Influence of Chirality on the Electromagnetic Wave Propagation: Unbounded Media And Chirowaveguides
Priyá Dilipa Gaunço Dessai
Dissertation submitted to obtain the Master Degree in Electrical and Computer Engineering
Jury President: Professor José Manuel Bioucas Dias Supervisor: Professor Carlos Manuel dos Reis Paiva Co- Supervisor: Professor António Luís da Silva Topa Member Professor Sérgio de Almeida Matos
December 2011 Abstract
When a chiral medium interacts with the polarization state of an electromag- netic plane wave and couples selectively with either the left or right circularly polarized component, we call this property the optical activity.
Since the beginning of the 19th century, the study of complex materials has intensi ed, and the chiral and bi-isotropic (BI) media have generated one of the most interesting and challenging subjects in the electromagnetic research groups in terms of theoretical problems and potential applications.
This dissertation addresses the theoretical interaction between waves and the chiral media.
From the study of chiral structures it is possible to observe the eect of the polarization rotation, the propagation modes and the cuto frequencies. The re ection and transmission coe cients between a simple isotropic media (SIM) and chiral media are also analyzed, as well as the relation between the Brewster angle and the chiral parameter.
The BI planar structures are also analyzed for a closed guide, the parallel-plate chirowaveguide, and for a semi-closed guide, the grounded chiroslab. From these structures we can investigate the surface modes as a function of chirality, which will lead us to understand the physical aspects of the chirowaveguides.
Keywords: Chiral media; optical activity; polarization; chirowavguides; bi-isotropic planar structures; re ection and transmission; Brewster angle
i Resumo
Desde o início do século XIX que o estudo de materiais complexos tem aumen- tado, sendo que os meios bi-isotrópicos e quirais geraram temas de estudos muito interessantes e desa antes dentro das comunidades cientí cas quanto à resolução de problemas teóricos, bem como ao estudo das suas aplicações práticas.
Uma onda electromagética plana ao passar por um meio quiral, vai provocar uma rotação de polarização sobre o plano. A onda adquire uma rotação circular esquerda e uma circular direita, a este fenómeno dá-se o nome de actividade
óptica.
Esta dissertação tem como objectivo analisar propriedades dos meios quirais, como o efeito da rotação de polarização, modos de propagação e frequências de corte. Também é abordado o estudo de transmissão e re exão numa interface dieléctrica-quiral, onde se determinam coe cientes de transmissão e re exão e é referida a relação entre o ângulo de Brewster e o parâmetro quiral.
A propagação guiada em meios quirais é abordada através do estudo de es- truturas bi-isotrópicas planares, como é o caso de um guia fechado (um plano assente sobre placas condutoras), e o caso de um guia semi-aberto (um guia quiral assente sobre um plano condutor e em contacto com o ar).
Palavra chave: Meio quiral; actividade óptica, polarização, guias de onda quirais, estruturas biisotrópicas planas, re exão e transmissão;
ângulo de Brewster
ii AKNOWLEDGEMENTS
I would like to express my gratitude to my supervisor Professor Carlos Paiva for suggesting me this dissertation, for his guidance and valuable critics and suggestions.
I also would like to thank to professor António Topa for his availability and precious advices.
To Filipa Prudêncio i would like to thank for being so patience and helpful.
I want to make a special reference to José Pedro Salreta, Abel Camelo, Pedro
Rodrigues, David Sousa and last but not least, Tiago Moura. You have all supported me along my journey at IST and your friendship, fellowship and company are truly remarkable.
Finally, i want to thank all my family, specially my father, mother, sister and
Freddy for their patience and aection.
iii Contents
Abstract i keywords i
Resumo ii
Palavra chave ii
Sumario ii
Acknowledgements iii
List of Symbols xii
1 Introduction 1
1.1 State of the art ...... 2
1.2 Motivation and Objectives ...... 5
1.3 Structure of the dissertation ...... 7
1.4 Main contributions ...... 8
2 Bi-isotropic and chiral media properties 11
iv 2.1 Introduction ...... 12
2.2 Chiral Media ...... 13
2.2.1 Wave eld postulates ...... 14
2.2.2 Polarization ...... 17
2.2.3 Polarization Rotation ...... 23
3 Re ections and Transmissions between a simple planar inter-
face and chiral media 27
3.1 Transmission and Re ection ...... 29
3.1.1 The Brewster Angle ...... 38
4 A method for the analysis of bi-isotropic planar waveguides 41
4.1 Introduction ...... 42
4.1.1 Chiral media ...... 47
4.1.2 Parallel-plated chirowaveguide ...... 48
4.1.2.1 Grounded chiroslab ...... 54
5 Conclusions 63
References 68
v List of Figures
2.1 The polarization vector p(a)changes direction in the sense of ro- tation on the ellipse is changed ...... 22
2.2 Polarization Rotation in chiral media ...... 25
3.1 Re ected and transmitted waves at an oblique incidence on a
semi-in nite chiral medium ...... 29
3.2 Re ection coe cients R11 for ε1 = 1e ε2 = 4 ...... 36
3.3 Re ection coe cients R22 for ε1 = 1e ε2 = 4 ...... 36
3.4 Transmission coe cients T11 for ε1 = 1e ε2 = 4 ...... 37
3.5 Transmission coe cients T22 for ε1 = 1e ε2 = 4 ...... 38
3.6 Transmission coe cients T12 for ε1 = 1e ε2 = 4 ...... 38
4.1 A parallel-plate chirowaveguide ...... 48
4.2 Propagation of the odd modes for χ = 0 ...... 50
4.3 Propagation of the odd modes for χ = 0.5 ...... 51
4.4 Propagation of the odd modes for χ = 1 ...... 51
4.5 Propagation of the even modes for χ = 0 ...... 52
4.6 Propagation of the even modes for χ = 0.5 ...... 52
4.7 Propagation of the even modes for χ = 1 ...... 53
vi 4.8 Variation of β with χ ...... 54
4.9 Grounded chiroslabguide ...... 55
4.10 Surface modes, χ = 0 ...... 60
4.11 Hybrid modes, χ = 0,5...... 60
4.12 Hybrid modes, χ = 1 ...... 61
vii List of Tables
2.1 Classi cation of bi isotropic medium ...... 13
2.3 Conditions of polarization ...... 22
3.1 Re ection coe cients R12 = R21 for ε1 = 1e ε2 = 4 ...... 37
viii ix Nomenclature
BI Bi-isotropic
LP Linear Polarization
LCP Left Circularly Polarized
RCP Right Circularly Polarized
TM Transverse Magnetic
TE Transverse Electric
TEM Transverse Electro-Magnetic
SIM Simple Isotropic Media
PEC Perfect Electric Conductor
EH Electric and Magnetic x xi List Of Symbols
αA Damping coe cient
αi Azimuthal angle of the incident wave
αr Azimuthal angle of the re ected wave
√ β Parameter containing εµ ± χ
δm Algebric parameter from modal equation
Permittivity
ε0 vacuum permittivity
εm Permittivity of the media
ζ Chiral parameter
η Wave Impedance
xii η0 Vacuum wave impedance
η± Positive/negative Wave Impedance
θm Angle perpendicular to the place of incidence
θi Angle of incidence
ϑm Algebric parameter from modal equation
κ Magneto electric eect
λc Cuto frequency
λn Eigenvalue
µ Permeability
µ0 Vacum Permeability
µm Permeability of the media
ν Algebric parameter from modal equation
ξ −iχ chiral parameter
xiii σm Algebric parameter from modal equation
ς Distance
τm Coupling coe cients from modal matrix
χ Chirality
χm Magnetic Susceptibility
ψ Angle of the plane of polarization
ω Angular velocity
Γ Relation between modal equation parameters
∆ εrµr − ξζ
Φ Angle of polarization
Ψ Angle from the plane of polarization
A Time-harmonic vector
Ac Real time-harmonic vector
xiv As Real time-harmonic vector a complex vector
ar real time-vector =Ac
ai complex vector =As a∗ complex conjugate vector
ac real vector
as real vector
B Magnetic Flux Density
C Coupling Matrix
D Electric Flux Density
d Point in the z axis
E0 Initial Electric Field
Ei Electric eld of the incident wave
xv Eik Component of the incident Electric eld
Et Component of transmitted Electric eld
H Magnetic Field Intensity
H0 Initial Magnetic Field Intensity
Ht⊥ Component of transmitted Magnetic eld
hs Transverse wave number
I Identity matrix
i Imaginary unit
k0 Vacuum wave number
k± Propagation constant kˆ Versor k
M Modal matrix
n Refractive index
xvi nef Eective refractive index p Real vector
< Real complex
R Radius
Rmm Re ection coe cient, with mm = 11, 12, ..., 21, 22... r Complex vector
2×π TT = ω
Tmm Transmission coe cient, with mm = 11, 12, ..., 21, 22... t Time variable
t0 Thickness of the chiral slab
u+ Right-hand circularly polarized unit vector
u− Left-hand circularly polarized unit vector
uz Direction of propagation of unit vector
xvii Y1 Chiral admittance of the dielectric medium
Y2 Chiral admittance of the chiral medium
xviii Chapter 1
Introduction
The present chapter it is done a brief overview about the chiral media since the beginning of its investigation in the early 19th century until the present time.
The motivation and objectives of this dissertation are de ned and a detailed information about the organization of the work is reviewed, chapter by chapter.
1 1.1 State of the art
A chiral media fall into the class of bi-isotropic (BI) media and when the light of a linearly polarized plane will rotate as it passes through the medium, interacts with the state of an electromagnetic waves and couples selectively with either left or right circularly polarized component, we call this property the optical activity.
This manifestation of spatial dispersion occurs because the polarization of a medium at a given point depends on the eld, not only in that point, but also in its surroundings.
Although the chiral media belongs to BI medium and presents the same prop- erties concerning to the characteristic waves and the polarization rotation, not every BI media is chiral [1]. A new era in physics began in the early 19th century, when Arago (1811) rst saw the manifestation of optical activity in a quartz crystal. He observed that the quartz crystal rotates the plane of polarization of a linearly polarized light which has passes along the crystal optic axis. Later, Biot (1812) proved that the optical activity was dependent on the thick- ness of the crystal plate and on the light wavelength [2]. Fresnel (1821), showed that a linearly polarized light ray of a crystal quartz separates into two circularly polarized rays of light. He argued that the dif- ference in the two wave velocities is the cause of the optical activity. He also tried to justify the dierent phase velocities for the two circularly polarized rays. He stated that the dierent phase velocities could result from a particular constitution of the refracting medium or integral molecules which established a dierence between the sense of right to left or vice-versa. Pasteur(1840's), began the study of the crystal structure of the materials and their relation with the optical activity. He postulated that molecules are three- dimensional objects and that the optical activity of a medium is caused by the chirality of its molecules. Hertz (1888), it was natural to look for the rotatory power in the materials that would be eective at these wavelengths, the main question was to know how did
2 the wavelengths aected the rotatory power. Lindman was the rst to look for the optical activity in radio waves. In 1914, he made his rst experiment where he studied the wave interaction with collections of randomly - oriented small wire helices, in order to create an arti cial chiral media where he was able to prove the existence of polarization rotation. In 1920 published his work which introduced a new approach for the study of chirality, when he devised macroscopic models of chiral media by using wire spirals instead of chiral molecules, also demonstrated the phenomenon of optical activity using microwaves instead of the light and has been often cited and his report followed in the microwave community of experimental chirality [3].
Winkler (1965), was able to develop Lindman's results over wider frequency band. He also observed that a chiral arrangement of a set of irregular tetrahedra did not rotate the plane of polarization [4]. The following year, Tinoco and
Freedman performed an experiment using oriented helices, and con rmed the chirality hypothesis and gave further results on the frequency dependence of the rotation [5].
Kong (1975) wrote a book where he gathered many information and references about the general bi-anisotropic media, from which the BI media degenerates
[6].
More recently, Engheta and Michelson (1982), did some studies about the tran- sition radiation at chiral-achiral interface. ,
In 1990, the concept of chirowaveguide was de ned by Pelet and Engheta. Since then, several studies have been done on this type of structure, such as the dispersion diagrams, and their application to optical devices, printed circuits antennas or communication system [7].
3 In 2003, Tretyakov discussed the possibility of realizing negative refraction by chiral nihility. The authors rst proposed the idea to fabricate a metamaterial composed of chiral particles, such as helical wires [8].
In 2004, Pendry discussed the possibility to achieve negative refraction in chiral metamaterials. He analyzed the conditions to realize negative refraction in chiral metamaterials and showed that they are simpler than for the regular metamaterials, which require both electric and magnetic resonances to have ε negative and negative µ. In chiral metamaterials, as mentioned above neither ε nor µ needs to be negative. As long as the chiral parameter χ is large enough, negative ε can be obtained in chiral metamaterials. Pendry then proposed a practical model of a chiral metamaterial working in the microwave regime with twisted Swiss rolls as elemental structures [9].
Now days, important work is being developed in the chiral media , chiral meta- materials are one of the most interesting subjects that are being explored. The fact that oer a simpler route to negative refraction, with a strong chirality, with neither ε nor µ negative required, because the chirality can replace these condi- tions, this subject has been constantly approached by researchers. So it becomes important to analyze the chiral media in order to develop new applications.
The chiral metamaterials with large optical activity have also been proposed and made for polarization control applications at microwave and optical fre- quencies. Research groups have been studying chiral metamaterial design with strong tunable optical activity in a relatively wide frequency bands with low transmission losses, makes it a very e cient material for tunable polarization rotators [10].
Also chiral printed circuits are being explored, there is an example of a four- port cascaded circuit model, which is mentioned as chiral cascaded circuit, is presented to represent an isotropic and lossless chiral media. Such a model is
4 based on the concept of transmission line and characteristic transformers.
Such a circuit model provides an e cient way to realize chiral media using transmission-line circuits and may nd potential applications in microwave tech- nologies [circuito].
1.2 Motivation and Objectives
There are certain areas within the electromagnetic research of today containing potential for diverse new applications in engineering. One of the most interesting
elds to be explored are the novel materials eects.
The progress of theoretical understanding of the wave-material interaction has been increasing since the 1990's, and the electromagnetic phenomena has been studied in order to solve new solutions for the problems, specially related to microwaves.
In Maxwell's theory of macroscopic electromagnetism, material media are de- scribed phenomenologically by constitutive relations.
Depending on the particular form of the constitutive relations, a medium can be characterized as homogeneous, inhomogeneous, isotropic, anisotropic, bi- anisotropic.
The constitutive relations of a bi-anisotropic medium relates D to both E and
B and H to both E and B.
When all four tensors become scalars quantities, the medium may be called
BI, which is the simplest case for reciprocal bi-anisotropic medium. And where exists magneto electric coupling of the elds, but the properties of the material are independent of the directions in space [11].
Chiral materials have been intensively studied since it is widely believed that they can be used to produce novel microwave devices and structures. Appli-
5 cations for chiral materials are, for instance, polarization transformers, phase shifters and devices that correct the cross polarization in lens antennas.
Some experiments have been done along the years such as the chirosorbTM , that introduced a novel synthetic material, which was invisible to electromagnetic energy and has properties which are independent of polarization in the back scatter direction.
This experiment is useful for radar identi cation and inverse scattering prob- lems, the location and shape of targets can be detected from a knowledge of waves re ected or scattered from the target boundaries. These boundaries can be looked on as variations or discontinuities of electrical parameters. If the re-
ected or scattered waves can be reduced signi cantly, the location and shape of the boundaries, and consequently targets, will not be detected. In other words, the targets will become 'invisible' [12].
This work aims to understand the theoretical interaction between waves and chiral media. From the study of chiral structures it is possible to observe the eect of polarization rotation, the propagation modes and cuto frequencies.
The re ection and transmission of dielectric/chiral interface are also analyzed as well as the Breweter's angle in uence with chirality.
BI structures are also observed as the parallel-plate and gounded chiroslab which is importance once it gives us more information and knowlege for chirowave structures.
The motivation of this work is to understand the theoretical concept of chiral media, in order to provide knowledge to pursue potential applications of the chiral materials to optical devices or waveguides and printed-circuits in the microwave and millimeter wave regime.
6 1.3 Structure of the dissertation
The present dissertation is composed by ve chapters.
The introduction referres to a general description of the chiral media since the discovery of the chiral properties until the evolution of the present days.
This section explains when was the chiral media rst observed and what studies have been done to understand their properties and characteristics.
Starting in the early nineteenth century where Arago (1811) started to observe the phenomenon of optical activity in a crystal quartz until the actual days where many experiments and studies have been done in several areas for the chiral media, such as the polymer science and manufacture of arti cial dielectric, for the application at the microwave or millimeter wavelengths.
In this chapter is also expressed the motivations and objectives of this disserta- tion, the reason for studying chiral media and some speci c properties of chiral media that will be addressed.
The second chapter the basic notions of the BI media are presented.
Applying Maxwell's equation in chiral media, properties of this media behavior are reviewed such as the optical activity, the wave eld postulates, the charac- teristic waves and the left and right polarizations describing mathematically.
All the knowledge of these properties will allow us to understand some results of the following chapters.
In the third chapter it is represented the mathematical problem of re ection and transmissions through a simple isotropic media (SIM) - Chiral media. The study of this chiral media is based on a Cartesian coordinate system (x, y, z).
When a plane wave is incident upon a boundary between a dielectric and a chiral medium splits into two transmitted waves proceeding into the chiral medium, and a re ected wave propagating back into the dielectric.
7 The study of re ection and transmission between SIM and chiral media will be done by determining the re ection and transmission coe cients.
In this chapter, it is also possible to observe the in uence of chirality over
Brewster's angle.
In the fourth chapter a study of a method for the analysis of a BI structure for planar waveguides is done . This method is general for all the BI homogeneous layered waveguides, this method is based on a 2 × 2 coupling matrix eigenvalue problem and it will be solved for a parallel-plate and also grounded chiral slab.
The parallel-plate is a simpler problem to be solved since the structure is sym- metrical, for a grounded chiral slab the structure is more complex since it is consider the Perfect electromagnetic conductor (PEC) on the ground, and the top of the chiral slab it is in contact with the air.
For both of the structures the modal equations is obtained in order to observe the propagation surface modes, and the cuto frequency is
In the fth and last chapter, all the results from the second, third and fourth chapter are discussed and the main conclusions explained so that the most interesting results may be useful to further works.
1.4 Main contributions
The research done in the electromagnetic theory has been increasing its relevance on the study of complex materials as the chiral, pseudo-chiral, omega and all the bi-anisotropic media in general.
The chiral media, is a reciprocal BI media that can be de ned and described through Maxwell electromagnetic equations. Although this subject has been studied since the 19th century, many properties and new applications are being studied in the actual days.
8 This work aims to give an overview about chiral media in general. The concept of optical activity, polarization, waveguides and characteristic waves is analyzed based on Maxwell's equations.
Apart from describing chiral media characteristics, the concept of re ection and transmission of a monochromatic plane wave upon a simple isotropic media
(SIM)/chiral interface is also observed. This is an interesting problem since it can be related to a chiral optical ber with a dieletric core.
Adopting a four-parameter model in the EH set of constitutive relations char- acterizing a BI medium examples of applications are given in order to analyze the surface modes of a parallel-plate chirowaveguide and a grounded chiroslab.
The study of concepts and application of chiral theoretical basic problems allows to complement and resume chiral properties, and hopefully will contribute to the continuity of more and more complex studies.
9 10 Chapter 2
Bi-isotropic and chiral media properties
In the present chapter basic notions between the bi-isotropic medium (BI) and the electromagnetic eld will be presented. A homogeneous BI medium can be split into wave elds, each of which sees the BI medium as an isotropic medium, which becomes easier to solve electromagnetic problems. The chiral media be- longs to BI medium and presents the same properties such as, the characteristic waves and the polarization rotation.
11 2.1 Introduction
The BI materials have the special optical property that they can twist the polarization of light in either refraction or transmission, which is called the optical activity.
This does not mean all materials with twist eect fall in the BI class. The twist eect of the class of BI materials is caused by the chirality and non-reciprocity of the structure of the media, in which the electric and magnetic eld of an electromagnetic wave (or simply, light) interact in an unusual way.
The BI media are birefringent which explains the two eigenvalues with dierent propagation factors.
The BI media can be described electromagnetically by the constitutive relations presented as
D = E + ξ H (2.1) B = ζ H − µ E
Where √ ξ = (κ + iχ) 0 µ0 and √ ζ = (κ − iχ) 0 µ0
The dielectric response of the material is contained in the permittivity = ε0ε which corresponds to the electric parameter and permeability µ = µ0µ, which corresponds to the magnetic parameter.
The i emphasizes the frequency domain character of the equations, and comes from the time-harmonic convention exp(−iwt), and the free-space parameter √ 0µ0.
12 The chirality parameter is represented by χ, it measures the degree of the hand- edness of the material and κ describes the magneto electric eect.
In many books, κ is considered the chirality of the material, but in this case, it will be represented by χ.
It is possible to observe the several classi cations of a medium according to the parameters of chirality and reciprocity, in Table 2.1.
This work, however, the study will be focused on the Pasteur medium, which is chiral and reciprocal.
nonchiral chiral (χ = 0) (χ 6= 0) reciprocal simple isotropic Pasteur medium (κ=0) medium or chiral nonreciprocal Tellegen general bi-isotropic (κ6= 0) medium medium Table 2.1: Classi cation of bi isotropic medium
2.2 Chiral Media
A chiral media, is said to be a macroscopically continuous medium composed of equivalent chiral object uniformly distributed and randomly oriented. Its main property relies on the fact that the object does not have a mirror image in rotation or translation. An object of this sort must have the property of handedness that is left-handed polarized or right-handed polarized [1].
A homogeneous BI media can be split into partial elds, the wave elds, and each one of them can be seen as an isotropic medium, which becomes easier to solve electromagnetic problems.
13 The BI constitutive relations and the chiral media, the constitutive relations are the same, (2.1) and can be related to Maxwell equations as
∇ × E = i ω B (2.2) ∇ × H = −i ω D and considering the constitutive relations of the BI media (2.1), we can rewrite
Maxwell's equations (2.2), for both electric and magnetic eld in frequency domain as it is shown below
∇ × E = iωµ H − i ω ζE (2.3) ∇ × H = −i ω E + i ω ξ H
2.2.1 Wave eld postulates
One of the mains aspects of the homogeneous unbounded electromagnetic wave propagation, is the de nition of their characteristic waves.
In the case of the chiral media, it is important to analyze the two characteristic waves to determine its polarization.
In order to verify the rotation of polarization, the electric and magnetic eld vectors E and H will be de ned with two other eld quantities. The wave elds will be decomposed in parameters represented as plus and minus, which combined will represent the total eld as
E = E+ + E− (2.4) H = H+ + H−
14 Considering chiral media as equivalent isotropic media, we will obtain two waves, which will be designated as positive and negative waves.
Satisfying the Maxwell equations of an achiral media, where there is no coupling, it is possible to obtain the positive and negative wave as
∇ × E+ = i ω µ+ H+ ”positive” wave (2.5) ∇ × H+ = −i ω + E+
∇ × E− = i ω µ− H− ”negative” wave (2.6) ∇ × H− = −i ω − E−
Considering the constitutive relations from the equation (2.1), we consider an equivalent isotropic media with the parameters ε+, ε−, µ+, µ− , the medium parameters will satisfy the conditions written below [13]
√ D+ = ε E+ + ξ ε0µ0 H+ = ε+ E+ (2.7) √ B+ = ζ ε0µ0 E+ + µ H+ = µ+ H+
√ D− = E− + ξ ε0µ0 H− = − E− (2.8) √ B− = ζ ε0µ0 E− − µ H− = µ− H−
After eliminating the eld vectors we can observe that the equivalent parameters
±, and µ± must satisfy the conditions below
(ε − ε+)(µ − µ+) − ξζ = 0 (2.9) (ε − ε+)(µ − µ+) − ξζ = 0
Also, the wave eld vectors must satisfy relations which can be written in the
15 form
E± = i η± H± (2.10)
The wave impedance parameters are de ned as
ξ µ+−µ η+ = −i = −i +− ζ (2.11)
ξ µ−−µ η− = i = i −− ζ
Since the electric eld and magnetic eld are interrelated through equation
(2.10), certain restrictions for the parameters arise. Inserting (2.10) with the
H+ of (2.5), one obtains
1 2 (2.12) 5 × H+ + i ω + E+ = −i (∇ × E+ + ω + η+ H+) = 0 η+ which should coincide with the equation from (2.5). This leads to the following relation
r µ± η± = (2.13) ε±
Replacing (2.11) and (2.15)
2 2 (µ± − µ) µ± µ± (2.14) η± = − 2 = = ξζ ζ ± + µ±−µ
Considering (2.13) one has
rµ η = η = η = (2.15) ± − ε
16 So the two characteristic waves have a constant of propagation stated in the following equation
k± = n±k0 (2.16)
Where k+corresponds to the positive wave and k− to the negative wave.
2.2.2 Polarization
Since the wave elds components of a plane wave do not couple in a homogeneous medium, we can analyze them as an independent plane wave which can be written as the expressions below
E±(r)= E±exp(i k± · r) (2.17)
H±(r)= H±exp(i k± · r) (2.18) it is assumed that components propagate in the same direction de ned by the real unit vector kˆ as mentioned
k± = k±kˆ = (n±k0)kˆ (2.19)
Unlike the simple isotropic media, the solutions from (2.17) and (2.18) are only possible for certain polarizations (circular), which coincide with those of the wave elds [13].
De ning a distance
ς = kˆ · r (2.20)
17 The characteristic waves may be written as
E+(r) = [E+exp( i χ k0 ς)]exp( i n k0 ς) (2.21) E−(r) = [E−exp(−i χ k0 ς)]exp( i n k0 ς)
In this case, the Maxwell equations from (2.5), (2.6), will obtain the following values
k± × E± = ωµ±H± (2.22) k± × H± = −ωε±E± where E and H become
k± ˆ ˆ E± = − ω ε (k × H±) E± = −η±(k × H±) ± → (2.23) k± 1 H± = (kˆ × E±) H± = (kˆ × E±) ω µ± η±
Since the TEM waves correspond to no electric nor magnetic eld in the direction of propagation
T EM wave → kˆ · E± = kˆ · H± = 0 (2.24)
The orthogonal relations will be given by
E+ · H+ = 0 (2.25) E− · H− = 0
From (2.10) and (2.15), of electric and magnetic eld vectors of the wave elds components, from Maxwell equations
18 i H+ = − E+ η (2.26) i H− = η E−
Based on (2.23), the electric eld satis es
ˆ k × E+ = −iE+ (2.27) ˆ k × E− = iE− where the E eld is orthogonal to the direction of propagation
E+ · E+ = 0 (2.28) E− · E− = 0
According to (2.28), each characteristic wave has circular polarization (CP). To determine which wave corresponds to the right or left polarization, let us con- sider a real vector notation, which will be suitable for describing time-harmonic vectors, which are real vectors rotating along an ellipse in a plane.
A(t) = Accos(ωt) + Assin(ωt)
It is possible to establish a relation between a complex vector with two real vectors a = ar + iai, where a real time-harmonic vector is de ned as
A(t) = <{a exp(−iωt)}
= <{(ar + iai)[cos(ωt) − i sin(ωt)]}
= arcos(ωt) + aisin(ωt)
19 so and . Inverting the relation above 2×π , one gets Ac = ar As = ai T = ω
T a = A(0) + iA( ) = A +iA (2.29) 4 c s
∗ The complex conjugate of a complex vector, a = ac − ias, corresponds to the time-harmonic vector A(−t) which means that the sense of rotation along the ellipse is reversed from the A(t).
This a vector it will be very useful to determine the polarization of vector A(t).
Some notes to be kept in mind are the fact that
Ac × As = 0 the vectors may be parallels or one of them might be null.
Ac × As 6= 0 the vectors will de ne the rotation of the vector A(t).
If we consider the Linear Polarization (LP),
Ac × As = 0, since Ac = ar and As = ai, we obtain a LP, if ar × as = 0.
By other hand the vectors
∗ a × a = (ar + iai) × (ar − iai) = −i(ar × ai) = −2i(ar × ai)
∗ ∴ ar × ai = 0 ⇔ a × a = 0 Then we are able to de ne
∗ ∴ LP → a × a = 0 (2.30)
For the Ac × As 6= 0 case it is possible to obtain the elliptical and the circular polarization.
20 For the particular case of the CP, we consider
2 2 2 2 2 2 | A(t) | =| Ac | cos (ωt)+ | As | sin (ωt) + (Ac · As)sin(2ωt) = R where R represents the radius.
For t=0, the| Ac |= R;
For t= T , the . 4 | As |= R Then, | Ac |= | As |= R where CP is
2 2 2 CP →| A(t) | = R + (Ac · As)sin(2ωt) = R which implies
Ac · As = 0
∴ CP → a · a = 0 (2.31) then
2 2 a · a = ar × iai · (ar + iai) = |ar| − |ai| + 2i(arai)
Therefore,
2 2 a · a = 0 implies that | ar | =| ai | = 0.
As mentioned before Ac = ar and As = ai , where we can conclude that PC corresponds to (2.31).
From this section of circular polarization in chiral media, we can resume it through the Table 2.3
21 polarization condition acronym linear a × a∗ LP circular a · a = 0 CP ellyptical other EP
Table 2.3: Conditions of polarization
The direction of rotation can be obtained through real-valued vector p which gives information about the polarization corresponding to a complex vector a
a × a∗ p(a) = (2.32) ja · a∗ p(a) points into the right-hand normal direction of the ellipse of the complex vector a , and its length is simply related to the axial ratio of the ellipse, e as it is seen in the Figure (2.1)
Figure 2.1: The polarization vector p(a)changes direction in the sense of rota- tion on the ellipse is changed
Inserting the electric eld E± from (2.27) in a of (2.32) it results in
∗ E± × E± E± × (k × E∗±) (2.33) p(E±) = i ∗ = = ±k iE± · E± E± · E±∗
This means that the wave eld E+ is a right-hand circularly polarized (RCP)
22 vector with respect to the direction of propagation u. On the other hand E− is left hand (LCP) vector, since it is right-handed when looking in the −u direction.
2.2.3 Polarization Rotation
In the magneto plasmonics and iron it is observable the Faraday rotation eect, which is an interaction between light and a magnetic eld in a medium.
The Faraday eect causes a rotation of the plane of polarization which is lin- early proportional to the component of the magnetic eld in the direction of propagation.
The chiral media it is observable the optical activity. Although the both eects cause a rotation in the polarization, the rst one has a non-reciprocal eect, while the second has a reciprocal eect [13].
The polarization rotation in chiral media it is only possible due to the circular birefringence, which means that we get circularly polarized TEM waves, which
”positive” wave → k+ = n+k0 = (n + χ)k0 → RCP
”negative” wave → k+− = n−k0 = (n − χ)k0 → LCP
We assume that the direction of propagation along the positive z axis, i.e. u = uz. Taking linearly polarized electric eld with an amplitude vector E sat- isfying E.uz =0 we can decompose in into two circularly polarized unit vector when looking in the direction of uz
1 u+ = √ (ux − iuy) 2
23 1 u− = √ (ux + iu) 2
where u+ is a RCP and u−is a LCP unit vector when looking in the uz.
A plane wave polarized along ux at z=0 is de ned by
E E = uxE = (u+ + u−)√ 2 or
E E E(z = 0) = 0 (ˆx+iˆy) + 0 (ˆx−iˆy) (2.34) 2 2
Taking into account the direction of propagation, the eld z = d, is given by
E E E(z = d) = 0 (ˆx+iˆy)exp(in k d) + 0 (ˆx−iˆy)exp(in k d) (2.35) 2 + 0 2 − 0
To simplify, one can de ne
1 φ = (k + k )d = (n + n )k d 2 + − + − 0
1 Ψ = (k − k )d = (n − n )k d 2 + − + − 0 where
φ + Ψ = k+d = n+k0d exp(in+k0d) = exp(iφ) × exp(iΨ) ⇒ φ − Ψ = k−d = n−k0d exp(in−k0d) = exp(iφ) × exp(−iΨ)
Where (2.35) can be replaced by
24 E E(d) = 0 [(ˆx+iˆy)exp(iΨ) + (ˆx−iˆy)exp(−iΨ)] exp(iφ) (2.36) 2
E = 0 {xˆ[exp(iΨ) + exp(−iΨ)] + iˆy[exp(iΨ) − exp(−iΨ)]} exp(iφ) 2
= E0xˆ[cos(Ψ) − ˆysin(Ψ)]exp(iφ)
From (2.36) we can observe that exists rotation of polarization on Ψ angle, shown in the Figure 2.2
z = 0 → E = ˆxE0
z = d → E = E0[ˆx cos(Ψ) − ˆysin(Ψ)]exp(iφ)
Figure 2.2: Polarization Rotation in chiral media
25 26 Chapter 3
Re ections and Transmissions between a simple planar interface and chiral media
When a plane wave is incident upon a boundary between a dielectric and a chiral medium it splits into two transmitted waves proceeding into the chiral medium, and re ected wave propagating back into the dielectric.
In this chapter it will be analyzed the behavior of the re ection and transmission of a monochromatic wave who is obliquely incident upon the interface between simple isotropic media (SIM) and chiral.
27 Introduction
A plane wave is considered to be a good approximation at a large distance of their sources and it is also a simple solution of Maxwell's equations, to represent wave propagation in chiral media.
In the case of a incident plane wave passing in a dieletric/chiral interface, the best way to mathematically formulate the problem of re ection and transmission is using a cartesian coordinate system (x,y,z).
Since the chiral medium is isotropic, there is no preferred direction of propa- gation, so usually the monochromatic plane wave propagates along the positive z-axis of the Cartesian system.
In order to calculate the amplitude of the re ected and transmitted waves and their polarization properties the boundary conditions must be applied to the electric and magnetic elds at planar interface, as shown in Figure 3.
The chiral media constitutive relations as mentioned on the previous chapter are given by
√ D = ε0εE + i ε0µ0χH (3.1) √ B = µ0µH−i ε0µ0χE
28 3.1 Transmission and Re ection
Figure 3.1: Re ected and transmitted waves at an oblique incidence on a semi- in nite chiral medium
The SIM interface has a permittivity ε1 and permeability µ1 and the chiral medium is represented by their constitutive relations presented in (3.1)
Considering the wave vector of the incident de ned as
ki = kiˆui where
ˆui = sin(θi)ˆx + cos(θi)ˆz
In free space propagation the waves are TEM , so they can be decomposed in parallel and perpendicular components along the plane of incidence. Since the
29 two transmitted waves are circularly polarized, the electric and magnetic eld can be written as [15].
Ei = Ei0exp(iki.r) = Ei0exp[iki(xsinθi + zcosθi] (3.2)
With
Ei0 = Ei⊥ + Eik = Ei⊥ˆy + Eik(cosθiˆx − sinθiˆz) (3.3) and
Hi = Hi0exp(iki.r) = Hi0exp[iki(xsinθi + zcosθi)] (3.4) where
Hi0 = Y1(ˆui × Ei0) = Y1[Eikˆy − Ei⊥(cosθiˆx − sinθiˆz) (3.5)
where Y1 is the chiral admittance of the dielectric medium, given by
r ε1 Y1 = Y0 (3.6) µ1
The wave vector of the re ected wave is
kr = kr ˆur where
ˆur = sin(θr)ˆx − cos(θr)ˆz and
30 Et1 = E01(cos θ1ex + sin θ2ez + iey) (3.7)
Ht1 = −iY2E01(cos θ1ex + sin θ1ez + iey) (3.8)
Et2 = Et2(cos θ2ex + sin θ2ez − iey) (3.9)
Ht2 = −iY2Et2(cos θ2ex + sin θ2ez − iey) (3.10)
Since the two transmitted waves are circularly polarized, they can be written as
Et = E01exp[ih1(z cos θ1 − x sin θ1)] + E02exp[ih2(z cos θ2 − x sin θ2)] (3.11)
Ht = H01exp[ih1(z cosθ1 − x sin θ1) + H02exp[ih2(z cos θ2 − x sin θ2)] (3.12) where
E01 = E01(cos θ1ex + sin θ2ez + iey) (3.13)
−1 (3.14) H01 = −iZ2 E01(cos θ1ex + sin θ1ez + iey) and
31 E02 = E02(cos θ2ex + sin θ2ez + −iey) (3.15)
−1 (3.16) H02 = −iZ2 E02(cos θ2ex + sin θ2ez − iey)
Here Y2 is the chiral admittance of the chiral medium, given by
r ε2 Y2 = Y0 (3.17) µ2
The parameters from the incident wave are known. In order to nd the complex- constant amplitude vectors of the re ected and transmitted waves, the boundary conditions for the tangential x and y components of the electric and magnetic
elds, have to be applied at the interface
(Ei + Er) × zˆ = Et × ˆz (3.18)
(Hi + Hr) × zˆ = Ht × ˆz (3.19)
The conditions above (3.18) and (3.19) can only be applied if
ki sinθi = kr sinθr = k01sinθ1 = k02sinθ2 (3.20) which is Snell's law.
Based on (3.20) relations obtained are given by
Eik cosθi + Erk cosθi = E01cosθ1 + E02cosθ2 (3.21)
32 Ei⊥ + Er⊥ = i (E01 − E02) (3.22)
Y1(Eik − Erk) = Y2(E01 + E02) (3.23)
Y1(Ei⊥ − Et⊥) cosθi = iY2(E01cosθ1 − E02cosθ2) (3.24)
That can also be represented by
0 −cosθi cosθ1 cosθ2 Er⊥ Ei⊥cosθi −1 0 i −i E E rk i⊥ = 0 Y1 Y2 Y2 Et1 Y1E ik Y1cosθi 0 iY2cosθ1 −iY2cosθ2 Et2 Y1Ei⊥cosθi (3.25)
The expressions for Er⊥ , Erk, E01, E02 can be written according to the compo- nents of the incident waves as expressed below, where the re ection and trans- mission coe cients matrix can obtained.
Er⊥ R11 R12 Ei⊥ = (3.26) Erk R21 R22 Eik
33 Et1 T11 T12 Ei⊥ = (3.27) Et2 T21 T22 Eik
For the re ection coe cient matrix, one has
2 2 2 (Y1 − Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θ1 − cosθ1cosθ2) (3.28) R11 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2)
−2Y1Y2(cosθ1 − cosθ2)cosθi R12 = R21 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2) (3.29)
2 2 2 (Y1 − Y2 )(cosθ1 + cosθ2)cosθi − 2Y1Y2(cos θ1 − cosθ1cosθ2) (3.30) R22 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2)
For the transmission coe cient matrix, one gets
−2iY1cosθi(Y1cosθ2 − Y2cosθi) (3.31) T11 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2)
2Y1cosθi(Y1cosθi − Y2cosθ2) (3.32) T12 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2)
34 2iY1cosθi(Y1cosθ1 − Y2cosθi) (3.33) T21 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2)
2Y1cosθi(Y1cosθi + Y2cosθ1) (3.34) T22 = 2 2 2 (Y1 + Y2 )(cosθ1 + cosθ2)cosθi + 2Y1Y2(cos θi + cosθ1cosθ2)
When the incident wave falls normally on the interfaces, i.e., θi = 0, the expres- sions above, get reduced to
1 − (Y1Y2) R11 = R22 = (3.35) 1 + (Y1Y2)
−i T11 = −iT22 = (3.36) 1 + (Y1Y2)
1 T12 = −iT21 = (3.37) 1 + (Y1Y2)
To visualize the relations of the equations from (3.28) to (3.34), three dimen- sional graphs are done with the chirality parameter χ and θi as the variables.
The permittivity from the media 1 and 2 are ε1 = 1 and ε2 = 2 and µ1 = µ2 = 1.
35 Figure 3.2: Re ection coe cients R11 for ε1 = 1e ε2 = 4
Figure 3.3: Re ection coe cients R22 for ε1 = 1e ε2 = 4
36 Table 3.1: Re ection coe cients R12 = R21 for ε1 = 1e ε2 = 4
For the transmission coe cients the same variables will be used which is χ and
θi , and also the same parameters ε1 = 1 e ε2 = 4.
Figure 3.4: Transmission coe cients T11 for ε1 = 1e ε2 = 4
37 Figure 3.5: Transmission coe cients T22 for ε1 = 1e ε2 = 4
Figure 3.6: Transmission coe cients T12 for ε1 = 1e ε2 = 4
3.1.1 The Brewster Angle
A monochromatic plane wave of arbitrary polarization, on re ection from a chiral medium, can become linearly polarized wave.
The angle of incidence at which this phenomenon occurs is called the Brew- ster angle. The plane containing the electric eld vector and the direction of propagation, is the plane of polarization. For a linear polarized wave the angle between the plane of polarization and the plane of incidence is called the az- imuthal angle. This angle ranges from π to π and is de ned to be positive − 2 2
38 whenever the direction of rotation of the plane polarization towards the plane of incidence and the direction of wave propagation form a right-handed screw
[?].
Considering αi and αr to be the azimuthal angles of the incident and re ected waves. In the equations below it can be shown that αi and αr can be complex angles
Ei⊥ tan αi = (3.38) Eik
Er⊥ tan αr = (3.39) Erk
The amplitudes of the parallel and perpendicular components of the incident and re ected waves are related to the equations (3.26) and (3.27) from the matrix of re ection and transmission. Using de nitions (3.38) and (3.39) and the matrix of re ection in (3.26), one gets
R12 + R11tan αi tan αr = (3.40) R22 + R21tan αi
The re ected wave is linearly polarized if the incident wave is incident upon the interface at Brewster angle (θB), otherwise, αr must be real constant for all αi [16].
When (84) is dierentiated with respect to αi, one gets
R11 R22 − R12 R21 = 0 (3.41)
Under this condition, equation (3.40) becomes
39 R12 R11 tan αr = = (3.42) R22 R21
And from equations (3.28) to (3.30) into (3.41), one will obtain
(1 − (Y Y )2)2 cos2θ (cosθ + cosθ )2 1 2 i 1 2 (3.43) 2 2 2 2 2 = 4(Y1Y2) (cos θi − cos θ1)(cos θi − cos θ2)
If θ1 and θ2 are written in terms of the angle of incidence θi, then it is possible to solve a numerical equation in terms of θi, since the angles of the transmitted waves can be expressed as
ki sin θi θ1 = arcsin (3.44) h1
ki sin θi θ2 = arcsin (3.45) h2
40 Chapter 4
A method for the analysis of bi-isotropic planar waveguides
In this chapter, it is described a general formalism for general bi-isotropic planar waveguides, and as an example it is studied theoretically the chirowaveguides.
In order to understand some of its properties and using the four-parameter model in the EH representation for the set of constitutive relations character- izing a bi-isotropic medium, a 2 × 2 coupling matrix eigenvalue-problems will be solved in a general description and later will be applied to a parallel-plate chirowaveguide and a grounded chiroslab. In this case it will be analyzed the modal equations and the cuto wavelengths for the guided hybrid modes.
41 4.1 Introduction
The problem of guided electromagnetic wave propagation in general bi-isotropic planar waveguides is described in terms of a linear operator formalism. Based on the transverse electromagnetic eld equations an eigenvalue problem is ad- dressed. An example of a bi-isotropic planar waveguide is a chirowaveguide.
The concept of chirowaveguide was de ned in the early 90's when the scienti c community showed a systematic interest in the electromagnetic properties and applications of these special isotropic materials.
A chirowaveguide can be a cylindrical waveguide lled with homogeneous isotropic chiral material. The electromagnetic chirality of the material inside the waveg- uide has several important features which were already analyzed, such as the re ection and transmission of guided electromagnetic waves, the eect of chiral material loss on guided electromagnetic modes, dispersion relations and cut-o frequencies.
It was also shown that the Helmholtz equations for the longitudinal components of electric and magnetic elds are always coupled and consequently in these waveguides, individual transverse electric (TE), transverse magnetic (TM), or transverse electromagnetic (TEM) modes cannot be supported [7].
The interest on this structure besides the academic one, resides in the fact that are some potential applications of chiral materials to integrate optical devices, optical waveguides and printed-circuit elements.
In this present case, from Maxwell curl equations for source free regions, the analysis of guided hybrid modes in a bi-isotropic layered structures is reduced to a 2 × 2 coupling matrix eigenvalue problem [17].
The structures to be studied are the parallel-plate chirowaveguide, consisting of two parallel perfectly conducting planes lled with a lossless, homogeneous,
42 isotropic chiral material, and the grounded chiroslab, where the chiral slab is in contact with the air, which introduces a more complex problem compared to the parallel-plate [22].
The main feature of the guides is that the propagation modes are always hybrid.
The application of the method
In this present section, the problem of surface waves in a chiral slab will be deter- mined from the derivation of simple closed-form expressions, than can be used to general bi-isotropic homogeneous layered waveguides. The time-harmonic variations of the form exp(−iwt) is considered. In the frequency domain, considering the Maxwell equations, ∇ × E = iωB (4.1) ∇ × H = −iωD we can represent the constitutive relations of the chiral media as,
√ D = ε εE + i ε µ χH 0 0 0 (4.2) √ B = µ0µH − i ε0 µ0 χE √ ∇ × E = iωµ0µH + ωχ ε0µ0 E (4.3) √ ∇ × H = −iωε0εE + ωχ ε0µ0 H
0 0 Considering normalized distances, one gets x = k0x, z = k0z The structure in nite and uniform in the y direction ∂ and , ∂y = 0 ∇ = ∂xˆx−inzˆ one has
ˆx ˆy ˆz ∼ ∼ ∼ ∂A = ∂Az y ∇ × A = ∂ 0 in xˆ(−inAy) +y ˆ(inAx − ) +z ˆ( ) ∼ ∂x ∂x ∂x
Ax Ay Az
43 From rot E: ∇ × E = iωµ0µH + k0χE
x : −inEy = iωµ0µHx + k0χEx (4.4)
∂E y : − z + inE = iωµ µH + k χE (4.5) ∂x x 0 y 0 y
∂E z : y = iωµ µH + k χE (4.6) ∂x 0 z 0 z
From rot H: ∇ × H = iωε0εE + k0χH
x : −inHy = −iωε0εEx + k0χHx (4.7)
∂H y : − z + inH = −iωε εE + k χH (4.8) ∂x x 0 y 0 y
∂H z : y = −iωε εE + k χH (4.9) ∂x 0 z 0 z
From (4.4),one obtains n χ Hx = − Ey + i Ex (4.10) µZ0 µZ0 From (4.6), one gets
∂Ey i χ Hz = − + i Ez (4.11) ∂x Z0µ Z0µ
44 From (4.7), one gets Z n χZ E = 0 H − i 0 H (4.12) x ε y ε x From (4.9), one obtains ∂H Z χZ E = i y 0 − i 0 H (4.13) z ∂x ε ε z
Equations (4.10) and (4.11) will be replaced in equation (4.8) in order to obtain two dierential equations depending on y
2 ∂ Ey 2 2 2 = −2iωµ0µk0χHy + −k (εµ + χ ) + n Ey (4.14) ∂x02 0
The same will be done to equations (4.7) and (4.8) which will be replaced in equation (4.5), where one obtains
2 ∂ Hy 2 2 2 = 2iωε0εk0χEy + −k (εµ + χ ) + n Hy (4.15) ∂x02 0
In order to obtain homogeneous layers from the results above, one has
∂2 u (x0) = Cu (x0) (4.16) ∂2x0 where C is the coupling matrix.
" # " #" # ∂2E k2(εµ + χ2) − n2 2iωµ µk χ E y = 0 0 0 y (4.17) 2 2 2 2 ∂ Hy −2iωε0εk0χ k0(εµ + χ ) − n Hy
Since this problem is reduced to a 2 × 2 coupling matrix, we can obtain two eigenvalues of C
det (C − λI) = 0
2 2 2 k0(εµ + χ ) − n 2iωµ0µk0χ λ 0 det − = 0 2 2 2 2iωε0εk0χ k0(εµ + χ ) − n 0 λ
45 2 2 2 2√ 2 2 2 (4.18) λn = k0εµ + k0χ ± 2k0 εµχ − n = k± − n
represents the wave guided propagation and 2 2 . λn k± = k0n±
Introducing modal matrix M for C, where M is [20] " # 1 1 M = τ1 τ2
Considering the following transformation 0 u(x) = M φ(x ) ∧ Φ = [φ1, φ2]
The equation (4.16) is reduced to
2 ∂ φ(x) −1 = −M C M φ(x0) (4.19) ∂x
−1 with M C M = diag(λ1, λ2) The coupling coe cients from the modal matrix are given by
λ1 − C11 C21 τn = = (4.20) C12 λ2 − C22 Hence, from equation (4.18) and (4.17), one obtains r ±k0 ε τs = −i (4.21) ωµ0 µ
From equations (4.4) to (4.9) the Ex and Hx will be de ned as n Ex = (ξEy + µZ0Hy) (4.22) M n Hx = − (εEy + ςZ0Hy) (4.23) M
The same will be done to Ez and Hz i ∂Ey ∂Hy Ez = ξ + µ (4.24) M ∂x0 ∂x0 i ∂Ey ∂Hy Hz = − ε + ζ (4.25) M ∂x0 ∂x0
Where 4 = εrµr − ξζ
46 The eld components are de ned as
Ey = φ1 + φ2 (4.26)
Hy = τ1φ1+τ2φ2 (4.27)
4.1.1 Chiral media
For εµ 6= χ2 and χ2 6= 0, only hybrid modes can propagate in the BI planar waveguides. TTo solve a wave-guiding problem, apart from the knowledge of the whole struc- ture, it is necessary to study the boundary conditions. As stated before, a chiral media is a lossless and reciprocal bi-isotropic media. For the reciprocity condition, one has [18] For the reciprocity condition, one has ξ = −ζ (4.28)
For a lossless BI medium , ξ = ζ∗ (4.29) where the ∗ denotes a complex conjugate. Although the four-parameter model is chiral, a bi-isotropic medium with (4.28) and (4.29), should be referred as chiral medium instead of lossless chiral reciprocal medium, in which ξ = −iχ For the chiral media, the transverse wave number is given by
2 2 2 (4.30) hs = β± + n where n is the eective refractive index and is given by n = k where k represents k0 the longitudinal wavenumber.
And β± depends if s = 1 or s = 2 and also depends on chirality.
47 √ β± = εµ ± χ (4.31)
4.1.2 Parallel-plated chirowaveguide
As an example of application of this general bi-isotropic media, it will be used a closed waveguide, which is the parallel-plated chirowaveguide.
The parallel-plated chirowaveguide consists of two parallel perfectly conducting planes of in nite length in the x and z directions, and lled with lossless, homo- geneous, isotropic chiral media, as shown in Figure 4.1 described by equations
(4.2).
Figure 4.1: A parallel-plate chirowaveguide
48 Observing the Figure 4.1, the direction of propagation is along z axis and the
eld quantities are all independent of y axis.
Due to the perfectly conducting planes, placed at x0 = 2t0 and x0 = 0, one must
0 0 0 0 0 0 impose that Ey(x = 0) = Ez(x = 0) = 0 and Ey(x = 2t ) = Ez(x = 2t ) = 0. By imposing these boundary conditions, a set of algebraic equations for the coe cients A± will be obtained.
φ (x0) = A [sin(h x0) + cos(h x0)] 1 + + + (4.32) 0 0 0 φ2(x ) = A−[sin(h−x ) − cos(h−x )]
The symmetry of the structure, allows the propagating modes to be divided into even and odd modes. Considering the odd modes φ (x0) = A sin(h x0) 1 + + (4.33) 0 0 φ2(x ) = A−sin(h−x )
The perfectly conducting plane, at x0 = t0 , should have 0 0 Ey(x = t ) = 0 (4.34)
0 0 Ez(x = t ) = 0 (4.35)
Imposing the boundary conditions for these eld components, a homogeneous set of algebraic equations for the coe cients A± in (4.33) and 4.24 is obtained. For the odd modes, one has
0 0 sin(h+t ) sin(h−t ) A+ 0 (4.36) = h+ 0 h− 0 cos(h+t ) − cos(h−t ) A 0 ε+ ε− − Considering the determinant of coe cients zero, a nontrivial solution is ob- tained. This leads to the modal equation for the propagating modes. h− 0 0 h+ 0 sin(h+t) − cos(h−t ) − sin(h−t ) cos(h+t ) = 0 (4.37) ε− ε+ This expression can also be de ned as
49 ε h + ε h ε h − ε h − + + − sin[(h + h )t0] − − + + − sin[(h − h )t0] = 0 (4.38) 2 + − 2 + −
From the expression mentioned above, it is possible to determine the odd prop- agation modes,
The gure (4.2) represents the TE and HE modes, while the gures (4.3) for
χ = 0 and (4.4) χ = 1 represent the hybrid modes.
Figure 4.2: Propagation of the odd modes for χ = 0
50 Figure 4.3: Propagation of the odd modes for χ = 0.5
Figure 4.4: Propagation of the odd modes for χ = 1
For the even modes, one has
0 0 cos(h+t ) cos(h−t ) A+ 0 (4.39) = h+ 0 h− 0 sin(h+t ) − sin(h−t ) A 0 ε+ ε− − h− 0 0 h− 0 cos(h+t) − sin(h−t ) − cos(h−t ) − sin(h−t ) = 0 (4.40) ε− ε−
51 that can also be given through
ε h + ε h ε h − ε h − + + − sin[(h + h )t0] − − + + − sin[(h − h )t0] = 0 (4.41) 2 + − 2 + −
Figure 4.5: Propagation of the even modes for χ = 0
Figure 4.6: Propagation of the even modes for χ = 0.5
52 Figure 4.7: Propagation of the even modes for χ = 1
From the equation (4.41) we can observe the even propagation modes where g- ure (4.5) representsχ = 0, the gure (4.6) represents χ = 0.5 and (4.7) represents
χ = 1.
At the cuto, , which implies (4.30) will be 2 2 . β = 0 hs = n±
Considering ε± = ε ± ycχ, one can make ε+h+ = ε−h− in (4.37) and (4.40) modal equations, where the expression obtained is
0 sin[(β+ + β−)t ] = 0 (4.42)
t n = √ (4.43) λc 4 εµ with n = 1, 2, 3, ...
If we consider the parameter t as a xed element, we can observe how the is λ n in uenced by the χ parameter.
For frequency f = 100 GHz, and t = 1 mm one has
53 t . λ ⇒ t = 0.33s
Figure 4.8: Variation of β with χ
Observing n as a function of χ, in gure (4.8) we can observe two propagation modes.
Considering the previous graphs from the propagation modes, where n depends of t , from the Figures (4.3), (4.4), (4.6) and (4.7) , we can observe that for λ t if we intersect the propagation modes, we will obtain two propagation λ = 0.33s modes as observed in (4.8).
4.1.2.1 Grounded chiroslab
In this type of structure, the chiral slab is in contact with air a t thickness And
lled with lossless, homogeneous, isotropic chiral media described by equation
(4.2) .
54 Figure 4.9: Grounded chiroslabguide
For the chiral layer, 0 < x0 < t0, one has
0 0 φ1(x) = A[sin(h+x ) + Γcos(h+x )] (4.44) 0 0 φ2(x) = A[r sin(h−x ) − Γcos(h−x )]
The propagation modes Ey and Hy are de ned by,
0 0 φ1 + φ2, 0 < x < t Ey = (4.45) 0 0 0 B exp[−αA(x − t)], x > t
0 0 τ1φ1 + τ2φ2, 0 < x < t Hy = (4.46) 0 0 0 −iC exp[−αA(x − t)], x > t with dEy and dHy represented by dx dx
0 0 0 0 dE φ1 + φ2, 0 < x < t y = (4.47) dx 0 0 0 −αAB exp[−αA(x − t )], x > t
55 0 0 0 0 dH τ1φ1 + τ2φ2, 0 < x < t y = (4.48) dx 0 0 0 0 iCαA exp[−αA(x − t )], x > t
The coupling coe cients for the hybrid modes are given by
τs = −iys (4.49)
r ε y = ± (4.50) s µ
Where once again, the plus and minus sign represent s=1 and s=2, respectively.
Due to the perfectly conducting plate at x0 = 0, one should have
0 Ey(x = 0) = 0 (4.51)
0 Ez(x = 0) = 0 (4.52)
The r represented on the (4.44) can be obtained through (4.52), where one obtains
h β r = + − (4.53) h−β+
For the air region, x0 > t0, one has
0 0 Ey(x) = B exp[−αa(x − t )] (4.54) 0 0 Hy(x) = −iC exp[−αa(x − t )]
56 where αa represents an attenuation coe cient given by
2 αa = n − 1
0 0 Considering the continuity of Ez and Hz at x = t , it will be possible to determine the modal equation as well as Γ mentioned on (4.32)
0 0 0 0 From Ez (x = t ) = Ez (x = t ), the following parameters will be de ned
0 0 (4.55) η1 = αa M y1sin(h1t ) − h1(χ − µy1)cos(h1t )
0 0 (4.56) ρ1 = αa M sin(h1t ) − h1(χ − µy1)y1cos(h1t )
0 0 (4.57) ν1 = αa M y1cos(h1t ) + h1(χ − µy1)sin(h1t )
0 0 (4.58) σ1 = αa M y1cos(h1t ) + h1(χ − µy1)sin(h1t )
0 0 0 0 From Hz (x = t ) = Hz (x = t ) one has
0 0 (4.59) η2 = αa M y2sin(h2t ) − h2(χ − µy2)cos(h2t )
0 0 (4.60) ρ2 = αa M sin(h2t ) − h2(χ − µy2)y2cos(h2t )
0 0 (4.61) ν2 = αa M y2cos(h2t ) + h2(χ − µy2)sin(h2t )
0 0 (4.62) σ2 = αa M y2cos(h2t ) + h2(χ − µy2)sin(h2t )
57 Thus, the modal equation will be written in the following form
(η1 + rη2)(σ1 − σ2) − (ρ1 + rρ2)(ν1 − ν2) = 0 (4.63)
And Γ is de ned as
η + rη ρ + rρ Γ = − 1 2 = − 1 2 (4.64) ν1 − ν2 σ1 − σ2
Replacing (4.55)-(4.62) the modal equation (4.63) can be simpli ed, after some algebraic manipulation and rewritten in the form
2 2 (4.65) 2[γ1 − (1 − y1)]δ1 + (1 + r )γ2δ2 − 2(ϑ1 − rϑ2) = 0 where (s=1,2), and
2 0 0 (4.66) γ1 = (1 + y1) cos(h1t )cos(h2t )
2 0 0 (4.67) γ2 = (1 + y2) sin(h1t )sin(h2t )
2 2 2 0 0 (4.68) ϑ1 = y1(αa M −h1β−) sin(h1t )cos(h2t )
2 2 2 0 0 (4.69) ϑ2 = y2(αa M −h2β+) sin(h2t )cos(h1t )
Through (4.65) it is possible to determine the cuto wavelength λcof any hybrid mode.
At cuto αa = 0 , hence δs = 0 for s = 1, 2, therefore (4.65) will be reduced to
ϑ1 = r ϑ2 from which one gets
58 f1(λc) + f2(λc) = 0 (4.70) where
f1 = q1β−sin q1cos q2 (4.71)
f2 = q2β+sin q2cos q1 (4.72) and where (s=1,2)
t q 2 (4.73) qs = 2π β± − 1 λc
In (4.73) the plus and minus sing corresponds to s=1 and s=2.
For χ = 0 the cuto frequencies are given by
t TM m = √ (4.74) λc 2 εµ − 1
t TE 2m + 1 = √ (4.75) λc 2 εµ − 1 with m = 0, 1, 2, ....
For numerical calculation the following values will be considered, ε = 9 and
µ = 1 and χ will have several values.
For numerical calculation the values considered are ε = 9 and µ = 1. For Figure
8, χ = 0 where the surface modes are obtained and for Figure 9, χ = 0.5 where we can observe the hybrid modes.
59 Figure 4.10: Surface modes, χ = 0
Figure 4.11: Hybrid modes, χ = 0,5
60 Figure 4.12: Hybrid modes, χ = 1
61 62 Chapter 5
Conclusions
In this chapter all the conclusions and results from the previous chapters are pointed out, as well as future work suggestions.
It will be done an analysis of the concepts that were addressed and the possible causes and consequences will be determined or discussed.
63 In this dissertation the main objective was to study the chiral media and its main properties and also to characterize the propagation modes of some of the theoretical problems that were studied, like the parallel-plate chirowaveguide and the grounded chiroslab.
In the rst chapter, it was referred the historical content of the chiral media, it is possible to understand when did the studies began and what were the properties observed and what experiences have been done along the years, since the begin- ning of the nineteenth century when Arago (1811) rst saw the phenomenon of optical activity in a quartz of crystal , until the present days where there are still doing experiments in the chiral media.
After describing the work of each chapter, the main contributions of this thesis are addressed, explaining the intents to complement and resume the chiral prop- erties, and hopefully to contribute to the continuity of more and more complex studies.
In second chapter important concepts are referred which de ne the bi-isotropic media and chiral medium through their constitutive relations and also the char- acteristic waves were demonstrated through wave eld postulates as well as the polarization and its rotation. It was shown that the polarization rotation in chiral media it is only possible due to the circular birefringence, and from a chiral media CLP and CRP waves are obtained .
The third chapter addresses a re ection/transmission problem between a SIM/chiral interface. This study is very interesting once it allow us to observe the propa- gation modes in dielectric chiral media which can be related to a chiral optical
ber with a dieletric core.
In general, when a monochromatic plane wave is incident obliquely upon the interface, it is obtained the re ected wave on the dielectric media and two prop- agation waves inside the chiral medium.
64 The rst step that was done, was to determine the re ection and transmission coe cients by studying the boundary conditions of the tangential components for E and H, which is basically applying the Snell's law.
In the fourth chapter it is studied and analyzed bi-isotropic planar structures, as the chirowaveguides, and theorical problems are solved for a closed chirowaveg- uide, the parallel-plate and a semi opened one, the grounded chiroslab. This study is done in order to observe the propagation modes of each structure and also to allow us to understand the physical concept that relies on both of the structures.
The rst thing to be done was to de ne the equations in a four parameter model representing EH that characterized the homogeneous bi-isotropic planar structures through Maxwell equations.
The following step was to determine the boundary conditions to determine the modal equations. Once the modal equations are obtained, it was possible to observe the propagation modes of both structures.
For the closed parallel-plate, the gures are separated in even and odd modes.
In gures 4.3-4.5 the χ = 0, 5 and for 4.6-4.7 the χ = 1 for the odd and even modes respectively. From the gures, we can observe that the propagation modes are hybrid, while in χ = 0 for the same gures it is possible to obtain the propagation modes which are the TE and TM.
Observing the gures mentioned above we can also conclude that the cuto frequency does not depend on chirality, since they start always from the same point, when changing the χ value, and from the equation (4.43), the cuto frequency does not have the χ parameter.
For the semi-opened structure, a grounded chiroslab, the same procedure was done, and the modal equation, more complex, was obtained.
The graphs form gures 4.10-4.11 we can observe 6 surface modes are all hybrid.
65 In the achiral case, for χ = 0 the propagations modes are TM and TE.
One should note that , there is a fundamental hybrid mode, EH0 with no cuto t/λc = 0 as shown in gures with χ = 0.5 and χ = 1 .
Comparing the propagation modes from the parallel-plate, we can conclude that they are dierent from the grounded chiroslab and not a superposition of one to another.
From this work we can conclude that the chiral media is an important additional parameter for the researchers, since it's possible to change it's value just like the ε and µ parameters. This fact gives more exibility to the chiral media and may help in the study and development of other materials or devices that use this media.
Perspectives of future work
Although important aspects of this dissertation were mentioned, there is still many studies that can be done to complement this work. This media represents an extensive subject of study.
The use of chiral media in optical ber, where the idea is when the chiral coating is properly designed, the backscattering of the linearly polarized plane wave almost disappears in some speci c frequency bands [24]. Based on the planar dielectric/chiral interface, more studies with dierent media can be done, as a study on a chiral interface between two dielectric media.
Development of more studies for the other media: pseudo-chiral, omega, bian- isotropic media and metamaterials, where some chiral metamaterials oer a simpler route to negative refraction, since in chiral metamaterials with a strong chirality, with neither ε nor µ negative required, because the chirality can replace these conditions.
66 And also the study of ber-reinforced plastic composite cylinder coated by chiral media. This cylinder can be treated as consisting of multilayered bi-isotropic and anisotropic materials. As the chiral coating is properly designed, the backscat- tering of the linearly polarized plane wave almost disappears in some speci c frequency bands [25].
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