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Influence of Chirality on the Electromagnetic Wave Electrical and Computer Engineering 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 intensied, 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 reection and transmission coecients 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; reection 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 desaantes dentro das comunidades cientícas 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 reexão numa interface dieléctrica-quiral, onde se determinam coecientes de transmissão e reexão 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, reexão 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 Waveeld postulates . 14 2.2.2 Polarization . 17 2.2.3 Polarization Rotation . 23 3 Reections and Transmissions between a simple planar inter- face and chiral media 27 3.1 Transmission and Reection . 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 Reected and transmitted waves at an oblique incidence on a semi-innite chiral medium . 29 3.2 Reection coecients R11 for "1 = 1e "2 = 4 ............ 36 3.3 Reection coecients R22 for "1 = 1e "2 = 4 ............ 36 3.4 Transmission coecients T11 for "1 = 1e "2 = 4 .......... 37 3.5 Transmission coecients T22 for "1 = 1e "2 = 4 .......... 38 3.6 Transmission coecients T12 for "1 = 1e "2 = 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 Classication of bi isotropic medium . 13 2.3 Conditions of polarization . 22 3.1 Reection coecients R12 = R21 for "1 = 1e "2 = 4 ........ 37 viii ix Nomenclature BI Bi-isotropic CP Circular Polarization 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 coecient αi Azimuthal angle of the incident wave αr Azimuthal angle of the reected wave p β Parameter containing εµ ± χ δm Algebric parameter from modal equation Permittivity "0 vacuum permittivity "m 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 #m 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 & Distance τm Coupling coecients from modal matrix χ Chirality χm Magnetic Susceptibility Angle of the plane of polarization ! Angular velocity Γ Relation between modal equation parameters ∆ "rµr − 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 k^ Versor k M Modal matrix n Refractive index xvi nef Eective refractive index p Real vector < Real complex R Radius Rmm Reection coecient, with mm = 11; 12; :::; 21; 22::: r Complex vector 2×π TT = ! Tmm Transmission coecient, 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 dened 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.
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