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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. 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