guided and leaky modes of circular phdthesis by kim

8/9/2019 Guided and Leaky Modes of Circular PhDThesis by Kim

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Guided and Leaky Modes of Circular

Open Electromagnetic Waveguides:

Dielectric, Plasma, and

Metamaterial Columns

Ki Young Kim


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Thesis for the Degree of Ph.D.





Ki Young Kim

Department of Electronics, Major in Wave Propagation Engineering

The Graduate School

December 2004

The Graduate School

Kyungpook National University


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Ki Young Kim

Department of Electronics, Major in Wave Propagation Engineering

The Graduate School

Supervised by Professor Heung-Sik Tae

Approved as a qualified thesis of Ki Young Kim

for the degree of Ph.D.

by the Evaluation Committee

December 2004

Chairman _

_ _ _ _ _

The Graduate School Council, Kyungpook National University


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ABSTRACT





by Ki Young Kim

Ph.D. in Electronics (Major in Wave Propagation Engineering)

Kyungpook National University, Daegu, Korea, 2004.

This thesis numerically investigates the guided and leaky mode characteristics of

circular open electromagnetic waveguides made of dielectrics, plasma, and

metamaterials that are assumed to be isotropic and homogeneous. The leaky or

complex modes are analyzed using complex propagation constants rigorously

obtained using Davidenko’s method.

Chapter 1 provides an introduction and addresses the background, motivation,

and general contents of this thesis.Chapter 2 then investigates the guided and leaky mode characteristics of circular

dielectric rod waveguides using specific numerical examples. The guided mode

characteristics of circular dielectric rod waveguides are first briefly reviewed. The

leaky mode characteristics of the circularly symmetric modes below the guided

mode cutoffs are then identified and classified as a nonphysical mode, antenna

mode(s), reactive mode, and spectral gap. In addition, the leaky TM0n and TE0n

modes are compared. The effects of the dielectric constant and radius of the

waveguide on the modal propagation characteristics are then briefly discussed.

Finally, the lossy effects on the modal characteristics, such as mode coupling

phenomena and the creation of a new transition region between the guided and

leaky modes, are identified.

In Chapter 3, the guided and leaky mode characteristics of plasma column

waveguides are investigated using specific numerical examples. The guided mode

characteristics of plasma columns are first briefly reviewed using frequency

independent and frequency dispersive material models. The leaky mode

characteristics of a plasma column are then analyzed using several normalized


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plasma frequencies. The physical validity of the numerically obtained complexpropagation constants is also evaluated using a steepest descent plane (SDP)

analysis.

In Chapter 4, the guided mode characteristics of metamaterial column waveguides

are investigated and compared using frequency independent and frequency

dispersive material models The frequency dispersive material models considered

are a double plasmonic model and Pendry’s model. The guided modes with the

frequency independent material model are analyzed using several combinations of

artificial material parameters. With the frequency dispersive material models,

guided modes of TE-like modes, as well as TM-like modes are found to exist, and

the plasma frequencies are associated with the existence of specific modes. The

leaky mode characteristics of metamaterial column waveguides when using the

dispersive material models are found to be similar to the leaky mode

characteristics of plasma column waveguides, except for the low frequency

behavior.

Finally, Chapter 5 summarizes the main results of this thesis and suggests some

areas for further research.

Thesis Supervisors: Professor Heung-Sik Tae and Professor Jeong-Hae Lee.


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Table of Contents

Abstract i

Table of Contents iii

List of Figures vii

List of Tables xxi

1. Thesis Introduction 1

1.1. Development of Electromagnetic Wave Technologies ························ 1

1.2. Open Electromagnetic Waveguiding Structures ·································· 4

1.3. Guided and Leaky Waves ······································································· 9

1.4. Thesis Overview ······················································································· 13

References ············································································································ 16

2. Guided and Leaky Modes of Circular Dielectric Rod Waveguides

192.1. Dielectric Media ························································································ 19

2.2. Dielectric Waveguides ············································································· 20

2.3. Guided Modes of Circular Dielectric Rod Waveguides – A Review · 23

2.4. Existence of Leaky TM0n Mode Below Guided Mode Cutoffs ·········· 28

2.5. Comparison Between Leaky TM0n and TE0n Modes ··························· 35

2.6. Effect of Design Parameters: Dielectric Constant and Radius ··········· 41

2.7. Lossy Effects on Modal Propagation Characteristics ·························· 44

2.7.1. Mode Coupling Phenomena of Leaky Modes ························· 45

2.7.2. Attenuation Constants of Guided Modes ································ 50

2.7.3. New Transition Region Between Guided and Leaky Modes ··· 51

2.8. Conclusions ······························································································· 53

References ············································································································ 54


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3. Guided and Leaky Modes of Plasma Column Waveguides 57

3.1. Plasma Media ···························································································· 57

3.2. Plasma Column Waveguides ·································································· 60

3.3. Guided Modes of Plasma Column Waveguides – A Review ············· 64

3.4. Leaky Mode Characteristics of Plasma Column Waveguides ··········· 68

3.5. Suggested Electrically Reconfigurable Self Sustained Plasma Column

Leaky Wave Antennas – Basic Concept ················································ 85

3.6. Conclusions ······························································································· 87

References ············································································································ 88

4. Guided and Leaky Modes of Metamaterial Column Waveguides 93

4.1. Metamaterials – Overview ······································································ 94

4.1.1. Ideas and Realizations ································································ 94

4.1.2. Terminologies and Basic Properties ········································· 96

4.1.3. Trends and Perspectives ····························································· 100

4.1.4. Effective Medium Approach ····················································· 102

4.2. Metamaterial Waveguiding Structures ················································· 102

4.3. Guided Mode Characteristics of Metamaterial Columns Waveguides 103

4.3.1. Frequency Independent Metamaterials ··································· 106

4.3.2. Dispersive Metamaterials: Identical Plasmonic Model ········· 114

4.3.3. Dispersive Metamaterials: Dissimilar Plasmonic Model ······· 117

4.3.4. Dispersive Metamaterials: Pendry’s Model ····························· 122

4.4. Leaky Mode Characteristics of Metamaterial Column Waveguides · 127

4.4.1. Frequency Independent Metamaterials ··································· 128

4.4.2. Dispersive Metamaterials: Dissimilar Plasmonic Model ······· 132

4.4.3. Dispersive Metamaterials: Pendry’s Model ····························· 138

4.5. Conclusions ······························································································· 141

References ············································································································ 142

5. Summary, Perspectives, and Future Work 153

5.1. Thesis Summary ······················································································· 153

5.2. Perspectives and Future Work ······························································· 155

References ············································································································ 157


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Appendix A. Flexible Circular Dielectric Waveguide 159

Appendix B. Summary of Characteristic Equations 173

Appendix C. Davidenko’s Method of Complex Root Search 179

Appendix D. Classifications of Complex Modes 187

Appendix E. Steepest Descent Plane (SDP) Analysis 191

Abstract in Korean 197

Acknowledgement 201


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List of Figures

Figure 1.1: Portraits of Scottish physicist J. C. Maxwell (left, 1831-1879) and

German physicist H. R. Hertz (right, 1857-1894). J. C. Maxwell theoretically

predicted the existence of electromagnetic waves in 1864, while H. R. Hertz

experimentally verified their existence in 1888. ······················································ 2

Figure 1.2: Plane electromagnetic wave propagation. The electric and magnetic

fields generate each other, and the vector of the electric and magnetic fields and

propagation direction of the electromagnetic waves are all perpendicular. ······ 2

Figure 1.3: Artistic drawings of modern applications of electromagnetic waves. (a)

Satellite communications among earth station, satellites, and space shuttles, where

the communications are performed using electromagnetic waves. (b) “The

photonic micropolis”, generated by Professor John D. Joannopoulos’s group at MIT,

composed of various photonic devices, such as waveguides and cavities. ········· 3

Figure 1.4: Two typical examples of closed waveguides: (a) rectangular waveguide

and (b) circular waveguide. An electromagnetic wave is guided in the interior

region of the waveguiding structure. The interior and outer regions of thewaveguides are isolated electrically by the conductor shielding. The modes are

determined by the operating frequency and physical dimensions a and b. ······· 5

Figure 1.5: Earth-ionosphere waveguiding system as example of closed waveguide.

An electromagnetic wave is reflected off the ionosphere layer composed of a

plasma layer. Meanwhile, the plasma acts as a conducting medium when the

incident wave frequency is higher than the ionosphere’s plasma frequency. A

simple explanation of electromagnetic wave propagation with plasma is given in

Chapter 3. ··················································································································· 5

Figure 1.6: Examples of open planar electromagnetic waveguides for opticalintegrated circuits: (a) strip waveguide, (b) embedded strip waveguide, (c) rib

waveguide, and (d) inverted rib waveguide. Note that2 1 0

n n n> > , where0

n is

the refractive index of the surrounding free space region. The guided and leaky

dispersion characteristics of the waveguides are governed by the refractive indexes

of the dielectric materials and waveguide dimensions, w and t. ························· 7


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Figure 1.7: Examples of open electromagnetic waveguides for planar microwave ormillimeter wave frequencies: (a) microstrip line, (b) slot line, (c) coplanar

waveguide, and (d) image guide. The guided and leaky dispersion characteristics

of the waveguides are governed by the dielectric constantr

ε and waveguide

dimensions, w , t , h , and s. ·························································································· 7

Figure 1.8: Optical fiber as example of electromagnetic surface waveguide: left,

optical fibers and right, principle of light transmitting through optical fiber by total

internal reflection. The incident angles α , β , and γ shown in the figure are less

than the critical angle. The electromagnetic waves are guided in the interior high

refractive index region by the total internal reflections. ········································ 8

Figure 1.9: Reflection and refraction of electromagnetic wave on surface of two

dissimilar materials. α is the angle of incidence. ···················································· 9

Figure 1.10: Reflection and refraction of electromagnetic wave with several

incident angles. ············································································································ 9

Figure 1.11: Geometrical optic descriptions of simplest guided surface and leaky

wave along dielectric film. The refractive index of the film is assumed to be higher

than that of the surrounding free space. (a) Guided mode with total internal

reflection. The electromagnetic wave energy is confined to the waveguide region.(b) Leaky mode configuration. Electromagnetic energy is constantly leaked into the

free space region. ········································································································ 10

Figure 1.12: Radiation from semi-infinite leaky waveguide: (a) Leaky waveguide is

fed by closed waveguide. When viewed from point z’ , the amplitude of the field

increases as the distance from the waveguide increases. (b) Expected field pattern

along x direction (transverse direction). ································································· 11

Figure 1.13: First page of the patent (US patent 2,402,622) for earliest leaky wave

antenna invented by William W. Hansen in 1940s. ··············································· 12

Figure 1.14: Circular open boundary electromagnetic waveguide. The inner region

( r a< ) is considered as a conventional dielectric, plasma, and metamaterial in

Chapters 2, 3, and 4, respectively. ············································································ 14


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Figure 2.1: Field quantities in dielectric body in presence of external field. a E

:applied field, P

: polarization,d

E

: depolarization, andtot

E

: total field. The

electric dipole polarization P

and applied fielda

E

generally proceed in the

same direction in natural dielectric media, resulting in dielectric constants that are

greater than unity. ······································································································ 20

Figure 2.2: Light caught in falling water: original principle of optical fibers. In 1854,

J. Tyndall showed that light could propagate in falling water, which is the earliest

official identification of the principle of optical fibers, and even earlier than

Maxwell’s equations. ·································································································· 21

Figure 2.3: Examples of circular dielectric rod waveguides: (a) Bundle of optical

fibers carrying visible light waves and (b) flexible waveguide in Q-band. Relatively

low radiation losses with bent or flexible waveguides can be achieved by properly

selecting the design parameters, such as the dielectric materials and radius of the

rod. In this case, the radiation loss is mainly affected by the propagating power

distributions, which in turn are controlled by the dispersion characteristics. The

full text is included in Appendix A. ········································································· 22

Figure 2.4: Guided and leaky mode regions. The guided mode region is between

2 2r r µ ε and

1 1r r µ ε . However, below the lower boundary of

2 2r r µ ε there are

leaky modes with complex propagation constants, which is the main concern ofthis and the following chapters. Meanwhile, above the upper boundary of

1 1r r µ ε

there are superslow guided modes, which will be mentioned in Chapter 4. As far

as the guided mode of a conventional dielectric waveguide is concerned, the leaky

mode regions and superslow mode regions are both forbidden. ······················· 24

Figure 2.5: Dispersion characteristics of circular dielectric rod waveguides with

circularly symmetric modes, i.e. , 0m = : (a) TM0n mode and (b) TE0n mode. The

radius and dielectric constants of the rod are assumed to be 10.0 mm and 4.0,

respectively. The cutoff frequencies for each guided mode are the same for the

same order of mode, i.e., n. ························································································ 26

Figure 2.6: Hybrid dipole modes with azimuthal eigenvalue of 1m = : (a) HE1n

mode and (b) EH1n mode. Unlike the case of the circularly symmetric modes, the

guided mode cutoff frequencies are no longer identical between the HE1n and EH1n

modes. Note that the HE11 mode has no guided mode cutoff. ···························· 27


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Figure 2.7: Normalized phase constant of circular dielectric rod waveguide. Thedielectric constant and radius of the rod were assumed to be 5.0 and 5.0 mm,

respectively. Leaky modes exist below the guided mode cutoffs. ······················ 32

Figure 2.8: Normalized leakage constant of circular dielectric rod waveguide. The

dielectric constant and radius of the rod were assumed to be 5.0 and 5.0 mm,

respectively. Below the guided mode cutoff frequencies, nonzero values were

generated for the normalized leakage constants. ·················································· 32

Figure 2.9: Enlarged scale of normalized phase constant in Figure 2.5. ············ 33

Figure 2.10: Leaky mode characteristics of circular dielectric rod waveguide for

four lower-order TM0n modes. The dielectric constant and radius of the rod were

assumed to be1

10.0r

ε = and 10.0 mma = , respectively. (a) Normalized phase

constant and (b) normalized leakage constant. The arrows in (a) depict the cutoffs

for the guided modes. ································································································ 36

Figure 2.11: Leaky mode characteristics of circular dielectric rod waveguide for

four lower-order TE0n modes. The dielectric constant and radius of the rod were

assumed to be1

10.0r

ε = and 10.0 mma = , respectively. (a) Normalized phase

constant and (b) normalized leakage constant. The arrows in (a) depict the cutoffs

for the guided modes. ································································································

37

Figure 2.12: Enlarged scaled plot of the Figure 2.10 (a). Normalized phase constant

of TM0n mode near unity. The arrows depict the cutoffs for the guided modes. 40

Figure 2.13: Enlarged scaled plot of the Figure 2.11 (a). Normalized phase constant

of TE0n mode near unity. The arrows depict the cutoffs for the guided modes. 40

Figure 2.14: Normalized phase constant of circular dielectric rod waveguide with

two different rod radii, while dielectric constant of rod was fixed at1

5.0r

ε = .

··········································································································································· 42

Figure 2.15: Enlarges scale of normalized phase constant in Figure 2.12. Only the

TM02 mode is shown. ································································································· 42

Figure 2.16: Normalized phase constant of circular dielectric rod waveguide with two

different rod dielectric constants, while radius of rod was fixed at 5.0 mma = . ········· 43


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Figure 2.17: Enlarged scale of normalized phase constant in Figure 2.14. Only the

TM02 mode is shown. ································································································· 43

Figure 2.18: Leaky mode characteristics of circular lossy dielectric rod waveguide.

(a) Normalized phase constant and (b) normalized leakage constant. The arrows in

(a) represent the nonphysical improper complex roots, the dashed lines are for the

lossless references, and the dotted and solid lines are for the cases of tan 0.005δ =

and tan 0.01δ = , respectively. However, in this scale, the lossy effects are rarely

distinguishable. The other non-arrowed branches are the proper physical guided

modes and real part of the normalized complex propagation constant due to the

finite value of the loss tangent. ················································································· 46

Figure 2.19: Enlarged scale of encircled regions in Figure 2.18 (a), where regions A

and B correspond to guided TM01 mode and leaky TM03 mode, respectively. In the

guided mode region ( A), the normalized phase constant becomes lower when the

loss tangent is higher. Conversely, in the leaky mode region (B), the normalized

phase constant becomes higher when the loss tangent is higher. This leaky mode

property is valid near the border between the guided and leaky mode regions.

When the frequency is much lower in a deep reactive region, the normalized phase

constant of a higher loss tangent can take a lower value, implying the existence of

crossing points between the curves of normalized phase constants with dissimilarloss tangents. Unfortunately, in this case, this property was unclear. ················ 47

Figure 2.20: Coupling of modes due to finite value of loss tangent. (a) No coupling

and (b) coupling. Mode coupling occurs when the dielectric loss is above a certain

critical value. ··············································································································· 48

Figure 2.21: Attenuation characteristics of circular lossy dielectric rod waveguide:

(a) Normalized phase constants, where arrowed branches represent physical

proper guided modes and (b) normalized leakage (or attenuation) constants. Below

the guided mode cutoff,0

/ k α is the normalized leakage constant, while above the

guided mode cutoff, 0/ k α is the normalized attenuation constant. ················ 49

Figure 2.22: Attenuation constants in Figure 2.21 (b). Comparison of attenuation

constants obtained using perturbation method and Davidenko’s method. ······ 50


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Figure 2.23: Transition region between guided and leaky mode regions. Thearrowed region is the transition region between the guided and leaky mode

regions considered here. The spectral gap region where the normalized phase

constants are greater than unity is not the transition region between the guided

and leaky mode regions. ··························································································· 51

Figure 2.24: Enlarged scale of transition region. In (a), in the lossless case, the

dispersion curves for the guided and leaky modes continue without any

discontinuity, however, when the dielectric loss is introduced, the dispersion

curves are split. ··········································································································· 52

Figure 3.1: Three examples of space plasma. Plasma makes up 99% of all visible

matter in the universe. (a) The aurora, or northern lights, flickering in the

uppermost reaches of earthʹs atmosphere. (b) X-ray image of the sun. (c) The solar

wind generates an immense sheet of electrical current that spirals like a ballerina ʹs

skirt as the sun rotates. Although naturally-occurring plasma is rare on earth, there

are many man-made examples. ················································································ 58

Figure 3.2: Examples of plasma columns: (a) missile surrounded by plasma sheath

due to friction between air and thermal ignition, and (b) ordinary fluorescent tube

as effective experimental prototype plasma antenna. ··········································· 61

Figure 3.3: Cross-sectional view of plasma column with radius a. The length of the

column extends infinitely, while the density of the plasma is assumed to be

uniform and the dielectric constant assumed to be isotropic for simplicity. ····· 61

Figure 3.4: Dielectric constant of plasma column with respect to normalized

plasma frequencies of 1.0 p

k a = , 2.0 p

k a = , and 3.0 p

k a = used in Chapter 3. If

the operating normalized frequency0

k a is higher than the normalized plasma

frequency, the dielectric constant will be positive according to the expression of

( )2

01 /

rp pk a k aε = − . ····································································································· 63

Figure 3.5: Guided dispersion characteristics of plasma columns. (a) Frequency

independent dielectric constant case. The dielectric and magnetic constants of the

plasma were assumed to be1

4.0r

ε = − and1

1.0r

µ = , respectively, and the radius

of the plasma column was 10.0 mma = . (b) Dispersive dielectric constant case

using equation in (3.4). The effective plasma frequency was assumed to be

10.0 p

k a = . ·················································································································· 67


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Figure 3.6: Leaky mode characteristics of plasma column with normalized plasmafrequency of 1.0

pk a = for TM0n mode. (a) Normalized phase constant and (b)

normalized leakage constant. The inset in (a) is an enlarged scale of the negative

normalized phase constants. ····················································································· 70

Figure 3.7: Leaky mode characteristics of plasma column with normalized plasma

frequency of 2.0 p

k a = for TM0n mode. (a) Normalized phase constant and (b)




frequency of 3.0 p

k a = for TM0n mode. (a) Normalized phase constant and (b)




frequency of 1.0 p

k a = for TE0n mode. (a) Normalized phase constant and (b)

normalized leakage constant. ··················································································· 73


frequency of 2.0 p


normalized leakage constant. ···················································································

74


frequency of 3.0 p



Figure 3.12: Complex propagation constants of plasma column with normalized

plasma frequency of 1.0 p

k a = on steepest descent plane. (a) “a” mode for TM0n

mode in Figure 3.6 and (b) “a” mode for TE0n mode in Figure 3.9. The leaky TM0n

mode solutions have a physical meaning throughout the frequency range, however,

the leaky TE0n mode solutions lose their physical meaning below 2.522 GHz,

marked by “ A” in (b). ································································································· 77


frequency of 1.0 p

k a = for HE1n mode. (a) Normalized phase constant and (b)




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Figure 3.14: Leaky mode characteristics of plasma column with normalized plasmafrequency of 2.0

pk a = for HE1n mode. (a) Normalized phase constant and (b)




frequency of 3.0 p

k a = for HE1n mode. (a) Normalized phase constant and (b)




frequency of 1.0 p

k a = for EH1n mode. (a) Normalized phase constant and (b)



frequency of 2.0 p




frequency of 3.0 p



Figure 3.19: Examples of radiation regions with plasma column waveguides. The

normalized phase and leakage constants are taken from the TM0n modes in Figure

3.6, i.e. , 1.0 p

k a = . (a) “a” mode, (b) “b” mode, and (c) “c” mode. ······················ 86

Figure 4.1: Professor V. G. Veselago of Moscow Institute of Physics and Technology

who predicted the novel properties of metamaterials in 1967. In his paper entitled

“The electrodynamics of substances with simultaneously negative values of ε and

µ” in Soviet Physics Uspekhi , Professor Veselago investigated the extraordinary

properties of metamaterials, such as the reversal of Snell’s law, a reversed Doppler

effect, and reversed Čerenkov radiation. ································································ 94

Figure 4.2: Negative effective permittivity can be obtained from thin metallic wires

and photonic crystals, while negative effective permeability can be obtained from

split ring resonators. (a) Thin metallic wires for negative permittivity, (b) diamond

geometry 3D wire mesh photonic crystals for negative permittivity, and (c) spit

ring resonators (SRRs) for negative permeability. ················································· 95


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Figure 4.3: Arrays of the thin wires plus split ring resonators. Simultaneousnegative permittivity and permeability can be achieved at certain frequency bands.

(a) First experimental embodiment of effective double negative material index by

UCSD group and (b) its three-dimensional extension. ········································· 96

Figure 4.4: (a) Right-handed and (b) left-handed rules. Metamaterials are also

called left-handed materials, as their fields are governed by left-handed rules. E

,

H

, S

, and k

are the electric field, magnetic field, Poynting’s vector, and wave

propagation vector, respectively. While k

and S

are parallel in conventional

media, they are anti-parallel in metamaterials. ····················································· 97

Figure 4.5: Positive and negative refractions. Angles α and β are the incidence

and refraction angles, respectively. Refer to the positive refraction in Figure 1.9 for

a comparison. Metamaterials are also called materials with a negative refractive

index (NRI). ················································································································· 98

Figure 4.6: Reversed energy flows inside and outside metamaterial open

waveguide. Metamaterials are also called backward (BW) wave materials or

materials with a negative phase velocity (NPV). ··················································· 98

Figure 4.7: Classification of materials according to signs of material constants and

behavior of incidence at air (conventional material) – material interface. ·········

99

Figure 4.8: Basic elements of various transmission line models: (a) Conventional

transmission line (right-handed (RH) transmission line), (b) left-handed (LH)

transmission line, (c) conventional lossy transmission lines, and (d) CRLH

(Composite Right/Left Hand) transmission line. L , C , R , and G are the inductance

per unit length, capacitance per unit length, resistance per unit length, and

conductance per unit length, respectively. The subscripts R and L stand for “right”

and “left”, respectively. The concept of the classical RH transmission lines, i.e. , (a)

and (c), has been well established for a long time. The LH transmission lines in (b)

were also proposed a long time ago to describe the backward wave property in

backward wave devices. The CRLH transmission lines are very useful to describe

the wave propagation along metamaterials. ························································ 101

Figure 4.9: Examples of light (electromagnetic wave) manipulations by three-

dimensional photonic crystals. ··············································································· 101


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Figure 4.16: Dielectric and magnetic constants of dispersive material followingexpression in (4.11). Note that the dielectric and magnetic constants are identical.

··········································································································································· 114

Figure 4.17: Dispersion curve for TM0n/TE0n mode of plasma column with various

normalized plasma frequencies. The dotted and dashed line indicates the border of

the SP mode and OS mode, while the vertical dotted line depicts the high

frequency cutoff ( / 2 p

k a ) of the principal mode. ············································· 115

Figure 4.18: Dispersion curve for HE1n/EH1n mode of metamaterial column

waveguides when using identical plasmonic model with several normalized

plasma frequencies. ·································································································· 116

Figure 4.19: Dispersion curve for HE2n/EH2n mode of metamaterial column

waveguides when using identical plasmonic model with several normalized

plasma frequencies. ·································································································· 116

Figure 4.20: Dielectric and magnetic constants of metamaterial in (4.12) based on

reference. The normalized electric and magnetic constants of the metamaterial

were assumed to be 2.34 pe

k a = and 1.98 pm

k a = . (a) Normalized frequency span

from 0.0 to 20.0 and (b) normalized frequency span from 1.0 to 3.0. ················ 118

Figure 4.21: TM-like dispersion curves for three lowest guided modes when using

dissimilar plasmonic model in (4.12): (a) TM0n mode and (b) HE1n mode. ······ 120

Figure 4.22: TE-like dispersion curves for three lowest guided modes when using

dissimilar plasmonic model in (4.12): (a) TE0n mode and (b) EH1n mode. ········ 121

Figure 4.23: Dielectric and magnetic constants of metamaterial: (a) Material

constants from 0 to 100 GHz and (b) same material constant span from 0 to 12 GHz.

As the frequency increases, the dielectric and magnetic constants approach 1.0 and

0.44, respectively. The dielectric and magnetic constants are simultaneously

negative in the shaded region, i.e. , from 4 to 6 GHz. From 6 to 10 GHz, the

dielectric constants are negative, yet the magnetic constants are positive. Above 10

GHz, both material constants are positive, yet below unity, which is similar to the

properties of artificial dielectrics. ··········································································· 123


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Figure 4.24: Guided dispersion characteristics of metamaterial column for TM-likemodes when using Pendry’s model: (a) TM0n mode, (b) HE1n mode, and (c) HE2n

mode. ························································································································· 125

Figure 4.25: Guided dispersion characteristics of metamaterial column for TM-like

modes when using Pendry’s model: (a) TE0n mode, (b) EH1n mode, and (c) EH2n

mode. ························································································································· 126

Figure 4.26: Normalized phase and leakage constants of metamaterial column for

TM0n mode. The column radius 10.0 mma = and the materials constants are

1 4.0

r ε = − and

1 1.0

r µ = − . ···················································································· 130

Figure 4.27: Normalized phase and leakage constants of conventional dielectric rod

waveguide for TM0n mode, as comparison. The rod radius 10.0 mma = and the

dielectric and magnetic constants are1

4.0r

ε = + and1

1.0r

µ = + . ·················· 131

Figure 4.28: Real and complex solutions for metamaterial column waveguide when

using dissimilar plasmonic model in (4.19). ························································· 133

Figure 4.29: Leaky mode characteristics of metamaterial columns when using

forms of material constants in (4.19) for TM0n modes: (a) Normalized phase

constants and (b) normalized leakage constants. ················································

134

Figure 4.30: Negative normalized phase and leakage constants of TM0n mode at

lower frequency. (a) TM01 mode. (b) TM02 mode. The bifurcated point is marked by

“ A” in (b). ··················································································································· 136

Figure 4.31: Leaky mode characteristics of metamaterial column when using

plasmonic forms of material constants in (4.19) for TE0n modes: (a) Normalized

phase constants and (b) normalized leakage constants. ······································· 137

Figure 4.32: Leaky mode characteristics of metamaterial column for the TM0n

modes when using Pendry’s model governed by the expression (4.20): (a)

Normalized phase constants and (b) normalized leakage constants. ················· 139

Figure 4.33: Leaky mode characteristics of metamaterial column for the TE0n modes

when using Pendry’s model governed by the expression (4.20): (a) Normalized

phase constants and (b) normalized leakage constants. ······································· 140


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Figure A.1: Geometry of circular dielectric waveguides in cylindrical geometry: (a)rod waveguide and (b) tube waveguide. r is the radius of the dielectric rod

waveguide;1

r and2

r is the inner and outer radii of the dielectric tube

waveguide, respectively. ························································································· 161

Figure A.2: Dispersion curves of various circular PTFE waveguides. ············· 163

Figure A.3: Fractional power flow ratios in each region of PTFE waveguides: (a)

dielectric region (b) air core region, and (c) free space region. ·························· 166

Figure A.4: Experimental setups. (a) Picture and (b) schematic. ······················ 168

Figure A.5: Dielectric losses of various flexible waveguides. ···························· 170

Figure A.6: Radiation losses of various flexible waveguides. ···························· 170

Figure B.1: Geometry and its cylindrical coordinate system ( ), ,r z θ associatedwith circular open wave guiding structure embedded in free space region. ·· 174

Figure D.1: Interface between two different media and its coordinate systems. 187

Figure E.1: Transformation concept from complex propagation constant plane ontoSDP, and associated relations.

z k is the complex propagation constant of the

guiding structure and Φ is the steepest decent variable. The physical validity of

the field solutions can be judged on the SDP. ···················································· 192

Figure E.2: Interface between two dissimilar media. The propagation and

transverse directions are the +z- and +x- directions, respectively. Φ is the complex

observation angle. ···································································································· 192

Figure E.3: Riemann sheets of complex z

k plane: (a) Top (proper) sheet and (b)

bottom (improper) sheet. ························································································· 193

Figure E.4: Steepest descent plane showing properties of waves in each partition.

The field solution mapping into the shaded region is physically valid. ·········· 196

Figure E.5: Steepest descent plane showing slow (unshaded) and fast (shaded)

waves. ························································································································ 196


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CHAPTER 1.

Thesis Introduction

Chapter 1 provides an introduction with a brief review of the developmentof academic and engineering electromagnetic wave technologies, including(1) open electromagnetic waveguides and (2) guided and leaky modes, asthe fundamental concepts underlying this thesis. Finally, brief summariesare given of Chapters 2, 3, 4, and 5, along with the basic motivation for thisstudy.

1.1. Development of Electromagnetic Wave Technologies

In 1864, the existence of electromagnetic wave propagation through space wastheoretically predicted for the first time by the Scottish physicist J. C. Maxwell1) (Figure 1.1, left) in his mathematical derivations of electromagnetic fields. Soonafterwards, in 1888, the German physicist H. R. Hertz2) (Figure 1.1, right)conducted an important experiment using a spark gap oscillator that provedMaxwell’s theory to be correct.

1) See, L. Campbell and W. Garnett, The Life of James Clerk Maxwell, Macmillan, London, 1882 (1997digital preservation). It is available at http://www.sonnetusa.com/bio/maxwell.asp.

2) C. Süsskind, “Heinrich Hertz: A short life,” IEEE Transactions on Microwave Theory and Techniques , vol.36, no. 5, pp. 802-805, May 1988.

R. S. Elliot, “The history of electromagnetics as Hertz would have known it,” IEEE Transactions on Microwave Theory and Techniques , vol. 36, no. 5, pp. 806-823, May 1988. J. D. Kraus, “Heinrich Hertz – Theorist and Experimenter,” IEEE Transactions on Microwave Theory andTechniques , vol. 36, no. 5, pp. 824-829, May 1988. J. H. Bryant, “The first century of microwaves – 1886 to 1986,” ”IEEE Transactions on Microwave Theoryand Techniques , vol. 36, no. 5, pp. 830-858, May 1988.


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Figure 1.1: Portraits of Scottish physicist J. C. Maxwell (left, 1831-1879) and Germanphysicist H. R. Hertz (right, 1857-1894). J. C. Maxwell theoretically predicted the existence ofelectromagnetic waves in 1864, while H. R. Hertz experimentally verified their existence in1888. (Source: http://freelektronik.free.fr/LEKTRONIK/annexes/biograph.htm)

Figure 1.2: Plane electromagnetic wave propagation. The electric and magnetic fieldsgenerate each other, and the vector of the electric and magnetic fields and propagationdirection of the electromagnetic waves are all perpendicular.(Source: http://micro.magnet.fsu.edu/primer/lightandcolor/electromagintro.html)


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The mathematical plane electromagnetic waves in Figure 1.2 have transverseelectric and magnetic components that are perpendicular to each other and aimedin a propagation direction also perpendicular to their transverse electric andmagnetic components. In addition, electromagnetic waves can carry their energy orinformation with them from one point to another.Science and modern technologies have flourished based on the discovery ofelectromagnetic waves. For example, modern technologies involvingelectromagnetic waves include satellite communications, high-energy particleaccelerators, electron microscopy, various active and passive microwave / opticaldevices, and many more modern complex high-precision devices and systems.Artistic representations of satellite communications and a collection of modernelectromagnetic wave-manipulating devices (electromagnetic crystal devices) areshown in Figure 1.3.The efficient transfer of information or energy from one point to another in achosen direction is performed by specially designed electromagnetic structures ormedia called, electromagnetic waveguides. As such, electromagnetic waveguidesare very important devices as regards carrying electromagnetic energy or signals ina certain direction, and as a basic part of various other RF/ microwave / millimeterwave / optical devices, such as filters, directional couplers, power dividers,resonators, and so on.

(a) (b)

Figure 1.3: Artistic drawings of modern applications of electromagnetic waves. (a) Satellitecommunications among earth station, satellites, and space shuttles, where thecommunications are performed using electromagnetic waves. (Source:http://www.hardlines.co.uk/gall_aw/tec_07.htm) (b) “The photonic micropolis”, generated by Professor John D. Joannopoulos’s group at MIT, composed of various photonic devices,such as waveguides and cavities. (Source: http://ab-initio.mit.edu/)


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x

y

z

b

a

x

a

z

(a) (b)

Figure 1.4: Two typical examples of closed waveguides: (a) rectangular waveguide and (b)circular waveguide. An electromagnetic wave is guided in the interior region of thewaveguiding structure. The interior and outer regions of the waveguides are isolatedelectrically by the conductor shielding. The modes are determined by the operatingfrequency and physical dimensions a and b.

Figure 1.5: Earth-ionosphere waveguiding system as example of closed waveguide. Anelectromagnetic wave is reflected off the ionosphere layer composed of a plasma layer.Meanwhile, the plasma acts as a conducting medium when the incident wave frequency ishigher than the ionosphere’s plasma frequency. A simple explanation of electromagneticwave propagation with plasma is given in Chapter 3.(Source: http://www.arsc.edu/science/ionosphere.html)


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Planar-type open electromagnetic waveguides have already been extensivelystudied for a long time (See, e.g., ref. [4] and references therein.). These waveguidesinclude image guides, microstrip lines, coplanar waveguides, and slot lines formicrowave or millimeter wave transmission, and strip guides, embedded stripguides, rib guides, and inverted rib guides for integrated optics. A list of examplesof planar-type open waveguides is shown in Figures 1.6 and 1.7. The guided andleaky dispersion characteristics of these planar-type waveguides have also beenextensively studied, as they are core parts of modern applications, such ascommunication devices and systems.Another type of open electromagnetic waveguiding structure that is not planar is acircular open electromagnetic waveguiding structure. The representativeapplication of a circular open waveguiding structure is an optical fiber [5].Figure 1.8 shows an optical fiber and its method of propagating total internalreflection. As such, current advancements in high-speed internet services andcommunications are due to optical fibers.Open electromagnetic waveguides usually have guided modes with a purely realpropagation constant and leaky modes with a complex propagation constant. Theguided mode and leaky mode characteristics of the planar open electromagneticwaveguides in Figures 1.6 and 1.7 are relatively well known, as they are practicallyemployed in many microwave / optical integrated circuits. Guided dispersioncharacteristics have also been studied for usage in various optical applications.

However, the leaky mode characteristics of circular open waveguiding structuresare relatively unknown in spite of their broad spectrum of potential uses. Therefore,these leaky mode characteristics are one of the major concerns of this thesis.


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Figure 1.6: Examples of open planar electromagnetic waveguides for optical integratedcircuits: (a) strip waveguide, (b) embedded strip waveguide, (c) rib waveguide, and (d)inverted rib waveguide. Note that 2 1 0n n n> > , where 0n is the refractive index of thesurrounding free space region. The guided and leaky dispersion characteristics of thewaveguides are governed by the refractive indexes of the dielectric materials and waveguidedimensions, w and t.

r ε

r ε

r ε r ε

Figure 1.7: Examples of open electromagnetic waveguides for planar microwave ormillimeter wave frequencies: (a) microstrip line, (b) slot line, (c) coplanar waveguide, and (d)image guide. The guided and leaky dispersion characteristics of the waveguides aregoverned by the dielectric constant r ε and waveguide dimensions, w , t , h , and s.


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Figure 1.8: Optical fiber as example of electromagnetic surface waveguide: left, optical fibersand right, principle of light transmitting through optical fiber by total internal reflection.The incident angles α , β , and γ shown in the figure are less than the critical angle. Theelectromagnetic waves are guided in the interior high refractive index region by the totalinternal reflections.(Source: http://www.hitachi-cable.co.jp/ewc/smenu091.htm)(Source: http://kottan-labs.bgsu.edu/teaching/workshop2001/chapter1.htm)


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1.3. Guided and Leaky Waves

Generally, once electromagnetic waves are incident upon the surface of twodifferent dielectric media, reflection and refraction phenomena occur between thetwo media due to the dissimilar refractive indexes. Figure 1.9 shows the reflectionand refraction phenomena that occur with geometrical optics. At a certain criticalangle of incidence or above, the transmitted ray disappears and all theelectromagnetic energy becomes confined to the incidence region, as shown inFigure 1.10. This is total internal reflection, and the fields in the upper regions ofFigure 1.9 or 1.10 exist in an evanescent form, i.e. , decaying exponentially in the +y-direction. This is the principle of a guided surface wave [6], and optical fibers and

other guided planar integrated optical circuits use this type of electromagneticwave.

Figure 1.9: Reflection and refraction of electromagnetic wave on surface of two dissimilarmaterials. α is the angle of incidence.

Figure 1.10: Reflection and refraction of electromagnetic wave with several incident angles.


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E

Figure 1.12: Radiation from semi-infinite leaky waveguide: (a) Leaky waveguide is fed by

closed waveguide [8]. When viewed from point z’ , the amplitude of the field increases as thedistance from the waveguide increases. (b) Expected field pattern along x direction(transverse direction) [9].

The mathematically improper character of leaky waves is derived from theirsource-free condition. Various basic articles and recent reviews are useful tounderstand the general physics of leaky waves [7-10].The earliest and most widely utilized engineering application of the leaky wavephenomenon is leaky wave antennas [8]. Figure 1.13 shows the structure of the firstleaky wave antenna [11], and this application of leaky waves is still being activelystudied [12]. Meanwhile, other leaky wave applications in themicrowave/millimeter/submillimeter wave band are low-loss waveguides [13],microwave applicators [14], dielectric resonators [15], leaky wave filters [16],directional couplers [17], and so on.


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Figure 1.13: First page of patent (US patent 2,402,622) for earliest leaky wave antennainvented by William W. Hansen in 1940s [11].


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1.4. Thesis Overview

The modal propagation characteristics, including the leaky mode characteristics, ofplanar dielectric waveguides are relatively well known. However, the leaky wavecharacteristics of circular geometry have rarely been investigated in spite of theirpotential importance in various applications due to their geometrical simplicity.Accordingly, this thesis examines the electromagnetic guided and leaky modesalong an open boundary-type cylindrical waveguiding structure composed of (1) adielectric column, (2) plasma column, and (3) metamaterial column, respectively.The dispersion property of a waveguide is an integral part of the variouswaveguide characteristics in terms of practical applications and the actual

dispersion physics. Practically, dispersion is the core component, as it determinesthe fundamental characteristics of the field distribution, mode classification, wavevelocity characteristics, such as the phase and group velocities, and so on. Thesecharacteristics are also important when designing waveguides / transmission linesor other relevant electromagnetic devices and improving applications. In thepreceding sections, the concept of an open-type waveguide was introduced, alongwith guided and leaky mode characteristics. A column structure is one of thesimplest nonplanar open electromagnetic guiding structures and the compositionmaterials considered are all nonconducting. A typical example of this structure isan optical fiber composed of dielectrics. Plasmas and metamaterials can also beused for open electromagnetic structures, therefore, this thesis focuses on the

unusual material indexes of plasmas and metamaterials, such as their negativematerial constants and frequency dispersive natures. More complex materialproperties, such as nonlinearity, inhomogeneities, anisotropy are excluded.Complex propagation constants from complex characteristic equations areobtained using Davidenko’s method that is known to be an efficient way of findinga complex root. (See Appendix C.)As such, Chapter 1 provides a brief review of some of the fundamentalelectromagnetic concepts underlying this thesis, such as open / closed waveguidesand guided / leaky modes, and gives an overview of the rest of the thesis.Chapters 2, 3, and 4 then deal with dielectric, plasma, and metamaterial columnswith open boundaries, respectively. Figure 1.14 shows the column structure used inthis thesis, where the radius of the column is a , the dielectric constants (relativepermittivity) of the inner and outer regions are designated by

1r ε and

2r ε ,

respectively, and the magnetic constants (relative permeability) of the inner andouter regions are designated by

1r µ and

2r µ , respectively. Detailed descriptions

of the material constants are given in the following chapters.


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Figure 1.14: Circular open boundary electromagnetic waveguide. The inner region ( r a< ) isconsidered as a conventional dielectric, plasma, and metamaterial in Chapters 2, 3, and 4,respectively.

Table 1.1: Contents of thesis. Chapter 2 Chapter 3 Chapter 4

Geometry Circular open

Media Dielectric Plasma Metamaterial

Material

DispersionConstant Mostly Dispersive Constant / Dispersive

Loss Lossless/Lossy Lossless Lossless

Mode Circularly symmetricCircularly symmetric /

Hybrid

Circularly symmetric /

Hybrid

Propagation

type

Guided: Review

Leaky: Original work

Guided: Review


Guided: Original work



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Table 1.1 shows the contents of this thesis.In Chapter 2, the guided and leaky mode characteristics of conventional circulardielectric rod waveguides are analyzed for circularly symmetric modes. Thedielectric constant and loss tangent are assumed to be frequency independent.Although the guided mode characteristics are well known, several modalproperties are repeated with specific design parameters. The leaky modes of thistype of waveguide are rigorously identified as the nonphysical mode, antennamode, reactive mode, and spectral gap. Two kinds of circularly symmetric modeare compared and the effects of the design parameters, such as the dielectricconstant and radius of the waveguide, are investigated, especially for the leakymodes. Lastly, the lossy effects on the modal propagation characteristics, such asleaky mode coupling and the creation of the transition region, are examined.In Chapter 3, the guided and leaky mode characteristics of open plasma columnsare investigated, where the plasma columns are composed of dispersive andfrequency-independent materials. The guided mode characteristics of the plasmacolumns with a frequency-independent dielectric constant and dispersiveplasmonic dielectric constants are reviewed using specific numerical examples.Meanwhile, the leaky mode characteristics of the plasma columns are examinedwith a dispersive plasmonic dielectric constant. Discrete solutions for the complexpropagation constants of the plasma columns are found in all the spectral ranges.The leaky mode characteristics of the plasma columns are investigated for several

normalized plasma frequencies. Possible applications for electrically reconfigurableplasma column leaky wave antennas are suggested from the leaky modecharacteristics.In Chapter 4, the guided and leaky mode characteristics of open metamaterialcolumns are investigated. The metamaterials considered in this thesis have (1)frequency-independent and (2) frequency dispersive models. The guided modecharacteristics of the metamaterial columns with frequency-independent materialconstants are investigated and several unusual characteristics, such as superslowwaves, backward waves, mode bifurcations, and guided mode coupling, are found.Plus, the guided mode characteristics of the metamaterial columns with dispersivematerials are also examined. Meanwhile, the leaky mode characteristics of the

metamaterial columns are examined based on the material parameters used in theguided mode analysis.In conclusion, Chapter 5 gives a summary of the entire thesis and suggests furtherresearch work related to this thesis.


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References[1] K. S. Packard, “The origin of waveguides: A case of multiple rediscovery,” IEEE

Transactions on Microwave Theory and Techniques, vol. 32, no. 9, pp. 961-969, September1984.

[2] a) S. F. Mahmoud, Electromagnetic Waveguides: Theory and Applications , Peter Peregrinus,1991.b) F. Olyslager, Electromagnetic Waveguides and Transmission Lines , Oxford UniversityPress, 1999.c) T. Rozzi and M. Mongiardo, Open Electromagnetic Waveguides , The Institution ofElectrical Engineers, 1997.

[3] K. Rawer, Wave Propagation in the Ionosphere , Kluwer Academic Publishers, 1993.

[4] a) S. K. Koul, Millimeter Wave and Optical Dielectric Integrated Guides and Circuits , JohnWiley and Sons, Inc., 1997.b) Special Issue on Open Guided Wave Structures, IEEE Transactions on MicrowaveTheory and Techniques , vol. 29, no. 9, September 1981.

[5] a) D .B. Keck, Selected Papers on Optical Fiber Technology , SPIE Press, 1992.b) D. Gloge, Optical Fiber Technology , IEEE Press, 1976.c) C. K. Kao, Optical Fiber Technology II , IEEE Press, 1981.

[6] a) R. E. Collin, Field Theory of guided Waves , 2nd ed., Chapter 11, IEEE Press, 1991.b) Surface Waves, IRE Transactions on Antennas and Propagation (Special Supplement) , vol.7, no. 5, pp. S132-S243, December 1959.

[7] T. Tamir, Integrated Optics , Springer-Verlag, 1979.[8]

a) L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas I: Rectangular

waveguides,” IRE Transactions on Antennas and Propagation , vol. 7, no. 4, pp. 307-319,October 1959.b) L. O. Goldstone and A. A. Oliner, “Leaky-wave antennas II: Circular waveguides,”IRE Transactions on Antennas and Propagation , vol. 9, no. 3, pp. 280-290, May 1961.

[9] J. –S. Myung, Guidance and leakage by open dielectric waveguides for millimeter waves , Ph.D.Thesis, Polytechnic Institute of New York, 1982.

[10] a) T. Tamir and A. A. Oliner, “Guided complex waves Part 1. Fields at an interface,”Proceeding of the IEE , vol. 110, no. 2, pp. 310-324, February, 1963.b) T. Tamir and A. A. Oliner, “Guided complex waves Part 2. Relation to radiationpatterns,” Proceeding of the IEE , vol. 110, no. 2, pp. 325-334, February, 1963.c) A. A. Oliner, “Types and basic properties of leaky modes in microwave and

millimeter-wave integrated circuits,” IEICE Transactions on Electronics , vol. E83-C, no. 5,pp. 675-686, May 2000.d) A. A. Oliner, “Leaky-wave antennas,” in R. C. Johnson, Antenna Engineering Handbook ,3rd ed., New York: McGraw-Hill, 1993.e) A. A. Oliner, “Leaky waves: Basic properties and applications,” 1997 Asia Pacific Microwave Conference , vol. 1, pp. 397-400, December 2-5, 1997.


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f ) H. Shigesawa and M. Tsuji, “Basic properties of leaky modes in printed-circuittransmission lines,” International Conference on Mathematical Methods in ElectromagneticTheory , vol. 1, pp. 93-98, September 10-13, 2002. g) A. A. Oliner, “Historical perspectives on microwave field theory,” IEEE Transactionson Microwave Theory and Techniques , vol. 32, no. 9, pp. 1022-1045, September 1984.h) A. Ishimaru, Electromagnetic Wave Propagation, Radiation, and Scattering , pp. 46-52,Prentice Hall, 1991.i) S. –T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of opendielectric waveguides: Part I – Mathematical formulations,” IEEE Transactions on Microwave Theory and Techniques , vol. 29, no. 9, pp. 843-855, September 1981. j) S. –T. Peng and A. A. Oliner, “Guidance and leakage properties of a class of open

dielectric waveguides: Part II – New physical effects,” IEEE Transactions on MicrowaveTheory and Techniques , vol. 29, no. 9, pp. 855-869, September 1981.[11] W. W. Hansen, “Radiating Electromagnetic Wave Guide,” US Patent 2,402,622.[12] a) W. Hong and Y. –D. Lin, “Single-conductor strip leaky-wave antenna,” IEEE

Transactions on Antennas and Propagation , vol. 52, no. 7, pp. 1783-1789, July 2004.b) W. Hong, T. –L. Chen, C. –Y. Chang, J. –W. Sheen, and Y. –D. Lin, “Broadbandtapered microstrip leaky-wave antenna,” IEEE Transactions on Antennas and Propagation ,vol. 52, no. 7, pp. 1783-1789, July 2004.

[13] a) M. Miyagi and S. Nishida, “Transmission characteristics of dielectric tube leakywaveguide,” IEEE Transactions on Microwave Theory and Techniques , vol. 28, no. 6, pp.536-541, June 1980.b) M. Miyagi and S. Nishida, “A proposal of low-loss leaky waveguide for

submillimeter waves transmission,” IEEE Transactions on Microwave Theory andTechniques , vol. 28, no. 4, pp. 398-401, April 1980.

[14] a) G. d’Ambrosio and M. D. Migliore, “Numerical and experimental analysis of leaky-wave microwave applicators,” IEEE Transactions on Antennas and Propagation , vol. 52, no.6, pp. 1429-1433, June 2004.b) G. Sauvé, M. Moisan, Z. Zakrzewski, and C. A. Bishop, “Sustaining long linearuniform plasmas with microwaves using a leaky-wave (Troughguide) field applicator,”IEEE Transactions on Antennas and Propagation , vol. 43, no. 3, pp. 248-256, March 1995.

[15] T. Wang, H. An, K. Wu, J. –J. Laurin, and R. G. Bosisio, “Spectral-domain analysis ofradiating cylindrical dielectric resonator for wireless communications,” IEEETransactions on Microwave Theory and Techniques , vol. 43, no. 12, pp. 2959-2964, December

1995.[16] E. G. Cristal, “Analytical solution to a waveguide leaky-wave filter structure,” IEEETransactions on Microwave Theory and Techniques , vol. 11, no. 3, pp. 182-190, May 1963.

[17] a) D. V. Petrov, “Directional coupler using a leaky wave of an anisotropic waveguide,”IEEE Photonics Technology Letters , vol. 8, no. 3, pp. 381-383, March 1996.


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b) D. –C. Niu, T. Yoneyama, and T. Itoh, “Analysis and measurement of NRD-guideleaky wave coupler in Ka-band,” IEEE Transactions on Microwave Theory and Techniques ,vol. 41, no. 12, pp. 2126-2132, December 1993.c) J. –S. Myung, Guidance and leakage by open dielectric waveguides for millimeter waves ,Chapter 5, Ph.D. Thesis, Polytechnic Institute of New York, 1982.


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CHAPTER 2.

Guided and Leaky Modes of

Circular Dielectric Rod

Waveguides

The guided and leaky mode characteristics of circular dielectric rodwaveguides are investigated. The guided mode characteristics are reviewedusing specific numerical examples, while the leaky modes are identified below the guided mode cutoff frequencies and classified as nonphysical,reactive, antenna mode regions, and spectral gaps for circularly symmetricmodes. The effects of the dielectric constant and radius on the leaky mode

characteristics are examined. Lossy effects, such as mode couplingphenomena and the creation of new transition regions between the guidedand leaky modes, are also discussed.

2.1. Dielectric Media

All materials are composed of positively charged atoms and negatively chargedelectrons. The molecules in materials are electrically neutral if no external field isapplied. The applied electric fields displace the charges that make up the dielectric

body, then the resulting dipole moments and induced dipole fields can diminishthe applied electric fields, see Figure 2.1. This effect accumulates and results in acollective response that can be macroscopically modeled by modifying the materialparameters from their vacuum values so that

0r ε ε ε = , where ε is the permittivity,

r ε is the relative permittivity or dielectric constant, and 0ε is the permittivity ofthe free space given by 12

08.854 10 / F mε −= × . Detailed descriptions of the internal

process in a dielectric body are already available in many textbooks.


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a E

a

d

tot

Figure 2.1: Field quantities in dielectric body in presence of external field. a E

: applied field, P

: polarization, d E

: depolarization, and tot E

: total field [1]. The electric dipolepolarization P

and applied field a E

generally proceed in the same direction in naturaldielectric media, resulting in dielectric constants that are greater than unity.

(See e.g., [1]). A similar (although not the same) procedure can be applied tomagnetic media and magnetic fields, where the relative permeability (magneticconstants)

r µ is obtained [1]. Together, the dielectric and magnetic constants are

two of the most important media parameters for describing the interactions between electromagnetic fields (or waves) and material media. Typical naturaldielectrics have dielectric constants that are above unity, and, to date, dielectricmaterials are most widely used in RF to optical frequency devices [2]. Many typicaldielectric constants can be found in existing literature [3].

2.2. Dielectric Waveguides

Waveguiding using dielectric structures dates back to the 19th century. In 1854, J.Tyndall1) demonstrated that light could be guided in free-falling water flows in hispaper entitled “On some phenomena connected with the motion of liquids”. SeeFigure 2.2. This is the earliest official publication regarding the principle of opticalfibers, and even earlier than Maxwell’s equations in 1864.

1) J. Tyndall, “On some phenomena connected with the motion of liquids,” Proceedings of the RoyalInstitution of Great Britain , 1854.


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Figure 2.2: Light caught in falling water: original principle of optical fibers. In 1854, J.Tyndall showed that light could propagate in falling water, which is the earliest officialidentification of the principle of optical fibers, and even earlier than Maxwell’s equations.(Source: http://stereo.thurstons.org/out_and_about.htm)

Thereafter, from the late 19th century through the early 20th century, J. Larmor 1) ,Lord Rayleigh2) , D. Hondros3) , P. Debye3) , H. Zhan4) , O. Schriever5) , et al.investigated the use of a dielectric rod as a waveguiding structure conceptually aswell as experimentally [4]. As a result, circular dielectric rod waveguides areregarded as one of the simplest nonplanar guiding structures and extensively usedfor microwave, millimeter wave, and optical frequencies [5, 6]. However, the mostsuccessful application of a circular dielectric rod is optical fibers [7], which were

1) J. Larmor, “Electric vibrations in condensing systems,” Proc. London Math. Soc , vol. 26, p. 119, Dec.1894.

2) Lord Rayleigh, “On the passage of electric waves through tubes, or the vibrations of dielectriccylinders,” Phil. Mag. , vol. XLIII, pp. 125-132, 1897.

3) D. Hondros and P. Debye, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. d. Phys., vol.32, ser. 4, p. 465, 1910.

4) H. Zhan, “Über den Nachweis Elektromagnetische Wellen an dielektrischen Drähten,” Ann. d. Phys.,vol. 49, ser. 4, p. 907, 1916.

5) O. Schriever, “Elektromagnetische Wellen an dielektrischen Drähten,” Ann. d. Phys., vol. 63, ser. 4, p.645, 1920.


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proposed in the 1960s and are now widely used for high-speed and high-capacityoptical communications. Similar to optical fibers, most circular dielectricwaveguide applications use guided modes that have purely real propagationconstants. Mathematical derivations of the real propagation constants can beobtained from the physical concepts of the total internal reflection (TIR), as shownin Figure 1.11 (a). Figure 2.3 shows two examples of circular dielectric rodwaveguides used with optical and millimeter wave frequencies. The mainadvantages of these waveguides are easy fabrication and analysis, as they can beeasily made by the well-established extrusion process and their modal guidedwave solutions are already recognized in existing literature.However, there are also leaky modes below the guided mode cutoff frequency [8],yet the characteristics of the leaky modes of circular dielectric waveguides arerelatively unknown compared with those of the guided modes.Accordingly, this chapter investigates the leaky mode properties of circulardielectric waveguides without any approximations. In addition, the lossy effects,such as mode coupling phenomena and the creation of a new transition region

between the guided and leaky modes, are also analyzed.

(a) (b)

Figure 2.3: Examples of circular dielectric rod waveguides: (a) Bundle of optical fiberscarrying visible light waves. (Source: http://www.hitachi-cable.co.jp/ewc/smenu091.htm) and(b) flexible waveguide in Q-band [6]. Relatively low radiation losses with bent or flexiblewaveguides can be achieved by properly selecting the design parameters, such as thedielectric materials and radius of the rod. In this case, the radiation loss is mainly affected bythe propagating power distributions, which in turn are controlled by the dispersioncharacteristics. The full text is included in Appendix A.


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2.3. Guided Modes of Circular Dielectric Rod Waveguides – A Review The mode classification of circular dielectric rod waveguides is not a recentresearch topic, as the guided mode of circular dielectric rod waveguides hasalready been extensively analyzed in existing literature [9].As such, this section provides a brief review of the guided mode characteristicsusing a specific numerical example. The procedure used to derive the characteristicequation for the waveguides and associated matters can be found in Appendix B.The characteristic equations for circular dielectric rod waveguides can be expressedas follows:

2

1 2 1 21 2 1 22 2

1 1 2 2 1 1 2 2 0 1 2

( ) ( ) ( ) ( ) 1 1( ) ( ) ( ) ( )

m m m mr r r r

m m m m

J k a K k a J k a K k a mk J k a k K k a k J k a k K k a k a k k ε ε µ µ β

′ ′ ′ ′ + + = +

(2.1)

where1r

ε and2r

ε are the dielectric constants (relative permittivity) of thedielectric and free space regions, respectively,

1r µ and

2r µ are the magnetic

constants (relative permeability) of the dielectric and free space regions,respectively, the dielectric and free space regions are represented by subscripts “1”and “2”,

2 1 21.0

r r r ε µ µ = = = is assumed for this situation, i.e. , a dielectric rod

embedded in free space, ( )m

J ⋅ and ( )m

K ⋅ are an ordinary Bessel function of thefirst kind and modified Bessel function of the second kind, respectively, m is theazimuth

guided and leaky modes of circular phdthesis by kim

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