Modeling Optical Metamaterials with Strong Spatial Dispersion
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Fakultät für Physik Institut für theoretische Festkörperphysik Modeling Optical Metamaterials with Strong Spatial Dispersion M. Sc. Karim Mnasri von der KIT-Fakultät für Physik des Karlsruher Instituts für Technologie (KIT) genehmigte Dissertation zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN (Dr. rer. nat.) Tag der mündlichen Prüfung: 29. November 2019 Referent: Prof. Dr. Carsten Rockstuhl (Institut für theoretische Festkörperphysik) Korreferent: Prof. Dr. Michael Plum (Institut für Analysis) KIT – Die Forschungsuniversität in der Helmholtz-Gemeinschaft Erklärung zur Selbstständigkeit Ich versichere, dass ich diese Arbeit selbstständig verfasst habe und keine anderen als die angegebenen Quellen und Hilfsmittel benutzt habe, die wörtlich oder inhaltlich über- nommenen Stellen als solche kenntlich gemacht und die Satzung des KIT zur Sicherung guter wissenschaftlicher Praxis in der gültigen Fassung vom 24. Mai 2018 beachtet habe. Karlsruhe, den 21. Oktober 2019, Karim Mnasri Als Prüfungsexemplar genehmigt von Karlsruhe, den 28. Oktober 2019, Prof. Dr. Carsten Rockstuhl iv To Ouiem and Adam Thesis abstract Optical metamaterials are artificial media made from subwavelength inclusions with un- conventional properties at optical frequencies. While a response to the magnetic field of light in natural material is absent, metamaterials prompt to lift this limitation and to exhibit a response to both electric and magnetic fields at optical frequencies. Due tothe interplay of both the actual shape of the inclusions and the material from which they are made, but also from the specific details of their arrangement, the response canbe driven to one or multiple resonances within a desired frequency band. With such a high number of degrees of freedom, tedious trial-and-error simulations and costly experimen- tal essays are inefficient when considering optical metamaterials in the design of specific applications. Therefore, to be able to discuss metamaterials on equal footing as natural materials and to consider them in the design of functional applications, the homogeniza- tion of optical materials is of utmost importance. Such effective description consists of mapping the optical response of an actual metamaterial to a set of spatially averaged, effective material parameters of a continuum. This step requires that the building blocks from which the metamaterials are made of are small and arranged with sufficient density in space in comparison to the operating wavelength. Often, local material laws have been considered in this mapping process, i.e., metama- terials are frequently modelled at the effective, i.e. the homogeneous level, by an electric permittivity, magnetic permeability, and in case of optical activity, terms that express magneto-electric coupling. Such description is borrowed from natural materials at optical frequencies, where the characteristic length scale is in the subnanometer range. Meta- materials, however, possess a characteristic length that is only slightly smaller than the wavelength of light. Thus, the spatial variations of the fields begin to become important and a local description is not enough to adequately describe the metamaterial at the effective level. In this thesis, we lift this limitation and consider nonlocal constitutive relations in the homogenization process for a realistic modelling of optical metamaterials. Nonlocality means that the effective response of a material at every point depends on the fieldsof light at some distant points or, alternatively, on spatial derivatives of the fields at the same point, or both. We focus on periodic metamaterials with centrosymmetric unit cells with a non-negligible period-to-wavelength ratio and show the importance of retaining nonlocality in the effective description of metamaterials. After introducing the necessary mathematical background, we discuss the physical origin of nonlocality, which in the spatial Fourier domain translates to spatial dispersion, i.e., to a generalized permittivity that depends on the wave vector of light. This can lead to an artificial magnetic response and ultimately to a negative effective index of refraction, and even beyond. Then, the aforementioned generalized permittivity is expanded into a Taylor polynomial of the wave vector up to the fourth order. Dispersion relations describ- ing light propagation in bulk metamaterials that are characterized by such constitutive relations are derived. We discuss the additional mode that emerges with nonlocality. We further, derive the appropriate interface conditions from first principles, in order to study how light couples from one media to another. With the interface conditions at hand, the Fresnel matrices, which ultimately allow us to analytically derive the reflection and transmission coefficients from a slab, are derived. vii viii Finally, we apply this formalism to three metamaterial examples. We show that a nonlocal description captures the properties of actual metamaterials much more accurate than the ordinary local description. Based on the scattering parameter retrieval, the ef- fective material parameters are retrieved from different structures, where the referential reflection and transmission coefficients are numerically calculated with a very highpreci- sion. In the first example, we studied an all-dielectric and isotropic material madeofan array of spheres arranged in a cubic lattice. We find that the optical features such asthe presence of the Brewster angle are better captured with a nonlocal description, especially at frequencies close to the first photonic band gap. In the second example, we investigated the fishnet metamaterial. It has a negative effective refractive index in the studied fre- quency range. We find that a nonlocal description allows to predict the optical properties at oblique incidence, where a local description failed to do so. Further, in the retrieved effective material parameters within the local approach, an unphysical anti-Lorentzian in the permittivity arises. This could be lifted when a nonlocal description is considered. In the third and last example, we studied a wire medium structure, that is a prototypical metamaterial that supports a nonlocal optical response. For this material, a phenomeno- logical approach with nonlocality already exists. We first show that the existing model fundamentally differs from the nonlocal model we have been proposing in this thesis, which suggests that homogenization is not unique and multiple models for an effective description may be used to explain the optical response of a specific metamaterial. We finalize this work by showing the limits of homogenization, and the drawbacks of the proposed retrieval method. In summary, we demonstrate that the nonlocal constitutive relations can describe the optical response much better than local constitutive relations would do. The general for- mulation we choose here can be extended to other kinds of nonlocal constitutive relations and renders our approach applicable to a wide class of centrosymmetric metamaterials. Table of Contents List of Figures xi List of Symbols xi 1 Introduction 1 2 Mathematical Background 9 2.1 Generalized Functions (Distributions) ..................... 10 2.1.1 The spaces D of test functions and D0 of distributions ....... 11 2.1.2 Lebesgue spaces ............................ 13 2.2 Sobolev spaces and weak formulation ..................... 16 2.2.1 The Sobolev Spaces Hk, H(div), and H(curl) ............ 16 2.2.2 Essence of weak formulation ...................... 20 2.3 Generalized Fourier transforms ........................ 23 2.3.1 Fourier transform of functions and the desire for an extended defi- nition .................................. 23 2.3.2 Fourier transform of test functions and of distributions ....... 25 2.3.3 Important Properties of Fourier transforms ............. 26 2.4 Chapter summary and discussion ....................... 28 3 Theoretical Background: Spatial dispersion 31 3.1 Multipoles in macroscopic media ....................... 33 3.2 Constitutive relations and spatial dispersion ................. 36 3.2.1 Field equations and linear constitutive relations ........... 36 3.2.2 Effective medium theory with spatial dispersion ........... 38 3.2.3 Weak spatial dispersion approximation ................ 43 3.3 Causality and Passivity with spatial dispersion ................ 47 3.4 Chapter Summary and concluding remarks .................. 50 4 Homogenization of metamaterials 53 4.1 Dispersion relation and additional modes ................... 54 4.1.1 Dispersion relation of WSD ...................... 55 4.1.2 Dispersion relation of the SYM model ................ 57 4.1.3 Dispersion relation of the SSD model ................. 59 4.2 Additional Interface Conditions ........................ 62 4.2.1 Weak solutions ............................. 64 4.2.2 Derivation of the interface conditions ................. 66 4.3 Establishing the Fresnel Equations ...................... 72 4.3.1 Illumination with TE-polarized light ................. 73 4.3.2 Illumination with TM-polarized light ................. 76 4.3.3 Half-space problem and further remarks ............... 79 4.4 Basic homogenization models ......................... 79 4.4.1 Maxwell-Garnett mixing rule ..................... 79 4.4.2 Nonlocal wire medium model ..................... 82 ix x TABLE OF CONTENTS 4.5 Chapter summary and concluding remarks .................. 85 5 Effective parameter retrieval 87 5.1 S-parameter optimization approach .....................