arooj hameed

8/9/2019 AROOJ Hameed

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Reflection and Transmission of Electromagnetic Waves

from an interface of Chiral and/or Chiral-Nihility

metamaterial in Non-integer Dimension Space

Arooj Hameed

Supervised by

Dr. Q. A. Naqvi

Department of ElectronicQuaid-i-Azam University, Islamabad, 45320, Pakistan.

February 2014

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Certificate of Approval

It is certified that the research work presented in this thesis, entitled “Reflection and Trans-

mission of Electromagnetic Waves from an interface of Chiral and/or Chiral-Nihility metamaterial

in Non-integer Dimension Space” was conducted by Mr. Arooj Hameed under the supervision of

Dr. Q. A . Naqvi.

Dr. Q. A. Naqvi. (Advisor), Dr. Farhan Saif ,

Associate Professor, Chairman,

Department of Electronic . Department of Electronic.

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Dedication

This thesis is dedicated to my first teacher, my father

Abdul Hameed

who taught me that, everything worthwhile in life is work. But if it puts a smile on

your face, it doesnt feel like work.

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Acknowledgements

First of all it is obligatory to thank Almighty ALLAH, the most Merciful and the Benefi-

cent, who gave me health, thoughts and co-operative people to enable me for achieving this

goal. I offer my praises to Holy Prophet MUHAMMAD (S.A.W) and his companions who

laid the foundation of Modern civilization and paved the way for social, moral, political,

economical, cultural and physical revolution.

I would like to thank Dr. Q. A. Naqvi, my supervisor, for his valuable suggestions and

consistent support during my work. He guided me in right direction which made me able

to accomplish my work. He showed a remarkable patience and believed in my ability to

complete the task. I pray for his health and prosperous life. I would also like to thank

Dr. Farhan Saif (Chairman) for his appreciation and support. I would like to thank man-

agement of Quaid-i-Azam University, Islamabad for providing me such a good environment

and labs. May ALLAH make me able to fulfill my duty and commitment towards the in-

stitute sincerely and devotedly. I am also grateful to my mother, my Uncles Mr. Abdul

Aziz, Mr. Abdul Majeed, Mr. Tariq Pervaiz and Mr. Bashart Mehmood, my brother’s

Mr. Umer Hameed, Mr. Usman and Mr. Wahab, Finally, I would like to thank my all

teachers, colleagues and seniors especially to Dr. A. A. Rizvi, Dr. Musarat Abbas, Dr.

Aqeel Ashraf, Dr. A. A. Sayed, Dr. Hassan, Omar, Abbas, Usman, Zeeshan and Waqas

for their counseling and motivation in my academic activities. May ALLAH give them all

successful and peaceful lives.

Arooj Hameed

February 2014

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Fractals provide a workable new middle ground between the excessive geometric

order of Euclid and the geometric Choas of roughness and fragmentation.

(Mandelbrot, 1989)

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Abstract

Fractional space formulation concept is useful in solving many real time problems in the

area of physics. Mandelbrot introduced the concept of “Fractal” for complex structure

which cannot be described by the Euclidean geometry [?]. Fractal has found an important

place in science as a representation of some of the unique geometrical features occurring in

nature. Many shapes (geometries) in nature that are highly complex like the roots of the

trees, dust particles, geometry of clouds or even galaxies in space are considered as fractional

dimension. Hence fractals are an excellent way of describing highly complex shapes which

cannot be described by the Euclidean geometry. Fractional calculus has gained a significant

footing to study various problems regarding fractals which is very useful to solve many

real time problems regarding electromagnetic (EM) theory in fractional space. To model

these complex structures at both microscopic and macroscopic level, fractional dimension

D is used to characterized these fractals [?, ?]. Media of high complexity can be defined

by small number of parameters. Self-similarity is another important feature of fractals. In

order to get maximum benefits of these fractal models, it is vital to simplify the theories of

electromagnetics in fractional spaces. In this dissertation, behavior of electromagnetic waves

in Chiral and/or Chiral-Nihility interface in non-integer Dimension Space is studied. The

coefficients of reflection and transmission for the chiral-chiral interface in fractional space,

when the fractal interface is excited by circularly polarized plane wave, are presented. A

new parameter D, is introduced and effect of dimension on tunneling and rejection of power

is also studied. It is found that dimension of media has strong effect on tunneling and

rejection of power. This analysis is further extended to chiral-nihility and reflection and

transmission of electromagnetic waves in Chiral-Nihility interface for non-integer dimension

Space is studied and effect of dimension on tunneling and rejection of power is studied. It is

found that dimension of media has effect on tunneling and rejection of power. The classical

results are recovered when integer-dimension space is considered i.e., D = 2.

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Contents

1 Introduction 1

1.1 Naturaly existing Fractals . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Fractional Dimensional Space and Electromagnetic Theory in Fractional Space 2

1.3 Axiomatic Basis for Non-Integer Dimension space . . . . . . . . . . . . . . . 3

1.4 Chiral medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.5 Chiral Nihility Medium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.6 Thesis Plan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 General background of Electromagnetics wave propagation in Non-integer

dimensional space 6

2.1 wave propagation in fractional space . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Parallel Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.2 Perpendicular Polarization . . . . . . . . . . . . . . . . . . . . . . . . 10

2.2 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 Electromagnetic Wave Behavior Of Chiral-Chiral interface in Fractional

Space 12

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

4 Chiral nihility-Chiral nihility interface in fractional space 25

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4.1 Chiral Nihility-Chiral Interface In fractional space . . . . . . . . . . . . . . . 26

4.1.1 Impedance matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4.1.2 Impedance mis-matching . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Chiral-Chiral Nihility Interface In fractional space . . . . . . . . . . . . . . . 33

4.2.1 Impedance matching . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.2.2 Impedance mis-matching . . . . . . . . . . . . . . . . . . . . . . . . . 35

4.3 Chiral Nihility-Chiral Nihility Interface In fractional space . . . . . . . . . . 39

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5 Conclusion and Future Work 44

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List of Figures

2.1 Parallel polarized wave at dielectric fractal-fractal interface. . . . . . . . . . 8

2.2 Perpendicular polarized wave at dielectric fractal-fractal interface. . . . . . . 10

3.1 Reflection and transmission in fractal chiral-chiral media: Incident RCP

(solid lines), reflected LCP (dotted dashed line), reflected RCP (double doted

dashed line), refracted LCP (dashed line), refracted RCP (long dashed line). 13

3.2 Co component of reflected power verses angle of incidence for non-integer

dimension,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, κ2 = 0.25 . 17

3.3 Cross component of reflected power verses angle of incidence for non-integer

dimension,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, κ2 = 0.25 . 18

3.4 Co component of reflected power verses angle of incidence for non-integer

dimension,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, κ2 = 0.75 . 19

3.5 Cross component of reflected power verses angle of incidence for non-integer

dimension,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, κ2 = 0.75 . 20

3.6 Co component of reflected power verses chirality of the media 1 for non-integer

dimension,when µ1 = µ2 = 1, for impedance mismatch κ2 = 0.25, θinc = 45◦ . 21

3.7 Cross component of reflected power verses chirality of the media 1 for non-

integer dimension,when µ1 = µ2 = 1, for impedance mismatch κ2 = 0.25, θinc =

45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


dimension,when µ1 = µ2 = 1, for impedance matching κ2 = 0.25, θinc = 45◦ . 22

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3.9 Cross component of reflected power verses chirality of the media I for non-

integer dimension,when µ1 = µ2 = 1, for impedance matching κ2 = 0.25, θinc =

45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22


dimension,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, θinc = 45◦ . 23

3.11 Cross component of reflected power verses chirality of the media 2 for non-

integer dimension,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, θinc =

45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23


dimension,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, θinc = 45◦ . 24

3.13 Cross component of reflected power verses chirality of the media 2 for non-integer dimension,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, θinc =

45◦ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

4.1 Reflection and transmission in chiral nihility- fractal chiral nihility medium

( Wave vector representation ): Reflected backward wave (double dotted

dashed line), refracted backward wave ( dotted line). . . . . . . . . . . . . . 26

4.2 Co-component of reflected power verses kappa 1, for impendence matching. . 27

4.3 Cr-component of reflected power verses kappa 1, for impendence matching. . 274.4 Co-component of reflected power verses kappa 2, for impendence matching. . 28

4.5 Cr-component of reflected power verses kappa 2, for impendence matching. . 28

4.6 Co-component of reflected power verses kappa 1, for impendence mis-matching. 29

4.7 Cr-component of reflected power verses kappa 1, for impendence mis-matching. 30



4.10 Co-component of reflected power verses theta, for impendence mis-matching. 31

4.11 Cr-component of reflected power verses theta, for impendence mis-matching. 32


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4.16 Co-component of reflected power verses kappa 1, for impendence miss-matching. 36




4.20 Co-component of reflected power verses theta, for impendence mis-matching. 38

4.21 Cr-component of reflected power verses theta, for impendence mis-matching. 38


4.23 Cr-component of reflected power verses kappa 1, for impendence matching. . 404.24 Co-component of reflected power verses kappa 2, for impendence matching. . 40


4.26 Co-component of reflected power verses theta, for impendence matching. . . 41

4.27 Cr-component of reflected power verses theta, for impendence matching. . . 42

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Publication

• A. Hameed∗, M. Omar, A. A. Syed and Q. A. Naqvi,“POWER TUNNELING ANDREJECTION FROM CHIRAL-CHIRAL INTERFACE IN FRACTIONAL SPACE”

(Submited).

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Chapter 1

Introduction

The behavior of electromagnetic waves in fractal media has been studied theoretically . Mo-

tivation of this work is to simplify and mutate the problems related to complex structures

using self-similarity property of fractals. Highly complex media can be expressed using less

number of parameters, where fractional dimension, D is one of those important parameter

.Earlier, such structures were categorized using numerical and experimental methods. How-

ever, using the concept of fractional space, it is possible to find the analytical solution of

fractal structures.

1.1 Naturaly existing Fractals

There are some structures occurs in nature which are examples of fractals like., mountains,

roots of trees, snow , dust particles, floor of sea, trees and even our galaxy. First it is very

important to ponder what is fractal. If one takes an example of a leave. Many leaves to

gather makes a branch, branch makes a tree and many trees together makes forest. Hence

one can say that the basic unit of a forest is leave. Similarly, our planet earth is made of

sand. Many planets together forms galaxy. Hence one can say that a sand particle is basic

unit of galaxy.

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Concept of fractal is equally important in the field of biology, as there are many biological

examples of fractals which includes Bronchial Tube, Endoplasmic Reticulum, Arteries, Mito-

chondrial Membrane (inner and outer), Blood vessels, Bronchial Tube, Alveolar Membrane,

lungs, alveoli and Brain. All these anatomical structure exhibits non-integer dimension.

Fractal dimension of our bran is 2.73− 2.79. Similarly, fractal dimension of Arteries is 2.7.Previously, they treated cell membranes as simple surface or plane.

1.2 Fractional Dimensional Space and Electromagnetic Theory in

Fractional Space

The idea of fractional dimension space is used to transform many real world problems, as

it is easy to characterize them using fractional space expression. It is significant to specify

here that world is a good example of fractal. The dimension of world is not accurately 3,but

it is about 3±10−6 [?].There are many structures like., Sierpinski carpet , Menger sponge,etc., are good exam-

ples of fractals. If one wants to find the behavior of EM waves for these fractals. They can

not use the expressions of fields in integer dimension space., hence it is very important to

find the fields expressions in fractional space.

Solution of Laplace’s and Poisson’s equation for non- integer dimension space has been

discussed in [14, 15]. ].Moreover, the equations of motion for non- integer dimension space

has also been expressed [11].Solution of EM waves in non-integer dimension space was pro-

posed by Zubair et al.[16-21]. Also Faraday’s and Ampere’s laws for non- integer dimension

space were derived by Martin et al. [13].

The reflection and transmission of electromagnetic waves at an interface for non-integer

dimension space, will enable us to examine slabs and behavior of waves in wave guide for

non integer dimension.

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1.3 Axiomatic Basis for Non-Integer Dimension space

Axiomatic basis for non-integer space are provided by Stillinger’s [4]. Four topologies are

used in order to generate any fractional dimensional space. Where D represents the fractional

dimension space. Assume S D denote non-integer dimension space having points x, y, ...,

then the topological arrangement can be described by following axioms:

Axiom1: S D is a metric space.

Axiom2: S D is dense in itself.

Axiom3: S D is metrically unbounded.

Axiom4: For any two points y, z ∈ S D, and any ϵ > 0, there exists an x ∈ S D such that:

(i) p(x, y) + p(x, z ) = p(y, z )

(ii) | p(x, y) − p(x, z ) < εp(y, z )|

The complete implication of Axiom 4 is that two points in S D are joined by a continuous

line surrounded in that space. So, star domain in S D will be shrinkable. A method with

which any function with non -integer dimension is transformed into integer dimension, was

introduced by Muslis [14],. Using this relation, transformation of non-integer to integer

dimension, Zubair et al. presented solution to wave equations for non-integer dimensionspace [17].

1.4 Chiral medium

Chiral media have been known for a long time by researchers. An object is chiral which

has a non- superimpose able mirror image of itself, It bears a phenomena of optical activity

and circular dichroism [10-24]. The turning of the plane of linearly polarized wave about

the direction of propagation of EMT wave is due to chirality of media. Chiral media arecategorized by right and left circularly polarized waves with different phase velocities and

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refractive index. The two refractive indices of chiral media are,

n± =√

ϵµ± κ

The polarization vectors of left circularly polarized (LCP) waves in chiral media are as

follow,

F±L = x̂ ± ik+z

k+ ŷ − iky

k+ẑ

Likewise, the polarization vectors of right circularly polarized (RCP) waves in chiral media

are as follow,

F±R = x̂ ∓ ik−z

k− ŷ +

ikyk−

ẑ

The superscript “+” and “-” denote two eigen waves i.e., LCP and RCP waves respectively.

. Therefore, it is inspiring to study the reflection and transmission of EM waves for such

merged materials at fractal interface.

constitutive relation for a chiral media are as follow,

D = ϵE + iκH (1.1)

B = µH− iκE (1.2)

By using these constitutive relations in Maxwell’s equations with no source,

∇× E = iωB (1.3)∇× H = −iωD (1.4)

we get the following dispersion relation for the two wavenumbers :

k± = ω(√

µϵ± κ) (1.5)

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where κ is the chirality of media and ”-” and ”+” denote two eigen waves i.e., LCP and

RCP waves respectively.

1.5 Chiral Nihility Medium

The concept of chiral nihility was introduced by Lakhtakia [?]. Nihility is assumed as a

special case of chiral medium for which both the permeability (µ = 0) and permittivity

(ϵ = 0) approaches to zero simultaneously at certain frequencies. The constitutive relations

for chiral nihility medium are,

k±(1,2) = ±ωκ (1.6)k+z(1,2) = −k−z(1,2) (1.7)k+(1,2) = −k−(1,2) (1.8)

1.6 Thesis Plan

The thesis is divided into five chapters. In Chapter 2, wave propagation and the expres-sions to planar wave equations in fractional dimension spaces is presented. In Chapter 3,

transmission and reflection coefficients for chiral-fractal interface are derived and effect of

dimension on power rejection and tunneling is discussed. It is also shown that if integer

dimension is considered in the proposed expressions, exact classical results can be recovered

. This is helpful for other multiple interfaces and will enable us to examine slabs and be-

havior of waves in wave guide for non integer dimension. In Chapter 4, reflection coefficient

for fractal chiral nihility interface are investigate and some new expressions of reflection and

transmission coefficient are found. Future work and conclusions are discussed in Chapter 5.

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Chapter 2

General background of

Electromagnetics wave propagation inNon-integer dimensional space

Electic and magnetic field for non-integer dimension space., can be found out from Helmotz

equation, [?]:

∇2DE + β 2E = 0 (2.1)

where D is dimension of the media, ∇2D is the scalar Laplacian operator first given by Palmerfor fractional space [?],and defined as:

∇2D = ∂ 2

∂x2 +

β 1 − 1x

∂

∂x +

∂ 2

∂y2 +

β 2 − 1y

∂

∂y

+ ∂ 2

∂z 2 +

β 3 − 1z

∂

∂z (2.2)

In order to describe the measure of distribution of space, the parameters β 1, β 2 and β 3 are

used . where 0 < β i ≤ 1,, for i = 1, 2, 3. Total dimension D = β 1 + β 2 + β 3. If same

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fractionality exhibits in all directions then, the value of D = 3β . In cartesian coordinates

system,

E = x̂E x + ŷE y + ẑE z (2.3)

Hence the electric field in D dimensional space is [16],

E x = xm1ym2z m3 [F 1J m1(γ xx) + F 2Y m1(γ xx)]

[F 3J m2(γ yy) + F 4Y m2(γ yy)] [F 5J m3(γ zz ) + F 6Y m3(γ zz )](2.4)

where m1 = 1−β 1/2, m2 = 1−β 2/2 and m3 = 1−β 3/2. J m1(γ xx) and Y m1(γ xx) are Bessel

functions of order m1. Standing waves are represented using Bessel function of first andsecond kind. Similarly solutions for E z and E y are obtained .

2.1 wave propagation in fractional space

Consider two dielectric-fractal media [?]. (ϵ1, µ1, 1 < D1 ≤ 2) and (ϵ2, µ2, 1 < D2 ≤ 2),where the fractionality exist in ẑ -direction and boundary is assumed to be infinite. To

study propagation of wave in fractional space,two cases can be considered. i.e., reflection

and transmission at an oblique angle of incidence, two components of electric field are .

i) Parallel polarization

ii) Perpendicular polarization

2.1.1 Parallel Polarization

If a parallel polarized plane wave is incident at a dielectric fractal interface, Figure 2.1 shows

the geometry of the problem . The electric and magnetic field equations are as follow

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Figure 2.1: Parallel polarized wave at dielectric fractal-fractal interface.

Einc = E i (x̂ cos θinc − ẑ sin θinc)exp(−iβ1 sin θincx) AD (2.5)Eref = E i (x̂ cos θref + ẑ sin θref ) R∥ exp

(−iβ1 sin θref x) BD (2.6)

Etra = E i (x̂ cos θtra − ẑ sin θtra) T∥ exp(−iβ2 sin θtrax) C D (2.7)Hinc = E iŷ

1

η1exp(−iβ1 sin θincx) E D (2.8)

Href = E i−ŷ R∥η1

exp(−iβ1 sin θref x) F D (2.9)

Htra = E iŷT∥η2

exp(−iβ2 sin θtrax) GD (2.10)

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where,

AD = (β 1 cos θincd)n1H 2n1(β 1 cos θincd) (2.11a)

BD = (β 1 cos θincd)n1H 1n1(β 1 cos θincd) (2.11b)

C D = (β 2 cos θtrad)n2H 2n2(β 2 cos θtrad) (2.11c)

E D = (β 1 cos θincd)nh1H 2nh1(β 1 cos θincd) (2.11d)

F D = (β 1 cos θincd)nh1H 1nh1(β 1 cos θincd) (2.11e)

GD = (β 2 cos θtrad)nh2H 2nh2(β 2 cos θtrad) (2.11f)

where β 1 = ω√

µ1ϵ1 and β 2 = ω√

µ2ϵ2 are wave numbers. η1 = √ µ1/ϵ1 and η2 = √ µ2/ϵ2are the wave impedance of the two fractal media . Hankel functions of first and second kind

of order n are used to represent backward and forward traveling waves. [?]. The value of

ni = |3 − Di|/2 and nhi = |Di − 1|/2 for i = 1, 2, and D is dimension of media. T andΓ represents the unknown transmission and reflection coefficients. According to boundary

conditions, continuity of tangential components of electromagnetic fields across the interface

(z = d), i.e.,

E t

inc + E t

ref = E t

tra (2.12)H tinc + H

tref = H

ttra (2.13)

The required transmission and reflection coefficients are as follow,

R∥ = η2 cos θinc(ADF D + BDE D)

η2 cos θtraC DF D + η1 cos θincBDGD(2.14)

T∥ = −η1 cos θincADGD + η2 cos θtraC DE D

η2 cos θtraC DF D + η1 cos θincBDGD(2.15)

Brewster’s angel (θinc = θbrw) can be determined by taking Eq. (2.14) equal to zero,

R∥ = −η1 cos θbrwADGD + η2 cos θtraC DE D

η2 cos θtraC DF D + η1 cos θbrwBDGD= 0 (2.16)

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Einc = E iŷ exp(−iβ1 sin θincx) AD (2.18)

Eref = E iŷR⊥ exp(−iβ1 sin θref x) BD (2.19)

Etra = E iŷE iT⊥ exp(−iβ2 sin θtrax) C D (2.20)

Hinc = E i (−x̂ cos θinc + ẑ sin θinc) 1η1

exp(−iβ1 sin θincx) E D (2.21)

Href = E i (x̂ cos θref + ẑ sin θref ) R⊥

η1exp(−iβ1 sin θref x) F D (2.22)

Htra = E i (−x̂ cos θtra + ẑ sin θtra) T⊥η2

exp(−iβ2 sin θtrax) GD (2.23)

By following same procedure as for parallel polarization, transmission and reflection coeffi-cients are given by [?],

R⊥ = −η1 cos θtraADGD + η2 cos θincC DE D

η2 cos θincC DF D + η1 cos θtraBDGD(2.24)

T⊥ = η2 cos θinc(ADF D + BDE D)

η2 cos θincC DF D + η1 cos θtraBDGD(2.25)

2.2 Summary

Transmission and Reflection coefficients at dielectric fractal interface were discussed in this

chapter. The effect of non-integer dimension on the magnitude of transmission and reflection

coefficients was investigated. The purpose of discussion was to give the basic concept of wave

propagation for fractional space, which would be helpful to understand work discuss in later

chapters.

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Chapter 3

Electromagnetic Wave Behavior Of

Chiral-Chiral interface in FractionalSpace

3.1 Introduction

In this chapter, the expressions for transmission and reflection coefficients are derived for

fractal interface of chiral-chiral and chiral nihility-chiral nihility are discussed. The general

solution of plane waves for non-integer dimension is used to formulate the expressions for

electric and magnetic fields. Classical results are formulated from fractional results, when

integer dimension space is considered. Section 3.2 covers the coefficients of reflection and

transmission at chiral-fractal chiral interface. Reflection and transmission coefficients at

chiral nihility-fractal chiral nihility interface are discussed in section 3.3 and summary of

the chapter is concluded in section 3.4. (ϵ1, µ1, κ1, D1) , where 0 < D1 < 1. While the

half space z > d, is filled with chiral media having constitutive parameters (ϵ2, µ2, κ2) andD2 = 2. Figure 3.1 shows the geometry of chiral-fractal chiral interface,

For simplicity, it is assumed that the fractionality of media 1 exists only in ẑ direction and

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Figure 3.1: Reflection and transmission in fractal chiral-chiral media: Incident RCP (solid lines),reflected LCP (dotted dashed line), reflected RCP (double doted dashed line), refracted LCP(dashed line), refracted RCP (long dashed line).

time dependency exp(iωt) is omitted throughout the paper [?]. If right circularly polarized(RCP) plane wave is incident at fractal chiral-chiral interface, the field expression for left

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fractal half space in term of eigen vectors are expressed as,

Einc = x̂− ik−z1k−1

ŷ + ikyk−1

ẑ exp(−ikyy)(k−z1z )neH 2ne(k−z1z ) (3.1)Eref = AD

x̂ + i

k−z1k−1

ŷ + ikyk−1

ẑ

exp(−ikyy)(k−z1z )neH 1ne(k−z1z )

+BD

x̂ − i k

+z1

k+1ŷ − i ky

k+1ẑ

exp(−ikyy)(k+z1z )neH 1ne(k+z1z ) (3.2)

Hinc = − iη1

x̂ − ik

−z1

k−1ŷ + i

kyk−1

ẑ

exp(−ikyy)(k−z1z )nhH 2nh(k−z1z ) (3.3)

Href = − iη1

[AD

x̂ + i

k−z1k−1

ŷ + ikyk−1

ẑ

exp(−ikyy)(k−z1z )nhH 1nh(k−z1z )

−BDx̂ − ik+z1

k+1 ŷ − ikyk+1 ẑ

exp(−ikyy)(k

+z1z )

nh

H 1nh(k

+z1z )] (3.4)

Hankel function of first and second kind of order ne and nh are used for propagation of waves

in ẑ direction. Where ne = |3 − D1|/2, nh = |D1 − 1|/2 and D1 is the dimension of media1 [?]. ηi =

√ µi/ϵi, for i = 1, 2 is the impedance of both media. The fields expressions for

integer dimension right half space can be expressed as,

Etra = C D

x̂ − ik

−z2

k−2ŷ + i

kyk−2

ẑ

exp−i(kyy + k−z2z )

+DD

x̂ + i k+z2

k+2ŷ − i ky

k+2ẑ

exp−i(kyy + k+z2z ) (3.5)

Htra = − iη2

[C D

x̂ − ik

−z2

k−2ŷ + i

kyk−2

ẑ

exp−i(kyy + k−z2z )

−DD

x̂ + ik+z2k+2

ŷ − i kyk+2

ẑ

exp−i(kyy + k+z2z )] (3.6)

where,

k

±

(1,2) = ω(

µ

±

(1,2)ϵ

±

(1,2) ± κ(1,2)) (3.7)

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κ(1,2), is chirality of media, where subscripts 1, 2 represents different media. AD, BD, C D

and DD are unknown reflection and transmission coefficients to be determine. By imposing

boundary conditions i.e.,

E tinc(z = d) + E tref (z = d) = E

ttra(z = d) (3.8)

H tinc(z = d) + H tref (z = d) = H

ttra(z = d) (3.9)

the unknown coefficients AD, BD, C D and DD are as follows:

AD = k−1

X D(a1a3η

21 + b1b3η

22)(k

−z1k

+1 − k−1 k+z1)(k−z2k+2 + k−2 k+z2)

+η1η2(a3b1 + a1b3)(−k+2 (k−1 k−z2k+z1 + k−z1(k−z2k+1 − 2k−2 k+z1))+(−2k−1 k−z2k+1 + k−2 (k−z1k+1 + k−1 k+z1))k+z2) (3.10)

BD = −2k+1

X D(k−1 k

−z1(a1a2η

21 − b1b2η22)(k−z2k+2 + k−2 k+z2)

−η1η2(a2b1 − a1b2)(k−2 k−z12k+2 + k−1 2k−z2k+z2)) (3.11)C D =

1

X Dη2k

−2 [(k

−z1

2k+2 k

+1 + k

−12k+z2k

+z1)(a2b1 − a1b2)(a3η1 − b3η2)

+(k−z1k

+2 k

−1 k

+z1 + k

−1 k

+z2k

−z1k

+1 )(η1(a2a3b1 + a1a3b2 + 2a1a2b3)

+η2(2a3b1b2 + a2b1b3 + a1b2b3))]

DD = 1

X Dη2k

+2 [(k

−z1

2k−2 k

+1 − k−1 2k−z2k+z1)(a2b1 − a1b2)(a3η1 + b3η2)

+(k−z1k−2 k

−1 k

+z1 − k−1 k−z2k−z1k+1 )(η1(a2a3b1 + a1a3b2 + 2a1a2b3)

−η2(2a3b1b2 + a2b1b3 + a1b2b3))] (3.12)X D = k

−1 ((a2a3η

21 + b2b3η

22)(k

−z1k

+1 + k

−1 k

+z1)(k

−z2k

+2 + k

−2 k

+z2)

+η1η2(a3b2 + a2b3)(k−z1(k

+2 (k

−z2k

+1 + 2k

−2 k

+z1) + k

−2 k

+1 k

+z2)

+k−1 (k−z2k

+2 k

+1 + (2k

−z2k

+1 − k−2 k+z1)k+z2))) (3.13)

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where,

a1 = (k−z1 ∗ d)neH 2ne(k−z1 ∗ d) (3.14a)

a2 = (k−z1 ∗ d)neH 1ne(k−z1 ∗ d) (3.14b)

a3 = (k+z1 ∗ d)neH 2ne(k+z1 ∗ d) (3.14c)

b1 = (k−z1 ∗ d)nhH 2nh(k−z1 ∗ d) (3.14d)

b2 = (k−z1 ∗ d)nhH 1nh(k−z1 ∗ d) (3.14e)

b3 = (k+z1 ∗ d)nhH 2nh(k+z1 ∗ d) (3.14f)

In the next section,integer dimension space results are recovered using proposed fractional

space expressions for D1 = 2.

3.2 Simulation Results

The classical results can be find out by inserting integer values of dimension i.e., D1 = 2. By

setting D1 = 2, the order of Hankel function becomes, ne = nh = 1/2. For large arguments,

the expression of Hankel function of first kind is given as,

H 11/2(z ) ≃

2

πz e j(z) (3.15)

and the expression of the Hankel function of second kind is,

H 21/2(z ) ≃

2

πz e− j(z) (3.16)

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Figure 3.2: Co component of reflected power verses angle of incidence for non-integer dimen-sion,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, κ2 = 0.25

by inserting (3.15) and (3.16) in (3.10-3.13),

AD=2 = e−2ik

−

z1d

X D=2[k+2 (η1 − η2)2(k−1 k−2 k+1 ) − (−4η1η2(k−z2k−z1 +

(η1 + η2)2k−1 k

−z2)k

+z1) + ((η1 − η2)2k−2 k−z1 − 4η1η2k−1 k−z2)

+2η1η2(2(k+z1k

−z1k

−2 k

+2 + k

−z2k

+z2k

+1 k

−1 ) + k

+z1k

−z2k

−1 k

+2

k+1 − (k−z2(η1 − η2)2k−1 k−2 k+z1)k+z2)] (3.17)

BD=2 = −2k+

1 k

−

z1e

−i(k−z1+k+z1)d

(η

2

1 − η2

2)(k

+

2 k

−

z2 + k

−

2 k

+

z2)X D=2(3.18)

C D=2 = 4η2k

−2 k

−z2e

−i(k−z1−k−

z2)d(η1 + η2)(k+2 k

+z1 + k

+1 k

+z2)

X D=2(3.19)

DD=2 = −4η2k−z1k

+2 e

−i(k−z1−k+z2)d(η1 − η2)(k−z2k+1 − k−2 k+z1)

X D=2(3.20)

X D=2 = k+2 ((η1 − η2)2k−z1k−z2k+1 + (4η1η2k−2 k−z1

+(η1 + η2)2k−1 k

−z2)k

+z1) + (((η1 + η2)

2k−2 k−z1

+4η1η2k−1 k

−z2)k

+1 + (η1

−η2)

2k−1 k−2 k

+z1)k

+z2. (3.21)

These were also discussed by Taj et al. [?], for LCP incident in integer-dimensional space

of chiral-chiral interface at z = 0. For θinc, relations are taken from [?]. Both co and cross

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Figure 3.3: Cross component of reflected power verses angle of incidence for non-integer dimen-sion,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, κ2 = 0.25

component of reflected power for non-integer dimension is studied. For each component,

two cases are disused first for impedance mismatching (η1 ̸= η2) and second for impedancematching (η1 = η2). Different colors are used for plots, it can be seen in all the plots that

for integer values of dimension i.e., D1 = 2, classical results are recovered which are the

same as those obtained by Faiz et al. for chiral-chiral interface for integer dimension [ ?].

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Figure 3.4: Co component of reflected power verses angle of incidence for non-integer dimen-sion,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, κ2 = 0.75

Angular dependence of reflected power is shown in Figure 3.2 through Figure 3.5. Figure

3.2, shows the co component of reflected power for (η1 ̸= η2) in fractional space. For bothmatching and mismatching power rejection (|RCO |+ |RCR | = 1) occurs at θinc = 90◦. Wherethe values of constitutive relations are taken arbitrary as ϵ1 = 4, ϵ2 = 1, for impedance

mismatching and ϵ1 = 1, ϵ2 = 1 for impedance matching. The effect of dimension can be seen

on the co component of reflected power. There is no effect of dimension on the magnitude of

reflected power for θinc = 0◦

−19◦ but from θinc = 19

◦

−90◦ magnitude of reflected power is

decreases as dimension of the media decreases and finally it approaches to 1 at θinc = 90◦ as

shown in Figure 3.2. Results are taken for different values of dimension. Cross component

of reflected power are shown in Figure 3.3, the magnitude of cross component of reflected

power increases as the dimension of the media decrease for all values of θinc and finally all

plots approaches to 0 at θinc = 90◦. Angular dependence of reflected power for (η1 = η2)

are shown in Figure 3.4 and Figure 3.5. In the case of impedance mating for non integer

dimension power rejection occurs after θinc = 19◦ as in the case of integer dimension. Co

component of reflected power for (η1 = η2) are shown in Figure 3.4. Figure 3.5 shows thecross component of reflected power for (η1 = η2) in fractional space.

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Figure 3.5: Cross component of reflected power verses angle of incidence for non-integer dimen-sion,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, κ2 = 0.75

The variation in magnitude of co and cross components for (η1 ̸= η2) and (η1 = η2)in fractional space verses chirality of media 1, are shown in Figure 3.6 through Figure

3.9. Rejection and tunneling of power with different dimensions for wide range of chirality

parameter has been observed for fixed value of θinc = 45◦. Power rejection is observed for

whole range of chirality except from κ1 = 1.2−2.7, where power tunneling is observed shownin Figure 3.6. Dimension of the media has strong effect on tunneling, with the decrease in

dimension tunneling effect also decreases. Where as the magnitude of cross component of

reflected power increase with decrease in dimension for all range of chirality of media 1 as

shown in Figure 3.7. Case of impedance matching for chirality of media 1 variation with non

integer space is shown in Figure 3.8 and Figure 3.9. Minor reflection of power is observed

from κ1 = 0.75 − 1.25. For all range of chirality of media 1, magnitude of co componentof reflected power decreases as the dimension decreases except for κ1 = 0.75 − 1.25, butpower rejection holds for all values of dimension shown in Figure 3.8. Figure 3.9, shows the

variation of magnitude of cross component of reflected power, as the dimension decreases

magnitude of cross component of reflected power increases.

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Figure 3.6: Co component of reflected power verses chirality of the media 1 for non-integer dimen-sion,when µ1 = µ2 = 1, for impedance mismatch κ2 = 0.25, θinc = 45

◦

Figure 3.10 through Figure 3.13 shows change in magnitude of co and cross components

of reflected power verses chirality parameter of media 2 with non integer dimension space,

where κ1 and θ◦inc are kept constant. Case of impedance mismatching for κ2 variation with

non integral space is shown Figure 3.10 through Figure 3.11. Power rejection is observed

form κ2 = 0.5 − 1.5. From κ2 = 0 − 2.25, magnitude of co component of reflected powerdecreases as the dimension decreases shown in Figure 3.10. Where as the magnitude of

cross component of reflected power increase with decrease in dimension for all range of

chirality of media 2 as shown in Figure 3.11. Total rejection of power for small values κ2 is

observed, from κ2 = 0.4 − 1.6 magnitude of co component of reflected power decreases asthe dimension decreases shown in Figure 3.12. Figure 3.13 shows the change in magnitude

of cross component of reflected power verses κ2, as the dimension decreases magnitude of

cross component of reflected power increases for all range of chirality of media 2.

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Figure 3.7: Cross component of reflected power verses chirality of the media 1 for non-integerdimension,when µ1 = µ2 = 1, for impedance mismatch κ2 = 0.25, θinc = 45

◦

Figure 3.8: Co component of reflected power verses chirality of the media 1 for non-integer dimen-sion,when µ1 = µ2 = 1, for impedance matching κ2 = 0.25, θinc = 45

◦

height=2.1in,width=4.8in]3.eps

Figure 3.9: Cross component of reflected power verses chirality of the media I for non-integerdimension,when µ1 = µ2 = 1, for impedance matching κ2 = 0.25, θinc = 45

◦

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Figure 3.10: Co component of reflected power verses chirality of the media 2 for non-integerdimension,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, θinc = 45

◦

Figure 3.11: Cross component of reflected power verses chirality of the media 2 for non-integer

dimension,when µ1 = µ2 = 1, for impedance mismatch κ1 = 0.25, θinc = 45

◦

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Figure 3.12: Co component of reflected power verses chirality of the media 2 for non-integerdimension,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, θinc = 45

◦

Figure 3.13: Cross component of reflected power verses chirality of the media 2 for non-integer

dimension,when µ1 = µ2 = 1, for impedance matching κ1 = 0.25, θinc = 45

◦

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Chapter 4

Chiral nihility-Chiral nihility

interface in fractional space

Chiral nihility is assumed as special case of chiral media with both permittivity (ϵ = 0) and

permeability (µ = 0) are zero. Under the assumption of chiral nihility, relations for waves

number simplifies to,

k±(1,2) = ±ωκ (4.1)

k+z(1,2) = −k−z(1,2) (4.2)k+(1,2) = −k−(1,2) (4.3)

An interesting property of chiral nihility is that it produces a backward wave as one of their

two polarizations as shown in Figure 5.2. Backward wave is a reflected wave that lies on

the same side of the normal to the boundary as the incident wave.

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Figure 4.1: Reflection and transmission in chiral nihility- fractal chiral nihility medium ( Wavevector representation ): Reflected backward wave (double dotted dashed line), refracted backwardwave ( dotted line).

4.1 Chiral Nihility-Chiral Interface In fractional space

4.1.1 Impedance matching

The variation in magnitude of co and cross components for (η1 = η2) in fractional space

verses chirality of media 1, are shown in Figure 4.2 through Figure 4.10. Rejection and

tunneling of power with different dimensions for wide range of chirality parameter has been

observed for fixed value of θinc = 45◦. Dimension of the media has strong effect on tunneling.

The magnitude of co-component of reflected power verses chirality of media 1 as shown in

Figure 4.2. Where as the magnitude of cross-component of reflected power verses chirality

of media 1 as shown in Figure 4.3. Change in magnitude of co and cross component of

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verses chirality of media 2 are given in Figure 4.4 and Figure 4.5

Figure 4.2: Co-component of reflected power verses kappa 1, for impendence matching.

Figure 4.3: Cr-component of reflected power verses kappa 1, for impendence matching.

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4.1.2 Impedance mis-matching

The variation in magnitude of co and cross components for (η1 ̸= η2) in fractional spaceverses chirality of media 1, are shown in Figure 4.6 through Figure 4.11. Rejection and

tunneling of power with different dimensions for wide range of chirality parameter has beenobserved for fixed value of θinc = 45

◦. The magnitude of co-component of reflected power

verses chirality of media 1 as shown in Figure 4.6. Where as the magnitude of cross-

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component of reflected power verses chirality of media 1 as shown in Figure 4.7. Change in

magnitude of co and cross component of verses chirality of media 2 are given in Figure 4.8

and Figure 4.9. Figure 4.10 and Figure 4.11 shows the change in magnitude of co and cross

components of reflected power verses incedent angle θinc

Figure 4.6: Co-component of reflected power verses kappa 1, for impendence mis-matching.

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Figure 4.7: Cr-component of reflected power verses kappa 1, for impendence mis-matching.


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Figure 4.10: Co-component of reflected power verses theta, for impendence mis-matching.

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Figure 4.11: Cr-component of reflected power verses theta, for impendence mis-matching.

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4.2 Chiral-Chiral Nihility Interface In fractional space

4.2.1 Impedance matching








verses chirality of media 2 are given in Figure 4.14 and Figure 4.15


33


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4.2.2 Impedance mis-matching

The variation in magnitude of co and cross components for (η1 ̸= η2) in fractional spaceverses chirality of media 1, are shown in Figure 4.16 through Figure 4.21. Rejection and


observed for fixed value of θinc = 45◦. The magnitude of co-component of reflected power

verses chirality of media 1 as shown in Figure 4.16. Where as the magnitude of cross-

component of reflected power verses chirality of media 1 as shown in Figure 4.17. Change

in magnitude of co and cross component of verses chirality of media 2 are given in Figure

4.18 and Figure 4.19. Figure 4.20 and Figure 4.21 shows the change in magnitude of co and

cross components of reflected power verses incedent angle θinc

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Figure 4.20: Co-component of reflected power verses theta, for impendence mis-matching.

Figure 4.21: Cr-component of reflected power verses theta, for impendence mis-matching.

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4.3 Chiral Nihility-Chiral Nihility Interface In fractional space








verses chirality of media 2 are given in Figure 4.24 and Figure 4.25.Change in magnitude of

co and cross component of verses θinc are given in Figure 4.27 and Figure 4.28.


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Figure 4.26: Co-component of reflected power verses theta, for impendence matching.

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Figure 4.27: Cr-component of reflected power verses theta, for impendence matching.

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4.4 Summary

The coefficients of reflection and transmission for the chiral Nihility interface in fractional

space were studied, when the fractal interface is excited by circularly polarized plane wave. A

new parameter D, is introduced and effect of dimension on tunneling and rejection of power

were also studied. Major focus of studies was to found the effect of dimension on tunneling

and rejection of power. It is found that dimension of media has strong effect on tunneling

and rejection of power. The classical results are recovered when integer-dimensional space

is considered instead of non-integer dimension.

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Chapter 5

Conclusion and Future Work

In this dissertation, wave propagation in Chiral and/or Chiral Nihility is being studied for

non-integer dimension space. Some new expressions are found analytically. In first problem,

The expressions for transmission and reflection coefficients are derived for fractal interface

of chiral-chiral. The general solution of plane waves for non-integer dimension is used to

formulate the expressions for electric and magnetic fields. Behavior of electromagnetic waves

in Chiral-Chiral interface in non-integer Dimension Space is studied. The coefficients of

reflection and transmission for the chiral-chiral interface in fractional space, when the fractal

interface is excited by circularly polarized plane wave, are presented. A new parameter D,

is introduced and effect of dimension on tunneling and rejection of power is also studied. It

is found that dimension of media has strong effect on tunneling and rejection of power.

In second problem, this analysis is further extended to chiral-nihility and reflection and

transmission of electromagnetic waves in Chiral-Nihility interface for non-integer dimension

Space is studied and effect of dimension on tunneling and rejection of power is studied. It is

found that dimension of media has effect on tunneling and rejection of power. The classical

results are recovered when integer-dimension space is considered i.e., D = 2. Moreover,

these solutions can also be used for other fractal cubes.

arooj hameed

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