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DESIGN,FABRICATION, AND CHARACTERIZATION OF DOUBLE-NEGATIVE METAMATERIALS FOR PHOTONICS Zur Erlangung des akademischen Grades eines DOKTORS DER NATURWISSENSCHAFTEN von der Fakultat¨ fur¨ Physik der Universitat¨ Karlsruhe (TH) genehmigte DISSERTATION von Diplom-Physiker Gunnar Dolling aus Cuxhaven Tag der mundlichen¨ Prufung:¨ 20. Juli 2007 Referent: Prof. Dr. Martin Wegener Korreferent: Prof. Dr. Kurt Busch . Publications Parts of this thesis have already been published: In scientific journals: • G. Dolling, M. W. Klein, M. Wegener, A. Schadle,¨ S. Burger, and S. Linden, Nega- tive beam displacements from negative-index photonic metamaterials, in preparation (2007). • G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, Design-related losses of double-fishnet negative-index photonic metamaterials, Opt. Express, submitted (2007). • M. Wegener, G. Dolling, and S. Linden, Plasmonics: Backward waves moving for- ward, Nature Mater. 6, 475 (2007). • G. Dolling, M. Wegener, and S. Linden, Realization of a three-functional-layer negative- index photonic metamaterial, Opt. Lett. 32, 551 (2007). • G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, Negative-index metamaterial at 780 nm wavelength, Opt. Lett. 32, 53 (2007). • G. Dolling, M. Wegener, and S. Linden, Der falsche Knick im Licht: Metamaterialien mit negativem Brechungsindex, invited paper, Phys. Unserer Zeit 38, 24 (2006). • S. Linden, C. Enkrich, G. Dolling, M. W. Klein, J. F. Zhou, T. Koschny, C. M. Souk- oulis, S. Burger, F. Schmidt, and M. Wegener, Photonic metamaterials: Magnetism at optical frequencies, invited paper, IEEE J. Sel. Top. Quant. 12, 1097 (2006). • G. Dolling, M. Wegener, A. Schadle,¨ S. Burger, and S. Linden, Observation of mag- netization waves in negative-index photonic metamaterials, Appl. Phys. Lett. 89, 231118 (2006). • G. Dolling, S. Linden, and M. Wegener, Metamaterialien: Licht im Ruckw¨ artsgang¨ , invited paper, Phys. Unserer Zeit 37, 157 (2006). • G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, Low-loss neg- ative-index metamaterial at telecommunication wavelengths, Opt. Lett. 31, 1800 (2006). iv Publications • G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, Simultaneous Negative Phase and Group Velocity of Light in a Metamaterial, Science 312, 892 (2006). • G. Dolling, M. Wegener, S. Linden, and C. Hormann, Photorealistic images of objects in effective negative-index materials, Opt. Express 14, 1842 (2006). • G. Dolling, C. Enkrich, M. Wegener, J. F. Zhou, C. M. Soukoulis, and S. Linden, Cut- wire pairs and plate pairs as magnetic atoms for optical metamaterials, Opt. Lett. 30, 3198 (2005). At international conferences and workshops (only own presentations): • G. Dolling, M. Wegener, C. M. Soukoulis, and S. Linden, Negative-index metama- terials have reached the visible, invited talk, Progress In Electromagnetics Research Symposium (PIERS), Beijing, China, March 26-30, 2007. • G. Dolling, Negative-index metamaterials have reached the visible, invited talk, Work- shop on Nano-Photonics, Berlin, Germany, October 18-20, 2006. • G. Dolling, C. Enkrich, M. Wegener, C. M. Soukoulis, and S. Linden, From “Mag- netic Atoms” to Low-Loss Negative-Index Metamaterials at Telecommunication Wave- lengths, talk QWE3, Quantum Electronics and Laser Science Conference (QELS), Long Beach, CA, U.S.A., May 21-26, 2006. Contents Zusammenfassung vii 1 Introduction 1 2 Fundamentals of linear optics 5 2.1 Electromagnetic wave propagation ...................... 5 2.1.1 Maxwell’s equations ......................... 5 2.1.2 Refractive index ........................... 7 2.1.3 Refraction at an interface ....................... 9 2.1.4 Some examples for optics with ¹ 6= 1 . 11 2.2 Natural materials ............................... 16 2.2.1 Lorentz oscillator ........................... 16 2.2.2 Drude model ............................. 17 2.2.3 Metallic nanoparticles ........................ 19 2.2.4 Magnetism .............................. 21 2.3 Metamaterials ................................. 21 2.3.1 Transmission lines .......................... 22 2.3.2 Resonant structures .......................... 24 2.3.3 Wood anomalies ........................... 25 3 Some aspects of negative refraction 27 3.1 Negative refraction in anisotropic materials . 27 3.2 Negative refraction in photonic crystals ................... 30 3.3 Negative refraction in thin, homogeneous, and isotropic materials . 31 3.3.1 Physics of a metallic mirror ..................... 31 3.3.2 Thin metal film ............................ 33 3.3.3 Thin dielectric film .......................... 34 4 Fabrication, characterization, and simulation 37 4.1 Fabrication .................................. 38 4.1.1 Scanning-electron microscope .................... 38 4.1.2 Electron-beam lithography ...................... 40 4.1.3 Sample preparation .......................... 41 4.2 Optical characterization ............................ 42 v vi Contents 4.2.1 Reflectance and angle-resolved transmittance spectroscopy . 43 4.2.2 Michelson interferometer ....................... 44 4.3 Numerical methods .............................. 45 4.3.1 MicroWave Studio .......................... 46 4.3.2 JCMsuite ............................... 47 4.3.3 Calculating an interferogram ..................... 47 4.3.4 Retrieval of the refractive index and the impedance . 48 5 From split-ring resonators to double-negative metamaterials 51 5.1 Split-ring resonators .............................. 51 5.2 Cut-wire pairs ................................. 53 5.2.1 Experimental results ......................... 54 5.3 Double-fishnet design ............................. 57 5.3.1 Influence of the hole shape ...................... 62 6 Experimental results 67 6.1 Simultaneous negative phase and group velocities of light . 67 6.1.1 Theory of negative group velocities . 69 6.1.2 Experimental data .......................... 72 6.2 Low-loss negative-index metamaterial .................... 75 6.3 Negative-index metamaterial operating in the visible . 79 6.4 Observation of magnetization waves ..................... 83 6.5 First steps towards three-dimensional photonic metamaterials . 87 7 Conclusions 95 A Transfer-matrix method for oblique incidence of light 99 Bibliography 102 Acknowledgements 111 Zusammenfassung Elektromagnetische Wellen sind ein wesentlicher Bestandteil unseres Alltags: Das mensch- liche Auge reagiert beispielsweise so auf das Licht, dass wir unsere Umgebung wahrnehmen konnen.¨ Die Bildentstehung basiert dabei auf der Brechung von Lichtstrahlen: Wenn eine elektromagnetische Welle auf eine Grenzflache¨ zwischen zwei Medien mit unterschiedlichen Brechungsindizes trifft, wird die Welle gebrochen und andert¨ ihre Ausbreitungsrichtung. Durch diesen Effekt kann Licht fokussiert und somit Objekte abgebildet werden. Ahnlich¨ wird in modernen Mikroskopen und Teleskopen die Brechung ausgenutzt, um den Mikro- kosmos oder weit entfernte Galaxien zu studieren. Neben der Brechung spielt heutzutage jedoch auch die Reflexion von elektromagnetischen Wellen eine wichtige Rolle: Unsere moderne Telekommunikation basiert beispielsweise haufig¨ auf Glasfaserkabeln. In diesen Kabeln wird das Licht durch Totalreflexion zwischen zwei Medien mit unterschiedlichen Brechungsindizes gefuhrt.¨ Die genannten Beispiele und Phanomene¨ basieren alle auf dem Brechungsindex und der resultierenden Brechung und Reflexion von Licht. Das Konzept des Brechungsindexes beruht dabei auf der Tatsache, dass die Wellenlange¨ des Lichts (einige hundert Nanometer) viel großer¨ als typische Großenskalen¨ (unter einem Nanometer bei Atomen) in naturlichen¨ Materialien ist. Aus diesem Grund kann eine elek- tromagnetische Welle die feinen Details (zum Beispiel Atome) im Material nicht auflosen.¨ Dieser Umstand vereinfacht die Beschreibung der Wechselwirkung zwischen Licht und dem Material erheblich: Wir mussen¨ nur den Brechungsindex n des Materials oder, um genau zu sein, die (elektrische) Permittivitat¨ " und die (magnetische) Permeabilitat¨ ¹ kennen. Die Permittivitat¨ beschreibt dabei die Wechselwirkung zwischen dem Material und dem elek- trischen Feld der Welle, wahrend¨ die Permeabilitat¨ die Wechselwirkung des Materials mit dem magnetischen Feld der Welle beschreibt. In der Natur existieren jedoch keine Mate- rialien mit ¹ 6= 1 bei optischen Frequenzen, so dass nur die Permittivitat¨ beziehungsweise der Brechungsindex der wichtige Parameter in der Optik ist. Diese Tatsache wird zudem p durch die bekannte Formel fur¨ den Brechungsindex wiedergegeben: n = + ". Trotzdem existieren viele Anwendungen bei optischen Frequenzen, obwohl naturliche¨ Materialien nur eine Manipulation des elektrischen Feldes des Lichtes ermoglichen.¨ Die Beeinflussung des elektrischen Feldes in der Optik ist jedoch so, als wurde¨ man nur eines seiner beiden Augen verwenden. Im Jahr 1967 beschaftigte¨ sich Victor Veselago mit der Frage, was passieren wurde,¨ wenn die Permeabilitat¨ von eins verschieden ware.¨ Er fand heraus, wenn sowohl die Per- meabilitat¨ und die Permittivitat¨ negativ sind, dass dann auch der Brechungsindex negativ werden wurde.¨ In solchen Medien waren¨ die Doppler-Verschiebung und die Cherenkov- vii viii Zusammenfassung Strahlung umgekehrt. Trotz der interessanten Effekte blieb seine Idee eine Sonderlichkeit, da keine naturlichen¨ Materialien mit ¹ 6= 1 oberhalb von Gigahertz Frequenzen oder hoher¨ bekannt waren. Daher war es eine faszinierende Idee, als Sir John Pendry im Jahr 1999 kunstliche¨ Materialien vorschlug, die ¹ 6= 1 aufweisen konnen.¨ Seine Idee
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