UNIVERSIDADE TÉCNICA DE LISBOA
INSTITUTO SUPERIOR TÉCNICO
Metamaterials with Negative
Permeability and Permittivity:
Analysis and Application
José Manuel Tapadas Alves
Dissertation submitted for obtaining the degree of
Master in Electrical and Computer Engineering
Jury
President: Professor José Manuel Bioucas Dias
Supervisor: Professor Carlos Manuel Dos Reis Paiva
Co-Supervisor: Professor António Luís Campos da Silva Topa
Member: Professor Sérgio de Almeida Matos
October, 2010
Abstract
In this dissertation we study and analyze the electromagnetic phenomena of media with negative permeability and permittivity, called DNG metamaterials, and how this leads into some
physical phenomena such as the appearance of backward waves and the emergence and implications of negative refraction.
Two simple DNG waveguiding structures are also studied: the DPS-DNG interface and the
DNG slab. As a DNG medium is necessarily dispersive, the utilization of a known dispersive
model, the Lorentz Dispersive Model, is used in the analysis of the DPS-DNG interface in
order to obtain physical signicant results. The appearance of super-slow modes in the DNG
slab propagation is also a subject of interest.
Finally we address the lens design using DNG metamaterials. The dependence on the refractive
index of this design process is evidenced. The particular structure of the DNG Veselago's at
lens is also analyzed in order to study a potentially practical application of DNG metamaterials
in optics and the implications of dealing with such materials as this lens structure overruns
some conventional limitations, allowing propagating waves to be brought to a single point
focus producing an image that has sub-wavelength detail.
Keywords
Double Negative Media, Metamaterials, Negative Refraction, Backward Waves, Planar Waveguides, Lens Design, Superlens, Microwaves, Photonics
i
Sumário
Nesta dissertação são estudados e analisados os fenómenos electromagnéticos associados aos
meios com permeabilidade e permitividade negativas, designados por meios duplamente negativos (DNG), e como esta característica leva ao aparecimento de alguns fenómenos sicos,
como por exemplo o surgimento de ondas regressivas, e o aparecimento, e implicações, de um
índice de refracão negativo.
São também estudadas duas estruturas simples de propagação guiada, mas utilizando meios
DNG: a interface DPS-DNG e a placa DNG. Como um meio DNG é necessáriamente dispersivo,
a utilização de um modelo dispersivo conhecido, como o modelo de Lorentz, é usado para a
análise da interface DPS-DNG, com vista a obter resultados sicamente signicativos.
O
aparecimento de modos super-lentos na propagação na placa DNG é também um assunto em
análise.
Finalmente é focado o estudo do desenho de lentes usando metamateriais DNG. É evidenciada a
dependência deste processo de desenho em relação ao índice de refracção. Estudamos também
a estrutura particular de uma lente DNG chamada lente plana de Veselago com vista a analisar
uma potencial aplicação e implicações deste tipo de materiais, já que este tipo de lente supera
algumas limitações de lentes convencionais permitido que as ondas propagadas sejam focadas
num único ponto produzindo uma imagem com um detalhe ao nível de comprimentos inferiores
ao comprimento de onda.
Palavras-Chave
Meios Duplamente Negativos, Metamateriais, Refracção Negativa, Ondas Regressivas, Guias
de Onda Planares, Desenho de Lentes, Superlentes, Microondas, Fotónica
ii
Acknowledgements
I would like to express my gratitude to both Professor Carlos Paiva and Professor António
Topa for the continuous support on the development of this work.
Without the help, the
suggestions, comments and the share of knowledge from these two professors the realization
of this dissertation would not be possible.
I also want to thank my family and friends who have always supported me.
My last acknowledgment goes to my colleagues who are working at the 4th Floor's work-room
of the IST's North Tower for the helpful and cheerful moments that have provided me during
the development of this work.
iii
Contents
Abstract
i
Keywords
i
Sumário
ii
Palavras-Chave
ii
Acknowledgements
iii
List of Figures
ix
List of Tables
x
Nomenclature
xi
List of Symbols
xii
1 Introduction
1
1.1
Historical Background
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Motivation and Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
iv
1.3
Structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
1.4
Main Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2 Electromagnetics of Double Negative (DNG) Media
11
2.1
Medium Characterization
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.2
Negative Refraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
2.3
The Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
28
2.3.1
A DNG interval using the Lorentz Model
. . . . . . . . . . . . . . . . .
31
2.3.2
A DNG interval using the Drude Model
. . . . . . . . . . . . . . . . . .
33
2.4
Group Velocity and Phase Velocity . . . . . . . . . . . . . . . . . . . . . . . . .
34
2.5
Kramers-Kronig Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
3 Guided Wave Propagation in DNG Media
3.1
3.2
3.3
Propagation on a Planar DNG-DPS Interface
39
. . . . . . . . . . . . . . . . . . .
39
3.1.1
Modal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
3.1.2
Surface Mode Propagation . . . . . . . . . . . . . . . . . . . . . . . . . .
43
3.1.2.1
Neglecting Losses in the LDM (ΓL
= 0)
3.1.2.2
Considering Losses in the LDM (ΓL
. . . . . . . . . . . . .
= −0.05 × ωpe )
44
. . . . . .
49
. . . . . . . . . . . . . . . . . . . . . .
54
3.2.1
Modal Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
3.2.2
Surface Mode Propagation . . . . . . . . . . . . . . . . . . . . . . . . . .
58
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Propagation on a DNG Slab Waveguide
v
4 Lens Design Using DNG Materials
69
4.1
Optical Path and the Lens Contour . . . . . . . . . . . . . . . . . . . . . . . . .
69
4.2
The Veselago's Flat Lens . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
72
4.3
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
5 Conclusions
81
5.1
Summary
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
Future Work
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
81
84
86
vi
List of Figures
1.1
Photo of a nonlinear tunable metamaterial. The close-up photo square shows
a split-ring resonator with variable-capacity diode.
(Source: Ilya, Shadrivov,
Australian National University, Nonlinear Physics Centre, Australia, 2008)
?
. .
4
. .
5
1.2
Metamaterial at lens consisting of an array of 3 by 20 by 20 unit cells. [ ]
2.1
Material Classication
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
The permittivity in the complex plan . . . . . . . . . . . . . . . . . . . . . . . .
21
2.3
Spatial Representation of the elds, the energy ux and the propagation constant for a DPS and a DNG medium . . . . . . . . . . . . . . . . . . . . . . . .
26
2.4
Scattering of a wave that incises on a DPS-DNG interface
27
3.1
The planar interface between a DPS and a DNG medium, here represented by
. . . . . . . . . . . .
a dashed line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Lorentz lossless dispersive model for
3.3
Relative refraction index (nr
DNG interface
=
εr,L
and
µr,L
. . . . . . . . . . . . . . . . .
β(ω),
Dispersion relation,
3.5
Attenuation constants
α1
the DPS-DNG interface
45
√ n ), using the lossless LDM, on the DPSε0 µ0
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
39
using the lossless LDM, on the DPS-DNG interface
46
.
47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
and
α2
for the TE modes, using the lossless LDM, on
vii
3.6
Attenuation constants
α1
α2
and
for the TM modes, using the lossless LDM,
on the DPS-DNG interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7
Lorentz dispersive model for
εr,L
3.8
Relative refraction index (nr
=
and
µr,L
. . . . . . . . . . . . . . . . . . . . .
Dispersion relation,
β(ω),
α1
3.10 Attenuation constants
the DPS-DNG interface
3.11 Attenuation constants
DPS-DNG interface
49
√ n ), using the lossy LDM, on the DPS-DNG
ε0 µ0
interface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.9
48
using the lossy LDM, on the DPS-DNG interface.
50
.
51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
52
and
α1 ,
α2 ,
for the TE modes, using the lossy LDM, on
for the TM modes, using the lossy LDM, on the
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.12 Variation of the electric eld,
Ey (t = 0, x, z),
on the DPS-DNG Interface . . . .
3.13 A DNG slab waveguide immersed on a DPS media
. . . . . . . . . . . . . . . .
52
53
54
3.14 The representation of the modal solutions (red dots) given by the intersection
of the curves for a DPS slab with
ε1 = µ 1 = 1
and
ε2 = µ2 = 2.
. . . . . . . . .
60
3.15 The representation of the modal solutions (red dots) given by the intersection
of the curves for a DNG slab with
ε1 = µ1 = 1 , ε2 = µ2 = −1.5
and
V = 0.5
.
61
. .
62
3.16 The representation of the modal solutions (red dots) given by the intersection
of the curves for a DNG slab with
ε1 = µ1 = 1 , ε2 = µ2 = −1.5
3.18 Modal solutions (red dots) for a DNG slab with
with
(i) V =
µ1
|µ2 | and
(ii) V =
π
2
ε1 = µ 1 = 1
,
and
V =3
ε2 = µ2 = −2,
. . . . . . . . . . . . . . . . . . . . . . . . . .
3.17 Dispersion diagram for a DNG slab with
ε1 = µ 1 = 1
ε2 = µ2 = −1.5
. . .
63
3.19 Dispersion diagram for a DNG slab with
ε1 = 2, µ1 = 1 , ε2 = −1 and µ2 = −1.5
65
3.20 Dispersion diagram for a DNG slab with
ε1 = 2, µ1 = 2 , ε2 = −1 and µ2 = −1.5
65
viii
and
62
4.1
Lens contour and optical path representation
4.2
The lenses contours for dierent refraction indexes,
4.3
Passage of light waves through a Veselago at lens,
focused image,
4.4
i.f.:
the internal focus point
. . . . . . . . . . . . . . . . . . .
n = −2.5, −1.5, 100, 1.5, 2.5
A:
the image source,
69
71
B:
. . . . . . . . . . . . . . . . . . . .
73
Evanescent eld variation in the presence of the Veselago's at lens. . . . . . . .
78
ix
List of Tables
3.1
Simulation parameters for the Lorentz Dispersive Model, on the DPS-DNG
interface structure
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
x
44
Nomenclature
BW
Backward Waves
DNG
Double Negative Medium
DPS
Double Positive Medium
ENG
Epsilon Negative Medium
LDM
Lorentz Dispersive Model
MNG Mu Negative Medium
NRI
Negative Refraction Index
TE
Transverse Electric
TM
Transverse Magnetic
xi
List of Symbols
Transverse attenuation constant in
αi
medium
i
B
Magnetic ux density
β
Propagation constant
c
Velocity of light
d
Thickness of dielectric slab
χe
Electric susceptibility
χm
Magnetic susceptibility
D
Electric ux density
E
Electric eld intensity
Ex
Electric eld (x-axis)
xii
Ey
Electric eld (y-axis)
Ez
Electric eld (z-axis)
ε
Electric permittivity
ε0
Electric permittivity (Real Part)
ε00
Electric permittivity (Imaginary Part)
ε0
Electric permittivity (Vaccum)
εi
Electric permittivity (Medium
η
Wave impedance
ζ
Normalized wave impedance
H
Magnetic eld intensity
Hx
Magnetic eld (x-axis)
Hy
Magnetic eld (y-axis)
Hz
Magnetic eld (z-axis)
hi
Transverse wavenumber (Medium
k
Wave vector
xiii
i)
i)
k
Wavenumber
k0
Wavenumber (Vaccum)
kx
Transverse wavenumber (x-axis)
ky
Transverse wavenumber (y-axis)
kz
Transverse wavenumber (z-axis)
ki
Wavenumber (Medium
S
Poyting Vector
Sav
Time-averaged of the Poynting vector
µ
Magnetic permeability
µ0
Magnetic permeability (Real Part)
µ00
i)
Magnetic permeability (Imaginary
Part)
µ0
Magnetic permeability (Vaccum)
µi
Magnetic permeability (Medium
n
Refractive index
xiv
i)
n0
Refractive index (vaccum)
n0
Refractive index (Real Part)
n00
Refractive index (Imaginary Part)
ni
Refractive index (Medium
vp
Phase Velocity
vg
Group Velocity
ω
Angular Frequency
ΓL
Lorentz damping coecient
χL
Lorentz coupling coecient
χe
Electric susceptibility
Mi
Magnetization eld
Zi
Impedance (Medium
t
Transmission Coecient
r
Reection Coecient
Ts
Overall Transmission Coecient
xv
i)
i)
Rs
Overall Reection Coecient
xvi
Chapter 1
Introduction
1.1
Historical Background
The study of the fundamental theories about the true nature of electricity have been challenging scientists for centuries. The rst empirical observations and written documents about
electric physical phenomena have their origins in ancient Egypt, from about 3000 B.C.E.,
which referred to the study of electric shocks produced by sh, who are described as the
Thunderers of Nile. These kind of phenomena have also fascinated and inuenced the stud-
?
ies made by the following civilizations (Greeks, Roman, Arabic, ...) [ ]. Ancient writers, such
as Pliny the Elder (23 C.E.) and Scribonius Larges (47 C.E.), wrote about the eect of electric
shocks delivered by shes and concluded about the guiding phenomenon of theses shocks
?
along conducting objects [ ]. Some ancient cultures also observe that some materials, as they
were rubbed against fur, could small attract objects.
Based on this observation Thales of
Miletos (600 B.C.E.) wrote some results about the nature of static electricity, where some
amber objects, after being rubbed, rendered magnetic properties in contrast with other ma-
?
terials that needed no rubbing, such as magnetite [ ]. Even thought Thales was incorrect by
believing that the nature of the attraction phenomenon was magnetic, later on science could
1
prove that there was in fact a direct link between magnetism and electricity.
The recognition about a connection between both the electric and magnetic phenomena was
made by André-Marie Ampère and Hans Christian Ørsted in the beginning of the XIX century
?
[ ] This electromagnetic unication theory, rst observed by Michael Faraday but extended by
James Clerk Maxwell, and then partially reformulated by Oliver Heaviside and Heinrich Hert,.
is one of the key accomplishments of XIX century mathematical physics. After Maxwell's publication of his Treatise of Treatise on Electricity and Magnetism (1873 C.E.), electricity and
magnetism were no longer two separate physical phenomena. He experimentally demonstrated
that: a) Electric charges attract or repel one another with a force inversely proportional to
the square of the distance between them: unlike charges attract, like ones repel; b) Magnetic
poles (or states of polarization at individual points) attract or repel one another in a similar way and always come in pairs: every north pole is yoked to a south pole; c) An electric
current in a wire creates a circular magnetic eld around the wire, its direction depending
on that of the current; and d) current is induced in a loop of wire when it is moved towards
or away from a magnetic eld, or a magnet is moved towards or away from it, the direction
of current depending on that of the movement. The equations obtained by Maxwell, along
with the Lorentz force law (that was also derived by Maxwell under the name of Equation
for Electromotive Force, fully describe classical electromagnetism. These equations have also
been the starting point for the development of relativity theory by Albert Einstein and are
still fundamental to physics and engineering. These equations show the existence of electromagnetic waves, propagating in vacuum and in matter, and seemingly dierent phenomena
like radio waves, visible light, and X-rays are then understood, by interpreting them all as
propagating electromagnetic waves with dierent frequency which is of major scientic and
?
engineering importance even nowadays [ ]
As a consequence of the development of the comprehension of electromagnetism many researchers have explored the interaction between electromagnetic elds and specic media. Articial electromagnetic materials, with negative permeability and permittivity, have pr oven to
2
have extraordinary electromagnetic properties. The study of these kind of articial materials
appears in the end of the XIX century, when Bose published his work on the rotation of the
?
plane of polarization by man-made twisted structures in 1898 [ ]. Lindman studied articial
?
chiral media formed by a collection of randomly oriented small wire helices in in 1914 [ ]. After
wards, there were several other investigators in the rst half of the XX century who studied
various man-made materials. In the 1950s and 1960s, articial dielectrics were explored by
?
Kock, and its application for lightweight microwave antenna lenses [ ]. The `bedofnails' wire
?
grid medium was used in the early 1960s to simulate wave propagation in plasmas [ ]. The
research on these kind of articial materials increased as the development of various potential
?
device and component applications appear [ ].
?
Veselago published a paper in 1967 [ ], but it was only translated to English in 1968, where
he considered a homogeneous isotropic electromagnetic material in which the permittivity
and permeability assumed negative real values.
He studied the uniform wave propagation
?
that kind of material, which he named as left-handed (LH) material [ ].
He concluded
that, in such medium, the direction of the Poynting vector of the wave is the opposite of
its phase velocity, suggesting that this isotropic medium supports a so called backwardwave propagation and that its refractive index can be negative. Since such materials were
not available until recently, the interesting concept of negative refraction, and its various
electromagnetic and optical consequences, suggested by Veselago had received little attention.
?
This was until Smith inspired by the work of Pendry [ ] constructed a composite medium
in the microwave regime by arranging periodic arrays of small metallic wires and split-ring
resonators and demonstrated the anomalous refraction at the boundary of this medium, which
?
is the result of negative refraction in this articial medium [ ].
3
Figure 1.1: Photo of a nonlinear tunable metamaterial. The close-up photo square shows a
(Source: Ilya, Shadrivov, Australian National University, Nonlinear Physics Centre, Australia, 2008)
split-ring resonator with variable-capacity diode.
The negative refractive index propriety of DNG metamaterials could be used to bring radiation
?
to a focus with a at metamaterial lens, as proposed by Veselago [ ] and then expanded by
?
Pendry [ ]. The advantage of a at lens in comparison to a conventional curved lens is that
the focal length could be varied simply by adjusting the distance between the lens and the
electromagnetic wave source. These lens could be constructed using the split-ring resonator
conguration in a periodic array of metallic rings and wires, based on work by researchers at
? ?].
the University of California at San Diego [ ,
?
A photograph of the at lens array of DNG
metamaterial cells, constructed by NASA [ ] is showed in Figure 1.2.
4
?
Figure 1.2: Metamaterial at lens consisting of an array of 3 by 20 by 20 unit cells. [ ]
For microwave radiation at wavelengths about 10 times a cell length, this conguration provides
negative eective values of electric permittivity and magnetic permeability, resulting in a
negative value for the index of refraction.
The NASA Glenn Research Center testing have
demonstrated that appears a reversed refraction eect with focusing of the microwave radiation
and nite element models are being developed and an optics ray tracing code in order to create
new lens designs and to develop new congurations that are more amenable for operation
at higher frequencies.
These research intends to achieve the applications of a at lens for
?
biomedical imaging and detection and other applications [ ].
5
1.2
Motivation and Objectives
With the introduction of these new physical properties of DNG metamaterials, the study
and interpretation of the associated results is in fact very attractive and challenging. There
are many established physical concepts that must be re-interpreted in order to comply with
this new paradigm and there is also the probability of nding new eects associated with
this kind of materials, since there is a whole new set of resulting physical phenomena. In this
dissertation we have the possibility to associate and consolidate the more conventional and well
known electromagnetic concepts but now, with the introduction of the DNG metamaterials,
in a more generalized perspective, as we study the physical eects found even on simple
guiding structures. As up to today the demonstrations and experiments of the new physical
phenomena associated with DNG metamaterials have lead to the construction of new types
of microwave structures whose applications to mobile communication systems have attracted
a lot of attention from the scientic community. These metamaterials could help improve the
performance of several communication devices, such as antennas, and a lot of eort is being
made on the the design of antennae using this kind of periodic structures.
This kind of material also have implications on lens design.
As classical electrodynamics
impose a resolution limit when imaging using conventional lenses, since this fundamental
limit, called the diraction limit, in its ultimate form, is attributed to the nite wavelength
of electromagnetic waves, the introduction of metamaterial lenses is also a subject of great
interest since no longer the resolution is restricted by the wavelength of the propagated light
waves.
Conventional lenses focuses only the propagating waves, resulting in an imperfect
image of the object. The ner spatial details (which are smaller than a wavelength) of the
object, carried by the evanescent waves, are lost due to the strong attenuation these waves
?
experience when traveling from the object to the image. As predicted by Pendry [ ], with DNG
metamaterial lenses, the evanescent waves are amplied by just the right amount. These waves
can be brought to a focus at the same position as an object's radiative eld, thereby producing
6
an image that has sub-wavelength detail.
As this kind of materials promise, for optical and microwave, new applications such as, for
example, new types of beam stirrers, modulators, band-pass lters, high resolution lenses,
microwave couplers, and antenna radomes, the study and research on these results is in fact
very encouraging and motivating for anyone who looks into addressing this subject of DNG
metamaterials.
The main objective of this dissertation is the analysis and study of wave propagation in DNG
metamaterial guides, and also the application of this kind of materials in lens design, taking
advantage of its particular electromagnetic properties to achieve results that are not present
in conventional lenses.
We try to understand the new physical phenomena that are associated with double negative
media and the eects when applied to well known propagation guide structures. The study
of lens design using DNG materials is also addressed in order to verify the dierent results
between these kind of lens against the physical limitations of common DPS lenses.
1.3
Structure
The rst chapter of this dissertation has the single purpose of introducing and situating the
reader in the subject that is addressed in this work.
In order to do so a brief historical
background analysis is included at rst, where key researchers, publications an results are
mentioned, chronologically, in order to understand the evolution of the research process that
eventually reached to the object of study in this work. In this introductory chapter we also
expose the main motivations and objectives of this dissertation, as well as this explanation of
the work's structure.
In the second chapter we study the electromagnetic phenomena associated with DNG metamaterials. After formulating the classication of a specic medium as DNG, the implications
7
of having a negative permittivity and permeability leads into studying the characterization
of the medium and the physical phenomena such as the appearance of backward waves and
the emergence and implications of negative refraction. A dispersive analysis is also introduced
in this chapter as we study the Lorentz Dispersive Model and nd a possible frequency interval in which a material can act as DNG. The introduction of dispersion helps us infers
about the nature of behavior of both the phase and group velocities when dealing with DNG
metamaterials.
The third chapter deals with the guided wave propagation with DNG materials.
We have
chosen to study two simple structures, the DPS-DNG interface and te DNG slab. A modal
analysis was made for both wave guiding structures and also numerical simulations, with the
respective interpretations.
As a DNG medium is necessarily dispersive, the utilization of a
known dispersive model, the Lorentz Dispersive Model, is used in the analysis of the DPSDNG interface, and the results with and without the introduction of losses are compared. The
appearance of super-slow modes in the DNG slab propagation is also a subject of analysis on
this subject, consequence of having a phase velocity that is smaller than the outer medium in
which a DNG slab is immersed. The existence of these super-slow modes enables the propagation on the DNG slab even if we use a less dense medium for the slab and this phenomenon
is studied as also. Even though we are using simple wave guiding structures and a somehow
elementary study when addressing the DNG guided propagation it proves to be an ecient
mechanism in order to evidenced this new physical problems and paradigm.
The fourth chapter is dedicated to the study of lens design using DNG metamaterials. First
we address a way to achieve a desired contour for a lens using physical concepts as the optical
path. The dependence of the refractive index on this process evidences the implications that
having a NIR medium as the material for designing lenses. The particular structure of the
Veselago's at lens, that is basically a DNG slab, is also analyzed in this chapter in order to
study a potentially practical application of DNG metamaterials in optics and the implications
of dealing with such materials. The conventional limitations of lens design when dealing with
8
sub-wavelenght detail are overrun by this DNG at lens as with DNG metamaterial lenses,
the evanescent waves are amplied by just the right amount allowing the waves to be brought
to single point focus at the same position as an object's radiative eld on the other side of the
lens and producing an image that has sub-wavelength detail. These results are also studied in
this chapter.
Finally, in the fth chapter, main conclusions are exposed and some developing potential
applications and further investigation hypothesis of the subjects addressed in this dissertation
are introduced.
1.4
Main Contributions
The main contribution of this dissertation is the analysis of known electromagnetic phenomena
but introducing the DNG metamaterials proprieties and concepts into the study of these physical subjects, hopefully helping further research on this kind of eld. The particular physical
phenomena that are generated by the usage of these materials when dealing with waveguides
or even with the design of lenses can provide a better comprehension of the potential of interest
when designing structures, being it communication devices or other physical components that
take advantage of the DNG media proprieties.
9
10
Chapter 2
Electromagnetics of Double Negative
(DNG) Media
Electromagnetic waves interact with the inclusions of particulate composite materials, inducing
magnetic and electric moments, which aects the macroscopic eective permittivity of the
bulk composite medium. Nowadays, metamaterials can be synthesized by articial fabricated
inclusions in an arbitrary host surface or host medium which provides the designer a wide set
of degrees of freedom, such as the host's size and shape and the composition's density and
alignment of the inclusions, in order to create a specic electromagnetic response that is not
found individually in each of the constituents.
2.1
Medium Characterization
Let us consider a specic material that is characterized by the two electromagnetic macroscopic
constitutive parameters: the electrical permittivity
ε
and the magnetic permeability
µ.
As
opposed to the response in vacuum, the response of materials to external elds generally
depends on the frequency of the eld, which reects the fact that a material's polarization
11
does not respond instantaneously to an applied eld. For this reason both the permittivity and
the permeability are often treated as complex functions of the frequency of the applied eld,
?
since complex numbers allow the specication of magnitude and phase [ ]. These parameters
can both be described as follows:
with
with
ε0 , ε00 <.
ε = ε0 + ε00
(2.1)
µ = µ0 + µ00
(2.2)
And:
µ0 , µ00 <
We can now proceed to the classication of the medium by analyzing the value of both
ε0 and µ0
(the real parts of the permittivity and permeability). A medium with both the permittivity and
permeability greater than zero (<(ε)
(DPS),
> 0 , <(µ) > 0
) is called a
Double Positive Medium
designation in which most naturally occurring media fall into (i.e.
dielectrics).
medium with the permittivity less than zero and the permeability greater than zero (<(ε)
0 , <(µ) > 0
) is called an
A
<
Epsilon Negative Medium (ENG), characteristic than can be
found, for certain frequency regimes, in many plasmas. A medium with permittivity greater
than zero and the permeability less than zero (<(ε)
> 0 , <(µ) < 0
) is designated by
Mu
Negative Medium (MNG), characteristic which, for certain frequency regimes, is exhibited
by some gyrotropic materials. A medium with both permittivity and permeability less than
zero (<(ε)
< 0 , <(µ) < 0
) is designated as a
Double Negative Medium (DNG),
this
characteristic has only been demonstrated, up to this date, in articially constructed materials
?
[ ].
Figure 2.1 shows the location of each medium qualication in a diagram whose axis is
formed by
ε0 = <(ε)
and
µ0 = <(µ).
12
Figure 2.1: Material Classication
Let us now consider a generic media were both the constitutive parameters can be written as
functions of the frequency:
D = ε0 ε(ω)E
(2.3)
B = µ0 µ(ω)H
(2.4)
Let us now consider that the electric eld is polarized along the
wave propagates in the
z-axis
x-axis and the electromagnetic
direction. We can write the expressions for both the electric
and magnetic elds in the time and z-axis domain:
E = xbE0 exp[i(kz − ωt)]
(2.5)
H = ybH0 exp[i(kz − ωt)]
(2.6)
13
Where the complex wave number
k, is given by:
k = kb
z
with
k = nk0
(where
n
(2.7)
is the refraction index).
The vacuum wave-number
k0 ,
is given by:
ω
√
k0 = ω ε0 µ0 =
c
where
c
(2.8)
is the speed of light.
Let us now consider the
Maxwell Equations:
∇×E=−
∂B
∂t
∇×H=J+
To allow us to transform both
E
and
H
∂D
∂t
(2.9)
(2.10)
∇·D=ρ
(2.11)
∇·E=0
(2.12)
from the time domain to the frequency domain we
use the following Fourier transform pair:
Tω (r, ω) =
ˆ
+∞
−∞
tω (r, t) exp[iωt]dt
14
(2.13)
1
tω (r, t) =
2π
where
ˆ
+∞
−∞
Tω (r, ω) exp[−iωt]dt
(2.14)
k = xb
x + yb
y + zb
z.
Applying (2.13) and (2.14) to (2.9)-(2.12) we obtain:
∇ × E = iω B(ω)
(2.15)
∇ × H = J(ω) − iω D(ω)
(2.16)
∇·D=ρ
(2.17)
∇·E=0
(2.18)
In order to express the spatial dependence of the eld quantities in (2.9)-(2.12) in the algebraic
form, we introduce the three-dimensional Fourier transform pair, which allows us to obtain
the Maxwell Equations in the wave number domain (or k-space):
Tk (r, k) =
tk (r, ω) = (
where
ˆ
+∞
−∞
1 3
)
2π
k = xb
x + yb
y + zb
z , dk = dkx dky dkz
Finally we can now work in the (k
− ω)
ˆ
tk (r, t) exp[−ik.r]dr
+∞
−∞
and
Tk (r, t) exp[−ik.r]dk
(2.19)
(2.20)
k.r = kx xb + ky yb + kz zb.
space by subjecting all eld quantities to a four-fold
Fourier transform given by the following transform pair:
15
Tk−ω (r, k) =
1
tk−ω (r, ω) = ( )4
2π
where
ˆ
+∞
−∞
ˆ
+∞
−∞
k = xb
x + yb
y + zb
z , dk = dkx dky dkz
tk (r, t) exp[iωt − ik.r]drdt
(2.21)
Tk−ω (r, ω) exp[ik.r − iωt]dkdω
(2.22)
and
k.r = kx xb + ky yb + kz zb.
By using (2.21) and (2.22) to transform (2.9)-(2.12) we now obtain:
ik × E = iω B
(2.23)
ik × H = J − iω D
(2.24)
−ik · D = ρ
(2.25)
−ik · E = 0
(2.26)
J
Assuming the inexistence of the conduction current ( =0) we now have from (2.23) and (2.24):
k × E = ω B = ωµ0 µH
(2.27)
k × H = −ω D = −ωε0 εE
(2.28)
Now from (2.5) we can write:
16
k × E = ybωµ0 µH0 exp[i(kz − ωt)]
(2.29)
k × H = −b
xωε0 εE0 exp[i(kz − ωt)]
(2.30)
As we assumed, the electric eld
polarized along the
y-axis
E
is polarized along the
x-axis
and the magnetic eld is
in a way that the electromagnetic waves propagate trough the
z-axis, in the direction of k.
Assuming that the media is isotropic we can state that:
k·E = k · H = 0
(2.31)
?
So, from (2.25)-(2.28) now we can say that, as in[ ]:
|k|E0 − ωµ0 µH0 = 0
(2.32)
−ωε0 εE0 + |k|H0 = 0
(2.33)
Or in its matricial form:



|k|




−ωµ0 µ   E0   0 
= 

−ωε0 ε
|k|
H0
0
(2.34)
As we are not trying to nd the solution were there are neither an electric nor a magnetic eld
(E0
= H0 = 0)
we will equal the matrix determinant to zero:
|k|2 − ω 2 µ0 µε0 ε = 0
17
(2.35)
From (2.7) and with (2.8) we can write:
|k|2 =
This will allow us to dene the
ω2
µε = k02 µε
c2
(2.36)
wave impedance, the ratio between the transverse components
?
of the electric and magnetic elds , [ ]:
η=
E0
ωµµ0
|k|
=
=
H0
|k|
ωεε0
(2.37)
Now we can dene the frequency defendant refraction index n from (2.36) , (2.7) and (2.8) :
n=
And we can also dene the
√
µε
(2.38)
normalized wave impedance, the relation between the intensities
of the electric and the magnetic eld:
η
ζ=
=
η0
with
r
n
µ
µ
= =
ε
ε
n
η0 being the free space intrinsic wave impedance.
(2.39)
As we have seen before, the polarization
does not respond instantaneously to an applied eld. This causes dielectric loss, which can
be expressed by a permittivity and permeability that is both complex and frequency dependent. Real materials are not perfect insulators either, i.e. they have non-zero direct current
?
conductivity [ ]. Taking both aspects into consideration, we can dene a complex refraction
index:
n = n0 + in00
where
n0
is the refractive index indicating the phase velocity coecient and
18
(2.40)
n
00
is called the
extinction coecient, which indicates the amount of absorption loss when the electromagnetic wave propagates through the material. Both
?
n0
[ ]. Let us now dene, based on (2.40) and (2.7), the
and
n00
are dependent on the frequency
phase velocity vp , of an electromagnetic
wave:
vp =
ω
ω
c
=
= 0
<(k)
k0 <(n)
n
(2.41)
We can now write the complex amplitude equations for both the electric and the magnetic
eld using (2.40):
E = xbE0 exp[ink0 z] = xbE0 exp[−n00 k0 z] exp[in0 k0 z]
H = yb
E0
E0
exp[ink0 z] = yb
exp[−n00 k0 z] exp[in0 k0 z]
ζη0
ζη0
(2.42)
(2.43)
The Time-Average Poynting Vector, which can be thought of as a representation of the energy
ux of the electromagnetic eld, is given by:
1
2
Sav = <(E × H∗ )
(2.44)
Using the expressions (2.42) and (2.43) on (2.44) we obtain:
Sav
In this case the value of
n00
|E0 |2
1
= zb
<
exp[−2n00 k0 z]
η0
ζ
(2.45)
needs to be always positive in order to verify energy extinction
along with the propagation of the wave on the z-axis, as expected since we are dealing with a
passive media where:
19
lim |E| ≤ E0
(2.46)
z→∞
Now to take conclusions about the direction of the power ux we need to analyze the sign
of
S.
As we can see from (2.45) it depends on the sign of the real part of the normalized
impedance's value. From (2.39) we have:
0
0
n0 µ + n00 µ00
n
n + in00
1
=<
=< 0
=
<
ζ
µ
µ + iµ00
µ0 2 + µ00 2
(2.47)
As we have seen above, we are dealing with a passive media, so, as we concluded from (2.46),
we have:
00
00
n00 > 0 → k 00 > 0 → µ , ε > 0
From (2.48), and by knowing that we are dealing with DNG media (µ
(2.48)
0
0
, ε < 0),
we can easily
verify that in (2.47) the divisor is always positive but we really can't conclude, at this moment,
about the sign of
S
because the sign of (2.47) may depend on the sign of
n
0
(present at its
dividend). In (2.38) we have established a relation between the refraction index and both the
permittivity and permeability, so we will use that in order to infer about the nature of
n = nµ nε =
We will now study the permittivity
ε
√ √
µ ε
in the complex plan using polar coordinates.
important to notice that we have chosen to study
ε
0
n.
(2.49)
(It is
but the analysis is exactly the same for
the permeability).
ε = ρε exp[iθε ]
20
(2.50)
?
Graphically represented, as proposed in [ ], by Figure 2.2.
Figure 2.2: The permittivity in the complex plan
We can also dene the permittivity dependent part of the refraction index in polar coordinates:
nε =
Using (2.50) we have for
√
0
00
ε = nε + inε =
√
θ
ρε exp i
2
(2.51)
ε:
ρε =
p
ε0 2 + ε00 2
(2.52)
0
0
ε
ε
cos(θε ) =
=√ 0
2
ρε
ε + ε00 2
00
(2.53)
00
ε
ε
sin(θε ) =
=√ 0
2
ρε
ε + ε00 2
21
(2.54)
As we are dealing with a passive DNG media (ε
hπ
θε =
We can also dene
nε =
nε
√
0
2
<0
,π
and
00
ε > 0)
we have for
θε :
i
(2.55)
by:
θε
θε
0
00
00
0
− i cos
ε = nε + inε = i(nε − inε ) = iρε sin
2
2
nε
And from (2.55) we can obtain the argument of
(2.56)
(by dividing it by 2):
θε h π π i
=
,
2
4 2
(2.57)
If we use the following trigonometric relations:
cos
sin
θε
2
θε
2
r
=
r
=
1 + cos(θε )
2
(2.58)
1 − cos(θε )
2
(2.59)
nε
From (2.52)-(2.54) we can now write the argument of
cos
sin
θε
2
θε
2
12
=
2
12
=
2
s
1+ √
ε0
(2.60)
ε0 2 + ε00 2
s
1− √
depending only on the permittivity:
ε0
(2.61)
ε0 2 + ε00 2
From (2.57) we now that for this specic interval both
cos
θε
2
and
sin
θε
2
must be greater
than 0 so we must choose the positive root. Knowing this and with (2.51), (2.52)-(2.54) and
22
(2.60)-(2.61) we can now write:
0
r
nε =
00
r
nε =
0
|ε |
2
|ε0 |
2
s
4
1+
s
4
1+
ε00
ε0
2 v
u 1 + sgn(ε0 )
u
ur
00 2
t
1 + εε0
ε00
ε0
2 v
u 1 − sgn(ε0 )
u
ur
00 2
t
1 + εε0
We know that we are dealing with DNG media so
0
sgn(ε ) = −1.
(2.62)
(2.63)
We can now easily see by
the result in (2.62) and (2.63) , and for the interval that we have dened for
θε
2 , that
00
0
nε > nε
.
Let us now consider the limit case where there are no losses:
00
ε =0
(2.64)
From (2.62) and (2.63) we obtain:
0
nε = 0
00
nε =
q
|ε0 |
(2.65)
(2.66)
And with these results in (2.65) and (2.66) we can write:
q
nε = i |ε0 |
(2.67)
As we have mentioned before a similar result can be obtained for the magnetic permeability
by using an analogous process:
23
q
nµ = i |µ0 |
(2.68)
With these two last results and using the denition in (2.49) we can now easily obtain the
refraction index for a DNG media:
q
q
q
0
0
n = nµ nε = i |µ |i |ε | = − |µ0 ε0 |
(2.69)
This proves that for a lossless DNG material the refraction index is negative.
Let us now
consider the losses to create a more general solution:
0
00
0
00
0
00
00
00
0
0
0
00
00
0
n = nµ nε = n + in = (nε + inε )(nµ + inµ ) = −(nε nµ − nε nµ ) + i(nε nµ + nε nµ )
As we have seen from (2.62) and (2.63),
00
0
nε > n ε
0
00
and the same happens for the permeability
00
0
nµ > nµ
as the demonstration process is analogous so
00
0
so with the result in (2.70):
0
n = −(nε nµ − nε nµ ) < 0
00
0
00
(2.70)
00
0
n = (nε nµ + nε nµ ) > 0
(2.71)
(2.72)
The results in (2.71) and (2.72) are indeed very important because they not only corroborate
the result in (2.48) that states that there is an extinction of the eld along the propagation
axis (as
since
0
00
n > 0)
n <0
but it also gives us the nal conclusion about the direction of the power ux
so from (2.45) and (2.47) we can say that:
S · zb > 0
24
(2.73)
As
0
n < 0
we can also state from (2.41) that we are dealing with medium with
phase velocity
negative
as its direction is the opposite from the energy ow and attenuation, from
(2.73) and from the fact that we are dealing with a passive media. Let us now analyze the
refraction index for the general case (with losses). In polar coordinates we have:
n=
√
θε + θµ
√ √
ρn exp(iθn ) = ρε ρµ exp i
2
We saw that the condition was valid for
so it is easy to see from (2.74) that
nε
arg(n)
and
nµ
the argument was in the interval
(2.74)
h
pi pi
4, 2
i
is also between those values. The refraction index
on a DNG medium is in fact negative and we can now relate it with the propagation constant:
k = k · zb = nk0 · zb = zb(n k0 + in k0 )
0
As
0
n k0 < 0
00
(2.75)
we can see that the direction of propagation is the opposite compared with the
energy ux:
0
k · zb < 0
(2.76)
From (2.73) and (2.76) we can create a graphical representation of both the electric and
magnetic elds with the energy ux vector and the propagation constant for a DPS medium
and for a DNG medium and compare the results, represented in Figure 2.3.
25
Figure 2.3: Spatial Representation of the elds, the energy ux and the propagation constant
for a DPS and a DNG medium
Here we can see that, from these two types of medium, both the Poynting vector and the
propagation constant shares the same axis but not the same direction because in the DPS
media there is a right-handed trihedral formed by
designated
Backward Waves [?] (BW),
0
[E0 , H0 , k ] .
From these results appears the
electromagnetic waves that present a propagation
direction that is the opposite of the associated power ux..
2.2
Negative Refraction
As we have seen on the previous section, the phase velocity for wave propagation in a DNG
media is negative and this has important implications.
Let us consider the scattering of a wave that incises on a DPS-DNG interface as shown in
Figure 2.4.
26
Figure 2.4: Scattering of a wave that incises on a DPS-DNG interface
Now we assume that we have a DNG medium, with a negative refraction index (n2
< 0),
in
the area with blue background (x < 0 and z > 0) and a DPS media, with a positive refraction
index (n2
> 0,
in x > 0 and z > 0. We also assume that the losses on both the DPS and the
DNG materials can be neglected.
The
Snell's law of reection
assures us that the angle of reection is equal to the angle of
incidence:
θr = θ i
(2.77)
If we consider an uniform plane wave incising obliquely on a plane boundary (z=0) between
materials with dierent constitutive parameters (and refraction indexes
n1 , n2 ),
and enforcing
the boundary conditions at the interface, we can also obtain, from the Snell's law of reection,
?
the relation between the angle of the transmitted wave and the angle of the incident wave [ ],
which is given by:
27
sin(θt )
n1
=
sin(θi )
n2
(2.78)
If we now consider the situation represented by the previous , where there is a DNG material with a negative refraction index
n2
we see that, for obtaining the correct angle of the
transmitted wave one must write (2.78) in the following form:
n1
θt = sgn(n2 ) arcsin
sin(θi )
|n2 |
(2.79)
We must note that if the refraction index of a medium is negative, according the Snell's Law,
the refracted angle should also become negative and then, as we have seen in the previous
section, the direction of the energy ux, given by
given by
k.
S, is the opposite of the wave propagation,
It's also important to notice that we are considering the solution where
00
n > 0,
as we have mentioned in the previous section, because we are dealing with a passive media.
But if we have chosen to use
00
n < 0,
according to Snell's Law we would not have a negative
refracted angle but a positive one instead, which is the same result as if the transmitted wave
was propagating in a DPS material, with one very important dierence, as we have mentioned
before, that the energy ux was then propagating in the direction of the interface (and the
source) which is the opposite of a causal direction and makes no sense for a passive media.
2.3
The Lorentz Model
The temporal response of a chosen polarization eld component
i
to the same component of
the electric eld, assuming that the electric charges can move in the same direction as the
electric eld, can be described by a material model called the
Lorentz Model [?].
This
model is derived from the description of the electron's motion in terms of a damped harmonic
oscillator:
28
d2
d
Pi + ΓL Pi + ω02 Pi = ε0 χL Ei
dt2
dt
(2.80)
Where the rst term describes the acceleration of the electric charges, the second one describes
the reduction of the oscillation's amplitude in terms of the damping coecient
ΓL
and the
third term describes the restoring forces of the system. On the right hand side of the equation
χL
is called the
coupling coecient.
The response in the frequency domain, using the operators used on the previous section, is
given by:
−ω 2 Pi (ω) − iωΓL Pi (ω) + ω02 Pi (ω) = ε0 χL Ei (ω)
We know that the electric susceptibility
χe ,
(2.81)
a measure of how easily it polarizes in response
to an electric eld, is given by:
χe =
P
ε0 E
(2.82)
With both (2.81) and (2.82) we can obtain the Lorentz frequency dependent electric susceptibility:
χe,Lorentz (ω) =
Pi (ω)
χL
= 2
ε0 Ei (ω)
ω0 − iωΓL − ω 2
(2.83)
The electric permittivity is given by:
ε = ε0 (1 + χe )
So with (2.83) and (2.84) we can obtain now the Lorentz electric permittivity:
29
(2.84)
εLorentz (ω) = ε0 1 +
χL
ω02 − iωΓL − ω 2
(2.85)
There are also other models which are particular cases of the Lorentz Model when we are
making certain assumptions:
If the term that is related to the charge acceleration is very small when compared with both
the damping and the restoring forces term then we can neglect it, obtaining from (2.81) the
Debye Model:
−iωΓd Pi (ω) + ω02 Pi (ω) = ε0 χd Ei (ω)
χe,Debye (ω) =
(2.86)
χd
ω02 − iωΓL
(2.87)
When we have the case where the restoring forces are neglectful then we obtain from (2.81)
the
Drude Model:
−ω 2 Pi (ω) − iωΓD Pi (ω) = ε0 χD Ei (ω)
χe,Drude (ω) =
The couple coecient
χL (χd
or
χD
sented by the plasmas frequency as
(2.88)
χD
−iωΓD − ω 2
(2.89)
depending on the model that is used) is normally repre-
χL = ωp2 .
We have made our analysis of the Lorentz Model in terms of the electric polarization eld, but
the same kind of process can be made in terms of the magnetization eld
polarization) and the magnetic susceptibility
χm .
analysis, is then given by:
30
Mi
(instead of the
The magnetic permeability, using similar
µLorentz (ω) = µ0
2.3.1
Mi (ω)
1+
Hi (ω)
= µ0 1 +
χL
ω02 − iωΓL − ω 2
(2.90)
A DNG interval using the Lorentz Model
Let us consider the following expressions for the real parts of both the (relative) permittivity
?
and (relative) permeability obtained using the Lorentz Model [ ]:
2 (ω 2 − ω 2 ) + ω 2 Γ2 + (ω 2 − ω 2 )2
ωpe
0e
0e
Le
2 − ω 2 )2 + ω 2 Γ2
(ω0e
Le
(2.91)
2 (ω 2 − ω 2 ) + ω 2 Γ2 + (ω 2 − ω 2 )2
ωpm
0m
0m
Lm
2 − ω 2 )2 + ω 2 Γ2
(ω0m
Lm
(2.92)
<(εr,L (ω)) =
<(µr,L (ω)) =
We now want to obtain a frequency interval, which we will represent as
[ω − , ω + ]
where both
parameters have negative real parts, by denition of a DNG media. This can be formulated
as the following conditions:
[ωε− , ωε+ ] −→ < (εr,L (ω)) < 0
(2.93)
[ωµ− , ωµ+ ] −→ < (µr,L (ω)) < 0
(2.94)
First we will try to nd this interval for the frequencies where the permittivity is negative and
we can assume that for the permeability the computation is analogous.
Initially we must nd the limit in which the permittivity becomes negative by nding where
the real part becomes zero:
2
2
< (µr,L (ω)) = 0 ⇒ ωpe
(ω02 − ω 2 ) + ω 2 Γ2Le + (ω0e
− ω 2 )2 = 0
31
(2.95)
2
2
2 2
4
ω 4 − ω 2 (2ω0e
+ ωpe
− Γ2Le ) + (ωpe
ω0e + ω0e
)=0
(2.96)
We can nd the zeros by applying the Quadratic Formula to (2.96) from which we obtain the
following result:
2
4
2ω 2 = 2ω0e
+ ωpe
− Γ2Le ±
q
4 − 4ω 2 Γ2 − 2ω 2 Γ2
Γ4Le + ωpe
pe Le
0e Le
(2.97)
Admitting that there are no losses, by considering that the oscillation amplitude does not
decrease in time (ΓL
= 0),
we can simplify expression (2.97):
2
2
2
2ω 2 = 2ω0e
+ ωpe
± ωpe
(2.98)
And now we have both the positive and negative solutions:



ω − = ω0e



(2.99)


q


 ω+ = ω2 + ω2
pe
0e
From this result, and by doing the same kind of computation for the permeability, we can
conclude that there are in fact two frequency intervals, one for the permittivity
one for the permeability
[ωµ− , ωµ+ ],
[ωε− , ωε+ ]
and
where they assume negative values:



ω − = ω0e


 ε


q


 ω+ = ω2 + ω2
ε
pe
0e
,



ω − = ω0m


 µ
(2.100)


q


 ω+ = ω2 + ω2
µ
pm
0m
So we are in the presence of a DNG medium when the frequencies are in the following interval:
32
[ω − , ω + ] = [ωe− , ωe+ ] ∩ [ωµ− , ωµ+ ]
(2.101)
Admitting that (2.101) it is not an empty set, we can nally write the interval in which, using
the Lorentz Dispersive model for both the permittivity and the permeability, the material acts
as a DNG medium:
[ω − , ω + ] = max ωε− , ωµ− , min ωε+ , ωµ+
2.3.2
(2.102)
A DNG interval using the Drude Model
Let us now consider the Drude Model, a particularization of the Lorentz Model that also allows
negative permeabilities and permittivities but neglects the restoring forces (of the harmonic
model), and apply a similar process as we have done in the previous section. First we separate
the real and imaginary parts of the model's expression.
As we have done in the previous
section, we will do the analysis for the permittivity as for the permeability the process is
analogous. The real and imaginary parts of the Drude Model permittivity is given by:
<(εr,D (ω)] =
Γ2De ω 2 + ω 4 − χDe ω 2
Γ2De ω 2 + ω 4
(2.103)
−iχDe ΓDe ω
Γ2De ω 2 + ω 4
(2.104)
=(εr,D (ω)] =
As we have done for the Lorentz Model, the wanted spectral interval can be found when
equaling to zero the real part of the model:
2
<(εr,D = 0 ⇒ ω 4 + ω 2 (Γ2De − ωpe
)=0
33
(2.105)
Using again the quadratic formula for nding the zeroes on (2.105) we obtain the following
expression:
2
2
2ω 2 = ωpe
− Γ2De ± Γ2De − ωpe
(2.106)
And now we have both the positive and negative solutions for this model:



ω− = 0



(2.107)


q


 ω + = ω 2 − Γ2
pe
De
From this result, and by doing the same kind of computation for the permeability, we can
again obtain, as we have done in the previous section, the two frequency intervals, one for
the permittivity
[ωε− , ωε+ ]
and one for the permeability
[ωµ− , ωµ+ ],
where they assume negative
values:



ω− = 0


 ε


q


 ω + = ω 2 − Γ2
pe
ε
De
,



ω− = 0


 µ
(2.108)


q


 ω + = ω 2 − Γ2
pm
µ
Dm
So the frequency interval in which the media is DNG, when using the Drude Model, is given
by:
[ω − , ω + ] = 0, min ωε+ , ωµ+
2.4
(2.109)
Group Velocity and Phase Velocity
?
The expression for the time-averaged energy density of a plane wave is [ ]:
34
∂(ωε) 2
1
∂(ωµ)
2
ε0
U=
|E| + µ0
|H|
4
∂ω
∂ω
(2.110)
From (2.110) we know that:
If we multiply (2.111) by
µ
∂(ωε)
>0
∂ω
(2.111)
∂(ωµ)
>0
∂ω
(2.112)
and (2.112) by
following expression, that we will call
A,
and then add them together we obtain the
which will be useful further in this section:
∂(ωε)
∂(ωµ)
∂(ωε)
∂(ωµ)
A = µε + ωµ
+ µε + ωε
= 2µε + ω µ
+ε
∂ω
∂ω
∂ω
∂ω
Since we are dealing with a DNG medium, where
ε, µ < 0,
(2.113)
we can conclude from (2.113) that
A < 0.
We know from (2.35) and (2.39) that for an isotropic medium we have:
k2 = ω 2 µ0 µ0 Deriving it in order of the frequency
ω
(2.114)
we obtain:
∂ ω 2 µε
∂(ωε)
∂(ωµ)
∂(k2 )
= µ 0 ε0
= µ0 ε0 ω 2εµ + ωµ
+ ωε
= µ0 ε0 ωA
∂ω
∂ω
∂ω
∂ω
(2.115)
From (2.45) we know that:
k = nk0 = n
35
ω
c
(2.116)
So we can also write (2.115) as:
∂(k)
∂(k2 )
∂(k)
ω ∂(k)
= 2k
= 2nk0
= 2n
∂ω
∂ω
∂ω
c ∂ω
The
Phase Velocity, vp , and the Group Velocity, vG
vp =
(2.117)
are given by:
ω
ω
c
=
= 0
<(k)
k0 <(n)
n
(2.118)
∂ω
∂k
(2.119)
vG =
Using (2.118) and (2.119) on (2.117) we obtain:
∂(k2 )
1 1
= 2ω
∂ω
vp vG
Since
A<0
this implies that
∂(k2 )
∂ω
<0
(2.120)
and from (2.210), for a lossy DNG medium, we can
conclude that, on a dispersive DNG medium, the group velocity and the phase velocity have
dierent signs.
For dispersive media is also easy to prove that the group velocity and phase velocity have
dierent values (since for a non-dispersive media
vp = vG ).
∂(k)
∂(ω n) 1
1
=
=
∂ω
∂ω c
c
If we derive (2.116) we obtain:
∂n
n+ω
∂ω
(2.121)
From (2.118) and (2.119) we can now write (2.122) as:
1
1
1 ∂n
=
+ ω
vG
vp
c ∂ω
36
(2.122)
That shows us that
vp = vG
is only possible when there is no frequency dependence of the
refraction index.
2.5
Kramers-Kronig Relations
Some general relations were developed to relate the real part of an analytic function to a
integral that contains it's imaginary part, and vice-versa.
Applying this relations to the dielectric function
ˆ
1
<[ε(ω)] = 1 +
2π
−2ω
=[ε(ω)] =
π
+∞
0
ˆ
+∞
0
ε(ω)
?
we obtain [ ]:
Ω=[ε(ω)]
dΩ
Ω2 − ω 2
(2.123)
<[ε(ω)] − 1
dΩ
Ω2 − ω 2
(2.124)
Named after Ralph Kronig and Hendrik Kramer, they are known as the
Kramers-Kronig
Relations.
The real part of
ε(ω),
(2.123), is related with the refraction index and the imaginary part,
(2.124), is related with the eld's extinction (as we have seen in the previous section). In the
computation of these integrals the Cauchy Principal Value method is used.
Equation (2.123) allows us to obtain the refraction index prole and chromatic dispersion, phenomenon where the phase velocity and the group velocity depend on frequency, of a medium
by knowing only it's frequency dependent losses, which can be measured over a large spectral
range. This is a very important result because it demonstrates that
there is an interdepen-
dency between losses and dispersion.
Equation (2.124) gives a not so useful result. We can use it to obtain the eld extinction by
knowing the refraction index but it is very dicult to measure this index over a wide frequency
range.
37
38
Chapter 3
Guided Wave Propagation in DNG
Media
3.1
Propagation on a Planar DNG-DPS Interface
In this section we will study the propagation of electromagnetic waves on a planar interface
between a DPS and a DNG medium, which is represented in Figure 3.1.
Figure 3.1: The planar interface between a DPS and a DNG medium, here represented by a
dashed line.
39
3.1.1
Modal Equations
Let us consider that the propagation direction is given by the z-axis, the transverse direction
by the x-axis and the y-axis as the transverse innite direction, where there is no variation
of both the electric and magnetic elds. Since the surface is homogeneous along the z-axis,
solutions to the wave equation can be taken as:
E(x, t) = Em (x) exp[i(βz − ωt)]
(3.1)
H(x, t) = Hm (x) exp[i(βz − ωt)]
(3.2)
Or, in the time harmonic form of the elds:
E(x, t) = Em (x) exp[iβz]
(3.3)
H(x, t) = Hm (x) exp[iβz]
(3.4)
We can now plug the general eld solutions (3.3) and (3.4) into the
Homogeneous Wave
Equation, for the Transverse Electric (TE) mode, given by:
∇2 E + ω 2 εµE = 0
(3.5)
∂2
∂2
E + 2 E + (k02 n2i )E = 0
2
∂x
∂z
(3.6)
∂2
E + k02 n2i + β 2 E = 0
2
∂x
(3.7)
40
Where
ni
is the refraction index of the medium
i
(given by
ni =
√
εi µi ) and β
the propagation
constant.
Since the term between parenthesis in equation (3.7) is constant in x we are dealing with a
constant coecient dierential equation that could have the following solution:
Ey (x) = E0 exp[ihi x] + E0 exp[−ihi x]
Where we can also dene
hi
(3.8)
as the transverse wave number of the medium
i
(given by
h2i =
k02 n2i − β 2 ).
In order to maintain wave guiding on the interface, the elds must be evanescent and decay
with distance away from the separation surface.
constant to be in the range of
k 0 ni < β ,
This requirement causes the propagation
therefore the propagation constant in both the
regions is complex and it is given by:
hi = ±iαi
Where
αi ,
(3.9)
the attenuation constant, is given by:
αi2 = β 2 + k02 n2i
The sign of
hi
(3.10)
is chosen in such way that the eld decays with distance away from the interface
so the resulting elds on both the DPS and DNG regions (that we will call
Admitting that the interface is on
x = 0,
Ey (x) =
the eld's expressions are given by:



E0 exp[−α1 x]








1 and 2 respectively).
E0 exp[α2 x]
41
,
x>0
(3.11)
,
x<0
With
α1 , α2
> 0.
We can now use Faraday's Law, from the Maxwell Equations, to compute the magnetic eld:
∇ × E = iωµH
(3.12)
1
∇×E
iωµ
(3.13)
H=
And for this case of TE propagation mode, expression (3.13) can be written as:
H=
1 ∂Ey
zb
iωµ ∂x
(3.14)
From Eq. (3.13) we can now obtain the expressions for the magnetic eld on both regions:
Hz (x) =











iE0 α1
ωµ1
exp[−α1 x]
−
x>0
(3.15)
−iE0 α2
ωµ2
exp[α2 x]
Applying the boundary conditions at the interface (x
magnetic eld components (Hz (0)
,
= Hz (0)+ )
,
= 0),
x<0
and assuring the continuity of the
we can write:
iE0 α1
−iE0 α2
exp[−α1 0] =
exp[α2 0]
ωµ1
ωµ2
(3.16)
α1
α2
=−
µ1
µ2
(3.17)
Now we have obtained the modal equation for the
mode given by:
42
Transverse Magnetic ()
propagation
α2 µ1 + α1 µ2 = 0
(3.18)
By applying a similar computation process to both the wave equation and eld expressions
for the TM modes, and using the following equation from the Maxwell Equations:
∇ × H = −iωεE
(3.19)
We can also obtain the modal equation for the TE mode:
α2 ε1 + α1 ε2 = 0
(3.20)
With these results (3.18) and (3.20) we are now able to infer if there is propagation along the
interface. Since we now that both
α1 , α2 > 0
α1 = −
and
µ1 , ε1 > 0,
from (3.18) :
µ1
α2 > 0 =⇒ µ2 < 0
µ2
(3.21)
ε1
α2 > 0 =⇒ ε2 < 0
ε2
(3.22)
And from (3.20):
α1 = −
So, from the implications on (3.21) and (3.22), we can conclude that it is in fact possible to
have propagation on an interface between a DPS medium and a DNG medium (ε2 ,
3.1.2
µ2 < 0).
Surface Mode Propagation
We will now use the Lorentz Dispersive Model (LDM), which was introduced on the previous
chapter, to study the solutions of the modal equations.
permittivity and permeability are given, as seen before, by:
43
The model frequency dependent
2
ωpe
εr,L (ω) = 1 + 2
ω0e − iωΓL − ω 2
(3.23)
2
ωpm
µr,L (ω) = 1 + 2
ω0m − iωΓL − ω 2
(3.24)
These models will be used to describe the frequency dependence of the parameters on the
DNG medium (region
2),
and have chosen the following values for the plasma's frequencies
ωpm , ωpe , central frequencies ω0e , ω0m , damping coecient ΓL , as well as the parameters of the
DPS medium
ε1,r , µ1,r .
The simulation parameters are represented at Table 2.1.
Parameter
Value
ωpe
ωpm
ω0e
ω0m
ΓL
ε1,r
µ1,r
2π × 7 × 109 rad.s−1
2π × 6 × 109 rad.s−1
2π × 2.5 × 109 rad.s−1
2π × 2.3 × 109 rad.s−1
0.05 × ωpe
1
1
Table 3.1: Simulation parameters for the Lorentz Dispersive Model, on the DPS-DNG interface
structure
3.1.2.1 Neglecting Losses in the LDM (ΓL = 0)
First lets analyze the variation of the parameters
εr,L (ω)
presented in Figure 3.2.
From Figure 3.1 we can easily identify three regions:
•
region (1) where:
ε<0
and
µ < 0 (DNG),
•
region (2) where:
ε<0
and
µ > 0 (ENG)
•
region (3) where:
ε>0
and
µ < 0 (DPS).
44
,
and
µr,L (ω).
The representation is
Figure 3.2: Lorentz lossless dispersive model for
εr,L
and
µr,L
With relation (3.27), and using the Lorentz dispersive model, we can now analyze the variation
of the refraction index with frequency, represented on Figure 3.3.
We can nd in Figure 3.3 that we have the three regions, as mentioned before. As expected, on
the DNG region, we have a negative refraction index and on the DPS region we have a positive
refraction index. On the ENG region, as the permittivity is negative and the permeability is
positive, we have a purely imaginary refraction index. The eect that
n
varies with frequency
(except in vacuum, where all frequencies travel at the same speed, c) is known as
dispersion.
In regions of the spectrum where the material does not absorb, the real part of the refractive
index tends to increase with frequency, as seen in Figure 3.3.
curve of the refractive index is a complex form given by the
Near absorption peaks, the
KramersKronig relations,
and can decrease with frequency. The real and imaginary parts of the complex refractive index
are related through use of the KramersKronig relations (one can determine a material's full
complex refractive index as a function of wavelength from an absorption spectrum of the
45
Figure 3.3: Relative refraction index (nr
=
√ n ), using the lossless LDM, on the DPS-DNG
ε0 µ0
interface
material).
Using equations (3.18), (3.20) and (3.11) we can now establish a relation that expresses the
variation of the propagation constant
β
with frequency, called the
dispersion relation.
The
dispersion relation describe the interrelations of wave properties such as wavelength, frequency,
velocities, refraction index, attenuation coecient. For the TE mode, the relation is given by:
v
u
2
u µ2 (ω)ε2 (ω) − µ1 ε1 µ2 (ω)
2
u
µ
1
β(ω) = t
k0
µ2 (ω)2
1−
µ2
(3.25)
v
u
2
u µ2 (ω)ε2 (ω) − µ1 ε1 ε2 (ω)
u
ε2
1
β(ω) = t
k0
2
1 − ε2 (ω)
ε2
(3.26)
1
And for the TM mode:
1
46
Using the Lorentz Dispersive Model, and the expressions in (3.26) and (3.25), we can now
obtain the dispersion relation graphical representation for both the TE and the TM modes,
presented in Figure 3.4.
Figure 3.4: Dispersion relation,
β(ω),
using the lossless LDM, on the DPS-DNG interface
From both Figures 2.4 and 2.5 we can see that when
µ2 (ω)2 µ2 = 1
1
of
β(ω)
(or
ε2 (ω)2 ε2 = 1 )
the value
1
goes to innity which represents an unphysical solution, as an innite propagation
constant, at a given frequency, is not a valid electromagnetic phenomenon.
The graphical representation of the attenuation constants, for both the TE and TM modes, is
represented in Figure 3.5 and 3.6.
47
Figure 3.5: Attenuation constants
α1
and
α2
for the TE modes, using the lossless LDM, on
α1
and
α2
for the TM modes, using the lossless LDM, on
the DPS-DNG interface
Figure 3.6: Attenuation constants
the DPS-DNG interface
48
The attenuation constants have, for this frequency interval, a positive real part, as a condition
to have propagation along the interface and exponential attenuation as we move away from
it, as stated in (3.21) and (3.22). Since we are not dealing with losses, the imaginary parts of
both
α1
and
α2
are both zero, in the intervals where we have propagation.
3.1.2.2 Considering Losses in the LDM (ΓL = −0.05 × ωpe )
Le us now consider a lossy structure using the Lorentz Dispersive mode.
The constitutive
parameters are graphically represented in Figure 3.7.
Figure 3.7: Lorentz dispersive model for
εr,L
and
µr,L
From Figure 3.7 we can identify that the three regions are approximately the same from the
previous structure when we were neglecting losses, and this happens since we are dealing with
a small value for
ΓL
. The positive imaginary parts are the result of a negative damping
present on the Lorentz's Model.
If we consider a relative refraction index given by:
49
nr = √
n
ε0 µ 0
(3.27)
For the lossy situation, a representation of the refraction index can be obtained and is shown
in Figure 3.8.
Figure 3.8: Relative refraction index (nr
=
√ n ), using the lossy LDM, on the DPS-DNG
ε0 µ0
interface.
In Figure 3.8 we also have the representation of the same three regions, as mentioned before.
As we expected on the DNG region we have a negative refraction index and on the DPS
region we have a positive refraction index. From the variation
n
on the ENG region, where
we also have a negative real component of the refraction index, we can take an important
conclusion. The existence of a negative real refraction index on this ENG region proves that a
DNG medium has always the designation of
(Negative Refraction Index) but a NRI medium
does not have to be DNG, as we can see when considering losses and dispersion.
The representation of the dispersion relation,
β(ω),
50
using the Lorentz Dispersive Model, and
considering losses, is showed in Figure 3.9.
Figure 3.9: Dispersion relation,
β(ω),
using the lossy LDM, on the DPS-DNG interface.
From the graphical representation of the dispersion relation in Figure 3.9 we can see that
the value of
β(ω)
no longer goes to innity (as it does when neglecting losses, Figure 3.4).
β(ω) experience a signicant increase in the range
ε2 (ω)2 µ2 (ω)2 =
1
(or asymptotes (
= 1) but they can now
2
2
µ
ε
Both the real and the imaginary parts of
of frequencies where there were
1
1
represent valid physical solutions as the propagation constant is no longer innite.
The representation of the attenuation constants for both the TE and TM modes are depicted
in Figure 3.10 and Figure 3.11.
51
Figure 3.10: Attenuation constants
α1
and
α2 ,
for the TE modes, using the lossy LDM, on
the DPS-DNG interface
Figure 3.11:
Attenuation constants
DPS-DNG interface
α1 ,
for the TM modes, using the lossy LDM, on the
52
The attenuation constants have also, for this frequency interval, a positive real part, as a
condition to have propagation along the interface and exponential attenuation as we move
away from it, as stated in (3.21) and (3.22) but now the imaginary parts of both
α1
and
α2
are always negative, condition that is needed in order to have propagation along the z-axis.
From the expressions in (3.11) we can also have a graphical representation of the electric eld's
variation along the x-axis dimension. This is shown in Figure 3.12. The variation of the eld
shows us that the eld intensity increases as we approach
x = 0,
this is a representation of the eld in the interface (which is at
as we expected, because
x = 0)
and the attenuation as
we get further from it. From modal equations (3.18) and (3.20) we can also verify that the
slope of the eld branches is also inuenced by the values of both the permittivities and the
permeabilities of the DPS and DNG media.
Figure 3.12: Variation of the electric eld,
Ey (t = 0, x, z),
53
on the DPS-DNG Interface
3.2
Propagation on a DNG Slab Waveguide
3.2.1
Modal Equations
In this section we will study the propagation of electromagnetic waves on a DNG slab waveguide represented by Figure 3.13.
Figure 3.13: A DNG slab waveguide immersed on a DPS media
For the TE modes, and as we have done in the previous chapter, we have the following wave
equation:
∂2
E + k02 n2i + β 2 E = 0
2
∂x
The solutions form this equation, considering that
−d < x < d,
Ey = A cos(h1 x) + B sin(h1 x)
with
h1 = ω 2 µ1 ε1 + β 2
(3.28)
can take the form of:
(3.29)
.
For this structure we have presented for the slab, we want that the electric eld decays with
distance as we get away from the slab, so the evanescence of the electric eld can be represented
by:
54



C exp(ih2 x)



Ey (x) =




 D exp(ih2 x)
,
x≥d
(3.30)
,
x ≤ −d
Where the transverse wave number is dened by:
h2 = ±jα2
With the attenuation constant,
α2 ,
(3.31)
given by:
α22 = β 2 − k22 = β 2 − ω 2 ε2 µ2
(3.32)
Placing this attenuation constant in (3.30) we can now establish for the evanescent elds the
following expressions:
Ey (x) =



C exp(−α2 x)








D exp(α2 x)
,
x≥d
(3.33)
,
x ≤ −d
From the result on (3.30) we can see that there are two kinds of solutions:
•
one even solution, given by the
•
one odd solution, given by the
cos(h1 x)
sin(h1 x)
term,
term.
We will show the manipulation only for the odd mode since the procedure is the same for the
even mode.
We can now represent the electric eld, inside and outside the slab, by the following relations:
55
Ey (x) =












B sin(h1 x) exp(iβz)
,
|x| ≤ d
C exp(−α2 x) exp(iβz)











D exp(α2 x) exp(iβz)
,
x≥d
,
x ≤ −d
(3.34)
Obtaining the magnetic eld expression can be done by using Faraday's Law, from the
Maxwell's Equations:
∇ × E = iωµH
H=
1
iωµ
(3.35)
∂Ey
∂Ey
−
x
b+
zb
∂z
∂x
(3.36)
Applying this equation on the resultant eld expression on (3.34) we obtain the magnetic eld
for this structure:
Hy (x) =












 











Bβ
sin(h1 x)b
x+
− ωµ
1
Cβ
− ωµ
exp(−α2 x)b
x+
2
iCα2
ωµ2
Dβ
exp(α2 x)b
x−
− ωµ
2
cos(h1 x)b
z exp(iβz)
,
|x| ≤ d
exp(−α2 x)b
z exp(iβz)
,
x≥d
exp(α2 x)b
z exp(iβz)
,
x≤d
ih1B
ωµ1
iCα2
ωµ2
Applying the boundary conditions at the interface (x
magnetic eld components
zb,
and assuming that
= d),
(3.37)
assuring the continuity of the
B = A = E1
and
C = D = E2 ,
we can
obtain, from (3.37):
B sin(h1 d) − C exp(−α2 d) = 0
56
(3.38)
−h1
µ2
cot(h1 d) = α2
µ1
(3.39)
We call to this result in (3.39) the asymmetric or odd TE modal equation, as we have used
the odd solution of the wave equation.
Repeating the same kind of algebraic manipulation procedure to the even solution of the wave
equation we obtain the even or symmetric TE modal equation:
h1
µ2
tan(h1 d) = α2
µ1
(3.40)
Achieving the results for the both the odd and even TE modal equations for the slab structure:



−h1 d µµ12 cot(h1 d) = α2 d








h1 d µµ12 tan(h1 d) = α2 d
(Odd Modes)
(3.41)
(Even Modes)
Using the same procedure to obtain the TM modes we get:



−h1 d εε12 cot(h1 d) = α2 d








h1 d εε21 tan(h1 d) = α2 d
(Odd Modes)
(3.42)
(Even Modes)
We can now simplify the modal equations by making the following substitutions:
a = α2 d
(3.43)
b = h1 d
(3.44)
57
Obtaining for the TE modes he following relations:



a = − µµ21 b cot(b)








a=
µ2
µ1 b tan(b)
(assymetric mode)
(3.45)
(symmetric mode)
The relation between the normalized propagation's constants is given by:
a2 + b2 = V 2
Where
V,
(3.46)
the normalized frequency, is given by:
√
V = k0 d ε2 µ2 − ε1 µ1
(3.47)
The intersection of the curve(3.46) with the modal equations will represent the modal solutions
for these modes in the slab.
3.2.2
Surface Mode Propagation
We will now study the surface modes on the DNG slab. From (3.29) we can easily nd that the
transverse propagation constant
h1
can take real values if
√
β < ω ε1 µ 1
and imaginary values if
√
β > ω ε1 µ 1
and, for the analysis of the slab, we know that assuming either imaginary or real
h1
we will maintain the surface mode conditions where we have the wave diminish
values for
with distance from the slab. Let us now assume that
B = −ib,
if we consider the following
relations:
tan(ix) = i tanh(x)
(3.48)
cot(ix) = −i coth(x)
58
We can now rewrite equations (3.45) and (3.46):
a=−
µ2
B coth(B)
µ1
(3.49)
a=−
µ2
B tanh(B)
µ1
(3.50)
a2 = B 2 + V 2
(3.51)
We can now nd the numerical solutions for the modes graphically. These solutions can be
found as the result of the intersection of the curves obtained from the modal equations (3.50)
and (3.51) and the curve from (3.46) or (3.51), as we have said before.
At rst we will consider the DPS situation where
ε1 = µ 1 = 1
and
ε2 = µ2 = 2,
the graphical
solution is shown on Figure 3.14, where the horizontal positive semi-axis represents the transverse propagation constant
we have previously called
b
and the negative semi-axis represent its imaginary value, that
B.
On Figure 3.15 we have the modal solution's representation, but now considering a DNG slab
with
ε1 = µ1 = 1
and
ε2 = µ2 = −1.5.
As we can see from the Figures 3.14 and Figure 3.15, for the DNG slab there are also solutions
with imaginary values of
b, that we dened as B .
since the phase velocity, given by
vp =
These modes are called
super-slow modes,
ω
β , assumes such values that:
vp < √
The graphical solution for a dierent value of
V
c
ε2 µ 2
is shown in Figure 3.16.
Form Figure 3.16 we can also see positive modal solutions, with
59
(3.52)
b
being real, as we have
Figure 3.14: The representation of the modal solutions (red dots) given by the intersection of
the curves for a DPS slab with
ε1 = µ 1 = 1
and
ε2 = µ2 = 2.
seen on the DPS slab, represented on Figure 3.14. These positive-b surface modes are called
slow-modes since the value of the phase velocity assumes values on the interval:
√
c
c
< vp < √
ε2 µ 2
ε1 µ 1
Since we are now dealing with a DNG medium for the slab,
(3.53)
ε2 , µ2 < 0,
this inverses the
signal of the modal equations in such way that the slopes of the tangents and cotangents are
changed, and we also have some slow-modes that, for a given range of frequencies, can have
more than one solution for the same
h1 d value,
as we can see on Figure 3.16. The slow/super-
slow transitions for multiple solutions of the same mode can be described by the next relations.
For the even modes:
cos2 (b) +
µ1
|µ2 |
2
2
sin (b) + b tan(b) = 0
60
(3.54)
Figure 3.15: The representation of the modal solutions (red dots) given by the intersection of
the curves for a DNG slab with
ε1 = µ1 = 1 , ε2 = µ2 = −1.5
and
V = 0.5
And for the odd modes:
2
sin (b) +
µ1
|µ2 |
2
2
cos (b) + b cot(b) = 0
(3.55)
The representation of the dispersion diagram for the DNG dielectric slab, shown on Figure
3.17.
Here we can see the two dashed lines that represent transition limits dened by functions of
βd(k0 d).
The rst one, given by
modes on the slab, where
k0 d =
h1 d = 0.
√ βd
µ1 ε1 , represents the cuto condition of the surface
The second limit line, from the relation
k0 d =
√ βd , gives
µ2 ε2
us the transition border from a slow-mode to a super-slow surface mode, as we can see from
Figure 3.17 where the fundamental mode is a super-slow odd mode represented by a red curve.
This super-slow mode, from Figure 3.17, becomes a slow-mode when
61
V =
µ1
|µ2 | and propagates
Figure 3.16: The representation of the modal solutions (red dots) given by the intersection of
the curves for a DNG slab with
until
V =
ε1 = µ1 = 1 , ε2 = µ2 = −1.5
V =3
π
2 as we can see from Figure 3.18.
i)
ii)
Figure 3.18: Modal solutions (red dots) for a DNG slab with
with
and
(i) V =
µ1
|µ2 | and
(ii) V =
ε1 = µ 1 = 1
,
ε2 = µ2 = −2,
π
2
On this DNG slab structure, from the results on Figure 3.18, we can conclude that there is a
direct relation between the constitutive parameters and the resultant dispersive diagram, as
the point from which the fundamental mode transitions from a super-slow mode to a slowmode depends on the value of both
µ1
and
µ2 .
On the previous situation, that we have used to
62
Figure 3.17: Dispersion diagram for a DNG slab with
ε1 = µ 1 = 1
and
ε2 = µ2 = −1.5
generate the results on both Figure 3.16 and Figure 3.17, we have assumed that
and that
µ1 < |µ2 |,
µ1 ε1 < µ2 ε2
however, if we consider a case where the slab's inner medium is less dense
than the outer medium,
µ1 ε1 > µ2 ε2 ,
we obtain dierent and important results.
From the expression (3.50), where we dened the normalized frequency, we can easily nd that
if we consider
µ1 ε1 > µ2 ε2
we obtain:
V2 <0
(3.56)
From this result, and still considering the situation where the outer medium is more dense
than the slab's inner medium, we have from (3.49) :
b2 + a2 < 0
Considering that, in order to have propagation one must satisfy the condition:
63
(3.57)
a2 ≥ 0
(3.58)
So now we can conclude, from equations (3.57) and (3.58), that the following relation must be
veried in order to have propagation on the slab:
b2 < 0
(3.59)
These conditions can only be true if we are in the presence of super-slow modes, as we can
see from Figure 3.19, which verify (3.58) , (3.59) and
B 2 + V 2 ≥ 0.
From this result we can
say that the propagation on less dense interior medium, as stated by the inequality (3.56), is
only possible if we are in the presence of super-slow modes and this is a phenomenon that is
veried when using a DNG slab.
We will now analyze the dispersion diagrams for this situations but considering the inuence
of the constitutive parameters, as we have mentioned before. As we have done for the denser
inner medium, we will rst consider the case where
|µ2 | > µ1 .
The dispersion diagram is
shown in Figure 3.19.
Here we can see that there is propagation of two super-slow modes where, as we increase in
frequency, or
k0 =
ω
c , both transverse propagation constants,
β
tend for the same value. Both
these modes have a null cuto frequency, one being a conventional mode, the even one, and a
limited odd mode.
The dispersion diagram where
|µ2 | < µ1
is shown on Figure 3.20 in order to compare with the
results obtained in Figure 3.19.
64
Figure 3.19: Dispersion diagram for a DNG slab with
ε1 = 2, µ1 = 1 , ε2 = −1 and µ2 = −1.5
Figure 3.20: Dispersion diagram for a DNG slab with
ε1 = 2, µ1 = 2 , ε2 = −1 and µ2 = −1.5
65
On this situation we can see that only one even super-slow mode propagates and on a limited
frequency band.
We can also notice that, for all the frequency band in which the mode
propagates, there are always two modal solutions and these tend to the same value as we
increase the frequency.
The point where the double-solutions intersect represents the limit
from which there is no surface mode propagation on the DNG slab.
3.3
Conclusions
In this chapter we have studied and presented the propagation in a DNG metamaterial medium
by analyzing the physical phenomena and implications of having both a negative magnetic
permeability and electric permittivity. We have shown that in this kind of media the existence
of waves that propagate in a antiparticle direction of the power ux is noticed, called the
Backward Waves, and that we are in the presence of a Negative Index of Refraction material,
which implies some modications in the interpretation of Snell's Law. A dispersive analysis
is also made using the Lorentz Dispersive Model, and for a particularization called the Drude
Model, and using these models we have also shown that it is possible to nd a DNG interval
even when considering dispersion. From this introduction of losses we have also concluded that
both the group and phase velocities have dierent values and, for this kind of DNG media,
they even have opposite directions.
We have also presented, in this chapter the study of two wave guiding structures using DNG
metamaterials: the DPS-DNG interface and the DNG slab. This is important because these
kind of waveguides present particular physical eects that could be used in wave propagation
structures and even replace some most common DPS guides. The dispersive models mentioned
before where used in the study of theses structures.
First we have showed that it is possible to have both TE and TM surface mode wave propagation on a DPS-DNG interface. This kind of propagation mode is new and does not exist
in other more conventional DPS wave guiding structures.
66
We have also found that, when
not neglecting losses, the feature of being a NIR medium can be applied to all DNG media
but the NIR designation is not exclusive of DNG materials since there are other non DNG
frequency bands where the medium also acts as a NIR. When dealing with the propagation of
this surface waves we have also seen that it permits large attenuation outside the interface.
Finally we have analyzed the guided propagation on a DNG slab, whose electromagnetic
proprieties can be of large interest for the practical application of DNG materials to the
construction of waveguides.
In this structure there's also the possibility of having surface
wave mode propagation but the most important result is the propagation of super-slow modes
that are a consequence of having a phase velocity that is smaller than the outer medium in
which a DNG slab is immersed (there is also slow-mode propagation that exhibits a double
modal solution for some frequency bands). The existence of these super-slow modes enables the
propagation on the DNG slab even if we use a less dense medium for the slab (i.e., medium
2) when compared to the outer medium (i.e.
medium 2),
ε1 µ1 > ε2 µ2 .
This phenomenon
is veried, fullling the propagation conditions, only on double negative materials.
analyzing the dispersion relation diagrams for the
for a medium with
|µ2 | > µ1
ε1 µ1 > ε2 µ2
When
structure we could see that
the are two super-slow modes and for a medium with
µ1 > |µ2 |
only one super-slow mode propagates on a limited frequency band, but that there are, in this
band, always two possible modal solutions.
The study of DNG media characterization and the application of DNG materials in the presented wave guiding structures could help the understanding of implications and capabilities
of using this kind of material on propagation structures.
67
68
Chapter 4
Lens Design Using DNG Materials
4.1
Optical Path and the Lens Contour
Let us consider that there are light rays emanating from a source at point
are being transmitted in the
θ
O
and that they
direction. In order to convert these light rays to plane waves
we must use a lens to assure that the optical paths for the dierent directions are equal as
they reach a plane wavefront. This can be represented by Figure 4.1.
Figure 4.1: Lens contour and optical path representation
69
Considering, in this 2D representation, a plane wavefront dened by the line formed with
points
P1
and
P2 ,
we can state that the two optical paths must be equal, one from
(where we have free space propagation), and one from
propagation from
c
to
P2 ).
O
to
c,
O
to
P2
O
to
P1
(where there is free space
and the propagation in a medium with a refraction index of
n
from
This equality can be represented by the following expression:
OP1 = R = OC + nCP2
(4.1)
R = d + n[R cos(θ − d)]
(4.2)
Or in polar coordinates:
R=
Where
n
d(1 − n)
[1 − n cos(θ)]
(4.3)
is the refraction index of the material of the lens.
Also from Figure 4.1 we can establish the Cartesian coordinates:



x=







 y=
R
d
cos(θ)
(4.4)
R
d
sin(θ)
Where,
R p 2
1−n
= x + y2 =
d
1 − n cos(θ)
(4.5)
From expression (4.5) we can also establish a direct relation between the coordinates and only
the refraction index:
70
p
x2 + y 2 =
1−n
1 − n √ 2x
(4.6)
y2
1
=
n2 − 1
(n + 1)2
(4.7)
x +y 2
And after some algebraic manipulation:
x−
n
n+1
2
−
From this expression we can achieve the lens contour in order to verify the equality we have
shown in (4.1). We can also see that expression (4.7) is in fact an elliptical formula, which
degenerates on a circumference as the refractive index approaches
n = 0.
In a matter of fact
the 2D lens contours are also commonly called as circles [19]. Dierent lens' contours, for
dierent values of
n,
are represented in Figure 4.2.
Figure 4.2: The lenses contours for dierent refraction indexes,
From (4.2) we can see that there is an asymptote in
n=
n = −2.5, −1.5, 100, 1.5, 2.5
1
cos(θ) , making the contours hyperbolic.
We can also see from Figure 4.2 that the curvature is the opposite depending on
n being either
positive or negative.
After we have calculated the optical path we can now analyze the design in terms of focal
71
length.
The focal length can be seen as a measure of how strongly the lens converges or
diverges light, or in geometric terms, the distance over which initially transmitted rays are
brought to a focus. It can be given, as
f,
?
by the following expression [ ]:
Rc f = 1 − n
(4.8)
From this expression we can see that this length depends on the refractive index,
the radius-of-curvature of the lens surface,
cylindrical lens with
so, if we consider
n
n = −1
Rc .
n,
and
We can note, as an example, that a concave
has the same focusing properties as a convex lens with
n = +3
to be negative, a lens with that properties can alter the trajectory of
transmitted waves as if the material possessed a much larger index.
4.2
The Veselago's Flat Lens
As we have seen from the expression (4.7) and in Figure 4.2, as the refraction index tends
to large values (or even innity), the contour tends to a straight line, which can be called
?
as a at lens [ ]. Knowing that such a large refractive index does not have any important
?
practical application [ ], a functional at lens was proposed by Victor Georgievich Veselago
?
?
in 1968 [ ]. In Veselago's paper [ ] he proposed that a planar slab, composed by a material
with the refractive index
n = −n0 ,
with
n0
being the refractive index of the medium in which
the slab was immersed, would focus the light waves emitted by a source to a single point. This
can be showed by a simple application of Snell's law, using a structure with two consecutive
?
boundaries. This structure is called the Veselago's at lens [ ] and a graphical representation
is shown in Figure 4.3.
72
Figure 4.3: Passage of light waves through a Veselago at lens,
image,
i.f.:
A: the image source, B: focused
the internal focus point
This lens geometry and structure, which converts a diverging beam to a converging one, and
vice-versa, creates the existence of a particular point called the internal focus, represented in
Figure 4.3. Knowing that the optical path from the external focus point to the internal focus
point must be zero, we can also processed to the computation of the lens contour, as we have
done in the previous section. From Figure 4.3 we can state that, in order to have an equality
of optical paths, one must have:
r1 + nr2 = d1 + nd2
r22 = r12 + (d1 + d2 )2 − 2r1 (d1 + d2 ) cos(θ)
where,
n is the refractive index of the lens.
(4.9)
(4.10)
Assuming that the optical path from focus to focus
is zero one must have:
d1 + nd2 = 0
73
(4.11)
In polar coordinates, after some manipulation:
(n + 1)
r1
d1
2
− 2n cos(θ)
r1
d1
+ (n − 1) = 0
(4.12)
As done in (4.4), using the Cartesian co-ordinates we obtain the following equation for the
lens contour:
x−
n
n+1
2
+ y2 =
1
(n + 1)2
Which is the expression of a circumference centered at
(4.13)
n
n+1 , 0 , and when
n = −1
we also
obtain a at lens. The equality imposed by expression (4.4) clearly implies that one must have
a NIR medium, which include all DNG media.
Let us now consider the general expression for the impedance of a specic medium:
r
Zi =
µ 0 µi
ε 0 εi
(4.14)
If we consider that the slab's material has, for the relative permittivity and permeability,
both
(ε0r
εr = µr = −1,
= µ0r = 1).
we can state that this DNG medium is a perfect match to free space
From this result, one of the conclusions is that there will not be reections
at the interfaces between the lens and freespace and even at the far boundary interface there
is again an impedance match, and the light is again perfectly transmitted to vaccum.
If the propagation is done in the
zb axis,
in order to have all the energy transmitted through
the slab it is required that we have a propagation constant:
kz0
r
ω
− kx2 − ky2
=−
c2
With the overall Transmission coecient being:
74
(4.15)
0
T = tt =
where
exp(ikz0 d)
d is the thickness of the slab.
r
ω
2
2
= exp −i
− kx − ky d
c2
(4.16)
The choice of the propagation constant is done in order to
maintain causality and this phase correction is what grants the lens the capability of refocusing
?
the image by canceling the phase of the transmitted wave as it propagates from its source [ ].
Let us consider a TE wave propagating in the vaccum, medium 1, with the following eld
expression:
E1 = exp(ikz z + ikx x − iωt)
(4.17)
where the propagation constant is:
r
kz = i kx2 + ky2 −
with
kx2 + ky2 >
ω2
c2
(4.18)
ω
. From this eld expression in (4.17) we can easily identify that we are
c2
dealing with an exponentially evanescent eld. At the interface, between media 1 and 2, the
waves experience both transmission (into medium 2) and the reection (back to medium 1).
It is also important to notice that in order to maintain causality the elds must decay as the
get away from the interface, so the eld expression for the transmitted wave can be:
Et2 = t exp(ikz0 z + ikx x − iωt)
(4.19)
And for the reected wave, the following expression:
Er1 = r exp(−ikz z + ikx x − iωt)
where the propagation constant is given by:
75
(4.20)
kz0
with
ε2
and
µ2
r
ω
= i kx2 + ky2 − ε2 µ2 2
c
being the permittivity and permeability of the slab, and also having
(4.21)
kx2 + ky2 >
ε2 µ2 cω2 .
When matching the wave elds at the interface from medium 1 to medium 2 we obtain the
reection and transmission coecients,
t
and
r:
t=
2µkz
µkz + kz0
(4.22)
r=
µkz − kz0
µkz + kz0
(4.23)
And for the transmission and reection coecients of the transition from inside medium 2 to
medium 1:
t0 =
2kz0
µkz + kz0
(4.24)
r0 =
kz0 − µkz
kz0 + µkz
(4.25)
Now in order to obtain the expression for the transmission of light through both the interfaces
?
one must sum the multiple scattering events, from [ ]:
Ts = tt0 exp(ikz0 d) + tt0 r02 exp(3ikz0 d) + tt0 r03 exp(5ikz0 d) + (...)
Ts =
tt0 exp(ikz0 d)
1 − r02 exp(2ikz0 d)
76
(4.26)
(4.27)
Considering the DNG situation (with
ε = µ = −1),
and using (4.22)-(4.27), we can compute
the limit to this values of permittivity and permeability in order to nd the overall transmission
coecient.
The solution for this special kind of structure is calculated asymptotically as
approaches
−1:
n
limµ→−1,ε→−1 (T s) =
= limµ→−1,ε→−1
= limµ→−1,ε→−1
tt0 exp(ikz0 d)
1−r02 exp(2ikz0 d)
2kz0
2µkz
µkz +kz0 µkz +kz0
exp(ikz0 d)
0
kz −µkz 2
1− k0 +µk
exp(2ikz0 d)
z
z
=
(4.28)
!
=
= exp(−ikz0 d) = exp(−ikz d)
This result in (4.29) is very important. As another consequence of having a negative index of
refraction, we have waves of the form
form
exp(kz ) inside the lens.
exp(−kz ),
outside the lens, that couple to waves of the
So, even if the waves decay outside the lens, they are
amplied
on the inside of it, recovering an image on the opposite side of the lens, from the source, and
all done by the transmission process. On Figure 4.4 we can see the evolution of the evanescent
eld variation in the presence of the Veselago's at lens.
Since the waves decay in amplitude and not in phase, as they get further from the source, the
lens focus the image by amplifying these waves rather than correcting the phase. This is a
proof that this medium does in fact amplify the evanescent waves, and so, with this kind of
lens, both the propagating and evanescent waves contribute to the resolution of the resulting
?
image [ ].
As we have stated before, as the result of the perfect matched impedance, there will be no
reected wave on the interface as we can also see by the asymptotic analysis of the overall
reection coecient:
77
Figure 4.4: Evanescent eld variation in the presence of the Veselago's at lens.
limµ→−1,ε→−1 (Rs) =
(4.29)
= limµ→−1,ε→−1 r +
tt0
exp(ikz0 d)
02
1−r exp(2ikz0 d)
=0
Which conrms that all the energy is transmitted between the media transitions.
4.3
Conclusions
In this chapter we have presented the optical lens design using the concept of optical path,
and we have particularized the design process with the usage of DNG materials. In order to
do so we have studied the situation where a DNG slab is used in order to produce an high
resolution lens, which is called the Veselago's at lens.
From the concept of optical path we have obtained an expression that enables us to infer
about the geometrical form of the lens contour and about its dependence on the value of the
refractive index of the lens material. The curvature of the lens contour can be concave if we are
dealing with positive refraction indexes and convex if we are dealing with negative refraction
78
index materials. The at lens contour is obtained when using large values of
unpractical, or if
n → −1.
n,
which is really
The concept of focal lenght is also introduced into the lens design
and we have seen that for a lens made of a DNG material, we can obtain the same focal lenght
as we would if a DPS material was used, but with the implication of having a much smaller
refraction index.
Then we introduce the Veselago DNG at lens. This lens structure consists of a planar DNG
slab, composed by a material with the refractive index
n = −n0 ,
with
n0
being the refractive
index of the medium in which the slab was immersed, that can focus the light waves emitted by
a source to a single point. This phenomenon is achieved by a simple ray tracing problem using
Snell's Law. When considering that the slab's material has, for the relative permittivity and
permeability, both
εr = µr = −1
we could see that there was a perfect impedance match for
both interfaces between the DNG slab and the medium in which it was immersed. From this
result, one of the conclusions is that there will not be reections at the interfaces between the
lens and free space and even at the far boundary interface there is again an impedance match,
and the light is again perfectly transmitted to vaccum to a single point. The DNG material
properties creates a physical phenomenon where we have waves of the form
the lens, that couple to waves of the form
exp(kz )
exp(−kz ),
outside
inside the lens. So, even if the waves decay
outside the lens, they amplied inside of it, recovering an image on the opposite side of the
lens, from the source. These two results are the responsible of granting the lens a capability
of refocusing the image by canceling the phase of the transmitted wave as it propagates from
its source.
This introduction into the lens design using DNG materials is important as its particular
physical properties could enable the creation of high resolution lenses and it is a proof of
practical application of DNG media in optics and engineering.
79
80
Chapter 5
Conclusions
In this chapter main conclusions are exposed as well as some developing potential applications and further investigation hypothesis of the subjects addressed in this dissertation are
introduced.
5.1
Summary
In the second chapter we study the electromagnetic phenomena associated with DNG metamaterials. After formulating the classication of a specic medium as DNG, the implications
of having a negative permittivity and permeability lead into studying the characterization of
the medium and the physical phenomena. We have shown that in this kind of media the existence of waves that propagate in a antiparalell direction of the power ux is noticed, called the
Backward Waves, and that we are in the presence of a Negative Index of Refraction material,
which implies some modications in the interpretation of Snell's Law. A dispersive analysis
is also made using the Lorentz Dispersive Model, and for a particularization called the Drude
Model, and using these models we have also shown that it is possible to nd a DNG interval
even when considering dispersion. From this introduction of losses we have also concluded that
81
both the group and phase velocities have dierent values and, for this kind of DNG media,
they even have opposite directions.
The third chapter deals with the guided wave propagation with DNG materials.
We have
chosen to study two simple structures, the DPS-DNG interface and te DNG slab. A modal
analysis was made for both waveguiding structures and also numerical simulations, with the
respective interpretations. First we have showed that it is possible to have both TE and TM
surface mode wave propagation on a DPS-DNG interface. This kind of propagation mode is
new and does not exist in other more conventional DPS wave guiding structures. We have
also found that, when not neglecting losses, the feature of being a NIR medium can be applied
to all DNG media but the NIR designation is not exclusive of DNG materials since there
are other non DNG frequency bands where the medium also acts as a NIR. When dealing
with the propagation of this surface waves we have also seen that it permits large attenuation
outside the interface.
Finally we have analyzed the guided propagation on a DNG slab,
whose electromagnetic proprieties can be of large interest for the practical application of DNG
materials to the construction of waveguides. In this structure there's also the possibility of
having surface wave mode propagation but the most important result is the propagation of
super-slow modes that are a consequence of having a phase velocity that is smaller than the
outer medium in which a DNG slab is immersed (there is also slow-mode propagation that
exhibits a double modal solution for some frequency bands). The existence of these super-slow
modes enables the propagation on the DNG slab even if we use a less dense medium for the
slab (i.e., medium 2) when compared to the outer medium (i.e.
medium 2),
ε1 µ1 > ε2 µ2 .
This phenomenon is veried, fullling the propagation conditions, only on double negative
materials. When analyzing the dispersion relation diagrams for the
could see that for a medium with
with
µ1 > |µ2 |
|µ2 | > µ1
ε1 µ1 > ε2 µ2
structure we
the are two super-slow modes and for a medium
only one super-slow mode propagates on a limited frequency band, but that
there are, in this band, always two possible modal solutions.
The fourth chapter is dedicated to the study of lens design using DNG metamaterials. We have
82
presented the optical lens design using the concept of optical path, and we have particularized
the design process with the usage of DNG materials. In order to do so we have studied the
situation where a DNG slab is used in order to produce an high resolution lens, which is called
the Veselago's at lens. From the concept of optical path we have obtained an expression that
enables us to infer about the geometrical form of the lens contour and about its dependence
on the value of the refractive index of the lens material. The curvature of the lens contour can
be concave if we are dealing with positive refraction indexes and convex if we are dealing with
negative refraction index materials. The concept of focal lenght is also introduced into the lens
design and we have seen that for a lens made of a DNG material, we can obtain the same focal
lenght as we would if a DPS material was used, but with the implication of having a much
smaller refraction index. Then we introduced the Veselago DNG at lens. This lens structure
consists of a planar DNG slab, composed by a material with the refractive index
with
n0
n = −n0 ,
being the refractive index of the medium in which the slab was immersed, that can
focus the light waves emitted by a source to a single point. This phenomenon is achieved by a
simple ray tracing problem using Snell's Law. When considering that the slab's material has,
for the relative permittivity and permeability, both
εr = µr = −1 we could see that there was a
perfect impedance match for both interfaces between the DNG slab and the medium in which
it was immersed. From this result, one of the conclusions is that there will not be reections at
the interfaces between the lens and free space and even at the far boundary interface there is
again an impedance match, and the light is again perfectly transmitted to vaccum to a single
point. The DNG material properties creates a physical phenomenon where we have waves of
the form
exp(−kz ),
outside the lens, that couple to waves of the form
exp(kz )
inside the lens.
So, even if the waves decay outside the lens, they amplied inside of it, recovering an image on
the opposite side of the lens, from the source. These two results are the responsible of granting
the lens a capability of refocusing the image by canceling the phase of the transmitted wave
as it propagates from its source. This is indeed a very important result since no longer the
resolution is restricted by the wavelength of the propagated light waves, as we can found in
83
conventional lens structures.
5.2
Future Work
When dealing with DNG media propagation the physical implications are more profound than
one could nd at rst sight. One could study the physical aspects of DNG media in motion,
the non linear eects of a DNG medium and the study of the anisotropic properties of media,
since we have dealt with the commonly used isotropic model.
We have also addressed two simple DNG waveguiding structures but there is a large set of
known DPS guides in which one could replace or add, one or several, DNG media, leading to
new results and guiding structures understanding.
Since no longer the resolution is restricted by the wavelength of the propagated light waves,
as we can found in conventional lens structures, the design of DNG metamaterial lens can also
lead to the development of the so called Superlens structures.
84
85
86
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