Metamaterials for Photonic Applications Natalia Dubrovina

Metamaterials for photonic applications Natalia Dubrovina

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Natalia Dubrovina. Metamaterials for photonic applications. Other [cond-mat.other]. Université Paris Sud - Paris XI, 2014. English. NNT : 2014PA112088. tel-01522399

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UNIVERSITE PARIS-SUD ECOLE DOCTORALE : Ondes et M atiere Institut d’Electronique Fondamentale (IEF)

DISCIPLINE PHYSIQUE

THÈSE DE DOCTORAT Soutenue le 14.05.2014 par

Natalia Dubrovina

M etamaterials for photonic applications

Directeur de thèse : Anatole LUPU Chargé de recherche CNRS, Insititut d’Electronique Fondamentale Composition du jury : Président du jury : André DE LUSTRAC Professeur, Université Paris Ouest Rapporteurs : Andrei LAVRINENKO Professeur, Technical University of Denmark Yannick DE WILDE Directeur de recherche, Institut Langevin, ESPCI-ParisTech Examinateurs : Alexandre BOUHELIER Chargé de recherche, Institut Carnot de Bourgogne Anatoly ZAYATS Professeur, King's College London

this thesis I dedicate to the memory of my father, V yatcheslav

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Acknowledgments

This thesis would not have been written without number of people to whom I am deeply thankful and to whom I would like to express my greatest gratitude. First of all and foremost I want to acknowledge my advisor Anatole Lupu for suggesting this topic to me and for being such a remarkable mentor. Without his guidance and support through my PhD years this work would never have been possible. His advanced ideas and confidence in obtained results were always encouraging me to continue the study. He has taught me not only how to make precise optical measurements, how to retrieve maximum information from a graph, how to make oral presentations, how to be creative keeping feet on the ground and many other scientific tricks, but also he was the one who had helped me a lot with accommodation in France. I wish to thank my committee members for being generous with their expertise. I would like to offer my special thanks to the reviewers Andrei Lavrinenko and Yannick de Wilde for the time they have spent reading the manuscript and for their remarks and suggestions. I would like to thank the other members of the committee Alexandre Bouhelier and Anatoly Zayats for finding the time in their schedule and André de Lustrac for accepting the role of the jury president. Their comments and questions were very beneficial in the complementation of the manuscript. This thesis would hardly been completed without help of Aloyse Degiron who has aided me both with simulation part of the work and with experimental aspects of the samples fabrication. His perfectionism in a good sense is an exemplar to me and a high standard to strive for. I am very grateful for the stimulating discussions we have had during three and a half years and all experience I have earned from him. I would particularly like to acknowledge all clean room technical staff of In st itu t d ’El ect ronqu e Fon dament al e (CTU-MINERVE) for the trainings and technological recommendations. I owe my deepest gratitude to Jean-René Coudevylle who has helped me a lot on all levels of the lithography process. I also have had a chance to work with Nathalie Isac and I thank her for sharing with me her experience in optical lithography fabrication and her positive humor. I still have her funny notices in my clean room notebook with the recipe “how to become a winner”. It is a pleasure to extend my appreciation to Xavier Leroux from whom I have greatly benefited in handling of Nanobeam machine. The fine adjustments he has helped me to make were the key point to win nanosized structures where separation distance between two elements could be no more than 30 nm. I am deeply grateful to the collaborators from the group PHODEV of Laboratoi re de Phot on iqu e et d e N anost ru ctu res for the proposed samples with split ring resonators and their i Acknowledgments optical characterization, to Bruno Gallas from Inst itu t d es N an oSciences de Pari s for the ellipsometry measurements and to the collaborators from L aborat oi re de N an otechn ol ogie et In st rumen tat ion Opti qu e for the near field measurements. I have had the support and encouragement from the CRIME group members of In stit ut d ’El ect ronqu e Fondament al e to whom I would also like to express my appreciation. I appreciate the feedback offered by Nawaz Burokur during the preparation to my viva voce and I thank him for his patience, time and constructive comments. I thank Tatiana Teperik for bringing me to sport classes; Alexandre Selier for his optimistic mood that he was spreading in the office every single day; Dylan Germain for his support and readiness to listen; Jianjia, Simon, Quynh, Paul-Henri, Rasta for making my days amusing and light. I would also like to thank my lovely friends: Lan, Maxim, Igor, Dmitry, Sergey N., Antoine, Giannis, Adam, Pierre, John, Andy, Paola, Leonid, Egor, Niraj, Andrey, Sergey M… Special thank I would like to send to Petr for checking on everyday basis the progress in manuscript writing. I owe my gratitude to all my family and especially to my grandmother, Galina, for giving me the direction in my education. Thanks to her I found myself in the place surrounded by brilliant teachers who have opened me the doors to science. Lastly but not least I want to thank Andrey Fedyanin, head of the Laboratory of N an ophotoni cs and M etamateri al s of M oscow State Uni versit y, without whom I would never have considered coming to France.

ii Abstract

The subject of the PhD thesis deals with metamaterials for photonic applications. The main objective is to investigate the potential of metallic metamaterials for building optical functions at near infrared optical frequencies ( λ = 1.5 µm). A significant part of the work is focused on the engineering of the metamaterials effective index associated with localized plasmon resonances. Two configurations of particular importance for fabrication technology are considered. • Free space light propagation, with the incident electromagnetic wave interacting with single metafilms at either normal or oblique incidence. • Guided wave configuration, with single metamaterial layer placed on top of dielectric waveguide. For the free space configuration, the validity of the effective medium approach was investigated both numerically and experimentally with the example of metamaterials composed of either gold cut wires on glass substrate or split ring resonators and continuous wires on silicon substrate. On the basis of these examples it was shown that the metafilm behavior is indeed analogous to that of a homogeneous layer. The thickness of this layer is that of the deposited metal. The validity of this conclusion was verified with respect to a number of criteria consistent with the Maxwell-Garnett approximation. It was shown in particular that near the resonance frequency the effective index of the metafilm layer can reach very high values neff ≈ 10 that cannot be attained with natural materials. The effective medium approach developed for a single metamaterial layer in free space configuration was further extended to a guided wave configuration. The objective is to achieve an efficient control over the flow of the light in the waveguide using effective index variation induced by metamaterial resonances. The possibility of achieving a significant effective index variation with a silicon slab waveguide covered by 200−50−50 nm gold cut wires was investigated by numerical modeling and confirmed by experimental results. The magnitude of local index variation in the vicinity of the resonance frequency deduced from experimental data is as high as ± 1.5. The possibility to control the local effective index at the nanoscale can be used in transformation optics applications. The hybrid metamaterial guided wave configuration may become a promising alternative to the bulk multi-layers metamaterial structures in the near infrared domain.

Key words : metamaterials; nanophotonics; surface plasmons; integrated optics. iii

Résumé

L’objet de cette étude concerne l’exploration, à la fois sur le plan théorique et expérimental, de la possibilité d’utilisation des métamatériaux pour des applications dans le domaine de la photonique aux longueurs d’onde télécoms ( λ = 1.5 µm). L’un des principaux objectifs adressés dans le cadre de la thèse est de réaliser l’ingénierie de l’indice effectif en utilisant les résonances des plasmons de surface localisés des métamatériaux métallo-diélectriques. Deux cas particulièrement importants du point de vue de la réalisation technologique sont considérés. • Propagation en espace libre quand une onde lumineuse sous incidence normale ou oblique interagit avec une surface diélectrique recouverte d’une monocouche de métamatériaux. • Propagation dans une configuration guide d’onde avec une monocouche de métamatériaux à la surface d’un guide d’onde en Silicium. Les résultats des modélisations et les mesures expérimentales montrent que les propriétés optiques d’une mono-couche de métamatériau peuvent être décrites par celle d’une couche homogène avec un certain indice effectif. L’épaisseur de cette couche est égale à celle des motifs métalliques, à condition qu’elle soit inférieure à quelques dizaines de nm. Pour des faibles facteurs de remplissage en surface, l’indice de réfraction d’une telle couche suit l’approximation de Maxwell- Garnett. Cet indice effectif ne dépend pas de l’angle d’incidence ni de l’orientation de la polarisation de la lumière (perpendiculaire ou dans le plan d’incidence). Au voisinage de la fréquence de résonance pour un facteur de remplissage de métamatériau de 20% en surface on obtient un indice de réfraction très élevé : neff ≈ 10. Cet indice de réfraction est plusieurs fois supérieur à celui qu’on trouve dans des matériaux naturels. L’adaptation de cette approche à une configuration guidée consiste à utiliser une structure hybride composée d’une couche de métamatériau à la surface d’un guide d’onde en Silicium. Les travaux réalisés ont permis de démontrer la possibilité d’effectuer l’ingénierie de l’indice effectif et de contrôler le niveau des pertes d’un tel guide d’onde hybride en utilisant des métamatériau métallo- diélectriques à base des fils d’Au de 200−50−50 nm. Le contraste d’indice au voisinage de la ligne de la résonance donné par des modélisations et confirmé expérimentalement est de ± 1.5, soit plus que ce que l’on peut obtenir avec un guide Silicium gravé. Ce résultat représente une première démonstration sur le plan international de fonctionnement des métamatériaux en configuration guidée.

v Résumé

De plus, en contrôlant l’orientation des motifs de métamatériaux, on peut réaliser un indice anisotrope. Les résultats obtenus ouvrent des perspectives très prometteuses pour la réalisation de dispositifs en optique guidée utilisant les transformations d’espace.

M ots -cle fs : métamatériaux ; nanophotonique ; plasmons de surface localisés ; optique intégrée.

Table of contents

Abbreviations and units...... xi Chapter 1 Introduction...... 1 1.1 Fundamental of metamaterials...... 1 1.1.1 Maxwell's equations...... 2 1.1.2 Electromagnetic constitutive parameters ε and µ...... 4 1.1.3 Constitutive parameters and the sign of refractive index...... 5 1.1.4 Poynting vector...... 6 1.1.5 Snell's law...... 6 1.1.6 Negative index metamaterials: principle and applications...... 7 1.1.6.1 Perfect lensing using negative index metamaterials...... 8 1.2 Transformation optics...... 9 1.2.1 Fundamental of transformation optics ...... 10 1.2.2 Applications of transformation optics...... 12 1.2.3 Implementation of transformation optics in the microwave domain...... 13 1.2.4. Implementation of transformation optics in the optical domain ...... 14 1.2.5 Hybrid metal-dielectric metamaterials for transformation optics applications...... 15 1.3 Thesis motivation and manuscript organization...... 17 Chapter 2 Homogenization of metamaterials ...... 25 2.1 Effective medium approximations...... 25 2.1.1 Maxwell-Garnett approximation...... 25 2.1.1.1 Maxwell-Garnett formula for spherical inclusions...... 26 2.1.1.2 Clausius-Mossotti formula ...... 27 2.1.2 Bruggeman’s approximation...... 27 2.1.3 Numerical homogenization approaches...... 28 2.2 Metamaterial effective parameters...... 28 2.2.1 Nicolson-Ross-Weir retrieval method ...... 29 2.2.1.1 Retrieval of metamaterials effective parameters ...... 30 2.2.1.2 Single metafilm effective medium behavior...... 32 2.3 Closing remarks ...... 33 Chapter 3 Metamaterials fabrication technology and experimental characterization methods39 3.1 Fabrication of metamaterials...... 39

vii Table of contents

3.1.1 Fabrication flow sheet for free space configuration metamaterials ...... 40 3.1.1.1 Substrate cleaning...... 41 3.1.1.2 Resist spin-coating...... 41 3.1.1.3 Electron beam lithography...... 42 3.1.1.4 Development...... 43 3.1.1.5 Metal deposition...... 43 3.1.1.6 Lift-off process...... 44 3.1.2 Fabrication flow sheet for guided wave configuration metamaterials...... 44 3.1.2.1 The second level of electron beam lithography...... 45 3.1.2.2 Etching of waveguide edges...... 46 3.1.2.3 Final cleaning...... 46 3.2 SEM images of fabricated samples...... 47 3.3 Sample characterization...... 48 3.3.1 FTIR spectrometry for free space configuration...... 48 3.3.2 Ellipsometry measurements...... 50 3.3.3 Guided wave configuration...... 50 3.4 Conclusion ...... 51 Chapter 4 Free space configuration single metafilm effective properties in optical domain.....55 4.1 Single metafilm modeling...... 55 4.1.1 Single metafilm effective medium criteria...... 56 4.1.2 Single metafilm layer effective medium behavior...... 56 4.1.2.1 Determination of effective layer thickness ...... 58 4.1.2.2 Maxwell-Garnett approximation validity...... 60 4.1.2.3 Single metafilm optical length study...... 62 4.1.2.4 Metafilm oblique incidence behavior...... 64 4.2 Experimental study of single metafilm effective medium behavior...... 70 4.2.1 Cut wires arrays with different filling factors and resonance frequencies...... 70 4.2.1.1 Ellipsometry measurements ...... 75 4.2.2 Split ring resonators with continuous wires metamaterial structures ...... 76 4.3 Conclusions...... 82 Chapter 5 Engineering of metamaterial properties in a guided wave configuration...... 87 5.1 Basic elements of guided wave optics...... 88 5.2 One-dimensional array of cut wires on slab waveguide...... 89 5.2.1 Effective parameters of single cut wires chain...... 90 5.2.2 Resonance frequency engineering...... 92 viii Table of contents

5.2.2.1 Influence of dielectric environment on the resonance frequency...... 92 5.2.2.2 Influence of waveguide thickness on the metamaterial effective permittivity..96 5.2.2.3 Influence of cut wires orientation on the effective parameters...... 97 5.3 Single layer 2D array of cut wires on slab waveguide...... 98 5.4 Double layer 2D array of cut wires on slab waveguide...... 101 5.5 Choice of cut wires design for guided wave configuration ...... 104 5.6 Experimental study of hybrid metal-dielectric waveguides...... 105 5.6.1 Influence of cut wires transverse separation distance...... 105 5.6.1.1 Straight CWs loaded waveguides ...... 107 5.6.1.2 CWs loaded tapered waveguides...... 110 5.6.2 Homogeneous model for metamaterials in guided wave configuration ...... 113 5.6.3 Layered model for metamaterials in guided wave configuration ...... 115 5.6.3.1 Evolution of Lorenz parameters with separation distance between CWs...... 118 5.6.4 Engineering of composite waveguide effective properties by cut wires geometry design 120 5.6.4.1 Control of the resonance frequency by cut wires length...... 120 5.6.4.2 Control of anisotropy properties by CWs orientation angle...... 123 5.7 Conclusions ...... 130 Chapter 6 Summary and perspectives ...... 135 6.1 Summary...... 135 6.2 Perspectives ...... 137 6.2.1 Devices based on transformation optics ...... 137 6.2.2 Nano-antenna and near-field enhancement properties...... 138 6.2.3 Guiding the light by metamaterial slab...... 139 6.2.4 Engineering of metallic metamaterials losses...... 141 6.2.4.1 Dark modes...... 141 6.2.4.2 Gain media based on metamaterials...... 142 List of Publications...... 145 Journal publications...... 145 Conferences proceedings...... 145 Conferences and symposiums communications ...... 146

Abbreviations and units

Used abbreviations 2D two dimension 3D three dimension CW cut wire DFB distributed feedback EM effective medium EMA effective medium approach FF surface filling factor FSC free space configuration GWC guided wave configuration IR infrared LH left-handed MG Maxwell-Garnett MM metamaterials NIM negative refraction index metamaterials NIR near infrared NRI negative refraction index NRW Nicolson-Ross-Weir PhC photonic crystal PRI positive refraction index RH right-handed RI refractive index SRR split ring resonator TO transformation optics US ultra sound WG waveguide Used standard indications ε electric permittivity µ magnetic permeability n refractive index

xi Abbreviations and units

ρ surface filling factor C E electric field f frequency in THz C H magnetic field C k wave vector Used standard units of measurements dB decibel eV electronvolt kV kilovolt µm micrometer/micron min minute nA nanoampere nm nanometer pA picoampere pm picometer sccm standard cubic centimeter per minute sec second THz terahertz W watt Used abbreviations of chemicals, equipment and methods Au gold Cr chromium EBL electron beam lithography FTIR Fourier transform infrared spectroscopy HFSS High frequency structural simulator HMDS hexamethyldisilazane ICP inductively coupled plasma IPA isopropyl alcohol MIBK methyl isobutyl ketone NBL nanobeam lithography PMMA polymethyl methacrylate RIE reactive ion etching SEM scanning electron microscope xii Abbreviations and units

Si silicon SNOM scanning near field optical microscopy SOI silicon on insulator

xiii

Chapter 1 Introduction

1.1 Fundamental of metamaterials

Metamaterials (MMs) are artificial composite materials engineered to possess electromagnetic properties that cannot be found in nature. They are assembled from individual elements which typical size is considerably smaller than the working wavelength of electromagnetic wave (λ0 ). These elements are usually arranged into periodical arrays and called meta- at oms because of their small size and by the analogy with atoms in conventional bulk materials. The period of the MM pattern also does not exceed several tenths of wavelength. Because of the subwavelength dimensions, exact microscopic details of each meta-atom cannot be detected by electromagnetic wave. Electromagnetic response of such materials is an averaged value of the collective responses. MMs gain their remarkable properties from the shape of meta-atoms, their size, geometry, mutual orientation and arrangement. Using these properties many fascinating devices can be created such as perfect lenses, cloaks of invisibility, electromagnetic concentrators and others. First, the word met amaterial was used by R.W. Walser [Walser 2001]. The prefix meta is coming from Greek language and can be translated as beyond , aft er. It reflects the fact that MMs can react to external electromagnetic radiation by completely different way than conventional materials do. The history of these materials is used to be counted from the seminal work of V.G. Veselago published in 1968 [Veselago 1968]. He studied a hypothetical material with simultaneously negative dielectric permittivity ε and magnetic permeability µ. In this work Veselago predicted interesting properties of the materials, for instance, negative refraction. But for a long time his work rested unnoticed since materials that could provide negative permittivity and permeability at the same time have not been known. For years the work of Veselago had been nothing else than mathematical toy and had no practical interest. The situation has changed in the late 90s of XX century when technology and computing power became enough for designing and producing materials with negative refractive index. Thus in 2000 J.B. Pendry writes a theoretical work about perfect lenses [Pendry 2000]. The work is followed by pile of experimental articles [Smith 2000, Baena 2004, Ran 2005, Zhang 2005, Zhou 2006]. 1 CH APTER 1. IN TRODUCTION

In this chapter we provide a few emblematic examples of metamaterials applications and remind some fundamentals of their physics.

1.1.1 M axwell's equations

To understand better what MMs are, how they interact with external electromagnetic radiation and what new properties they can introduce it is necessary to make one step back and recall the basis of the general course of electromagnetism. C Electromagnetic field can be described by two microscopic vectors: electric field intensity e C and magnetic field intensity h that characterize electromagnetic field at a point in time and in space. The microscopic Maxwell's equations are valid for the vectors [Landau 1984]. After canonical ensemble averaging all microscopic values transform into macroscopic ones. For a medium in absence of sources of charges and currents Maxwell's equations take the following form: C C C 1 ∂B [∇× E]= − , c ∂Ct C C 1 ∂D [∇× H ]= , (1.1) C C c ∂t ∇ ⋅ D = 0, (C C) (∇ ⋅ H )= 0, C C C C C where electric field E is averaged value of e , magnetic field H is averaged value of h ; B is the C magnetic induction, D is the displacement field; c is speed of light in vacuum. Generally speaking the procedure of averaging can be performed only if the wavelength is much larger than the linear size of averaging volume which is not always true especially at optical frequencies (for example, non- local effects and non-linear effects). C C C C Pairs of vectors B and H and also D and E are bound among themselves by the constitutive relations through tensors of the dielectric permittivity εˆ(ω) and of the magnetic permeability µˆ(ω) : C C D = εˆ(ω)E, C C (1.2) B = µˆ(ω)H. Actually, these two constitutive tensors εˆ(ω) and µˆ(ω) 1 define how the medium responds to external electromagnetic waves. Knowing the constitutive parameters of a medium and solving

1 Further we shall omit the reference to the dependence of the constitutive parameters on the frequency of electromagnetic field though we always keep it in mind. 2 1.1 Fundamental of metamaterials

Maxwell's equations in combination with boundary conditions it is possible to predict the behavior of electromagnetic waves. Considering isotropic media from (1.1) and (1.2) one can obtain the wave equation for the electric field (the same can be done for the magnetic field as well): C C 1 ∂2 E ∆E − = 0, (1.3) v2 ∂t 2 c2 where v2 = is the square of wave velocity in the medium. The root square of the constitutive (εµ)2 parameters product is called refractive index (RI) of the medium and usually denoted by the letter n : n = ± εµ. (1.4) The general solution of the wave equation (1.3) is: C C C C C C E(r,t) = F(r − vt) + G(r + vt), (1.5) C C C where r is the radius-vector and t is the point in time; F and G are the right and left traveling functions, respectively. Most of electromagnetic problems can be interpreted using harmonic functions. The harmonic solution of the wave equation is: C C CC C i(kr −ωt) E(r,t) = E0e , (1.6) C C ω C 2π C where ω is the angular frequency, k is the wave vector ( k = m = ; m is the unit vector v λ perpendicular to surfaces of constant phase). The Maxwell’s equations (1.1) bring limitations of C C choosing vectors m and k : C C ω C [k × E]= µH, c (1.7) C C ω C [k × H ]= − εE. c C C C The last relations (1.7) show that the vectors k , E and H must be perpendicular to each other. But shall they present right-handed set of vectors or shall they present left-handed set, Maxwell’s equations do not give the answer. The signs of the constitutive parameters would determine the C C C mutual orientation of the vectors k , E and H . Exactly this fact was accented by Veselago in his famous work [Veselago 1968].

3 CH APTER 1. IN TRODUCTION

1.1.2 Electromagnetic constitutive parameters ε and µ

Now if one thinks in terms of the material (constitutive) parameters ε and µ all possible media can be pointed on the diagram (see Fig. 1).

Fig. 1. Permittivity and permeability diagram.

• The most often materials encountered in nature, e.g. dielectrics, have both ε and µ positive. • Materials with negative dielectric permittivity (ε < 0) are also widely known. For example noble metals behave as gases of free electrons below their electric plasma

frequency (ω pe ). That means that these metals exhibit a negative permittivity at

excitation frequencies below ω pe . The main problem of using these materials is related to the fact that metals are opaque substances. Arrays of thin metallic wires can be adopted to obtain negative permittivity with low losses due to reduced total volume of metal [Pendry 1996]. • Natural materials with negative magnetic ( µ < 0) permeability are less easy to find. Basically one can name only ferrites (ferrimagnetic materials) exited by electromagnetic

waves of frequencies below the magnetic plasma frequency (ω pm ). It is significant to note that natural magnetism at high frequencies (THz and higher) has not been

4 1.1 Fundamental of metamaterials

reported. J.B. Pendry proposed the solution to this problem by construction subwavelength split ring resonators (SRRs) that are basically metallic rings with gaps [Pendry 1999]. • Furthermore, there are no known natural materials that can posses both negative permittivity (ε < 0) and negative permeability ( µ < 0) at the same range of frequencies.

1.1.3 Constitutive parameters and the sign of refractive index

As it was already mentioned the material parameters play essential role in Maxwell’s equations: they determine how a substance with certain ε and µ responds to external electromagnetic field. The RI n of the substance depends on its constitutive parameters, the relative permittivity ε and the relative permeability µ , through the equation (1.4):

n = ± εµ. In case of one negative constitutive parameter (area 2 or 4 on the Fig. 1) the RI is an imaginary value. By consequence the substance is a lossy medium where propagating waves are evanescent. Though if both parameters either positive either negative (area 1 or 3 on the Fig. 1) the value of n is real, and the waves are propagative. The double-positive substances (area 1 on the Fig. 1) are already mentioned dielectrics that support usual forward propagation of the electromagnetic waves. In this C C C case vectors E , H and k form the right-handed (RH) triplet of vectors (Fig. 2a). The double- negative substances (area 3 on the Fig. 1) are left-handed (LH) materials or negative refractive index C C C (NRI) materials. In this case vectors E , H and k form a left-handed set and the wave propagation is called backward (Fig. 2b). a b

Fig. 2. The triplets (a) of right-handed set of vectors and (b) of left-handed set of vectors.

5 CH APTER 1. IN TRODUCTION

1.1.4 Poynting vector C The Poynting vector S represents the directional energy flux density, i.e. it is the rate of energy transferred per unit area. It can be shown that the Poynting vector can be given by following expression: C c C C S = [E × H ]. (1.8) 8π C C C Thus the direction of the energy flux rests the same and vectors E , H , S always form a right- handed set of vectors independent on the signs of constitutive parameters. That means that in spite of the term backward propagation for LH or a b NRI materials the energy flux (in other words the Poynting vector) is always directed forward (see Fig. 2W on the right).

And it is only the phase velocity that has opposite direction to the Poynting vector. It Fig. 2A. The triplets (a) of right-handed set of vectors and means that the wave front is moving in the (b) of left-handed set of vectors with direction of the opposite direction to the flow of the energy. C Poynting vector S .

1.1.5 Snell's law

The most distinctive representation of differences between positive refractive index (PRI) media and

NRI media is an implementation of Snell's law. If light travels from medium 1 with the RI n1 to the medium 2 with the RI n2 , the incidence angle α1 and the refraction angle α 2 (Fig. 3) are bound by the famous expression: n n 1 = 2 . (1.9) sinα 2 sinα1

Fig. 3. Negative and positive refraction angles.

6 1.1 Fundamental of metamaterials

If, for instance, the medium 2 has NRI, the refraction angle also has to be negative. And refracted ray travels in the same half of space from the normal vector as the incidence ray.

1.1.6 Negative index metamaterials: principle and applications

Often negative index MMs are double-negative materials. They are a pictorial example of controlling the light and turning it to the unusual course. NRI of these materials is a key property for creating perfect lenses or planar polarizers. The first NRI MM (NIM) operating at radio frequency range was experimentally created by simple combination of thin metallic wires (to efficiently obtain negative

ε r ) and metallic SRRs (responsible for negative µr ) [Shelby 2001]. The characteristic feature of that kind of NIM is that that for nearly all experimental realizations negative permeability µr is achieved for incidence angles different from zero (oblique incidence). This limitation caused by the orientation of SRRs related to the planar fabrication technology. The shifting of working range to the higher frequencies requires some tricks since a straightforward downsizing approach greatly complicates the technological realization. The big progress in the NIM operating in the infrared and optical domain is associated with the introduction of the concept of fishnet structure. Using this approach near-infrared NIMs were demonstrated by Zhang et al. [Zhang 2005]. In this work to construct NIMs two metallic plates with holes of subwavelength diameters were used. The plates are separated by dielectric layer and can provide resonance interaction. NIMs were also experimentally demonstrated in the optical domain at the resonance frequency around 200 THz [Shalaev 2005] and around 300 THz [Dolling 2005]. The structure used in these experiments can be viewed as a Babinet analog of Zhang fishnet. It consists of an array of paired metallic nanorods. Magnetic resonance in parallel metallic nanorods is similar to that of LC circuit with the metal rods providing the inductance L and the dielectric gaps between the rods acting as capacitive elements C . In this case, negative permittivity ε r is caused by symmetric mode dipole resonance of each metallic nanorod when negative permeability µr is the result of antisymmetric resonance occurring in the pair of nanorods. Moreover it was shown that essentially pair of cut wires has the equivalent behavior to the behavior of SRR [Podolskiy 2005, Dolling 2005]. SRR can be viewed as one magnetic coil with inductance L and capacitance C concentrated in a gap (Fig. 4). The oscillating magnetic field perpendicular to the plane of SRR provokes a current in the coil. At the frequency close to resonant one that is given by well known formula ωLC =1/ LC , the circulating current can give significant rise to a magnetic moment perpendicular to the plane of SRR. The magnetic moment must be directed in such a way to be able to counteract the external

7 CH APTER 1. IN TRODUCTION magnetic field (Lenz’s law) producing the negative permeability. Enlargement of the SRR gap leads to decreasing of capacitance C and accordingly to increasing of the resonance frequency ωLC . The same happens if the bottom part of SRR is dropped out. As a result of this transition from the SRR to the pair of CWs (Fig. 4) the Ohmic currents in the horizontal arms on the SRR are replaced by displacement currents on the pair of CWs [Dolling 2005]. Thereby for the same size of structure’s features the resonance frequency is increased which brings the idea of optical metamaterial structures closer to the reality. Though as the compensation the lattice constant a is decreased and this increases the ratio λ0 / a leading to the possible alienation of the structure composed from pairs of CWs from true NIMs. Although we discuss conditions that allow different structures to be counted as MMs more carefully in the second chapter.

Fig. 4. Adiabatic transition from split ring resonator to pairs of cut wires as "magnetic atoms" of optical metamaterials [Dolling 2005].

The last two examples of NIMs (two metallic plates with holes and the pairs of CWs) can support normal incidence. Aside NRI materials based on MMs there are also NRI materials based on photonic crystals structures. Here, negative refraction is a consequence of band-folding effects and not a consequence of constructional features of material [Parimi 2004, Berrier 2004, Luo 2002].

1.1.6.1 Perfect lensing using negative index metamaterials

A planar slab of NIM with sufficient thickness can act as a lens (Veselago lens). Double negative refraction that light rays undergo on their pass lead to creating non-inverted image of an object in the free space after output of the lens (Fig. 5a). Compared to conventional convex lens (Fig. 5b) the Veselago lens does not magnify objects or focus parallel rays. The image is a perfect restoration of the object. J.B. Pendry showed that NIM slab with RI n = -1 allows recognizing objects with subwavelength precision and thereby overcome the diffraction limit [Pendry 2000]. The reason that makes it possible is related to the growth of evanescent waves inside the NIM slab that contain information about sub-wavelength details of the object. In that way after the output end of the Veselago lens both parts of the light, propagating and evanescent, can be restored together (Fig. 5c). In contrast, the sub-wavelength information is completely lost after passing through the conventional lens (Fig. 5d). 8 1.1 Fundamental of metamaterials

It is significant to note two points. First of all, the growth of the evanescent field inside the NIM media does not affect the conservation energy law. As known evanescent waves carry no energy [Cai 2010]. Second, any realistic losses or anisotropy can entirely eliminate the desired effect of perfect imaging [Smith 2003]. Still all NIMs based on resonant particles are highly dissipative, lossy and strongly anisotropic. And for the best of our knoweledge far-field super lensing has not yet been demonstrated. a b

c d

Fig. 5. (a) Thick layer of metamaterial with negative refractive index focuses the light. (b) A conventional convex lens that is able to collect only propagating waves. (c) Inside negative index metamaterial slab evanescent waves are growing that means that the metamaterial lens fully restores the image behind the slab. (d) The evanescent waves weaken through the conventional lens.

Other application of NRI material involve focusing devices that can conceal the positive space between a source and its image [Pendry 2003]; subwavelength waveguides (WGs) with lateral dimension below diffraction limits [Wu 2003]; media that can support backward radiation [Grbic 2002], etc.

1.2 Transformation optics

Though NIMs played a very important role in the evolution of MM physics, MM applications are extremely versatile not uniquely limited to NIM. One of the very important applications of MMs is related to the transformation optics (TO). TO is using the invariance of Maxwell's equation in regard to Lorentz transformation. It is very powerful instrument for designing such devices as invisibility cloak (see Fig. 6a), concentrator (Fig. 6b), taper (Fig. 6c and Fig. 6d) and others. The great advantage of this approach is that it does not necessarily require a NRI. 9 CH APTER 1. IN TRODUCTION

The TO approach was initially introduced in two research articles simultaneously published in 2006 [Pendry 2006, Leonhardt 2006]. Both articles come in the same issue of “Science” and propose the design idea for the fascinating imagination cloaking devices. The approach considered by Pendry is based on Maxwell’s equations transformation, while that of Leonhardt is based on two-dimensional Helmholtz equation. a b

c d

Fig. 6. (a) Invisibility cloak [Pendry 2006]; (b) field concentrator [Yang 2011]; (c,d) taper [Tichit 2010].

Both approaches provided the formal theoretical demonstration for the ability to achieve invisibility using TO. It should be noted that the first attempts for designing cloaking devices are dating since late 1980s and early 1990s [Ung]. However these attempts remained unsuccessful and it was considered that cloaking is unrealizable. The advent of MMs turns on the dream for cloaking into reality. The idea is that cloaking principle is based on the implementation of particular distribution of the RI allowing the light to circumvent the hided object (Fig. 6a). This make possible to render a macroscopic objects invisible. For engineering the distribution of the RI mathematical tools based on coordinate transformation are used. From purely mathematical point of view for a given wavelength this approach can be applied to hide object of any size and any shape but practical demonstrations of the cloaking devices based on TO and using metallic type of MMs, for the moment, are essentially limited to the microwave domain of frequencies.

1.2.1 Fundamental of transformation optics

According to the Fermat's principle in free isotropic space the light path follows a straight trajectory. Now if one part of isotropic space is replaced by another material (or several materials) with different RI (RIs) the light path will be bent (or even curved). Usually TO gives an answer: how to vary RI in space to obtain the target light path. To understand the main idea better one should start with regular undistorted space where the light path is represented by a straight line (Fig. 7a). Plotting

10 1.2 Transformation optics the space grid (lattice) we can fix the light path to the local coordinate system. Now if one starts to distort the grid (distort the space) keeping the light path inseparably attached to the local coordinate frames, it is possible to shape the light trajectory in a desirable way (Fig. 7b) [Pendry 2012]. As mentioned, for practical purposes the inverse problem is of most interest. The problem to be solved is how the distorted space (lattice) can be transformed to the regular one. To solve this problem the first step is related to the proper choice of the distorted space that would satisfy some target purposes/functionalities. The next step consists in making a coordinate transformation from the distorted to the regular space. The Jacobi matrix associated with the coordinate transformation determines rules for the modification of constitutive parameters ε and µ . a b

Fig. 7. Transformation optics principle: (a) light path in undistorted space; (b) curved light path in distorted space [Pendry 2012].

This procedure is correct due to the invariance of Maxwell’s equations under coordinate transformation. Thus only parameters as ε and µ are affected by the transformation. In a new coordinate system constitutive parameters can be written as follows:

i' j' −1 i' j' ij ε ' = [det(Λ)] ΛiΛ jε , (1.10) i' j' −1 i' j' ij µ' = [det(Λ)] ΛiΛ j µ , where ε and µ are the permittivity and permeability, respectively, in the original coordinates, ε ' and µ' are the corresponding quantities in the transformed frame. Λ is given by the first derivatives of the coordinate transformation: ∂x j' Λj' = . (1.11) j ∂x j Note, that TO can be applied not only to Maxwell’s equations, but also to the Helmholtz equation or to any equation of physics that is invariant under the space transformation.

11 CH APTER 1. IN TRODUCTION

1.2.2 Applications of transformation optics

Aside various cloaking devices that had been already discussed [Schurig 2006_b] TO approach was also applied to design field concentrators [Rahm 2008_a], perfect absorbers [Narimanov 2009], beam bends and beam expanders [Rahm 2008_b], polarization rotators [Kwon 2008], flat lenses [Vakil 2011], etc. Here below we briefly highlight some of these applications.

Cloaking devices As mentioned, cloaking devices have played a strategic role in the history of MMs. In particular their intriguing functionalities gave an impetus to the intensive development of MMs and TO. Cloak of invisibility is a device that can conceal an object from external electromagnetic detection (Fig. 6a). It is a MM shell that leads the light rays around area being hidden. There is a variation of cloak: carpet of invisibility [Li 2008]. The difference from the cloak is that the object placed on a surface disappears from an external observer when being covered by the carpet. In such a way the cancellation of the refracted light occurs only in the half-space above the object.

Perfect absorbers For modern engineering absorbers play very important role. Though nowadays absorbers can be made by different ways most of them suffer from limitations related to the angle of incidence. Narimanov proposed omnidirectional absorber designed on TO principles (Fig. 8). It consists from a symmetrical MM shell that bends the light towards the center of the shell and a core where the energy is being absorbed.

Fig. 8. Cutout views of the spherical and cylindrical optical "black holes" [Narimanov 2009].

Field concentrators Field concentrator is a device that focuses incident electromagnetic waves on a given area. It can play important role in designing solar cells. The example represented in Fig. 6b shows the strength of TO approach that allows to chose transformation area having almost any possible shape (with rotational symmetry/or even without symmetries).

12 1.2 Transformation optics

1.2.3 Implementation of transformation optics in the microwave domain

The implementation of the TO approach requires the ability for an independent tailoring of the permittivity and the permeability (see Eq.1.10). Transformation optical devices are typically implemented with MMs because they require sophisticated anisotropic material parameters that cannot be readily found in conventional media [Schurig 2006_b, Kanté 2009, Liu 2009, Cai 2007, Valentine 2009, Gabrielli 2009]. This is especially true for designs based of nonconformal transformations which often exhibit permittivities ε and permeabilities µ much smaller and/or larger than 1 [Liu 2012]. To obtain such parameters, it is often necessary to use MMs made of subwavelength metallic resonators featuring dramatic variations of ε and µ near the resonance wavelength. For the time being most of the experimental demonstrations for the TO applications using metallic MMs were performed in the microwave domain. For instance, the first experimental demonstration of cloaking device based on TO approach and operating at microwave frequencies in TE polarization was done in 2006 [Schurig 2006_b]. The variation of the magnetic permeability was achieved through the excitation of magnetic resonances of SRRs. The design is not sensitive to the object being cloaked. As it can be seen from Fig. 9 ten layers of SRRs are enough to achieve essential effect of concealing the inner area of the cape. A similar approach but with a substantially different design was used by [Kanté 2009_a]. The difference is that cloaking effect is achieved this time by using only electric resonances.

Fig. 9. Invisibility cloak operating at radio frequency range [Schurig 2006_b].

13 CH APTER 1. IN TRODUCTION

1.2.4. Implementation of transformation optics in the optical domain

The experimental demonstrations for the TO applications in the optical domains are for the moment essentially limited to all dielectric MMs. The two essential factors preventing the use of metallic MMs in the optical domain are related to the planar technology limitations for fabricating a stack with a great number of precisely aligned MM layers [Valentine 2008, Liu 2008, Ghasemi 2012] and the losses due to the absorption of the metal resonator elements. There is also a more specific issue concerning the engineering of the magnetic permeability µ in the optical domain. The limitation comes from the increasing importance of kinetic energy of the electrons in the metal in comparison with the magnetic energy and beyond 200 THz the metal cannot be considered as ideal conductor anymore [Zhou 2005]. The resonance frequency deviates from the linear regime (where the resonant frequency is proportional to the characteristic size of the resonant elements). The enumerated issues inherent to metallic metamaterials motivated an intensive research on TO using all dielectric MMs. Most of the time the approach is based on using of the photonic crystal (PhC) type structures operating essentially in a homogenization regime. PhCs are periodical, but not necessary artificial, structures. An example of one-dimensional PhC is shown in the Fig. 10a. The dispersion relation for 1D PhC is displayed in the Fig. 10b. As it can be seen, there is a strong analogy between the band structure of electrons in natural crystals and the band gap in PhCs. a b

Fig. 10. (a) 1D photonic crystal where a is its period; (b) dispersion curve of photonic crystal.

For instance the light propagation is forbidden in the frequency range corresponding to the photonic band-gap and is allowed for photons with energy above or below the band gap. For an operation well below the band gap, the period of the structure is much smaller than the wavelength. The behavior of the PhC structure is, in this case, essentially similar to that of a MM. The light propagation inside PhC is then essentially determined by the average values of effective parameters

(ε eff and µeff ). Accordingly this regime of PhCs is called homogenization regime. PhCs in homogenization regime are not resonant so MMs based on PhCs can operate over a large bandwidth with virtually no material losses. The MMs based on the PhC geometry are generally planar structures and they are not used in free space but rather in a guided optics configuration.

14 1.2 Transformation optics

The index variation in TO structures based on homogenized PhCs is usually achieved by means of a spatial modulation of the density of nanoholes or nanopillars. By using this approach examples of graded PhC nanolens [Spadoti 2010] or mode adaptor [Markov 2012] were successfully demonstrated. Another interesting example of TO approach applied to homogenized PhC is a fabrication of a carpet cloak for visible light in silicone nitride WG [Gharghi 2011]. The carpet does not require an extreme variation of constitutive parameters because the working incidence angle is limited to a half space. It does not work for arbitrary angle as cloak does and can be implemented by using only isotropic media. The great advantage of using dielectrics is their low-loss behavior in the optical domain. The issue however is that the variation of the effective RI will be always bounded from the upper side by the RI of dielectric material and from the lower side by that of the air which serves as a surrounding cladding. The effective RI of PhC structure will be somewhere in between of the two RIs from which PhC is made of. For the same reason it would be also difficult to achieve significant anisotropy required for most of TO applications. To resume the situation, the advantage of metallic MMs is that potentially they can provide a higher index variation with stronger anisotropy. However metallic MMs are extremely lossy at optical frequencies. The fabrication of nanoscale size elements multilayer structures required for TO applications is also highly laborious. While concerning dielectric MMs, they present generally negligible losses but have a lower range of index variation and the implementation of index anisotropy is more challenging. The explored approach intended to circumvent these issues is detailed below.

1.2.5 Hybrid metal-dielectric metamaterials for transformation optics applications

The problem of absorption in metals has generated an intense effort to develop active MMs and plasmonic components where losses are compensated by a gain medium [Poutrina 2009, Xiao 2010, Fang 2009, Fang 2010, Dong 2010, Lagarkov 2010]. Another promising approach consists in creating hybrid photonic structures in which metallic parts are coupled with dielectric (and almost lossless) WGs [Nikolajsen 2004, Degiron 2007, Oulton 2009, Dai 2009, Kim 2010, Wu 2010]. In this configuration, useful functionalities are obtained by allowing just enough light to interact with the metallic parts of the system. The remaining part of the energy propagates in the dielectric WG, thereby, considerably mitigating the losses. To date, the potential of hybrid structures has been mostly explored in the context of integrated plasmonic circuits. There is much less literature on hybrid MMs combining

15 CH APTER 1. IN TRODUCTION subwavelength metallic inclusions and dielectric WGs, except few works where it was shown that dielectric films incorporating metallic MMs can support resonant surface modes [Maier 2004, Maier 2005, Reinhard 2010]. In the present work we address the potential of hybrid MMs in guided wave optics in the near infrared (NIR) range. We consider the case of a composite guiding structure made of a single sheet of metamaterial above silicon WG (Fig. 11). In such a configuration most of the light is essentially confined in the silicon slab while only the evanescent tail interacts with the MM layer. In other words, the MM layer acts mostly as a perturbation. Its role is to modify the effective index of the composite WG structure. Such a solution allows to significantly reduce the propagation losses since the main part of the electromagnetic energy do not interact directly with the metallic part of the metamaterial. Another important advantage of the guided wave configuration is that it helps to avoid the necessity of fabrication multi-layered MM structures. The transposition of the light flow parallel to the MM layer eliminates the technological limitations concerning MM optical thickness. The fabrication process would require only one single layer of lithographically patterned elements. a c

Fig. 11. (a) Geometry of the mode adapter considered in this study. (b) Transition from the large to the narrow waveguide using a mode adapter. The y-component of the electric field is shown in the x-y plane located halfway through the silicon slab. (c) Evolution of the transverse mode profile using the mode adapter. Shown y-z cross-sections illustrate the power of the mode at x = 0.25 µm, x = 1.75 µm, x = 3.25 µm and x = 4.75 µm.

16 1.2 Transformation optics

Validation of this approach in the framework of a phenomenological TO layer model was performed by Ghasemi et al. [Ghasemi 2010]. For this example a 30 nm thick TO layer placed on the top of 170 nm thick silicon WG and bearing a taper mode adapter function was considered. The distribution of the electric field in the middle of the silicon slab is represented in the Fig. 11b. The evolution of the transverse mode profile is shown in the Fig. 11c. An evident conclusion that can be drawn from the presented results is that a thin film of TO can indeed allow controlling light propagating in the underneath guiding layer. In the mentioned study the TO layer is treated as a homogeneous anisotropic media. The problem of how to perform the transition from a model dealing with some local constitutive parameters to the design of a MM unit cell meeting given ε and µ was not addressed. Solving this problem and investigation of the feasibility of MMs for the NIR domain using the hybrid metal- dielectric approach greatly impacted the orientations of the present work.

1.3 Thesis motivation and manuscript organization

The subject of the PhD thesis work deals with the MMs for photonic applications. The main objective is to investigate the potential of metallic MMs for building optical functions at 1.5 µm. A significant part of the work is focused on the engineering of the MM effective index associated with localized plasmon resonances. Two examples of particular importance for the reasons of the relative simplicity of technological realization are considered:

• Single MM layer on a dielectric substrate effective parameters ε and µ engineering for a free space light propagation.

• Single MM layer on a dielectric WG effective parameters ε and µ engineering for a guided wave light propagation. Concerning the first point, the introduction of the effective parameters for a single MM layer is still a controversial subject up-to-date [Simovski 2007, Simovski 2011, Menzel 2009, Kanté 2009_b]. The intend of this work is to provide a further insight on the subject and contribute to the clarification of the situation. While concerning the second point, the engineering of the MM effective parameters in a guided wave configuration represents a new and unexplored so far approach that was investigated in this thesis. The minimization of metal related losses by using of a hybrid metal-dielectric approach is specifically addressed as well as the engineering of the resonance frequency of the localized plasmons.

17 CH APTER 1. IN TRODUCTION

Aside the problem related to the metal losses, another important issue is related to the extremely reduced dimensions of the resonating MM structures. It become especially critical when considering the implementation of the MMs on high index material WGs such as Si or InP. The experimental demonstration of the feasibility of the MM structure compliant with the operation at 1.5 µm wavelength is being particularly addressed. The manuscript is organized as follows. The subject introduction and definition of thesis objectives are described in the first chapter. The Chapter 2 presents an overview of the homogenization methods for compound materials and retrieval procedures allowing determination of the effective permittivity ε and effective permeability µ . The description of the methods for MM technological fabrication and experimental optical characterization is performed in the Chapter 3. The theoretical and experimental investigation of the effective parameters engineering for the case of a single MM layer on a dielectric substrate for a free space light propagation configuration is presented in the Chapter 4. The Chapter 5 deals with the theoretical and experimental investigation of the MM effective parameters in a guided wave configuration using hybrid metal-dielectric approach. The summary and perspectives for the future work are detailed in the Chapter 6.

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[Walser 2001] R.M. Walser, “Electromagnetic metamaterials,” Proc. SPIE 4467, 1 (2001)

23 CH APTER 1. IN TRODUCTION

[Weis 2009] P. Weis, O. Paul, C. Imhof, R. Beigang, M. Rahm, “Strongly birefringent metamaterials as negative index terahertz wave plates,” Appl. Phys. Lett. 95, 171104 (2009)

[Wu 2003] B.-I. Wu, T.M. Grzegorczyk, Y. Zhang, J.A. Kong, “Guided modes with imaginary transverse wave number in a slab waveguide with negative permittivity and permeability,” J. Appl. Phys. 93, 9386 (2003)

[Wu 2010] M. Wu, Z. Han, V. Van, “Conductor-gap-silicon plasmonic waveguides and passive components at subwavelength scale,” Opt. Exp. 18, 11728 (2010).

[Xiao 2010] S. Xiao, V.P. Drachev, A.V. Kildishev, X. Ni, U.K. Chettiar, H.-K. Yuan, V.M. Shalaev, “Loss-free and active optical negative-index metamaterials,” Nature 466, 735 (2010)

[Yang 2011] J. Yang, M. Huang, C. Yang, G. Cai , “A metamaterial acoustic concentrator with regular polygonal cross section,” J. Vib. Acoust. 133, 061016 (2011)

[Zhang 2005] S. Zhang, W. Fan, N.C. Panoiu, K.J. Malloy, R.M. Osgood, S.R.J. Brueck, “Experimental demonstration of near-infrared negative-index metamaterials,” Phys. Rev. Lett. 95, 137404 (2005)

[Zhou 2005] J. Zhou, T. Koschny, M. Kafesaki, E.N. Economou, J.B. Pendry, C.M. Soukoulis, “Saturation of the magnetic response of split-ring resonators at optical frequencies,” Phys. Rev. Lett. 95, 223902 (2005)

[Zhou 2006] J. Zhou, L. Zhang, G. Tuttle, T. Koschny, C.M. Soukoulis, “Negative index materials using simple short wire pairs,” Phys. Rev. B 73, 041101(R) (2006)

Chapter 2 Homogenization of metamaterials

2.1 Effective medium approximations

One of the important fundamental questions is how to describe complex structures such as MMs by macroscopic parameters. The straightforward approach is to solve macroscopic Maxwell’s equations with an inhomogeneous permittivity and permeability by using numerical techniques such as finite time domain or finite spectral domain methods. These numerical approaches demand however huge computer capacities and memory resources and are lacking to provide an insight on the MM macroscopic properties. A common and useful practice is to describe MMs as a homogeneous effective medium (EM) with an effective electric permittivity ε eff and magnetic permeability µeff . Such an approach is generally justified if the spatial dimensions of the constituent components of some composite material are sufficiently small compared to the wavelength. Among the variety of existing homogenization methods the Maxwell Garnett and Bruggeman’s approximations are of particular importance for the present study. They are briefly described in the next sections. As known, Maxwell Garnett and Bruggman’s models are both first-order approximations in volume fraction. The main difference between them is how the components of composite media are considered: as a mixture of different elements or as guest elements embedded in a host media.

2.1.1 M axwell-Garnett approximation

Maxwell-Garnett (MG) approximation is commonly used to describe the macroscopic properties of a composite material formed by a mixture of two or more components [Garnett 1904]. The MG approach is also known as quasistatic or Clausius-Mossotti approximation [Levy 1997]. The approach is usually used when one of the components can be considered as a host and the rest of the components can be treated as inclusions. Many MM structures that can be viewed as composed from the air as a host material, and containing metal resonant elements as inclusions, also fall in this category of compound structures. The MG approximation, as most of other EM models, assumes that the macroscopic system is homogeneous and uses mean field theory approach to account for

25 CH APTER 2. H OM OGEN IZATION OF… the averaged interactions of dipole moments induced by inclusions. The validity of the MG approximation requires the fulfillment of the following conditions [Koledintseva 2006]: • the complex media needs to be electrodynamically isotropic; • none of the constitutive parameters of the mixture do not depend on the intensity of electromagnetic field; • parameters of mixture need to be time independent; • inclusions are separated by distance greater than their characteristic size; • the characteristic size of inclusions must be smaller than the wavelength in the EM (quasistatic approximation); • inclusions should be randomly oriented. These conditions fix the limits of validity for the MG approximation. For instance, in the optical domain the usage of the MG approximation may be inappropriate since the size of MM particles becomes comparable with the wavelength and anisotropy starts to play an essential role.

2.1.1.1 M axwell-Garnett formula for spherical inclusions

Let us consider a mixture of spherical-like inclusions with permittivity ε i randomly sited in the

2 environment (host medium) with permittivity ε e . Let f be a volume fraction occupied by inclusions and then host medium takes volume 1− f (Fig. 12).

Fig. 12. Spherical-like inclusions in the dielectric host can be replaced by an effective medium within the limits of M axwell-Garnett approximation.

Making assumption that whole material can be described as a homogeneous media with effective permittivity ε eff one can rewrite formula (1.2) in the form: C C

D = ε eff E. (2.1) The averaged electric field and electric induction can be written by weighing the fields with the corresponding volume factors:

2 For simplicity sake here we consider dielectric inclusions 26 2.1 Effective medium approximations

C C C D = fε i ei + (1− f )ε eee , C C C (2.2) E = fei + (1− f )ee , C where we assume that ε i and ee to be constant. Introducing the ratio between external and internal C C fields ei = Aee the effective permittivity can be written: fε A + ε (1− f ) ε = i e . (2.3) eff fA + (1− f )

For spherical particles field ratio between internal and external fields: A = 3ε e /(ε i + 2ε e ). Then the effective permittivity can be rewritten as:

ε i − ε e ε eff = ε e + 3 fε e . (2.4) ε i + 2ε e − f (ε i − ε e ) Formula (2.4) is known as the MG formula. This formula also can be adopted, for instance, for ellipsoidal inclusions [Barrera 1993, Salski 2012].

2.1.1.2 Clausius-M ossotti formula

Clausius-Mosssotti formula gives the relation between microscopic parameters (e.g. molecular polarisability α ) and macroscopic parameters (e.g. the dielectric constant ε ) [Choy 1999]. The C C C C dipole moment of a molecule p is given by p = αEL ( EL is local electric field). Averaging this relation and knowing the relation between dipole moment and external field for spherical inclusions one can get Clausius-Mossotti formula (or Lorenz-Lorentz formula):

ε eff − ε e nα = , (2.5) ε eff + 2ε e 3ε e where n is the number density of the dipoles. Note that for the averaging process to be valid it is necessary that the inclusions are separated from each other by sufficient space. Namely this formula is derived in assumption that each inclusion experiences the same local field.

2.1.2 Bruggeman’s approximation

Bruggeman’s model assumes that material is composed of two types of spherical-like grains with different complex values of permittivities. For essential Bruggeman’s model can be viewed as an improvement of MG approximation but it treats the two composites in a symmetrical way. This is in contrast to MG approximation where guest and host formalism is used [Sihvola 1999]. It can be shown that Bruggeman’s model is equivalent to a medium with two inclusions of very different sizes, so that any of two inclusions of the same size are well separated. The whole assembly can be considered as a diluted composite [Torquato 2001]. We can assume that material surrounding the 27 CH APTER 2. H OM OGEN IZATION OF…

inclusions can be replaced by the EM (homogenized medium) with an effective permittivity ε eff . The homogenized medium here is considered as a background. The Bruggemann’s approximation then can be easily generalized to include any number of components N :

N ≈ ’ ∆ ε i − ε eff ÷ ƒ fi ∆ ÷ = 0, (2.6) i « ε i + 2ε eff ◊

th where fi is fractional ratio of i component.

2.1.3 Numerical homogenization approaches

The detailed above EM approximation methods allow the description of the macroscopic behavior for many composite optical media. These homogenization methods hold true especially when the size of inclusions is much smaller than the wavelength and anisotropy effects can be neglected. However, these methods fail in the case of complex geometry composite media. In this case, numerical homogenization methods have to be applied. In numerical approaches the medium is usually divided into small cells and the fields are solved for a finite number of points. The process can be simplified by making assumption that the media is having some periodicity. Indeed, large amount of complex media are periodic or at least close to periodic [Bakhalov 1989].

2.2 M etamaterial effective parameters

Let us consider a situation where homogenization conditions for MM structure are fulfilled and it can be described as an EM. The effective parameters ε and µ can be obtained from the macroscopic response of MM structure to an electromagnetic field. One of the most common techniques used to extract effective parameters is the Nicolson-Ross-Weir (NRW) method. It uses measured or calculated values of scattering parameters (complex values of reflection and transmission coefficients) from a composite slab layer under test. The method is based on the inversion of the Fresnel-Airy formulas. The RI and wave impedance are extracted by replacement of the real composite layer of thickness h by a uniform continuous medium of the same thickness whose ε and µ have to be found. The adaptation of the NRW method to MM structures by Smith et al. [Smith 2002] greatly contributed to the popularization of this technique that becomes widely used for retrieving effective parameters. However, despite the widespread use of the NRW method in the metamaterial community, the accuracy of the retrieval procedure by this method is not always 28 2.2 M etamaterial effective parameters granted and the validity of its using in the context of metafilms is still a subject of debate [Simovski 2007, Scher 2009, Holloway 2009, Simovski 2011].

2.2.1 Nicolson-Ross-Weir retrieval method

Smith et al. described the electromagnetic scattering properties of a thin MM film in terms of RI n and wave impedance z [Smith 2002]. The impedance z of the electromagnetic wave is given by the ratio of the transverse components of the electric and magnetic fields:

− E0 (x) z = − . (2.7) H 0 (x) It can be shown that the last expression is equivalent to the following one:

µ z = . (2.8) ε This means that it is only the matter of choice which pairs of parameters to use: n and z , or ε and µ . Knowing n and z it is always possible to make transition to constitutive parameters ε and µ : ε = n / z and µ = nz . Recall that all these parameters are frequency dependent and they may be complex values. a b

Fig. 13. Approximation of effective media for metamaterial layer. (a) M etamaterial layer under incidence electromagnetic wave. (b) Effective medium approximation. h* and h are thickness of metamaterial layer and effective media layer, respectively. i is incidence wave, r is reflected wave and t is transmitted wave.

Let us consider an infinite slab of a MM that can be described by effective parameters ε and µ (see Fig. 13a). Let h* be the physical thickness of MM layer. The thickness of the equivalent EM layer is h . The effective parameters should be defined in such a way that in the far field an observer cannot see the difference between MM layer and EM layer. Then the transmission coefficient t for normal plane incidence wave is given by the expression:

29 CH APTER 2. H OM OGEN IZATION OF…

» i ≈ 1 ’ ÿ ikh t −1 = …cos(nkh) − ∆ z + ÷sin(nkh)Ÿe , (2.9) 2 « z ◊ ⁄ and the refractive coefficient r : r 1 ≈ 1 ’ = − i∆ z − ÷sin(nkh)eikh , (2.10) t 2 « z ◊ where k = 2π / λ is the wavenumber of incidence plane (λ is the wavelength in vacuum). From this system of two last equations wave impedance and effective refractive index can be found:

(1+ r) 2 − t 2e2ikh z = ± , (2.11) (1− r)2 − t 2e2ikh

≈ ≈ 1 ’ ’ ∆ cos −1 ∆ [1− (r 2 − t 2e 2ikh ]÷ ÷ ∆ « 2teikh ◊ ÷ Im(n) = ± Im , (2.12) ∆ kh ÷ ∆ ÷ « ◊

≈ ≈ 1 ’ ’ ∆ cos −1 ∆ [1− (r 2 − t 2e 2ikh ]÷ ÷ ∆ « 2teikh ◊ ÷ 2πm Re(n) = ± Re + , (2.13) ∆ kh ÷ kh ∆ ÷ « ◊ where m is an integer. As evident from Eqs. (2.12-2.13) z and n are comlex functions with multiple solution branches. This leads to certain difficulties for an unambiguous solution determination. Imposing additional constrains from the general physical considerations can reduce the ambiguity for the appropriate solution determination. For example, in the case of passive materials Re(z) and Im(n) can take only positive values. The situation is more complex what concerns Re(n) , but the usage of some tips can greatly help in making the proper choice for the branch. The reliability of the NRW method is generally good in the case of thin MM layers but becomes worse when thickness is increased. With the increase of film thickness the number of solution branches augment, they are less separated and the retrieval procedure may become unreliable. Aside the proper choice of the branch, the accuracy and validity of the NRW method are generally greatly depending of the very validity of the effective medium assumption. This point needs to be thoroughly addressed for each particular study and it will be discussed in more details in the next section.

2.2.1.1 Retrieval of metamaterials effective parameters

Despite certain issues inherent to the NRW method, it became very popular among scientists, primary because of its relative simplicity. Indeed, by assuming that effective medium approach 30 2.2 M etamaterial effective parameters

(EMA) is valid, it is sufficient to find the complex coefficients of reflection and transmission to determine MM effective parameters ε and µ . A huge amount of works, dealing with MM effective parameters retrieval, is based on such approach. Here on the figure one can see the impressive growth of the number of articles that are using Smith adapted formula or at least discussing it. Nowadays around 250 articles per year are dedicated to MM studies, and approximately one third of these articles is explicitly using the idea of MM homogenization. The NRW is a well-established method providing valuable results. Here we are not attempting an exhausting overview of NRW method, but rather to show the variety of studies where it was applied. The method was successfully applied to fishnet structures [Zhou 2008, Kafesaki 2007], to nanowires [Zhou 2008_a], to elliptical holes in double gold film with dielectric interlayer [Zhang 2006], to metamaterials composed from short-slabs and continuous wires [Gundogdu 2008], to pillar nanoparticles [Ekinci 2008], to pairs of nanopillars [Grigorenko 2006], to SRRs [Linden 2004, Yen 2004], et c. The relation between effective parameters ε , µ of MMs and their geometry parameters also was investigated by the NRW method [Ekinci 2008, Grigorenko 2006]. The method was successfully used for studies of multi-layered structures, for instance, of MM structure composed of several SRR layers [Liu 2008]. One of the most important parameters entering in retrieval procedure is the EM thickness h . The determination of the MM effective thickness is relatively easy in the case of a multi-layered structure. Loosely the MM structure thickness is equal to the number of periods multiplied by the length of the MM unit cell along the propagation direction. The validity of this approach was verified in several studies using convergency criterion. Recall that homogenization approach requires that effective parameters of a multi-layered metafilms should not depend of the number of layers. It implies convergency of the effective parameters for a multi-layered MM structure when the number of layers is sequentially increased. For instance a good convergency of the effective parameters was observed for a multi-layered continuous wires structure studied by [Smith 2004]. The definition of effective MM thickness is however more problematic for a single MM layer which spatial extent in the propagation direction is much smaller than the wavelength. This case corresponds to the most commonly encountered in practice situation where the MM structure is formed by a periodic arrangement of resonant elements on a substrate surface. This is also the case for our study. Among the few attempts to address this problem we can note those done by [Zhang 31 CH APTER 2. H OM OGEN IZATION OF…

2005] and [Kanté 2009]. For instance in the work of [Kanté 2009] the behavior of a single MM layer formed by gold nanowires and split ring resonators (SRRs) was investigated by Fourier transform infrared (FTIR) experiments in the mid IR domain. By exploiting the interferogram of the asymmetric cavity formed by MMs on silicon substrate it was concluded that it is indeed possible to assign an effective thickness and index of refraction to a single MM layer. The effective thickness deduced from experimental results is equal to twice the physical thickness of the gold MM layer. However it could be noted that interferometric measurements are intrinsically sensitive to optical path difference. By consequence it is difficult to unambiguously dissociate thickness and RI by this method. Furthermore, the thickness of a single MM layer reported in the literature differs according to authors. So for MM structure operating in the optical domain composed of gold SRRs embedded in PC403 spacer on top of a glass substrate, the effective MM layer thickness was founded to be that of the spacer [Liu 2008]. It turns out that effective thickness of MM layer is, in this case, even not related to that of the metal. Another example concerns a MM layer composed of vertically oriented gold SRRs and continues wires [Rockstuhl 2008]. Here the height of both elements is 300 nm, but the effective thickness of the MM layer was founded to be 400 nm. These examples show that to date there is no an established point of view concerning the determination of the effective thickness in the case of single MM layer. Moreover, the very validity of assigning an effective thickness to a metafilm is a subject of highly critical remarks [Simovski 2007, Scher 2009, Holloway 2009, Simovski 2011]. An analysis of these critical remarks is performed in the next section.

2.2.1.2 Single metafilm effective medium behavior

While concerning the description of a single MM layer as an EM, the essential point of criticism is that such an approach is non-physical and leads to misinterpretations. The principal argument is that effective bulk material properties are not uniquely defined for a metafilm. The demonstration of this statement is performed on the example of a surface covered by an array of scattering particles [Simovski 2011] (Fig. 14). The array scattering is determined by the electro-dipole, magneto-dipole and quadrupole polarizabilities of scattering particles and by the array period. It is stated that the reflection and transmission properties of such a surface are little influenced by the particle thickness that doesn’t determine, in this case, the phase shift of the wave across the layer. The conclusion drawn is that while the geometry of the scatterers and the array constant are uniquely defined, the thickness of the equivalent layer is not and can be arbitrary fixed to any value within reasonable limits. To obviate the issue with the effective parameters alternative definitions dealing with “surface susceptibilities” are introduced [Holloway 2009, Scher 2010].

32 2.2 M etamaterial effective parameters

Fig. 14. Representing a metafilm as an effective medium of thickness d [Holloway 2009].

Despite the severe critical remarks, it seems nonetheless prematurely interring the idea of using EMA for the case of single MM layer, in particular what concerns operation in the optical domain. As evident from the previous remarks, the central point is the effective thickness determination and the main critical argument is that this can be done in an arbitrary way. There are however some shortcomings in these arguments. The phase variation across a composite material layer is influenced by both constituent components, and not only by the scattering one. In other words, by considering metal particles as inclusions and surrounding medium as host material, there is no any reason to neglect the host contribution to the phase shift of the transmitted wave. As obvious, the relative host contribution is by far more important when the concentration of the inclusions is small. By following this reasoning, in the limit of very small inclusions concentration the effective thickness of the metafilm should be equal to that of the host layer. Note, that such a treatment of the EM has nothing exceptional. It is commonly used in ellipsometry to describe the behavior of a fractionally covered surface, which is treated as an effective film having a certain thickness and RI. Of course, to be valid, the model of EM introduced in such a way should be self-consistent. For instance, neither the effective permittivity ε nor permeability µ should depend of the incidence angle if anisotropy effects are absent. This EMA should also meet other criteria, such as the validity of the MG approximation etc.

2.3 Closing remarks

In the following chapters we provide through numerical modeling and experimental results of the validity of the EMA in the case of a single MM layer. There are several reasons why this study is important for this thesis work:

• Engineering of MM layer properties for free space configuration.

• Adaptation of the free space EM model to the guide wave configuration.

• Engineering of MM properties in a guided wave configuration. Some explanatory remarks are necessary before ending this chapter. Many important aspects of the homogenization approach in the context of MMs are intentionally left out since do not

33 CH APTER 2. H OM OGEN IZATION OF… directly or very little concern thesis subject. So many aspects related for example to other homogenization methods or retrieval techniques are not discussed. Most of the retrieval performed in the thesis is based on the NRW method, though in certain situations alternative methods mostly similar to that of [Popa 2005] and [Andryieuski 2009] are used to verify the consistency of obtained results. For same reasons, aspects related to the problematic effective parameters evolution under the transition from the single to a multi-layered MM structure is not discussed. The interested reader may find a detailed description of these topics in [Rockstuhl 2008, Simovski 2011, Andryieuski 2012].

References

[Andryieuski 2009] A. Andryieuski, R. Malureanu, A.V. Lavrinenko, “Wave propagation retrieval method for metamaterials: Unambiguous restoration of effective parameters,” Phys. Rev. B 80, 193101 (2009)

[Andryieuski 2012] A. Andryieuski, S. Ha, A.A. Sukhorukov, Y.S. Kivshar, and A.V. Lavrinenko, “Bloch-mode analysis for retrieving effective parameters of metamaterials,” Phys. Rev. B 86, 035127 (2012)

[Bakhvalov 1989] N. Bakhvalov, G. Panasenko, H omogeni sati on: averagin g processes in peri odi c med ia: mathemat ical problems in t he mechani cs of composi te mat erials, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht (1989)

[Barrera 1993] R.G. Barrera, J. Giraldo, W.L. Mochán, “Effective dielectric response of a composite with aligned spheroidal inclusions,” Phys. Rev. B 47, 8528 (1993)

[Belov 2003] P.A. Belov, R. Marqués, S.I. Maslovski, I.S. Nefedov, M. Silveirinha, C.R. Simovski, S.A. Tretyakov, “Strong spatial dispersion in wire media in the very large wavelength limit,” Phys. Rev. B 67, 113103 (2003)

[Choy 1999] T.C. Choy, Ef fect i ve medi um theory: prin ciples and appli cat i on s, Oxford University Press, Inc. (1999)

[Ekinci 2008] Y. Ekinci, A. Christ, M. Agio, O.J.F. Martin, H.H. Solak, J.F. Löffler, “Electric and magnetic resonances in arrays of coupled gold nanoparticle in-tandem pairs,” Opt. Exp. 16, 13287 (2008)

[Garnett 1904] J.C.M. Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. London, Ser. B 203, 385 (1904)

[Grigorenko 2006] A.N. Grigorenko, “Negative refractive index in artificial metamaterials,” Opt. Exp. 31, 2483 (2006)

[Gundogdu 2008] T.F. Gundogdu, N. Katsarakis, M. Kafesaki, R.S. Penciu, G. Konstantinidis, A. Kostopoulos, E.N. Economou, C.M. Soukoulis, “Negative index short-slab pair and continuous wires metamaterials in the far infrared regime,” Opt. Exp. 16, 9173 (2008)

35 CH APTER 2. H OM OGEN IZATION OF…

[Holloway 2009] C.L. Holloway, A. Dienstfrey, E.F. Kuester, J.F. O’Hara, A.K. Azadd, A.J. Taylor, “A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional, equivalent of metamaterials,” Metamaterials 3, 100 (2009)

[Kafesaki 2007] M. Kafesaki, I. Tsiapa, N. Katsarakis, T. Koschny, C.M. Soukoulis, E.N. Economou, “Left-handed metamaterials: the fishnet structure and its variations,” Phys. Rev. B 75, 235114 (2007)

[Koledintseva 2006] M.Y. Koledintseva, R.E. DuBroff, R.W. Schwartz, “A Maxwell Garnett model for dielectric mixtures containing conducting particles at optical frequencies,” PIER 63, 223 (2006)

[Kanté 2009] B. Kanté, J.-M. Lourtioz, A. de Lustrac, “Infrared metafilms on a dielectric substrate,” Phys. Rev. B 80, 205120 (2009)

[Levy 1997] O. Levy, D. Stroud, “Maxwell Garnett theory for mixtures of anisotropic inclusions: application to conducting polymers,” Phys. Rev. B 56, 8035 (1997)

[Linden 2004] S. Linden, C. Enkrich, M. Wegener, J. Zhou, T. Koschny, C.M. Soukoulis, “Magnetic response of metamaterials at 100 Terahertz,” Science 306, 1351 (2004)

[Liu 2008] N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31 (2008)

[Popa 2005] B.-I. Popa, S.A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005)

[Rockstuhl 2008] C. Rochstuhl, T. Paul, F. Lederer, T. Pertsch, T. Zentgraf, T.P. Meyrath, H. Giessen, “Transition from thin-film to bulk properties of metamaterials,” Phys. Rev. B 77, 035126 (2008)

[Salski 2012] B. Salski, “The extension of the Maxwell Garnett mixing rule for dielectric composites with nonuniform orientation of ellipsoidal inclusions,” PIERL 30, 173 (2012)

[Scher 2009] A.D. Scher, E.F. Kuester, “Extracting the bulk effective parameters of a metamaterial via the scattering from a single planar array of particles,” Metamaterials 3, 44 (2009)

[Sihvola 1999] A. Sihvola, El ect romagn eti c mi xi ng formu las and appli cat i on s, The Institution of Electrical Engineers (1999)

36 References

[Simovski 2007] C.R. Simovski, S.A. Tretyakov, “Local constitutive parameters of metamaterials from an effective-medium perspective,” Phys. Rev. B 75, 195111 (2007)

[Simovski 2011] C.R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011)

[Smith 2002] D.R. Smith, S. Schultz, P. MarkoŠ, C.M. Soukoulis, “Determination of effective permittivity and permeability of metamaterials from reflection and transmission coefficients,” Phys. Rev. B 65, 195104 (2002)

[Torquato 2001] S. Torquato, S. Hyun, “Effective-medium approximation for composite media: realizable single-scale dispersion,” J. Appl. Phys. 89, 1725 (2001)

[Yen 2004] T.J. Yen, W.J. Padilla, D.C. Vier, D.R. Smith, J.B. Pendry, D.N. Basov, X. Zhang, “Terahertz magnetic response from artificial materials,” Science 303, 1494 (2004)

[Zhang 2005] S. Zhang, W. Fan, N.C. Panoiu, K.J. Malloy, R.M. Osgood, S.R.J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005)

[Zhang 2006] S. Zhang, W. Fan, K.J. Malloy, S.R.J. Brueck, N.C. Panoiu, R.M. Osgood, “Demonstration of metal-dielectric negative-index metamaterials with improved performance at optical frequencies,” J. Opt. Soc. Am. B 23, 434 (2006)

[Zhou 2008_a] J. Zhou, T. Koschny, C.M. Soukoulis, “An efficient way to reduce losses of left- handed metamaterials,” Opt. Exp. 16, 11147 (2008)

[Zhou 2008_b] J. Zhou, T. Koschny, M. Kafesaki, C.M. Soukoulis, “Size dependence and convergence of retrieval parameters of metamaterials,” Photon. Nanostruct. – Fundam. Appl. 6, 96 (2008)

Chapter 3 M etamaterials fabrication technology and experimental characterization methods

3.1 Fabrication of metamaterials

One of the main goals of this work is the experimental investigation of MMs behavior at the NIR optical domain. For this purpose several samples were fabricated and characterized, they are destined either for free space or guided wave configuration experiments. Before the discussion of the obtained results it is important to understand where are the challenges for their technological fabrication and what experimental characterization techniques were used. One of the serious issues concerning the technological realization of MMs operating in this optical domain is related to the extremely small size of MM elements and their separation distances. NIR range optical domain extends approximately from 120 THz to 400 THz (in wavelength from 750 nm to 2.4 microns; in wave numbers 4200 cm-1 to 13300 cm-1, and in energy from 0.52 eV to 1.65 eV) (see Fig. 15). That basically means that typical dimension of MM structure shall not exceed several tens of nanometers. Furthermore, the technological realization of metallic MMs in guided wave configuration (GWC) is essentially double-step process that requires accurate matching of two lithographical levels.

Fig. 15. Near infrared range of frequencies [Lal 2007].

In this chapter we present the most employed experimental techniques and characterization methods, namely, electronic lithography, deposition of metal, dielectric etching, spectral characterization of WGs and some other associated techniques. The fabrication process as well as

39 CH APTER 3. M ETAM ATERIALS FABRICATION … technology characterization measurements were performed using In st i tu t d’Elect ronque Fondamen tal e (IEF) clean room MINERVE facilities.

3.1.1 Fabrication flow sheet for free space configuration metamaterials

The technological realizations of this study essentially concern gold MM structures fabricated on the top of either silicon or glass substrates. For the given spectral range the largest dimension of MM particles does not exceed several tenth of micron. The most common way to realize such structures is electron beam lithography (EBL). The technological flow for the fabrication of a single MM layer can be divided into 6 principal steps (see Fig. 16): • substrate cleaning, • coating the substrate with appropriate resist, • lithography act, • resist development, • metal deposition, • final cleaning of the finished sample. These steps are described further in more details.

a b c

f e d

Fig. 16. Fabrication process with positive resist: (a) cleaned dielectric substrate; (b) the substrate covered by positive resist; (c) electron beam lithography exposure process; (d) gained resist mask on top of the substrate; (e) the substrate and resist mask covered by metal layer; (f) finished sample with metal particles on it after lift-off process.

40 3.1 Fabrication of metamaterials

3.1.1.1 Substrate cleaning

Well-prepared and clean substrate is a pledge of a good sample. In our case two different dielectric substrates were used: silicon on insulator (SOI) substrate with silicon thickness of 500 µm, and glass substrate (SiO2) with thickness of 300 µm. The substrates are usually cleaned with acetone to remove salts and isopropyl alcohol (IPA) to remove organic bases and other compounds. IPA also works as water-drying chemical. In some cases the substrates required additional treatment after liquid cleaning. Thus, for instance, to remove upper oxidized silicon layer from the top of SOI plate oxygen plasma cleaning was used. The standard duration time is about 10–15 min. While the glass substrates demanded more careful ultrasound (US) cleaning in acetone bath during 15–30 min. Before the next step cleaned substrates are put on a hot plate for 5–10 min, the temperature is 170–190˙C. This step is not imperatively necessary but it helps to dry out the water from the substrates’ surface before resist coating.

3.1.1.2 Resist spin-coating

Depending on the required design different resists may be used. Most of the time we worked with PMMA A6 (6% of polymethyl methacrylate and 94% of anisole) or ZEP520A (composed of 11% of methyl styrene and chloromethyl acrylate copolymer and 89% of anisole) electronic resists. Both resists are positive, that means that after exposure they become soluble and mask for EBL is formed by unexposed areas. A simple spinning holder is used to deposit resist on a substrate. The acceleration, the speed and the working time of the spin-coating tournette were chosen based on the desired resist thickness. The higher speed of the tournette is or the longer spinning time is, the thinner resist layer will be. The resulting resist thickness can be also varied by dilution the resist with A-Thinner (anisole). The main idea is to obtain resist thickness approximately three times larger than the desired thickness of metal particles. In our case the thickness of gold particles is intent to be around 50 nm, so the thickness of deposited resist must be no less than 150 nm. To obtain the correct thickness of the resist PMMA A6 needed to be diluted with anisole in weight proportion 1:13. A better adhesion between electronic resist and substrate surface can be achieved by using standard HDMS (hexamethyldisilazane) primer. However, most of the time this additional layer was required only for ZEP520A resist. Use of PMMA resist showed no need in additional HDMS primer both for SOI substrates or glass substrates.

3 Diluted in this proportion PMMA A6 basically becomes PMMA A3: 3% of solid compounds and 97% of anisole. 41 CH APTER 3. M ETAM ATERIALS FABRICATION …

As it is well known glass substrates are liable to buildup of negative charge on the substrate surface that can cause electron beam (e-beam) deflection, and thus pattern/mask distortion. To solve the surface charging problem during EBL process the glass substrates must be covered by finalizing layer of water soluble conducting polymer Espacer 300Z. Besides its primary function Espacer 300Z helps with determination of the resist covered side of glass substrate since it has light blue hue whereas glass substrate and resist are transparent that complicates the determination of the covered side. In total, depending on the taken substrate and chosen resist each substrate can be covered by one (SOI substrate covered by PMMA resist), two (SOI substrate covered by HDMS primer and ZEP resist; or glass substrate covered by PMMA resist and Ecpaser 300Z) or three layers (glass substrate covered by HDMS primer, ZEP resist and Ecpaser 300Z) of chemicals. Each layer needs to be fixed by soft post-backing procedure. The only exception is HDMS primer that may be directly covered by next resist layer without additional post-backing. Below one can find a table with all process parameters for two mentioned resists as well as for HDMS and Espacer 300Z. Dilution Spin-coating Post-backing Resist with anisole Speed, rpm Time, sec Temperature, ˙C Time, min HDMS - 6000 60 170 / - 3 / - PMMA A6 1:1 3000 60 170 15 ZEP520A - 6000 60 190 3 Espaser 300Z - 6000 30 90 3

3.1.1.3 Electron beam lithography

Prepared substrate can be proceeded for EBL. In general EBL is a method of printing nano-sized structures that is derived from electronic microscopy. This form of lithography is maskless that means that literally it is a direct writing line by line of desired design on the sample. The design must be programmed in advance and divided to writing fields and subfields, primitive blocks for writing. Consequently, one of the important limitations of the method is throughput, in particular the timing of the process. And hence one must always keep in mind possible movements of the holder/platform or electron beam instability through time and try to make necessary corrections of the design or writing process. For instance, writing of nano-sized MMs can take around 10-15 minutes, when writing of continuous WGs takes several hours. Such long writing time may cause significant displacements in the design along the stitches’ lines between adjacent writing fields. For the realization of our samples Raith 150 and Nanobeam NB4 machines were used. First samples were fabricated by means of Raith 150 EBL machine. The acceleration voltage for Raith 42 3.1 Fabrication of metamaterials

150 varies from 200 V to 30 kV, useable current – from 4 pA to 10 nA. Writing time of our first samples with only MM patterns was around 20 – 30 min, which was acceptable, though writing time for the samples with long WGs (that will be discussed later) was expected to be overly long. That is why for WGs writing or the later samples Nanobeam NB4 machine was used. Nanobeam NB4 allows to work at high acceleration voltage from 30 kV to 100 kV and useable current from 2 nA to 50 nA. Good electronic beam and mechanical stabilization systems keep the position drifting below 150 nm per hour and stitching error below 25 nm. Another important advantage of Nanobeam lithography (NBL) is its capability of automatic localization of alignment marks, which is extremely useful tool when fabricating multi-layered structures such as WGs covered by MM. Parameters that were used for the realization of the samples depending on the chosen resist are reported in the table below. PMMA A6 Resist being used ZEP 520 EBL Raith 150 NBL NB4 Voltage, kV 20 80 80 Current 9.09 pA 2.2 nA 2.2 nA Dose factor 1.8-2.4 8-9.3 4

3.1.1.4 Development

Development is one of the crucial steps of the process since at this point every detail is important and plays significant role on the result: developer, dilution ratio, time, and even the style of development. For PMMA resist we use solution of methyl isobutyl ketone (MIBK) and IPA, volume proportion is MIBK:IPA=1:3. Development time is 50 sec. To stop development the sample is placed in IPA for 25 sec. ZEP resist requires additional development steps: first, the sample is placed into ZED for 30 sec; then solution MIBK : IPA (1:3) for 30 sec; and finally IPA for 30 sec. If the sample is based on a glass substrate, before the development process the Espaser 300Z needs to be removed. It can be done by simple placing the sample into the water for several seconds (usually from 7 to 10 sec). At the end of this step we gain a substrate covered by programmed resist mask.

3.1.1.5 M etal deposition

The metal deposition shall be done preferably right after the development process. The deposition is effectuated using e-beam evaporation system Plassys MEB550S. Before the gold (Au), thin chromium (Cr) film of 2 nm is deposited. Cr serves for better adhesion between silicon or silica surface and Au particles. After Cr layer, the main layer of gold is deposited. The usual thickness of 43 CH APTER 3. M ETAM ATERIALS FABRICATION … gold layer is 50 nm. Since the thickness of the layers is negligible the deposition speed must be low in order to keep the process stable and adiabatic. In our case the speed is kept around 0.1 nm/s by automatic controlling of the beam current.

3.1.1.6 Lift-off process

The final step of the simple MM fabrication process on top of glass or SOI substrate is lift-off process. The aim of this step is to remove completely the resist mask without damaging the nanostructures. Basically the step consists in placing the sample in a solvent for several hours. The sample should be placed in a position close to the vertical one and be faced down. This is important to avoid sticking of waste gold film to the sample/substrate. PMMA resist can be well washed out it 24 hour acetone bath. ZEP resist usually demands additional bath in methyl ethyl ketone compound (butanone). For nanoparticles arranged in very dense matrixes (with distances between particles less than 40-50 nm) ultra sound (US) bath for 5-10 sec can be required.

3.1.2 Fabrication flow sheet for guided wave configuration metamaterials

Since our study is dedicated not only to the metallic MMs in free space configuration (FSC) but also to the metallic MMs on top of dielectric WGs in GWC, in this section we present the fabrication process of such WGs. The fabrication of MM on top of silicon WGs basically repeats the same steps that were discussed above but with an additional level of lithography process and the process of WG etching: • substrate cleaning, • coating the substrate with appropriate resist, • first lithography level destined to form mask for MM nanostructures as well as auxiliary marks for the second lithography level, • resist development, • metal deposition, • lift-off process, • coating the sample with MM nanostructures on top by appropriate resist for the second lithography level, • second lithography level destined to form mask for WG pattern, • resist development, • etching process,

44 3.1 Fabrication of metamaterials

• cleaning of the finished sample. As one can see, first, MM layer is deposited on substrate and only after that WGs are added. This is done to simplify the matching procedure of two lithographical levels. For this reason the initial mask of MM design is characterized by presence of additional marks which are used for the further matching of MM design with WGs or localization of MM structures during scanning electron microscope (SEM) imaging (Fig. 17). The bottom left mark is considered as zero position point during the fabrication process of all samples. All other aspects of fabrication metallic nanostructures remain the same as it was discussed in the previous section 3.1.1.

Fig. 17. Scheme of additional marks placement: (1) auxiliary marks are used for matching the first and the second level of lithography; (2) the middle marks; (3) auxiliary marks that are helpful for localization metamaterial structures during SEM imaging.

3.1.2.1 The second level of electron beam lithography

When the first level is accomplished, namely after gold deposition and lift-off process, new layer of resist is coated right over the gold MM particles. After that the sample is exposed second time by NBL machine. Attention must be paid to the matching of the precedent MM layer and the new- programmed pattern with WGs. At this point it is important to match marks of both layers that can be done with the help of the NB machine function for automatic marks localization. The NB machine detects the metallic auxiliary marks (marks 1 on the Fig. 17) and based on their position the WGs are being written. This step can be time consuming due to considerable area that needs to be exposed. To minimize this area only thin lines nearby the edges of WGs are being exposed. Thus the WGs are actually formed by 6 µm width hollows (see Fig. 18a). The principal scheme of a WG is shown on the Fig. 18b. MM area is placed on top of the main WG with width of 10 µm. The approximate length of the whole WG is around 5 mm. A tapered region is conceived as spatial filter element of the WG: it is placed before the MM region

45 CH APTER 3. M ETAM ATERIALS FABRICATION … and serves for selecting fundamental mode of the electromagnetic wave. The light is brought into the main WG through WG with width of 3 µm and length about 2 mm. a b

Fig. 18. The schematic of a waveguide. (a) Front section of the waveguide: the waveguide is formed by two 6 µm width hollows; the depth of hollows is around 200 nm. (b) Top view of the waveguide; the yellow rectangle represents the metamaterial area; the red arrow represents the incoming light, the green arrow represents the outgoing light.

3.1.2.2 Etching of waveguide edges

The etching of WG edges is effectuated on ICP RIE machine (inductively coupled plasma reactor for reactive ion etching) from STS. It allows to achieve high etch rates which is important for obtaining vertical walls of WGs. The duration of the process, which is used for etching 200 nm height WGs at the room temperature, is only 18 sec. Other system parameters are following: base pressure 5 mTorr, power 2000 W, chamber gases are C4F8 (200 sccm) and SF6 (200 sccm). This mixture of gases is well functioning for Si etching; the working gas is SF6 and C4F8 is the passivation gas. The presence in the chamber both etch and passivation gases together is responsible for very smooth sidewall surfaces [Park 2010], besides on such short time period there is no reason to divide the process into smaller ones.

3.1.2.3 Final cleaning

The final cleaning of the sample can be done by placing the sample in acetone bath for several hours. If chosen resist was ZEP additional wash in butanone may be required. To completely remove resist layer oxygen plasma processing of about 5 min may be applied if needed. Though whenever possible it is better to avoid plasma cleaning since it affects the top surface of gold nanoparticles [Gun 2009] and may decrease their height. We have found by means of a contact profilometer Dektak 8 that the difference in height of metallic particles before and after 10 min plasma cleaning can make around 12 nm.

46 3.2 SEM images of fabricated samples

3.2 SEM images of fabricated samples

In this section we present some scanning electron microscope (SEM) images to give an idea of the samples that were fabricated for the further study. Two sets of samples were fabricated in framework of this study: one set of gold MM patterns on top of glass and silicon substrates for FSC measurements and one set of gold MM patterns on top of silicon WGs for GWC measurements4. Let us name the sets by the letters A and B, correspondingly. The set A of gold MM consists from two samples: one on top of glass substrate that was fabricated in the IEF clean rooms by myself and the second one on top of silicon substrate was fabricated in the LPN (L aboratoi re d e Phot on iqu e et d e N an ost ructures) clean rooms by our collaborators. The first sample was dedicated for normal incidence excitation (IEF) and the second one - for oblique incidence excitation (LPN). These samples described in more details in the Chapter 4, section 4.2. Examples of SEM images for the set A are presented on the Fig. 19a and Fig. 19b. On the Fig. 19a one can observe some contamination between cut wires rows. It may be caused either by the auxiliary layer of Espaser 300Z that is used for SEM imaging of the structures on top of glass substrate, either by some resist remains. In both cases the contamination is optically transparent at the working range of frequencies. a b

Fig. 19. (a) Sample fabricated in the IEF clean rooms: gold cut wires on top of glass substrate. Dimensions of gold cut wires are 550×70×50 nm, the transversal distance between particles is 90 nm.(b) Sample fabricated in the LPN clean rooms: gold SRRs with continuous wires in between. The length of a SRR is 355 nm.

4 Here we note, that however in this manuscript we present only two sets of samples, to obtain fine and appropriate samples 9 sets were fabricated for different tests. 47 CH APTER 3. M ETAM ATERIALS FABRICATION …

The set B with silicon WGs and gold MM particles on top of SOI substrates was fabricated in the IEF clean rooms by myself. The set includes three samples. Each sample has several tens of WGs with different MM patterns or without it. The samples are described in the Chapter 5. Examples of SEM images for the set B are presented on the Fig. 20. a b

Fig. 20. Example of a sample from the set B fabricated in the IEF clean rooms. (a) The top view of waveguide, metamaterial area can be seen. (b) Larger SEM image of the same metamaterial area consistent from rotated gold cut wires, dimensions of cut wires are 200×50×50 nm, transversal separation between cut wires is 100 nm.

3.3 Sample characterization

In order to verify simulation results on real structures basic linear spectral measurements were performed. The set A was examined using standard Fourier transform infrared (FTIR) spectroscopy and angle resolved FTIR spectroscopy setups. Besides of linear spectra set A was characterized using ellipsometry measurements. The set B was characterized by optical bench setup. All setups are shortly described below.

3.3.1 FTIR spectrometry for free space configuration

The Fig. 21 represents scheme of a standard FTIR setup. The setup is based on a Michelson’s interferometer with a coherent light source and a detection system. It allows to collect simultaneously spectral data in a wide spectral range. Characterization of the samples was performed in the IEF clean rooms by myself by means of the machine FTIR Varian 610. This machine allows to measure transmission spectra at normal incidence and does not support angle resolved measurements. The Fig. 22 represents scheme of angle resolved FTIR setup. The constructive difference from the basic FTIR setup described previously is an additional goniometric platform that allows to

48 3.3 Sample characterization change the incidence angle from 0° to 70°. The angle resolution is 0.3°. A sample can be placed in the middle of the goniometric platform. The platform can serve for measurements of transmission or reflection spectra depending on the position of the detector. The setup allows to control the polarization of incidence wave due to the polarizer (P on the Fig. 22), which opens the door to full spectral analysis of the sample. Spectral measurements for the second sample from the set A were performed by our collaborators from LPN.

Fig. 21. Standard FTIR setup based on M ichelson interferometer. M oving mirror allows controlling the path difference. Signal is collected by detector and then computer processing allows to obtain the resulting transmission spectra from the raw data.

Fig. 22. Angle resolved FTIR setup. The sample can be placed in the middle of goniometric platform that allows to vary the light incidence angle. D – diaphragm; P – polarizer; h – hole. Angle range 0°–70°, resolution ±0.3°, 0.5 cm -1.

49 CH APTER 3. M ETAM ATERIALS FABRICATION …

3.3.2 Ellipsometry measurements

Ellipsometry is highly sensitive and accurate method to examine different surfaces as well as interfaces. The main idea is based on studying effects of polarization changes after interaction of the light with material surface (see Fig. 23). The changes are characterized by amplitude ratio of incident and reflected waves and the phase difference. Ellipsometry can give information about layers which thickness is less than wavelength of the probing wave. Since ellipsometry deals not with absolute values but rather with ratio or difference of values, it is very strong and accurate tool for studying material interfaces. As a result it gives effective parameters of the media under studying.

Fig. 23. Scheme of set up for ellipsometry measurements. Linearly polarized light is reflected from the sample surface, changes in polarization are detected to determine the sample response.

3.3.3 Guided wave configuration

For transmission measurements in GWC the optical bench setup represented on the Fig. 24 was used. The characterization of the set B was performed by myself in the IEF. One of three available tunable semiconductor lasers served as a laser light source. The usage of three lasers gives the possibility to work in a broad spectral range: the wavelength can be varied from 1.25 µm to 1.64 µm (240 – 180 THz). An optical fiber brings a linearly polarized light to the entrance face of a WG. The coupling between the fiber and the WG’s end occurs due to integrated fiber lens. The WG can be chosen by adjusting the position of a moveable platform that holds the sample. The position of the platform can be controlled by a microscope placed over the sample. The output light is collected by a 32× objective with 0.6 numerical aperture and is measured either with a large area IR detector or coupled by means of a second 20× objective into a 100 µm diameter multimode fiber connected to a broadband optical component tester CT400 from Yenista Optics. This assembling allows to perform a fast and accurate acquisition of the hybrid MMs WG transmission spectra with a resolution about 1 pm.

50 3.3 Sample characterization

Fig. 24. Scheme of optical bench setup. Optical fiber with integrated lens focuses the light into the entrance of the waveguide. After the waveguide transmitted light is collected by the detector. The signal processing is done by means of a computer.

3.4 Conclusion

All basic experimental techniques are described in this chapter. The fabrication of MMs is a fine process that demands high accuracy and attentiveness. We presented the procedure of the fabrication MM structures such as gold nano cut wires with typical dimensions of 200×50×50 nm and separation distance between elements 50 nm. Two dielectric substrates are used: glass substrate and SOI substrate. The presented techniques allow to gain MM structures for FSC and for GWC experiments. Another part of this chapter shortly shows main setups for linear optical characterization of MM for both FSC and GWC study. Main parameters of the setups are listed. The working spectral range of each setup is centered around 1.5 µm. The next chapters of the manuscript are discussing the actual results obtained in the framework of the thesis project.

References

[Gun 2009] J. Gun, D. Rizkov, O. Lev, M.H. Abouzar, A. Poghossian, M.J. Schöning, “Oxygen plasma-treated gold nanoparticle-based field-effect devices as transducer structures for bio-chemical sensing,” Microchimica Acta 164, 395 (2009)

[Lal 2007] S. Lal, S. Link, N.J. Halas, “Nano optics from sensing to waveguiding,” Nat. Photon. 1, 641 (2007)

[Park 2010] S.-Y. Park, S.G. Di Giacomo, R. Anisha, P.R. Berger, P.E. Thompson, I. Adesida,

“Fabrication of nanowires with high aspect rations utilized by dry etching with SF6:C4F8 and self- limiting thermal oxidation on Si substrate,” J. Vac. Sci. Technol. B 28, 763 (2010)

Chapter 4 Free space configuration single metafilm effective properties in optical domain

As discussed in the previous chapters, MM macroscopic behavior can in many situations be described using EMA. The description of MM properties in terms of effective permittivity ε and permeability µ revealed to be a very useful approach, namely what concerns applications of MMs in TO. Numerous demonstrations of TO inspired devices in the microwave domain are based on the usage of multi-layered MM structures. The transposition of these concepts in the optical domain faced, however, with serious issues of technological, material and also of fundamental character. Fundamental type issues are related to the validity of EMA when considering metal MM structures in the optical domain. In particular this concerns the description of single metafilm behavior as an EM with unambiguously assigned equivalent thickness and RI. Though validity of such an approach is strongly contested [Scher 2009, Holloway 2009, Simovski 2011], other results let presume that this may be still possible [Zhang 2005, Kanté 2009, Rockstuhl 2008, Menzel 2009]. The objective of the study presented in this chapter is to address this point through modeling and experimental investigation of single metafilm effective properties in the optical domain in FSC. The study may provide guidelines necessary for the engineering of single MM layer effective properties in view of real world photonic applications.

4.1 Single metafilm modeling

All numerical modeling results presented in the chapter were performed by means of HFSS (high frequency structural simulator) commercial program [HFSS]. This program from Ansys is based on finite element solver. Usually it is used for antenna design or other complex circuits operating at radio frequencies. For our purposes we supplement the standard HFSS material database with frequency-dependent complex dielectric permittivity of materials used for numerical modeling . The dielectric permittivities of Au, Si or SiO2 materials are taken from Palik’s Handbook [Palik 1991].

55 CH APTER 4. FREE SPACE CON FIGURATION …

4.1.1 Single metafilm effective medium criteria

First of all it is important to emphasize that all presented studies are performed in a linear approximation. The effective parameters of the MM structure do not depend on the intensity of the electromagnetic field. The parameters of the structure are also considered as time independent. The characteristic size of MM elements is typically much smaller than the working wavelength. To prove that a metafilm can be indeed described as a homogeneous layer, it is necessary to verify that its behavior meets the following conditions: • Effective magnetic permeability µ ≈ 1 except the resonance region (condition of non- magnetic behavior ). • Linearity of the dielectric permittivity variation with MM surface filling factor (FF) ρ : the more material is placed on the unit surface the stronger electromagnetic answer it must give (validity of MG approximation). • Linearity of the optical length variation with respect to the deposited metal thickness. In other words h ∝ h*. If it is not true then the effective parameters of EM cannot be determined. This condition can be viewed as an analog to the request of the effective parameters convergence with increasing number of MM layers. • Invariance of the MM layer dielectric permittivity with respect to the incidence angle: as soon as the effective parameters are determined they must remain unchanged for all possible wave processes, including oblique incidence waves. Further we verify the validity of the criteria for an array of gold cut wires (CWs) on top of a silicon substrate by computer simulations as well as by experimental methods.

4.1.2 Single metafilm layer effective medium behavior

For our study we consider the example of a single layer two-dimensional array of CWs on a silicon substrate. It represents probably the most elementary type of MMs that can be used for building more complex geometries of MMs. For instance CWs with continuously changing dimensions can be used as building elements for TO devices. Another great advantage of the CWs structure is its is essentially non-magnetic behavior with µ ≈ 1 due to the absence of notable coupling between electric and magnetic resonances in a zero-order approximation. The schematic of 2D array of CWs considered in the study is displayed in Fig. 25a. As mentioned, dielectric permittivity of Au used for numerical modeling is that given by [Palik 1991]. For the sake of simplicity we consider a substrate with a RI of 3.45 that does not vary with the wavelength. This index value is that of the silicon at 1.5 µm (≈200 THz).

56 4.1 Single metafilm modeling

a b

Fig. 25. (a) 2D array of gold cut wires on top of a silicon substrate in free space configuration; s- polarized incidence wave; the electric field is along long axis of gold cut wires; the incidence angle of the light is α. (b) Equivalent model: the silicon substrate covered by effective medium layer.

The EM model of a CWs structure is schematically shown in Fig. 25b. To calculate metafilm effective permittivity ε eff and permeability µeff from the complex reflection and transmission coefficients r and t , respectively, we use the NRW retrieval method extended to the case of oblique incidence geometry [Menzel 2008, Lupu 2011]. We note the first media with index 1 and the last media with index l . In the simplest case we are discussing, the first media is air, and the last media is silicon semi-infinite substrate. We consider a plane wave oblique incidence of angle α1 with electric field perpendicular to the incidence plane. The RI of a film with thickness h can be written as following5:

≈ ’2 ∆ β ÷ 2 2 (4.1) neff = ∆ ÷ + n1 sin α1. « k0h ◊

Here β is defined through its cosine

p p (1− r 2 + t'2 ) cos β = 1 l (4.2) p1 (1− r) + pl (1+ r) t' and is a multiple branches function: 2πm β = ±arccos(cosβ ) + , (4.3) k0 h where m is integer. The sign of β must be chosen in such a way that imaginary part of RI rests positive Im(neff ) > 0 (requirement for passive media).

5 From the beginning we add to the RI n an index eff that indicates that after simulations exactly this formula for RI will determine the effective RI of the EM approximating MM film. 57 CH APTER 4. FREE SPACE CON FIGURATION …

The parameter p :

ε 1 p = cosα = cosα (4.4) µ z is the medium characteristic admittance, i.e. the inverse of the medium impedance z for polarization perpendicular to the incidence plane.

The impedance of the layer zeff is given by the expression: 1 1 1+ r cos β − 1 pl p p t' z 2 = 1 l , (4.5) eff cos2 α 1− r eff p cos β − p p l 1 l t' where

2 ≈ n ’ ∆ 1 ÷ 2 (4.6) cosα eff = 1− ∆ ÷ sin α1. « neff ◊ Since we consider passive MMs, the EM layer must also stay passive. For this the impedance must satisfy the following condition: Re(zeff ) > 0 . As it was discussed in the Chapter 2, the effective impedance and the effective RI can be used for calculation of the effective constitutive parameters ε eff and µeff :

neff ε eff = , (4.7) zeff

µeff = neff ⋅ zeff . (4.8)

It is only the matter of choice which pair of parameters to use ε eff and µeff or neff and zeff . We remind that, all formulas above are valid for the case when the electric field is perpendicular to the incidence plane (s-polarized wave). The case, when the electric field is in the incidence plane (p-polarized wave), will be discussed in the subsection 4.1.2.4. For the normal incidence, when α1 = 0, the formulas (4.1 – 4.8) are also correct. Now we may proceed to the verification of the validity of EM criteria.

4.1.2.1 Determination of effective layer thickness

The first point of our analysis concerns the determination of the thickness to be assigned to the metafilm. As mentioned, outside the resonance region, the effective permeability obtained by the retrieval procedure must be consistent with the condition µ ≈ 1 [Smith 2012]. In the MG approximation the MM layer represents a mixture of CWs and air. Here we make the hypothesis that its thickness h* equals that of the deposited metal. 58 4.1 Single metafilm modeling

a b

Fig. 26. 2D array of gold cut wires on top of silicon substrate. (a) Sketch of normal incidence excitation geometry with the electric field vector along the long axis of gold cut wires. (b) M ain design parameters of the metamaterial pattern: L, W and h (not depicted) are the length, the width and the height of

cut wires; d and d* are transverse and longitudinal separations between particles, simultaneously; A1 and A2 are the two periods.

To verify this assertion a 2D array of gold CWs was modeled under normal incidence excitation (Fig. 26) by means of HFSS. The length of cut wires is L = 200 nm, the width of cut wires

6 is W = 50 nm. The height h is varying from 20 nm to 100 nm . The periods in longitudinal A1 and transverse A2 directions are equal to 300 nm. The surface FF ρ that represents the density of 2D array, in this case, is equal to ρ = LW / A1 A2 = 0.11. This corresponds to the case of weak interaction between adjacent CWs (only 11% of the surface is covered by CWs). Indeed, the transverse separation between gold nanorods is 250 nm. It is much larger than the typical particle size in this direction. The longitudinal separation is 100 nm. The longitudinal separation has a much less influence on the structure behavior. The resulting effective permeability is shown on the Fig. 27. It can be seen that outside of the resonance region the condition µ ≈ 1 is well satisfied in the whole range of metal thickness variation, thus validating our hypothesis about the thickness of the effective layer. The region where µ differs from unity corresponds to the well-known magnetic antiresonance that is always concomitant with the electrical resonance [Koschny 2003]. Though we have deal with antiresonances that can point on the violation of the causality principle, note, that the deviation of Re(µ) from unit is minor and occurs only in small region close to the resonance, especially when h ≤ 60 nm. The shift toward higher frequencies of the resonance with metal thickness is likely to be related to the decrease of the asymmetric bound supermode effective index for higher metal

6 Further we denote the dimensions of CWs using the form: L×W× h . 59 CH APTER 4. FREE SPACE CON FIGURATION … thickness [Sarid 1981, Berini 2009]. The gold material dispersion also brings contribution to supermode effective index variation

Fig. 27. Real and imaginary parts of magnetic permeability µ obtained for the array of cut wires 200×50×h nm where h varies from 20 nm to 100 nm. The asymptotes of the real and imaginary parts are Re(µ)=1 and Im(µ)=0, correspondingly.

Thus the obtained results confirm the validity of the first criterion. It means that the thickness of the effective layer is equal to the height of the metal particles. This assertion is maintained for the rest of the modeling studies.

4.1.2.2 M axwell-Garnett approximation validity

The second criterion implies linearity of the dielectric permittivity variation with MM surface FF. Increasing the FF would obviously result in an adjacent CWs interaction enhancement that could alter the validity of MG approximation. To verify the MG approximation validity we vary the FF. The variation of the FF can be achieved by changing the main design parameters of the structure in several ways. The parameters of the structure under the study are: the length of CWs L , the width W , the transversal and longitudinal periods A1 , A2 and both separation distances d * , d . These parameters can be changed separately or simultaneously in some combinations. In the set of the 6 variable design parameters only 4 of them are independent. For instance, we can chose L , W ,

A1 and A2 as independent parameters. By consequence, we can change independently each of these 4 parameters while keeping the rest fixed. On the Fig. 28 one can find different schemes for parameters variation considered in this study which all provide essentially similar results. The results obtained at normal incidence for 200×50×10 nm CWs with electric field orientation along the longitudinal axis are shown on the Fig. 29. In this example the FF variation is obtained by changing the transverse separation between adjacent CWs. This corresponds to the

60 4.1 Single metafilm modeling configurations represented on the Fig. 28g. For better viewing the displayed results represent the effective permittivity normalized by the FF: ε norm = ε eff / ρ . a

b c d e

f g

Fig. 28. Schemes of filling factor variation. (a) Basic structure of cut wire array. (b), (c), (d), and (e)

Respectively, variation of each parameter L, W, A1 and A2 independently when other three parameters are fixed. The varying parameter is marked by red color. (f) Variation of L when longitudinal separation d * is fixed. (g) Variation of W when transverse separation d is fixed. The varying parameter is marked by red color, the fixed parameter – by green color.

As it can be seen, the normalized permittivity is little dependent on the FF. For both real and imaginary part the maximal variation of permittivity is less than 16%, while the FF variation attains 450%. The shift of the resonance toward higher frequencies with the increase of the FF is in agreement with similar MM structures reported by [Weber 2011]. This shift is due to the enhancement of coupling between CWs by dipolar interaction. As it will be shown in the following, the variation of the normalized permittivity would be even lower without the frequency shift. The relative independence of the normalized permittivity is conserved when the variation of the FF is performed in a different manner. This point was verified for FF variation obtained by 61 CH APTER 4. FREE SPACE CON FIGURATION … changing the longitudinal separation distance or the width of the CWs, the rest of parameters being fixed. Similar behavior holds also for higher metafilm thickness. Note also the high values of normalized permittivity exceeding 1000. Even for a FF of 10% this means that the effective index is around 10 at the resonance.

Fig. 29. Real and imaginary parts of normalized effective permittivity. Dimensions of cut wires are 200×50×10 nm. The curves presented for different filling factors from 7 % to 33 %. The filling factor varies by changing the transversal period of the structure. M etal thickness is 10 nm.

4.1.2.3 Single metafilm optical length study

Genuine MM effective parameters have to be independent of the MM thickness. For the microwave domain this condition is readily verified by varying the number of MM layers. A convergence of the retrieved MM effective parameters is expected with the increase of the number of layers. In the case of a single metafilm layer this means that effective parameters have to be independent on the metal thickness. The equivalent condition is to have a linear variation of the optical length with h . To verify this condition we consider several patterns with gold CWs of dimensions 200×50× h nm on top of silicon substrate. The metal thickness h varies from 10 nm to 100 nm. The longitudinal lattice period is set to A1 =300 nm. We consider normal incidence wave with the vector of electric field oriented along longitudinal axis of CWs as before. The RI used to calculate the optical length corresponds to the maximal value of the effective index at the resonance frequency. As for the permittivity, its value is also normalized by the FF. The simulations were performed for several FFs. The variation of the FF is achieved through the variation of the transverse separation distance between CWs and corresponds to the configuration represented in the Fig. 28g. The variation of the optical length (normalized by the FF) as a function of the thickness h for different FFs is shown in the Fig. 30a. The displayed results show that the

62 4.1 Single metafilm modeling variation of the optical length with metal thickness is approximately linear for thin metal particles. Strong deviations from linearity occur for metal thickness above 25 nm, especially for higher FFs. a b

Fig. 30. (a) Optical length for different CWs densities, normalized by FF. (b) Real and imaginary parts of effective permittivity for different metal thickness; the permittivity is normalized on the corresponding surface filling factor.

To understand the deviation from linear dependence we compare the spectral characteristics of the normalized permittivity for different CWs thickness. The results corresponding to 10, 25 and 50 nm CWs thickness whit a fix FF ρ = 11 % are represented in the Fig. 30b. The shift of the resonance frequency and disparity of effective permittivity variation are much more important as compared to the case of FF variation at fixed metal thickness shown in the Fig. 29. To avoid the frequency contribution we propose to fix the resonance frequency when changing the metal thickness. Tuning the design parameters of gold CWs can do this. First, the length of CWs is to be adjusted to maintain the given resonance frequency. Let the new length of CW be L' . Then the new width W ' of CWs can be found based on the following rule:

W '= ρA1 A2 / L' . Obviously, parameters A1 and A2 are not changing for fixed ρ . In our study the resonance frequency is fixed to 148 THz. This is the resonance frequency of the pattern with CWs 200×50×25 nm and periods A1 = A2 =300 nm. For instance, for the unit cell of 300×300 nm2 to obtain the FF ρ = 11 % the design parameters of gold CW must be following:

138×72×10 nm, or 245×41×50 nm, or 265×38×80 nm, et c. The optical length determined for the case of a resonance frequency fixed to 148 THz using CWs tuning procedure is shown on the Fig. 31a. The dependence is much more linear in this case, especially when the FF is low. The linear character of the dependence is maintained up to a metal thickness of 60 nm even for high enough FF ρ = 22 %. This result further proves the validity of the MG EM approximation applied to a single metafilm layer.

63 CH APTER 4. FREE SPACE CON FIGURATION …

The normalized permittivities for CWs having a resonance at 148 THz are represented on the Fig. 31a. The variation of the normalized permittivity is less than 12 % in this case, while it attains 248 % for the case corresponding to the Fig. 30b. This result deserves a special attention. It means that for our particular configuration when the surface of the resonant element and the resonance frequency are fixed, it is the resonance frequency that determines for the essential the effective permittivity. As soon as the resonance frequency is fixed the properties of the MM layer do not vary significantly with its thickness. On this point the obtained result agrees with the assertion of [Simovski 2011] discussed in the second chapter of this thesis. At the same time the obtained results are also confirming our statement that EM thickness can be determined in a unique way and is not arbitrary. This apparent contradiction stems from the fact that when considering a MG approximation, the host contribution to the dielectric function should also to be taken into account. a b

Fig. 31. (a) Optical length for different CWs densities, normalized by FF, for the resonance frequency fixed to 148 THz. (b) Real and imaginary parts of normalized permittivity for different metal thickness but fixed resonance frequency 148 THz.

As already mentioned, the genuine EM should not depend of the wave excitation process: oblique incidence, evanescent waves, etc [Simovski 2011]. This last point is addressed in the next section

4.1.2.4 M etafilm oblique incidence behavior

The last condition demands independency of the effective parameters functions on incidence angle of electromagnetic waves. We consider only the cases when the projection of the electric field on the substrate surface is parallel to the long axis of CWs. Thus the resonance frequency is kept around 150 THz. The sketch of s-polarized light configuration when electric field is perpendicular to the incidence plane is shown in Fig. 32a. P-polarization configuration when the light electric field lies in the incidence plane is shown in Fig. 32b.

64 4.1 Single metafilm modeling

The case of s-polarized light was considered in the beginning of the section 4.1.2. Here we remind the formula (4.5): 1 1 1+ r cos β − 1 pl p p t' z 2 = 1 l . (4.9a) eff ,s cos2 α 1− r eff p cos β − p p l 1 l t' a b

Fig. 32. Sketch of (a) s-polarization oblique incidence configuration and (b) p-polarization oblique incidence configuration. The projection of the electric vector on the sample surface is parallel to the long axis of cut wires.

It is possible to make extension for the case when the incidence light is p-polarized. If we consider a homogeneous EM layer (see Fig. 33), the formula (4.5) for the impedance of metafilm layer takes the following form: 1− r ql cos β − q1ql z 2 = cos2 α t' , eff , p eff 1 1 1+ r (4.9b) cos β − ql q1ql t' The indices 1 and l , as before, designate the first and last semi-infinite media, respectively. Parameter q is given by the expression similar to the formula (4.4):

µ q = cosα = z cosα. (4.10) ε All other variables are determined analogues to the section 4.1.2: it is sufficient to replace p by q .

65 CH APTER 4. FREE SPACE CON FIGURATION …

Fig. 33. Propagation of electromagnetic wave through a homogeneous layer. S-polarization of the light when the electric field is perpendicular to the incidence plane, p-polarization of the light when the electric field is lying in the plane of incidence.

The first point of our study addresses the case of s-polarized incident wave. For s-polarization the electric field orientation along the CWs longitudinal axis does not vary with the incidence angle. We consider gold CWs with dimensions 200×50×25 nm on top of silicon semi-infinite substrate which surface FF ρ = 11 %. The reflection and transmission spectra for 0°, 45° and 85° of incidence angle are shown in the Fig. 34a. It can be observed that in agreement with Fresnel reflection laws, the reflectivity increases with the incidence angle [Born 1999]. At the same time an apparent decrease of the resonance variation for high incidence angles is observed in transmission and reflection spectra. The metafilm effective parameters are found using NRW retrieval procedure. The effective permeability behavior shown in the Fig. 34b is same as for the previously considered examples corresponding to normal incidence and does not change with the incidence angle. As it can be seen from the Fig. 34c, the effective permittivity also does not dependent of the incidence angle. It is important to emphasize the excellent agreement with the EM model that holds up to very high incidence angles (85° and higher). The effective dielectric permittivity is fitting very well a Lorentz dispersion law. It should be noted also that outside the resonance region the effective permittivity and respectively the effective index are moderately high for low FFs. As it can be seen from Fig. 34d, at low frequency the effective index neff ≈ 3.7 is comparable to that of silicon. By consequence, for high incidence angles considered in our example, the refraction angle inside the metafilm should substantially deviate from the normal axis. It follows thus that the independence of the metafilm effective parameters with the incident angle cannot be attributed to the near normal axis propagation as it was suggested by [Menzel 2009]. Here SRRs single metafilm was investigated for different incident angles in the THz spectral domain.

66 4.1 Single metafilm modeling

a b

c d

Fig. 34. Cut wires with dimensions 200×50×25 nm on top of silicon. Filling factor ρ = 1 1 % . (a) Reflection (R) and transmission (T) spectra for s-polarized light for different incidence angles. (b) Retrieved effective permeability. (c) Retrieved effective permittivity. (d) Retrieved effective index.

In contrast, at the resonance frequency of 148 THz the effective index reaches a very high value neff ≈ 8.8 that otherwise is not attainable in nature existing materials. The presented results show that for the particular investigated geometry, when the electric field is oriented along the main CWs symmetry axis, the metafilm behavior is indeed analogous to that of a homogeneous layer. The effective layer thickness is that of deposited metal. The next point of our study addresses the case of p-polarized incident wave. In contrast to the previous case of s-polarization, now the electric field orientation along the CWs longitudinal axis vary with the incidence angle. Furthermore, under oblique incidence there is an additional component of polarization that is oriented along z -axis of the CWs. By consequence for p- polarization anisotropy of effective parameters is expected. The reflection and transmission spectra obtained by HFSS modeling for incidence angles up to 60° are displayed in the Fig. 35a. As for the case of s-polarization, a maximum in reflection is observed at the resonance frequency. Far from the resonance the reflection is significantly lower and tends to that of a silicon substrate. Since the incidence angles are below the Brewster angle of the silicon substrate α < α Br ≈ 73.8°, the reflectivity decreases with the incidence angle [Born 1999].

67 CH APTER 4. FREE SPACE CON FIGURATION …

For the time being we are not aware of any analytical anisotropy retrieval procedure that would allow an appropriate treatment for this case. It is nevertheless tempting to apply NRW method as it is and analyze the obtained results. The effective permittivity obtained in such a way for different incident angles is shown in the Fig. 35b. Despite presumably using inappropriate retrieval procedure, an astonishing result is obtained. As it can be seen from the Fig. 35b, the effective permittivity for p- polarization also doesn’t dependent of the incidence angle. As evident, the effective permittivity is the same for both s- and p-polarization, meaning that single metafilm layer is having a genuine EM behavior. Of course, the obtained results are requiring a more sound theoretical treatment in order to properly take into account anisotropy effects. The lacks of the used retrieval procedure are manifest when considering magnetic permeability results shown in the Fig. 35c. The magnetic permeability is substantially different from unity, even far from resonance. Such a behavior is a pure artifact caused by CWs anisotropy. a

b c

Fig. 35. (a) Reflection (R) and transmission (T) spectra for p-polarized light for different incidence angles below the Brewster angle for silicon substrate. (b) Retrieved effective permittivity and (c) permeability. Cut wires with dimensions 200×50×25 nm on top of silicon. FF ρ = 1 1 % .

One last point that it would be interesting to verify, is the consistency of the EM model in the vicinity of the Brewster angle. The transmittance and reflectance calculated by HFSS modeling for p-polarization at 60°, 75° and 85° of incidence angle are represented in Fig. 36a. It can be observed 68 4.1 Single metafilm modeling that at 75° of incidence angle, which is near to the Brewster angle for silicon substrate, the reflection outside the resonance region is very low. This is fully consistent with the Brewster angle behavior.

The effective index of a 25 nm equivalent metafilm is relatively close to that of the silicon ( neff ≈ 3.7 at 50 THz). In contrast, for same 75° incidence angle a maximum in reflection is observed at the resonance frequency. This is again consistent with the fact that Re(neff ) ≈ 8.8 near the resonance frequency. Since the imaginary index component is also very high Im(neff ) ≈ 8.4,. even for a very low thickness the wave is strongly attenuated. The metafilm behavior at the resonance frequency is then similar to that of a semi-infinite substrate. In other words, the absorption near the resonance is so high, that the wave doesn’t see much difference whether there is thin film or substrate. This conclusion is further proved when examining metafilm behavior beyond the Brewster angle for 85° incidence angle. This time outside the resonance region the reflection is again similar to that of the silicon substrate. The observed result is in total agreement with the reflection of a thin layer with neff ≈ 3.7 on a silicon substrate. However, in contrast to all previous results a minimum in reflection is observed for the resonance frequency. The obtained result is again consistent with the behavior of a high index and highly absorbent thin film mentioned above. The Brewster angle of a substrate with a refractive neff ≈ 8.8 is around 83.5°. By consequence the reflection is very low near this incidence angle. As in the previous cases, the effective permittivity is identical below, at, or beyond the Brewster angle. All this arguments strongly support our conclusion about the possibility of describing single metafilm behavior as an EM. The effective permittivity of the metafilm can be readily engineered through the control of its resonance frequency and CWs geometrical parameters using MG approach. a b

Fig. 36. (a) Reflection (R) and transmission (T) spectra for p-polarized light for different incident angles in the vicinity of the Brewster’s angle for silicon substrate. (b) Retrieved effective permittivity for CWs with dimensions 200×50×25 nm on top of silicon. FF ρ = 1 1 % .

69 CH APTER 4. FREE SPACE CON FIGURATION …

4.2 Experimental study of single metafilm effective medium behavior

This section is dedicated to the experimental study of single MM layers in FSC. The simulation results obtained in the previous paragraph require a further experimental verification. For this purpose two samples were fabricated (the set A). Details of the fabrication process can be found in the Chapter 3. The first experimental sample with different design parameters 2D CWs arrays on top of a glass substrate was fabricated in the IEF clean rooms. The glass substrate is chosen since it allows characterizing the sample by standard FTIR spectroscopy means. The CWs structures are designed to have three different FF: 5%, 10% and 20%. Since the complex dielectric permittivity of the gold deposited by our evaporation setup, does not necessary coincide with the literature data, we implemented in our design a variation of the design parameters to account for an eventual drift of the resonance frequency. Thus, even if the complex permittivity of the real gold considerably differs from the modeling one, at least one of the designs will fall in the target frequency range: 100 THz – 300 THz. The second sample was fabricated in the LPN clean rooms. The sample represents a 3×3 mm2 array of gold SRRs and continuous wires on top of silicon substrate. The sample was designed for angle-resolved FTIR, which explains the very large dimensions of the array.

4.2.1 Cut wires arrays with different filling factors and resonance frequencies

The gold CWs arrays sample was designed to have three FFs: 5%, 10% 20% and five resonance frequencies: 150, 175, 200, 225 and 250 THz. The design of the structures was performed using HFSS modeling results by considering a gold thickness of h = 50 nm. The schematic of the sample is shown in the Fig. 37a. SEM image of an experimentally realized structure with a FF ρ= 20 % is shown in the Fig. 37b. The fabricated CWs structure is of good quality and suitable to be used for further experimental investigations.

70 4.2 Experimental study of single metafil effective medium behavior

a b

Fig. 37. (a) Schematic of sample design. Each colored area represents a cut wires array, which design

parameters adjusted to fit a given filling factor ρ and target resonance frequency f0. (b) Example of SEM

image for cell 3.3 with ρ = 20 % and f0=200 THz.

The CWs structures nominal and experimental design parameters are summarized in Table 1. The experimental dimensions were determined from SEM images at different locations of an array and represent an average value. The agreement between nominal and experimental parameters is good. For almost all structures the absolute difference between nominal and experimentally found dimensions does not exceed 10 nm. The accuracy is somewhat lower for structures with high FF ρ= 20 %. Here the width of the CWs differs by around 20 nm from the nominal value. This issue seems to be related to an over exposure of the area due to higher density of elements. However, as it was shown in the previous section, the CWs width does not significantly impact the resonance frequency.

71 CH APTER 4. FREE SPACE CON FIGURATION …

ρ/f 250 THz 225 THz 200 THz 175 THz 150 THz 0

a 3.1 3.2 3.3 3.4 3.5 e r t a e P

m Design Exp Design Exp Design Exp Design Exp Design Exp 380 387 450 460 540 556 630 635 700 - 20 L % W 71 83 60 78 50 72 43 63 39 -

A1 800 805 800 795 800 805 800 806 800 -

A2 169 164 169 169 169 166 169 172 169 -

a 2.1 2.2 2.3 2.4 2.5 e r t a e

m Design Exp Design Exp Design Exp Design Exp Design Exp

320 319 380 394 440 444 520 515 600 - 10 L % W 69 76 58 71 50 57 42 48 37 -

A1 800 805 800 810 800 798 800 802 800 -

A2 275 279 275 275 275 274 275 273 275 -

a 1.1 1.2 1.3 1.4 1.5 e r t

a e P

m Design Exp Design Exp Design Exp Design Exp Design Exp L 300 302 340 342 400 410 470 472 530 - 5% W 67 77 59 58 50 51 43 54 38 -

A1 800 800 800 804 800 804 800 811 800 -

A2 500 507 500 495 500 489 500 504 500 -

Table 1. Design and experimental parameters of unit cell and cut wires dimensions. The experimental parameters determined from SEM images. All parameters are given in nanometers. Each block corresponds to an array of the sample. L, W, A1, A2 are main design parameters of the patterns.

The experimental realization of structures designed to operate at f0 = 150 THz resonance frequency also failed for doze exposure issues. This is related to the fact that the whole sample was exposed by NBL machine using the dose factor fixed to 1.1 and was not individually adapted to each CWs structure. Thus for very thin CWs (matrixes 1.5, 2.5 and 3.5) such a dose factor appears to be very large. Since all other steps of the fabrication procedure (such as development and lift-off process) are the same, it can be concluded that dose factor is having the most important role.

72 4.2 Experimental study of single metafil effective medium behavior

a e

b f

c g

d h

Fig. 38. (a-d) FTIR experimental and HFSS modeling transmission spectra for cut wires arrays. Array structures 1.1-1.4, correspondingly. (e-h) SEM images of the related cut wires structures.

73 CH APTER 4. FREE SPACE CON FIGURATION …

The fabricated sample was characterized by FTIR measurements performed in IEF clean rooms. The objective of FTIR characterizations is to experimentally verify the position of the CWs resonance frequencies with respect to the target values provided by modeling results. FTIR experimental and HFSS modeling transmission spectra for CWs arrays with same FF ρ = 5% but different resonance frequencies are presented in the Fig. 38(a-d). The SEM images corresponding to these structures are shown in the Fig. 38 (e-h), respectively. As expected the resonance frequency shifts towards lower frequencies when the length of CWs is increased. As evident from comparison of experimental and modeling results, the experimental resonance frequencies are very near to those given by HFSS modeling. In addition to the CWs resonance, HFSS transmission spectra reveal also the presence of a marked narrow dip occurring at same frequency 270 THz for all structures. This dip is due to the opening of the first diffraction order and is determined by the structure periodicity and the glass substrate index. The diffraction frequency

f d = 270 THz given by HFSS modeling results is close to that given by standard diffraction formula: Ω λd = nsub A1 = 1.45⋅800nm ≈ 1160 nm f d ≈ 258 THz. It can be observed that the presence of diffraction order in the vicinity of CWs resonance frequency alters the FTIR spectral response and can induce an error on the exact determination of the resonance position. For this reason it is necessary to work outside the diffraction region. For subsequent studies only array structures 1.3, 1.4, 2.3, 2.4, 3.3, 3.4 can be interesting, where the resonance frequency is far enough from f d . The position of the resonance frequencies obtained from HFSS simulations and from experimental FTIR measurements for different measured structures is represented in Fig. 39. The agreement between numerical and experimental results, especially for low FF ρ =5 % is good. This fact can serve as indirect evidence that material parameters of gold taken from Palik’s Handbook [Palik 1998] are in a good agreement with the parameters of the gold in our experimental realizations. The difference of the spectral shape that can be observed between FTIR experimental and HFSS modeling results is related to the non-identical conditions of light incidence. HFSS transmission spectra are obtained for a normal incidence plane wave. FTIR measurements are performed by using a microscope objective with incidence beam that is not parallel but converging. By consequence there is no full agreement between experimental and modeling spectral response (see Fig. 38). To obtain a better agreement between experimental and modeling results the sample was transmitted for further studies to In st it ut d es N an oSci en ces d e Pari s (INSP) for ellipsometry characterizations.

74 4.2 Experimental study of single metafil effective medium behavior a b c

Fig. 39. Comparison of the resonance position for each array structure obtained from HFSS simulations and experimental FTIR measurements. (a) ρ = 5 %, (b) ρ = 10 %, (c) ρ = 20 %.

4.2.1.1 Ellipsometry measurements

This technique was chosen as it provides an independent determination of the RI and film thickness and does not require some additional assumptions. The effective permittivity obtained from experimental ellipsometry measurements and that provided by HFSS modeling results is shown in Fig. 40. The ellipsometry measurements were performed in a wide frequency range extending from 120 THz to 400 THz. They confirmed the validity of the EM model. The equivalent thickness of 50nm corresponds to of deposited gold. As it can be seen there is a good qualitative and quantitative agreement between experimental and modeling results. In addition, ellipsometry measurements allowed to experimentally verifying the proportionality of the effective permittivity to the FF that confirmed the validity of the MG approximation for single layer CWs structure.

Fig. 40. Real and imaginary parts of effective permittivity for structure 1.3 with filling factor ρ=5%. Incidence angle 20°.

75 CH APTER 4. FREE SPACE CON FIGURATION …

4.2.2 Split ring resonators with continuous wires metamaterial structures

The validity of the EMA was further verified on a second sample with different from CWs design. The MM structure is composed of gold SRRs and continuous wires placed on top of silicon substrate. Structure design parameters as well as experimentally measured dimensions are provided in Fig. 41a. The size of the unit cell is 600×600 nm2. The particular feature of this sample is its very large area: 3×3 mm2. Such a large size of the structure is motivated by the requirements for the angle resolved FTIR measurements. This structure was fabricated in LPN using electron beam lithography (Fig. 41b). The thickness of the deposited gold is 40 nm. a b

Fig. 41. (a) Structure design, nominal and measured parameters; (b) Structure SEM image.

The sample characterizations were performed using LPN angle resolved FTIR setup represented schematically in Fig. 22. The reflection and transmission spectra for p- and s- polarization at different incident angles are summarized in Fig. 42 and Fig. 43. It can be observed that according to the orientation of the electric field two marked resonances are observed in the spectral region: around 2 µm (150 THz) and 3.5 µm (85 THz). Fields and currents distributions obtained by HFSS modeling for these frequencies are shown in Fig. 44. It follows that the lowest frequency resonance at 85 THz corresponds to the fundamental resonance of SRRs element, while that at 150 THz to the first higher order SRRs resonance. The goal of this study is to provide an experimental evidence for the validity of EMA for the investigated single layer MM structure. FTIR measurements provide data on the intensity of reflection and transmission coefficients but do not contain phase information required to perform the retrieval procedure. One solution would be in this case to consider MM structure as an EM film, approximate its effective permittivity by some dispersion law, calculate film reflectance and transmittance and compare them with the experimental data. The adjustment of the dispersion law parameters can then be performed by a fitting procedure. Though such a solution is in principle possible, it may eventually require fitting an important number of parameters entering the dispersion laws. The convergence of solution search may not be granted.

76 4.2 Experimental study of single metafil effective medium behavior a b

c d

Fig. 42. Reflection spectra for p- and s-polarizations and incident angle variation from 0° (blue lines) to 60° (red lines). An arrow shows the orientation of the electric field. (a,b) p-polarization light incidence. (c,d) s- polarization light incidence. a b

c d

Fig. 43. Transmission spectra for p- and s-polarizations and incident angle variation from 0° (blue lines) to 60° (red lines). An arrow shows the orientation of the electric field. (a,b) p-polarization light incidence. (c,d) s-polarization light incidence.

77 CH APTER 4. FREE SPACE CON FIGURATION … a b

c d

Fig. 44. (a,b) HFSS fields distribution for 85THz and 150THz, respectively. (c,d) HFSS currents distribution for 85 THz and 150 THz, respectively.

To obviate this difficulty we consider a somewhat different approach. It consists to adjust HFSS model to fit the experimental spectral response. Then from HFSS modeling data it is possible to obtain through retrieval procedure the effective MM parameters and inject them into a thin film model for final validity verification. As a first guess we consider a structure with nominal design parameters and [Palik 1998] literature data for gold. The first guess HFSS modeling and FTIR experimental results are displayed in Fig. 45a. It can be observed that the general experimental behavior is well reproduced by the HFSS modeling, except the position of the resonance frequency given by HFSS that is by 15 THz shifted toward lower frequencies. As it will be detailed in the next chapter, this is related to the presence of low index interface in contact with the gold element. In our case it is reasonably to consider the presence of some native silicon oxide on the substrate surface. Such thin layers of silicon oxide can arise on the surface of silicon under conditions of humid air at room temperatures [Morita 1990]. The typical thickness of silicon native oxide is around 5nm. The introduction of 5nm silicon native oxide layer in the HFSS model allows obtaining a very good fit with the experimental data (see Fig. 45b). 78 4.2 Experimental study of single metafil effective medium behavior

a b

Fig. 45. Reflection, transmission and absorption spectra: FTIR (cyan, red, brown colors, respectively), HFSS (blue, magenta, black colors, respectively). (a) HFSS model with metamaterials on bare silicon substrate; (b) HFSS model with additional interlayer of 5nm silicon oxide.

a b

c d

Fig. 46. (a) HFSS modeling results for transmission (green), reflection (blue) and absorption (red) spectra. (b) Real part (blue) and imaginary part (red) of the effective permittivity calculated from HFSS data. (c) Experimental FTIR transmittance (black), reflectance (cyan), absorption (magenta) and plane waves effective film model transmittance (green), reflectance (blue) and absorption (red) using HFSS data retrieved effective permittivity. (d) Experimental FTIR transmission (black), reflection (cyan), absorption (magenta) spectra and plane waves effective film model transmittance (green), reflectance (blue) and absorption (red) after averaging.

79 CH APTER 4. FREE SPACE CON FIGURATION …

It could be noted that HFSS model is using a semi-infinite silicon substrate while FTIR measurements are performed for a sample fabricated on a 500 µm thick silicon substrate. The averaging of Fabry-Perot interference fringes modify absolute transmission and reflection levels and could be taken into account. This required the introduction of an additional intermediary step in the modeling procedure that looks as follows: • Complex transmission and reflection coefficients are calculated with refined HFSS model taking into account 5nm of native oxide (Fig. 46a). • NRW retrieval procedure is used to find the effective dielectric permittivity for 40nm thick equivalent layer (Fig. 46b). • The effective permittivity is introduced in the model of a 40 nm equivalent layer on a finite silicon substrate thickness and its spectral response is calculated by a plane wave model (Fig. 46c). • Averaged spectral response of equivalent film is compared to FTIR experimental results (Fig. 46d). This procedure is used to calculate equivalent layer spectral response for s- and p-polarizations at different incident angles. Experimental FTIR results as well as calculated using equivalent EM model for 5°, 30° and 60° incidence angles are shown in Fig. 47 and Fig. 48 for s- and p- polarizations, respectively. A very good agreement between experimental and modeling can be observed. This proves the validity of a single medium approach for the investigated single layer MM structures composed of SRRs and continuous wires. It is worthful to note that the effective permittivity obtained by retrieval procedure has two contributions: Drude and Lorentz dispersion law:

2 2 Ω p Ω p ε (ω) = ε b + f1 2 2 − f 2 2 . (4.11) ω12 −ω − iω12ωr ω + iω12ωc,strips The Drude component corresponds to the dispersion law of continuous gold wires. The Lorenz component corresponds to the dispersion law of SRRs. Their combination provides a particular dispersion characteristic [Kanté 2009]. Despite its unusual shape the dispersion is fully causal and satisfies the Kramers-Kronig relation.

80 4.2 Experimental study of single metafil effective medium behavior

a b c

Fig. 47. Reflection, transmission and absorption spectra of the sample obtained using FTIR (cyan, black, magenta colors, respectively) and HFSS (blue, green, red colors, respectively) for different incident angles. S-polarization light incidence. a b c

Fig. 48. Reflection, transmission and absorption spectra of the sample obtained using FTIR (cyan, black, magenta colors, respectively) and HFSS (blue, green, red colors, respectively) for different incident angles. P-polarization light incidence.

The effective permittivity and effective RI retrieved form HFSS modeling for s-polarization for different incident angles are shown in Fig. 49 and Fig. 50, respectively. The curves nicely coincide for all angles up to 89° (here we present angles up to 60°). Despite the very different dispersion characteristic, the EM behavior is again verified. Note also, that the use of such MM structures can provide effective RIs around 8 when the working frequency is close to the resonance region. a b

Fig. 49. (a) Real part and (b) imaginary part of effective permittivity.

81 CH APTER 4. FREE SPACE CON FIGURATION …

a b

Fig. 50. (a) Real part and (b) imaginary part of effective refractive index.

4.3 Conclusions

This chapter addresses the problem of the validity of EMA when applied to the case of a single MM layer in the optical domain for a free space light propagation configuration. The validity of the EMA was investigated using HFSS numerical modeling on the example of gold CWs array on silicon substrate and also for a MMs composed of gold SRRs and continuous wires on silicon substrate. On the base of these examples it was shown that metafilm behavior is indeed analogous to that of a homogeneous layer. The thickness of this layer is that of the deposited metal. The validity of this conclusion was verified with respect to a number of criteria consistent with the Maxwell-Garnett EM model. Namely, it was shown that outside the resonance region the magnetic permeability µ ≈ 1 and that in accordance with the MG approximation the effective permittivity is proportional to the FF of the MMs. Also, it was found that for small metal thickness and FFs the metafilm optical length is proportional to the metal thickness. The linearity of the optical length with metal thickness turns to be greatly improved when considering tuned MMs having the same resonance frequency. Finally, it was established that metafilm permittivity is independent of the incident angle and is also presumably independent of the state of light polarization. The last assertion requires an additional analysis in order to address anisotropy issues. Nevertheless, the qualitative analysis of transmission and reflection behavior in the vicinity of the Brewster’s angle for p-polarized light is found to be consistent with EM model. The realization of experimental sample with gold CWs arrays on silicon substrate allowed to verify the general validity of the HFSS modeling results that were found to be in a good agreement with FTIR experimental results for transmission and reflection. The ellipsometry measurements further confirmed the validity of the EMA. The experimentally found thickness of the EM layer was

82 4.2 Experimental study of single metafil effective medium behavior found to correspond to metal thickness. The experimentally determined effective permittivity showed a good qualitative a quantitative agreement with that obtained from modeling. The general behavior of the effective permittivity is following a Lorentz dispersion law. The analysis of angular resolved FTIR transmission and reflection spectra for a single layer of MMs composed of gold SRRs and continuous wires of silicon substrate allowed to confirm further for this structure the validity of the EMA. In agreement with theory predictions the effective permittivity can be described by the sum of two contributions: Drude and Lorentz dispersion laws.

References

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[Born 1999] M. Born, E. Wolf, Pri ncipl es of Opt ics: Electromagn et ic Theory of Propagati on , In terf eren ce and Di ff racti on of L ight , Cambridge University Press, Cambridge (1999)

[Dubrovina 2012] N. Dubrovina, L.O. le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. de Lustrac, A. Vial, G. Lerondel, A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Appl. Phys. A 109, 901 (2012)

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[Lupu 2011] A. Lupu, N. Dubrovina, R. Ghasemi, A. Degiron, A. de Lustrac, “Metal-dielectric metamaterials for guided wave silicon photonics,” Opt. Exp. 19, 24746 (2011)

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[Simovski 2011] C.R. Simovski, “On electromagnetic characterization and homogenization of nanostructured metamaterials,” J. Opt. 13, 013001 (2011)

[Weber 2009] D. Weber, P. Albella, P. Alonso-González, F. Neubrech, H. Gui, T. Nagao, R. Hillenbrand, J. Aizpurua, A. Pucci, “Longitudial and transversal coupling in infrared gold nanoantenna array: long range versus short range interaction regimes,” Opt. Exp. 19, 15047 (2011)

[Zhang 2005] S. Zhang, W. Fan, N. C. Panoiu, K. J. Malloy, R. M. Osgood, and S. R. J. Brueck, “Experimental Demonstration of Near-Infrared Negative-Index Metamaterials,” Phys. Rev. Lett. 95, 137404 (2005).

Chapter 5 Engineering of metamaterial properties in a guided wave configuration

The previous chapter addressed the problematic of EM behavior for single layer MM structures operating in free space light configuration in the optical domain. The objective of the study presented in this chapter is to extend EMA to a GWC. The interest is motivated by the advantages offered by the planar technology for the GWC, which may constitute a promising alternative to the multi-layered MM approach in the optical domain. Indeed, in a GWC light propagation occurs along the MM structure. Therefore the length of propagation through MMs is proportional to N y , where

N y is the number of unit cells in the horizontal direction and corresponds to the number of MM layers (Fig. 51). The number of unit cells can be very large. Such structure may be considered as an analog of multi-layered MMs with N y >> 1. From this point of view investigation of MMs in GWC represent an interesting object for the experimental study of multi-layered MMs. The hybrid metamaterial GWC may become a promising alternative to the bulk multi-layered metamaterial structures in the NIR domain. The aim of the present study is to achieve an efficient control over the flow of light in the WG underneath the metamaterial layer. By means of this MM layer we intend to obtain a significant variation of the effective refraction index in order to efficiently mold the flow of light in the WG. The possibility for controlling at the nanoscale the local effective index can be used in TO applications.

Fig. 51. M etamaterial structure on top of dielectric substrate. The number of unit cells in the direction of the light propagation Ny is not technology limited.

Furthermore, as previously mentioned, the MMs in GWC may serve as building blocks in TO devices for photonic applications. To attain these objectives, the following points are theoretically and experimentally addressed and results presented in this study:

87 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

• Investigation of the EMA for MMs in a guided wave configuration.

• Establishing of guiding rules for engineering of MM effective properties.

• Demonstration of the technological feasibility and operation of the MMs in a guided wave configuration in the spectral domain around 1.5 µm.

5.1 Basic elements of guided wave optics

Before considering MMs in GWC we provide here some basics of WG theory. This is not attempt to provide a detailed description of guided wave optics but just bring out some insight on the theory. Our consideration is limited to the case of homogeneous slab dielectric WG that is represented by structure confined only in one dimension and homogeneous rib WG that is confined in two directions (see Fig. 52a and Fig. 52b). a b

c d

Fig. 52. (a) Schematic of slab waveguide. Waveguide is confined in z direction. The plane wave propagates along y direction. (b) Schematic of 2D rib waveguide confined in x and z directions. (c) TE mode: the electric field is along x-axis; (d) TM mode: the magnetic field is along x-axis.

To find the possible modes that such WG can support one needs to solve the wave equation (5.1)7 in the system with boundary conditions (condition of continuity of the fields on the WG boundaries). C C 1 ∂ 2 E ∆E − = 0, (5.1) v 2 ∂t 2 Since WG is homogeneous along y direction solution to the wave equation can be written in the form: C C

E = Em (x, z)exp(i(ωt − βy)), (5.2)

7 In general, the same equation and all further analysis can be done for the magnetic field analogously. 88 5.1 Basic elements of guided wave optics

Where β is propagating constant: 2π β = n , (5.3) λ eff

neff is effective RI of a propagating mode, λ is wavelength in vacuum. Then C C 2 2 ∆Em (x, z) + β ((n / neff ) −1)Em (x, z) = 0, (5.4) C

Solutions of the equation (5.4) for the specific WG (Em (x, z),neff ) pq are called modes of the WG, where p and q indicate the order of the mode and show how many nodes the electric field has along x or z axis, correspondingly. Broadly speaking, a mode denotes certain field pattern that can propagate through the system independently. It can be shown that the lowest mode 00 ( p =0, q =0) has the highest effective RI. The WG that can support only one mode is called single-mode WG. Plane WGs can support two modes: TE mode (transverse electric), when there is no electric field in the direction of propagation (Fig. 52c), and TM mode (transverse magnetic), when there is no magnetic field in the direction of propagation (Fig. 52d). Dielectric WGs limited in two directions (Fig. 52b) can support both modes: TE and TM. For TE mode there is a non-zero component of the magnetic field along y axis, and for TM mode there is a non-zero component of the electric field along the same axis.

5.2 One-dimensional array of cut wires on slab waveguide

In this section we focus our attention on a hybrid MMs made of one-dimensional array of subwavelength gold CWs on top of a silicon slab (see Fig. 53a). The CWs represent probably the most elementary type of MMs used for building more complex geometry MMs. Their greatest advantage is the essentially non-magnetic behavior with µ ≈ 1 due to the absence of notable coupling between the electrical and magnetic resonances. In order to obtain the resonance frequency around 200 THz, which corresponds to 1.5 µm wavelength, we fix the CWs dimensions to 200×50×50 nm. Due to the aspect ratio of the cut wires, it is expected that the hybrid MM structure exhibit a strong anisotropy [Knight 2011]. The degree of anisotropy can be controlled through the orientation of the CWs with respect to the propagation direction. This can be useful for designing TO devices requiring anisotropy of MM properties.

89 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

5.2.1 Effective parameters of single cut wires chain

In the case under study the CWs pattern has only one unit cell along the direction of the wave propagation (Fig. 53a). The separation between two adjacent cut wires in longitudinal direction is set to 100 nm (longitudinal period is 300 nm). This corresponds to a moderate coupling between the chain elements. For the sake of simplicity we consider slab WG with a core RI of 3.45 surrounded by air on both sides. This index value is that that of the silicon at 1.5 dm. The dielectric permittivity of gold used for numerical modeling is taken from Palik’s Handbook [Palik 1998]. The transmission and reflection spectra of this structure are calculated by means of HFSS software. To retrieve effective parameters, the single MM layer EM model given in the Chapter 4 should be extended to the GWC. a b

Fig. 53. (a) Sketch of a slab waveguide with a chain of gold cut wires on the top, TE polarization of the light. (b) Equivalent model for the case, TE polarization.

In the GWC depicted in Fig. 53a, the light is guided by the silicon slab before incoming on the CWs structure. The appropriate approach for the GWC consists in approximating the entire structure – that is, the silicon layer and the gold CWs on the top of it – as a single dielectric slab guiding the signal by total internal reflection (Fig. 53b). We note that a similar treatment has already been performed in the THz regime for an array of SRRs embedded in a very thin dielectric slab [Reinhard 2010]. The thickness of the effective media slab is considered the same as that of silicon slab. The length of the EM slab is equal to the width of CWs. The difference with respect to the equivalent model for FSC shown in Fig. 25b is that here it is the silicon slab that should be considered as the first/last media. .

To calculate the effective permittivity ε eff and permeability µeff from the complex reflection and transmission coefficients r and t , respectively, we use NRW retrieval method. It can also be applied to a GWC since guided mode has a planar wavefront. The formulas for the guided propagation are analogues to the formulas from the section 4.1.2. Here we do not produce them again. The difference with respect to the Eqs. (4.9a) and (4.9b) is that the effective index of the WG slab neff _ slab could be used as first/last media in the expression for the impedance z . Then the

90 5.2 One-dimensional array of cut wires on slab waveguide impedance for TE mode can be described by the Eq. (4.9a) and for TM mode by the Eq. (4.9b), where the incidence angle α1 = 0. a b

c d

Fig. 54. (a) Transmission (red and cyan), reflection (blue and green) and loss (black and magenta) spectra for chain of cut wires 200×50×50 nm on top of plane silicon waveguide; lines in red, blue and black are HFSS simulation data; lines in cyan, green and magenta are EM model data obtained using Lorentz dispersion law. (b) Transmission (red and green) and reflection (blue and cyan) phase difference for the same structure; blue and red lines are HFSS simulation data, green and cyan lines are model data. (c) Effective refractive index of unloaded waveguide (green) and real (blue) and imaginary (red) parts of refractive index of effective media slab (d) Retrieved effective permittivity: real part (blue and cyan) and imaginary part (red and magenta); blue and red lines are data obtained from applying the retrieval procedure to HFSS data; cyan and magenta lines are described by Lorentz dispersion law.

The Fig. 54 shows the comparison between the HFSS simulation data and that calculated using slab EM model. The Fig. 54a and Fig. 54b represent the transmission, reflection, loss spectra and the phase difference, respectively, for the single chain of CWs 200×50×50 nm placed on top of 100 nm silicon slab. The loss is calculated by subtracting the transmittance and reflectance from the incidence power and represents the sum of absorption and dipole emission in the studied case. As it can be seen, the broad peak appearing in the spectra indicates that the light propagating in silicon WG is coupled to the CWs surface plasmon resonance. The presence of the energy dissipation mechanism related to the poor conductivity of gold at such high frequencies prevents one from attaining high resonance quality factors. The quality factors for the investigated cut wires, determined in a conventional way – f0 divided by the half-width of ε"eff ( f ) – are typically around

10. Here f0 is the resonance frequency.

91 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

The Fig. 54d represents the retrieved effective permittivity for the single chain of CWs. One can see that HFSS data are in a good agreement with the plane wave modeling results using EMA and Lorentz dispersion law. The obtained results confirm thus the validity of the considered model treating the case of a one-dimensional array of CWs in a GWC. Further we discuss it in more details.

5.2.2 Resonance frequency engineering

As it can be seen from the Fig. 54a the resonance frequency of probed CWs with dimensions of 200×50×50 nm in GWC is 168 THz. To decrease the losses at the working frequency the resonance frequency must be slightly above or below 200 THz. In our work we choose the resonance frequency to be above 200 THz in order to obtain values of the effective RI higher than that of silicon slab WG. There are several methods to shift the resonance frequency and to control the interaction between propagating waves and CWs localized surface plasmons. The straightforward way for shifting resonance frequency is to downscale the dimensions of CWs. For example, to achieve a resonance frequency around 250 THz the length of CWs needs to be reduced in this case to around 100 nm. Although the solution of using shorter length cut wires is in principle viable, an accurate control of the dimensions of such small structures is technologically challenging. Maintaining high aspect ratio of CWs is also important for the implementation of an anisotropic RI [Knight 2011]. . To circumvent these issues and relax the technological constraints, a different approach is considered. As known, due to the strong localization of the surface plasmons near the metal- dielectric interface, their properties strongly depend on the permittivity of the dielectric adjacent to the metal. By consequence the resonance frequency of surface plasmons is higher in the presence of a low index dielectric [Kelly 2003]. We used this property for engineering CWs resonance frequency. This method constitutes a promising alternative to the downscaling of CWs.

5.2.2.1 Influence of dielectric environment on the resonance frequency

We start with the problem of the resonance frequency engineering. The idea consists in modifying the dielectric environment of the localized surface plasmons. For this purpose we introduce an intermediary layer of 10 nm of silica between the silicon slab and the metal (Fig. 55b). As discussed in the previous chapter, such thin films of silicon oxide can rise naturally on top of silicon. In complement to this we consider the configuration where in addition to the 10 nm silica interlayer, the CWs are recovered with silica overclad. For symmetry reasons the same silica is used as underclad in this case. The corresponding situation is represented on the Fig. 55c. The interest for

92 5.2 One-dimensional array of cut wires on slab waveguide such a structure is motivated by the technological considerations related to the fabrication of multi- layered MMs. a b c

Fig. 55. Sketch of the slab waveguide with a chain of cut wires on the top. An arrow indicates direction of light propagation. Cut wires: (a) in contact with silicon slab; (b) separated from silicon slab by SiO2; (c) separated from silicon slab and embedded in SiO2.

The resulting transmission, reflection and loss spectra calculated for 100nm Si slab thickness corresponding to the configurations depicted Fig. 55(a-c) are shown in Fig. 56(a-c), respectively. By examining these results, we see that the position of the resonance frequency depends on the CWs environment. The position of the minimum of transmission is 168 THz for CWs in contact with the silicon slab. The resonance frequency shifts to 250 THz for CWs separated by 10 nm silica interlayer from silicon slab. The resonance frequency is however shifted down to 209 THz for CWs additionally embedded in silica. This evolution demonstrates the possibility of resonance frequency engineering through the control of CWs dielectric environment. a b c

Fig. 56. Intensity of transmission (dashed red), reflection (dotted blue) and loss (continuous green) coefficients for the 200×50×50nm cut wires. Cut wires (a) in contact with 100 nm silicon slab; (b) separated from silicon slab by SiO2; (c) separated from silicon slab and embedded in SiO2.

The frequency variation of the CWs meta-slab real ε 'eff and imaginary ε"eff components of the effective complex permittivity for the cases corresponding to the configurations represented in the Fig. 55 (a-c), are shown in Fig. 57(a-c), respectively. As it can be observed, between 100THz and

300 THz there is a strong variation of the ε"eff with the frequency f that can be approximated by a Lorentzian type dispersion formula.

93 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… a b c

Fig. 57. The real (dotted blue) and imaginary (dashed red) components of the complex effective permittivity of the 200×50×50 nm cut wires. Cut wires: (a) in contact with 100 nm silicon slab; (b) separated from silicon slab by SiO2; (c) separated from silicon slab and embedded in SiO2.

The frequency variation of the CWs meta-slab real imaginary components of the effective index is shown on the Fig. 58(a-c), respectively. As expected, for a Lorentzian type dispersion the RI is high just below the resonance frequency and low (eventually even below 1, meaning strong reflection) just above the resonance. This opens up the ability, as a function of the target index value, to engineer the effective RI by an appropriate positioning of the cut wire resonance with respect to the operating frequency. a b c

Fig. 58. The real (dotted blue) and imaginary (dashed red) components of the effective index of the 200×50×50nm cut wires. The green continuous curve corresponds to the effective index of a 100 nm thick

silicon slab. Cut wires: (a) in contact with 100 nm silicon slab; (b) separated from silicon slab by SiO2; (c)

separated from silicon slab and embedded in SiO2.

Based on these considerations, we now discuss how to obtain a maximal index contrast with respect to that of a silicon slab. At the same time it is highly desirable to avoid the adverse effects of losses inherent to the absorption of metal cut wires, which can be very high at the resonance frequency. To minimize the losses, it is preferable to operate not at the exact resonance but in its near vicinity and find an optimal trade-off between index variation and loss. To achieve a low loss operation at 200 THz, corresponding to the 1.5µm telecommunication wavelength, the resonance frequency should be shifted out of this spectral domain by a few half- widths of the absorption line. For example, if we wish to attain a resonance frequency of 250 THz

94 5.2 One-dimensional array of cut wires on slab waveguide on a silicon slab, one already mentioned solution is to decrease the length of the CWs. In order to keep the same metal filling ratio, the period of the CWs chain is reduced in the same proportion as the CWs length. For the case of CWs in contact with silicon slab corresponding to the Fig. 55a the CWs length is tuned to 100 nm and the period is 150 nm. For the case of CWs embedded in silica corresponding to the case of the Fig. 55c, the CWs length is tuned to 160 nm. The period is then 240 nm. The variation of the real and imaginary components of the effective complex permittivity and RI corresponding to the configurations shown in Fig. 55(a-c) are represented in Fig. 59. The obtained results show that the dielectric response is very similar for all considered configurations. This indicates that at a fixed resonance frequency, the dielectric response is essentially determined by the “concentration” of MM elements and is in fact little sensitive to the environment. In other words, changing the host material around the CWs does not influence their response if one maintains their resonance at the same frequency and if the volume concentration of the CW “meta-atoms” remains constant. This observed behavior is quite similar to that examined in the Chapter 4 for single MM layer, which effective permittivity is also essentially determined by the resonance frequency (see Fig. 31b). a b

Fig. 59. (a) The real (continuous curves) and imaginary (dashed curves) components of the cut wires loaded slab permittivity. Triangles - 100×50×50 nm cut wires in contact with the Si slab; circles – 200×50×50

nm cut wires separated by 10 nm SiO2 from silicon slab; squares – 160×50×50nm cut wires embedded in SiO2 . (b) The real (continuous curves) and imaginary (dashed curves) components of the cut wires loaded slab effective index. Same symbols used as in (a). The black dotted curve corresponds to that of a 100nm thick Si slab effective index.

In this context the choice of the optimal design is dictated by technological facility and CWs high aspect ratio considerations. For these reasons a composite slab of CWs located on a 10nm thick silica interlayer seems to be the optimal solution. This corresponds to the configuration depicted in Fig. 55b. It may be however an interest in using silica embedded CWs configuration

95 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… provided that the use of multiple CWs layers instead of one provides higher effective index contrast. The verification of this hypothesis is provided a few sections later in this chapter.

5.2.2.2 Influence of waveguide thickness on the metamaterial effective permittivity

The effective parameters of the hybrid WG with CWs on top are also influenced by the silicon WG thickness. The waveguide mode is less confined for a thinner slab that results in a higher interaction with CWs localized surface plasmons. To show this, three cases with different silicon slab thickness are considered. a b c

Fig. 60. Intensity of transmission (red), reflection (blue) and loss (black) coefficients for a 200×50×50 nm cut wires chain placed on the top of silicon slab with thickness: (a) 100 nm; (b) 150 nm; (c) 200 nm.

The resulting transmission, reflection and loss spectra calculated for 100 nm, 150 nm and 200 nm Si slab thickness are shown in Fig. 60(a), 60(b) and 60(c), , respectively. The inspection of the obtained results shows that the position of the resonance frequency depends on the slab thickness. The position of the minimum of transmission shifts from 169 THz to 158 THz as the slab thickness is increased from 100 nm to 200 nm. These results indicate that the localized surface plasmons are sensitive to the mode effective index of the slab and not only to its index of refraction. a b c

Fig. 61. The real (blue) and imaginary (red) components of the complex effective permittivity of a single 200×50×50 nm cut wires chain placed on the top of silicon slab with thickness: (a) 100 nm; (b) 150 nm; (c) 200 nm.

96 5.2 One-dimensional array of cut wires on slab waveguide

The frequency variation of the CW chain real and imaginary components of the effective complex permittivity ε for the case of 100 nm, 150 nm and 200 nm slab thickness are represented in Fig. 61(a), 61(b) and 61(c), respectively. It can be observed, the magnitude of the effective permittivity variation is strongly dependent of the silicon slab thickness. This dependence is for essential determined by the amount of field propagating in the WG and interacting with CWs elements. The control of the light confinement in the CWs region represents thus a powerful tool for engineering the magnitude of the effective index variation induced by the CWs resonance. The interaction between CWs elements and guided mode should also depend on the orientation of the CWs with respect to the electric field and mode polarization.

5.2.2.3 Influence of cut wires orientation on the effective parameters

Fig. 62. Scheme of hybrid wave guide when cut wires are rotated by 90°. TE-polarized mode. The electric field is perpendicular to the long axis of cut wires. a b c

Fig. 63. The real (blue) and imaginary (red) components of the complex effective permittivity of silicon waveguide with 200×50×50 nm and 90° rotated cut wires placed on the top. The thickness of silicon slab is: (a) 100 nm; (b) 150 nm; (c) 200 nm.

Due to the important aspect ratio of the cut wires, it is expected that the hybrid MM structure exhibits a strong anisotropy [Knight 2011]. Rotated by 90° CWs (Fig. 62) should not display any resonance behavior for TE-polarized WG mode at frequencies under interest. We have verified that the response of a periodic chain of 200×50×50 nm cut wires rotated by 90° with respect to the current configuration indeed does not produce any noticeable resonance and that the effective index of the rotated structure is similar to that of a Si slab (Fig. 63) Hence by changing the rotational angle

97 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… of gold CWs it is possible to tune the effective permittivity which opens the door to realization TO devices in GWC.

5.3 Single layer 2D array of cut wires on slab waveguide

The next step of our analysis consists in expanding the developed approach to the case of a MM system with several unit cells along the propagation direction. In this case, the CWs form a 2D array on top of the Si slab, as shown in Fig. 64. As it was mentioned in the end of the previous section, here and further we consider WGs with silica interlayer.

Fig. 64. Sketch of the slab waveguide with several coupled chains of cut wires with SiO2 interlayer.

We consider the case of a MM structure with 10 unit cells along the propagation direction. As before, the thickness of silica interlayer is 10 nm. The dimensions of CWs are 200−50−50 nm. The longitudinal separation d * =100 nm. We performed HFSS numerical simulations for CWs separated by d =50 nm, d =100 nm and d =150 nm. The transmission, absorption and reflection coefficients obtained in the case of 100 nm and 150 nm thick silicon slab are represented in Fig. 65(a-c) Fig. 65(d-f), respectively. There is a marked difference between the reflection spectra of the slab WG loaded with a single chain of CWs and with a 2D array of strongly coupled CWs ( d =50 nm). In the last case no pronounced maximum is observed in reflection. The reflection is low and does not exceed 11% and 3% for the case of structures on top of 100 nm or 150 nm thick silicon slabs, respectively. This is much lower as compared to the reflection of single CWs chain displayed in the Fig. 56b. When the transversal separation distance d between adjacent wires is increased to 100 nm or 150 nm, the coupling between them is decreased. A maximum in reflection is progressively building in the vicinity of the resonance. The observed phenomena are more pronounced in the case of 100 nm thick silicon slab where the interaction with the chains of CWs is higher. The observed behavior presents great similarities with that of a chain of coupled PhC nano-cavities [Olivier 2001]. The transmission of strongly coupled cavities in the vicinity of the resonance is akin to an EM WG with a high group index. The reflection level is very low. When the coupling between the cavities becomes weaker, an interference effect occurs leading to the appearance of pronounced peaks in reflection and dips in transmission. Although in the case of the present study the involved

98 5.3 Single layer 2D array of cut wires on slab waveguide interactions are more complex than for crystal nano-cavities, most of the observed results can be fairly explained by the model of coupled plasmonic nano-cavities. In the case of a slab WG loaded with strongly coupled CWs, its behavior is mostly similar to that of an EM slab WG. a b c

d e f

Fig. 65. Intensity of transmission (red), reflection (blue) and loss (black) coefficients for 2D array of cut wires with dimensions 200×50×50 nm placed on 10nm silicon dioxide on the top of 100 nm silicon slab. Transversal separation is: (a) d = 50nm; (b) d = 100nm; (c) d = 150nm. (d), (e), (f) are the same but for the thickness of silicon slab 150 nm: (d) d = 50nm; (e) d = 100nm; (f) d = 150.

This assertion for an EM behavior is also supported by the distribution of the electrical field in the structure provided by our numerical HFSS simulations. The Fig. 66 shows the snapshots of the wave propagation at 200 THz in three cases for a chain of cut wires separated by (a) d =50 nm, (b) d =100 nm and (c) d =150 nm on a 10 nm layer of silicon dioxide on the top of a 100 nm silicon slab. The field distribution across the coupled chain region is much more uniform in the case of a strong coupling corresponding to the 50nm of separation between the chains. Conversely, stronger field localization around the CWs is observed when the separation between the adjacent chains is increased. The exchange of energy in this case preferentially occurs between the chain and the slab and not between the adjacent chains. The two essential conditions for the validity of the EMA are: (i) have a strong coupling between adjacent wires and (ii) the action of metamaterial layer could be small enough to be considered as perturbation. It could be noted that these conditions for EM validity in GWC are at some amount in contradiction with the objective high index variation induced by MM resonances. Nonetheless, as it will be demonstrated in the following by modeling and experimental results, it is

99 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… always possible to achieve a very strong local effective index variation, though the general behavior of the structure is not homogeneous in this case. a

Fig. 66. Cross section view for the electric field distribution of the propagated wave at 200 THz for 10 coupled 200×50×50 nm periodic cut-wire chains separated by (a) d = 50nm, (b) d = 100nm and (c) d = 150nm placed on 10 nm silicon dioxide on the top of 100 nm silicon slab.

To retrieve the effective dielectric permittivity in the case of a slab loaded with chains of coupled CWs, the application of the NRW retrieval method using the equation (2.13), becomes impractical. The large optical thickness of the MM region does not allow to unambiguously determine the corresponding branch of the inverse cosine function. To find the effective index we use an alternative method based on the analysis of field spatial distribution in the composite slab WG. The method is essentially similar to that reported by [Andryieuski 2009].The results obtained for 10 unit cells with wires separated by d =50 nm, d =100 nm and d =150 nm in the case of 100 nm and 150 nm thick silicon slabs are

100 5.3 Single layer 2D array of cut wires on slab waveguide represented in the Fig. 67a and Fig. 67b, respectively. The general behavior of the effective index variation is qualitatively similar to that of a single chain of CWs though the index contrast with respect to that of a silicon slab is reduced by approximately a factor of two. The index contrast is higher in the case of stronger coupled wires than that for smaller separation distance. For 50 nm of transversal separation distance the index contrast at 200 THz is around 0.5 for 100 nm Si slab and around 0.25 for 150 nm Si slab. Such an index contrast can be sufficient to be used in integrated optics applications. The main issue with the considered system is related to the level of propagation losses, which is of the order of few dB per 10 unit cells. This corresponds to a propagation distance of around one micron, implying that these structures are best suitable for extremely compact designs, which are the real targets of this technology. a b

Fig. 67. (a) The real components of effective index for a single (magenta) or coupled 200×50×50 nm cut wires chains with separation distance: d = 50 nm (blue), d = 100 nm (red), d = 150 nm (cyan) placed on a 10 nm silicon dioxide on the top of silicon slab of 100 nm. The green curve corresponds to the effective index of the silicon slab alone. (b) Same as in (a) for 150 nm silicon slab thickness.

5.4 Double layer 2D array of cut wires on slab waveguide

The next step of our analysis is to verify whether using multiple layers of CWs on the top of a slab WG will result in a further increasing of the effective index variation. To answer this question, we consider the case of double layer CWs structure with 10 unit cells along the propagation direction (Fig. 68). We compare this case with the case of one-layered CWs hybrid WG. For both configurations a 10 nm thick silica layer is inserted between the silicon slab and the CWs and their dimensions are 200×50×50 nm. The whole structure is embedded in silica from the upper and lower side. We performed HFSS numerical simulations for CWs with the transversal separation d =50 nm. The vertical separation between layers is also 50 nm. This corresponds to the condition of strong coupling between adjacent CWs elements, that is necessary to achieve an EM behavior. The

101 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… longitudinal distance separation between the CWs is 100 nm. The transmission, reflection and loss spectra obtained in the case of 100 nm and 150 nm thick Si slabs are displayed in the Fig. 69. a b

Fig. 68.Sketch of the slab waveguide with an array of CWs on the top. Light propagation direction is indicated by an arrow. The E field is parallel to the layers. (a) Single CWs layer; (b) double CWs layer.

An inspection of the results reveals that the transmittance and losses of a single or double layers array of strongly coupled CWs are almost identical. They are also qualitatively similar to that of a single chain of CWs displayed in the Fig. 65a and in the Fig. 65d except that a great reduction of reflection and losses is observed for both the single and double layers of strongly coupled CWs. In the last case no pronounced maximum is observed in reflection. The reflection level is also much lower and does not exceed 23% and 7% for the case single and double layers structures on top of 100 nm thick silicon slabs, respectively. This is considerably lower as compared to the reflection of single chain displayed in Fig. 56c. The effective index for the case of double layer structure was determined from spatial field distribution, in the same manner as for single layer structure (see section 5.3). The resulting frequency dependent effective index variation is shown in the Fig. 70. As evident, the index variation obtained with a single or double layer of CWs are quasi identical. The presence of a second layer of CWs does not provide a further improvement of the effective index variation. The essential reason for such a behavior is related to the screening of the electromagnetic field by the first CWs layer that is directly coupled to the slab WGs. The screening effect is clearly visible from the snapshot of field distribution shown in the Fig. 71. The amount of field coupled to the upper CWs layer is much lower and the interaction with the upper layer is very limited. This explains the small influence of this upper layer on transmission/reflection properties and effective index behavior. The conclusion that can be drawn is that the presence of the second layer is mostly negligible and de facto useless.

102 5.4 Double layer 2D array of cut wires on slab waveguide

a b

c d

Fig. 69. Intensity of transmission (dashed red), reflection (dotted blue) and losses (continuous

green) spectra for a 200×50×50 nm silica embedded cut wires separated by 10 nm SiO2 from silicon slab. (a) Single layer of cut wires, thickness of silicon slab is 100 nm. (b) Single layer of cut wires, silicon slab thickness is 150 nm. (c) Double cut wires layer, 100 nm silicon slab. (d) Double cut wires layer, 150nm silicon slab. a b

Fig. 70. The real components of the effective index for a single (dotted blue) or double (dashed red)

layers of coupled array of 200×50×50 nm silica embedded cut wires separated by 10 nm SiO2 from silicon slab. The green continuous curve corresponds to the effective index of the silicon slab alone. (a) The thickness of silicon slab is 100 nm; (b) the thickness of silicon slab is 150 nm.

103 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

Fig. 71. Cross section view for the electric field distribution of the propagated wave for 10 coupled 200×50×50 nm periodic cut-wire chains placed on 10 nm silicon dioxide on the top of 150 nm silicon slab: (a) operation at 150 THz (below the resonance frequency); (b) operation at 300 THz (above the resonance frequency).

5.5 Choice of cut wires design for guided wave configuration

Our numerical simulations shown that an array of gold coupled CWs over a slab WG leads to a significant variation of the slab effective index in the vicinity of the resonance. We state that the optimal geometry of CWs is 200×50×50 nm. Such CWs being placed on top of silicon WG covered by thin silica layer can serve for the purposes of TO. Such hybrid WGs provide high magnitude of the effective permittivity function at the same time allow maintaining important aspect ratio of CWs. The last fact leaves an open room for possible variation of the dimensions of CWs or their rotational angle, which can be useful for designing of TO devices. However, we underline the important result obtained from our numerical study: using of multiple MM layers does not provide higher index contrast. Thus we limit our further consideration by one layer of MM on top of dielectric WG. That case does not blank any significant raise of the effective parameters.

104 5.5 Choice of cut wires design for guided wave configuration

These results demonstrate that the dispersion and guiding properties of hybrid WGs can be carefully controlled with planar metallo-dielectric MMs on the top, which paves the way for new optical functionalities.

5.6 Experimental study of hybrid metal-dielectric waveguides

In previous sections the effective permittivity ε eff and the effective RI neff of a composite slab was determined using adapted NRW retrieval method for a single chain of gold CWs. To determine the effective RI neff of 2D arrays of CWs on top of dielectric planar WG we used the alternative method based on spatial field distribution in the composite slab MM WG. We showed that an array of gold coupled CWs (1D or 2D) over the slab WG leads to a significant variation of the slab effective index. The effective index contrast can achieve the magnitude of ± 1. The next step is to experimentally demonstrate the hybrid metal-dielectric WG at NIR frequencies. Here we discuss the potential of hybrid MMs WGs for guided optics applications. We consider the case of a composite guiding structure made of a single film of MM over a high index WG, such as silicon WG. The MM under the study takes the simplest form of 2D CW arrays. We aim to examine experimentally the interaction between adjacent CWs as well as the influence of the CW length and their rotational angle (anisotropy properties) on the resonance properties. We also will discuss the possibility of practical tailoring of the effective parameters of hybrid metal-dielectric WGs. All samples used for this subsection were fabricated in the clean rooms of IEF and grouped in the set of samples B. They are shortly described in the end of the Chapter 3.

5.6.1 Influence of cut wires transverse separation distance

The design of the first experimental sample fabricated for the study of hybrid metal-dielectric WG is schematically depicted in Fig. 72. The sample is divided in three areas corresponding to different transversal separation d between CWs. By changing the transverse separation one can control the interaction between adjacent CWs influencing the resonance properties of the hybrid WGs. Each of three areas includes 10 tapered WGs loaded with CWs array (Fig. 72b). The design parameters of CWs area are presented in Fig. 72c. In order to obtain the resonance frequency around 200 THz, which corresponds to 1.5 µm wavelength, we fix the CWs dimensions to 200×50×50 nm. The separation distance between two adjacent elements along the light propagation direction is set to either d =50, 100 or 150 nm. The separation of two adjacent cut wires along the longitudinal axis,

105 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… which is perpendicular to the light propagation direction, is set to d* = 100nm. The number of CWs in longitudinal direction is N x =30, in transversal direction N y =10. The light travels along y direction. The tapered transitions from a 3 µm wide input WG down to 0.6 µm wide single-mode WG, then to the 10 µm wide output WG are used to insure fundamental mode propagation (see Fig. 72b). This point is especially critical since at a 10 µm width the silicon WG is intrinsically multimode. a b

Fig. 72. (a) Sketch of the silicon slab waveguide with an array of CWs on the top. Light propagation direction along y-axis. (b) The waveguide consists from 3 areas: input waveguide, tapered region and principal waveguide with metamaterial particles on the top. (c) Scheme of metamaterial area.

We use a non-intentionally doped SOI wafers from SOITEC with a 200 nm thick top silicon film separated from the silicon substrate by a 2 µm buried SiO2. The technology fabrication process of these hybrid WGs is described in the Chapter 3. At the end of the fabrication process the sample was in details inspected using scanning electron microscope (SEM). The Fig. 73 shows SEM images of the fabricated sample. The precision of the WG alignment with respect to the CWs array is of the order of one hundred nanometers. Such a precision is by far sufficient in our case where the WG width is 10 µm and the CWs array extent across the WG is 9 µm (Fig. 73a). On the Fig. 73b one can see an enlarged image of the CWs area. The experimental dimensions of the CWs are 198×50×50 nm with longitudinal separation of 102 nm and transverse separation of 100 nm. These dimensions are in very good agreement with designed parameters. The Fig. 73c shows the region of the tapered transitions. The walls of fabricated WGs are well vertical (see Fig. 73d). In complement to tapered WGs, we also included in our design 10 µm wide straight WGs loaded with same type CWs arrays. Straight WGs without CWs also included in our design served for reference purposes to calibrate the end-fire setup or to normalize transmission spectra.

106 5.6 Experimental study of hybrid metal-dielectric waveguides

The sample was characterized using the end-fire setup described in the Chapter 3. Each WG was measured for two incoming light polarizations: TE and TM. The results are discussed in the next sections. a b

c d

Fig. 73. SEM images of the fabricated sample: (a) top view of metamaterial area; (b) enlarged view of CWs array with nominal dimensions 200×50×50 nm, d=100 nm, d*=100 nm; (c) region of tapered transitions; (c) the cut-off view of a waveguide.

5.6.1.1 Straight CWs loaded waveguides

We start the study with the straight silicon WGs loaded by gold 200×50×50 nm CWs on the top. The Fig. 74 shows the TE- and TM-polarization transmission spectra of a 10 µm wide multimode straight WG having a separation distance between adjacent CWs d = 150 nm. The measurements are not corrected for the coupling losses. As it can be seen from the Fig. 74a, for the TE-polarized light a marked dip in transmission is observed. Its spectral position around 225 THz is in a good agreement with the CWs resonance frequency obtained using HFSS numerical modeling. As it can be observed from Fig. 74b, no such a dip in transmission is present for TM-polarized light, i.e. when the electric field is perpendicular to the layers interface and the orientation of the CWs. These results let presuming that the transmission dip observed for TE polarization is indeed caused by the resonant excitation of the gold CWs.

107 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

a b

c d

Fig. 74. TE (a, c) and TM (b, d) transmission spectra of a 10 µm wide straight cut wires loaded waveguides with either air (a, b) or index liquid n = 1.44 as upper interface (c, d).

To further verify this assumption we covered the investigated structure with Cargille n = 1.44 RI liquid. As it can be seen from the Fig. 74c, for TE-polarized light the dip position is red-shifted to 215 THz. Such a variation of the resonance frequency is induced by the change of the dielectric environment (see results of the section 5.2.2.1). While concerning TM-polarization, the transmission spectrum displayed in the Fig. 74d still does not show any notable manifestation of resonant absorption. Besides, it is worth to note that without MM particles on the top of the WG there is no dip in the transmission spectrum for both polarizations, TE or TM. For instance, one can see spectra for the WG without CWs covered by liquid index n = 1.44 on the Fig. 75. Transmission level of straight WGs for TE polarization is around -14 dB, for TM polarization transmission level is around -12 dB. These results confirm the efficient excitation of the CWs resonance in a GWC for TE polarization. Thus we associate the dip in transmission spectra with the excitation of CWs surface plasmon resonances.

108 5.6 Experimental study of hybrid metal-dielectric waveguides

a b

Fig. 75. (a, b) TE and TM transmission spectra, respectively, of a 10 µm wide straight waveguide without cut wires on the top covered by index liquid n = 1.44.

The noisy aspect of the transmission spectra displayed in the Fig. 74 is due to the presence of high frequency oscillations known as Fabry-Perot resonances. These oscillations are due to the reflections experienced by light when propagating inside the cavity formed by the WG facets. Their study carries useful information on the presence of index discontinuities or losses, their localization and the magnitude of the perturbation. The important length (up to several millimeters) of our WGs and the high resolution of about 1 pm of the end-fire setup used for the sample characterization are greatly favoring such analysis.

To process the data we perform the following procedure. At each frequency f0 the narrow frequency window ∆f is chosen to apply the Fourier transformation. We find the oscillations’ ~ ~ frequencies and magnitudes of Fourier components ( fi and Ai , correspondingly). The frequency

f0 varies over all experimental range 180–240 THz. As the result the found Fourier components can be plotted in the form of 2D plot (Fig. 75). The x axis represents the characteristic length ~ calculated from Fourier transformation and associated with fi , the y axis indicates the frequency ~ f0 of incoming light, and colors of the plot represent the magnitudes of Fourier components Ai . In general the 2D plot depicts the dispersion of the group velocity inside the WG, i . e. the dispersion of the group RI. Thus, measuring only transmission spectra one can obtain information not only on the reflection and losses occurring in the system but also on the modal composition and group index of the propagating waves.

109 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… a b

c d

Fig. 76. Fourier analysis of transmission spectra. (a) and (b) 2D plots for straight waveguides without metamaterial on the top for TE and TM polarization, respectively. (c) and (d) 2D plots for straight waveguides with metamaterial on the top for TE and TM polarization, respectively.

The Fig. 76a and Fig. 76b show the Fourier diagrams for the straight WGs without CWs on the top, for TE and TM polarization of incoming wave, respectively. One can easily distinguish well- pronounced features corresponding to the reflections between the input and the output ends of the

WG. At each frequency f0 TE mode shows two significant Fourier components corresponding to the single LWG and double length 2LWG of the WG. TM mode has only one significant Fourier component corresponding to the single length of the WG. In contrast, as evident from TE polarization Fourier diagrams for straight WGs loaded with CWs that are shown in Fig. 76c, the appearance of complementary features indicates the presence of additional reflections. We can find the correspondence between Fourier components and additional reflections caused by CWs array.

The additional features are corresponding to L1 : the distance between the WG input end and the

CWs area, and L2 : the distance between CWs area and the exit end of the WG, as well as to the combination or integer multiples of L1 and L2 . Notice that the drop in intensity of LWG Fourier component visible around 215 THz is due to the resonant absorption of CWs at this frequency.

5.6.1.2 CWs loaded tapered waveguides

Now we consider CWs loaded tapered WGs. The CWs dimensions are 200×50×50 nm, the transverse separation between CWs is d = 50, 100 or 150 nm. The Fig. 77 shows experimental

110 5.6 Experimental study of hybrid metal-dielectric waveguides transmission spectra corresponding to these cases. Only TE-polarization results are presented, since TM transmission spectra do not reveal any noticeable resonant absorption. One important feature that can be observed from the spectra in the Fig. 77 is that the transmission level strongly depends of the separation distance d between adjacent elements. For the same amount of CWs, the transmission level drops by almost 20 dB when separation distance d varies from 50 to 150 nm. By covering the CWs on top of the WGs with Cargille liquid having the RI n =1.44, a red shift in the resonance frequency towards 200 THz can be observed. The observed behavior is in good agreement with similar results obtained for straight WGs loaded with CWs. The shift of the resonance frequency provides an unambiguous proof of the excitation of localized surface plasmons. These experimental data are also consistent with HFSS modeling results shown in the Fig. 78. The agreement is particular good in the case where Air is the upper interface. For the case when CWs are covered with index liquid the qualitative agreement is good but the shift of the resonance position toward lower frequencies is much higher as compared with the experimental results. Aside the decrease of transmission level for higher separation distance d between CWs, another general trend evidenced by modeling results is the increase of reflection with d . a b c

d e f

Fig. 77. Transmission spectra for CWs loaded tapered waveguides with (a-c) air or (d-f) index liquid n = 1.44 as upper interface for different transverse separations: d = 50, 100, 150 nm. The position of the cut wires resonance frequency for the upper row of images indicated by arrows.

111 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

a b c

d e f

Fig. 78. Transmission and reflection spectra for tapered waveguides loaded with cut wires with either air (a–c) or n=1.44 refractive index liquid as upper interface (d – f). Transverse separations: (a, d) d = 50 nm; (b, e) d = 100 nm; (c, f) d = 150 nm.

The experimental verification of this reflection behavior can be obtained through the analysis of Fourier diagrams. We performed it for the non-covered with index liquid CWs WGs that better matches experimental data. The magnitude of Fourier components corresponding to design characteristic length L1 , L2 and the total WG length L1 + L2 for different separation distance d are traced in Fig. 78. The intensity of Fourier components is given in normalized units on a logarithmic scale (dB) but does not provide directly the reflection coefficient for each characteristic length. Nonetheless, an estimation of the reflection can be easily achieved since the facets reflection coefficient corresponding to L1 + L2 is readily obtained knowing the WG effective index. For a 200 nm thick and 10 µm wide SOI WG the TE effective is around 2.6. The reflection coefficient is then around 20%. This corresponds to a reflection level of -7 dB. Shifting down by this value all the graphs in the Fig. 79 provides a reasonable estimation of reflection coefficients. It means that at 180 THz, where the influence of the resonance absorption is essentially negligible, the reflection coefficient from CWs region is -17 dB for d =50 nm and it grows up to -12 dB for a higher separation distance. Note also that reflection coefficients for L1 practically coincide with that for L2 . This further verify the validity of the method since reflection from either left or right side of CWs region should be the same. The only substantial difference between modeling and experiment concerns the resonance region. In contrast to the modeling results, experimental data show a drop in reflection in the

112 5.6 Experimental study of hybrid metal-dielectric waveguides frequency region around 215 THz. This discrepancy is due to the fact that for experiment it is not the directly reflected but transmitted light that is collected. The unavoidable attenuation caused by absorption in CWs region explains this difference. Note also that facets reflection corresponding to

L1 + L2 is stronger attenuated by the resonant absorption since for this Fourier component light experiences two additional travels across the CWs region. To summarize, the good agreement with experimental results proves the validity of HFSS modeling data. a b c

Fig. 79. Fourier components corresponding to characteristic lengths L1, L2 and total waveguide

length L1+L2 for different separation distance: (a) d=50 nm; (b) d=100 nm; (c) d=150 nm.

5.6.2 Homogeneous model for metamaterials in guided wave configuration

To explain the observed dependence of the composite WG transmittance with the CWs separation distance d we consider the equivalent waveguide model depicted in the Fig. 80b. The essential of this model, which consists in approximating composite WG region with same thickness equivalent slab, is presented in the section 5.2.1 for the case of a one-dimensional array of CWs on slab WG. The only difference with respect to the situation represented in Fig. 53b is that in the actual case the ~ length is equal to the total length of two-dimensional CWs array: L = N y ⋅W + (N y −1) ⋅ d (see Fig. 72c for notations). a b

Fig. 80. (a) Sketch of the hybrid CWs loaded waveguide. (b) Homogeneous effective slab model.

The transmission and reflection of the equivalent homogeneous WG was determined using the characteristic matrix for a plane wave propagating in a layered optical medium [Born 1999]. The bare silicon WG effective index was used to account for the input and output semi-infinite media

113 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… and the composite CWs WG region was modeled as a homogeneous layer with Lorentzian dispersion law: 4πΩ2 ( ) p . ε ω = ε b + 2 2 (5.5) ω −ω0 − 2iγω

Here ε b is the silicon slab WG effective permittivity, ω0 is resonance frequency, γ is the damping constant related to the half-width of the Lorentz line, and Ω p is the plasma frequency. The parameters of the Lorentzian dispersion relation were determined by a fitting procedure using transmission and reflection data provided by HFSS numerical modeling. For the most this method used to find effective parameters is similar to that reported by [Popa 2005]. The transmission and reflection spectra obtained by this way shown a good agreement with HFSS results for 50 nm and 100 nm of separation distance d , but failed to reproduce the reflection peak at 250 THz for a larger d =150 nm (Fig. 81). Note that data are plotted in a logarithmic (dB) scale and only the agreement in the upper part of the graphs is meaningful. For instance the difference in the reflection level of -20 dB and -30 dB doesn’t matter since only very little amount of the energy is carried by the light wave. a b c

Fig. 81. Transmission and reflection spectra for different separation distance: (a) d=50 nm; (b) d=100 nm; (c) d=150 nm. HFSS modeling: reflectance (magenta), transmittance (red). Homogeneous approach plane waves fit: reflectance (cyan), transmittance (blue).

Composite slab WG effective index determined from HFSS data using described procedure are shown on the Fig. 82. For the sake of comparison the bare silicon WG effective index also represented in same figure. In conformity with model predictions, composite slab WG effective index is higher than that of silicon slab below the resonance frequency and become lower above the resonance frequency.

114 5.6 Experimental study of hybrid metal-dielectric waveguides

a b c

Fig. 82. Real (blue) and imaginary (red) parts of the composite slab effective index for different separation distance: (a) d=50 nm; (b) d=100 nm; (c) d=150 nm. Bare silicon slab waveguide effective index traced in green.

It can be observed also that the effective index variation does not vary so much with the separation distance d . This result is better visible when tracing differential effective index variation represented in Fig. 83. It can be observed that the magnitude of the real part of the effective index variation in the vicinity of the resonance is of the order of 10-1. The imaginary part of the effective index variation increases with the separation distance, but the homogeneous model doesn’t provide any insight on the possible origins of such a behavior. It also fails to take into account the increase of reflection around 250THz. A more elaborated model is necessary to account for these effects. a b c

Fig. 83. Real (blue) and imaginary (red) parts of the composite slab differential effective index variation for different separation distance: (a) d=50 nm; (b) d=100 nm; (c) d=150 nm.

5.6.3 Layered model for metamaterials in guided wave configuration

The reflection peak at 250 THz indicates the presence of Bragg reflection interference effects. To verify this assumption we considered a sliced (layered) waveguide equivalent model represented in Fig. 84b. The transmission and reflection coefficients and the parameters of the Lorentzian dispersion relation were determined by taking EM slice thickness equal to that of the gold cut wires.

115 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

a b

Fig. 84. (a) Sketch of the hybrid CWs waveguide. (b) Layered (sliced) equivalent model.

The transmission and reflection coefficients for the equivalent sliced model are displayed in the Fig. 85. The agreement with HFSS modeling results is very good whatever is the separation distance. a b c

Fig. 85. Transmission and reflection coefficients for different parameter d: (a) d=50 nm; (b) d=100 nm; (c) d=150 nm. HFSS modeling: reflectance (magenta), transmittance (red). Layered approach plane waves fit: reflectance (cyan), transmittance (blue).

The validity of layered model can be further verified using experimental data. The parameters of Lorentz dispersion law are this time adjusted to fit transmission measurements. The only additional parameter introduced in the fitting procedure is the correction for the absolute value of transmission level, which is not provided by experimental measurements. Experimental and fitted results represented in the Fig. 86 are showing a good agreement, whatever is the separation distance d . The trend for the increase of reflection with separation distance d is also reproduced. The reflection and transmission levels given by HFSS modeling and those found from the fit of experimental data are also well consistent. The last point to verify is the consistency of the Lorentz dispersions characteristics obtained by fitting either HFSS or experimental data. a b c

Fig. 86. Experimental (red) and layered model transmission (blue) and reflection (green) spectra for cut wires loaded tapered waveguides with different d: (a) d=50 nm; (b) d=100 nm; (c) d=150 nm.

116 5.6 Experimental study of hybrid metal-dielectric waveguides

Real and imaginary parts of the composite slab differential effective index variation obtained from experimental and HFSS modeling results are displayed on the Fig. 87. We note the very good agreement for the differential index variation obtained by either fitting experimental or modeling results. However, in contrast to the homogeneous model behavior shown in the Fig. 83, the situation corresponding to the layered model is totally different. Here the magnitude of the effective index variation induced by the CWs resonances is strongly dependent of the separation distance between the CWs. For a separation distance d = 150nm the experimentally found magnitude of the real part of the differential effective index variation is as high as ± 0.7 in the vicinity of the resonance frequency. The magnitude of the differential index variation is even higher when CWs area is covered with n = 1.44 Cargille index liquid. The corresponding results are presented in the Fig. 88. For a separation distance d = 150 nm the experimentally found magnitude of the real part of the differential effective index variation is as high as ± 1.5 in the vicinity of the resonance (Fig. 88c). a b c

Fig. 87. Real (measurements blue, HFSS cyan) and imaginary (measurements red, HFSS magenta) parts of the composite slab differential effective index variation for different transverse separations between cut wires: (a) d = 50 nm, (b) d = 100 nm, (c) d = 150 nm. The dimensions of cut wires are 200×50×50 nm. Upper interface air.

a b c

Fig. 88. Real (measurements blue, HFSS cyan) and imaginary (measurements red, HFSS magenta) parts of the composite slab differential effective index variation for different transverse separations between cut wires: (a) d = 50 nm, (b) d = 100 nm, (c) d = 150 nm. The dimensions of cut wires are 200×50×50 nm. Cargille index liquid n =1.44 for upper interface.

117 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

The much lower magnitude of the effective index variation found for the homogeneous model as compared to the layered one is essentially related to the question what we are meaning by effective index in each case. In the homogeneous model the effective index means an average index defined for the entire CWs area. In contrast, for the layered model it means a local index defined only for each CWs slice area. It follows that when increasing separation distance d , the local index associated with the slice area can reach very high (and also very low) values in the vicinity of the CWs resonance. Silicon WG slab sections, which effective index is different, separate the slice areas. The composite WG behavior obviously deviate from that of homogeneous EM when the separation distance is increased and resulting periodic index profile matches Bragg interference condition. The substantial difference, as compared to the conventional dielectric PhC structures, is the resonant character of the local effective index. The presented so far results suggest that the local effective index, associated with the resonant element, is strongly dependent of the interaction with neighbors. The interaction between adjacent CWs elements is lower when the separation distance is higher. The tendency for the increase of the local effective index with separation distance d is well visible from the Fig. 87 and Fig. 88. The evolution of this trend with further increase of separation distance is addressed in the next section.

5.6.3.1 Evolution of Lorenz parameters with separation distance between CWs

The investigation of the evolution of the Lorentz dispersion law parameters for higher separation distance d is performed on the base of HFSS modeling data. Transmission and reflection spectra for d = 50, 100, 150, 200, 250 and 300 nm are shown on the Fig. 89. One can see that HFSS simulation data can be approximated with very good precision by the layered EM model. Results for local effective index variation, obtained by using fitting procedure, are shown in the Fig. 90. The increase of the magnitude of local effective index variation is confirmed, though a trend to growth saturation is also observed. To understand the reasons of this behavior it is instructive to analyze the evolution of parameters entering in the Lorentz dispersion formula. The dependence of the half-width of the Lorentz line of the transverse separation distance d is shown in the Fig. 91a. It turns out that the variation of the Lorentzian half-width is approximately inversely proportional to the separation distance. For high separation distance the Lorentzian half- width tends to an asymptotic value of around 3 THz. The observed behavior is due to the decrease of the interaction between adjacent CWs with distance. For lower interactions the quality factor of resonance become higher. The resonance frequency doesn’t seem to experience a strong dependence from the separation distance. The variation of the resonance frequency displayed in the Fig. 91b is around few percents.

118 5.6 Experimental study of hybrid metal-dielectric waveguides a b c

d e f

Fig. 89. Transmission (red and magenta) and reflection (blue and cyan) spectra of hybrid waveguides with CWs on the top for different parameter d : (a) d = 50 nm; (b) d = 100 nm; (c) d = 150 nm; (d) d = 200 nm; (e) d = 250 nm; (f) d = 300 nm. The dimensions of cut wires are 200×50×50 nm. Blue and red lines correspond to the spectra obtained from HFSS simulations. Cyan and magenta lines correspond to the spectra obtained from the layered effective media slab model. The y -axis is scaled logarithmically. a b c

d e f

Fig. 90. Real (blue) and imaginary (red) parts of the composite slab local effective index variation for different transverse separations between cut wires: (a) d = 50 nm; (b) d = 100 nm; (c) d = 150 nm; (d) d = 200 nm; (e) d = 250 nm; (f) d = 300 nm. The dimensions of cut wires are 200×50×50 nm. Upper interface air.

119 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… a b c

Fig. 91. Dependence of Lorentz parameters on the transverse separation distance between adjacent cut wires: (a) half-width, (b) central frequency, and (c) strength factor.

The dependence of the Lorentz line strength factor shown in the Fig. 91c is significantly more important. The growth of the strength factor also shows a tendency to saturation for higher separation distance. Recall that this parameter is proportional to the squared plasma frequency (see Eq. 5.5). The observed behavior may be explained by the increase with separation distance of Bragg interference effects that results in a more efficient interaction of light waves with metal electrons. The resulting effect is then similar to the increase of the electrons concentration. For a separation distance d = 300nm the magnitude of the real part of local effective index varies from 3 to -2. The resulting contrast of effective index variation is twice higher as compared to that of etched silicon WG. These results show that using metamaterials in a GWC indeed allows an efficient engineering of the effective index.

5.6.4 Engineering of composite waveguide effective properties by cut wires geometry design

5.6.4.1 Control of the resonance frequency by cut wires length

The design of experimental sample dedicated to the study of CWs length influence on the composite WG effective properties is schematically depicted in the Fig. 92. The sample is divided in three areas corresponding to different transversal separation d between CWs. Each area of the sample has a different number of CWs chains along the direction of light propagation: N y = 20, 40 or 60. The length of CWs varies from 100 nm to 200 nm with a step of 11 nm. The thickness of deposited gold is 50 nm. The geometry of the WGs is the same as for the first sample from the set B. In complement to CWs loaded WGs, additional tapered and straight WGs not loaded with CWs are also included in the design for reference or calibration purposes. At the total there are 126 WGs implemented in this sample design.

120 5.6 Experimental study of hybrid metal-dielectric waveguides

Fig. 92. Sketch of the experimental sample and main design parameters. a b c

d e f

Fig. 93. SEM view of cut wires with different parameters. (a) 20 chains of cut wires; (b) 40 chains of cut wires; (c) 60 chains of cut wires. Transverse separation distance d between chains of cut wires is (d) 50 nm, (e) 100 nm, or (f) 150 nm.

At the end of the fabrication process all 90 CWs arrays areas on the sample were inspected and characterized by SEM imaging. The Fig. 93 shows a collection of few SEM images illustrating different types of CWs arrays structures implemented on the sample. The CWs array can be constituted of 20 longitudinal periods (Fig. 93a), 40 periods (Fig. 93b) or 60 periods (Fig. 93c). The separation distance between CWs is 50 nm (Fig. 93d), 100 nm (Fig. 93e) or 150 nm (Fig. 93f). The experimental parameters of CWs arrays are close to nominal design values, the relative error does not exceed several percents. 121 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL… a b c d

Fig. 94. Transmission spectra for cut wires loaded tapered waveguides. Cut wires transverse separation d = 50 nm. Cut wires length is: (a) 156 nm; (b) 178 nm; (c) 189 nm; (d) 200 nm. a b c d

Fig. 95. Transmission spectra for cut wires loaded tapered waveguides. Cut wires transverse separation d = 150 nm. Cut wires length is: (a) 156 nm; (b) 178 nm; (c) 189 nm; (d) 200 nm. a b c

Fig. 96.Experimentally determined resonance frequency variation with cut wires length for different separation distances: (a) d=50nm; (b) d=100nm; (c) d=150nm. a b c

Fig. 97. HFSS modeling determined resonance frequency variation with cut wires length for different separation distances: (a) d = 50 nm; (b) d = 100 nm; (c) d = 150 nm.

The sample was characterized using the end-fire setup. Figures 93 and 94 shows transmission spectra for tapered WGs loaded with different length CWs arrays. The transverse separation d between CWs is either 50 nm (Fig. 94) or 150 nm (Fig. 95). The transmission spectra for separation

122 5.6 Experimental study of hybrid metal-dielectric waveguides distance d = 100 nm were measured as well, but are not presented because of essentially similar behavior. All displayed spectra are measured for TE polarization. TM spectra, as for the previous sample, do not reveal any resonant absorption. As expected, the resonance frequency is red shifting when the length of CWs increases: for L =156 nm the resonance frequency is f0 ≈ 230 THz, for

L =200 nm the resonance frequency is f0 ≈ 185 THz. Note, that the position of the resonance does not depend on the transverse separation distance d . The experimental results for the dependence of the resonance frequency of CWs length are summarized in the Fig. 96. Similar results obtained from HFSS modeling are shown in the Fig. 97. Both sets of data are in a very good agreement. As for the previously investigated sample the resonance frequency practically doesn’t depend of the CWs separation distance. The experimental results also show that the resonance frequency doesn’t depend on the length of CWs array along the propagation direction. This confirms the EM behavior of composite WGs.

5.6.4.2 Control of anisotropy properties by CWs orientation angle

The following step of our study addresses the engineering of composite WG anisotropy properties. This functionality is highly desirable in view of TO applications. The control of anisotropy properties is intended to be achieved through the orientation of CWs with respect to the direction of light propagation. The design of experimental sample dedicated to the study anisotropy properties is schematically depicted in Fig. 98. Three types of CWs design are considered for this study: • all CWs are rotated by the same angle and in the same direction (Fig. 98a); • half of the CWs array is rotated in one direction and the other half is rotated by the same angle but in opposite directions (Fig. 98b); • two neighbor CWs in a column are rotated by the same angle but in opposite directions (Fig. 98c). The reasons for such a design are explained later. The transverse separation distance between CWs d =50 nm or 100 nm. The length of CWs is varying from 156 nm to 200 nm with the step of 11 nm. The width of CWs is 50 nm, the height of deposited gold is 50 nm. The number of rows is

N x ≈ 30, the number of chains is N y = 20. As it was shown in the previous section the higher N y value will not change the resulting effective parameters of MMs. Rotational angle of CWs is changing from 0° to 60° by step of 15°. In complement to CWs loaded tapered WGs, additional tapered and straight WGs not loaded with CWs are also included in the design for reference or calibration purposes. In total the third sample of the set B includes 150 WGs with CWs on the top and 45 tapered or straight WGs without CWs on the top.

123 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

a b c

Fig. 98. Three different patterns that were used for the sample fabrication.

The sample was fabricated under same conditions as those used for the first two samples from the set B. SEM images of MM areas are presented on the Fig. 99. Unlike to the previous samples the third sample revealed unexpected problem. The CWs rotated in one direction were washed out from the sample during the lift-off process. At the same time the fabrication of CWs rotated in the opposite direction was totally successful (Fig. 99b). Furthermore, these defects were not related to a particular direction of rotational angle. On some patterns these defects appeared for counterclockwise rotated CWs (for instance, Fig. 99c), while on other patterns for clockwise rotated CWs (Fig. 99b). Also, the higher the rotational angle is, the higher probability of defects is. However, this type of defects was totally absent when all the CWs elements were rotated in the same direction or not rotated at all (Fig. 99a and Fig. 99d). The problem is most likely connected to the pixelization of the programmed pattern by NanoBeam machine. The successful writing of rotated CWs could be determined by the first line in the pattern: CWs rotated the same direction as the first line were written correctly, whereas CWs rotated to the opposite direction show tendency to be washed out from the sample. We proposed a solution to the problem: we separate CWs rotated in different direction to different pattern levels. Thus NanoBeam machine writes first, for instance, the CWs rotated to the left and only after it proceeds to the CWs rotated to the right. This simple method allowed us to obtained a good sample 3.2 for our further studies of anisotropy properties. The Fig. 100 shows SEM images of different type patterns from the sample 3.2. An enlarged view of fabricated CWs arrays is shown in the Fig. 101.

124 5.6 Experimental study of hybrid metal-dielectric waveguides

a b c d

Fig. 99. Electronic lithography writing defects on the sample with rotated gold cut wires: some rotated cut wires were washed out during the lift-off process. a b c d

Fig. 100 SEM images of the sample 3.2 with rotated cut wires. Cut wires areas show no gaps in the patterns. a b c

Fig. 101. SEM view of cut wires: (a) all cut wires are rotated in the same direction; (b) the bottom cut wires of the metamaterial pattern are rotated counterclockwise and the upper cut wires of the structure are rotated clockwise; (c) adjacent lines of cut wires are rotated in opposite directions.

125 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

a b c

d e

Fig. 102. TE transmission spectra for the cut wires 178×50×50 nm with d = 100 nm for different rotational angles from (a) 0° to (e) 60°. a b

Fig. 103. TE polarization HFSS modeling results for composite waveguides loaded with rotated cut wires. (a) Variation of the resonance frequency with rotational angle. (b) Variation of the minimal level of transmission coefficient at the resonance frequency on the rotational angle of cut wires.

The TE polarization experimentally measured transmission spectra for composite WGs with different CWs rotation angles are shown in Fig. 102. The CWs array pattern corresponds to that represented in Fig. 101c. The CWs dimensions are 178×50×50 nm. The transverse separation d =100 nm. A marked dip in transmission related to the resonant absorption of CWs is present for all the rotation angles. The drift of the resonance frequency is mostly due to the variation of the CWs length related to minor technological imperfections persisting for the electronic lithography writing of oblique elements. Aside the perturbations induced by the technological issues, the general behavior of rotated CWs composite WGs is in a good agreement with HFSS modeling predictions displayed in the Fig. 103. The last figure shows that resonance frequency is essentially determined by 126 5.6 Experimental study of hybrid metal-dielectric waveguides the CWs length and only little affected by the rotational angle. The resonant absorption is highly efficient up to a rotation angle of 60°, and then rapidly decreases for higher rotation angles. Note that because of the limitations inherent to HFSS software, it was possible to model only the WGs with CWs pattern corresponding to that represented in the Fig. 98c. On the Fig. 104 one can see the comparison of TE polarization transmission spectra for composite WGs with the same dimensions of CWs 178×50×50 nm, d =100 nm but for different designs corresponding to that depicted in the Fig. 98. Angle of rotation of CWs is 30°. It can be observed that the dip in transmission spectrum shown in the Fig. 104c, corresponding to the configuration represented in the Fig. 98c, is significantly more pronounced as compared to the dips in transmission spectra observed for the other configurations shown in Fig. 104(a, b) and corresponding to the configurations represented in the Fig. 98(a, b), correspondingly. At the resonance frequency the difference in transmission level is around 15 dB. Also as it is evident from the Fig. 104(a, b) the transmission spectra are practically identical for two first configurations corresponding to that represented in the Fig. 98(a, b). a b c

Fig. 104. TE transmission spectra for different metamaterial patterns with cut wires 178×50×50 nm and d = 100 nm. The rotation angle is 30° for all cases.

To explain the observed behavior we can use analogy with half transparent plates model (Fig. 105). The case when adjacent CWs are rotated in opposite directions can be viewed as an analogy of half transparent plates system represented in Fig. 105a. The characteristic feature of this configuration is that part of the rays are backward reflected. When applied to our case, it means that variation of the local effective index induced by resonant elements makes them act as partially transparent mirrors. The presence of backward reflections leads to interference effects and results in more efficient resonant absorption. The situation is greatly similar to that considered in 5.6.1.2. For the case when adjacent CWs are rotated in the same direction there is no additional backward reflection and resonant absorption is lower.

127 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

a b

Fig. 105. Analogy with half-transparent plates. (a) When cut wires are rotated in the same direction there is no additional backward reflection. (b) When the adjacent cut wires are rotated in opposite directions additional backward in reflection occurs.

The observed difference in transmission properties depending on the mutual orientation of adjacent CWs represents an unambiguous proof of anisotropy effect manifesting through the reflection induced by local index variation. The component of polarization vector of a single rotated C CW along y axis pxy (see Fig. 106) is responsible for re-emission of the light sideward. Hence, the degree of anisotropy can be controlled by the rotation angle of CWs. In a zero order approximation it can be considered that the cross term of effective permittivity is proportional to the sine of the rotational angle.

C Ω pxy ∝ sin β ε xy ∝ sin β

Fig. 106. Scheme of a single rotated cut wire with components of the vector of polarization. The component of effective permittivity in this case is proportional to the sine of rotation angle β .

The second evidence of anisotropy effects in hybrid WGs with rotated CWs on the top can be obtained through observation of TM transmission spectra. Up to now our experimental investigations focused on the composite WG effective properties for TE polarization. For the given frequency range no efficient excitation of CWs resonances can obtained when electric field is perpendicular to their long axis. The situation becomes however different for the case of rotated CWs. It is important to remind, that the wave modes for 2D waveguides are actually not pure TE or C

TM modes, but quasi-TE or quasi-TM. Quasi TE mode has magnetic field component H y along C C C the wave vector k . Quasi TM mode has electric field component E y along the vector k (see Fig.

128 5.6 Experimental study of hybrid metal-dielectric waveguides

107). The component H y in TE polarization does not induce additional effects for any configuration of MM pattern since gold CWs are not magnetic (Fig. 107a). On the contrary, the C component E y in TM polarization provides excitation of localized surface plasmons in the case of rotated CWs (see Fig. 107b). The resonance frequency must be exactly the same as for TE polarization since the resonance frequency of localized surface plasmons is determined by the length of the resonant elements. a b

Fig. 107. (a) TE mode of 2D dielectric waveguide and single rotated cut wire. (b) TM mode of 2D dielectric waveguide and single rotated cut wire.

The experimental transmission spectra shown in the Fig. 108 are totally confirming this assertion. We consider the case of composite WG with a CWs pattern corresponding to that depicted in the Fig. 98c. The dimensions of CWs are 178×50×50 nm, d = 100 nm and the rotation angle is 45°. As it can be seen from the results presented in the Fig. 108b, a marked dip is observed for TM polarization transmission. The magnitude of the transmission dip and the resonance frequency are the same as for TE polarization shown the in the Fig. 108a. The obtained results are interesting in view of polarization conversion applications and illustrate the great diversity of metamaterials operation in a GWC.

a b

Fig. 108. Transmission spectra of loaded tapered waveguides with cut wires: (a) TE-mode; (b) TM - mode. Rotation angle of cut wires is 45°.

129 CH APTER 5. EN GIN EERIN G OF M ETAM ATERIAL…

5.7 Conclusions

This chapter is considering the EMA in a GWC. The interest is motivated by the advantages offered by the planar technology for the GWC, which may constitute a promising alternative to the multi- layered MMs approach in the optical domain. The aim of the study is to achieve an efficient control over the flow of light in the WG using effective index variation induced by MM resonances. The engineering of the resonance frequency was in particular addressed. The retained solution consists to use a thin interlayer of low index dielectric in order to conciliate high resonance frequency requirement with that of important aspect ratio necessary for anisotropy control. The EMA developed for a single MM layer in FSC was extended to a GWC. It served for the determination of composite WG effective parameters. The investigation of the EMA for MMs in a GWC allowed the establishing of main guiding rules for the engineering of MM effective properties. The possibility for achieving a significant effective index variation using silicon slab WG covered with 200×50×50 nm cut wires was demonstrated by numerical modeling. The CWs represent probably the most elementary type of MMs used for building more complex geometry MMs. Their greatest advantage is the essentially non-magnetic behavior with µ ≈ 1 due to the absence of notable coupling between the electrical and magnetic resonances. The experimental realization and characterization of such a composite WG operating in the spectral domain around 1.5 µm shown to be in a good agreement with modeling predictions. The effective index variation achieved for this structure in the limits of validity of homogeneous approach is of the order of ± 10-1. A much higher magnitude can be obtained when considering local index variation induced by the resonant element. The magnitude of local index variation in the vicinity of the resonance frequency deduced from experimental data is as high is ± 1.5. In can be almost twice higher when the separation distance between CWs is further increased. The performed analysis identified the two main factors explaining this behavior: 1. Increase of the resonance quality factor due to the lower interaction with neighbor elements. 2. Interference phenomena producing an effect similar to the increase of the plasma frequency. The further validation of the EMA in a GWC was performed on the example of the composite WG with different length for CWs array. The control of the resonance frequency through the length of CWs resonant element was experimentally demonstrated. The problem of anisotropic effective index was addressed through the realization and experimental investigation of a composite WG loaded with rotated CWs array. The experimentally

130 5.7 Conclusions observed difference in transmission properties depending of the mutual orientation of adjacent CWs represents an unambiguous proof of anisotropy effect. The degree of anisotropy can be controlled by the rotational angles of CWs angle. In a zero order approximation it can be considered that the cross term of effective permittivity is proportional to the sine of the rotational angle. In contrast to the previous designs, composite WGs with rotated CWs array can operate not only in TE, but also in TM polarization. This effect was experimentally demonstrated. The obtained results are interesting in view of polarization conversion applications and illustrate the great diversity of metamaterials operation in a GWC. The possibility for controlling at the nanoscale the local effective index can be used in TO applications.

131

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[Ghasemi 2010] R. Ghasemi, P.-H. Tichit, A. Degiron, A. Lupu, A. de Lustrac, “Efficient control of a 3D optical mode using a thin sheet of transformation optical medium,” Opt. Exp. 18, 20305 (2010)

[Kanté 2009] B. Kanté, D. Germain, A. de Lustrac, “Experimental demonstration of a nonmagnetic metamaterial cloak at microwave frequencies,” Phys. Rev. B 80, 201104 (2009)

[Kelly 2003] K.L. Kelly, E. Coronado, L.L. Zhao, G.C. Schatz, “The Optical Properties of Metal Nanoparticles: The Influence of Size, Shape, and Dielectric Environment,” J. Phys. Chem. B 107, 668 (2003)

[Knight 2011] M.W. Knight, H. Sobhani, P. Nordlander, N.J. Halas, “Photodetection with active optical antennas,” Science 332, 702 (2011)

[Olivier 2001] S. Olivier, C. Smith, M. Rattier, H. Benisty, C. Weisbuch, T. Krauss, R. Houdré, U. Oesterlé, “Miniband transmission in a photonic crystal coupled-resonator optical waveguide,” Opt. Lett. 26, 1019 (2001)

[Palik 1998] E.D. Palik, H andbook of Opt ical Constan ts of Solid s, Academic Press, New York (1998)

[Popa 2005] B.-I. Popa, S.A. Cummer, “Determining the effective electromagnetic properties of negative-refractive-index metamaterials from internal fields,” Phys. Rev. B 72, 165102 (2005)

[Reinhard 2010] P.O. Reinhard, R. Beigang, M. Rahm, “Experimental and numerical studies of terahertz surface waves on a thin metamaterial film,” Opt. Lett. 35, 1320 (2010)

133

Chapter 6 Summary and perspectives

In the preceding chapters we spoke about MMs at NIR optical regime, their practical realization as well as EMA in regard to such structures. This chapter is dedicated to the discussion of the main results obtained in the frame of the thesis work. We show the central points outcoming from the performed examination in free space and GWCs of experiment. We also discuss feasible applications and show some possible perspectives of MM structures in the NIR range of frequencies.

6.1 Summary

The main aim of the work was investigation of possible application of MM at NIR optical frequencies when typical size of elements becomes extremely small which causes whole bunch of technological issues and issues in theoretical description of such structures. We considered two types of MMs: 1) MM in FSC when the incidence electromagnetic wave interacts with single metafilms and the incidence angle varies from 0° to 89°; 2) MM in GWC when electromagnetic wave bound in dielectric WG interacts with MM placed on top of the WG. For MM in FSC we reviewed EMA. The four criteria for single metafilms were chosen. We state that the criteria are satisfactory cut-off statements that allow to recognize MM structures that can be described by EM theory. The criteria bring us to several important conclusions. First of all, we saw that the thickness of MM single layer can be described by EM layer and the thickness of EM layer equals to the real thickness of metallic particles. By this, we exclude former vagueness in the determination of this parameter. We showed that EM layer with the thickness equaled to the thickness of MM particles is essentially non-magnetic far from the MM resonance. The changes in effective permeability at the resonance rest negligible for the single MM layer with thickness h ≤ 60 nm. We showed that the effective permittivity linearly dependent on the surface FF ρ irrespectively of which design parameter is varying: L , W , A1 or A2 . It was demonstrated that the optical length of EM layer is linearly dependent on the thickness of metallic particles h when h ≤ 30 nm. However, if one fixes the resonance frequency of MM particles by adjusting the design parameters, the linearity remains considerable even for high parameters h up to 60 nm. The FF at that can be varied in wide range from several percents up to 135 CH APTER 6. SUM M ARY AN D PERSPECTIVES

20–30 %. Hence the resonance frequency is substantial parameter in the determination of the effective permittivity function. In other words, when the resonance frequency is fixed for the same FF the shape of the resonant elements or surrounding media do not influence the effective permittivity. Of course, this is valid only for thin metafilm layers when h ≤ 60 nm. Multilayered MM films are considered in the following section. Note, that the effective RI for considered MM in FSC can achieve the values of 10 close to the resonance region, which is a very significant value. Using simple EM model independence of the effective permittivity on the incidence angle was demonstrated for s-polarized waves. The incidence angle can be varied from 0° up to 85°. Moreover, the analysis of transmission and reflection behavior in the vicinity of the Brewster’s angle for p-polarized light is found to be also consistent with EM layer model. We showed experimentally the validity of EM model for two different types of MM patterns: simple gold CWs on top of glass substrate and gold SRRs with continuous wires on top of silicon substrate. Good agreement between experimental and simulation data were observed. The effective permittivity functions retrieved from experimental data do not dependent on the incidence. We considered MM in GWC. This experimental configuration allows studying multilayered MM structures and it is a transitional step to fabricating real TO devices for communicational purposes. The technique of MM realization in GWC was developed. We are capable to fabricate patterns of CWs with typical dimensions around 40 nm in the direction of the light propagation and transversal separation of the same order. The CWs can be aligned parallel to each other and perpendicular to the WG walls or rotated at the chosen angle and assembled in different designs. Proposed technique of technological realization is a step towards real hybrid MM-dielectric devices at NIR range of frequencies. We introduce the EM model to describe MMs in GWC. The method allows to estimate local effective RI of the hybrid metal-dielectric WGs. We studied the influence of the orientation of MM particles, their dimensions, the WG thickness and surrounding media on the effective parameters of EM. We showed the possibility of the optical response engineering by changing listed parameters. We observed experimentally the remarkable dip in the transmission spectra when the electromagnetic wave had TE polarization. The characteristic frequency was the same as it was predicted from HFSS simulations. It was shown that high values of local effective RI could be achieved in this case. TM polarization of the electromagnetic waves, as it was expected, does not interact with resonant particles at the chosen frequency range so the transmission spectra rests the same for WGs with and without MM on the top.

136 6.1 Summary

We emphasize that the effective RI cannot be increased by simple increasing the number of MM layers on top of the WG, since top layers of MM cannot be efficiently excited by the electromagnetic wave that is bound in the dielectric WG. In this case the evanescent tales of electromagnetic wave mostly interacts only with the lowest layer of MM, which was shown by HFSS simulations. The additional layers of MM do not change considerably the effective RI. The local effective RI can be controlled by changing of transversal separation between metallic particles. When the transversal separation is small d ≤ 100 nm the hybrid WG still can be approximated by homogeneous model where whole MM region of the WG is replaced by homogeneous WG slab with effective constitutive parameters. While for high values of d the interference effects must be considered. The higher is d the higher is quality factor of EM layers in layered EM model, the higher is local RI. The last fact leads to the rise of interference effects in the hybrid WG. We showed in practice and by computer simulations the possibility to obtain high values of RI contrast near the resonance region: ∆n = ± 1.5. Changing of the rotational angle of CWs brings us close to the practical realization of anisotropic MM layers. We observed the rise of additional wave components when the CWs are rotated. That indicates that now the hybrid WG is anisotropic. For instance, the component of electric field along the direction of light propagation presenting for TM polarized waves leads to the appearance of the dip in transmission spectra. Also we showed that in zero approximation the cross term of the effective anisotropic permittivity is changing proportionally with sin β, where β is the angle of rotation of CWs.

6.2 Perspectives

In this section we present some perspective directions of further development of the results discussed in this work.

6.2.1 Devices based on transformation optics

As we discussed in the Chapter 1, one of the main trends in MM use is their application for TO devices. The realization of TO devices requires high values of the effective RI as well as possibility to vary RI in a wide range. In this work we showed theoretically and experimentally the means to make anisotropic MMs in GWC at NIR range of frequencies, which is an essential condition for development of TO devices. The hybrid WG with anisotropic MM on the top can serve as an 137 CH APTER 6. SUM M ARY AN D PERSPECTIVES elementary cell for TO devices. If one knows how to build the anisotropic permittivity function, it is possible to proceed to the demonstration of devices such as integrated tapers for communicational purposes.

6.2.2 Nano-antenna and near-field enhancement properties

MM on top of WG can be used also as assembly of plasmonic antenna. Part of energy can be redistributed. On the Fig. 109b one can see a far field image of MM area made by IR camera when the excitation frequency is close to the resonance frequency (around 197 THz). To efficiently excite CWs TE mode of the light was used. On the figures one can observe a radiation coming from the metamaterial area. Similar images were obtained for Yagi-Uda antennas in GWC [Arango 2012]. a b c

Fig. 109. (a) Image made by IR camera placed above metamaterial area on top of SOI waveguide. The metamaterial area consistent from cut wires with dimensions 200×50×50 nm and transverse separation distance d = 150 nm. The camera is focused on the waveguide. (b) The same waveguide when the IR camera is focused on the metamaterial area. (c) SEM image of metamaterial area with cut wires with dimensions 200×50×50 nm and transverse separation distance d = 50 nm (left), SNOM image of the same area (middle) and HFSS field distribution in the metamaterial area top view (right). The white arrows indicate the direction of light propagation.

This antenna property can be used for several applications. Firstly, MM on top of a WG can be a functional element that helps to extract/insert light from/to the WG. Recently it was shown that a single gold CW could couple up to 20% of the input light into the WG [Arango 2012]. Therefore MM in GWC can be used, for instance, for multilevel WG structures where MM area of the first WG radiates energy that can be caught by MM area of the second WG. Secondly, MM on top of WGs can be used as biological or chemical sensors. It is possible due to the high sensitivity of MM resonance to the dielectric environment. And thirdly, such structures can be used as resonant photodetectors. Moreover, scanning near field optical microscopy (SNOM) was carried out at L aborat oi re d e N an ot echn ol ogi e et In st rumentati on Opt iqu e (LNIO). The experiment, performed in the same spectral range, revealed for TE polarized light a strong enhancement of the electric field confined in the region between the ends of the adjacent CWs (Fig. 109c, middle). These results are

138 6.2 Perspectives in a very good agreement with HFSS numerical simulations (Fig. 109c, right). The enhancement of the electric field between rows of gold CWs is very predictable, accepting the general electromagnetism theory: the skin depth of gold at optical frequencies is around several nanometers. As it can be seen from both experimental and modeling data the repartition of the electric field along the propagation direction across the MM area is quite uniform. The enhancement of the electric field can be used for exaltation of nonlinear optical effects.

6.2.3 Guiding the light by metamaterial slab

It is also interesting to consider MM structures as main propagating material for the light. The hybrid WG can be approximated by EM homogeneous WG slab only when the interaction between MM and the dielectric WG slab is weak. The metallic particles in this case behave as a small perturbation. It is also possible to consider pure MM WG without an additional silicon slab (see Fig. 110a). By that the interaction between dielectric WG and metallic particles is excluded completely from the model, and the interaction between metallic MM elements becomes dominant. There are works dedicated to theoretical study of wave modes that can be supported by MM WGs with given functions of ε and µ [Liu 2008] and to some interesting effects in such WGs, for instance, slowing down of light pulses [Zentgraf 2010]. Here we answer the question: can MM metafilm guide electromagnetic waves in the NIR range of frequencies. a b

Fig. 110. (a) Sketch of metamaterial waveguide. Light propagates through metamaterial. (b) Homogeneous effective medium model can be applied in this case.

We consider 2D array of gold CWs 200×50×50 nm in between of two silicon plates with the thickness of 200 nm. Silicon plates serve to bring the light into and collect it after the MM WG. The

WG is embedded in silica ( n = 1.4). The longitudinal period A1 = 300 nm. The Fig. 110b shows the transmission, reflection and loss spectra when the transverse separation d = 50 nm for TE polarization of the light. The losses are defined by subtraction reflectance and transmittance from the unit and represent radiation and absorption in MM film. Absorption in auxiliary silicon WGs is negligible in this case. One can see that when frequency f > 220 THz ( λ < 1.35 dm) MM WG behaves like a mirror. For smaller frequencies the reflectance diminishes whereas the transmittance contrarily increases: the metamaterial film starts to guide electromagnetic waves. The resonance in losses spectra can be observed with the resonance frequency of 220 THz.

139 CH APTER 6. SUM M ARY AN D PERSPECTIVES

a b

Fig. 111. (a) Transmission (red curve), reflection (green curve) and loss (blue curve) spectra for 10 chains of gold cut wires with transverse separation between particles d = 50 nm. (b) The real (green) and imaginary (red) components of the effective refraction index of 10 chains of cut wires for d = 50 nm (round marks), d = 100 nm (square marks) or d = 1 50 nm (triangle marks); dashed gray line represents the refractive index of surrounding dielectric medium (n = 1.4).

From complex reflection and transmission spectra one can retrieve the effective RI using homogeneous EM model presented on the Fig. 110b. The thickness of EM equals to the thickness of silicon slabs and the EM length equals to the separation distance between silicon slabs filled with MM. Extracted RI n for d = 50, 100 and 150 nm is shown in the Fig. 111b. Frequency range is limited from 100 THz to 230 THz where homogeneous model can be applied for all three transverse separations. The effective RI matches with RI determined from spatial field distribution in the region of the metallic film. In the case of strong interaction between adjacent cut wires

( d = 50 nm) the contrast between RIs of MM WG and surrounding SiO2 is 0.66 at 200 THz. That is 1.4 times higher than in cases of weaker interaction with d = 100 nm and 1.9 times higher than in case when d = 150 nm. Note, that in the region of long wavelengths the transmittance differs from the unit, which indicates to non-resonance dissipations in the system (Fig. 111a). This energy can be efficiently redirected along the direction of wave propagation by increasing the thickness of MM film i.e. by adding MM layers. Contrarily to the case examined in the Chapter 5 here one can expect a noticeable increase in the contrast of effective RI when full thickness of MM WG is less than the thickness of auxiliary silicon WGs. We fix d to be 50 nm (strong coupling between metallic particles) and change the number of MM layers up to 4 layers. The separation distance between layers in stack is set to 50 nm. The refraction, transmission and loss spectra are presented in Fig. 112a. Now one can see that increasing of number of layers leads to increasing of transmission level in the long-wave region, as it was expected. The transmission and reflection spectra do not change a lot near the resonance frequency, while the contrast of RI at 200 THz is remarkably higher for multilayered MM WG than for 1-layered film (see Fig. 112b). The increasing of RI comes mostly from the phase changing of transmitted wave. The clear tendency to saturation of the RI contrast can be observed,

140 6.2 Perspectives

when the full thickness of MM WG becomes higher than the thickness of SiO2 WG (3 layers and more). c d

Fig. 112. (a) Transmission (red curve), reflection (green curve) and loss (blue curve) spectra for 3D array of gold cut wires with transverse separation between particles d = 50 nm; distance between layers is 50 nm. N y=10. 1 layer (round marks), 2 layers (square marks), 3 layers (triangle marks) or 4 layers (rhombus marks); (b) Contrast in RI for different number of layer at 200 THz.

Consideration of pure MM WG without silicon slab can give us a possibility to increase contrast of RI variation by multiplying layers of gold particles involved in the process of light propagation. It was shown that when increasing the number of layers, the WG exhibit a higher effective RI without additional losses.

6.2.4 Engineering of metallic metamaterials losses

One of the main problems of metallic MM rests the presence of losses. Though we showed that using simple MM consistent from 2D array of gold CWs could result in relatively high contrast in RI, the losses play significant role. There are two kinds of losses that are related to the considered MMs: Joule losses, associated with heating of the conductor by the electric current, and extrinsic radiation losses. It is practically impossible to exclude heat losses from a system. Though there is an approach to reduce radiation losses concerned with the idea of dark plasmon modes. However we shall note that in some devices presence of high losses is a necessity, for example, systems with distributed feedback (DFB).

6.2.4.1 Dark modes

All MMs we discussed in this work are formed by symmetrical translation of the unit cell. Spatial symmetry breaking can lead to appearance of dark plasmon modes (also known as trapped modes) characterized by very high quality factors [Fedotov 2007]. These modes have a pure near-field nature and are weakly coupled to free space. Special asymmetry can be a result of some translation shifts while reproducing the unit cell, or of an unbalance in the unit cell or even in a single element. Dark 141 CH APTER 6. SUM M ARY AN D PERSPECTIVES modes were reported in several structures, for instance, in coupled CWs [Christ 2008], in a single- dolmen structure [Gallinet 2013], in z-shaped atoms [Dhouibi 2013], in non-concentric ring/disk nanocavities [Hao 2009], et c. Dark modes can arise from high-order multipole resonances (spherical nanoparticles) or from interaction between bright modes of coupled nanoparticles. Most of the time dark modes are related to the excitation of Fano resonances. Fano resonance appears as constructive and destructive interference of a narrow resonance with a broad spectral line or continuum. [Luk’yanchuk 2010]. We propose to excite dark modes in MM structures in order to minimize the radiative losses.

6.2.4.2 Gain media based on metamaterials

DFB lasers have an active media periodically structured as a diffraction grating. It can be achieved by simple placing the diffraction grating on top of an active media layer. Conventional lasers use two discrete mirrors to form the optical cavity, when DFB lasers use the grating as a selective frequency element and as a mirror reflecting the light inside the cavity. DBF lasers are constructed in such a way to produce a very narrow band of wavelengths, which is determined by the grating period. So they give as output a single longitudinal lasing mode. The structure of DFB diode laser developed by LPN group can be seen on the Fig. 113a. As a DFB element chromium (Cr) grating is used. a b

Fig. 113. (a) Sketch of distributed feedback laser based on metallic Bragg grating; (b) Sketch of proposed distributed feedback laser based on metamaterials consistent from gold cut wires.

Since MMs considered in this work have shown high level of losses (high value of imaginary part of the effective RI), it is possible to replace Cr grating by the MMs (see Fig. 113b). MM design gives the means to engineer the complex effective RI. As it was shown MM resonances are very sensitive to the environment.

142

References

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[Christ 2008] A.Christ, O.J.F. Martin, Y. Ekinci, N.A. Gippius, S.G. Tikhodeev, “Symmetry breaking in a plasmonic metamaterial at optical wavelength,” Nano Lett. 8, 2171 (2008)

[Dhouibi 2013] A. Dhouibi, S.N. Burokur, A. Lupu, A. de Lustrac, A. Priou, “Excitation of trapped modes from a metasurface composed of only z-shaped meta-atoms,” App. Phys. Lett. 103, 184103 (2013)

[Fedotov 2007] V.A. Fedotov, M. Rose, S.L. Prosvirnin, N. Papasimakis, N.I. Zheludev, “Sharp trapped-mode resonances in planar metamaterials with a broken structural symmetry,” Phys. Rev. Lett. 99, 147401 (2007)

[Gallinet 2013] B. Gallinet, O.J.F. Martin, “Refractive index sensing with subradiant modes: a framework to reduce losses in plasmonic nanostructures,” ACS Nano 7, 6978 (2013)

[Hao 2009] F. Hao, P. Nordlander, Y. Sonnefraud, P. van Dorpe, S.A. Maier, “Tunability of subradiant dipolar and Fano-type plasmon resonances in metallic ring/disk cavities: implications for nanoscale optical sensing,” ACS Nano 3, 643, (2009)

[Liu 2008] N. Liu, H. Guo, L. Fu, S. Kaiser, H. Schweizer, and H. Giessen, “Three-dimensional photonic metamaterials at optical frequencies,” Nat. Mater. 7, 31 (2008)

[Luk’yanchuk 2010] B. Luk’yanchuk, N.I. Zheludev, S.A. Maier, N.J. Halas, P. Nordlander, H. Giessen, C.T. Chong, “The Fano resonance in plasmonic nanostructures and metamaterials,” Nat. Mater. 9, 707 (2010)

[Zentgraf 2010] T. Zentgraf, J. Valentine, N. Tapia, J. Li, X. Zhang, “An optical “Janus” device for integrated photonics,” Adv. Mater. 22, 2561 (2010)

143

List of Publications

Journal publications

/1/ N. Dubrovina, L. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. de Lustrac, A. Vial, G. Lerondel, and A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” Applied Physics A 109, 901-906 (2012).

/2/ R. Ghasemi, N. Dubrovina, P.đH. Tichit, A. Degiron, A. Lupu, and A. de Lustrac, “Transformation optics and infrared metamaterials for optical devices,” Applied Physics A 109, 819-823 (2012).

/3/ A. Lupu, N. Dubrovina, R. Ghasemi, A. Degiron, and A. de Lustrac, “Metalþdielectric metamaterials for guided wave silicon photonics,” Opt. Exp. 19, 24746þ24761 (2011).

Conferences proceedings

/1/J. Beltran, N. Dubrovina, R. Salas-Montiel, H. Marquez-Becerra, A. de Lustrac, M. Fevrier, A. Apuzzo, G. Lerondel, A. Lupu, S. Blaize, "Optical near field imaging of localized surface plasmons modes in metallic nanostructures integrated on dielectric waveguides," SPIE Optics + Photonics, San Diego, California, USA, 25-29 August 2013. (poster) Proc. SPIE 8809, 880932 (2013).

/2/A. de Lustrac, R. Ghasemi, N. Dubrovina, P.-H. Tichit, A. Degiron, A. Lupu, “Transformation optics and 3D metamaterials for infrared applications,” Metamaterials 2013, Bordeaux, France, 16-19 September, 2013 (oral). IEEEXplore-20130406-035931. ISBN 978-952-67611-5-2.

/3/N. Dubrovina, R. Salas-Montiel, S. Blaize, A. de Lustrac, G. Lerondel, A. Lupu, “Guided wave metamaterial configurations for application in the near IR domain,” Metamaterials 2013, Bordeaux, France, 16-19 September, 2013 (poster). IEEEXplore-20130630-054822. ISBN 978-952-67611-5-2

/4/A. Lupu, N. Dubrovina, X. Le Roux, R. Salas-Montiel, S. Blaize, G. Lerondel, A. de Lustrac, “Nanoscale engineering of the waveguide local effective index by metamaterial resonances: toward transformation optics applications,” ICTON 2013, 15th International Conference on Transparent Optical Networks, Cartagena, Spain, June 23-27, 2013. (Invited Talk). IEEE Conference Publications ICTON 2013, pp. 1-4. ISSN 2161-2056

/5/N. Dubrovina, X. Le Roux, S. Blaize, A. de Lustrac, G. Lerondel, A. Lupu, “Metal-dielectric metamaterials for guided wave optics applications,” SPIE Photonics West, San Francisco, California, USA, 2-7 February 2013. (oral) Proc. SPIE 8627, 86270Y (2013).

145 List of Publications

/6/N. Dubrovina, X. Le Roux, S. Blaize, A. de Lustrac, G. Lerondel, A. Lupu, “Metal-dielectric metamaterials for silicon photonic applications,” The 5th International Photonics and OptoElectronics Meetings (POEM 2012), Wuhan, China, 1-2 November 2012. (oral) Proc. IONT OSA, Washington, DC, paper ITh5A.5, (2012).

/7/A. Lupu, N. Dubrovina, R. Ghasemi, A. Degiron, A. De Lustrac, “Metal-dielectric metamaterials for guided optics applications,” SPIE Photonics Europe, Brussels, Belgium, 16-19 April 2012. (oral) Proc. SPIE 8423, 842306 (2012).

Conferences and symposiums communications

/1/N. Dubrovina, X. Le Roux, A. de Lustrac, A. Lupu, “Contrôle de l’indice effectif d’un guide d’onde par des métamatériaux : vers l’optique de transformation,” Assemblée Générale du GDR ONDES “Interférences d’ondes”, l’Université de Bourgogne, Dijon, 28-30 octobre 2013. (poster)

/2/N. Dubrovina, B. Gallas, M. Warenghem, A. Lupu, “Détermination par méthode éllipsométrique des paramètres effectifs d’une mono-couche de métamatériau,” Colloque National Métamatériaux (CNM 2013), Orsay, France, 7-8 octobre 2013. (poster)

/3/N. Dubrovina, X. Le Roux, A. de Lustrac, A. Lupu, “Métamatériaux métallo-diélectriques pour l’optique guidée,” Colloque National Métamatériaux (CNM 2013), Orsay, France, 7-8 octobre 2013. (poster)

/4/N. Dubrovina, A. Lupu, X. Le Roux and A. de Lustrac, “Plasmonic engineering of local effective index for transformation optics applications,” 1st EOS Topical Meeting on Optics at the Nanoscale (ONS'13), Capri, Italy, 12-14 September 2013. (oral)

/5/N. Dubrovina, X. Le Roux, S. Blaize, A. de Lustrac, G. Lerondel, A. Lupu, “Effective index engineering in hybrid metamaterials on SOI waveguides,” SPIE Optics + Optoelectronics, Prague, Czech Republic, 15-18 April 2013. (oral)

/6/N. Dubrovina, P. Ghenuche, N. Bardou, N. Burokur, A. de Lustrac, S. Collin, G. Lerondel, A. Lupu, “ Single metafilm behavior in optical domain: the determinative role of the metamaterial resonance frequency on the effective parameters,” SPIE Optics + Optoelectronics, Prague, Czech Republic, 15-18 April 2013. (oral)

/7/N. Dubrovina, N. Burokur, R. Ghasemi, A. Degiron, A. de Lustrac, and A. Lupu, “Approche milieu effectif dans le domaine optique appliquée à une monocouche de métamatériau sur substrat diélectrique,” Les Journées Scientifiques 2013 d’URSI-France, Paris, 26-27 mars 2013. URSI-F2013 pp. 235- 237. (poster)

/8/N. Dubrovina, X. Le Roux, A. de Lustrac, A. Lupu, “Métamatériaux métallo-diélectriques pour l’optique intégrée,” Les Journées Scientifiques 2013 d’URSI-France, Paris, France, 26-27 mars 2013. URSI- F2013 pp. 7-10. (oral)

146 List of Publications

/9/R. Ghasemi, N. Dubrovina, P.-H. Tichit, A. Degiron, A. Lupu, and A. de Lustrac, “Optique de transformation et Métamatériaux pour l’infrarouge,” Les Journées Scientifiques 2013 d’URSI-France, Paris, 26-27 mars 2013. URSI-F2013 pp. 11-14. (oral)

/10/N. Dubrovina, X. Le Roux, S. Blaize, A. de Lustrac, G. Lerondel, A. Lupu, “Effective index engineering using metal-dielectric metamaterials for silicon photonics applications,” META’13, the 4th International Conference on Metamaterials, Photonic crystals and Plasmonics, Sharjah, UAE, 18- 22 March 2013. (oral)

/11/N. Dubrovina, R. Ghasemi, X. Le Roux, A. Degiron, A. de Lustrac, A. Lupu, “Nanoscale engineering of SOI waveguide local effective index with metamaterial resonances : toward transformation optics applications,” 21e Colloque Alain Bouyssy, Orsay, France, 14 février 2013. (poster)

/12/N. Dubrovina, X. Le Roux, R. Salas-Montiel, S. Blaize, A. de Lustrac, G. Lerondel, A. Lupu, “Utilisation des métamatériaux métallo-diélectriques pour des applications photonique guidée,” Journées thématiques RF/millimétrique et optique intégrée, GDR Ondes, Grenoble, France, 17-18 Janvier 2013. (oral)

/13/N. Dubrovina, L. O. Le Cunff, X. Le Roux, N. Burokur, A. De Lustrac, S. Blaize, A. Vial, G. Lerondel, and A. Lupu, “Metal-dielectric metamaterials effective medium behavior in the near- infrared domain,” Journées thématiques sur Modélisation du visible au THz & sur la Plasmonique moléculaire, GDR Ondes, Troyes, France, 20-21 Novembre 2012. (oral)

/14/N. Dubrovina, R. Ghasemi, A. de Lustrac and A. Lupu, “Metal-dielectric metamaterials for integrated optics applications,” 9th International ETOPIM, Marseille, France, September 2-7, 2012. (oral)

/15/R. Ghasemi, P.-H. Tichit, N. Dubrovina, X. Leroux, A. Degiron, A. Lupu, A. de Lustrac, “Transformation optics in integrated Circuits,” Workshop on "Plasmonics for Light Detection or Emission", Ecole Polytechnique, Paris, France, May 18th, 2012. (poster)

/16/N. Dubrovina, A. Lupu, A. Degiron, R. Ghasemi and A. de Lustrac, “Effective slab waveguide behaviour of metal metamaterial multilayers structure in the near infrared domain,” E-MRS 2012 Spring Meeting, Strasbourg, France, May 14-18, 2012. (poster)

/17/A. Lupu, P. Ghenuche, S. Collin, N. Bardou, N. Dubrovina, S. N. Burokur, R. Ghasemi, and A. De Lustrac, “Analysis of angle resolved spectroscopic measurements of a single metafilm on a dielectric substrate,” META’12, the 3rd International Conference on Metematerials, Photonic crystals and Plasmonics, Paris, France, 19-22 April, 2012. (poster)

/18/R. Ghasemi, N. Dubrovina, P.đH. Tichit, A. Lupu, and A. De Lustrac, “Transformation optics and infrared metamaterials for optical devices,” META’12, the 3rd International Conference on Metematerials, Photonic crystals and Plasmonics, Paris, France, 19-22 April 2012, (Invited Talk)

/19/N. Dubrovina, L. Le Cunff, N. Burokur, R. Ghasemi, A. Degiron, A. De Lustrac, A. Vial, G. Lerondel, and A. Lupu, “Single metafilm effective medium behavior in optical domain: Maxwell-Garnett approximation and beyond,” META’12, the 3rd International Conference on Metematerials, Photonic crystals and Plasmonics, Paris, France, 19-22 April 2012. (oral)

147 List of Publications

/20/ N. Dubrovina, R. Ghasemi, A. Degiron, A.Lupu, A. de Lustrac, “Metalþdielectric metamaterials for guided optics applications,” Summer School of Plasmonics 2, Porquerolles Island, Côte d'Azur, France, 3-7 octobre 2011. (poster)

148