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CHARACTERIZATION OF SPHERICAL METAMATERIALS
BY
HUANG NINGYUN
B.ENG. COMMUNICATION ENGINEERING
XIAN JIAOTONG UNIVERSITY, 2000
A THESIS SUBMITTED
FOR THE DEGREE OF MASTER OF ENGINEERING
DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING
NATINAL UNIVERSITY OF SINGAPORE
2004
@ National University of Singapore, All Right Reserved 2004
i
ACKNOWLEDGEMENT
I indeed feel great pleasure to express my sincere gratitude to my supervisor Professor Li
Le-Wei, for his valuable guidance, advice, strong support and many useful opinions and
suggestions on many parts of the research work in the process of my postgraduate
program. I would like to particularly acknowledge and thank Prof. Li who has given me
continual help and great encouragement, which have always inspired me to work harder
and have given me confidence.
I am also very grateful to Professor Leong Mook Seng and Professor Yeo Tat Soon for
their kind encouragement and helpful suggestions.
My gratitude is also extended to my fellow laboratory members and many friends in
Microwave Division, for their kind help and advice when I encountered some difficulties
in the project.
Last but not least, I take this opportunity to express my deep thanks to my beloved
parents and younger sister for their encouragement and support.
Huang Ningyun
15 March 2004
Table of Contents
1
TABLE OF CONTENTS
ABSTRACT ……………………………………………………………………………..I
ACKNOWLEDGEMENT ……………………………………..………………………II
1
2
INTRODUCTION....................................................................................................... 1
1.1
PROBLEM DESCRIPTION ....................................................................................... 1
1.2
MOTIVATION AND OBJECTIVES ............................................................................ 2
1.3
OUTLINE OF THE THESIS ...................................................................................... 2
1.4
ORIGINAL CONTRIBUTION.................................................................................... 3
BACKGROUND INTRODUCTION OF METAMATERIAL............................. 5
2.1
NEGATIVE REFRACTIVE INDEX ............................................................................ 5
2.2
NEGATIVE REFRACTION MAKES A PERFECT LENS ............................................... 8
2.3
LATERAL DISPLACEMENT
OF
A
GAUSSIAN BEAM REFLECTED FROM
A
GROUNDED METAMATERIAL ........................................................................................... 9
3
2.3.1
Numerical Results ....................................................................................... 9
2.3.2
Simulation Results..................................................................................... 14
SPHERICAL DYADIC GREEN’S FUNCTIONS............................................... 17
3.1
INTRODUCTION .................................................................................................. 17
3.2
FUNDAMENTAL FORMULATION .......................................................................... 19
3.2.1
Eigenfunction Expansion of DGF in Unbounded Media.......................... 19
Table of Contents
2
3.2.2
DGFs for Spherical Multilayered Media.................................................. 23
3.2.3
The Recurrence Matrix Equations for the Coefficients of Scattering DGFs
24
3.3
4
SCATTERING OF SPHERE ................................................................................ 30
4.1
Rayleigh Scattering................................................................................... 30
4.1.2
Mie Scattering........................................................................................... 32
SCATTERING BY TWO SPHERES .......................................................................... 36
METAMATERIAL SPHERE................................................................................ 44
5.1
INTRODUCTION .................................................................................................. 44
5.2
CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERE ................................. 44
5.2.1
Introduction............................................................................................... 44
5.2.2
Using Two Methods to Obtain the Coefficients ........................................ 45
5.2.3
Coefficients in the Special Case of Metamaterial..................................... 48
5.2.4
Calculation of the Electrical Field ........................................................... 51
5.3
6
INTRODUCTION .................................................................................................. 30
4.1.1
4.2
5
CONCLUSIONS .................................................................................................... 29
CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERE .................................... 54
5.3.1
Introduction............................................................................................... 54
5.3.2
The Calculation of the Coefficients........................................................... 55
5.3.3
Coefficients in the Special Case of Metamaterial..................................... 57
5.3.4
Calculation of the Electrical Field ........................................................... 61
METAMATERIAL SPHERICAL SHELL.......................................................... 65
Table of Contents
6.1
INTRODUCTION .................................................................................................. 65
6.2
CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL SHELL ................. 65
6.2.1
Introduction............................................................................................... 65
6.2.2
The Calculation of the Coefficients........................................................... 66
6.2.3
Coefficients in the Special Case of Metamaterial..................................... 70
6.2.4
Calculation of the Electrical Field ........................................................... 73
6.3
7
3
CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERICAL SHELL .................... 76
6.3.1
Introduction............................................................................................... 76
6.3.2
The Calculation of the Coefficients........................................................... 77
6.3.3
Coefficients in the Special Case of Metamaterial..................................... 80
6.3.4
Calculation of the Electrical Field ........................................................... 83
MULTI-SPHERICAL LAYERS OF METAMATERIALS ............................... 87
7.1
INTRODUCTION .................................................................................................. 87
7.2
CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL MULTILAYERS .... 87
7.2.1
Introduction............................................................................................... 87
7.2.2
The Calculation of the Coefficients........................................................... 88
7.2.3
Coefficients in the Special Case of Metamaterial..................................... 93
7.2.4
Calculation of the Electrical Field ........................................................... 95
7.3
CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERICAL MULTILAYERS ....... 99
7.3.1
Introduction............................................................................................... 99
7.3.2
The Calculation of the Coefficients......................................................... 100
7.3.3
Coefficients in the Special Case of Metamaterial................................... 104
4
Table of Contents
7.3.4
8
Calculation of the Electrical Field ......................................................... 106
SIMULATION RESULTS ................................................................................... 110
8.1
SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERE ............. 110
8.2
SIMULATION RESULTS
AND
DISCUSSION
OF
METAMATERIAL SPHERICAL SHELL
114
8.3
SIMULATION RESULTS
AND
DISCUSSION
OF
METAMATERIAL SPHERICAL
MULTILAYERS ............................................................................................................. 118
9
SUMMARY ........................................................................................................... 125
REFERENCES.............................................................................................................. 127
Abstract
i
Abstract
Materials possessing both negative permittivity and permeability simultaneously, i.e.,
left-handed materials or metamaterials were firstly introduced in 1968 by V.G.Veselago.
This special kind of materials demonstrates very different characteristics from the
conventional materials. In this thesis, the electromagnetic fields produced by a dipole in
the presence of metamaterial spheres are analyzed.
Firstly, some properties of metamaterial, including negative refractive index and lateral
displacement of a Gaussian beam reflected from a grounded metamaterial, are introduced
and discussed. Secondly, spherical dyadic Green’s functions in such a material are
analyzed, and the eigenfunction expansion of DGF and the recurrence matrix equations
for the coefficients of scattering DGF are emphasized. Thirdly, the scattering of normal
material sphere is introduced. Scattering by two separate spheres is analyzed in particular.
Finally, the spherical metamaterial objects are analyzed. Three cases are considered
respectively: the single sphere, the spherical shell and the multi-layer sphere. Finally,
simulation results are given and discussed.
List of Figure
1
List of Figures
Fig 2.1: A medium with negative refractive index bend light to a negative angle with the
surface normal. Light formerly diverges from a point source and then set in reverse
and converges back into a point. The light reaches a focus for a second time after it
released from the medium........................................................................................... 8
Fig 2.2: Configuration of a Gaussian beam incident upon a slab with thickness
d = d 2 − d1
.................................................................................................................. .10
Fig 2.3: Time-averaged power density on the xz plane for a 30° incidence of a Gaussian
beam upon a grounded slab of thickness d = 6λ with ε1 = ε 0 and µ = µ0 . This is
the simulation result by using the formula developed in this thesis ......................... 15
Fig 2.4: Reference time-averaged power density on the xz plane for a 30° incidence of a
Gaussian beam with ε1 = ε 0 and µ = µ0 . This is the simulated result by using the
formula developed in reference [6]........................................................................... 15
Fig 2.5: Time-averaged power density on the xz plane for a 30° incidence of a Gaussian
beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 . This is
the simulation result by using the formula developed in this thesis ......................... 16
Fig 2.6: Time-averaged power density on the xz plane for a 30° incidence of a Gaussian
beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 . This is
the simulated result by using the formula developed in reference [6] ...................... 16
List of Figure
2
Fig 3.1: Geometry of a spherically multilayered medium ................................................ 18
Fig 8.1: Structure of the metamaterial sphere. ................................................................ 110
Fig 8.2: Near field of metamaterial sphere when k2 = k1 = k0 , a = 1λ , b = 2λ ............ 112
Fig 8.3: Near field of metamaterial sphere when k2 = 2k1 , a = 1λ , b = 2λ . ................. 112
Fig 8.4: Near field of metamaterial sphere when k2 = 4k1 , a = 1λ , b = 2λ .................. 113
Fig 8.5: Near field of metamaterial sphere when k2 = − k1 , a = 1λ , b = 2λ ............. …113
Fig 8.6: Near field of metamaterial sphere when k2 = −2k1 , a = 1λ , b = 2λ . ............... 114
Fig 8.7: Structure of the metamaterial spherical shell. ................................................... 115
Fig 8.8: Near field of metamaterial spherical shell when k2 = k1 , a1 = 4λ , a2 = 2λ ... 116
Fig 8.9: Near field of metamaterial spherical shell when k2 = 2k1 . a1 = 4λ , a2 = 2λ .. 116
Fig 8.10: Near field of metamaterial spherical shell when k2 = − k1 , a1 = 4λ , a2 = 2λ .117
Fig 8.11: Near field of metamaterial spherical shell when k2 = −2k1 , a1 = 4λ , a2 = 2λ 117
Fig 8.12: Structure of the spherical mutilayers ( m = 5 ). ................................................ 118
Fig 8.13: Near field of 5-layer metamaterial spheres when k2 = k1 , a1 = 4λ , a2 = 3.5λ ,
a3 = 2.5λ , a4 = 2λ , b = 5λ . ................................................................................... 119
List of Figure
3
Fig 8.14: Near field of 5-layer metamaterial spheres when k2 = 2k1 , a1 = 4λ , a2 = 3.5λ ,
a3 = 2.5λ , a4 = 2λ , b = 5λ ................................................................................... 119
Fig 8.15: Near field of 5-layer metamaterial spheres when k2 = −k1 , a1 = 4λ , a2 = 3.5λ ,
a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 10 . ....................................................................... 120
Fig 8.16: Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ ,
a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 ....................................................................... 120
Fig 8.17: Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 4λ , a2 = 3.5λ ,
a3 = 2.5λ , a4 = 2λ , b = 5λ ., n = 30 ...................................................................... 121
Fig 8.18: Near field of 5-layer metamaterial spheres when k2 = − k1 , a1 = 5λ , a2 = 4λ ,
a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 ....................................................................... 121
Fig 8.19: Near field of
5-layer metamaterial spheres when k2 = −2k1 ,
a1 = 4λ ,
a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 ..................................................... 122
Fig 8.20: Near field of
5-layer metamaterial spheres when k2 = −2k1 ,
a1 = 4λ ,
a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 30 ..................................................... 122
Fig 8.21: Near field of 5-layer metamaterial spheres when k2 = −2k1 , a1 = 5λ , a2 = 4λ ,
a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 ........................................................................ 123
List of Symbols
List of Symbols
E:
Electric field
H:
Magnetic field
k:
Wave number
k0 :
Wave number in free space
λ:
Wave length]
λ0 :
Wave length in free space
f
:
Frequency
ω:
Angular frequency
ε:
Permittivity
nc :
Refractive index
µ:
Permeability
εr :
Relative permittivity
µr :
Relative permeability
ε0 :
Permittivity in free space
µ0 :
Permeability in free space
Chapter1
1
Introduction
1 INTRODUCTION
1.1 PROBLEM DESCRIPTION
In 1968, metamaterial, a kind of artificial materials with the electric permittivity and
magnetic permeability simultaneously negative was firstly introduced by V.G. Veselago,
who predicted that if such a material could be found, it would exhibit very unusual
electromagnetic scattering phenomena. In recent years, the electromagnetic properties of
this kind of special materials have attracted much attention. Such metamaterial has
optical properties that would be impossible to be found in a conventional material and
could make novel antennas or perfect optical lenses theoretically. Metamaterials can be
engineered to have specific electromagnetic behaviors that are physically impossible for
natural materials. Now, in this thesis, we focus on the electromagnetic properties of a
spherically multilayered medium, which is metamaterial. The spectral-domain
electromagnetic Dyadic Green’s Function is constructed for defining the electromagnetic
fields in the multilayered media. The scattering dyadic Green’s function in each layer is
constructed and the coefficients of the function are derived. The general solution can be
applied to specific geometries, e.g., two-, three- and four-layered media which are
frequently employed in practical cases. Boundary conditions are also used to solve this
problem. Results and discussions are given in succession.
1
Characterization of Spherical Metamaterials
Chapter1
Introduction
2
1.2 MOTIVATION AND OBJECTIVES
The research aims to obtain electromagnetic radiation characteristics of a dipole in each
layer of the spherically multilayered media. The metamaterial exhibits different
properties from other conventional materials, such as opposite electromagnetic wave
phase velocity. Accordingly, the media made of metamaterial will show a different
electromagnetic radiation pattern. This is what we are interested in. Metamaterial’s
special properties are also very beneficial in potential practical uses. The simulation
results of models with different layers are given and discussed.
1.3 OUTLINE OF THE THESIS
The thesis is divided into 8 chapters, as follows:
Chapter 2 gives the basic information about this thesis by providing an overview of
background knowledge for the metamaterials.
The Chapter 3 gives thorough description about spherical dyadic Green’s functions,
which are used in this project. Some necessary implementations of the theories used in
the thesis are also provided here.
In Chapter 4, detailed description regarding the scattering by spheres is provided. The
scattering of two spheres is emphasized.
2
Characterization of Spherical Metamaterials
Chapter1
Introduction
3
Chapter 5 analyzes the fields associated with a single sphere, using the spherical dyadic
Green’s function. Two circumstances are considered: the source locates outside the
sphere and inside the sphere, respectively. The specific case, i.e., the metamaterial sphere
case, is also studied.
Chapter 6 analyzes the fields associated with a spherical shell. Similar to those in Chapter
5, two cases are considered where the source locates outside the sphere shell and inside
the sphere shell, respectively, and also examined is the specific case of metamaterial
sphere.
In Chapter 7, the general case is considered where spherical multilayer is involved, with
the source set to be outside the layers and inside the layers, respectively. The
metamaterial case is considered.
In Chapter 8, the simulation results of the single sphere and spherical multi layers are
given.
Chapter 9 gives a summary of the whole thesis.
1.4 ORIGINAL CONTRIBUTION
Conference paper:
3
Characterization of Spherical Metamaterials
Chapter1
Introduction
4
Le-Wei Li, Ningyun Huang, Qun Wu and Zhong-Cheng Li, "Macroscopic Characteristics
of Electromagnetic Waves Radiated by a Dipole in the Presence of Metamaterial Sphere
({Invited})", Proc. of 5th Asia-Pacific Engineering Research Forum on Microwaves and
Electromagnetic Theory, Kyushu University, Fukuoka, Japan, July 29-30, 2004
4
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
5
2 BACKGROUND INTRODUCTION OF METAMATERIAL
After metamaterial was first introduced in 1968 by Veselago[3], many scientists have
tried to realize it in the real world. A composite medium, based on a periodic array of
interspaced conducting, nonmagnetic copper split ring resonators and continuous wires,
which exhibits a frequency region in the microwave regime with simultaneously negative
values of effective permeability µ and permittivity ε , forms a metamaterial. It has been
predicted that such phenomena as the Doppler Effect, the Cerenkov radiation, and the
Snell’s law are inverted. The shape of the ring can be both circular and square. The rings
and wires are on opposite sides of the boards. The structure is improved for better
material characteristics in the following years.
In this chapter some basic properties of the metamaterial are presented. The negative
refractive index of metamaterials is introduced firstly; then the lateral displacement of
Gaussian beam reflected from grounded metamaterial is introduced afterwards.
2.1 NEGATIVE REFRACTIVE INDEX
The real part of the refractive index of a nearly transparent and passive medium is usually
taken to have only positive values. Through an analysis of a current source radiating into
a metamaterial, it can be determined that the sign of the real part of the refractive index is
actually negative. The regime of negative index leads to unusual electromagnetic wave
5
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
6
propagation and merits further exploration. The negative refractive index can be derived
from the analysis of the equation of the electric field and the equation of the work. The
general solution to the one-dimensional wave equation with an arbitrary current density,
in frequency domain, can be written as [7]:
E (ω ) = − z (ω )
j (ω )
,
c
(2.1)
where the generalized impedance can be defined [7] as:
z (ω ) =
µ (ω )
,
n(ω )
(2.2)
where n is refractive index. We require that the source on average do positive work on
the fields. Thus we define the quantity
P (ω ) =
j (ω )
1
z (ω )
2
c
2
,
(2.3)
which should be greater than zero [7]. In order to make sure the positive work, we get
z (ω ) > 0. In a metamaterial, since µ < 0 , we conclude that the solution with n < 0 leads
to the correct interpretation that the currents perform work on the fields.
The existence of negative refractive index is consistent with causality, which introduces
the constraints:
d [ε (ω )ω ]
dω
> 1 and
d [ µ (ω )ω ]
dω
> 1,
(2.4)
valid for nearly transparent media. Wave propagation and wave interaction with current
sources in metamaterial are therefore necessarily complicated by the implicit frequency
dependence of the material parameters, and even simple geometries can lead to
6
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
7
mathematical or numerical complexity. For nearly transparent media, Im [ z (ω )] can be
neglected since it is obtained from the reference [7]:
d (nω ) 1 d
ωµ 1 d (ωε ) 1 d (ωµ )
=
+
(ωε z +
) = z
.
2 dω
2 dω
dω
z
z dω
Applying Eq. (2.4) to Eq. (2.5), we have:
velocity of a wave, defined as vg =
(2.5)
d (nω ) z + z −1
>
> 1. Furthermore, the group
dω
2
c
, must therefore always be positive and less
d (nω )
dω
than c in either normal media or metamaterial. At last, we can find that at some
frequency range, the wave propagates with a negative index of refraction.
We can also get the same result from analyzing the composite medium which is made use
of an array of metal posts to create a frequency region with ε < 0 . The structure is
interspersed with an array of split ring resonators (SRRs) having a frequency region with
µ < 0, which is introduced in the beginning of this chapter. The thin wire medium can be
described by the dielectric function [7]:
ε (ω ) = 1 −
ω 2p
,
ω2
(2.6)
where the plasma frequency ω p is related to the geometry of the wire array.
7
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
8
We can assume that the wire and the SRR arrays do not interact directly, the index of the
refraction of this structure can be presented as: n(ω ) = ε (ω ) µ (ω ). Thus, it can be
obtained that µ and ε are simultaneously negative in the region between ω 0 and ω b , and
there exist propagating modes in this region.
2.2 NEGATIVE REFRACTION MAKES A PERFECT LENS
The sharpness of the image is determined by the wavelength of the light for a
conventional lens. A slab of material with negative refractive index, i.e., a metamaterial
slab, can focus all the Fourier components of a 2D image, even including those do not
propagate in a radiative manner. With the current technology, this kind of super-lenses
can be realized in the microwave band provided that the material has no transmission loss.
Figure 2.1 A medium with negative refractive index bend light to a negative angle with the surface
normal. Light formerly diverges from a point source and then set in reverse and converges back to a
point. The light reaches a focus for a second time after it released from the medium.
8
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
9
Assuming that n = −1, we can seen from the figure that it obeys Snell’s laws of refraction
at the surface as light inside the medium makes a negative angle with the surface normal.
Another characteristic we should notice of this structure is the double focusing effect
revealed. Evanescent waves emerge from the far side of the medium enhanced in
amplitude by the transmission process, so the medium can also cancel the decay of
evanescent waves. This does not violate energy conservation, because evanescent waves
do not transport energy. Several developments in technology make such a lens a practical
possibility, at least in some regions of the spectrum.
2.3 LATERAL DISPLACEMENT OF A GAUSSIAN BEAM REFLECTED FROM A
GROUNDED METAMATERIAL
A dramatic negative lateral shift can be observed when a Gaussian beam reflected from a
grounded metamaterial, which is distinctly different from a shift made by a conventional
grounded slab [6].
2.3.1 NUMERICAL RESULTS
The waves inside and outside the material can be solved analytically from Maxwell’s
equations and the boundary conditions at the interfaces, then, the field values in all
regions can be clearly determined.
A system as shown in Fig. 2.2 has the following incident wave in region “I”:
∞
Eiy = ∫ dk x ei ( kx x + k0 z z )ψ (k x ),
−∞
(2.7)
9
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
10
where
ψ (k x ) =
g
2 π
e− g
2
( k x − kix )2 / 4
.
(2.8)
Y
I ( ε 0 , µ0 )
0
II ( ε1 , µ1 )
d1
θi
III ( ε 2 , µ2 )
d2
X
Figure 2.2 Configuration of a Gaussian beam incident upon a slab with thickness d = d 2 − d1
ˆ ix + zk
ˆ 0 sin θ i + zk
ˆ iz = xk
ˆ 0 cos θ i , where θ i is
The Gaussian beam is centered about k i = xk
the incident angle.
In region “I”, we can obtain the expression of the E 0 y , which is the sum of the incident
wave and reflection wave [6]:
E
∞
0y
= ∫ dkxψ (kx )(eik0 z z + Re−ik0 z z )eikx x .
−∞
(2.9)
Because
10
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
(2.10)
∇ × E = jωµ H
Hx = −
11
1 ∂E y
,
jωµ ∂z
(2.11)
and
1 ∂E y
,
jωµ ∂x
Hz =
(2.12)
we can obtain:
∞
− k0z
−∞
ωµ 0
H 0 x = ∫ dk xψ (k x )
∞
kx
−∞
ωµ 0
H 0 z = ∫ dk xψ (k x )
(e ik0 z z − Re −ik0 z z )e ik x x ,
(2.13a)
(e ik0 z z + Re −ik0 z z )e ik x x .
(2.13b)
In region “II”, we have
∞
E1 y = ∫ dk xψ (k x )( Ae ik1 z z + Be −ik1 z z )e ik x x ,
−∞
∞
− k1 z
−∞
ωµ1
H 1x = ∫ dk xψ (k x )
( Ae ik1 z z − Be −ik1 z z )e ik x x ,
(2.14)
(2.15a)
and
∞
kx
−∞
ωµ1
H 1z = ∫ dk xψ (k x )
( Ae ik1 z z + Be −ik1 z z )e ik x x .
(2.15b)
In region “III”, we have
∞
E 2 y = ∫ dk xψ (k x )Te ik 2 z z +ik x x ,
−∞
(2.16)
thus we can obtain:
∞
− k2z
−∞
ωµ 2
H 2 x = ∫ dk xψ (k x )
Te ik 2 z z +ik x x ,
(2.17a)
11
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
∞
kx
−∞
ωµ 2
H 2 z = ∫ dk xψ (k x )
Te ik 2 z z + ik x x .
12
(2.17b)
The coefficients R , A , B and T can be obtained by matching the boundary conditions
for the tangential components of electric and magnetic fields at x = d1 and x = d 2 .
We apply the boundary conditions:
nˆ × E1 = nˆ × E2 ,
(2.18a)
nˆ × H1 = nˆ × H 2 ;
(2.18b)
Because nˆ = zˆ , then:
zˆ × H 0 z = zˆ × H1z = zˆ × H 2 z = 0 .
(2.19)
Thus we have [6]
when z = d1 ,
zˆ × E0 y = zˆ × E1 y ,
(2.20a)
zˆ × H 0 x = zˆ × H1x ;
(2.20b)
zˆ × E1 y = zˆ × E2 y ,
(2.21a)
zˆ × H1x = zˆ × H 2 x .
(2.21b)
when z = d 2 ,
From (2.20) and (2.21), four equations are derived [6]:
e ik0 z d1 + Re −ik0 z d1 = Ae ik1 z d1 + Be −ik1 z d1 ,
− k0z
ωµ 0
(e ik0 z d1 − Re −ik0 z d1 ) =
− k1 z
ωµ1
( Ae ik1 z d1 − Be −ik1 z d1 ) ,
(2.22a)
(2.22b)
12
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
Ae ik1 z d 2 + Be −ik1 z d 2 = Te ik 2 z d 2 ,
− k1 z
( Ae ik1 z d 2 − Be −ik1 z d 2 ) =
ωµ1
− k2z
ωµ 2
13
(2.22c)
Te ik 2 z d 2 .
(2.22d)
By solving (2.22a-d), we have:
R01 + R12 e i 2 k1 z ( d 2 − d1 ) i 2 k0 z d1
e
,
1 + R01 R12 e i 2 k1 z ( d 2 − d 1)
(2.23a)
A=
2e − i ( k1 z − k0 z ) d1
,
(1 + p 01 ) 1 + R01 R12 e i 2 k1 z ( d 2 − d1 )
(2.23b)
B=
2 R12 e −i ( k1 z − k0 z ) d1 e i 2 k1 z d 2
,
(1 + p 01 ) 1 + R01 R12 e i 2 k1 z ( d 2 − d1 )
(2.23c)
R=
[
]
[
]
4e ik0 z d1 e ik1 z ( d 2 − d1 ) e −ik 2 z d 2
T=
,
(1 + p 01 )(1 + p12 ) 1 + R01 R12 e i 2 k1 z ( d 2 − d1 )
[
]
(2.23d)
where
R01 =
1 − p 01
1 − p12
, R12 =
,
1 + p12
1 + p12
p 01 =
µ 0 k1 z
µk
, p12 = 1 2 z .
µ1 k 0 z
µ 2 k1 z
(2.24a)
(2.24b)
Now we consider the special case that ε 1 = −ε 0 , µ1 = − µ 0 , d 1 = 0 and d 2 = d . Because
region 2 is perfectly conducting and ε 2 is close to infinity, we can simplify the
coefficients R , A , B and T in this special case
k = ω εµ ,
(2.25)
so
k1z = −k0 z ,
(2.26)
13
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
14
and
p01 = 1 , R01 = 0 .
(2.27)
Because k2 z = ω ε 2 µ 2 is close to infinity, p12 is close to infinity, and
R12 = −1 .
(2.28)
We have [6]
R = R12 e i 2 k1 z d = R12 e − i 2 k0 z d = e −i 2 k0 z d ,
A=1,
B = R12 e i 2 k1 z d = R12 e −i 2 k0 z d = e − i 2 k0 z d ,
T=0.
2.3.2
(2.29a)
(2.29b)
(2.29c)
(2.29d)
SIMULATION RESULTS
The time averaged power density for the TE case is given as:
Sn =
2
2
1
Re ( Eny H nz* ) + Re ( Eny H nx* ) ,
2
(2.30)
where n =0,1,2 denotes the regions.
Fig. 2.3 and Fig. 2.5 are the simulation results obtained in this thesis, Fig. 2.4 and Fig. 2.6
are the simulation results in [6]. Fig 2.3 and Fig. 2.4 are the time-averaged power density
for the normal material. Fig. 2.5 and Fig. 2.6 are the time-averaged power density for the
metamaterial.
14
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
15
10
8
x/λ
6
4
2
0
-2
-2
-1
0
1
2
z/λ
3
4
5
6
7
8
9
10
°
Figure 2.3 Time-averaged power density on the xz plane for a 30 incidence of a Gaussian beam
upon a grounded slab of thickness d = 6λ with
ε1 = ε 0
and
µ = µ0 . This is the simulated result by
using the formula developed in this thesis
Figure 2.4 Reference time-averaged power density on the xz plane for a
beam with
ε1 = ε 0
and
µ = µ0 .
30° incidence of a Gaussian
This is the simulated result by using the formula developed in
reference [6]
15
Characterization of Spherical Metamaterials
Chapter 2
Background and Introduction of Metamaterial
16
2
0
x/λ
-2
-4
-6
-8
-10
-2
-1
0
1
2
3
4
5
6
7
8
9
10
z/λ
Figure 2.5 Time-averaged power density on the xz plane for a
upon a grounded slab of thickness d = 6λ with
ε1 = −ε 0
30° incidence of a Gaussian beam
and µ = − µ0 . This is the simulated result
by using the formula developed in this thesis
°
Figure 2.6 Time-averaged power density on the xz plane for a 30 incidence of a Gaussian beam
upon a grounded slab of thickness d = 6λ with
ε1 = −ε 0
and µ = − µ0 . This is the simulated result
by using the formula developed in reference [6]
16
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
17
3 SPHERICAL DYADIC GREEN’S FUNCTIONS
3.1 INTRODUCTION
Green’s function is named in honor of English mathematician and physicist George
Green (1793-1841). Green’s function is basically distribution due to a point source, a
solution to a linear differential equation and a building block that can be used to construct
many useful solutions. The linear differential equation is derived from the boundary
value problem due to a unit excitation source .The exact form of the Green’s function
depends on the differential equation, the geometrical shape and the type of the boundary
conditions present. The solution involves all the electromagnetic field phenomena. The
quasi-static solution is a special case of the general Dyadic Green’s Function at the zero
frequency.
In general, there are two typical applications when the Dyadic Green’s Function (DGF) is
applied to solve boundary-value problems. One of them is: for a particular structure due
to an arbitrarily assumed distributed source, a DGF is employed as an electromagnetic
(EM) response of the dielectric medium involved, and then the EM field can be
formulated. The other application is: we can use a DGF to derive the parameters matrices
for the unknown source current distribution. Except for a few simple geometries, the
Green’s functions are difficult to be obtained, with each new geometry requiring a new
formulation. This is the main limitation of Green’s function technique.
17
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
18
This chapter will present a detailed description of electromagnetic dyadic Green’s
function in spherically multilayered media.
Figure 3.1 shows the spherically N -layered geometry. The transmitter with an arbitrary
electric current distribution J s or an arbitrary magnetic current distribution M s is
located in the s th (source) layer ( s = 1, 2,..., N ). The receiver lies in the f th (field) layer
( f = 1, 2,..., N ) of the spherically N -layered system.
Figure 3.1 Geometry of a spherically multilayered medium
18
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
19
3.2 FUNDAMENTAL FORMULATION
3.2.1 EIGENFUNCTION EXPANSION OF DGF IN UNBOUNDED MEDIA
Electromagnetic problems can always be solved by beginning with the analysis of
Maxwell’s equations. The Maxwell’s functions in dyadic form can be given as:
∇ × E = iωµ0 H,
(3.1a)
∇ × H = J − iωε 0 E,
(3.1b)
∇ ⋅ J = iωρ ,
(3.1c)
∇ × (ε 0E ) = ρ ,
(3.1d)
∇ × ( µ0 H ) = 0.
(3.1e)
From (3.1), the vector field equations can be found from,
∇ × ∇ × E ( R ) − k 2 E ( R ) = iωµ J ( R ) ,
(3.2a)
∇ × ∇ × H ( R ) − k 2H ( R ) = ∇ × J ( R ) .
(3.2b)
We consider a general case, that is, a structure of a spherically N -layered geometry
which is shown in Figure 3.1. Throughout the analysis, a time dependence exp ( −iω t ) for
the fields is assumed. The electric and magnetic current sources J s and M s lie in the s th
layer ( s = 1, 2,...N ) of the medium. From (3.2), we can get that the expressions of the
electromagnetic radiation fields- E f and H f in the f th layer ( f = 1, 2,...N ) can be found
from:
∇ × ∇ × E f − k 2f E f = iωµ f J f δ fs − ( ∇ × M ) f δ fs ,
(3.3a)
19
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
∇ × ∇ × H f − k 2f H f = iωε f M f δ fs + ( ∇ × J ) f δ fs ,
20
(3.3b)
where δ fs is the Kronecker delta function, and ε f , µ f are the permittivity, permeability
of the medium, respectively. The propagation constant in the f th layer of the medium is:
iσ
k f = ω µ f ε f 1 + f
ωε f
,
(3.4)
where σ f denotes the conductivity of the medium.
Moreover, the electric and magnetic fields, E f and H f , due to the electric and magnetic
current sources J s and M s , can be expressed by the electric type and magnetic type of
dyadic Green’s functions G (e fs ) ( R, R′ ) and G (mfs ) ( R, R′ ) , respectively:
E f ( R ) = iωµ f ∫∫∫ G (e fs ) ( R, R′ ) ⋅ J s ( R′ ) dV ′,
(3.5a)
H f ( R ) = iωµ f ∫∫∫ G (mfs ) ( R, R′ ) ⋅ M s ( R′ ) dV ′,
(3.5b)
Vs
Vs
where Vs indicates the volume occupied by the sources in the s th layer, R is the position
vector of the field point and R′ is the position vector of the source point.
The relationship between the dyadic Green’s function of the electric and the dyadic
Green’s function of the magnetic type is subject to the reciprocity theorem. The magnetic
DGF can be obtained from the electric DGF by simple replacements of E → H, H → −E,
J → M, M → −J , µ → ε and ε → µ . Only DGF of the electric type will be considered
to avoid unnecessary repetition.
20
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
21
Substituting (3.5a) into (3.3a), we have:
∇ × ∇ × G (e fs ) ( R, R′ ) − k 2f G (e fs ) ( R, R′ ) = Iδ ( R , R ′ ) δ fs ,
(3.6)
where I is the unit dyadic, δ ( R, R′ ) is the Dirac delta function. The electric type of
dyadic Green’s function G (e fs ) ( R, R′ ) satisfies the boundary conditions at the spherical
interfaces r = a j ( j = 1, 2,....N − 1 ):
( f +1) s
fs
,
rˆ × G (e ) = rˆ × G e
1
µf
rˆ × ∇ × G (e fs ) =
1
µ f +1
( f +1) s
rˆ × ∇ × G e
.
(3.7a)
(3.7b)
Based on the scattering superposition method, the dyadic Green’s function can be
separated into two parts: the unbounded dyadic Green’s function and the scattering
dyadic Green’s function, as follows:
G (e fs ) ( R, R′ ) = G 0 e ( R, R′ ) δ fs + G (esfs ) ( R, R′ ) ,
(3.8)
where the superscript ( fs ) is the layers where the field point and source point locate,
respectively, while the subscript e indicates the electric type of the dyadic Green’s
function, and the subscript s identifies the scattering dyadic Green’s function.
The unbounded dyadic Green’s function G 0 e ( R, R′ ) corresponds to the contribution of
the direct waves from an unbounded medium, i.e., an infinite homogeneous space. The
scattering dyadic Green’s function G (esfs ) ( R, R′ ) represents an additional contribution of
21
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
22
the multiple reflected and transmitted waves due to the presence of the spherical
boundary of the dielectric layered media.
Using the contour integration method in the complex h -plane, the dyadic Green’s
function in the unbounded medium can be represented in terms of the normalized
spherical vector wave functions as a result of the residue theorem. The unbounded DGF
>
under the spherical coordinate system is expressed for r r ′ as:
<
G 0e ( R, R′) =
ˆˆ
ik
rr
δ ( R − R′ ) + s
2
ks
4π
∞
n
2n + 1 (n − m)!
∑∑ ( 2 − δ ) n(n + 1) (n + m)!
0
m
n =0 m=0
M e ( ks ) M′e ( ks ) + N e ( k s ) N′e ( ks ) , r ≥ r ′,
o mn
o mn
o mn
o mn
×
(1)
(1)
M eo mn ( ks ) M′oe mn ( ks ) + N oe mn ( k s ) N′oe mn ( ks ) , r ≤ r ′,
(1)
(1)
(3.9)
where the prime indicates the coordinates ( r ′,θ ′, φ ′ ) of the current source J f , m and n is
the eigen-value parameters, M e mn is the ΤΕ -wave spherical vector wave function that
o
represents the electric field of the ΤΕ mn modes, and N e mn is the ΤΜ -wave spherical
o
vector wave function that represents the electric field of the ΤΜ mn modes. The
superscript (1) of the vector wave functions denotes the third-type spherical Bessel
function or the first-type spherical Hankel function. M e mn and N e mn can be expressed as:
o
o
sin
m
jn ( kr ) Ρ mn ( cos θ )
mφθˆ
M e mn ( k ) = ∇ × φ e mn ( k ) r = ∓
o
o
cos
sin θ
d Ρ m ( cos θ ) cos
mφφˆ,
− jn ( kr ) n
sin
dθ
(3.10a)
22
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
23
and
N e mn ( k ) =
o
cos
1
n(n + 1)
jn ( kr ) Ρ mn ( cos θ )
mφ rˆ
∇ × ∇ × φ e mn ( k ) r =
o
sin
k
kr
m
sin
m m
1 d rjn ( kr ) d Ρ n ( cos θ ) cos
mφθˆ ∓
mφφˆ ,
+
Ρ n ( cos θ )
sin
cos
sin θ
kr
dr
dθ
(3.10b)
where
r = rrˆ
φ
e
o mn
(k ) = jn (kr )Ρ mn ( cos θ )
(3.11a)
cos
( nφ ) ,
sin
(3.11b)
with the subscripts e and o denoting the even and odd modes, respectively.
3.2.2 DGFS FOR SPHERICAL MULTILAYERED MEDIA
The scattering dyadic Green’s function should have the similar form of the unbounded
dyadic Green’s function, and the multiple reflection and transmission effects should be
considered as well. The electromagnetic fields always consist of the radial wave modes
that propagate outwards and inwards under the spherical coordinate. Assuming that the
current source is located in the s th layer, we can obtain the scattering dyadic Green’s
function in the f th layer among the multi layers. Now we can express the scattering DGF
utilizing the spherical Bessel and Hankel function and Kroncker delta as follows:
23
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
G es( fs ) (R, R′) =
{(1 − δ
N
f
iks
4π
∞
n
∑∑ (2 − δ
n=0 m=0
0
)
24
2n + 1 (n − m)!
n(n + 1) (n + m)!
)M (1)
(k f ) (1 − δ s1 ) AMfs M′e mn (ks ) + (1 − δ sN ) BMfs M′e(1)mn (ks )
e
o mn
o
o
+ (1 − δ fN )N (1)
(k f ) (1 − δ s1 ) ANfs N′e mn (ks ) + (1 − δ sN ) BNfs N′e(1)mn (ks )
e
o mn
o
o
1
1
fs
N
fs
(1)
+ (1 − δ f )M e mn (k f ) (1 − δ s )CM M′e mn (ks ) + (1 − δ s ) DM M′e mn (ks )
o
o
o
(3.12)
}
+ (1 − δ 1f )N e mn (k f ) (1 − δ s1 )CNfs N′e mn (ks ) + (1 − δ sN ) DNfs N′e(1)mn (ks ) ,
o
o
o
where AMfs, N , BMfs, N , CMfs, N and DMfs , N ( s, f = 1, 2,..., N ) are the coefficients of the
scattering DGF to be solved, and the superscript N is the number of the layers of the
multi-layer medium. Same as the expression of the unbounded dyadic Green’s function,
the superscript (1) indicates that the third-type spherical Bessel function or the first-type
spherical Hankel function should be chosen in the function of the spherical wave vector
functions: M e mn and N e mn . Because the normal first-type spherical Bessel function can be
o
o
used to represent both outgoing and incoming waves, it should also be chosen for the rest
of the vector wave functions.
3.2.3 THE RECURRENCE MATRIX EQUATIONS FOR THE COEFFICIENTS OF
SCATTERING DGFS
We should derive the coefficients of the scattering dyadic Green’s function from the
boundary conditions. The boundary conditions can be expressed in terms of the dyadic
Green’s function shown in (3.7a) and (3.7b). We can rewrite them in the form of linear
equation system, which can also be replaced by the coefficient matrix equations as:
24
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
ℑ ff CMfs
ff AMfs
fs
s
fs
δ
D
ℑ
+
(
)
B
ff
M
f
ff M +
=
fs
fs
∂ ff AN
C
∂ℑ
ff N
∂
fs
fs
s
∂ ff BN ∂ℑ ff ( DN + δ f )
25
( f +1) f
+ δ fs +1 ) ℑ
( f +1) f CM
( f +1) s
( f +1) f
ℑ( f +1) f DM
( f +1) f BM
, (3.13a)
+
( f +1) f
( f +1) s
s
+ δ f +1 ) ∂ℑ( f +1) f CN
( f +1) f ( AN
∂ℑ( f +1) f DM( f +1) f
∂ ( f +1) f BN( f +1) s
( f +1) f
(A
( f +1) s
M
and
∂ ff AMfs
k f ∂ ff BMfs k f
+
µ f ff ANfs µ f
fs
ff BN
∂
∂ℑ ff CMfs
fs
s
∂ℑ ff ( DM + δ f ) k f +1
=
fs
C
ℑ
ff
N
µ f +1
ℑ D fs + δ s
f )
ff ( N
+ δ fs +1 )
∂ ( f +1) f BM( f +1) s
( f +1) s
+ δ fs +1 )
( f +1) f ( AN
( f +1) s
( f +1) f BN
( f +1) f
(A
( f +1) s
M
(3.13b)
∂ℑ( f +1) f CM( f +1) f
k f +1 ∂ℑ( f +1) f DM( f +1) f
,
+
µ f +1 ℑ( f +1) f C N( f +1) f
( f +1) f
ℑ( f +1) f DM
where
ℑil = jn ( ki al ) ,
(3.14a)
= hn(1) ( ki al ) ,
(3.14b)
il
∂ℑil =
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
(3.14c)
i l
∂
il
(1)
1 d ρ hn ( ρ )
=
ρ
dρ
.
(3.14d)
ρ = ki al
The coefficient matrix equations are the results of simplifying the complicated algebraic
calculation of the linear equation system.
Equations (3.13a) and (3.13b) can be simplified as:
25
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
AM( f,+N1) s + δ fs +1
1
= H ,V
( f +1) s
BM , N
TFf
AMfs , N RFfH ,V
fs + H ,V
BM , N TFf
26
CMfs, N
,
fs
s
DM , N + δ f
(3.15a)
and
CM( f,+N1) s RPfH ,V
( f +1) s = H ,V
DM , N TPf
AMfs , N
1
fs + H ,V
BM , N TPf
CMfs, N
,
fs
s
DM , N + δ f
(3.15b)
where
RPfH =
µ f k f +1∂ ( f +1) f ff − µ f +1k f ∂ ff
µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff
( f +1) f
,
(3.16a)
RFfH =
µ f k f +1∂ℑ( f +1) f ℑ ff − µ f +1k f ∂ℑ ff ℑ( f +1) f
,
µ f k f +1∂ℑ( f +1) f ff − µ f +1k f ℑ( f +1) f ∂ ff
(3.16b)
RPfV =
µ f k f +1 ( f +1) f ∂ ff − µ f +1k f ff ∂
µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂
,
(3.16c)
RFfV =
µ f k f +1ℑ( f +1) f ∂ℑ ff − µ f +1k f ℑ ff ∂ℑ( f +1) f
,
µ f k f +1ℑ( f +1) f ∂ ff − µ f +1k f ∂ℑ( f +1) f ff
(3.16d)
TPfH =
µ f k f +1 (ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f
µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff
TFfH =
TPfV =
TFfV =
µ f k f +1 (∂ℑ( f +1) f
µ f k f +1∂ℑ( f +1) f
( f +1) f
ff
( f +1) f
( f +1) f
− ∂ℑ( f +1) f
)
,
(3.16e)
,
(3.16f)
,
(3.16g)
,
(3.16h)
( f +1) f
( f +1) f
− µ f +1k f ℑ( f +1) f ∂
( f +1) f
ff
( f +1) f
− ℑ( f +1) f ∂
µ f k f +1 (∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂
µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂
µ f k f +1 (ℑ( f +1) f ∂
µ f k f +1ℑ( f +1) f ∂
( f +1) f
)
ff
( f +1) f
)
( f +1) f
( f +1) f
− µ f +1k f ∂ℑ( f +1) f
AMNs, N CM1s , N
Ns = 1s = 0.
BM , N DM , N
)
ff
(3.16i)
26
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
27
The subscripts F and P indicate the centrifugal and centripetal waves, respectively.
T( HP, F ) f and R(HP , F ) f represent the centripetal and centrifugal transmission (corresponding to
T ) and reflection (corresponding to F ) contributions from ΤΕ waves (corresponding to
the superscript H). T(VP , F ) f and R(VP , F ) f represent the centripetal and centrifugal
transmission and reflection contributions from ΤΜ waves (corresponding to the
superscript V ).
Now a method of formulating the coefficients of the scattering dyadic Green’s function
for spherical arbitrary multilayered medium will be provided. However, the matrix
equations for the coefficients are coupled. Hence, the de-coupling of these equations is
needed to obtain the general expression of the coefficients of the scattering dyadic
Green’s function. From (3.15a) and (3.15b), we have:
AM( f,+N1) s + δ fs +1
( f +1) s
CM , N
1
H ,V
BM( f,+N1) s TFf
=
DM( f,+N1) s RPfH ,V
TPfH ,V
RFfH ,V
TFfH ,V AMfs, N
⋅ fs
1 CM , N
TPfH ,V
,
+ δ
BMfs , N
fs
M ,N
D
s
f
(3.17)
where the symbols T( HP ,,FV ) f and R(HP,,VF ) f represent the equivalent transmission and reflection
coefficients given by (3.15a) and (3.15b), respectively.
Given the following coefficient matrix, parameter matrix and the operators are:
A fs
CMfs , N = Mfs, N
CM , N
BMfs , N
,
DMfs, N
(3.18a)
27
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
f
M ,N
T
1
TFfH ,V
= H ,V
RPf
TPfH ,V
28
,
1
TPfH ,V
H ,V
RFf
TFfH ,V
1 0
A11 =
, and
0 0
(3.18b)
0 0
A 22 =
.
0 1
(3.18c)
CMf 1, N = TMf −, N1 ⋅⋅⋅ TM1 , N ( C11
M , N + A 22 ) ,
(3.19a)
Now we can obtain the relations for s = 1:
for s ≠ 1 and N :
{
}
CMfs , N = TMf −, N1 ⋅⋅⋅ TMs , N TMs −,1N ⋅⋅⋅ TM1 , N C1Ms , N + H ( f − s − 1) A 22 − H ( f − s ) A11 , (3.19b)
and for s = N :
CMfN, N = TMf −, N1 ⋅⋅⋅ TM1 , N C1MN, N − H ( f − N ) A11 ,
(3.19c)
where the step function H ( x − x0 ) is given as:
1, x ≥ x0
.
H ( x − x0 ) =
0, x < x0
(3.20)
To obtain the results of the coefficients from equations (3.19a), (3.19b) and (3.19c), we
assume that the field point locates in the last layer, i.e. f = N . The coefficients in the
first and last layers are easy to obtain from the equation (3.16i). Then, the rest of the
coefficients can be found directly by using the recurrence relations in (3.19) once again.
So far, the general expressions of the scattering dyadic Green’s function for the spherical
multilayered media have been provided.
28
Characterization of Spherical Metamaterials
Chapter 3
Spherical Dyadic Green’s Function
29
3.3 CONCLUSIONS
In this chapter, we presented the general expression of the dyadic Green’s function for
spherical multilayered medium, including the unbounded dyadic Green’s function and the
scattering dyadic Green’s function. The complete set of general coefficients of the
scattering dyadic Green’s function is also provided analytically. Furthermore, only the
electric type of dyadic Green’s functions is presented in this thesis in order to avoid
unnecessary repetition. The magnetic type of the dyadic Green’s functions can be easily
obtained by making the simple change of
E → H, H → −E, J → M, M → −J ,
µ → ε and ε → µ . The dyadic Green’s function for cylindrically layered media has also
been derived in [21].
29
Characterization of Spherical Metamaterials
Chapter 4
Scattering of Sphere
30
4 SCATTERING OF SPHERE
4.1
INTRODUCTION
4.1.1 RAYLEIGH SCATTERING
Rayleigh scattering represents the scattering property of the electromagnetic wave due to
a particle much smaller than the wavelength. We can assume a spherical particle with
radius a , permittivity ε and permeability µ locates at the origin of the coordinate. The
incident electric field is polarized in the z direction:
ˆ 0 eikx .
Ein = zE
(4.1)
The solution of the scattered field can be given as:
−iωµ Ileikr
E sc =
4π r
2
i 2 i
i
i
ˆ
ˆ
r
2
cos
1
sin
+
+
+
+
θ
θ
θ
,
kr kr
kr kr
(4.2a)
and
−ikIleikr i
ˆ
H sc = φ
+ 1 sin θ ,
4π r kr
(4.2b)
where Il (polar distance) will be decided by the value of E0 and ε .
Because of kr > 1. From (4.2a) and (4.2b), we have:
ε − ε 2 2 a ikr
Eθ = − s
k a E0 e sin θ ,
r
ε s + 2ε
(4.9a)
and
Hφ =
ε
E.
µ θ
(4.9b)
For the perfectly conducting sphere, the inner electric field of the sphere Ein is zero. From
(4.4) and (4.6a), we have:
Il = −i 4π ka 3
ε
E.
µ 0
(4.10)
4.1.2 MIE SCATTERING
Mie scattering is brought forward when the radius of the sphere is bigger than the
Rayleigh scattering case. The solutions of the sphere scattering problem due to planar
waves can be obtained precisely by applying the boundary conditions. For convenience,
we should introduce Debye potentials: π e and π m , then the spherical wave can be
divided to ΤΜ wave and ΤΕ wave due to rˆ . The ΤΜ wave due to rˆ includes [25]
A = rπ e ,
(4.11a)
32
Characterization of Spherical Metamaterials
Chapter 4
Scattering of Sphere
1 ∂
∂
H = ∇ × A = θˆ
π e − φˆ π e .
sin θ ∂φ
∂θ
33
(4.11b)
The ΤΕ wave due to rˆ includes:
Z = rπ m ,
1 ∂
∂
E = ∇ × Z = θˆ
π m − φˆ π m .
sin θ ∂φ
∂θ
(4.12a)
(4.12b)
Debye potentials satisfy the following Helmholtz function under the spherical coordinate:
π
+ k 2 ) e = 0,
π m
(4.13)
1 ∂2
1
∂
∂
1
∂2
∇ =
+
sin θ
+
.
r ∂r 2 r 2 sin θ ∂θ
∂θ r 2 sin 2 θ ∂φ 2
(4.14)
(∇
2
where
2
Applying the Maxwell’s equations and (4.13), we have [25]
Er =
Eθ =
i ∂2
2
2 rπ e + k rπ e ,
ωε ∂r
(4.15a)
i 1 ∂2
1 ∂
πm,
rπ e +
ωε r ∂r ∂θ
sin θ ∂φ
(4.15b)
1
∂2
∂
rπ e −
πm,
∂θ
ωε r sin θ ∂r ∂φ
(4.15c)
i ∂2
2
2 rπ m + k rπ m ,
ωµ ∂r
(4.15d)
i 1 ∂2
1 ∂
πe,
rπ m +
ωµ r ∂r ∂θ
sin θ ∂φ
(4.15e)
Eφ =
i
Hr = −
Hθ = −
33
Characterization of Spherical Metamaterials
Chapter 4
Scattering of Sphere
Hφ = −
34
1
∂2
∂
π e.
rπ m −
ωµ r sin θ ∂r ∂φ
∂θ
i
(4.15f)
Assume a sphere located in the origin of the coordinate has a radius of a , permittivity of
ε and permeability of µ . The incident planar wave is given as:
ˆ 0 eikz = xE
ˆ 0 eik cosθ ,
E = xE
(4.16a)
ε
E0 eik cosθ .
µ
H = yˆ
(4.16b)
The incident wave propagates in the z direction.
We have
∞
eikr cosθ = ∑ ( −i )
n=0
−n
( 2n + 1) jn ( kr ) Pn ( cosθ ),
(4.17)
Thus the incident wave can be expanded as:
Er = E0 sin θ cos φ eikr cosθ
=−
iE0 cos φ
( kr )
2
∞
∑ ( −i ) ( 2n + 1) Jˆ ( kr ) P ′ ( cosθ ),
−n
n
n =1
(4.18)
n
where
Jˆn ( kr ) = krjn ( kr ) .
(4.19)
From (4.4), π e and π m can be obtained as follows [25]:
E cos φ
πe = − 0
ωµ r
( −i ) ( 2n + 1) Jˆ kr P ′ cosθ ,
) n(
)
∑
n(
n ( n + 1)
n =1
∞
−n
(4.20a)
34
Characterization of Spherical Metamaterials
Chapter 4
Scattering of Sphere
E sin φ
πm = 0
kr
35
( −i ) ( 2n + 1) Jˆ kr P ′ cosθ .
) n(
)
∑
n(
n ( n + 1)
n =1
−n
∞
(4.20b)
The Debye potentials of scattered field can be expressed as:
π es = −
π ms =
E0 cos φ
ωµ r
∞
∑ a Hˆ ( kr ) P ′ ( cosθ ),
(4.21a)
E0 sin φ ∞
′
bn Hˆ (1)
∑
n ( kr ) Pn ( cos θ ).
kr n =1
(4.21b)
n =1
(1)
n
n
n
where
(1)
Hˆ (1)
n ( kr ) = krhn ( kr ) .
(4.22)
The inner field of the sphere can also be expressed by Debye potentials as:
E0 cos φ ∞ ˆ
∑ c J ( k r ) P ′ ( cosθ ),
ωµ s r n =1 n n s n
(4.23a)
E0 sin φ
ks r
(4.23b)
π ei = −
π mi =
∞
∑ d Jˆ ( k r ) P ′ ( cosθ ).
n =1
n
n
s
n
The unknown coefficients can be obtained by applying the boundary conditions at the
surface of r = a, we have
−n
−i ) ( 2n + 1) − ε s µ Jˆn′ ( ka ) Jˆn ( ks a ) + εµ s Jˆn ( ka ) Jˆn′ ( k s a )
(
an =
i
,
n ( n + 1)
ε s µ Hˆ n(1)′ ( ka ) Jˆn ( ks a ) − εµ s Hˆ n(1) ( ka ) Jˆn′ ( k s a )
(4.24a)
−n
−i ) ( 2n + 1) − ε s µ Jˆn ( ka ) Jˆn′ ( k s a ) + εµ s Jˆn′ ( ka ) Jˆn ( ks a )
(
bn =
,
i
n ( n + 1)
ε s µ Hˆ n(1) ( ka ) Jˆn′ ( ks a ) − εµ s Hˆ n(1)′ ( ka ) Jˆn ( ks a )
(4.24b)
i εsµ
( −i ) ( 2n + 1) i
cn =
,
n ( n + 1)
ε s µ Hˆ n(1)′ ( ka ) Jˆn ( ks a ) − εµ s Hˆ n(1) ( ka ) Jˆn′ ( ks a )
(4.24c)
−n
35
Characterization of Spherical Metamaterials
Chapter 4
Scattering of Sphere
36
− εµ s
( −i ) ( 2n + 1) i
dn =
.
n ( n + 1)
ε s µ Hˆ n(1) ( ka ) Jˆn′ ( ks a ) − εµ s Hˆ n(1)′ ( ka ) Jˆn ( ks a )
−n
(4.24d)
When the sphere satisfies ka b
G e 0 (R, R′) = −
ik
1 ˆˆ
RRδ (R − R′) + 1
2
k1
4π
∑C
mn
M (1) (k1 )M′(k1 ) + N (1) (k1 )N′(k1 ) , (5.5a)
ik
1 ˆˆ
RRδ (R − R′) + 1
2
k1
4π
∑C
mn
M (k1 )M′(1) (k1 ) + N(k1 )N′(1) (k1 ) , (5.5b)
m,n
when R < b
G e 0 (R, R′) = −
m,n
45
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
46
with M e mn and N e mn given in equations (3.10a) and (3.10b).
o
o
The boundary conditions at the surface of the sphere ( R = a ) of electric dyadic Green
functions are expressed as follows:
Rˆ × G e1 (R, R′) = Rˆ × G e 2 (R, R′),
(5.6a)
1 ˆ
1 ˆ
R × ∇ × G e1 (R, R′) =
R × ∇ × G e 2 (R, R′).
µ1
µ2
(5.6b)
Because R = a < b , the electric free space DGF should use the equation (5.5b).
According to the functions of the boundary conditions, we may obtain that the
21
21
coefficients BM11 , B11
N , DM , DN must satisfy the following equations:
jn ( ρ1 ) + BM11 hn(1) ( ρ1 ) = DM21 jn ( ρ 2 ),
(5.7a)
(1)
k1 [ ρ1 jn ( ρ1 )]′
k2 21 [ ρ 2 jn ( ρ 2 )]′
11 [ ρ1hn ( ρ1 )]′
+ BM
=
DM
,
µ1
ρ1
ρ1
ρ2
µ2
[ ρ1 jn ( ρ1 )]′
ρ1
+ B11
N
[ ρ1hn(1) ( ρ1 )]′
ρ1
= DN21
[ ρ 2 jn ( ρ 2 )]′
ρ2
,
(1)
21 k 2
[ jn ( ρ1 ) + B11
jn ( ρ 2 ),
N hn ( ρ1 )] = DN
µ1
µ2
k1
(5.7b)
(5.7c)
(5.7d)
where
ρ1 = k1a, ρ 2 = k2 a.
(5.8)
By solving the equations from (5.7a) to (5.7d), the expressions of the four coefficients are
obtained as:
BM11 =
k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k2 µ1 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′
,
k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.9a)
46
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
47
B11
N =
k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′
,
k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.9b)
DM21 =
k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
,
k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.9c)
DN21 =
k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
.
k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.9d)
The recurrence matrix equations for the coefficients of scattering dyadic Green’s function
introduced in Chapter 3 will be used to obtain the coefficients. Applying
N = 2 and
s = 1 to equation (3.16i) and equation (3.17), we have:
21
11
BMN
= DMN
= 0,
1
( f +1)1 T HV
BMN
Ff
( f +1)1 = HV
DMN RPf
TPfHV
RFfHV
f1
TFfHV BMN
⋅
,
1
1
f
1 DMN + δ f
TPfHV
(5.10a)
(5.10b)
where TFfHV , TPfHV , RFfHV , RPfHV are determined from equation (3.16a) to equation (3.16h).
For f = 1, the equation (5.10b) can be rewritten as follows:
21
MN
B
and
=
21
DMN
=
RFHV1
11
+ HV ( DMN
+ 1) ,
TF 1
(5.11a)
RPHV
1
11
11
1
BMN
+ HV ( DMN
+ 1) .
HV
TP1
TP1
(5.11b)
1
TFHV
1
11
MN
B
From the above two equations together with equation (5.10a), the following equations
will be derived:
11
BMN
= − RFHV1 ,
(5.12a)
47
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
21
DMN
=
and
1
TPHV
1
(1 − R
HV
F1
48
RPHV
1 ).
(5.12b)
By applying f = 1 to the expressions of TFfHV , TPfHV , RFfHV , and RPfHV , the coefficients are
obtained:
BM11 = − RFH1 =
k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k2 µ1 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′
,
k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.13a)
V
B11
N = − RF 1 =
k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ1 jn ( ρ1 )[ ρ 2 jn ( ρ 2 )]′
,
k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.13b)
DM21 =
k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
1
H
H
−
=
1
R
R
( F1 P1 ) k µ ρ h(1) ( ρ )[ ρ j ( ρ )]′ − k µ ρ j ( ρ )[ ρ h(1) ( ρ )]′ , (5.13c)
TPH1
2 1 1 n
1
2 n
2
1 2 2 n
2
1 n
1
DN21 =
k1µ 2 ρ 2 hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ 2 ρ 2 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
1
V
V
−
=
1
R
R
( F1 P1 ) k µ ρ h(1) ( ρ )[ ρ j ( ρ )]′ − k µ ρ j ( ρ )[ ρ h(1) ( ρ )]′ . (5.13d)
TPV1
1 2 1 n
1
2 n
2
2 1 2 n
2
1 n
1
It is obvious that the above four equations are the same as the results derived from the
boundary conditions.
5.2.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL
Now let us consider the metamaterial case, i.e., k2 = − k1 and µ 2 = − µ1 . We know
(−1) k
z
jn ( z ) = ∑
k = 0 k !Γ ( n + k + 1) 2
∞
n+2k
,
(5.14)
thus we have
j (− z ), When n is even,
jn ( z ) = n
− jn (− z ), When n is odd.
(5.15)
Based on the properties of the Bessel functions, we have the following equations:
48
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
49
1
[ ρ1 jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1[ jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1 [ jn −1 ( ρ1 ) − jn +1 ( ρ1 )] ,
2
(5.16a)
1
[ ρ 2 jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2 [ jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2 [ jn −1 ( ρ 2 ) − jn +1 ( ρ 2 )] .
2
(5.16b)
When n is even, n + 1 and n − 1 are both odd, we have:
[ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′.
(5.17a)
When n is odd, then n + 1 and n − 1 are both even, we have:
[ ρ 2 j n ( ρ 2 )]′ = −[ ρ1 j n ( ρ1 )]′ .
(5.17b)
21
21
After applying above two equations to the expressions of BM11 , B11
N , DM , DN , the results
of the coefficients in the special case can be obtained.
For BM11 , when n is even:
k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
− k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
= − (1)
,
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
BM11 =
(5.18a)
when n is odd:
− k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
,
= − (1)
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
BM11 =
(5.18b)
therefore
BM11 = −
2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
,
h ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
(1)
n
(5.19)
49
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
50
where n can be either odd or even. .
For B11
N , when n is even:
k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
−k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
= − (1)
,
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
BN11 =
(5.20a)
when n is odd:
−k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
= − (1)
,
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
BN11 =
(5.20b)
therefore
B11
N = −
2 jn ( ρ1 )[ ρ1 jn ( ρ1 )]′
= An ,
h ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
(1)
n
(5.21)
whenever n is even or odd.
For DM21 , when n is even:
DM21 =
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
− k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[hn(1) ( ρ1 )]′
= − n(1) 1 1 n 1
,
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[hn(1) ( ρ1 )]′
(5.22a)
when n is odd:
50
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
DM21 =
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[hn(1) ( ρ1 )]′
= n(1) 1 1 n 1
.
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[hn(1) ( ρ1 )]′
51
(5.22b)
For DN21 , when n is even:
DN21 =
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
− k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
= − n(1) 1 1 n 1
,
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
(5.23a)
when n is odd:
DN21 =
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
h(1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
= n(1) 1 1 n 1
.
hn ( ρ1 )[ ρ1 jn ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
DM21 = DN21 .
It is obvious that
(5.23b)
(5.24)
5.2.4 CALCULATION OF THE ELECTRICAL FIELD
Rectangular coordinate can be transformed to spherical coordinate as:
xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ.
(5.25)
For the infinitesimal horizontal electric dipole introduced in the beginning of section 5.2,
because θ ′ = 0, φ ′ = 0 , we have:
xˆ = θˆ .
(5.26)
Therefore, we have
51
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
52
n(n + 1)
, m =1
mpnm (cosθ )
= 2
,
lim
θ →0
sin θ
0, otherwise
(5.27a)
n(n + 1)
, m =1
dpnm (cosθ )
.
lim
= 2
θ →0
dθ
0, otherwise
(5.27b)
Applying m = 1 to equation (5.3), we have:
Cmn = 2 ×
2n + 1
1
.
×
n(n + 1) n(n + 1)
(5.28)
From the above three equations, the following equation can be obtained:
mpnn (cos θ )
dp m (cosθ )
2n + 1
= Cmn × lim n
=
.
θ →0
θ →0
sin θ
dθ
n(n + 1)
Cmn × lim
(5.29)
Additionally, we have:
dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′.
(5.30)
Now the electrical fields will be calculated in three cases.
For R > b :
E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ,
(5.31)
then derived from equations (3.10a), (3.10b) and (5.29), the result can be obtained as:
E1 (R ) = −
k1ωµ1c ∞ 2n + 1
i jn ( ρb ) + An hn(1) ( ρb ) M o(1)1n (k1 )
∑
4π n =1 n(n + 1)
{
}
(5.32)
+ ( ρb jn ( ρb ))′ + Bn ( ρ h ( ρb ))′ N (k1 ) ρ b ,
(1)
b n
(1)
e1n
where ρb = k1b.
For a < R < b :
52
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
53
E 2 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ
=−
k1ωµ1c ∞ 2n + 1
i{hn(1) ( ρb )[M o1n (k1 ) + An M o(1)1n (k1 )]
∑
4π n =1 n(n + 1)
(5.33)
+ ρb hn(1) ( ρb ) ′ Ν e1n (k1 ) + Bn N (1)
e1n ( k1 ) ρ b .
For R < a :
E3 (R ) = iωµ2 ∫∫∫ G e 2 (R, R′)iJ (R′)dV ′ = iωµ2 cG e 2 (R, R′)i xˆ = iωµ2 cG e 2 (R, R′)iθˆ
=−
(5.34)
[ ρb hn(1) ( ρb )]′
k1ωµ2 c ∞ 2n + 1 (1)
• Dn N e1n (k2 ) .
hn ( ρb ) • Cn M o1n (k2 ) +
∑
4π n =1 n(n + 1)
ρb
In the metamaterial case, the above expression can be written as:
E3 (R ) = iωµ2 ∫∫∫ G e 2 (R , R ′)iJ (R ′)dV ′ = iωµ2 cG e 2 (R, R′)i xˆ = iωµ2 cG e 2 (R, R′)iθˆ
=
[ ρb hn(1) ( ρb )]′
k1ωµ1c ∞ 2n + 1 (1)
(
)
(
)
•
−
+
• Dn N e1n (− k1 ) .
h
ρ
C
M
k
∑
n
b
n
o1n
1
4π n =1 n(n + 1)
ρb
(5.35)
We use the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b),
o
o
together with the properties of the Bessel function shown in equations (5.15), (5.17a) and
(5.17b), we have:
when n is even,
M e mn ( k ) = M e mn ( −k ) ,
(5.36a)
N e mn ( k ) = − N e mn ( −k ) ,
(5.36b)
M e mn ( k ) = −M e mn ( −k ) ,
(5.36c)
N e mn ( k ) = N e mn ( −k ) .
(5.36d)
o
o
when n is odd,
o
o
o
o
o
o
53
Characterization of Spherical Metamaterials
Chapter 5
5.3
Metamaterial Sphere
54
CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERE
5.3.1 INTRODUCTION
The single layered sphere is centered at (0, 0, 0). An infinitesimal horizontal electric
dipole with current moment c
points in the
x -direction and locates at
R′ = 0,θ ′ = 0, φ ′ = 0 (the center of the sphere).
J (R′) = c
δ ( R′ − 0)δ (θ ′ − 0)δ (φ ′ − 0)
xˆ.
R′2 sin θ ′
(5.37)
Scattering dyadic Green’s functions can be expressed generally as in equation (3.12).
Applying this equation to special case where N = 2 , s = 2, we can get the scattering
DGFs of region 1 (outside the sphere) and region 2 (inside the sphere):
′
G12
es ( R , R ) =
ik2
4π
G es22 (R, R′) =
∑C
mn
m,n
ik2
4π
∑C
m,n
AM12 M (1) (k1 )M′(k2 ) + AN12 N (1) (k1 )N′(k2 ) ,
(5.38a)
CM22 M (k2 )M′(k2 ) + CN22 N(k2 )N′(k2 ) ,
(5.38b)
mn
where
Cmn = (2 − δ 0 )
2n + 1 (n − m)!
,
n(n + 1) (n + m)!
(5.39)
22
22
and the coefficients AM12 , A12
N , CM , C N need to be solved for.
54
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
55
5.3.2 THE CALCULATION OF THE COEFFICIENTS
The recurrence matrix equations for the coefficients of scattering dyadic Green’s function
introduced in the chapter 3 will be used to obtain the coefficients. Green functions of
region 1 and region 2 are presented as follows respectively:
′
G e1 (R, R′) = G12
es ( R , R ),
(5.40a)
Ge2 (R, R′) = Ge0 (R, R′) +Ges22 (R, R′),
(5.40b)
where the electric free space DGF is:
because R > 0
G e 0 (R, R′) = −
1 垐
ik
RRδ (R − R ′) + 2
2
k2
4π
∑C
m,n
mn
M (1) (k2 )M′(k2 ) + N (1) (k2 )N′(k2 ) , (5.41)
where M e mn and N e mn are given in equations (3.10a) and (3.10b), and Cmn is shown in
o
o
equation (5.39).
Applying N = 2 and s = 2 to equation (3.16i) and equation (3.17), we have:
22
12
AMN
= CMN
= 0,
1
f +1) 2
(
2
AMN + δ f +1 TFfHV
= HV
( f +1) 2
CMN
RPf
TPfHV
(5.42a)
RFfHV
f2
TFfHV AMN
⋅ C f 2 ,
1 MN
TPfHV
(5.42b)
where TFfHV , TPfHV , RFfHV , RPfHV are determined from equation (3.16a) to equation (3.16h).
For f = 1, the equation (5.40b) can be rewritten as follows:
55
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
22
AMN
+1 =
22
CMN
=
and
1
TFHV
1
12
AMN
+
RFHV1 12
CMN ,
TFHV
1
RPHV
1 12
12
1
AMN
+ HV CMN
.
HV
TP1
TP1
56
(5.43a)
(5.43b)
From the above two equations together with equation (5.42a), the following coefficients
are derived:
12
AMN
= TFHV
1 ,
22
CMN
=
and
HV
TFHV
1 RP1
.
TPHV
1
(5.44a)
(5.44b)
Apply f = 1 to the expressions of TFfHV , TPfHV , RFfHV , and RPfHV . Then the results of the
coefficients are obtained as follows:
AM12 = TFH1 =
k2 µ1 ρ1hn(1) ( ρ 2 )[ ρ 2 jn ( ρ 2 )]′ − k2 µ1 ρ1 jn ( ρ 2 )[ ρ 2 hn(1) ( ρ 2 )]′
,
k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k1µ 2 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
(5.45a)
V
A12
N = TF 1 =
k2 µ1 ρ1 jn ( ρ 2 )[ ρ 2 hn(1) ( ρ 2 )]′ − k2 µ1 ρ1hn(1) ( ρ 2 )[ ρ 2 jn ( ρ 2 )]′
,
k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ − k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 jn ( ρ 2 )]′
(5.45b)
CM22 =
TFH1 RPH1 k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ − k1µ 2 ρ 2 hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
, (5.45c)
=
TPH1
k1µ 2 ρ 2 j n ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ − k2 µ1 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′
C N22 =
TFV1 RPV1 k2 µ1 ρ 2 hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ − k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′
=
. (5.45d)
TPV1
k1µ 2 ρ1hn(1) ( ρ1 )[ ρ 2 j n ( ρ 2 )]′ − k2 µ1 ρ 2 jn ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
56
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
57
5.3.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL
Now let us consider the case that the sphere is metamaterial, i.e., k2 = − k1 and µ 2 = − µ1 .
We consider the same case as before where current distribution is located outside the
sphere. Because:
(−1) k
z
jn ( z ) = ∑
k = 0 k !Γ ( n + k + 1) 2
∞
n+2k
,
(5.46)
we have
j (− z ), When n is even,
jn ( z ) = n
− jn (− z ), When n is odd.
(5.47)
Together with the properties of the Bessel functions, we have the following equations:
1
[ ρ1 jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1[ jn ( ρ1 )]′ = jn ( ρ1 ) + ρ1 [ jn −1 ( ρ1 ) − jn +1 ( ρ1 )] ,
2
(5.48a)
1
[ ρ 2 jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2 [ jn ( ρ 2 )]′ = jn ( ρ 2 ) + ρ 2 [ jn −1 ( ρ 2 ) − jn +1 ( ρ 2 )] ,
2
(5.48b)
where ρ1 and ρ 2 have been given in equation (5.8).
Then, it is easy to get the results shown below. When n is even, n + 1 and n − 1 are both
odd, we have:
[ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′.
(5.49a)
When n is odd, then n + 1 and n − 1 are both even, we have:
[ ρ 2 j n ( ρ 2 )]′ = −[ ρ1 j n ( ρ1 )]′ .
(5.49b)
Based on the properties of the Hankel function.
57
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
58
We have:
hn(1) ( z ) =
j− n ( z ) − jn ( z ) e − iπ n
i sin ( nπ )
( z < ∞, arg z < π ).
(5.50)
Because
j− n ( z ) = ( −1) jn ( z ) ,
n
(5.51)
the following can be obtained:
hn(1) ( ρ 2 ) = hn(1) ( ρ1 ) ,
when n is even,
(5.52a)
j ( ρ ) − jn ( ρ 2 ) e− iπ n ′ j− n ( ρ 2 ) ′ − jn ( ρ 2 ) ′ e − iπ n
[hn(1) ( ρ 2 )]′ = − n 2
=
i sin ( nπ )
i sin ( nπ )
j ( ρ ) − jn ( ρ1 ) e
= − −n 1
i sin ( nπ )
− iπ n
′
(1)
= −[hn ( ρ1 )]′,
[ ρ 2 hn(1) ( ρ 2 )]′ = hn(1) ( ρ 2 ) + ρ 2 [hn(1) ( ρ 2 )]′ = hn(1) ( ρ1 ) + ρ1[hn(1) ( ρ1 )]′
= [ ρ1hn(1) ( ρ1 )]′;
hn( ) ( ρ 2 ) =
1
when n is odd,
j− n ( ρ1 ) + jn ( ρ1 ) e −iπ n
i sin ( nπ )
,
(5.52b)
(5.52c)
(5.53a)
j− n ( ρ 2 ) − jn ( ρ 2 ) e− iπ n ′ j− n ( ρ 2 ) ′ − jn ( ρ 2 ) ′ e − iπ n
′
[hn ( ρ 2 )] =
=
i sin ( nπ )
i sin ( nπ )
(5.53b)
− iπ n ′
− j ( ρ ) − jn ( ρ1 ) e
= −n 1
,
i sin ( nπ )
(1)
(1)
[ ρ 2 hn ( ρ 2 )]′ = hn(1) ( ρ 2 ) + ρ 2 [hn(1) ( ρ 2 )]′ =
j− n ( ρ1 ) + jn ( ρ1 ) e − iπ n
i sin ( nπ )
j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n ′
+ ρ1
.
i sin ( nπ )
(5.53c)
58
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
59
22
22
After applying above equations to the expressions of AM12 , A12
N , CM , C N , the results of the
coefficients in the special case can be obtained.
For AM12 , when n is even:
− k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
A =
− k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
12
M
hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′ − jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
,
= − (1)
hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
(5.54a)
when n is odd:
AM12 =
k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1 jn ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
h (1) ( ρ )[ ρ j ( ρ )]′ − jn ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′
,
= n(1) 2 1 n 1
hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
(5.54b)
where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a).
For A12
N , when n is even:
AN12 =
− k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′
k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′
j ( ρ )[ ρ h (1) ( ρ )]′ − hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′
,
= − n 1 1 n(1) 1
jn ( ρ1 )[ ρ1hn ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′
(5.55a)
when n is odd:
A12
N =
k1µ1 ρ1 jn ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ − k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1 jn ( ρ1 )]′
− k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′
j ( ρ )[ ρ h(1) ( ρ )]′ − hn(1) ( ρ 2 )[ ρ1 jn ( ρ1 )]′
,
= − n 1 2 n(1) 2
jn ( ρ1 )[ ρ1hn ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 jn ( ρ1 )]′
(5.55b)
59
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
60
where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a). It is
obvious that
AM12 = A12
N ,
(5.56)
wherever n is even or odd.
For CM22 , when n is even:
CM22 =
−k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
k1µ1 ρ1 j n ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′
2hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
;
=−
j n ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′
(5.57a)
when n is odd:
CM22 =
−k1µ1 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′ − k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
−k1µ1 ρ1 j n ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ − k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′
h(1) ( ρ )[ ρ h(1) ( ρ )]′ + hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′
= n 1 2 n(1) 2
,
j n ( ρ1 )[ ρ1hn ( ρ1 )]′ + hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′
(5.57b)
where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a).
For C N22 , when n is even:
C
22
N
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
.
=
− k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ − k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
2hn(1) ( ρ1 )[ ρ1hn(1) ( ρ1 )]′]′
;
= − (1)
hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
(5.58a)
when n is odd:
60
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
C N22 =
k1µ1 ρ1hn(1) ( ρ 2 )[ ρ1hn(1) ( ρ1 )]′ + k1µ1 ρ1hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′
k1µ1 ρ1hn(1) ( ρ1 )[ ρ1 j n ( ρ1 )]′ + k1µ1 ρ1 jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
h(1) ( ρ )[ ρ h(1) ( ρ )]′ + hn(1) ( ρ1 )[ ρ 2 hn(1) ( ρ 2 )]′
= n (1) 2 1 n 1
,
hn ( ρ1 )[ ρ1 j n ( ρ1 )]′ + jn ( ρ1 )[ ρ1hn(1) ( ρ1 )]′
61
(5.58b)
where [ ρ 2 hn(1) ( ρ 2 )]′ and hn(1) ( ρ 2 ) are expressed in equations (5.53c) and (5.53a). It is
obvious that
CM22 = CN22 ,
(5.59)
wherever n is even or odd.
5.3.4 CALCULATION OF THE ELECTRICAL FIELD
Rectangular coordinates can be transformed to spherical coordinates as:
xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ .
(5.60)
For the infinitesimal horizontal electric dipole introduced in the beginning of 5.3,
because θ ′ = 0, φ ′ = 0 , we have:
xˆ = θˆ .
(5.61)
Because
n(n + 1)
, m =1
mpnm (cos θ )
,
= 2
lim
θ →0
sin θ
0, otherwise
(5.62a)
n(n + 1)
, m =1
dpnm (cos θ )
.
= 2
θ →0
dθ
0, otherwise
(5.62b)
lim
Applying m = 1 to equation (5.39), we have:
61
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
Cmn = 2 ×
2n + 1
1
×
.
n(n + 1) n(n + 1)
62
(5.63)
From three equations above, the following equation will be obtained:
mpnn (cos θ )
dp m (cosθ )
2n + 1
.
= Cmn × lim n
=
θ →0
θ
→
0
sin θ
dθ
n(n + 1)
Cmn × lim
(5.64)
Additionally, we have:
dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′.
(5.65)
Now the electrical fields will be calculated in three cases. For R > a :
E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R , R ′)i xˆ = iωµ1cG e1 (R , R ′)iθˆ,
(5.66)
then derived from equations (3.10a), (3.10b) and (5.40a), the electric field can be
obtained as:
E1 (R ) = −
[ k2 rjn (k2 r )]′ A12 N (1) (k ) .
k2ωµ1c ∞ 2n + 1
12
(1)
j
(0)
A
(
k
)
lim
i
M
+
∑
1
1
n
M
o1n
N
e1n
r →0
4π n =1 n(n + 1)
k2 r
(5.67)
Because we still have:
1, n = 0
jn (0) =
,
0, otherwise
2
, n =1
[k2 rjn (k2 r )]′
lim
,
= 3
r →0
k2 r
0, otherwise
(5.68a)
(5.68b)
the electric field can be then represented as follows:
E1 (R ) = −
k2ωµ1c 12 (1)
AN N e11 (k1 ) .
4π
(5.69)
In the metamaterial case, the above expression can be written as:
62
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
E1 (R ) =
k1ωµ1c 12 (1)
AN N e11 (k1 ) .
4π
63
(5.70)
We use the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b),
o
o
together with the properties of the Hankel functions as given in equations (5.52a), (5.52c),
(5.53a) and (5.53c), the following can be then derived:
when n is even,
M (1)
( k ) = M (1)e mn ( −k ) ,
e
mn
(5.71a)
N (1)
( k ) = −N (1)e mn ( −k ) ,
e
mn
(5.71b)
o
o
o
o
when n is odd, equations (5.53a) and (5.53c) should be applied.
For R < a :
E 2 (R ) = iωµ 2 ∫∫∫ G e 2 (R, R ′)iJ (R ′)dV ′ = iωµ 2 cG e 2 (R , R′)i xˆ = iωµ 2 cG e 2 (R , R′)iθˆ,
(5.72)
then derived from equations (3.10a), (3.10b) and (5.40b), the electric field can be
obtained as:
E2 (R ) = −
k2ωµ 2 c ∞ 2n + 1
22
i jn (0) M (1)
∑
o1n ( k 2 ) + CM M o1n ( k 2 ) +
4π n =1 n(n + 1)
{
[ k rj (k r )]′ N (1) (k
lim 2 n 2
r →0
k2 r
e1n
+
C
k
N
)
(
)
e1n
2
2 .
(5.73)
22
N
Applying the equation (5.68a) and (5.68b) to (5.73), the following result can be given:
E2 (R ) = −
k2ωµ 2 c (1)
N e11 (k2 ) + CN22 N e11 (k2 ) .
4π
(5.74)
In the metamaterial case, the above expression can be written as:
E 2 (R ) =
k1ωµ 2 c (1)
N e11 (− k1 ) + CN22 N e11 (− k1 ) .
4π
(5.75)
63
Characterization of Spherical Metamaterials
Chapter 5
Metamaterial Sphere
64
We use the expressions of N e mn ( k ) , i.e., equations (3.10b), together with the properties
o
of the Bessel functions given in equations (5.47), (5.49a) and (5.49b), the following can
then be derived:
when n is even,
N e mn ( k ) = − N e mn ( −k ) ,
(5.76a)
when n is odd,
N e mn ( k ) = N e mn ( −k ) .
(5.76b)
o
o
o
o
64
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
65
6 METAMATERIAL SPHERICAL SHELL
6.1
INTRODUCTION
The geometry of the spherically two-layered medium is considered as a single sphere
with a coating layer superimposed by an unbounded homogeneous medium. In this
chapter, the inner sphere is air. The radius of the outer sphere is a1 , the radius of the inner
sphere is a2 . The two-layered sphere (spherical shell) under two different circumstances
will be studied. One is for the current distribution located outside the spherical shell, the
other is for the current distribution located in the center of the spherical shell. In each
instance, the general case will be first analyzed, and then the special case: metamaterial
sphere shell.
6.2
CURRENT DISTRIBUTION LOCATED OUTSIDE THE SPHERICAL SHELL
6.2.1 INTRODUCTION
The spherical shell is centered at (0, 0, 0). An infinitesimal horizontal electric dipole with
the current moment c pointes in the x -direction and locates at R ′ = b,θ ′ = 0, φ ′ = 0,
which is at the top of the spherical shell. Thus we have:
J (R′) = c
δ ( R′ − b)δ (θ ′ − 0)δ (φ ′ − 0)
xˆ.
b 2 sin θ ′
(6.1)
65
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
66
Scattering dyadic Green’s functions can be expressed generally as in equation (3.12).
Applying this equation to a special case where N = 3 and s = 1, we can obtain the
scattering DGFs of Region 1 (outside the spherical shell), Region 2 (between the outer
sphere and the inner sphere) and Region 3 (inside the inner sphere):
′
G11
es ( R , R ) =
G es21 (R, R′) =
ik1
4π
∑C
ik1
4π
∑ C { B
mn
m,n
(1)
′(1) (k1 ) ,
BM11M (1) (k1 )M′(1) (k1 ) + B11
N N ( k1 ) N
M (1) (k2 )M′(1) (k1 ) + BN21N (1) (k2 )N′(1) (k1 )
21
M
mn
m,n
}
(6.2a)
(6.2b)
+ D M (k2 )M′ (k1 ) + D N (k2 )N′ (k1 ) ,
21
M
′
G 31
es ( R , R ) =
(1)
ik1
4π
21
N
∑C
m,n
mn
(1)
(1)
DM31M (1) (k3 )M′(1) (k1 ) + DN31N (1) (k3 )N′(1) (k1 )
(6.2c)
where
Cmn = (2 − δ 0 )
2n + 1 (n − m)!
,
n(n + 1) (n + m)!
(6.3)
21
21
21
21
31
31
and the coefficients BM11 , B11
N , BM , BN , DM , DN , DM , DN need to be solved for.
6.2.2 THE CALCULATION OF THE COEFFICIENTS
The electric dyadic Green functions of Region 1-3 are presented as follows respectively:
′
G e1 (R, R′) = G e 0 (R, R ′) + G11
es ( R , R ),
(6.4a)
Ge2 (R, R′) = Ges21(R, R′),
(6.4b)
′
Ge3(R, R′) = G31
es (R, R ),
(6.4c)
where the electric free space DGF is:
66
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
67
when R > b
G e 0 (R, R′) = −
1 ˆˆ
ik
RRδ (R − R ′) + 1
2
k1
4π
∑C
mn
M (1) (k1 )M′(k1 ) + N (1) (k1 )N′(k1 ) , (6.5a)
1 ˆˆ
ik
RRδ (R − R ′) + 1
2
k1
4π
∑C
mn
M (k1 )M′(1) (k1 ) + N(k1 )N′(1) (k1 ) , (6.5b)
m,n
when R < b
G e 0 (R, R′) = −
m,n
where M e mn and N e mn are given in equations (3.10a) and (3.10b).
o
o
The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering
dyadic Green’s functions. Applying N = 3 and s = 1 to equations (3.16i) and (3.17), we
have:
31
11
BMN
= DMN
= 0,
1
f +1)1
(
BMN TFfHV
( f +1)1 = HV
DMN RPf
TPfHV
RFfHV
f1
TFfHV BMN
⋅
Df1 +δ1 ,
f
1 MN
HV
TPf
(6.6a)
(6.6b)
where TFfHV , TPfHV , RFfHV , RPfHV are determined from equations (3.16a) to (3.16h), f will be
either 1 or 2. For f = 2, equation (6.6b) can be rewritten as follows:
31
BMN
=
(6.7a)
RPHV2 21
1
21
BMN + HV ( DMN
+ 0) ,
HV
TP 2
TP 2
(6.7b)
RFHV1
11
+ 1) ,
( DMN
TFHV
1
(6.7c)
TFHV
2
31
DMN
=
for f = 1,
RFHV2
( DMN21 + 0 ) ,
TFHV
2
1
21
BMN
=
1
TFHV
1
21
BMN
+
11
BMN
+
67
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
21
DMN
=
RPHV
1
11
11
1
BMN
+ HV ( DMN
+ 1) .
HV
TP1
TP1
68
(6.7d)
From the above four equations together with equation (6.6a), the following equations will
be derived:
RFH2,V TFH1 ,V + RFH1,V TPH1 ,V
,
TPH1 ,V + TFH1 ,V RPH1,V RFH2,V
BM11, N = −
BM21, N =
DM31, N =
RFH2,V RPH1,V RFH1,V − RFH2,V
,
TPH1 ,V + TFH1 ,V RPH1,V RFH2,V
1
H ,V
P2
T
DM21, N =
RPH2,V BM21, N + DM21, N ,
1
H ,V
P1
T
RPH1,V BM11, N + 1 ,
(6.8a)
(6.8b)
(6.8c)
(6.8d)
where
RFH2 =
µ2 k3∂ℑ32 ℑ22 − µ3k2∂ℑ22 ℑ32
,
µ2 k3∂ℑ32 22 − µ3k2 ℑ32 ∂ 22
(6.9a)
RFV 2 =
µ2 k3ℑ32 ∂ℑ22 − µ3k2 ℑ22 ∂ℑ32
,
µ2 k3 ℑ32 ∂ 22 − µ3k2∂ℑ32 22
(6.9b)
TFH1 =
µ1k2 (∂ℑ21 21 − ℑ21∂ 21 )
,
µ1k2 ∂ℑ21 11 − µ2 k1ℑ21∂ 11
(6.9c)
TFV1 =
µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 )
,
µ1k2 ℑ21∂ 11 − µ2 k1∂ℑ21 11
(6.9d)
RFH1 =
µ1k2∂ℑ21ℑ11 − µ2 k1∂ℑ11ℑ21
,
µ1k2 ∂ℑ21 11 − µ2 k1ℑ21∂ 11
(6.9e)
RFV1 =
µ1k2 ℑ21∂ℑ11 − µ2 k1ℑ11∂ℑ21
,
µ1k2 ℑ21∂ 11 − µ2 k1∂ℑ21 11
(6.9f)
68
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
69
TPH1 =
µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 )
,
µ1k2 ℑ11∂ 21 − µ2 k1∂ℑ11 21
(6.9g)
TPV1 =
µ1k2 (∂ℑ21 21 − ℑ21∂ 21 )
,
µ1k2 ∂ℑ11 21 − µ2 k1ℑ11∂ 21
(6.9h)
RPH1 =
µ1k2 ∂ 21
µ1k2 ℑ11∂
,
(6.9i)
,
(6.9j)
RPV1 =
µ1k2 21∂
µ1k2 ∂ℑ11
− µ2 k1∂ 11
21 − µ 2 k1∂ℑ11
11
− µ2 k1 11∂
21 − µ 2 k1ℑ11∂
11
21
21
21
21
TPH2 =
µ2 k3 (ℑ32 ∂ 32 − ∂ℑ32 32 )
,
µ2 k3 ℑ22 ∂ 32 − µ3k2 ∂ℑ22 32
(6.9k)
TPV2 =
µ2 k3 (∂ℑ32 32 − ℑ32 ∂ 32 )
,
µ2 k3∂ℑ22 32 − µ3k2 ℑ22 ∂ 32
(6.9l)
RPH2 =
µ2 k3∂ 32
µ2 k3ℑ22 ∂
RPV 2 =
µ2 k3 32 ∂
µ2 k3∂ℑ22
− µ3k2∂ 22
32 − µ3 k 2 ∂ℑ22
22
− µ3k2 22∂
32 − µ3 k 2 ℑ22 ∂
22
32
,
(6.9m)
,
(6.9n)
32
32
32
where
ℑ32 = jn (k3 a2 ), ℑ22 = jn (k2 a2 ), ℑ21 = jn (k2 a1 ), ℑ11 = jn (k1a1 ),
32
= hn(1) (k3a2 ),
∂ℑ32 =
22
= hn(1) (k2 a2 ),
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
21
= hn(1) (k2 a1 ),
∂ℑ22 =
3 2
∂ℑ21 =
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
2 1
11
= hn(1) (k1a1 ),
(6.10a)
(6.10b)
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
(6.10c)
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
(6.10d)
2 2
∂ℑ11 =
1 1
69
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
∂
32
=
(1)
1 d ρ hn ( ρ )
ρ
dρ
(1)
1 d ρ hn ( ρ )
∂ 21 =
ρ
dρ
,
∂
22
=
ρ = k3 a2
,
ρ = k2 a1
70
(1)
1 d ρ hn ( ρ )
ρ
dρ
(1)
1 d ρ hn ( ρ )
∂ 11 =
ρ
dρ
,
(6.10e)
.
(6.10f)
ρ = k2 a2
ρ = k1a1
Because the structure is a spherical shell, we have additionally:
ε 3 = ε1 ,
µ3 = µ1 ,
k3 = k1.
(6.11)
The same results can be obtained from equation (3.19a):
CMf 1, N = TMf −, N1 ⋅⋅⋅ TM1 , N ( C Mf 1, N + A 22 ) .
6.2.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL
Now let us consider the metamaterial case, that is,
k2 = − k1 = − k3 ,
µ2 = − µ1 = − µ3 .
(6.12)
The equations (6.9a) to (6.9n) can be written as:
RFH2 =
RFV 2 =
− µ1k1∂ℑ32 ℑ22 + µ1k1∂ℑ22 ℑ32 ∂ℑ32 ℑ22 − ∂ℑ22 ℑ32
=
,
− µ1k1∂ℑ32 22 + µ1k1ℑ32 ∂ 22 ∂ℑ32 22 − ℑ32 ∂ 22
− µ1k1ℑ32 ∂ℑ22 + µ1k1ℑ22 ∂ℑ32 ℑ32 ∂ℑ22 − ℑ22 ∂ℑ32
=
,
− µ1k1ℑ32 ∂ 22 + µ1k1∂ℑ32 22 ℑ32 ∂ 22 − ∂ℑ32 22
TFH1 =
TFV1 =
− µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21
=
− µ1k1∂ℑ21 11 + µ1k1ℑ21∂ 11 ∂ℑ21
− µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 ) ℑ21∂
=
− µ1k1ℑ21∂ 11 + µ1k1∂ℑ21 11 ℑ21∂
− ℑ21∂
11 − ℑ21∂
21
− ∂ℑ21
11 − ∂ℑ21
21
21
(6.13a)
(6.13b)
,
(6.13c)
,
(6.13d)
11
21
11
70
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
71
RFH1 =
− µ1k1∂ℑ21ℑ11 + µ1k1∂ℑ11ℑ21 ∂ℑ21ℑ11 − ∂ℑ11ℑ21
,
=
− µ1k1∂ℑ21 11 + µ1k1ℑ21∂ 11 ∂ℑ21 11 − ℑ21∂ 11
(6.13e)
RFV1 =
− µ1k1ℑ21∂ℑ11 + µ1k1ℑ11∂ℑ21 ℑ21∂ℑ11 − ℑ11∂ℑ21
=
,
− µ1k1ℑ21∂ 11 + µ1k1∂ℑ21 11 ℑ21∂ 11 − ∂ℑ21 11
(6.13f)
TPH1 =
− µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 )
ℑ ∂
= 21
− µ1k1ℑ11∂ 21 + µ1k1∂ℑ11 21 ℑ11∂
TPV1 =
RPH1 =
RPV1 =
TPH2 =
TPV2 =
RPH2 =
RPV 2 =
− ∂ℑ21
21 − ∂ℑ11
21
− µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21
=
− µ1k1∂ℑ11 21 + µ1k1ℑ11∂ 21 ∂ℑ11
− ℑ21∂
21 − ℑ11∂
− µ1k1∂ 21
− µ1k1ℑ11∂
∂ 21
ℑ11∂
− ∂ 11
21 − ∂ℑ11
∂
− 11∂
21 − ℑ11∂
− µ1k1 21∂
− µ1k1∂ℑ11
+ µ1k1∂ 11
21 + µ1k1∂ℑ11
11
21
=
21
+ µ1k1 11∂
21 + µ1k1ℑ11∂
11
21
21
=
21
21
∂ℑ11
11
11
− µ1k1 (ℑ32 ∂ 32 − ∂ℑ32 32 )
ℑ ∂
= 32
− µ1k1ℑ22 ∂ 32 + µ1k1∂ℑ22 32 ℑ22 ∂
− ∂ℑ32
32 − ∂ℑ22
− µ1k1 (∂ℑ32 32 − ℑ32 ∂ 32 )
∂ℑ
= 32
− µ1k1∂ℑ22 32 + µ1k1ℑ22 ∂ 32 ∂ℑ22
− ℑ32 ∂
32 − ℑ22 ∂
− µ1k1∂ 32
− µ1k1ℑ22 ∂
− µ1k1 32 ∂
− µ1k1∂ℑ22
+ µ1k1∂ 22
32 + µ1k1∂ℑ22
22
+ µ1k1 22 ∂
32 + µ1k1ℑ22 ∂
22
32
=
32
32
32
=
∂ 32
ℑ22∂
32
∂
∂ℑ22
,
(6.13g)
,
(6.13h)
,
(6.13i)
,
(6.13j)
21
21
21
21
21
21
21
21
32
32
− ∂ 22
32 − ∂ℑ22
22
− 22∂
32 − ℑ22 ∂
22
32
,
(6.13k)
,
(6.13l)
,
(6.13m)
,
(6.13n)
32
32
32
32
32
32
32
where
ℑ32 = jn (k1a2 ), ℑ22 = jn (−k1a2 ), ℑ21 = jn (−k1a1 ), ℑ11 = jn (k1a1 ),
32
= hn(1) (k1a2 ),
22
= hn(1) (−k1a2 ),
∂ℑ32 = [ ρ jn ( ρ ) ]′
,
ρ = k1a2
21
= hn(1) (− k1a1 ),
11
∂ℑ22 = [ ρ jn ( ρ )]′
(6.14a)
= hn(1) (k1a1 ),
(6.14b)
,
(6.14c)
ρ =− k1a2
71
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
∂ℑ21 = [ ρ jn ( ρ ) ]′
∂
∂
32
21
= ρ hn(1) ( ρ ) ′
= ρ hn(1) ( ρ ) ′
72
∂ℑ11 = [ ρ jn ( ρ ) ]′
,
ρ =− k1a1
∂
,
22
ρ = k1a2
∂
,
ρ =− k1a1
= ρ hn(1) ( ρ ) ′
11
(6.14d)
,
ρ = k1a1
,
(6.14e)
.
(6.14f)
ρ =− k1a2
= ρ hn(1) ( ρ ) ′
ρ = k1a1
We still have the following results.
j (− z ), When n is even,
jn ( z ) = n
− jn (− z ), When n is odd.
(6.15a)
When n is even:
[ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′ ρ =− ρ ,
1
(6.15b)
2
when n is odd:
[ ρ 2 jn ( ρ 2 )]′ = − [ ρ1 jn ( ρ1 )]′ ρ =− ρ .
1
(6.15c)
2
When n is even:
hn(1) ( ρ 2 ) = hn(1) ( ρ1 )
ρ1 =− ρ2
[ ρ 2 hn(1) ( ρ 2 )]′ = [ ρ1hn(1) ( ρ1 )]′
,
ρ1 =− ρ 2
(6.15d)
(6.15e)
;
when n is odd:
hn(1) ( ρ 2 ) =
(1)
[ ρ 2 hn ( ρ 2 )]′ =
j− n ( ρ1 ) + jn ( ρ1 ) e − iπ n
i sin ( nπ )
j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n
i sin ( nπ )
(6.15f)
,
ρ1 =− ρ 2
j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n ′
+ ρ1
i sin ( nπ )
.
(6.15h)
ρ1 =− ρ 2
72
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
73
It is obvious that we have the following results in this special case:
RFH2 = RFV 2 , TFH1 = TFV1 , RFH1 = RFV1 , TPH1 = TPV1 ,
RPH1 = RPV1 , TPH2 = TPV2 , RPH2 = RPV 2 ,
(6.16)
hence,
BM11 = B11
BM21 = BN21 ,
N ,
31
31
21
21
DM = DN , DM = DN .
(6.17)
6.2.4 CALCULATION OF THE ELECTRICAL FIELD
Rectangular coordinates can be transformed to spherical coordinates as:
xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ,
(6.18)
For the infinitesimal horizontal electric dipole introduced in the beginning of 6.2,
because θ ′ = 0, φ ′ = 0 , we have:
xˆ = θˆ .
(6.19)
Because
n(n + 1)
, m =1
mpnm (cos θ )
,
= 2
lim
θ →0
sin θ
0, otherwise
(6.20a)
n(n + 1)
, m =1
dpnm (cos θ )
= 2
,
lim
θ →0
dθ
0, otherwise
(6.20b)
Apply m = 1 to equation (6.3), we have:
Cmn = 2 ×
2n + 1
1
.
×
n(n + 1) n(n + 1)
(6.21)
From the above three equations, the following equation can be obtained:
73
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
mpnn (cos θ )
dp m (cos θ )
2n + 1
= Cmn × lim n
=
.
θ →0
θ
→
0
sin θ
dθ
n(n + 1)
Cmn × lim
74
(6.22)
Additionally, we have:
dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′.
(6.23)
Now the electrical fields will be calculated of each region. When R > b :
E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R, R ′)i xˆ = iωµ1cG e1 (R , R ′)iθˆ, (6.24)
together with equation (6.5a), then we have:
E1 (R ) = −
k1ωµ1c ∞ 2n + 1
i jn ( ρb ) + BM11 hn(1) ( ρb ) M (1)
∑
o1n ( k1 )
4π n =1 n(n + 1)
{
′ (1)
(1)
+ ( ρb jn ( ρb ) )′ + B11
N ( ρ b hn ( ρ b ) ) N e1n ( k1 ) ρ b ,
(6.25)
where ρb = k1b. When a1 < R < b :
E 2 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R , R ′)i xˆ = iωµ1cG e1 (R , R ′)iθˆ, (6.26)
together with equation (6.5b), then we have:
E 2 (R ) = −
k1ωµ1c ∞ 2n + 1
i hn(1) ( ρb ) M o1n (k1 ) + BM11M (1)
∑
o1n ( k1 )
4π n =1 n(n + 1)
{
(1)
+ ( ρ h ( ρb ) )′ N e1n (k1 ) + B11
N N e1n ( k1 ) ρ b .
(6.27)
(1)
b n
When a2 < R < a1 :
E3 (R ) = iωµ 2 ∫∫∫ G e 2 (R, R ′)iJ (R ′)dV ′ = iωµ2 cG e 2 (R , R ′)i xˆ = iωµ 2 cG e 2 (R , R′)iθˆ, (6.28)
applying the related equations to the above equation, we have:
74
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
E3 ( R ) = −
k1ωµ2 c ∞ 2n + 1
i hn(1) ( ρb ) DM21M o1n (k2 ) + BM21M o(1)1n (k2 )
∑
4π n =1 n(n + 1)
75
{
+ ( ρ h ( ρb ) )′ DN21N e1n (k2 ) + BN21N (1)
e1n ( k 2 ) ρ b .
(6.29)
(1)
b n
In the metamaterial case, the above expression can be written as:
E3 ( R ) =
k1ωµ1c ∞ 2n + 1
i hn(1) ( ρb ) DM21M o1n (− k1 ) + BM21M (1)
∑
o1n ( − k1 )
4π n =1 n(n + 1)
{
+ ( ρ h ( ρb ) )′ DN21N e1n (−k1 ) + BN21N (1)
e1n ( − k1 ) ρ b .
(6.30)
(1)
b n
We use the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b),
o
o
together with the properties of the Bessel functions and the Hankel functions given in
equations (6.15a)-(6.15h), the following can then be derived:
when n is even,
M e mn ( k ) = M e mn ( −k ) ,
(6.31a)
N e mn ( k ) = − N e mn ( −k ) ,
(6.31b)
M (1)
( k ) = M (1)e mn ( −k ) ,
e
mn
(6.31c)
N (1)
( k ) = −N (1)e mn ( −k ) ,
e
mn
(6.31d)
M e mn ( k ) = −M e mn ( −k ) ,
(6.31e)
N e mn ( k ) = N e mn ( −k ) .
(6.31f)
o
o
o
o
o
o
when n is odd,
o
o
o
o
o
o
Equations (6.15f) and (6.15h) should also be applied in the odd case.
When R < a2 :
E 4 (R ) = iωµ3 ∫∫∫ G e 3 (R, R ′)iJ (R′)dV ′ = iωµ3cG e3 (R , R′)i xˆ = iωµ3cG e 3 (R , R ′)iθˆ,
(6.32)
75
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
76
applying the related equations to the above equation, we have:
E 4 (R ) = −
k1ωµ3c ∞ 2n + 1 (1)
ihn ( ρb ) DM31M o1n (k3 ) + ( ρb hn(1) ( ρb ) )′ DN31N e1n (k3 ) ρb . (6.33)
∑
4π n =1 n(n + 1)
Because k1 = k3 , the above expression can be written as:
E 4 (R ) = −
6.3
k1ωµ1c ∞ 2n + 1 (1)
i hn ( ρb ) DM31M o1n (k1 ) + ( ρb hn(1) ( ρb ) )′ DN31N e1n (k1 ) ρb . (6.34)
∑
4π n =1 n(n + 1)
CURRENT DISTRIBUTION LOCATED INSIDE THE SPHERICAL SHELL
6.3.1 INTRODUCTION
The spherical shell is centered at (0, 0, 0). An infinitesimal horizontal electric dipole with
a current moment c pointing in the x -direction and located at R′ = 0,θ ′ = 0, φ ′ = 0 (the
center of the spherical shell), let:
J (R′) = c
δ ( R′ − 0)δ (θ ′ − 0)δ (φ ′ − 0)
xˆ.
R′2 sin θ ′
(6.35)
Scattering dyadic Green’s functions can be expressed generally in equation (3.12).
Applying this equation to special case where N = 3 and s = 3, we can obtain the
scattering DGFs of Region 1 (outside the spherical shell), Region 2 (between the outer
sphere and the inner sphere) and Region 3 (inside the inner sphere):
′
G13
es ( R , R ) =
G es23 (R, R′) =
ik3
4π
ik3
4π
∑C
mn
m ,n
AM13M (1) (k1 )M′(k3 ) + AN13 N (1) (k1 )N′(k3 ) ,
∑ C { A
23
M
mn
m ,n
M (1) (k2 )M′(k3 ) + AN23 N (1) (k2 )N′(k3 )
}
(6.36a)
(6.36b)
+ C M (k2 )M′(k3 ) + C N(k2 )N′(k3 ) ,
23
M
23
N
76
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
′
G 33
es ( R , R ) =
ik3
4π
∑C
mn
m,n
CM33M (k3 )M′(k3 ) + C N33 N(k3 )N′(k3 ) ,
77
(6.36c)
where
Cmn = (2 − δ 0 )
2n + 1 (n − m)!
,
n(n + 1) (n + m)!
(6.37)
and the coefficients AM13, N , AM23, N , CM23, N , CM33, N need to be solved for.
6.3.2 THE CALCULATION OF THE COEFFICIENTS
The electric dyadic Green functions of Regions 1-3 are presented as follows respectively:
′
G e1 (R, R′) = G13
es ( R , R ),
(6.38a)
Ge2 (R, R′) = Ges23(R, R′),
(6.38b)
′
Ge3(R, R′) = Ge0 (R, R′) +G33
es (R, R ),
(6.38c)
where the electric free space DGF is:
because R > 0
G e 0 (R, R′) = −
ik
1 ˆˆ
RRδ (R − R ′) + 3
2
4π
k3
∑C
m ,n
mn
M (1) (k3 )M′(k3 ) + N (1) (k3 )N′(k3 ) , (6.39)
where M e mn and N e mn are given in equations (3.10a) and (3.10b), and Cmn is shown in
o
o
equation (6.37).
The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering
dyadic Green’s function. Applying N = 3 and s = 3 to equation (3.16i) and equation
(3.17), we have:
33
13
AMN
= CMN
= 0,
(6.40a)
77
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
1
f +1)3
(
3
AMN + δ f +1 TFfHV
= HV
( f +1)3
CMN
RPf
TPfHV
RFfHV
f3
TFfHV AMN
⋅ C f 3 ,
1 MN
TPfHV
78
(6.40b)
where TFfHV , TPfHV , RFfHV , RPfHV are determined from equation (3.16a) to equation (3.16h),
and f will be either 1 or 2. For f = 2, equation (6.40b) can be rewritten as follows:
AM33, N + 1 =
CM33, N =
for f = 1,
AM23, N =
CM23, N =
1
RFH2,V 23
CM , N ,
TFH2,V
AM23, N +
TFH2,V
(6.41a)
RPH2,V 23
1
AM , N + H ,V CM23, N ,
H ,V
TP 2
TP 2
(6.41b)
RFH1,V 13
CM , N ,
TFH1 ,V
(6.41c)
RPH1,V 13
1
AM , N + H ,V CM13, N .
H ,V
TP1
TP1
(6.41d)
1
TFH1 ,V
AM13, N +
From the above four equations together with equation (6.40a), the following coefficients
will be derived:
13
M ,N
A
TFH1 ,V TFH2,V TPH1 ,V
= H ,V
,
TP1 + TFH1 ,V RPH1,V RFH2,V
23
M ,N
A
=
CM23, N =
CM33, N =
AM13, N
TFH1 ,V
,
RPH1,V AM13, N
TPH1 ,V
(6.42a)
(6.42b)
,
1
RPH2,V AM23, N + CM23, N ,
TPH2 ,V
(6.42c)
(6.42d)
where
78
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
79
RFH2 =
µ2 k3∂ℑ32 ℑ22 − µ3k2∂ℑ22 ℑ32
,
µ2 k3∂ℑ32 22 − µ3k2 ℑ32 ∂ 22
(6.43a)
RFV 2 =
µ2 k3ℑ32 ∂ℑ22 − µ3k2 ℑ22 ∂ℑ32
,
µ2 k3 ℑ32 ∂ 22 − µ3k2∂ℑ32 22
(6.43b)
TFH1 =
µ1k2 (∂ℑ21 21 − ℑ21∂ 21 )
,
µ1k2 ∂ℑ21 11 − µ2 k1ℑ21∂ 11
(6.43c)
TFV1 =
µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 )
,
µ1k2 ℑ21∂ 11 − µ2 k1∂ℑ21 11
(6.43d)
TFH2 =
µ2 k3 (∂ℑ32 32 − ℑ32 ∂ 32 )
,
µ2 k3∂ℑ32 22 − µ3k2 ℑ32 ∂ 22
(6.43e)
TFV2 =
µ2 k3 (ℑ32∂ 32 − ∂ℑ32 32 )
,
µ2 k3ℑ32 ∂ 22 − µ3k2∂ℑ32 22
(6.43f)
TPH1 =
µ1k2 (ℑ21∂ 21 − ∂ℑ21 21 )
,
µ1k2 ℑ11∂ 21 − µ2 k1∂ℑ11 21
(6.43g)
TPV1 =
µ1k2 (∂ℑ21 21 − ℑ21∂ 21 )
,
µ1k2∂ℑ11 21 − µ2 k1ℑ11∂ 21
(6.43h)
RPH1 =
µ1k2 ∂ 21
µ1k2 ℑ11∂
,
(6.43i)
,
(6.43j)
TPH2 =
µ2 k3 (ℑ32 ∂ 32 − ∂ℑ32 32 )
,
µ2 k3ℑ22 ∂ 32 − µ3 k2 ∂ℑ22 32
(6.43k)
TPV2 =
µ2 k3 (∂ℑ32 32 − ℑ32 ∂ 32 )
,
µ2 k3∂ℑ22 32 − µ3k2 ℑ22 ∂ 32
(6.43l)
RPV1 =
RPH2 =
µ1k2 21∂
µ1k2 ∂ℑ11
µ2 k3∂ 32
µ2 k3ℑ22 ∂
− µ2 k1∂ 11
21 − µ 2 k1∂ℑ11
11
− µ2 k1 11∂
21 − µ 2 k1ℑ11∂
11
− µ3 k2∂ 22
32 − µ3 k 2 ∂ℑ22
22
21
21
21
21
32
,
(6.43m)
32
79
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
RPV 2 =
µ2 k3 32 ∂
µ2 k3∂ℑ22
− µ3 k2 22∂
32 − µ3 k 2 ℑ22 ∂
22
80
32
,
(6.43n)
32
where
ℑ32 = jn (k3 a2 ), ℑ22 = jn (k2 a2 ), ℑ21 = jn (k2 a1 ), ℑ11 = jn (k1a1 ),
32
= hn(1) (k3 a2 ),
∂ℑ32 =
22
= hn(1) (k2 a2 ),
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
21
= hn(1) (k2 a1 ),
∂ℑ22 =
3 2
∂ℑ21 =
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
∂
32
(1)
1 d ρ hn ( ρ )
∂ 21 =
ρ
dρ
= hn(1) (k1a1 ),
(6.44b)
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
(6.44c)
1 d [ ρ jn ( ρ ) ]
,
ρ
dρ
ρ =k a
(6.44d)
2 2
∂ℑ11 =
2 1
(1)
1 d ρ hn ( ρ )
=
ρ
dρ
11
(6.44a)
1 1
,
∂
ρ = k3 a2
,
ρ = k2 a1
22
(1)
1 d ρ hn ( ρ )
=
ρ
dρ
(1)
1 d ρ hn ( ρ )
∂ 11 =
ρ
dρ
,
(6.44e)
,
(6.44f)
ρ = k2 a2
ρ = k1a1
Additionally, because the structure is a spherical shell, we have:
ε 3 = ε1 ,
µ3 = µ1 ,
k3 = k1.
(6.45)
The same results can be obtained from equation (3.19c):
CMfN, N = TMf −, N1 ⋅⋅⋅ TM1 , N C1MN, N − H ( f − N ) A11.
6.3.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL
Now let us consider the metamaterial case, that is,
k2 = − k1 = − k3 ,
µ2 = − µ1 = − µ3 .
(6.46)
The equations (6.43a) to (6.43n) can be written as:
80
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
RFH2 =
RFV 2 =
81
− µ1k1∂ℑ32 ℑ22 + µ1k1∂ℑ22 ℑ32 ∂ℑ32 ℑ22 − ∂ℑ22 ℑ32
=
,
− µ1k1∂ℑ32 22 + µ1k1ℑ32 ∂ 22 ∂ℑ32 22 − ℑ32 ∂ 22
− µ1k1ℑ32 ∂ℑ22 + µ1k1ℑ22 ∂ℑ32 ℑ32 ∂ℑ22 − ℑ22 ∂ℑ32
=
,
− µ1k1ℑ32 ∂ 22 + µ1k1∂ℑ32 22 ℑ32 ∂ 22 − ∂ℑ32 22
TFH1 =
TFV1 =
TFH2 =
TFV2 =
TPH1 =
TPV1 =
RPH1 =
RPV1 =
TPH2 =
TPV2 =
RPH2 =
− µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21
=
− µ1k1∂ℑ21 11 + µ1k1ℑ21∂ 11 ∂ℑ21
− ℑ21∂
11 − ℑ21∂
21
− µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 )
ℑ ∂
= 21
− µ1k1ℑ21∂ 11 + µ1k1∂ℑ21 11 ℑ21∂
− ∂ℑ21
11 − ∂ℑ21
− ℑ32 ∂
22 − ℑ32 ∂
− µ1k1 (ℑ32 ∂ 32 − ∂ℑ32 32 )
ℑ ∂
= 32
− µ1k1ℑ32 ∂ 22 + µ1k1∂ℑ32 22 ℑ32 ∂
− ∂ℑ32
22 − ∂ℑ32
− µ1k1∂ 21
− µ1k1ℑ11∂
∂ 21
ℑ11∂
− ∂ 11
21 − ∂ℑ11
∂
− 11∂
21 − ℑ11∂
− µ1k1 21∂
− µ1k1∂ℑ11
+ µ1k1 11∂
21 + µ1k1ℑ11∂
11
21
=
21
21
21
=
21
∂ℑ11
− µ1k1 (ℑ32 ∂ 32 − ∂ℑ32 32 )
ℑ ∂
= 32
− µ1k1ℑ22 ∂ 32 + µ1k1∂ℑ22 32 ℑ22 ∂
− µ1k1 (∂ℑ32 32 − ℑ32 ∂ 32 )
∂ℑ
= 32
− µ1k1∂ℑ22 32 + µ1k1ℑ22 ∂ 32 ∂ℑ22
− µ1k1∂ 32
− µ1k1ℑ22 ∂
+ µ1k1∂ 22
32 + µ1k1∂ℑ22
22
32
32
=
∂ 32
ℑ22∂
11
11
(6.47d)
32
,
(6.47e)
,
(6.47f)
22
21
,
(6.47g)
,
(6.47h)
,
(6.47i)
,
(6.47j)
21
21
21
21
21
21
− ∂ℑ32
32 − ∂ℑ22
32
− ℑ32 ∂
32 − ℑ22 ∂
32
− ∂ 22
32 − ∂ℑ22
22
,
22
− ∂ℑ21
21 − ∂ℑ11
− ℑ21∂
21 − ℑ11∂
(6.47c)
32
21
21
,
11
32
− µ1k1 (∂ℑ21 21 − ℑ21∂ 21 ) ∂ℑ21
=
− µ1k1∂ℑ11 21 + µ1k1ℑ11∂ 21 ∂ℑ11
+ µ1k1∂ 11
21 + µ1k1∂ℑ11
21
32
− µ1k1 (ℑ21∂ 21 − ∂ℑ21 21 )
ℑ ∂
= 21
− µ1k1ℑ11∂ 21 + µ1k1∂ℑ11 21 ℑ11∂
(6.47b)
11
21
− µ1k1 (∂ℑ32 32 − ℑ32∂ 32 )
∂ℑ
= 32
− µ1k1∂ℑ32 22 + µ1k1ℑ32 ∂ 22 ∂ℑ32
11
21
(6.47a)
21
32
,
(6.47k)
,
(6.47l)
,
(6.47m)
32
32
32
32
32
81
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
RPV 2 =
− µ1k1 32 ∂
− µ1k1∂ℑ22
+ µ1k1 22 ∂
32 + µ1k1ℑ22 ∂
22
32
32
=
32
∂
∂ℑ22
82
− 22∂
32 − ℑ22 ∂
22
32
,
(6.47n)
32
where
ℑ32 = jn (k1a2 ), ℑ22 = jn (−k1a2 ), ℑ21 = jn (−k1a1 ), ℑ11 = jn (k1a1 ),
32
= hn(1) (k1a2 ),
22
= hn(1) (−k1a2 ),
∂ℑ32 = [ ρ jn ( ρ ) ]′
∂ℑ21 = [ ρ jn ( ρ ) ]′
∂
∂
32
21
21
= hn(1) (− k1a1 ),
∂ℑ22 = [ ρ jn ( ρ )]′
,
ρ = k1a2
= ρ hn(1) ( ρ ) ′
, ∂
22
ρ = k1a2
,
ρ =− k1a1
= hn(1) (k1a1 ),
(6.48b)
,
(6.48c)
ρ =− k1a2
∂ℑ11 = [ ρ jn ( ρ ) ]′
,
ρ =− k1a1
= ρ hn(1) ( ρ ) ′
11
∂
= ρ hn(1) ( ρ ) ′
11
(6.48a)
,
(6.48d)
ρ = k1a1
,
(6.48e)
.
(6.48f)
ρ =− k1a2
= ρ hn(1) ( ρ ) ′
ρ = k1a1
We still have the results represented in equation (6.15a)-(6.15h).
It is obviously that we can get the following results in this special case:
RFH2 = RFV 2 , TFH1 = TFV1 , TFH2 = TFV2 , TPH1 = TPV1 ,
RPH1 = RPV1 , TPH2 = TPV2 , RPH2 = RPV 2 ,
(6.49)
Therefore, we have
AM13 = AN13 , AM23 = AN23 ,
33
33
23
23
CM = CN , CM = C N .
(6.50)
82
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
83
6.3.4 CALCULATION OF THE ELECTRICAL FIELD
Before we calculate the electric field, we should know some equations first. Rectangular
coordinates can be transformed to spherical coordinates as:
xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ .
(6.51)
Because θ ′ = 0, φ ′ = 0 , we have:
xˆ = θˆ .
(6.52)
Because
n(n + 1)
, m =1
mpnm (cos θ )
,
lim
= 2
θ →0
sin θ
0, otherwise
(6.53a)
n(n + 1)
, m =1
dpnm (cos θ )
,
= 2
θ →0
dθ
0, otherwise
(6.53b)
lim
Applying m = 1 to equation (6.37), we have:
Cmn = 2 ×
2n + 1
1
.
×
n(n + 1) n(n + 1)
(6.54)
From three equations as above, the following equation will be obtained:
mpnn (cos θ )
dp m (cosθ )
2n + 1
= Cmn × lim n
=
.
θ →0
θ →0
sin θ
dθ
n(n + 1)
Cmn × lim
(6.55)
Moreover, we have:
dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′.
(6.56)
Now the electrical fields will be calculated in three cases. For R > a1 :
83
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R ′)iJ (R ′)dV ′ = iωµ1cG e1 (R , R′)i xˆ = iωµ1cG e1 (R , R′)iθˆ,
84
(6.57)
then derived from equations (3.10a), (3.10b) and (6.38a), the result can be obtained as:
E1 (R ) = −
[ k3rjn (k3r )]′ A13N (1) (k ) .
k3ωµ1c ∞ 2n + 1
13
(1)
(0)
(
)
lim
+
i
M
j
A
k
n
∑
M
o1n
1
N
e1n
1
r →0
4π n =1 n(n + 1)
k3 r
(6.58)
Because we still have:
1, n = 0
jn (0) =
,
0, otherwise
(6.59a)
2
, n =1
[k3rjn (k3r )]′
= 3
lim
,
r →0
k3 r
0, otherwise
(6.59b)
2n + 1
3
= ,
n(n + 1) n =1 2
(6.59c)
the electric field can be then represented as follows:
E1 (R ) = −
k3ωµ1c 13 (1)
AN N e11 (k1 ) .
4π
(6.60)
Because k1 = k3 , the above expression can be written as:
E1 (R ) = −
k1ωµ1c 13 (1)
AN N e11 (k1 ) .
4π
(6.61)
For a2 < R < a1 :
E 2 (R ) = iωµ 2 ∫∫∫ G e 2 (R, R′)iJ (R′)dV ′ = iωµ 2 cG e 2 (R, R′)i xˆ = iωµ 2 cG e 2 (R, R′)iθˆ,
(6.62)
then derived from equations (3.10a), (3.10b) and (6.38b), the result can be obtained as:
84
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
E 2 (R ) = −
k3ωµ2 c ∞ 2n + 1
i jn (0) CM23M o1n (k2 ) + AM23M (1)
∑
o1n ( k 2 ) +
4π n =1 n(n + 1)
{
k3 rjn (k3 r ) ]′ 23
[
C N
lim
r →0
85
k3 r
N
(k2 ) .
e1n ( k 2 ) + A N
23
N
(6.63)
(1)
e1n
Applying the equations (6.59a), (6.59b) and (6.59c) to (6.63), the following result can be
given:
E 2 (R ) = −
k3ωµ2 c 23
CN N e11 (k2 ) + AN23 N (1)
e11 ( k 2 ) .
4π
(6.64)
In the metamaterial case, the above expression can be written as:
E 2 (R ) =
k1ωµ1c 23
CN N e11 (−k1 ) + AN23N (1)
e11 ( − k1 ) .
4π
(6.65)
We use the expressions of N e mn ( k ) , i.e., equations (3.10b), together with the properties
o
of the Bessel function and the Hankel function given in equations (6.15a)-(6.15h),
follows can then be derived:
when n is even,
N e mn ( k ) = −N e mn ( −k ) ,
o
(6.66a)
N (1)
( k ) = −N (1)e mn ( −k ) ,
e
mn
(6.66b)
N e mn ( k ) = N e mn ( −k ) .
(6.66c)
o
when n is odd,
o
o
o
o
equations (6.15f) and (6.15h) should also be applied in the odd case.
When R < a2 :
E3 (R ) = iωµ3 ∫∫∫ G e 3 (R, R′)iJ (R′)dV ′ = iωµ3cG e3 (R, R′)i xˆ = iωµ3cG e 3 (R, R′)iθˆ,
(6.67)
Then derived from equations (3.10a), (3.10b) and (6.38c), we have:
85
Characterization of Spherical Metamaterials
Chapter 6
Metamaterial Spherical Shell
E3 ( R ) = −
k3ωµ3c ∞ 2n + 1
i jn (0) M o(1)1n (k3 ) + CM33M o1n (k3 ) +
∑
4π n =1 n(n + 1)
{
k3 rjn (k3 r ) ]′ (1)
[
N (k ) + C 33N
lim
r →0
86
k3 r
e1n
3
N
e1n ( k3 ) .
(6.68)
Applying the equations (6.59a), (6.59b) and (6.59c) to (6.68), we have:
E3 ( R ) = −
k3ωµ3c (1)
N e11 (k3 ) + CN33N e11 (k3 ) .
4π
(6.69)
Because k1 = k3 , the above expression can be written as:
E3 ( R ) = −
k1ωµ1c (1)
N e11 (k1 ) + CN33N e11 (k1 ) .
4π
(6.70)
86
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
7
7.1
87
MULTI-SPHERICAL LAYERS OF METAMATERIALS
INTRODUCTION
In this chapter, the general case will be considered. The structure was represented in
Figure 3.1. One should notes that there is air in the layers n, n − 2, n − 4, n − 6...... ,
therefore, the structure is a periodical structure or a spherical photonic bandgap structure.
The multi-spherical layers under two different circumstances will be studied. One is for
the current distribution located outside the spherical multilayers, the other is for the
current distribution located in the center of the spherical multilayers. In each instance, the
normal case will be first analyzed, and then the special case: spherical layers of
metamaterials.
7.2
CURRENT DISTRIBUTION LOCATED
OUTSIDE
THE
SPHERICAL
MULTILAYERS
7.2.1 INTRODUCTION
The spherical multilayers has a center at (0, 0, 0). An infinitesimal horizontal electric
dipole with current moment c pointing in the x -direction and located at
R ′ = b,θ ′ = 0, φ ′ = 0, which is at the top of the spherical multilayers. Thus we have:
J (R′) = c
δ ( R′ − b)δ (θ ′ − 0)δ (φ ′ − 0)
xˆ.
b 2 sin θ ′
(7.1)
87
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
88
Scattering dyadic Green’s functions can be expressed generally in equation (3.12).
Applying this equation to the case where s = 1, we can get the scattering DGFs of each
region.
For f = 1,
′
G11
es ( R , R ) =
∞
n
ik1
4π
∑∑ C
ik1
4π
∞
n =0 m=0
mn
(1)
BM11M (1)
(7.2a)
′(1)
(k1 )M′e(1)mn (k1 ) + B11
e
N N e mn ( k1 ) N e mn ( k1 ) .
o mn
o
o
o
For f ≠ 1, N ,
G (esf 1) (R, R′) =
n
∑∑ C
n =0 m =0
mn
{ B
M (1)
(k f )M′e(1)mn (k1 ) + BNf 1N (1)
(k f )N′e(1)mn (k1 )
e
e
o mn
o
o mn
o
f1
M
}
(7.2b)
+ D M e mn (k f )M′ (k1 ) + D N e mn (k f )N′ (k1 ) .
o
o
f1
M
(1)
e
o mn
f1
N
(1)
e
o mn
For f = N ,
G es( N 1) (R, R′) =
ik1
4π
∞
n
∑∑ C
n =0 m=0
mn
DMN 1M e (k N )M′e(1) (k1 ) + DNN 1N e (k N )N′e(1) (k1 )
o mn
o mn
o mn
o mn
(7.2c)
where
Cmn = (2 − δ 0 )
2n + 1 (n − m)!
,
n(n + 1) (n + m)!
(7.3)
f1
f1
f1
f1
N1
N1
the coefficients BM11 , B11
N , BM , BN , DM , DN , DM , DN need to be determined, and the
first superscript of DMN 1 , DNN 1 and the subscript of k N all indicate the layer number N .
7.2.2 THE CALCULATION OF THE COEFFICIENTS
The electric dyadic Green functions of each region are presented as :
f = 1,
′
G e1 (R, R′) = G e 0 (R, R′) + G11
es ( R , R ),
(7.4a)
88
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
Gef (R, R′) = Gesf 1(R, R′),
f ≠ 1, N ,
GeN (R, R′) = GesN1(R, R′),
f = N,
89
(7.4b)
(7.4c)
where the electric free space DGF is:
when R > b
G e 0 (R, R′) = −
1 垐
ik
RRδ (R − R′) + 1
2
4π
k1
∑C
mn
M (1) (k1 )M′(k1 ) + N (1) (k1 )N′(k1 ) , (7.5a)
ik
1 ˆˆ
RRδ (R − R′) + 1
2
k1
4π
∑C
mn
M (k1 )M′(1) (k1 ) + N(k1 )N′(1) (k1 ) , (7.5b)
m,n
when R < b
G e 0 (R, R′) = −
m,n
where M e mn and N e mn are given in equations (3.10a) and (3.10b).
o
o
The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering
dyadic Green’s functions. The equations (3.18a), (3.18b), (3.18c) and (3.19a) should be
applied:
A fs
CMfs , N = Mfs, N
CM , N
f
M ,N
T
1
TFfH ,V
= H ,V
RPf
TPfH ,V
1 0
A11 =
, and
0 0
BMfs , N
,
DMfs, N
,
1
H ,V
TPf
H ,V
RFf
TFfH ,V
0 0
A 22 =
.
0 1
CMf 1, N = TMf −, N1 ⋅⋅⋅ TM1 , N ( C11
M , N + A 22 ) .
89
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
90
We can first assume that the field point is located in the last layer: f = N . Because the
coefficients satisfy the equation (3.16i):
AMNs, N CM1s , N
Ns = 1s = 0,
BM , N DM , N
the coefficients in the first and the last layers can be found. After obtaining the
coefficients in the first and last layers, the rest of the coefficients can be derived from the
equations (3.18a), (3.18b), (3.18c) and (3.19a).
Now we apply the method mentioned above to the s = 1 case. First, assume f = N , the
following can be obtained:
0 BM11, N 0 0
0 BMN 1, N
N −1
1
=
⋅⋅⋅
+
T
T
.
M ,N
M ,N
N1
0 DM11, N 0 1
0 DM , N
(7.6)
From equation (3.16i):
N1
11
BMN
= DMN
= 0.
(7.7)
TN = TMN ,−N1 TMN ,−N2 ⋅⋅⋅ TM1 , N ,
(7.8)
Let
then equation (7.6) can be written as:
0 TN11 TN12 0 BM11, N
0
0 D N 1 =
.
T
T
N
N
0
1
,
M
N
21
22
(7.9)
From above equation, we have:
TN11 BM11, N + TN12 = 0,
(7.10a)
DMN 1, N = TN 21 BM11, N + TN 22 ,
(7.10b)
90
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
91
then:
BM11, N = −
DMN 1, N = −
TN12
,
TN11
TN 21TN12
+ TN 22 .
TN11
(7.11a)
(7.11b)
The coefficients of the first and the last layers are given. We will study with the rest of
the coefficients. For f ≠ 1, N , from equations (3.19a) and (7.11a):
0 BM11, N 0 0
0 BMf 1, N
f −1
1
= TM , N ⋅⋅⋅ TM , N
+
.
f1
0
0 0 1
0 DM , N
(7.12)
Let
Tf = TMf −, N1 TMf −, N2 ⋅⋅⋅ TM1 , N ,
(7.13)
together equation (7.11a), then equation (7.12) can be written as:
TN12
0 BMf 1, N Tf11 Tf12 0 −
TN11 .
=
f1
T
T
f
f
0
D
21
22
M ,N
1
0
(7.14)
From above equation, we have:
BMf 1, N = −
Tf11TN12
+ Tf12 ,
TN11
(7.15a)
DMf 1, N = −
Tf 21TN12
+ Tf 22 .
TN11
(7.15b)
TMf , N is represented as equation (3.18b). TFfH ,V , RFfH ,V , TPfH ,V , RPfH ,V were given as equations
from (3.16a) to (3.16h):
91
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
µ f k f +1∂ ( f +1) f ff − µ f +1k f ∂ ff
µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff
RPfH =
( f +1) f
,
( f +1) f
RFfH =
µ f k f +1∂ℑ( f +1) f ℑ ff − µ f +1k f ∂ℑ ff ℑ( f +1) f
,
µ f k f +1∂ℑ( f +1) f ff − µ f +1k f ℑ( f +1) f ∂ ff
RPfV =
µ f k f +1 ( f +1) f ∂ ff − µ f +1k f ff ∂
µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂
TPfH =
( f +1) f
,
( f +1) f
µ f k f +1ℑ( f +1) f ∂ℑ ff − µ f +1k f ℑ ff ∂ℑ( f +1) f
,
µ f k f +1ℑ( f +1) f ∂ ff − µ f +1k f ∂ℑ( f +1) f ff
RFfV =
µ f k f +1 (ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f
µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff
µ f k f +1 (∂ℑ( f +1) f
µ f k f +1∂ℑ( f +1) f
TFfH =
( f +1) f
ff
− ℑ( f +1) f ∂
TFfV =
µ f k f +1 (ℑ( f +1) f ∂
µ f k f +1ℑ( f +1) f ∂
ℑil = jn ( ki al ) ,
il
( f +1) f
ff
− ∂ℑ( f +1) f
)
( f +1) f
,
( f +1) f
)
( f +1) f
− µ f +1k f ℑ( f +1) f ∂
µ f k f +1 (∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂
µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂
TPfV =
where
92
,
ff
( f +1) f
)
,
( f +1) f
( f +1) f
− µ f +1k f ∂ℑ( f +1) f
)
,
ff
= hn(1) ( ki al ) ,
∂ℑil =
1 d [ ρ jn ( ρ ) ]
,
dρ
ρ
ρ =k a
i l
∂
il
(1)
1 d ρ hn ( ρ )
=
ρ
dρ
because
there
, as given in equations from (3.14a) to (3.14d). Additionally,
ρ = ki al
is
air
in
layers
N , N − 2, N − 4, N − 6......
,
when
f = N , N − 2, N − 4, N − 6...... :
ε f = ε1 ,
µ f = µ1 ,
k f = k1.
(7.16)
92
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
93
7.2.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL
Now let us consider the metamaterial case, that is, when f = N − 1, N − 3, N − 5......2 :
µ f = − µ1 .
k f = −k1 ,
(7.17)
Whatever f is, we have
µ f k f +1 = µ f +1k f = − µ1k1.
(7.18)
The equations from (3.16a) to (3.16h) can then be written as:
RPfH =
RFfH =
RPfV =
RFfV =
TPfH =
TFfH =
TPfV =
TFfV =
− µ1k1∂
( f +1) f
− µ1k1ℑ ff ∂
ff
( f +1) f
+ µ1k1∂
ff
( f +1) f
+ µ1k1∂ℑ ff
( f +1) f
− µ1k1∂ℑ( f +1) f ℑ ff + µ1k1∂ℑ ff ℑ( f +1) f
− µ1k1∂ℑ( f +1) f
ff
+ µ1k1ℑ( f +1) f ∂
− µ1k1
ff
+ µ1k1
( f +1) f
− µ1k1∂ℑ ff
∂
( f +1) f
( f +1) f
+ µ1k1ℑ ff ∂
( f +1) f
− µ1k1ℑ( f +1) f ∂ℑ ff + µ1k1ℑ ff ∂ℑ( f +1) f
− µ1k1ℑ( f +1) f ∂
− µ1k1 (ℑ( f +1) f ∂
− µ1k1ℑ ff ∂
( f +1) f
− µ1k1 (∂ℑ( f +1) f
− µ1k1∂ℑ( f +1) f
− µ1k1 (∂ℑ( f +1) f
− µ1k1∂ℑ ff
( f +1) f
− µ1k1 (ℑ( f +1) f ∂
− µ1k1ℑ( f +1) f ∂
( f +1) f
ff
)
=
( f +1) f
− ℑ( f +1) f ∂
− ℑ( f +1) f ∂
+ µ1k1ℑ ff ∂
( f +1) f
ff
− ∂ℑ( f +1) f
( f +1) f
+ µ1k1ℑ( f +1) f ∂
( f +1) f
( f +1) f
+ µ1k1∂ℑ( f +1) f
+ µ1k1∂ℑ ff
( f +1) f
ff
ff
)
=
ff
( f +1) f
)
=
( f +1) f
− ∂ℑ( f +1) f
+ µ1k1∂ℑ( f +1) f
( f +1) f
ff
)
=
ff
∂
ff
=
=
=
=
∂
( f +1) f
ℑ ff ∂
ff
( f +1) f
−∂
ff
( f +1) f
− ∂ℑ ff
( f +1) f
,
(7.19a)
,
(7.19b)
∂ℑ( f +1) f ℑ ff − ∂ℑ ff ℑ( f +1) f
∂ℑ( f +1) f
( f +1) f
∂ℑ ff
∂
ff
ff
( f +1) f
− ℑ( f +1) f ∂
−
ff
∂
( f +1) f
− ℑ ff ∂
( f +1) f
ff
ℑ( f +1) f ∂ℑ ff − ℑ ff ∂ℑ( f +1) f
ℑ( f +1) f ∂
ℑ( f +1) f ∂
ff
− ∂ℑ( f +1) f
,
(7.19c)
,
(7.19d)
ff
( f +1) f
− ∂ℑ( f +1) f
( f +1) f
− ∂ℑ ff
( f +1) f
− ℑ( f +1) f ∂
( f +1) f
ff
− ℑ( f +1) f ∂
ff
∂ℑ( f +1) f
( f +1) f
− ℑ( f +1) f ∂
∂ℑ ff
( f +1) f
− ℑ ff ∂
( f +1) f
− ∂ℑ( f +1) f
( f +1) f
− ∂ℑ( f +1) f
ff
ℑ ff ∂
∂ℑ( f +1) f
∂ℑ( f +1) f
ℑ( f +1) f ∂
ℑ( f +1) f ∂
ff
( f +1) f
, (7.19e)
( f +1) f
( f +1) f
, (7.19f)
, (7.19g)
( f +1) f
. (7.19h)
When f = N − 1, N − 3, N − 5......2 :
93
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
94
′
ℑ ff = jn (− k1a f ), ∂ℑ ff = [ ρ jn ( ρ ) ] ρ =− k a ,
1 f
ff = hn(1) (− k1a f ), ∂ ff = ρ hn(1) ( ρ ) ′
.
k
a
ρ
=−
1 f
(7.20a)
When f = N , N − 2, N − 4, N − 6...... :
′
ℑ( f +1) f = jn (−k1a f ), ∂ℑ( f +1) f = [ ρ jn ( ρ )] ρ =− k a ,
1 f
( f +1) f = hn(1) (−k1a f ), ∂ ( f +1) f = ρ hn(1) ( ρ ) ′
.
=−
k
a
ρ
1 f
(7.20b)
We still have the following results.
j (− z ), When n is even,
jn ( z ) = n
− jn (− z ), When n is odd.
(7.21a)
When n is even:
[ ρ 2 jn ( ρ 2 )]′ = [ ρ1 jn ( ρ1 )]′ ρ =− ρ ,
1
(7.21b)
2
when n is odd:
[ ρ 2 jn ( ρ 2 )]′ = − [ ρ1 jn ( ρ1 )]′ ρ =− ρ .
1
(7.21c)
2
When n is even:
hn(1) ( ρ 2 ) = hn(1) ( ρ1 )
ρ1 =− ρ2
[ ρ 2 hn(1) ( ρ 2 )]′ = [ ρ1hn(1) ( ρ1 )]′
,
ρ1 =− ρ 2
(7.21d)
;
(7.21e)
when n is odd:
94
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
j− n ( ρ1 ) + jn ( ρ1 ) e − iπ n
hn ( ρ 2 ) =
i sin ( nπ )
(1)
95
,
(7.21f)
ρ1 =− ρ2
j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n ′
j− n ( ρ1 ) + jn ( ρ1 ) e− iπ n
[ ρ 2 hn ( ρ 2 )]′ =
+ ρ1
i sin ( nπ )
i sin ( nπ )
(1)
.
(7.21g)
ρ1 =− ρ 2
It is obvious that we can get the following results in this special case:
RFfH = RFfV , TFfH = TFfV ,
H
V
H
V
RPf = RPf , TPf = TPf ,
(7.22)
BM11 = B11
BMf 1 = BNf 1 ,
N ,
f1
f1
N1
N1
DM = DN , DM = DN .
(7.23)
hence
7.2.4 CALCULATION OF THE ELECTRICAL FIELD
Rectangular coordinate can be transformed to spherical coordinate as:
xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ,
(7.24)
For the infinitesimal horizontal electric dipole introduced in the beginning of 7.2, because
θ ′ = 0, φ ′ = 0 , we have:
xˆ = θˆ .
(7.25)
Because
n(n + 1)
, m =1
mpnm (cos θ )
= 2
,
θ →0
sin θ
0, otherwise
lim
(7.26a)
95
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
n(n + 1)
, m =1
dpnm (cos θ )
,
= 2
lim
θ →0
dθ
0, otherwise
96
(7.26b)
Substituting m = 1 into equation (7.3), we have:
Cmn = 2 ×
2n + 1
1
×
.
n(n + 1) n(n + 1)
(7.26c)
From the three equations above, the following equation will be obtained:
mpnn (cos θ )
dp m (cosθ )
2n + 1
= Cmn × lim n
=
.
θ →0
θ
→
0
sin θ
dθ
n(n + 1)
Cmn × lim
(7.27)
Additionally, we have:
dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′.
(7.28)
The electrical fields will be calculated of each region. When R > b :
E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ. (7.29)
Applying the expression of G e1 (R, R′) to the above equation, we have:
E1 (R ) = −
k1ωµ1c ∞ 2n + 1
i jn ( ρb ) + BM11hn(1) ( ρb ) M (1)
∑
o1n ( k1 )
4π n =1 n(n + 1)
{
+ ( ρb jn ( ρb ) )′ + BN11 ( ρb hn(1) ( ρ b ) )′ N e(1)1n (k1 ) ρ b ,
(7.30)
where ρb = k1b. When a1 < R < b :
E 2 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ, (7.31)
Together with the equation (7.5b), we have:
96
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
E 2 (R ) = −
k1ωµ1c ∞ 2n + 1
i hn(1) ( ρb ) M o1n (k1 ) + BM11M o(1)1n (k1 )
∑
4π n =1 n(n + 1)
97
{
(1)
+ ( ρ h ( ρb ) )′ N e1n (k1 ) + B11
N N e1n ( k1 ) ρ b .
(7.32)
(1)
b n
For a f < R < a f −1 , where f ranges from 2 to N − 1 :
E f +1 (R ) = iωµ f ∫∫∫ G ef (R, R′)iJ (R′)dV ′ = iωµ f cG ef (R, R′)i xˆ = iωµ f cG ef (R, R′)iθˆ, (7.33)
apply the related equations to the above equation, we have:
E f +1 (R ) = −
k1ωµ f c
4π
∞
2n + 1
∑ n(n + 1) i{h
(1)
n
n =1
( ρb ) DMf 1M o1n (k f ) + BMf 1M (1)
o1n ( k f )
+ ( ρ h ( ρb ) )′ DNf 1N e1n (k f ) + BNf 1N (1)
e1n ( k f ) ρ b .
(7.34a)
(1)
b n
when f = N − 2, N − 4, N − 6...... , the following expression will be obtained:
E f +1 (R ) = −
k1ωµ1c ∞ 2n + 1
i hn(1) ( ρb ) DMf 1M o1n (k1 ) + BMf 1M (1)
∑
o1n ( k1 )
4π n =1 n(n + 1)
{
+ ( ρ h ( ρb ) )′ DNf 1N e1n (k1 ) + BNf 1N (1)
e1n ( k1 ) ρ b .
(7.34b)
(1)
b n
In the metamaterial case, when f = N − 1, N − 3, N − 5......2, equation (7.34a) can be
rewritten as:
E f +1 (R ) =
k1ωµ1c ∞ 2n + 1
i hn(1) ( ρb ) DMf 1M o1n (− k1 ) + BMf 1M (1)
∑
o1n ( − k1 )
4π n =1 n(n + 1)
{
+ ( ρ h ( ρb ) )′ DNf 1N e1n (− k1 ) + BNf 1N (1)
e1n ( − k1 ) ρ b .
(7.34c)
(1)
b n
From the expressions of M e mn ( k ) and N e mn ( k ) , i.e., equations (3.10a) and (3.10b),
o
o
together with the properties of the Bessel function and the Hankel function shown in
equations (7.21a)-(7.21g), the following can then be obtained:
97
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
when n is even,
M e mn ( k ) = M e mn ( −k ) ,
(7.35a)
N e mn ( k ) = − N e mn ( −k ) ,
(7.35b)
M (1)
( k ) = M (1)e mn ( −k ) ,
e
mn
(7.35c)
N (1)
( k ) = −N (1)e mn ( −k ) ,
e
mn
(7.35d)
M e mn ( k ) = −M e mn ( −k ) ,
(7.35e)
N e mn ( k ) = N e mn ( −k ) .
(7.35f)
o
o
o
o
o
o
o
when n is odd,
98
o
o
o
o
o
When R < aN −1 :
E N +1 (R ) = iωµ N ∫∫∫ G eN (R, R′)iJ (R′)dV ′ = iωµ N cG eN (R, R′)i xˆ
= iωµ N cG eN (R, R′)iθˆ.
(7.36)
Applying the related equations to the above equation, we have:
E N +1 (R ) = −
k1ωµ N c ∞ 2n + 1
i{hn(1) ( ρb ) DMN 1M o1n (k N )
∑
4π n =1 n(n + 1)
+ ( ρ h ( ρb ) )′ DNN 1N e1n (k N ) ρb .
(7.37a)
(1)
b n
Because k1 = k N , (7.37a) can be rewritten as:
E N +1 (R ) = −
k1ωµ1c ∞ 2n + 1
i{hn(1) ( ρb ) DMN 1M o1n (k1 )
∑
4π n =1 n(n + 1)
+ ( ρ h ( ρb ) )′ DNN 1N e1n (k1 ) ρb .
(7.37b)
(1)
b n
98
Characterization of Spherical Metamaterials
Chapter 7
7.3
Muti-spherical Layers of Metamaterials
CURRENT
DISTRIBUTION
LOCATED
INSIDE
99
THE
SPHERICAL
MULTILAYERS
7.3.1 INTRODUCTION
The spherical multilayers has a center at (0, 0, 0). An infinitesimal horizontal electric
dipole with current moment c pointing in the x -direction and located at
R′ = 0,θ ′ = 0, φ ′ = 0 (the center of the spherical multilayers), let:
J (R′) = c
δ ( R′ − 0)δ (θ ′ − 0)δ (φ ′ − 0)
xˆ.
R′2 sin θ ′
(7.38)
Scattering dyadic Green’s functions can be expressed generally in equation (3.12).
Applying this equation to the special case that s = N , we can get the scattering DGFs of
each region. For f = 1,
G1esN (R, R′) =
ik N
4π
∑C
mn
m,n
AM1N M (1) (k1 )M′(k N ) + AN1N N (1) (k1 )N′(k N ) .
(7.39a)
For f ≠ 1, N ,
G esfN (R, R′) =
ik N
4π
∑ C { A
fN
M
mn
m,n
M (1) (k f )M′(k N ) + ANfN N (1) (k f )N′(k N )
}
(7.39b)
+ C M (k f )M′(k N ) + C N(k f )N′(k N ) .
fN
M
fN
N
For f = N ,
G esNN (R, R′) =
ik N
4π
∑C
m,n
mn
CMNN M (k N )M′(k N ) + CNNN N(k N )N′(k N ) ,
(7.39c)
where
Cmn = (2 − δ 0 )
2n + 1 (n − m)!
,
n(n + 1) (n + m)!
(7.40)
99
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
100
the coefficients AMN 1 , ANN 1 , AMfN , ANfN , CMfN , CNfN , CMNN , CNNN need to be determined, and the
superscript of coefficients and the subscript of k N all indicate the layer number N .
7.3.2 THE CALCULATION OF THE COEFFICIENTS
The electric dyadic Green functions of each region are presented as follows respectively:
f = 1,
f ≠ 1, N ,
G e1 (R, R′) = G1esN (R, R′),
(7.41a)
Gef (R, R′) = GesfN (R, R′),
(7.41b)
GeN (R, R′) = Ge0 (R, R′) +GesNN (R, R′),
f = N,
(7.41c)
where the electric free space DGF is:
because R > 0
G e 0 (R, R′) = −
ik
1 ˆˆ
RRδ (R − R′) + N
2
4π
kN
∑C
m,n
mn
M (1) (k N )M′(k N ) + N (1) (k N )N′(k N ) , (7.42)
where M e mn and N e mn are given in equation (3.10a) and (3.10b), and Cmn is shown as
o
o
equation (7.40).
The equations introduced in Chapter 3 will be used to obtain the coefficients of scattering
dyadic Green’s functions. Because s = N , equations from (3.18a), (3.18b), (3.18c) and
(3.19c) should be applied:
A fs
CMfs , N = Mfs, N
CM , N
BMfs , N
,
DMfs, N
100
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
1
TFfH ,V
= H ,V
RPf
TPfH ,V
f
M ,N
T
1 0
A11 =
, and
0 0
101
,
1
TPfH ,V
H ,V
RFf
TFfH ,V
0 0
A 22 =
.
0 1
CMfN, N = TMf −, N1 ⋅⋅⋅ TM1 , N C1MN, N − H ( f − N ) A11.
We can first assume that the field point is located in the last layer: f = N . Because the
coefficients satisfy equation (3.16i):
AMNs, N CM1s , N
Ns = 1s = 0,
BM , N DM , N
the coefficients in the first and the last layers can be found. After obtaining the
coefficients in the first and last layers, the rest of the coefficients can be derived from the
equations from (3.18a), (3.18b), (3.18c) and (3.19c).
Firstly, by assuming f = N , we have:
AMNN, N
NN
CM , N
0
N −1
1
= TM , N ⋅⋅⋅ TM , N
0
AM1N, N
1N
CM , N
0 1 0
.
−
0 0 0
(7.43)
From equation (3.16i):
NN
1N
AMN
= CMN
= 0.
(7.44)
TN = TMN ,−N1 TMN ,−N2 ⋅⋅⋅ TM1 , N ,
(7.45)
Let
then the equation (7.43) can be written as:
101
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
0
C NN
M ,N
0 TN11 TN12 AM1N, N
=
0 TN 21 TN 22 0
102
0 1 0
−
.
0 0 0
(7.46)
Deriving the above equation, we have:
TN11 AM1N, N − 1 = 0,
(7.47a)
CMNN, N = TN 21 AM1N, N ,
(7.47b)
then:
AM1N, N =
1
,
TN11
(7.48a)
CMNN, N =
TN 21
.
TN11
(7.48b)
The coefficients of the first and the last layers now are given. We will study the rest of
the coefficients. For f ≠ 1, N , from equations (3.19c) and (7.48a):
AMfN, N
fN
CM , N
0
AM1N, N
f −1
1
=
⋅⋅⋅
T
T
M ,N
M ,N
0
0
0
.
0
(7.49)
Let
Tf = TMf −, N1 TMf −, N2 ⋅⋅⋅ TM1 , N ,
(7.50)
together with equation (7.48a), then equation (7.49) can be rewritten as:
AMfN, N
fN
CM , N
1
, 0
0 Tf11 Tf12
TN11
.
=
0 Tf 21 Tf 22
0
0
(7.51)
Deriving the above equation, we have:
102
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
103
AMfN, N =
Tf11
,
TN11
(7.52a)
CMfN, N =
Tf 21
.
TN11
(7.52b)
TMf , N is represented in equation (3.18b). TFfH ,V , RFfH ,V , TPfH ,V , RPfH ,V were given in equations
from (3.16a) to (3.16h):
RPfH =
µ f k f +1∂ ( f +1) f ff − µ f +1k f ∂ ff
µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff
( f +1) f
,
( f +1) f
RFfH =
µ f k f +1∂ℑ( f +1) f ℑ ff − µ f +1k f ∂ℑ ff ℑ( f +1) f
,
µ f k f +1∂ℑ( f +1) f ff − µ f +1k f ℑ( f +1) f ∂ ff
RPfV =
µ f k f +1 ( f +1) f ∂ ff − µ f +1k f ff ∂
µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂
RFfV =
TPfH =
TPfV =
TFfV =
,
( f +1) f
µ f k f +1ℑ( f +1) f ∂ℑ ff − µ f +1k f ℑ ff ∂ℑ( f +1) f
,
µ f k f +1ℑ( f +1) f ∂ ff − µ f +1k f ∂ℑ( f +1) f ff
µ f k f +1 (ℑ( f +1) f ∂ ( f +1) f − ∂ℑ( f +1) f
µ f k f +1ℑ ff ∂ ( f +1) f − µ f +1k f ∂ℑ ff
TFfH =
( f +1) f
µ f k f +1 (∂ℑ( f +1) f
µ f k f +1∂ℑ( f +1) f
( f +1) f
ff
− ℑ( f +1) f ∂
µ f k f +1 (ℑ( f +1) f ∂
µ f k f +1ℑ( f +1) f ∂
( f +1) f
ff
− ∂ℑ( f +1) f
,
( f +1) f
)
( f +1) f
− µ f +1k f ℑ( f +1) f ∂
µ f k f +1 (∂ℑ( f +1) f ( f +1) f − ℑ( f +1) f ∂
µ f k f +1∂ℑ ff ( f +1) f − µ f +1k f ℑ ff ∂
)
( f +1) f
,
ff
( f +1) f
)
,
( f +1) f
( f +1) f
− µ f +1k f ∂ℑ( f +1) f
)
,
ff
103
Characterization of Spherical Metamaterials
Chapter 7
where
Muti-spherical Layers of Metamaterials
ℑil = jn ( ki al ) ,
il
104
= hn(1) ( ki al ) ,
∂ℑil =
1 d [ ρ jn ( ρ ) ]
,
dρ
ρ
ρ =k a
i l
∂
il
(1)
1 d ρ hn ( ρ )
=
ρ
dρ
because
there
, as shown in equations from (3.14a) to (3.14d). Additionally,
ρ = ki al
is
air
in
layers
N , N − 2, N − 4, N − 6......
of
,
when
f = N , N − 2, N − 4, N − 6...... :
ε f = ε1 ,
µ f = µ1 ,
k f = k1.
(7.53)
7.3.3 COEFFICIENTS IN THE SPECIAL CASE OF METAMATERIAL
Let us consider the metamaterial case, that is, for f = N − 1, N − 3, N − 5......2 :
µ f = − µ1 .
k f = −k1 ,
(7.54)
It is obvious that whatever f is,
µ f k f +1 = µ f +1k f = − µ1k1.
(7.55)
The equations from (3.16a) to (3.16h) can then be written as:
RPfH =
RFfH =
RPfV =
− µ1k1∂
( f +1) f
− µ1k1ℑ ff ∂
ff
( f +1) f
+ µ1k1∂
ff
( f +1) f
+ µ1k1∂ℑ ff
( f +1) f
− µ1k1∂ℑ( f +1) f ℑ ff + µ1k1∂ℑ ff ℑ( f +1) f
− µ1k1∂ℑ( f +1) f
ff
+ µ1k1ℑ( f +1) f ∂
− µ1k1
ff
+ µ1k1
( f +1) f
− µ1k1∂ℑ ff
∂
( f +1) f
=
ff
∂
( f +1) f
+ µ1k1ℑ ff ∂
( f +1) f
ff
=
=
∂
( f +1) f
ℑ ff ∂
ff
( f +1) f
−∂
ff
( f +1) f
− ∂ℑ ff
( f +1) f
,
(7.56a)
,
(7.56b)
∂ℑ( f +1) f ℑ ff − ∂ℑ ff ℑ( f +1) f
∂ℑ( f +1) f
( f +1) f
∂ℑ ff
∂
ff
ff
( f +1) f
− ℑ( f +1) f ∂
−
ff
∂
( f +1) f
− ℑ ff ∂
( f +1) f
ff
,
(7.56c)
104
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
RFfV =
TPfH =
TFfH =
TPfV =
TFfV =
− µ1k1ℑ( f +1) f ∂ℑ ff + µ1k1ℑ ff ∂ℑ( f +1) f
− µ1k1ℑ( f +1) f ∂
− µ1k1 (ℑ( f +1) f ∂
− µ1k1ℑ ff ∂
( f +1) f
( f +1) f
− µ1k1 (∂ℑ( f +1) f
− µ1k1 (∂ℑ( f +1) f
− µ1k1∂ℑ ff
ff
− µ1k1 (ℑ( f +1) f ∂
ff
( f +1) f
ff
)
=
( f +1) f
− ℑ( f +1) f ∂
− ℑ( f +1) f ∂
+ µ1k1ℑ ff ∂
( f +1) f
)
=
ff
( f +1) f
)
=
( f +1) f
− ∂ℑ( f +1) f
( f +1) f
+ µ1k1∂ℑ( f +1) f
ff
( f +1) f
− µ1k1ℑ( f +1) f ∂
− ∂ℑ( f +1) f
+ µ1k1ℑ( f +1) f ∂
( f +1) f
( f +1) f
+ µ1k1∂ℑ( f +1) f
+ µ1k1∂ℑ ff
( f +1) f
− µ1k1∂ℑ( f +1) f
ff
)
=
=
105
ℑ( f +1) f ∂ℑ ff − ℑ ff ∂ℑ( f +1) f
ℑ( f +1) f ∂
ff
− ∂ℑ( f +1) f
,
ℑ( f +1) f ∂
( f +1) f
− ∂ℑ( f +1) f
ℑ ff ∂
( f +1) f
− ∂ℑ ff
( f +1) f
− ℑ( f +1) f ∂
( f +1) f
ff
− ℑ( f +1) f ∂
ff
∂ℑ( f +1) f
( f +1) f
− ℑ( f +1) f ∂
∂ℑ ff
( f +1) f
− ℑ ff ∂
( f +1) f
− ∂ℑ( f +1) f
( f +1) f
− ∂ℑ( f +1) f
ff
∂ℑ( f +1) f
∂ℑ( f +1) f
ℑ( f +1) f ∂
ℑ( f +1) f ∂
ff
(7.56d)
ff
( f +1) f
, (7.56e)
( f +1) f
( f +1) f
, (7.56f)
, (7.56g)
( f +1) f
. (7.56h)
When f = N − 1, N − 3, N − 5......2 :
′
ℑ ff = jn (− k1a f ), ∂ℑ ff = [ ρ jn ( ρ ) ] ρ =− k a ,
1 f
ff = hn(1) (− k1a f ), ∂ ff = ρ hn(1) ( ρ ) ′
.
ρ
=−
k
a
1 f
(7.57a)
When f = N , N − 2, N − 4, N − 6...... :
′
ℑ( f +1) f = jn (−k1a f ), ∂ℑ( f +1) f = [ ρ jn ( ρ )] ρ =− k a ,
1 f
( f +1) f = hn(1) (−k1a f ), ∂ ( f +1) f = ρ hn(1) ( ρ ) ′
.
=−
ρ
k
a
1 f
(7.57b)
Equations (7.21a)- (7.21g) should be applied to (7.57).
We can obtain easily the following results in this special case:
H
V
H
V
RFf = RFf , TFf = TFf ,
H
V
H
V
RPf = RPf , TPf = TPf ,
(7.58)
105
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
106
therefore,
AM1N = AN1N , AMfN = ANfN ,
fN
fN
NN
NN
CM = CN , CM = CN .
(7.59)
7.3.4 CALCULATION OF THE ELECTRICAL FIELD
Rectangular coordinate can be transformed to spherical coordinate as:
xˆ = sin θ cos φ rˆ + cos θ cos φθˆ − sin φφˆ.
(7.60)
For the infinitesimal horizontal electric dipole introduced in the beginning of 7.3, for
θ ′ = 0, φ ′ = 0 , we have:
xˆ = θˆ .
(7.61)
Because
n(n + 1)
, m =1
mpnm (cos θ )
,
= 2
lim
θ →0
sin θ
0, otherwise
(7.62a)
n(n + 1)
, m =1
dpnm (cos θ )
.
= 2
θ →0
dθ
0, otherwise
(7.62b)
lim
Applying m = 1 to equation (7.40), we have:
Cmn = 2 ×
2n + 1
1
×
.
n(n + 1) n(n + 1)
(7.62c)
From the three equations as above, the following equation will be obtained:
mpnn (cos θ )
dp m (cosθ )
2n + 1
= Cmn × lim n
=
.
θ →0
θ →0
sin θ
dθ
n(n + 1)
Cmn × lim
(7.63)
Additionally, we have:
106
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
dV ′ = R′2 dR′ sin θ ′dθ ′dφ ′.
107
(7.64)
The electrical fields will be calculated in each region. When R > a1 :
E1 (R ) = iωµ1 ∫∫∫ G e1 (R, R′)iJ (R′)dV ′ = iωµ1cG e1 (R, R′)i xˆ = iωµ1cG e1 (R, R′)iθˆ. (7.65)
Applying the expression of G e1 (R, R′) to the above equation, we have:
k N rjn (k N r ) ]′ 1N (1)
[
k N ωµ1c ∞ 2n + 1
1N
(1)
E1 (R ) = −
i jn (0) AM M o1n (k1 ) + lim
AN N e1n (k1 ) . (7.66)
∑
r
0
→
4π n =1 n(n + 1)
kN r
Because we still have:
1, n = 0
jn (0) =
,
0, otherwise
(7.67a)
2
, n =1
[k3rjn (k3r )]′
= 3
lim
,
r →0
k3 r
0, otherwise
(7.67b)
2n + 1
3
= ,
n(n + 1) n =1 2
(7.67c)
the electric field can be then represented as follows:
E1 (R ) = −
k N ωµ1c 1N (1)
AN N e11 (k1 ) .
4π
(7.68)
Because k1 = k N , (7.68) can be rewritten as:
E1 (R ) = −
k1ωµ1c 1N (1)
AN N e11 (k1 ) .
4π
(7.69)
107
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
a f < R < a f −1 ,
For
f
ranges
from
108
2
to
N −1
:
E f (R ) = iωµ f ∫∫∫ G ef (R, R′)iJ (R′)dV ′ = iωµ f cG ef (R, R′)i xˆ = iωµ f cG ef (R, R′)iθˆ, (7.70)
applying the related equations to the above equation, we have:
E f (R ) = −
k N ωµ f c
4π
2n + 1
∞
∑ n(n + 1) i{ j (0) C
n
n =1
[ k rj (k r )]′ C fN N
lim N n N
r →0
kN r
N
fN
M
M o1n (k f ) + AMfN M (1)
o1n ( k f ) +
+
N
(
)
(
)
k
A
k
e1n
f
f .
fN
N
(7.71)
(1)
e1n
Applying equation (7.67), the following result can be obtained:
E f (R ) = −
k N ωµ f c
4π
CNfN N e11 (k f ) + ANfN N (1)
e11 ( k f ) .
(7.72a)
When f = N − 2, N − 4, N − 6...... , the following expression will be derived:
E f (R ) = −
k1ωµ1c
C NfN N e11 (k1 ) + ANfN N (1)
e11 ( k1 ) .
4π
(7.72b)
In the metamaterial case, when f = N − 1, N − 3, N − 5......2, equation (7.34a) can be
written as:
E f (R ) =
k1ωµ1c
CNfN N e11 (−k1 ) + ANfN N (1)
e11 ( − k1 ) .
4π
(7.72c)
We use the expressions of N e mn ( k ) , i.e., equation (3.10b), together with the properties of
o
the Bessel function and the Hankel function given in equation (7.21), we have:
when n is even,
N e mn ( k ) = −N e mn ( −k ) ,
(7.73a)
N (1)
( k ) = −N (1)e mn ( −k ) ,
e
mn
(7.73b)
N e mn ( k ) = N e mn ( −k ) .
(7.73c)
o
o
o
when n is odd,
o
o
o
108
Characterization of Spherical Metamaterials
Chapter 7
Muti-spherical Layers of Metamaterials
109
E N (R ) = iωµ N ∫∫∫ G eN (R, R′)iJ (R′)dV ′ = iωµ N cG eN (R, R′)i xˆ
(7.74)
When R < aN −1 :
= iωµ N cG eN (R, R′)iθˆ.
Applying the related equations to the above equation, we have:
E N (R ) = −
k N ωµ N c ∞ 2n + 1
NN
i jn (0) M (1)
∑
o1n ( k N ) + CM M o1n ( k N )
4π n =1 n(n + 1)
{
[ k rj (k r )]′ N (1) (k
+ lim N n N
r →0
kN r
e1n
N
)+C
NN
N
N e1n (k N ) .
(7.75)
From equation (7.67), we obtain:
E N (R ) = −
k N ωµ N c (1)
N e11 (k N ) + CNNN N e11 (k N ) .
4π
(7.76a)
Because k1 = k N , the above expression can be derived as:
E N (R ) = −
k1ωµ1c (1)
N e11 (k1 ) + CNNN N e11 (k1 ) .
4π
(7.76b)
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Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
8
110
SIMULATION RESULTS
In this chapter, the simulation results are presented and discussed. Here we consider the
case that every other layer is air. The chapter is divided into three parts. The first part is
about the case of metamaterial sphere, i.e., m = 2, where m is the number of the layers.
Metamaterial spherical shell ( m = 3 ) is studied in the second part. The last part is
concentrated on the case m = 5 .
8.1
SIMULATION RESULTS AND DISCUSSION OF METAMATERIAL SPHERE
a
1
2
k2
k1
Figure 8.1 Structure of the metamaterial sphere
Regarding the truncation number n for the expansion expressions of the field
components, it is dependent upon the electric radius of the sphere: the bigger the radius is;
the larger is the number needed. We usually truncate the series based on experience and
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Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
111
numerical test. Here, n is from 1 to 20 when the radius of the sphere is 5λ or less. CPU
frequency used in our simulation is 800MHz, the simulations ( n = 30 ) each took half an
hour to finish.
The bright circles in the figures from Fig. 8.2 to Fig. 8.6 exist because of the
discontinuity of the free space dyadic Green’s function in R = b . When k2 = k1 , as shown
in Figure 8.2, region 2 can also be considered as air. After coming out from the dipole,
the radiated energy will spread very quickly while not focus. When k2 = 2k1 , as shown in
Figure 8.3, the refractive angles are less than those in the k2 = k1 case, therefore, some of
the refractive waves are focused near the surface of the sphere in the opposite side. The
smaller the angles between the incident waves and z -axis are, the more likely the
refractive waves will be focused. When k2 = 4k1 , as shown in Figure 8.4, the angles
between the refractive waves and the normal are even less than those in the k2 = 2k1 case.
Therefore, there are more refractive waves that will be focused. We then find that the
density of the focused area when k2 = 4k1 is much more than that in the k2 = 2k1 case.
Now let us consider metamaterial cases. When k2 = − k1 , as shown in Figure 8.5, because
of the negative refraction, i.e., the incident waves and the refractive waves are in the
same side of the normal, the refractive waves focus near the metamaterial spherical
surface quickly. When k2 = −2k1 , as shown in Figure 8.6, the angles between the
refractive waves and the normal are even smaller than that in the k2 = −k1 case, together
111
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
112
with the negative refraction, the focused area is much larger and the density is much less
than that in the k2 = −k1 case.
Figure 8.2 Near field of metamaterial sphere when k2 = k1 = k0 , a = 1λ , b = 2λ .
Figure 8.3 Near field of metamaterial sphere when k2 = 2k1 , a = 1λ , b = 2λ .
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Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
113
Figure 8.4 Near field of metamaterial sphere when k2 = 4k1 , a = 1λ , b = 2λ .
Figure 8.5 Near field of metamaterial sphere when k2 = − k1 , a = 1λ , b = 2λ .
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Characterization of Spherical Metamaterials
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Simulation Results
114
Figure 8.6 Near field of metamaterial sphere when k2 = −2k1 , a = 1λ , b = 2λ .
We can find that the density of the focused area when k2 = 4k1 even can be comparable
with the case when k2 = − k1 .
8.2
SIMULATION RESULTS
AND
DISCUSSION
OF
METAMATERIAL
SPHERICAL SHELL
Regarding the truncation number n for the expansion expressions of the field
components, it is dependent upon the electric radius of the sphere: the bigger the radius is;
the larger is the number needed. We usually truncate the series based on experience and
114
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
115
numerical test. Here, n is from 1 to 30 when the radius of the sphere is 10λ or less. CPU
used in our simulation is 2GHz, the simulations ( n = 30 ) each took six hours to finish.
Current distribution is located outside the spherical shell. When k2 = k1 = k3 , as shown in
Figure 8.8, the radiated energy spreads from the dipole. When k2 = 2k1 = 2k3 , as shown
in Figure 8.9, some of the refractive waves are focused, but most of the refractive waves
radiate. When k2 = −k1 = − k3 , because of the negative refraction, the waves radiate after
they are focused near the interface between region 1 and region 2. The refractive waves
focus again very near the interface between region 2 and region 3 when they arrive at the
region 3. After the second focus, the waves spread again.
a1
1
a2
3
2
k3
k2
k1
Figure 8.7 Structure of the metamaterial spherical shell
115
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
116
1
Figure 8.8 Near field of metamaterial spherical shell when k2 = k1 , a1 = 4λ , a2 = 2λ .
Figure 8.9 Near field of metamaterial spherical shell when k2 = 2k1 . a1 = 4λ , a2 = 2λ .
116
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
117
Figure 8.10 Near field of metamaterial spherical shell when k2 = − k1 , a1 = 4λ , a2 = 2λ .
Figure 8.11 Near field of metamaterial spherical shell when k2 = −2k1 , a1 = 4λ , a2 = 2λ .
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Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
118
Because the focus is very near to the interface, the radiate waves will spread around near
the interface, as shown in Figure 8.10. When k2 = −2k1 = −2k3 , as shown in Figure 8.11,
the refractive waves are focused on the area that is far away from the interface more than
on the area in the k2 = −k1 = − k3 case. Then, the refractive waves will focus again when
leaving region 3 and entering region 2.
8.3
SIMULATION RESULTS
AND
DISCUSSION
OF
METAMATERIAL
SPHERICAL MULTILAYERS
Here we consider the m = 5 case, i.e., region 1, 3 and 5 are air.
a1
a2
a3
1
2
3
5
k5
4
a4
k4
k3
k2
k1
Figure 8.12 Structure of the spherical mutilayers ( m = 5 )
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Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
119
Figure 8.13 Near field of 5-layer metamaterial spheres when k2 = k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ .
Figure 8.14 Near field of 5-layer metamaterial spheres when k2 = 2k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ .
119
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
120
Figure 8.15 Near field of 5-layer metamaterial spheres when k2 = − k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 10 .
Figure 8.16 Near field of 5-layer metamaterial spheres when k2 = − k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 .
120
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
121
Figure 8.17 Near field of 5-layer metamaterial spheres when k2 = − k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 30 .
Figure 8.18 Near field of 5-layer metamaterial spheres when k2 = − k1 ,
a1 = 5λ , a2 = 4λ , a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 .
121
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
122
Figure 8.19 Near field of 5-layer metamaterial spheres when k2 = −2k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 20 .
Figure 8.20 Near field of 5-layer metamaterial spheres when k2 = −2k1 ,
a1 = 4λ , a2 = 3.5λ , a3 = 2.5λ , a4 = 2λ , b = 5λ , n = 30 .
122
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
123
Figure 8.21 Near field of 5-layer metamaterial spheres when k2 = −2k1 ,
a1 = 5λ , a2 = 4λ , a3 = 3λ , a4 = 2λ , b = 5.5λ , n = 30 .
The truncation number n for the expansion expressions of the field components is
dependent upon the electric radius of the sphere: the bigger the radius is; the larger is the
number needed. The series are usually truncated based on experience and numerical test.
Here, n is from 1 to 30 when the radius of the sphere is 10λ or less. The computer used
in our simulation has a CPU frequency of 2GHz, each of the simulations ( n = 30 ) took
nine hours to finish.
When k2 = k4 = k1 , as shown in Figure 8.13, the waves radiated from the dipole will not
focus. When k2 = k4 = 2k1 , as shown in Figure 8.14, refractive waves focus in each
region, with the density of the focused area becoming less. Most of the refractive waves
radiate. When k2 = k4 = −k1 , as shown in Figures 8.17 and 8.18, because of the negative
123
Characterization of Spherical Metamaterials
Chapter 8
Simulation Results
124
refraction, the refractive waves focus on the area near the interface in every region, and
then they radiate. We can find from the figures that the waves focus very well in this case.
When k2 = k4 = −2k1 , as shown in Figures 8.20 and 8.21, the angles between the
refractive waves and the normal in this case are much smaller than the angles in the case
k2 = k4 = −k1 . The refractive waves and the incident waves are in the same side of the
normal because of the negative refraction. Therefore, the waves focus slower than the
k2 = k4 = −k1 case, and much more energy will be radiated because they do not been
focused.
From the discussion above, we know that the spherical metamaterials are able to focus
waves. The smaller the absolute value of k is, the stronger the focus capability is. We
also see that the normal spheres can also focus waves to some extent, especially when k
is big, which is contrary to the metamaterial cases. Because it causes singularity when the
dipole is located in the center of the spheres, the results of these cases have already been
studied, so we only pay attention to the cases when the dipole is located outside the
spheres in this chapter.
124
Characterization of Spherical Metamaterials
Chapter 9
Summary
125
9
SUMMARY
In this degree thesis project, metamaterial spherical multilayers are studied by using the
spherical dyadic Green’s function. Basic theories of metamaterials, the Green’s functions
and scattering of spheres are described. The special and general structures of spherical
multilayers are examined.
Metamaterials are very different materials from the conventional materials. They behaves
with very interesting characteristics. Metamaterials have negative refractive index. They
can maketheoretically perfect lens. Electromagnetic waves travel in them with quite
different routes from the normal materials.
Spherical dyadic Green’s functions are described in Chapter 3. Eigenfunctions used for
dyadic Green’s functions in unbounded media, i.e., free space DGF, are first given. Then,
the scattering dyadic Green’s functions for spherical multilayered media are described.
To obtain the coefficients of the DGF, the recurrence matrix equations for the coefficients
of scattering dyadic Green’s functions are given.
The scattering of normal material spheres is introduced in Chapter 4. Rayleigh scattering
and Mie scattering are described firstly. These two kinds of scattering are the basis of
understanding the scattering of spheres. In succession, scattering of two separate spheres
as a special example is analyzed in particular.
125
Characterization of Spherical Metamaterials
Chapter 9
Summary
126
Three cases of the spherical multilayers are considered respectively: the single sphere, the
spherical shell and the spherical multilayers (general case). Both the conventional
materials and the metamaterials are studied. Two circumstances where the dipole is
located outside the structure, and the dipole locates inside the structure are examined,
respectively. The analysis and discussion of the simulation results are given afterwards.
This project can be extended to cylindrical metamaterials case, which is a very promising
area.
126
Characterization of Spherical Metamaterials
Reference
127
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Characterization of Spherical Metamaterials
[...]... (field) layer ( f = 1, 2, , N ) of the spherically N -layered system Figure 3.1 Geometry of a spherically multilayered medium 18 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 19 3.2 FUNDAMENTAL FORMULATION 3.2.1 EIGENFUNCTION EXPANSION OF DGF IN UNBOUNDED MEDIA Electromagnetic problems can always be solved by beginning with the analysis of Maxwell’s equations The... of the refractive index of a nearly transparent and passive medium is usually taken to have only positive values Through an analysis of a current source radiating into a metamaterial, it can be determined that the sign of the real part of the refractive index is actually negative The regime of negative index leads to unusual electromagnetic wave 5 Characterization of Spherical Metamaterials Chapter 2... xz plane for a 30 incidence of a Gaussian beam upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 and µ = − µ0 This is the simulated result by using the formula developed in reference [6] 16 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 17 3 SPHERICAL DYADIC GREEN’S FUNCTIONS 3.1 INTRODUCTION Green’s function is named in honor of English mathematician and... geometry requiring a new formulation This is the main limitation of Green’s function technique 17 Characterization of Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 18 This chapter will present a detailed description of electromagnetic dyadic Green’s function in spherically multilayered media Figure 3.1 shows the spherically N -layered geometry The transmitter with an arbitrary... simulation results of the single sphere and spherical multi layers are given Chapter 9 gives a summary of the whole thesis 1.4 ORIGINAL CONTRIBUTION Conference paper: 3 Characterization of Spherical Metamaterials Chapter1 Introduction 4 Le-Wei Li, Ningyun Huang, Qun Wu and Zhong-Cheng Li, "Macroscopic Characteristics of Electromagnetic Waves Radiated by a Dipole in the Presence of Metamaterial Sphere... incidence of a Gaussian This is the simulated result by using the formula developed in reference [6] 15 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 16 2 0 x/λ -2 -4 -6 -8 -10 -2 -1 0 1 2 3 4 5 6 7 8 9 10 z/λ Figure 2.5 Time-averaged power density on the xz plane for a upon a grounded slab of thickness d = 6λ with ε1 = −ε 0 30° incidence of a Gaussian... [7]: ε (ω ) = 1 − ω 2p , ω2 (2.6) where the plasma frequency ω p is related to the geometry of the wire array 7 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 8 We can assume that the wire and the SRR arrays do not interact directly, the index of the refraction of this structure can be presented as: n(ω ) = ε (ω ) µ (ω ) Thus, it can be obtained that... Spherical Metamaterials Chapter 3 Spherical Dyadic Green’s Function 22 the multiple reflected and transmitted waves due to the presence of the spherical boundary of the dielectric layered media Using the contour integration method in the complex h -plane, the dyadic Green’s function in the unbounded medium can be represented in terms of the normalized spherical vector wave functions as a result of the... 1, 2, , N ) are the coefficients of the scattering DGF to be solved, and the superscript N is the number of the layers of the multi-layer medium Same as the expression of the unbounded dyadic Green’s function, the superscript (1) indicates that the third-type spherical Bessel function or the first-type spherical Hankel function should be chosen in the function of the spherical wave vector functions:... a Dipole in the Presence of Metamaterial Sphere ({Invited})", Proc of 5th Asia-Pacific Engineering Research Forum on Microwaves and Electromagnetic Theory, Kyushu University, Fukuoka, Japan, July 29-30, 2004 4 Characterization of Spherical Metamaterials Chapter 2 Background and Introduction of Metamaterial 5 2 BACKGROUND INTRODUCTION OF METAMATERIAL After metamaterial was first introduced in 1968 by ... layer ( f = 1, 2, , N ) of the spherically N -layered system Figure 3.1 Geometry of a spherically multilayered medium 18 Characterization of Spherical Metamaterials Chapter Spherical Dyadic Green’s... contribution of 21 Characterization of Spherical Metamaterials Chapter Spherical Dyadic Green’s Function 22 the multiple reflected and transmitted waves due to the presence of the spherical boundary of. .. limitation of Green’s function technique 17 Characterization of Spherical Metamaterials Chapter Spherical Dyadic Green’s Function 18 This chapter will present a detailed description of electromagnetic