Spoof Plasmons in Corrugated Metamaterials

Transcription

1 International Journal of Research Stuies in Electrical an Electronics Enineerin (IJRSEEE) Volume, Issue, 16, PP 8-1 ISSN (Online) DOI: Spoof Plasmons in Corruate Metamaterials Tatjana Gric a,b, Michael Caa a a Dalhousie University, Halifax, Canaa b Vilnius Geiminas Technical University, Vilnius, Lithuania Abstract:To obtain the eienmoes of the electromanetic fiel at a perioically corruate metamaterial we consier a theory of spoof plasmons propaatin on metamaterial perforate with planar perioic rooves. We present the riorous moelin an analysis of surface waves at the bounary of the corruate metamaterial structure. Particularly, we pay special attention to the ways of controllin the properties of surface waves. Moreover, we analyze the epenence of the ispersion characteristics on the metamaterial esin. Keywors:Surface plasmons; Surface waves; Terahertz; Metamaterials 1. INTRODUCTION The surface collective chare ensity oscillations in aition to volume plasmon moes are allowe by the presence of a bounary in a metal[1].hybri photon-plasma excitations, known as surfaceplasmonpolaritons (SPPs) emere once these surface moes couple to liht[, 3]. In a previous paper [4], we provie a etaile analysis of the ispersion relation of SPPs on a perioically corruate semiconuctor surface. As an outcome of this stuy, a low-loss propaation of spoof SPPs has been achieve by an optimum esin of the surface structure. It was foun that by increasin the lattice constant or by reucin the roove epth, the investiate structure can provie a low uiin attenuation. Althouh the analysis of the ratin semiconuctor structure provies useful information on how the corruation affects the ispersion relation of SPP moes, it oes not provie the full SPP moe solution, incluin the presence of the anisotropic material, i. e. hyperbolic metamaterial. The purpose of the present paper is to present such stuy. In Section we formulate the mathematical mechanism of propaation of plasmons in corruate metamaterials an ive the main steps for the erivation of a perturbative treatment that allows us to obtain the complex propaation parameter an the full propaation characteristics of SPPs supporte by lossy metamaterialratins. The metho can be consiere to be an extension to semiconuctor interfaces of the solution presente in [4] for solvin the isotropic problem of the interfaces. In Section 3, we use this metho to obtain analytical results that correspon to the hyperbolic metamaterial situations consiere in [5, 6].Our simulations show the ispersion characteristics ifferent from those obtaine for SPPs propaatin alon metallic corruate interfaces.. PROPAGATION OF PLASMONS IN CORRUGATED TRANSPARENT CONDUCTING OXIDES Let us consier a corruate surface in which an array of rooves exists. This structure is shown in Fi. 1. The surface is continuous in the x-irection an there are two ifferent materials in the z-y plane: the metamaterial occupies the lower reion an above there is a ielectric as a surrounin meium. Such an interface can be moelle as a three-layer structure, consistin of a homoeneous anisotropic layer of a thickness h see fiure 1 escribin corruations, place between a metamaterial an a ielectric. It is possible to approximate the central layer as an anisotropic effective meium ue to the perio bein much smaller than the wavelenth of the electromanetic fiel. It shoul be pointe out that the investiation of the optical properties of the metamaterial structure (Fi. 1) is carrie out via exact mathematical moelin. ARC Pae 8

2 Tatjana Gric& Michael Caa Fi1. Geometry of structure metamaterial surface In orer to fin the ielectric parameters of the effective meium, consier a perioic assembly of parallel plates. The effective ielectric constants of such an assembly are as follows [7] x y z a a a/ a / () Here, is the permittivity of the surrounin meia, - is the permittivity of the material, fillin the roves, f / f m _ met m _ met _ met / _ met (1) is the permittivity of the metamaterialwith f m _ met tm _ met / t an f _ met t _ met / t bein the fillin ratio of the metal sheet an ielectric, respectively [8]. By expanin the spoof plasmon wave vector presente in [9], i. e. rearranin it in terms of ielectric parameters of the effective meium, one can obtain the followin expression: k Here, k k tan k h (3) y k f / c is the vacuum wave vector, h is the rove epth in the metamaterial, k is the wave vector of the wave propaatin on the rooves efine as [1]: k k Where 1, (4) l s k Re is the skin epth with f m m f [8]. After substitutin Eqs. () an (4) into Eq. (3) one obtains the followin expression of the wave vector f _ f met m _ met hk a a k tan _ met m _ met k k (5) Where ε i 1 1 ak f f ls i 1 1 a 1/ Re _ met _ met m _ met m _ met Eq. (5) is the ispersion relation for spoof plasmons supporte by the corruate surface of a metamaterial. International Journal of Research Stuies in Electrical an Electronics Enineerin (IJRSEEE) Pae 9

3 Tatjana Gric& Michael Caa 3. EXAMPLE: HYPERBOLIC METAMATERIAL In orer to unerstan the properties of the reporte SPPs, we provie a simple example of polaritons in corruate metamaterials at THz frequencies. To illustrate the properties of SPPs, we plot the wave vector k(equation (5)) as a function of the frequency. The case of the hyperbolic metamaterial is consiere [8]. It is assume that the structure is surroune by air, i.e. 1. The ispersion curves of spoof SPPs on structure surfaces with = μm are shown in Fiure 3(a). It is of particular interest to analyze the effect of the roove epth on the ispersion curves of spoof SPPs. For this reason, two ifferent roove epths, i.e. h = 3 an.5 are sueste for the stuy. It shoul be pointe out that several values of the roove with are consiere for each roove epth. As seen from Fiure 3, the asymptotic frequency ecreases with an increase in roove epth. We thus state that small corruation amplitues are neee for obtainin the rowth of the asymptotic frequency. This is also the case for the roove with. However, the roove epth rastically affects asymptotic frequency compare to the roove with. We note that by varyin the roove epth, it is possible to control the properties of SPPs. The losses of these spoof SPPs as a function of frequency are presente in Fiure 3(b). It can be observe that the loss of spoof SPPs rows sinificantly with an increase in frequency. For instance, in the caseof a =.4 an h =.5, the attenuation coefficient of the spoof SPPs is 13 m 1 forf =.1 THz, an it increases up to 38 m 1 for f =.13 THz an 8 for f =.14 THz, respectively. The losses in our stuy are assine by investiatin a skin effect, i.e. introucin the skin epth, i.e. a measure of how eep electromanetic waves can penetrate into a material. The analysis of the effect of the lattice constant () on the ispersion of spoofspps is of particular importance. In Fiure 4(a), we have outline a situation with the ispersion curves for spoof SPPs on corruate surfaces with ifferent lattice constants = 1, 15, an μm, respectively. The roove parameters are a = 8μm an h = 6 μm for all cases. The losses of spoof SPPs for three cases are plotte in Fiure 4(b).As seen from Fiure 4(b), a smaller lattice constant correspons to a larer loss of spoof SPPs for a iven frequency. As shown in Fiure 4(b), the lattice constant rastically affects losses of the spoof SPPs. For a iven frequency, an increase of the lattice constantmay results in a sinificant reuction of the loss of spoof SPPs. With the nee for acompact, reliable, an flexible THz system for various applications, a low-loss THzwave uiin system is essential. Fiure3. (a) Dispersion curves for spoof SPPs. (b) Attenuation coefficients of spoof SPPs,lattice constant = μm. International Journal of Research Stuies in Electrical an Electronics Enineerin (IJRSEEE) Pae 1

4 Tatjana Gric& Michael Caa Fiure4. Dispersion curves (a) an attenuation coefficients (b) of spoof SPPs for ifferent latticeconstants = 1, 15, an μm, respectively. Parameters of rooves: a = 8μm an h = 6 μm. It is of particular interest to analyze the effect of the employe metamaterial on the ispersion curves of spoof SPPs. For this reason four three ifferent metamaterials, i. e. Au/HfO [11], Au/Al O 3 [1], A/MF [13] are sueste for the stuy. As seen from Fi. 5, the smallest asymptotic frequency is achieve employin the metamaterial havin ielectric with the smallest permittivity, i. e. HfO. We can imaine that by chanin the ielectric, i. e. by consierin a metamaterial with appropriate plasma frequency, the resonant frequencies can be controlle within a broa rane of values, thus tailorin the structure for a esire spectral reion. Fiure5.Dispersion curves (a) an attenuation coefficients (b) of spoof SPPs for ifferent metamaterials. Lattice constant = μm. 4. CONCLUSION In conclusion, we have introuce a perturbative treatment to fin propaation constants associate to SPPs moes propaatin alon the corruate metamaterial interface. The methocan be reare as an extension to semi conuctor inters faces presente in [4]. Moreover, the propose methooloy has been use to investiate how a rectanular corruation affects the propaation constants. The simulation results presente here corroborate the conclusions obtainethrouh an analysis base on the stuy of isotropic semiconuctor ratin interface [4] an allow us to provie the vivi emonstrations of phenomena occurrin in metamaterial interfaces, such as the ecrease of the asymptotic frequency. REFERENCES [1] R. H. Ritchie, Plasma losses by fast electrons in thin films, Phys. Rev. 16 (1957) 874. [] V. M. Aranovich, D. L. Mills, Surface Polaritons: Electromanetic Waves at Surfaces an Interfaces, 1 st E., North-Hollan, Amsteram, 198. [3] E. N. Economou, Surface plasmons in thin films, Phy. Rev. 18 (1969) 539. [4] T. Gric, M.S. Wartak, M. Caa, J.J. Woo, O. Hess, J. Pistora,Spoof plasmons in corruate semiconuctors, J. Electromanet. Wave 9 (15) [5] A.Poubny,I.Iorsh,P.Belov, Y.Kivshar, Hyperbolic metamaterials, Nat. Photonics 7 (13) [6] L. Ferrari, C. Wu, D. Lepae, X. Zhan, Z. Liu,Hyperbolicmetamaterials an their applications, Pro. Quant. Electron. 4 (15) 1-4. [7] M. Born, E. Wolf, Principles of Optics (Cambrie, 1999). [8] Y.Xian, J.Guo, X.Dai, S.Wen, D.Tan,Enineere surface Bloch waves in rapheme-base hyperbolic metamaterials, Opt. Express (14) [9] A.Rusina, M.Durach, M. I.Stockman, Theory of spoof plasmons in real metals, Appl. Phys. A 1 (1) [1] A.Rusina, M.Durach, K. A.Nelson, M. I.Stockman,Nanoconcentration of terahertz raiation in plasmonic waveuies, Opt. Express 16 (8) [11] I. Iorsh, A. Orlov, P. Belov, Y. Kivshar, Interface moes in nanostructure metal-ielectric metamaterials, Appl. Phys. Lett. 99 (11) [1] J. Kim, V. P. Drachev, Z. Jacob, G. V. Naik, A. Boltasseva, E. E. Narimanov, V. M. Shalaev, Improvin the raiative ecay rate for ye molecules with hyperbolic metamaterials, Opt. Express (1) International Journal of Research Stuies in Electrical an Electronics Enineerin (IJRSEEE) Pae 11

5 Tatjana Gric& Michael Caa [13] T. U. Tumkur, Lei Gu, J. K. Kitur, E. E. Narimanov, M. A. Noinov, Control of absorption with hyperbolic metamaterials, Appl. Phys. Lett. 1 (1) AUTHORS BIOGRAPHY Prof. Tatjana Gric, Dr.Gric s research career has been focuse on the investiation of waveuie evices (waveuie moulators, filters etc.), namely on proposin their electroynamical analysis. Another major oal of her stuies is plasmonics as the examination of the interaction between electromanetic fiel an free electrons in a metal. Moreover, the broa scope of research carrie out by Dr.Gric has inclue investiations into the new fascinatin properties of novel materials. Dr.Gric also has a recor of effective teachin in the rank of Associate Professor. Dr.Gric has publishe extensively in her fiel of investiation with more than 5 peer-reviewe papers in top journals in physics, electroynamics, an optics. It is worth notin that her recent publication rate is ettin even hiher with her bein the first author. Prof. Michael Caa, Dr.Caa stuie with A.M. Prokchorov, a Nobel Prize Laureate for Physics. A former Canaa Research Chair in Interate Active Photonics, he has attracte more than C$1 million in research funs; publishe more than 5 referee articles, conference papers an inustrial reports; supervise more than hihly qualifie personnel an set up a hih-tech company. Dr.Caa is a principal investiator in a European Union nanotechnoloy trainin project that has recently attracte more than C$ million in funin. International Journal of Research Stuies in Electrical an Electronics Enineerin (IJRSEEE) Pae 1