akhtar thesis 230110

8/9/2019 Akhtar Thesis 230110

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FRACTIONAL

TRANSMISSION LINES AND WAVEGUIDES

IN ELECTROMAGNETICS

Akhtar Hussain

In Partial Fulllment of the Requirements

for the Degree of

Doctor of Philosophy

Department of Electronics

Quaid-i-Azam University

Islamabad, Pakistan

2009


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FRACTIONAL

TRANSMISSION LINES AND WAVEGUIDES

IN ELECTROMAGNETICS

by

Akhtar Hussain

In Partial Fulllment of the Requirements

for the Degree of

Doctor of Philosophy



Islamabad, Pakistan

2009


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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



Islamabad, Pakistan

Submitted through

Dr. Qaisar Abbas Naqvi

Chairman

Department of ElectronicsQuaid-i-Azam University

Islamabad, Pakistan


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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 eld 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 eld of my research.


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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. I

pay thank to my beloved sisters and brothers who always showed their concern aboutmy 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. 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 want to acknowledge my wife specially, who gave methe company and served me with delicious snacks during very long sittings for the

compilation of my research work.

Akhtar Hussain


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To

My Family


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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 eld 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 beeninvestigated. 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.


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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 . . . . . . . . . . . . . . . . . . . . . . . . 51CHAPTER IV: Fractional Chiro Waveguide and the Concept of

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

4.1. Reection 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 elds 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


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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 cylindricalreector, 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.


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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 dα f (x)/dx α ,

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

by modern examples of applications in physics, signal processing, uid 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 (PI λ Dµ ) controllers [4]. Numerous applications

have demonstrated that fractional PID-controllers (PI λ Dµ −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 ux prole more closely to actual


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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 tothe scientists, fractional calculus can be the language of twenty rst 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 denition 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

elds 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.


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

a general problem, the canonical cases are solved rst. 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, thisconcept 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


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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 arecipe 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 satises 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., Lα 1 .L α 2 = L α 2 .L α 1 = L α 1 + α 2 .

IV. The operator L α 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


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

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

in the space C n . That is, L : C n →C n where C n is a n-dimensional vector space overthe eld of complex numbers. Once a linear operator such as L is given to us, a recipefor constructing the fractional operator Lα can be described as follows [13]

1. One nds the eigenvectors and eigenvalues of the operator L in the space C n so

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

and eigenvalues of the operator L in C n respectively.

2. Provided A m s form a complete orthogonal basis in the space C n , any vector in

this space can be expressed in terms of linear combination of A m s. Thus, an

arbitrary vector G in space C n can be written as

G =n

m =1

gm A m

where gm s are the coefficients of expansion of G in terms of A m s.

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

operator Lα can be dened to have the same eigenvectors A m s, but with the

eigenvalues as ( am )α , i.e.,

L α .A m = ( am )α A m

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

C n , one gets

L α .G = L α .n

m =1gm A m =

n

m =1gm L α . A m =

n

m =1gm (am )α A m

The above equation essentially denes 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.


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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 dened 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 eld F as a function of three spatial variables

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

curl F =∂F z∂y −

∂F y∂z

x̂ +∂F x∂z −

∂F z∂x

ŷ +∂F y∂x −

∂F x∂y

ẑ (1.1)

where F x , F y , and F z are the Cartesian components of vector F , and x̂ , ŷ , and ẑ 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

F k

{F (x,y,z )

}= F̃ (kx , ky , kz )

= ∞

−∞

∞

−∞

∞

−∞

F (x,y,z )exp( −ikx x −iky y −ikz z)dxdy dz(1.2)

F k {curl F (x,y,z )}= ∞

−∞

∞

−∞

∞

−∞

curl F (x,y,z )exp( −ikx x −iky y −ik z z)dxdy dz= ik × F̃ (kx , ky , kz ) (1.3)

where a tilde over the vector F̃ 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 F̃ . It is suggested that, in order to fractionalize the curl operator,

one should rst fractionalize the cross product operator ( ik×) in the k -domain. Clearlythe cross product operator is an operator that gets two vectors as its inputs and gives

one vector as its output, e.g., k ×Ũ = W̃ . However, if one picks the rst vector k , then


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the operator ( k×) can be considered as a linear operator which takes one vector (e.g.,Ũ ) as its input and gives out one vector (e.g., W̃ ) as an output. Both vectors Ũ and W̃

are three-dimensional vectors in the k -domain. Thus, the linear operator ( k×) belongsto 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 thenext 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., ẑ k . The eigenvaluesand (normalized) eigenvectors of the operator ( ẑ k ×) in the k -domain are obtained as

A 1 = x̂ k −iŷ k√ 2 , a1 = i

A 2 = x̂ k + iŷ k

√ 2 , a2 = −iA 3 = ẑ k , a3 = 0 (1 .4)

where x̂ k , ŷ k , and ẑ k are the unit vectors in the k -domain.

The vector Ũ = Ũ x x̂ k + Ũ y ŷ k + Ũ z ẑ k upon which the fractional cross product

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

as

Ũ = g1A 1 + g2 A 2 + g3 A 3 (1.5)

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

g1 =Ũ x + iŨ y

√ 2 , g2 =Ũ x −iŨ y√ 2 , g3 = Ũ z


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Operation of cross product can be written as

(ẑ k ×)Ũ = a1g1 A 1 + a2g2A 2 + a3g3 A 3 (1.6a)

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

equation (1.6a) and hence can be expressed as

(ẑ k ×)α Ũ = ( a1 )α g1 A 1 + ( a2)α g2A 2 + ( a3)α g3A 3= (+ i)α

(Ũ x + iŨ y )( x̂ −iŷ k )2

+ ( −i)α (Ũ x

−iŨ y )( x̂ + iŷ k )

2 + (0)α

Ũ z ẑ k (1.6b)

For example, if one takes Ũ = x̂ k , equation (1.6b) becomes

(ẑ k ×)α x̂ k = (i)α 1√ 2

x̂ k −i ŷ k√ 2 + ( −i)α 1√ 2

x̂ k + iŷ k√ 2

= cosαπ2

x̂ k + sinαπ2

ŷ k (1.7)

which provides the fractional cross product of ( ẑ k ×)α acting on vector x̂ k . As observedfrom equation (1.7), when α = 1, one obtain the conventional (ordinary) cross product

ẑ k ×x̂ k = ŷ k . When α = 0, one then obtains ( ẑ k ×)0 x̂ k = x̂ k = I .x̂ k , i.e., the identityoperator operating on x̂ k . 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 appliedto the case of ik × F̃ in the k-domain. If the resulting expression can then be inverseFourier transformed back into the ( x,y,z )-domain, the nal result may be called the

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

operator in electromagnetics is addressed.


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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 and permeability µ) for the harmonic time

dependence exp( −iωt ) as

curl H = −iω Ecurl E = 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 × (η H̃ ) = −Ẽ1ik

ik × (Ẽ ) = η H̃ik . (η H̃ ) = 0

ik . Ẽ = 0 (1 .9)

where η = µ/ , k = ω√ µ . Ẽ (kx , ky , kz ) and H̃ (kx , ky , kz ) are the spatial Fouriertransforms 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 rst two equations in (1.9). it gives

1(ik)α +1

(ik×)α +1 (η H̃ ) = − 1(ik)α (ik×)α Ẽ1

(ik)α +1(ik×)α +1 Ẽ =

1(ik)α

(ik×)α (η H̃ ) (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


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as

1ik

ik × 1

(ik)α(ik×)α (η H̃ ) = −

1(ik)α

(ik×)α Ẽ (1.11a)1ik ik ×

1(ik)α (ik×)

α ˜E =

1(ik)α (ik×)

α

(η ˜H ) (1.11b)

It can be shown that the following equations also hold

ik . 1

(ik)α(ik×)α (η H̃ ) = 0 (1 .11c)

ik . 1

(ik)α(ik×)α Ẽ = 0 (1 .11d)

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

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

elds dened by

Ẽ fd = 1

(ik)α(ik×)α Ẽ (1.12a)

η H̃ fd = 1

(ik)α(ik×)α (η H̃ ) (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

E fd = 1

(ik)αcurlα E (1.13a)

ηH fd = 1

(ik)αcurlα (ηH ) (1.13b)

From equations (1.13), it can be seen that

α = 0⇒ E fd = E , ηH fd = ηH

and α = 1⇒ E fd = ηH , ηH fd = −E

which shows that for α = 0 , (E fd , ηH fd ) gives the original solution while ( E fd , ηH fd )

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


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all values of α between zero and unity, ( E fd , ηH fd ) 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 elds as expressed with the subscript fd for these elds.

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 elds 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 + iα 2 reveals that the fractional solutions


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may be represented as fractional dual solutions only if α 1 and α 2 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 chiralnihility [41-42].

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

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

for oblique incidence on an impedance innite surface and derived the reection coef-

cients of the fractional boundary in terms of the original reection coefficients. They

found that impedance of the fractional reecting surface is anisotropic and gave theimpedance 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 reection from a bi-isotropic slab backed by a PEC surface

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 reection

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 rst. 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


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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 eldpattern 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 reecting boundaries have been

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

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

solutions have been termed as the solutions of reection 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 (TE z) and transverse magnetic (TM z) incidences have beendiscussed separately.

Study related to the reection 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 eld 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


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

lled 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 lled 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.


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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 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 = ky ŷ + kz ẑ. Generic expressions for the electric eld E and the

magnetic eld H corresponding to this wave can be written as

E = [E 0x x̂ + E 0y ŷ + E 0z ẑ] exp(iky y + ikz z) (2.1a)

H = [H 0x x̂ + H 0y ŷ + H 0 z ẑ]exp(iky y + ikz z) (2.1b)

where k = ω√ µ = k2y + k2zAs per recipe described in chapter I, in order to write fractional dual solutions between

(E , ηH ) and ( ηH , −E ), we need to write the eld vectors in terms of eigen vectors of the cross product operator k̂×= 1k (ky ŷ + kz ẑ)× as

E = [P 1 A 1 + P 2 A 2 + P 3 A 3 ]exp(iky y + ikz z) (2.2a)

ηH = [Q1 A 1 + Q2 A 2 + Q3 A 3 ]exp(iky y + ikz z) (2.2b)


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where A 1 , A2 , and A 3 are the normalized eigen vectors of the cross product operator

k̂×. The normalized eigen vectors and the corresponding eigen values of the crossproduct operator k̂× are as given below

A 1 = 1√ 2 x̂ − ik (kz ŷ −ky ẑ) , a1 = + i (2.3a)

A 2 = 1√ 2 x̂ +

ik

(kz ŷ −ky ẑ) , a2 = −i (2.3b)A 3 =

ik

(ky ŷ + kz ẑ), a3 = 0 (2 .3c)

In equation (2.2), quantities P 1 , P 2 , and P 3 are the coefficients of expansion and are

given below

P 1 = 1√ 2 E 0x + ik (kz E 0y −ky E 0 z) (2.3d)

P 2 = 1√ 2 E 0x −

ik

(kz E 0y −ky E 0 z ) (2.3e)

P 3 = √ 2 −ik (ky E 0y + kz E 0z ) (2.3f )while Q1 , Q2 , and Q3 are required co-efficients for the magnetic eld and may be

obtained by the symmetry.

Fractional dual solutions ( E fd , ηH fd ) to the Maxwell equations, corresponding

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

following relations

E fd = ( k̂×)α E= [( a1 )α P 1 A 1 + ( a2 )α P 2 A 2 + ( a3 )α P 3 A 3 ]exp(iky y + ikz z) (2.4a)

ηH fd = ( k̂×)α ηH= [( a1 )α Q1 A 1 + ( a2 )α Q2 A 2 + ( a3 )α Q3 A 3 ] exp(iky y + ikz z) (2.4b)

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

duality theorem, i.e.,

ηH fd = ( k̂×)E fd (2.4c)


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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 k̂ =1k (−ky ŷ + kz ẑ). Associated electric and magnetic elds are given by

E = x̂ exp(−iky y + ikz z) (2.5a)ηH =

kzk

ŷ + kyk

ẑ exp(−iky y + ikz z) (2.5b)

Fractional dual solutions can be obtained using the following relations

E fd = 1√ 2 [P 1 (a1 )

α A 1 + P 2 (a2 )α A 2 + P 3 (a3 )α A 3 ]exp(−iky y + ikz z)ηH fd = ( k̂×)E fd

where A 1 A2 , and A 3 are the eigen vectors and a1 , a2 , and a3 are the corresponding

eigen values of the operator k̂× = 1k (−ky ŷ + kz ẑ)×. Quantities P 1 , P 2 , and P 3 arethe coefficients of expansion. Hence the fractional dual solutions can be written as

E fd = cosαπ2

x̂ + kzk

sinαπ2

ŷ + kyk

sinαπ2

ẑ

exp(−iky y + ikz z) (2.6a)ηH

fd =

−sin

απ

2x̂ +

kzk

cosαπ

2ŷ +

kyk

cosαπ

2ẑ

exp(−iky y + ikz z) (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, elds given in equation (2.6)can be regarded as fractional dual elds between the original and dual to the original

elds of the plane wave propagating in an oblique direction k̂ = 1k (−ky ŷ + kz ẑ). It

may also be noted from equation (2.6) that fractional dual elds represent a planewave propagating in the same direction as the original wave. However its transverse

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

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

different kinds of planar boundaries placed at y = 0.


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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

k ref

Eref

Hinc

kinc

Einc

Figure 2.1. Normal incidence on a PEC plane

The electric and magnetic elds associated with the incident and reected plane waves

are given below

E inc = x̂ exp(−iky ) (2.7a)ηH inc = ẑ exp(−iky ) (2.7b)

Eref

= −ˆx exp( iky ) (2.7c)

ηH ref = ẑ exp( iky ) (2.7d)

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

elds and are given below

E = −x̂2i sin(ky) (2.8a)


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ηH = ẑ2cos(ky) (2.8b)

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

as

E incfd = cosαπ2

x̂ + sinαπ2

ẑ exp(−iky ) (2.9a)ηH incfd = −sin

απ2

x̂ + cosαπ2

ẑ exp(−iky ) (2.9b)

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

E ref fd = −exp( iαπ ) cosαπ2

x̂ + sinαπ2

ẑ exp( iky ) (2.10a)

ηH ref fd = exp( iαπ ) −sin απ2 x̂ + cos απ2 ẑ exp( iky ) (2.10b)

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

reected elds as

E fd = E incfd + Eref fd

ηH fd = ηHincfd + ηHref fd

which give

E fd = −exp iαπ2

cosαπ2

x̂ + sinαπ2

ẑ 2i sin ky + απ2

(2.11a)

ηH fd = exp iαπ2 −sin

απ2

x̂ + cosαπ2

ẑ 2cos ky + απ2

(2.11b)

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

α = 0⇒

(E fd , ηH fd ) = ( E , ηH )

and α = 1⇒

(E fd , ηH fd ) = ( ηH , −E )

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

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


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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 dened by the ratio of transverse components of corresponding

electric and magnetic elds as

Z fd = −E fdxH fdz

= E fdzH fdx

= iη tan ky + απ

2

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

dual boundary and can be written as

Z fd = iη tanαπ2

(2.12)

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

dual boundary is Z fd = 0, i.e., PEC surface while for α = 1, the impedance is Z fd = ∞,i.e., PMC surface. For 0 < α < 1, the reecting 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 k̂ inc = 1k (−ky ŷ + kz ẑ) hits a planar PEC boundary placed at y = 0. Due tothe PEC boundary, reected wave is produced in direction described by the vector

k̂ ref = 1k (ky ŷ + kz ẑ) as shown in Figure 2.2. In case of oblique incidence, the easy

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

and parallel polarization components. For the eld conguration shown in Figure 2.2,

perpendicular polarization can also be referred to as transverse electric polarization

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

magnetic polarization to the z-direction (i.e., T M z polarization). The elds of the two


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polarizations are related through the duality theorem. Let us study T E z and TM z

polarizations separately.

Figure 2.2 Oblique incidence on a PEC plane

Case 1: Transverse electric ( T E z ) polarization

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

shown in Figure 2.2. The electric and magnetic elds for the incident and reected

waves can be written as

E inc = x̂ exp(−iky y + ikz z) (2.13a)ηH inc =

kzk

ŷ + kyk

ẑ exp(−iky y + ikz z) (2.13b)E ref = −x̂ exp( iky y + ikz z) (2.13c)

ηH ref = −kzk

ŷ + kyk

ẑ exp( iky y + ikz z) (2.13d)

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

E incfd = cosαπ2

x̂ + kzk

sinαπ2

ŷ + kyk

sinαπ2

ẑ

exp(−iky y + ikz z) (2.14a)ηH incfd = −sin

απ2

x̂ + kzk

cosαπ2

ŷ + kyk

cosαπ2

ẑ

exp(−iky y + ikz z) (2.14b)


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Fractional dual solutions corresponding to the reected elds may be obtained using

equation (2.4), i.e.,

E fd = [(a1 )α P 1 A 1 + ( a2 )α P 2 A 2 + ( a3 )α P 3 A 3 ]exp(iky y + ikz z)

ηH fd = ( k̂×)E fd

where A 1 A2 , and A 3 are the eigen vectors and a1 , a2 , and a3 are the corresponding

eigen values of the operator k̂ ref × = 1k (ky ŷ + kz ẑ)×. Quantities P 1 , P 2 , and P 3 arethe coefficients of expansion. Therefore, we can write

E ref fd = exp( iαπ ) −cosαπ2

x̂ + kzk

sinαπ2

ŷ − kyk

sinαπ2

ẑ

exp( iky y + ikz z) (2.14c)

ηH ref fd = exp( iαπ ) −sinαπ2

x̂ − kzk

cosαπ2

ŷ + kyk

cosαπ2

ẑ

exp( iky y + ikz z) (2.14d)

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

as sum of the incident and the reected elds as

E fd = 2 −iC α sin ky y + απ

2 x̂ + kzk S α cos ky y +

απ2 ŷ

−ikyk

S α sin ky y + απ

2ẑ exp i kz z +

απ2

(2.15a)

ηH fd = 2 −S α cos ky y + απ

2x̂ −i

kzk

C α sin ky y + απ

2ŷ

+kyk

C α cos ky y + απ

2ẑ exp i kz z +

απ2

(2.15b)

where

C α = cos απ2S α = sin

απ2

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

of α at an observation point ( ky y, k z z) = ( π/ 4, π/ 4).


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Figure 2.3 Plots of fractional dual T E z polarized elds at a point ( ky y, k z z) =

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


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

i.e., for α = 0

E fdx = E x , ηH fdx = ηH x

E fdy = E y , ηH fdy = ηH y

E fdz = E z , ηH fdz = ηH z

and for α = 1

E fdx = ηH x , ηH fdx = −E xE fdy = ηH y , ηH fdy = −E yE fdz = ηH z , ηH fdz = −E z

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

and magnetic elds as

Z fdxz = −E fdxH fdz

= iη kky

tan ky y + απ

2

Z fdzx = E fdzH fdx

= iηkyk

tan ky y + απ

2

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

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

zfd = Z fd /η as given below

zfd = kky

x̂ẑ + kyk

ẑ x̂ zTEfd (2.16)

where

zTEfd = i tanαπ

2, for 0

≤α

≤1

It may be noted that

zfdxz = kky

zTEfd , zfdzx = ky

k zTEfd

Since zfdxz = zfdzx , so it can be interpreted that impedance of the fractional dual PECboundary for oblique incidence is anisotropic in nature.


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Case 2: Transverse magnetic (TM z ) polarization

Plane wave reection 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 ( TM z polarization case)

Electric and magnetic elds shown in the gure can be written as

E inc = −kzk ŷ − kyk ẑ exp(−iky y + ikz z) (2.17a)ηH inc = x̂ exp(−iky y + ikz z) (2.17b)

E ref = −kzk

ŷ + kyk

ẑ exp( iky y + ikz z) (2.17c)

ηH ref = x̂ exp( iky y + ikz z) (2.17d)

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

elds in the region y > 0 may be obtained as

E fd = 2 −iS α sin ky y + απ

2x̂ −

kzk

C α cos ky y + απ

2ŷ

+ ikyk

C α sin ky y + απ

2ẑ exp i kz z +

απ2

(2.18a)

ηH fd = 2 C α cos ky y + απ

2x̂ −i

kzk

S α sin ky y + απ

2ŷ


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+kyk

S α cos ky y + απ

2ẑ exp i kz z +

απ2

(2.18b)

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

theorem, i.e.,

α = 0⇒

(E fd , ηH fd ) = ( E , ηH )

and α = 1⇒

(E fd , ηH fd ) = ( ηH , −E )

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

dual surface can be written as

zfd = kky

x̂ẑ + kyk

ẑ x̂ zTMfd (2.19)

where

zTMfd = i tanαπ2

, 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 T E z and T M z polarizations.

2.3 Reection from a planar impedance boundary

Let the planar interface is placed at y = 0 and has a nite 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.


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2.3.1 Transverse electric ( T E z ) incidence

Consider a plane wave with T E z polarization incident on an impedance planar

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

incident and reected elds can be written as

E inc = x̂ exp(−iky y + ikz z) (2.20a)ηH inc =

kzk

ŷ + kyk

ẑ exp(−iky y + ikz z) (2.20b)E ref = Γ te x̂ exp( iky y + ikz z) (2.20c)

ηH ref = Γ tekzk

ŷ − kyk

ẑ exp( iky y + ikz z) (2.20d)

where Γ te is the reection coefficient as given below

Γte = −1 −zb

k yk

1 + zbk yk

(2.21)

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

E fd = x̂C α exp −iky y −iαπ2

+ Γ te exp iky y + iαπ2

+ ŷS αkzk

exp −iky y −iαπ2 −Γte exp iky y + i

απ2

+ ẑS α ky

kexp −iky y −i απ2 + Γ te exp iky y + i

απ2

exp i kz z + απ2

(2.22a)

ηH fd = −x̂S α exp −iky y −iαπ2 −Γte exp iky y + i

απ2

+ ŷC αkzk

exp −iky y −iαπ2

+ Γ te exp iky y + iαπ2

+ ẑC αkyk

exp −iky y −iαπ2 −Γte exp iky y + i

απ2

exp i kz z + α π2 (2.22b)

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

E t = E fdx x̂ + E fdzẑ

ηH t = ηH fdx x̂ + ηH fdzẑ


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Plots of these tangential elds 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 elds while dashed lines are for the corresponding magnetic elds.

In Figure 2.5, zero values of the electric eld for ( α, z b) = (0 , 0) and ( α, z b) = (1 , 100)

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

eld for (α, z b) = (1 , 0) and ( α, z b) = (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 theboundary 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 elds for T E z polarization in the presence of

an impedance boundary


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Impedance of the fractional dual surface can be written as

zfd

= kky

x̂ẑ + kyk

ẑ x̂ zTEfd (2.23)

where

zTEfd =zb

k yk + i tan

απ2

1 + zbk yk i tan

απ2

, for 0 ≤α ≤1

2.3.2 Transverse magnetic ( T M z ) incidence

Now consider the case of T M z -polarized wave incident on an impedance boundary

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

elds for this polarization can be written as

E inc = −kyk

ŷ − kzk

ẑ exp(−iky y + ikz z) (2.24a)ηH inc = x̂ exp(−iky y + ikz z) (2.24b)

E ref = Γ tm −kyk

ŷ + kzk

ẑ exp(−iky y + ikz z) (2.24c)ηH ref = Γ

tmx̂ exp(

−ik

yy + ik

zz) (2.24d)

where Γ tm is the reection coefficient given by

Γtm =k yk −zb

k yk + zb

(2.25)

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

the duality on the elds of equation (2.22) subject to the replacement of the reection

coefficient Γte by Γtm and hence can be written as

E fd = x̂S α exp −iky y −iαπ2 −Γtm exp iky y + i

απ2

−ŷC αkzk


+ Γ tm exp iky y + iαπ2

− ẑC αkyk

exp −iky y −iαπ2 −Γtm exp iky y + i

απ2


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exp i kz z + απ2

(2.26a)

ηH fd = x̂C α exp −iky y −iαπ2


+ ŷS αkz

kexp

−iky y

−iαπ

2 −Γtm exp iky y + i

απ

2+ ẑS α

kyk



exp i kz z + απ2

(2.26b)

In order to validate the elds 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, z b) = (0 , 0) and ( α, z b) =

(1, 100), boundary conditions for PEC surface are satised while for ( α, z b) = (1 , 0)

and ( α, z b) = (0 , 100), boundary conditions for PMC surface are satised.

Figure 2.6 Fractional dual tangential elds for T M z polarization


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Impedance of the fractional dual impedance surface can be written as

zfd = kky

x̂ẑ + kyk

ẑ x̂ zTMfd (2.27)

where

zTMfd =zb kk y + i tan

απ2

1 + zb kk y i tanαπ2

, for 0 ≤α ≤1 (2.27a)

It may be noted that in case of T M z polarization, impedance of the fractional dual

impedance surface is different from T E z 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 gure it can further be noted that for normalized impedance zb = 0, behaviors

for the two polarizations become same, i.e., zTEfd = zTMfd 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 TE z and TM z polarizations


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The reected elds ( E ref d , ηH ref d ) corresponding the transformed incident elds

(E incd , ηH incd ) can be written using the PEC boundary conditions and nally the inverse

transformation [45] may be used to get the elds reected from the PEMC surface as

Eref

H ref = 1(Mη)2 + 1 Mη −η1η Mη Eref d

H ref d (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 ( T E z ) incidence

Consider a plane wave with T E z -polarization is incident upon a PEMC boundary

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

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

E inc = x̂ exp(

−iky y + ikz z)

ηH inc =kzk

ŷ + kyk

ẑ exp(−iky y + ikz z)Applying the transformation given in equation (2.29), duality transformed elds cor-

responding to the incident elds can be written as

E incd = Mηx̂ + −kzk

ŷ + kyk

ẑ exp(−iky y + ikz z) (2.31a)

ηH incd = −x̂ + Mη −kzk

ŷ + kyk

ẑ exp(−iky y + ikz z) (2.31b)Fields reected from the planar PEC boundary, when the incident wave dened by

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

E ref d = − Mηx̂ + −kzk

ŷ + kyk

ẑ exp( iky y + ikz z) (2.32a)

ηH ref d = −x̂ + Mη −kzk

ŷ + kyk

ẑ exp( iky y + ikz z) (2.32b)


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– 34 –

The elds reected from the PEMC boundary can be written by applying the inverse

transformation given in equation (2.30) and are given below

E ref =1 −(Mη)21 + ( Mη)2

x̂ − 2Mη

1 + ( Mη)2 −kzk

ŷ + kyk

ẑ exp(−iky y + ikz z) (2.33a)ηH ref = k̂ ref ×E ref (2.33b)

where

k̂ ref = 1k

(ky ŷ + kz ẑ)

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

E ref = cos(2θ)x̂ + sin(2 θ)kzk

ŷ − kyk

ẑ exp( iky y + ikz z) (2.34a)

H ref = −sin(2θ)x̂ + cos(2 θ)kzk

ŷ − kyk

ẑ exp( iky y + ikz z) (2.34b)

Total elds in the region y > 0 can be written as sum of the incident and reected

elds and are given below

E = exp( ik z z) 2x̂ cos(ky y)sin2

(θ) −i sin(ky y)cos2

(θ)+

kzk

ŷ sin(2θ) cos(ky y) + i sin(ky y)

−kyk

ẑ sin(2θ) cos(ky y) + i sin(ky y) (2.35a)

ηH = exp( ik z z) −sin(2θ)x̂ cos(ky y) + i sin(ky y)+

kzk

2ŷ cos(ky y)cos2 (θ) −i sin(ky y)sin2 (θ)

+kyk 2ẑ cos(ky y)sin

2

(θ) −i sin(ky y)cos2

(θ) (2.35b)

At θ = π/ 2, above equations reduce as

E = −2i exp( ikz z)x̂ sin(ky y) (2.36a)ηH = 2 exp( ikz z)

kzk

ŷ cos(ky y) −ikyk

ẑ sin(ky y) (2.36b)


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– 35 –

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

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

ŷ ×(ηH + tan θE ) = 0

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

E fd = exp i kz z + απ

2x̂ (C θ + C α )cos ky y +

απ2

+ ( C θ −C α ) i sin ky y + απ

2

+ ŷkzk

(S θ + S α )cos ky y + απ

2+ ( S θ −S α ) i sin ky y +

απ2

−ẑ

ky

k(S θ

−S α )cos ky y +

απ

2+ ( S θ + S α ) i sin ky y +

απ

2 (2.37a)

η0 H fd = exp i kz z + απ

2

−x̂ (S θ + S α )cos ky y + απ

2+ ( S θ −S α ) i sin ky y +

απ2

+ ŷkzk

(C θ + C α )cos ky y + απ

2+ ( C θ −C α ) i sin ky y +

απ2

−̂zkyk

(C θ −C α )cos ky y + απ

2+ ( C θ + C α ) i sin ky y +

απ2

(2.37b)

where

C θ = cos 2θ − απ

2S θ = sin 2θ −

απ2

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

ratio of the elds at y = 0 as

zfd

= kky

zT E fdxz x̂ẑ + kyk

zT E fdzx ẑˆx , 0 ≤α ≤1 (2.38)where

zT E fdxz =C α (C θ + C α ) + iS α (C θ −C α )C α (C θ −C α ) + iS α (C θ + C α )

(2.38a)

zT E fdzx =C α (S θ −S α ) + iS α (S θ + S α )C α (S θ + S α ) + iS α (S θ −S α )

(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.


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– 36 –

2.4.2 Transverse magnetic ( T M z ) incidence

Consider a plane wave with T M z -polarization is incident upon a PEMC boundary

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

reected from the PEMC surface can be written as

E ref = −cos(2θ) −kzk

ŷ + kyk

ẑ −sin(2θ)x̂ exp( iky y + ikz z) (2.39a)

H ref = −sin(2θ)kzk

ŷ − kyk

ẑ −cos(2θ)x̂ exp( iky y + ikz z) (2.39b)

Fractional dual solutions for the problem can be written as

E fd = exp i kz z + απ2

−x̂ (S θ −S α )cos ky y + απ

2+ ( S θ + S α ) i sin ky y +

απ2

+ ŷkzk

(C θ −C α )cos ky y + απ

2+ ( C θ + C α ) i sin ky y +

απ2

−̂zkyk

(C θ + C α )cos ky y + απ

2+ ( C θ −C α ) i sin ky y +

απ2

(2.40a)

ηH fd = exp i kz z + απ

2

−x̂ (C θ −C α )cos ky y + απ

2 + ( C θ + C α ) i sin ky y + απ

2

−ŷkzk

(S θ −S α )cos ky y + απ

2+ ( S θ + S α ) i sin ky y +

απ2

+ ẑkyk

(S θ + S α )cos ky y + απ

2+ ( S θ −S α ) i sin ky y +

απ2

(2.40b)

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

zfd = kky

zTMfdxz x̂ẑ + kyk

zTMfdzx ẑˆx , 0 ≤α ≤1 (2.41)

where

zTMfdxz =C α (S θ −S α ) + iS α (S θ + S α )C α (S θ + S α ) + iS α (S θ −S α )

(2.41a)

zTMfdzx =C α (C θ + C α ) + iS α (C θ −C α )C α (C θ −C α ) + iS α (C θ + C α )

(2.41b)


47/109

– 37 –

It may be noted here that

zT M fdxz = zT E fdzx , and zT M fdzx = zT E 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 tan 2 θ while zfdzx changes from

cot 2 θ to 1 as the value of the fractional parameter α changes from 0 to 1. Further

we can see from the gure 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 , cot 2 θ)α = 0 .5

⇒ (zfdxz , zfdzx ) = (1 , 1)

α = 1 ⇒ (zfdxz , zfdzx ) = (tan 2 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.


48/109

– 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


49/109

– 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 eld 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 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 innite extent and the direction of propagation is considered as positive z-axis.

Figure 3.1. Geometry of parallel plate waveguide


50/109

– 40 –

Electric and magnetic elds 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

∂x 2 + ∂ 2

∂y 2 + ∂ 2

∂z 2 is the Laplacian operator and k = ω√ µ is the wavenumber. Taking z-dependance as exp( iβz ), equation (3.1) can be reduced to two

dimensional vector Helmholtz equation as

∇2xy E (x, y ) + h2E (x, y ) = 0 (3 .2a)

∇2xy H (x, y ) + h2H (x, y ) = 0 (3 .2b)

where h2 = k2 −β 2 , β is the propagation constant.Since propagation is along z-direction and the waveguide dimensions are consid-

ered innite 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 eld components only. The transverse eld components may be obtained

using the axial components of the elds and Maxwell equations. So scalar Helmholtzequations for the axial components can be written as

d2E zdy2

+ h2E z = 0 (3 .3c)

d2H zdy2

+ h2H z = 0 (3 .3d)


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– 41 –

General solution of the above equations is

E z = An cos(hy) + Bn sin(hy) (3.3e)

H z = C n cos(hy) + D n sin(hy) (3.3f )

where An , B n , C n , 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 ( E z , H z ), that is

E x = 1h2

iβ ∂E z∂x

+ ik∂ηH z

∂y (3.4a)

E y = 1h2iβ ∂E z

∂y −ik ∂ηH z

∂x (3.4b)

H x = 1h2

iβ ∂H z∂x −

ikη

∂E z∂y

(3.4c)

H y = 1h2

iβ ∂H z∂y

+ ikη

∂E z∂x

(3.4d)

where

η = µ is impedence of the medium inside the guideIn 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. TM z and TE z cases have

been discussed separately.


52/109

– 42 –

Case 1: Transverse magnetic ( T M z ) mode solution

Suppose a transverse magnetic ( TM z ) 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 eld is given

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

ẑE z (y, z ) = ẑAn sin (hy)exp( iβz )

= −̂zAn2i

[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 elds can be written

as

ŷE y (y, z ) = ŷiβ h

An cos (hy)exp( iβz )

= ŷiβ h

An2

[exp(−ihy + iβz ) + exp( ihy + iβz )] (3.5b)x̂ηH x (y, z ) = −x̂

ikh

An cos (hy)exp( iβz )

= −x̂ ikh An2 [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 = E 1 + E 2 (3.6a)

ηH = ηH 1 + ηH 2 (3.6b)

where (E 1 , ηH 1) are the electric and magnetic elds associated with one plane wave

and are given below

E 1 = An

2iẑ +

iβ h

ŷ exp(−ihy + iβz ) (3.7a)ηH 1 = −x̂

ikh

An2

exp(−ihy + iβz ) (3.7b)


53/109

– 43 –

while (E 2 , ηH 2 ) are the electric and magnetic elds associated with the second plane

wave and are given below

E 2 =

An2 −iẑ +

iβ

h ŷ exp( ihy + iβz ) (3.8a)

ηH 2 = −x̂ikh

An2

exp(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 elds inside the waveguide

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

(E 1 , ηH 1) = −ikh

An2

(E inc , ηH inc )

and

(E 2 , ηH 2 ) = −ikh

An2 (E

ref , ηH

ref )

provided that h = ky and β = kz .

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

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


54/109

– 44 –

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

parallel plate PEC waveguide can be written as

E fd =

−ik

h An

−iS α sin hy +

απ

2x̂

− β

kC α cos hy +

απ

2ŷ

+ ihk

C α sin hy + απ

2ẑ exp i βz +

απ2

(3.9a)

ηH fd = −ikh

An C α cos hy + απ

2x̂ −i

β k

S α sin hy + απ

2ŷ

+hk

S α cos hy + απ

2ẑ exp i βz +

απ2

(3.9b)

where

C α = cosαπ2

S α = sinαπ2

3.2.1 Behavior of elds inside the fractional parallel plate PEC waveguide

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

waveguide, electric and magnetic eld 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 eld plots

while magnetic elds are shown by dashed lines. From the gure we see that eld 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 eld lines are perpendicular to the guide walls and there

are no magnetic eld 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 eld lines

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


55/109

– 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 eld while dashed lines are for the magnetic eld

3.2.2 Mode behavior for higher values of the fractional parameter

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

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

α = 0

⇒

E fd = E , ηH fd = ηH ,

E fdz = 0 , H fdz = 0 , zfd = 0 (3 .10a)α = 1⇒ E fd = ηH , ηH fd = −E ,

E fdz = 0 , H fdz = 0 , zfd = ∞ (3.10b)α = 2⇒ E fd = −E , ηH fd = −ηH ,


56/109

– 46 –

E fdz = 0 , H fdz = 0 , zfd = 0 (3 .10c)α = 3⇒ E fd = −ηH , ηH fd = E ,

E fdz = 0 , H fdz = 0 , zfd = ∞ (3.10d)α = 4⇒ E fd = E , ηH fd = ηH ,

E fdz = 0 , H fdz = 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 conguration and guide walls nature upon α(a) α = 0 (b ) α = 1 ( c) α = 2 (d ) α = 3


57/109

– 47 –

From the gure, 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 eld conguration 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 eld conguration 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 elds inside the parallel plate PEC waveguide are pro-

portional to the elds 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 reecting boundary discussed in

chapter 2.

Case 2: Transverse electric (T E z ) 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 H z while E z = 0

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

equation (3.4) so that the electric and magnetic elds may be considered as the eldsof two TEM plane waves bouncing back and forth between the two conducting plates.

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

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

reection from the planar PEC boundary placed at y = 0 with −ikh C n as the constant


58/109

– 48 –

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

E fd = −ikh

C n −iC α sin hy + απ

2x̂ +

β k

S α cos hy + απ

2ŷ

−ihk S α sin hy +

απ2 ẑ exp i βz +

απ2 (3.11a)

ηH fd = −ikh

C n −S α cos hy + απ

2x̂ −i

β k

C α sin hy + απ

2ŷ

+hk

C α cos hy + απ

2ẑ exp i βz +

απ2

(3.11b)

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

elds have been plotted in Figure 3.5 which shows the behavior of eld lines in the

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

Figure 3.5 T E z eld lines in yz-plane at different values of α ; solid lines are for the

electric eld while dashed lines are for the magnetic eld


59/109

– 49 –

3.3 Fractional parallel plate impedance waveguide

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

uide are proportional to the elds in the region y > 0 in the presence of a reecting

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 elds inside the fractional

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

agating through a parallel plate waveguide whose walls have nite impedance Z w .

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

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

boundary conditions as given below

E z |(y =0) = Z w H x |(y =0) (3.12a)E z |(y = b) = −Z w H x |(y = b) (3.12b)

Solution for the electric and magnetic elds 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 elds 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 elds given in equation (2.24)

through the constant of proportionality given as

C = −kh

An2

(F + i)

where

F = −izwkh

zw = Z w

η


60/109

– 50 –

Hence the fractional dual solutions for the TM z mode can be written as

E fd = B −ikh −iS α F cos hy + απ

2+ sin hy +

απ2

x̂

+β k C α F sin hy +

απ2 −cos hy +

απ2 ŷ

+ihk

C α F cos hy + απ

2+ sin hy +

απ2

ẑ

exp i βz + απ

2 (3.13a)

ηH fd = B −kh −iC α F sin hy + απ

2 −cos hy + απ

2x̂

+β k

S α F cos hy + απ

2+ sin hy +

απ2

ŷ

− ihk S α F sin hy + απ2 −cos hy + απ2 ẑexp i βz +

απ2

(3.13b)

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

portional to the fractional dual elds ( T M z 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 eld 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 gure that for zw = 0, eld patterns match with

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

of the wall changes from the zero value, the eld 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 eld patterns with α is similar as in the case of PEC guide.


61/109

– 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 eld while dashed lines are for the magnetic eld

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 thecondition 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.


62/109

– 52 –

The PEMC boundary conditions are given below

n × ηH + tan θE = 0 , n. D −tan θB = 0 (3 .14)

where

tan θ = M η

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 elds inside a fractional parallel plate

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

waveguide using the same transformation. Therefore relation between the elds 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 T M z mode solution for a parallel plate PEC waveguide as

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

proportional to the fractional dual solutions of the T M z polarized wave in the presence

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

C = −ikh

An

Hence the fractional dual solutions (transformed from the T M z mode solution) inside

the parallel plate PEMC waveguide can be written as

E fd = −ikh

An

−x̂ (S θ −S α )cos hy + απ

2+ ( S θ + S α ) i sin hy +

απ2


63/109

– 53 –

+ ŷβ k

(C θ −C α )cos hy + απ

2+ ( C θ + C α ) i sin hy +

απ2

−̂zhk

(C θ + C α )cos hy + απ

2+ ( C θ −C α ) i sin hy +

απ2

exp i βz + απ

2 (3.15a)

ηH fd = −ikh

An

−x̂ (C θ −C α )cos hy + απ

2+ ( C θ + C α ) i sin hy +

απ2

−ŷβ k

(S θ −S α )cos hy + απ

2+ ( S θ + S α ) i sin hy +

απ2

+ ẑhk

(S θ + S α )cos hy + απ

2+ ( S θ −S α ) i sin hy +

απ2

exp i βz + απ

2 (3.15b)

where

C θ = cos 2θ − απ

2S θ = sin 2θ −

απ2

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

to the fractional dual solutions for T M z 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 eld lines for the fractional parallel plate PEMC

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

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 gure that for θ = π/ 2, the eld patterns match with the fractional parallel plate

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

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

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


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– 55 –

CHAPTER IV

Fractional Chiro Waveguide and

the Concept of Fractional Transmission Lines

In chapter II, fractional duality has been studied for d

akhtar thesis 230110

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