FRACTIONAL

TRANSMISSION LINES AND WAVEGUIDES

IN ELECTROMAGNETICS

Akhtar Hussain

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

2010

FRACTIONAL

TRANSMISSION LINES AND WAVEGUIDES

IN ELECTROMAGNETICS

by

Akhtar Hussain

In Partial Fulfillment of the Requirements

for the Degree of

Doctor of Philosophy

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

2010

– ii –

CERTIFICATE

It is to certify that Mr. Akhtar Hussain has carried out the work contained in

this dissertation under my supervision.

Dr. Qaisar Abbas Naqvi

Associate Professor

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

Submitted through

Dr. Qaisar Abbas Naqvi

Chairman

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

– iii –

Acknowledgments

In the name of Allah, the Most Gracious and the Most Merciful. Thanks to the

Almighty Allah Who blessed me with his countless blessings. I offer my praises to

Hazrat Muhammad (S. W. A.), Who taught us to unveil the truth behind the natural

phenomena which gave us motivation for research.

I would speak the role of my supervisor, Dr. Qaisar A. Naqvi, in the completion

of this work. He showed a remarkable patience and believed in my ability to complete

the task. His constant guidance, support and encouragement is highly acknowledged.

I would also thank National Center of Physics, Dr. Q. A. Naqvi and Electronics

Department for arranging the visit of Prof. Kohei Hongo (Toho university Japan)

and Prof. Masahiro Hashimoto (Osaka-Electrocommunication University Japan) to

Quaid-i-Azam university. Their visits proved to be a source of inspiration for me.

Thanks are due to the Higher Education Commission of Pakistan which provided me

the opportunity by starting the programme of indigenous PhD scholarships. I am also

thankful to Prof. Nader Engheta (University of Pennsylvania USA) for introducing a

very interesting field of fractional paradigm in electromagnetism. His idea of fractional

curl is the main source of inspiration to start my research work. I also thank Prof.

Elder I. Veliev, chairman, 12th International Conference on Mathematical Methods

in Electromagnetic Theory (MMET08), Odessa, Ukraine for inviting me to present a

research paper in the conference.

I enjoyed the company of very joyful friends like Ahsan Ilahi, Amjad Imran, Fazli

Manan, Maj. Muhammad Naveed, Shakeel Ahmad, and Abdul Ghaffar in which

Amjad Imran was the most cheerful person. I found Ahsan Illahi the most cool and

caring towards his friends. I am thankful for his useful technical discussions and

software support when needed. I must thank my friends Muhammad Faryad and

Husnul Maab for their useful technical contribution in the field of my research.

– iv –

I think, my parents are the best teachers I have ever had. They taught me

to respect others and helped me to build what has brought me this far. They can

take all the credit for much of what I have achieved and what I will achieve in the

future. My daughters Noor-ul-Huda and Imaan Akhtar have been praying for the

successful and timely completion of my research wok. Their affection and prays are

dually acknowledged. I pay thank to my beloved sisters and brothers who always

showed their concern about my studies.

Last but not the least, the never-ending understanding and encouragement from

my beloved wife is the main reason for keeping me optimistic in the face of many

hardships. I want to acknowledge her for giving me the company and serving the

delicious snacks during very long sittings for the compilation of my research work.

Akhtar Hussain

– v –

To

My Family

– vi –

Abstract

Fractional curl operator has been utilized to derive the fractional dual solutions for

different planar boundaries. Perfect electric conductor (PEC), impedance, and perfect

electromagnetic conductor (PEMC) planar boundaries have been investigated and the

behavior of fractional dual solutions is studied with respect to the fractional parameter.

The knowledge of fractional dual solutions has been extended by studying the fractional

parallel plate waveguides, fractional transmission lines and the fractional rectangular

waveguides. Fractional parallel plate waveguides with PEC, impedance, and PEMC

walls as original problems have been studied for the field distribution inside the guide

region and transverse impedance of the guide walls. The investigations have also been

given for the fractional parallel plate chiro waveguides. Fractional transmission lines

of symmetric and non-symmetric nature have been analyzed for their intermediate

behavior and the impedance matching condition has been derived in terms of the

fractional parameter. The fractional rectangular impedance waveguide has also been

investigated. The fractional dual solutions and impedance have been compared with

the reference results which have been found in good agreement for limiting values of

the fractional parameter.

– vii –

CONTENTS

Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

CHAPTER I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1. Conditions for fractionalization of an operator . . . . . . . . . . . . . . . . . . . . . 4

1.2. Recipe for fractionalization of a linear operator . . . . . . . . . . . . . . . . . . . . 4

1.3. Fractional curl operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4. Fractional cross product operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.5. Fractional duality in electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

CHAPTER II: Fractional Dual Solutions for Planar Boundaries . 15

2.1. Fractional dual solutions for a travelling plane wave . . . . . . . . . . . . . . 15

2.2. Planar perfect electric conductor (PEC) interface . . . . . . . . . . . . . . . . . 18

2.2.1 Normal incidence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.2.2 Oblique incidence on a planar PEC boundary . . . . . . . . . . . . . . . 20

2.3. Reflection from a planar impedance boundary . . . . . . . . . . . . . . . . . . . . 26

2.3.1. Transverse electric (TEz) incidence . . . . . . . . . . . . . . . . . . . . . . . . 27

2.3.2. Transverse magnetic (TMz) incidence . . . . . . . . . . . . . . . . . . . . . . 29

2.4. Reflection from a planar perfect electromagnetic conductor (PEMC)

boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

2.4.1. Transverse electric (TEz) incidence . . . . . . . . . . . . . . . . . . . . . . . . 33

2.4.2. Transverse magnetic (TMz) incidence . . . . . . . . . . . . . . . . . . . . . . 36

CHAPTER III: Fractional Parallel Plate Waveguides . . . . . . . . . . . . . . 39

3.1.General wave behavior along a guiding structure . . . . . . . . . . . . . . . . . . . 39

3.2. Fractional parallel plate PEC waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.2.1. Behavior of fields inside the fractional parallel plate PEC

waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

– viii –

3.2.2. Mode behavior for higher values of the fractional parameter 45

3.2.3. Transverse impedance of walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.3. Fractional parallel plate impedance waveguide . . . . . . . . . . . . . . . . . . . . 49

3.4. Fractional parallel plate PEMC waveguide . . . . . . . . . . . . . . . . . . . . . . . . 51

CHAPTER IV: Fractional Chiro Waveguide and the Concept of

Fractional Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . 55

4.1. Reflection from a chiral-achiral interface . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2. Fractional parallel plate chiro waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.3. The concept of fractional transmission lines . . . . . . . . . . . . . . . . . . . . . . . 65

4.3.1. Fractional symmetric transmission line . . . . . . . . . . . . . . . . . . . . . 66

4.3.2. Fractional non-symmetric transmission line . . . . . . . . . . . . . . . . . 70

4.3.3. Multiple-sections fractional non-symmetric transmission line 74

CHAPTER V: Fractional Rectangular Impedance Waveguide . . . . 76

5.1. General theory of rectangular waveguides . . . . . . . . . . . . . . . . . . . . . . . . . 76

5.2. Field formulation for the rectangular impedance waveguide . . . . . . . 78

5.3. Fractional rectangular impedance waveguide . . . . . . . . . . . . . . . . . . . . . . 81

5.3.1. Behavior of fields inside the fractional rectangular

impedance waveguide . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.3.2. Surface impedance of walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

5.3.3. Power transferred through a cross section . . . . . . . . . . . . . . . . . 88

CHAPTER VI: Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

– ix –

List of Publications

[1] A. Hussain and Q. A. Naqvi, Fractional curl operator in chiral medium and

fractional nonsymmetric transmission line,Progress in Electromagnetic Research

PIER 59, pp: 199-213, 2006.

[2] A. Hussain, S. Ishfaq and Q. A. Naqvi, Fractional curl operator and fractional

waveguides, Progress in Electromagnetic Research, PIER 63, pp: 319-335, 2006.

[3] S. A. Naqvi, Q. A. Naqvi, and A. Hussain, Modelling of transmission through a

chiral slab using fractional curl operator, Optics Communications, 266, pp: 404-

406, 2006.

[4] A. Hussain, M. Faryad, and Q. A. Naqvi, Fractional curl operator and fractional

chiro-waveguide, Journal of Electromagnetic Waves and Applications, Vol. 21,

No. 8, pp: 1119-1129, 2007.

[5] A. Hussain, Q. A. Naqvi, and M. Abbas, Fractional duality and perfect electro-

magnetic conductor (PEMC), Progress in Electromagnetics Research, PIER 71,

pp: 85-94, 2007.

[6] A. Hussain and Q. A. Naqvi, Perfect electromagnetic conductor (PEMC) and

fractional waveguide, Progress in Electromagnetics Research, PIER 73, pp: 61-

69, 2007.

[7] A. Hussain, M. Faryad and Q. A. Naqvi, Fractional waveguides with impedance

walls, Progress in Electromagnetic Research C, PIERC 4, pp: 191-204, 2008.

[8] A. Hussain, M. Faryad, and Q. A. Naqvi, Fractional dual parabolic cylindrical

reflector, 12th International Conference on Mathematical Methods in Electro-

magnetic Theory, Odessa, Ukraine, June 29-July 02, 2008.

[9] A. Hussain and Q. A. Naqvi, Fractional rectangular impedance waveguide, Prog-

ress in Electromagnetics Research, PIER 96, pp: 101-116, 2009.

– 1 –

CHAPTER I

Introduction

Fractional calculus is a branch of mathematics that deals with operators hav-

ing non-integer and/or complex order, e.g., fractional derivatives and fractional inte-

grals. Fractional derivatives/integrals are mathematical operators involving differenti-

ation/integration of arbitrary (non-integer) real or complex orders such as dαf(x)/dxα,

where α can be taken to be a non integer real or even complex number [1]. In a

sense, these operators effectively behave as the so-called intermediate cases between

the integer-order differentiation and integration. Fractional Fourier transform is one

of the examples of fractional operators and has many applications in the field of optics

and signal processing [2]. Indeed, recent advances of fractional calculus are dominated

by modern examples of applications in physics, signal processing, fluid mechanics, vis-

coelasticity, mathematical biology, and electrochemistry. For example, fractance as

a generalization of resistance and capacitance has been introduced in [3]. Fractance

represents the electrical element with fractional order impedance and can behave as

a fractional integrator of order 1/2. Another example in the area of control theory is

that all proportional-integral-derivative (PID) controllers are special cases of the frac-

tional proportional-integral-derivative (PIλDµ) controllers [4]. Numerous applications

have demonstrated that fractional PID-controllers (PIλDµ−controllers) perform suffi-

ciently better for the control of fractional order dynamical systems than the classical

PID-controllers. Odhoham and Spanier [5] suggested the replacement of classical inte-

ger order Fick’s law of diffusion, which describe the diffusion of electro-active species

towards electrodes, by the fractional order integral law for describing generalized diffu-

sion problems ranging from electro-active species to the atmospheric pollutants. The

concept of fractional divergence as introduced in reactor description may in future

lead to the development of reactor criticality concept based on fractional geometrical

buckling [6]. This enables to describe the reactor flux profile more closely to actual

– 2 –

which can be utilized to maintain efficient correction and control. Fractional diver-

gence may be used to describe several anomalous effects presently observed in diffusion

experiments, e.g., non-linearity effects and its explanation are the challenges which are

hard to meet through integer order theory or by probabilistic methods. According to

the scientists, fractional calculus can be the language of twenty first century for phys-

ical system description and controls [6]. The ”ifs and buts”, related to this calculus

as today, is due to our own limitations and understanding. This will have a clearer

picture tomorrow when products based on this subject will be used in the industry.

Prof. Nader Engheta, University of Pennsylvania USA, initiated work on bringing

the tools of fractional derivatives/fractional integrals into the theory of electromag-

netism [8-17]. He termed this special area of electromagnetics as fractional paradigm

in electromagnetic theory. He introduced the definition for fractional order multipoles

[10] of electric charge densities and proved that the fractional multipoles effectively

behave as intermediate sources bridging the gap between the cases of integer order

point multipoles such as point monopoles, point dipoles, and point quadrupoles etc.

He formulated the electrostatic potential distribution for the fractional multipoles in

front of the structures like dielectric spheres [8], perfectly conducting wedges and cones

[9] and termed the methods as ”fractional image methods”. Using the fractional order

integral relation, fractional dual solutions to the scalar Helmholtz equation have been

derived and discussed in [11]. It has been determined that fractional dual solutions to

the scalar Helmholtz equation may represent the generalized solutions in between the

fields radiated by a two dimensional source (i.e., line source) and a one dimensional

source (i.e., plate source). Naqvi and Rizvi [25] used the fractional order integral re-

lation for correlating the fractional solutions of [11] and determined the intermediate

solutions of the fractional solutions for a line source and a plate source. Lakhtakia [18]

derived a theorem which shows that the new set of solutions for time harmonic Fara-

day and Ampere Maxwell equations with sources can be obtained using a differential

operator which commutes with the curl operator.

– 3 –

During mathematical treatment in fractional paradigm, often in order to solve

a general problem, the canonical cases are solved first. Second step is to derive an

operator that can transform one canonical case into the other and then one may think

of the possibility of any solution between the two canonical cases. Schematically, this

concept has been shown in Figures 1.1a and 1.1b for a linear operator L.

Figure 1.1a. A symbolic bock diagram representing a problem and its two canonical

cases

Figure 1.1b. A bock diagram symbolizing the fractional paradigm

– 4 –

Figure 1.1a shows the block diagram of a classical electromagnetic problem in which

two canonical solutions of the problem are shown while fractional paradigm of the

problem is shown in Figure 1.1b which shows possible intermediate solutions of the

two canonical cases. The conditions for fractionalization of a linear operator and a

recipe for fractionalization of a linear operator has been discussed in [13-16] and also

given in the next subsections. The new fractionalized operator, which may symbolically

be denoted by Lα with the fractional parameter α, under certain conditions, can be

used to obtain the intermediate cases between the canonical case 1 and case 2. The

two cases may be connected through the number of intermediate cases.

1.1 Conditions for fractionalization of an operator

A linear operator L may be a fractionalized operator (i.e. Lα) that provides the

intermediate solutions to the original problems, if it satisfies the following properties:

I. For α = 1, the fractional operator Lα should become the original operator L,

which provide us with case 2 when it is applied to case 1.

II. For α = 0, the operator Lα should become the identity operator I and thus the

case 1 can be mapped onto itself.

III. For any two values α1 and α2 of the fractionalization parameter α, Lα should

have the additive property in α , i.e., Lα1 .Lα2 = Lα2 .Lα1 = Lα1+α2 .

IV. The operator Lα should commute with the operator involved in the mathematical

description of the original problem.

1.2 Recipe for fractionalization of a linear operator

Let us consider a class of linear operators (or mappings) where the domain and

range of any linear operator of this class are similar to each other and have the same

– 5 –

dimensions. In other words, any linear operator of this class, which can be generically

shown by L, should map an element from the space Cn into generally another element

in the space Cn. That is, L : Cn → Cn where Cn is a n-dimensional vector space over

the field of complex numbers. Once a linear operator such as L is given to us, a recipe

for constructing the fractional operator Lα can be described as follows [13]

1. One finds the eigenvectors and eigenvalues of the operator L in the space Cn so

that L.Am = amAm where Am and am for m = 1, 2, 3, ..., n are the eigenvectors

and eigenvalues of the operator L in Cn respectively.

2. Provided Ams form a complete orthogonal basis in the space Cn, any vector in

this space can be expressed in terms of linear combination of Ams. Thus, an

arbitrary vector G in space Cn can be written as

G =n∑

m=1

gmAm

where gms are the coefficients of expansion of G in terms of Ams.

3. Having obtained the eigenvectors and eigenvalues of the operator L, the fractional

operator Lα can be defined to have the same eigenvectors Ams, but with the

eigenvalues as (am)α, i.e.,

Lα.Am = (am)αAm

When this fractional operator Lα operates on an arbitrary vector G in the space

Cn, one gets

Lα.G = Lα.

n∑m=1

gmAm =n∑

m=1

gmLα. Am =n∑

m=1

gm(am)αAm

The above equation essentially defines the fractional operator Lα from the knowledge

of operator L and its eigenvectors and eigenvalues. In next section, above recipe has

been applied to fractionalize the curl operator.

– 6 –

1.3 Fractional curl operator

The concept of fractional curl operator was introduced in 1998 by Engheta [13].

The curl operator falls in the class of linear operators defined earlier and hence can

be fractionalized. It may be noted that the curl operation becomes a cross product

operation in the Fourier domain as explained below:

Consider a three-dimensional vector field F as a function of three spatial variables

in (x, y, z) coordinate system. Curl of this vector can be written as

curlF =(

∂Fz

∂y− ∂Fy

∂z

)x +

(∂Fx

∂z− ∂Fz

∂x

)y +

(∂Fy

∂x− ∂Fx

∂y

)z (1.1)

where Fx, Fy, and Fz are the Cartesian components of vector F, and x, y, and z are

the unit vectors in the space domain. The next step is to apply the spatial Fourier

transform, from the space domain (x, y, z) into the k-domain (kx, ky, kz), on vectors F

and curl F. Assuming that the spatial Fourier transforms of these two vector functions

exist, the Fourier transform can be written as

Fk {F(x, y, z)} = F(kx, ky, kz)

=∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞F(x, y, z) exp (−ikxx− ikyy − ikzz)dx dy dz

(1.2)

Fk {curlF(x, y, z)} =∫ ∞

−∞

∫ ∞

−∞

∫ ∞

−∞curlF(x, y, z) exp (−ikxx− ikyy − ikzz)dx dy dz

= ik× F(kx, ky, kz) (1.3)

where a tilde over the vector F denotes the spatial Fourier transform of vector F.

Hence in the k-domain, the curl operator can be written as a cross product of vector

ik with the vector F. It is suggested that, in order to fractionalize the curl operator,

one should first fractionalize the cross product operator (ik×) in the k-domain. Clearly

the cross product operator is an operator that gets two vectors as its inputs and gives

one vector as its output, e.g., k×U = W. However, if one picks the first vector k, then

– 7 –

the operator (k×) can be considered as a linear operator which takes one vector (e.g.,

U) as its input and gives out one vector (e.g., W) as an output. Both vectors U and W

are three-dimensional vectors in the k-domain. Thus, the linear operator (k×) belongs

to the class of linear operators mentioned earlier, and therefore it can be fractionalized.

Thus fractionalization of curl operator is equivalent to fractionalization of this cross

product operator. With the method described in section 1.2, fractionalization of the

cross product operator as (ik×)α can be proposed in the k-domain as explained in the

next section.

1.4 Fractional cross product operator

The procedure for fractionalization of a linear operator presented earlier can be

used here to obtain the fractionalized cross product operator shown by the symbol

(k×)α. For an illustrative example, let us take the case where the vector k in the op-

erator (k×) is the unit vector along the z-axis in the k-domain, i.e., zk. The eigenvalues

and (normalized) eigenvectors of the operator (zk×) in the k-domain are obtained as

A1 =xk − iyk√

2, a1 = i

A2 =xk + iyk√

2, a2 = −i

A3 = zk, a3 = 0 (1.4)

where xk, yk, and zk are the unit vectors in the k-domain.

The vector U = Uxxk + Uyyk + Uz zk upon which the fractional cross product

operation has to be performed must be written in terms of eigen vectors of the operator

as

U = g1A1 + g2A2 + g3A3 (1.5)

where g1, g2, and g3 are the coefficients of expansion and can be written as

g1 =Ux + iUy√

2, g2 =

Ux − iUy√2

, g3 = Uz

– 8 –

Operation of cross product can be written as

(zk×)U = a1g1A1 + a2g2A2 + a3g3A3 (1.6a)

Fractionalization of the cross product means fractionalization of the eigen values in

equation (1.6a) and hence can be expressed as

(zk×)αU = (a1)αg1A1 + (a2)αg2A2 + (a3)αg3A3

= (+i)α (Ux + iUy)(x− iyk)2

+ (−i)α (Ux − iUy)(x + iyk)2

+ (0)αUz zk (1.6b)

For example, if one takes U = xk, equation (1.6b) becomes

(zk×)αxk =[(i)α 1√

2

(xk − iyk√

2

)+ (−i)α 1√

2

(xk + iyk√

2

)]

= cos(απ

2

)xk + sin

(απ

2

)yk (1.7)

which provides the fractional cross product of (zk×)α acting on vector xk. As observed

from equation (1.7), when α = 1, one obtain the conventional (ordinary) cross product

zk × xk = yk. When α = 0, one then obtains (zk×)0xk = xk = I.xk, i.e., the identity

operator operating on xk. For other values of α between zero and unity, one gets the

intermediate or fractional cases of cross product operation.

The above fractionalization of the cross product can then, in principle, be applied

to the case of ik× F in the k-domain. If the resulting expression can then be inverse

Fourier transformed back into the (x, y, z)-domain, the final result may be called the

fractional curl of vector F, i.e., curlαF. In next section application of fractional curl

operator in electromagnetics is addressed.

– 9 –

1.5 Fractional duality in electromagnetics

One of the potential applications of fractional curl operator can be the fractional-

ization of the duality theorem in electromagnetism. Consider the source-free Maxwell

equations in vacuum (with permittivity ε and permeability µ) for the harmonic time

dependence exp(−iωt) as

curlH = −iωεE

curlE = iωµH

div H = 0

div E = 0 (1.8)

Applying the spatial Fourier transform on both sides of the above equations leads to

1ik

[ik× (ηH)

]= −E

1ik

[ik× (E)

]= ηH

ik. (ηH) = 0

ik. E = 0 (1.9)

where η =√

µ/ε, k = ω√

µε. E(kx, ky, kz) and H(kx, ky, kz) are the spatial Fourier

transforms of the vectors E(x, y, z) and H(x, y, z) respectively.

Let us apply the fractional cross product operator 1(ik)α (ik×)α on both sides of

the first two equations in (1.9). it gives

1(ik)α+1

(ik×)α+1 (ηH) = − 1(ik)α

(ik×)αE

1(ik)α+1

(ik×)α+1 E =1

(ik)α(ik×)α(ηH) (1.10)

Since cross product operator holds the property of commutation, it can be shown that

(ik×)α+1 = (ik×)1(ik×)α = (ik×)α(ik×)1, and thus equation (1.10) can be rewritten

– 10 –

as

1ik

ik×[

1(ik)α

(ik×)α (ηH)]

= −[

1(ik)α

(ik×)αE]

(1.11a)

1ik

ik×[

1(ik)α

(ik×)α E]

=[

1(ik)α

(ik×)α(ηH)]

(1.11b)

It can be shown that the following equations also hold

ik.

[1

(ik)α(ik×)α (ηH)

]= 0 (1.11c)

ik.

[1

(ik)α(ik×)α E

]= 0 (1.11d)

Comparing equation (1.11) with equation (1.9) reveals that, since E(kx, ky, kz) and

H(kx, ky, kz) are solutions to the source-free Maxwell equations in the k-domain, the

fields defined by

Efd =[

1(ik)α

(ik×)α E]

(1.12a)

ηHfd =[

1(ik)α

(ik×)α (ηH)]

(1.12b)

are also a new set of solutions to the source-free Maxwell equations. Inverse Fourier

transforming these back into the (x, y, z)-domain, we obtain the new set of solutions

as

Efd =[

1(ik)α

curlα E]

(1.13a)

ηHfd =[

1(ik)α

curlα (ηH)]

(1.13b)

From equations (1.13), it can be seen that

α = 0 ⇒ Efd = E, ηHfd = ηH

and α = 1 ⇒ Efd = ηH, ηHfd = −E

which shows that for α = 0, (Efd, ηHfd) gives the original solution while (Efd, ηHfd)

gives dual to the original solution of the Maxwell equations for α = 1. Therefore for

– 11 –

all values of α between zero and unity, (Efd, ηHfd) provides the new set of solutions

which can effectively be regarded as the intermediate solutions between the original

solution and dual to the original solution. These solutions are also called the fractional

dual fields as expressed with the subscript fd for these fields.

Various investigations have been made in exploring the role of fractional duality in

electromagnetics. The study related to the fractional dual solutions to Maxwell equa-

tions in homogeneous chiral medium is given in [26]. Field decomposition approach

of [48] has been used and it is determined that orientation of the polarization ellipse

of fractional fields is rotated by an angle απ/2 with respect to the original solutions.

Study relating the fractional duality in metamaterials with negative permittivity and

permeability is given in [27] and it is proved that fractional dual solutions in double

negative (DNG) medium are similar to the ordinary or double positive (DPS) medium

with an additional multiplication factor of απ. Application of the complex and higher

order of the fractional curl operator in electromagnetics has been discussed in [28].

It has been found that the fractional dual solutions are periodic with respect the

fractional parameter α and the period is 4. The period has four subranges and the

fractional solutions for any subrange act as original solution for the next subrange as

explained below:

Further it is concluded that transverse impedance is periodic with period 2 such that

if the fractional dual surface acts as an inductive surface in one subrange, it will act

as a capacitive surface in the next subrange. The study corresponding to the complex

value of the fractional parameter α = α1 + iα2 reveals that the fractional solutions

– 12 –

may be represented as fractional dual solutions only if α1 and α2 falls in two different

fractional ranges shown above. Hussain and Naqvi introduced the concept of fractional

transmission lines and fractional waveguides [29-35]. Recently Naqvi has extended the

investigations to fractional duality in the chiral medium having property of chiral

nihility [41-42]. It may be noted that more than one paths are possible between the

given two canonical cases. This means that the set of intermediate solutions between

the two canonical cases may be different depending upon the value and sign of the

fractional parameter as shown in Figure 1.2.

Figure 1.2. An example showing that the two canonical cases may be connected

through different intermediate paths

Veliev and Engheta has addressed the problem of reflection from a fractional dual

boundary [19]. They obtained the fractional dual solutions in terms of fixed solutions

for oblique incidence on an impedance infinite surface and derived the reflection coef-

ficients of the fractional boundary in terms of the original reflection coefficients. They

found that impedance of the fractional reflecting surface is anisotropic and gave the

impedance boundary conditions for the new boundary. The more generalized form of

these boundaries has been discussed and given in [21]. The fractional curl operator has

been applied to study the reflection from a bi-isotropic slab backed by a PEC surface

– 13 –

in [23] in which it is shown that order of the fractional curl operator can be used to

control the twist polarizer effect.

In view of the interesting role of fractional curl operator in illustrating the polar-

ization of the propagating wave and effective impedance of the boundary in reflection

problems attracted me to investigate the role and utility of fractional curl operator

in microwave engineering. ”Fractional transmission lines and waveguides in electro-

magnetics” has been chosen as the topic of research. In order to discuss fractional

transmission lines and fractional waveguides, the two canonical cases must be men-

tioned first. Discussion in this thesis deals with the two canonical cases which are

related through the principle of duality. That is, the two canonical cases are related

through the curl operator. So by fractionalizing the curl operator, one can get new set

of transmission lines and waveguides which may be regarded as intermediate step of

the two canonical cases related through the duality theorem. The transmission lines

and waveguides which are intermediate step of the two cases have been termed as

the fractional transmission lines and fractional waveguides. Answers to the following

questions are targeted: What would be the meaning of fractional transmission lines

and fractional waveguides in electromagnetics? How to derive the expressions which

govern fractional transmission lines and waveguides? What is the behavior of field

pattern inside the fractional waveguides? What is impedance of walls of the fractional

waveguides? How the power density is distributed across the cross section of the frac-

tional waveguides? To answer these questions, the thesis has been organized in the

following manners:

In chapter II, The fractional dual solutions for the travelling plane wave in a

lossless, homogeneous, and isotropic medium are derived. Then the fractional dual

solutions for the standing waves in the presence of the reflecting boundaries have been

discussed. Planar boundaries of perfect electric conductor (PEC), impedance, and

perfect electromagnetic conductor (PEMC) have been considered. The fractional dual

– 14 –

solutions have been termed as the solutions of reflection from fractional PEC, frac-

tional impedance, and fractional PEMC boundaries. Dependence of the impedance

of the fractional boundaries with respect to the fractional parameter has been stud-

ied. Transverse electric (TEz) and transverse magnetic (TMz) incidences have been

discussed separately.

Study related to the reflection of a plane wave from the planar boundaries has

been extended for the parallel plate waveguides and given in chapter III. The resulting

waveguides have been termed as the fractional parallel plate waveguides. Focus of this

chapter is to study the field distribution inside the fractional parallel plate waveguides.

Fractional parallel plate PEC, fractional parallel plate impedance, and fractional par-

allel plate PEMC waveguides have been investigated. Dependence of impedance of

walls of the fractional waveguides upon the fractional parameter has been discussed.

Chapter IV deals with the fractional dual solutions in the chiral medium. In this

chapter behavior of the chiral-achiral interface has been studied with respect to the

fractional parameter α. Fractional parallel plate waveguides having PEC walls and

filled with a chiral medium has been investigated. The concept of fractional transmis-

sion lines has also been discussed in this chapter. Transmission lines of symmetric and

non-symmetric nature have been considered.

In chapter V, fractional rectangular impedance waveguides have been investigated.

The rectangular waveguide having impedance walls and filled with a homogenous,

lossless, and isotropic material has been considered. Field distribution in the transverse

plane of the waveguide, impedance of walls of the guide and power density distribution

in the cross sectional plane have been investigated. The special case of fractional

rectangular waveguide having PEC walls has also been discussed.

The thesis has been concluded in chapter VI.

– 15 –

CHAPTER II

Fractional Dual Solutions for Planar Boundaries

In this chapter, fractional dual solutions to the Maxwell equations for different

planar boundaries have been derived. Perfect electric conductor (PEC), impedance,

and perfect electromagnetic conductor (PEMC) boundaries have been considered for

discussion. The behavior of fractional dual solutions with respect to fractional pa-

rameter is studied and dependence of the impedance of fractional dual boundary on

fractional parameter has been noted. In each case, planar boundary is placed at y = 0

and the region y > 0 is occupied by a lossless, homogeneous, and isotropic medium

having permittivity ε and permeability µ.

2.1 Fractional dual solutions for a travelling plane wave

Let us consider an electromagnetic plane wave propagating in a direction described

by the wave vector k = kyy + kz z. Generic expressions for the electric field E and the

magnetic field H corresponding to this wave can be written as

E = [E0xx + E0yy + E0z z] exp(ikyy + ikzz) (2.1a)

H = [H0xx + H0yy + H0z z] exp(ikyy + ikzz) (2.1b)

where k = ω√

εµ =√

k2y + k2

z

As per recipe described in chapter I, in order to write fractional dual solutions between

(E, ηH) and (ηH,−E), we need to write the field vectors in terms of eigen vectors of

the cross product operator k× = 1k (kyy + kz z)× as

E = [P1A1 + P2A2 + P3A3] exp(ikyy + ikzz) (2.2a)

ηH = [Q1A1 + Q2A2 + Q3A3] exp(ikyy + ikzz) (2.2b)

– 16 –

where A1, A2, and A3 are the normalized eigen vectors of the cross product operator

k×. The normalized eigen vectors and the corresponding eigen values of the cross

product operator k× are as given below

A1 =1√2

[x− i

k(kzy − kyz)

], a1 = +i (2.3a)

A2 =1√2

[x +

i

k(kzy − kyz)

], a2 = −i (2.3b)

A3 =i

k(kyy + kz z), a3 = 0 (2.3c)

In equation (2.2), quantities P1, P2, and P3 are the coefficients of expansion and are

given below

P1 =1√2

[E0x +

i

k(kzE0y − kyE0z)

](2.3d)

P2 =1√2

[E0x − i

k(kzE0y − kyE0z)

](2.3e)

P3 =√

2[−i

k(kyE0y + kzE0z)

](2.3f)

while Q1, Q2, and Q3 are required co-efficients for the magnetic field and may be

obtained by the symmetry.

Fractional dual solutions (Efd, ηHfd) to the Maxwell equations, corresponding

to the original field solutions given in equation (2.1), may be obtained by using the

following relations

Efd = (k×)αE

= [(a1)αP1A1 + (a2)αP2A2 + (a3)αP3A3] exp(ikyy + ikzz) (2.4a)

ηHfd = (k×)αηH

= [(a1)αQ1A1 + (a2)αQ2A2 + (a3)αQ3A3] exp(ikyy + ikzz) (2.4b)

It may be noted that the fields in fractional dual solutions are also related through the

duality theorem, i.e.,

ηHfd = (k×)Efd (2.4c)

– 17 –

In order to give more insight to the fractional dual solutions of Maxwell equations,

let us consider a plane wave propagating in direction described by the vector k =

1k (−kyy + kz z). Associated electric and magnetic fields are given by

E = x exp(−ikyy + ikzz) (2.5a)

ηH =[kz

ky +

ky

kz]

exp(−ikyy + ikzz) (2.5b)

Fractional dual solutions can be obtained using the following relations

Efd =1√2

[P1(a1)αA1 + P2(a2)αA2 + P3(a3)αA3] exp(−ikyy + ikzz)

ηHfd = (k×)Efd

where A1 A2, and A3 are the eigen vectors and a1, a2, and a3 are the corresponding

eigen values of the operator k× = 1k (−kyy + kz z)×. Quantities P1, P2, and P3 are

the coefficients of expansion. Hence the fractional dual solutions can be written as

Efd =[cos

(απ

2

)x +

kz

ksin

(απ

2

)y +

ky

ksin

(απ

2

)z]

exp(−ikyy + ikzz) (2.6a)

ηHfd =[− sin

(απ

2

)x +

kz

kcos

(απ

2

)y +

ky

kcos

(απ

2

)z]

exp(−ikyy + ikzz) (2.6b)

It may be noted that for α = 0 above set of expressions yield result (E, ηH) and for

α = 1 it yields (ηH,−E). For α between zero and unity, fields given in equation (2.6)

can be regarded as fractional dual fields between the original and dual to the original

fields of the plane wave propagating in an oblique direction k = 1k (−kyy + kz z). It

may also be noted from equation (2.6) that fractional dual fields represent a plane

wave propagating in the same direction as the original wave. However its transverse

fields have been rotated by an angle (απ/2).

Now let us derive fractional solutions for the reflection of a plane wave from

different kinds of planar boundaries placed at y = 0.

– 18 –

2.2 Planar perfect electric conductor (PEC) interface

Consider a planar PEC interface which is placed at y = 0. Let us discuss the

fractional dual solutions for normal as well as oblique incidence of a plane wave at the

PEC interface.

2.2.1 Normal incidence

Assume that the PEC interface is excited by a normally incident unit amplitude

plane wave as shown in Figure 2.1.

PEC boundary

z

y

Href

kref

Eref

Hinc

kinc

Einc

Figure 2.1. Normal incidence on a PEC plane

The electric and magnetic fields associated with the incident and reflected plane waves

are given below

Einc = x exp(−iky) (2.7a)

ηHinc = z exp(−iky) (2.7b)

Eref = −x exp(iky) (2.7c)

ηHref = z exp(iky) (2.7d)

Total fields in the region y > 0 can be written as a sum of the incident and reflected

fields and are given below

E = −x2i sin(ky) (2.8a)

– 19 –

ηH = z2 cos(ky) (2.8b)

Fractional dual solutions for the incident wave can be written by using equation (2.4)

as

Eincfd =

[cos

(απ

2

)x + sin

(απ

2

)z]exp(−iky) (2.9a)

ηHincfd =

[− sin

(απ

2

)x + cos

(απ

2

)z]exp(−iky) (2.9b)

Similarly fractional dual solutions for the reflected wave can be written as

Ereffd = − exp(iαπ)

[cos

(απ

2

)x + sin

(απ

2

)z]exp(iky) (2.10a)

ηHreffd = exp(iαπ)

[− sin

(απ

2

)x + cos

(απ

2

)z]exp(iky) (2.10b)

Fractional dual solutions in the region y > 0 can be written as sum of the incident and

reflected fields as

Efd = Eincfd + Eref

fd

ηHfd = ηHincfd + ηHref

fd

which give

Efd = − exp(iαπ

2

) [cos

(απ

2

)x + sin

(απ

2

)z]2i sin

(ky + α

π

2

)(2.11a)

ηHfd = exp(iαπ

2

) [− sin

(απ

2

)x + cos

(απ

2

)z]2 cos

(ky + α

π

2

)(2.11b)

It may be noted from equations (2.8) and (2.11) that for

α = 0 ⇒ (Efd, ηHfd) = (E, ηH)

and α = 1 ⇒ (Efd, ηHfd) = (ηH, −E)

This means that the fractional dual solutions given by equation (2.11) represent the

original field solution for α = 0 while for α = 1, equation (2.11) represents the solution

– 20 –

which is dual to the original solution. For the range 0 < α < 1, equation (2.11)

gives the solutions which are intermediate step of the original solution and dual to the

original solution and hence may be called as fractional dual solutions.

Wave impedance is defined by the ratio of transverse components of corresponding

electric and magnetic fields as

Zfd = −Efdx

Hfdz=

Efdz

Hfdx= iη tan

(ky +

απ

2

)

At y = 0, this becomes impedance of the new reflecting boundary called the fractional

dual boundary and can be written as

Zfd = iη tan(απ

2

)(2.12)

From equation (2.12), it can be interpreted that for α = 0, impedance of the fractional

dual boundary is Zfd = 0, i.e., PEC surface while for α = 1, the impedance is Zfd = ∞,

i.e., PMC surface. For 0 < α < 1, the reflecting surface behaves as an intermediate

step between PEC and PMC surface that depends upon fractional parameter α.

2.2.2 Oblique incidence on a planar PEC boundary

Consider a unit amplitude plane wave propagating in direction described by the

vector kinc = 1k (−kyy + kz z) hits a planar PEC boundary placed at y = 0. Due to

the PEC boundary, reflected wave is produced in direction described by the vector

kref = 1k (kyy + kz z) as shown in Figure 2.2. In case of oblique incidence, the easy

way to solve Maxwell equations is to break the fields into perpendicular polarization

and parallel polarization components. For the field configuration shown in Figure 2.2,

perpendicular polarization can also be referred to as transverse electric polarization

to the z-direction (i.e., TEz polarization), while parallel polarization as transverse

magnetic polarization to the z-direction (i.e., TMz polarization). The fields of the two

– 21 –

polarizations are related through the duality theorem. Let us study TEz and TMz

polarizations separately.

Figure 2.2 Oblique incidence on a PEC plane

Case 1: Transverse electric (TEz) polarization

Let us first consider an incident wave having transverse electric polarization as

shown in Figure 2.2. The electric and magnetic fields for the incident and reflected

waves can be written as

Einc = x exp(−ikyy + ikzz) (2.13a)

ηHinc =[kz

ky +

ky

kz]

exp(−ikyy + ikzz) (2.13b)

Eref = −x exp(ikyy + ikzz) (2.13c)

ηHref =[−kz

ky +

ky

kz]

exp(ikyy + ikzz) (2.13d)

Fractional dual solutions for the incident wave are same as given in equation (2.6), i.e.,

Eincfd =

[cos

(απ

2

)x +

kz

ksin

(απ

2

)y +

ky

ksin

(απ

2

)z]

exp(−ikyy + ikzz) (2.14a)

ηHincfd =

[− sin

(απ

2

)x +

kz

kcos

(απ

2

)y +

ky

kcos

(απ

2

)z]

exp(−ikyy + ikzz) (2.14b)

– 22 –

Fractional dual solutions corresponding to the reflected fields may be obtained using

equation (2.4), i.e.,

Efd = [(a1)αP1A1 + (a2)αP2A2 + (a3)αP3A3] exp(ikyy + ikzz)

ηHfd = (k×)Efd

where A1 A2, and A3 are the eigen vectors and a1, a2, and a3 are the corresponding

eigen values of the operator kref× = 1k (kyy + kz z)×. Quantities P1, P2, and P3 are

the coefficients of expansion. Therefore, we can write

Ereffd = exp(iαπ)

[− cos

(απ

2

)x +

kz

ksin

(απ

2

)y − ky

ksin

(απ

2

)z]

exp(ikyy + ikzz) (2.14c)

ηHreffd = exp(iαπ)

[− sin

(απ

2

)x− kz

kcos

(απ

2

)y +

ky

kcos

(απ

2

)z]

exp(ikyy + ikzz) (2.14d)

Fractional dual solutions corresponding to the fields in the region y > 0 can be written

as sum of the incident and the reflected fields as

Efd = 2[−iCα sin

(kyy +

απ

2

)x +

kz

kSα cos

(kyy +

απ

2

)y

−iky

kSα sin

(kyy +

απ

2

)z]

exp[i(kzz +

απ

2

)](2.15a)

ηHfd = 2[−Sα cos

(kyy +

απ

2

)x− i

kz

kCα sin

(kyy +

απ

2

)y

+ky

kCα cos

(kyy +

απ

2

)z]

exp[i(kzz +

απ

2

)](2.15b)

where

Cα = cos(απ

2

)

Sα = sin(απ

2

)

The fields given in equation (2.15) have been plotted in Figure 2.3 for different values

of α at an observation point (kyy, kzz) = (π/4, π/4).

– 23 –

Figure 2.3 Plots of fractional dual TEz polarized fields at a point (kyy, kzz) =

(π/4, π/4) (a) real parts (b) imaginary parts

– 24 –

From Figure 2.3, it can be seen that fractional dual fields satisfy the principle of duality,

i.e., for α = 0

Efdx = Ex, ηHfdx = ηHx

Efdy = Ey, ηHfdy = ηHy

Efdz = Ez, ηHfdz = ηHz

and for α = 1

Efdx = ηHx, ηHfdx = −Ex

Efdy = ηHy, ηHfdy = −Ey

Efdz = ηHz, ηHfdz = −Ez

Wave impedance is defined by ratio of the transverse components of the electric

and magnetic fields as

Zfdxz = −Efdx

Hfdz= iη

k

kytan

(kyy +

απ

2

)

Zfdzx =Efdz

Hfdx= iη

ky

ktan

(kyy +

απ

2

)

At y = 0, these impedances become impedance of the new reflecting boundary called

the fractional dual boundary and can be written in terms of normalized impedance

zfd = Zfd/η as given below

zfd

=[

k

kyxz +

ky

kzx

]zTEfd (2.16)

where

zTEfd = i tan

(απ

2

), for 0 ≤ α ≤ 1

It may be noted that

zfdxz =k

kyzTEfd , zfdzx =

ky

kzTEfd

Since zfdxz 6= zfdzx, so it can be interpreted that impedance of the fractional dual PEC

boundary for oblique incidence is anisotropic in nature.

– 25 –

Case 2: Transverse magnetic (TMz) polarization

Plane wave reflection geometry for the transverse magnetic polarization from PEC

plane placed at y = 0 is shown in Figure 2.4.

Figure 2.4 Oblique incidence on PEC plane (TMz polarization case)

Electric and magnetic fields shown in the figure can be written as

Einc =[−kz

ky − ky

kz]

exp(−ikyy + ikzz) (2.17a)

ηHinc = x exp(−ikyy + ikzz) (2.17b)

Eref =[−kz

ky +

ky

kz]

exp(ikyy + ikzz) (2.17c)

ηHref = x exp(ikyy + ikzz) (2.17d)

Using the similar procedure as in the above section, fractional dual solutions for the

fields in the region y > 0 may be obtained as

Efd = 2[−iSα sin

(kyy +

απ

2

)x− kz

kCα cos

(kyy +

απ

2

)y

+iky

kCα sin

(kyy +

απ

2

)z]

exp[i(kzz +

απ

2

)](2.18a)

ηHfd = 2[Cα cos

(kyy +

απ

2

)x− i

kz

kSα sin

(kyy +

απ

2

)y

– 26 –

+ky

kSα cos

(kyy +

απ

2

)z]

exp[i(kzz +

απ

2

)](2.18b)

It may be noted from equation (2.18), that the fractional dual fields satisfy the duality

theorem, i.e.,

α = 0 ⇒ (Efd, ηHfd) = (E, ηH)

and α = 1 ⇒ (Efd, ηHfd) = (ηH, −E)

Using the field components given in equation (2.18), impedance of the fractional

dual surface can be written as

zfd

=[

k

kyxz +

ky

kzx

]zTMfd (2.19)

where

zTMfd = i tan

(απ

2

), for 0 ≤ α ≤ 1

Comparing equation(16) and (19), it can be deduced that transverse impedance of

the fractional dual PEC surface is anisotropic and dependance of the impedance on

fractional parameter is same for TEz and TMz polarizations.

2.3 Reflection from a planar impedance boundary

Let the planar interface is placed at y = 0 and has a finite nonzero impedance Z.

By developing the fractional dual solutions of Maxwell equations for the geometry, it

is of interest to see the behavior of the fractional dual impedance boundary. Trans-

verse electric and transverse magnetic incidences are discussed separately in different

subsections.

– 27 –

2.3.1 Transverse electric (TEz) incidence

Consider a plane wave with TEz polarization incident on an impedance planar

boundary having the normalized impedance zb = Z/η and placed at y = 0. The

incident and reflected fields can be written as

Einc = x exp(−ikyy + ikzz) (2.20a)

ηHinc =[kz

ky +

ky

kz]

exp(−ikyy + ikzz) (2.20b)

Eref = Γtex exp(ikyy + ikzz) (2.20c)

ηHref = Γte

[kz

ky − ky

kz]

exp(ikyy + ikzz) (2.20d)

where Γte is the reflection coefficient as given below

Γte = −1− zbky

k

1 + zbky

k

(2.21)

Now fractional dual solutions in the region y > 0 can be written as

Efd =[xCα

{exp

(−ikyy − i

απ

2

)+ Γte exp

(ikyy + i

απ

2

)}

+ ySαkz

k

{exp

(−ikyy − i

απ

2

)− Γte exp

(ikyy + i

απ

2

)}

+ zSαky

k

{exp

(−ikyy − i

απ

2

)+ Γte exp

(ikyy + i

απ

2

)}]

exp[i(kzz + α

π

2

)](2.22a)

ηHfd =[−xSα

{exp

(−ikyy − i

απ

2

)− Γte exp

(ikyy + i

απ

2

)}

+ yCαkz

k

{exp

(−ikyy − i

απ

2

)+ Γte exp

(ikyy + i

απ

2

)}

+ zCαky

k

{exp

(−ikyy − i

απ

2

)− Γte exp

(ikyy + i

απ

2

)}]

exp[i(kzz + α

π

2

)](2.22b)

Tangential fields at the boundary y = 0 may be written as

Et = Efdxx + Efdzz

ηHt = ηHfdxx + ηHfdzz

– 28 –

Plots of these tangential fields are given in Figure 2.5 for two values of normalized

impedance that is, zb = 0 and zb = 100. It may be noted that normalized impedance

zb = 0 gives PEC case while zb = 100 gives PMC case. Solid lines show the plots of

tangential electric fields while dashed lines are for the corresponding magnetic fields.

In Figure 2.5, zero values of the electric field for (α, zb) = (0, 0) and (α, zb) = (1, 100)

show the boundary conditions for the PEC surface while zero values of the magnetic

field for (α, zb) = (1, 0) and (α, zb) = (0, 100) show the boundary conditions for the

PMC. For any value of the normalized impedance zb between 0 and ∞, the dual to

the impedance boundary will be an admittance boundary and fractional dual of the

boundary would be an intermediate step between the impedance boundary and the

admittance boundary and hence may be called fractional impedance boundary.

Figure 2.5 Fractional dual tangential fields for TEz polarization in the presence of

an impedance boundary

– 29 –

Impedance of the fractional dual surface can be written as

zfd

=[

k

kyxz +

ky

kzx

]zTEfd (2.23)

where

zTEfd =

zb

(ky

k

)+ i tan

(απ2

)

1 + zb

(ky

k

)i tan

(απ2

) , for 0 ≤ α ≤ 1

2.3.2 Transverse magnetic (TMz) incidence

Now consider the case of TMz-polarized wave incident on an impedance boundary

having normalized impedance zb. The incident and reflected electric and magnetic

fields for this polarization can be written as

Einc =[−ky

ky − kz

kz]

exp(−ikyy + ikzz) (2.24a)

ηHinc = x exp(−ikyy + ikzz) (2.24b)

Eref = Γtm

[−ky

ky +

kz

kz]

exp(−ikyy + ikzz) (2.24c)

ηHref = Γtmx exp(−ikyy + ikzz) (2.24d)

where Γtm is the reflection coefficient given by

Γtm =ky

k − zb

ky

k + zb

(2.25)

Fractional dual solutions in the region y > 0 for this case can be written by applying

the duality on the fields of equation (2.22) subject to the replacement of the reflection

coefficient Γte by Γtm and hence can be written as

Efd =[xSα

{exp

(−ikyy − i

απ

2

)− Γtm exp

(ikyy + i

απ

2

)}

− yCαkz

k

{exp

(−ikyy − i

απ

2

)+ Γtm exp

(ikyy + i

απ

2

)}

− zCαky

k

{exp

(−ikyy − i

απ

2

)− Γtm exp

(ikyy + i

απ

2

)}]

– 30 –

exp[i(kzz + α

π

2

)](2.26a)

ηHfd =[xCα

{exp

(−ikyy − i

απ

2

)+ Γtm exp

(ikyy + i

απ

2

)}

+ ySαkz

k

{exp

(−ikyy − i

απ

2

)− Γtm exp

(ikyy + i

απ

2

)}

+ zSαky

k

{exp

(−ikyy − i

απ

2

)+ Γtm exp

(ikyy + i

απ

2

)}]

exp[i(kzz + α

π

2

)](2.26b)

In order to validate the fields given in equation (2.26), tangential components at

y = 0 have been plotted in Figure 2.6. In contrast to Figure 2.5, Figure 2.6 shows

that the fractional dual solutions given by equation(2.26) satisfy the conditions of

fractional dual impedance boundary. That means for (α, zb) = (0, 0) and (α, zb) =

(1, 100), boundary conditions for PEC surface are satisfied while for (α, zb) = (1, 0)

and (α, zb) = (0, 100), boundary conditions for PMC surface are satisfied.

Figure 2.6 Fractional dual tangential fields for TMz polarization

– 31 –

Impedance of the fractional dual impedance surface can be written as

zfd

=[

k

kyxz +

ky

kzx

]zTMfd (2.27)

where

zTMfd =

zb

(kky

)+ i tan

(απ2

)

1 + zb

(kky

)i tan

(απ2

) , for 0 ≤ α ≤ 1 (2.27a)

It may be noted that in case of TMz polarization, impedance of the fractional dual

impedance surface is different from TEz polarization. To show the difference, impedan-

ces given in equation (2.27a) and (2.23a) have been plotted as shown in the Figure 2.7.

From the figure it can further be noted that for normalized impedance zb = 0, behaviors

for the two polarizations become same, i.e., zTEfd = zTM

fd which is as for the case of

fractional dual PEC boundary discussed earlier.

Figure 2.7 Behavior of impedance of the fractional dual impedance surface with

respect to α for TEz and TMz polarizations

– 32 –

2.4 Reflection from a planar perfect electromagnetic conductor (PEMC)

boundary

Perfect electric conductor (PEC) and perfect magnetic conductor (PMC) are basic

concepts in electromagnetics. Lindell has recently introduced the concept of perfect

electromagnetic conductor (PEMC) as generalization of PEC and PMC [44]. It is well

known that PEC boundary may be defined by the conditions

n×E = 0, n.B = 0

While PMC boundary may be defined by the boundary conditions

n×H = 0, n.D = 0

The PEMC boundary conditions are of the more general form

n× (H + ME) = 0, n.(D−MB) = 0 (2.28)

where M denotes the admittance of the PEMC boundary. It is obvious that PMC

corresponds to M = 0, while PEC corresponds to M = ±∞. It may be noted that

PEMC boundary is non-reciprocal. Non-reciprocity of the PEMC boundary can be

demonstrated by showing that the polarization of the plane wave reflected from PEMC

surface is rotated. Problems involving PEMC boundaries with the admittance param-

eter M and air or other isotropic medium can be transformed to problems involving

PEC or PMC boundaries using duality transformation [45]. The incident wave fields

may be transformed using the following duality transformation

[Einc

d

Hincd

]=

[Mη η− 1

η Mη

] [Einc

Hinc

](2.29)

where η is the impedance of the medium facing the PEMC boundary.

– 33 –

The reflected fields (Erefd , ηHref

d ) corresponding the transformed incident fields

(Eincd , ηHinc

d ) can be written using the PEC boundary conditions and finally the inverse

transformation [45] may be used to get the fields reflected from the PEMC surface as[

Eref

Href

]=

1(Mη)2 + 1

[Mη − η

1η Mη

] [Eref

d

Hrefd

](2.30)

Another generalization of PEC and PMC reveals from the concept of fractional curl

operator, i.e., (∇×)α. The boundary is known as fractional dual interface with PEC

and PMC as the two special situations of the fractional dual interface [13]. In this

section, intermediate situations between the PEMC boundary and dual to the PEMC

boundary (DPEMC) using the idea of fractional curl operator would be discussed.

Behavior of the impedance dealing with intermediate situations is of interest.

2.4.1 Transverse electric (TEz) incidence

Consider a plane wave with TEz-polarization is incident upon a PEMC boundary

plane placed at y = 0. Electric and magnetic fields for the incident wave are similar

to as equation (2.13), i.e.,

Einc = x exp(−ikyy + ikzz)

ηHinc =[kz

ky +

ky

kz]

exp(−ikyy + ikzz)

Applying the transformation given in equation (2.29), duality transformed fields cor-

responding to the incident fields can be written as

Eincd =

[Mηx +

(−kz

ky +

ky

kz)]

exp(−ikyy + ikzz) (2.31a)

ηHincd =

[−x + Mη

(−kz

ky +

ky

kz)]

exp(−ikyy + ikzz) (2.31b)

Fields reflected from the planar PEC boundary, when the incident wave defined by

the fields given in equation (2.31) hits the boundary, can be written as

Erefd = −

[Mηx +

(−kz

ky +

ky

kz)]

exp(ikyy + ikzz) (2.32a)

ηHrefd =

[−x + Mη

(−kz

ky +

ky

kz)]

exp(ikyy + ikzz) (2.32b)

– 34 –

The fields reflected from the PEMC boundary can be written by applying the inverse

transformation given in equation (2.30) and are given below

Eref =[1− (Mη)2

1 + (Mη)2x− 2Mη

1 + (Mη)2

(−kz

ky +

ky

kz)]

exp(−ikyy + ikzz) (2.33a)

ηHref = kref ×Eref (2.33b)

where

kref =1k

(kyy + kz z)

The quantity Mη can be represented in terms of angle as Mη = tan θ, where θ = π/2

represents Mη = 0, that is PEC boundary and θ = 0 represents Mη = ∞, that is

PMC boundary. Hence equation (2.33) may be written in alternate form as

Eref =[cos(2θ)x + sin(2θ)

(kz

ky − ky

kz)]

exp(ikyy + ikzz) (2.34a)

Href =[− sin(2θ)x + cos(2θ)

(kz

ky − ky

kz)]

exp(ikyy + ikzz) (2.34b)

Total fields in the region y > 0 can be written as sum of the incident and reflected

fields and are given below

E = exp(ikzz)[2x

{cos(kyy) sin2(θ)− i sin(kyy) cos2(θ)

}

+(

kz

k

)y sin(2θ)

{cos(kyy) + i sin(kyy)

}

−(

ky

k

)z sin(2θ)

{cos(kyy) + i sin(kyy)

}](2.35a)

ηH = exp(ikzz)[−sin(2θ)x

{cos(kyy) + i sin(kyy)

}

+(

kz

k

)2y

{cos(kyy) cos2(θ)− i sin(kyy) sin2(θ)

}

+(

ky

k

)2z

{cos(kyy) sin2(θ)− i sin(kyy) cos2(θ)

}](2.35b)

At θ = π/2, above equations reduce as

E = −2i exp(ikzz)x sin(kyy) (2.36a)

ηH = 2 exp(ikzz)[(

kz

k

)y cos(kyy)− i

(ky

k

)z sin(kyy)

](2.36b)

– 35 –

which are the relations for the PEC boundary. It may also be noted that at y = 0

equation (2.35) satisfies the PEMC boundary conditions, i.e.,

y × (ηH + tan θE) = 0

Fractional dual solutions in the region y > 0 can be written as

Efd = exp[i(kzz +

απ

2

)]

[x

{(Cθ + Cα) cos

(kyy +

απ

2

)+ (Cθ − Cα) i sin

(kyy +

απ

2

)}

+y(

kz

k

) {(Sθ + Sα) cos

(kyy +

απ

2

)+ (Sθ − Sα) i sin

(kyy +

απ

2

)}

−z(

ky

k

) {(Sθ − Sα) cos

(kyy +

απ

2

)+ (Sθ + Sα) i sin

(kyy +

απ

2

)}](2.37a)

η0Hfd = exp[i(kzz +

απ

2

)]

[−x

{(Sθ + Sα) cos

(kyy +

απ

2

)+ (Sθ − Sα) i sin

(kyy +

απ

2

)}

+y(

kz

k

) {(Cθ + Cα) cos

(kyy +

απ

2

)+ (Cθ − Cα) i sin

(kyy +

απ

2

)}

−z(

ky

k

) {(Cθ − Cα) cos

(kyy +

απ

2

)+ (Cθ + Cα) i sin

(kyy +

απ

2

)}](2.37b)

whereCθ = cos

(2θ − απ

2

)

Sθ = sin(2θ − απ

2

)

Normalized impedance of the fractional dual PEMC surface may be obtained from the

ratio of the fields at y = 0 as

zfd

=[

k

kyzTEfdxzxz +

ky

kzTEfdzxzx

], 0 ≤ α ≤ 1 (2.38)

where

zTEfdxz =

[Cα(Cθ + Cα) + iSα(Cθ − Cα)Cα(Cθ − Cα) + iSα(Cθ + Cα)

](2.38a)

zTEfdzx =

[Cα(Sθ − Sα) + iSα(Sθ + Sα)Cα(Sθ + Sα) + iSα(Sθ − Sα)

](2.38b)

This shows that both the components of the normalized impedance of the fractional

dual PEMC boundary have different behavior with respect to the fractional parameter.

– 36 –

2.4.2 Transverse magnetic (TMz) incidence

Consider a plane wave with TMz-polarization is incident upon a PEMC boundary

plane placed at y = 0. Following the treatment similar to the last section, fields

reflected from the PEMC surface can be written as

Eref =[− cos(2θ)

(−kz

ky +

ky

kz)− sin(2θ)x

]exp(ikyy + ikzz) (2.39a)

Href =[− sin(2θ)

(kz

ky − ky

kz)− cos(2θ)x

]exp(ikyy + ikzz) (2.39b)

Fractional dual solutions for the problem can be written as

Efd = exp[i(kzz +

απ

2

)]

[−x

{(Sθ − Sα) cos

(kyy +

απ

2

)+ (Sθ + Sα) i sin

(kyy +

απ

2

)}

+y(

kz

k

) {(Cθ − Cα) cos

(kyy +

απ

2

)+ (Cθ + Cα) i sin

(kyy +

απ

2

)}

−z(

ky

k

) {(Cθ + Cα) cos

(kyy +

απ

2

)+ (Cθ − Cα) i sin

(kyy +

απ

2

)}](2.40a)

ηHfd = exp[i(kzz +

απ

2

)]

[−x

{(Cθ − Cα) cos

(kyy +

απ

2

)+ (Cθ + Cα) i sin

(kyy +

απ

2

)}

−y(

kz

k

) {(Sθ − Sα) cos

(kyy +

απ

2

)+ (Sθ + Sα) i sin

(kyy +

απ

2

)}

+z(

ky

k

) {(Sθ + Sα) cos

(kyy +

απ

2

)+ (Sθ − Sα) i sin

(kyy +

απ

2

)}](2.40b)

Normalized impedance of the fractional dual PEMC surface may be obtained as

zfd

=[

k

kyzTMfdxzxz +

ky

kzTMfdzxzx

], 0 ≤ α ≤ 1 (2.41)

where

zTMfdxz =

[Cα(Sθ − Sα) + iSα(Sθ + Sα)Cα(Sθ + Sα) + iSα(Sθ − Sα)

](2.41a)

zTMfdzx =

[Cα(Cθ + Cα) + iSα(Cθ − Cα)Cα(Cθ − Cα) + iSα(Cθ + Cα)

](2.41b)

– 37 –

It may be noted here that

zTMfdxz = zTE

fdzx, and zTMfdzx = zTE

fdxz

Plots of these impedances are given in Figure (2.8). Figure 2.8a shows variation along

the α axis while Figure 2.8b shows the variation of the impedance components with

the admittance parameter θ. Figure 2.8a shows that for values of θ between π/2 and

0, the impedance component zfdxz changes from 1 to tan2 θ while zfdzx changes from

cot2 θ to 1 as the value of the fractional parameter α changes from 0 to 1. Further

we can see from the figure that for θ = π/2, impedance of the fractional dual PEMC

boundary represents the fractional dual PEC boundary. Behavior of the impedance of

fractional dual PEMC boundary along the admittance axis as seen from Figure 2.8b,

for values of θ between π/2 and 0, can be described as

α = 0 ⇒ (zfdxz, zfdzx) = (1, cot2 θ)

α = 0.5 ⇒ (zfdxz, zfdzx) = (1, 1)

α = 1 ⇒ (zfdxz, zfdzx) = (tan2 θ, 1)

That means at α = 0, impedance component zfdxz becomes independent of the admit-

tance parameter θ and the same is true for zfdzx at α = 1. For α = 0.5, impedance of

the fractional dual PEMC boundary becomes independent of the admittance param-

eter θ. As the admittance parameter approaches the limiting values of π/2 and 0, the

two impedance components approach the same values equal to the case of PEC surface

and PMC surface respectively. Hence Figure 2.8b shows that for (α, θ) = (0, π/2) and

(α, θ) = (1, 0), the PEMC surface behaves as a PEC surface and for (α, θ) = (1, π/2)

and (α, θ) = (0, 0), the PEMC surface behaves as a PMC surface. This is also in

accordance with the published literature.

– 38 –

Figure 2.8a Transverse impedance of the fractional dual PEMC surface versus α

Figure 2.8b Transverse impedance of the fractional dual PEMC surface versus θ

– 39 –

CHAPTER III

Fractional Parallel Plate Waveguides

In this chapter, discussion of previous chapters has been extended to parallel plate

waveguides with PEC, impedance, and PEMC walls. Parallel plate waveguides with

fractional dual solutions have been termed as the fractional parallel plate waveguides.

The effect of fractional parameter on field distribution inside the guide is discussed.

Transverse impedance of the walls of fractional guide has been determined.

3.1. General wave behavior along a parallel plate guiding structure

Consider a waveguide consisting of two parallel plates separated by a dielectric

medium with constitutive parameters ε and µ. One plate is located at y = 0, while

other plate is located at y = b as shown in Figure 3.1. The plates are assumed to be

of infinite extent and the direction of propagation is considered as positive z-axis.

Figure 3.1. Geometry of parallel plate waveguide

– 40 –

Electric and magnetic fields in the source free dielectric region must satisfy the following

homogeneous vector Helmholtz equations

∇2E(x, y, z) + k2E(x, y, z) = 0 (3.1a)

∇2H(x, y, z) + k2H(x, y, z) = 0 (3.1b)

where ∇2 = ∂2

∂x2 + ∂2

∂y2 + ∂2

∂z2 is the Laplacian operator and k = ω√

µε is the wave

number. Taking z-dependance as exp(iβz), equation (3.1) can be reduced to two

dimensional vector Helmholtz equation as

∇2xyE(x, y) + h2E(x, y) = 0 (3.2a)

∇2xyH(x, y) + h2H(x, y) = 0 (3.2b)

where h2 = k2 − β2, β is the propagation constant.

Since propagation is along z-direction and the waveguide dimensions are consid-

ered infinite in xz-plane. So x-dependence can be ignored in the above equations.

Under this condition, equation (3.2) becomes ordinary second order differential equa-

tion as

d2E(y)dy2

+ h2E(y) = 0 (3.3a)

d2H(y)dy2

+ h2H(y) = 0 (3.3b)

As a general procedure to solve waveguide problems, the Helmholtz equation is solved

for the axial field components only. The transverse field components may be obtained

using the axial components of the fields and Maxwell equations. So scalar Helmholtz

equations for the axial components can be written as

d2Ez

dy2+ h2Ez = 0 (3.3c)

d2Hz

dy2+ h2Hz = 0 (3.3d)

– 41 –

General solution of the above equations is

Ez = An cos(hy) + Bn sin(hy) (3.3e)

Hz = Cn cos(hy) + Dn sin(hy) (3.3f)

where An, Bn, Cn, and Dn are constants and can be found from the boundary condi-

tions.

Using Maxwell curl equations, the transverse components can be expressed in

terms of longitudinal components (Ez,Hz), that is

Ex =1h2

(iβ

∂Ez

∂x+ ik

∂ηHz

∂y

)(3.4a)

Ey =1h2

(iβ

∂Ez

∂y− ik

∂ηHz

∂x

)(3.4b)

Hx =1h2

(iβ

∂Hz

∂x− ik

η

∂Ez

∂y

)(3.4c)

Hy =1h2

(iβ

∂Hz

∂y+

ik

η

∂Ez

∂x

)(3.4d)

where

η =√

µ

εis impedence of the medium inside the guide

In the proceeding part of this chapter, parallel plate waveguides with PEC, impedance,

and PEMC walls have been considered and the fractional dual solutions have been

determined and analyzed.

3.2. Fractional parallel plate PEC waveguide

In this section, parallel plate waveguide with PEC walls is the one problem while

parallel plate waveguide with PMC walls is the other problem. According to Maxwell

equations, these two problems are related through the curl operator. Using fractional

curl operator, the waveguide which may be regarded as intermediate step of the waveg-

uides with PEC walls and PMC walls has been studied. TMz and TEz cases have

been discussed separately.

– 42 –

Case 1: Transverse magnetic (TMz) mode solution

Suppose a transverse magnetic (TMz) mode is propagating inside the waveguide

shown in Figure 3.1. Let plates of the waveguide are perfect electric conductor (PEC)

and z-axis is the direction of propagation. Axial component of the electric field is given

by the solution of equation (3.3e) for PEC boundaries as

zEz(y, z) = zAn sin (hy) exp(iβz)

= −zAn

2i[exp(−ihy + iβz)− exp(ihy + iβz)] (3.5a)

where h = nπb and An ia an arbitrary constant that depends upon initial conditions.

Using (3.4), the corresponding transverse components of the fields can be written

as

yEy(y, z) = yiβ

hAn cos (hy) exp(iβz)

= yiβ

h

An

2[exp(−ihy + iβz) + exp(ihy + iβz)] (3.5b)

xηHx(y, z) = −xik

hAn cos (hy) exp(iβz)

= −xik

h

An

2[exp(−ihy + iβz) + exp(ihy + iβz)] (3.5c)

Fields inside the waveguide may be considered as combination of two TEM plane waves

bouncing back and forth obliquely between the two conducting plates, i.e.,

E = E1 + E2 (3.6a)

ηH = ηH1 + ηH2 (3.6b)

where (E1, ηH1) are the electric and magnetic fields associated with one plane wave

and are given below

E1 =An

2

(iz +

iβ

hy)

exp(−ihy + iβz) (3.7a)

ηH1 = −xik

h

An

2exp(−ihy + iβz) (3.7b)

– 43 –

while (E2, ηH2) are the electric and magnetic fields associated with the second plane

wave and are given below

E2 =An

2

(−iz +

iβ

hy)

exp(ihy + iβz) (3.8a)

ηH2 = −xik

h

An

2exp(ihy + iβz) (3.8b)

Propagation through the parallel plate waveguide in terms of two TEM plane waves

is shown in Figure 3.2.

Figure 3.2 Plane wave representation of the fields inside the waveguide

Comparing equation (3.7) and (3.8) with (2.17), it may be noted that the fields

(E1, ηH1) =(− ik

h

An

2

)(Einc, ηHinc)

and

(E2, ηH2) =(− ik

h

An

2

)(Eref , ηHref)

provided that h = ky and β = kz.

This means that solution of the parallel plate PEC waveguide is proportional to

the solution of the reflection problem in the region y > 0 for a planar PEC boundary

– 44 –

at y = 0 . Hence from the knowledge of chapter 2, fractional dual solutions inside the

parallel plate PEC waveguide can be written as

Efd =(− ik

hAn

)[−iSα sin

(hy +

απ

2

)x− β

kCα cos

(hy +

απ

2

)y

+ih

kCα sin

(hy +

απ

2

)z]

exp[i(βz +

απ

2

)](3.9a)

ηHfd =(− ik

hAn

)[Cα cos

(hy +

απ

2

)x− i

β

kSα sin

(hy +

απ

2

)y

+h

kSα cos

(hy +

απ

2

)z]

exp[i(βz +

απ

2

)](3.9b)

where

Cα = cos(απ

2

)

Sα = sin(απ

2

)

3.2.1 Behavior of fields inside the fractional parallel plate PEC waveguide

In order to study the behavior of fields inside the fractional parallel plate PEC

waveguide, electric and magnetic field lines are plotted in the yz-plane and are shown

in Figure 3.3. These plots are for the mode propagating through the guide at an angle

π/6 so that β/k = cos(π/6), h/k = sin(π/6). Solid lines show the electric field plots

while magnetic fields are shown by dashed lines. From the figure we see that field lines

are partially parallel and partially perpendicular to the guide walls for non-integer

values of α. This shows that walls of the waveguide can be considered as intermediate

step between the PEC and PMC walls. For limiting values of α, the behavior is as

follows: For α = 0, electric field lines are perpendicular to the guide walls and there

are no magnetic field lines in the yz-plane which shows that the walls are PEC and

the mode is transverse magnetic. For α = 1, it can be seen that magnetic field lines

are perpendicular to the guide walls while there are no electric field lines which shows

– 45 –

that the walls are PMC and the propagating mode is the transverse electric. These

patterns are also in accordance with [47].

Figure 3.3 Field lines in yz-plane at different values of α; solid lines are for the

electric field while dashed lines are for the magnetic field

3.2.2 Mode behavior for higher values of the fractional parameter

Let us note the modal configuration for higher order values of the fractional pa-

rameter α. It may be noted from equation (3.9) and (3.6) that

α = 0 ⇒ Efd = E, ηHfd = ηH,

Efdz 6= 0, Hfdz = 0, zfd = 0 (3.10a)

α = 1 ⇒ Efd = ηH, ηHfd = −E,

Efdz = 0, Hfdz 6= 0, zfd = ∞ (3.10b)

α = 2 ⇒ Efd = −E, ηHfd = −ηH,

– 46 –

Efdz 6= 0, Hfdz = 0, zfd = 0 (3.10c)

α = 3 ⇒ Efd = −ηH, ηHfd = E,

Efdz = 0, Hfdz 6= 0, zfd = ∞ (3.10d)

α = 4 ⇒ Efd = E, ηHfd = ηH,

Efdz 6= 0, Hfdz = 0, zfd = 0 (3.10e)

In above equations, zfd represents transverse impedance of plates of the fractional

guide. These behaviors are shown in Figure 3.4.

Figure 3.4 Dependence of modal configuration and guide walls nature upon α

(a) α = 0 (b) α = 1 (c) α = 2 (d) α = 3

– 47 –

From the figure, it can be interpreted that if one starts with a transverse magnetic

mode propagating through a parallel plate waveguide with PEC walls, α = 1 gives the

solution for a transverse electric mode propagating through a parallel plate waveguide

with PMC walls. Increasing value of α from 1 to 2 further gives the rotation of π/2

in the field configuration which represents the transverse magnetic mode and walls of

the waveguide are also become PMC. These changes in the behavior continue with

increasing integer values of α and the field configuration is repeated at α = 4. Hence

it may be deduced that behavior of solutions with respect to the fractional parameter

is periodic with period 4.

3.2.3 Transverse impedance of walls

It has been seen that the fields inside the parallel plate PEC waveguide are pro-

portional to the fields in the region y > 0 in the presence of a planar PEC boundary

at y = 0. Therefore transverse impedance of the walls of the fractional parallel plate

PEC waveguide would be same as the planar PEC reflecting boundary discussed in

chapter 2.

Case 2: Transverse electric (TEz) mode solution

Solution for the transverse electric mode propagating through a parallel plate

PEC waveguide may be obtained by solving the equation (3.3f) for Hz while Ez = 0

for this case. Field solutions may be obtained by using PEC boundary conditions and

equation (3.4) so that the electric and magnetic fields may be considered as the fields

of two TEM plane waves bouncing back and forth between the two conducting plates.

Similar to the transverse magnetic case, fields in the transverse electric case are also

proportional to the fields in the region y > 0 for the problem of a transverse electric

reflection from the planar PEC boundary placed at y = 0 with(− ik

h Cn

)as the constant

– 48 –

of proportionality. Hence fractional dual solutions have the same proportionality, i.e.,

Efd =(− ik

hCn

)[−iCα sin

(hy +

απ

2

)x +

β

kSα cos

(hy +

απ

2

)y

−ih

kSα sin

(hy +

απ

2

)z]

exp[i(βz +

απ

2

)](3.11a)

ηHfd =(− ik

hCn

)[−Sα cos

(hy +

απ

2

)x− i

β

kCα sin

(hy +

απ

2

)y

+h

kCα cos

(hy +

απ

2

)z]

exp[i(βz +

απ

2

)](3.11b)

where Cn is an arbitrary constant and depends upon the initial conditions. These

fields have been plotted in Figure 3.5 which shows the behavior of field lines in the

yz-plane. The simulation data is same as of Figure 3.3.

Figure 3.5 TEz field lines in yz-plane at different values of α; solid lines are for the

electric field while dashed lines are for the magnetic field

– 49 –

3.3 Fractional parallel plate impedance waveguide

In the last section, it is seen that the fractional fields inside a parallel plate waveg-

uide are proportional to the fields in the region y > 0 in the presence of a reflecting

PEC boundary at y = 0. A parallel plate waveguide with impedance walls repre-

sented by the fractional dual solutions can be termed as the fractional parallel plate

impedance waveguide. In order to study the behavior of fields inside the fractional

parallel plate impedance waveguide, let us consider a transverse magnetic mode prop-

agating through a parallel plate waveguide whose walls have finite impedance Zw.

Geometry of the waveguide under consideration is same as shown in Figure 3.1. The

electric and magnetic fields inside the impedance waveguide must satisfy the impedance

boundary conditions as given below

Ez|(y=0) = ZwHx|(y=0) (3.12a)

Ez|(y=b) = −ZwHx|(y=b) (3.12b)

Solution for the electric and magnetic fields inside the parallel plate impedance waveg-

uide can be written by using the impedance boundary conditions. Similar to the case of

parallel plate PEC waveguide, the fields may be represented in terms of two travelling

plane waves bouncing back and forth between the two plates. Fields inside the par-

allel plate impedance waveguide are also related to the fields given in equation (2.24)

through the constant of proportionality given as

C =(−k

h

An

2

)(F + i)

where

F = −izw

(k

h

)

zw =Zw

η

– 50 –

Hence the fractional dual solutions for the TMz mode can be written as

Efd = B

(−ik

h

) [−iSα

{F cos

(hy +

απ

2

)+ sin

(hy +

απ

2

)}x

+(

β

k

)Cα

{F sin

(hy +

απ

2

)− cos

(hy +

απ

2

)}y

+(

ih

k

)Cα

{F cos

(hy +

απ

2

)+ sin

(hy +

απ

2

)}z]

exp[i(βz +

απ

2

)](3.13a)

ηHfd = B

(−k

h

) [−iCα

{F sin

(hy +

απ

2

)− cos

(hy +

απ

2

)}x

+(

β

k

)Sα

{F cos

(hy +

απ

2

)+ sin

(hy +

απ

2

)}y

−(

ih

k

)Sα

{F sin

(hy +

απ

2

)− cos

(hy +

απ

2

)}z]

exp[i(βz +

απ

2

)](3.13b)

Since the fields inside the fractional parallel plate impedance waveguide are pro-

portional to the fractional dual fields (TMz case )in the region y > 0 in the presence

of a planar impedance boundary at y = 0, so the transverse impedance is same as

given in equation (2.27).

The electric and magnetic field line plots for the fractional parallel plate impedance

waveguide in the yz-plane have been shown in Figure 3.6. The plots are for the nor-

malized impedance zw = 0 and zw = 2, other parameters of the simulation are same as

for Figure 3.5. It can be seen from the figure that for zw = 0, field patterns match with

the patterns of fractional parallel plate PEC waveguide. As the normalized impedance

of the wall changes from the zero value, the field lines have both the components par-

allel as well as perpendicular to walls of the guide even for α = 0. Further the shift in

the field patterns with α is similar as in the case of PEC guide.

– 51 –

Figure 3.6 Field lines in the yz-plane at different values of α for zw = 0 and zw = 2;

solid lines are for the electric field while dashed lines are for the magnetic field

3.4 Fractional parallel plate PEMC waveguide

Let us consider a parallel plate waveguide whose walls are of perfect electromag-

netic conductor (PEMC). Geometry of the Figure 3.1 may be considered subject to the

condition that its walls are PEMC having admittance M . Parallel plate PEMC waveg-

uide with fractional dual solutions may be termed as fractional parallel plate PEMC

waveguide. Solutions for the PEMC waveguide may be obtained from the general

solutions given in equation (3.3) and (3.4) by using the PEMC boundary conditions.

– 52 –

The PEMC boundary conditions are given below

n×[ηH + tan θE

]= 0, n.

[D− tan θB

]= 0 (3.14)

where

tan θ = Mη

It may be noted that θ = π/2 corresponds to the PEC boundary and θ = 0 corresponds

to the PMC boundary. As discussed in chapter 2, solutions of a PEMC boundary can

be written by applying the transformation given in equations (2.28) and (2.29) to

the solutions of a PEC boundary. Similarly fields inside a fractional parallel plate

PEMC waveguide can be written from the fields inside a fractional parallel plate PEC

waveguide using the same transformation. Therefore relation between the fields inside

a fractional parallel plate PEMC waveguide and the fractional dual solutions in the

region y > 0 for a planar PEMC boundary is same as for the case of parallel plate

PEC waveguide and the planar PEC boundary.

Let us consider a TMz mode solution for a parallel plate PEC waveguide as

given in equation (3.9). The fields inside the fractional PEMC waveguide must be

proportional to the fractional dual solutions of the TMz polarized wave in the presence

of a PEMC boundary through the same constant of proportionality, i.e.,

C =(− ik

hAn

)

Hence the fractional dual solutions (transformed from the TMz mode solution) inside

the parallel plate PEMC waveguide can be written as

Efd =(− ik

hAn

)

[−x

{(Sθ − Sα) cos

(hy +

απ

2

)+ (Sθ + Sα) i sin

(hy +

απ

2

)}

– 53 –

+y(

β

k

) {(Cθ − Cα) cos

(hy +

απ

2

)+ (Cθ + Cα) i sin

(hy +

απ

2

)}

−z(

h

k

) {(Cθ + Cα) cos

(hy +

απ

2

)+ (Cθ − Cα) i sin

(hy +

απ

2

)}]

exp[i(βz +

απ

2

)](3.15a)

ηHfd =(− ik

hAn

)

[−x

{(Cθ − Cα) cos

(hy +

απ

2

)+ (Cθ + Cα) i sin

(hy +

απ

2

)}

−y(

β

k

) {(Sθ − Sα) cos

(hy +

απ

2

)+ (Sθ + Sα) i sin

(hy +

απ

2

)}

+z(

h

k

) {(Sθ + Sα) cos

(hy +

απ

2

)+ (Sθ − Sα) i sin

(hy +

απ

2

)}]

exp[i(βz +

απ

2

)](3.15b)

where

Cθ = cos(2θ − απ

2

)

Sθ = sin(2θ − απ

2

)

Since the fields inside a fractional parallel plate PEMC waveguide are proportional

to the fractional dual solutions for TMz mode in the region y > 0 in the presence

of a PEMC boundary at y = 0, so the transverse impedance is same as given in

equation (2.59).

Plots of electric and magnetic field lines for the fractional parallel plate PEMC

waveguide in the yz-plane have been shown in Figure 3.7. The plots are for the fields

inside the PEMC waveguide having admittance of the walls as θ = π/2 and θ = π/3,

other parameters of the simulation are same as for the Figure 3.3. It can be seen from

the figure that for θ = π/2, the field patterns match with the fractional parallel plate

PEC waveguide. It may be noted that in fractional parallel plate PEMC waveguide,

the electric and magnetic fields have their both components parallel and perpendicular

to the guide plates in the yz plane for all the values of α.

– 54 –

Figure 3.7 Field lines in the yz-plane at different values of α for θ = π/2 and

θ = π/3; solid lines are for the electric field while dashed lines are for the magnetic

field

– 55 –

CHAPTER IV

Fractional Chiro Waveguide and

the Concept of Fractional Transmission Lines

In chapter II, fractional duality has been studied for different planar boundaries

while chapter III contains the same discussion for different parallel plate waveguides. In

present chapter, fractional dual solutions for a planar chiral-achiral interface have been

derived when it is excited by a uniform plane wave. Secondly fractional dual solutions

for a parallel plate PEC waveguide filled with a chiral medium are determined. The

concept of fractional transmission lines is also addressed.

4.1 Reflection from a chiral-achiral interface

Consider a chiral-achiral interface located at z = 0 as shown in Figure 4.1. The

region z < 0 is a lossless, isotropic, and reciprocal chiral medium while region z > 0

consists of a simple dielectric medium. The interface is excited by a uniform plane

Figure 4.1 Reflection from a chiral-achiral interface

– 56 –

wave. According to field decomposition approach [48], electric field E and the magnetic

field H may be pictured as consisting of two parts, i.e., ( E+,H+) and (E−,H−). The

two parts are termed as wavefields. Simple expressions for the wavefields can be written

as

E+ =12

(E + iηH)

E− =12

(E− iηH)

H+ =12

(H− i

ηE

)

H− =12

(H +

i

ηE

)

The electric and magnetic fields corresponding to two wavefields for the incident

wave can be written as

Einc± (z) = Einc

± (0) exp(ik±z) (4.1a)

η±Hinc± (z) = z×Einc

± (z) = ±iEinc± (z) (4.1b)

where k± = k(1± κr) are wave numbers for the two wavefields. k = ω√

µε is the net

wave number and κr is the relative chirality parameter. The wavefields vectors may

be defined for the propagation along positive z-direction as

Einc± (0) = Einc

± (0)(

x∓ iy2

)

This means that each wavefield sees chiral medium as achiral medium with equivalent

constitutive parameters (ε+, µ+) and (ε−, µ−). Medium parameters of the equiva-

lent isotropic media are related to the parameters of chiral medium by the following

relations

ε± = ε(1± κr)

µ± = µ(1± κr)

η± =√

µ±ε±

= η

– 57 –

Upon reflection from the chiral-achiral interface, left circularly polarized (LCP) wave

with wave number k− becomes right circularly polarized (RCP) wave with wave num-

ber k+ and vice versa. Hence the reflected fields corresponding to the incident wave-

fields can be written as

Eref± (z) = R±Einc

± (0) exp(−ik∓z) (4.2a)

ηHref± (z) = ±iR±Einc

± (0) exp(−ik∓z) (4.2b)

where R± are the reflection co-efficients for positive and negative incident wave-

fields. For a reciprocal chiral medium (intrinsic impedance η) interface with any

other isotropic medium(intrinsic impedance η1), the reflection co-efficient becomes

R± = R = η1−ηη1+η .

Since the wavefields (Einc± , η±Hinc

± ) represent two independent plane waves prop-

agating in positive z-direction, the fractional dual solutions can be written as

Einc±fd = (z×)αEinc

± = (±i)αEinc± (4.3a)

ηHinc±fd = z×Einc

±fd = ±i(±i)αEinc± (4.3b)

Similarly fractional dual solutions for the reflected wave can be written in terms of the

two wavefields as

Eref±fd = (−z×)αEref

± = (∓i)αEref± (4.3c)

ηHref±fd = −z×Eref

±fd = ∓i(∓i)αEref± (4.3d)

Fractional dual solutions in the region z < 0 may be written by adding the fractional

dual fields for incident and the reflected waves as

Efd(z) =[Einc

+ (0) exp(ikκrz) + Einc− (0) exp (−iαπ) exp(−ikκrz)

][exp

{i(kz +

απ

2

)}+ R exp

{−i

(kz +

απ

2

)}](4.4a)

ηHfd(z) = i[Einc

+ (0) exp(ikκrz) + Einc− (0) exp (−iαπ) exp(−ikκrz)

][exp

{i(kz +

απ

2

)}−R exp

{−i

(kz +

απ

2

)}](4.4b)

– 58 –

For the special case, when the achiral medium is a PEC then R = −1. Hence the

fractional fields given in equation (4.4) take the following form as

Efd(z) = 2i[Einc

+ (0) exp(ikκrz) + Einc− (0) exp (−iαπ) exp(−ikκrz)

]

sin(kz +

απ

2

)(4.5a)

ηHfd(z) = 2i[Einc

+ (0) exp(ikκrz) + Einc− (0) exp (−iαπ) exp(−ikκrz)

]

cos(kz +

απ

2

)(4.5b)

Wave impedance can be defined by taking ratio of the field components as

Zfdxy =Exfd

Hyfd= iη tan

(kz +

απ

2

)

Zfdyx = −Eyfd

Hxfd= iη tan

(kz +

απ

2

)

Normalized impedance of the chiral-PEC interface can be obtained by putting z = 0

as

zfd

= [zfdxzxz + zfdzxzx] , 0 ≤ α ≤ 1 (4.6)

where

zfdxz = zfdzx = i tan(απ

2

)(4.6a)

which is same as for the achiral-PEC interface discussed in chapter II.

4.2. Fractional parallel plate chiro waveguide

A parallel plate waveguide having PEC walls and filled with a lossless, isotropic,

and reciprocal chiral medium is termed as parallel plate chiro waveguide [46]. Hence

the chiro waveguide with fractional dual solutions may be termed as fractional parallel

plate chiro waveguide. Consider a parallel plate waveguide consisting of two perfect

electric conducting plates separated by a distance b. Geometry of the Figure 3.1 may

– 59 –

be considered with the difference that the medium inside the guide is considered as

chiral instead of ordinary dielectric. Other parameters are same as for chapter III, The

medium inside the waveguide may be described by the constitutive relations as given

below

D = εE + iξcB (4.7a)

H = iξcE + B/µ (4.7b)

where ε, µ and ξc are called permittivity, permeability, and cross susceptibility of the

medium respectively.

It is well known that in conventional parallel plate waveguide filled with homoge-

nous achiral material, the field configuration can be obtained by superposing two plane

waves in a suitable manners. These waves propagate with the same bulk wave num-

ber. In achiral waveguides, there are only two component waves and the direction

of these waves depend upon the frequency ω and the dimension b. In parallel plate

chirowaveguide, however, due to the fact that chiral medium supports double mode

propagation (k+ and k−), there are four component waves. Two of the waves are right

circularly polarized (RCP) propagating with a wave number k+ and the other two

are left circularly polarized (LCP) propagating with a wave number k− as shown in

Figure 4.2. Component waves means the modes of propagation.

Consider R1, R2, L1, and L2 to be the amplitudes of the component waves

ER1, ER2, EL1, and EL2 respectively. These waves can be expressed as follows [46]

ER1 = R1 exp[ik+(z cos θ + y sin θ)]eR1 (4.8a)

ER2 = R2 exp[ik+(z cos θ − y sin θ)]eR2 (4.8b)

EL1 = L1 exp[ik−(z cosϕ + y sin ϕ)]eL1 (4.8c)

EL2 = L2 exp[ik−(z cosϕ− y sin ϕ)]eL2 (4.8d)

– 60 –

where eR1, eR2, eL1, and eL2 are the circular basis unit vectors for the right and left

circularly waves in the direction of θ and ϕ as shown in Figure 4.2. Since the guided

wave propagates in the z direction with the propagation constant β, the z component

of the bulk mode wave numbers must be equal to β, i.e.,

k+ cos θ = k− cos ϕ = β (4.9a)

Figure 4.2 Schematics of propagation through a parallel plate chiro-waveguide

Now let

k+ sin θ = h+, k− sin ϕ = h− (4.9b)

Using equation (4.9), field expressions in (4.8) can be written as

ER1 = R1 exp[i(βz + h+y)]eR1 (4.10a)

ER2 = R2 exp[i(βz − h+y)]eR2 (4.10b)

EL1 = L1 exp[i(βz + h−y)]eL1 (4.10c)

EL2 = L2 exp[i(βz − h−y)]eL2 (4.10d)

– 61 –

where the direction vectors are as given below

eR1 =1√2

[x + i

β

k+y − i

h+

k+z]

(4.11a)

eR2 =1√2

[x + i

β

k+y + i

h+

k+z]

(4.11b)

eL1 =1√2

[x− i

β

k−y + i

h−k−

z]

(4.11c)

eL2 =1√2

[x− i

β

k−y − i

h−k−

z]

(4.11d)

Magnetic fields corresponding to the electric fields given in equation (4.10) can be

written using the duality theorem as

ηHR(1,2) = kR(1,2) ×ER(1,2) = (i)ER(1,2) (4.12a)

ηHL(1,2) = kL(1,2) ×EL(1,2) = (−i)EL(1,2) (4.12b)

where

kR1 =(

h+

k+y +

β

k+z)

(4.13a)

kR2 =(−h+

k+y +

β

k+z)

(4.13b)

kL1 =(

h−k−

y +β

k−z)

(4.13c)

kL2 =(−h−

k−y +

β

k−z)

(4.13d)

It may be noted that

|kR1| = |kR2| = |kL1| = |kL2| = 1

Hence using equation (4.10), equations for total electric and magnetic fields can be

written as

E = ER1 + EL1 + ER2 + EL2 (4.14a)

ηH = ηHR1 + ηHL1 + ηHR2 + ηHL2

= i(ER1 −EL1 + ER2 −EL2) (4.14b)

– 62 –

As it has been seen that the field inside the parallel plate chiro waveguide can be rep-

resented in terms of four independent plane waves. Hence the fractional dual solutions

(Eifd, ηHifd) inside the waveguide may be obtained by using the following relations

Eifd =1

(ik)α[(∇×)αEi] = (ki×)αEi (15a)

ηHifd =1

(ik)α[(∇×)αηHi] = (ki×)αηHi, i = R1, R2, L1, L2 (15b)

which give

ER(1,2)fd = (i)αER(1,2) (4.16a)

EL(1,2)fd = (−i)αEL(1,2) (4.16b)

It may be noted that both the right circular components and the left circular compo-

nents have been rotated by an angle απ2 in the counterclockwise direction. Fractional

dual solutions (Efd, ηHfd) corresponding to the total fields (E, ηH) may be obtained

by the linear combination of (Eifd, ηHifd), that is

Efd = ER1fd + EL1fd + ER2fd + EL2fd

= (i)αER1 + (−i)αEL1 + (i)αER2 + (−i)αEL2 (4.17a)

ηHfd = ηHR1fd + ηHL1fd + ηHR2fd + ηHL2fd

= i[(i)αER1 − (−i)αEL1 + (i)αER2 − (−i)αEL2

](4.17b)

It can be seen from equation (4.14) and (4.17) that

α = 0 ⇒ Efd = E, ηHfd = ηH

α = 1 ⇒ Efd = ηH, ηHfd = −E

which shows that the fields given in equation (4.17) satisfy the duality principle for

the limiting values of α, that is for α = 0, (Efd, ηHfd) yields an original field solution

and for α = 1, (Efd, ηHfd) yields dual to the original field solution. Hence it may be

– 63 –

deduce that for intermediate values 0 < α < 1, fields given in equation (4.17) are the

fractional dual solutions.

Using equation (4.10), the fractional dual solutions given in (4.17) can be written

as

Efd = exp(iβz)[R1 exp

{i(h+y − απ

2

)}eR1 + L1 exp

{i(h−y +

απ

2

)}eL1

+R2 exp{−i

(h+y +

απ

2

)}eR2 + L2 exp

{−i

(h−y − απ

2

)}eL2

](4.18a)

ηHfd = (i) exp(iβz)[R1 exp

{i(h+y − απ

2

)}eR1 − L1 exp

{i(h−y +

απ

2

)}eL1

+ R2 exp{−i

(h+y +

απ

2

)}eR2 − L2 exp

{−i

(h−y − απ

2

)}eL2

](4.18b)

In order to meet the boundary conditions, the relation between the amplitudes of the

component waves should be as

R2 = −L1, L2 = −R1

Using these relations in equation (4.18) and re-arranging in a suitable manners gives

Efd = exp (iβz) L1

[exp

{i(h−y +

απ

2

)}eL1 − exp

{−i

(h+y +

απ

2

)}eR2

]

+ exp (iβz)R1 exp (−iαπ)[exp

{i(h+y +

απ

2

)}eR1 − exp

{−i

(h−y +

απ

2

)}eL2

](4.19a)

ηHfd = (−i) exp (iβz)L1

[exp

{i(h−y +

απ

2

)}eL1 + exp

{−i

(h+y +

απ

2

)}eR2

]

+ (i) exp (iβz) R1 exp (−iαπ)[exp

{i(h+y +

απ

2

)}eR1 + exp

{−i

(h−y +

απ

2

)}eL2

](4.19b)

– 64 –

Now since

k± = k(1± κr), h± = h(1± κr),

soh+

k+=

h−k−

=h

k

Using this fact in equation (4.19), the fractional dual solutions can be written in terms

of x, y, z co-ordinates as

Efd = −i√

2 exp(iβz) sin(hy +

απ

2

)

[L1 exp(−ihκry) + exp(−iαπ)R1 exp(ihκry)

]x

+1√2

exp(iβz)[{

L1 exp(−ihκry)(−iβ

k−

)+ exp(−iαπ)R1 exp(ihκry)

(iβ

k+

)}

exp{

i(hy +

απ

2

)}

+{

L1 exp(−ihκry)(−iβ

k+

)+ exp(−iαπ)R1 exp(ihκry)

(iβ

k−

)}

exp{−i

(hy +

απ

2

)}]y

− i√

2 exp(iβz)(

ih

k

)sin

(hy +

απ

2

)

[L1 exp(−ihκry)− exp(−iαπ)R1 exp(ihκry)

]z (4.20a)

ηHfd = (i)√

2 exp(iβz) cos(hy +

απ

2

)

[L1 exp(−ihκry)− exp(−iαπ)R1 exp(ihκry)

]x

− i1√2

exp(iβz)[{

L1 exp(−ihκry)(

iβ

k−

)+ exp(−iαπ)R1 exp(ihκry)

(iβ

k+

)}

exp{

i(hy +

απ

2

)}

+{

L1 exp(−ihκry)(−iβ

k+

)+ exp(−iαπ)R1 exp(ihκry)

(−iβ

k−

)}

exp{

i(hy +

απ

2

)}]y

– 65 –

− (i)√

2 exp(iβz)(−ih

k

)cos

(hy +

απ

2

)

[L1 exp(−ihκry) + exp(−iαπ)R1 exp(ihκry)

]z (4.20b)

The transverse impedance can be found by using the transverse components of the

fraction dual fields as

Zfdxz =Exfd

Hzfd= iη tan

(hy +

απ

2

) k

h

Zfdzx = −Ezfd

Hxfd= iη tan

(hy +

απ

2

) h

k

Normalized impedance of the walls of fractional chiro-waveguide can be represented

by putting y = 0 as

zfd

=[k

hzfdxzxz +

h

kzfdzxzx

], 0 ≤ α ≤ 1 (4.21)

where

zfdxz = zfdzx = i tan(απ

2

)(4.21a)

which is similar to the case of parallel plate PEC waveguide filled with an ordinary

dielectric medium.

4.3. The concept of fractional transmission lines

When a plane electromagnetic wave of any polarization propagates through a

plane parallel structure (propagating medium and a plane reflecting boundary), the

propagation defined by Maxwell equations in the medium can be analyzed using the

theory of transmission lines. If the medium of propagation is an isotropic simple

medium, the propagation can be analyzed as a single scalar transmission line also called

a symmetric transmission line. A plane wave propagating through a plane-parallel

structure of bi-isotropic medium can be analyzed in terms of two non-interacting

– 66 –

scalar transmission lines with two eigen waves much in the same way as simple isotropic

medium. Since in this case propagation constant of the incident and reflected wave are

different to each other, the line is called non-symmetric transmission line. In this part

of the chapter, fractional dual solutions for the symmetric and non-symmetric trans-

mission lines have been derived. The transmission lines described by their fractional

dual solutions may be regarded as the fractional transmission lines. The fractional

transmission lines may be considered as generalization of the short circuit lines and

the open circuit transmission lines. Condition for impedance matching of the fractional

transmission lines network is also given in terms of the fractional parameter.

4.3.1. Fractional symmetric transmission line

Consider a uniform transmission line having characteristic impedance Zc and ter-

minated at a load ZL as shown in Figure 4.3.

Figure 4.3 Terminated transmission line

z-axis has been considered as direction of propagation of voltage wave and current wave

along the transmission line. The differential equations describing the propagation of

– 67 –

voltage and current wave may be written as

1iβ

d

dzV = ZcI (4.22a)

1iβ

d

dzZcI = V (4.22b)

where L and C are the inductance and capacitance per unit length along the line. The

operating frequency is ω, β = ω√

LC is the propagation constant, and Zc =√

L/C is

the intrinsic impedance of the medium.

The coupled differential equations given in equations (4.22) can be decoupled to

give rise to second order differential equations as

1(iβ)2

d2

dz2V (z) = V (z) (4.23a)

1(iβ)2

d2

dz2I(z) = I(z) (4.23b)

Solutions of these differential equations may be written as

V (z) = V + [exp(iβz) + Γ exp(−iβz)] (4.24a)

ZcI(z) = V + [exp(iβz)− Γ exp(−iβz)] (4.24b)

where

Γ =ZL − Z

ZL + Z

and V + is the amplitude of the voltage wave travelling in the positive z-direction.

Input impedance of this line at any point along the line can be defined as

Zin(z) =V (z)I(z)

= Zc

[ZL + iZc tan(βz)Zc + iZL tan(βz)

](4.25)

Let us derive fractional dual solutions of the voltage and current wave equations.

The transmission line represented by these fractional dual solutions may be regarded

– 68 –

as fractional transmission line. As can be seen from equations (4.22) (V, ZcI) and

(ZcI, V ), are two solutions of the transmission line equations. It may be noted that

one solution is dual to the other one. let us Operate 1(iβ)2α

d2α

dz2α on both sides of

equations (4.22) as[

1(iβ)2α

d2α

dz2α

]1iβ

d

dzV =

[1

(iβ)2α

d2α

dz2α

]ZcI (4.26a)

[1

(iβ)2α

d2α

dz2α

]1iβ

d

dzZcI =

[1

(iβ)2α

d2α

dz2α

]V (4.26b)

Since the differential operator[

1(iβ)2α

d2α

dz2α

]and d

dz has the property of commutation so

equations (4.26) can be written as

1iβ

d

dz

[1

(iβ)2α

d2α

dz2α

]V =

[1

(iβ)2α

d2α

dz2α

]ZcI (4.27a)

1iβ

d

dz

[1

(iβ)2α

d2α

dz2α

]ZcI =

[1

(iβ)2α

d2α

dz2α

]V (4.27b)

Hence the new set of solutions that satisfies the differential equations governing the

transmission line can be written as

Vfd =1

(iβ)2α

d2α

dz2αV (4.28a)

ZcIfd =1

(iβ)2α

d2α

dz2αZcI (4.28b)

It can be seen from the above equation that for α = 0, original solution (V,ZcI) is

obtained and for α = 1/2, dual to the original solution (ZcI, V ) of the transmission

line is obtained. This mean that for 0 < α < 1/2, solution set (Vfd, ZcIfd) may be

regarded as intermediate step between the original solution and dual to the original

solution. Solution set (Vfd, ZcIfd) may also be termed as fractional dual solutions of

the transmission line.

Using equation (4.24) in equation (4.28), fractional dual solutions of the trans-

mission line can be written as

Vfd =V +

i2α[exp(iβz + iαπ) + Γ exp(−iβz − iαπ)] (4.29a)

ZcIfd =V +

i2α[exp(iβz + iαπ)− Γ exp(−iβz − iαπ)] (4.29b)

– 69 –

Values of the fractional dual voltage and current wave have been plotted versus frac-

tional parameter α for resistive load of zL = 3 as shown in Figure 4.4.

Figure 4.4 Absolute values of voltage and current at load versus α; zL = 3, βz = π/4

From the figure, it may be deduced that at the limiting values of α, duality principle

is satisfied, i.e.,

α = 0 ⇒ Vfd = V, ZcIfd = ZcI

α = 1/2 ⇒ Vfd = ZcI, ZcIfd = V

The input impedance at any point along the fractional transmission line is given below

Z infd(z) =

Vfd(z)Ifd(z)

= ZcZL + iZc tan(βz + απ)Zc + iZL tan(βz + απ)

(4.30)

For α = 0, above equation becomes

Z infd = Zc

ZL + iZc tan(βz)Zc + iZL tan(βz)

– 70 –

which is the relation for input impedance at any point for original transmission line.

Now for α = 12

Z infd = Zc

ZL + iZc cot(βz)Zc + iZL cot(βz)

which is the relation for input admittance at any point for dual to the original trans-

mission line. So, α = 0 represents the original transmission line and α = 12 represents

the dual to the original transmission line, while 0 < α < 12 represents the fractional

dual transmission line.

Normalized load impedance of the fractional dual transmission line can be defined

by putting z=0 in equation (4.16) as

zLfd =zL + i tan(απ)1 + izL tan(απ)

, 0 ≤ α ≤ 1/2 (4.31)

Plot of the fractional dual load impedance versus α has been given for different values

of the original load impedance as shown in Figure 4.5 which shows that dual of a

transmission line with impedance load is a line terminated at an admittance load and

vice versa. As a particular case, for a short circuited transmission line (zL = 0), dual

line is an open circuited line (zL = ∞) while fractional dual transmission line is the

line which has been terminated by a load which is intermediate between these two

limiting cases as shown in the Figure 4.5.

4.3.2. Fractional non-symmetric transmission line

As it is seen in section 4.1 that solution of the Maxwell equations in a chiral

medium gives rise to two circularly polarized waves. One of the waves is right circularly

polarized (RCP) while the other is left circularly polarized (LCP). These RCP and LCP

components move with different phase velocities and may be represented in terms of

wave numbers k+ and k− respectively.

– 71 –

Figure 4.5 Load impedance versus α for a fractional dual transmission line

For the two circularly polarized TEM eigen waves depending only upon z-coordi-

nate, the source free Maxwell equations can be written as:

z×E′±(z) = iωµ±H±(z) (4.32a)

z×H′±(z) = −iωε±E±(z) (4.32b)

where the primes denote differentiation with respect to z. In terms of circular polar-

ization (CP) unit vectors u± satisfying z × u± = ±iu±, the two wavefields can be

written as

E′± = u±E±

H′± = u±H±

where u+ and u− are the unit vectors showing directions of the right circular and left

circular polarization respectively. Using these values, scalar form of equation (4.32)

– 72 –

can be written as

E′±(z) = iωµ±{∓iH±(z)} (4.33a)

∓iH ′±(z) = iωε±E±(z) (4.33b)

They resemble the transmission line equations (4.22), so that if one identify the electric

field with voltage V± = E±, the current must be recognized as I± = ∓iH± [43]. The

positive and negative wave-fields are represented by their respective voltage and current

components.

The mathematical model of the transmission line equivalent of equations (4.33)

may be written as

V±(z) = V±(0) exp(iβ±z) + ΓV±(0) exp(−iβ∓z) (4.34a)

ZcI±(z) = V±(0) exp(iβ±z)− ΓV±(0) exp(−iβ∓z) (4.24b)

where

Γ =ZL − Z

ZL + Z

is the reflection coefficient and β± is the wave number corresponding to the two wave-

fields. Now voltage and current for the total field can be written as

V (z) =V+(0){exp(iβ+z) + Γ exp(−iβ−z)}+

V−(0){exp(iβ−z) + Γ exp(−iβ+z)}

= V1 + V2 + V3 + V4 (4.35a)

ZI(z) =V+(0){exp(iβ+z)− Γ exp(−iβ−z)}+

V−(0){exp(iβ−z)− Γ exp(−iβ+z)}

= ZI1 + ZI2 + ZI3 + ZI4 (4.35b)

– 73 –

The non-symmetric transmission lines represented by the fractional dual solutions can

be regarded as the fractional non-symmetric transmission line. The fractional dual

solutions of voltage and current equations (4.35) can be written using (4.28) as

Vfd(z) = exp(−iαπ)V+(0)[exp{i(β+z + απ)}+ Γ exp{−i(β−z + απ)}]

+ exp(−iαπ)V−(0)[exp{i(β+z + απ)}+ Γexp{−i(β+z + απ)}] (4.36a)

ZIfd(z) = exp(−iαπ)V+(0)[exp{i(β+z + απ)} − Γ exp{−i(β−z + απ)}]

+ exp(−iαπ)V−(0)[exp{i(β−z + απ)} − Γ exp{−i(β+z + απ)}] (4.36b)

For α = 0 above equation gives original solution (V, ZcI), for α = 1/2 it gives dual to

the original solution (ZcI, V ) and the solutions for 0 < α < 1/2 may be regarded as

fractional dual solutions.

Similar to the case of symmetric transmission line, input impedance of the frac-

tional non-symmetric transmission line can be defined as

Zfd =Vfd(z)Ifd(z)

Using the following relations

β+ = β + βκr β− = β − βκr

equation (4.36) can be written as

Vfd(z) =1

(ZL + Zc)exp(−iαπ){V+(0) exp(iβκrz) + V−(0) exp(−iβκrz)}

2 cos(βz + απ){ZL + iZc tan(βz + απ)} (4.37a)

ZIfd(z) =1

(ZL + Z)exp(−iαπ){V+(0) exp(−iβκrz) + V−(0) exp(−iβκrz)}

2 cos(βz + απ){iZL tan(βz + απ) + Z} (4.37b)

Hence input impedance of the fractional dual line can be written as

Z infd = Z

ZL + iZ tan(βz + απ)Z + iZL tan(βz + απ)

, 0 ≤ α ≤ 1/2 (4.38a)

– 74 –

Normalized load impedance zLfd of the fractional line can be obtained in terms of nor-

malized load impedance zL = ZL

Zcof the original line by putting z=0 in equation (4.38a)

as

zLfd =zL + i tan(απ)1 + izL tan(απ)

, 0 ≤ α ≤ 1/2 (4.38b)

which is same as the case of uniform symmetric transmission line, i.e., dual of a trans-

mission line with impedance load is a line terminated at an admittance load and vice

versa.

4.3.3. Multiple-sections fractional non-symmetric line

Let us consider a transmission line that is connected to another transmission line

of length L which is terminated by a load ZL. Let Z1 and Z2 are intrinsic impedances

of line-1 and line-2 respectively while Z2in is the input impedance of line-2 at the

junction of two lines. Input impedance of the whole transmission line network can be

written as

Zin = Z1Z2in + iZ1 tan(β1z)Z1 + iZ2in tan(β1z)

(4.39a)

where

Z2in = Z2ZL + iZ2 tan(β2L)Z2 + iZL tan(β2L)

(4.39b)

Using the same treatment as done in in the last section, input impedance of the

fractional transmission line network can be written as

Zinfd = Z1Z2infd + iZ1 tan(β1z + απ)Z1 + iZ2infd tan(β1z + απ)

(4.40a)

where

Z2infd = Z2ZL + iZ2 tan(β2L + απ)Z2 + iZL tan(β2L + απ)

(4.40b)

Condition for impedance matching of the transmission line network described by (4.40)

is

Z2infd = Z2ZL + iZ2 tan(β2L + απ)Z2 + iZL tan(β2L + απ)

= Z1 (4.41a)

– 75 –

The value of fractional parameter α, in terms of the impedances, required for the

impedance matching is given below

α =1π

[β2L− arctan

(Z − 2(Z1 − ZL)i(Z2

2 − Z1ZL)

)](4.41b)

Hence for α = β2Lπ , equation (4.40) represents the input impedance of the circuit, as if

there is no transmission line having characteristic impedance Z2. Therefore α = β2Lπ is

the condition of impedance matching for the network of fractional transmission lines.

– 76 –

CHAPTER V

Fractional Rectangular Impedance Waveguide

This chapter deals with the fractional impedance rectangular waveguide. The rect-

angular waveguide has impedance walls and filled with an ordinary dielectric medium.

Variations of field distribution inside the guide and impedance of the walls of the frac-

tional waveguide with respect to fractional parameter have been studied. Variations in

power density distribution in the transverse plane with respect to fractional parameter

is also the matter of interest.

5.1. General theory of rectangular waveguide

Consider a waveguide having a rectangular cross section of size a× b in xy-plane

as shown in Figure 5.1. The medium inside the guide is a lossless, homogeneous, and

isotropic having permittivity ε and permeability µ. The guide is considered infinitely

long along z-axis.

Figure 5.1. Geometry of the rectangular wave guide

For the sack of simplicity, Walls of the guide are considered to have anisotropic

impedance under special conditions so that the modal solution exist. The general

– 77 –

impedance matrix for our geometry is

[Z1 Z2

Z3 Z4

]=

[ ∓[Ex/Hz]y=0,b ± [Ez/Hx]y=0,b

∓[Ez/Hy]x=0,a ± [Ey/Hz]x=0,a

](5.1)

The compatibility relation for the modal solution is as below [52]

Z1Z3 − Z2Z3 + Z2Z4 = 0 (5.1a)

Alternately, equation (5.1a) can be written as

Z3

Z4− Z2Z3

Z1Z4+

Z2

Z1= 0 (5.1b)

Let us consider a special case where the impedance matrix is of the form

[Z1 Z2

Z3 Z4

]=

[∞ Zw

Zw ∞]

(5.2)

so that TMz(Hz = 0) mode propagation through the guide is possible. As discussed in

the previous chapters, it is required to solve Helmholtz equations for axial components

only. The transverse components of the electric and magnetic fields can be found using

Maxwell equations. Scalar Helmholtz equation for the axial component of the electric

field may be written as

∂2Ez(x, y)∂x2

+∂2Ez(x, y)

∂y2+ k2

cEz(x, y) = 0

where k2c = k2 − β2, k = ω

√µε is the wave number, ω is angular frequency, and β is

the propagation constant. Using method of separation of variables, axial component

of the electric field may be written as

Ez(x, y) = X(x)Y (y) (5.3)

where

X(x) = Am sin(kxx) + Bm cos(kxx) (5.3a)

Y (y) = An sin(kyy) + Bn cos(kyy) (5.3b)

– 78 –

where Am, Bm, An, and Bn are constants and can be found from the boundary

conditions. Once axial component of electric field is found, other field components can

be written using Maxwell curl equations as

Ex(x, y) =1k2

c

(iβ

∂Ez(x, y)∂x

)=

iβ

k2c

Y (y)X′(x) (5.4a)

Ey(x, y) =1k2

c

(iβ

∂Ez(x, y)∂y

)=

iβ

k2c

X(x)Y′(y) (5.4b)

Hx(x, y) = − 1k2

c

(ik

η

∂Ez(x, y)∂y

)= − 1

k2c

ik

ηX(x)Y

′(y) (5.4c)

Hy(x, y) =1k2

c

(ik

η

∂Ez(x, y)∂x

)=

1k2

c

ik

ηY (y)X

′(x) (5.4d)

Using boundary conditions on the general solutions given by equations (5.3) and (5.4),

particular solutions for the rectangular waveguide have been derived in the next sec-

tion.

5.2 Field formulation for the rectangular impedance waveguide

For a TMz mode propagating through the impedance rectangular waveguide,

particular solution for the axial component corresponding to the general solution given

in equation (5.3) can be written using the impedance boundary condition given in

equation (5.2) as

Ez(x, y) =Amn

2

[{(1− FxFy)Cx−y − (Fx − Fy)iSx−y

}

−{

(1 + FxFy)Cx+y − (Fx + Fy)iSx+y

}](5.5)

where Amn are constants that depend upon initial conditions and

Fx = zw

(kkx

k2c

), zw =

Zw

η

Fy = zw

(kky

k2c

)

Cx−y = cos(kxx− kyy)

Cx+y = cos(kxx + kyy)

Sx−y = sin(kxx− kyy)

Sx+y = sin(kxx + kyy)

– 79 –

while dispersion relations for the possible values of kx

kcand ky

kccan be written as

i tan(kxa) =2Fx

1 + F 2x

, i tan(kyb) =2Fy

1 + F 2y

Now equation (5.4) can be used to write the transverse electric and magnetic field

components as

Ex(x, y) =Amn

2

(βkx

k2c

) [{−(1− FxFy)iSx−y + (Fx − Fy)Cx−y

}

+{

(1 + FxFy)iSx+y − (Fx + Fy)Cx+y

}](5.6a)

Ey(x, y) =Amn

2

(βky

k2c

) [{(1− FxFy)iSx−y − (Fx − Fy)Cx−y

}

+{

(1 + FxFy)iSx+y − (Fx + Fy)Cx+y

}](5.6b)

ηHx(x, y) =Amn

2

(kky

k2c

) [{(1− FxFy)iSx−y − (Fx − Fy)Cx−y

}

+{

(1 + FxFy)iSx+y − (Fx + Fy)Cx+y

}](5.6c)

ηHy(x, y) =Amn

2

(kky

k2c

) [{(1− FxFy)iSx−y − (Fx − Fy)Cx−y

}

−{

(1 + FxFy)iSx+y − (Fx + Fy)Cx+y

}](5.6d)

Components of electric and magnetic field which are tangential to the walls of the

guide plate at x = 0 and x = a can be written as

Et = Ey(x, y)y + Ez(x, y)z (5.7a)

Ht = Hy(x, y)y (5.7b)

In order to validate the fields given in equation (5.6), fields of equation (5.7) have been

plotted for different values of the normalized impedance zw as shown in Figure 5.2.

Simulation data is for the mode propagating through the guide at an angle φz =

π/6(β/k = cos(π/6), kc/k = sin(π/6)

)with z-axis in the yz-pane and angle φx =

π/4(kx/kc = cos(π/4), ky/kc = sin(π/4)

)with the x-axis in the xy-plane. The plots

are along the line of observation (kxx, kyy, kzz) = (kxx, π/4, π/4) in the transverse

xy-plane for the square guide with a = b = λ/2. Figure 5.2a shows the electric field

plots while magnetic fields are shown in Figure 5.2b. Field values may be noted at

kxx = 0 and kxx = π, which represent the locations of the plates at x = 0

– 80 –

(a)

(b)

Figure 5.2. Plots of fields tangential to the plates at x = 0 and x = a taking

kyy = π/4, βz = π/4, β/k = cos(π/6), kc/k = sin(π/6), kx/kc = cos(π/4), ky/kc =

sin(π/4); (a) electric field (b) magnetic field

– 81 –

and x = a. It can be seen that, at α = 0, the tangential electric field is zero for

zw = 0 (i.e., PEC plates) while magnetic field is zero for zw >> 0, e.g., zw = 100

(i.e., PMC plates). Hence equation (5.6) represents the fields inside the guide having

impedance walls with normalized impedance zw, which may be converted to PEC and

PMC waveguide solution for the limiting values of zw.

5.3 Fractional rectangular impedance waveguide

Fractional rectangular waveguide can be modelled by determining the fractional

dual solutions inside the rectangular impedance waveguide. Fields given in equa-

tion (5.6) can be written in terms of four independent plane waves. Re-introducing

the z-dependance (eiβz), electric and magnetic fields of the four plane waves can be

written as

E1 =Amn

4k2c

B1

(−βkxx− βkyy − k2c z

)exp

{i(−kxx− kyy + βz)

}(5.8a)

E2 =Amn

4k2c

B2

(βkxx− βkyy + k2

c z)exp

{i(−kxx + kyy + βz)

}(5.8b)

E3 =Amn

4k2c

B3

(−βkxx + βkyy − k2c z

)exp

{i(kxx− kyy + βz)

}(5.8c)

E4 =Amn

4k2c

B4

(βkxx + βkyy − k2

c z)exp

{i(kxx + kyy + βz)

}(5.8d)

ηH1 =kAmn

4k2c

B1 (−kyx + kxy) exp{

i(−kxx− kyy + βz)}

(5.8e)

ηH2 =kAmn

4k2c

B2 (−kyx− kxy) exp{

i(−kxx + kyy + βz)}

(5.8f)

ηH3 =kAmn

4k2c

B3 (kyx + kxy) exp{

i(kxx− kyy + βz)}

(5.8g)

ηH4 =kAmn

4k2c

B4 (kyx− kxy) exp{

i(kxx + kyy + βz)}

(5.8h)

where

B1 = 1 + FxFy + Fx + Fy (5.9a)

B2 = 1− FxFy + Fx − Fy (5.9b)

– 82 –

B3 = 1− FxFy − Fx + Fy (5.9c)

B4 = 1 + FxFy − Fx − Fy (5.9d)

Using eigen values and eigen vectors of the cross product operators of the propagation

direction vectors, fractional dual solutions (Eifd, ηHifd) corresponding to the four plane

waves can be written as

E1fd =Amn

4k2c

B1 exp(iβz)[cos

(απ

2

){−βkxx− βkyy − k2

c )z}

+ sin(απ

2

){kkyx− kkxy

}]

exp[−i

(kxx +

απ

2

)]exp

[−i

(kyy +

απ

2

)](5.10a)

E2fd =Amn

4k2c

B2 exp(iβz)[cos

(απ

2

){βkxx− βkyy + k2

c )z}− sin

(απ

2

){kkyx + kkxy

}]

exp[−i

(kxx +

απ

2

)]exp

[i(kyy +

απ

2

)](5.10b)

E3fd =Amn

4k2c

B3 exp(iβz)[cos

(απ

2

){−βkxx + βkyy − k2

c )z}

+ sin(απ

2

){kkyx + kkxy

}]

exp[i(kxx +

απ

2

)]exp

[−i

(kyy +

απ

2

)](5.10c)

E4fd =Amn

4k2c

B4 exp(iβz)[cos

(απ

2

){βkxx + βkyy − k2

c )z}− sin

(απ

2

){kkyx− kkxy

}]

exp[i(kxx +

απ

2

)]exp

[i(kyy +

απ

2

)](5.10d)

ηH1fd =Amn

4k2c

B1 exp(iβz)[sin

(απ

2

) {−βkxx− βkyy − k2

c )z}− cos

(απ

2

){kkyx− kkxy

}]

exp[−i

(kxx +

απ

2

)]exp

[−i

(kyy +

απ

2

)](5.10e)

ηH2fd =Amn

4k2c

B2 exp(iβz)[− sin

(απ

2

){βkxx− βkyy + k2

c )z}− cos

(απ

2

){kkyx + kkxy

}]

exp[−i

(kxx +

απ

2

)]exp

[i(kyy +

απ

2

)](5.10f)

– 83 –

ηH3fd =Amn

4k2c

B3 exp(iβz)[sin

(απ

2

) {βkxx− βkyy + k2

c )z}

+ cos(απ

2

){kkyx + kkxy

}]

exp[i(kxx +

απ

2

)]exp

[−i

(kyy +

απ

2

)](5.10g)

ηH4fd =Amn

4k2c

B4 exp(iβz)[sin

(απ

2

) {βkxx + βkyy − k2

c )z}

+ cos(απ

2

){kkyx− kkxy

}]

exp[i(kxx +

απ

2

)]exp

[i(kyy +

απ

2

)](5.10h)

Fractional dual solutions of the total electric and magnetic field inside the guide can

be written as

Efd = E1fd + E2fd + E3fd + E4fd

ηHfd = ηH1fd + ηH2fd + ηH3fd + ηH4fd

which give

Efdx =Amn

k2c

exp(iβz){−βkxCα − kkySα

}

[(Cy+αFx − iSy+α)(Cx+α − FyiSx+α)

](5.11a)

Efdy =Amn

k2c

exp(iβz){−βkyCα + kkxSα

}

[(Cy+α − FxiSy+α)(Cx+αFy − iSx+α)

](5.11b)

Efdz = −Amn exp(iβz)Cα

[(Cy+αFx − iSy+α)(Cx+αFy − iSx+α)

](5.11c)

ηHfdx =Amn

k2c

exp(iβz){−βkxSα + kkyCα

}

[(Cy+α − FxiSy+α)(Cx+αFy − iSx+α)

](5.11d)

ηHfdy =Amn

k2c

exp(iβz){−βkySα − kkxCα

}

[(Cy+αFx − iSy+α)(Cx+α − FyiSx+α)

](5.11e)

ηHfdz = −AmnSα exp(iβz)[(Cy+α − FxiSy+α)(Cx+α − FyiSx+α)

](5.11f)

– 84 –

where

Cx+α = cos(kxx +

απ

2

)

Sx+α = sin(kxx +

απ

2

)

Cy+α = cos(kyy +

απ

2

)

Sy+α = sin(kyy +

απ

2

)

The fields given in equation (5.11) have been plotted in Figure 5.3. Validity of the

fractional fields defined by equations (5.11) may be noted from Figure 5.3 which shows

that fractional dual fields satisfy the principle of duality for the limiting values of α,

i.e., for α = 0, (Efd, ηHfd) represents the original solution and for α = 1, (Efd, ηHfd)

represents dual to the original solution. For the range 0 < α < 1, (Efd, ηHfd) are the

intermediate step between the original and dual to the original solutions and hence

may be called as the fractional dual solutions. Further from Figure 5.3, we see that

for α = 0, Ez 6= 0 and Hz = 0 which shows the transverse magnetic mode, while for

α = 1, Ez = 0 and Hz 6= 0 which shows the transverse electric mode.

In order to validate the dependance on impedance of the walls (i.e., zw = Zw/η),

tangential electric and magnetic fields at the wall at x = 0 of the fractional rect-

angular impedance waveguide have been plotted versus α for different values of the

original impedance of walls, i.e., (zw = 0, 1, 2, 100) as shown in Figure 5.4. Figure 5.4a

shows the plots for tangential electric fields at an observation point((kxx, kyy, βz) =

(0, π/4, π/4))

and the corresponding magnetic fields are shown in Figure 5.4b. It can

be seen from the figures that tangential electric field is zero only at((α, zw) = (0, 0)

or (α, zw) = (1, 100))

, i.e., PEC walls while tangential magnetic field is zero at((α, zw) = (1, 0) or (α, zw) = (0, 100)

), i.e., PMC walls case.

– 85 –

Figure 5.3 Fractional fields versus α at (kxx, kyy, βz) = (π/4, π/4, π/4), other pa-

rameters are same as in Figure 5.2

– 86 –

(a)

(b)

Figure 5.4. Plots of tangential fractional dual fields, (a) electric field (b) magnetic

field, taking x = 0 and other parameters are same as in Figure (5.2)

– 87 –

5.3.1 Behavior of fields inside the fractional rectangular impedance waveg-

uide

In order to study the behavior of field lines inside the fractional rectangular waveg-

uide, the field plots are given in the transverse xy-plane as shown in Figure 5.5. Solid

lines show the electric field while magnetic field is shown by the dashed lines.

Figure 5.5 Field lines ; solid lines show electric field and dashed lines show the

magnetic field. Simulation parameters are same as in Figure 5.2

From these figures, it can be seen that electric field lines are perpendicular and magnetic

field lines are parallel to the guide plates when the walls meet the conditions of PEC, i.e,((α, zw) = (0, 0) or at (α, zw) = (1, 10)

)while magnetic field lines are perpendicular

– 88 –

and electric field lines are parallel to the guide walls when the walls meet the conditions

of PMC, i.e.,((α, zw) = (0, 10) or at (α, zw) = (1, 0)

). This is also in accordance with

[36].

5.3.2 Surface impedance of walls

Surface impedance matrix given in equation (5.1) for the fractional impedance

rectangular waveguide can be found using ratios of the fractional dual fields of equa-

tion (5.11). That is[

z1fd z2fd

z3fd z4fd

]=

[ ∓[Efdx/Hfdz]y=0,b ± [Efdz/Hfdx]y=0,b

∓[Efdz/Hfdy]x=0,a ± [Efdy/Hfdz]x=0,a

](5.12)

For the limiting values of the fractional parameter, equation (5.12) may be written as;

α = 0 ⇒[

z1fd z2fd

z3fd z4fd

]=

[∞ zw

zw ∞]

(5.12a)

and α = 1 ⇒[

z1fd z2fd

z3fd z4fd

]=

[1/zw 00 1/zw

](5.12b)

It may be noted that the compatibility relation is satisfied for both the cases such that

α = 0 represents the transverse magnetic mode solution while α = 1 represents the

transverse electric mode solution. For intermediate values of the fractional parameter

α, the impedance matrix has all the four components with non zero finite values

and hence represents the coupled mode solutions. Further it may be noted that for

zw = 0, equation (5.12a) represents the TMz mode propagating through the PEC

waveguide while equation (5.12b) represents the TEz mode propagating through the

PMC waveguide which is also in accordance with the published literature.

5.3.3 Power transferred through a cross section

The time averaged power density at any point of the transverse plane (i.e., xy-

plane) of the fractional rectangular impedance waveguide can be obtained using the

Poynting vector theorem as

Pav(x, y, z) =12Re[EfdxH∗fdy − EfdyH∗fdx] (5.13)

– 89 –

where H∗fdy shows the complex conjugate of Hfdy and so on. Contour plots for the

power density given by equation (5.13) have been plotted for different values of the

fractional parameter as shown in Figure 5.6. Variation in the power distribution at

the transverse plane may be noted.

Figure 5.6 Time averaged Power distribution over the cross section for different

values of α

Time averaged power density at the center point of the cross sectional face has been

plotted in Figure 5.7 which shows the relative maxima at α = 0.5. This shows that

one may use the fractional curl operator to illustrate the pattern of transmitted power

through the waveguide. The average power density at the cross sectional plane can

– 90 –

be obtained by integrating the local power density given in equation (5.13) over the

whole cross section as

P(z) =∫ π

0

∫ π

0

12Re[EfdxH

∗fdy − EfdyH

∗fdx]d(kxx)d(kyy) (5.14)

This power density has been plotted for the entire range of α as in Figure 5.8 which

shows that the average power density through the cross sectional plane remains fairly

constant for the whole range of the fractional parameter.

Figure 5.7 Time averaged power density at the center of the transverse plane, i.e.,

at (kxx, kyy) = (π/2, π/2)

– 91 –

Figure 5.8 Average power density at the transverse plane

– 92 –

CHAPTER VI

Conclusions

The research work carried out in the thesis has been concluded and summarized

as follows:

By selecting order of the fractional curl operator, impedance of the reflecting

boundary as well as polarization of the incident wave can be illustrated. For the case

of a planar PEC surface as an original surface, impedance of the fractional dual surface

is isotropic for normal incidence while it is anisotropic for the case of oblique incidence.

The impedance of the fractional dual surface is a function of the fractional parameter

so that fractional dual is a surface whose impedance is the intermediate step of the

PEC and PMC surface. Further this impedance is same for transverse electric (TEz)

and transverse magnetic (TMz) polarizations, that is, independent of the polarization

of incident field.

When the original reflecting surface is an impedance surface, the fractional dual

surface has anisotropic impedance which is a function of impedance of the origi-

nal surface, the fractional parameter, and type of polarization of the incident wave.

Impedance of the fractional impedance surface is complex and behavior of both com-

ponents is same with respect to α. When the original reflecting surface is a PEMC

surface, the fractional dual PEMC surface has anisotropic impedance such that be-

haviors of the two components are different with respect to α.

Fractional parallel plate waveguide model represents that if TMz mode is prop-

agating through a waveguide with impedance walls then its dual solution means a

TEz mode propagating through the waveguide with admittance walls. The fractional

waveguides means a hybrid mode propagation through the waveguide having the walls

whose impedance is an intermediate step between the impedance walls and the ad-

mittance walls. Field behavior inside the fractional parallel plate PEC waveguides,

– 93 –

impedance waveguides, and PEMC waveguides are same as the field in the region

y > 0 in presence of the corresponding fractional reflecting boundary planes placed at

y = 0. Fractional dual solutions in the chiral region for a chiral-achiral interface yields

the result similar to the ordinary dielectric-dielectric region. Similar is the relation

between a parallel plate chiro waveguide and the parallel plate dielectric waveguide.

Fractional transmission lines can be modelled using the fractional order differen-

tial operator. Fractional transmission line model is a generalization of the transmission

line having input impedance which is intermediate step of the input impedance of the

original line and input admittance of the original line. As a special case, when the

original line is a short circuit line, the fractional line has a complex load and it is an

intermediate step of the short circuit transmission line and the open circuit transmis-

sion line. Fractional non-symmetric transmission line shows the behavior similar to

the fractional symmetric transmission line.

In case of rectangular waveguide, It has been seen that the relative power density

distribution at the the cross sectional plane changes with varying α. For example, at

center of the cross sectional plane, the relative power density distribution is maximum

at α = 0.5. However the average power density at the cross sectional plane remains

fairly constant for all values of α between 0 and 1. Hence it may be concluded that

the fractional curl operator may be used to illustrate the power distribution pattern

over the cross section of the guide.

– 94 –

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