a cylindrical cavity, a detailed study is given in the following. In a spherical cavity, the vector solutions to Helmholtz equation with boundary conditions were Elm ðr Þ ¼ fl ðrÞLYlm ð^r Þ Hlm ðr Þ ¼ gl ðrÞLYlm ð^r Þ With boundary conditions,   1 ^r  Elm ðr Þ þ curl Hlm ðr Þ ¼0 jwe r¼R and ^r  ½curl Elm ðr Þjr¼R ¼ 0 correspond to the vanishing of the tangential components of the electrified and the normal component of the magnetic field on the surface. The boundary conditions determine the characteristic frequencies w of oscillation.

4.3 Cylindrical terahertz DRA In the core of a cylinder cavity, we solve Helmholtz equation in cylindrical coordinates to obtain the basis functions. For expanding the e.m. fields and the Dirac fields specifically, the basic functions satisfy  
2  Et 2 0 r þk Ht With boundary conditions determined by the few that E? vanishes to t ¼ 0, and t ¼ d, H? vanishes to t ¼ 0 and t ¼ d, and Et vanishes to r ¼ Rð@Ht =@rÞ and to r ¼ R. Doing the results in an expansion of the e.m. field as P ! ! E ðt; rÞ ¼ Re cE ðn; mÞc Enm ðr ÞejwE ðn;mÞt P þ Re dE ðn; mÞg Enm ðr ÞejwH ðn;mÞt !

!

And likewise, for H ðt; rÞ: The boundary conditions completely determine the characteristic frequencies wE ðn; mÞ; wH ðn; mÞ of oscillations as well as the basis function cE ; gE ; etc.

Terahertz dielectric resonator antennas design and modeling

61

(For that matter it should be noted that giving a boundary of any shape whether spherical or cylindrical or any other, in order get the correct basis function for expanding with e.m. fields or the Dirac field, we express the Laplacian in the Helmholtz equation in a coordinate say q1 is such that q1 ¼ constant that is defining the boundary of the surface.) To solve the Dirac equation by the same method, we choose any of their basis functions, say fun ðr Þg; that vanish on the cylindrical boundary and expand the Dirac wave function as Y ðr Þ ¼

1 X

c n u n ðr Þ

n¼1

where c n C 4 and resultant into the time-independent Dirac equation:  !   ! a
ir þ bm Y ðr Þ ¼ EYðr Þ We then multiply this equation by um ð r Þ and  integrate over the cylindrical volume to obtain Eigen-matrix equation for C n and E; thus our term-dependent Dirac wave function has the form X Y ðt; r Þ ¼ expð
iEk tÞcðnk Þ un ðr Þ n;k

And in the second quantization process, cðnk Þ become the certain and annihilation operators of electron and positron. Then the interaction energy ð
 !  ! ! HI ðtÞ ¼ e yðt; rÞT a
1 A t; r yðt; rÞd 3 r defines a cylindrical form in the electron–positron photon creation–annihilation operators to which the theory of the Dirac series, Feynman diagram, etc. can all be applied to calculate mathematical properties or the quantum fields. The quantization process for the Dirac operator cðnk Þ involves introduction to the canonical anticommutation relation (CAR): 0

0

fyðr Þ; yþ ðr Þg ¼ d3 ðr
r Þ X
0 0 0 are un ðr Þun ðr Þ ¼ dðr
r Þ; it follows that there AR Since n o n ðk Þ ðlÞþ ¼ dkl dn;m in show, where an expression for the interaction Hamiltonian cn ; cm between the e.m. field of the photons and the Dirac fields of electron and positron comparing of bosonic and fermionic operators are the canonical commutator rules of the bosonic operators and the CAR(anticommutator) of the fermionic operators can be used to calculate all the matrix elements of this interaction Hamiltonian and hence all the quantum statistical properties of the image field formed. This interaction Hamiltonian may also be used to calculate radiative correlation to the Dirac wave function and hence to the Dirac current field and correlation to the far-field

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Terahertz dielectric resonator antennas

radiation e.m. field pattern, and hence correlation to the mean fluctuations of the radiation quantum field in any state. Remarks: The Eigen functions of r2 for a cylindrical boundary are

ppt am ½nr  ðcos ðmjÞ; sin ðmjÞÞ  sin Jm R d or Jm

ppt am ½nr  ðcos ðmjÞ; sin ðmjÞÞ  cos R d 0

where m, n, p is curve integers am ½n are either the zeros of the Jm ðnÞ or Jm ðnÞ, with Jm ðnÞ being the Bessel function. The corresponding Eigen value determines the oscillation frequency: ! pp2 a ½n2 1=2 m þ wðmnpÞ ¼ R2 d The frequencies of the Dirac equation obtained using their basis function will however be different.

4.4 Conical terahertz DRA In fact, it is the conical cavity that most accurately represents the shape of the eye retina photo receptors (Figure 4.4). The conical cavity is regarded as a section of the sphere. Let us to denote the cone angle. Then the appropriate basis functions for expansion are obtained by solving the Helmholtz equation:
2  r þ k2 y ¼ 0 in spherical polar coordinates in the region 0  r  R; jqj  a; 0  j  2p with vanishing boundary conditions on the surfaces fr ¼ R; jqj  a; 0  j  2pg: And f0  r  R; q ¼ a; 0  j  2pg. These Eigen functions can be obtained by forming linear combinations of those for the spherical cavity case and the choosing the linear combination. Coefficients to that the previous boundary conditions, the previous cases yield the possible Eigen-frequencies oscillations.

4.5 Conclusion In this chapter, THz DRAs of various shapes and geometries have been explained, i.e., rectangular, spherical, cylindrical and conical DRAs at optical frequencies. Their coherent states have been given more importance. The current density

Terahertz dielectric resonator antennas design and modeling

63

Silicon (resonating element) Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Lumped port

Figure 4.4 Conical terahertz DRA

fluctuations have controlled to its optimized threshold value for getting desired far fields. These quantum antennas can be named THz nanoantennas or optical antennas. These are similar to radio waves and microwave antennas, and their purpose is to convert the energy of free propagating radiation to localized energy and vice versa. Optical antennas exploit the unique properties of metal nanostructures, which behave as strongly coupled plasmas at optical frequencies. Applications are optical sensors, lasers, solid-state lighting, photovolatics, scanner and imaging and in telecommunication, short (S), conventional (C) and long (L) bands of the optical communication spectrum. The inter-chip wireless communication, including infrared and THz technologies, can be used. Here, properties of the dielectric materials were assumed as constant and the Drude model was considered for the metal used (Ag), transmit/couple optical energy from/into plasmonic circuits. The time-dependent current across the full depth of the Si wafer is calculated by numerically solving the pulse-propagation equation. The coherent polarizations occur due to the simultaneous generation of heavy-hole–light-hole wave packets. Nanoantennas enable optical wireless links, which would present much less absorption losses, largely outperforming conventional plasmonic waveguides. Replacing waveguide networks by nano-antenna chip-to-chip and intra-chip links also provide more on-chip space that can be used to house other circuitry, hence enabling further miniaturization.

References [1] P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon., vol. 1, no. 3, pp. 438–483, 2009. [2] G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Herna´ndezFigueroa, “Dielectric resonator antenna for applications in nanophotonics,” Opt. Express, vol. 21, no. 1, pp. 1234–1239, 2013.

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[3] W. T. Sethi, H. Vettikalladi, H. Fathallah, and M. Himdi, “Dielectric resonator nantennas for optical communication,” Optical Communication Technology, Pedro Pinho, IntechOpen, doi: 10.5772/intechopen.69064. [4] G. N. Malheiros-Silveira and H. E. Herna´ndez-Figueroa, “Wireless optical coupling evaluation in a dielectric resonator nanoantenna,” OSA Continuum, vol. 1, no. 3, pp. 805–811, 2018, https://doi.org/10.1364/OSAC.1.000805. [5] A. Alu` and N. Engheta, “Wireless at the nanoscale: Optical interconnects using matched nano antennas,” Phys. Rev. Lett., vol. 104, no. 21, p. 213902, 2010. [6] D. M. Solı´s, J. M. Taboada, F. Obelleiro, and L. Landesa, “Optimization of an optical wireless nanolink using directive nanoantennas,” Opt. Express, vol. 21, no. 2, pp. 2369–2377, 2013.

Chapter 5

Surface plasmon polytrons (SPP) into terahertz DRA

Abstract Surface plasmonic resonance occurs at interface of metal and substrate at terahertz frequencies. The terahertz’s wavelength is located between the microwave and the infrared region of the electromagnetic spectrum. Optical wavelength falls after infrared region. Terahertz DRA (TDRA) makes use of surface plasmon polytron (SPP) phenomenon for the excitation of dielectric resonator antenna (DRA). The Drude model helps to characterize TDRA, when switching from microwave to terahertz frequencies. Drude’s scattering is function of frequency, resistance and conductance at photonic wavelength. Here, photon’s generation is classified by bosonic fields or fermionic fields. The nonlinearity of photons is described by Dirac second quantized field equations. Feynman path integral gives know-how about particle’s moment fluctuations. Correlation coefficient manipulation provides proper beam formation. Photon’s spin is the main cause of nonlinearity in terahertz antennas. Optical spectrum has frequencies from 1013 to 1015 Hz. Optical DRA can couple optical energy into plasmonic resonance and vice versa. Plasmon resonance thus becomes the main cause of excitation to TDRA. Terahertz antennas that involve SPP phenomenon for radiations can provide highly directive radiation pattern and high gain.

5.1 Introduction In 1939, Richtmyer found that dielectric objects can resonate and radiate into free space based on boundary conditions air–dielectric interface [1–5]. In 1985, SA Long experimentally proved. Hence, dielectric resonators (DRs) are very efficient radiators. DR materials are commercially available with vast range of permittivity ranging from 10 to 1,600 F/m. An antenna designer can exercise choice for exciting various higher order modes depending on material’s permittivity and dimensions of DR at optical frequency [6–11]. It consists of electrons and positrons that are solved by Dirac second quantized field equations based on quantum electrodynamics to describe photon spin. Here, electromagnetic (e.m.) field is produced by linear superposition of creation and annihilation operators of photon fields [12–16]. The electron positron creation and

66

Terahertz dielectric resonator antennas DR TMM

Proximity coupling

Gaussian beam LASER

SiO2 LASER beam Nanostrip

Figure 5.1 Terahertz DRA design with CDRA, laser and SPP annihilation operator fields plus photon’s creation and annihilation operators’ fields are total fields produced in terahertz DRA (TDRA) as shown in Figure 5.1. The current densities of fields are obtained by quadratic functions of the Dirac field operators. This way, the current density in turn produces quantum e.m. fields. These fields can be described by retardation potentials or Dirac wave function [21,22]. Hence computations of mean and mean-square fluctuations of quantum e. m. fields produced by TDRA plus free photon e.m. fields in any state shall provide the complete solution (Figures 5.2 and 5.3). The terahertz RDRA (TRDRA) is excited by plasmonic resonance (Figure 5.4). Terahertz input is given into an SiO2 substrate thorough silver nano waveguide. Energy band gap of semiconductor materials is targeted to accelerate by input excitation. Transient photocurrents are thus produced [23–25]. Accelerated charge carriers are thus generated. Thus, oscillations in TRDRA also known as resonance are created at THz frequency. This results into surface plasmon polytron (SPP) and radiations into space, and it takes place on the basis of boundary conditions of TRDRA. Thus, these radiations are governed by the rule of plasmonic. Using higher order modes concept, dimension scaling can become possible. It is a device that can convert terahertz radiations into localized energy and vice versa. Proximity feed is used to excite TDRA. The Poynting theorem describes power dissipated by a time harmonic system. Nano waveguide (silver) is inserted into a substrate to create SPP and SPR (surface plasmon resonance). Then through capacitive coupling, near field is coupled to TDRA for generating far-field radiations.

5.2 Working principle of TDRA 1. 2. 3.

Light in, light out. Input excitation by semiconductor LASER. SPP and SPR plasmonic resonances take place into interface of substrate and ground.

Surface plasmon polytrons (SPP) into terahertz DRA Mega

MMW

VHF 300 MHz

UHF 3 GHz

Peta

Sub-MMw

Visible

Infrared

Microwave

T - rays

SHF

EHF

30 GHz

300 GHZ

FIR 1 THZ

10 THz

MIR 100 THz

NIR 385 THz

(780 nm)

Figure 5.2 Frequency spectrum used in terahertz DRA

Microstrip nano feed Dielectric resonator

Excitation port Substrate

Ground plane

0

500

1e+003 (nm)

Figure 5.3 Terahertz rectangular DRA with Gaussian input

z

E Field (y/m) 2.7357E+005 2.5538E+005 2.3719E+005 2.1900E+005 2.0082E+005 1.8263E+005 1.6444E+005 1.4625E+005 1.2806E+005 1.0987E+005 9.1683E+004 7.3494E+004 5.5305E+004 3.7116E+004 1.8928E+004 7.3875E+002

x y 0

15

30 (um)

Figure 5.4 Terahertz RDRA fields

67

68

Terahertz dielectric resonator antennas

4. 5. 6. 7. 8. 9. 10. 11.

Electromagnetic coupling takes place to TRDRA. Capacitive coupling takes place due to proximity feed. It forms photon clouds and fluctuations due to spin effect of photons. Screening of electronic–hole takes place near band gap due to accelerated charge carriers. Initially very small band-gap energy was observed. Laser provided higher photon energy than initial band gap energy. Laser accelerated photons due to excitation known as plasmonic resonance. Laser created clouds of charge carrier (screening of fields takes place). Quick change in field takes place thus generated transient currents.

5.3 Terahertz CDRA design and simulations Design of TCDRA is given in Figure 5.5. Resonant frequency ( f ) formulations for TDRA (cylindrical) is given in the following: "    2 # 6:324c d d pffiffiffiffiffiffiffiffiffiffiffiffi 0:27 þ 0:36 f ¼ þ 0:002 2h 2h 2pd er þ 2 Skin effect plays an important role at terahertz frequencies. The penetration of e.m. fields into metal ground plane takes place. Election oscillations cause skin effect. Drude’s scattering is a function of frequency, resistance and conductance. Dielectric resonator antenna (DRA) is an efficient radiator. It can excite an infinite number of resonant modes. Knowledge of resonant modes can predict antenna behavior. Dissipation factor or dielectric loss can be predicted based on Drude’s model. Photons generation can be called as fermionic or bosonic fields. Beam formation can be worked on the basis of correlation coefficients. Radiation or emissions are based on Dirac second quantized field equation. Feynman path integral is used to define photon density and moments fluctuations. Photon spin is the main cause of nonlinearity in quantum antennas. Beaming of these nonlinear photons fields requires the treatment of quantum mechanics for getting a solution of transmitting Silicon (resonating element)

h d

Silver nanostrip

h2 s

Silicon dioxide (substrate)

x z

h3 h1

Silver (ground)

Input by LASER Gaussian beam

Figure 5.5 TCDRA with Gaussian input to silver nano waveguide

Surface plasmon polytrons (SPP) into terahertz DRA

69

Table 5.1 Dimensions terahertz CDRA S. no.

Parameter

Value

1 2 3 4 5 6 7 8

Radius r Height h Permittivity Input frequency Radiated frequency Gain Substrate Dimension of the substrate

0.250 mm 0.325 mm 11.9 F/m 473 THz 192 THz 10.5 dB Arlon 4.64.5 mm

quantum states from one location to another. The Dirac equation can provide a generation of photons and its fields to constitute currents. This current density becomes the main factor to the introduction of quantum e.m. field. This field can be estimated on the basis of the solution of retarded potential. Solution of resonant modes can be worked out making the use of linear super position of photon fields by creation and annihilation. Here, we apply standard commutation rule for photons and anticommutations rule for electron, positron/hole. Figures 5.1–5.10 shows terahertz DRA radiation parameters. Table 5.1 shows design dimension of cylindrical DRA.

5.4 Terahertz DRA main features 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Resonant and light scattering. Antenna structures and resonant frequency. Higher order resonances and fundamental resonance. Near field and far fields. Focused electron beam. Feed gap between waveguide and DRA (proximity feed). The feed gap and impedance match (source and DRA). Proper impedance-match, high-energy transfer between a localized source and the TDRA. The collective excitation corresponds dipole moment (super directive and super radiance). Physical dimensions are reduced by the square root of the dielectric constant, i.e., an er of DR increased to 100, which can reduce dimensions by a factor of 10. These DRAs operate via displacement current in a low-loss high-permittivity dielectric, resulting in reduced energy dissipation in the resonators. At optical regime, radiation penetrates into metals and gives rise to plasma oscillations. Best efficiency the internal energy dissipation of any antenna must be minimized. Radio wave and microwave technologies predominantly make use of antennas to manipulate e.m. fields.

70

Terahertz dielectric resonator antennas dB 5.31 2.81 0.308 –2.19 –4.69 –7.19 –9.69 –12.2 –14.7 –17.2 –19.7 –22.2 –24.7 –27.2 –29.7 –32.2 –34.7

z Theta

y

Phi x

Far field (F=193.5) (1) z

Type Farfield Approximation enabled (kR >> 1) Component Abs Output Gain Frequency 193.5 THz –2.884 dB Rad. effic. –2.896 dB Tot. effic. 5.308 dB Gain

y x

Figure 5.6 Radiation in TCDRA

z

y

Theta

Far field (Fs68Macro_633,000nn) Type Farfield Approximation enabled (kR >> 1) Component Abs Directivity Output Frequency 193.5 THz 11.02 Die.

Phi

x

11 10.3 9.65 8.96 8.27 7.58 6.89 6.2 5.51 4.82 4.13 3.44 2.76 2.07 1.38 0.689 0

z

y x

Figure 5.7 Radiation pattern of TCDRA (super directive)

15. 16.

17.

THz interfacing efficiently between propagating radiation and localized fields. The fabricated device comprising an arrangement of TiO2 near resonance introduces a progressive phase delay to the wave front, resulting in a clearly observable beam deflection. The use of low-loss DRs at THz to concentrate radiation from far-field modes into a volume opens a new paradigm alternative to purely plasmonic

Surface plasmon polytrons (SPP) into terahertz DRA

71

1 0.8 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1

0

5e–05

0.0001

0.00015

0.0002

0.00025

Time (ns)

Figure 5.8 Gaussian beam input to TCDRA

–10

S11

–15

dB

–20 1

(192.63, –33.143)

–25

–30

–35 180

185

190

195 200 Frequency (THz)

205

210

Figure 5.9 Reflection coefficient of nano-CNDRA at 193 GHz

18.

(metallic) antennas. This is the precursor to purely DRAs with minimal energy dissipation. DRA dimensions are in order of a wavelength of light, demanding fabrication accuracies better than 10 nm.

72

Terahertz dielectric resonator antennas Gain (IEEE), φ = 0.0, Max. Value (subrange) 5.7

5.6

5.5

5.4

5.3

5.2

5.1

5 192

193

194

195

196

197

198

199

200

Frequency (THz)

Figure 5.10 Gain 5.6 dBi of nano-CDRA at 193 GHz 19. 20. 21. 22. 23.

24. 25. 26. 27. 28. 29.

THZ DRA has applications short-range optical communications, ultrafast computation and molecular sensing. Here CDRA with a diameter of 180 nm, a height of 50 nm is used. Optical DRA 65% of the power is absorbed by the array, as plasmonic and dielectric loss. Antenna radiation is due to the resonance of surface waves that propagate on the antenna. SPR is the resonant oscillation of conduction electrons at the interface between negative and positive particles, wp is the frequency of bulk longitudinal electron excitations, the plasma frequency. The fundamental limit is given by the tunneling time (SPP at interface). A resonant plasmonic antenna contributes to the local density of states in the junction. Here, h is Planck’s constant yielding a time limit on the order of 4 fs, the limit is given by the circuit time constant RC. The emitted power as a function of the current fluctuations. The emitted power is on the order of 10 pW. The coupling between the antenna mode and the propagating plasmon mode has to be optimized.

Surface plasmon polytrons (SPP) into terahertz DRA 30. 31. 32. 33.

34. 35.

36. 37. 38.

39. 40. 41. 42. 43. 44.

45. 46. 47. 48. 49. 50. 51.

73

The figure of merit is quality measure of TDRA. Surface plasmon polaritons are mixed electronic and e.m. waves. The magnetic field of an SPP is parallel to the surface and perpendicular to its direction of propagation. The surface plasmon polaritons are solutions of Maxwell’s equations in which the effects of retardation, non-propagating, collective vibrations of the electron plasma near the metal surface. Scanning near-field optical microscopy did it become possible to measure the surface polariton field. If the contribution of the antenna mode is larger than the contribution of the non-radiative modes, it becomes possible to avoid quenching (absorption) and therefore to increase the radiative efficiency. Antenna radiation is due to the resonance of surface waves that propagate on the antenna. Sensor development using SPP and studies about kinetics of interaction between biomolecules are the most investigated applications in the field. The contributions from SPPs and from other surface waves to the antenna radiation. It is found that for antennas with long arms that support higher order resonances, SPPs provide a dominant contribution to the antenna radiation. For sensor, measure the surface polariton field at interface. Generation of surface plasmon polaritons with a focused laser beam. Get electric-field intensity distribution on a silver surface. Polytrons at a spot size of the focused beam. In antenna, contribution to the local density of states. Spatially averaged enhancement factor in the junction barrier. The antenna produces a narrow emission spectrum and provides an enhancement of the efficiency. The efficiency defined as the number of emitted photons due to plasmon leakage per electron. The fields emitted by a fluctuating current density in the tunneling gap. The power spectral density of the current fluctuations is given as hI2i. Optical antennas deal with light emission, based on surface plasmons. Photon emission in the visible by electrically driven optical antennas has been reported demonstrating a spectral control and also a directional control. SPR-based sensor. The TDRA is modeled as a resonant plasmonic cavity. w2p 2 ðwÞ ¼ 1  w2 ; kspp is the wave number   w 212ðwÞ 1=2 ksp ¼ c 21 þ 2ðwÞ wp wsp ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 21 þ 1

74

Terahertz dielectric resonator antennas

5.5 Mathematical formulations used in TDRA Following sequence of activities takes place in quantum antennas: 1. 2. 3. 4. 5. 6. 7.

Light in (LASER). Input excitations is Gaussian beam from LASER (nonlinear). Nano waveguide feed is used at optical frequency (light–matter interaction). SPP takes place at an SiO2 substrate and ground metal interface. Nano-DRA is excited by SPP through proximity feed due to capacitive coupling. Radiated frequency is lower to input laser frequency ( fp < LASER fr for propagation). Electrons, positrons and photons fields become a radiated beam of quantum antenna at terahertz frequencies with clouds of photons spins field that becomes as input excitation (SPP) to nano-DRA.

This is a very important and exciting phenomenon in quantum antenna. The mathematical aspect of its working principle is input excitation to radiated power of quantum. With the illumination of femtosecond laser pulses, electron-hole pairs are created in the photoconductive material when the laser pulses have higher photon energy than the band-gap energy of the photoconductive material, and the quick change in the electric field at the gap results in a transient current and finally an e. m. pulse in the THz frequency range is radiated. Mean of photons along with spin field included during radiation needs to develop quantum mechanics control for desired radiation orientation characteristics. Silver nanostrip or nano waveguide is used in the model. For coherently coupled photon sources whose emission direction can be dynamically steered by phase control. Feed gap is used for impedance matching. Optical antenna operates on “light in light out” mode: ðwÞ ¼ 0 ðwÞ þ i00 ðwÞ wp ðwÞ ¼ 1  2 0 w pffiffiffiffiffiffiffiffiffiffi þ iw 2 ne wp ¼ m  o w2 r  r  Eðr; wÞ  2 ðr; wÞEðr; wÞ ¼ 0 c   i x r ¼ oEz e kx  wt Mean=fluctuations ¼

hyjAu ðxÞAv ðx0 Þjyi ðcontrolled to thresholdÞ hyjAu1 ðx1 ÞAu2 ðx2 Þ    Aun ðxn Þjyi

mo Ð J^ u ðpÞeipx 4 where Au ðxÞ ¼ 4p p2  io dp ; hc/¼E, where is the photon energy; wP is the plasma volume frequency; G is the damping constant; w is the center frequency where w > wP: Drude’s model is used to characterize metal transport properties when switching from microwave to optical frequency. It is dispersive in nature. Loss

Surface plasmon polytrons (SPP) into terahertz DRA

75

tangent is a dielectric dissipation factor: tand ¼

00 0

At THz frequency, the penetration of e.m. fields into metal ground plane takes place. This causes electron oscillations or plasmonic resonance or surface wave plasmonic resonance. Skin depth is a function of frequency, resistance and conductance. rffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r 1 þ ðrwÞ2 þ rw d¼ wm rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jwo mo mvz ; surface impedance Zo ¼ s0 P Z¼ ; j I j2 Silver nano waveguide dimensions are given as ml Lðlo Þ ¼ 2eff ; m is the resonant modes order. leff ¼

2Lðlo Þ lo ¼ heff m

The length of nano waveguide will be shorter at optical frequency as compared to microwave. Transcendental equation gives solution to determine resonant frequency. ffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi kz tanðkz z=2Þ ¼ ðr  1Þk02  kz2 ; when propagating in the z direction. Resonant frequency is given by the following transcendental equation: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ko2 ¼ kx2 þ ky2 þkz2 ; p ffiffiffiffiffiffiffiffiffi w m  and k ¼ mp; k ¼ np; k ¼2p/l; x y 0 o o a b   s ¼ o z ei kxx  wt :

where ko2 ¼

Surface charge density is given as follows: rðt; rÞ ¼ ejyðt; rÞj2 Current density is given as follows: i ! ! ! ieh h yðt; rÞr y ðt; rÞ  y ðt; rÞryðt; rÞ ; J ðt; rÞ ¼  2m yðt; rÞ ! wave function iℏ

@y ðt; rÞ ℏ2 2 e2 r yðt; rÞ  Z yðt; rÞ ¼ 2m r dt

76

Terahertz dielectric resonator antennas Electric scalar vector potential is given as follows:   rr0 0 ð jyj t  C ; r e ∅ ðt; rÞ ¼  d 3 r0 4 pe0 jrr0 j Magnetic vector potential:  ð  iehmo rr0 ; r Aðt; rÞ ¼  jyj  8 pm C     rr0 0 rr0 0  r ry t dr30  ry t  C C

where yðt; rÞ satisfies Schro¨dinger’s equation that gives a solution of wave function for single electrons: @yðt; rÞ h2 2 Ze2 ry ðt; rÞ  yðt; rÞ ¼ 2m r dt This provides discrete dynamic solution. If there are N electrons in joint wave function then wave function can be described as N N X @y h2 X Ze2 ¼ r2 y  y im 1¼1 @t jn j 1¼1

Hence, current density can be expressed as J ðxÞ ¼ eyðxÞ lm yðsÞ where lm ¼ ^o ^m This satisfies Dirac second quantized field equations. This is obtained in a quadratic function of the Dirac field operators. This current density produces quantum e.m. field. Quantum-retarded potential: ð m J ðt  jr  r0 j; r0 3 0 d r Aðt; rÞ ¼ 4pjr  r0 j where Aðt; rÞ is the bosonic field. It is expressed as quadratic function field operators. Quantum antenna radiated fields: hyjAu ðxÞAv ðx0 Þjyi fluctuation fields ð m J^ m ðpÞeipx 4 d A u ðx Þ ¼ o 4p p2  10 P

Surface plasmon polytrons (SPP) into terahertz DRA

77

5.6 Terahertz DRA applications There are many advanced applications at THz frequencies as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

Imaging for pattern recognition with a higher resolution of images, biosensors spectroscopy. Large band width for 5G and beyond, 6G, IOT, LIDAR and sensors. Compact size for wearable, secured communication due to encryption. Higher order modes for dimension scaling, i.e. miniaturization. Defense and space application, i.e. detection of size of aircraft. On-chip communication (chip-to-chip high speed communication). It can detect any metallic object carried by any person at airport without manual search. At airport advanced landing system during a bad weather. Rectenna for energy harvesting, i.e. renewable energy source. Nanoantenna can receive direct optical energy. Biological sensor detection and biomedical applications in imaging. Comprehensive integrated border management system. The advantages of nanoantenna rectennas are as follows: it can work whole day (day and night), and not sensitive to weather conditions, drones, e-cars and sensors.

5.7 Conclusion TDRA has very sharp beam width, super directive and high gain. They are having compact size and very high operating frequency, i.e. THz frequency. They are good candidates for high-resolution imaging, scanning, LiDAR and sensors. Necessity of fabrication and testing facilities at such a high frequency in THz range is needed. Correlation coefficient manipulation is used in beam formation in quantum antenna. It is observed that the spin of photons becomes a main function of nonlinearity at optical frequency. There is strong need to establish relationship between radiated frequencies, SPPs resonance frequencies and input excitation frequency. SPP wavelength is ten times smaller to free-space wavelength. Hence, radiated frequency shall be smaller to input excitation frequency in plasmonic antennas. Terahertz medical sensing and ultrafast data transfer and ultrawide-band antenna. Plasmon frequency must be smaller to input excitation frequency for propagation. Used for high-speed chip-to-chip communications and biomedical sensing.

References [1] R. S. Yaduvanshi and H. Parthasarathy, Rectangular Dielectric Resonator Antennas: Theory and Design, Springer, New Delhi, 2016. [2] M. Hayasi, Quantum Information Theory: Mathematical Foundation, Springer, Berlin Heidelberg, 2017.

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[3] C. A. Balanis, Antenna Theory: Analysis and Design, Wiley, New York, NY, 2005. ¨ chsner and H. Altenbach (Editor), Properties and Characterization of [4] A. O Modern Materials (Advanced Structured Materials), Springer, Singapore, 2017. [5] S. A. Long, Dielectric Resonator Antenna Handbook, Artech House, Norwood, MA, 2007. [6] P. Bhardwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon., vol. 1, pp. 438–483, 2009. [7] D. H. Youn, M. Seol, J. Y. Kim, et al. “TiN nanoparticles on CNT–graphene hybrid support as noble-metal-free counter electrode for quantum-dotsensitized solar cells,” ChemSusChem, vol. 6, no. 2, pp. 261–267, 2013. [8] R. P. Feynman and A. R. Hibbs, Quantum Mechanics and Path Integration, McGraw-Hill, New York, NY, 1965. [9] L. Novotny and N. Van Hulst, “Antennas for light,” Nat. Photonics, vol. 5, no. 2, p. 83, 2011. [10] W. T. Sethi, H. Vettikalladi, and H. Fathallah, “Dielectric resonator Nano antenna at optical frequencies,” In Information and Communication Technology Research (ICTRC), 2015 International Conference on, IEEE, pp. 132–135, 2015. [11] F. J. Rodrı´guez-Fortun˜o, A. Espinosa-Soria, and A. Martı´nez, “Exploiting metamaterials, plasmonics and nanoantennas concepts in silicon photonics,” J. Opt., vol. 18, no. 12, p. 123001, 2016. [12] L. Zou, W. Withayachumnankul, C. M. Shah, et al. “Dielectric resonator nanoantennas at visible frequencies,” Opt. Express, vol. 21, no. 1, pp. 1344– 1352, 2013. [13] X. Wu, F. Tian, W. Wang, J. Chen, M. Wu, and J. X. Zhao, “Fabrication of highly fluorescent graphene quantum dots using L-glutamic acid for in vitro/ in vivo imaging and sensing,” J. Mater. Chem. C, vol. 1, no. 31, pp. 4676– 4684, 2013. [14] A. E. Krasnok, D. S. Filonov, C. R. Simovski, Y. S. Kivshar, and P. A. Belov, “Experimental demonstration of superdirective dielectric antenna,” Appl. Phys. Lett., vol. 104, no. 13, p. 133502, 2014. [15] N. Shinohara and H. Matsumoto, “Experimental study of large rectenna array for microwave energy transmission,” IEEE Trans. Microwave Theory Tech., vol. 46, no. 3, pp. 261–268, 1998. [16] G. Varshney, A. Verma, V. S. Pandey, R. S. Yaduvanshi, and R. Bala, “A proximity coupled wideband graphene antenna with the generation of higher order TM modes for THz applications,” Opt. Mater., vol. 85, pp. 456–463, 2018. [17] W. Chen, R. L. Nelson, D. C. Abeysinghe, and Q. Zhan, “Optimal plasmon focusing with spatial polarization engineering,” OPN Opt. Photonics News, pp. 36–41, 2009. [18] P. Biagioni, J.-S. Huang, and B. Hecht, “Adult teratoma of the testicle metastasizing as adult teratoma,” Rep. Prog. Phys., vol. 75, pp. 1–40, 2012.

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[19] Y. Zhao and A. Alu, “Optical nanoantennas and their applications,” IEEE Radio Wirel. Symp. RWS, pp. 58–60, 2013. [20] S. Abadal, E. Alarco´n, A. Cabellos-Aparicio, M. Lemme, and M. Nemirovsky, “Graphene-enabled wireless communication for massive multicore architectures,” IEEE Commun. Mag., vol. 51, no. 11, pp. 137–143, 2013. [21] M. Schnell, P. Alonso-Gonza´lez, L. Arzubiaga, et al., “Nanofocusing of mid-infrared energy with tapered transmission lines,” Nat. Photonics, vol. 5, no. 5, pp. 283–287, 2011. [22] J. Wen, S. Romanov, and U. Peschel, “Excitation of plasmonic gap waveguides by nanoantennas,” Opt. Express, vol. 17, no. 8, p. 5925, 2009. [23] J.-S. Huang, T. Feichtner, P. Biagioni, and B. Hecht, “Impedance matching and emission properties of optical antennas in a nanophotonic circuit,” Nano Lett., vol. 9, no. 5, pp. 1897–1902, 2008. [24] L. Zou, W. Withayachumnankul, C. M. Shah, et al., “Efficiency and scalability of dielectric resonator antennas at optical frequencies,” IEEE Photonics J., vol. 6, no. 4, pp. 1–10, 2014. [25] G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Herna´ndezFigueroa, “Dielectric resonator antenna for applications in nanophotonics,” Opt. Express, vol. 21, no. 1, pp. 1234–1239, 2013.

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Chapter 6

Terahertz conical dielectric resonator antenna—design, simulation and implementations

Abstract Terahertz conical dielectric resonator antennas (DRAs) have been designed, simulated and analyzed at a terahertz spectrum, i.e. 10 THz. A lower terahertz frequency band is useful for high data speed communication, whereas a terahertz frequency upper band is used for optical sensors or sensing applications. The freespace wireless communications also use terahertz DRAs (TDRAs). A unique and novel geometry for terahertz dual-band operations has been investigated with mathematical formulations. The comprehensive analyses in a terahertz regime have been worked out with a reflection coefficient (S11), a voltage standing wave ratio, a radiation pattern and efficiency using a conical TDRA. The conical TDRA is designed at 10 THz frequency with 4.9 dBi, gain and bore sight radiation pattern. The equivalent circuit of conical TDRA has been developed at given frequencies.

6.1 Introduction Optical antennas such as dipoles, graphene patch, silver and gold patch antennas and many other metallic nano antennas have been theoretically demonstrated by many research groups such as Bhardwaj, A Alu and Novotny [1–3]. Dielectric materials for use of antennas have been available since 1939 developed by Ritchmeyer. Dielectric resonator antenna (DRA) has high efficiency, temperature stability, ease of fabrication and low losses at microwave and optical frequencies. They have been used so far to develop classical antennas at microwave frequencies for applications in mobile personal communication, radar, satellite and sensors. Many researchers such as SA Long, MM Antaar, A Kishk and R Mongia, Guha and Ravi have contributed in this fields since 1980 [4–6]. These dielectric materials are ceramics having permittivity, er, ranging from 9 to 100 (dielectric material permittivity). Reduced dimensions and efficient radiations are two very important properties of DRAs. DRA’s main characteristics are a wide range of operations from microwave to optical, simple and efficient, built with high dielectric constant ceramics for miniaturization, ease of fabrication and highly stable at high operating temperature. Optical DRAs have been trending in current era research and still they

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Terahertz dielectric resonator antennas

are infant in photonics science [2,3]. Nanophotonics is a branch of science that deals with transmission and reception of signals at terahertz and optical frequencies and falls in nanometer dimensions [4,5]. A nanoantenna has two important features, miniaturization and super directivity [6–8]. So far researchers have worked on metallic antennas such as Yagi–Uda, dipole antenna and Bowtie antenna, but they suffer from losses at terahertz and optical frequencies. Nanotechnology is a multidisciplinary area of science that merges physics, chemistry, materials, biology, medicine and engineering branches for innovations on current trends of optical DRAs in a nano regime [7,9,10]. The physics in optical antennas involves a dominance behavior of skin effects, surface plasmon politrons (SPP) and light–matter interaction (creation and annihilation of photons) [11–13]. It is well known that light can penetrate through opaque objects such as clothes, plastic suitcases at optical frequencies; hence detection and scanning becomes possible in concealed items [14]. Light–matter interaction at an optical domain is an important parameter in optical devices, and their mathematical parts are described in the next section. In a terahertz DRA (TDRA), an SPP phenomenon occurs due to strong light– matter interactions [15–17]. Total electric dipole moment in scattering can be computed using tensor analysis. This way radiations of desired wave vector with polarization, scattering, suppression, forward and backward scattering can thus be computed mathematically. Optical electromagnetic fields are realized by quantum wave operator fields. It can be precisely defined as an ensemble of an infinite number of quantum harmonic oscillators. It can be analyzed as a superposition of constant plane waves with coefficients being creation and annihilation operators in a boson Fock space. At an optical spectrum, it is better to talk about quantum state of wave field like a coherent state of photons. The light interacts with electrons, atoms and molecules of a material medium it travels. At terahertz and optical frequencies, the material medium is treated as a plasma. Having a definite permittivity, permeability and conductivity of the medium diffraction, reflection and refraction is solved based on linear and nonlinear properties [1,18,19]. Nonlinear dispersion relation can be developed between an oscillation frequency of plasma, electromagnetic fields and wave vector. The frequency of radiation by DRA and frequency of plasma oscillations are different. The plasmon frequency is lesser than input excitation frequency due to the absorption of photonic phenomenon. In optical antennas, radiated resonant frequency is always lesser than input excitation frequency. The light–matter interaction at optical frequency is more subtle. The simple way to model shall be a treating matter as a second quantized electron positron field using Dirac’s relativistic wave equation with wave function being an operator field and then to include photon interaction with it [20–22]. An SPP phenomenon can be better described by a Drude model at a metal dielectric interface as metal behaves as a plasma or gas at optical frequency. The total Hamiltonian of the photons and matter fields can thus be defined by creation and annihilation operators, Dirac’s fields and electron–positron–photons creation and annihilation. Thus interaction Hamiltonian is used to compute the amplitudes of scattering and electromagnetic wave fields patterns based on surface current density. Mean and covariance of electromagnetic fields shall be based on state of

Terahertz conical dielectric resonator antenna

83

photon–electron positron fields due to Dirac currents in terms of operators. The resonant frequency of cone optical DRA will depend upon the volume and shape of DRA. Resonant modes excitation shall be mainly governed by aspect ratio, cavity dimensions, boundary conditions and feed position of DRA [23–25]. Since light is in fact a quantum electromagnetic field, when it falls on conical cavity, it starts immediately interacting with the matter within the conical cavity. A coarse approximation of this matter is to regard as a sea of electrons and positrons and then apply the formalism of quantum electrodynamics to describe its interaction with the incident photons, both before the light hits the conical cavity matter directed based on boundary conditions. The field in the cavity is actually a planar quantum electromagnetic field in a certain state and we wish to calculate the probability that such a field will be formed. The penetration of radiation into silver metals gives rise to SPP. The electromagnetic response due to collective electron oscillations (Plasmon’s) takes place. Plasmons also become excitation to cone DRA. SPPs are surface modes in silver metal nanostructures at interface. Optical antennas are compact in size in RF and microwave. There are many differences in classical and optical antenna concepts. Although metals are not perfect conductors at optical frequencies, free electron gas or plasma is. The optical antennas are fed by localized light emitters, not by real currents, also known as a local density of electromagnetic states (LDOS). These LDOS can be solved using a Green’s function tensor. Developing these devices hardware are still in infant stage and involves huge cost with limited fabrication facilities worldwide. At optical frequency DR design becomes compact and it can be useful for imaging, scanning and other biomedical healthcare sensor applications such as a UV sensor, environment sensor and oxygen saturation sensor. A conical optical DRA integrated with photodiodes can work as a retinal implant as it can provide all functionalities of cone-type photo receptors used in a retinal surface of human eye. The study has revealed that photons are received by retina using available arrays of cones and rods present in a retina at central part and periphery. These DRAs can very well operate at a vision spectrum of human eyes, which lies in between 430 and 750 THz. Similarity optical DRAs can set new trends of large bandwidth for high-speed terahertz communication and wireless optical communication. Being small in size, DRAs can fit into integrated circuits for chip-to-chip communications in system-on-chip and network-on-chip circuits and Internet-of-Things sensors for healthcare. Also 0.1–10 THz is lying as a vacant frequency band; hence it can be used for high-speed communication. Other application areas are light detection and ranging, three-dimensional holography, THz technology, sensing, photovoltaics, optical processing, security and biotechnology [23–29].

6.2 Design structure of conical THz DRAs The conical DRA design structures are shown in Figure 6.1 at 10 THz using silicon (er ¼ 11.9) as radiating elements, i.e. DRAs. DRAs are placed at the top of the substrate with optimized design dimensions. The proposed conical DRAs are

84

Terahertz dielectric resonator antennas Silicon (resonation element)

Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Laser Gaussain beam

Figure 6.1 Conical terahertz DRA design architecture excited using a rectangular nano silver feed line of width (wÞ with the stub of desired dimensions. The optimum value of the stub length has been chosen for good impedance match. The dispersive properties of hsilver used
in the nano waveguide i are selected using the Drude model as
Ag ¼
o
1  fp2 =ff ðf þ igÞg . Here, f is the operating frequency,
Ag is the permittivity of silver, fp is the plasma frequency, g is the collision frequency and it is the inverse of relaxation time,
1 is the offset of real part of the dielectric constant and
o ¼ 8:85  1012 F=m. The cone DRA is designed with a ground plane and it will look like a biconical DRA due to image effect. The volume of a cone DRA is given as ¼ pr2 ðh=3Þ. The finite value of Planck’s constant in a TDRA becomes a source of radiation, in terms of quanta. In conical cavity, this matter is to regard as a sea of electrons and positrons and then applies the formalism of quantum electrodynamics to describe its interaction with incident photons, both before the light hits the cavity matter directed based on boundary conditions. In bosonic creation and annihilation operators, they satisfy the canonical communication relations and finally compute the mean and covariance of the electromagnetic field in a state. Hence, due to Planck’s constant in quantum antennas, radiation energy comes in quanta. The mean-square fluctuations are at least of the order of magnitude of Planck’s constant. The mathematical derivation of quantum antenna states has been done to differentiate it from classical antennas. These conical DRAs can also be used in biomedical applications, sensors, chip-to-chip communications and high-speed telecommunications. Thus, discrete particle distributions of electrons, positrons and photons do matter and this discreetness is also responsible for producing quantum fluctuation and formation of particular state or statistical moments. The polynomial in the photon and electron, positron creation and annihilation operators hence determine their statistical moments, i.e. mean, variance, etc. in any given state of the electron positron photon field in the case of optical DRAs.

6.3 Model-1 multiband conical TDRA Resonant frequency fr ¼ 10 THz, l ¼ c/f

Terahertz conical dielectric resonator antenna

85

Cone DRA Substrate Ground Z X

Figure 6.2 Conical 10-THz DRA with Gaussian beam laser input excitation (silicon radiating element at 10 THz) 0 –5

S11 (dB)

–10 –15 –20 S11

–25 –30 6

7

8

9

10

11

12

Frequency (THz)

Figure 6.3 S11 simulated conical terahertz DRA

Substrate (SiO2): 1.5*lo ¼ 1.5*30 ¼ 45 mm and height of cone DRA ¼ lo/3 ¼ 30/3 ¼ 10 mm Substrate dimensions: length*breadth*height ¼ 45*45*10 mm3 Ground plane (silver): 1.5*lo ¼ 1.5*30 mm ¼ 45 mm and height of a DRA (silicon) ¼ lo/3 ¼ 30/3 ¼ 10 mm Nanostrip line or waveguide (silver Ag): length*width*height ¼ 22.5*3* 0.3 mm3 Cone DRA (silicon): h ¼ lo/10 ¼ 30/10 ¼ 3 mm (height of cone DRA), Radius of base in cone DRA (r) ¼ 3 mm TDRA aspect ratio (h/r) ¼ 1 Figure 6.2 shows conical DRA with substrate and ground. The aspect ratio of cone DRA is used to excite resonant modes, i.e. fundamental or higher order modes; here it is chosen in between 0.5 and 2.5. Thus by varying aspect ratio, higher order modes can be excited. Here, this TDRA has dual band applications as shown in Figure 6.3. Figure 6.4 has a radiation pattern of TDRA with gain that is 4.9 dBi

86

Terahertz dielectric resonator antennas 90 60

120

5 0

30

150

–5

Gain (dB)

–10 –15 –20

180

0

–15 –10 –5

330

210 f=0

–0

f = 90

5

300

240 270

Figure 6.4 Radiation pattern of conical TDRA 7 6

VSWR

5 4 3 2 1 VSWR 0 6

7

8

9 10 11 Frequency (THz)

12

13

Figure 6.5 VSWR of conical TDRA

broadside. Figures 6.5 and 6.6 are voltage-standing wave ratio (VSWR) and input impedance at operating frequencies. Figure 6.7 shows a better efficiency of TDRA. Figure 6.8 shows current density J. Figure 6.9 shows E-fields. The ground plane material is assigned as silver (Ag). The substrate of the material, silicon dioxide was used and placed on silver ground plane (Ag). The loss tangent material for a

Terahertz conical dielectric resonator antenna

87

Z11 ()

50

45

40

35 Z11 6

7

8

9 10 11 Frequency (THz)

12

13

Figure 6.6 Z11 of conical TDRA with multibands

Radiation efficiency

1.00 0.95 0.90 0.85 0.80

f= 0, q= 0

6

7

8

9 10 11 Frequency (THz)

12

13

Figure 6.7 Radiation efficiency of TDRA substrate was chosen to minimize the losses. Input excitation as Gaussian pulse from laser was assigned as input and coupled to silver nano waveguide for proximity feed. This silver nano waveguide is made up of silver nano wire. For perfect match, we designed nanostrip waveguide (Ag), whose lower part is connected to ground plane and top part is inserted into substrate (height h1) near to DRA, as proximity feed. The width, length and height of the nano feed (h2) have been chosen for matching of the design to 50-W impedance. A silicon cone DRA was placed on the top of a substrate as a radiating element. The silver nano waveguide is inserted into a substrate to create SPP and surface plasmon resonance (SPR). Light–matter interaction shall result into a generation of plasmonic frequency due to proximity feed. The plasmon frequency, fp, is responsible for the excitation of resonant modes into DRA [30,31]. This will

88

Terahertz dielectric resonator antennas Jsurf (A/m) 1.0001E+003 7.8187E+002 6.1127E+002 4.7789E+002 3.7361E+002 2.9209E+002 2.2835E+002 1.7852E+002 1.3957E+002 1.0911E+002 8.5305E+001 6.6691E+001 5.2139E+001 4.0762E+001 3.1868E+001 2.7914E+001

0

20

40 (µm)

Figure 6.8 E-Fields J current density

z

E field (V/m) 1.6620e+008 1.5110e+008 1.3600e+008 1.2089e+008 1.0579e+008 9.0688e+007 7.5585e+007 6.0482e+007 4.5379e+007 3.0276e+007 1.5174e+007 7.0629e+004

x

0

300

600 (µm) μµ

Figure 6.9 E-Fields of TDRA

cause DRA to radiate at resonant frequency. The quantum-mechanical model is required to show radiated states of cone DRA at terahertz l frequencies. Silver nano waveguide is used for the excitation of conical DRA with laser Gaussian beam. Figures 6.1–6.9 have shown simulated design and corresponding results obtained (Table 6.1).

Table 6.1 Comparison table Dimensions er

Reference

Mathematical analysis

Antenna geometry

Frequency Feeding technique

[3]

No

Conical

[2]

No

Conical annular ring

Proposed work

Yes

Conical TDRA

Microstrip h ¼ 5 mm line r ¼ 7 mm 2.72–11.36 Microstrip h ¼ 10 line r¼6 10 THz Silver nano r ¼ 3 mm waveguide h ¼ 3 mm 8.3 GHz

Multiband Bandwidth

10

No

0.9 GHz

20

Yes

8.64 GHz

11.9 Yes

High, dual band 300 GHz

90

Terahertz dielectric resonator antennas

6.4 Mathematical modeling of terahertz conical DRA The theory of terahertz conical DRA has been developed with (6.1)–(6.24), which describe the concept of terahertz radiations when excited with Gaussian beam input. In these equations, the case have been theoretically analyzed for statistical moments or terahertz radiated states are formed in conical TDRAs. Let h be the height of the cone and r, its radius. We may thus assume the conical region to be specified by the following equation:
a 0  z  d; x2 þ y2  a2 z2 ; a ¼ tan 2 where a is the cone angle, and this cone has its apex at origin and its surface is a revolution of a line passing through the origin making an angle a=2 with the z-axis. Using spherical polar coordinate system to expand the field as ^ þ Ej ðr; q; jÞ^ j Eðr; q; jÞ ¼ Er ðr; q; jÞ^r þ Eq ðr; q; jÞq

(6.1)

likewise, for the magnetic field the boundary conditions are Er ¼ Ej ¼ 0; Hq ¼ 0; q ¼

a 2

current into the PEC boundary conditions, the vanishing of the tangential component of the electric field and the normal component of magnetic field on the conical surface. Further if the top of the cone at z ¼ h is again PEC, then the boundary conditions corresponding to this is Eq ¼ Ej ¼ 0; Hz ¼ 0; z ¼ h ^ Note that ^z ¼ cos ðqÞ^r  sin ðqÞq

(6.2)

Hence, the boundary condition can also be expressed as Hr cos ðqÞ  sin ðqÞHq ¼ 0; r  cos ðqÞ ¼ h

(6.3)

We now write down the Maxwell curl equations in the spherical polar coordinate system. Quantum image processing via the Belkin filter, 0 1 ^ sin ðqÞ ^ =r ^r =r2 sin ðqÞ q=r j B C @ @ @ C curl E ¼ detB @ A (6.4) @r @q @j Er rEq r sin ðqÞEj   ^ þ Hj j ^ ¼ jwm Hr^r þ Hq q

Terahertz conical dielectric resonator antenna

91

Similarly, 0

curl H

^ sin ðqÞ ^r =r2 sin ðqÞ q=r B @ @ ¼ detB @ @r @q Hr rHq   ^ þ Ej j ^ ¼ jw
Er^r þ Eq q

1 ^ =r j C @ C A @j r sin ðqÞHj

Writing out the components explicitly gives ðsin ðqÞ HjÞ;q  Hq;j ¼ jw mr sin ðqÞHr   Er;j  sin ðqÞ r  Ej ; r ¼ jw mr sin ðqÞHq ðrEq Þ; r  Er;q ¼ jwm rHj

(6.5)

Hence, its dual with E ! H; H ! E and e $ m. Also these Maxwell curl equations imply the Helmholtz equation:  2  (6.6) r þ k 2 ðE; H Þ ¼ 0; k 2 ¼ w2 m
Helmholtz equations in spherical polar coordinate for that, we note that @^r ¼ 0; @r ^ @q ¼ 0; @r ^ @j ¼ 0; @r

@^r ^ @^r ¼ q; ¼ sin ðqÞ^ j; @q @j ^ ^ @q @q ¼ ^r ; ¼ cos ðqÞ^ j; @q @j ^ ^ @j @j ^ ¼ 0; ¼ ^x cos ðjÞ  ^y sin ðjÞ ¼  r @q @j

^ ¼ ^r sin ðqÞ  qcos ðqÞ Thus; r2 ðEr^r Þ ¼ ðr2 Er Þ ^r þ 2ðrEr ; rÞ^r þ Er r2^r ^ þ r2  ðsin qÞ1 Er;q j ^ With ðrEr ; rÞ^r ¼ r2 Er;q q

(6.7)

And ^ @ 2^r @ q ¼ ^r ¼ 2 @q @q ^ @ 2^r @j ^ ¼  sin q sin q^r þ cos ðqÞq ¼ sin q @j @j2 show that cotq @ 2^r 1 @ 2^r 1 @ 2^r þ þ 2 2 r2 @q r2 @q r2 sin ðqÞ @j2 @ 2^r ^ 1 1 ^ q  2 ^r þ 2 ð^r  cotqÞq ¼ r r @j2 2 ^r ¼  2 r

r2 r ¼

(6.8)

92

Terahertz dielectric resonator antennas Thus,

    2Er 2 2 ^þ ^ r2 ðEr^r Þ ¼ ^r r2 Er  2 þ 2 Er;q q Er;j j r r r2 sin ðqÞ

(6.9)

Next,       ^ ¼ r2 E q q ^ ^ þ E q r2 q ^ þ 2 rEq; r q r2 Eq q And   Eq;j ^ ¼  2Eq;q ^r þ cos ðqÞ^ j rEq; r q 2 2 r r ðsinðqÞÞ2 And further, ^ ^ ^ 1 @2q cotq @ q 1 @2q þ þ 2 2 2 2 r @q r @q r2 ðsinðqÞÞ @j2
 cotq 1^ 1 ^ þ ¼ 2 ^r  2 q ðsin ðqÞcos ðqÞÞ^r  ðcosðqÞÞ2 q 2 r r r2 ðsinðqÞÞ 2cotq 1 ^ q ¼  2 ^r  2 r r ðsinðqÞÞ2

^ r2 q¼

(6.10) Thus,   2 2 cotq 1 ^  2 Eq;q^r þ ^ ¼ ðr2 Eq Þq ^ r2 Eq q Eq;j cos ðqÞ^ j þ Eq 2 ^r  q 2 ðsinðqÞÞ2 r2 r r2 ðsinðqÞÞ2 r !     Eq;q Eq cotq Eq 2 cotq ^ r2 Eq  ^ þ q ¼ 2^r þ þ j Eq;j r2 r2 sin ðqÞ r2 r2 ðsinðqÞÞ2

Finally       ^ ¼ r2 E j j ^ þ 2 rEj r j ^ þ E j r2 j ^ r2 Ej j

(6.11)

Now 

   1 ^ ^ ¼  2 Ej;j ^r  sin ðqÞ þ qcos r2 Ej ; r j ðq Þ r ^ j ^¼ And r2 j r2 ðsinðqÞÞ2

Thus     1 1 ^ ^ ¼ ^r  2 Ej;j sin ðqÞ þ q  2 Ej;j cos ðqÞ r Ej j r r ! Ej 2 ^ r Ej  þj r2 ðsinðqÞÞ2 2





(6.12)

Terahertz conical dielectric resonator antenna

93

^ and j ^ components, The Helmholtz equation for E gives, on equating the ^r , q the following three equations:   2Er Eq;q Eq cot q Ej 2  2 þ k 2 Er ¼ 0; þ r Er  2  2 2 2 r r r r 2Er;q Eq Ej;j 2  2 þ k 2 Ej ¼ 0; r Eq þ 2  r r r2 ðsinðqÞÞ2 Ej 2 2cot q 2 Er;j þ 2 Eq;j ¼ 0 þ r Ej  r sin ðqÞ r2 ðsinðqÞÞ2 r2 sin ðqÞ Likewise, with E replaced by H, it is hard to solve this Helmholtz equation taking into account the earlier Maxwell curl equations. Before delving into the special case of these equations, we simplify matters assuming that the entire electromagnetic field is replaced by a single scalar field yðr; q; jÞ within the cone. The Helmholtz equation is  1 2 1 y;r þ y;rr þ 2 cos ðqÞy;q þ y;qq þ 2 y;jj þ k 2 y ¼ 0 r r r

(6.13)

With the boundary conditions that y vanishes on all the surface of the cone, we can equivalently express this equation as 2 y y þ y;rr  L2 2 þ k 2 y ¼ 0 r ;r r where 1 @ @ 1 @2 L ¼  sin ðqÞ þ sin ðqÞ @q @q ðsinðqÞÞ2 @j2

!

2

(6.14)

is the angular part of the Laplacian or what is known in quantum mechanics as the square of the angular momentum operator separation of variable gives us Yðr; q; jÞ ¼ RðrÞYlm ðq; jÞ where Ylm are the spherical harmonics that satisfy the eigen relations: L2 Ylm ¼ lðl þ 1ÞYlm ; Lz Ylm ¼ mYlm ; Lz ¼ i

@ @j

(6.15)

Thus, the radial component of the wave function R(r) satisfies   0 00 2rR ðrÞ þ r2 R ðrÞ þ k 2 r2  lðl þ 1Þ þ RðrÞ ¼ 0 We substitute it into this equation: RðrÞ ¼ ra F ðrÞto get    0 0 00 2r ara1 F ðrÞ þ ra F ðrÞ þ r2 aða  1Þr2 F ðrÞ þ 2ara1 F ðrÞ þ ra F ðrÞ þ ðk 2 r2  lðl þ 1Þra F ðrÞÞ ¼ 0

94

Terahertz dielectric resonator antennas Or equivalently,   00 0 r2 F ðrÞ þ F ðrÞð2r þ 2arÞ þ F ðrÞ aða þ 1Þ þ k 2 r2  lðl þ 1Þ ¼ 0 Choosing a ¼ 1/2 gives us  2 ! 0 F ð r Þ 1 ¼0 r 2 F ðr Þ þ r þ F ðr Þ k 2 r 2  l þ r 2 00

(6.16)

which is the Bessel equation of the order l þ 1=2: Its solution is F ðrÞ ¼ jlþ1=2 ðkrÞ And hence, RðrÞ ¼ r1=2 jlþ1=2 ðkrÞ The general solution to the three-dimensional Helmholtz equation then can be expressed as a superposition: X Yðr; q; jÞ ¼ cðl; mÞr1=2 jlþ1=2 ðkrÞ Ylm ðq; jÞ i0;jmjl

The first boundary condition is
a  X
a  cðl; mÞjlþ1=2 ðkrÞYlm ; j ¼ 0 0 ¼ r; ; j ¼ 2 2 lm Or equivalently writing the spherical harmonics in terms of modified Legendre polynomials: Ylm ðq; jÞ ¼ Plm ðcos qÞexpðimjÞ

(6.17)

This Boundary condition gives us
X a jlþ1=2 ðkrÞ ¼ 0; 8 m 2 z; 0  z  L cðl; mÞPlm cos 2 l pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where L ¼ h2 þ a2 is the length of the side of cone. The other boundary condition is Y that vanishes on the circular lid at z¼h, 0  q  a2 : 0 ¼ ðh  tanq; q; jÞ So that X a cðl; mÞPlm ðcos qÞjlþ1=2 ðkhtanqÞ ¼ 0; 8 m 2 z; 0  q  2 l

(6.18)

By multiplying these equations by linearly independent function of r and q; respectively, and integrating over the appropriate range, we get a sequence of linear homogeneous equations for cðl; mÞ; l  0foreach m that therefore have a nontrivial solution if the corresponding infinite determinant vanishes this, which gives as a

Terahertz conical dielectric resonator antenna

95

nonlinear equation for k from which one can in principle obtain the characteristics frequencies of oscillation. We now come back to the electromagnetic field case and analyze it in a different way using the multiple-expansion method based on vector-valued functions. Let electric field be given by Elm ¼ fl ðrÞLYlm ðq; jÞ where L ¼ ir  r is the usual angular momentum vector in quantum mechanics then ^r  Elm ¼ 0:

(6.19)

Also, the Helmholtz equation for Elm and the fact that r commutes with L implies that 2

2 0 lðl þ 1Þ 00 fl ðrÞ þ fl ðrÞ  f l ðr Þ þ k 2 f l ðr Þ ¼ 0 r r2 Or equivalently   00 0 r2 fl ðrÞ þ 2rfl ðrÞ þ k 2 r2  lðl þ 1Þ fl ðrÞ ¼ 0 which has solutions fl ðrÞ ¼ hl ðkrÞ where hl are the Hankel functions. Elm also satisfies the Gauss equation: divElm ¼ 0

(6.20)

Since ^r :L ¼ 0 And r  Ly ¼ ir  ðr  ryÞ ¼ ir  ðr  ðryÞÞ ¼ 0 Since r  r ¼ 0, we can repeat the same analysis for magnetic field by defining hlm ¼ hl ðkrÞLYlm ðq; jÞ Thus, the general solution to the Helmholtz equation in spherical polar coordinate can be expressed as X Eðw; rÞ ¼ ½cðl; mÞhl ðkrÞLYlm ð^r Þ þ d ðl; mÞcurlðhl ðkrÞLYlm ð^r ÞÞ=jw
 lm X ½d ðl; mÞhl ðkrÞLYlm ð^r Þ  cðl; mÞcurlðhl ðkrÞLYlm ð^r ÞÞ=jwm H ðw; rÞ ¼ lm

(6.21)

96

Terahertz dielectric resonator antennas It is a simple matter to verify that these fields satisfy Maxwell’s curl equations: curlE ¼ jwmH; curlH ¼ jw
E By making use of the relations,   divðhl ðkrÞLYlm ð^r ÞÞ ¼ 0; r2 þ k 2 ðhl ðkrÞLYlm ð^r ÞÞ ¼ 0

For our cone problem, the coefficient cðl; mÞ, d ðl; mÞ and characteristics oscilpffiffiffiffiffi lation frequency w ¼ k= me are obtained from the following boundary conditions:
a 
a  Er r; ; j ¼ 0; Hq r; ; j ¼ 0; 0  r  L; 0  j  2p 2 2 (6.22) a Ez ðh  tanq; q; jÞ ¼ 0; 0  q  ; 0  j  2p 2 Denote the characteristics frequencies by w½n; n ¼ 1; 2; 3 . . . and for each n, we have eigen vector ðcðl; m; nÞÞh ; ðd ðl; m; nÞÞn : Then the general solution in a time domain can be expressed as " X Eðt; r; q; jÞ ¼ Re ½cðl; m; nÞhl ðk ½nrÞLYlm ð^r Þexpðjw½ntÞ lm;n

þd ðl; m; nÞcurlðhl ðk ½nrÞLYlm ð^r Þexpðjw½ntÞÞ=jw½n
 n ð X ynþ1 ðxÞ ¼  S ðx  yÞgm yk ðyÞAmnk ðyÞd 4 y k¼0

Amðnþ1Þ ðxÞ ¼

n ð X

Gðx  yÞyk ðyÞ am ynk ðyÞd 4 y

(6.23)

k¼0

For n ¼ 0; 1; 2 . . . ; yð0Þ ðxÞ; Amð0Þ ðxÞ describe the free fields, i.e. in the absence of electron–positron and photon interaction and for n ¼ 1; 2; 3 . . . ; yðnÞ ðxÞ; AmðnÞ ðxÞ describe the higher order interaction terms. In this way, continuing up to any order N of accuracy, photon radiation field can be expressed as N X en AmðnÞ ðxÞ (6.24) A m ðx Þ ffi n¼0

Equations (6.1)–(6.24) have been developed to impart a complete view of quantum radiations in conical DRAs at higher frequencies (terahertz and optical frequencies). Table 6.1 shows a comparative analysis of the previous work. The proposed model has good bandwidth and mathematical analysis as novelty.

6.5 Equivalent electrical circuit of conical terahertz DRA This frequency-dependent resistance is also called a dynamic resistance of the circuit. Figure 6.10 is built as an equivalent electrical circuit of conical TDRA at

Terahertz conical dielectric resonator antenna

R1

97

X1 R3

R2

Z1 Zin Z2

Z3

X2

X3

Figure 6.10 Equivalent circuit of conical terahertz DRA

R1

R1

X1

R3

R2

Z1

X1 R3

R2

Z1

Zin Z2

X2

Z3

X3

First stage third-order high-pass filter

Z2

X2

Z3

X3

Second stage third-order high-pass filter

Figure 6.11 Expanded form of equivalent electrical circuit of conical terahertz DRA fundamental mode and Figure 6.11 is developed for showing higher order resonant modes. The conical TDRA input impedance of the circuit: Zin ¼ Zinreal þ Zinimaginary ðwÞ Zinreal 6¼ f ðwÞ where Zinreal means only real part of input impedance that is called only resistive part of input impedance Conical TDRA-resistive part of input impedance is a function of frequency dependent: Zinreal ¼ resistance ¼ f ðwÞ Resonant frequency of this circuit depends on R, L and C but input impedance of this circuit is a function of R, L, C and frequency: Zin ¼ f ðR; L; C; wÞ The impedance depends only on the real part of Zin and real part of Zin is dependent only on the frequency of the signal. The quality factor has been

98

Terahertz dielectric resonator antennas

estimated from the equivalent circuit; hence bandwidth, resonant frequency, components values can thus be obtained at terahertz frequency. Quality factor ðQÞ ¼ 2p  Q¼

maximum energy stored per cycle power dissipatetd per cycle

wr ðw 2  w 1 Þ vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u      sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2  

u L2 R23 1 1 2 L2 L2 R23 1 1 2 L2 1 1 1 2 u   R22 þ R 2 R3  þ 4 L2 R23  L22 þ

 u  C 2  C  R2 C þ C  C R 2 R 3  C C3 C1 C3 C1 C1 C3 C32 C1 C1 C32 u 1 1 3 1 3 3 u  

u 1 1 u 2 L2 R23  L22 þ u t : C1 C3

¼

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R1 C1  R2 C1  L2 C3 þ R2 R3 C1 C3 þ fL2 C3 þ R1 C1 C3 ðR2 þ R3 Þg R1 C1 L2 C3  R3 C1 L2 C3

6.6 Conclusion The proposed terahertz conical DRA has highest bandwidth as shown in comparative Table 6.1 and excellent gain. Hence, it is best suitable for high-speed communication requirements. Gaussian beam input excitation was used to compensate phase nonlinearity with specifications as beam power: P1¼1.0 W, wavelength: 632.47 nm, propagation vector: (0/0/1), focus point distance from source center: 1.27E06 m and linear polarization. Conical TDRA can be used to generate higher order modes depending upon a proper selection of aspect ratio; hence higher gain can be obtained at higher order modes as compared to fundamental mode. The design, dimensions, radiation pattern of THz conical DRAs, along with S11 and other antenna parameters have been obtained in Figures 6.1–6.9, and Figures 6.10 and 6.11 are drawn as equivalent R, L, C circuit based on which dynamic impedance is found to be frequency dependent. They can provide wide bandwidth and high speed data rate. TDRA can be used for optical communication. New geometry-based terahertz antenna has opened up new space for fast speed communication antennas. The equivalent circuit of TDRA is a novel field of analysis for terahertz or optical antennas.

References [1] A. A. Kisk, Y. Yin and A. W. Glisson “Conical DRA for wireless applications,” IEEE J., 2002. [2] V. Gaurav, R. S. Yaduvanshi, and V. S. Pandey, “Conical shape DRA for ultra wide band applications,” IEEE Conference, 2015. [3] S. Dash, T. Khan, and B. Kanaujia, “Conical dielectric resonator antenna with improved gain and bandwidth for X-band applications,” International Journal of Microwave and Wireless Technologies, vol. 9, no. 8, pp. 1749– 1756, 2017. [4] L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics, vol. 5, no. 2, pp. 83–90, 2011.

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[5] P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys., vol. 75, p. 024402, 2012. [6] L. Zou, W. Withayachumnankul, C. Shah, et al., “Dielectric resonator nano antennas at visible frequencies,” Opt. Express, vol. 21, no. 1, pp. 1344–1352, 2013. [7] G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Herna´ndezFigueroa, “Dielectric resonator antenna for applications in nano photonics,” Opt. Express, vol. 21, no. 1, pp. 1234–1239, 2013. [8] K. M. Luk and K. W. Leung, Dielectric Resonator Antennas, Research Studies Press LTD, England, 2003. [9] Y. Zhao, N. Engheta, and A. Alu`, “Effects of shape and loading of optical nano antennas on their sensitivity and radiation properties,” J. Opt. Soc. Am., vol. B28, no. 5, pp. 1266–1274, 2011. [10] E. Ozbay, “Plasmonic: merging photonics and electronics at nanoscale dimensions,” Science, vol. 311, pp. 189, 2006. [11] C. Balanis, Antenna Theory: Analysis and Design, 3rd ed., Wiley Interscience, New York, NY, 2005. [12] A. A. Kishk and Y. M. M. Antar, “Dielectric resonator antennas,” Antenna Engineering Handbook, 4th ed., John L. Volakis, New York, NY. [13] A. Petosa, Dielectric Resonator Antennas Handbook, Artech House, Norwood, MA, 2007. [14] A. Bonakdar and H. Mohseni, “Impact of optical antennas on active optoelectronic devices,” Nanoscale, vol. 6, pp. 10961–10974, 2014. [15] R. K. Mongia and P. Bhartia, “Dielectric resonator antennas—are view and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter Wave Comput. Aided Eng., vol. 4, no. 3, pp. 230–247, 1994. [16] S. A. Maier, Plasmonic—Fundamentals and Applications, Springer, 2007. [17] A. Mehmood, O. H. Karabey, and R. Jakoby, “Dielectric resonator antenna with tilted beam,” IEEE Antennas Wirel. Propag. Lett., 2016. [18] S. Fakhte, H. Oraizi and L. Matekovits, “High gain rectangular dielectric resonator antenna using uniaxial material at fundamental mode,” IEEE Trans. Antennas Propag., vol. 65, no. 1, pp. 342347, 2017. [19] R. S. Yaduvanshi and H. Parthasarathy, “Coupled solution of Boltzmann transport equation, Maxwell’s and Navier–Stokes equations,” IJACSA, 2010. [20] R. Cicchetti, A. Faraone, E. Miozzi, R. Ravanelli, and O. Testa, “A high gain mushroom-shaped dielectric resonator antenna for wideband wireless applications,” IEEE Trans. Antennas Propag., vol. 64, no. 7, pp. 2848–2861, 2016. [21] L. Zou, W. Withayachumnankul, C. Shah, et al., “Efficiency and scalability of dielectric resonator antennas at optical frequencies,” IEEE Photonics J., vol. 6, pp. 1–10, 2014. [22] J. D. Jackson, Classical Electrodynamics, Wiley, New York, NY, 1962. [23] R. S. Yaduvanshi and H. Parthasarathy, Rectangular Dielectric Resonator Antennas: Theory and Design, Springer, New Delhi, 2016.

100 [24] [25] [26] [27]

[28] [29]

[30]

[31]

Terahertz dielectric resonator antennas R. S. Yaduvanshi and V. Gaurav, Nano Dielectric Resonator for 5G Applications, CRC Press, Boca Raton, FL, 2020. S. Fakhte, “High gain rectangular dielectric resonator antenna using uniaxial material at fundamental mode,” IEEE Trans. Antennas Propag., 2017. M. Humayun and O. de Koo, Retinal Prosthesis—A Clinical Guide to Successful Implementation, Springer, Cham, 2017. P. Mu¨hlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science, vol. 308, no. 5728, pp. 1607–1609, 2005. E. Cubukcu, E. A. Kort, K. B. Crozier, and F. Capasso, “Plasmonic laser antenna,” Appl. Phys. Lett., vol. 89, no. 9, p. 093120, 2006. R. K. Mongia and P. Bhartia, “Dielectric resonator antenna—A review and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter Wave Comput. Aided Eng., vol. 4, pp. 230–247, 1994. A. Petosa and A. Ittipiboon, “Dielectric resonator antennas: A historical review and the current state of the art,” IEEE Antennas Propag. Mag., vol. 52, pp. 91–116, 2010. S. K. K. Dash, T. Khan, and A. De, “Dielectric resonator antennas: An application oriented survey,” Int. J. RF Microwave Comput. Aided Eng., vol. 27, no. 3, 2017.

Chapter 7

Cylindrical terahertz and optical DRA—design and analysis

Abstract Terahertz cylindrical dielectric resonator antennas (DRAs) have been studied and implemented using computer simulation technology. The theoretically analyses have also been performed at terahertz frequency. Mathematical formulation for the Poynting vector and far fields radiated pattern for terahertz antennas have been developed. Their simulation results along with theoretical concepts have been presented. The work carried out in this chapter is much useful for high-speed communication applications and retinal artificial photoreceptors applications. This optical DRAs work can also be used for sensing, scanning, imaging and chip-tochip communications. The cylindrical optical DRAs have been built at 521 THz with a gain of 4.04 dBi. Another optical DRA has also been developed at 10 THz with 6-dBi gain. Simulated results on both optical DRAs have been included in this chapter. Antenna parameters such as reflection coefficient (S11), radiation pattern, voltage standing wave ratio, impedance (Z11) plots along with other antenna results are included in this chapter. These are compact antennas and suitable for 5G and beyond networks for providing better connectivity with lower RF exposer levels.

7.1 Introduction The radiation pattern is based on input to feed, current density fluctuations and retardation potentials. The fluctuations in quantum antennas must be controlled to a minimum threshold value. The radiated field is a state of jointly coherent for the bosons (i.e. photons) and for the fermions (positrons–electrons). These are basic concepts of radiations into dielectric resonator antennas (DRAs) at terahertz and optical frequencies. Hence, interaction between these bosonic and fermionic fields causes the electromagnetic (e.m.) field to change and induce surface currents that radiate out in space. This change in e.m. fields must be optimized to a threshold value, i.e. to obtain a desired variance of quantum radiation pattern fluctuations, controlling current density fluctuations. These DRAs are useful for high-speed communications and optical sensors for biomedical applications.

102

Terahertz dielectric resonator antennas

Terahertz and optical cylindrical dielectric resonator antennas (CDRAs) are simple, efficient, fascinating, futuristic, promising and an interdisciplinary subject, whose applications are in mobile and wireless communications that have capability to deliver ultrahigh-speed data rates [1]. It can be used for energy harvesting, sensing, imaging, microscopy and scanning in biomedical applications. They can be very useful to harvest energy from natural sources such as sunlight. Nanoantennas, light antennas, photonic antennas, optical antennas and quantum antennas are all synonyms used for nanometer-size antennas that operate at optical wavelength and their physics is different from classical microwave antennas [2]. Light–matter interaction is a well-known phenomenon in photonic antennas. Scattering light particle takes place, and mathematical solution for scattering e.m. waves can be provided with coupled Boltzmann–Maxwell’s equation or using Drude’s model theory [3]. This phenomenon of light–matter interaction is used in photonic or optical antennas. Photonic antennas are nanocircuits and thus can be used for chip-to-chip communication in electronic printed circuit boards and for inhouse communications. Photonic antennas can operate in coherent states. Mario Agio and Andrea Alu` have accomplished tremendous work in the field of optical antennas [4–7]. They can be well described based on Drude’s model, where metal behavior is different at optical frequencies. e.m. Signal at optical frequency can partially penetrate through metals. SPP (surface plasmon polytrons) are emissions from light–matter interactions and dominated by florescence as coherent absorptions–emissions as dipole moment scattering, which is described by Green’s function tensor.

7.2 Model 2 TCDRA at 10-THz resonant frequency 7.2.1 1.

Design computations

Here, all the measurements are given in terms of free-space wavelength (lo) Resonant frequency ¼ 10 THz Impendence ¼ 50 W We know, l¼

c f

where c is the speed of light in free space ¼ 3  108 m/s lo ¼

2.

3  108 ¼ 0:3  104 m 10  1012 ¼ 30 mm

Substrate (SiO2) ! 1.5  lo ¼ 1.5  30 mm ¼ 45 mm

Cylindrical terahertz and optical DRA—design and analysis

3.

4.

5.

103

i.e. length and breadth of substrates (by the help of basic model given we know the dimensions of substrate, ground, nanostrip waveguide and nano-DRA). The height of substrate is given by ¼ lo/3 ¼ 30/3 mm ¼ 10 mm (use double height of substrate for high gain) So, length  breadth  height ¼ 45  45  10 mm3 Ground (silver) ! 1.5  lo ¼ 1.5  30 mm ¼ 45 mm i.e. length and breadth of silver ground. We use an equal height of ground and substrate (given by basis model) ¼ 10 mm So, length  breadth  height ¼ 45  45  10 mm3 Nanostrip waveguide (silver): ! Length of nanostrip ¼ 1/2  length of substrate or ground ¼ 1/2  45 ¼ 22.5 mm Width and height of nanostrip waveguide is according to matching, but the width of nanostrip should be less than or equal to width of nano-DRA used. Nano-DRA (silver) ! Here, the fixed height of dielectric resonator in terms of wavelength (lo) ¼ lo/10 ¼ 30/10 mm ¼ 3 mm (height of DRA) Now, by the variation of the diameter (D) of the dielectric resonator we can achieve resonance in this nanoantenna for resonance, we have to put the nanoDRA having an aspect ratio between 0.5 and 2.5 (height of DRA/radius of DRA). We know the relation between resonance frequency and dimension of DRA. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2   C 4 p þ 0 2 Fmnp ¼ pffiffiffiffiffiffiffiffiffiffiffi 2 2 rmr 3a h p ¼ 1 (for fundamental) a ¼ side of DRA and h0 ¼ 2  height of DRA According to basic model, height is fixed at lo/10 ¼ 3 mm As an aspect ratio between 0.5 and 2.5, we select the radius of cylinder ¼ 4 mm And the diameter of cylinder ¼ a ¼ 8 mm The height of cylinder ¼ 3 mm; so, h0 ¼ 2  height of DRA ¼ 6 mm  aspect ratio of DRA ¼ 3/4 ¼ 0.75 We use silicon DRA with er ¼ 11.9 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2  2 3  108 4 1 Fmnp ¼ pffiffiffiffiffiffiffiffiffiffi þ 3  8  106 6  106 2 11:9 ’10 THz (resonant frequency)

104

Terahertz dielectric resonator antennas

7.3 Terahertz antennas detailed description Both microwave and photonic antennas have common features such as resonant frequency, impedance, current, rower, gain, reflection coefficient (S11), scattering parameters, voltage-standing wave ratio (VSWR), bandwidth, beam width, directivity, polarization, permittivity, permeability, dielectric constant, efficiency, quality factor, resonant modes, axial ratio, aspect ratio, frequency ratio, low profile, directive gain (DG), magnetic dipole, electric dipole, dipole moment, higher order modes, Poynting vector, magnetic vector potential, wave number, boundary conditions, image effect, Bessel’s function, Green’s function, frequency spectrum, conductivity, wavelength (lffiffiffiffiffiffiffiffiffiffiffi ¼ C=f ); group p   delay, loss tangent, tan (@o ), freespaceimpedance h ¼ mo =o ¼ 377 W , end-fire radiation pattern, broadside radiation pattern, power gain, multiband, correlation, circular polarization, RHCP and LHCP (right-hand circular polarization and left-hand circular polarization), MIMO (multi-input–multi-output), envelopment correlation coefficient, DG, mean effective gain, channel capacity loss, total active reflection coefficient, surface current, volume current and bore sight radiation pattern. This process in microwave antennas just involves expressing the transverse components of the E–H fields in terms of the longitudinal components of the E and H fields. Computing the coefficients of the longitudinal components using the specified field pattern of the cylindrical and cone-shaped microwave antennas is essential to determine operating resonant modes. Indeed photonic case, today we know that the e.m. field is neither a ray nor a wave field. It is actually a quantum wave operator field. The mechanism of the photonic antenna is described by scattering. The distortion of the surface current is due to the source of the light scattering. The surface current on the surface is proportional to the total tangential magnetic field and the scattering amplitude is also proportional to the total tangential magnetic field on the f surface antennas, which are solved based on volume currents instead of surface currents. Displacement currents are introduced due to dynamic inductances and resistance becomes a function of frequency in photonic antennas. Drude’s model solution is obtained by taking all scattering parameters of SPPs. In currents induced by e.m. fields, it is easy enough to expand the whole scattering process by calculating the reemitted light with the well-known formula for the vector potential. The secondorder resonance shows the familiar quadruple emission pattern. More precisely, it is an ensemble of an infinite number of quantum harmonic oscillators or in fact, a superposition of plane or constrained plane waves with coefficients being creation and annihilation operators in a boson Fock space. It is therefore not quite precise to talk about the wave-field pattern on the photonic antennas but rather specify the quantum state of the wave field like a coherent state, or a number state and then compute the statistics of the wave operator field. On the surface, the action of creation and annihilation operators on coherent states or on photon number states is used. After we do such an analysis, we can ask the more interesting question: how does light within the cylindrical cavity resonator interact

Cylindrical terahertz and optical DRA—design and analysis

105

with the matter field? Specially, when light propagates within the cylindrical eye cavity, it interacts with electrons, atoms and molecules of the fluid medium of which the cavity is composed of fluid and plasma medium. In classical wave field theory, we would model this fluid medium as a plasma composed of a definite permittivity, permeability and conductivity and would analyze the propagation of the e.m. field in this plasma using the Vlasov equations that are in fact the coupled Boltzmann equation for the particle distribution function of the plasma and the Maxwell equation for e.m. field. The result of this analysis using linearized perturbation theory would be dispersion relation between the oscillation frequencies of the plasma and the e.m. field and the wave vector. However, at the quantum level, the description of the interaction between light and matter is more subtle. A simplified analysis would be to model the matter as just the second quantized electron–positron field using Dirac’s relativistic wave equation with the wave function being an operator field and then to include the photon interaction term in the usual way. In this way, the total Hamiltonian of the photon and matter field splits into three terms: one, the Hamiltonian of the e.m. field described as a quadratic form in the photon creation and annihilation operators within the cavity; two, the Hamiltonian of the Dirac field with cylindrical cavity boundary conditions on the wave function described as a quadratic form in the electron–positron creation and annihilation operators; and three, the Hamiltonian of the interaction between the photon and electron–positron field described as a quadratic form in the electron– positron creation–annihilation operators multiplied with a linear form in the photon creation–annihilation operators. This interaction Hamiltonian can be used to compute the amplitudes for scattering but since we are primarily interested in the statistics of the e.m. wave field pattern on the retinal surface, we shall describe this interaction using the Dirac current density expressed as a quadratic form in the electron–positron creation–annihilation operator fields that drives the photon field using the wave equation for the e.m. field in the presence of a current density. Thus, we shall by perturbation theory be able to calculate the change in the e. m. field pattern on the retinal surface caused by the Dirac current in terms of operators and then by assuming a definite state of the photon–electron–positron field, we shall be able to compute the mean and covariance of the e.m. field on the retinal screen surface in this state. The second quantized Dirac fermionic field can be expressed on the other hand in terms of fermionic creation and annihilation operators. In the photon case, the antiparticle of a photon is again a photon and so its creation operator at a given momentum helicity is the adjoint of the corresponding annihilation operator. On the other hand, the antiparticle of the electron is another particle, the positron, and hence the Dirac field should be expressed as a superposition of electron annihilation operators and positron creation operators. The creation of a positron of positive energy is according to Dirac equivalent to annihilating an electron of negative energy. Thus, the Dirac field is expressed as the superposition, given a quantum matter field, either bosonic or fermionic, it satisfies a classical wave equation of the Klein–Gordon type, or the 3D wave type or the Dirac type with certain boundary conditions. This second quantized field can therefore be expressed as a superposition of eigen functions corresponding to the

106

Terahertz dielectric resonator antennas

boundary conditions with coefficients being particle creation and annihilation operators. If the field is bosonic, these creation and annihilation operators will satisfy CCRs while if the field is fermionic, they will satisfy CCRs. If H corresponds to the Hamiltonian of the first quantized theory, then we can write the second quantized Hamiltonian of the field as when the electron–positron field interacts with the bosonic radiation field, the Dirac current density acquires extra terms involving coupling between the fermionic and bosonic components. We analyze this interaction in what follows. The Dirac equation in the presence of the radiation field is given by the free Dirac current, i.e. in the absence of interactions with the photon field. We have already indicated how to compute the far-field radiation pattern produced by this field and how to evaluate the moments of this field. Specifically, if G(x, y) denotes causal Green’s function for the wave operator, then the e.m. four potentials produced by the Dirac current are given. We can consider a fermionic coherent state rather than a fermionic number state. Such a state is parameterized by a Grassmannian vector variable. By imposing such restrictions, we can calculate easily the moments of the current density field and hence of the radiated field is a state that is jointly coherent for the bosons (i.e. photons) and for the fermions. We observe that the perturbation to the current density of the Dirac field caused by the interactions between the electron–positron field and the photon field is given up to first order in the photon field and second order in the fermionic field by an expression of the form. This expression is manifestly trilinear in the operators. Specifically, it is quadratic in the electron–positron field and linear in the photon field, totally yielding a trilinear term. It can be expressed as entanglement of the fermionic modes caused by interaction with the photon radiation field. When the electron–positron field interacts with the bosonic radiation field, the Dirac current density acquires extra terms involving coupling between the fermionic and bosonic components. We analyze this interaction in what follows. The Dirac equation is used in the presence of the radiation field. From this expression, it is clear that the photon operators tend to couple the other modes of the electron–positron field and hence produce additional terms in the far-field radiation, a homogeneous polynomial of degree three in the electron–positron– photon operators with the photon operators appearing with a total degree of m and the electron–positron operators appearing with a total degree of 2 m. We can also evaluate the previous moment in a joint coherent state.

7.4 Theory of terahertz cylindrical DRA and mathematical formulations Magnetic and electric dipole and dipole moment formations are an important phenomenon before radiations are achieved in photonic antennas. At a nanometer scale or optical frequency the device has modeled as in the following in comparison with the microwave antenna using required mathematical derivations to establish the fact.

Cylindrical terahertz and optical DRA—design and analysis

107

Let d be the height of the cylindrical cavity and a its radius as shown terahertz cylindrical DRA (TCDRA) design. Assume that all its walls, including the top and bottom surfaces, are perfect conductors. At the classical scale, using the fundamental identities resulting from the Maxwell curl equations relating the transverse components of the e.m. field to the longitudinal component is given as follows:     1 jwm r? Hz  ^z (7.1) E ? ¼ @z 2 r ? Ez  h h2     1 jwm H? ¼ @z 2 r? Hz  (7.2) r? Ez  ^z h h2 Also the longitudinal components of the Maxwell curl equation give us r?  E? ¼ jwm Hz^z

(7.3)

r?  H? ¼ jw Ez^z

(7.4)

In terms of the cylindrical coordinate components,    g jwm @f Hz E r ¼  2 @r E z  h h2    g  jwm @r Hz E f ¼  2 r @f Ez þ h h2    g jw Hr ¼  2 @r Hz  r @f Ez h h2    g jw r @f E z Hf ¼  2 @r Hz  h h2

(7.5) (7.6) (7.7) (7.8)

where g is to be replaced by the operator @=@z ; it should be noted that the values assumed by g are pp=d, where p is an integer in order that the tangential components of the electric field and the normal component of the magnetic field vanish on the top and bottom surfaces of the CDRA. We note that ^ ^ þ Ef f E? ðr; f; zÞ ¼ Er r And likewise for H? also   ^ f ^  @r þ @f r? ¼ r r

(7.9)

(7.10)

We are using the abbreviation @x ¼

@ ; xi ¼ r; f; z @x

(7.11)

108

Terahertz dielectric resonator antennas Then using (7.1)–(7.4), we get  2  r? þ h2 ðEz ; Hz Þ ¼ 0

(7.12)

We note that application of the boundary conditions, namely Ez ; Hf ; Hr vanish at r, while Er ; Ef ; Hz vanish at r ¼ 0; d; i.e. the tangential components of the electric field and the normal components of the magnetic field vanish on all the boundaries gives us for the TM modes, h2 ¼ h2 ðEmn Þ ¼

pp2 am ½n2 2 ¼ w ð E; mnp Þ   m a2 d

(7.13)

where m; n; p are integers and am ½n; n ¼ 1; 2; . . . are the roots of the Bessel function Jm ðxÞ; and for the TE modes, h2 ¼ h2 ðH; mnÞ ¼

pp2 bm ½n2 ¼ wðH; mnpÞ2 m  2 a d

(7.14)

where bm ½n2 ¼ 1; 2 . . . are the roots of Jm0 ðxÞ ¼ 0; thus, we obtain the following expansions: 

        am ½nr ppz Ez ðt; r; f; zÞ ¼ Jm cos Re cðE; mnpÞexp j mf  w Emnp t a d mnp  X     Re cðE; mnpÞexp jw Emnp t mmnp ðrÞ X

mnp

(7.15)     X b ½nr ppz Hz ðt; r; f; zÞ ¼ Jm m sin ReðcðH; mnpÞexpðjðmf  wðH; mnpÞtÞÞÞ a d mnp X      Re cðH; mnpÞexp jw Hmnp t vmnp ðrÞ mnp

(7.16)

From these expressions, we can easily derive the corresponding expansions for the tangential components of the e.m. field in the time domain: X 1

        E? ¼ ðEmn Þ Re c Emnp exp jw Emnp t @z umnp ðrÞ h mnp            P m  mnp ðHmn Þ2 Re jw Hmnp c Hmnp exp jw Hmnp t r? nmnp ðrÞ  ^z h (7.17) 2

And likewise H? ¼

X 1

        ðHmn Þ2 Re c Hmnp exp jw Hmnp t @z nmnp ðrÞ h mnp            P  þ mnp ðHmn Þ2 Re jw Emnp c Emnp exp jw Emnp t r? umnp ðrÞ  ^z h (7.18)

Cylindrical terahertz and optical DRA—design and analysis

109

Note that the characteristic frequencies of oscillation of the TM modes are ! 2 pp2 1   1=2 am ½n w Emnp ¼ ðmÞ (7.19) þ a2 d 2 And those for the TE mode are 



w Hmnp ¼ ðmÞ

1=2

! bm ½n2 pp2 1 þ a2 d 2

(7.20)

Thus, we can write        Re c Emnp exp jw Emnp t yEmnp ðrÞ mnp  X       Re c Hmnp exp jw Hmnp t cEmnp ðrÞ þ

Eðt; r; f; zÞ ¼ Eðt; rÞ ¼

X

(7.21)

mnp

Equations (7.1)–(7.21) have been used to mathematically formulate E and H fields of TDRA. Where, now yEmnp ðrÞ and cEmnp ðrÞ are C3 —vector-valued complex function of the position variable. The first summation before is the contribution to the total electric field coming from the TM components, while the second summation is the contribution to the total electric field coming from the TE components. Likewise,  X       Re c Emnp exp jw Emnp t yHmnp ðrÞ H ðt; r; f; zÞ ¼ H ðt; rÞ ¼ mnp  X       Re c Hmnp exp jw Hmnp t cHmnp ðrÞ þ mnp

(7.22) Note that  am ½nr  expðjmfÞJm umnp ðrÞ ¼ cos d a   ppz b ½nr  expðjmfÞJm m nmnp ðrÞ ¼ sin d a ppz



(7.23) (7.24)

Thus, pp ppz a ½nr m @z umnp ðrÞ ¼  sin Jm expðjmfÞ d d a

(7.25)

The vector-valued function yEmnp ; cEmnp ; yH ðmnpÞ; cH ðmnpÞ possess the usual orthogonality properties. After appropriate normalization of these function, using

110

Terahertz dielectric resonator antennas

the orthonormality of the vector-valued functions, we can express the total energy in the e.m. field within the cavity C as  ð m ð 2 H ðt; rÞj2 d 3 r HF ¼ Eðt; rÞj d 3 r þ 2 2 (7.26) cavity X c          

  w Emnp c Emnp c Emnp þ w Hmnp c Hmnp Þ c Hmnp mnp

Hence it is clear that after quantization, in   order to obtain the correct ; t with time as time dependence of the coefficients, iec E mnp      it should   vary  exp iw Emnp twhilec Hmnp ; t should vary as exp iw Hmnp t ; we must enforce the bosonic commutation relations h  i   c Emnp ; c Em0 0 n0 p0 Þ ¼ d½m  m0 d½n  n0 d½p  p0  h  i (7.27)   c Hmnp ; c Hm0 0 n0 p0 Þ ¼ d½m  m0 d½n  n0 d½p  p0  with all the other commutators vanishing. We then easily obtain using these commutation relations that        

 dc Emnp ; t ¼ i HF ; c Emnp ; t ¼ iw Emnp c Emnp ; t (7.28) dt     

    dc Hmnp ; t ¼ i HF ; c Hmnp ; t ¼ iw Hmnp ; t c Hmnp ; t (7.29) dt On solving which, the desired time dependence of these coefficients is obtained. These commutation relations can alternatively be obtained by requiring that the Maxwell equations follow from the Heisenberg equations for the e.m. field operators. We shall now briefly discuss the computation of the far-field radiation pattern. The surface current density induced on the sidewalls of the cavity is given by r  H ðt; a; f; zÞ Jss ðt; f; zÞ ¼ ^ ^ ¼ Hf ðt; a; f; zÞ^z þ Hz ðt; a; f; zÞf

(7.30)

That induced on the bottom surface is Jsb ðt; r; fÞ ¼ ^z  H ðt; r; f; 0Þ ^ þ Hf ðt; r; f; 0Þ^ ¼ Hr ðt; r; f; 0Þf r

(7.31)

And that on the top surface is ^ þ Hf ðt; r; f; d Þ^ r Jst ðt; r; f; d Þ ¼ Hr ðt; r; f; d Þf

(7.32)

Cylindrical terahertz and optical DRA—design and analysis

111

By using the formulas ^ ¼ ^x  sin ðfÞ þ ^y  cos ðfÞ ^ ¼ ^x  cos ðfÞ þ ^y  sin ðfÞ; f r

(7.33)

We can evaluate the far-field-radiated magnetic vector potential as m



jkr exp  Aðw; rÞ ¼ 4p r

ð

Js ðw; r0 Þexpðjk^r  r0 ÞdS ðr0 Þ

(7.34)

SC

where SC denotes the boundary surface of the CDRA. The far-field quantum e.m. field radiated the CDRA is easily seen to be a linear function of    by      c Emnp ; c Emnp ; c Hmnp ; c Hmnp ; m; n; p  Z. This is because the magnetic field is linear in these observables and hence the surface current density on the CDRA is also linear observables. Thus, we can express the radiated e.m. field in the form X      c Emnp F1 ðmnp; rÞexp iw Emnp t

Eðt; rÞ ¼

mnp

    

 1 ðmnp; rÞexp iw Emnp t þ c Emnp Þ F X      c Hmnp F2 ðmnp; rÞ  exp iw Hmnp t þ mnp

      2 ðmnp; rÞ  exp iw Hmnp t þ c Hmnp Þ F

(7.35)

And likewise for the radiated magnetic field: H ðt; rÞ ¼

X      c Emnp G1 ðmnp; rÞexp iw Emnp t mnp

      1 ðmnp; rÞexp iw Emnp t þ c Emnp Þ G X      c Hmnp G2 ðmnp; rÞ  exp iw Hmnp t þ mnp

      2 ðmnp; rÞ  exp iw Hmnp t þ c Hmnp Þ G

(7.36)

where Fk ; Gk ; k ¼ 1; 2 are complex 3  1 vector-valued functions of position only. The far-field electric field pattern has only the 1=r dependence and hence we can express it in the from

112

Terahertz dielectric resonator antennas  Xh      r c Emnp exp iw Emnp t  Q1 ðmnp; ^r Þ c mnp   i    r  ^ ð mnp; r Þ þ c Emnp  exp iw Emnp t  Q c 1   Xh     r þ r1 c Hmnp exp iw Hmnp t  Q2 ðmnp; ^r Þ c mnp   i    r  ^ þ c Hmnp  exp iw Emnp t  Q ð mnp; r Þ c 2

Eðt; rÞ ¼ r1

(7.37)

with a similar expression for the far-field magnetic field pattern. In fact, using the Maxwell equation curlE ¼ jwmH It easily follows that the far-field magnetic field pattern is given by  Xh      r c Emnp exp iw Emnp t  ^r  Q1 ðmnp; ^r Þ H ðt; rÞ ¼ r1 h1 c mnp    i   r   1 ðmnp; ^r Þ ^r  Q þ c Emnp  exp iw Emnp t  c   Xh     r r1 h1 c Hmnp exp iw Hmnp t  ^r  Q2 ðmnp; ^r Þ c mnp þc



Hmnp



  i  r   2 ðmnp; ^r Þ ^r  Q  exp iw Emnp t  c

(7.38)

where rffiffiffi m h¼  From these expressions, it is immediate that the far-field time-averaged quantum pointing vector field pattern is given by   "X h    2^r 2 r c Emnp Þ c Emnp Q1 ðmnp; ^r Þj2 P ðr Þ ¼ h mnp i     (7.39) þ c Hmnp c Hmnp Q2 ðmnp; ^r Þj2 We can calculate the quantum average of the fields and Poynting vector in any state of the photons, for example in a coherent state of the CDRA. In such a state

Cylindrical terahertz and optical DRA—design and analysis

113

jfðuÞi; we have     (7.40) hfðuÞ c Emnp fðuÞi ¼ u Emnp ;    (7.41) hfðuÞ c Emnp Þ fðuÞi ¼ u Emnp ;      where u ¼ u Emnp ; u Hmnp is an infinite dimensional complex vector that parameterizes the state of the photons in the quantum e.m. field within the CDRA. In order to calculate the covariance of quantum fluctuations in the e.m. field in a coherent state, or the average value the Poynting vector in a coherent state or more generally, the higher order moments of the field in a coherent state, we require identities as      hfðuÞ c Emnp Þ c Emnp fðuÞi ¼ Emnp j2  tl hfðuÞ Prk¼1 cðEmk nk pk Þ qk Psl¼1 Em0l n0l p0l jfðuÞi  tl ¼ Prk¼1 uðEmk nk pk Þqk Psl¼1 u Em0l n0l p0l

(7.42)

These identities should be combined with the commutation rules to ensure that the moments of these field creation and annihilation operators taken in any order can be expressed as linear combinations of the moments of these operators in the normal order, i.e., in each turn all the creation operators appear to the left of all the annihilation operators. It should be noted that these moment can also be computed in a state of the field corresponding to a finite number of photons in prescribed number states. For example, the state.  k  jyi ¼jN1 ðm1 n1 p1 Þ; . . . ; Nk ðmk nk pk Þ ¼  Nj mj nj pj i

(7.43)

j¼1

Represents a state in which the (mj nj pj Þth-mode photon is in the Nj -th number state. Thus we have   





qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     cðmnpÞ Nj mj nj pj i¼d mmj d nnj d ppj Nj mj nj pj Nj mj nj pj 1i (7.44) And     cðmnpÞ Nj mj nj pj i ¼ j1ðmnpÞ; Nj mj nj pj i Provided that ðmnpÞ 6¼ (mj nj pj Þ and otherwise qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi         c mj nj pj Nj mj nj pj i ¼ Nj mj nj pj þ 1 Nj mj nj pj þ 1i

(7.45)

(7.46)

Equations (7.23)–(7.46) have been used to formulate state of quantized fields of TCDRA.

114

Terahertz dielectric resonator antennas

7.5 Optical CDRA description A coarse approximation of this matter is to regard as a sea of electrons and positrons and then apply the formalism of quantum electrodynamics to describe its interaction with the incident photons, both before the light hits the cylindrical cavity matter directed based on boundary conditions. The e.m. response is then dictated by collective electron oscillations (plasmons) characteristic of a strongly coupled plasma [1–6]. In CAR and CCR, the momentum of the current density field and hence of the radiated field is a state that is jointly coherent for the bosons (photons) and for the fermions (electrons and positrons). Hence, e.m. fields are realized by quantum wave operator fields. Here, the electrons and positrons interaction with photons or the quantum light–matter field in a fermionic bosonic coherent state (scattering, distortions, SPP and Hamiltonian as the amplitudes for scattering). The parameters of the fermionic and bosonic annihilation operators define fermionic and bosonic as Dirac field (momentum–spin, photonic wave field). The Dirac current (displacement type, tangential magnetic field), combined effect of electrons positrons and photons, eigen solution to the Dirac wave equation using 44 Dirac a-matrices satisfying the anti-commutation relation express the Dirac current density (e.m. wave field pattern), find far-field magnetic vector potential (retardation potentials), determine far-field radiation pattern (quantum wave operator field) and Berezin weight function parameterized by q (Grassmann variables). Minimizing the quantum fluctuation correlation in the pattern is used for possible matching the average antenna pattern to the desired pattern (boson Fock space) [7]. The mean and mean square fluctuations of quantum e.m. fields (antenna and free photons) are analyzed. Radiations are super directive (emitted vector potential, photons with different momenta and helicities, e.m. four potential fields), and radiated field is a state of jointly coherent for the bosons (i.e. photons) and for the fermions (positrons–electrons) [8]. Using the previous formulae, we calculate the expected value of  sj  0  0 0 (7.47) Prj¼1 c Epj qj rj Pm k¼1 Ep k qk rk in the previous state jyi: The photon field inside the CDRA interacts with matter in the form of electrons and positrons. The free Dirac field as i@t yðt; rÞ ¼ ðða; irÞ þ bmÞyðt; rÞ

(7.48)

where a1 ; a2 ; a3 ; b are the Dirac 4  4 matrices mutually anticommuting with each having the identity as its square. yðt; rÞ is a four-component operator wave field. We must solve this equation within the cavity with the boundary condition that y vanishes on the boundary surface of the cavity. Thus, we take our basis functions as   ppz am ½nr expðimfÞsin ; ðmnpÞZ3 (7.49) cmnp ðrÞ ¼ N ðmnpÞJm a d

Cylindrical terahertz and optical DRA—design and analysis

115

where N ðmnpÞ are appropriate normalizing constants. We expand the wave field as X yðt; rÞ ¼ ½aðmnpsÞcmnp ðrÞuðmnpsÞexpðiwð1mnpÞtÞ mnps

 mnp ðrÞvðmnpsÞexpðiw2mnpÞt þbðmnpsÞ c

(7.50)

Using the orthonormality of the functions cmnp and their gradients, we find that the condition that y satisfy the Dirac equation is that 2 3 ð    mnp ðrÞ a; ircmnp ðrÞ d 3 r þ bm5u ðmnpsÞ wð1mnpÞuðmnpsÞ ¼ 4 c c

(7.51) And 2 3 ð   wð2mnpÞvðmnpsÞ ¼ 4 cmnp ðrÞ a; ir c mnp ðrÞ d 3 r þ bm5v ðmnpsÞ c

(7.52) This implies that on defining ð   mnp ðrÞ ir PðmnpÞ ¼ c c mnp ðrÞd 3 r

(7.53)

c

which implies by integration by parts that PðmnpÞ is a real vector that wð1mnpÞmðmnpsÞ ¼ ½ða; PðmnpÞÞ þ bm uðmnpsÞ

(7.54)

wð2mnpÞvðmnpsÞ ¼ ½ðða; PðmnpÞÞ þ bm vðmnpsÞ

(7.55)

Titanium oxide (resonating element)

Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Lumped port

Figure 7.1 Terahertz CDRA design with silver nano waveguide

116

Terahertz dielectric resonator antennas

RC

LSUB

HC HSUB

LN WN

HG

LG WG/WSUB

Figure 7.2 Terahertz CDRA with dimensions

S11

–6.00

Name

X

Y

m1 474.4444 –10.1643 m2 524.4444 –10.0981

HFSSDesign1 Curve Info dB (S11) Setup1: Sweep

–8.00 –10.00

m2

m1

dB (S11)

–12.00 –14.00 –16.00 –18.00 –20.00 –22.00 450.00

475.00

500.00 Freq (THz)

525.00

550.00

Figure 7.3 S11 (reflection coefficient of TCDRA) From which it follows that wð2mnpÞ ¼ wð1mnpÞ ¼ wðmnpÞ is the positive eigenvalue of the 4  4 matrixða; PðmnpÞÞ þ bm; i:e: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wðmnpÞ ¼ m2 þ PðmnpÞ2 (7.56) Then, we have that mðmnpsÞ, s ¼ 1; 2 are two orthnormal eigenvectors of ða; PðmnpÞÞ þ bm corresponding to the eigenvalue wðmnpÞ while vðmnpsÞ are two

Cylindrical terahertz and optical DRA—design and analysis 8.75

VSWR

117

HFSSDesign1 Curve Info dB (VSWR(1)) Setup1: Sweep

Name X Y m1 500.0000 1.9132 m2 503.3333 1.6539

dB (VSWR(1))

7.50

6.25

5.00

3.75

2.50

m1

1.25 450.00

475.00

m2

500.00 Freq (THz)

525.00

550.00

Figure 7.4 VSWR of TCDRA

4.00

HFSSDesign1 Curve Info dB(Gain Total) Setup1: Sweep φ=′0˚′ θ =′0˚′

GAIN Name

X

Y

m1 500.0000 2.8038

m1

dB (GainTotal)

2.00

0.00

–2.00

–4.00

–6.00 450.00

475.00

500.00 Freq (THz)

525.00

550.00

Figure 7.5 Gain vs frequency of TCDRA

orthonormal eigenvectors of (a; PðmnpÞÞ  bm corresponding to the same eigenvalue. When the quantum e.m. field within the resonator cavity interacts with the electron–positron field, the total second quantized Hamiltonian is given by H ðtÞ ¼ HF þ HD þ HI ðtÞ

(7.57)

118

Terahertz dielectric resonator antennas X

Name

149.94

Z11

Y

m1 500.0000 61.9750 m2 500.0000 –2.7477

HFSSDesign1 Curve Info re(Z11) Setup1: Sweep

im(Z11)

125.00

Setup1: Sweep

Y1

100.00 75.00

m1

50.00 25.00 m2

0.00 –23.42 450.04

462.50

475.00

487.50

500.00 Freq (THz)

512.50

525.00

537.50

549.10

Figure 7.6 Impedance vs frequency of TCDRA Name Theta

Ang

Radiation pattern 1

Mag

HFSSDesign1 Curve Info dB(Gaintotal) Setup1: Sweep Freq = 500,000GHz′ I =′0˚′ dB(Gaintotal) Setup1: Sweep Freq = 500,000 GHz′ I =′90˚′

0 m1

m1 0.0000 0.0000 2.8038

30

–30 0.00 –5.00

–60

60

–10.00 –15.00

90

–90

–120

120

150

–150 –180

Figure 7.7 Radiation pattern (f—0 and 90 ) of TCDRA where HF ¼

X mnp

           

w Emnp c Emnp c Emnp þ w Hmnp c Hmnp Þ c Hmnp

is the Hamiltonian of the free constrained e.m. field, ð yðrÞ ðða; irÞ þ bmÞyðrÞd 3 r HD ¼ P   ¼ mnp wðmnpÞðaðmnpsÞ aðmnpsÞ þ bðmnpsÞ bðmnpsÞÞ is the Hamiltonian of the free constrained Dirac field and ð  HI ¼ e yðrÞ a; Aðt; rÞyðrÞd 3 r

(7.58)

(7.59)

Cylindrical terahertz and optical DRA—design and analysis Radiation efficiency

HFSSDesign1

1.00

abs(Radiation efficiency)

119

Curve Info

abs(Radiation Efficiency) Setup1: Sweep φ = ′0˚ ′θ = ′0˚′

0.75

0.50

0.25

0.00 450.00

475.00

500.00 Freq (THz)

525.00

550.00

Figure 7.8 Radiation efficiency vs frequency of TCDRA

Silicon (resonating element)

Silver nanostrip

Silicon dioxide (substrate)

Lumped port

Silver (ground)

Figure 7.9 TCDRA 10-THz design

is the interaction Hamiltonian between the Dirac field of electrons and positrons. We can express it in terms of the photon and electron–position creation and annihilation operators as follows. First note that since divE ¼ 0 within the resonator and we are adopting the coulomb gauge for which divA ¼ 0; it follows for the Maxwell theory that r2 F ¼ 0 and hence F ¼ 0 where F is the electric scalar potential. This is because the equation divE ¼ 0 implies that there is zero-charge density inside. Hence the electric field and magnetic vector potential are related by Eðt; rÞ ¼ @t Aðt; rÞ Equations (7.47)–(7.60) have been used to define Dirac fields.

(7.60)

120

Terahertz dielectric resonator antennas

0

15

30 (Pm)

0

20

40 (Pm)

Figure 7.10 TCDRA (cylindrical DRA) TiO2 at 10 THz (radius ¼ 2 mm and height ¼ 3 mm)

0.00

@S11

HFSSDesign1 Curve Info

Name X Y m1 8.4000 –34.5138 m2 10.2000 –22.2780 m3 11.7000 –18.6079

dB (S11)

Setup1: Sweep

–5.00

dB (S11)

–10.00 –15.00 m3

–20.00 m2

–25.00 –30.00 m1

–35.00 7.00

8.00

10.00

9.00

11.00

12.00

Freq (THz)

Figure 7.11 S11 at 10-THz TCDRA

z

Cylindrical dielectric resonator

T

y Excitation port Pni

x

Substrate

Microstrip nano silver feed Ground plane

Figure 7.12 TCDRA super directive nature

Cylindrical terahertz and optical DRA—design and analysis GAIN1 7.50 5.00

Name

X

121

HFSSDesign1

m2

Curve Info

Y

dB (GainTotal)

m1 10.2000 4.5514 m2 9.8000 6.8311

Setup1: Sweep φ = ′0q’θ = ′0q′

m1

dB (GainTotal)

2.50 0.00 –2.50 –5.00 –7.50 –10.00 –12.50 7.00

8.00

10.00

9.00

11.00

12.00

Freq (THz)

Figure 7.13 Gain vs frequency 6.8 dBi of TCDRA

Radiation pattern 1 0

Name Theta Ang Mag m1 360.0000 –0.0000 6.0082

–30

HFSSDesign1 Curve Info dB (GainTotal) Setup1: Sweep Freq = ′10,000 GHz ′ φ = ′0q ′ dB (GainTotal) Setup1: Sweep Freq = ′10,000 GHz ′ φ = ′90q ′

30 2.00 –6.00

xdb20Beamwidth(3) 21.1406

36.2132

60

–60 –14.00 –22.00

90

–90

–120

120

–150

150 –180

Figure 7.14 Radiation pattern co- and cross-polarization at 10 THz of TCDRA

Design formulation of resonant frequency of CDRA is given in the following: r  r 2 c 6:324 pffiffiffiffiffiffiffiffiffiffiffiffiffi 0:27 þ 0:36 þ 0:02 (7.61) fr ¼ 2pr 2r þ 2 2h 2h where h is height of CDRA and r is the radius of DRA to get resonant frequency fr, c is the velocity of light and permittivity of material used is defined by er in the previous formulation. Equation (7.61) is used as formulation to compute resonant frequency of TCDRA at dominant mode.

122

Terahertz dielectric resonator antennas Smith chart 1

Name Freq Ang Mag RX m1 8400.0000 94.7017 0.0188 0.9962+0.0374i

120 130 140

110

100

90

80

1.00

0.50

HFSSDesign1

70

Curve Info S11 Setup1: Sweep

60 50

2.00

40 30

150 160 0.20

20

5.00

10

170 0.00 180 0.00

0.20

0.50

m1 1.00

2.00

5.00

0 –10

–170 –160

–5.00

–0.20

–20 –30

–150 –140 –0.50 –130 –120 –1.00 –110 –100 –90

–40

–2.00

–80

–70

–50

–60

Figure 7.15 Smith chart of TCDRA VSWR 16.00 14.00

HFSSDesign1

X Y Name m1 8.4000 0.3267 m2 10.2000 1.3391 m3 11.7000 2.0486

Curve Info dB(VSWR(1))

Setup1: Sweep

dB (VSWR(1))

12.00 10.00 8.00 6.00 4.00 m3

2.00

m2

m1

0.00 7.00

8.00

9.00

10.00

11.00

12.00

Freq (THZ)

Figure 7.16 VSWR of TCDRA The equations developed for defining the Poynting vector of CDRA are given in the following. The vector-valued functions possess the usual orthogonality properties. After an appropriate normalization of these functions, using the orthonormality of the vector-valued functions, we can express the total energy in the e.m. field within the cavity C as CDRA optical has been developed using nano-DRA of TiO2 (titanate) materials at terahertz frequency with silver nano feed, which have been developed. Proximity feed has been used to excite higher order modes in a photonic antenna.

Cylindrical terahertz and optical DRA—design and analysis Radiation efficiency

HFSSDesign1

1.00 abs (radiation efficiency)

123

0.80 0.60 0.40 Curve Info abs(radiation efficiency) Setup1: Sweep φ = ′0q ′ θ = ′0q ′

0.20 0.00 7.00

8.00

9.00

11.00

10.00

12.00

Freq (THz)

Figure 7.17 Efficiency vs frequency of TCDRA

Z11 50.00 47.50

Name

X

Y

HFSSDesign1

m2

m1

m1 9.3500 48.2811 m2 10.7000 49.0443

Curve Info dB20(Z11) Setup1: Sweep

dB20 (Z11)

45.00 42.50 40.00 37.50 35.00 32.50 7.00

8.00

9.00

10.00

11.00

12.00

Freq (THz)

Figure 7.18 Z11 of TCDRA

Visible

THz

Microwave

IR

106

107

108

109

1010

1011

1012

1013

Frequency (Hz) mm Waves

Figure 7.19 Frequency spectrum

1014

1015

1016

124

Terahertz dielectric resonator antennas

S

H (0)

N

E (0)

Figure 7.20 Origin of magnetic and electric dipoles, which turns into antenna radiation pattern z

Dielectric resonator

Microstrip nano feed

Substrate

Ground plane

x

y

Figure 7.21 Optical CDRA at 521 THz with proximity feed and LASER input

y z

x

Figure 7.22 Optical CDRA Input to silver nano waveguide is fed through laser as Gaussian beam to excite SPP into an SiO2 (silicon oxide) substrate. Drude’s nonlinear model has been used to define SPP. The electrons and positrons interaction solution is solved by Dirac second quantized field equations. This is based on quantum electrodynamics to

Cylindrical terahertz and optical DRA—design and analysis S11

Name X Y m1 474.4444 –10.1897 m2 521.1111 –10.0453

125

HFSSDesign1ANSOFT Curve Info dB (S11) Setup1:Sweep

–7.50 m2

m1

–10.00

dB (S11)

–12.50 –15.00 –17.50 –20.00 –22.50 –25.00 –27.50 450.00

475.00

500.00 Freq (thz)

525.00

550.00

Figure 7.23 S11 of optical CDRA

0 30

I= 0

30

I= 180

60

60

90

90 –150

–50

120

0

120

150

150 180 Theta / Degree vs dB

Figure 7.24 Radiation pattern of a photonic antenna at 521 THz, gain 4.04 dBi describe photon spin. Here, e.m. field is produced by linear superposition of creation and annihilation operators of photon fields. The electron positron creation and annihilation operator fields plus photons creation and annihilation operators fields are total fields produced in a photonic antenna. The current densities of fields are obtained by quadratic functions of the Dirac field operators. The current density produced quantum e.m. field is described by the retardation potentials as Dirac wave function. Hence computations of mean and mean square fluctuations of quantum e.m. fields produced by antenna plus free photon e.m. fields in any state

126

Terahertz dielectric resonator antennas A/m

4,160 3,600 3,200 2,800 2,400 2,000 1,600 1,200 800 400 0 y x

z

Figure 7.25 H-Field

v/m 1.37e+06 1.2e+06 1e+06 8e+05 6e+05 4e+05 2e+05 0

y

z

Figure 7.26 E-Field

1 0.8 Mag 0.6 0.4 0.2 0 –0.2 –0.4 –0.6 –0.8 –1

Time in nano seconds

Figure 7.27 Excitation used in optical CDRA (laser output)

x

Cylindrical terahertz and optical DRA—design and analysis z

y

127

3.63 3.4 3.17 2.95 2.72 2.49 2.27 2.04 1.81 1.58 1.36 1.13 0.987 0.68 0.453 0.227 0

T

Pni x

z x

y

Figure 7.28 3D super directive radiations in optical CDRA (highly directive) at 521 THz V/m(log) 1.04e+08 1e+07 1e+06 1e+05 10,000 1,000 100 0

z y

x

Figure 7.29 Current (electric) in optical CDRA at 521 THz

will provide the complete solution. Dirac current moments are combined state of electrons positrons and photons fields. The interaction of free-photon fields and radiated fields of electrons positrons takes place to introduce spin fields known as current moments. Drude’s modeling can provide solution at optical frequencies. At optical frequency, it can penetrate through metals and give rise to excitations of free elections. At optical frequencies scattering, absorption and local field enhancement takes place. Higher order resonant modes can be excited in optical antennas, which can result into super directivity. The characteristics of photonic antennas are listed in the following:

128

Terahertz dielectric resonator antennas A/m (log) 2.55e+05

10,000 1,000 100 10 1 0.1 0

z y

x

Figure 7.30 Current (magnetic) in photonic CDRA S-Parameters [impedance view] 1i

S11 (57.75 Ω)

0.6i

2i

0.4i 2i

0.2i

5i

1i

20i 0.0

0.2 0.4

2

5

20

–20i –1i

–0.2i

–2i

–5i

–0.4i –2i –0.6i –1i

Figure 7.31 Smith chart showing the impedance of photonic antenna at 521 THz

1. 2. 3.

Light in light out is principle of working. Input excitation is provided by semiconductor LASER. Surface plasmonic resonance phenomenon takes place into substrate and metal interface.

Cylindrical terahertz and optical DRA—design and analysis

129

Table 7.1 Terahertz CDRA dimensions table Parameters

Materials

Dimensions (nm)

LSUB WSUB HSUB LG WG HG LN WN RC HC

Silicon dioxide (er ¼ 4)

900 900 100 900 900 100 450 67 64 60

Silver (er ¼ 1) Silver (er ¼ 1) Titanium oxide (er ¼ 8.29)

Table 7.2 TCDRA 10 THz dimensions table Parameters

Materials

Dimensions (mm)

LSUB WSUB HSUB LG WG HG LN WN RC HC

Silicon dioxide (er ¼ 4, mr ¼ 1 and mass density ¼ 2,220)

45 45 10 45 45 10 22.5 3 4 3

Silver (er ¼ 1, mr ¼ 0.99998 and mass density ¼ 10,500) Silver (er ¼ 1, mr ¼ 0.99998 and mass density ¼ 10,500) Silicon (er ¼ 11.9, mr ¼ 1, loss tangent ¼ 0.001 and mass density ¼ 2,330)

Table 7.3 Dimensions of optical CDRA at 521 THz S. no

Name

Materials Permittivity Length (mm)

Width (mm)

Height (mm)

1 2 3 4

Ground plane Substrate Microstrip nano feed Cylindrical DRA

Silver Arlon Silver Silicon

4.5 4.6 0.34 –

0.175 0.175 0.02 0.325

4. 5. 6.

1 2.17 1 11.9

4.6 4.5 2.3 R ¼ 0.255

e.m. Coupling takes place to QRDRA through plasmonic resonance. Capacitive coupling takes place due to proximity feed mechanism. Screening of electronic–hole takes place near band gap due to accelerated charge carriers.

130

Terahertz dielectric resonator antennas

H Field [A/m] 2.5616E+004 2.3911E+004 2.2206E+004 2.0501E+004 1.8796E+004 1.7091E+004 1.5386E+004 1.3681E+004 1.1976E+004 1.0271E+004 8.5657E+003 6.8607E+003 5.1556E+003 3.4506E+003 1.7456E+003 4.0524E+001

0

15

30 (um)

Figure 7.32 TDRA fields H

E Field [V/m] 3.8641E+006 3.6066E+006 3.3490E+006 3.0915E+006 2.8340E+006 2.5764E+006 2.3189E+006 2.0614E+006 1.8038E+006 1.5463E+006 1.2888E+006 1.0313E+006 7.7372E+005 5.1619E+005 2.5866E+005 1.1274E+003

0

15

30 (um)

Figure 7.33 TDRA fields E E Field [V/m] 1.8641E+006 1.1374E+006 6.9399E+005 4.2344E+005 2.5896E+005 1.5764E+005 9.6186E+004 5.8689E+004 3.5809E+004 2.1849E+004 1.3331E+004 8.1342E+003 4.9631E+003 3.0283E+003 1.8477E+003 1.1274E+003

0

15

Figure 7.34 TDRA fields scalar

30 (um)

Cylindrical terahertz and optical DRA—design and analysis

131

H Field [A/m] 1.0016E+004 6.9368E+003 4.8043E+003 3.3273E+003 2.3044E+003 1.5960E+003 1.1054E+003 7.6554E+002 5.3020E+002 3.6720E+002 2.5432E+002 1.7613E+002 1.2199E+002 8.4484E+001 5.8512E+001 4.0524E+001

0

15

30 (um)

Figure 7.35 TDRA fields scalar H

Jsurf [A/m] 3.1753E+004 1.8280E+004 1.0524E+004 6.0587E+003 3.4880E+003 2.0080E+003 1.1560E+003 6.6552E+002 3.8314E+002 2.2058E+002 1.2699E+002 7.3106E+001 4.2087E+001 2.4229E+001 1.3949E+001 8.0304E+000

0

25

50 (um)

Figure 7.36 TDRA surface current J

7. 8. 9. 10. 11.

Initially very small band-gap energy was observed. LASER provided higher photon energy then initial band-gap energy. LASER accelerated photons due to excitation. LASER created clouds of charge carriers due to which screening of fields takes place. Quick change in fields takes place to generate transient currents.

132

Terahertz dielectric resonator antennas Skin effect role is important at optical frequency and it is given as follows: sffiffiffiffiffiffirffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2r d¼ 1 þ ðrw 2Þ2 þ rw 2 wm

(7.62)

In these previous formulations, (7.62) shows that resistance becomes the function of frequency. Drude’s model can provide characterization of noble metals at optical frequencies based on (7.63) formulation: eðwÞ ¼ e1 

w2p w2 þ jgw

(7.63)

Light–matter interaction is presented in the previous equation, where skin effect is presented and it is frequency dependent. Metal shall behave as a transparent material to optical frequency, as it does not reflect e.m. wave at an optical spectrum similar to microwave regime. Light gets inserted into metals. Plasmon resonance takes place due to light–matter interaction and hence it is clear that after quantization, in order to obtain the correct time dependence of the coefficients, i.e. c(Emnp; t) should vary with time as exp(iw(Emnp)t) while c(Hmnp; t) should vary as exp(iw (Hmnp)t), we must enforce the bosonic commutation relations on solving which the desired time dependences of these coefficients are obtained. These commutation relations can alternatively be obtained by requiring that the Maxwell equations follow from the Heisenberg equations for the e.m. field operators. We shall now briefly discuss the computation of the far-field radiation pattern. The surface current density is induced on the sidewalls of the cavity. Here, surface current denotes the boundary surface of the CDRA. The far-field photonic or quantum e.m. field radiated by the CDRA is easily seen to be a linear function of c(Emnp); c(Emnp); c(Hmnp); c(Hmnp). This is because the magnetic field is linear in these observables and hence the surface current density on the CDRA is also linear in these observables. The optical CDRA and TCDRA parameters have been obtained by simulations on HFSS software and these results are shown in Figures 7.1–7.31. Tables 7.1–7.3 are design dimensions. Figures 7.32–7.36 are fields as shown later for E and H and current density J.

7.6 Conclusion High efficiency, super directivity and large bandwidth with compact size are important parameter of an optical CDRA. Mathematical formulations at terahertz frequency have been broadly developed to make the subject more interesting. They are best in communication having razor-sharp beam with highly effective communication even in lossy environments. Comprehensive investigations on optical CDRA have been made. This chapter will be very useful to develop retinal photoreceptors and rectennas if integrated with photo diodes. They can be used as an implant antenna as photoreceptors in retinal photoreceptors applications.

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References [1] J. W. Choong, N. Nefedkin, and A. Krasnok, “Collectively driven optical nanoantennas,” Phys. Rev. A, vol. 103, 043714, 2021. [2] P. Bharadwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photon, vol. 1, pp. 438–483, 2009. [3] G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Herna´ndezFigueroa, “Dielectric resonator antenna for applications in nanophotonics,” Opt. Express, vol. 21, no. 1, 2013. [4] W. T. Sethi, H. Vettikaladi, H. Fathallah, and M. Himdi, “Dielectric resonator nantennas for optical communication,” Optical Communication Technology, Pedro Pinho, IntechOpen, doi: 10.5772/intechopen.69064. [5] G. N. Malheiros-Silveira and H. E. Herna´ndez-Figueroa, “Wireless optical coupling evaluation in a dielectric resonator nanoantenna,” OSA Continuum, vol. 1, no. 3, pp. 805–811, 2018, https://doi.org/10.1364/OSAC.1.000805. [6] A. Alu` and N. Engheta, “Wireless at the nanoscale: Optical interconnects using matched nanoantennas,” Phys. Rev. Lett., vol. 104, no. 21, p. 213902, 2010. [7] D. M. Solı´s, J. M. Taboada, F. Obelleiro, and L. Landesa, “Optimization of an optical wireless nanolink using directive nanoantennas,” Opt. Express, vol. 21, no. 2, pp. 2369–2377, 2013. [8] S. Weinberg, The Quantum Theory of Fields, vol. 1, Cambridge University Press, New York, NY, 1995.

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Chapter 8

Spherical terahertz and optical DRA—design and implementations

Abstract In this chapter, terahertz spherical DRA (dielectric resonator antenna) has been simulated for radiation parameters and its mathematically modeling is carried out. The terahertz DRA has been simulated at 511 THz. This frequency corresponds to a visible spectrum, and suitable for LiDAR and retinal photoreceptor applications. One more model has been developed at 10 THz. The spherical DRA is built with silicon (er ¼ 11.9) materials, and analyses on exciting higher order modes have been carried out. It has a gain of 18 dBi, in which multipole generation has been observed. Two separate models (silicon and titanate) at optical frequencies have also been developed and validated with 2.4-dB gain using computer simulation technology. Multi-input–multi-output spherical DRA has also been realized at 10 THz to get excellent directivity. Mathematical formulation on super directivity is also formulated. The proximity coupled feed has been used with laser input, and Gaussian beam excitation. The silver nano waveguides have been used in the modeling to provide feed to terahertz DRAs of different geometries, i.e. spherical DRA, etc. LASER input interacts with silver metal and dielectric SiO2 substrate. Hence, in terahertz DRA, SPP waves are generated due to light–matter interaction, which will give rise to plasmon frequency generation. This plasmon frequency is always kept lesser than LASER input frequency for forward propagation, i.e. so as to enable it to propagate in forward direction. The radiated field is a state of jointly coherent for the bosons (i.e. photons) and for the fermions (positrons–electrons).

8.1 Introduction Dielectric resonator antennas (DRAs) can efficiently operate right from microwave to an optical frequency spectrum [1,2]. DRAs have high efficiency, low-loss, high permittivity, design simplicity, ease of fabrication, design flexibility and high stability in a wide frequency spectrum. They can operate at higher frequencies such as terahertz and optical frequencies because of low conductor losses as compared to metallic antennas. Quantum electromagnetic (e.m.) fields can be described as understanding of light since Sir Isaac Newton started with first assuming light to be corpuscles, namely small particles of different colors corresponding to the different frequencies. This idea was first propounded by Newton and came to be known as

136

Terahertz dielectric resonator antennas

the corpuscular theory of light [3–5]. Several years later, based on experiments involving interference of light, Huygens came to understand that light is to be regarded as being composed not of particles but rather is a wave and that any light field can be regarded as a superposition of plane waves of different frequencies and wave numbers. This meant that during the process of interference involving computation of the total intensity of light coming from different sources, one should not add the respective intensities, but rather the amplitudes, then square the resultant amplitude and then form its time average [6–8]. This meant that in the resultant total intensity, there would appear cross terms that are oscillatory, which lead us to the phenomenon of interference. Later on, Fermat propounded the ray theory of light in order to explain Snell’s theorems on reflection and refraction of light, according to which a light ray always follows the path of minimum time [9–11]. The wave nature of light as propounded by Christian Huygens gained more support when Maxwell unified electromagnetism with light by showing that the basic equations of electromagnetism imply that the electric and magnetic fields in space satisfy the wave equation with a velocity of propagation being equal to that of light. This fundamental discovery gave firm support to the wave theory of light [12,13]. However, with the birth of the quantum theory in the early part of the nineteenth century, it became clear that light comes in discrete packets called quanta and each quantum of radiation carries an energy proportional to its frequency and that the intensity of light is proportional to the number of quanta [14,15]. These quanta came to be known as photons and it appeared therefore that with the advent of the quantum theory, physicists had reverted back to Newton’s corpuscular theory of light. However, while Planck was busy creating his quantum theory of light, simultaneously Einstein wrote some beautiful papers proving that light can behave sometimes as a particle and sometimes as a wave, specifically showing that the variance of fluctuations in the energy of black-body radiation has two components: a particle component in agreement with Planck’s quantum hypothesis and a wave component in agreement with Rayleigh’s theory on the number of “modes” of a wave field within a given energy shell. The particle theory of light got further support from Einstein’s special theory of relativity giving the relationship between the energy and momentum of a particle of any given mass and in particular for light particles having zero rest mass. However, the crucial breakthrough into the wave– particle duality matter and in particular of light came with de Broglie’s discovery of the fundamental relationship between the wavelength of a wave and the momentum of the associated particle. The de Broglie theory was applicable to all of matter not just light. It gave conclusive evidence that even particles like electrons exhibit wave–particle duality, i.e. they sometimes behaved like particles and sometimes like waves. The final crunch describing the wave–particle duality came with the wave equation for particles discovered by Erwin Schro¨dinger using which he was able to calculate the energy levels of an electron bound to a nucleus in terms of the eigenvalues of a partial differential equation. The corresponding eigen functions were complex wave fields that Schro¨dinger could not immediately interpret. These eigen functions approximately corresponded to the kind of “matter–waves” predicted by Louis de Broglie having frequency determined by Planck’s quantum

Spherical terahertz and optical DRA—design and implementations

137

hypothesis relating frequency of waves to the energy of the quanta and having wavelength determined by de Broglie’s theory relating the wavelength of waves to the momenta of the particles. Schro¨dinger’s wave equation was a nonrelativistic wave equation in that it could be “derived” by assuming the nonrelativistic relationship between energy and momentum and then assigning operators to energy and momentum. The correct interpretation of the eigen functions of Schro¨dinger’s equations was provided about a year later by Max Born with the suggestion that the modulus square of the wave function gave the probability density of the particle to be present at a given point in space at a given time. Equivalently, this modulus square could be interpreted as the number density of particles or the intensity of the wave at that point in space at that time. In fact, Schro¨dinger’s equation could also be generalized to determine the wave functions and energy levels for any particle with a prescribed energy–momentum relationship by replacing the energy and momentum with appropriate operators and applying both sides of this relationship to a wave function. Thus, Schro¨dinger’s formalism could be applied to determine the kind of de Broglie waves associated with a particle having any given energy– momentum relation. The de Broglie wave solutions then have the interpretation that their modulus square represents the intensity/probability density of particles in space at a given time. In special relativity, the energy momentum relationship for a relativistic particle is a quadratic form in the energy and momenta unlike the Newtonian-nonrelativistic case where it is linear in the energy and quadratic in the momenta. Consequently the Schro¨dinger equation is a linear PDE that is of the first order in time and second order in spatial derivatives. The nonrelativistic Schro¨dinger equation as a consequence then leads to a unitary evolution of the wave function with time and with the additional pleasing property that evolution from time zero to time t1 þ t2 can be expressed as the composition of evolution from time zero to time t1 followed by evolution from time t1 to time t1 þ t2. In other words, Schro¨dinger evolution follows the semigroup property. This is in contrast to the case of special relativity where in the wave equation is quadratic in time leading to the wave function at a given time being a superposition of a forward-propagating solution and a backward-propagating solution. Thus, the total probability will be conserved in Schro¨dinger’s nonrelativistic theory but not so in the relativistic theory. This difficulty was finally resolved by Paul Dirac who factorized the energy– momentum relationship of special relativity into linear factors in energy and momentum using 4  4 complex anticommuting matrices and it led to Dirac’s relativistic theory of the electron which along with Maxwell’s equations is at the heart of quantum electrodynamics describing the interaction of light with matter. Dirac’s equation being linear in the space and time derivatives is a truly relativistic wave equation for quantum theory because first it respects Lorentz invariance in that space and time are treated on an equal footing and that we can find a spin or representation of the Lorentz group under which Dirac’s equation remains invariant and second it is first order in time and therefore preserves the unitary semigroup property of the evolution guaranteeing, therefore, conservation of total probability. Light as we understand today is a second quantized e.m. field in which the vector potential components in the spatial frequency domain represent creation and

138

Terahertz dielectric resonator antennas

annihilation operators of photons with different momenta and helicities. This is owing to the fact that the total energy of the e.m. field can in the Coulomb gauge be expressed as a quadratic functional of the spatial frequency components of the vector potential and each spatial frequency component of the vector potential, according to Maxwell’s equations, evolves harmonically with time with a frequency being w, where k is the wave vector. Such a picture is in perfect accord with the Hamiltonian theory of a collection of independent quantum harmonic oscillators. However, this model of light can actually be traced back to the work of Satyendranath Bose who proposed a statistical method for deriving Planck’s law of black-body radiation wherein we are interested in distributing a total amount of energy E having N quanta at frequency n so that E ¼ Nhn, amongst p oscillators with the quanta being regarded as indistinguishable particles. When an oscillator has n such quanta, we say in the modern language of quantum mechanics that it has been excited to the n energy level by an application of n creation operators to the vacuum. In this way, the entire field of black-body radiation is simply a collection of quantum harmonic oscillators and if we maximize the entropy, i.e. the total number of ways of distributing this energy then we end up with the famous Bose– Einstein statistics that gives us the relative number of photons at each frequency. This in turn enables us to determine the intensity of black-body radiation as a function of frequency. Summarizing the modern point of view, the entire photon field described as an operator e.m. four potential field is just a superposition of plane waves of different wave vectors whose coefficients are creation and annihilation operators at the different wave vectors and a given state of the photon field is actually a linear combination of number states wherein a number state is specified by specifying the number of photons that occupy each state of definite wave-vector/ momentum and definitive helicity/spin. A particular kind of photon state called a coherent state is a state constructed by an appropriate linear combination of this kind that turns out to be an eigen state for all the photon annihilation operators. This model for the quantum e.m. field implicitly contains both the particle nature and the wave nature of light. The particle nature is contained in the presence of the creation and annihilation operators while the wave nature is contained in the plane waves that act as carrier signal fields for the creation and annihilation operators. Second quantization means a quantization of a classical field theory. It is called second quantization for the following reasons: The first quantization is simply a wave equation like the three-dimensional wave equation, the Schro¨dinger wave equation for a single or an infinite number of quantum particles, the Klein Gordon equation or the Dirac equation. If such a classical wave equation is quantized, then it describes an infinite number of quantum particles. This can be seen clearly from the following example: Take the Klein–Gordon equation that is the wave equation described before corresponding to Einstein’s energy momentum relation with the energy and momenta replaced by appropriate operators. We expand the solution wave field as a three-dimensional Fourier series within a cube of side length L. The coefficient of this Fourier series then becomes quantum operators each one and its adjoint, i.e. Hermitian conjugate describes a single quantum particle. Thus, a second

Spherical terahertz and optical DRA—design and implementations

139

quantized field can equivalently be described by a countably infinite number of quantum particles. For bosonic quantum fields like the Maxwell e.m. photon field, the state of the field can be described, for example by specifying how many particles are occupying each state of definite momentum and helicity or equivalently by specifying the momentum and helicity of each of the particles. In this case, there can be zero, one or more than one particle having a specified momentum and helicity. In the case of fermionic fields like the second quantized Dirac field, we cannot have more than one particle having a definitive value of momentum and spin. In this chapter, terahertz spherical DRA characteristics equation has been developed for possible resonant frequency oscillations and compared with microwave working of spherical DRAs [16]. The far-field formulations have also been developed. Surface plasmon resonance has excited terahertz spherical DRA. E and H fields parameters have been obtained in terahertz spherical DRA. All relevant parameters are listed in Figures 8.1–8.8 and dimensions are given in Tables 8.1 and 8.2. The role of high-order multiples in the light scattering can be worked out for the development of meta-surfaces and metamaterials in terahertz range. The study

Titanium dioxide (resonating element) Silver nanostrip Teflon (substrate) Silver

Figure 8.1 TiO2 spherical DRA at 511 THz Z

350q

Figure 8.2 Silicon terahertz spherical DRA at 511-THz E fields

140

Terahertz dielectric resonator antennas

Terahertz spherical DRA

Silver nano waveguide

Gaussian beam

Figure 8.3 10-THz spherical DRA (radius ¼ 3 mm and height ¼ 6 mm)

S11

0.00

HFSSDesign1 Curve info dB (S11) Setup1: Sweep

–5.00 –10.00 dB (S11)

–15.00 –20.00

–25.00 –30.00

Name X Y m1 7.6667 –30.6318 m2 8.6667 –25.4193 m3 10.5556 –20.5234 m4 12.1111 –30.4801

–35.00 5.00

7.50

10.00 Freq (THz)

12.50

15.00

Figure 8.4 Reflection coefficient of spherical DRA at 10 THz

Gain

20.00

HFSSDesign1 Curve info dB (Gain total) Setup1 : Sweep φ = '0º' θ = '0º'

15.00

dB (Gain total)

10.00 5.00 0.00

–5.00

–10.00 –15.00 –20.00 5.00

7.50

10.00 Freq (THz)

12.50

Figure 8.5 Gain of spherical DRA at higher order mode in high

15.00

Spherical terahertz and optical DRA—design and implementations Name m1

q 360.0000

Ang

Mag

–0.0000

4.7920

141

HFSSDesign1

Radiation pattern 1 0

Curve info dB (Gain Total) Setup1 : Sweep Freq = '10 000GHz' f = '0º' dB (Gain Total) Setup1 : Sweep Freq = '10 000GHz' f = '90º'

30

–30 3.50 –0.50 –60

60 –4.50 –8.50

–90

90

–120

120

150

–150 –180

Figure 8.6 Radiation pattern of terahertz spherical DRA Name m1

Freq 7666.6667

Ang Mag 33.1525 0.0294

RX 1.0499 + 0.0338i

110

Smith chart 1 90 100 80 1.00

HFSSDesign1

Curve info S11 Setup1 : Sweep

70

120

60 2.00 50

130 0.50 140

40

150

30

160 0.20

5.00 20

170

10

180 0.00 0.00

0.20

0.50

1.00

2.00

5.00

0

–170

–10 –5.00 –20

–160 –0.20

–30

–150 –140 –130 –0.50 –120 –110

–40 –2.00 –50 –60 –100

–1.00 –90

–80

–70

Figure 8.7 Smith chart of terahertz spherical DRA 10 THz of excitation of both electric and magnetic multipole resonances at optical frequency is an important phenomenon, which has dependence on geometry, size and material characteristics. Scattering of light by nanosize particles can result into multi-resonance. Higher order resonances up to third-order multipole excitations can be investigated for useful applications. Total electric dipole moment in scattering can be computed using tensor analysis. Interference, polarization and

142

Terahertz dielectric resonator antennas VSWR

14.00

HFSSDesign1 Curve info dB (VSWR(1)) Setup1 : Sweep

12.00

dB (VSWR(1))

10.00 8.00 6.00 4.00 2.00 0.00 5.00

Name Y X m1 10.5556 1.6404 m2 12.1111 0.5200 m3 7.6667 0.5110 m4 8.6667 0.9317

7.50

10.00 Freq (THz)

12.50

15.00

Figure 8.8 VSWR of terahertz spherical DRA

Table 8.1 TiO2 terahertz spherical DRA dimensions S. no

Dimensions

Materials

Value

Permittivity/ loss tangent

1 2 3 4 5

Ground plane Substrate Nanostrip waveguide Resonator radius Resonator height

Silver Teflon Silver Titanium dioxide Titanium dioxide

900  900  100 nm3 900  900  100 nm3 450  67  10 nm3 32 nm 64 nm

1 0.0006 1 8.29 8.29

Table 8.2 Design dimensions of optical spherical resonator antenna Name of components

Materials used

All dimensions (mm)

LSUB WSUB HSUB LG WG HG LM WM RS HS

Substrate Silicon dioxide (er ¼ 4)

42 42 8 42 42 8 6.5 0.5 3 6

Ground plane Silver (er ¼ 1) Nanostrip Silver (er ¼ 1) DRA used Silicon (er ¼ 11.9)

collision integral are few important phenomena. This way, radiations of desired wave vector with polarization by scattering, suppression, forward and backward scattering can be obtained. These investigations can be applied to develop nanoantennas in optical spectrum. These DRAs have applications as sensors, chipto-chip communication, retinal photo receptors, scanning and imaging devices, etc.

Spherical terahertz and optical DRA—design and implementations

143

The mathematical modeling of terahertz spherical DRA is established using (8.1)– (8.16). Two DRAs and one multi-input–multi-output (MIMO) cases at terahertz frequency have been developed and validated along with super directivity derivations. No similar work exists for making its comparative study.

8.2 Design of terahertz spherical DRA at 511 THz Case-1 TiO2 spherical DRA at 511 THz Case-2 10-THz spherical DRA

8.3 Mathematical formulations of terahertz spherical DRA The multipole electric field can be expressed as  X 1 curlðhl ðkrÞLYlm ð^r ÞÞ cðlmÞhl ðkrÞLYlm ð^r Þ þ d ðlmÞ Eðw; rÞ ¼ jwe lm And the multipole magnetic field as  X 1 cðlmÞcurlðhl ðkrÞLYlm ð^r ÞÞ þ d ðlmÞhl ðkrÞLYlm ð^r Þ  H ðw; rÞ ¼ jwm lm where L ¼ ir  r

(8.1)

This is because hl ðkrÞLYlm ð^r Þ satisfies the Helmholtz equation and is orthogonal to the radial direction ^r : Further, it is easily verified that div ðhl ðkrÞLYlm ð^r ÞÞ ¼ 0 because ^r  L ¼ 0 and divðLYlm ð^r ÞÞ ¼ 0. So all the Maxwell equations in free space are satisfied by the multipole fields. For a detailed analysis of multipole field, see JD Jackson, “Classical electrodynamics,” Wiley. It should be noted that hl ðkrÞLYlm ð^r Þ is purely transverse, i.e. its radial components vanish, while curlð hl ðkrÞLYlm ð^r ÞÞ satisfies the Helmholtz equation but has both nonvanishing radial and transverse components. Its divergence obviously vanishes because the divergence of a curl is zero. The Maxwell equations are satisfied by multipole fields because curlðcurlðhl ðkrÞLYlm ð^r ÞÞÞ ¼ rðdivðhl ðkrÞLYlm ð^r ÞÞÞ  r2 ðhl ðkrÞLYlm ð^r ÞÞ ¼ k 2 hl ðkrÞLYlm ð^r Þ It should be noted that hl ðxÞ are Hankel functions constructed as hl ðxÞ ¼ x1=2 Jlþ1=2 ðxÞ where Jv ðxÞ satisfies Bessel’s equation:   x2 Jv 00 ðxÞ þ xJv 0 ðxÞ þ x2  v2 Jv ðxÞ ¼ 0

144

Terahertz dielectric resonator antennas This can be verified by directly substituting yðhl ðkrÞLYlm ð^r ÞÞ into the following Helmholtz equation  2  x þ k2 y ¼ 0

Expressed in spherical-polar coordinate and deriving from that the radial equation for hl ðkrÞ. It can also be obtained by the method of separation of variables. The frequency of oscillation of the e.m. field within this spherical cavity gets quantized because of the requirement that the radial component of magnetic field ^ ^j components of the electric field, are and the tangential components, i.e. q; required to vanish on the spherical surface, i.e. at r ¼ R now, ^r  LYlm ¼ 0; r  curlðhl ðkrÞLYlm ð^r ÞÞ ¼ divðhl ðkrÞr  LYlm ð^r ÞÞ

(8.2)

Since curl r ¼ 0 Also, ^r  LYlm

¼ ¼

i^r  ðr  rÞYlm ir@r Ylm þ irrYlm ¼ irrYlm

Thus,   r  curlðhl ðkrÞLYlm ð^r ÞÞ ¼ div ihl ðkrÞr2 rYlm ð^r Þ The boundary conditions imply that ^r  H ¼ 0; ^r  E ¼ 0; r ¼ R Hence X lm



   1 ^r  ðcurlðhl ðkrÞLYlm ð^r ÞÞÞ  cðlmÞhl ðkrÞð^r Þ  LYlm ð^r Þ þ d ðlmÞ jwe r¼R

¼0 And multipole magnetic field as X ½cðlmÞ^r  curlðhl ðkrÞLYlm ð^r ÞÞjr¼R ¼ 0 lm

Now 0

^r  curlðf ðrÞg ð^r ÞÞ ¼ ^r  f ðrÞ^r  g ð^r Þ þ ^r  ðf ðrÞr  g ð^r ÞÞ

Spherical terahertz and optical DRA—design and implementations

145

Applied to our problem, this because  0  ^r  curlðhl ðkrÞLYlm ð^r ÞÞ ¼ ^r  khl ðkrÞ^r  LYlm ð^r Þ þ ^r  ðhl ðkrÞr  ðLYlm ð^r ÞÞÞ 0 ¼ kkhl ðkrÞLYlm ð^r Þ þ ^r  ðhl ðkrÞr  ðLYlm ð^r ÞÞÞ (8.3) ir  Lyð^r Þ ¼ r  ðr  ryð^r ÞÞ ¼ ¼ r2 yð^r Þr  ðr; rÞryð^r Þ þ ðryð^r Þ; rÞr  3ryð^r Þ ¼ rr2 yð^r Þ  r@r r yð^r Þ  2ryð^r Þ ¼ rr2 yð^r Þ  ryð^r Þ

(8.4)

Thus, 

 hl ðkrÞ ^r  curlðhl ðkrÞLYlm ð^r ÞÞ ¼  khl ðkrÞ þ LYlm ð^r Þ r 0

Thus, our boundary conditions reduce to    d ðlmÞ hl ðkrÞ 0 khl ðkrÞ þ LYlm ð^r Þ ¼ 0; r ¼ R; cðlmÞhl ðkrÞ^r  LYlm ð^r Þ  jwe r cðlmÞrr2 Ylm ð^r Þ ¼ 0; r ¼ R These yield 0

cðlmÞ ¼ 0; kRhl ðkRÞ þ hl ðkRÞ ¼ 0 Thus, oscillation frequency ðlnÞ; n ¼ 1; 2; 3; . . .; the roots of the equation are   wR 0 ¼0 wRhl ðwRcÞ þ hl c And the solution for the multipole field within the spherical DRA is given by the previous superpositions. The far-field radiation pattern generated by the surface current density on the sphere: The relevant multipole fields are "    # X d ðlmnÞ wðlnÞr expðjwðlnÞtÞcurl hl LYlm ð^r Þ Eðt; rÞ ¼ Re jwðlnÞ c "lmn    # X wðlnÞr LYlm ð^r Þ d ðlmnÞexpðjwðlnÞtÞcurl hl H ðt; rÞ ¼ Re c lmn where r ¼ ðr; q; jÞ; 0  r  R; 0  q < p; 0  j < 2p (8.5) This e.m. field satisfies the Maxwell equations and all the relevant boundary conditions on the perfectly conducting spherical surface. We now evaluate the

146

Terahertz dielectric resonator antennas

far-field radiation pattern produced by the surface currents: the surface current density is given by JS ðt; R^r Þ ¼ ^r  H ðt; R^r Þ "    # X wðlnÞR ^r LYlm ð^r Þ ¼ Re d ðlmnÞexpðjwðlnÞtÞ hl c lmn Now, for a function yð^r Þ or ^r ¼ ðq; jÞ only, we have that ^r  LYlm ð^r Þ ¼ i^r  ðr  ryð^r ÞÞ ¼ ir ryð^r Þ Since ^r  ryð^r Þ ¼ 0 because ryð^r Þ has only q and j components. Then "   #  X wðlnÞR rYlm ð^r Þ lRd ðlmnÞexpðjwðlnÞtÞ hl JS ðt; rÞ ¼ Re c lmn "    # (8.6) X wðlnÞR ¼  RIm rYlm ð^r Þ d ðlmnÞexpðjwðlnÞtÞ hl c lmn And hence, the far-field magnetic vector potential in the time domain generated by this surface current density is given by ! ð 0 m r R^r ^r ; R^r 0 dS ð^r 0 Þ JS t  þ Aðt; rÞ ¼ c 4p c S

"

    mR3 X wðlnÞR jwðlnÞr ¼ Im  exp  expðjwðlnÞtÞ d ðlmnÞhl 4p lmn c c ð

R^r ^r exp jwðlnÞ c

0

! rYlm ð^r ÞdWð^r 0 Þ

S

Not that on the surface of the sphere X Re ½Aln ðrÞexpðjwðlnÞt dS ð^r 0 Þ ¼ R2 dWð^r 0 Þ ¼ ln

Note that the characteristics oscillation frequencies wðlnÞ do not depend upon m. We now compute the energy of radiation field within the spherical cavity. This expression would enable us to quantize the field inside in terms of cavity creation and annihilation operators: ðh i e m U¼ (8.7) jEðt; rÞj2 þ jH ðt; rÞj2 d 3 r 2 2 B

Spherical terahertz and optical DRA—design and implementations

147

where B is the volume region of cavity: B ¼ fðr; q; jÞ: 0  r  R; 0  q < p; 0  j < 2pg And S is its surface: S ¼ fðR; q; jÞ: 0  q < p; 0  j < 2pg 1 X e jd ðlmnÞj2 ¼ 2 lmn 2 wðlnÞ2 e2

!ð     2   curl hl wðlnÞr LYlm ð^r Þ  drdS ð^r Þ   c B

2

ð  wðlnÞr  1Xm 2  LYlm ð^r Þ  drdS ð^r Þ þ jd ðlmnÞj  hl 2 lmn 2 c

(8.8)

B

Now using the orthonormality of the spherical harmonics as well as their eigen function property for operator L2 ; we see that   2 2 ðR   ð        hl wðlnÞr LYlm ð^r Þ  drdS ð^r Þ ¼ lðl þ 1Þ  hl wðlnÞr LYlm ð^r Þ  r2 dr     c c B

0

And    2 ð ð   curl hl wðlnÞr LYlm ð^r Þ  drdS ð^r Þ ¼ jcurlyð^r Þj2 d 3 r   c B

B

where   wðlnÞr LYlm ð^r Þ y ðr Þ ¼ h l c Now, jcurlyj2 ¼ ðcurly ; curlyÞ ¼ divðy  curlyÞ þ ðy ; curlcurlyÞ The integral of the divergence term on the right over B by virtue of Gauss’s integral there in once we note that ^r  curly ¼ 0 where r ¼ R in view of the vanishing boundary condition for tangential component of the electric field on surface of the surface. In fact, this condition was one of the defining conditions for the computing the characteristic oscillation frequency wðlnÞ: Further, ! 2 w ð ln Þ y curlcurl y ¼ r2 y ¼ c Since as observed earlier, divy ¼ 0 by the Maxwell equation that states that the magnetic field has a vanishing divergence. Thus, we get finally after combining

148

Terahertz dielectric resonator antennas

all these equations that the total cavity field energy is given by Xm lðl þ 1Þf ðl; nÞjd ðlmnÞj2 U¼ 2 lmn

(8.9)

where  ðR    wðlnÞr 2 2   r dr f ðl; nÞ ¼ hl  c

(8.10)

0

And this quadratic equation in the d ðlmnÞ can be immediately quantized using the harmonics oscillator method. Equations (8.1)–(8.10) have been developed for the theoretical analysis of terahertz spherical DRA.

8.4 Results and discussions Simulations results of terahertz spherical DRAs have been validated and results such as S11, radiation pattern, VSWR, E and H fields are given in Figures 8.9–8.17.

8.4.1

Super directivity in spherical DRA

Equations (8.11)–(8.16) have been developed to define the concept of super directivity in spherical DRA at terahertz frequency. Fermionic Dirac wave operator field can be defined as X yðt; rÞ ¼ ½aðmnpsÞexpðjwðmnpÞtÞU ðmnps; rÞ mnps

þbðmnpsÞ expðjwðmnpÞtÞX~ ðmnps; rÞV ðmnpsÞ

Name m1 m2 m3

x

y

483.3333 527.7778 511.1111

–9.8962 –10.1172 –28.1755

S11

(8.11)

HFSSDesign1 Curve info dR (S11) Setup1 : Sweep

–10.00

dB (S11)

–15.00

–20.00

–25.00

–30.00 450.00

475.00

500.00 Freq (THz)

525.00

550.00

Figure 8.9 Reflection coefficient of silicon spherical optical DRA at 511 THz

Spherical terahertz and optical DRA—design and implementations

149

0 –30

30 2.80 2.10

–60

60 1.40 0.70

90

90

–120

120

150

–150 –180

Figure 8.10 Radiation pattern with max gain 2.83 dBi of optical spherical DRA at 511 THz

The Dirac four-current density corresponding to this fermionic wave field is given by J m ðt; rÞ ¼ eyðt; rÞ am yðt; rÞ; am ¼ g0 gm

(8.12)

The e.m. four quantum potential of radiation generated by this current is given by the standard retarded potential formula given as follows: 

ð A ðt; rÞ ¼ k  J m

m

 jr  r0 j 0 3 0 ; r d r =jr  r0 j; t c

C

k¼

m 4p

The far-field magnetic vector potential is given by   ð   k r r0  J t  þ ^r  ; r0 d 3 r0 ðAr ðt; rÞ: r ¼ 1; 2; 3Þ ¼ Aðt; rÞ ¼ c r c C

(8.13)

150

Terahertz dielectric resonator antennas

110

100

120 130

Smith chart 90 80 1.00

Curve info S11

70

Setup1 : Sweep

60 2.00

0.50

50 40

140 150 160

30 5.00 20

0.20

170

10

180 0.00 0.00

0.20

0.50

1.00

2.00

5.00

0

–170

–10

–160 0.20

–5.00 –20

–150

–30

–140

–40 –2.00

–0.50

–130 –120

–110

1.00 –100 –90

–50

–60 –80

–70

Figure 8.11 Smith chart showing an impedance of terahertz spherical DRA

Figure 8.12 Laser beam input to terahertz spherical DRA at 511 THz

And in the frequency domain, this can be expressed as  ð w

k ^r  r0 d 3 r0 J ðw; r0 Þexp j Aðw; rÞ ¼ r c

Spherical terahertz and optical DRA—design and implementations

151

Phase = 220deg

Figure 8.13 Silicon optical spherical DRA showing vector fields at 511 THz, E-field vector

Z

Phase = 180q

Figure 8.14 E-Field scalar terahertz spherical DRA By the Fourier transform of the current density operator field, we get ð  J ðw; rÞ ¼ e yðw0  w; rÞ a  yðw0 ; rÞdw0 By the convolution theorem for Fourier transform, we get X yðw; rÞ ¼ ½aðmnpsÞU ðmnpsÞdðw  wðmnpÞÞX ðmnp; rÞ mnps

  þ bðmnpsÞd w þ wðmnpÞX~ ðmnp; rÞ 

(8.14)

152

Terahertz dielectric resonator antennas E Field (V/m) 7.0000E+007 6.3000E+007 5.8800E+007 5.4600E+007 5.0400E+007 4.6200E+007 4.2000E+007 3.7800E+007 3.3600E+007 2.9400E+007 2.5200E+007 2.1000E+007 1.6800E+007 1.2600E+007 8.4000E+006 4.2000E+006 2.5474E–003

Figure 8.15 E Fields and input excitation of terahertz spherical DRA H Field (A/m) 1.7126E+005 1.5475E+005 1.4444E+005 1.3412E+005 1.2380E+005 1.1349E+005 1.0317E+005 9.2852E+004 8.2535E+004 7.2218E+004 6.1901E+004 5.1584E+004 4.1267E+004 3.0951E+004 2.0634E+004 1.0317E+004 2.8313E–003

Figure 8.16 H-Field vector of terahertz spherical DRA H Field (A/m) 1.7126E+005 1.5475E+005 1.4444E+005 1.3412E+005 1.2380E+005 1.1349E+005 1.0317E+005 9.2852E+004 8.2535E+004 7.2218E+004 6.1901E+004 5.1584E+004 4.1267E+004 3.0951E+004 2.0634E+004 1.0317E+004 2.8313E–003

Figure 8.17 H-Field scalar of terahertz spherical DRA

Spherical terahertz and optical DRA—design and implementations Hence,

"

ð

J ðw; rÞ ¼ e dw

0

X

153

~ ðmnpsÞX~ mnp ðrÞdðw0  w  wðmnpÞÞ aðmnpsÞ U

mnps

þ bðmnpsÞV~ ðmnpsÞXmnp ðrÞdðw0  w þ wðmnpÞÞ " X aðmnpsÞU ðmnpsÞXmnp ðrÞdðw0  mnps

wðmnpÞÞbðmnpsÞ V ðmnpsÞX~ mnp ðrÞdðw0 þ wðmnpÞÞ ¼ e

X

aðmnpsÞ aðm0 n0 p0 s0 ÞU ðm0 n0 p0 s0 ÞX~ mnp ðrÞ

mnpsm0 n0 p0 s0

Xm0 n0 p0 ðrÞdðwðm0 n0 p0 Þ  wðmnpÞ  wÞ  ~ ðmnpsÞV ðm0 n0 p0 s0 ÞXm0 n0 p0 ðrÞ þ aðmnpsÞ bðm0 n0 p0 s0 Þ U X~ mnp ðrÞdðwðm0 n0 p0 Þ þ wðmnpÞ þ wÞÞ þ bðmnpsÞaðm0 n0 p0 s0 ÞV~ ðmnpsÞU ððm0 n0 p0 s0 Þ Xmnp ðrÞX~ m0 n0 p0 ðrÞdðwðm0 n0 p0 Þ þ wðmnpÞ  wÞÞ  þ bðmnpsÞ; bðm0 n0 p0 s0 Þ V~ ; ðmnpsÞ; V ; ððm0 n0 p0 s0 Þ

X~ mnp ðrÞX~ m0 n0 p0 ðrÞdðwðmnpÞ  wðm0 n0 p0 Þ  wÞÞ

(8.15)

It is now a simple matter to evaluate the far-field vector potential and hence the far-field e.m. field and hence also the far-field Poynting vector and determine a state jn⟩ of the electron–positron field such that if S ðw; rÞ ¼ Pðw; ^r Þ^r =r2 denote the far-field Poynting vector, then for a given frequency (i.e. one among the set wðmnpÞ  wðm0 n0 p0 Þ such that hnjPðw; ^r Þjni is a maximum for a given direction, etc. To compute this quantum average, all we require all the matrix elements hhjaðmnpsÞ aðm0 n0 p0 s0 Þjhi; hhjaðmnpsÞ bðm0 n0 p0 s0 Þjhi; hhjbðmnpsÞaðm0 n0 p0 s0 Þjhi; hhjbðmnpsÞbðm0 n0 p0 s0 Þ jhi

(8.16)

8.5 MIMO (multi-input–multi-output) spherical DRA MIMO DRAs are directive antennas for enhanced gain and diversity. The aspect ratio of spherical DRA is (h/r) ¼ 7/3.5 ¼ 2. Figures 8.18 and 8.19 have shown two spherical DRA connected with independent feed but common ground plane. The isolation has been measured between two. Results simulated using HFSS are shown in Figures 8.20–8.25.

154

Terahertz dielectric resonator antennas

Spherical DRA Nano feed

Gaussian beam input

Figure 8.18 MIMO spherical DRA top view z

y 0

10

20 (Pm)

Figure 8.19 MIMO spherical DRA side view S11

–2.50

HFSSDesign1 Curve info

dB (S 11 Setup1 : Sweep

)

–5.00 –7.50 –10.00 dB (S11)

–12.50

–15.00 –17.50 –20.00 –22.50

Name m1 m2 m3 m4

–25.00 5.00

X 8.2000 9.8000 11.5000 12.7000

Y –19.6511 –14.7939 –24.5187 –14.4691

7.50

10.00

12.50

15.00

Freq (THz)

Figure 8.20 S11 of MIMO spherical DRA S12

–10.00

HFSSDesign1 Curve info dB (S12) Setup1 : Sweep

–15.00 –20.00 dB (S12)

–25.00 –30.00 –35.00 –40.00 –45.00 5.00

7.50

10.00 Freq (THz)

12.50

Figure 8.21 S12 MIMO spherical DRA

15.00

Spherical terahertz and optical DRA—design and implementations Name

T

Ang

Radiation pattern 5 0

Mag

m1 360.0000 –0.0000 3.6801

–30

4.00

HFSSDesign1 Curve info

xdb20Beamwidth(3)

dB (Gain Total) Setup1 : Sweep Freq = '10 000GHz' φ = '0º' dB (Gain Total) Setup1 : Sweep Freq = '10 000GHz' φ = '0º'

30

0.50

89.6753 52.8760

60

–60 –3.00 –6.50

90

–90

120

–120

–150

150 –180

Figure 8.22 MIMO spherical DRA radiation pattern

120 130 140

Smith chart 3 90 100 80 110 1.00

0.50

HFSSDesign1 Curve info S11

70

Setup1 : Sweep

60 2.00 50

S12 Setup1 : Sweep

40 30

150 160

5.00 20

0.20

10

170 180 0.00 0.00

0.20

0.50

2.00

0.00

5.00

0

–170

–10

–160 –0.20

–5.00 –20

–150

–30

–140

–40

–130 –0.50 –120 –110

–100

–1.00 –90

–80

155

–2.00 –50 –60 –70

Figure 8.23 MIMO spherical DRA smith chart

156

Terahertz dielectric resonator antennas

12.50

VSWR

HFSSDesign1

Name X Y 8.2000 1.8150 m1 m2 11.5000 1.0338

Curve info dB (VSWR(1)) Setup1 : Sweep

dB (VSWR(1))

10.00

7.50

5.00

2.50

0.00 5.00

7.50

10.00

15.00

12.50

Freq (THz)

Figure 8.24 MIMO spherical DRA VSWR

Diversity gain

HFSSDesign1 Curve info dB (diversityGain) MIMO Calculation 2

20.000

dB (Diversity gain)

19.998 19.995 19.993 19.990 19.988 19.985 19.983 8.50

9.00

9.50

10.00

10.50

11.00

11.50

Freq (THz)

Figure 8.25 MIMO spherical DRA diversity gain Model-3 terahertz spherical DRA designed at 8.5 THz

8.6 Conclusion Microwave optical spherical DRA at 511 THz with 2.45-dB gain has been simulated using HFSS for optical applications such as imaging, sensor and scanning. Theoretical modeling of radiation pattern has been formulated in (8.1)–(8.10). THz DRA with a high gain of 18 dBi and a frequency of 10 THz has been developed for communication applications. It is having super directivity. These spherical DRAs have applications such as biomedical sensors, scanning, imaging, chip-to-chip communications and retinal photoreceptors at optical frequencies. MIMO DRA has advantages of diversity reception and super directivity. Results at 8.5 THz frequency have also been seen in Figures 8.26–8.28.

Spherical terahertz and optical DRA—design and implementations 90 10

120

60

5 0

150

30

–10 –15

0

180

–10 –5 210

0

330

φ=0 φ = 90

5

300

240

10

270

Figure 8.26 Radiation pattern at 8.5 THz

0 –5 –10 S11 (dB)

Gain (dB)

–5

–15 –20 S11 –25 –30 6

7

8

9 10 Frequency (THz)

11

Figure 8.27 S11 at 8.5 THz

12

157

158

Terahertz dielectric resonator antennas

Z11 (:)

50

45

40

35 6

7

8

9 10 Frequency (THz)

11

12

Figure 8.28 Z11 at 8.5 THz

References [1] R. S. Yaduvanshi and H. Parthasarathy, Rectangular Dielectric Resonator Antennas: Theory and Design, Springer, New Delhi, 2016. [2] R. S. Yaduvanshi and V. Gaurav, Nano Dielectric Resonator for 5G Applications, CRC Press, Boca Raton, FL, 2020. [3] L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics, vol. 5, no. 2, pp. 83–90, 2011. [4] P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys., vol. 75, p. 024402, 2012. [5] L. Zou, W. Withayachumnankul, C. Shah, et al., “Dielectric resonator nanoantennas at visible frequencies,” Opt. Express, vol. 21, no. 1, pp. 1344– 1352, 2013. [6] G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Hernandez Figueroa, “Dielectric resonator antenna for applications in nanophotonics,” Opt. Express, vol. 21, no. 1, pp. 1234–1239, 2013. [7] M. W. Knight, H. Sobhani, P. Nordlander, and N. J. Halas, “Photodetection with active optical antennas,” Science, vol. 332, no. 6030, pp. 702–704, 2011. [8] P. Mu¨hlschlegel, H.-J. Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science, vol. 308, no. 5728, pp. 1607–1609, 2005. [9] Y. Zhao, N. Engheta, and A. Alu`, “Effects of shape and loading of optical nano antennas on their sensitivity and radiation properties,” J. Opt. Soc. Am., vol. B28, no. 5, pp. 1266–1274, 2011.

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[10] S. A. Maier, Plasmonic—Fundamentals and Applications, Springer, New York, NY, 2007. [11] E. Ozbay, “Plasmonics: merging photonics and electronics at nano scale dimensions,” Science, vol. 311, p. 189, 2006. [12] L. Zou, W. Withayachumnankul, C. Shah, et al., “Efficiency and scale ability of dielectric resonator antennas at optical frequencies,” IEEE Photon. J., vol. 6, p. 110, 2014. [13] J. D. Jackson, Classical Electrodynamics, Wiley, New York, NY, 1962. [14] A. Bonakdar and H. Mohseni, “Impact of optical antennas on active optoelectronic devices,” Nanoscale, vol. 6, pp. 10961–10974, 2014. [15] P. D. Terekhova, K. V. Baryshnikovab, Y. A. Artemyevb, A. Karabchevskya, A. S. Shalinb, and A. B. Evlyukhinb, “Optical multipole resonances of non-spherical silicon nanoparticles and the influence of illumination direction,” Proc. of SPIE Vol. 10528 1052802-1, SPIE OPTO, San Francisco, CA, United States, 2018. [16] D. Storozhenko, A Nepomnyaschiy, and A. V. Deev, “High directivity in the narrow band of spherical dielectric antennas in GHz and THz ranges,” Semiconductors, vol. 53, no. 14, pp. 1967–1969, 2019.

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Chapter 9

Rectangular terahertz DRA—design, simulation and implementations

Abstract In this, design and implantations of a terahertz rectangular dielectric resonator antenna (DRA) using an HFSS simulator has been presented. The aspect-ratio has played a deterministic role for exciting desired multiple resonant modes at terahertz frequencies. The design has large bandwidth hence can be used for high-speed communications and terahertz sensors. It is simple due to rectangular geometry, efficient due to use of DR, compact due to terahertz frequency and unique using silicon as a radiator. Mathematical modeling of 10-THz frequency using a terahertz DRA is presented along with simulations results of S11 below
10 dB, gain of 4.89 dBi with good radiation pattern, which have been validated. DRAs at photonic wavelength have also been developed for 511-THz frequency with all desired parameters and simple geometry. These DRAs fall within visible frequency; hence they can be used as artificial retinal photoreceptors. A DR material has advantages over metallic antennas as they can operate with stability from microwave to optical regime. These optical antennas can be suitable for wireless transmission of TBPS data rate. Also, coherent states and control at optical frequencies have been formulated mathematically with unique characteristics at terahertz frequency.

9.1 Introduction It is well known that a dielectric resonator antenna (DRA) is an efficient radiator within microwave frequency to optical frequency regime. Dielectric materials were first suggested by Richtmyer in 1939 for use as an antenna-radiating element, later SA Long experimentally used DRA for practical implementations [1]. Since then, RK Mongia, A Kisk, YMM Antaar, Guha, Van Bladel, Pan, Petosa, Leung, Yaduvanshi and many other researchers haves done research work on DRAs [1–11]. Yaduvanshi and Parthasarathy have published on rectangular DRAs (RDRAs) in Springer 2016 on RDRA. A nano-DRA for 5G applications is published in CRC Press 2020 by Yaduvanshi, Varshney. Terahertz is a new field of applications for high-speed communications due to its large bandwidth characteristics, i.e., 300 GHz to 10 THz. This band is still lying vacant for telecommunication application [1–5]. Terahertz

162

Terahertz dielectric resonator antennas

technology has effectively miniaturized the size of DRA. They consume less power due to small size. Hence, terahertz DRAs (TDRAs) are in much demand nowadays for its high-speed communications and sensing applications to be used in Internet on Things networks. A frequency spectrum of microwave, terahertz and photonics is given in Figure 9.1. The design of TDRA is shown in Figure 9.2 and dimensions are shown in Table 9.1 for a 10-THz DRA. Figures 9.3–9.10 show simulated results. Here, most of our work shall be in the range of 1011–1015 Hz (THz, IR and optical spectrum). The DR materials used were Al2O3, TMM, sapphire, BaSrTiO2 at microwave frequencies and silicon, graphene, TiO2, arlon and teflon at terahertz and optical

THz science and technology Electronics

Photonics THz Visible gap

RF and microwave HF, VHF, UHF, SHF 100

103 kilo

106 mega

109 giga

1012 tera

X-ray

1015 peta

Y-ray

1018 exa

1021 zetta

Freq (Hz)

Figure 9.1 Microwave, terahertz and photonic science (spectrum)

BR

LSUB

LR

HR HSUB

LN WN

HG LG

WG/WSUB

Figure 9.2 Design of TRDRA (CST model) Table 9.1 Dimension table Name

Material

Permittivity and loss tangent

Dimension (nm)

Substrate top Substrate bottom Nano waveguide CDRA Gaussian beam

Silicon dioxide Silver Silver Titanium dioxide LASER pulses

4 and 0 1 and 0 1 and 0 8.29 and 0.01 500 THz

900*900*100 900*900*100 450*67*10 L ¼ 64, B ¼ 64, L ¼ 60 133.5*100

Rectangular terahertz DRA—design, simulation and implementations

163

Terahertz radiations

Silicon radiation SiO2 Silver

Gaussian beam laser

Figure 9.3 Terahertz DRA side view with flow (TRDRA)

S11

0.00

Name m1 m2 m3 m4

–5.00

X 7.4500 8.3000 10.1500 11.6000

HFSSDesign1 Curve Info

Y –26.0926 –29.3306 –28.5944 –22.1923

dB (S ) 11 Setup 1 : Sweep

dB (S11)

–10.00 –15.00 –20.00

m4 m1

–25.00

m3

m2

–30.00 –35.00 7.00

8.00

9.00

Freq (THz)

10.00

11.00

12.00

Figure 9.4 Terahertz RDRA gain plot with max gain 7.38 dBi (TRDRA)

6.00

Gain Name

X

Y

m1

10.0000

4.8900

GainDesign1 Curve info

m1

dB (Gain total) Setup 1 : Sweep φ = '0º' θ = '0º'

dB (Gain total)

4.00 2.00 0.00 –2.00 –4.00 –6.00 –8.00 7.00

8.00

9.00

Freq (THz)

10.00

Figure 9.5 Gain of rectangular TDRA

11.00

12.00

164

Terahertz dielectric resonator antennas

abs (Radiation efficiency)

1.00

Radiation efficiency Name

X

Y

m1

10.0000

0.9186

HFSSDesign1

m1

0.90 0.80 0.70 0.60 Curve info

abs (Radiation efficiency) Setup 1: Sweep φ = '0º' θ = '0º'

0.50 7.00

8.00

9.00

Freq (THz)

10.00

11.00

12.00

Figure 9.6 Efficiency of rectangular TDRA Name

m1

Theta

Ang

Radiation pattern 0

Mag

360.0000 –0.0000 4.8900

HFSSDesign1 Curve info

dB (Gain total) Setup 1: Sweep Freq = '10, 000GHz' φ = '0º' dB (Gain total) Setup 1: Sweep Freq = '10, 000GHz' φ = '90º'

30

–30 2.00 –1.00

60

–60 –4.00 –7.00

–90

90

–120

120

–150

150 –180

Figure 9.7 Radiation pattern of rectangular DRA z

E Field (Y/m) 8.9013E+005 8.3085E+005 7.7156E+005 7.1228E+005 6.5300E+005 5.9371E+005 5.3443E+005 4.7515E+005 4.1587E+005 3.5658E+005 2.9730E+005 2.3802E+005 1.7873E+005 1.1945E+005 6.0167E+004 8.8348E+002

0

15

30 (mm)

Figure 9.8 E field of rectangular DRA

xdb20Beamwidth(3)

23.5344 34.3375

Rectangular terahertz DRA—design, simulation and implementations

165

z

H Field (A/m) 5.5743E+003 5.2032E+003 4.8321E+003 4.4609E+003 4.0898E+003 3.7167E+003 3.3476E+003 2.9764E+003 2.6053E+003 2.2342E+003 1.8631E+003 1.4920E+003 1.1208E+003 7.4971E+002 3.7859E+002 7.4697E+000

0

15

30 (mm)

Figure 9.9 H Fields at 10 THz z

H Field (A/m) 1.5743E+004 1.4694E+004 1.3645E+004 1.2596E+004 1.1547E+004 1.0498E+004 9.4480E+003 8.3998E+003 7.3587E+003 6.3017E+003 5.2526E+003 4.2036E+003 3.1546E+003 2.1055E+003 1.0565E+003 7.4597E+000

0

15

30 (mm)

Figure 9.10 H-Field vector spectrum. DRAs are efficient in radiating elements throughout the entire frequency spectrum. The permittivity of these materials is available from 10 to 1,600. Hence, due to high permittivity, the materials help to reduce device sizes as it is inversely proportional to size, i.e. high permittivity corresponds to small size. They have design flexibility as DRAs command two different aspect-ratio height/base-length or height/ base-width. They have capability of generating higher order modes along with fundamental modes depending upon aspect-ratios. It is generally chosen between 0.5 and 2.5 for generating resonant modes of our choice. Higher order modes can have higher gain and better directivity as compared to fundamental mode. Terahertz regime can

166

Terahertz dielectric resonator antennas

offer higher bandwidth. Hence terahertz is suitable for higher speed communication. This terahertz band is lying vacant thus can be used for high-speed communication. Due to skin effect, metal behaves like plasma and Drude’s analysis will be required to assess scattering properties. Wave-port analysis to a terahertz device may not give accurate results; however, Gaussian pulse along with sliver nano waveguide shall give accurate results. The Gaussian beam at terahertz is used for phase matching, and proximity feed is used to accommodate electrical effects of surface plasmon polariton (SPP) length. Plasma frequency of each metal is different and for propagation it is always kept lower than laser input excitation frequency. The Dirac equation is used to compute radiated fields at optical frequencies. The development of quantum antenna has been a slow process and operators are used for mathematical analysis of quantum antennas. There is great contribution of Bhardwaj, Deutsch and Novotny in the development of light antenna or optical antennas [12–17]. Propagation loss is the main disadvantage at higher frequencies. These losses can be minimized by making use of DRAs at terahertz and optical frequencies. The quantum cascade lasers can be used at 1–5 THz, 90 mW, narrow width line and single-mode operation source to feed. Noble metal silver is used as waveguide at higher frequency in place of microstrip line feed. With this antenna, TBPS of data rate can be achieved. This high data rate is good for 5G and beyond communication systems. Microwave antennas have limited capability to deliver data speed only up to few GBPS. These terahertz antennas can be placed in chip (CMOS). Of course, they will be costly to develop, but cost can be reduced if they are to be produced at large industrial scale. Initially light antennas were to be operated only near field. By making use of DR, their gain can be enhanced and they are able to produce far fields and most suitable for long distance wireless communications at THz and optical frequencies. TDRAs of cylindrical, conical and spherical shape at optical frequency and dimension can be very useful for the development of artificial retinal photoreceptors as these DRA photonic antennas are efficient [18,19].

9.2 Propagation of light The understanding of light since Sir Isaac Newton started with first assuming light to be corpuscles, namely small particles of different color corresponding to the different frequencies. This idea was first propounded by Newton and came to be known as the corpuscular theory of light. Several years later, based on experiments involving interference of light, Huygens came to understand that light is to be regarded as being composed not of particles but rather is a wave and that any light field can be regarded as a superposition of plane waves of different frequencies and wave numbers. This meant that during the process of interference involving computation of the total intensity of light coming from different sources, one should not add the respective intensities, but rather the amplitudes, then square the resultant amplitude and then form its time average. This meant that in the resultant total intensity, there would appear cross terms that are oscillatory, which lead us to the

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167

phenomenon of interference. Later, Fermat propounded the ray theory of light in order to explain Snell’s theorems on reflection and refraction of light, according to which a light ray always follows the path of minimum time. The wave nature of light as propounded by Christian Huygens gained more support when Maxwell unified electromagnetism with light by showing that the basic equations of electromagnetism imply that that the electric and magnetic fields in space satisfy the wave equation with the velocity of propagation being equal to that of light. This fundamental discovery gave firm support to the wave theory of light. However, with the birth of the quantum theory in the early part of the nineteenth century, it became clear that light comes in discrete packets called quanta and each quantum of radiation carries an energy proportional to its frequency and that the intensity of light is proportional to the number of quanta. These quanta came to be known as photons and it appeared therefore that with the advent of the quantum theory, physicists had reverted back to Newton’s corpuscular theory of light. However, while Planck was busy creating his quantum theory of light, simultaneously Einstein wrote some beautiful papers proving that light can behave sometimes as a particle and sometimes as a wave, specifically showing that the variance of fluctuations in the energy of black-body radiation has two components, a particle component in agreement with Planck’s quantum hypothesis and a wave component in agreement with Rayleigh’s theory on the number of “modes” of a wave field within a given energy shell. The particle theory of light got further support from Einstein’s special theory of relativity giving the relationship between the energy and momentum of a particle of any given mass and in particular for light particles having zero rest mass. However, the crucial breakthrough into the wave– particle duality matter and in particular of light came with de Broglie’s discovery of the fundamental relationship between the wavelength of a wave and the momentum of the associated particle. The de Broglie theory was applicable to all of matter not just light. It gave conclusive evidence that even particles like electrons exhibit wave–particle duality, i.e., they sometimes behaved like particles and sometimes like waves. The final crunch describing the wave–particle duality came with the wave equation for particles discovered by Schro¨dinger using which he was able to calculate the energy levels of an electron bound to a nucleus in terms of the eigen values of a partial differential equation. The corresponding eigen functions were complex wave fields that Schro¨dinger could not immediately interpret. These eigen functions approximately corresponded to the kind of “matter-waves” predicted by Louis de Broglie having frequency determined by Planck’s quantum hypothesis relating frequency of waves to the energy of the quanta and having wavelength determined by de Broglie’s theory relating the wavelength of waves to the momenta of the particles. Schro¨dinger’s wave equation was a nonrelativistic wave equation in that it could be “derived” by assuming the nonrelativistic relationship between energy and momentum and then assigning operators to energy and momentum. The correct interpretation of the eigen functions of Schro¨dinger’s equations was provided about a year later by Max Born with the suggestion that the modulus square of the wave function gave the probability density of the particle to be present at a given point in space at a given time. Equivalently, this modulus

168

Terahertz dielectric resonator antennas

square could be interpreted as the number density of particles or the intensity of the wave at that point in space at that time. In fact, Schro¨dinger’s equation could also be generalized to determine the wave functions and energy levels for any particle with a prescribed energy–momentum relationship by replacing the energy and momentum with appropriate operators and applying both sides of this relationship to a wave function. Thus, Schro¨dinger’s formalism could be applied to determine the kind of de Broglie waves associated with a particle having any given energy– momentum relation. The de Broglie wave solutions then have the interpretation that their modulus square represents the intensity/probability density of particles in space at a given time. In special relativity, the energy–momentum relationship for a relativistic particle is a quadratic form in the energy and momenta unlike the Newtonian-nonrelativistic case where it is linear in the energy and quadratic in the momenta. Consequently, the Schro¨dinger equation is a linear PDE that is of the first order in time and second order in spatial derivatives. The nonrelativistic Schro¨dinger equation as a consequence then leads to a unitary evolution of the wave function with time and the additional pleasing property that evolution from time zero to time t1 þ t2 can be expressed as the composition of evolution from time zero to time t1 followed by evolution from time t1 to time t1 þ t2. In other words, Schro¨dinger evolution follows the semigroup property. This is in contrast to the case of special relativity wherein the wave equation is quadratic in time leading to the wave function at a given time being a superposition of a forward-propagating solution and a backward-propagating solution and the loss of unitarily of the evolution. Thus, the total probability will be conserved in Schro¨dinger’s nonrelativistic theory but not so in the relativistic theory. This difficulty was finally resolved by Paul Dirac who factorized the energy–momentum relationship of special relativity into linear factors in energy and momentum using 4  4 complex anticommuting matrices and it led to Dirac’s relativistic theory of the electron which along with Maxwell’s equations is at the heart of quantum electrodynamics describing the interaction of light with matter. Dirac’s equation being linear in the space and time derivatives is a truly relativistic wave equation for quantum theory because first it respects Lorentz invariance in that space and time are treated on an equal footing and that we can find a spin or representation of the Lorentz group under which Dirac’s equation remains invariant and second it is first order in time and therefore preserves the unitary semigroup property of the evolution guaranteeing therefore conservation of total probability. Light as we understand today is a second quantized electromagnetic field in which the vector potential components in the spatial frequency domain represent creation and annihilation operators of photons with different momenta and helicities. This is owing to the fact that the total energy of the electromagnetic field can in the Coulomb gauge be expressed as a quadratic functional of the spatial frequency components of the vector potential and each spatial frequency component of the vector potential, according to Maxwell’s equations, evolves harmonically with time with a frequency being w, where k is the wave vector. Such a picture is in perfect accord with the Hamiltonian theory of a collection of independent quantum harmonic oscillators. However, this model of light can actually be traced back to the work of

Rectangular terahertz DRA—design, simulation and implementations

169

Satyendranath Bose who proposed a statistical method for deriving Planck’s law of black-body radiation wherein we are interested in distributing a total amount of energy E having N quanta at frequency n so that E ¼ Nhn, amongst p oscillators with the quanta being regarded as indistinguishable particles. When an oscillator has n such quanta, we say in the modern language of quantum mechanics that it has been excited to the nth energy level by an application of n creation operators to the vacuum. In this way, the entire field of black-body radiation is simply a collection of quantum harmonic oscillators and if we maximize the entropy, i.e., the total number of ways of distributing this energy then we end up with the famous Bose–Einstein statistics that gives us the relative number of photons at each frequency. This in turn enables us to determine the intensity of black-body radiation as a function of frequency. Summarizing the modern point of view, the entire photon field described as an operator electromagnetic four-potential field is just a superposition of plane waves of different wave vectors whose coefficients are creation and annihilation operators at the different wave vectors and a given state of the photon field is actually a linear combination of number states wherein a number state is specified by the number of photons that occupy each state of definite wave-vector/momentum and definitive helicity/spin. A particular kind of photon state called a coherent state is a state constructed by an appropriate linear combination of this kind that turns out to be an eigen state for all the photon annihilation operators. This model for the quantum electromagnetic field implicitly contains both the particle nature and the wave nature of light. The particle nature is contained in the presence of the creation and annihilation operators, while the wave nature is contained in the plane waves that act as carrier signal fields for the creation and annihilation operators. Second quantization means a quantization of a classical field theory. It is called second quantization for the following reasons: the first quantization is simply a wave equation like the three-dimensional wave equation, the Schro¨dinger wave equation for a single or an infinite number of quantum particles, the Klein Gordon equation or the Dirac equation. If such a classical wave equation is quantized, then it describes an infinite number of quantum particles. This can be seen clearly from the following example. Take the Klein–Gordon equation that is the wave equation described before corresponding to Einstein’s energy momentum relation with the energy and momenta replaced by appropriate operators. We expand the solution wave field as a three-dimensional Fourier series within a cube of side length L. The coefficient of this Fourier series then becomes quantum operators, each one of which and its adjoint, i.e., Hermitian conjugate describes a single quantum particle. Thus, a second quantized field can equivalently be described by a countable infinite number of quantum particles. For bosonic quantum fields like the Maxwell electromagnetic photon field, the state of the field can be described, for example by specifying how many particles are occupying each state of definite momentum and helicity or equivalently by specifying the momentum and helicity of each of the particles. In this case, there can be zero, one or more

170

Terahertz dielectric resonator antennas

than one particle having a specified momentum and helicity. In the case of fermionic fields like the second quantized Dirac field, we cannot have more than one particle having a definitive value of momentum and spin [20,21].

9.3 Design and simulation of terahertz dielectric resonator antenna Figure 9.1 shows the frequency spectrum and Figure 9.2 shows the design of terahertz rectangular antenna based on dimensions available in Table 9.1, respectively. An SPP phenomenon takes place at light–matter interaction and Gaussian beam is given as an input to feed. Silver nano waveguide has been used as feed. The resonant frequency of TRDRA can be computed based on the following formulations given, fmnl is resonant frequency, where a, b, d are dimensions and m, n, l are indices in x, y, z directions, c is the velocity of light and er is the permittivity of material used for radiating element: ffiffiffiffiffiffi ffi pcffiffiffiffiffiffiffi fmn‘ ¼ 2pckpmn‘ m 2r ¼ 2p m 2r r

r

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 ‘p2 þ b þ d  a

9.4 Synthesis of a terahertz rectangular DRA at optical frequency and its radiation theory 1.

ih [qa ; pb  ¼ 2p dða; bÞ

2.

where q is the position vector; p is the time-derivative moment vector; b ¼ 1=kT and T is temperature. ð
bH Þ r ¼ Trexp ðexpð
bH ÞÞ; mean field energy in Gibbs state ¼ r is given by Tr ðrH) and the Tr ðrp ) classically we would get Ð

expð
bH Þpa d N qd N p Ð ¼0 expð
bH ÞdqN dpN

ffiffi k are annihilation and creation operators ak ¼ akpþip 2 ak
ipk pffiffiffi ; k ¼ 1; 2; 3 2   where ak ; am ¼ ðh=2pÞdðk; mÞ is the Heisenberg commutation relation. N P ak ak þ Nh Electromagneticfield ¼ H ¼ 12 4p ; field energy. ak ¼

k¼1

Quantam states or eigen states of H . . ; nN ¼ 0; 1; 2; 3; . . . jn i ¼ jn1 ; n 2 ; n3 ; . . . ; nN i; n1 ; n2 ; n3 ; .  h N þ n1 þ n2 þ . . . þ nN Hjn i ¼ 2p 2

Rectangular terahertz DRA—design, simulation and implementations

171

Thus, the quantum ak in the Gth state is

! ð
bNhÞ ð
bhÞ X  exp hnjak  exp a am 4p 2p m m ! ð
bNhÞ X ð
bhÞ X nm dðn; m
ek Þ ¼ 0 ðni ¼ exp n exp 4p 2p ek ¼ ½0; 0; 0; 1; 0

m

Here   Tr p2k expð
bhÞ 
bh  1  2   Tr pk þ p2 k
ak ak
ak ak e ¼
2 z q ðb Þ zq ðbÞ ¼ quantam partition   function

1 h expð
bhÞ ¼ Tr ak ak þ 2zq ðbÞ 4p !  X   1 ð
bNhÞ 1 h bh X ¼ exp nm n nk þ 2 2p exp
4p 2zq ðbÞ 4p m !   ð X a2k þ p2k a2k dqN dpN =zc ðbÞ exp
b 2 k where zc ðbÞ is quantum part (Figures 9.11–9.14).

9.5 Mathematical analysis of resonant modes excited into a terahertz rectangular DRA Consider a cavity resonator of one angstrom size, i.e., a cube with each side of length a ¼ 10
10 m: The Maxwell equations in such a cube have solutions of the

Dielectric resonator Microstrip nanofeed

Substrate

Ground plane

Figure 9.11 Optical RDRA at 484 THz

172

Terahertz dielectric resonator antennas Name m2

HFSSDesign1

XY Plot 1

X Y 484.4444 –25.3988

Curve info dB (S11) Setup1: Sweep

–10.00 –12.50

db (S11)

–15.00 –17.50 –20.00 –22.50 m2

–25.00 –27.50 450.00

475.00

525.00

500.00 Freq (thz)

550.00

Figure 9.12 S11 at 484 THz HFSSDesign1

XY Plot 17

X Y 500.0000 0.7808

Name m1

Curve info RadiationEfficiency Setup1: Sweep Phi="Odeg" Theta="Odeg"

m1

Radiation efficiency

0.78

0.76

0.74

0.72

0.70

0.68 450.00

475.00

500.00 Freq (thz)

525.00

550.00

Figure 9.13 Radiation efficiency at 484 THz from Ar ðt; x; y; zÞ ¼

X

cðmnp; tÞur ; mnpðx; y; zÞ; r ¼ 1; 2; 3

(9.1)

mnp

where ur ; mnp are spatial functions obtained by integrating the electric field w:r:t: n mpx mpx o n npy npy n ppy ppz o cos ; sin  cos ; sin  cos ; sin a a a a a a Multiplied by some constants depending on the indies ðm; n; pÞ: We may, without loss of generality, assume those as normalized so that ð ur ; mnpðrÞ u s ; m0 n0 p0 ðrÞd 3 r ¼ drs dmm0 dnn0 dpp0 C

Rectangular terahertz DRA—design, simulation and implementations Radiation pattern 0

HFSSDesign1 Curve info dB(GainTotal) Setup1 : Sweep Freq='500000GHz' Phi='0deg' dB(GainTotal) Setup1 : Sweep Freq='500000GHz' Phi='0deg'

30

–30

173

0.00 –5.00 60

–60 –10.00 –15.00

90

–90

120

–120

150

–150 –180

Figure 9.14 Radiation pattern of TRDRA at 484 THz The dependence of cðmnp; tÞ on t is expðiwðmnpÞtÞ where w ðmnpÞ are the characteristic frequencies of oscillation: pc pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ n2 þ p2 ; m; n; p ¼ 1; 2 (9.2) wðmnpÞ ¼ a which are of the order of magnitude: w¼

pc a

The electric field is X E r ¼ @t Ar ¼ cðmnp; tÞiwðmnpÞur ; mnpðrÞ mnp

The magnetic field is B ¼ curlA Which is of the order of magnitude jcðmnp; tÞj=a where by cðmnp; tÞ we actually mean that it is an average in a coherent state. The total electric field energy within the cavity C is  ð 0

Ej2 d 3 r UE ¼ 2 C which has components of the order of magnitude 0 ðmnpÞcðmnp; tÞj2 a3 ¼ 0 wðmnpÞ2 a3 jcðmnp; tÞj2 w

174

Terahertz dielectric resonator antennas The total magnetic field energy within the cavity is ð


1 2 3 UB ¼ ð2m0 Þ Bj d r c

which have components of the order of magnitude

a

cðmnp; tÞ 2 a3

¼ cðmnp; tÞ 2

m0 a m0 The ratio of the orders of magnitude of the electric field energy and the magnetic field energy within the cavity therefore has the order of magnitude: UE w2 a2  m0 0 wðmnpÞ2 a2  2  1 UB c As expected, the canonical commutation relations are h  0 i  ih   0 d3 r
r ¼ Ar ðt; rÞ; @t As t; r 2p

(9.3)

These fields are i h drs dmm0 dnn0 dpp0 cr ðmnp; tÞ; wðmnpÞcs ðm0 n0 p0 ; tÞ ¼ 2p So that the eigenvalues of cr ðmnp; tÞ cr ðmnp; tÞ are positive integer multiples of h=2pwðmnpÞ: This means that the field energy within the cavity, when a finite number of modes are excited, assumes eigenvalues that are of the same order of magnitude as positive integer multiples of hw=2p as expected by Planck’s quantum theory of radiation. This fact also yields the result that jcðmnp; tÞj is of the order of pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi magnitude of h=ð2pwÞ: Now we come to the question of computing the order of the magnitude of Poynting vector power flux at a given radial distance R from the quantum cavity antenna caused by the surface current density induced by the magnetic field on the antenna surface. The magnetic field on the surface and hence the corresponding induced surface current density p both the order of magnitudes of jcðmnp; tÞj=a ffiffiffiffiffiffiffiffihave ffi which is of the order of a
1 h=w: Therefore, the far-field magnetic vector potential at a distance of R from the cavity pffiffiffiffiffiffiffiffi ffi is of the order of magnitude (use the retarded potential formula) ða=RÞ h=w and hence the corresponding pffiffiffiffiffiffiffiffiffi far-field radiated magnetic field is of the order of magnitude ðw=cÞp ða=R Þ ffiffiffiffiffiffiffiffiffi h=w while the 2 Þffiffiffiffi h=w: Actually, these near-field magnetic field of the order of magnitude ða=Rp expressions for the magnetic field must be multiplied by N where N is a positive integer corresponding to the largest modal eigenvalue of the operators ð2pwðmnpÞ=hÞcðmnp; tÞ cðmnp; tÞ: The far-field Poynting vector has the order of magnitude of B2 c=2m0 which is of the following order:

Rectangular terahertz DRA—design, simulation and implementations 

c 2m0



175

rffiffiffiffi!2      pffiffiffiffi w  a h h w a2  ¼ N N c R w 2m0 c R2

And the total power radiated outward by this quantum antenna in the far-field zone is thus of the order of magnitude:   2  h a w P¼N ¼ 2m0 c Now we look at the order of magnitude of the power radiated in the far-field zone by the Dirac field of electrons and positrons within the cavity. The Dirac equation is  m  ig @m
m yðxÞ ¼ 0 Or more precisely in arbitrary units,  

    ih ih 2 @t
c a;
r
bmc yðxÞ ¼ 0 2p 2p the appearances of the constants h; m; c are explicitly shown. Now the

Here,

yðxÞj2 c is the probability density of the electron that must integrate to unity over the cavity volume. Thus yðxÞ is of the order of magnitude a
3=2

. The Dirac current density J m ¼ ey g0 gm y has the same order of magnitude as e yðxÞj2 c that is ec=a3=2 : Therefore, the far-field magnetic vector potential at a radial distance of R from the cavity is, in accordance with the retarded potential theory, of the order    3 ec a eca3=2  ¼ 3=2 R R a The electric field in the far-field zone is then of the order E w

eca3=2 R

(9.4)

where w is the characteristic oscillation frequency of the Dirac current. The magnetic field is of the order rffiffiffi eca3=2 a ¼ ec B  a
1  R R If P is the characteristic momentum of the electrons and positrons in a given state, for example P may be the average momentum of an electron in a given state, then according to de Broglie, P is of the order h=a since a is the order of the electron wavelength. Then the electron energy is of the order of sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 (9.5) Ee ¼ c m2 c2 þ P2  c m2 c2 þ 2 a

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Terahertz dielectric resonator antennas

And the characteristic frequency of oscillation of the Dirac wave field is then given by w¼

Ee h

The Poynting vector corresponding to the power radiated by the Dirac field in the far-field zone then has the order of magnitude:   B2 e2 c3 a ¼ c3 0 w2 ea3 þ (9.6) S  c 0 E 2 þ m0 m0 Equations (9.1)–(9.6) have been derived for formulating rectangular quantum or optical antennas.

9.6 Terahertz optical RDRA at 484 THz The application of optical antennas in photodetectors is particularly promising. The main reason is that an optical antenna increases the absorption cross-section and hence the light flux that impinges on a detector, giving it a clear signal-to-noise advantage. There is an upper limit to the performance of optical components. A minimum volume must be ensured to guarantee the necessary number of modes for information to be processed. The limit is valid for multiple scattering events and for linear interactions. The minimum volume scales with the dielectric constant of a material, and hence it can be reduced by using metallic structures such as optical antennas (Figure 9.15).

9.6.1

Approximate analysis of a rectangular terahertz DRA and its controlled electromagnetic fields

The TDRA is assumed to be the cuboid shape; let region be defined as ½0; a  ½0; b  ½0; d : In the rectangular cavity of a, b and d dimensions, it is assumed to be comprising photons initially due to feed and electrons and positrons Z Z

Microstrip nanofeed

Dielectric resonator Substrate

X

Ground plane

X 0

15

30(um)

Figure 9.15 Optical rectangular DRA top and side view

Rectangular terahertz DRA—design, simulation and implementations

177

due to substrate or DRA materials. Light–matter interaction takes place. The mathematical model is developed for this cavity under light–matter interaction. The exact equations governing the quantum fields corresponding to these particles are (a) the Maxwell equations for the four-vector potential driven by the Dirac field current and (b) the Dirac field equations driven by an interaction between the Dirac field and the Maxwell field four vector potential. Value and correlations in a given quantum coherent state of the photons and electrons–positrons within the cavity are close to specified values. These exact field equations are Am ðxÞ ¼ m0 ey ðxÞam yðxÞ; x ¼ ðt; rÞðða;
irÞ þ bm0 Þyðt; rÞ þ eAm ðxÞam yðxÞ ¼ i@t yðt; rÞ where am ¼ g0 gm ; b ¼ g0 2

And gm are the Dirac-gamma matrices. Note that ðg0 Þ ¼ I4 and hence a0 ¼I4 : The boundary conditions under which we need to solve these Maxwell–Dirac equations are that the Dirac operator wave field yðxÞ; the tangential components of the electric field For ¼ Ar;0
A0;r0 r ¼ 1; 2; 3 and the normal components of the magnetic field Frs ¼ As;r
Ar;s ; 1 r < s 3 must vanish on the boundaries of the cavity. In particular, the free Dirac field must have an expansion as X cðmnp; tÞmmnp ðrÞ yð0Þ ðt; rÞ ¼ mnp

where m; n; p run over positive integers and  pffiffiffi   2 2 mpx npy ppz mmnp ðrÞ ¼ pffiffiffiffiffiffiffiffi sin sin sin a b d abd Substituting this into the free Dirac equation, without any electromagnetic interactions, we get X

  X  i@t cðmnp; tÞmmnp ðrÞ ¼ a;
ir þ bm0 : cðmnp; tÞmmnp ðrÞ

mnp

mnp

From which we derive on taking the inner products on both sides with ukls ðrÞ and using the orthonormality of this set of functions over the cavity volume: ð hukls ; umnp i ¼ ukls ðrÞumnp ðrÞd 3 r ¼ dKm dln dsp B

178

Terahertz dielectric resonator antennas

where B is the cavity volume B ¼ ½0; a  ½0; b  ½0; d ; The following sequence of differential equations: X i@t cðkls; tÞ ¼ hukls
i@x umnp ia1 cðmnp; tÞ þ hukls ;
i@y umnp ia2 cðmnp; tÞ mnp

þhukls ;
i@z umnp ia3 cðmnp; tÞ þ m0 bcðkls; tÞ

Now we evaluate mp ða 2 kpx mpx sin cos dx hukls
i@x umnp i
idln dsp a a a a 0

¼ a1 ðk; mÞdln dsp where  a1 ðk; mÞ ¼

   ða mpx 2imp kpx cos dx
2 sin a a a 0

Likewise, hukls;
i@y umnp i ¼ a2 ðl; nÞdkm dsp ; And hukls;
i@z umnp i ¼ a3 ðs; pÞdkm dln Combining all these equations gives us finally, i@t cðkls; tÞ ¼

X

a1 ðk; mÞa1 cðmls; tÞ þ

m

þ

X

a2 ðl; nÞa2 cðkns; tÞ

n

X

a3 ðs; pÞa3 cðklp; tÞ þ m0 bcðkls; tÞ

p

Note that a1 ; a2 ; a3 ; b are 4  4 Hermitian matrices while cðmnp; tÞ is a 4  1 complex vector. Arranging the 4  1 vectors cðmnp; tÞm; n; p 1 in lexicographic order to give an infinite vector c(t) and likewise defining a block structured infinitedimensional Dirac Hamiltonian matrix H0 by

Rectangular terahertz DRA—design, simulation and implementations X H0 ¼ a1 ðk; mÞðI4  eðklsÞÞa1 ðI4  eðmlsÞT Þ

179

klsm

X

a2 ðl; nÞðI4  eðklsÞÞa2 ðI4  eðknsÞT Þ

klsm

X

a3 ðs; pÞðI4  eðklsÞÞa3 ðI4  eðklpÞT Þ þ m0 b  I

klsp

where we may choose eðmnpÞ; m; n; p 1 as any orthonormal basis for l2 ðZþ Þ; the Hilbert space of all one-sided square summable infinite sequences and define X cðmnp; tÞeðmnpÞ c ðt Þ ¼ mnp

By orthonormal, we mean that eðklsÞT eðmnpÞ ¼ dkm dln dsp Thus, the free Dirac equation in the RDRA has been put in “standard” block matrix form: i

dcðtÞ ¼ H0 cðtÞ dt

The general solution to which can be expressed as X d ðnÞ  cn expð
iEðnÞtÞ c ðt Þ ¼ n where cn ; n 1 forms an orthonormal basis for l2 ðZþ Þ and the d ðnÞ0 s are arbitrary complex numbers such that X

d ðnÞj2 ¼ 1 n 0

EðnÞ s are the (energy) eigenvalues of infinite-dimensional Hermitian H0 : detðH0
EðnÞIÞ ¼ 0 The average energy of the free Dirac field of electrons and positrons within the cavity is then X EðnÞd ðnÞ d ðnÞ hcðtÞ; H0 cðtÞi ¼ n It is easy to see as in the case of the Dirac equation in free space that if EðnÞ is an eigenvalue of H0 then so is
EðnÞ where the EðnÞ0 s may be taken as positive, Hence if cen is an eigenvector of H0 corresponding to the eigenvalue
EðnÞ and cpn

180

Terahertz dielectric resonator antennas

is an eigenvector corresponding to the eigenvalue
EðnÞ; then the solution can be expressed as X  c ðt Þ ¼ de ðnÞcen expðiEðnÞtÞ þ dp ðnÞ cpn expðiEðnÞtÞ n

Therefore, it is plausible in the second quantized theory to look upon the de ðnÞ0 s as annihilation operators of the electrons and the dp ðnÞ0 s as the creation operators of the positrons. The actual Dirac wave function yðt; rÞ in the absence of electromagnetic interactions is then X  d ðk Þcek ðmnpÞumnp ðrÞexpð
iEðk ÞtÞ yðt; rÞ ¼ kmnp e þdp ðk Þ cpk ðmnpÞumnp ðrÞexpðiEðk ÞtÞ A simple calculation then shows that the second quantized Hamiltonian of the free Dirac field of electrons and positrons within the cavity is given by ð HD0 ¼ yðt; rÞ ðða;
irÞ þ bm Þyðt; rÞd 3 r B

ð

yðt; rÞ i@t yðt; rÞd 3 r

¼ B

X

¼

 Eðk Þ de ðk Þ de ðk Þ
dp ðk Þdp ðkÞ Þ

k

Now from the basic anticommutation relations for the Dirac field, we have   fyðt; rÞ; y t; r0 Þ ¼ d3 ðr
r0 ÞIg And this immediately implies the following anticommutation relations for the electron and positron creation and annihilation operators: fde ðk Þde ðmÞ g ¼ dkm ; fdp ðk Þ; dp ðmÞ g ¼ dkm with all the other anticommutators vanishing. This completes our description of the free Dirac field of electrons and positrons within the RDRA. Using these anticommutation relations, we immediately get that the scalar potential is zero since there are no charges while the magnetic vector potential admits and expansion obtained from E ¼
@t A As Aðt; rÞ ¼

X k

½bðk Þwk ðrÞ expð
iwðk ÞtÞ þ bðk Þ wk ðrÞ expðiwðk ÞtÞÞ

Rectangular terahertz DRA—design, simulation and implementations

181

where now wk ðrÞ has three components that are calculated from the expansion of Ez and the relationship between the transverse and longitudinal components of the electric field within the cavity. We note that the third, i.e., z component of wk ðrÞ where the index k is identified with the modal triplet ðmnpÞ is proportional to sin

mpx a

sin

npy b

cos

ppz d

in view of the boundary conditions on the electric field is the fact that each mode of the magnetic vector potential is proportional to the electric field ð
jwA ¼ EÞ bðk Þ ¼ bðmnpÞ that is identified with a photon annihilation operator while bðk Þ with a photon creation operator. They satisfy the canonical commutation relations ½bðk Þ; bðmÞ  ¼ dkm Formally, we can compute both the free Dirac current density yðt; rÞ am yðt; rÞ of electrons and positrons with the cavity and the surface current density on the RDRA walls induced by the tangential components of the quantum magnetic B ¼ curl A and obtain the far-field radiation pattern generated by both of these cavity current components. Obviously, this far-field radiation pattern will have its first component being a quadratic form in the electron–positron creation and annihilation operators de ðk Þ; dp ðk Þ; de ðk Þ ; dp ðk Þ while the second component will be linear in the photon creation–annihilation operators bðk Þ; bðk Þ and therefore, in principle, we can compute all the statistical moments of the radiation field in a joint coherent state of the photons, electrons and positrons. However, this picture of the far-field quantum radiation pattern is incomplete because it does not take into account the cavity surface current density terms caused by perturbation in the Maxwell field caused by its interaction with the Dirac field. We shall now indicate an approximate first-order calculation between the Maxwell field and the Dirac field. We denote the free Dirac field within the cavity derived before by yð0Þ ðt; rÞ and the corresponding momentum space wave function cðmnp  tÞ by cð0Þ ðmnp  tÞ: Likewise, we denote the free Maxwell field within the cavity by Að0Þ . Let dA denote the perturbation to the Maxwell field caused by the Dirac current and dy; dcðmnp; tÞ the perturbation to the Dirac field caused by the Maxwell current. Then, clearly if S ðx
yÞ denotes the electron propagator and Dðx
yÞ, the photon propagator, we have been using (9.1) and (9.2), approximately, ð dAm ðt; rÞ ¼ m0 e Dðt
t0 ; r
r0 Þyð0Þ ðt0 ; r0 Þam yð0Þ ðt0 ; r0 Þdt0 d 3 r0 ð dyðt; rÞ ¼ e S ðt
t0 ; r
r0 ÞAðm0Þ ðt0 ; r0 Þam yð0Þ ðt0 ; r0 Þdt0 d 3 r0

182

Terahertz dielectric resonator antennas

where we substitute for yð0Þ and Aðm0Þ the expressions given in (9.3) and (9.4). Then, X yð0Þ am yð0Þ ðt; rÞ ¼ ½de ðkÞ c ek ðmnpÞexpðiEðk ÞtÞ kmnpk 0 m0 n0 p

þ dp ðk Þc pk ðmnpÞexpð
iEðk ÞtÞÞ  am ½de ðk 0 Þcek 0 ðm0 n0 pÞexpð
iEðk 0 ÞtÞ    þdp k 0 Þ cpk ðm0 n0 pÞexpðiEðk 0 ÞtÞ umnp ðrÞum0 n0 p0 ðrÞ ¼

X

½de ðkÞ de ðk 0 Þc ek ðmnpÞam cek 0 ðm0 n0 p0 Þ

expðiEðk Þ
Eðk 0 ÞtÞumnp ðrÞum0 n0 p0 ðrÞ þ

X   de kÞ dp ðk 0 Þc ek ðmnpÞam cpk 0 ðm0 n0 p0 Þ expðiEðk Þ þ Eðk 0 ÞtÞumnp ðrÞum0 n0 p0 ðrÞ X

dp ðk Þde ðk 0 Þc pk ðmnpÞam cek 0 ðm0 n0 p0 Þ

expð
iEðk Þ þ Eðk 0 ÞtÞumnp ðrÞum0 n0 p0 ðrÞ X

 dp ðk Þdp kÞ c pk ðmnpÞam cek 0 ðm0 n0 p0 Þ

expð
iEðk Þ
Eðk 0 ÞtÞumnp ðrÞum0 n0 p0 ðrÞ We see that the frequencies of the Dirac current that generate the perturbation to the quantum electromagnetic field are Eðk Þ  Eðk 0 Þ; k; k 0 ¼ 1; 2; . . . or more precisely, these are divided by Planck’s constant. Here Eðk Þ was obtained by solving the free Dirac eigenvalue equation inside the rectangular cavity with zero boundary conditions. The Eðk Þ0 s were obtained as the eigen values of the Dirac Hamiltonian. From basic principles of special relativity, it is easy to see that these Eðk Þ0 s are of the order qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi c m20 c2 þ P2

Rectangular terahertz DRA—design, simulation and implementations

183

where  P ¼ 2

h 2p

2 

mp 2 np 2 pp 2 þ þ a b d

with m; n; p being positive integers determined by the mode of oscillation of the field within the cavity. Now this current is of the general form X   d ðk Þ d ðk 0 Þfkk; ðt; rÞ þ d ðk Þd ðk 0 Þgkk; ðt; rÞ þ d ðk Þ d ðk 0 Þ g k ; ðt; rÞ yð0Þ am yð0Þ ðt; rÞ k;k 0

where the d ðk Þ0 s are the annihilation operators of the electrons and positrons and their adjoints d ðk Þ are the corresponding creation operators. The functions fkk 0 gkk 0 are components that are easily seen to be expressible as superpositions of space–time 0 sinusoids with the temporal frequencies being exp iðEðk Þ  Eðk ÞÞumnp ðrÞum0 n0 p0 ðrÞ and these components are easily seen to be expressible as superpositions of space–time sinusoids with the temporal frequencies being Eðk Þ  Eðk 0 Þ or their negatives and the spatial frequencies, i:e:; 0 0 wave numbers being mp=a; np=b; pp=d; m p=a; p0 p=b; p p=d. Let ð m J ðk Þ ¼ e yð0Þ ðxÞam yð0Þ ðxÞexp ð
ik  xÞd 4 x ð    ¼ yð0Þ ðt; rÞam yð0Þ ðt; rÞexp
i k 0 t
K  r dtd 3 r where   k ¼ ðk m Þ ¼ k 0 ; K denotes the space–time four-dimensional Fourier transform of the unperturbed Dirac four current density. Then, we can write down the space–time Fourier transform of the correction dAm ðxÞ; x ¼ ðt; rÞ to the electromagnetic four potential caused by this Dirac current as ð dAm ðk Þ ¼ dAm ðxÞ  exp ð
ik  xÞd 4 x ¼

2

m0 J m ðk Þ=k 2 ; k 2 ¼ km k m ¼ ðk 0 Þ
jK j2

in units where c ¼ 1: It should be noted that by the convolution theorem for Fourier transforms, if yð0Þ ðk Þ denotes the space–time Fourier transform of yð0Þ ðxÞ; then ð J m ðk Þ ¼ ð2pÞ
4 yð0Þ ðk 0
k Þam yð0Þ ðk 0 Þd 4 k 0 and hence, the perturbation to the electromagnetic four potential in the space– time Fourier domain, i:e: in four momentum space of the photon, can be

184

Terahertz dielectric resonator antennas

expressed as dAm ðk Þ ¼

m0 ð2pÞ4 k 2

!ð

yð0Þ ðk 0
k Þam yð0Þ ðk 0 Þd 4 k 0

The unperturbed electromagnetic fields is in the Coulomb gauge, i:e:; div Að0Þ ¼ 0 and also since there is no charge/current for the unperturbed field, the unperturbed electric scalar potential is a matter field that is identically zero, i:e:; Að0Þ0 ¼ 0: Hence we are guaranteed that the unperturbed electromagnetic potentials also satisfy the Lorentz gauge conditions, i:e:; divAð0Þ þ @t Að0Þ0 ¼ 0: This means that while computing the perturbations to the electromagnetic potentials caused by currents coming from the Dirac field, we can safely work in the Lorentz gauge. Likewise, the change in the Dirac field caused by interaction with the electromagnetic field within the cavity is given up to first-order perturbation theory by ð dyðxÞ ¼ dyðxÞ ¼ e S ðx
x0 ÞAðm0Þ ðx0 Þam yð0Þ ðx0 Þd 4 x0 ð ¼ e S ðx
x0 ÞAðr0Þ ðx0 Þar yð0Þ ðx0 Þd 4 x0 ð  ¼
e S ðx
x0 Þ a; Að0Þ ðx0 Þ yð0Þ ðx0 Þd 4 x0 ð  X   0   0  0 0 0 ¼
e S t
t ; r
r ½bðk Þ a; wk r expð
iwðk Þt0 Þ þ bðkÞ a; wk r Þ expðiwðk Þt  k " #  0  0  0 X 0 0 de ðk Þcek ðmnpÞumnp r expð
iEk Þt þ dp ðk Þ cpk ðmnpÞumnp r exp iE ðk Þt dt d 3 r0  kmnp  0 ð    0  0    0 0 X 0 0 S t
t ; r
r a; wk r cek 0 ðmnpÞumnp r exp
i wðk Þ þ E k t dt0 d 3 r0
e bðk Þde k 0

kk mnp


e

X

kk 0 mnp


e

X

kk 0 mnp


e

X

kk 0 mnp

0 

bðk Þdp ðk Þ

ð   0    0 0   0 0 S t
t ; r
r ða; wk r0 Þ cpk 0 ðmnpÞumnp r exp
i wðk Þ
E k t dt0 d 3 r0

ð   0    0 0 0 0 bðkÞ de ðk Þ S t
t ; r
r ða; wk ðr0 ÞÞcek 0 ðmnpÞumnp r exp i wðk Þ
E k t dt0 d 3 r0 ð    0    0 0   0 0 bðkÞ dp ðk 0 Þ S t
t ; r
r a; wk r0 Þ cpk 0 ðmnpÞumnp r exp i wðk Þ þ E k t dt0 d 3 r0

From this expression, it is clear that the characteristic frequencies of the interaction term between the electromagnetic potentials and the Dirac field and hence the characteristic frequencies of the perturbation in the Dirac field caused by electromagnetic interaction are wðk Þ  E: In terms of the compact notation introduced earlier, namely using the same symbol d ðk Þ for both electron and positron annihilation operators and likewise d ðk Þ for both electron and positron creation operators, we can write ð  hX  0  0 0 0 0 0 bðk Þd k h1 kk t ; r dyðxÞ ¼ S t
t ; r
r   0 0   0  0 0 0 0  þ b ðk Þd k 0 Þ h2 kk t ; r þ b kÞ d k h3 kk t ; r   0  0 0 i 0  þ d k 0 Þ d k h4 kk t ; r dt0 d 3 r0

Rectangular terahertz DRA—design, simulation and implementations

185

0 0 0 are built by superposing the functions where the functions hm kk t ; r    0    exp i 6¼ wðk Þ  E k t a; wk ðrÞck 0 ðmnpÞumnp ðrÞ and the same expression with wk ðrÞis replaced by its complex conjugate wk ðrÞ : Here, the symbol ck 0 ðmnpÞ stands for either cek 0 ðmnpÞ or cpk 0 ðmnpÞ: In particular, this expression shows that the perturbation to the Dirac field 0 caused by electromagnetic interactions has frequencies wðk Þ  E k ; namely linear combinations of the unperturbed electromagnetic characteristic frequencies and the unperturbed Dirac characteristic frequencies. This represents a new feature of our model. Before proceeding further, observe that we can write in the dimensional momentum/space–time frequency domain:

 ð 0Þ dyðk Þ ¼ S ðk ÞF eAðmÞ am yð0Þ ðk Þ where  
1 S ðk Þ ¼ k 0
ða; K Þ
bm0 þ i0 is the electron propagator in the four-momentum domain k ¼ ðk m Þ ¼ ðk 0 ; K Þand control of the quantum electromagnetic field and the Dirac field of electrons and positrons within the rectangular cavity by means of a classical electromagnetic field coming from a probe inserted into the cavity: let AcðmÞ ðxÞ denote the classical electromagnetic four potential from the laser and JðcmÞ ðxÞ be the classical current density coming from the probe insertion. The relevant equations are Am ¼
em0 y am y þ m0 JðcmÞ ; ðða;
irÞ þ bmÞy ¼ ½
eða; AÞ
eða; Ac Þy The first-order perturbative solution to these equations with x ¼ ðt; rÞ is yðxÞ ¼ yð0Þ ðxÞ þ dyðxÞ; Ar ðxÞ ¼ Arð0Þ ðxÞ þ dAr ðxÞ; r ¼ 1; 2; 3; where yð0Þ ðxÞ ¼

X

de ðk Þcek ðmnpÞumnp ðrÞ exp ð
iEðk ÞtÞ

kmnp

þdp ðk Þ cpk ðmnpÞumnp ðrÞ exp ðiEðk ÞtÞ Arð0Þ ðxÞ ¼

X  bðk Þwrk ðrÞ exp ð
iwðk ÞtÞ þ bðk Þ w
r k ðrÞ exp ðiwðk Þt Þ k

186

Terahertz dielectric resonator antennas ð h i dyðxÞ ¼
e Se ðx
yÞ a; Að0Þ ðyÞ þ ða; Ac ðyÞÞ yð0Þ ðyÞd 4 y ð h i ¼
e Se ðx
yÞ ar Aðr0Þ ðyÞ þ ar Acr ðyÞ yð0Þ ðyÞd 4 y ¼ dy1 ðxÞð þ dyctr ðxÞ;



ð

dAr ðxÞ ¼
em0 Dðx
yÞðy ar yÞðyÞd y þ m0 Dðx
yÞJrc ðyÞd 4 y ð ð  
em0 Dðx
yÞ yð0Þ ar yð0Þ ðyÞd 4 y þ m0 Dðx
yÞJrc ðyÞd 4 y 4

where the classically controllable part of the Dirac field as ð yctr ðxÞ ¼
e Se ðx
yÞar Aðr0Þ ðyÞyð0Þ ðyÞd 4 y And this component contains a classical field component Acr and a quantum field component yð0Þ ; while the part of the Dirac field perturbation that is not controllable is ð dy1 ðxÞ ¼
e Se ðx
yÞar Aðr0Þ ðyÞyð0Þ ðyÞd 4 y On the other hand, the controllable part of the electromagnetic field is purely classical: ð dAr;ctr ðxÞ ¼ m0 Dðx
yÞJrc ðyÞd 4 y If we go one step further in the perturbation series, then we get an additional term in the controllable part of the electromagnetic field so that the previous equation gets modified to ð dAr;ctr ðxÞ ¼ m0 Dðx
yÞJrc ðyÞd 4 y ð 
em0 Dðx
yÞdyctr ðyÞ ar yð0Þ þ dy1 ðyÞd 4 y ð 
em0 Dðx
yÞ yð0Þ þ dy1 ðyÞdyctr ðyÞd 4 y Note that in this analysis, the perturbation parameter is the electron charge e and if we neglect 0ðe2 Þ terms, then the previous expression for the controllable part of the electromagnetic field simplifies to ð dAr;ctr ðxÞ ¼ m0 Dðx
yÞJrc ðyÞd 4 y ð
em0 Dðx
yÞdyctr ðyÞ ar yð0Þ ðyÞd 4 y ð em0 Dðx
yÞyð0Þ ðyÞar dyctr ðyÞd 4 y

Rectangular terahertz DRA—design, simulation and implementations

187

In the particular case of the RDRA considered here, we find that the controllable part of the Dirac field has the following expansion: ð dyctr ðxÞ ¼
e Se ðx
yÞar Acr ðyÞyð0Þ ðyÞd 4 y ð

  0 0 0 0 ¼
e Se t
t ; r
r ar Acr t ; r "

 0  X 0 de ðk Þcek ðmnpÞumnp r exp
iEðk Þt kmnp

  0  i 0 þ dp kÞ cpk ðmnpÞumnp r exp iEðk Þt dt0 d 3 r0 Now define the following Fourier components of the control classical laser generated electromagnetic field w:r:t: the cavity boundary conditions and the energy spectrum of the free Dirac field in the cavity after ð   0   0 0 0 0 0 0 Acr t ; r umnp r exp
iK  r exp iwt d 3 r dt ¼ CA;r ðw; K jm; n; pÞ Then, we can express the previous controllable part of the Dirac field in the following form in the spatial–temporal Fourier domain: ð dyctr ðt; rÞ  expðiðwt
K  rÞdtd 3 r X
ed e ðk ÞS ðw; K Þcek ðmnpÞar Cr ðw ¼ kmnp


Eðk Þ; K jmnpÞ
ed e ðkÞ S ðw; K Þcpk ðmnpÞar Cr ðw þ Eðk Þ; K jmnpÞ X ¼
eS ðw; K Þ de ðk Þcek ðmnpÞar Cr ðw
Eðk Þ; K jmnpÞ kmnp



þ dp kÞ cpk ðmnpÞar Cr ðw þ Eðk Þ; K jmnpÞ The controllable part of the Dirac four current density is then given up to firstorder perturbation terms by ðx ¼ ðt; rÞÞ dJ m ðt; rÞ ¼
eyð0Þ ðxÞam dyctr ðxÞ
edyctr ðxÞ am yð0Þ ðxÞ And it is immediately clear from the previous expression that the far-field radiated electromagnetic potential generated by this controllable current field can be expressed in the following form:

188

Terahertz dielectric resonator antennas ð   0 0 0 0 m dAR ðt; rÞ ¼ D t
t ; r
r dJ m t ; t dt0 d 3 r0   0 ð  X   s
w0
E k 0 ; K 0 jm0 n0 p0 Þ de ðk Þ de k Cr ðw
Eðk Þ; K jmnpÞC kmnp   0 0 F m rs t; r w; K; w ; K ; mnpk; m0 n0 p0 k 0 dwd 3 Kdw0 d 3 K 0  0   0   0  plus three other similar terms involving de ðk Þ dp k ; de k ; dp ðk Þdp k : In compact notation, the expected value of this controllable far-field pattern can be expressed as a Hermitian quadratic form in the complex numbers Cr ðw; K jmnpÞ;w  R; K R3 ; m; n; p  Zþ : These complex numbers are controllable since they represent in some sense the spatiotemporal components of the Fourier components of the classical control electromagnetic field Acm :

9.7 Conclusion TDRAs are useful for high-speed communications. TDRAs are compactly efficient and they have ultralarge bandwidth. DRs are robust and reliable as they can operate at entire spectrum, i.e., microwave to optical regime. They have large design flexibility and customized aspect-ratio. At optical frequency, TDRAs can be used for retinal photoreceptors. LiDAR can be a beautiful application of terahertz antennas and sensors. Terahertz devices shall have compact size and less power consumption.

References [1] R. D. Richtmyer, “Dielectric resonators,” J. Appl. Phys., vol. 10, pp. 391–398, 1939. [2] S. A. Long, M. McAllister, and L. C. Shen, “The resonant cylindrical dielectric cavity antenna,” IEEE Trans. Antennas Propag., vol. 31, pp. 406–412, 1983. [3] M. Gastine, L. Courtois, and J. J. Dormann, “Electromagnetic resonances of free dielectric spheres,” IEEE Trans. Microwave Theory Tech., vol. 15, no. 12, pp. 694–700, 1967. [4] R. S. Yaduvanshi and H. Parthasarathy, Rectangular Dielectric Resonator Antennas: Theory and Design, Springer, New Delhi, 2016. [5] J. Van Bladel, “On the resonances of a dielectric resonator of very high permittivity,” IEEE Trans. Microwave Theory Tech., vol. 23, no. 2, pp. 199–208, 1975. [6] R. S. Yaduvanshi and V. Gaurav, Nano Dielectric Resonator for 5G Applications, CRC Press, Boca Raton, FL, 2020. [7] K. M. Rajesh and B. Prakash, “Dielectric resonator antenna—A review and general design relation for resonant frequency and bandwidth,” Int. J. Microwave Millimetre Wave Comput. Aided Eng., vol. 4, no. 3, pp. 230–247, 1994.

Rectangular terahertz DRA—design, simulation and implementations

189

[8] Y. M. Pan, K. W. Leung, and L. Guo, “Compact laterally radiating dielectric resonator antenna with small ground plane,” IEEE Trans. Antennas Propag., vol. 65, no. 8, pp. 4305–4310, 2017. [9] Y. M. M. Antar, “Antennas for wireless communication: Recent advances using dielectric resonators,” IET Circuits Devices Syst., vol. 2, no. 1, pp. 133–138, 2008. [10] A. Petosa and A. Ittipiboon, “Dielectric resonator antennas: A historical review and the current state of the art,” IEEE Antennas Propag. Mag., vol. 52, no. 5, pp. 91–116, 2010. [11] K. W. Leung, E. H. Lim, and X. S. Fang, “Dielectric resonator antennas: From the basic to the aesthetic,” Proceedings of the IEEE, vol. 100, no. 7, pp. 2181–2193, 2012. [12] P. A. Martin and F. Rothen, Many-Body Problems and Quantum Field Theory, Springer, Berlin Heidelberg, 2002. [13] P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express, vol. 15, pp. 14266–14274, 2007. [14] L. Novotny, “Optical antennas: A new technology that can enhance lightmatter interactions,” Front. Eng., vol. 39, no. 4, pp. 100–120, 2012. [15] S. Lepeshov and A. Gorodetsky, “All-dielectric optically tunable metasurface for terahertz phase and amplitude modulation,” J. Phys. Conf. Ser., vol. 1461, pp. 012203, 2020 IOP Publishing, doi:10.1088/1742-6596/1461/ 1/012203. [16] M. Hayasi, Quantum Information Theory, Springer, Berlin Heidelberg, 2017. [17] P. Bhardwaj, B. Deutsch, and L. Novotny, “Optical antennas,” Adv. Opt. Photonics, vol. 1, pp. 438–483, 2009. [18] M. Humayun and L. O. de Koo, Retinal Prosthesis—A Clinical Guide to Successful Implementation, Springer, 2018. [19] Y. Yang, Y. Yamagami, X. Yu, et al., “Terahertz topological photonics for on-chip communication,” Nature Photonics, vol. 14, pp. 446–451, 2020. [20] J. W. Choong, N. Nefedkin, and A. Krasnok, “Collectively driven optical nanoantennas,” Phys. Rev. A, vol. 103, 043714, 2021. [21] S. Weinberg, The Quantum Theory of Fields, vol. 1, Oxford University Press, New York, NY, 1995.

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Chapter 10

Equivalent circuit analysis on terahertz and optical dielectric resonator antennas (DRAs)

Abstract In this chapter, an equivalent R–L–C circuit for terahertz and optical dielectric resonator antenna (DRA) has been developed. The behavior of characteristics modes in response to resonance frequency, input impedance and quality factor has been formulated. Higher order resonant modes in terahertz DRA has been depicted with circuit concepts. The terms, such as quality factor, bandwidth, resonance frequency and selectivity in quantum circuits, have been comprehensively investigated in quantum circuits input impedance that is complex and nonlinear, the real part of which is a function of frequency. The dynamic impedance at terahertz frequency involves skin effect. The authors have developed R, L, C (resistor (R), inductor (L) and capacitor (C)) equivalent circuit for terahertz and optical DRAs. These investigations are useful for the theoretical modeling of optical DRAs that shall help designers during the design stage. Their comprehensive analysis and study of quality factor, bandwidth, resonant frequency and selectivity in quantum circuits have been illustrated. The mathematical formulations for antenna and absorber conditions have been worked out. Maximum resistance condition in R, L, C circuit-favored absorber designs and minimum resistance conditions have been found suitable for antenna functions. These investigations also depict the phenomenon of higher order resonant modes in terahertz DRA. These resonant modes can be modulated to get merged modes or controlled separated modes to provide wide bandwidth or narrow bandwidth in antenna designs.

10.1 Introduction Optical antenna has a finite quantity of planks constant value. The quantum antennas’ radiations can be correlated with black-body radiations or light emitted. An RLC-parallel circuit is equivalent for the classical dielectric resonator antenna (DRA), and the transfer function of the same is used to approximate a given function of frequency generated by a quantum antenna. The solution for error between quantum function and classical function can be found by minimizing the error function. Surface plasmon-mediated hot carrier generation is widely utilized for the manipulation of the electron–photon interactions in many types of

192

Terahertz dielectric resonator antennas

optoelectronic devices. The spectrum of quantum antenna can be defined in terms of Planck’s constant. The RLC circuit response of a classical circuit can be transformed to quantum circuit response by integrating over s-domain quantum spectrum omega– s-domain classical omega over a derivative of omega and differentiating with respect to R, L, C values. This will give us an average value, where mean and covariance are most important in a quantum antenna for radiated fields. At quantum level, electromagnetic fields are realized by quantum wave operator fields, and it can be precisely defined as an ensemble of an infinite number of quantum harmonic oscillators. It can be analyzed as a superposition of constant plane wave with coefficients being creation and annihilation operators in a boson Fock space. At an optical spectrum, it is better to talk about quantum state of wave fields like coherent state of photons. It is a process of light–matter interaction in an optical antenna to introduce a new term known as SPP (surface plasmon polytrons). The light interacts with electrons, atoms and molecules of materials of a medium it travels through. At optical frequency, a material medium is treated as plasma having definite permittivity, permeability and conductivity for the propagation of electromagnetic fields in it. The quantum antenna circuit, with input excitation, the potential drop across impedances Z2 and Z3 are the same since they are in parallel connection. Also, there is some potential drop in impedance Z1 . But currents in impedance Z2 and Z3 are different because Z2 and Z3 have different values: Z2 ¼ R2 þ jwL2 ; Z3 ¼ R3  I2 ¼

V V ; I3 ¼ Z2 Z3

j ; wC3

where magnitudes of Z2 and Z3 impedances vary due to variable frequency term in Z2 and Z3 : At a particular high frequency, the impedances Z2 and Z3 become equal; as a result, the current flowing through each of them also becomes equal. The currents in Z2 and Z3 remain equal for a very small duration of time as it happens only at particular high frequency. This phenomenon is due to the Dirac-delta function. This high frequency at which currents in Z2 and Z3 are the same is called first tuning frequency. After this frequency, the currents I2 and I3 in impedances Z2 and Z3 , respectively, are the same in magnitude but the opposite in phase. So, the resultant of I2 and I3 is zero and the net current only flows through impedance Z1 , given by Z1 ¼ R1  j=wC1 (Z1 branch is frequency dependent). At high frequency, the magnitude of impedance Z1 is very low resulting in very high current and power. The circuit impedance is dependent only upon Z1 , which is then called the dynamic impedance of the circuit. The current flowing through Z1 is dependent upon potential drop across Z1 and given by i1 ¼ CðdV1 =dtÞ since dt is very small time, so the current is very high resulting in a high selectivity of the circuit. This current i1 is known as surface current or displacement current of the circuit. In the RLC circuit of quantum antenna, the input impedance of the circuit is frequency dependent and also of very small-time interval due to high-frequency

Equivalent circuit analysis on terahertz and optical DRA

193

analysis. This is due to the phenomena of the Dirac-delta function. In this time duration, current is maximum and depends upon impedance Z1 that is frequency dependent. Since the impedance Z1 is very small at high frequency, so the corresponding current through Z1 is also very high. As a result, power is very high and signal strength is very high. Both signal capturing and delivering are very high. This is the main advantage of a quantum antenna circuit. During the initial tuning of this circuit, the current is the same but the opposite in phase, so their resultant is zero. In this condition, the inductor current behaves like a surface current that is proportional to area. The surface current in inductor is present due to skin effect. In the quantum antenna, the input impedance and the dynamic impedance of the circuit are frequency dependent. As the frequency increases, the dynamic impedance decreases, which further reduces the power loss. Since the real part of input impedance of a quantum antenna circuit consumes the real power, imaginary term of input impedance does not consume any power at all. And the other elements like L and C transfer power from one cycle to other cycle among each other. These circuits are very effective for 5G technologies because these technologies need very less power loss and high selectivity at high frequency. As L and C store more power, the quality factor of the circuit is very high: Q ¼ ðfr =bandwidthÞ. The quality factor is very high, which implies the bandwidth of the system is very low. But in a quantum antenna circuit, the resonance frequency is very high due to which the bandwidth of the system is not very small. The design analysis of an antenna needs a strong mathematical background. An RLC circuit can be realized by a simple solution for any antenna [1]. In this chapter, HFSS NDRA modeling has been developed with equivalent circuit design approach [2–8]. An approximate RLC circuit model has been introduced [9]. This narrates the main mode of propagating as an equivalent transmission line by making use of an aperture-coupled slot. Lumped impedances have been derived to accurately describe the functions from source to end points. The reactance’s account for the reactive power due to feed and the termination and resistances exactly show the radiated waves in space. This technique can be used to compute an input impedance of NDRA with aperture-coupled loading [10]. Very less research on modeling an NDRA equivalent circuit has been done so far; however, most of the literature is available on a patch antenna equivalent circuit [8,11–13]. The equivalent circuit representation gives correct results for both the load and the internal impedance of NDRA. The modeling of NDRA impedances and radiation fields has been presented with higher order and fundamental modes. Circuit bandwidth, resonance and other radiation field parameters have been represented using an equivalent circuit model. Higher order modes can be accurately modeled with this approach. In this approach, circuit models of resonant modes can be used to predict radiation patterns and other field behavior of NDRA [14]. In this chapter, a physics-based circuit for resonant modes has been developed with simple and absolutely accurate form. This equivalent circuit model can extrapolate bandwidth, resonant frequency, impedance, quality factor and radiation pattern of NDRA. This is beneficial in designing and analyzing of different NDRA structures. The study of resonance can simplify designing of NDRAs. This method of analysis has linked the NDRA circuit models to its radiated fields for the first time in NDRA research [15].

194

Terahertz dielectric resonator antennas

10.2 Quantum DRA-equivalent circuit mathematical analysis for mixed circuits 10.2.1 Impedance (Zin) Equivalent circuits of optical antenna are given in Figure 10.1. In the previous circuit, the resistance and reactance values are as follows: X1 ¼ 

1 1 ; X2 ¼ wL2 and X3 ¼  wC1 wC3

Input impedance of this circuit Zin ¼ Z1 þ Z2 kZ3 ¼ Z1 þ

Z2 Z3 Z2 þ Z3

where Z1 ¼ R1 þ jX1 , Z2 ¼ R2 þ jX2 and Z3 ¼ R3 þ jX3 Zin ¼ R1 þ jX1 þ

ðR3 þ jX3 ÞðR2 þ jX2 Þ R2 þ jX2 þ R3 þ jX3

Zin ¼ R1 þ jX1 þ

R2 R3  X2 X3 þ jðX2 R3 þ R2 X3 Þ R2 þ R3 þ jðX3 þ X2 Þ

Zin ¼ R1 þ jX1 þ Zin ¼ R1 þ jX1 þ

(10.1)

fR2 R3  X2 X3 þ jðX2 R3 þ R2 X3 ÞgfR2 þ R3  jðX3 þ X2 Þg ðR 2 þ R 3 Þ2 þ ðX 3 þ X 2 Þ2

ðR2 R3  X2 X3 ÞðR2 þ R3 Þ þ ðX2 R3 þ R2 X3 ÞðX3 þ X2 Þ þ jfðR2 þ R3 ÞðX2 R3 þ R2 X3 Þ  ðX3 þ X2 ÞðR2 R3  X2 X3 Þg ðR2 þ R3 Þ2 þ ðX3 þ X2 Þ2

" Zin ðreal partÞ ¼ R1 þ

# ðR2 R3  X2 X3 ÞðR2 þ R3 Þ þ ðX2 R3 þ R2 X3 ÞðX3 þ X2 Þ ðR2 þ R3 Þ2 þ ðX3 þ X2 Þ2

R1

X1 R3

R2

Z1 Zin Z2

X2

Z3

X3

Figure 10.1 Equivalent R, L, C circuit for optical antennas

(10.2)

Equivalent circuit analysis on terahertz and optical DRA " Zin ðimaginary partÞ ¼ X1 þ

ðR2 þ R3 ÞðX2 R3 þ R2 X3 Þ  ðX3 þ X2 ÞðR2 R3  X2 X3 Þ ðR2 þ R3 Þ2 þ ðX3 þ X2 Þ2

195

#

(10.3)

Resonance frequency imaginary part of input impedance is equal to zero: Zin ¼ Zin ðreal partÞ þ Zin ðimaginary partÞ At resonance imaginary part of Zin ¼ 0 X1 þ

ðR2 þ R3 ÞðX2 R3 þ R2 X3 Þ  ðX3 þ X2 ÞðR2 R3  X2 X3 Þ ðR2 þ R3 Þ2 þ ðX3 þ X2 Þ2

¼0

(10.4)

n o   R2 R3 X2 þR22 X3 þR23 X2 þR2 R3 X3  R2 R3 X2 X22 X3 þR2 R3 X3 X32 X2 þX1 ðR2 þR3 Þ2 þ ðX3 þX2 Þ2 ¼0   R22 X3 þR23 X2 þX22 X3 þX32 X2 þX1 R22 þR23 þ2R2 R3 þX22 þX32 þ2X2 X3 ¼0 R22 ðX1 þX3 ÞþR23 ðX1 þX2 ÞþX22 ðX1 þX3 ÞþX32 ðX1 þX2 Þþ2X1 ðR2 R3 þX2 X3 Þ¼0   ðR22 þX22 ÞðX1 þX3 Þþ R23 þX32 ðX1 þX2 Þþ 2X1 ðR2 R3 þX2 X3 Þ¼0

Substituting the value of X1 ; X2 and X3 in the previous equation, 

     1 1  2 2 L2  R2 R3  þ  R2 þ w2 L22  ¼0 C3 wC1 wC3 wC1  2        w L2 C1  1 R23 w2 C32 þ 1 1 1 1  2 2 L2 2 2 þ R ¼0   þ w L R R   2 3 2 2 C3 wC1 w C1 C 3 wC1 w2 C32 wL2 

1 wC1



R23 þ

1 w2 C32



Multiplying w3 both sides in previous equation,        2w2 ðw2 L2 C1  1Þ R23 w2 C32 þ 1 1 1  2 L2 2 2 2 R ¼0  þ w   þ w L R R  2 3 2 2 C1 C3 C1 C3 C1 C32  4         w L2 C1 R23 C32  1 þ w2 L2 C1  R23 C32 1 1 1 1 2w2 L2 4 2 2 2 þ w  ¼0 þ w L   R   R R  2 3 2 2 C1 C3 C1 C3 C1 C3 C1 C32         1 1 L2 R2 1 1 2 L2 1 w4 L2 R23  L22 þ þ R2 R3  ¼0 þ w2 2  3  R22   C3 C1 C3 C1 C3 C1 C1 C32 C3 C1

(10.5) w ½ A þ w ½ B   C ¼ 0 4

2

Comparing the previous two equations, we get A, B and C as  A ¼ L2 R23  L22 C¼

     1 1 L2 R2 1 1 2 L2 ; B ¼ 2  3  R22  and þ þ R2 R3  C1 C3 C1 C3 C1 C3 C3 C1

1 C1 C32

Let w2 ¼ p p2 ½A þ p½B  C ¼ 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B  B2 þ 4AC P¼ ; where w2 ¼ p 2A

(10.6)

196

Terahertz dielectric resonator antennas

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi B B2 þ4AC 2A vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u      sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2  

u 2 L2 R23 1 1 2 L2 1 1 1 u  L2  R3 R2 1 þ 1  2 R R  L2  R22 þ R2 R3  þ4 L2 R23 L22 þ   u 2 3 2 2 C3 C3 C1 C3 C1 C1 C3 C1 C1 C3 C3 C1 C32 C1 u C1 C32 u  

u 1 1 u ¼u 2 L2 R23 L22 þ C1 C3 u u : u t

w¼

(10.7) Thus, resonant frequency of this circuit is depending on R, L and C, but input impedance of this circuit is a function of R, L, C and frequency. Hence, impedance is frequency dependent: Zin ¼ f ðR; L; C; wÞ

(10.8)

Zin ¼ Zinreal ðwÞ þ Zinimaginary ðwÞ

(10.9)

Hence, terahertz DRA input impedance of the circuit, Zin ¼ Zinreal þ Zinimaginary ðwÞ

(10.10)

Zinreal 6¼ f ðwÞ

(10.11)

where Zinreal means only real part of input impedance that is called only resistive part of input impedance. On the other hand, terahertz DRA resistive part of input impedance is a function of frequency: Zinreal ¼ resistance ¼ f ðwÞ

(10.12)

10.2.2 The frequency-dependent resistance is also called dynamic resistance of the circuit Zd ¼ R1 þ

ðR 2 R 3  X 2 X 3 ÞðR 2 þ R 3 Þ þ ðX 2 R 3 þ R 2 X 3 ÞðX 3 þ X 2 Þ ðR 2 þ R 3 Þ2 þ ðX 3 þ X 2 Þ2

1 1 , X2 ¼ wC2 and X3 ¼ wC in the previous equation X1 ¼ wC 1 3    R2 1 R2 R3  CL23 ðR2 þ R3 Þ þ wL2 R3  wC  wL 2 wC3 3 Zd ¼ R1 þ  2 1 ðR2 þ R3 Þ2 þ wL2  wC 3   2  2 w L2 C3 1 3 C3 R2 R2 R3  CL23 ðR2 þ R3 Þ þ w L2 RwC wC3 3 Zd ¼ R1 þ  2 2 2 2 C3 1 ðR2 þ R3 Þ þ w LwC 3

(10.13)

Equivalent circuit analysis on terahertz and optical DRA 

197



R2 R3 C2 ðR2 þR3 Þw2 C32 þðw2 L2 R3 C3 R2 Þðw2 L2 C3 1Þ L

3

w2 C32 2 w2 C3 ðR2 þR3 Þ2 þðw2 L2 C3 1Þ2 w2 C32

Zd ¼ R1 þ

 n o  L2 2 R1 w2 C32 ðR2 þ R3 Þ2 þ ðw2 L2 C3  1Þ þ R2 R3  ðR2 þ R3 Þw2 C32 þ ðw2 L2 R3 C3  R2 Þðw2 L2 C3  1Þ C3 Zd ¼ w2 C32 ðR2 þ R3 Þ2 þ ðw2 L2 C3  1Þ2

Zd ¼ f ðwÞ or Rin ¼ f ðwÞ

10.2.3 Two resonant modes, i.e. fundamental and higher order 1 X1 ¼  wC1 1 , X2 ¼ wL2 and X3 ¼  wC 3

Yl ¼ Y2 þ Y3 Yl ¼

1 1 þ R2 þ jX2 R3 þ jX3

Yl ¼

1 1 þ R2 þ jwL2 R3  wCj

Yl ¼

R2  jwL2 R3 þ þ R2 þ jwL2 R3 

Yl ¼

R22

R2 þ þ w2 L22

Yl ¼ Gl þ jBl

3

j wC3 j wC3

R3 1 R23 þ 2 2 w C3

2 6 þ j6 4

1 wC3 1 R23 þ 2 2 w C3

3 

R22

wL2 7 7 þ w2 L22 5

(10.14)

For the first or fundamental resonant mode imaginary part of ðYl Þ ¼ 0; it means Bl ¼ 0 1 wC3

R23 þ w21C2



3

1 wC3

R23

þ

1 w2 C32

¼

wC3 2 R3 w2 C32 þ

1

wL2 ¼0 R22 þ w2 L22

R22 ¼

wL2 þ w2 L22 R22

wL2 þ w2 L22

198

Terahertz dielectric resonator antennas     C3 R22 þ w2 L22 ¼ L2 R23 w2 C32 þ 1   w2 L22 C3  L2 R23 C32 ¼ L2 þ R22 C3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi L2 þ R22 C3 w1 ¼ 2 L2 C3  L2 R23 C32

(10.15)

10.2.4 Second resonant mode YT ¼ Y1 þ Yl YT ¼ Y1 þ ðGl þ jBl Þ YT ¼ YT ¼

1 þ ðGl þ jBl Þ R1  wCj 1 R1 þ wCj 1 R21 þ w21C2

þ ðGl þ jBl Þ

1

 YT ¼

2

R1 w C12 þ Gl R21 w2 C12 þ 1

 þ j Bl þ

!

1 wC1

R21 þ w21C 2 1

For the second resonant mode or over all resonant mode imaginary part of YT ¼ 0 Bl þ wC3 2 R3 w2 C32 þ C3 2 2 R3 w C32 þ

wL2 þ 1 þ w2 L22 L2  2 þ 1 R2 þ w2 L22 

R22

wC1 2 R1 w2 C12 þ wC1 2 R1 w2 C12 þ C1 2 2 R1 w C12 þ

1 1 1

¼0 ¼0 ¼0

On solving the previous equation the value of resonant frequency for higher order mode (wr) (Figure 10.2): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u      sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2  

2 u L2 R23 1 1 2 L2 1 1 1 u  L2  R3 R2 1 þ 1  2 R R  L2    R22 þ R2 R3  þ4 L2 R23 L22 þ 2 3 u 2 2 2 2 C3 C3 C1 C3 C1 C1 C3 C1 C1 C3 C3 C1 C3 C1 C1 C3 u u  

wr ¼ u 1 1 2 L2 u 2 L R þ 2 3 2 u t : C1 C3

(10.16) Quality factor ðQÞ ¼ 2p

maximum energy stored per cycle power dissipatetd per cycle

For the series case Z ¼ RþjX   ¼ Rþj(XL XC )    Q¼

jX L j jX C j ¼ R R

Equivalent circuit analysis on terahertz and optical DRA

R1

199

X1 R3

R2

Y1 YT Y2

Y3

X2

X3

Yl

Figure 10.2 Higher order modes analysis

For the parallel case Y ¼ G þ jB ¼ G þ jðBc BL Þ Q ¼ jBGL j ¼ jBGC j   . According to this, Z in ¼ Rin þ jXin ¼ Zin ðrealÞ þ Zin ðimaginaryÞ

X þ ðR2 þR3 ÞðX2 R3 þR2 X3 ÞðX3 þX2 ÞðR2 R3 X2 X3 Þ 2 2

1

ð þR Þ þ ð X þX Þ R 2 3 3 2

Quality factor ðQÞ ¼

ðR2 R3 X2 X3 ÞðR2 þR3 ÞþðX2 R3 þR2 X3 ÞðX3 þX2 Þ

R 1 þ

2 2 ðR þR Þ þðX þX Þ 2

3

3

(10.17)

2

n o ðR2 þ R3 ÞðX2 R3 þ R2 X3 Þ  ðX3 þ X2 ÞðR2 R3  X2 X3 Þ þ X1 ðR2 þ R3 Þ2 þ ðX3 þ X2 Þ2 n o Q¼ ðR2 R3  X2 X3 ÞðR2 þ R3 Þ þ ðX2 R3 þ R2 X3 ÞðX3 þ X2 Þ þ R1 ðR2 þ R3 Þ2 þ ðX3 þ X2 Þ2 Q¼

  R22 X3 þR23 X2 þX22 X3 þX32 X2 þX1 R22 þR23 þ2R2 R3 þX22 þX32 þ2X2 X3   R22 R3 þR23 R2 R2 X2 X3 X2 R3 X3 ðX22 R3 þX32 R2 þX2 X3 R2 þX2 X3 R3 ÞþR1 R22 þR23 þ2R2 R3 þX22 þX32 þ2X2 X3

Q¼

R22 ðX3 þX1 ÞþR23 ðX2 þX1 ÞþX22 ðX3 þX1 ÞþX32 ðX2 þX1 Þþ2X1 fR2 R3 þX2 X3 g R22 ðR3 þR1 ÞþR23 ðR2 þR1 ÞþX22 ðR3 þR1 ÞþX32 ðR2 þR1 Þþ2R1 ðR2 R3 þX2 X3 Þ2X2 X3 R2 2X2 X3 R3 ðR22 þX22 ÞðX3 þX1 ÞþðR23 þX32 ÞðX2 þX1 Þþ2X1 ðR2 R3 þX2 X3 Þ ðR3 þR1 ÞðR22 þX22 Þþ ðR2 þR1 ÞðR23 þX32 Þþ2R2 ðR1 R3 X2 X3 Þþ2X2 ðX3 R1 X3 R3 Þ         1 1  2 1 1 2 L2 R2 þw2 L22 þ wL2  R23 þ 2 2    R2 R3  wC1 wC3 wC1 wC C3 w C3     1   Q¼  2  1 L R R3 2 1 þ2wL2 ðR3 þR1 Þ R2 þw2 L22 þ ðR2 þR1 Þ R23 þ 2 2 þ2R2 R1 R3  þ C3 wC1 wC3 w C3    2  2 2 2     1 1 1  2 w L2 C1 1 w C3 R3 þ1 2 L2 R2 þw2 L22 þ   þ R R  2 3 C3 w C1 C3 wC1 wC1 w2 C32  2 2 2      Q¼  2  w C R þ1 L R1 R3 2 3 3 2 2 þ2R þ2L ðR3 þR1 Þ R2 þw L2 þ ðR2 þR1 Þ R R  þ 2 1 3 2 2 C3 C1 C3 w2 C3

Q¼

200

Terahertz dielectric resonator antennas Multiplying w3 numerator and denominator, 

      2w2 1 1  2 1  2 L2 R2 þ w2 L22 þ þ w L2 C1  1 w2 C32 R23 þ 1  R2 R3  2 C1 C3 C1 C C1 C3      3  Q¼   2  R2 þ R1  2 2 2 L2 R1 R3 2 3 2 3 3 w ðR3 þ R1 Þ R2 þ w L2 þ w þ w C3 R3 þ 1 þ 2w R2 R1 R3  þ 2w L2 2 C3 C1 C3 C3         2 1 1 L2 1 1 R 2 L2 1 þw2 2 R22  3  w4 L2 R23 L2 þ þ R2 R3  C1 C3 C1 C3 C1 C1 C3 C3 C1 C32        Q¼   L2 R1 R3 R2 þR1 þ2L2 þw w5 L22 ðR3 þR1 Þ þw3 R22 ðR3 þR1 ÞþR23 ðR2 þR1 Þþ2R2 R1 R3  þ 2 C3 C1 C3 C3 w2

(10.18)

10.3 Higher order resonant modes Terahertz antenna circuit behaves like a high-pass filter because the circuit output zero for low frequency and output is finite due to high frequency. Only one-stage two-mode exists in antenna circuit, but the higher stage odd and even harmonics of w increase (Figure 10.3). For the first stage of a high-pass filter, we calculate first mode and second mode resonator values. Consider that the previous circuit input and output are represented by Xi and Xo : R3 sC3 þ1 1 þ1 Using the Laplace domain Z1 ¼ R1 sC sC1 , Z2 ¼ R2 þ sL2 and Z3 ¼ sC3 Zp Xo ðsÞ Z2 Z3 Z2 Z3 ¼ ¼ where Zp ¼ ðZ2 þ Z3 Þ Xi ðsÞ Z1 þ Zp Z1 ðZ2 þ Z3 Þ þ Z2 Z3 Xo ðsÞ 1 ¼ ¼ Xi ðsÞ 1 þ ZZ1 p

1 1þ

R1 sC1 þ1 sC1



ðR2 þsL2 Þ



¼

1 1þ

ðR1 sC1 þ1ÞfðR2 þR3 þsL2 ÞsC3 þ1g sC1 ðR2 þsL2 ÞðR3 sC3 þ1Þ

R3 sC3 þ1 sC3

ðR2 þR3 þsL2 ÞsC3 þ1

R1

R1

X1 R3

R2

Z1

X1 R3

R2

Z1

Zin Z2

X2

Z3

X3

First stage third-order high-pass filter

Z2 X2

Z3

X3

Second stage third-order high-pass filter

Figure 10.3 Higher order resonant modes, i.e., sixth-order high-pass filter (multistage high-pass filter)

Equivalent circuit analysis on terahertz and optical DRA

201

Xo ðsÞ 1 ¼ Xi ðsÞ 1 þ s3 R1 C31 L2 C3 þs2 fL2 C2 3 þR1 C1 C3 ðR2 þR3 ÞgþR1 C1 sþ1 s R3 C1 L2 C3 þs ðL2 C3 þR2 R3 C1 C3 ÞþR2 C1 s

The value of s ¼ jw, transfer function Xo ðjwÞ 1 ¼ 3R C L C Þ 1 1 2 3 Xi ðjwÞ 1 þ 1w2 fL22 C3 þR1 C1 C3 ðR2 þR3 ÞgþjðR1 C1 ww 3 w ðL2 C3 þR2 R3 C1 C3 ÞþjðwR2 C1 w R3 C1 L2 C3 Þ

In the previous transfer function for very low frequency (ideally ¼ 0), gain is zero. And for very high frequency (ideally ¼ ?), gain is a finite value, it means that for low frequency it does not pass the signal but for the high frequency, it passes the signal. Hence it behaves as a high-pass filter. For the calculation of different modes of operation to simplify the transfer function, Xo ðjwÞ 1 1 ¼ ¼ aþjb aþcþjðbdÞ Xi ðjwÞ 1 þ cþjc 1 þ 2 2 c þd

where a ¼ 1  w2 fL2 C3 þ R1 C1 C3 ðR2 þ R3 Þg; b ¼ R1 C1 w  w3 R1 C1 L2 C3

  c ¼ w2 ðL2 C3 þ R2 R3 C1 C3 Þ; d ¼ wR2 C1  w3 R3 C1 L2 C3

Xo ðjwÞ 1

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

X ðjwÞ ¼ r 2  2 i bdÞ þ 1 þ c2aþc 2 2 2 þd c þd For a high value of gain b  d ¼ 0 and a þ c ¼ 0. There two frequencies are arising:     R1 C1 w  w3 R1 C1 L2 C3  wR2 C1  w3 R3 C1 L2 C3 ¼ 0 And the second condition is 1  w2 fL2 C3 þ R1 C1 C3 ðR2 þ R3 Þg  w2 ðL2 C3 þ R2 R3 C1 C3 Þ ¼0 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1 C 1  R2 C 1 w1 ¼ R1 C1 L2 C3  R3 C1 L2 C3 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 w2 ¼ L2 C3 þ R2 R3 C1 C3 þ fL2 C3 þ R1 C1 C3 ðR2 þ R3 Þg

202

Terahertz dielectric resonator antennas

10.4 Bandwidth (BW) of terahertz DRA ðw2  w1 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 ¼ L2 C3 þ R2 R3 C1 C3 þ fL2 C3 þ R1 C1 C3 ðR2 þ R3 Þg rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R1 C1  R2 C1  R1 C1 L2 C3  R3 C1 L2 C3 The input impedance depends only on real part of Zin and the real part of Zin is dependent on frequency of the signal. ffi vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u      sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi     2  

u L2 R23 1 1 2 L2 L2 R23 1 1 2 L2 1 1 1 2 2 u 2  L2   R þ R R  þ 4 L R þ   2 3 2 u  2  C  R2 C þ C  C R2 R3  C 2 3 2 C3 C1 C3 C1 C1 C3 C3 C32 C1 C1 C32 u 1 1 3 1 3 u  

u 1 1 u þ 2 L2 R23  L22 u t : C1 C3 Q¼

wr ¼ ðw2  w1 Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 R1 C1  R2 C1  R1 C1 L2 C3  R3 C1 L2 C3 L2 C3 þ R2 R3 C1 C3 þ fL2 C3 þ R1 C1 C3 ðR2 þ R3 Þg

(10.19)

10.5 Simulated results based on MATLAB“ Simulated results based on MATLAB are given in Figures 10.4–10.8.

10.6 Design development and evaluation of NDRA Nano rectangular DRA design and implementation: the design step of nano-DRA is given in Figure 10.9.

10.6.1 Resonant frequency of TRDRA formulations Resonant frequencies of TRDRA formulations are shown in Figures 10.10–10.14. sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  2ffi   2 2 ckmn‘ c mp np ‘p þ þ  fmn‘ ¼ pffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffi 2p mr 2r 2p mr 2r a b d

10.7 Synthesis of NDRA radiation theory 1.

ih [qa ; pb  ¼ 2p dða; bÞ where q is the position vector, p is the time derivative moment vector.

b ¼

1 ; where T is the temperature: kT

Equivalent circuit analysis on terahertz and optical DRA Magnitude of Im (Zin)

Dynamic impedance of NDRA Magnitudeof Re(Zin)

7,000

6,500

6,000 0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

0.2

10

2

3

9

2

3

4 5 6 Frequency (rad/s)

3

8

9

1

2

3

9

10 ×105

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

4 2

1

2

3

5

×10

Dynamic impedance of Quantum antenna

2,040

8

Quality factor 3

0 0

10

7

2

6

7

4 5 6 Frequency (rad/s)

4

×105

2

1

2

Quality factor 1

0 0

10

4

(a)

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Img of Zin Mgof Im(Zin)

0.3

2,020

2,000 0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

0.2 0.1 0 0

10

Magnitude of input Zin

2,040

1

2

3

×105

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Quality factor 1

×10–4 MG of Q

1.5

2,020

1 0.5

2,000 0

MG of Q

2

2

3

4 5 6 Frequency (rad/s)

7

8

9

0 0

10

1

2

3

×105

Quality factor 2

×10–4

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Quality factor 3 4

1

0 0

(b)

1

MG of Q

Magnitude of Re (Zin)

8

Quality factor 2

×10

0 0

Magnitude of Re (Zin)

7

MG of Q

MG of Q

6

MG of Q

Magnitudeof Re(Zin)

–5

4 5 6 Frequency (rad/s)

1

6

Magnitude of input Zin

1

0 0

×105

Magnitude of input Zin

6,000 0

Img Zin

0.3

7,000

6,500

203

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

2

0 0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Figure 10.4 Magnitude plot of all factors of a quantum antenna for the values of: (a) R1 ¼ 3 530 W, R2 ¼ 3 560 W, R3 ¼ 3 590 W; (b) R1 ¼ 1 000 W, R2 ¼ 1 035 W, R3 ¼ 1 065 W; (c) L2 ¼ 1 210e6H, C1 ¼ 25e15F, C3 ¼ 143e15F, R1 ¼ 3 530 W, R2 ¼ 3 560 W, R3 ¼ 3 590 W; (d) L2 ¼ 1 210e6H, C1 ¼ 25e15F, C3 ¼ 143e15F with R1 ¼ 1 000 W, R2¼1 035 W, R3¼1 065 W

Terahertz dielectric resonator antennas Dynamic impedance of quantum antenna

7,090

200

7,089.5

100

7,089

7,088.5 0

Img Zin

300 MG of Im(Zin)

Magnitudeof Re(Zin)

204

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

0 0

10

1

2

3

×105

7

8

9

10 ×105

Quality factor 1

Magnitude of input impedance (Zin)

7,096

4 5 6 Frequency (rad/s)

0.04

MG of Q

7,094

0.02

7,092 7,090 0

0

1

2

3

8

9

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Quality factor 3 4

0.04 0.02 0 0

0

10 ×105

MG of Q

Mg of Q

7

Quality factor 2

0.06

2

0

1

2

3

(c)

4 5 6 Frequency (rad/s)

7

8

9

10

0

1

2

3

5

×10

Dynamic impedance of quantum antenna

2,035

Magnitudeof Re(Zin)

4 5 6 Frequency (rad/s)

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Img part of Zin Mg of Im (Zin)

200

2,034.8

2,034.6 0

100

0 1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10

0

1

2

3

×105

Magnitude of input (Zin)

2,050

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Quality factor 1

2,045

Mg of Q

Mg of Zin

0.1

0.05

2,040 2,035 0

1

2

3

7

8

9

Mg of Q

0.05 0.0 0

0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Quality factor 3

15

0.1

(d)

0

10 ×105

Quality factor 2

0.15 Mg of Q

4 5 6 Frequency (rad/s)

10

5

0

1

2

3

4 5 6 Frequency (rad/s)

7

8

9

10

0

1

2

×105

Figure 10.4

(Continued )

3

4 5 6 Frequency (rad/s)

7

8

9

10 ×105

Equivalent circuit analysis on terahertz and optical DRA

205

Dynamic impedance (Zd)

7,100 7,000 6,900 6,800 Mag (Zd)

6,700 6,600 6,500 6,400 6,300 6,200 6,100 0

1

2

3

4

5

6

7

8

9

Frequency (rad/sec)

10 ×105

Figure 10.5 Dynamic impedance

Img part of input impedance

0.35 0.3 IMG value

0.25 0.2

0.15 0.1

0.05 0 0

1

2

3

4 5 6 Freqency (rad/sec)

7

8

9

10 ×105

8

9

10 ×105

Figure 10.6 Dynamic impedance (Img)

Magnitude of input impedance (Zin)

7,100 7,000 6,900 MG value of Zin

6,800 6,700 6,600 6,500 6,400 6,300 6,200 6,100

0

1

2

3

4

6 5 Freqency (rad/sec)

7

Figure 10.7 Dynamic impedance vs frequency

5

×10–5

Quality factor (Q)

4.5 4 MG value of Q

3.5 3

2.5 2

1.5 1 0.5 0 0

1

2

3

4

5 6 Freqency (rad/sec)

7

8

9

10 ×105

Figure 10.8 Quality factor vs frequency z

HR Ws

Ls

LSUB/LG

LP

Hsub

WP WSUB/WG Y

0

15

30 (mm)

Figure 10.9 Nano-RDRA (rectangular DRA) S11

Y Name X m1 474.4444 –10.1897 m2 521.1111 –10.0453

HFSSDesign1 Curve info dB (S11)

Setup 1: Sweep

–7.50 –10.00

m2

m1

–12.50 dB (S11)

–15.00 –17.50 –20.00 –22.50 –25.00 –27.50 450.00

475.00

500.00 Freq (THz)

Figure 10.10 S11

525.00

550.00

Equivalent circuit analysis on terahertz and optical DRA

207

r ¼ TrexpðbHÞ ðexpðbH ÞÞ, mean-field energy in Gibbs state ¼ r is given by Tr ðrH) and the Tr ðrp ) classically we would get

2.

Ð

expðbHÞpa d N qd N p Ð ¼0 expðbHÞdqN dpN

ak ¼

ak þ ipk pffiffiffi annihilation and creation operators 2

ak ¼

ak  i pk pffiffiffi ; k ¼ 1; 2; 3 . . . 2



 h dðk; mÞ; Heisenberg commutation relation ak ; am ¼ 2p

Electromagneticfield ¼ H ¼

N 1X Nh ; field energy ak ak þ 2 k¼1 4p

Quantam states or eigen states of H

Hjni ¼ n1 ; n2 ; n3 ; . . . ; nN i ; n1 ; n2 ; n3; . . . ; nN ¼ 0; 1; 2; 3; . . .   h N Hjni ¼ þ n1 ; þn2 þ . . . þ nN 2p 2

Name m1 m2

Freq

Ang

Mag

RX

504444.4444 29.7136 0.0533 1.0955 + 0.0580i 461111.1111 3.0539 0.4227 2.4558 + 0.1347i

110 120

Smith chart 90 100 80 1.00

HFSSDesign1

Curve Info S11 Setup1: Sweep

70 60 2.00 50

130 0.50 140

40

150

30

160 0.20

5.00 20 10

170 0.00 180 0.00

0.20

0.50

1.00

2.00

5.00

0 –10

–170 –160 0.20

–5.00 –20

–150

–30 –40

–140 –130 –0.50

–2.00 –50

–120 –110

–60

1.00 –100

–90

–80

–70

Figure 10.11 Smith chart showing impedance

208

Terahertz dielectric resonator antennas Radiation pattern 0

HFSSDesign1 Curve info dB (Gain total) Setup1 : Sweep Freq='5, 00, 000 GHz' φ = '0º'

30

–30

dB (Gain total) Setup1 : Sweep Freq='5, 00, 000 GHz' φ = '90º'

0.00 –5.00 –60

60 –10.00 –15.00

–90

90

–120

120

–150

150 –180

Figure 10.12 Radiation pattern

dB (Gain total) 4.6831e+000 2.9388e+000 1.1944e+000 –5.4991e–001 –2.2943e+000 –4.0386e+000 –5.7829e+000 –7.5273e+000 –9.2716e+000 –1.1016e+001 –1.2760e+001 –1.4505e+001 –1.6249e+001 –1.7993e+001 –1.9738e+001 –2.1482e+001 –2.3226e+001

z θ

x φ

Figure 10.13 Three-dimensional radiation pattern

Equivalent circuit analysis on terahertz and optical DRA Name m1 m2 m3 m4

X

Gain

Y

468.8889 –15.0786 490.0000 –28.2458 497.7778 –14.6564 501.1111 –11.0892

209

HFSSDesign1 Curve info dB (Gain total) Setup1 : Sweep φ = '0º' θ = '0deg'

–5.00

dB (Gain total)

–10.00

–15.00

m4

m3

m1

–20.00

–25.00

–30.00 450.00

475.00

500.00

Freq (THz)

525.00

Figure 10.14 Gain vs frequency

Thus, the quantum ak in the Gth state is ðbNhÞ ðbhÞ X  exp njak  exp a am 4p 2p m m

!

! ðbNhÞ X ðbhÞ X nm dðn; m  ek Þ ¼ 0 ðni ¼ exp n exp 4p 2p m ek ¼ ½0; 0; . . . ; 0; 1; . . . ; 0

Here   bh  1     Tr p2k þ p2 k  ak ak  ak ak e Tr p2k expðbhÞ ¼  2 zq ðbÞ zq ðbÞ ¼ Quantam partition function    1 h  Tr ak ak þ expðbhÞ ¼ 2zq ðbÞ 4p !     1 ðbNhÞ X 1 h bh X exp nm ¼ n nk þ 2 2p exp  4p 2zq ðbÞ 4p m ! ð X ða2 þ p2 Þ k k exp b a2k dqN dpN =zc ðbÞ 2 k where zc ðbÞ is a quantum part.

550.00

210

Terahertz dielectric resonator antennas

10.8 Drude’s model "

eAg

fp2 ¼ e0 ea þ f ðf þ igÞ

#

gcollision frequency ea ¼ 5 ðoffset of real frequencyÞ fp plasmon frequency g p ¼ 4:35 THz e0 absolute permitvity k0 —free-space wave number er dielectric constant c—velocity of light. In order of Planck’s constant in quantum antennas, radiation energy comes in quanta. Thus, discrete particle distributions of electrons, positrons and photons do matter, and this discreetness is also responsible for producing quantum fluctuation and the radiation field pattern waves are " # fp2 ¼ 129:17 þ j3:28 eAg ¼ e0 ea þ f ðf þ igÞ e0 ¼ 8:85  1012 F=m ea ¼ 5 er ¼ 1:4  1016 rad=s ðhighÞ due to collision photo gas particle er ¼ 1:4  1016 rad=s ðlowÞ ðcollective motion of electronÞ Larentz disperstion model Drude–Larentz model for dispersion in dielectrics, sffiffiffiffiffiffiffiffi ne2 ep ¼ e0 m Optical antennas, sub-wavelength devices that directly convert an electric mode of energy w, where is the reduced Planck constant and w ¼ 2pf is the particular frequency, which are the modes [16–40].

10.9 MATLAB program r1¼3530; r2¼3560; r3¼3590;

Equivalent circuit analysis on terahertz and optical DRA

211

l2¼1210e-9; c1¼25e-12; c3¼143e-12; w¼1:1:1000; p1¼r1*(w.^2*c3^2.*(r2þr3)^2þ(w.^2*l2*c3-1).^2) p2¼((r2*r3-l2/c3).*(r2þr3)*w.^2*c3^2þ(w.^2*l2*r3*c3-r2).*(w.^2*l2*c3-1)) p3¼(w.^2*c3^2*(r2þr3)^2þ(w.^2*l2*c3-1).^2) r¼(p1þp2)./p3 % plot(w,r) x1¼-1./w*c1; x2¼w*l2; x3¼-1./w*c3; q1¼((r2þr3).*(r3*x2þr2*x3))-((x2þx3).*(r2*r3-x2.*x3)); q2¼(r2þr3).^2þ(x2þx3).^2; q¼x1þ(q1./q2) % plot(w,q) z¼abs(rþ1i*q) % plot(w,z) g¼q./r % plot(w,g) k1¼((r.^2þx2.^2).*(x1þx3))þ((r3.^2þx3.^2).*(x1þx2))þ(2*x1).*(r2*r3þx2. *x3); k2¼((r1þr3)*(r2.^2þx2.^2))þ((r1þr2)*(r3.^2þx3.^2))þ2*r2*(r1*r3-x2.*x3)þ (2*x2).*(r1*x3-x3*r3); k¼k1./k2 % plot(w,k) h1¼w.^4*(l2*r3^2-l2*(1/c1þ1/c3))þw.^2*(l2/(c3^2)-r2^2*(1/c1þ1/c3)-r3^2/ (c1)-2/c1*(r2*r3-l2/c3)-1/c1*c3^2); h2¼w.^5*((l2^2*(r1þr3)))þw.^3*(r2^2*(r1þr3)þr3^2*(r1þr2)þ2*r2*(r1*r3-l2/ c3)þ2*l2*(-r1/c1þr3/c3))þw*(r2þr1/c3^2) h¼abs(h1./h2) plot(w,h)

10.10 Conclusion Reflection coefficient, radiation pattern and impedance of NDRA have been evaluated using R, L, C circuit as an equivalent circuit of NDRA. The measured and simulated results of NDRA have also been obtained. The work done on equivalent circuit configuration has been carried out for the first time in this manner and theoretical model thus developed will help designers in the realization of NDRA circuits.

References [1] M. Alam and Y. Massoud, “RLC ladder model for scattering in single metallic nanoparticles,” IEEE Trans. Nanotechnol., vol. 5, no. 5, pp. 491–498, 2006.

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[2] R. K. Mongia and P. Bhartia, “Dielectric resonator antennas—Are view and general design relations for resonant frequency and bandwidth,” Int. J. Microwave Millimeter Wave Comput. Aided Eng., vol. 4, no. 3, pp. 230–247, 1994. [3] M. Alam and Y. Massoud, “An accurate closed-form analytical model of single nanoshells for cancer treatment,” Proc. 2005 48th Midwest Symp. Circuits Syst., pp. 1928–1931, 2005. [4] J. L. West and N. J. Halas, “Applications of nanotechnology to biotechnology: Commentary,” Curr. Opin. Biotechnol., vol. 11, pp. 215–217, 2000. [5] C. Oubre and P. Nordlander, “Optical properties of metallodielectric nanostructures calculated using the finite difference time domain method,” J. Phys. Chem. B, vol. 108, pp. 17740–17747, 2004. [6] P. J. Flatau, “Fast solvers for one dimensional light scattering in the discrete dipole approximation,” Opt. Express, vol. 12, pp. 3149–3155, 2004. [7] M. Alam and Y. Massoud, “A closed-form analytical model for single nanoshells,” IEEE Trans. Nanotechnol., vol. 5, no. 3, pp. 265–272, 2006. [8] J. L.-W. Li, Z.-C. Li, H.-Y. She, S. Zouhdi, J. R. Mosig, and O. J. Martin, “A new closed-form solution to light scattering by spherical nanoshells,” IEEE Trans. Nanotechnol., vol. 8, no. 5, pp. 617–626, 2009. [9] L. A. Ambrosio and H. E. Herna´ndez-Figueroa, “RLC circuit model for the scattering of light by small negative refractive index spheres,” IEEE Trans. Nanotechnol., vol. 11, no. 6, pp. 1217–1222, 2012. [10] D. C. Tzarouchis, P. Yla¨-Oijala, and A. Sihvola, “Resonant scattering characteristics of homogeneous dielectric sphere,” IEEE Trans. Antennas Propag., vol. 65, no. 6, pp. 3184–3191, 2017. [11] M. Farhat, C. Rockstuhl, and H. Bagˇcı, “A 3D tunable and multi-frequency grapheme plasmonic cloak,” Opt. Express, vol. 21, pp. 12592–12603, 2013. [12] S. Ghadarghadr and H. Mosallaei, “Coupled dielectric nanoparticles manipulating metamaterials optical characteristics,” IEEE Trans. Nanotechnol., vol. 8, no. 5, pp. 582–594, 2009. [13] C. F. Bohren and D. R. Huffman, Absorption and Scattering by a Sphere, Wiley, New York, NY, 1983. [14] A. Garcıa-Etxarri, R. Go´mez-Medina, L. S. Froufe-Pe´rez, et al., “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express, vol. 19, pp. 4815–4826, 2011. [15] Y. Guo, T. Zhang, W.-Y. Yin, and X.-H. Wang, “Improved hybrid FDTD method for studying tunable graphene frequency-selective surfaces (GFSS) for THz-wave applications,” IEEE Trans. THz Sci. Technol., vol. 5, no. 3, pp. 358–367, 2015. [16] T. Bian, X. Gao, S. Yu, L. Jiang, J. Lu, and P. Leung, “Scattering of light from graphene-coated nanoparticles of negative refractive index,” Optik, vol. 136, pp. 215–221, 2017. [17] A. Vakil, Transformation Optics Using Graphene: One-Atom-Thick Optical Devices Based on Graphene, University of Pennsylvania, Philadelphia, PA, 2012.

Equivalent circuit analysis on terahertz and optical DRA

213

[18] Y. S. Cao, L. J. Jiang, and A. E. Ruehli, “An equivalent circuit model for graphene-based terahertz antenna using the PEEC method,” IEEE Trans. Antennas Propag., vol. 64, no. 4, pp. 1385–1393, 2016. [19] T. Christensen, A.-P. Jauho, M. Wubs, and N. A. Mortensen, “Localized plasmons in graphene-coated nanospheres,” Phys. Rev. B, vol. 91, Art. no. 125414, 2015. [20] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions: With Formulas, Graphs, and Mathematical Tables, vol. 55, Courier Corporation, North Chelmsford, MA, 1964. [21] L. Novotny and B. Hecht, Principles of Nano-Optics, Cambridge University Press, Cambridge, 2006. [22] H. W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl, and R. E. Smalley, “C60: Buckminsterfullerene,” Nature, vol. 318, pp. 162–163, 1985. [23] H. Nishihara, T. Simura, S. Kobayashi, et al., “Oxidation-resistant and elastic mesoporous carbon with single-layer graphene walls,” Adv. Funct. Mater., vol. 26, pp. 6418–6427, 2016. [24] H. Yang, Z. Hou, N. Zhou, B. He, J. Cao, and Y. Kuang, “Graphene encapsulated SnO2 hollow spheres as high-performance anode materials for lithium ion batteries,” Ceram. Int., vol. 40, pp. 13903–13910, 2014. [25] J.-S. Lee, S.-I. Kim, J.-C. Yoon, and J.-H. Jang, “Chemical vapor deposition of mesoporous graphene nanoballs for supercapacitor,” ACS Nano, vol. 7, pp. 6047–6055, 2013. [26] M. Naserpour, C. J. Zapata-Rodrı´guez, S. M. Vukovi´c, H. Pashaeiadl, and M. R. Beli´c, “Tunable invisibility cloaking by using isolated graphenecoated nanowires and dimers,” Sci. Rep., vol. 7, 2017, Art. no. 12186. [27] A Santos and V Garjo, An exact Solution of the Boltzmann Equation for a Binary Mixture, Taylor and Francis, 2006. (doi.org/10.1080/ 00411459208203789). [28] L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics, vol. 5, no. 2, pp. 83–90, 2011. [29] P. Biagioni, J.-S. Huang, and B. Hecht, “Nano antennas for visible and infrared radiation,” Rep. Prog. Phys., vol. 75, p. 024402, 2012. [30] L. Zou, W. Withayachumnankul, C. Shah, A. Mitchell, M. Bhaskaran, S. Sriram, and C. Fumeaux, “Dielectric resonator nano antennas at visible frequencies,” Opt. Express, vol. 21, no. 1, pp. 1344–1352, 2013. [31] G. N. Malheiros-Silveira, G. S. Wiederhecker, and H. E. Herna´ndezFigueroa, “Dielectric resonator antenna for applications in nano photonics,” Opt. Express, vol. 21, no. 1, pp. 1234–1239, 2013. [32] M. W. Knight, H. Sobhani, P. Norlander, and N. J. Halas, “Photodetection with active optical antennas,” Science, vol. 332, pp. 702–704, 2011. [33] P. Mu¨hlschlegel, H.-Eisler, O. J. F. Martin, B. Hecht, and D. W. Pohl, “Resonant optical antennas,” Science, vol. 308, no. 5728, pp. 1607–1609, 2005. [34] Y. Zhao, N. Engheta, and A. Alu`, “Effects of shape and loading of optical nano antennas on their sensitivity and radiation properties,” J. Opt. Soc. Am. B, vol. 28, no. 5, pp. 1266–1274, 2011.

214 [35] [36]

[37]

[38] [39] [40]

Terahertz dielectric resonator antennas E. Ozbay, “Plasmonics: merging photonics and electronics at nanoscale dimensions,” Science, vol. 311, p. 189, 2006. R. S. Yaduvanshi and H. Parthasarathy, “Coupled solution of Boltzmann transport equation, Maxwell’s and Navier Stokes equations,” IJACSA, vol. 3, 2010. L. Zou, W. Withayachumnankul, C. Shah, et al., “Efficiency and scalability of dielectric resonator antennas at optical frequencies,” IEEE Photon. J., vol. 6, pp. 1–10, 2014. R. S. Yaduvanshi and H. Parthasarathy, Rectangular DRA Theory and Design, Springer, 2016. R. S. Yaduvanshi and V. Gaurav, Nano Dielectric Resonator for 5G Applications, CRC Press, 2020. A. Bonakdar and H. Mohseni, “Impact of optical antennas on active optoelectronic devices,” Nanoscale, vol. 6, pp. 10961–10974, 2014.

Chapter 11

Optical DRA for retinal applications—next generation DRAs

Abstract Human eye retinas have millions of cones and rods cells, also known as photoreceptors, to receive photons from the nature. There are arrays of photoreceptors on retina periphery and central part of retina. These cells perform functions of photonic antennas, for example they receive light particles and convert it into electrical impulses. In this chapter, cones and cylindrical rods have been studied and analyzed as photonic wavelength antennas. These terahertz dielectric resonator antennas (DRAs) have been developed using HFSS and CST simulators. Their simulation modeling and theoretical modeling have been presented in this chapter. The work carried out in this chapter is much useful for the development of artificial retinal antennas for possible use in retinal prosthesis using terahertz DRAs and photodiodes as bio-photoreceptors. These DRAs at terahertz frequencies can also be used in applications of scanning, imaging and wideband ultrahigh-speed communication systems.

11.1 Introduction Light is reflected by natural objects when falls on any object, later it reaches eye cornea, retina, optical nerves for image formation into brain; thus reflected light particles are being captured by retina in the vision spectrum. Light enters through the eye lens and is focused on to retina through the white color ball known as a vitreous humor, which is a gel-like material and consists of 98% water. An eye retina has 1.26 million rods and approximately 5 million cones known as photoreceptors. Rods in retina have 496-nm wavelengths, which are responsible for black and white vision. Cone are color vision cells and they are in three wavelengths and corresponds to R, G, B basic color wavelengths. Cones have three different wavelengths of 420, 530 and 560 nm, respectively. These photoreceptors can absorb light in the frequency spectrum from 430 to 750 THz, also known as a vision spectrum [1,2]. These photoreceptors have capability to transduce light (photons) into electrical impulses (neurons). Photoreceptor cells have plasmonic membranes and vision pigment molecules called rhodopsin. These photons react with rhodopsin,

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phenomenon of electro-chemistry takes place and electrical current of very small magnitude is generated in the form of neurons. Neurons can communicate in chemical form as well as in electrical form. There are networks of millions of optic nerves that carry these neuron signals from retinal surface to brain. Brain performs processing at a very fast rate to develop images corresponding to visuals or scene captured by a human eye. Each photon-receptor cells constitute one pixel to develop pixel formats in the brain, and this will enable to develop an array of pixels for transforming it to a two-dimensional image. These photons are collected by both the retinas simultaneously and synchronously processed by the brain. These captured scenes or images are of temporal and spatial in nature. The phenomenon of real-time processing takes place at a rapid rate. Rods cells are responsible for creating black and white vision, whereas cones cells have capability to create color vision. Thus, rod cells create photopic vision and cone cell create scotopic vision. Rods provide peripheral vision and cones provide central vision. The diameter of rods cell is 1–3 mm and the length of rod is 6–50 mm in retina. Rods are in cylindrical shape and cones are in conical shape. The refraction index of eye tissue is 1.336, in a vision spectrum. The specific absorption rate of tissue is measured as 2 W/kg on 10 g. Permittivity of rod tissues is 69 F/m and conductivity is 1.53 S/m. Technical specifications of retinal membranes for rods and cones are given in Tables 11.1 and 11.2. Photoreceptors form the first part of vision process, for sending signals progressively to different cells before reaching the brain. Bipolar cells and ganglion cells form the important part of vision process. Bipolar cells are induced graded potential based on photoreceptor vision signal and ganglion cells process it as gated signal and transverse it to a bundle of optic nerves, which get terminated on a lateral geniculate nucleus (LGN). From an LGN the visual signal is sent to the primary visual cortex of brain to generate two-dimensional image that is also called visual scene. Ganglion cells play a crucial part in vision process to detect object movements contrast and intensity part of the seen image due to visual pigment present as then. Table 11.1 Rod cell parameters Count and shape

1.26 millions, cylindrical

Vision spectral Diameter Length Photocurrent Refraction index Wavelength Vision Specific absorption rate (SAR) Vitreous humor (
r ) Photopigment Density

430–750 THz 1–3 mm 5–60 mm 1 pA 1.336 496 nm Peripheral vision 2 W/kg 67 F/m Rhodopsin 1.60
103/mm2

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Table 11.2 Retina parameters Number of layers Number of incident photons Gated potential Grade potential Incident power Photoreceptors in center Photoreceptors in periphery

10 500 photons/mm/s Alternate signal Bipolar cell 0/P (10 to 70 mV) 7 mW Cones Rods

1 Pm Membrane shelves lined with rhodopsin or color pigment

Outer segment

Inner segment

Outer limiting membrane

5 Pm

Mitochondria

Nucleus

Synaptic body

Quantum antenna

Figure 11.1 Quantum antenna as a 5-mm length cylindrical rod

There are two types of retinal diseases: one is age-related macular degeneration and the other is retinitis pigments. These are caused by a degeneration of photopigment in old age. Worldwide, millions of people have been affected by these two abovementioned diseases resulting into visual impairment or blindness. This chapter proposes a technique of vision restoration by suggesting a quantum antenna as an artificial rod cell to be used as eye prosthesis (Figure 11.1). A quantum antenna is a unique technique with an integration of photodiode to provide required graded potential to bipolar cells, and the rest of the vision process will remain unaltered. The resolution obtained by using a quantum antenna will be higher than the previous work available. In our proposed work, capture of the image scene requirement of external camera has been eradicated. As the per literature survey Argus I, Argus II and Alpha eye prosthetics have been found very bulky and with low resolution. Cylindrical, spherical and conical antennas in

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MIMO or array pattern can be suitable candidates for designing artificial photoreceptors. These are given in Figures 11.2–11.4. Working principle of optical antennas: Light in (source LASER) light–matter interaction: the electrons and positrons interaction with photons or the quantum light–matter field in a fermionic bosonic coherent state (scattering, distortions, SPP (surface plasmon polytron) and Hamiltonian as the amplitudes for scattering) Define the fermionic and bosonic annihilation and creation operators. Describe the fermionic and bosonic as Dirac field (momentum-spin, photonic wave field). Compute the Dirac current (displacement type, tangential magnetic field, combined effect of electrons positrons and photons).

Silver divider Silicon (resonating element)

Silicon (resonating element) Silver nanostrip

Silver nanostrip

Silicon dioxide (substrate) Silver (ground)

Lumped port

Figure 11.2 MIMO conical DRA at photonic wavelength

Silicon Silver divider (resonating element)

Silver nanostrip

Silicon (resonating element)

Silver nanostrip

Silicon dioxide (substrate) Silver (ground)

Lumped port

Figure 11.3 MIMO spherical DRA at photonic wavelength

Optical DRA for retinal applications—next generation DRAs Silver divider

219

Silicon (resonating element)

Silicon (resonating element)

Silver nanostrip

Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Lumped port

Figure 11.4 MIMO CDRA at photonic wavelength Determine eigen solution to the Dirac wave equation using 4
4 Dirac-amatrices satisfying the canonical anticommutation relation for fermions and canonical commutation relation (CCR) for bosons: Express the Dirac current density (electromagnetic (e.m.) wave field pattern). Find far-field magnetic vector potential (retardation potentials). Determine far-field radiation pattern (quantum wave operator field). Berezin weight function parameterized by  q (relates to Grassmann variables). Minimize the quantum fluctuation correlation in the pattern for possible matching the average antenna pattern to the desired pattern (boson Fock space). The mean and mean-square fluctuations of quantum e.m. fields (antenna and free photons). In the super directive (emitted vector potential, photons with different momenta and helicities, e.m. four potential fields), radiated field is a state of jointly coherent for the bosons (i.e. photons) and for the fermions (positrons–electrons).

11.2 Optical antenna arrays basic requirements 1. 2. 3. 4. 5.

6.

Study of vision process. Use of biocompatible electronic devices for prosthesis. To develop an electronic device for a possible replacement of dead photoreceptors in retinal layer. Compatibility of electronic device output with bipolar cell graded potential. Major part of image processing involves cornea, lens, vitreous humor, bipolar cells, ganglion cells, rods, cones (photoreceptors) with epithelium as absorber, optic nerves and brain. Reflected light from natural objects falls on cornea travels through vitreous humor reaches to photoreceptors gets converted into electrical signal.

220 7. 8.

9. 10. 11. 12. 13. 14.

15. 16. 17. 18. 19. 20. 21.

22. 23.

24. 25. 26.

Terahertz dielectric resonator antennas These photoreceptors are formed by 126 million rods and cones in retina. Cones and rods have several layers of plasma membranes, and it has photosensitive vision pigment molecules. It has capability to absorb light by making use of rhodopsin and convert into electrical signal. Rods are responsible for black and white vision that is also known as peripheral vision. Rods are located at periphery of the retina. Cones are responsible for colored vision and located at the center of retina. Photoreceptors absorb, amplify incident light, transduce it to neural response for further communication to brain through network of optic nerves. Image is developed from the scenes in the brain, i.e. photons get converted into pixels. Photoreceptors are at initial stage in the vision process. In the second stage these signals are progressively processed by bipolar cells as graded potential. At the next stage, ganglion cells participate to convert the graded potential signals into gated potentials signals. Now these signals can be termed neural signals. Each ganglion cell gated potential signal travels to brain through optic nerves. Neurons have capability to communicate through chemical potential as well as through electrical potential. Temporal and spatial phenomena on real-time basis take place in ten layers of retina before developing an image in the brain. The resolution of eye cell depends on the number of photons received by quantum antennas in prosthesis. Array of quantum antennas can be used as retinal implants. Vision in human retinal is circularly polarized. Optical antenna: Light–matter interaction on the nanoscale, occurrence of plasmon resonances, volume currents, kinetic inductance, effective wavelength scaling, near-field intensity enhancement and Lorentzian resonance phenomenon takes place. The e.m. field is only one possible physical observable operator of the quantum state. The desired higher order modes can be excited by surface plasmon resonance (SPR), which is a sub-wavelength phenomenon and allows coexistence between different modes. Incident field, scattering field, total field and directivity are calculated. The permittivity of the metal can be described using Drude’s model. SPPs or resonance frequency, the quantum efficiency and the enhancement of field are major objectives of an optical antenna. We can incorporate some unknown real parameters into the Berezin linear combination of coherent states and estimate these parameters by minimizing the distance between the average value of the e.m. field generated by the fermions and the desired e.m. field pattern. If need be, we may modify this cost function to be minimized by constraining the higher order quantum statistical moments of the generated quantum e.m. field to be

Optical DRA for retinal applications—next generation DRAs hv

221

λspp

Hy2 Ex2 ε2Ez2 H Ex1 ε1Ez1 y1

Figure 11.5 Wave vector of SPP (kSPP ¼ 1:1k0)

27.

28.

29.

specified. An example of an application of this circle of ideas is to use a quantum antenna to generate a set of desired spatial patterns at a given set of frequencies. Let g be a Grassmannian variable that will be used to specify the coherent state of this fermion just as a complex number z is used to specify a coherent state of a single boson. g Anticommutes with itself, g and with a; a just as in bosonic situation, the complex number z that specifies the coherent state commutes with itself, with z and with the boson creation and annihilation operators. The current density field generated by this field as according to Dirac’s theory, a quadratic function of these operators, and hence the e.m. field generated by this current density according to the retarded potential formula, is also a quadratic function of these operators. Figure 11.5 shows a rectangular silicon terahertz dielectric resonator antenna (DRA) with LASER and silver FEED (side and top view).

Due to interaction between the surface charge density and the e.m. field results in the momentum of the SP mode in terahertz DRA cavity. Wave vector (SPP) being greater than that of a free-space photon of the same frequency is the freespace wave vector (k0 ¼ w/c). SPPs are generated on a metal–dielectric interface and a feedback mechanism allows SPP to resonate. The interaction of surface charge on the metal and dielectric at interface generates a strong capacitive region. The previous phenomenon is expressed as follows using flowchart mechanism: The terahertz antenna far field radiations Light in Gaussian pulse for phase matching. + The electrons and positrons or the quantum matter field in a fermionic coherent state. + Define the fermionic annihilation operator. + Describe the fermionic Dirac field. + Compute the Dirac current. + Determine eigen solution to the Dirac wave equation using 4
4 Dirac-amatrices satisfying the anticommutation relation. +

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Terahertz dielectric resonator antennas

Express the Dirac current density. + Find far-field magnetic vector potential. + Determine far-field radiation pattern. + Berezin weight function parameterized by  q. + Minimize the quantum fluctuation correlation in the pattern for possible matching the average antenna pattern to the desired pattern. + e.m. radiated fields (super directive; Figure 11.6) The enhancement effect is attributed to the resonance of SPPs. Near field is due to a resonant excitation of SPPs. The electric dipole source, SPP, is first launched by an emitter in the nanogap and then scattered, e.m. radiation, emission rates and the far-field radiation pattern; SPP imposes fundamental dominant mode and bounded. The electrooptical antennas working principal is based on electron plasmon coupling. Localized fields near optical antenna structures have spatial dimensions that approach the length scale of atomic/molecular quantum wave functions. The localization of optical antenna gives rise to photon momenta that are of the order of the momenta of electrons in matter and can therefore give rise to traditionally momentum-forbidden transitions use of optical sensing. It can absorb all of the incident radiation if each molecule is coupled to an optical antenna. The penetration of radiation into metals takes place. Owing to the finite electron

z y

Theta

Phi x

Figure 11.6 Super-directive radiations

Optical DRA for retinal applications—next generation DRAs

223

density, there is delay between the driving field and the electronic response, resulting in a skin depth that is typically larger than the diameter of the antenna elements. As a consequence, electrons in metals do not respond to the wavelength l of the incident radiation but to an effective wavelength leff, which is determined by a simple linear scaling rule:   l leff ¼ n1 þ n2 lp where n1 and n2 are geometric constants and lp is the plasma wavelength; according to this wavelength scaling rule, an optical half-wave antenna is not l/2 in length but a shorter length of leff/2. The difference between l and leff depends on geometric factors but is typically in the range of 2–5 for most of the metals used as optical antennas. Reducing the dimensions even further reaches the molecular scale, where the design of optical antennas could draw inspiration from biology. For example, in light-harvesting proteins, chlorophyll molecules arrange. Nonlinear optical antennas hold promise for generating and controlling this nonlinear response on a sub-wavelength scale. For example, it has been demonstrated that controlling the gap between the nanoparticles in a gold nanoparticle dimer allows the intensity of the nonlinear frequency conversion to be tuned over four orders of magnitude.

11.3 Optical antenna design To control and manipulation of optical fields at the nanometer scale, SPP mechanism is used. This is generated through proximity-coupled feed. The optical antennas are similar to their microwave antennas. Optical antennas have small size with resonant properties of nanostructures. Optical antennas surpass the diffraction limit in optical imaging. SPRs make optical antennas particularly efficient at selected frequencies. These are useful in biological sensing and detection applications. They have a feature of photo detection, light emission, sensing, heat transfer and spectroscopy. A terahertz DRA here is a cavity resonator consisting of an ensemble of electrons, positrons and photons (Figure 11.7).  The electron, positron field is the Dirac second quantized wave function y t; r ; i.e. a fermionic operator field,

while the second quantized photon field is the Maxwell potential field. The firstorder change in the current density dJm ðxÞ caused by the laser source and the interaction between the Dirac field with the quantum photon field is ðoÞþ

ð oÞ

dJm ðxÞ ¼ eyðxÞ am dyðxÞ  edyðxÞþ am yðxÞ

The notion of a fermionic coherent state and its application to the computation of the quantum statistical moments of the quantum e.m. field is generated by electrons and positrons in terahertz DRA.

224

Terahertz dielectric resonator antennas Silicon DR Graphene disk Sub

Ws

ro

x

lf

ri

y

e

ls Gro und

z

stra t

Wf

t

pu

In h

Figure 11.7 Terahertz DRA design using graphene disk and silicon annular DRA

11.4 Entanglement It is said to be in entanglement when states of photons are in coherent or synchronous spin. Particles remain connected so that actions performed on one instantaneously affect the state of the other, even when they are separated by large distances. Convert entanglement of photons into the entanglement of the atoms that absorb those photons. Let two ground-state atoms separated by a small distance excite those atoms to a very high-energy state, they interact over longer ranges and can thus shift each other’s energy levels. The two atoms are now entangled if both are in superposition due to laser pulse, i.e. Rydberg level and other in ground state. Hence, an interaction in two atoms that leads to indeterminate but correlated final states. How to identify SPP in optical antenna response is reflectance (reflected power) measurement on wavelength, which provide two dips: larger dip is antenna resonance (lower frequency) and smaller dip as SPR (at higher frequency). An optical antenna can provide focused optical energy when absorption quality factor is equal to radiation quality factor. This is achieved by an SiO2 surface on silver metal by substrate thickness-controlled radiations through constructive interference. Coupled mode theory is involved for field enhancement.

11.5 Modeling of optical antennas Laser output of coherent states is given inputs to optical antenna. Due to laser inputs, SPP phenomenon takes place and generates SPR. Multimode phenomenon as well as coupling of resonant modes takes place. Due to the interaction of light with matter scattering, absorption, i.e. creation and annihilation, happens. SPR is now coupled to DRA for possible radiations. This process can be described in more detail as given next.

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225

Creation and annihilation (add particle and remove particles from state) antenna resonance is dependent upon dimensions, material and loading, effective mode volume, effective antenna aperture to match laser input resulting into enhanced directivity, minimal loss (silver) at optimum quality factor, i.e. Qrad ¼ Qabs (SiO2 thickness); also less thickness more radiation and more thickness, more absorption and less radiation. It is possible to shape the second and higher order correlation functions of the field emitted by a quantum antenna in the far-field zone by designing its initial state. The designing states with two initial excitations lead to highly directional emission of photon pairs for the same antenna or even produce the effect of no radiating sources by suppressing the field in the far-field zone. The quantum antenna can produce multi-photon momentum entangled states. The desired spatial patterns of the correlation functions, also patterned higher order correlation functions of the emitted field, are introduced by a quantum antenna. The quantum antennas can be realized in a number of different ways, such as chains of semiconductor quantum dots, cold atoms in optical lattices. Laser-generated coherent photon: the second-order correlation function of the two-particle entangled state can produce photon pairs that are strongly correlated in momentum. We describe the procedure for designing the antenna state with the required correlation functions.

11.6 Light–matter interaction Localization of the emitted field inside a finite volume is something that one would really expect. In twin-photon generation, the coefficients provide a desired spatial pattern of the second-order correlation function. Effective excitation and photon extraction are particularly important, weak signals emitted by a single atom or molecule, and large efficiencies can be achieved by either planar dielectric antennas beaming with nanostructures that beam emission into a narrow angular distribution, accurate positioning of the quantum emitter, planar optical antenna that beams light emitted by a single molecule, high numerical aperture. The initial state corresponds to excited pairs of the dipoles located symmetrically on the opposite sides of the antenna. The wave function of the emitted field state is computed. The dipole moments are orthogonal to the antenna plane. Quantum interferences are essential for shaping the correlation functions (spatial correlation functions). Capture the mechanism of spontaneous emission and the effects stemming from it. Field correlation effects can still be successfully captured. The best known example is super radiance. The onset of cooperative effects and phase correlations leading to the formation of the super radiant field pulse can be quite accurately described. The correlations self-establish at the initial stage of cooperative emission. Simultaneously changing the shape and the initial state of the antenna, one can get a high directivity of the correlation function, which is a simple way to obtain directional correlations of emitted photons that are localized in a narrow region. In dipole orientation to produce polarization, spatiotemporal current (J) is produced or displacement current, inductive

226

Terahertz dielectric resonator antennas

impedance. Power is mostly transferred to the metal by absorption or excitation of surface plasmon polaritons, or the emission is suppressed by destructive interference with the image dipole. Excited electrons jump to higher state and come to lower state, and these are Rabi oscillations or surface plasmonic resonance. Matching the spatial profile of the emitter with that of a focused laser beam, the antenna allows for lower pump powers, which in turn implies that a better signalto-noise ratio is achieved. Results can be explained in terms of the image dipole induced in the reflector by the source. When the two dipoles radiate with an appropriate phase difference, which depends on the optical constants, they constructively interfere in the forward direction, which results in a beaming effect (beaming occurs in the range of l/6n  d1  l/4n). In momentum space structure or gradient field operator or field enhancement, constructive interference gives rise to the beaming effect. The simultaneous second-order correlation functions in the far-field zone for excitation Silicon (resonating element)

Silver nanostrip

Silicon dioxide (substrate)

Lumped port

Silver (ground)

Figure 11.8 Terahertz cylindrical DRA design and analysis

Silicon (resonating element)

Silver nanostrip

Silicon dioxide (substrate)

Silver (ground)

Lumped port

Figure 11.9 Terahertz conical DRA design and analysis

Optical DRA for retinal applications—next generation DRAs

227

S11 (dB)

–10

–20

–30

Rectangular Cylindrical Spherical Conical

–40

–50 450

475

500 Frequency (THz)

525

550

Figure 11.10 Reflection coefficient (S11) simulation results on different geometries of terahertz DRAs 10

Gain (dB)

0

–10

Rectangular Cylindrical Spherical Conical

–20 450

475

500 Frquency (THz)

525

550

Figure 11.11 Gain simulation results

of two states. The beaming of light from a single molecule in a semi-cone is only 20 wide, the emission pattern resembles a weakly focused Gaussian laser beam.

11.7 Theory of coupled resonant modes The coupling of cavity modes based on structure takes place. The field enhancement is another important parameter. The emitted power is inversely proportional to the excited state lifetime. The optical antenna can strongly direct the radiation in a particular direction. An equivalent circuit of a quantum antenna can be developed on the basis of coupled Boltzmann transport equation (BTE) solution. Nanoantennas are operating in the terahertz, infrared and visible spectral ranges.

228

Terahertz dielectric resonator antennas

Radiation efficiency (%)

80 75 70 65 60 Rectangular Cylindrical Spherical Conical

55 50 450

475

500 Frequency (THz)

525

550

Figure 11.12 Radiation efficiency simulation results 0 330

30

0 300

Gain (dB)

–10

60

–20 270

90

–20

–10

120

240

0 150

210 180

Figure 11.13 Radiation pattern co- and cross-radiation pattern simulation results

It is well known that an entangled state of emitters can lead to an entanglement of the emitted photons (i.e. the state of the field can be mapped into the emitter’s state and vice versa). This effect was suggested as a basis for a quantum memory device capable of storing entangled states of light. The entanglement of emitters in antennas can lead to intensity distributions. So-called timed Dick states bear information about the location of emitters and provide a special quantum mechanism that introduces a nonreciprocity of the antenna.

Optical DRA for retinal applications—next generation DRAs

229

11.8 Designs of terahertz DRAs simulation results for various shapes Designs of terahertz DRAs simulation results for various shapes are described in Figures 11.8–11.13.

11.9 Conclusion and applications Terahertz DRAs are most efficient antenna. They offer low loss at higher frequencies. They can provide super directivity and ultralarge bandwidth for wireless communications. Optical antennas are useful in sensing, scanning and they are super directive. They are also used in quantum information processing, spectroscopy, optics and secure communication. Higher order modes generation has many advantages such as mode merging for wideband, highly directive or super directive for higher gain and directivity with less beam width. Super-directive antennas are much useful for communication and radars. TDRA can be the best candidate for photoreceptors in applications to retinal prosthesis.

References [1] M. Humayun and L. Olmos de Koo, Retinal Prosthesis—A Clinical Guide to Successful Implementation, Springer, Cham, 2018. [2] Y. Yang, Y. Yamagami, X. Yu, et al., “Terahertz topological photonics for on-chip communication,” Nat. Photonics, vol. 14, pp. 446–451, 2020.

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Chapter 12

Conclusion and futuristic vision

Abstract The dielectric resonator antennas (DRAs) have capabilities to operate at low frequencies, microwave frequencies, terahertz frequencies as well as optical frequencies. These DRAs possess low loss, design flexibilities and efficient radiation mechanism even at high temperatures. DRAs have been recently been integrated with an Apple i-phone 12 mobile phone antenna at millimetric-wave frequencies for 5G applications, i.e., 28 GHz. These DRAs can operate higher order modes in addition to fundamental modes. They can provide reconfigurable antenna characteristics and wide bandwidths. DRAs have mode control mechanism. They can provide physical insights during design stage itself. They are versatile due to availability of many dielectric constants materials. They can use DR values ranging from 10 to 1,600. These DRAs can have two different aspect ratios to extend design flexibilities. Resonant modes merging is another important feature available to designer. DRA absorbers are also designed using the same concept of DR materials. Absorbers are used as artificial photosynthesis and energy-harvesting solar curtains can be developed using DR absorbers. These DRs can have materials such as TMM, PDMS, TiO2, graphene, graphite, sapphire, polymers and polymers composites. TMM materials are mostly suitable for millimetric-wave frequencies and microwave frequencies. Graphite, graphene and polymers composites are being used at terahertz frequencies. TiO2, sapphire and PDMS are used at optical frequencies. They are in use for developing photonic devices. Sapphire antennas can be used for esthetic design and used for wearable sensors such as rings. DRAs can also be used as LiDAR in autonomous cars and self-driving vehicles. DRAs have been used for artificial leaves for photosynthesis and rectennas for energyharvesting application. They are being used as absorbers. Solar curtain is another application for energy harvesting. These solar curtains shall become boon to green buildings, smart homes and smart cities. They can provide clean energy for domestic use. Designing retinal DRAs is another era of Hi-Fi. These can become futuristic DRAs/antennas arrays as a substitute of rods and cones in human eye retina as implants. They can very well be used for sight restoration or eye prosthesis. The other field of applications of DRAs can be agriculture and environmental. Namely, moisture sensor, rain sensor, temperature sensor, carbon monoxide gas sensor can

232

Terahertz dielectric resonator antennas

be developed using DRA. Scanning and imaging is another field where terahertz DRA can be used for service to the society.

12.1 Introduction In this book, design simulations and implementations have been included in microwave, millimetric wave, terahertz and optical spectrum using different DR materials and different shapes. Their theoretical background using mathematical derivations has also been comprehensively developed and nicely explained. These derivations will provide physical insight to design engineer, application engineer, industry personal, new reader, researcher, professional, academicians and UG/PG students. They have been simulated on computer simulation technology (CST) to provide graphical insights and few have been converted into prototype models to have feel of physical device and instrumentation testing. Figure 12.1 is the implementation of MIMO dielectric resonator antenna (DRA) to be used in X, K and Ku band radars and satellites. MIMO DRAs can be developed for diverse applications such as high gain and high isolation for diversity requirements. Many such MIMO DRAs can be developed for 5G and beyond applications by choosing a right kind of aspect-ratio and dimension of DRAs to achieve desired gain and isolation along with other

y'

x'

ab d

z y x

wg

S-Parameters (dB)

lg

0

S11 (simu)

–10

S11 (meas) S22 (meas)

–20

S12 (simu)

–30

S12 (meas)

–40 –50 –60 12

16

20

24

Freq (GHz)

Figure 12.1 MIMO DRA prototype developed for 12–24 GHz, gain 5.7 dBi for radar and satellite applications

Conclusion and futuristic vision

233

radiation parameters. Here, terahertz DRAs have been designed to operate at 511, 541 and 10 THz in different shapes such as rectangular DRA, spherical DRA, conical DRA and cylindrical DRA. They have been broadly supported with mathematical formulation on radiation mechanism and effective control on beam orientation. MIMO DRAs are very useful in high data rate communications.

12.2 Patient-centric healthcare system outline Patient-centric healthcare system is innovative approach developed for the wellbeing people living in society. In this system, various health parameters are being proposed to be monitored at regular interval of time or randomly. These sensors can be integrated as wearable just like a wrist watch. These measured parameters through a sensor are being wireless transmitted to a server for analytics. Wi-Fi connectivity is maintained between wearables and server. The server has microcontroller-based Arduino Uno system for acquiring biomarker data such as pulse, temperature or glucose level, etc. Human health is continuously monitored by using these sensors thus it develops biomedical history of person. The proposed model is developed for blood glucose monitoring. Thumb based tissue model DRAs have been proposed for glucose level monitoring. Due to change in glucose concentrations levels, shift in radiated resonant frequency is observed. The change in concentrations of glucose level is calibrated with frequency shifts. For data acquisition, Arduino-based Internet of Things (IoT) system is also integrated with machine learning algorithm for analytic purpose. Using thumb DRA with Arduino health monitoring system, alerts on biomarker deviations shall be recorded and reported to patients as well as doctors. Medical history using data acquisition system can be used for better treatment and illness diagnosis. With this system a patient is well educated about any illness beforehand as preliminary information received as biomarker alerts.

12.3 Thumb DRA sensors integrated with patient-centric healthcare system Use of an oximeter for checking oxygen saturation is common nowadays. Pulse rate is measured by electronic sensors. Body temperature is monitored using an IR sensor. Breathe analysis can be monitored using sensors. A moisture sensor is used for detecting rain, etc. Blood pressure using cuffless system can be developed. Blood glucose is an important biomarker for monitoring health of the human being. The idea to develop patient-centric healthcare system was inspired by studentcentric education system. Now students can select their choice of subjects, projects and teacher and college. Internet has created revolution in education system. Libraries have been integrated for facilitating and sharing of resources. Media and open network became discussion platforms like Quora, YouTube and may be other platforms. This could happen because of IT developments. This has embedded collaborations approaches. In this model, a patient and a doctor both are sharing database of healthcare system. Doctors and patients shall share information from

234

Terahertz dielectric resonator antennas

the history and records of biomarkers along with preinformation to a doctor and a patient both. The doctor will be able to understand the reason and cause of illness. It will help to localize and better diagnose any kind of illness and execute treatment fast. A patient can also take precautions in one’s life-style based on history biomarkers. Machine learning algorithm integration shall help to determine normal and deviated parameters. Machine learning model was trained on data acquisition parameters to get predictions about health or illness in future. Sensed output is DC signal and converted into DC values to be integrated further with Arduino Uno. A block diagram of Arduino connected sensors is given in Figures 12.2–12.4. The sensors attached as wearables transmit using Wi-Fi frequency are received by Arduino Wi-Fi modules. A thumb DRA is developed using permittivity analysis based on shift in frequency due to change in frequency. The permittivity of glucose shall change when the concentration of glucose level changes. This change in permittivity will cause shift in frequency. Hence based on calibration and signal processing, glucose level can be monitored and displayed.

12.4 Thumb DRA design and implementations The DRAs have been designed based on Cole–Cole model of a bio-characteristic of tissue dielectric properties, and DRA properties have been used in this CST model. Figure 12.5 shows a simulation diagram of a thumb DRA and it monitors for shift in frequency due to change in dielectric constant. Figures 12.2–12.4 have been developed for showing total patient-centric healthcare system using an Arduino microcontroller and a sensor connected with it using Wi-Fi links for periodic data acquisition, and machine learning algorithm is for data analysis and predictions to predetect any possible disease or illness in the patient. This data can be used by a concerned doctor for possible medication and timely treatment. DRAs with and without thumb have been simulated and results have been given in Figures 12.6– 12.11 with a proper label marked below of each figure.

#TRANSMITTER END

Wifi 2 To connect the transmitter end and the receiver end

Insulin sensor

Oximeter

Hardwired connection

Hardwired connection

Wifi 1 To connect glucose sensor to the arduino

ARDUINO with Wifi module connected (at wrist of user)

Hardwired connection

Wifi 1

Glucose sensor

Temperature sensor

Figure 12.2 Sensors developed are connected to Arduino microcontroller with wireless system

Conclusion and futuristic vision #RECIEVER END

235

ML ALGORITHM Step 1-: The algorithm will convert the text file to csv file

Wifi 2 To connect the transmitter end and the receiver end

Step 2-: SVM is used to give predictions (output). Output-: 0 1 - > No Disease 0 1 - > Diabetes 1 0 - > Cardiovascular Disease 1 1 - > Both Diseases

ARDUINO with Wifi module connected

Using CoolWin application to print the data received in a text file

Step 3-: Addition of new data to excel file. Step 4-: Output is written in text file. Calculation of output -: 1. Prediction given by ML Algorithm 2. Calculation of deviation form standard values using the formula given below -: |standard value - measured value| ×100 standard value 3. Calculation of deviation form average value : |average value - measured value| ×100 average value

Figure 12.3 Server connected to Arduino microcontroller with machine learning environment for feedback to patient

Wifi 2 #FEEDBACK SYSTEM

Will send the switch response back to the server for updating the parameters of the ML Algorithm.

Buzzer Hardwired connection Feedback received from the server

(sign of anomaly)

ARDUINO with Wifi module connected (at wrist of user)

Switch Hardwired connection

(the user presses the switch if the values are not anomalous in accordance to the general parameter readings of the user)

OLED The device will display the following readings-: 1. Prediction (output) 2. Deviation from the standard value 3. Deviation from the average value

Figure 12.4 Patient information system for health alerts

Tissue details are given in Figure 12.11 and an equivalent circuit of thumb tissue is given in Figure 12.10. The effects of glucose concentrations to provide shift in frequency have been given by S11 parameters and radiation parameters are given in Figures 12.6 and 12.9. The dimensions and material details have been

236

Terahertz dielectric resonator antennas

Figure 12.5 Design of thumb shape tissue DRA for blood glucose monitoring (TMM DR, tissue electrical properties have been used in design based on Cole–Cole model)

7.8208957

7.3731341

6.477612

6.9253731

6.029851

5.5820894

5.1343284

4.6865673

4.2388058

3.7910447

3.3432837

2.8955224

2

2.4477613

dB

S11 (without thumb) 0 –5 –10 –15 –20 –25 –30 –35 –40 –45

Frequency (GHz)

Figure 12.6 DRA S11 at frequency 7.37 GHz

7.8208957

7.3731341

6.9253731

6.477612

6.029851

5.5820894

5.1343284

4.6865673

4.2388058

3.7910447

3.3432837

2.8955224

2.4477613

2

dB

S11 (with thumb) 0 –5 –10 –15 –20 –25 –30

Frequency (GHz)

Figure 12.7 S11 of thumb DRA sensor at frequency 5.13 GHz

Conclusion and futuristic vision Radiation pattern (with thumb) 0° 5.00E+00

330°

30°

4.00E+00 3.00E+00

300°

60° 2.00E+00 1.00E+00 0.00E+00

270°

90°

240°

120°

210°

150° 180° Co

Cross

Figure 12.8 Radiation pattern of thumb DRA sensor

Radiation pattern (with thumb) 0° 5.00E+00

330°

4.00E+00

30°

3.00E+00

300°

60° 2.00E+00 1.00E+00 0.00E+00

270°

90°

240°

120°

210°

150° 180° Co

Cross

Figure 12.9 Thumb DRA sensor 3D radiation pattern

237

238

Terahertz dielectric resonator antennas Y

Prop.Dir Theta

Phi X

Figure 12.10 Equivalent circuit of thumb DRA tissues (showing electrical properties of skin, fat and muscle)

RSkin CSkin RFat CFat

HSkin

HFat

HMuscle RMuscle CMuscle

Figure 12.11 Actual view of thumb tissue skin, fat and muscles details

provided in Tables 12.1 and 12.2. Hence, other sensors can be developed to monitor real environments and real health biomarkers using DRAs. Hence, shift in frequency is calibrated with glucose concentration levels of human being in the proposed healthcare system.

Conclusion and futuristic vision

239

Table 12.1 DRA without thumb material details Component name

Material

Length (x-axis) (mm)

Breath (y-axis) (mm)

Height (z-axis) (mm)

Ground Substrate Feed RDRA Groove

Copper Rogers RT5880 Copper Alumina Alumina

60 60 4.65 20 15

60 60 40 20 20

0.035 3.04 0.7 12 2

Table 12.2 Thumb (tissue) DRA sensor material details Component name

Permittivity Length (x-axis) (mm)

Breath (y-axis) (mm)

Height (z-axis) (mm)

Skin Fat Blood Bone Nail

30 5.5 Cole–Cole 20 3

24 24 24 24 15.5

1 0.5 2.5 4 0.3

15 15 15 15 15

12.5 Conclusion This thumb DRA can wirelessly communicate Arduino environment based healthcare system for monitoring and acquisition of biomedical data of any person. In this manner biomarker deviations can be real-time monitored and communicated to any remote location. Thus, proposed healthcare system can be used for remote health monitoring and medication. Future plans are to develop a similar thumb DRA for sensing other noninvasive biomarkers sensors such as blood pressure, breathe analyzer, breathe pattern, heart rate, oxygen saturation, thyroid and other environmental gas-monitoring sensors.

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Appendix A

Case studies

In this annexure, DRAs prototyped have been developed for operating at different frequencies of applications, i.e. Wi-Fi, ISM band, WiMAX, communication satellites, and millimeter-wave frequencies. These DRAs have been developed using Roger substrates, Sapphire and TMM DR materials and SMA connectors for feeding RF inputs through feed. These prototyped DRA products have been fabricated, tested and then validated in echo-free environments. Anechoic chamber testing results are matched with simulated results (Cases A.1–A.15).

Case A.1 DRA implementation at 3.5 GHz for WiMAX applications

Case A.2 MIMO DRA at 4 and 3.5 GHz for WiMAX and satellite communication applications

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Terahertz dielectric resonator antennas

Case A.3 DRAs at 3.5 GHz for WiMAX applications

Case A.4 DRAs at 3.5 and 4 GHz mobile communication applications

Case A.5 DRA MIMO at 4 GHz for applications satellite communication applications

Case studies

243

Case A.6 DRAs under an anechoic chamber at 5.8 GHz for Wi-Fi applications

Case A.7 DRAs at 3.5 GHz for WiMAX applications

Case A.8 DRAs at 12 GHz for satellite communication applications

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Terahertz dielectric resonator antennas

Case A.9 DRAs at 6.5 GHz for satellite communication applications

Case A.10 DRAs under measurement with VNA at 4.6-GHz satellite communication applications

Case A.11 Rectangular DRAs at 22.5- and 2.4-GHz radar and Wi-Fi applications

Case studies

245

Case A.12 DRAs designed from 2.4 to 22.5 GHz frequencies for mobiles, satellites and radar applications

Case A.13 Sapphire DRAs at 28 GHz for millimeter-wave high data rate transmission

Case A.14 Vehicle DRA products prototyped at 3.6 GHz

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Terahertz dielectric resonator antennas

Case A.15 DRAs at 6-GHz frequency for sub-6 GHz, 5G communications

Appendix B

Terahertz absorbers

Absorbers have many applications for energy harvesting and metamaterials devices designs. Many optical sensors are optical absorbers that are used in our daily life. Retinal cones and rods are excellent examples of photonic device absorbers. Optical absorbers are new era technology electronic devices. These devices can generate renewable energy. Artificial photo synthesis is a new field of research. Electric leaves are new types of electronics for use energy resources. They can be embedded into high rise glass buildings for generating electrical energy. Graphene materials have many such applications of absorbers to be used in day-to-day life. These graphene curtains can thus generate electrical energy just like solar cells. The quality factor can be expressed as Q ¼ (wWe)/(Prad). Hence, DRA forming standing waves due to boundary conditions can act as absorbers. They have high-Q factor and high-energy storage; if accelerating charges are created in DRA absorbers, they behave like DRAs due to energy leakage into air. Thus the quality factor of absorbers is reduced to act as antennas. Hence, the Q factor of absorbers is high and that of DRAs is low. The eigen state at resonance is absorber cavity, and excited state of cavity is an antenna due to accelerating charge careers and simultaneously decreasing the quality factor as well. The feed used at THz DRA is Gaussian beam and feed is wave port for absorber. For making absorbers, imaginary part must be high. The second derivate of image impedance w.r.t. frequency must be negative.

B.1 Absorber characteristics High-Q factor and plane wave incidence are two important features of absorbers. Polarization sensitivity and incidence angle are two important properties of absorbers. Cavity, multilayer architecture, plane wave input excitations, absorptance, resonance are few important features. Standing wave formation, SPP help to develop absorbers. They are used for sensing, security, reducing radar cross section, energy harvesting, solar fuel, solar cells, solar house curtain, graphene curtains, graphene batteries and other applications. The complex imaginary part of impedance is made high to increase loss in all absorbers. Absorbers are frequency dependent and resonance is based on cavity dimensions. The complex permittivity and permeability play an important role to develop absorbers. Addition of magnetic

248

Terahertz dielectric resonator antennas

materials will significantly increase complex permeability of the absorber thus enhances capability of absorber. Reflection loss is measured in absorbers. The metallic ground is used for obtaining the reflection and then the electromagnetic wave is absorbed in the materials and frequency-selective resonant cells. A graphite/graphene disk stack is placed on the low-permittivity dielectric resonant cavity for obtaining the multiple generated resonance peaks at the nearby frequencies. The response of absorbers has been tuned by varying the chemical potential of graphene for generating higher order modes and playing with resonant modes (Figures B.1–B.3). This is directly related to dielectric loss value, sample thickness and impedance matching. Impedance matching is a concept used to maximize energy absorption by the load and minimize refection. This occurs when the input impedance at the air and composite interface is equivalent to that of free space; Zin ¼ Z0, where it is clear that the input impedance Zin is dependent on the composite electric and magnetic properties, er and mr, respectively, the sample thickness, d, and the frequency, w. The attenuation of the electromagnetic energy inside the sample is controlled by the attenuation factor ead, which works on attenuating (reducing) the wave amplitude with respect to traveling distance, d. The attenuation constant, a, is also dependent on the electric and magnetic properties of the material. The dielectric loss is the reason of absorption. The large height of a substrate or composites can absorb more power. Air impedance and absorber impedance must be matched for loss or absorption. Er

Ei

Hi ki

Hr

Hi

Ei

θi

ki

kr

θr

kr

Er

Composite εr

Hr TM TE Thickness

Perfect conductor

Figure B.1 Reflection loss in absorber a

d

a TiO2

h Silver

y

z

x

Figure B.2 TiO2 DRAs as optical absorbers

Terahertz absorbers

249

Substrate

D-3 D-4 Ex

k

Graphite sheet

D-1 D-2

Hy

Figure B.3 Terahertz absorbers (9.5 THz) using multilayers of graphite and graphene

B.2 Absorbers mathematical analysis The relationship is given as follows: loss tangent ¼ tan(@) ¼ e00 /e0 ; where e00 is the imaginary part of permittivity and e0 is the real part. Generally capacitive part is imaginary and loss tangent must be high for the absorber in dielectrics. The loss tangent is also dependent on the conductivity of the material if metals are used. High conductivity shall lead to high-loss tangent, hence a good absorber. Highquality factor is desired for good absorbers. (  )1=2    rffiffiffiffiffi  s 2 1=2 Zin mr wd pffiffiffiffiffiffiffiffi me mr er ; a ¼ w 1þ ¼ 1 ; tan h j Z0 er c 2 we Zin  Z0 : RLðdBÞ ¼ 20 log Zin  Z0 Absorption loss depends upon loss tangent, and loss tangent depends upon complex permittivity, frequency, conductivity and dielectric constant; tan(d) ¼ permittivity second derivative upon permittivity first derivative ¼ s upon w into permittivity. The absorber design can be with perfect match and dielectric constant layer can be placed from low to high and from high to low for perfect absorption. The absorber design can be built with a ground plane as metals and layers of dielectric materials in stacking. Zero reflection is absorber requirements.

B.3 Optical absorbers applications RF energy harvesting, imaging, scanning, rectenna, solar curtains, solar cells, attenuators, microwave absorbers, sensing, isolation, detection, etc. are applications. Graphene is a highly conductive material for electrical and thermal properties: light weight, transparent, flexible materials made of carbon single atoms and

250

Terahertz dielectric resonator antennas

two-dimensional materials. This is the graphene material that we are using as the tip of a pencil to write on a piece of paper. Graphene is used to design absorbers at THz frequencies. The absorber has broad applications of sensing, imaging, isolation, energy harvesting, etc. High-dielectric constant materials have high tangent loss, and negative imaginary parts of permittivity offer huge dielectric loss and positive imaginary parts of permittivity offer electrical energy to storage. Magnetic properties enhance absorber properties of materials used in design. The cavity gives plane wave inputs to all absorbers against antenna input that is Gaussian beam at THz. The resonance in cavity occurs at plane wave inputs, standing waves are formed and electrical energy is absorbed or stored into cavity. Boundary conditions are maintained for no leakage of power. Loss tangent increases in the case of an absorber and it is minimized in the case of antennas. The electrical energy stored in an absorber may be converted into heat due to the law of conservation of energy. At an optical window or a vision spectrum, human retina has cones and rods of multilayers that accept photons and convert these photons into neuron impulses, which are electrical in nature due to absorption properties of light. Hence multilayered absorbers are generally designed. Absorption high- and zero reflection coefficients are two important measurement parameters for optical absorbers.

Appendix C

Antenna measured values in anechoic chamber

Figures C.1 is hardware of 5.8 GHz antenna used in mobile communication.

Figure C.1 0.46 m, 25 dBi, antenna used for mobile towers Radiation plot of antenna at 5.8 GHz (Figures C.2–C.4).

–30°

5 0 –5 –10 –15 –20 –25 –30 –35 –40 –45

0°

30° co cross 60°

–60°

90° –90°

Figure C.2 Radiation pattern polar plot at 5.8 GHz

Terahertz dielectric resonator antennas 10

Amplitude (dB)

0 –10 co –20 cross –30

90°

60°

30°

0°

–30°

–60°

–50

–90°

–40

Degree

Figure C.3 Radiation pattern 2D plot at 5.8 GHz

40 30 20 10 0 –10 co –20

cross

–30 –40

Degree

Figure C.4 Directivity plot at 5.8 GHz

90°

60°

30°

0°

–30°

–60

–60°

–50 –90°

Amplitude (dB)

252

Antenna measured values in anechoic chamber

253

Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 90.0000 89.9100 89.8199 89.7299 89.6398 89.5498 89.4597 89.3697 89.2796 89.1896 89.0995 89.0095 88.9195 88.8294 88.7394 88.6493 88.5593 88.4692 88.3792 88.2891 88.1991 88.1091 88.0190 87.9290 87.8389 87.7489 87.6588 87.5688 87.4787 87.3887 87.2987 87.2086 87.1186 87.0285 86.9385 86.8484 86.7584 86.6683 86.5783 86.4882 86.3982 86.3082 86.2181 86.1281 86.0380 85.9480 85.8579

Amplitude (dB) 45.2769 45.2706 45.2515 45.2199 45.1758 45.1193 45.0505 44.9698 44.8774 44.7734 44.6583 44.5323 44.3958 44.2491 44.0927 43.9269 43.7521 43.5687 43.3772 43.1781 42.9716 42.7582 42.5385 42.3127 42.0813 41.8447 41.6033 41.3575 41.1077 40.8542 40.5974 40.3377 40.0753 39.8106 39.5439 39.2756 39.0058 38.7349 38.4632 38.1908 37.9181 37.6453 37.3726 37.1002 36.8283 36.5571 36.2869

Phase ( ) 153.6596 153.6586 153.6555 153.6504 153.6434 153.6344 153.6234 153.6107 153.5960 153.5797 153.5616 153.5419 153.5206 153.4979 153.4738 153.4484 153.4219 153.3943 153.3657 153.3362 153.3060 153.2751 153.2437 153.2119 153.1798 153.1476 153.1153 153.0830 153.0510 153.0192 152.9879 152.9570 152.9269 152.8974 152.8689 152.8414 152.8149 152.7897 152.7657 152.7432 152.7222 152.7028 152.6851 152.6693 152.6554 152.6435 152.6337

(Continues)

254

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 85.7679 85.6778 85.5878 85.4977 85.4077 85.3177 85.2276 85.1376 85.0475 84.9575 84.8674 84.7774 84.6873 84.5973 84.5073 84.4172 84.3272 84.2371 84.1471 84.0570 83.9670 83.8769 83.7869 83.6968 83.6068 83.5168 83.4267 83.3367 83.2466 83.1566 83.0665 82.9765 82.8864 82.7964 82.7064 82.6163 82.5263 82.4362 82.3462 82.2561 82.1661 82.0760 81.9860 81.8960 81.8059 81.7159

Amplitude (dB) 36.0177 35.7498 35.4833 35.2184 34.9553 34.6941 34.4349 34.1779 33.9232 33.6709 33.4212 33.1742 32.9301 32.6888 32.4506 32.2156 31.9838 31.7554 31.5305 31.3092 31.0916 30.8778 30.6680 30.4622 30.2605 30.0631 29.8701 29.6816 29.4978 29.3186 29.1444 28.9752 28.8111 28.6524 28.4991 28.3514 28.2096 28.0736 27.9439 27.8205 27.7036 27.5936 27.4906 27.3948 27.3066 27.2259

Phase ( ) 152.6262 152.6210 152.6182 152.6180 152.6204 152.6255 152.6335 152.6445 152.6584 152.6756 152.6961 152.7200 152.7474 152.7785 152.8134 152.8522 152.8952 152.9424 152.9940 153.0502 153.1113 153.1773 153.2485 153.3251 153.4073 153.4955 153.5899 153.6908 153.7984 153.9132 154.0354 154.1656 154.3041 154.4514 154.6080 154.7745 154.9514 155.1393 155.3391 155.5515 155.7773 156.0175 156.2731 156.5453 156.8353 157.1445

(Continues)

Antenna measured values in anechoic chamber

255

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 81.6258 81.5358 81.4457 81.3557 81.2656 81.1756 81.0855 80.9955 80.9054 80.8154 80.7254 80.6353 80.5453 80.4552 80.3652 80.2751 80.1851 80.0950 80.0050 79.9150 79.8249 79.7349 79.6448 79.5548 79.4647 79.3747 79.2846 79.1946 79.1046 79.0145 78.9245 78.8344 78.7444 78.6543 78.5643 78.4742 78.3842 78.2941 78.2041 78.1141 78.0240 77.9340 77.8439 77.7539 77.6638 77.5738

Amplitude (dB) 27.1495 27.0757 27.0047 26.9362 26.8704 26.8071 26.7463 26.6879 26.6319 26.5783 26.5269 26.4777 26.4307 26.3858 26.3429 26.3020 26.2631 26.2261 26.1909 26.1576 26.1260 26.0961 26.0678 26.0412 26.0162 25.9928 25.9709 25.9506 25.9318 25.9144 25.8987 25.8844 25.8717 25.8605 25.8510 25.8431 25.8369 25.8325 25.8300 25.8295 25.8310 25.8348 25.8409 25.8496 25.8609 25.8745

Phase ( ) 157.4743 157.8248 158.1956 158.5866 158.9976 159.4285 159.8790 160.3489 160.8382 161.3465 161.8738 162.4198 162.9843 163.5672 164.1681 164.7869 165.4233 166.0771 166.7480 167.4356 168.1397 168.8599 169.5959 170.3472 171.1136 171.8945 172.6895 173.4982 174.3199 175.1543 176.0007 176.8587 177.7275 178.6066 179.4954 179.6069 178.7007 177.7870 176.8662 175.9392 175.0065 174.0691 173.1274 172.1823 171.2273 170.2300

(Continues)

256

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 77.4837 77.3937 77.3036 77.2136 77.1236 77.0335 76.9435 76.8534 76.7634 76.6733 76.5833 76.4932 76.4032 76.3132 76.2231 76.1331 76.0430 75.9530 75.8629 75.7729 75.6828 75.5928 75.5028 75.4127 75.3227 75.2326 75.1426 75.0525 74.9625 74.8724 74.7824 74.6923 74.6023 74.5123 74.4222 74.3322 74.2421 74.1521 74.0620 73.9720 73.8819 73.7919 73.7019 73.6118 73.5218 73.4317

Amplitude (dB) 25.8902 25.9075 25.9262 25.9459 25.9663 25.9869 26.0074 26.0275 26.0466 26.0645 26.0808 26.0951 26.1070 26.1162 26.1224 26.1254 26.1248 26.1205 26.1123 26.1002 26.0841 26.0639 26.0399 26.0120 25.9806 25.9458 25.9079 25.8674 25.8247 25.7802 25.7343 25.6877 25.6383 25.5840 25.5245 25.4592 25.3880 25.3107 25.2270 25.1371 25.0410 24.9390 24.8314 24.7187 24.6013 24.4797

Phase ( ) 169.1861 168.0954 166.9581 165.7741 164.5437 163.2673 161.9454 160.5788 159.1684 157.7153 156.2211 154.6874 153.1160 151.5091 149.8691 148.1986 146.5005 144.7778 143.0338 141.2717 139.4950 137.7072 135.9118 134.1123 132.3121 130.5144 128.7224 126.9389 125.1666 123.4079 121.6649 119.9388 118.1906 116.3990 114.5695 112.7081 110.8207 108.9137 106.9933 105.0659 103.1378 101.2150 99.3031 97.4077 95.5337 93.6855

(Continues)

Antenna measured values in anechoic chamber

257

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 73.3417 73.2516 73.1616 73.0715 72.9815 72.8914 72.8014 72.7114 72.6213 72.5313 72.4412 72.3512 72.2611 72.1711 72.0810 71.9910 71.9009 71.8109 71.7209 71.6308 71.5408 71.4507 71.3607 71.2706 71.1806 71.0905 71.0005 70.9105 70.8204 70.7304 70.6403 70.5503 70.4602 70.3702 70.2801 70.1901 70.1001 70.0100 69.9200 69.8299 69.7399 69.6498 69.5598 69.4697 69.3797 69.2896

Amplitude (dB) 24.3548 24.2270 24.0972 23.9660 23.8344 23.7030 23.5726 23.4440 23.3180 23.1953 23.0767 22.9630 22.8548 22.7528 22.6536 22.5548 22.4563 22.3583 22.2609 22.1640 22.0680 21.9729 21.8790 21.7867 21.6961 21.6076 21.5215 21.4383 21.3583 21.2820 21.2097 21.1421 21.0796 21.0226 20.9718 20.9277 20.8910 20.8622 20.8415 20.8271 20.8182 20.8147 20.8161 20.8224 20.8332 20.8483

Phase ( ) 91.8672 90.0821 88.3332 86.6229 84.9529 83.3246 81.7389 80.1962 78.6965 77.2396 75.8248 74.4512 73.1177 71.8222 70.5303 69.2251 67.9099 66.5875 65.2609 63.9330 62.6062 61.2832 59.9661 58.6572 57.3582 56.0711 54.7973 53.5382 52.2949 51.0685 49.8597 48.6691 47.4971 46.3440 45.2099 44.0946 42.9981 41.9197 40.8509 39.7540 38.6253 37.4658 36.2765 35.0586 33.8133 32.5418

(Continues)

258

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 69.1996 69.1096 69.0195 68.9295 68.8394 68.7494 68.6593 68.5693 68.4792 68.3892 68.2991 68.2091 68.1191 68.0290 67.9390 67.8489 67.7589 67.6688 67.5788 67.4887 67.3987 67.3087 67.2186 67.1286 67.0385 66.9485 66.8584 66.7684 66.6783 66.5883 66.4983 66.4082 66.3182 66.2281 66.1381 66.0480 65.9580 65.8679 65.7779 65.6878 65.5978 65.5078 65.4177 65.3277 65.2376 65.1476

Amplitude (dB) 20.8676 20.8910 20.9183 20.9496 20.9847 21.0238 21.0670 21.1142 21.1659 21.2221 21.2833 21.3499 21.4222 21.5008 21.5866 21.6820 21.7865 21.8992 22.0190 22.1445 22.2746 22.4080 22.5433 22.6792 22.8142 22.9472 23.0767 23.2018 23.3215 23.4349 23.5417 23.6417 23.7351 23.8224 23.9046 23.9835 24.0583 24.1257 24.1825 24.2258 24.2531 24.2624 24.2526 24.2232 24.1747 24.1083

Phase ( ) 31.2455 29.9259 28.5843 27.2224 25.8419 24.4443 23.0314 21.6050 20.1668 18.7186 17.2621 15.7990 14.3310 12.8596 11.3810 9.8352 8.2027 6.4812 4.6688 2.7639 0.7657 1.3259 3.5103 5.7853 8.1480 10.5938 13.1169 15.7097 18.3636 21.0682 23.8122 26.5835 29.3690 32.1558 34.9309 37.7555 40.7237 43.8272 47.0533 50.3855 53.8034 57.2836 60.8005 64.3267 67.8351 71.2995

(Continues)

Antenna measured values in anechoic chamber

259

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 65.0575 64.9675 64.8774 64.7874 64.6973 64.6073 64.5173 64.4272 64.3372 64.2471 64.1571 64.0670 63.9770 63.8869 63.7969 63.7069 63.6168 63.5268 63.4367 63.3467 63.2566 63.1666 63.0765 62.9865 62.8965 62.8064 62.7164 62.6263 62.5363 62.4462 62.3562 62.2661 62.1761 62.0860 61.9960 61.9060 61.8159 61.7259 61.6358 61.5458 61.4557 61.3657 61.2756 61.1856 61.0955 61.0055

Amplitude (dB) 24.0259 23.9302 23.8242 23.7113 23.5951 23.4795 23.3680 23.2644 23.1680 23.0726 22.9774 22.8819 22.7859 22.6896 22.5934 22.4981 22.4047 22.3143 22.2283 22.1483 22.0759 22.0130 21.9615 21.9235 21.9013 21.8972 21.9110 21.9387 21.9788 22.0296 22.0897 22.1578 22.2327 22.3135 22.3994 22.4898 22.5846 22.6836 22.7872 22.8962 23.0116 23.1348 23.2676 23.4109 23.5593 23.7054

Phase ( ) 74.6961 78.0041 81.2064 84.2898 87.2454 90.0675 92.7541 95.3060 97.7875 100.2992 102.8321 105.3755 107.9189 110.4518 112.9648 115.4489 117.8963 120.3003 122.6554 124.9571 127.2024 129.3895 131.5176 133.5871 135.5997 137.5578 139.5279 141.5950 143.7583 146.0160 148.3659 150.8050 153.3298 155.9361 158.6190 161.3734 164.1935 167.0739 170.0091 172.9942 176.0253 179.0999 177.7825 174.5135 170.9469 167.0815

(Continues)

260

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 60.9155 60.8254 60.7354 60.6453 60.5553 60.4652 60.3752 60.2851 60.1951 60.1051 60.0150 59.9250 59.8349 59.7449 59.6548 59.5648 59.4747 59.3847 59.2946 59.2046 59.1146 59.0245 58.9345 58.8444 58.7544 58.6643 58.5743 58.4842 58.3942 58.3042 58.2141 58.1241 58.0340 57.9440 57.8539 57.7639 57.6738 57.5838 57.4937 57.4037 57.3137 57.2236 57.1336 57.0435 56.9535 56.8634

Amplitude (dB) 23.8411 23.9583 24.0490 24.1063 24.1249 24.1017 24.0362 23.9306 23.7894 23.6191 23.4270 23.2212 23.0096 22.7913 22.5470 22.2759 21.9805 21.6647 21.3335 20.9924 20.6470 20.3026 19.9643 19.6366 19.3234 19.0282 18.7540 18.5035 18.2783 18.0624 17.8470 17.6328 17.4204 17.2110 17.0060 16.8068 16.6151 16.4327 16.2613 16.1028 15.9588 15.8314 15.7223 15.6328 15.5520 15.4741

Phase ( ) 162.9247 158.4948 153.8226 148.9511 143.9350 138.8370 133.7230 128.6566 123.6936 118.8782 114.2410 109.7989 105.5555 101.4372 97.2854 93.1474 89.0754 85.1123 81.2909 77.6331 74.1513 70.8491 67.7232 64.7652 61.9627 59.3007 56.7626 54.3302 51.9823 49.6219 47.2222 44.8016 42.3761 39.9587 37.5599 35.1874 32.8464 30.5398 28.2679 26.0292 23.8203 21.6356 19.4680 17.3041 15.0531 12.6931

(Continues)

Antenna measured values in anechoic chamber

261

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 56.7734 56.6833 56.5933 56.5033 56.4132 56.3232 56.2331 56.1431 56.0530 55.9630 55.8729 55.7829 55.6928 55.6028 55.5128 55.4227 55.3327 55.2426 55.1526 55.0625 54.9725 54.8824 54.7924 54.7024 54.6123 54.5223 54.4322 54.3422 54.2521 54.1621 54.0720 53.9820 53.8919 53.8019 53.7119 53.6218 53.5318 53.4417 53.3517 53.2616 53.1716 53.0815 52.9915 52.9015 52.8114 52.7214

Amplitude (dB) 15.3978 15.3222 15.2467 15.1711 15.0956 15.0208 14.9473 14.8763 14.8091 14.7473 14.6926 14.6468 14.6066 14.5613 14.5082 14.4451 14.3706 14.2841 14.1858 14.0768 13.9586 13.8335 13.7042 13.5737 13.4454 13.3224 13.1999 13.0663 12.9209 12.7638 12.5962 12.4197 12.2368 12.0503 11.8633 11.6790 11.5007 11.3318 11.1755 11.0336 10.8935 10.7513 10.6070 10.4613 10.3152 10.1701

Phase ( ) 10.2392 7.7060 5.1077 2.4578 0.2313 2.9483 5.6837 8.4295 11.1799 13.9311 16.6818 19.4330 22.2471 25.2158 28.3165 31.5230 34.8082 38.1447 41.5060 44.8676 48.2076 51.5077 54.7537 57.9357 61.0480 64.0890 67.1312 70.2578 73.4371 76.6381 79.8326 82.9956 86.1066 89.1494 92.1124 94.9887 97.7752 100.4728 103.0855 105.6302 108.2122 110.8408 113.4962 116.1605 118.8180 121.4555

(Continues)

262

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 52.6313 52.5413 52.4512 52.3612 52.2711 52.1811 52.0910 52.0010 51.9110 51.8209 51.7309 51.6408 51.5508 51.4607 51.3707 51.2806 51.1906 51.1006 51.0105 50.9205 50.8304 50.7404 50.6503 50.5603 50.4702 50.3802 50.2901 50.2001 50.1101 50.0200 49.9300 49.8399 49.7499 49.6598 49.5698 49.4797 49.3897 49.2996 49.2096 49.1196 49.0295 48.9395 48.8494 48.7594 48.6693 48.5793

Amplitude (dB) 10.0277 9.8899 9.7587 9.6366 9.5257 9.4285 9.3472 9.2740 9.2035 9.1347 9.0669 8.9999 8.9338 8.8691 8.8067 8.7479 8.6943 8.6477 8.6101 8.5831 8.5599 8.5364 8.5104 8.4802 8.4448 8.4037 8.3572 8.3061 8.2519 8.1966 8.1427 8.0930 8.0456 7.9930 7.9327 7.8633 7.7842 7.6958 7.5991 7.4962 7.3897 7.2825 7.1782 7.0805 6.9900 6.8978

Phase ( ) 124.0623 126.6307 129.1552 131.6332 134.0646 136.4515 138.8016 141.2078 143.6991 146.2603 148.8764 151.5332 154.2169 156.9154 159.6176 162.3144 164.9985 167.6648 170.3104 172.9481 175.6935 178.5593 178.4733 175.4244 172.3148 169.1653 165.9963 162.8269 159.6743 156.5529 153.4743 150.4469 147.3992 144.2364 140.9845 137.6717 134.3263 130.9758 127.6455 124.3580 121.1323 117.9831 114.9209 111.9520 109.0361 106.0535

(Continues)

Antenna measured values in anechoic chamber

263

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 48.4892 48.3992 48.3092 48.2191 48.1291 48.0390 47.9490 47.8589 47.7689 47.6788 47.5888 47.4987 47.4087 47.3187 47.2286 47.1386 47.0485 46.9585 46.8684 46.7784 46.6883 46.5983 46.5083 46.4182 46.3282 46.2381 46.1481 46.0580 45.9680 45.8779 45.7879 45.6978 45.6078 45.5178 45.4277 45.3377 45.2476 45.1576 45.0675 44.9775 44.8874 44.7974 44.7074 44.6173 44.5273 44.4372

Amplitude (dB) 6.8019 6.7017 6.5974 6.4897 6.3801 6.2706 6.1636 6.0618 5.9684 5.8864 5.8164 5.7508 5.6877 5.6260 5.5650 5.5049 5.4460 5.3895 5.3368 5.2899 5.2511 5.2229 5.2048 5.1920 5.1822 5.1730 5.1631 5.1515 5.1381 5.1232 5.1080 5.0943 5.0843 5.0804 5.0803 5.0804 5.0776 5.0697 5.0555 5.0343 5.0069 4.9745 4.9395 4.9049 4.8743 4.8473

Phase ( ) 103.0183 99.9542 96.8838 93.8281 90.8054 87.8318 84.9196 82.0783 79.3140 76.6293 73.9809 71.2603 68.4775 65.6494 62.7929 59.9243 57.0590 54.2110 51.3925 48.6137 45.8827 43.2048 40.4954 37.6796 34.7724 31.7908 28.7530 25.6781 22.5855 19.4945 16.4231 13.3878 10.4030 7.4533 4.3983 1.2325 2.0221 5.3412 8.6987 12.0672 15.4198 18.7310 21.9772 25.1382 28.2031 31.3070

(Continues)

264

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 44.3472 44.2571 44.1671 44.0770 43.9870 43.8969 43.8069 43.7169 43.6268 43.5368 43.4467 43.3567 43.2666 43.1766 43.0865 42.9965 42.9065 42.8164 42.7264 42.6363 42.5463 42.4562 42.3662 42.2761 42.1861 42.0960 42.0060 41.9160 41.8259 41.7359 41.6458 41.5558 41.4657 41.3757 41.2856 41.1956 41.1056 41.0155 40.9255 40.8354 40.7454 40.6553 40.5653 40.4752 40.3852 40.2951

Amplitude (dB) 4.8201 4.7904 4.7565 4.7178 4.6744 4.6275 4.5791 4.5319 4.4893 4.4551 4.4305 4.4123 4.3983 4.3869 4.3772 4.3689 4.3626 4.3596 4.3617 4.3716 4.3927 4.4288 4.4779 4.5370 4.6033 4.6747 4.7498 4.8278 4.9087 4.9935 5.0839 5.1840 5.3017 5.4338 5.5755 5.7226 5.8713 6.0186 6.1626 6.3026 6.4394 6.5752 6.7194 6.8773 7.0422 7.2080

Phase ( ) 34.4946 37.7425 41.0253 44.3166 47.5900 50.8203 53.9839 57.0604 60.0321 62.8870 65.7445 68.6682 71.6415 74.6465 77.6635 80.6724 83.6531 86.5858 89.4520 92.2347 94.9244 97.6496 100.4638 103.3571 106.3168 109.3269 112.3690 115.4221 118.4636 121.4699 124.4166 127.3090 130.3125 133.4540 136.7257 140.1137 143.5980 147.1527 150.7460 154.3416 157.8997 161.3788 164.8694 168.5385 172.3789 176.3724

(Continues)

Antenna measured values in anechoic chamber

265

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 40.2051 40.1151 40.0250 39.9350 39.8449 39.7549 39.6648 39.5748 39.4847 39.3947 39.3047 39.2146 39.1246 39.0345 38.9445 38.8544 38.7644 38.6743 38.5843 38.4942 38.4042 38.3142 38.2241 38.1341 38.0440 37.9540 37.8639 37.7739 37.6838 37.5938 37.5038 37.4137 37.3237 37.2336 37.1436 37.0535 36.9635 36.8734 36.7834 36.6933 36.6033 36.5133 36.4232 36.3332 36.2431 36.1531

Amplitude (dB) 7.3692 7.5216 7.6627 7.7919 7.9111 8.0244 8.1397 8.2686 8.4061 8.5458 8.6822 8.8116 8.9319 9.0436 9.1495 9.2552 9.3689 9.5043 9.6587 9.8229 9.9887 10.1489 10.2982 10.4339 10.5563 10.6694 10.7803 10.9006 11.0222 11.1276 11.2008 11.2300 11.2090 11.1385 11.0258 10.8837 10.7284 10.5640 10.3748 10.1539 9.9006 9.6193 9.3185 9.0094 8.7044 8.4159

Phase ( ) 179.5088 175.3027 171.0554 166.8194 162.6506 158.6045 154.6907 150.6578 146.4642 142.1282 137.6788 133.1545 128.6018 124.0721 119.6179 115.2896 111.1254 106.8817 102.4182 97.7382 92.8586 87.8105 82.6376 77.3939 72.1392 66.9329 61.8282 56.5846 50.9644 45.0012 38.7613 32.3417 25.8610 19.4445 13.2065 7.2370 1.5930 3.9727 9.6736 15.4025 21.0520 26.5263 31.7502 36.6736 41.2709 45.5384

(Continues)

266

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 36.0630 35.9730 35.8829 35.7929 35.7029 35.6128 35.5228 35.4327 35.3427 35.2526 35.1626 35.0725 34.9825 34.8924 34.8024 34.7124 34.6223 34.5323 34.4422 34.3522 34.2621 34.1721 34.0820 33.9920 33.9020 33.8119 33.7219 33.6318 33.5418 33.4517 33.3617 33.2716 33.1816 33.0915 33.0015 32.9115 32.8214 32.7314 32.6413 32.5513 32.4612 32.3712 32.2811 32.1911 32.1011 32.0110

Amplitude (dB) 8.1554 7.9145 7.6783 7.4446 7.2135 6.9871 6.7692 6.5649 6.3799 6.2208 6.0939 5.9910 5.9035 5.8276 5.7614 5.7040 5.6566 5.6214 5.6022 5.6041 5.6324 5.6848 5.7544 5.8355 5.9239 6.0169 6.1137 6.2159 6.3271 6.4534 6.6025 6.7703 6.9472 7.1246 7.2963 7.4585 7.6114 7.7591 7.9098 8.0750 8.2518 8.4256 8.5831 8.7145 8.8147 8.8845

Phase ( ) 49.4901 53.3671 57.3009 61.2360 65.1224 68.9176 72.5876 76.1081 79.4634 82.6462 85.6677 88.7519 91.9566 95.2456 98.5827 101.9321 105.2603 108.5362 111.7331 114.8282 117.8604 121.0774 124.4882 128.0624 131.7641 135.5520 139.3816 143.2068 146.9820 150.6647 154.4322 158.5095 162.8703 167.4749 172.2701 177.1916 177.8318 172.8721 167.9949 163.1915 158.0496 152.5152 146.6449 140.5179 134.2295 127.8794

(Continues)

Antenna measured values in anechoic chamber

267

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 31.9210 31.8309 31.7409 31.6508 31.5608 31.4707 31.3807 31.2906 31.2006 31.1106 31.0205 30.9305 30.8404 30.7504 30.6603 30.5703 30.4802 30.3902 30.3001 30.2101 30.1201 30.0300 29.9400 29.8499 29.7599 29.6698 29.5798 29.4897 29.3997 29.3097 29.2196 29.1296 29.0395 28.9495 28.8594 28.7694 28.6793 28.5893 28.4993 28.4092 28.3192 28.2291 28.1391 28.0490 27.9590 27.8689

Amplitude (dB)

Phase ( )

8.9309 8.9660 9.0062 9.0441 9.0488 9.0040 8.9006 8.7383 8.5249 8.2740 8.0028 7.7285 7.4506 7.1265 6.7523 6.3343 5.8842 5.4165 4.9460 4.4861 4.0483 3.6356 3.2118 2.7730 2.3256 1.8774 1.4365 1.0104 0.6061 0.2298 0.1152 0.4536 0.7933 1.1311 1.4633 1.7855 2.0933 2.3824 2.6484 2.8879 3.1154 3.3396 3.5596 3.7737 3.9798 4.1752

121.5580 115.3326 109.2371 102.9412 96.2229 89.2027 82.0251 74.8388 67.7733 60.9215 54.3307 48.0036 41.7974 35.4640 29.1433 22.9728 17.0522 11.4394 6.1535 1.1831 3.5052 7.9867 12.4545 16.8642 21.1491 25.2665 29.1945 32.9289 36.4787 39.8629 43.1150 46.3735 49.6406 52.8736 56.0417 59.1247 62.1117 65.0000 67.7935 70.5045 73.2446 76.0375 78.8503 81.6553 84.4310 87.1613

(Continues)

268

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 27.7789 27.6888 27.5988 27.5088 27.4187 27.3287 27.2386 27.1486 27.0585 26.9685 26.8784 26.7884 26.6983 26.6083 26.5183 26.4282 26.3382 26.2481 26.1581 26.0680 25.9780 25.8879 25.7979 25.7079 25.6178 25.5278 25.4377 25.3477 25.2576 25.1676 25.0775 24.9875 24.8974 24.8074 24.7174 24.6273 24.5373 24.4472 24.3572 24.2671 24.1771 24.0870 23.9970 23.9070 23.8169 23.7269

Amplitude (dB)

Phase ( )

4.3569 4.5214 4.6656 4.7976 4.9243 5.0464 5.1638 5.2756 5.3802 5.4754 5.5584 5.6262 5.6831 5.7339 5.7804 5.8231 5.8621 5.8963 5.9237 5.9419 5.9476 5.9426 5.9306 5.9134 5.8923 5.8673 5.8377 5.8016 5.7562 5.6979 5.6264 5.5448 5.4554 5.3595 5.2579 5.1497 5.0335 4.9060 4.7628 4.6015 4.4261 4.2399 4.0459 3.8457 3.6401 3.4283

89.8352 92.4470 94.9975 97.5924 100.2668 102.9917 105.7402 108.4882 111.2150 113.9031 116.5388 119.1157 121.7483 124.4740 127.2652 130.0945 132.9354 135.7627 138.5536 141.2876 143.9563 146.6923 149.5248 152.4281 155.3752 158.3388 161.2914 164.2068 167.0601 169.8523 172.7343 175.7215 178.7926 178.0757 174.9091 171.7347 168.5797 165.4710 162.3775 159.1346 155.7375 152.1990 148.5363 144.7712 140.9294 137.0389

(Continues)

Antenna measured values in anechoic chamber

269

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 23.6368 23.5468 23.4567 23.3667 23.2766 23.1866 23.0965 23.0065 22.9165 22.8264 22.7364 22.6463 22.5563 22.4662 22.3762 22.2861 22.1961 22.1061 22.0160 21.9260 21.8359 21.7459 21.6558 21.5658 21.4757 21.3857 21.2956 21.2056 21.1156 21.0255 20.9355 20.8454 20.7554 20.6653 20.5753 20.4852 20.3952 20.3052 20.2151 20.1251 20.0350 19.9450 19.8549 19.7649 19.6748 19.5848

Amplitude (dB)

Phase ( )

3.2077 2.9718 2.7214 2.4653 2.2121 1.9701 1.7465 1.5459 1.3705 1.2188 1.0920 1.0052 0.9726 1.0031 1.0986 1.2541 1.4585 1.6968 1.9576 2.2505 2.5746 2.9245 3.2919 3.6676 4.0419 4.4058 4.7512 5.0846 5.4150 5.7411 6.0611 6.3721 6.6710 6.9543 7.2186 7.4624 7.6950 7.9189 8.1343 8.3408 8.5372 8.7220 8.8932 9.0486 9.1902 9.3220

133.1290 129.0887 124.7520 120.1169 115.1940 110.0078 104.5975 99.0153 93.3221 87.5666 81.5238 75.1489 68.5360 61.8068 55.0959 48.5315 42.2191 36.2300 30.5269 24.9056 19.4266 14.1597 9.1548 4.4414 0.0301 4.0839 7.9176 11.6128 15.2428 18.7777 22.1942 25.4761 28.6133 31.6016 34.4414 37.1614 39.8811 42.5948 45.2802 47.9179 50.4917 52.9883 55.3974 57.7115 60.0063 62.3340

(Continues)

270

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 19.4947 19.4047 19.3147 19.2246 19.1346 19.0445 18.9545 18.8644 18.7744 18.6843 18.5943 18.5043 18.4142 18.3242 18.2341 18.1441 18.0540 17.9640 17.8739 17.7839 17.6938 17.6038 17.5138 17.4237 17.3337 17.2436 17.1536 17.0635 16.9735 16.8834 16.7934 16.7034 16.6133 16.5233 16.4332 16.3432 16.2531 16.1631 16.0730 15.9830 15.8929 15.8029 15.7129 15.6228 15.5328 15.4427

Amplitude (dB)

Phase ( )

9.4454 9.5606 9.6677 9.7659 9.8541 9.9308 9.9945 10.0470 10.0897 10.1237 10.1493 10.1664 10.1744 10.1722 10.1582 10.1306 10.0903 10.0383 9.9753 9.9013 9.8162 9.7190 9.6086 9.4819 9.3361 9.1721 8.9905 8.7914 8.5747 8.3396 8.0847 7.8070 7.4978 7.1558 6.7809 6.3723 5.9290 5.4492 4.9304 4.3685 3.7480 3.0606 2.3067 1.4903 0.6232 0.2699

64.6738 67.0061 69.3127 71.5764 73.7818 75.9147 77.9887 80.0986 82.2392 84.3929 86.5421 88.6692 90.7570 92.7883 94.7489 96.7161 98.7242 100.7597 102.8081 104.8539 106.8807 108.8713 110.8076 112.7166 114.6786 116.6890 118.7394 120.8195 122.9176 125.0197 127.1099 129.1861 131.3480 133.6232 136.0194 138.5448 141.2075 144.0158 146.9788 150.1093 153.5616 157.4965 162.0421 167.3635 173.6688 178.7942

(Continues)

Antenna measured values in anechoic chamber

271

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 15.3527 15.2626 15.1726 15.0825 14.9925 14.9025 14.8124 14.7224 14.6323 14.5423 14.4522 14.3622 14.2721 14.1821 14.0920 14.0020 13.9120 13.8219 13.7319 13.6418 13.5518 13.4617 13.3717 13.2816 13.1916 13.1016 13.0115 12.9215 12.8314 12.7414 12.6513 12.5613 12.4712 12.3812 12.2911 12.2011 12.1111 12.0210 11.9310 11.8409 11.7509 11.6608 11.5708 11.4807 11.3907 11.3007

Amplitude (dB)

Phase ( )

1.1424 1.9177 2.4881 2.7066 2.4717 1.8184 0.8886 0.1650 1.2357 2.2648 3.2338 4.1589 5.0350 5.8586 6.6289 7.3470 8.0146 8.6339 9.2089 9.7560 10.2785 10.7763 11.2493 11.6972 12.1203 12.5184 12.8920 13.2495 13.5942 13.9256 14.2431 14.5461 14.8342 15.1066 15.3629 15.6073 15.8434 16.0705 16.2882 16.4957 16.6925 16.8777 17.0507 17.2134 17.3692 17.5176

169.7708 159.0888 146.6229 132.5941 118.1923 104.8136 93.3187 83.8540 76.1523 69.8373 64.5447 59.9704 55.9785 52.4621 49.3321 46.5158 43.9538 41.5977 39.4034 37.3132 35.3162 33.4091 31.5869 29.8433 28.1716 26.5643 25.0131 23.4853 21.9750 20.4890 19.0323 17.6080 16.2178 14.8623 13.5410 12.2288 10.9126 9.6010 8.3013 7.0194 5.7599 4.5263 3.3215 2.1273 0.9210 0.2885

(Continues)

272

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 11.2106 11.1206 11.0305 10.9405 10.8504 10.7604 10.6703 10.5803 10.4902 10.4002 10.3102 10.2201 10.1301 10.0400 9.9500 9.8599 9.7699 9.6798 9.5898 9.4998 9.4097 9.3197 9.2296 9.1396 9.0495 8.9595 8.8694 8.7794 8.6893 8.5993 8.5093 8.4192 8.3292 8.2391 8.1491 8.0590 7.9690 7.8789 7.7889 7.6988 7.6088 7.5188 7.4287 7.3387 7.2486 7.1586

Amplitude (dB) 17.6582 17.7903 17.9133 18.0266 18.1291 18.2216 18.3070 18.3849 18.4550 18.5169 18.5699 18.6132 18.6462 18.6686 18.6821 18.6870 18.6827 18.6691 18.6455 18.6114 18.5658 18.5083 18.4390 18.3579 18.2648 18.1595 18.0415 17.9102 17.7647 17.6040 17.4260 17.2303 17.0166 16.7843 16.5326 16.2607 15.9672 15.6502 15.3041 14.9266 14.5161 14.0705 13.5874 13.0637 12.4960 11.8792

Phase ( ) 1.4936 2.6873 3.8638 5.0180 6.1453 7.2579 8.3846 9.5188 10.6531 11.7805 12.8949 13.9903 15.0614 16.1151 17.1835 18.2630 19.3473 20.4301 21.5054 22.5674 23.6102 24.6369 25.6809 26.7419 27.8154 28.8966 29.9806 31.0622 32.1360 33.2027 34.2969 35.4232 36.5804 37.7672 38.9818 40.2224 41.4867 42.7777 44.1372 45.5821 47.1238 48.7758 50.5546 52.4803 54.5784 56.8872

(Continues)

Antenna measured values in anechoic chamber

273

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 7.0685 6.9785 6.8884 6.7984 6.7084 6.6183 6.5283 6.4382 6.3482 6.2581 6.1681 6.0780 5.9880 5.8979 5.8079 5.7179 5.6278 5.5378 5.4477 5.3577 5.2676 5.1776 5.0875 4.9975 4.9075 4.8174 4.7274 4.6373 4.5473 4.4572 4.3672 4.2771 4.1871 4.0970 4.0070 3.9170 3.8269 3.7369 3.6468 3.5568 3.4667 3.3767 3.2866 3.1966 3.1066 3.0165

Amplitude (dB)

Phase ( )

11.1998 10.4507 9.6269 8.7253 7.7487 6.7122 5.6569 4.6690 3.8950 3.5485 3.7765 4.5239 5.5869 6.7685 7.9469 9.0661 10.1164 11.0953 12.0039 12.8463 13.6276 14.3529 15.0272 15.6554 16.2468 16.8058 17.3345 17.8350 18.3092 18.7584 19.1841 19.5879 19.9740 20.3440 20.6985 21.0380 21.3631 21.6740 21.9712 22.2552 22.5287 22.7924 23.0463 23.2905 23.5251 23.7500

59.5138 62.5635 66.1653 70.4975 75.8057 82.4205 90.7543 101.2369 114.2265 129.2652 144.7783 158.8735 170.5135 179.6765 173.1891 167.5759 163.0260 159.2665 156.1127 153.4294 151.1167 149.1004 147.3244 145.7431 144.2987 142.9683 141.7397 140.6030 139.5494 138.5715 137.6628 136.8160 136.0085 135.2354 134.4972 133.7938 133.1252 132.4911 131.8912 131.3236 130.7745 130.2417 129.7269 129.2313 128.7559 128.3015

(Continues)

274

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

2.9265 2.8364 2.7464 2.6563 2.5663 2.4762 2.3862 2.2961 2.2061 2.1161 2.0260 1.9360 1.8459 1.7559 1.6658 1.5758 1.4857 1.3957 1.3057 1.2156 1.1256 1.0355 0.9455 0.8554 0.7654 0.6753 0.5853 0.4952 0.4052 0.3152 0.2251 0.1351 0.0450 0.0450 0.1351 0.2251 0.3152 0.4052 0.4952 0.5853 0.6753 0.7654 0.8554 0.9455 1.0355 1.1256

23.9654 24.1715 24.3702 24.5618 24.7463 24.9235 25.0933 25.2557 25.4104 25.5577 25.6990 25.8345 25.9641 26.0876 26.2048 26.3156 26.4198 26.5174 26.6095 26.6964 26.7778 26.8536 26.9237 26.9879 27.0461 27.0981 27.1447 27.1861 27.2220 27.2525 27.2775 27.2969 27.3105 27.3183 27.3205 27.3170 27.3080 27.2935 27.2734 27.2478 27.2167 27.1799 27.1369 27.0879 27.0330 26.9722

127.8690 127.4577 127.0581 126.6691 126.2919 125.9276 125.5773 125.2416 124.9216 124.6169 124.3209 124.0329 123.7538 123.4843 123.2253 122.9774 122.7411 122.5166 122.2991 122.0882 121.8845 121.6885 121.5009 121.3221 121.1526 120.9924 120.8385 120.6905 120.5489 120.4140 120.2863 120.1661 120.0539 119.9499 119.8521 119.7603 119.6748 119.5958 119.5236 119.4586 119.4010 119.3511 119.3083 119.2726 119.2439 119.2222

(Continues)

Antenna measured values in anechoic chamber

275

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

1.2156 1.3057 1.3957 1.4857 1.5758 1.6658 1.7559 1.8459 1.9360 2.0260 2.1161 2.2061 2.2961 2.3862 2.4762 2.5663 2.6563 2.7464 2.8364 2.9265 3.0165 3.1066 3.1966 3.2866 3.3767 3.4667 3.5568 3.6468 3.7369 3.8269 3.9170 4.0070 4.0970 4.1871 4.2771 4.3672 4.4572 4.5473 4.6373 4.7274 4.8174 4.9075 4.9975 5.0875 5.1776 5.2676

26.9057 26.8336 26.7559 26.6726 26.5825 26.4856 26.3822 26.2724 26.1564 26.0344 25.9065 25.7726 25.6309 25.4815 25.3246 25.1602 24.9886 24.8101 24.6247 24.4321 24.2302 24.0189 23.7981 23.5680 23.3287 23.0801 22.8224 22.5550 22.2750 21.9821 21.6759 21.3562 21.0226 20.6748 20.3125 19.9343 19.5361 19.1164 18.6740 18.2074 17.7149 17.1946 16.6443 16.0599 15.4325 14.7564

119.2076 119.2003 119.2003 119.2080 119.2247 119.2504 119.2845 119.3269 119.3771 119.4349 119.4998 119.5727 119.6572 119.7531 119.8597 119.9760 120.1012 120.2343 120.3742 120.5217 120.6838 120.8603 121.0499 121.2512 121.4625 121.6819 121.9074 122.1398 122.3898 122.6572 122.9406 123.2381 123.5474 123.8659 124.1902 124.5211 124.8731 125.2469 125.6416 126.0558 126.4877 126.9348 127.3934 127.8648 128.3684 128.9099

(Continues)

276

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

5.3577 5.4477 5.5378 5.6278 5.7179 5.8079 5.8979 5.9880 6.0780 6.1681 6.2581 6.3482 6.4382 6.5283 6.6183 6.7084 6.7984 6.8884 6.9785 7.0685 7.1586 7.2486 7.3387 7.4287 7.5188 7.6088 7.6988 7.7889 7.8789 7.9690 8.0590 8.1491 8.2391 8.3292 8.4192 8.5093 8.5993 8.6893 8.7794 8.8694 8.9595 9.0495 9.1396 9.2296 9.3197 9.4097

14.0249 13.2301 12.3616 11.4059 10.3452 9.1514 7.7738 6.1476 4.1677 1.6435 1.8223 7.2021 13.5599 6.9746 1.8797 1.3919 3.7563 5.5911 7.0793 8.3222 9.3815 10.2998 11.1147 11.8445 12.5016 13.0957 13.6342 14.1231 14.5669 14.9707 15.3427 15.6862 16.0030 16.2945 16.5622 16.8071 17.0299 17.2317 17.4156 17.5829 17.7344 17.8707 17.9923 18.0995 18.1928 18.2721

129.4940 130.1265 130.8153 131.5708 132.4078 133.3553 134.4906 135.9265 137.8817 140.8578 146.3033 160.4921 134.9403 75.1399 61.8093 56.5005 53.5279 51.5348 50.0429 48.8411 47.8220 46.9177 46.0695 45.2648 44.4976 43.7634 43.0585 42.3798 41.7246 41.0856 40.4381 39.7840 39.1279 38.4735 37.8238 37.1811 36.5474 35.9203 35.2772 34.6188 33.9495 33.2737 32.5951 31.9172 31.2432 30.5730

(Continues)

Antenna measured values in anechoic chamber

277

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

9.4998 9.5898 9.6798 9.7699 9.8599 9.9500 10.0400 10.1301 10.2201 10.3102 10.4002 10.4902 10.5803 10.6703 10.7604 10.8504 10.9405 11.0305 11.1206 11.2106 11.3007 11.3907 11.4807 11.5708 11.6608 11.7509 11.8409 11.9310 12.0210 12.1111 12.2011 12.2911 12.3812 12.4712 12.5613 12.6513 12.7414 12.8314 12.9215 13.0115 13.1016 13.1916 13.2816 13.3717 13.4617 13.5518

18.3378 18.3907 18.4315 18.4607 18.4789 18.4866 18.4840 18.4712 18.4466 18.4104 18.3635 18.3064 18.2400 18.1647 18.0812 17.9897 17.8875 17.7738 17.6495 17.5154 17.3724 17.2210 17.0620 16.8960 16.7197 16.5312 16.3313 16.1207 15.9001 15.6702 15.4317 15.1852 14.9276 14.6552 14.3683 14.0675 13.7530 13.4254 13.0849 12.7319 12.3634 11.9725 11.5585 11.1210 10.6592 10.1724

29.8836 29.1733 28.4466 27.7077 26.9609 26.2104 25.4605 24.7134 23.9449 23.1500 22.3331 21.4987 20.6516 19.7970 18.9405 18.0869 17.2116 16.3046 15.3703 14.4132 13.4389 12.4535 11.4639 10.4781 9.4736 8.4337 7.3618 6.2627 5.1417 4.0056 2.8625 1.7220 0.5698 0.6201 1.8458 3.1043 4.3913 5.7011 7.0261 8.3566 9.6973 11.0805 12.5103 13.9889 15.5177 17.0970

(Continues)

278

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

13.6418 13.7319 13.8219 13.9120 14.0020 14.0920 14.1821 14.2721 14.3622 14.4522 14.5423 14.6323 14.7224 14.8124 14.9025 14.9925 15.0825 15.1726 15.2626 15.3527 15.4427 15.5328 15.6228 15.7129 15.8029 15.8929 15.9830 16.0730 16.1631 16.2531 16.3432 16.4332 16.5233 16.6133 16.7034 16.7934 16.8834 16.9735 17.0635 17.1536 17.2436 17.3337 17.4237 17.5138 17.6038 17.6938

9.6599 9.1204 8.5499 7.9338 7.2666 6.5432 5.7576 4.9026 3.9699 2.9492 1.8264 0.5556 0.8964 2.5474 4.3683 6.1670 7.3864 7.3223 6.0973 4.4303 2.7573 1.2324 0.1159 1.2989 2.3369 3.2490 4.0517 4.7675 5.4222 6.0222 6.5721 7.0752 7.5344 7.9515 8.3281 8.6673 8.9808 9.2713 9.5397 9.7863 10.0112 10.2143 10.3951 10.5532 10.6927 10.8168

18.7259 20.4012 22.1255 23.9396 25.8706 27.9486 30.2110 32.7065 35.4993 38.6772 42.3673 46.8351 52.5196 60.1490 70.9514 86.7173 108.4998 132.8843 153.3681 168.1473 178.5822 173.8659 168.1682 163.6904 160.0445 156.9862 154.3561 151.9957 149.7722 147.6743 145.6934 143.8207 142.0478 140.3664 138.7684 137.2319 135.6844 134.1290 132.5779 131.0408 129.5252 128.0370 126.5807 125.1575 123.7105 122.2212

(Continues)

Antenna measured values in anechoic chamber

279

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

17.7839 17.8739 17.9640 18.0540 18.1441 18.2341 18.3242 18.4142 18.5043 18.5943 18.6843 18.7744 18.8644 18.9545 19.0445 19.1346 19.2246 19.3147 19.4047 19.4947 19.5848 19.6748 19.7649 19.8549 19.9450 20.0350 20.1251 20.2151 20.3052 20.3952 20.4852 20.5753 20.6653 20.7554 20.8454 20.9355 21.0255 21.1156 21.2056 21.2956 21.3857 21.4757 21.5658 21.6558 21.7459 21.8359

10.9261 11.0212 11.1023 11.1691 11.2213 11.2582 11.2796 11.2869 11.2816 11.2645 11.2363 11.1974 11.1478 11.0871 11.0144 10.9275 10.8275 10.7160 10.5942 10.4629 10.3226 10.1735 10.0151 9.8437 9.6569 9.4566 9.2448 9.0227 8.7915 8.5518 8.3036 8.0448 7.7673 7.4715 7.1602 6.8355 6.4995 6.1541 5.8001 5.4378 5.0588 4.6553 4.2322 3.7952 3.3507 2.9056

Phase ( ) 120.7024 119.1661 117.6229 116.0827 114.5546 113.0468 111.5327 109.9553 108.3253 106.6549 104.9564 103.2426 101.5262 99.8202 98.1278 96.3718 94.5414 92.6481 90.7047 88.7254 86.7260 84.7234 82.7365 80.7194 78.6060 76.4040 74.1239 71.7786 69.3841 66.9594 64.5266 62.0934 59.5541 56.8855 54.0892 51.1705 48.1391 45.0097 41.8022 38.5424 35.1701 31.5648 27.7038 23.5703 19.1546 14.4576

(Continues)

280

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

21.9260 22.0160 22.1061 22.1961 22.2861 22.3762 22.4662 22.5563 22.6463 22.7364 22.8264 22.9165 23.0065 23.0965 23.1866 23.2766 23.3667 23.4567 23.5468 23.6368 23.7269 23.8169 23.9070 23.9970 24.0870 24.1771 24.2671 24.3572 24.4472 24.5373 24.6273 24.7174 24.8074 24.8974 24.9875 25.0775 25.1676 25.2576 25.3477 25.4377 25.5278 25.6178 25.7079 25.7979 25.8879 25.9780

2.4668 2.0405 1.6299 1.2237 0.8346 0.4841 0.1943 0.0153 0.1328 0.1563 0.0949 0.0338 0.2231 0.4726 0.7755 1.1200 1.4914 1.8745 2.2548 2.6198 2.9694 3.3105 3.6432 3.9661 4.2768 4.5723 4.8491 5.1034 5.3321 5.5411 5.7350 5.9154 6.0829 6.2374 6.3780 6.5032 6.6111 6.6998 6.7711 6.8278 6.8721 6.9053 6.9281 6.9408 6.9428 6.9330

9.4933 4.2907 1.1245 6.9522 13.2757 20.0829 27.3114 34.8423 42.5084 50.1173 57.4854 64.5845 71.6063 78.4437 84.9843 91.1413 96.8605 102.1198 106.9230 111.2930 115.4239 119.4365 123.3044 127.0063 130.5269 133.8566 136.9911 139.9306 142.6902 145.4225 148.1556 150.8679 153.5394 156.1520 158.6898 161.1390 163.4881 165.7537 168.0582 170.4042 172.7746 175.1520 177.5186 179.8562 177.8530 175.6266

(Continues)

Antenna measured values in anechoic chamber

281

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

26.0680 26.1581 26.2481 26.3382 26.4282 26.5183 26.6083 26.6983 26.7884 26.8784 26.9685 27.0585 27.1486 27.2386 27.3287 27.4187 27.5088 27.5988 27.6888 27.7789 27.8689 27.9590 28.0490 28.1391 28.2291 28.3192 28.4092 28.4993 28.5893 28.6793 28.7694 28.8594 28.9495 29.0395 29.1296 29.2196 29.3097 29.3997 29.4897 29.5798 29.6698 29.7599 29.8499 29.9400 30.0300 30.1201

6.9088 6.8677 6.8124 6.7457 6.6695 6.5855 6.4947 6.3975 6.2938 6.1790 6.0474 5.9022 5.7466 5.5834 5.4150 5.2434 5.0698 4.8949 4.7125 4.5136 4.3019 4.0810 3.8546 3.6258 3.3974 3.1716 2.9495 2.7237 2.4812 2.2255 1.9605 1.6905 1.4190 1.1497 0.8853 0.6277 0.3697 0.0907 0.2068 0.5189 0.8413 1.1697 1.4999 1.8280 2.1509 2.4734

Phase ( ) 173.4401 171.1873 168.8718 166.5068 164.1075 161.6916 159.2788 156.8909 154.5512 152.2289 149.8254 147.3438 144.7940 142.1897 139.5490 136.8939 134.2508 131.6499 129.0616 126.3879 123.6287 120.7911 117.8876 114.9369 111.9641 109.0006 106.0843 103.1982 100.2409 97.2068 94.0993 90.9288 87.7135 84.4801 81.2639 78.1088 75.0210 71.8957 68.7159 65.4774 62.1839 58.8483 55.4939 52.1555 48.8799 45.7010

(Continues)

282

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 30.2101 30.3001 30.3902 30.4802 30.5703 30.6603 30.7504 30.8404 30.9305 31.0205 31.1106 31.2006 31.2906 31.3807 31.4707 31.5608 31.6508 31.7409 31.8309 31.9210 32.0110 32.1011 32.1911 32.2811 32.3712 32.4612 32.5513 32.6413 32.7314 32.8214 32.9115 33.0015 33.0915 33.1816 33.2716 33.3617 33.4517 33.5418 33.6318 33.7219 33.8119 33.9020 33.9920 34.0820 34.1721 34.2621

Amplitude (dB) 2.8273 3.2148 3.6325 4.0762 4.5413 5.0223 5.5137 6.0099 6.5089 7.0593 7.6800 8.3715 9.1325 9.9599 10.8472 11.7834 12.7520 13.7310 14.7680 15.9192 17.1113 18.1728 18.8203 18.8062 18.1641 17.1733 16.1046 15.1008 14.1529 13.2532 12.4050 11.6130 10.8823 10.2172 9.6212 9.0974 8.6485 8.2554 7.8951 7.5626 7.2556 6.9733 6.7167 6.4881 6.2904 6.1276

Phase ( ) 42.5185 39.2908 35.9974 32.6251 29.1706 25.6431 22.0679 18.4899 14.9696 11.4226 7.7447 3.8591 0.3167 4.8707 9.8928 15.4669 21.6525 28.4563 36.1180 45.3653 56.9276 71.4502 88.7639 107.0692 123.7290 137.2629 147.7283 155.9703 163.2640 169.7993 175.6311 179.1798 174.5683 170.4683 166.8166 163.5546 160.6293 157.7777 154.8445 151.8873 148.9536 146.0828 143.3063 140.6483 138.1272 135.7561

(Continues)

Antenna measured values in anechoic chamber

283

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 34.3522 34.4422 34.5323 34.6223 34.7124 34.8024 34.8924 34.9825 35.0725 35.1626 35.2526 35.3427 35.4327 35.5228 35.6128 35.7029 35.7929 35.8829 35.9730 36.0630 36.1531 36.2431 36.3332 36.4232 36.5133 36.6033 36.6933 36.7834 36.8734 36.9635 37.0535 37.1436 37.2336 37.3237 37.4137 37.5038 37.5938 37.6838 37.7739 37.8639 37.9540 38.0440 38.1341 38.2241 38.3142 38.4042

Amplitude (dB) 6.0030 5.9085 5.8380 5.7877 5.7552 5.7392 5.7397 5.7576 5.7951 5.8555 5.9440 6.0638 6.2101 6.3785 6.5654 6.7679 6.9840 7.2129 7.4553 7.7133 7.9994 8.3227 8.6758 9.0512 9.4406 9.8357 10.2287 10.6133 10.9857 11.3458 11.7088 12.0729 12.4148 12.7100 12.9363 13.0782 13.1311 13.1023 13.0102 12.8798 12.7255 12.5375 12.3120 12.0513 11.7622 11.4548

Phase ( ) 133.5146 131.1834 128.7430 126.2241 123.6577 121.0740 118.5030 115.9738 113.5147 111.1529 108.8344 106.3692 103.7573 101.0122 98.1515 95.1974 92.1770 89.1224 86.0704 83.0617 79.9861 76.6441 73.0174 69.0952 64.8764 60.3732 55.6143 50.6467 45.5349 40.3580 34.9126 28.9293 22.4082 15.3999 8.0150 0.4197 7.1856 14.6020 21.6667 28.2748 34.6562 41.0485 47.3552 53.4828 59.3532 64.9107

(Continues)

284

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 38.4942 38.5843 38.6743 38.7644 38.8544 38.9445 39.0345 39.1246 39.2146 39.3047 39.3947 39.4847 39.5748 39.6648 39.7549 39.8449 39.9350 40.0250 40.1151 40.2051 40.2951 40.3852 40.4752 40.5653 40.6553 40.7454 40.8354 40.9255 41.0155 41.1056 41.1956 41.2856 41.3757 41.4657 41.5558 41.6458 41.7359 41.8259 41.9160 42.0060 42.0960 42.1861 42.2761 42.3662 42.4562 42.5463

Amplitude (dB) 11.1409 10.8325 10.5411 10.2774 10.0417 9.8166 9.5967 9.3792 9.1634 8.9506 8.7432 8.5450 8.3604 8.1942 8.0496 7.9152 7.7845 7.6537 7.5211 7.3861 7.2501 7.1152 6.9849 6.8633 6.7551 6.6568 6.5610 6.4646 6.3661 6.2650 6.1621 6.0593 5.9595 5.8660 5.7831 5.7140 5.6533 5.5975 5.5446 5.4937 5.4445 5.3978 5.3550 5.3180 5.2896 5.2730

Phase ( ) 70.1247 74.9877 79.5107 83.7190 87.7657 91.8775 96.0181 100.1452 104.2203 108.2098 112.0867 115.8315 119.4324 122.8847 126.2313 129.6871 133.2481 136.8730 140.5209 144.1535 147.7357 151.2377 154.6352 157.9105 161.0527 164.2110 167.4696 170.7928 174.1447 177.4909 179.2008 175.9588 172.8071 169.7653 166.8477 164.0343 161.1632 158.2287 155.2548 152.2656 149.2846 146.3341 143.4346 140.6042 137.8584 135.2101

(Continues)

Antenna measured values in anechoic chamber

285

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 42.6363 42.7264 42.8164 42.9065 42.9965 43.0865 43.1766 43.2666 43.3567 43.4467 43.5368 43.6268 43.7169 43.8069 43.8969 43.9870 44.0770 44.1671 44.2571 44.3472 44.4372 44.5273 44.6173 44.7074 44.7974 44.8874 44.9775 45.0675 45.1576 45.2476 45.3377 45.4277 45.5178 45.6078 45.6978 45.7879 45.8779 45.9680 46.0580 46.1481 46.2381 46.3282 46.4182 46.5083 46.5983 46.6883

Amplitude (dB) 5.2707 5.2801 5.2985 5.3236 5.3537 5.3876 5.4249 5.4659 5.5112 5.5626 5.6220 5.6935 5.7779 5.8715 5.9709 6.0731 6.1756 6.2765 6.3747 6.4699 6.5627 6.6545 6.7498 6.8521 6.9565 7.0587 7.1547 7.2413 7.3164 7.3789 7.4291 7.4684 7.4996 7.5270 7.5551 7.5809 7.6012 7.6135 7.6164 7.6095 7.5934 7.5698 7.5412 7.5108 7.4825 7.4612

Phase ( ) 132.6100 129.9320 127.1830 124.3775 121.5310 118.6605 115.7838 112.9189 110.0835 107.2948 104.5687 101.8464 99.0124 96.0714 93.0335 89.9121 86.7241 83.4892 80.2299 76.9702 73.7354 70.5503 67.3724 64.0621 60.6232 57.0712 53.4272 49.7174 45.9726 42.2265 38.5143 34.8709 31.3290 27.8900 24.3956 20.8308 17.2165 13.5774 9.9408 6.3356 2.7905 0.6671 4.0121 7.2227 10.2807 13.2730

(Continues)

286

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 46.7784 46.8684 46.9585 47.0485 47.1386 47.2286 47.3187 47.4087 47.4987 47.5888 47.6788 47.7689 47.8589 47.9490 48.0390 48.1291 48.2191 48.3092 48.3992 48.4892 48.5793 48.6693 48.7594 48.8494 48.9395 49.0295 49.1196 49.2096 49.2996 49.3897 49.4797 49.5698 49.6598 49.7499 49.8399 49.9300 50.0200 50.1101 50.2001 50.2901 50.3802 50.4702 50.5603 50.6503 50.7404 50.8304

Amplitude (dB) 7.4468 7.4369 7.4295 7.4233 7.4176 7.4120 7.4070 7.4036 7.4031 7.4075 7.4196 7.4450 7.4831 7.5313 7.5870 7.6482 7.7132 7.7807 7.8499 7.9205 7.9926 8.0669 8.1454 8.2368 8.3410 8.4547 8.5748 8.6984 8.8228 8.9461 9.0663 9.1824 9.2940 9.4012 9.5052 9.6150 9.7335 9.8573 9.9831 10.1079 10.2291 10.3445 10.4528 10.5532 10.6459 10.7316

Phase ( ) 16.2946 19.3354 22.3833 25.4241 28.4424 31.4222 34.3469 37.2005 39.9678 42.6345 45.1942 47.7577 50.3676 53.0201 55.7088 58.4253 61.1586 63.8959 66.6227 69.3231 71.9801 74.5762 77.1024 79.6693 82.3157 85.0396 87.8359 90.6961 93.6082 96.5569 99.5236 102.4870 105.4236 108.3081 111.1154 113.9363 116.8433 119.8325 122.8961 126.0224 129.1955 132.3959 135.6008 138.7848 141.9208 144.9809

(Continues)

Antenna measured values in anechoic chamber

287

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 50.9205 51.0105 51.1006 51.1906 51.2806 51.3707 51.4607 51.5508 51.6408 51.7309 51.8209 51.9110 52.0010 52.0910 52.1811 52.2711 52.3612 52.4512 52.5413 52.6313 52.7214 52.8114 52.9015 52.9915 53.0815 53.1716 53.2616 53.3517 53.4417 53.5318 53.6218 53.7119 53.8019 53.8919 53.9820 54.0720 54.1621 54.2521 54.3422 54.4322 54.5223 54.6123 54.7024 54.7924 54.8824 54.9725

Amplitude (dB) 10.8123 10.8923 10.9806 11.0756 11.1748 11.2759 11.3771 11.4768 11.5740 11.6682 11.7595 11.8485 11.9366 12.0256 12.1218 12.2334 12.3585 12.4945 12.6392 12.7904 12.9459 13.1042 13.2636 13.4230 13.5817 13.7398 13.8975 14.0598 14.2403 14.4367 14.6452 14.8616 15.0816 15.3006 15.5144 15.7187 15.9101 16.0856 16.2434 16.3831 16.5056 16.6231 16.7387 16.8463 16.9402 17.0151

Phase ( ) 147.9370 150.7969 153.6889 156.6265 159.6049 162.6161 165.6494 168.6913 171.7257 174.7348 177.6992 179.4010 176.5866 173.8779 171.2428 168.5728 165.8586 163.0971 160.2881 157.4339 154.5399 151.6145 148.6688 145.7167 142.7744 139.8606 136.9953 134.1653 131.2472 128.2161 125.0618 121.7779 118.3627 114.8200 111.1595 107.3981 103.5591 99.6725 95.7737 91.9020 88.0973 84.2468 80.2566 76.1357 71.9016 67.5810

(Continues)

288

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 55.0625 55.1526 55.2426 55.3327 55.4227 55.5128 55.6028 55.6928 55.7829 55.8729 55.9630 56.0530 56.1431 56.2331 56.3232 56.4132 56.5033 56.5933 56.6833 56.7734 56.8634 56.9535 57.0435 57.1336 57.2236 57.3137 57.4037 57.4937 57.5838 57.6738 57.7639 57.8539 57.9440 58.0340 58.1241 58.2141 58.3042 58.3942 58.4842 58.5743 58.6643 58.7544 58.8444 58.9345 59.0245 59.1146

Amplitude (dB) 17.0670 17.0931 17.0925 17.0657 17.0148 16.9436 16.8568 16.7598 16.6586 16.5591 16.4615 16.3639 16.2653 16.1651 16.0633 15.9604 15.8570 15.7546 15.6544 15.5583 15.4682 15.3860 15.3139 15.2546 15.2098 15.1778 15.1569 15.1458 15.1434 15.1485 15.1605 15.1788 15.2029 15.2328 15.2685 15.3101 15.3581 15.4131 15.4789 15.5614 15.6589 15.7698 15.8925 16.0252 16.1667 16.3153

Phase ( ) 63.2084 58.8247 54.4745 50.2034 46.0547 42.0669 38.2723 34.6955 31.3515 28.1046 24.8799 21.6878 18.5403 15.4499 12.4295 9.4913 6.6470 3.9069 1.2801 1.2260 3.6050 5.8520 7.9633 9.9750 11.9852 13.9982 16.0123 18.0249 20.0326 22.0310 24.0148 25.9783 27.9152 29.8186 31.6814 33.4960 35.2546 36.9491 38.6079 40.3020 42.0364 43.8127 45.6317 47.4931 49.3954 51.3357

(Continues)

Antenna measured values in anechoic chamber

289

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 59.2046 59.2946 59.3847 59.4747 59.5648 59.6548 59.7449 59.8349 59.9250 60.0150 60.1051 60.1951 60.2851 60.3752 60.4652 60.5553 60.6453 60.7354 60.8254 60.9155 61.0055 61.0955 61.1856 61.2756 61.3657 61.4557 61.5458 61.6358 61.7259 61.8159 61.9060 61.9960 62.0860 62.1761 62.2661 62.3562 62.4462 62.5363 62.6263 62.7164 62.8064 62.8965 62.9865 63.0765 63.1666 63.2566

Amplitude (dB) 16.4697 16.6287 16.7910 16.9557 17.1218 17.2888 17.4563 17.6247 17.8050 17.9997 18.2068 18.4240 18.6492 18.8800 19.1139 19.3484 19.5811 19.8095 20.0314 20.2450 20.4488 20.6418 20.8238 20.9950 21.1649 21.3379 21.5115 21.6830 21.8498 22.0091 22.1588 22.2968 22.4215 22.5318 22.6273 22.7083 22.7756 22.8307 22.8759 22.9137 22.9472 22.9838 23.0268 23.0745 23.1255 23.1784

Phase ( ) 53.3101 55.3130 57.3372 59.3743 61.4139 63.4446 65.4530 67.4295 69.4510 71.5540 73.7444 76.0268 78.4042 80.8777 83.4462 86.1060 88.8503 91.6690 94.5485 97.4717 100.4183 103.3649 106.2858 109.1539 112.0398 115.0119 118.0714 121.2164 124.4421 127.7403 131.0993 134.5041 137.9368 141.3770 144.8028 148.1912 151.5194 154.7655 157.9095 160.9332 163.8216 166.6478 169.4765 172.3116 175.1548 178.0063

(Continues)

290

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 63.3467 63.4367 63.5268 63.6168 63.7069 63.7969 63.8869 63.9770 64.0670 64.1571 64.2471 64.3372 64.4272 64.5173 64.6073 64.6973 64.7874 64.8774 64.9675 65.0575 65.1476 65.2376 65.3277 65.4177 65.5078 65.5978 65.6878 65.7779 65.8679 65.9580 66.0480 66.1381 66.2281 66.3182 66.4082 66.4983 66.5883 66.6783 66.7684 66.8584 66.9485 67.0385 67.1286 67.2186 67.3087 67.3987

Amplitude (dB) 23.2320 23.2851 23.3366 23.3858 23.4320 23.4748 23.5141 23.5501 23.5830 23.6135 23.6424 23.6708 23.7000 23.7377 23.7863 23.8431 23.9060 23.9724 24.0401 24.1070 24.1710 24.2304 24.2835 24.3291 24.3661 24.3941 24.4127 24.4222 24.4231 24.4163 24.4031 24.3847 24.3667 24.3505 24.3342 24.3163 24.2953 24.2701 24.2398 24.2038 24.1619 24.1141 24.0605 24.0017 23.9384 23.8715

Phase ( ) 179.1353 176.2731 173.4117 170.5569 167.7156 164.8955 162.1050 159.3529 156.6479 153.9990 151.4146 148.9026 146.4691 144.0390 141.5617 139.0331 136.4509 133.8149 131.1270 128.3912 125.6136 122.8022 119.9672 117.1203 114.2747 111.4443 108.6441 105.8888 103.1930 100.5706 98.0344 95.5956 93.1763 90.7201 88.2327 85.7214 83.1948 80.6623 78.1342 75.6211 73.1341 70.6840 68.2814 65.9363 63.6578 61.4544

(Continues)

Antenna measured values in anechoic chamber

291

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 67.4887 67.5788 67.6688 67.7589 67.8489 67.9390 68.0290 68.1191 68.2091 68.2991 68.3892 68.4792 68.5693 68.6593 68.7494 68.8394 68.9295 69.0195 69.1096 69.1996 69.2896 69.3797 69.4697 69.5598 69.6498 69.7399 69.8299 69.9200 70.0100 70.1001 70.1901 70.2801 70.3702 70.4602 70.5503 70.6403 70.7304 70.8204 70.9105 71.0005 71.0905 71.1806 71.2706 71.3607 71.4507 71.5408

Amplitude (dB) 23.8020 23.7310 23.6598 23.5895 23.5216 23.4571 23.3975 23.3434 23.2943 23.2498 23.2093 23.1727 23.1396 23.1098 23.0833 23.0598 23.0395 23.0223 23.0084 22.9977 22.9906 22.9871 22.9875 22.9920 23.0010 23.0148 23.0336 23.0579 23.0882 23.1274 23.1756 23.2323 23.2974 23.3703 23.4509 23.5387 23.6335 23.7351 23.8432 23.9576 24.0780 24.2042 24.3360 24.4732 24.6157 24.7633

Phase ( ) 59.3333 57.3009 55.3625 53.5224 51.7843 50.1507 48.6035 47.0723 45.5525 44.0458 42.5541 41.0795 39.6240 38.1898 36.7792 35.3945 34.0380 32.7119 31.4184 30.1598 28.9383 27.7560 26.6149 25.5172 24.4650 23.4601 22.5048 21.6009 20.7468 19.9114 19.0863 18.2702 17.4619 16.6603 15.8646 15.0740 14.2882 13.5067 12.7293 11.9559 11.1866 10.4215 9.6611 8.9059 8.1565 7.4138

(Continues)

292

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 71.6308 71.7209 71.8109 71.9009 71.9910 72.0810 72.1711 72.2611 72.3512 72.4412 72.5313 72.6213 72.7114 72.8014 72.8914 72.9815 73.0715 73.1616 73.2516 73.3417 73.4317 73.5218 73.6118 73.7019 73.7919 73.8819 73.9720 74.0620 74.1521 74.2421 74.3322 74.4222 74.5123 74.6023 74.6923 74.7824 74.8724 74.9625 75.0525 75.1426 75.2326 75.3227 75.4127 75.5028 75.5928 75.6828

Amplitude (dB) 24.9157 25.0729 25.2347 25.4010 25.5716 25.7464 25.9254 26.1120 26.3127 26.5277 26.7569 27.0005 27.2586 27.5314 27.8191 28.1219 28.4402 28.7742 29.1244 29.4911 29.8747 30.2758 30.6949 31.1324 31.5889 32.0649 32.5610 33.0775 33.6151 34.1737 34.7536 35.3544 35.9755 36.6154 37.2720 37.9557 38.6874 39.4662 40.2863 41.1343 41.9832 42.7856 43.4690 43.9420 44.1215 43.9740

Phase ( ) 6.6788 5.9526 5.2368 4.5327 3.8423 3.1676 2.5107 1.8686 1.2310 0.5948 0.0428 0.6848 1.3341 1.9938 2.6670 3.3573 4.0681 4.8036 5.5680 6.3660 7.2029 8.0846 9.0176 10.0095 11.0688 12.2050 13.4292 14.7542 16.1944 17.7668 19.4907 21.3880 23.4840 25.8071 28.3886 31.2914 34.6353 38.5419 43.1649 48.6928 55.3389 63.3028 72.6809 83.3244 94.7268 106.0949

(Continues)

Antenna measured values in anechoic chamber

293

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 75.7729 75.8629 75.9530 76.0430 76.1331 76.2231 76.3132 76.4032 76.4932 76.5833 76.6733 76.7634 76.8534 76.9435 77.0335 77.1236 77.2136 77.3036 77.3937 77.4837 77.5738 77.6638 77.7539 77.8439 77.9340 78.0240 78.1141 78.2041 78.2941 78.3842 78.4742 78.5643 78.6543 78.7444 78.8344 78.9245 79.0145 79.1046 79.1946 79.2846 79.3747 79.4647 79.5548 79.6448 79.7349 79.8249

Amplitude (dB) 43.5361 42.8920 42.1325 41.3289 40.5270 39.7529 39.0196 38.3322 37.6917 37.0966 36.5446 36.0330 35.5589 35.1196 34.7126 34.3355 33.9862 33.6630 33.3640 33.0879 32.8333 32.5990 32.3831 32.1776 31.9807 31.7922 31.6117 31.4391 31.2741 31.1166 30.9664 30.8233 30.6871 30.5577 30.4349 30.3187 30.2089 30.1054 30.0081 29.9168 29.8315 29.7521 29.6785 29.6105 29.5483 29.4916

Phase ( ) 116.6443 125.8856 133.6853 140.1472 145.4725 149.8734 153.5359 156.6110 159.2173 161.4465 163.3696 165.0415 166.5052 167.7946 168.9366 169.9527 170.8603 171.6736 172.4043 173.0620 173.6546 174.1890 174.6749 175.1490 175.6176 176.0797 176.5345 176.9811 177.4191 177.8479 178.2672 178.6768 179.0763 179.4658 179.8450 179.7861 179.4274 179.0790 178.7408 178.4128 178.0948 177.7869 177.4889 177.2007 176.9222 176.6533

(Continues)

294

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 79.9150 80.0050 80.0950 80.1851 80.2751 80.3652 80.4552 80.5453 80.6353 80.7254 80.8154 80.9054 80.9955 81.0855 81.1756 81.2656 81.3557 81.4457 81.5358 81.6258 81.7159 81.8059 81.8960 81.9860 82.0760 82.1661 82.2561 82.3462 82.4362 82.5263 82.6163 82.7064 82.7964 82.8864 82.9765 83.0665 83.1566 83.2466 83.3367 83.4267 83.5168 83.6068 83.6968 83.7869 83.8769 83.9670

Amplitude (dB) 29.4404 29.3946 29.3542 29.3192 29.2894 29.2648 29.2454 29.2312 29.2221 29.2180 29.2190 29.2250 29.2360 29.2519 29.2729 29.2987 29.3295 29.3653 29.4060 29.4515 29.5021 29.5588 29.6249 29.7002 29.7842 29.8765 29.9769 30.0849 30.2003 30.3229 30.4523 30.5883 30.7307 30.8792 31.0338 31.1941 31.3600 31.5314 31.7080 31.8898 32.0765 32.2680 32.4642 32.6650 32.8702 33.0796

Phase ( ) 176.3938 176.1437 175.9029 175.6712 175.4486 175.2349 175.0300 174.8339 174.6465 174.4678 174.2975 174.1358 173.9825 173.8376 173.7011 173.5730 173.4532 173.3417 173.2387 173.1440 173.0578 172.9793 172.9061 172.8378 172.7738 172.7140 172.6580 172.6054 172.5562 172.5099 172.4665 172.4258 172.3875 172.3516 172.3179 172.2863 172.2566 172.2288 172.2027 172.1783 172.1555 172.1341 172.1143 172.0957 172.0785 172.0625

(Continues)

Antenna measured values in anechoic chamber

295

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 84.0570 84.1471 84.2371 84.3272 84.4172 84.5073 84.5973 84.6873 84.7774 84.8674 84.9575 85.0475 85.1376 85.2276 85.3177 85.4077 85.4977 85.5878 85.6778 85.7679 85.8579 85.9480 86.0380 86.1281 86.2181 86.3082 86.3982 86.4882 86.5783 86.6683 86.7584 86.8484 86.9385 87.0285 87.1186 87.2086 87.2987 87.3887 87.4787 87.5688 87.6588 87.7489 87.8389 87.9290 88.0190 88.1091

Amplitude (dB) 33.2932 33.5109 33.7324 33.9577 34.1867 34.4193 34.6552 34.8945 35.1369 35.3823 35.6307 35.8818 36.1356 36.3919 36.6506 36.9115 37.1745 37.4393 37.7060 37.9742 38.2438 38.5147 38.7866 39.0594 39.3328 39.6067 39.8807 40.1548 40.4285 40.7018 40.9743 41.2458 41.5159 41.7844 42.0510 42.3153 42.5770 42.8358 43.0914 43.3433 43.5912 43.8347 44.0734 44.3068 44.5347 44.7565

Phase ( ) 172.0478 172.0341 172.0216 172.0101 171.9997 171.9902 171.9816 171.9740 171.9672 171.9612 171.9561 171.9517 171.9481 171.9452 171.9430 171.9414 171.9405 171.9402 171.9406 171.9415 171.9429 171.9448 171.9473 171.9502 171.9536 171.9574 171.9616 171.9662 171.9712 171.9765 171.9821 171.9880 171.9942 172.0005 172.0071 172.0139 172.0208 172.0279 172.0351 172.0423 172.0496 172.0569 172.0642 172.0714 172.0786 172.0857

(Continues)

296

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 88.1991 88.2891 88.3792 88.4692 88.5593 88.6493 88.7394 88.8294 88.9195 89.0095 89.0995 89.1896 89.2796 89.3697 89.4597 89.5498 89.6398 89.7299 89.8199 89.9100 90.0000

Amplitude (dB) 44.9719 45.1803 45.3814 45.5748 45.7600 45.9365 46.1040 46.2620 46.4102 46.5481 46.6754 46.7917 46.8968 46.9902 47.0718 47.1412 47.1983 47.2429 47.2748 47.2941 47.3005

Horizontal port measured directivityFrequency 5.8 GHz Amplitude (dB) Azimuth ( ) 90.0000 36.6228 89.9100 36.6165 89.8199 36.5974 89.7299 36.5657 89.6398 36.5215 89.5498 36.4648 89.4597 36.3959 89.3697 36.3150 89.2796 36.2223 89.1896 36.1181 89.0995 36.0027 89.0095 35.8763 88.9195 35.7395 88.8294 35.5924 88.7394 35.4355 88.6493 35.2693 88.5593 35.0940 88.4692 34.9102 88.3792 34.7182 88.2891 34.5185 88.1991 34.3115 88.1091 34.0976

Phase ( ) 172.0927 172.0995 172.1062 172.1126 172.1188 172.1248 172.1305 172.1358 172.1409 172.1456 172.1500 172.1539 172.1575 172.1606 172.1633 172.1656 172.1674 172.1687 172.1696 172.1700 172.1699 Phase ( ) 98.5090 98.5079 98.5049 98.4999 98.4928 98.4838 98.4729 98.4601 98.4455 98.4291 98.4110 98.3913 98.3701 98.3474 98.3233 98.2979 98.2713 98.2437 98.2151 98.1856 98.1554 98.1246

(Continues)

Antenna measured values in anechoic chamber

297

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 88.0190 87.9290 87.8389 87.7489 87.6588 87.5688 87.4787 87.3887 87.2987 87.2086 87.1186 87.0285 86.9385 86.8484 86.7584 86.6683 86.5783 86.4882 86.3982 86.3082 86.2181 86.1281 86.0380 85.9480 85.8579 85.7679 85.6778 85.5878 85.4977 85.4077 85.3177 85.2276 85.1376 85.0475 84.9575 84.8674 84.7774 84.6873 84.5973 84.5073 84.4172 84.3272 84.2371 84.1471 84.0570 83.9670

Amplitude (dB) 33.8773 33.6509 33.4190 33.1818 32.9399 32.6935 32.4431 32.1891 31.9317 31.6714 31.4085 31.1433 30.8762 30.6074 30.3372 30.0659 29.7937 29.5211 29.2480 28.9749 28.7020 28.4294 28.1574 27.8861 27.6158 27.3466 27.0788 26.8125 26.5478 26.2849 26.0240 25.7652 25.5086 25.2545 25.0028 24.7538 24.5076 24.2643 24.0240 23.7869 23.5530 23.3224 23.0954 22.8720 22.6523 22.4364

Phase ( ) 98.0932 98.0614 98.0294 97.9972 97.9649 97.9327 97.9007 97.8690 97.8377 97.8070 97.7769 97.7476 97.7192 97.6917 97.6654 97.6402 97.6164 97.5940 97.5731 97.5538 97.5363 97.5205 97.5067 97.4950 97.4853 97.4779 97.4729 97.4702 97.4701 97.4726 97.4778 97.4859 97.4969 97.5110 97.5283 97.5488 97.5728 97.6003 97.6315 97.6665 97.7055 97.7485 97.7959 97.8477 97.9041 97.9653

(Continues)

298

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 83.8769 83.7869 83.6968 83.6068 83.5168 83.4267 83.3367 83.2466 83.1566 83.0665 82.9765 82.8864 82.7964 82.7064 82.6163 82.5263 82.4362 82.3462 82.2561 82.1661 82.0760 81.9860 81.8960 81.8059 81.7159 81.6258 81.5358 81.4457 81.3557 81.2656 81.1756 81.0855 80.9955 80.9054 80.8154 80.7254 80.6353 80.5453 80.4552 80.3652 80.2751 80.1851 80.0950 80.0050 79.9150 79.8249

Amplitude (dB) 22.2244 22.0166 21.8129 21.6134 21.4185 21.2280 21.0422 20.8612 20.6852 20.5142 20.3485 20.1881 20.0333 19.8841 19.7409 19.6036 19.4727 19.3482 19.2304 19.1194 19.0157 18.9193 18.8307 18.7500 18.6774 18.6093 18.5443 18.4823 18.4233 18.3671 18.3139 18.2633 18.2155 18.1703 18.1276 18.0874 18.0496 18.0141 17.9809 17.9499 17.9209 17.8941 17.8691 17.8461 17.8250 17.8057

Phase ( ) 98.0316 98.1031 98.1800 98.2627 98.3513 98.4462 98.5477 98.6561 98.7717 98.8950 99.0263 99.1661 99.3149 99.4733 99.6418 99.8210 100.0116 100.2145 100.4304 100.6603 100.9051 101.1660 101.4443 101.7413 102.0587 102.4012 102.7697 103.1639 103.5836 104.0285 104.4983 104.9927 105.5115 106.0542 106.6207 107.2106 107.8237 108.4595 109.1177 109.7980 110.4999 111.2229 111.9667 112.7307 113.5144 114.3172

(Continues)

Antenna measured values in anechoic chamber

299

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 79.7349 79.6448 79.5548 79.4647 79.3747 79.2846 79.1946 79.1046 79.0145 78.9245 78.8344 78.7444 78.6543 78.5643 78.4742 78.3842 78.2941 78.2041 78.1141 78.0240 77.9340 77.8439 77.7539 77.6638 77.5738 77.4837 77.3937 77.3036 77.2136 77.1236 77.0335 76.9435 76.8534 76.7634 76.6733 76.5833 76.4932 76.4032 76.3132 76.2231 76.1331 76.0430 75.9530 75.8629 75.7729 75.6828

Amplitude (dB) 17.7881 17.7722 17.7580 17.7455 17.7346 17.7253 17.7177 17.7117 17.7074 17.7048 17.7040 17.7051 17.7082 17.7133 17.7206 17.7303 17.7426 17.7577 17.7757 17.7971 17.8220 17.8509 17.8840 17.9218 17.9641 18.0107 18.0612 18.1152 18.1721 18.2316 18.2932 18.3563 18.4204 18.4848 18.5490 18.6122 18.6739 18.7334 18.7899 18.8429 18.8917 18.9356 18.9742 19.0069 19.0334 19.0532

Phase ( ) 115.1385 115.9777 116.8340 117.7068 118.5952 119.4985 120.4158 121.3462 122.2889 123.2427 124.2068 125.1800 126.1614 127.1499 128.1443 129.1436 130.1466 131.1521 132.1591 133.1664 134.1729 135.1773 136.1786 137.1854 138.2407 139.3508 140.5171 141.7408 143.0230 144.3650 145.7674 147.2312 148.7569 150.3447 151.9945 153.7059 155.4782 157.3100 159.1995 161.1445 163.1420 165.1888 167.2809 169.4138 171.5825 173.7818

(Continues)

300

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 75.5928 75.5028 75.4127 75.3227 75.2326 75.1426 75.0525 74.9625 74.8724 74.7824 74.6923 74.6023 74.5123 74.4222 74.3322 74.2421 74.1521 74.0620 73.9720 73.8819 73.7919 73.7019 73.6118 73.5218 73.4317 73.3417 73.2516 73.1616 73.0715 72.9815 72.8914 72.8014 72.7114 72.6213 72.5313 72.4412 72.3512 72.2611 72.1711 72.0810 71.9910 71.9009 71.8109 71.7209 71.6308 71.5408

Amplitude (dB) 19.0663 19.0724 19.0716 19.0640 19.0499 19.0297 19.0039 18.9731 18.9379 18.8993 18.8581 18.8121 18.7588 18.6974 18.6271 18.5472 18.4575 18.3576 18.2477 18.1279 17.9987 17.8607 17.7146 17.5614 17.4020 17.2377 17.0694 16.8984 16.7259 16.5530 16.3808 16.2106 16.0434 15.8801 15.7218 15.5695 15.4240 15.2863 15.1570 15.0318 14.9078 14.7851 14.6635 14.5431 14.4241 14.3065

Phase ( ) 176.0058 178.2488 179.4954 177.2328 174.9695 172.7112 170.4634 168.2311 166.0188 163.8306 161.6688 159.4785 157.2306 154.9334 152.5957 150.2270 147.8373 145.4368 143.0356 140.6436 138.2703 135.9246 133.6146 131.3473 129.1290 126.9646 124.8584 122.8133 120.8314 118.9141 117.0617 115.2739 113.5500 111.8884 110.2875 108.7450 107.2585 105.8253 104.4417 103.0653 101.6758 100.2768 98.8720 97.4646 96.0578 94.6547

(Continues)

Antenna measured values in anechoic chamber

301

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 71.4507 71.3607 71.2706 71.1806 71.0905 71.0005 70.9105 70.8204 70.7304 70.6403 70.5503 70.4602 70.3702 70.2801 70.1901 70.1001 70.0100 69.9200 69.8299 69.7399 69.6498 69.5598 69.4697 69.3797 69.2896 69.1996 69.1096 69.0195 68.9295 68.8394 68.7494 68.6593 68.5693 68.4792 68.3892 68.2991 68.2091 68.1191 68.0290 67.9390 67.8489 67.7589 67.6688 67.5788 67.4887 67.3987

Amplitude (dB) 14.1906 14.0766 13.9648 13.8554 13.7489 13.6456 13.5459 13.4503 13.3591 13.2730 13.1925 13.1180 13.0501 12.9895 12.9368 12.8926 12.8576 12.8321 12.8137 12.8019 12.7963 12.7964 12.8019 12.8126 12.8282 12.8485 12.8733 12.9025 12.9359 12.9736 13.0156 13.0619 13.1128 13.1684 13.2291 13.2951 13.3671 13.4455 13.5310 13.6246 13.7287 13.8427 13.9654 14.0955 14.2314 14.3718

Phase ( ) 93.2580 91.8702 90.4937 89.1307 87.7829 86.4520 85.1394 83.8464 82.5738 81.3225 80.0929 78.8855 77.7003 76.5374 75.3965 74.2772 73.1792 72.0918 70.9712 69.8130 68.6186 67.3892 66.1262 64.8310 63.5051 62.1500 60.7676 59.3596 57.9279 56.4745 55.0013 53.5104 52.0041 50.4843 48.9532 47.4130 45.8656 44.3130 42.7572 41.1934 39.5515 37.8089 35.9625 34.0097 31.9485 29.7775

(Continues)

302

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 67.3087 67.2186 67.1286 67.0385 66.9485 66.8584 66.7684 66.6783 66.5883 66.4983 66.4082 66.3182 66.2281 66.1381 66.0480 65.9580 65.8679 65.7779 65.6878 65.5978 65.5078 65.4177 65.3277 65.2376 65.1476 65.0575 64.9675 64.8774 64.7874 64.6973 64.6073 64.5173 64.4272 64.3372 64.2471 64.1571 64.0670 63.9770 63.8869 63.7969 63.7069 63.6168 63.5268 63.4367 63.3467 63.2566

Amplitude (dB) 14.5149 14.6592 14.8029 14.9444 15.0819 15.2140 15.3393 15.4567 15.5653 15.6647 15.7547 15.8359 15.9091 15.9757 16.0366 16.0892 16.1296 16.1544 16.1607 16.1462 16.1095 16.0501 15.9685 15.8662 15.7454 15.6091 15.4608 15.3043 15.1434 14.9820 14.8241 14.6732 14.5329 14.4010 14.2695 14.1375 14.0048 13.8716 13.7380 13.6048 13.4726 13.3427 13.2161 13.0941 12.9783 12.8701

Phase ( ) 27.4965 25.1062 22.6088 20.0084 17.3105 14.5225 11.6540 8.7162 5.7217 2.6848 0.3796 3.4562 6.5299 9.5863 12.6940 15.9545 19.3542 22.8743 26.4917 30.1793 33.9076 37.6455 41.3621 45.0282 48.6176 52.1083 55.4826 58.7279 61.8363 64.8041 67.6314 70.3216 72.8802 75.3720 77.8891 80.4217 82.9588 85.4898 88.0050 90.4954 92.9535 95.3727 97.7477 100.0747 102.3509 104.5750

(Continues)

Antenna measured values in anechoic chamber

303

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 63.1666 63.0765 62.9865 62.8965 62.8064 62.7164 62.6263 62.5363 62.4462 62.3562 62.2661 62.1761 62.0860 61.9960 61.9060 61.8159 61.7259 61.6358 61.5458 61.4557 61.3657 61.2756 61.1856 61.0955 61.0055 60.9155 60.8254 60.7354 60.6453 60.5553 60.4652 60.3752 60.2851 60.1951 60.1051 60.0150 59.9250 59.8349 59.7449 59.6548 59.5648 59.4747 59.3847 59.2946 59.2046 59.1146

Amplitude (dB) 12.7712 12.6832 12.6079 12.5471 12.5027 12.4730 12.4526 12.4400 12.4337 12.4324 12.4350 12.4405 12.4483 12.4579 12.4689 12.4814 12.4956 12.5120 12.5314 12.5548 12.5835 12.6190 12.6605 12.7030 12.7421 12.7737 12.7942 12.8003 12.7895 12.7600 12.7113 12.6436 12.5582 12.4572 12.3436 12.2208 12.0927 11.9634 11.8324 11.6887 11.5305 11.3574 11.1702 10.9702 10.7598 10.5417

Phase ( ) 106.7468 108.8674 110.9390 112.9647 114.9489 116.9513 119.0435 121.2213 123.4798 125.8136 128.2168 130.6830 133.2054 135.7769 138.3904 141.0387 143.7150 146.4131 149.1276 151.8537 154.5885 157.3300 160.1571 163.1688 166.3551 169.7014 173.1890 176.7950 179.5071 175.7467 171.9545 168.1611 164.3955 160.6837 157.0476 153.5042 150.0653 146.7378 143.4700 140.1335 136.7533 133.3620 129.9903 126.6658 123.4123 120.2488

(Continues)

304

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 59.0245 58.9345 58.8444 58.7544 58.6643 58.5743 58.4842 58.3942 58.3042 58.2141 58.1241 58.0340 57.9440 57.8539 57.7639 57.6738 57.5838 57.4937 57.4037 57.3137 57.2236 57.1336 57.0435 56.9535 56.8634 56.7734 56.6833 56.5933 56.5033 56.4132 56.3232 56.2331 56.1431 56.0530 55.9630 55.8729 55.7829 55.6928 55.6028 55.5128 55.4227 55.3327 55.2426 55.1526 55.0625 54.9725

Amplitude (dB) 10.3190 10.0950 9.8731 9.6566 9.4488 9.2528 9.0714 8.9072 8.7505 8.5957 8.4425 8.2911 8.1421 7.9962 7.8543 7.7179 7.5881 7.4666 7.3551 7.2553 7.1691 7.0985 7.0451 7.0035 6.9698 6.9428 6.9213 6.9046 6.8922 6.8839 6.8796 6.8798 6.8850 6.8962 6.9146 6.9417 6.9793 7.0277 7.0828 7.1413 7.2001 7.2565 7.3079 7.3525 7.3888 7.4163

Phase ( ) 117.1896 114.2444 111.4186 108.7137 106.1279 103.6570 101.2942 99.0287 96.7624 94.4617 92.1413 89.8145 87.4934 85.1885 82.9087 80.6613 78.4519 76.2844 74.1613 72.0834 70.0503 68.0600 66.1048 64.0936 61.9983 59.8278 57.5908 55.2964 52.9538 50.5718 48.1593 45.7246 43.2753 40.8183 38.3593 35.9030 33.4524 30.9469 28.2830 25.4672 22.5104 19.4263 16.2311 12.9433 9.5837 6.1740

(Continues)

Antenna measured values in anechoic chamber

305

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 54.8824 54.7924 54.7024 54.6123 54.5223 54.4322 54.3422 54.2521 54.1621 54.0720 53.9820 53.8919 53.8019 53.7119 53.6218 53.5318 53.4417 53.3517 53.2616 53.1716 53.0815 52.9915 52.9015 52.8114 52.7214 52.6313 52.5413 52.4512 52.3612 52.2711 52.1811 52.0910 52.0010 51.9110 51.8209 51.7309 51.6408 51.5508 51.4607 51.3707 51.2806 51.1906 51.1006 51.0105 50.9205 50.8304

Amplitude (dB) 7.4350 7.4458 7.4502 7.4506 7.4498 7.4471 7.4351 7.4100 7.3689 7.3103 7.2336 7.1397 7.0306 6.9093 6.7796 6.6459 6.5128 6.3852 6.2669 6.1480 6.0241 5.8948 5.7601 5.6211 5.4793 5.3367 5.1959 5.0596 4.9308 4.8127 4.7086 4.6215 4.5446 4.4722 4.4029 4.3356 4.2698 4.2054 4.1428 4.0831 4.0274 3.9776 3.9357 3.9040 3.8846 3.8719

Phase ( ) 2.7364 0.7073 4.1370 7.5351 10.8871 14.2768 17.8204 21.4892 25.2496 29.0655 32.9000 36.7175 40.4852 44.1751 47.7645 51.2370 54.5824 57.7961 60.8935 64.0208 67.1949 70.3910 73.5856 76.7572 79.8871 82.9596 85.9628 88.8881 91.7307 94.4889 97.1642 99.7654 102.4079 105.1311 107.9198 110.7586 113.6324 116.5269 119.4284 122.3249 125.2061 128.0638 130.8921 133.6875 136.4645 139.3565

(Continues)

306

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 50.7404 50.6503 50.5603 50.4702 50.3802 50.2901 50.2001 50.1101 50.0200 49.9300 49.8399 49.7499 49.6598 49.5698 49.4797 49.3897 49.2996 49.2096 49.1196 49.0295 48.9395 48.8494 48.7594 48.6693 48.5793 48.4892 48.3992 48.3092 48.2191 48.1291 48.0390 47.9490 47.8589 47.7689 47.6788 47.5888 47.4987 47.4087 47.3187 47.2286 47.1386 47.0485 46.9585 46.8684 46.7784 46.6883

Amplitude (dB) 3.8613 3.8499 3.8351 3.8152 3.7891 3.7566 3.7180 3.6746 3.6282 3.5815 3.5374 3.4948 3.4461 3.3880 3.3182 3.2357 3.1405 3.0336 2.9171 2.7939 2.6676 2.5423 2.4223 2.3086 2.1918 2.0699 1.9425 1.8098 1.6731 1.5340 1.3950 1.2588 1.1284 1.0071 0.8986 0.8029 0.7125 0.6253 0.5406 0.4580 0.3777 0.3002 0.2268 0.1590 0.0987 0.0482

Phase ( ) 142.3821 145.5236 148.7614 152.0742 155.4397 158.8351 162.2386 165.6297 168.9906 172.3065 175.5660 178.8496 177.7345 174.2150 170.6244 166.9960 163.3630 159.7563 156.2038 152.7289 149.3501 146.0804 142.9275 139.8473 136.7086 133.5281 130.3324 127.1465 123.9933 120.8927 117.8609 114.9107 112.0509 109.2869 106.6205 104.0065 101.3348 98.6154 95.8649 93.0995 90.3347 87.5844 84.8616 82.1771 79.5399 76.9569

(Continues)

Antenna measured values in anechoic chamber

307

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 46.5983 46.5083 46.4182 46.3282 46.2381 46.1481 46.0580 45.9680 45.8779 45.7879 45.6978 45.6078 45.5178 45.4277 45.3377 45.2476 45.1576 45.0675 44.9775 44.8874 44.7974 44.7074 44.6173 44.5273 44.4372 44.3472 44.2571 44.1671 44.0770 43.9870 43.8969 43.8069 43.7169 43.6268 43.5368 43.4467 43.3567 43.2666 43.1766 43.0865 42.9965 42.9065 42.8164 42.7264 42.6363 42.5463

Amplitude (dB)

Phase ( )

0.0101 0.0164 0.0352 0.0488 0.0590 0.0674 0.0747 0.0813 0.0870 0.0910 0.0918 0.0877 0.0763 0.0586 0.0379 0.0175 0.0003 0.0136 0.0213 0.0233 0.0202 0.0136 0.0058 0.0002 0.0034 0.0068 0.0132 0.0248 0.0427 0.0671 0.0974 0.1320 0.1686 0.2039 0.2343 0.2578 0.2768 0.2934 0.3091 0.3248 0.3406 0.3560 0.3697 0.3799 0.3840 0.3788

74.4326 71.8860 69.2453 66.5232 63.7340 60.8929 58.0162 55.1203 52.2218 49.3364 46.4790 43.6631 40.8744 37.9804 34.9729 31.8694 28.6899 25.4570 22.1950 18.9292 15.6846 12.4853 9.3536 6.3031 3.2048 0.0140 3.2474 6.5550 9.8824 13.2025 16.4884 19.7142 22.8566 25.8948 28.8134 31.7264 34.6948 37.7027 40.7320 43.7634 46.7769 49.7524 52.6705 55.5126 58.2618 60.9080

(Continues)

308

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 42.4562 42.3662 42.2761 42.1861 42.0960 42.0060 41.9160 41.8259 41.7359 41.6458 41.5558 41.4657 41.3757 41.2856 41.1956 41.1056 41.0155 40.9255 40.8354 40.7454 40.6553 40.5653 40.4752 40.3852 40.2951 40.2051 40.1151 40.0250 39.9350 39.8449 39.7549 39.6648 39.5748 39.4847 39.3947 39.3047 39.2146 39.1246 39.0345 38.9445 38.8544 38.7644 38.6743 38.5843 38.4942 38.4042

Amplitude (dB)

Phase ( )

0.3604 0.3301 0.2905 0.2438 0.1919 0.1360 0.0766 0.0137 0.0532 0.1256 0.2070 0.3056 0.4192 0.5441 0.6766 0.8137 0.9526 1.0913 1.2287 1.3645 1.4993 1.6424 1.8012 1.9706 2.1455 2.3211 2.4932 2.6585 2.8150 2.9619 3.1004 3.2356 3.3834 3.5416 3.7050 3.8690 4.0298 4.1846 4.3322 4.4728 4.6087 4.7445 4.8992 5.0776 5.2748 5.4864

63.5672 66.2877 69.0613 71.8771 74.7216 77.5792 80.4324 83.2621 86.0482 88.7700 91.4306 94.1655 96.9983 99.9244 102.9347 106.0155 109.1482 112.3095 115.4717 118.6036 121.6708 124.7397 127.9401 131.2708 134.7224 138.2783 141.9134 145.5947 149.2819 152.9285 156.4841 159.9289 163.4461 167.0702 170.7911 174.5902 178.4401 177.6948 173.8572 170.0953 166.4607 163.0016 159.5658 156.0523 152.4554 148.7775

(Continues)

Antenna measured values in anechoic chamber

309

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 38.3142 38.2241 38.1341 38.0440 37.9540 37.8639 37.7739 37.6838 37.5938 37.5038 37.4137 37.3237 37.2336 37.1436 37.0535 36.9635 36.8734 36.7834 36.6933 36.6033 36.5133 36.4232 36.3332 36.2431 36.1531 36.0630 35.9730 35.8829 35.7929 35.7029 35.6128 35.5228 35.4327 35.3427 35.2526 35.1626 35.0725 34.9825 34.8924 34.8024 34.7124 34.6223 34.5323 34.4422 34.3522 34.2621

Amplitude (dB) 5.7081 5.9362 6.1683 6.4029 6.6403 6.8829 7.1550 7.4721 7.8271 8.2121 8.6181 9.0350 9.4524 9.8604 10.2515 10.6217 11.0011 11.4050 11.8051 12.1678 12.4584 12.6476 12.7204 12.6812 12.5535 12.3734 12.1739 11.9519 11.7003 11.4196 11.1169 10.8039 10.4951 10.2059 9.9515 9.7463 9.5886 9.4665 9.3720 9.2999 9.2480 9.2166 9.2086 9.2295 9.2870 9.3932

Phase ( ) 145.0296 141.2321 137.4147 133.6162 129.8838 126.2716 122.6572 118.8653 114.8553 110.5952 106.0640 101.2575 96.1924 90.9122 85.4895 80.0247 74.3317 68.0768 61.2016 53.7099 45.6948 37.3515 28.9581 20.8207 13.2068 6.2952 0.3090 6.8921 13.3484 19.5683 25.4553 30.9370 35.9690 40.5329 44.6303 48.2959 51.9311 55.6616 59.4502 63.2551 67.0315 70.7337 74.3169 77.7387 80.9596 84.0320

(Continues)

310

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 34.1721 34.0820 33.9920 33.9020 33.8119 33.7219 33.6318 33.5418 33.4517 33.3617 33.2716 33.1816 33.0915 33.0015 32.9115 32.8214 32.7314 32.6413 32.5513 32.4612 32.3712 32.2811 32.1911 32.1011 32.0110 31.9210 31.8309 31.7409 31.6508 31.5608 31.4707 31.3807 31.2906 31.2006 31.1106 31.0205 30.9305 30.8404 30.7504 30.6603 30.5703 30.4802 30.3902 30.3001 30.2101 30.1201

Amplitude (dB) 9.5592 9.7758 10.0328 10.3216 10.6349 10.9681 11.3203 11.6952 12.1029 12.5721 13.1061 13.6721 14.2280 14.7242 15.1130 15.3630 15.4726 15.4730 15.4036 15.1912 14.7979 14.2427 13.5763 12.8584 12.1404 11.4598 10.8400 10.2943 9.7656 9.2056 8.6243 8.0350 7.4511 6.8847 6.3458 5.8416 5.3772 4.9377 4.4762 3.9960 3.5056 3.0139 2.5299 2.0617 1.6161 1.1986

Phase ( ) 87.3526 90.9728 94.8842 99.0672 103.4882 108.0979 112.8319 117.6134 122.3577 127.4295 133.3951 140.3633 148.3931 157.4426 167.3226 177.6966 171.8408 161.6351 151.7324 141.3890 130.9537 120.9580 111.7526 103.4633 96.0414 89.3419 83.1820 77.3721 71.6097 65.8576 60.2468 54.8556 49.7162 44.8245 40.1514 35.6513 31.2686 26.9021 22.4673 18.0586 13.7493 9.5839 5.5823 1.7451 1.9418 5.5027

(Continues)

Antenna measured values in anechoic chamber

311

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 30.0300 29.9400 29.8499 29.7599 29.6698 29.5798 29.4897 29.3997 29.3097 29.2196 29.1296 29.0395 28.9495 28.8594 28.7694 28.6793 28.5893 28.4993 28.4092 28.3192 28.2291 28.1391 28.0490 27.9590 27.8689 27.7789 27.6888 27.5988 27.5088 27.4187 27.3287 27.2386 27.1486 27.0585 26.9685 26.8784 26.7884 26.6983 26.6083 26.5183 26.4282 26.3382 26.2481 26.1581 26.0680 25.9780

Amplitude (dB) 0.8082 0.4106 0.0034 0.4072 0.8149 1.2131 1.5958 1.9576 2.2937 2.6013 2.9036 3.2073 3.5090 3.8050 4.0913 4.3638 4.6184 4.8513 5.0592 5.2560 5.4494 5.6384 5.8215 5.9966 6.1610 6.3120 6.4463 6.5608 6.6634 6.7606 6.8531 6.9408 7.0231 7.0985 7.1650 7.2199 7.2604 7.2905 7.3154 7.3367 7.3555 7.3718 7.3847 7.3926 7.3927 7.3820

Phase ( ) 8.9852 12.4998 16.0048 19.4488 22.7983 26.0348 29.1528 32.1566 35.0583 37.8820 40.7327 43.6081 46.4715 49.2962 52.0642 54.7654 57.3965 59.9601 62.4662 65.0116 67.6167 70.2521 72.8931 75.5203 78.1189 80.6788 83.1946 85.6670 88.1948 90.8119 93.4911 96.2077 98.9389 101.6649 104.3688 107.0370 109.6632 112.3608 115.1673 118.0558 120.9989 123.9701 126.9437 129.8962 132.8065 135.6660

(Continues)

312

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 25.8879 25.7979 25.7079 25.6178 25.5278 25.4377 25.3477 25.2576 25.1676 25.0775 24.9875 24.8974 24.8074 24.7174 24.6273 24.5373 24.4472 24.3572 24.2671 24.1771 24.0870 23.9970 23.9070 23.8169 23.7269 23.6368 23.5468 23.4567 23.3667 23.2766 23.1866 23.0965 23.0065 22.9165 22.8264 22.7364 22.6463 22.5563 22.4662 22.3762 22.2861 22.1961 22.1061 22.0160 21.9260 21.8359

Amplitude (dB)

Phase ( )

7.3628 7.3392 7.3133 7.2866 7.2592 7.2303 7.1978 7.1587 7.1094 7.0515 6.9885 6.9232 6.8568 6.7899 6.7216 6.6497 6.5709 6.4812 6.3820 6.2780 6.1727 6.0687 5.9674 5.8687 5.7712 5.6717 5.5669 5.4609 5.3609 5.2729 5.2011 5.1479 5.1133 5.0951 5.0887 5.0949 5.1215 5.1740 5.2551 5.3643 5.4983 5.6511 5.8150 5.9844 6.1658 6.3615

138.6107 141.6697 144.8158 148.0200 151.2529 154.4853 157.6893 160.8395 163.9383 167.1410 170.4606 173.8723 177.3485 179.1394 175.6206 172.1234 168.6736 165.2368 161.6514 157.9209 154.0667 150.1146 146.0940 142.0359 137.9711 133.9285 129.8076 125.4743 120.9517 116.2734 111.4816 106.6245 101.7527 96.9151 92.1434 87.2584 82.2270 77.1099 71.9741 66.8870 61.9107 57.0966 52.4828 48.0341 43.5788 39.1506

(Continues)

Antenna measured values in anechoic chamber

313

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 21.7459 21.6558 21.5658 21.4757 21.3857 21.2956 21.2056 21.1156 21.0255 20.9355 20.8454 20.7554 20.6653 20.5753 20.4852 20.3952 20.3052 20.2151 20.1251 20.0350 19.9450 19.8549 19.7649 19.6748 19.5848 19.4947 19.4047 19.3147 19.2246 19.1346 19.0445 18.9545 18.8644 18.7744 18.6843 18.5943 18.5043 18.4142 18.3242 18.2341 18.1441 18.0540 17.9640 17.8739 17.7839 17.6938

Amplitude (dB)

Phase ( )

6.5709 6.7913 7.0186 7.2476 7.4725 7.6871 7.8937 8.0980 8.3006 8.5007 8.6968 8.8866 9.0671 9.2352 9.3883 9.5309 9.6654 9.7928 9.9133 10.0265 10.1312 10.2258 10.3082 10.3774 10.4355 10.4846 10.5257 10.5594 10.5858 10.6041 10.6132 10.6108 10.5952 10.5682 10.5314 10.4861 10.4325 10.3708 10.3002 10.2193 10.1233 10.0117 9.8861 9.7478 9.5973 9.4349

34.7984 30.5650 26.4852 22.5843 18.8781 15.3732 11.9512 8.5423 5.1754 1.8764 1.3323 4.4328 7.4111 10.2575 12.9916 15.7351 18.4868 21.2259 23.9327 26.5888 29.1773 31.6833 34.0938 36.4854 38.9166 41.3697 43.8264 46.2678 48.6751 51.0295 53.3125 55.5349 57.7998 60.1060 62.4372 64.7760 67.1032 69.3985 71.6408 73.8106 75.9922 78.2277 80.5058 82.8127 85.1326 87.4471

(Continues)

314

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 17.6038 17.5138 17.4237 17.3337 17.2436 17.1536 17.0635 16.9735 16.8834 16.7934 16.7034 16.6133 16.5233 16.4332 16.3432 16.2531 16.1631 16.0730 15.9830 15.8929 15.8029 15.7129 15.6228 15.5328 15.4427 15.3527 15.2626 15.1726 15.0825 14.9925 14.9025 14.8124 14.7224 14.6323 14.5423 14.4522 14.3622 14.2721 14.1821 14.0920 14.0020 13.9120 13.8219 13.7319 13.6418 13.5518

Amplitude (dB)

Phase ( )

9.2601 9.0718 8.8651 8.6340 8.3795 8.1024 7.8030 7.4813 7.1366 6.7672 6.3689 5.9279 5.4414 4.9093 4.3316 3.7085 3.0414 2.3324 1.5858 0.7963 0.0123 0.7789 1.4070 1.7778 1.7957 1.4476 0.8112 0.0125 0.9510 1.9317 2.9044 3.8397 4.7223 5.5456 6.3079 7.0154 7.6881 8.3274 8.9329 9.5048 10.0431 10.5484 11.0211 11.4630 11.8854 12.2907

89.7356 91.9752 94.1946 96.4899 98.8629 101.3102 103.8262 106.4025 109.0280 111.6884 114.3856 117.2543 120.3530 123.7225 127.4104 131.4727 135.9759 140.9985 146.6369 153.2271 161.1394 170.6367 178.1703 165.5371 152.2199 139.2720 127.5627 117.3489 108.5433 101.0907 94.8117 89.4967 84.9526 81.0183 77.5641 74.4702 71.6157 68.9732 66.5220 64.2418 62.1133 60.1184 58.2402 56.4586 54.7263 53.0395

(Continues)

Antenna measured values in anechoic chamber

315

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 13.4617 13.3717 13.2816 13.1916 13.1016 13.0115 12.9215 12.8314 12.7414 12.6513 12.5613 12.4712 12.3812 12.2911 12.2011 12.1111 12.0210 11.9310 11.8409 11.7509 11.6608 11.5708 11.4807 11.3907 11.3007 11.2106 11.1206 11.0305 10.9405 10.8504 10.7604 10.6703 10.5803 10.4902 10.4002 10.3102 10.2201 10.1301 10.0400 9.9500 9.8599 9.7699 9.6798 9.5898 9.4998 9.4097

Amplitude (dB)

Phase ( )

12.6785 13.0485 13.4002 13.7334 14.0476 14.3427 14.6246 14.8959 15.1563 15.4054 15.6428 15.8679 16.0802 16.2793 16.4678 16.6484 16.8209 16.9849 17.1401 17.2859 17.4218 17.5473 17.6634 17.7723 17.8741 17.9683 18.0547 18.1327 18.2018 18.2614 18.3116 18.3541 18.3889 18.4157 18.4342 18.4439 18.4443 18.4346 18.4144 18.3845 18.3448 18.2950 18.2350 18.1641 18.0817 17.9872

51.4022 49.8167 48.2832 46.8012 45.3688 43.9824 42.6098 41.2440 39.8924 38.5607 37.2536 35.9745 34.7259 33.5095 32.3004 31.0844 29.8695 28.6629 27.4708 26.2985 25.1507 24.0314 22.9232 21.8029 20.6785 19.5575 18.4470 17.3532 16.2822 15.2395 14.2140 13.1765 12.1332 11.0913 10.0578 9.0394 8.0427 7.0743 6.1285 5.1733 4.2120 3.2509 2.2963 1.3548 0.4330 0.4620

(Continues)

316

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 9.3197 9.2296 9.1396 9.0495 8.9595 8.8694 8.7794 8.6893 8.5993 8.5093 8.4192 8.3292 8.2391 8.1491 8.0590 7.9690 7.8789 7.7889 7.6988 7.6088 7.5188 7.4287 7.3387 7.2486 7.1586 7.0685 6.9785 6.8884 6.7984 6.7084 6.6183 6.5283 6.4382 6.3482 6.2581 6.1681 6.0780 5.9880 5.8979 5.8079 5.7179 5.6278 5.5378 5.4477 5.3577 5.2676

Amplitude (dB)

Phase ( )

17.8796 17.7579 17.6220 17.4715 17.3059 17.1246 16.9267 16.7111 16.4763 16.2185 15.9362 15.6283 15.2931 14.9290 14.5335 14.1038 13.6356 13.1179 12.5434 11.9050 11.1930 10.3951 9.4946 8.4685 7.2818 5.8632 4.1156 1.8712 1.2142 6.0132 14.4186 9.0604 2.8210 0.9591 3.6487 5.7338 7.4347 8.8689 10.1063 11.1919 12.1573 13.0339 13.8378 14.5787 15.2644 15.9010

1.3317 2.2077 3.0901 3.9740 4.8544 5.7255 6.5809 7.4135 8.2214 9.0346 9.8565 10.6847 11.5161 12.3468 13.1718 13.9845 14.7815 15.5925 16.4279 17.2925 18.1924 19.1356 20.1323 21.1978 22.3601 23.7196 25.4360 27.8438 31.8555 41.0918 85.8166 173.2518 170.7911 165.3124 162.3738 160.4300 158.9802 157.8135 156.8253 155.9574 155.1730 154.4355 153.7362 153.0724 152.4418 151.8420

(Continues)

Antenna measured values in anechoic chamber

317

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 5.1776 5.0875 4.9975 4.9075 4.8174 4.7274 4.6373 4.5473 4.4572 4.3672 4.2771 4.1871 4.0970 4.0070 3.9170 3.8269 3.7369 3.6468 3.5568 3.4667 3.3767 3.2866 3.1966 3.1066 3.0165 2.9265 2.8364 2.7464 2.6563 2.5663 2.4762 2.3862 2.2961 2.2061 2.1161 2.0260 1.9360 1.8459 1.7559 1.6658 1.5758 1.4857 1.3957 1.3057 1.2156 1.1256

Amplitude (dB)

Phase ( )

16.4937 17.0466 17.5637 18.0527 18.5169 18.9579 19.3768 19.7749 20.1532 20.5126 20.8541 21.1814 21.4958 21.7974 22.0868 22.3641 22.6297 22.8836 23.1264 23.3602 23.5857 23.8027 24.0114 24.2118 24.4039 24.5877 24.7633 24.9323 25.0951 25.2515 25.4014 25.5448 25.6816 25.8115 25.9348 26.0526 26.1650 26.2721 26.3736 26.4693 26.5592 26.6431 26.7210 26.7938 26.8616 26.9242

151.2708 150.7262 150.2049 149.6909 149.1838 148.6871 148.2034 147.7347 147.2823 146.8474 146.4293 146.0153 145.6048 145.2010 144.8065 144.4235 144.0538 143.6991 143.3593 143.0239 142.6924 142.3671 142.0502 141.7436 141.4489 141.1675 140.8999 140.6377 140.3803 140.1293 139.8865 139.6531 139.4305 139.2199 139.0217 138.8297 138.6435 138.4641 138.2924 138.1297 137.9766 137.8342 137.7026 137.5780 137.4599 137.3489

(Continues)

318

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

1.0355 0.9455 0.8554 0.7654 0.6753 0.5853 0.4952 0.4052 0.3152 0.2251 0.1351 0.0450 0.0450 0.1351 0.2251 0.3152 0.4052 0.4952 0.5853 0.6753 0.7654 0.8554 0.9455 1.0355 1.1256 1.2156 1.3057 1.3957 1.4857 1.5758 1.6658 1.7559 1.8459 1.9360 2.0260 2.1161 2.2061 2.2961 2.3862 2.4762 2.5663 2.6563 2.7464 2.8364 2.9265 3.0165

26.9817 27.0338 27.0804 27.1213 27.1566 27.1867 27.2118 27.2316 27.2463 27.2557 27.2598 27.2584 27.2515 27.2391 27.2212 27.1980 27.1693 27.1353 27.0959 27.0512 27.0010 26.9447 26.8825 26.8144 26.7406 26.6611 26.5761 26.4856 26.3895 26.2867 26.1772 26.0611 25.9386 25.8100 25.6753 25.5346 25.3879 25.2335 25.0712 24.9013 24.7239 24.5392 24.3473 24.1483 23.9421 23.7263

137.2455 137.1504 137.0641 136.9871 136.9198 136.8600 136.8075 136.7625 136.7253 136.6959 136.6746 136.6618 136.6578 136.6622 136.6750 136.6959 136.7250 136.7621 136.8071 136.8602 136.9215 136.9929 137.0741 137.1648 137.2646 137.3732 137.4901 137.6150 137.7484 137.8940 138.0519 138.2213 138.4016 138.5921 138.7920 139.0007 139.2187 139.4522 139.7014 139.9654 140.2433 140.5341 140.8365 141.1495 141.4739 141.8190

(Continues)

Antenna measured values in anechoic chamber

319

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

3.1066 3.1966 3.2866 3.3767 3.4667 3.5568 3.6468 3.7369 3.8269 3.9170 4.0070 4.0970 4.1871 4.2771 4.3672 4.4572 4.5473 4.6373 4.7274 4.8174 4.9075 4.9975 5.0875 5.1776 5.2676 5.3577 5.4477 5.5378 5.6278 5.7179 5.8079 5.8979 5.9880 6.0780 6.1681 6.2581 6.3482 6.4382 6.5283 6.6183 6.7084 6.7984 6.8884 6.9785 7.0685 7.1586

23.5008 23.2657 23.0209 22.7666 22.5026 22.2291 21.9454 21.6487 21.3385 21.0145 20.6762 20.3235 19.9557 19.5725 19.1724 18.7513 18.3077 17.8402 17.3473 16.8273 16.2783 15.6981 15.0826 14.4234 13.7154 12.9536 12.1323 11.2451 10.2857 9.2488 8.1297 6.9251 5.6638 4.4275 3.3845 2.7841 2.8180 3.4396 4.4157 5.5321 6.6540 7.7170 8.6971 9.5899 10.3990 11.1308

142.1851 142.5715 142.9772 143.4012 143.8421 144.2982 144.7714 145.2746 145.8093 146.3759 146.9744 147.6047 148.2664 148.9590 149.6865 150.4690 151.3128 152.2234 153.2069 154.2707 155.4232 156.6743 158.0434 159.5784 161.3170 163.3055 165.6051 168.2983 171.4981 175.3606 179.8782 173.8162 165.9488 155.6791 142.5731 127.0005 110.6641 95.7988 83.6157 74.0426 66.6142 60.8004 56.1666 52.3947 49.2596 46.6025

(Continues)

320

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

7.2486 7.3387 7.4287 7.5188 7.6088 7.6988 7.7889 7.8789 7.9690 8.0590 8.1491 8.2391 8.3292 8.4192 8.5093 8.5993 8.6893 8.7794 8.8694 8.9595 9.0495 9.1396 9.2296 9.3197 9.4097 9.4998 9.5898 9.6798 9.7699 9.8599 9.9500 10.0400 10.1301 10.2201 10.3102 10.4002 10.4902 10.5803 10.6703 10.7604 10.8504 10.9405 11.0305 11.1206 11.2106 11.3007

11.7939 12.4025 12.9625 13.4780 13.9527 14.3895 14.7910 15.1594 15.4973 15.8106 16.1012 16.3705 16.6194 16.8488 17.0592 17.2511 17.4251 17.5836 17.7276 17.8578 17.9748 18.0789 18.1705 18.2498 18.3168 18.3715 18.4147 18.4471 18.4692 18.4815 18.4843 18.4780 18.4624 18.4361 18.3994 18.3529 18.2974 18.2334 18.1614 18.0818 17.9948 17.8979 17.7905 17.6734 17.5474 17.4131

44.3011 42.2434 40.3815 38.6824 37.1204 35.6750 34.3295 33.0704 31.8806 30.7248 29.5986 28.5020 27.4349 26.3971 25.3885 24.4088 23.4531 22.4941 21.5305 20.5663 19.6051 18.6503 17.7050 16.7724 15.8518 14.9158 13.9620 12.9951 12.0197 11.0406 10.0626 9.0907 8.1275 7.1445 6.1359 5.1066 4.0618 3.0071 1.9485 0.8923 0.1557 1.2232 2.3214 3.4456 4.5907 5.7508

(Continues)

Antenna measured values in anechoic chamber

321

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

11.3907 11.4807 11.5708 11.6608 11.7509 11.8409 11.9310 12.0210 12.1111 12.2011 12.2911 12.3812 12.4712 12.5613 12.6513 12.7414 12.8314 12.9215 13.0115 13.1016 13.1916 13.2816 13.3717 13.4617 13.5518 13.6418 13.7319 13.8219 13.9120 14.0020 14.0920 14.1821 14.2721 14.3622 14.4522 14.5423 14.6323 14.7224 14.8124 14.9025 14.9925 15.0825 15.1726 15.2626 15.3527 15.4427

17.2710 17.1217 16.9655 16.7994 16.6217 16.4331 16.2344 16.0261 15.8088 15.5831 15.3493 15.1043 14.8446 14.5706 14.2829 13.9819 13.6681 13.3420 13.0037 12.6504 12.2750 11.8771 11.4569 11.0144 10.5496 10.0627 9.5536 9.0195 8.4466 7.8320 7.1746 6.4741 5.7308 4.9465 4.1257 3.2741 2.3796 1.4583 0.5537 0.2622 0.8904 1.2305 1.2300 0.9181 0.3778 0.3107

6.9194 8.0887 9.2508 10.4288 11.6411 12.8842 14.1539 15.4450 16.7512 18.0648 19.3768 20.7019 22.0670 23.4715 24.9138 26.3914 27.9002 29.4348 30.9876 32.5657 34.2042 35.9108 37.6917 39.5526 41.4985 43.5334 45.6601 47.8894 50.2802 52.8727 55.7099 58.8426 62.3311 66.2467 70.6714 75.7061 81.5925 88.5904 96.9457 106.8277 118.1726 130.5161 143.0093 154.7391 165.3447 174.6894

(Continues)

322

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

15.5328 15.6228 15.7129 15.8029 15.8929 15.9830 16.0730 16.1631 16.2531 16.3432 16.4332 16.5233 16.6133 16.7034 16.7934 16.8834 16.9735 17.0635 17.1536 17.2436 17.3337 17.4237 17.5138 17.6038 17.6938 17.7839 17.8739 17.9640 18.0540 18.1441 18.2341 18.3242 18.4142 18.5043 18.5943 18.6843 18.7744 18.8644 18.9545 19.0445 19.1346 19.2246 19.3147 19.4047 19.4947 19.5848

1.0739 1.8556 2.6189 3.3422 4.0139 4.6286 5.1913 5.7160 6.2046 6.6583 7.0780 7.4642 7.8171 8.1367 8.4244 8.6894 8.9344 9.1605 9.3683 9.5577 9.7287 9.8807 10.0130 10.1287 10.2311 10.3213 10.4000 10.4676 10.5239 10.5684 10.6002 10.6190 10.6271 10.6260 10.6168 10.6002 10.5762 10.5446 10.5044 10.4543 10.3941 10.3256 10.2503 10.1692 10.0827 9.9908

Phase ( ) 177.2767 170.4407 164.6257 159.6502 155.3532 151.6011 148.2284 145.0772 142.1253 139.3548 136.7495 134.2946 131.9759 129.7805 127.6791 125.5811 123.4861 121.4052 119.3479 117.3220 115.3338 113.3881 111.4863 109.5565 107.5745 105.5558 103.5150 101.4656 99.4204 97.3912 95.3888 93.3795 91.2917 89.1394 86.9391 84.7075 82.4612 80.2169 77.9905 75.7857 73.5080 71.1463 68.7169 66.2371 63.7253 61.2004

(Continues)

Antenna measured values in anechoic chamber

323

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

19.6748 19.7649 19.8549 19.9450 20.0350 20.1251 20.2151 20.3052 20.3952 20.4852 20.5753 20.6653 20.7554 20.8454 20.9355 21.0255 21.1156 21.2056 21.2956 21.3857 21.4757 21.5658 21.6558 21.7459 21.8359 21.9260 22.0160 22.1061 22.1961 22.2861 22.3762 22.4662 22.5563 22.6463 22.7364 22.8264 22.9165 23.0065 23.0965 23.1866 23.2766 23.3667 23.4567 23.5468 23.6368 23.7269

9.8925 9.7866 9.6707 9.5450 9.4118 9.2727 9.1287 8.9804 8.8275 8.6689 8.5023 8.3239 8.1362 7.9423 7.7448 7.5458 7.3466 7.1476 6.9481 6.7433 6.5321 6.3207 6.1150 5.9201 5.7400 5.5773 5.4325 5.3036 5.1858 5.0855 5.0097 4.9635 4.9490 4.9654 5.0087 5.0725 5.1497 5.2412 5.3496 5.4755 5.6169 5.7702 5.9303 6.0910 6.2458 6.3930

58.6820 56.1895 53.6679 51.0431 48.3274 45.5357 42.6849 39.7939 36.8827 33.9725 31.0659 28.0456 24.8928 21.6167 18.2307 14.7522 11.2021 7.6045 3.9855 0.2779 3.6299 7.7304 12.0073 16.4344 20.9771 25.5928 30.2345 34.8704 39.6613 44.6314 49.7339 54.9103 60.0941 65.2170 70.2149 75.0333 79.7079 84.4034 89.0827 93.6969 98.1996 102.5504 106.7172 110.6770 114.4164 118.0586

(Continues)

324

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

23.8169 23.9070 23.9970 24.0870 24.1771 24.2671 24.3572 24.4472 24.5373 24.6273 24.7174 24.8074 24.8974 24.9875 25.0775 25.1676 25.2576 25.3477 25.4377 25.5278 25.6178 25.7079 25.7979 25.8879 25.9780 26.0680 26.1581 26.2481 26.3382 26.4282 26.5183 26.6083 26.6983 26.7884 26.8784 26.9685 27.0585 27.1486 27.2386 27.3287 27.4187 27.5088 27.5988 27.6888 27.7789 27.8689

6.5365 6.6779 6.8173 6.9540 7.0860 7.2108 7.3253 7.4258 7.5126 7.5890 7.6574 7.7190 7.7747 7.8241 7.8663 7.8995 7.9210 7.9289 7.9262 7.9159 7.9001 7.8803 7.8570 7.8300 7.7983 7.7583 7.7060 7.6447 7.5775 7.5068 7.4342 7.3608 7.2863 7.2099 7.1266 7.0307 6.9256 6.8141 6.6985 6.5806 6.4613 6.3409 6.2185 6.0874 5.9396 5.7780

121.6907 125.2870 128.8225 132.2740 135.6211 138.8468 141.9378 144.8950 147.8532 150.8418 153.8419 156.8336 159.7968 162.7114 165.5583 168.3196 171.0072 173.7494 176.5510 179.3945 177.7403 174.8754 172.0345 169.2418 166.5221 163.8511 161.1079 158.2983 155.4397 152.5534 149.6636 146.7972 143.9831 141.2513 138.5689 135.8249 133.0266 130.1891 127.3315 124.4767 121.6513 118.8845 116.2080 113.5904 110.9303 108.2289

(Continues)

Antenna measured values in anechoic chamber

325

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

27.9590 28.0490 28.1391 28.2291 28.3192 28.4092 28.4993 28.5893 28.6793 28.7694 28.8594 28.9495 29.0395 29.1296 29.2196 29.3097 29.3997 29.4897 29.5798 29.6698 29.7599 29.8499 29.9400 30.0300 30.1201 30.2101 30.3001 30.3902 30.4802 30.5703 30.6603 30.7504 30.8404 30.9305 31.0205 31.1106 31.2006 31.2906 31.3807 31.4707 31.5608 31.6508 31.7409 31.8309 31.9210 32.0110

5.6054 5.4245 5.2375 5.0460 4.8510 4.6530 4.4437 4.2094 3.9528 3.6773 3.3861 3.0828 2.7706 2.4525 2.1312 1.7989 1.4305 1.0276 0.5941 0.1349 0.3449 0.8387 1.3400 1.8414 2.3458 2.8960 3.4948 4.1356 4.8091 5.5021 6.1975 6.8749 7.5131 8.0986 8.6859 9.2768 9.8393 10.3345 10.7228 10.9754 11.0855 11.0728 10.9783 10.8638 10.7482 10.6147

105.4940 102.7381 99.9784 97.2372 94.5414 91.9225 89.3619 86.7633 84.1163 81.4181 78.6712 75.8846 73.0747 70.2658 67.4915 64.7539 61.9515 59.0544 56.0425 52.9017 49.6264 46.2221 42.7099 39.1299 35.5148 31.7295 27.6911 23.3342 18.5985 13.4372 7.8304 1.8033 4.5545 11.0968 18.0388 25.5845 33.7993 42.6601 52.0185 61.5933 71.0174 79.9290 88.0570 95.6834 103.2889 110.8482

(Continues)

326

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 32.1011 32.1911 32.2811 32.3712 32.4612 32.5513 32.6413 32.7314 32.8214 32.9115 33.0015 33.0915 33.1816 33.2716 33.3617 33.4517 33.5418 33.6318 33.7219 33.8119 33.9020 33.9920 34.0820 34.1721 34.2621 34.3522 34.4422 34.5323 34.6223 34.7124 34.8024 34.8924 34.9825 35.0725 35.1626 35.2526 35.3427 35.4327 35.5228 35.6128 35.7029 35.7929 35.8829 35.9730 36.0630 36.1531

Amplitude (dB) 10.4522 10.2569 10.0330 9.7908 9.5455 9.3146 9.1153 8.9443 8.7813 8.6115 8.4264 8.2240 8.0083 7.7879 7.5745 7.3818 7.2090 7.0393 6.8646 6.6813 6.4901 6.2953 6.1040 5.9256 5.7702 5.6469 5.5437 5.4522 5.3686 5.2915 5.2215 5.1614 5.1156 5.0901 5.0923 5.1280 5.1912 5.2750 5.3739 5.4837 5.6017 5.7273 5.8613 6.0071 6.1701 6.3578

Phase ( ) 118.2896 125.5165 132.4290 138.9429 145.0011 150.5768 155.7810 161.1645 166.7419 172.4248 178.1112 176.3027 170.9113 165.7898 160.9902 156.5403 152.1309 147.5443 142.8685 138.1916 133.5957 129.1514 124.9138 120.9219 117.1985 113.7108 110.1400 106.4599 102.7192 98.9659 95.2449 91.5968 88.0556 84.6485 81.3954 78.2070 74.8311 71.2725 67.5546 63.7053 59.7567 55.7436 51.7017 47.6652 43.6644 39.5284

(Continues)

Antenna measured values in anechoic chamber

327

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 36.2431 36.3332 36.4232 36.5133 36.6033 36.6933 36.7834 36.8734 36.9635 37.0535 37.1436 37.2336 37.3237 37.4137 37.5038 37.5938 37.6838 37.7739 37.8639 37.9540 38.0440 38.1341 38.2241 38.3142 38.4042 38.4942 38.5843 38.6743 38.7644 38.8544 38.9445 39.0345 39.1246 39.2146 39.3047 39.3947 39.4847 39.5748 39.6648 39.7549 39.8449 39.9350 40.0250 40.1151 40.2051 40.2951

Amplitude (dB) 6.5653 6.7789 6.9852 7.1717 7.3288 7.4504 7.5355 7.5884 7.6183 7.6261 7.5935 7.5083 7.3655 7.1678 6.9246 6.6500 6.3603 6.0719 5.8000 5.5387 5.2691 4.9912 4.7077 4.4228 4.1420 3.8715 3.6173 3.3858 3.1826 3.0049 2.8356 2.6715 2.5114 2.3549 2.2031 2.0577 1.9214 1.7973 1.6889 1.5983 1.5153 1.4344 1.3525 1.2676 1.1792 1.0878

Phase ( ) 35.0053 30.1016 24.8440 19.2812 13.4840 7.5408 1.5489 4.3971 10.2175 16.1416 22.3601 28.7461 35.1569 41.4531 47.5168 53.2641 58.6483 63.6574 68.3063 72.7997 77.2988 81.7432 86.0828 90.2796 94.3084 98.1562 101.8207 105.3089 108.6350 111.9002 115.2554 118.6703 122.1124 125.5520 128.9641 132.3277 135.6275 138.8531 141.9994 145.0987 148.3197 151.6582 155.0806 158.5526 162.0411 165.5153

(Continues)

328

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

40.3852 40.4752 40.5653 40.6553 40.7454 40.8354 40.9255 41.0155 41.1056 41.1956 41.2856 41.3757 41.4657 41.5558 41.6458 41.7359 41.8259 41.9160 42.0060 42.0960 42.1861 42.2761 42.3662 42.4562 42.5463 42.6363 42.7264 42.8164 42.9065 42.9965 43.0865 43.1766 43.2666 43.3567 43.4467 43.5368 43.6268 43.7169 43.8069 43.8969 43.9870 44.0770 44.1671 44.2571 44.3472 44.4372

0.9951 0.9036 0.8165 0.7379 0.6642 0.5887 0.5084 0.4218 0.3288 0.2301 0.1279 0.0248 0.0754 0.1687 0.2520 0.3317 0.4109 0.4908 0.5717 0.6529 0.7332 0.8108 0.8830 0.9471 0.9995 1.0388 1.0692 1.0928 1.1110 1.1248 1.1345 1.1399 1.1403 1.1346 1.1211 1.0975 1.0612 1.0128 0.9550 0.8905 0.8212 0.7491 0.6752 0.6002 0.5242 0.4466

168.9476 172.3147 175.5983 178.7863 177.9910 174.6612 171.2601 167.8242 164.3886 160.9862 157.6461 154.3931 151.2466 148.2209 145.2977 142.3272 139.3082 136.2668 133.2282 130.2161 127.2518 124.3542 121.5386 118.8175 116.2001 113.6397 111.0233 108.3600 105.6646 102.9524 100.2388 97.5390 94.8673 92.2373 89.6614 87.1502 84.6512 82.0657 79.3996 76.6626 73.8666 71.0252 68.1538 65.2694 62.3896 59.5325

(Continues)

Antenna measured values in anechoic chamber

329

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( )

Amplitude (dB)

Phase ( )

44.5273 44.6173 44.7074 44.7974 44.8874 44.9775 45.0675 45.1576 45.2476 45.3377 45.4277 45.5178 45.6078 45.6978 45.7879 45.8779 45.9680 46.0580 46.1481 46.2381 46.3282 46.4182 46.5083 46.5983 46.6883 46.7784 46.8684 46.9585 47.0485 47.1386 47.2286 47.3187 47.4087 47.4987 47.5888 47.6788 47.7689 47.8589 47.9490 48.0390 48.1291 48.2191 48.3092 48.3992 48.4892 48.5793

0.3662 0.2794 0.1833 0.0817 0.0217 0.1235 0.2209 0.3119 0.3952 0.4704 0.5380 0.5997 0.6585 0.7193 0.7792 0.8349 0.8837 0.9237 0.9539 0.9745 0.9864 0.9916 0.9928 0.9935 0.9984 1.0080 1.0199 1.0325 1.0447 1.0560 1.0662 1.0760 1.0866 1.0996 1.1170 1.1418 1.1794 1.2293 1.2892 1.3569 1.4306 1.5088 1.5905 1.6749 1.7617 1.8512

56.7162 53.9019 50.9703 47.9240 44.7744 41.5372 38.2322 34.8826 31.5145 28.1558 24.8356 21.5822 18.3974 15.1386 11.7911 8.3734 4.9079 1.4200 2.0624 5.5106 8.8961 12.1918 15.3730 18.4181 21.4072 24.4309 27.4783 30.5361 33.5895 36.6222 39.6177 42.5591 45.4301 48.2153 50.9005 53.4788 56.0556 58.6723 61.3267 64.0141 66.7278 69.4590 72.1971 74.9300 77.6442 80.3250

(Continues)

330

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 48.6693 48.7594 48.8494 48.9395 49.0295 49.1196 49.2096 49.2996 49.3897 49.4797 49.5698 49.6598 49.7499 49.8399 49.9300 50.0200 50.1101 50.2001 50.2901 50.3802 50.4702 50.5603 50.6503 50.7404 50.8304 50.9205 51.0105 51.1006 51.1906 51.2806 51.3707 51.4607 51.5508 51.6408 51.7309 51.8209 51.9110 52.0010 52.0910 52.1811 52.2711 52.3612 52.4512 52.5413 52.6313 52.7214

Amplitude (dB) 1.9438 2.0415 2.1534 2.2793 2.4158 2.5596 2.7074 2.8562 3.0032 3.1463 3.2836 3.4140 3.5373 3.6540 3.7734 3.8981 4.0234 4.1452 4.2593 4.3623 4.4517 4.5258 4.5841 4.6270 4.6565 4.6752 4.6882 4.7021 4.7153 4.7254 4.7311 4.7313 4.7256 4.7145 4.6989 4.6802 4.6603 4.6418 4.6273 4.6215 4.6284 4.6463 4.6738 4.7095 4.7523 4.8013

Phase ( ) 82.9575 85.5340 88.1601 90.8760 93.6831 96.5794 99.5598 102.6148 105.7314 108.8925 112.0774 115.2625 118.4215 121.5276 124.6758 127.9419 131.3203 134.7998 138.3632 141.9880 145.6464 149.3068 152.9349 156.4956 159.9543 163.2787 166.4778 169.6926 172.9297 176.1754 179.4135 177.3739 174.2059 171.1021 168.0813 165.1618 162.3603 159.6918 157.1697 154.7569 152.3511 149.9481 147.5495 145.1585 142.7796 140.4189

(Continues)

Antenna measured values in anechoic chamber

331

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 52.8114 52.9015 52.9915 53.0815 53.1716 53.2616 53.3517 53.4417 53.5318 53.6218 53.7119 53.8019 53.8919 53.9820 54.0720 54.1621 54.2521 54.3422 54.4322 54.5223 54.6123 54.7024 54.7924 54.8824 54.9725 55.0625 55.1526 55.2426 55.3327 55.4227 55.5128 55.6028 55.6928 55.7829 55.8729 55.9630 56.0530 56.1431 56.2331 56.3232 56.4132 56.5033 56.5933 56.6833 56.7734 56.8634

Amplitude (dB) 4.8560 4.9159 4.9811 5.0517 5.1282 5.2116 5.3053 5.4188 5.5511 5.7000 5.8635 6.0397 6.2265 6.4219 6.6242 6.8315 7.0421 7.2547 7.4680 7.6814 7.9078 8.1544 8.4176 8.6938 8.9789 9.2681 9.5566 9.8392 10.1107 10.3662 10.6014 10.8130 10.9994 11.1604 11.3100 11.4518 11.5804 11.6910 11.7794 11.8428 11.8795 11.8893 11.8739 11.8359 11.7794 11.7094

Phase ( ) 138.0836 135.7819 133.5232 131.3172 129.1746 127.1065 125.1004 123.0701 121.0003 118.8855 116.7216 114.5066 112.2402 109.9242 107.5630 105.1630 102.7339 100.2877 97.8395 95.4067 92.9201 90.3120 87.5681 84.6763 81.6276 78.4172 75.0458 71.5208 67.8576 64.0803 60.2222 56.3249 52.4370 48.6089 44.7328 40.7338 36.6256 32.4301 28.1772 23.9031 19.6491 15.4586 11.3744 7.4362 3.6789 0.1310

(Continues)

332

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 56.9535 57.0435 57.1336 57.2236 57.3137 57.4037 57.4937 57.5838 57.6738 57.7639 57.8539 57.9440 58.0340 58.1241 58.2141 58.3042 58.3942 58.4842 58.5743 58.6643 58.7544 58.8444 58.9345 59.0245 59.1146 59.2046 59.2946 59.3847 59.4747 59.5648 59.6548 59.7449 59.8349 59.9250 60.0150 60.1051 60.1951 60.2851 60.3752 60.4652 60.5553 60.6453 60.7354 60.8254 60.9155 61.0055

Amplitude (dB) 11.6311 11.5502 11.4736 11.4064 11.3476 11.2963 11.2516 11.2131 11.1805 11.1537 11.1328 11.1184 11.1108 11.1110 11.1199 11.1386 11.1685 11.2146 11.2842 11.3758 11.4878 11.6186 11.7668 11.9310 12.1098 12.3020 12.5063 12.7217 12.9469 13.1812 13.4236 13.6734 13.9312 14.2159 14.5335 14.8827 15.2619 15.6689 16.1012 16.5555 17.0274 17.5113 18.0004 18.4861 18.9586 19.4070

Phase ( ) 3.1859 6.2561 9.1237 11.9248 14.6680 17.3535 19.9797 22.5439 25.0423 27.4702 29.8226 32.0938 34.2780 36.3692 38.3611 40.2474 42.0215 43.7142 45.4034 47.0990 48.8071 50.5325 52.2792 54.0495 55.8448 57.6650 59.5086 61.3727 63.2525 65.1414 67.0310 68.9104 70.7712 72.6912 74.7191 76.8771 79.1889 81.6798 84.3766 87.3068 90.4980 93.9762 97.7627 101.8711 106.3023 111.0394

(Continues)

Antenna measured values in anechoic chamber

333

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 61.0955 61.1856 61.2756 61.3657 61.4557 61.5458 61.6358 61.7259 61.8159 61.9060 61.9960 62.0860 62.1761 62.2661 62.3562 62.4462 62.5363 62.6263 62.7164 62.8064 62.8965 62.9865 63.0765 63.1666 63.2566 63.3467 63.4367 63.5268 63.6168 63.7069 63.7969 63.8869 63.9770 64.0670 64.1571 64.2471 64.3372 64.4272 64.5173 64.6073 64.6973 64.7874 64.8774 64.9675 65.0575 65.1476

Amplitude (dB) 19.8206 20.1893 20.5062 20.7840 21.0228 21.2055 21.3173 21.3472 21.2912 21.1521 20.9395 20.6677 20.3530 20.0116 19.6581 19.3049 18.9618 18.6367 18.3355 18.0627 17.8141 17.5838 17.3701 17.1718 16.9881 16.8183 16.6618 16.5184 16.3879 16.2703 16.1657 16.0743 15.9964 15.9326 15.8832 15.8489 15.8304 15.8286 15.8465 15.8840 15.9395 16.0116 16.0987 16.1995 16.3128 16.4372

Phase ( ) 116.0432 121.2496 126.5719 132.1382 138.0788 144.3446 150.8481 157.4684 164.0638 170.4909 176.6243 177.6295 172.3273 167.4919 163.1193 159.1876 155.6647 152.5139 149.6980 147.1813 144.8271 142.5442 140.3287 138.1780 136.0905 134.0646 132.0997 130.1950 128.3499 126.5643 124.8375 123.1694 121.5596 120.0076 118.5132 117.0760 115.6956 114.3707 113.0396 111.6634 110.2397 108.7664 107.2414 105.6633 104.0310 102.3436

(Continues)

334

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 65.2376 65.3277 65.4177 65.5078 65.5978 65.6878 65.7779 65.8679 65.9580 66.0480 66.1381 66.2281 66.3182 66.4082 66.4983 66.5883 66.6783 66.7684 66.8584 66.9485 67.0385 67.1286 67.2186 67.3087 67.3987 67.4887 67.5788 67.6688 67.7589 67.8489 67.9390 68.0290 68.1191 68.2091 68.2991 68.3892 68.4792 68.5693 68.6593 68.7494 68.8394 68.9295 69.0195 69.1096 69.1996 69.2896

Amplitude (dB) 16.5717 16.7151 16.8662 17.0240 17.1876 17.3561 17.5286 17.7044 17.8830 18.0638 18.2468 18.4400 18.6470 18.8644 19.0883 19.3145 19.5382 19.7544 19.9578 20.1429 20.3048 20.4388 20.5413 20.6100 20.6439 20.6437 20.6115 20.5509 20.4662 20.3625 20.2451 20.1180 19.9797 19.8299 19.6694 19.4990 19.3201 19.1342 18.9432 18.7487 18.5524 18.3563 18.1618 17.9706 17.7841 17.6035

Phase ( ) 100.6009 98.8034 96.9518 95.0479 93.0940 91.0933 89.0499 86.9688 84.8556 82.7170 80.5599 78.3055 75.8847 73.2867 70.5024 67.5252 64.3518 60.9836 57.4282 53.6997 49.8204 45.8202 41.7358 37.6098 33.4874 29.4146 25.4343 21.5848 17.8974 14.3959 11.0965 7.9689 4.8832 1.8441 1.1327 4.0328 6.8441 9.5568 12.1634 14.6584 17.0387 19.3027 21.4505 23.4834 25.4038 27.2147

(Continues)

Antenna measured values in anechoic chamber

335

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 69.3797 69.4697 69.5598 69.6498 69.7399 69.8299 69.9200 70.0100 70.1001 70.1901 70.2801 70.3702 70.4602 70.5503 70.6403 70.7304 70.8204 70.9105 71.0005 71.0905 71.1806 71.2706 71.3607 71.4507 71.5408 71.6308 71.7209 71.8109 71.9009 71.9910 72.0810 72.1711 72.2611 72.3512 72.4412 72.5313 72.6213 72.7114 72.8014 72.8914 72.9815 73.0715 73.1616 73.2516 73.3417 73.4317

Amplitude (dB) 17.4302 17.2651 17.1092 16.9633 16.8283 16.7049 16.5937 16.4950 16.4059 16.3253 16.2528 16.1879 16.1304 16.0798 16.0359 15.9984 15.9673 15.9422 15.9232 15.9100 15.9027 15.9011 15.9053 15.9152 15.9309 15.9524 15.9798 16.0132 16.0527 16.0984 16.1505 16.2092 16.2761 16.3540 16.4423 16.5408 16.6489 16.7664 16.8929 17.0281 17.1717 17.3233 17.4827 17.6496 17.8238 18.0049

Phase ( ) 28.9197 30.5228 32.0281 33.4399 34.7621 35.9989 37.1541 38.2369 39.2961 40.3424 41.3758 42.3961 43.4034 44.3973 45.3775 46.3438 47.2956 48.2325 49.1541 50.0599 50.9493 51.8217 52.6766 53.5133 54.3312 55.1297 55.9080 56.6654 57.4012 58.1145 58.8045 59.4704 60.1227 60.7822 61.4507 62.1295 62.8197 63.5224 64.2386 64.9695 65.7160 66.4790 67.2594 68.0580 68.8757 69.7131

(Continues)

336

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 73.5218 73.6118 73.7019 73.7919 73.8819 73.9720 74.0620 74.1521 74.2421 74.3322 74.4222 74.5123 74.6023 74.6923 74.7824 74.8724 74.9625 75.0525 75.1426 75.2326 75.3227 75.4127 75.5028 75.5928 75.6828 75.7729 75.8629 75.9530 76.0430 76.1331 76.2231 76.3132 76.4032 76.4932 76.5833 76.6733 76.7634 76.8534 76.9435 77.0335 77.1236 77.2136 77.3036 77.3937 77.4837 77.5738

Amplitude (dB) 18.1926 18.3868 18.5872 18.7934 19.0052 19.2222 19.4443 19.6711 19.9023 20.1376 20.3766 20.6192 20.8649 21.1135 21.3697 21.6420 21.9300 22.2337 22.5527 22.8867 23.2351 23.5971 23.9719 24.3580 24.7538 25.1569 25.5644 25.9727 26.3771 26.7721 27.1512 27.5071 27.8317 28.1170 28.3555 28.5409 28.6690 28.7382 28.7497 28.7074 28.6174 28.4870 28.3244 28.1375 27.9339 27.7202

Phase ( ) 70.5708 71.4495 72.3496 73.2715 74.2155 75.1815 76.1697 77.1796 78.2110 79.2632 80.3352 81.4258 82.5336 83.6566 84.8111 86.0311 87.3254 88.7032 90.1748 91.7517 93.4462 95.2722 97.2447 99.3800 101.6956 104.2096 106.9407 109.9066 113.1236 116.6043 120.3556 124.3761 128.6535 133.1621 137.8621 142.6993 147.6084 152.5176 157.3546 162.0526 166.5558 170.8219 174.8231 178.5452 178.0149 174.8517

(Continues)

Antenna measured values in anechoic chamber

337

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 77.6638 77.7539 77.8439 77.9340 78.0240 78.1141 78.2041 78.2941 78.3842 78.4742 78.5643 78.6543 78.7444 78.8344 78.9245 79.0145 79.1046 79.1946 79.2846 79.3747 79.4647 79.5548 79.6448 79.7349 79.8249 79.9150 80.0050 80.0950 80.1851 80.2751 80.3652 80.4552 80.5453 80.6353 80.7254 80.8154 80.9054 80.9955 81.0855 81.1756 81.2656 81.3557 81.4457 81.5358 81.6258 81.7159

Amplitude (dB) 27.5021 27.2835 27.0606 26.8330 26.6017 26.3678 26.1327 25.8973 25.6628 25.4302 25.2004 24.9743 24.7525 24.5359 24.3249 24.1199 23.9215 23.7299 23.5455 23.3683 23.1987 23.0367 22.8824 22.7359 22.5973 22.4666 22.3437 22.2286 22.1215 22.0221 21.9305 21.8466 21.7705 21.7020 21.6411 21.5878 21.5420 21.5037 21.4728 21.4494 21.4334 21.4248 21.4235 21.4295 21.4429 21.4636

Phase ( ) 171.9535 169.2904 166.7317 164.2563 161.8699 159.5767 157.3792 155.2785 153.2746 151.3665 149.5523 147.8296 146.1955 144.6466 143.1795 141.7905 140.4760 139.2323 138.0558 136.9431 135.8907 134.8955 133.9544 133.0646 132.2231 131.4276 130.6756 129.9648 129.2931 128.6586 128.0595 127.4941 126.9607 126.4581 125.9848 125.5397 125.1216 124.7295 124.3624 124.0196 123.7002 123.4037 123.1293 122.8766 122.6451 122.4345

(Continues)

338

Terahertz dielectric resonator antennas

(Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 81.8059 81.8960 81.9860 82.0760 82.1661 82.2561 82.3462 82.4362 82.5263 82.6163 82.7064 82.7964 82.8864 82.9765 83.0665 83.1566 83.2466 83.3367 83.4267 83.5168 83.6068 83.6968 83.7869 83.8769 83.9670 84.0570 84.1471 84.2371 84.3272 84.4172 84.5073 84.5973 84.6873 84.7774 84.8674 84.9575 85.0475 85.1376 85.2276 85.3177 85.4077 85.4977 85.5878 85.6778 85.7679 85.8579

Amplitude (dB) 21.4928 21.5332 21.5845 21.6460 21.7173 21.7980 21.8876 21.9857 22.0920 22.2062 22.3279 22.4569 22.5929 22.7356 22.8848 23.0404 23.2020 23.3695 23.5427 23.7214 23.9054 24.0947 24.2889 24.4881 24.6919 24.9003 25.1132 25.3303 25.5516 25.7769 26.0061 26.2390 26.4755 26.7155 26.9588 27.2053 27.4549 27.7073 27.9626 28.2204 28.4807 28.7433 29.0080 29.2746 29.5431 29.8131

Phase ( ) 122.2429 122.0654 121.9006 121.7474 121.6047 121.4717 121.3477 121.2318 121.1236 121.0224 120.9278 120.8392 120.7564 120.6789 120.6064 120.5386 120.4753 120.4160 120.3608 120.3092 120.2612 120.2165 120.1750 120.1366 120.1010 120.0681 120.0379 120.0103 119.9850 119.9620 119.9413 119.9227 119.9061 119.8915 119.8788 119.8679 119.8587 119.8513 119.8454 119.8412 119.8384 119.8371 119.8371 119.8385 119.8412 119.8452

(Continues)

Antenna measured values in anechoic chamber (Continued) Vertical port reading for directivity: Frequency 5.8 GHz Elevation ( ) 85.9480 86.0380 86.1281 86.2181 86.3082 86.3982 86.4882 86.5783 86.6683 86.7584 86.8484 86.9385 87.0285 87.1186 87.2086 87.2987 87.3887 87.4787 87.5688 87.6588 87.7489 87.8389 87.9290 88.0190 88.1091 88.1991 88.2891 88.3792 88.4692 88.5593 88.6493 88.7394 88.8294 88.9195 89.0095 89.0995 89.1896 89.2796 89.3697 89.4597 89.5498 89.6398 89.7299 89.8199 89.9100 90.0000

Amplitude (dB) 30.0846 30.3573 30.6310 30.9054 31.1805 31.4559 31.7315 32.0069 32.2819 32.5562 32.8296 33.1018 33.3724 33.6411 33.9077 34.1717 34.4328 34.6908 34.9451 35.1954 35.4413 35.6824 35.9182 36.1485 36.3726 36.5903 36.8010 37.0044 37.1999 37.3871 37.5656 37.7350 37.8948 38.0447 38.1842 38.3130 38.4306 38.5369 38.6314 38.7139 38.7841 38.8419 38.8870 38.9193 38.9388 38.9453

Phase ( ) 119.8503 119.8565 119.8638 119.8721 119.8814 119.8917 119.9027 119.9146 119.9273 119.9407 119.9547 119.9693 119.9845 120.0001 120.0162 120.0326 120.0494 120.0663 120.0835 120.1007 120.1181 120.1354 120.1526 120.1697 120.1866 120.2032 120.2195 120.2355 120.2509 120.2659 120.2803 120.2940 120.3071 120.3194 120.3309 120.3416 120.3515 120.3604 120.3683 120.3753 120.3812 120.3861 120.3900 120.3927 120.3944 120.3949

339

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Appendix D

Dielectric materials and resources

Absorber: The quality factor is Q ¼ (wWe)/(Prad). Hence, absorbers are dielectric resonator antennas (DRAs) with standing wave and if these standing waves are converted into travelling waves, these absorbers are converted into antennas using excitation into DRAs. Absorbers have high-Q factor and high-energy storage; if accelerating charges are created in a DR absorber, they behave like DRAs due to energy leakage into air. And thus quality factor of an absorber is reduced to act as antennas. Hence, the Q factor of an absorber is high and that of a DRA is low. The eigen state at resonance is absorber cavity and excited state of cavity is an antenna due to accelerating charge careers and simultaneously decreasing the quality factor too. The feed used at THz fDRA is Gaussian beam and feed is wave port for an absorber. For making an absorber, imaginary part must be high and second derivate image impedance w.r.t. frequency must be negative.

Permittivity Supplier or manufacturer

1

6.3

3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

9.5

Countis Laboratories 12295 Charles Dr, Grass Valley, CA 95945, United States Tel: þ1 530-272-8334 Email: [email protected]

Terahertz dielectric resonator antennas

2

MgO–SiO2 (CD-6) MgO–SiO2–TiO2 (CD-9) MgO–TiO2–SiO2 (CD-13) MgO–TiO2 (CD-15) MgO–TiO2 (CD-16) MgO–CaO–TiO2 (CD-18) MgO–CaO–TiO2 (CD-20) MgO–CaO–TiO2 (CD-30) MgO–CaO–TiO2 (CD-50) MgO–CaO–TiO2 (CD-100) MgO–CaO–TiO2 (CD-140) Boron nitride (ECCOSTOCK@) Beryllium oxide (ECCOSTOCK@) Magnesium oxide (ECCOSTOCK@) Magnesium titanate (ECCOSTOCK@) Zirconia (ECCOSTOCK@) Titanium dioxide (rutile) (ECCOSTOCK@) Strontium titanate (ECCOSTOCK@)

342

S. no. Materials

13.0 15.0 16.0 18.0 20.0 30.0 50.0 100.0 140.0 4.0 6.0 9.0 10.0

Emerson & Cuming Microwave Products N.V. A unit of Laird Technologies Hong Kong Holdings (4) Ltd. Unit 2507-8, 25/F, Office Tower, Langham Place, 8 Argyle Street, Mongkok, Kowloon, Hong Kong Tel: þ852-2923 0600, þ852-2923 0605 Email: [email protected]

20.0 50.0 >100.0

(Continues)

19 20 21 22 23 24 25 26 27

29 30 31 32 33 34 35

9.2 16.0 20.0 37.0 80–100 6.0 20.0 37.0

Hiltek Microwave Limited 15200 Shady Grove Road Suite 350, Rockville, MD 20850, United States Tel: þ1 (301) 670-2833 Fax: þ1 (301) 670-2831 Email: www.hiltek.com Morgan Advanced Materials 150 Kampong Ampat, 05-06A KA Centre, Singapore 368324, Singapore Tel: þ65 6595 0000 Fax: þ65 6595 0005 Email: [email protected]

76.5 6.5 9.5 12.0

Dielectric materials and resources

28

Pacific Ceramics, Inc. Advanced Microwave Ceramic Materials 824 San Aleso Ave Sunnyvale, CA 94085, USA Tel: þ1 (408) 747-4600 Email: [email protected]

13.0 15.0 16.0 18.0 25.0

(Continues)

343

Magnesium manganese aluminum iron ferrite Magnesium titanate Lithium ferrite Zirconium tin titanate Titania ceramic MgSi (steatite) (D6) CaMgTi (Mg, Ca, Ti) (D20) ZrTiSn (Zr, Sn, Ti) (D36) BaSmTi (Ba, Sm, Ti) (D37) Titanate with other ingredients (PD-6) Titanate with other ingredients (PD-9) Titanate with other ingredients (PD-12) Titanate with other ingredients (PD-13) Titanate with other ingredients (PD-15) Titanate with other ingredients (PD-16) Titanate with other ingredients (PD-18) Titanate with other ingredients (PD-25)

(Continued)

36

38.0

37 38 39 40 41 42 43 44 45

Titanate with other ingredients (PD-38) Titanate with other ingredients (PD-50) Titanate with other ingredients (PD-100) Titanate with other ingredients (PD-160) Titanate with other ingredients (PD-270) Zr, Sn, Ti oxide (E2000) E3000 Ba Zn Ta oxide (E4000) Ba, Sm, Ti oxide (E5000) Ti, Zr, Nb, Zn Oxide (E6000)

50.0 98.0 160.0 270.0 37.0 34.0 30.0 78.0 45.0

Temex Components & Temex Telecom, USA Supplier 1 SM CREATIVE No 845, 2nd Cross, 7th Main HAL 2nd Stage, Indiranagar, Bangalore, 560 038, India Tel: þ91 (80) 25210268, þ91 (80) 41255492 Mobile: þ91 (98) 45410417 Email: [email protected] http://www.smcel.com S M Creative Electronics Ltd. #10, Electronic City, Sector-18 Gurgaon 122 015, Haryana Tel: þ91 124-4909850 Fax: þ91 124-2455 212 Email: [email protected] Supplier 2 SIMAL # 60 & 60/1, 18th Cross, 4th Main, Malleswaram, Bangalore, 560 055, India Tel: þ91 (80) 41532079, þ91 (80) 23444410 Mobile: þ91 (99721) 24165 Email: [email protected] http://www.simal.com.sg

(Continues)

Terahertz dielectric resonator antennas

Permittivity Supplier or manufacturer

344

S. no. Materials

46

Trans-Tech Skyworks Solutions, Inc. 5520 Adamstown Road Adamstown, MD 21710 6.3 Supplier SM Electronic Technologies Pvt. Ltd. 15.0 #1790, 5th Main, 9th Cross, RPC Layout, Vijayanagar 2nd Stage, Bangalore 560 040, India 16.0 Mr. Manjunath 29.0–30.7 Tel: þ91-80-23301030 Email: [email protected] 29.5–31.0

54 55 56

Cordierite (Mg, Al, Si) (D-4) Forsterite (Mg, Si, oxide) (D-6) Mg–Ti (D-15) Mg–Ti (D-16) Ba, Zn, Ta oxide (D-29) Ba, Zn, Ta oxide (perovskite) (D-87) Ba, Zn, Co, Nb (D-83) Zirconium titanate based (D-43) E-11 E-20 E-37

57 58 59 60 61 62

TE-21 TE-30 TE-36 TE-45 TE-80 TE-90

21.0 30.0 36.0 45.0 80.0 90.0

47 48 49 50 51 52

35.0 – 36.5 44.7–46.2 11.0 20.0 37.0

T-CERAM, RF & Microwave Okruzˇnı´ 1144, 500 03 Hradec Kra´love´, Czech Republic, EU Tel: þ420 774 406 438CZ 42196078 Email: [email protected] www.t-ceram.com Token Electronics Industry Co., Ltd. No. 137, Sec. 1, Chung Shin Rd., Wu Ku Hsiang, Taipei Hsien, Taiwan, R.O.C. Tel: þ886-2-2981 0109 Fax: þ886-2-2988 7487 http://www.token.com.tw Email: [email protected]

(Continues)

Dielectric materials and resources

53

4.5

345

346

(Continued) Permittivity Supplier or manufacturer

63

20.0

64 65 66 67 68 69 70 71 72 73 74

Mg-Ca-Ti (MDR20) Ta with other ingredients (MDR24) Ta with other ingredients (MDR30) Zn–Sn–Ti (MDR38) La-Ba-Ti (MDR45) DR-30 DR-36 DR-45 DR-80 RT6010, RT-6002 MCT-25 SMAT BaTiO3 ECCOSTOCK’SHIK

24.0 30.0

MCV Microwave 6640 Lusk Blvd, Suite A102 San Diego, CA 92121, United States Tel: þ1 858-450-0468 Fax: þ1 858-869-8404 www.mcv-microwave.com

38.0 45.0 30.0 36.0 45.0 80.0 10.2 25 27 14 10,20,30,40

TCI Ceramics, Inc. 18450 Showalter Rd., Hagerstown, MD 21742, United States Tel: þ1 301-766-0560 Fax: þ1 301-766-0566 Email: [email protected] www.tciceramics.com Trans-Tech

Terahertz dielectric resonator antennas

S. no. Materials

Appendix E

Dual-band graphene antenna design and implementation

Dual-band graphene antenna design and implementation are described in Figures E.1–E3 and Table E.1. Theory: Consider a cavity resonator of one angstrom size, i.e. a cube with each side of length a ¼ 1010 m: The Maxwell equations in such a cube have solutions of the form: X cðmnp; tÞur ; mnpðx; y; zÞ; r ¼ 1; 2; 3 Ar ðt; x; y; zÞ ¼ mnp

where ur ; mnp are spatial functions obtained by integrating the electric field w:r:t: n mpx mpxo n npy npyo ; sin  cos ; sin cos a a a a n ppy ppzo  cos ; sin a a Multiplied by some constants depending on the indices ðm; n; pÞ: We may, without loss of generality, assume those are normalized so that ð ur ; mnpðrÞ
u s ; m0 n0 p0 ðrÞd 3 r ¼ drs dmm0 dnn0 dpp0 C

The dependence of cðmnp; tÞ on t is expðiwðmnpÞtÞ where wðmnpÞ are the characteristic frequencies of oscillation: pcpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 þ n2 þ p2 ; m; n; p ¼ 1; 2; . . . wðmnpÞ ¼ a which are of the order of magnitude w¼

pc a

Terahertz dielectric resonator antennas Ws1 ls2

Reflector

l1

Dipole

Director

ls1

l2

Wf

l3

lf w ws2

y

x

z

Figure E.1 Graphene antenna

z x

y

hs2

Substrate-2 hf

hs1

Feed line

Substrate-1 Ground plane

hf

Figure E.2 The geometry of a graphene antenna

0 –10 –20 S11 (dB)

348

–30 –40 –50 –60 –70 15.0

15.5

16.0 16.5 Freq (GHz)

17.0

Figure E.3 Simulated response of the S11 parameter

Dual-band graphene antenna design and implementation

349

Table E.1 The dimensions of the proposed antenna geometry Parameters

Dimensions (mm)

Substrate-1 width (ws1 Þ Substrate-1 length (ls1 Þ Substrate-1 height (hs1 Þ Substrate-2 width (ws2 Þ Substrate-2 length (ws2 Þ Substrate-2 height (hs2 Þ Feed-line length (lf Þ Feed-line width (wf Þ Feed-line thickness (tf) Ground-plane width (ws1 Þ Ground-plane length (ls1 Þ Ground-plane height Reflector length (l1 Þ Dipole length (l2 Þ Director length (l3 Þ Width of reflector ¼ width of dipole ¼ width of director (wÞ Thickness of reflector ¼ thickness of dipole ¼ thickness of director (tÞ

110 110 3.2 70 70 1.2 37 5 0.035 110 110 0.035 71 65 56 4 1 nm

Materials name Silicon dioxide Silicon dioxide Silicon dioxide Silicon dioxide Silicon dioxide Silicon dioxide Silver Silver Silver PEC PEC PEC Graphene Graphene Graphene Graphene

(SiO2 Þ (SiO2 Þ (SiO2 Þ (SiO2 Þ (SiO2 Þ (SiO2 Þ

Graphene

The electric field is X cðmnp; tÞiwðmnpÞur ; mnpðrÞ E r ¼ @t Ar ¼ mnp

The magnetic field is B ¼ curlA Þj which is of the order of magnitude jcðmnp;t ; where by cðmnp; tÞ, we actually mean a its average in a coherent state. The total electric field energy within the cavity C is  ð  0 Ej2 d 3 r UE ¼ 2 C

which has components of the order of magnitude 0 =wðmnpÞcðmnp; tÞj2 a3 ¼ 0 wðmnpÞ2 a3 jcðmnp; tÞj2 The total magnetic field energy within the cavity is Ð UB ¼ ð2m0 Þ1 c Bj2 d 3 r which has components of the order of magnitude   cðmnp; tÞ2 a3  2 a      m ¼ cðmnp; tÞj m a 0 0

350

Terahertz dielectric resonator antennas

The relation of the orders of magnitude of the electric field energy and the magnetic field energy within the cavity therefore has the order of magnitude UE w2 a2  m0 0 wðmnpÞ2 a2  2  1 UB c As expected, the canonical commutation relations are   ih 3 0 d ðr  r 0 Þ ½Ar ðt; rÞ; @t As ðt; r Þ ¼ 2p These fields are "

# 0 0 0



cr ðmnp; tÞ; wðmnpÞcs ðm n p ; t Þ ¼



 h drs dmm0 dnn0 dpp0 2p

So that the eigenvalues of cr ðmnp; tÞ cr ðmnp; tÞ are positive integer multiples of h=2pwðmnpÞ: This means that the field energy within the cavity when a finite number of modes are excited assumes eigenvalues that are of the same order of magnitude as positive integer multiples of hw=2p as expected by Planck’s quantum theory of radiation. This fact pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi also fulfills the result that jcðmnp; tÞj is of the order of magnitude of h=ð2pwÞ: Now we come to the question of computing the order of magnitude of the Poynting vector power flux at a given radial distance R from the quantum cavity antenna caused by the surface current density induced by the magnetic field on the antenna surface. The magnetic field on the surface and hence the corresponding induced surface current pdensity ffiffiffiffiffiffiffiffiffi both have the order of magnitudes of jcðmnp; tÞj=a that is of the order a1 h=w: Therefore, the far-field magnetic vector potential at a distance R frompthe is of the order of magnitude (use the retarded potential ffiffiffiffiffiffiffifficavity ffi formula) ða=RÞ h=w and hence the corresponding far-field-radiated magnetic field pffiffiffiffiffiffiffiffiffi is of the order of magnitude ðw=c Þ ð a=R Þ h=w while the near-field magnetic field of pffiffiffiffiffiffiffiffiffi 2 Þffiffiffiffi h=w: Actually, these expressions for the magnetic the order of magnitude ða=Rp field must be multiplied by N where N is a positive integer corresponding to the largest modal eigenvalue of the operators ð2pwðmnpÞ=hÞcðmnp; tÞ cðmnp; tÞ: The far-field Poynting vector has the order of magnitude of B2 c=2m0 that is of the order rffiffiffiffi!2        pffiffiffiffi w a  h c h w a2 N ¼ N 2m0 c R w 2m0 c R2 And the total power radiated outward by this quantum antenna in the far-field zone is thus of the order of magnitude   2  h aw P¼N ¼ 2m0 c

Dual-band graphene antenna design and implementation

351

Now we look at the order of magnitude of the power radiated in the far-field zone by the Dirac field of electrons and positrons within the cavity. The Dirac equation is  m ig @m  m yðxÞ ¼ 0 Or more precisely in arbitrary units,

       ih ih 2 @t  c a;  r  bmc yðxÞ ¼ 0 2p 2p the appearances of the constants h; m; c are explicitly shown. Now  Here, yðxÞj2 c is the probability density of the electron that must integrate to unity over the cavity volume. Thus yðxÞ is of the order of magnitude a3=2 . The Dirac current density J m ¼ ey g0 gm y has the same order of magnitude as eyðxÞj2 c; which is ec=a3 =2 : Therefore, the far-field magnetic vector potential at a radial distance of R from the cavity is, in accordance with the retarded potential theory of the order    3 ec a eca3=2 ¼  3=2 R R a The electric field in the far-field zone is then of the order eca3=2 R where w is the characteristic oscillation frequency of the Dirac current. The magnetic field is of the order rffiffiffi 3=2 a 1 eca Ba  ¼ ec R R E w

If P is the characteristic momentum of the electrons and positrons in a given state, for example P may be the average momentum of an electron in a given state, then according to de Broglie, P is of the order h=a since a is the order of the electron wavelength. Then the electron energy is of the order sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2 Ee ¼ c m2 c2 þ P2  c m2 c2 þ 2 a And the characteristic frequency of oscillation of the Dirac wave field is then Ee h The Poynting vector corresponding to the power radiated by the Dirac field in the far-field zone then has the order of magnitude   B2 e2 c3 a S  c 0 E 2 þ ¼ c3 0 w2 ea3 þ m0 m0 w¼

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Appendix F

Miniaturization design techniques

Abstract A novel technique is presented for a miniaturization rectangular dielectric resonator antenna (DRA). The miniaturized DRA is built by applying the metallic strips on the sidewalls of the DRA. The specific format of the applied metallic strips excites the generation of fundamental and higher order modes. The frequency ratio and aspect-ratio can be manipulated to obtain the right choice of aspect-ratio and then destined resonant modes. The frequency ratio is tuned by the right size of applied metallic strips at right locations in destined geometries. A small frequency ratio is optimized design and implementation requirements.

F.1

Introduction

A dielectric resonator antenna (DRA) is efficient technology and extends deign flexibilities with many other advantages as compared to metal or patch antennas. The rectangular DRA is simple in design and has been used to show miniaturization and compactness (Figure F.1). Here, metallic strips have been used to show the size reduction of DRA. High-permittivity materials of DRAs can also be used in size reductions. The use of high-permittivity materials is one of the methods used for miniaturization. The limitation of using high-permittivity materials is that it reduces bandwidth. The techniques like increasing the aspect-ratio for the enhancement of bandwidth using the high-permittivity DR by merging of modes, applying the metallic coating, the fundamental and third-order orthogonal degenerate modes were excited. The simulated and measured results on miniaturization techniques have been validated and found closely matching with expectations and simulated ones. Radiation pattern, S11, fields, higher order modes and tuned results are presented in this appendix (Figures F.2–F.8 and Table F.1).

354

Terahertz dielectric resonator antennas Face-1

h2

Face-3

w2 z y

x

h

z x

y w1

Face-2

h1

Face-4

Figure F.1 Rectangular DRA with metallic strips painted on its side walls. Dimensions (mm), face: 1—front, 2—right, 3—back and 4—left, copper color strips—at front face and blue color strips—at back face, wg ¼ lg ¼ 50; a ¼ b ¼ h ¼ 12; lm ¼ 25; s ¼ 5; wm ¼ 2:2; ls ¼ 9; ws ¼ 1, w1 ¼ w2 ¼ 3:6; h1 ¼ 3; h2 ¼ 1:3; h ¼ 12 0

S11 (dB)

–10

–20

x TE 113

TE y111

TE y111

TE x111

Without strips

–30

With strips –40 3.5

4

4.5

y TE 113

5 5.5 6 Freq (GHz)

y TE 113

6.5

7

Figure F.2 Three resonant modes

Figure F.3 Electric fields inside DRA

Miniaturization design techniques

Figure F.4 E Fields, three half-wave variations

Figure F.5 Prototype model of rectangular DRA

S11 (dB)

0

–10

–20 Simu –30 3.5

4

Meas

4.5 5 5.5 6 Frequency (GHz)

6.5

7

Figure F.6 Measured vs simulated results of DRA

355

356

Terahertz dielectric resonator antennas 0º 30º

–30º –60º

60º

–90º

90º

–120º

120º

–150º

150º 180º

Figure F.7 Radiation pattern of rectangular DRA

0

S11 (dB)

–10 –20 –30 –40 3.5

Without strips 3.6, 1.3 9, 2.5 Four strips 4

4.5

5 5.5 6 6.5 Frequency (GHz)

7

7.5

Figure F.8 Tuning of DRA using different lengths of strip lines

F.2

Conclusion

The metallic strips are applied in a specific format to get the miniaturization in the rectangular DRA. The sidewalls of the DR are divided into four layers. Each layer is having the height h=4. The height of strips at the lower three layers is selected as h1 ¼ h=4. The height of strips at the top layer near to the top edges of the DR is initially selected as h2 ¼ h=8.

Table F.1 Results with metallic strips Antenna

Operating modes

fr (GHz)

LB

UB

LB

DR without strips

y TE111

y TE113

4.75

DR with strips

y TE111 x TE111

y TE113 x TE113

3.88 4.46

3-dB AR bandwidth (MHz) LB

UB

10-dB impedance bandwidth (MHz)

UB

LB

UB

S

M

S

M

S

M

S

M

6.80

0

–

0

–

*

130 (4.34– 4.47)

160 (4.33– 4.49)

60 (6.55– 6.61)

90 (6.54– 6.63)

330 (6.64– 6.97) 390 (6.33– 6.72)

–

6.43 6.66

490 (4.47– 4.96) 910 (3.77– 4.68)

1 030 (3.78– 4.81)

410 (6.37– 6.78)

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Appendix G

Gaussian beam feed process

How to apply Gaussian beam into the feed to terahertz dielectric resonator antennas (TDRAs) (silver nano waveguide) and TDRAs. The steps have been presented bit-by-bit with screenshots. This will be helpful to beginners for ease of CST modeling. The following figure is a CDRA model. Now we need to apply “Gaussian Beam”:

So, follow the steps shown in the following figure:

360

Terahertz dielectric resonator antennas Step 1: Follow the steps and finally click on the “Gaussian Beam.” After clicking a new window will open as shown in the following:

Step 2: Modify the propagation vector (x/y/z): 0/1/0: Press OK.

Gaussian beam feed process

361

Step3: Perform Transform operation as shown in the following:

This is the final model with Gaussian Beam applied, and now simulation can be started:

CST Model simulations with Gaussian Beam. With Gaussian Beam:

362

Terahertz dielectric resonator antennas 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z E Field ( f =192.68) (fsGBMacro_633.000 nm) Component Abs Frequency 192.68 THz Phase 191.25 Maximum 0 dB

y

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z E Field (f=195) (fsGBMacro_633.000 nm) Component Abs Frequency 195 THz Phase 191.25 Maximum 0 dB

y

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z E Field (f=200) (fsGBMacro_633.000 nm) Component Abs Frequency 200 THz Phase 191.25 Maximum 0 dB

y

x

Gaussian beam feed process

363 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z

y

H Field (f=192.68) (fsGBMacro_633.000 nm) Component Abs Frequency 192.68 THz Phase 191.25 Maximum 0 dB

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z H Field (f=195) (fsGBMacro_633.000 nm) Component Abs Frequency 195 THz Phase 191.25 Maximum 0 dB

y

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z H Field (f=200) (fsGBMacro_633.000 nm) Component Abs Frequency 200 THz Phase 191.25 Maximum 0 dB

y

x

364

Terahertz dielectric resonator antennas

Showing Super Directivity of DRA-radiated beam in the opposite direction of input applied (laser outputs):

11 10.3 9.65 8.96 8.27 7.58 6.89 6.2 5.51 4.82 4.13 3.44 2.76 2.07 1.38 0.689 0

Fer field (f=193.5) (fsGBMacro_633.000 nm) Type Farfield Approximation enabled (kR >> 1) Component Abs Output Directivity Frequency 193.5 THz Dir 11.02

z

y

x

Far-field radiations with Gaussian Beam input excitation:

S-Parameters (magnitude in dB)

–10

S11

–15

dB

–20 –25 –30 –35 –40 180

185

190

195 Frequency (THz)

200

205

210

Gaussian beam feed process

365 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

E field (f=192.68) (1) Component Abs Frequency 192.68 THz 0 Phase Maximum 0 dB

y

z

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

y

z E field (f=195) (1) Component Abs Frequency 195 THz 0 Phase Maximum 0 dB

x

H Field: 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

y

z H field (f=192.68) (1) Component Abs Frequency 192.68 THz 0 Phase Maximum 0 dB

x

366

Terahertz dielectric resonator antennas 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

y

z

H field (f=195) (1) Component Abs Frequency 195 THz 0 Phase Maximum 0 dB

x

Radiation pattern with lumped port:

5.31 2.81 0.308 –2.19 –4.69 –7.19 –9.69 –12.2 –14.7 –17.2 –19.7 –22.2 –24.7 –27.2 –29.7 –32.2 –34.7

Fer field (f=193.5) (1) Type Farfield Approximation enabled (kR >> 1) Component Abs Output Gain Frequency 193.5 THz Rad.effic. –2.884 dB Tot. effic –2.896 dB Gain 5.308 dB

Far-field radiation with lumped-port input excitation:

z y x

Gaussian beam feed process

367 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z

y

E field (f=192.68) (fsGBMacro_633.000 nm) Component Abs Frequency 192.68 THz Phase 191.25 Maximum 0 dB

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z E field (f=195) (fsGBMacro_633.000 nm) Component Abs Frequency 195 THz Phase 191.25 Maximum 0 dB

y

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z E field (f=200) (fsGBMacro_633.000 nm) Component Abs Frequency 200 THz Phase 191.25 Maximum 0 dB

y

x

368

Terahertz dielectric resonator antennas 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z H field (f=192.68) (fsGBMacro_633.000 nm) Component Abs Frequency 192.68 THz Phase 191.25 Maximum 0 dB

y

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z H field (f=195) (fsGBMacro_633.000 nm) Component Abs Frequency 195 THz Phase 191.25 Maximum 0 dB

y

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

z H field (f=200) (fsGBMacro_633.000 nm) Component Abs Frequency 200 THz Phase 191.25 Maximum 0 dB

y

x

Gaussian beam feed process

369 11 10.3 9.65 8.96 8.27 7.58 6.89 6.2 5.51 4.82 4.13 3.44 2.76 2.07 1.38 0.689 0

z

Fer field (f=193.5) (fsGBMacro_633.000nm) Type Farfield Approximation enabled (kR >> 1) Component Abs Output Directivity Frequency 193.5 THz Dir 11.02

y

x

With waveguide port:

S-Parameters (magnitude in dB)

–10

S11

–15

dB

–20 –25 –30 –35 –40 180

185

190

195 Frequency (THz)

200

205

210

370

Terahertz dielectric resonator antennas 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

E field (f=192.68) (1) Component Abs Frequency 192.68 THz Phase 0 Maximum 0 dB

y

z

x

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

y

z E field (f=195) (1) Component Abs Frequency 195 THz Phase 0 Maximum 0 dB

x

H Field:

0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

y

z H field (f=192.68) (1) Component Abs Frequency 192.68 THz 0 Phase Maximum 0 dB

x

Gaussian beam feed process

371 0 –4 –8 –12 –16 –20 –24 –28 –32 –36 –40

H field (f=195) (1) Component Abs Frequency 195 THz 0 Phase Maximum 0 dB

Fer field ( f =193.5) (1) Type Farfield Approximation enabled (kR >> 1) Component Abs Gain Output Frequency 193.5 THz Rad. effic. –2.884 dB Tot. effic. –2.896 dB Gain 5.308 dB

y

z

x

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Appendix H

Silicon dielectric resonator antenna at 5-THz frequency

 r  r 2  c 6:324 p ffiffiffiffiffiffiffiffiffiffiffiffi 0:27 þ 0:36 þ 0:02 fr ¼ 2pr er þ 2 2h 2h

H.1 THz DRA fabrication process Clean room for micromachining, spectroscopy, oscilloscope, spectrometer, dry and wet etching, angstrom engineering electron beam lithography, sputtering system, masking system, spin coater, bake plate, surface roughness tester, photo-resist chemical, microscope, spectrometer, signal generator, laser source, VNA and extension head (10 MHz to 500 THz). An SU-8 glue-type film or layer is used. It is used þ1-mm precision in fabrication. A .GDS II design file is compatible with foundry/silicon CDRA on a silver surface and an SiO2 substrate. We intend to test this using terahertz TDS in reflection mode. Terahertz timedomain spectroscopy is used for testing. The terahertz time-domain spectroscopy system operates both in transmission and reflection modes. It will be a costly process. We can process on a 300 -wafer, and so, we may end up many samples for testing. Use of complex Si binding process makes it hard. For terahertz, we use highresistivity Si (intrinsic, float zone Si). Also, we use materials like TiO2 for optical frequencies. By the sample size and integration (spot size 20 mm, 0.3–7.0 THz), not only by VNA, we can test in free-space mode at 5-THz frequency. The process may take about 4–6 weeks to fabricate once we finalize the design. To combine single crystal Si with SiO2, it is a very complicated process. For it to work at terahertz, Si has to be intrinsic Si (no impurities), and so, cannot bond to SiO2 by typical techniques (like anodic bonding) and needs a very thin cross-linking polymer (imagine it is like glue). This is the role of the thin SU-8. For terahertz, we use high-resistivity Si (intrinsic, float zone Si). The materials like TiO2 are used at optical frequencies. The use of silver nano feed line is important. Both free-space testing capabilities (spot size 20 mm, 0.3–7.0 THz) and VNA to directly probe and test at 5 THz are important testing capabilities. Once we finalize the design, we can

374

Terahertz dielectric resonator antennas

then get the GDSII file. To combine single-crystal Si with SiO2, it is very complicated. For it to work at terahertz, the Si has to be intrinsic Si (no impurities), and so, cannot bond to SiO2 by typical techniques (like anodic bonding) and needs a very thin cross-linking polymer (imagine it is like glue). This is the role of the thin SU-8 (Figure H.1 and Table H.1).

ws

r

ls

lf wf

zy

DR

h3 Substrate-2 hg

x

y

h

Substrate-3 h2 Substrate-1 Ground plane

h1

Figure H.1 The geometry of the proposed antenna Table H.1 The dimensions of the proposed antenna geometry Parameters

Dimensions (mm)

Materials name

Substrate width (ws Þ Substrate length (ls Þ Feed-line length (lf Þ Feed-line width (wf Þ Radius (rÞ Ground height (hg Þ Substrate-1 height (h1 Þ Substrate-2 height ðh2 Þ Substrate-3 height (h3 Þ Dielectric resonator (hÞ

50 50 26.5 3 12 0.035 1.6 0.2 0.1 12

Silicon dioxide Silicon dioxide Silver ðAgÞ Silver ðAgÞ Silicon ðSiÞ Silicon (Si) Silicon dioxide Silicon dioxide Silicon dioxide Silicon ðSiÞ

ðSiO2 Þ ðSiO2 Þ

ðSiO2 Þ ðSiO2 Þ ðSiO2 Þ

Silicon dielectric resonator antenna at 5-THz frequency 0 –5 –10 –15 –20 –25 –30 –35 –40 –45 –40

375

S-Parameters (magnitude in dB) S11

4

4.2

4.4

4.6

4.8

5 5.2 Frequency (THz)

5.4

5.6

Simulated S11 parameter at 4.92 THz

5.8

6

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Appendix I

DRA designing process

I.1 Design process of aperture coupled DRA Dielectric resonator antenna (DRA) has low-loss and high-radiation efficiency. A DRA can excite multiple modes simultaneously using aspect-ratio between 0.5 and 2.5. They are most stable due to dielectric materials. DRA material is available in different dielectric constant values, i.e. 10–100. They can generate TE/TM/TEM modes. They are best candidates for getting multimode and in turn multiband antennas. These are basic requirements for any vehicular wireless communication system. An antenna designer can build an antenna of his choice of operating frequencies. Let resonant frequency be 3.7 GHz, resonant mode be TE111 (fundamental mode), impedance 50 W and substrate used FR4,
r ¼ 4:4; height ¼ 0:8 mm: lo is the frequency space wavelength, and lg is the guided wavelength. Standard formulas used are lo ¼

C fo

lo lg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
r effective

(I.1)

Microstrip line dimensions (width and length) can be computed as 377p pffiffiffiffi ; 2 zo er rffiffiffiffiffiffiffiffiffiffiffiffi   20 er þ 1 er  1 0:11 ; 0:23 þ þ A¼ 60 2 er þ 1 er     w 8 eA 2 er  1 0:61  ½B  1  ln½2B  1 þ ¼ 2A ln½B  1 þ 0:39  ; e 2 d p 2 er er

B¼

Zo ¼ 50 W where w is the width of microstrip line, d is the thickness of FR-4 (0.8 mm). Width obtained for microstrip line where the length of microstrip line characteristic ¼ lo :

378

Terahertz dielectric resonator antennas lg 8 Ls ¼ slot ¼ 0.4 lg

Stub ¼

Ws ¼ slot ¼ 0.2 Ls (0.2  0.4 lg) W d

> 2 (aspect-ratio mode dependent)

Ground plane ¼ 4 times of length Infinite ground plane ¼ 4 times of width Finite ground plane 6hhL ðdesign DRA) 6h þ W ðdesign DRA) Let us now use 3.7 GHz as design frequency microstrip line. B ¼ A ¼ ¼ W ¼ d ¼

377p 377  3:14 pffiffiffiffiffiffiffi ¼ 5:64 pffiffiffiffi ¼ 2 zo er 2  50  4:4 rffiffiffiffiffiffiffiffiffiffiffiffi   Zo er þ 1 er  1 0:11 0:23 þ þ 60 2 er þ 1 er rffiffiffiffiffiffiffiffiffiffiffiffiffiffi   50 4:4 þ 1 4:4  1 0:11 0:23 þ ¼ 1:53 þ 60 2 4:4 þ 1 4:4     8eA 2 er  1 0:61  ½ B  1  ln ½ 2B  1   þ ln ð B  1 Þ þ 0:39  e2A  2 p 2 er er   8 e1:53 2  ½5:64  1  ln½2ð5:64Þ  1 2 ð e 1:53Þ  2 p   4:4  1 0:61 ln½5:64  1 þ 0:39  þ 2ð4:4Þ 4:4

¼ ð1:218Þ½4:64  2:33 þ ð0:386Þ½1:786 ) 3:5025 W ¼ 3:5025; w ¼ 3:5025  0:8 mm d w ¼ 2:8 mm Stub: fr ¼ 3:7 GHz; lo ¼ 81 mm ¼

C fr

3  108  103 ¼ 81 mm 3:7  109   lo 81 4:4 þ 12:8 ¼ 8:6 ) lg ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffi ¼ 27:62 mm er effective ¼ 2
r effective 8:6 lo ¼

) Length of stub ¼

lg 27:62 ¼ ¼ 3:4525 mm 8 8

DRA designing process Slot: Ls ¼ 0:4 lg ¼ 0:4  27:62 ¼ 11:048 mm Ws ¼ 0:2 ls ¼ 0:2  11:048 ¼ 2:2096 mm DRA dimensions: ðTE111 fundamental mode) Formula used for rectangular DRA designing, ffi sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp 2 np 2 ‘p2 ckmn‘ c þ þ  fmn‘ ¼ pffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffi 2p mr 2r 2p mr 2r a b d a ¼ 20 mm b ¼ 20 mm d ¼ 14 mm a ¼ 0:52:5 b b ¼ 0:52:5 fix d ¼ 14 mm; d fr fr ¼ Q¼ fh  fl bandwidth m ¼ n ¼ l ¼ 1;

379

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Appendix J

DRA design case study

List of equipment used in terahertz dielectric resonator antenna (DRA) and microwave DRA design and testing are as follows: A vector network analyzer comes with calibration kit, signal generator, spectrum analyzer, power meter, noise figure analyzer, cables and connectors, anechoic chamber, antenna test system, material test system, power source and power extenders. Resonant frequency formulations used in rectangular DRA design (Figures J.1–J.14):

DR Microstrip feedline Substrate

z

y

x Ground plane

Figure J.1 Cylindrical DRA

Figure J.2 CDRA with fields

382

Terahertz dielectric resonator antennas
r k02 ¼ kx2 þ ky2 þ kz2 k02 ¼ w20 m0
0

Ground plane DR Aperture Microstrip feedline

z

y

x Substrate

Figure J.3 Rectangular DRA with feed slot and stub components

Figure J.4 DRA E-field dipole formation

Silicon

SU-8 Gold Chromium Silicon

Figure J.5 Terahertz cylindrical DRA

DRA design case study

383

Silicon DR Graphene disk Sub s

z

y

e

lf

ri

ls

ws

ro

d

x

trat

pu

In

Gr

oun

wf t

h

Figure J.6 Terahertz ring DRA

x

y

s

ws

ls

wp

w

h2

wst l st

Graphene ribbons

lp

SiO2

z

lf

l wf h1

lg z

y x

Figure J.7 Terahertz graphene DRA

Thus, the resonant frequency can be calculated and analyzed using k0 . For an isolated rectangular DR in free space, the resonant frequency can be determined as given in the following equation (Figures J.15–J.19):

c ðf Þr m; n; p ¼ pffiffiffiffiffi 2p m


s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi mp2 np2 pp2  þ þ a b h

384

Terahertz dielectric resonator antennas

K

K

s E-field (x-axis) End-fire radiation

H-field

H-field (z-axis)

K

H-field d

Lf

K Direction of wave propagation on y-axis

z y x

Figure J.8 Terahertz fields vectors

DGS

Feed Stub

C1

L1

L2

C2

Feed Slot

L3

DRA

C3

C4 R1

Figure J.9 Equivalent DRA circuit with R, L, C representation

Figure J.10 DRA exciting resonant mode

L4

DRA design case study

Figure J.11 DRA exciting higher order modes

Figure J.12 Rectangular DRA fields z

z

Short magnetic dipole

–1 L3

+1

h

s

L1 Fundamental mode

2h 3

y Third-order mode

Figure J.13 Resonant modes field diagrams Cylindrical DRA resonant frequency formulation: "    2 # 6:324c d d pffiffiffiffiffiffiffiffiffiffiffiffi 0:27 þ 0:36 þ 0:002 f ¼ 2h 2h 2pd er þ 2

385

386

Terahertz dielectric resonator antennas

Figure J.14 Rectangular DRAs with metal strips for exciting higher order modes

0º 0 –30º –60º

30º 60º

–20

–90º

90º

–40

–120º

120º 150º

–150º 180º

Figure J.15 Radiation pattern

T

x

I

z

y

Figure J.16 3D radiation pattern of DRA

DRA design case study

Substrate

Feedline

DR

Figure J.17 Terahertz DRA with feed

Feedline

Path-2 Path-1 Excitation O/4

Figure J.18 Terahertz DRA feed

Uniform field propagation

Point of excitation

Figure J.19 Terahertz DRA fields

387

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Appendix K

Vector network analyzer process for calibration

Used case 40-GHz VNA (Figure K.1). 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

23.

24.

Switch on VNA. Click on calibration. Click on calibration wizard. Set frequency. Click on next. Select on port. Starting frequency range and stop frequency. View calibration/kit. Calibration response Port 1/Port 2. Set the calibration kit to 85052B (3.5 mm). Apply the open, short and broadband at the load side. First connect open then choose a male/female connector option. For an antenna as one-port select female option. Second connect short to load and similarly choose male/female. Third select broadband and select female for antenna as one port. Now remove the broadband and connect the antenna for the measurement. Check S parameters with the help of marker. Data (for .csv) and graphs (.jpg). For a two-port device, it will be a female option. Repeat the previous points 4–6. Rest all the same as was done for a single-port option. Draw an equivalent circuit with CST software and compare results with MATLAB“. In a post-processing window of CST, there is a tab with name network analysis and by using this tab, you can transform your desired model into lump components by mentioning the order of cascaded network along with specific frequency. Just import the measured S parameters (S1p file) from the VNA into Design Studio canvas of CST software. Then create the lumped circuit model and optimize R, L, C values. S parameters are the same (minimizing the difference S11). The same can be done with ADS, Microwave Office and even MATLAB.

390

Terahertz dielectric resonator antennas

Figure K.1 VNA for DRA measurements 25. 26. 27.

28. 29. 30. 31. 32. 33. 34. 35. 36. 37.

Mostly, the RLC approach will work for narrow band antennas. You get the s-parameters, exported from your VNA. On the Design Studio you ask for the item “Touchstone file block,” which is under the class “Data Import.” There you will point the path to your .s1p/.s2p file, which will show up as a box with one or two ports, depending on your measurement. It can then be connected to other circuit elements on your canvas—an RLC circuit model for return loss by MATLAB. You can draw the S11 on the Smith chart from result in project tree and can determine the imaginary and real parts; then you can predict the circuit. Drawing mode chart graph and also finding the coupling matrix of a coupled resonator filter using ADS. Theoretically, you need to approximate the antenna in terms of R, C and L. The input impedance of an antenna can be obtained from S11 since S11 ¼ (ZL  ZO)/(ZL þ ZO). For S11 magnitude and phase and given ZO, one can calculate ZL. By changing the frequency, one can get ZL as a function of frequency. An equivalent circuit that is normally a parallel R, L, C circuit shunted by the far field. SPP–nano-DRA–light consists of an electromagnetic wave that oscillates the electrons of metal. Once electrons have deviated from their mean position, they oscillate back and forth. When the frequency of incident wave matches with the free oscillation of metal, it generates local surface plasmon.

Glossary DR Q-Factor TE mode TM mode DRA HEM mode NDRA VSWR DWM CP MIMO AR FSS EBG ECC TARC DG MEG CCL DGS AMC SEM LHCP RHCP LP

dielectric resonator quality factor transverse electric mode transverse magnetic mode dielectric resonator antenna hybrid electromagnetic mode nano dielectric resonator antenna voltage standing wave ratio dielectric waveguide model circularly polarized multi-input–multi-output axial-ratio frequency selective surfaces electromagnetic band-gap envelope correlation coefficient total active reflection coefficient diversity gain mean effective gain channel capacity loss defected ground structures artificial magnetic conductor scanning electron microscope left-hand circularly polarized right-hand circularly polarized linearly polarized

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Index

absorber 341 absorber characteristics 247–9 absorber impedance 248 absorbers mathematical analysis 249 absorption loss 249 air impedance 248 anechoic chamber testing 241 antenna measured values, in anechoic chamber 251–339 artificial photo synthesis 247 bipolar cells 216–7 blood glucose monitoring 233 Boltzmann–Maxwell’s equation 102 Boltzmann Transport equation 43 bosonic operators 2, 61 bosonic or fermionic Fock space 9 canonical anticommutation relations (CAR) 2, 48 canonical communication relations 3, 6 canonical commutation rules 2 channel capacity loss (CCL) 12, 104 circular polarization (CP) 11, 104 classical wave field theory 46, 105 coherent state 8, 31, 46, 51, 59, 104, 113, 138, 169, 173, 349 Cole–Cole model 234, 236 communication satellites 241 computer simulation technology (CST) 232 conical DRA 233 conical terahertz DRA, 62–3 design architecture 84 equivalent circuit of 97

model-1 multiband, 84–9 radiation pattern of 86 VSWR of 86 corpuscles 29, 135, 166 corpuscular theory of light 29, 136, 166 coupled Maxwell’s equations 43 current density fluctuations 19, 62–3, 101 cylindrical cavity 9, 57, 60 cylindrical dielectric resonator antenna (CDRA) 4–6, 56, 103, 233, 381 with fields 381 model 359 optical 125 resonant frequency formulation 385 cylindrical terahertz and optical DRA 101–32 design computations 102–3 model 2 TCDRA at 10-THz resonant frequency 102 optical CDRA description 114–32 terahertz antennas detailed description 104–5 theory of terahertz cylindrical DRA and mathematical formulations 106–13 cylindrical terahertz DRA 54, 60–2 de Broglie’s theory 30, 137 design dimension 18, 26 dielectric materials and resources 341–6 dielectric resonator antennas (DRAs) 25–40, 68, 102, 135, 161, 191–211, 241, 341, 353

394

Terahertz dielectric resonator antennas

approximate analysis of rectangular quantum antenna 38–9 bandwidth (BW) of terahertz DRA 202 design case study 381–7 design development and evaluation of NDRA 202 resonant frequency of TRDRA formulations 202 designing process 377–9 design of terahertz dielectric resonator antenna 32–3 Drude’s model 210 higher order resonant modes 200–2 fabrication and testing 33 mathematical analysis of terahertz RDRA 34–8 MATLAB program 210–11 propagation of light 29–32 quantum DRA-equivalent circuit mathematical analysis for mixed circuits 194–200 frequency-dependent resistance 196–7 impedance (Zin) 194–6 second resonant mode 198–200 two resonant modes 197–8 simulated results based on MATLAB 202 synthesis of NDRA radiation theory 202–8 terahertz antenna far-field radiations 33–4 terahertz DRA simulation results 40 dielectric resonator antennas (DRAs) and synthesis 1–19 CDRA (cylindrical DRA) 4–6 functions of terahertz DRA 14–16 parameters of microwave and terahertz DRA 15–16 rectangular nano-DRA design parameters 17–18 design steps 17–18 terahertz MIMO DRA parameters 12–14

microwave DRAs vs optical DRA parameters 13 optical DRAs 13–14 radiated fields 14 terahertz or quantum devices characteristics 6–12 Drude’s model theory 11–12 radiation parameters 11 terahertz DRA or quantum DRA near fields/far fields 10–11 theory of TDRA 6–10 THz DRA model design parameters 16 dielectric resonators (DRs) 65 dimensions terahertz CDRA 69 Dirac current density 9, 37, 47, 57, 105–6, 114, 175, 181, 219, 351 Dirac equation 28, 32, 69, 106, 166, 168–9, 175, 179, 351 Dirac field 8, 105, 184 Dirac’s relativistic wave equation 9, 46, 57, 82, 105 diversity gain (DG) 12 DRA E-field dipole formation 382 Drude–Larentz model 210 Drude’s model 4, 82, 84, 210 Drude’s model theory 11–12, 103 dual-band graphene antenna design and implementation 347–51 dynamic impedance 192–3, 205 Dyson series 10, 57, 60 E-Field 126 E-Field scalar terahertz spherical DRA 151 E-Fields J current density 88 envelope correlation coefficient (ECC) 12 far-field magnetic vector potential 149, 219, 350–1 far-field Poynting vector 36, 153, 174, 350 far-field-radiated magnetic vector potential 111, 114

Index far-field radiation pattern 48, 50, 106, 110, 114, 132, 145–6, 181, 219, 222 feeding mechanism parameters 16 fermionic Dirac wave operator field 148 fermionic operators 2, 61 Feynman diagrams 2, 10, 56, 61 frequency-dependent resistance 96, 196–7 futuristic vision 231–9 patient-centric healthcare system outline 233 thumb DRA design and implementations 234–9 thumb DRA sensors integrated with patient-centric healthcare system 233–4 ganglion cells 216 Gaussian beam 34, 166, 170, 247, 250, 359–61 Gaussian beam feed process 359–71 Gaussian pulse excitation 3, 17 GDSII file 374 graphene 249–50 graphene antenna 348 graphene disk and silicon annular DRA terahertz DRA design using 224 Helmholtz equation 59–62, 91, 93–5, 143–4 H-Field vector 165 HFSS NDRA modeling 193 higher order modes analysis 199 higher order resonant modes 200–2 high-permittivity materials 353 high-Q factor 247 impedance matching 17, 74, 248 incident photons 3, 10, 53, 83, 84, 114 input impedance 86, 191–3, 202, 248 ISM band 241 Klein–Gordon equation 32, 138, 169

395

laser Gaussian input pulse, for nonlinear phase matching 58 laser-generated coherent photon 225 lateral geniculate nucleus (LGN) 216 left-hand circular polarization (LHCP), 104 light-imaging detection and ranging (LiDAR) 18, 188 light–matter interaction 103, 132 in terahertz dielectric resonator antennas 43–51 light–matter interaction theory, in quantum antenna 44–7 local density of electromagnetic states (LDOS) 83 machine learning algorithm 233–4 machine learning model 234 MATLAB program 210–11 Maxwell curl equations 91, 93, 107 Maxwell–Dirac equations 10, 56, 177 Maxwell equations 34, 38, 44, 58–9, 105, 110, 112, 132, 143, 145, 147, 171, 177, 347 Maxwell unified electromagnetism 29, 136, 167 mean effective gain (MEG) 12, 104 mean-square fluctuations 3, 11, 14, 59, 66, 84 metallic strips 353, 356 microwave antennas 28, 63, 104, 106, 166, 225 microwave DRAs 18 design and testing 381 millimeter-wave frequency 241 MIMO (multi-input–multi-output) spherical DRA 153–6 diversity gain 156 at photonic wavelength 218 radiation pattern 155 VSWR 156 MIMO CDRA, at photonic wavelength 218–19 MIMO dielectric resonator antenna (DRA) 232

396

Terahertz dielectric resonator antennas

miniaturization design techniques 353–7 model-1 multiband conical TDRA 84–9 multi-input–multi-output (MIMO) 143 multipole electric field 143 nano antenna 3 nanophotonics 3, 82 nano-RDRA (rectangular DRA) 17 radiation theory, synthesis of 202–8 nanostrip waveguide 17, 55 nanotechnology 3–4, 55, 82 optical absorbers 247 applications 249–50 optical antenna 11, 13, 43, 53, 55, 57, 63, 74, 81–3, 166, 176, 191, 218 optical antenna arrays basic requirements 219–23 optical CDRA 124, 132 3D super directive radiations in 127 excitation in 126 optical communication bands 55 optical DRA for retinal applications 215–29 designs of terahertz DRAs simulation results for various shapes 229 entanglement 224 light-matter interaction 225–7 modeling of optical antennas 224–5 optical antenna arrays basic requirements 219–23 optical antenna design 223–4 theory of coupled resonant modes 227–8 optical rectangular DRA 176 optical sensors 63, 101, 247 optical spherical resonator antenna design dimensions of 142 patient-centric healthcare system 233 patient information system, for health alerts 235 photon–electron–positron field 57

photon energy 26 photonic antennas 102, 104 photoreceptors 215 Planck’s quantum hypothesis 29, 136–7, 167 Planck’s quantum theory 36, 174, 350 plane wave incidence 247 plasmonics 29 plasmons 53, 83 Poynting theorem 66 propagation of light 166–70 proposed antenna geometry, dimensions of 349, 374 proposed healthcare system 238–9 quality factor 44, 341 quantum antenna 13, 28, 38, 63, 74, 166, 191, 193, 210, 217, 225, 227, 350 circuit 192 quantum communications 51 quantum computing 51 quantum electrodynamics 3, 10, 31, 53, 83, 114, 168 quantum electromagnetic fields 135 quantum entanglement, theory of 47–51 quantum fluctuation correlation 34, 114, 219, 222 quantum-mechanical analysis 13 quantum-mechanical model 88 quantum mechanics 11, 14, 31, 51, 58, 93, 138, 169 quantum theory 9, 29, 31, 57, 137, 167–8 radiated far fields 14, 47 radiated fields 14, 219 radiation efficiency simulation results 228 radiation pattern 11, 208, 386 co- and cross-radiation pattern simulation results 228 Rayleigh’s theory 29, 136, 167 rectangular DRA (RDRA) 16, 161, 233

Index design 379, 381 fields 385 with metal strips 386 prototype model of 355 radiation pattern of 356 rectangular nano-DRA design parameters 17–18 design steps 17–18 rectangular terahertz DRA 161–88 design and simulation of 170 efficiency of 164 E field of 164 gain of 163 mathematical analysis of resonant modes 171–6 propagation of light 166–70 radiation pattern of 164 synthesis of 170–1 terahertz optical RDRA at 484 THz 176–88 approximate analysis of 176–88 reflection loss, in absorber 248 resonant frequency formulations 381 resonant modes field diagrams 386 retina parameters 217 rhodopsin 215 right-hand circular polarization (RHCP), 104 RLC circuit 193 rod cell parameters 216 rods cells 216 Schro¨dinger’s equation 43–4, 76, 168 Schro¨dinger’s wave equation 30, 32, 137 sea of electrons and positrons 83, 114 short electric dipoles 4 short magnetic dipoles 4 signal-to-noise ratio (SNR) 11, 58 silicon dielectric resonator antenna, at 5-THz frequency 373–5 silver nano waveguide dimensions 75 small particles 29, 135, 166 S-parameters 13 spherical DRA 233

397

spherical optical DRA 13 spherical terahertz and optical DRA 135–58 design of terahertz spherical DRA at 511 THz 143 mathematical formulations of terahertz spherical DRA 143–8 MIMO (multi-input–multi-output) spherical DRA 153–6 results and discussions 148–53 super directivity in spherical DRA 148–53 super-directive antennas 229 super-directive radiations 222 surface current 192 surface plasmon polaritons 28, 166 surface plasmon polytrons (SPP), into terahertz DRA 65–77 mathematical formulations in TDRA 74–6 terahertz CDRA design and simulations 68–9 terahertz DRA applications 77 terahertz DRA main features 69–73 working principle of TDRA 66–8 surface plasmon resonance (SPR) 220 TE mode, 108–9 terahertz (THz) 25, 27, 53 communication bands 3 terahertz absorbers 247–50 characteristics 247–9 mathematical analysis 249 optical absorbers applications 249–50 terahertz antenna circuit 200 terahertz antennas 2 terahertz conical dielectric resonator antenna 81 design structure of 83–4 equivalent electrical circuit of 96–8 mathematical modeling of 90–6 model-1 multiband conical TDRA 84–9

398

Terahertz dielectric resonator antennas

terahertz conical DRA design and analysis 226 terahertz cylindrical DRA (TCDRA) 382 design, 107 with silver nano waveguide 115 design and analysis 226 with dimensions 116 dimensions table 129 gain vs frequency of 117 Gaussian beam input to 71 impedance vs frequency of 118 parameters 132 radiation efficiency vs frequency of 119 radiation in 70 radiation pattern of 70, 118 super directive nature 120 10-THz design 119 10 THz dimensions table 129 VSWR of 117 terahertz dielectric resonator antennas (TDRAs) 2, 26–7, 66, 82, 223, 229, 233, 359, 381 efficiency of 39 feed 387 fields 387 fields E 130 fields H 130 fields scalar 130 fields scalar H 131 frequency spectrum in 67 with laser input 28 radiated fields in 6 radiation pattern of 39 radiations 39 surface currents in 5 surface current J 131 terahertz dielectric resonator antennas design and modeling 53–63 conical terahertz DRA 62–3 cylindrical terahertz DRA 54, 60–2 mathematical formulations 58–60 terahertz DRA radiations, with Gaussian beam laser 27

terahertz fields vectors 384 terahertz graphene DRA 383 terahertz MIMO DRA parameters 12–14 microwave DRAs vs optical DRA parameters 13 optical DRAs 13–14 radiated fields 14 terahertz or quantum devices characteristics 6–12 Drude’s model theory 11–12 radiation parameters 11 terahertz DRA or quantum DRA near fields/far fields 10–11 theory of TDRA 6–10 terahertz rectangular DRA (TRDRA) 66, 162 with Gaussian input 67 fields 67 radiation pattern of 173 terahertz ring DRA 383 terahertz spectrum 7 terahertz spherical DRA H-Field scalar of 152 H-Field vector of 152 radiation pattern of 141 VSWR of 142 terahertz time-domain spectroscopy (TDS) 33, 40, 373 three-dimensional radiation pattern 208, 386 thumb (tissue) DRA sensor material details 239 thumb DRA design and implementations 234–9 thumb DRA sensor 3D radiation pattern 237 integrated with patient-centric healthcare system 233–4 radiation pattern of 237 thumb DRA tissues, equivalent circuit of 238 THz DRA fabrication process 373–5 THz DRA model design parameters 16 TiO2 DRAs 248

Index TM mode, 108–9 total active reflection coefficient (TARC) 12–13, 16 twin-photon generation 225 vector network analyzer 381 vector network analyzer process, for calibration 389–90 vertical port reading, for directivity 253–339

399

vision restoration, technique of 217 voltage-standing wave ratio (VSWR) 85–6 wave 29 Wi-Fi 241 Wi-Fi connectivity 233 WiMAX 241