Self-organization approach for THz polaritonic metamaterials

Transcription

1 Self-oganization appoach fo THz polaitonic metamateials A. Reyes-Coonado, 1,2, M. F. Acosta, 3 R. I. Meino, 3 V. M. Oea, 3 G. Kenanakis, 1, N. Katsaakis, 1, M. Kafesaki, 1, Ch. Mavidis, 1, J. Gacía de Abajo, E. N. Economou, 1 and C. M. Soukoulis 1, 1 Institute of Electonic Stuctue and Lase (IESL), Foundation fo Reseach and Technology-Hellas (FORTH), P.O. Box 13, Heaklion, Cete, Geece 2 Instituto de Física, Beneméita Univesidad Autónoma de Puebla, Apatado Postal J-, Puebla, Pue. 2, Mexico 3 Instituto de Ciencia de Mateiales de Aagón, CSIC-Univesidad de Zaagoza, E- Zaagoza, Spain Science Depatment, Technological Educational Institute of Cete, Heaklion, Cete, Geece Depatment of Mateial Science and Technology, Univesity of Cete, 3 Heaklion, Cete, Geece Instituto de Química Física, Rocasolano, Seano, 2 Madid, Spain Ames Laboatoy-USDOE, and Depatment of Physics and Astonomy, Iowa State Univesity, Ames, Iowa, USA *a.eyescoonado@gmail.com Abstact: In this pape we discuss the fabication and the electomagnetic (EM) chaacteization of anisotopic eutectic metamateials, consisting of cylindical polaitonic LiF ods embedded in eithe KCl o NaCl polaitonic host. The fabication was pefomed using the eutectics diectional solidification self-oganization appoach. Fo the EM chaacteization the specula eflectance at fa infaed, between 3 THz and THz, was measued and also calculated by numeically solving Maxwell equations, obtaining good ageement between expeimental and calculated specta. Applying an effective medium appoach to descibe the esponse of ou samples, we pedicted a ange of fequencies in which most of ou systems behave as homogeneous anisotopic media with a hypebolic dispesion elation, opening thus possibilities fo using them in negative efactive index and imaging applications at THz ange. 212 Optical Society of Ameica OCIS codes: (1.31) Metamateials; (1.) Anisotopic optical mateials; (22) Micostuctue fabication; (1.12) Atificially engineeed mateials; (1) Optical popeties; (1) Optical mateials. Refeences and links 1. R. Köhle, A. Tedicucci, F. Beltam, H. E. Beee, E. H. Linfield, A. G. Davies, D. A. Ritchie, R. C. Iotti, and F. Rossi, Teahetz semiconducto-heteostuctue lase, Natue 1, 1 1 (22). 2. S. W. Smye, J. M. Chambelain, A. J. Fitzgeald, and E. Bey, The inteaction between Teahetz adiation and biological tissue, Phys. Med. Biol., R1 R2 (21). 3. D. L. Woolad, J. O. Jensen, R. J. Hwu, and M. S. Shu, Teahetz Science and Technology fo Militay and Secuity Applications (Wold Scientific Publishing Co. Pte. Ltd., 2).. T. Edwads, Gigahetz and Teahetz Technologies fo Boadband Communications (Atech House Inc., 2). #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 13

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Naimanov, Optical hypelens: fa-field imaging beyond the diffaction limit, Opt. Expess 1, 2 2 (2). 2. A. Salandino and N. Engheta, Fa-field subdiffaction optical micoscopy using metamateial cystals: theoy and simulations, Phys. Rev. B, 3 (2). 2. A. Fang, T. Koschny, and C. M. Soukoulis, Optical anisotopic metamateials: negative efaction and focusing, Phys. Rev. B, 212 (2). 2. G. A. Wutz, W. Dickson, D. O Conno, R. Atkinson, W. Henden, P. Evans, R. Pollad, and A. V. Zayats, Guided plasmonic modes in nanood assemblies: stong electomagnetic coupling egime, Opt. Expess 1, (2). 2. N. Liu, H. Guo, L. Fu, S. Kaise, H. Schweize, and H. Giessen, Thee-dimensional photonic metamateials at optical fequencies, Nat. Mate., 31 3 (2). 2. J. Valentine, S. Zhang, T. Zentgaf, E. Ulin-Avila, D. A. Genov, G. Batal, and X. Zhang, Thee-dimensional optical metamateial with a negative efactive index, Natue, 3 3 (2). 2. J. K. Gansel, M. Thiel, M. S. Rill, M. Decke, K. Bade, V. Saile, G. von Feymann, S. Linden, and M. Wegene, Gold helix photonic metamateial as boadband cicula polaize, Science 32, 3 (2). 3. D. B. Buckel, J. R. Wendt, G. A. Ten Eyck, A. R. Ellis, I. Bene, and M. B. Sinclai, Fabication of 3D metamateial esonatos using self-aligned membane pojection lithogaphy, Adv. Mate. 22, 3 31 (2). 31. C. Rockstuhl, F. Ledee, C. Etich, T. Petsch, and T. Schaf, Design of an atificial thee-dimensional composite metamateial with magnetic esonances in the visible ange of the electomagnetic spectum, Phys. Rev. Lett., (2). 32. V. M. Oea, J. I. Peña, A. Laea, R. I. Meino, and P. B. Oliete, Engineeed self-oganized micostuctues using diectional solidification of eutectics, Ceamics Tans. 22, 1 1 (2). 33. V. M. Oea and A. Laea, NaCl-assisted gowth of micomete-wide long single cystalline fluoide fibes, Opt. Mate. 2, (2). 3. D. A. Pawlak, S. Tuczynski, M. Gajc, K. Kolodziejak, R. Diduszko, K. Rozniatowski, J. Smalc, and I. Vendik, How fa ae we fom making metamateials by self-oganization, Adv. Funct. Mate. 2, 1 2 (2). #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

3 3. J. Lloca and V. M. Oea, Diectionally solidified eutectic ceamic oxides, Pog. Mate. Sci. 1, (2). 3. V. M. Oea, J. I. Peña, P. B. Oliete, R. I. Meino, and A. Laea, Gowth of eutectic ceamic stuctues by diectional solidification methods, J. Cyst. Gowth (2), doi:.1/j.jcysgo E. D. Palik, Handbook of Optical Constants of Solids (Academic Pess Inc., 1). 3. V. M. Oea, A. Laea, R. I. Meino, M. A. Rebolledo, J. A. Valles, R. Goto, and J. I. Peña, Novel photonic mateials made fom ionic eutectic compounds, Acta Phys. Slovaca, 21 2 (2). 3. A. Laea and V. M. Oea, Poous cystal stuctues obtained fom diectionally solidified eutectic pecusos, J. Cyst. Gowth 3, 3 33 (2).. A. Laea, L. Conteas, R. I. Meino, J. Lloca, and V. M. Oea, Micostuctue and physical popeties of CaF2-MgO eutectics poduced by the Bidgman method, J. Mat. Res. 1, (2). 1. S. Foteinopoulou, M. Kafesaki, E. N. Economou, and C. M. Soukoulis, Two-dimensional polaitonic photonic cystals as teahetz uniaxial metamateials, Phys. Rev. B, 312 (2). 2. J. C. Maxwell Ganett, Colous in metal glasses and metal films, Phil. Tans. R. Soc. London Se. A 23, 3 2 (1). 3. A. Sihvola, Metamateials Handbook. Theoy and Phenomena of Metamateials, F. Capolino, ed. (CRC Pess, 2), Chap... A. Kichne, K. Busch, and C. M. Soukoulis, Tanspot popeties of andom aays of dielectic cylindes, Phys. Rev. B, 2 2 (1).. J. A. Staton, Electomagnetic Theoy (Wiley, 2).. W. T. Doyle, Optical popeties of a suspension of metal sphees, Phys. Rev. B 3, 2 (1).. R. Ruppin, Evaluation of extended Maxwell-Ganett theoies, Opt. Commun. 12, 23 2 (2).. P. A. Belov, R. Maqués, S. I. Maslovski, I. S. Nefedov, M. Silveiinha, C. R. Simovski, and S. A. Tetyakov, Stong spatial dispesion in wie media in the vey lage wavelength limit, Phys. Rev. B, 33 (23).. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, Phonon-polaiton excitations in photonic cystals, Phys. Rev. B, 2 (23).. K. C. Huang, P. Bienstman, J. D. Joannopoulos, K. A. Nelson, and S. Fan, Field expulsion and econfiguation in polaitonic photonic cystals, Phys. Rev. Lett., 12 (23). 1. Intoduction With the ecent emeging technology of THz souces, such as Quantum Cascade Lases (QCL) [1], the possibility of exploing and exploiting the THz egime of the electomagnetic spectum becomes moe and moe appealing. This exploitation is of exteme technological inteest as it can lead to a lage vaiety of potential applications, anging fom tissue imaging [2], secuity and sensing [3], communications [], and even astonomy []. The availability of THz adiation souces togethe with its immense technological potential applications highlight the need fo THz manipulation components, as polaizes, filtes, beam splittes, collimatos, lenses, etc., which can not be achieved using the appoaches employed in the optical egime, due to the non-stong-esponse of optical mateials in the THz domain. One possibility to ovecome this situation is to employ metamateials opeating in the THz egime. Due to the vaiety of extaodinay electomagnetic popeties that metamateials can possess (like negative efactive index, backwads popagation, [] etc.), and the associated possibilities that they offe (like, e.g. pefect in pinciple lensing []), along with the possibility to enginee thei electomagnetic popeties, can constitute a geat tool fo the manipulation of THz waves. As it is well known, the main functional component of most of todays metamateials is metal, and most of the fascinating metamateial popeties and possibilities ae based on the negative pemittivity esponse of the metal, esulting fom the esonant fee electon cuents [ ]. Besides that, it has been poposed that specific metamateial popeties, like atificial magnetism (leading to negative pemeability) and negative efactive index, can be achieved also using high-index dielectics instead of metals [ 13], whee the ole of the equied cuent is undetaken by the stong displacement cuent. A categoy of mateials that can combine both the advantages of the metals and the high index dielectics, and moeove, opeate in the THz egime, ae the so-called polaitonic mateials #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

4 [1]. Polaitonic mateials ae pola cystals whee an incident electomagnetic wave can excite lattice vibations (optical phonons) in the cystal. The coupling of the electomagnetic adiation with the tansvese optical phonons, which occus in the THz egime, can be descibed by a esonant electical pemittivity esponse of Loenz type, chaacteized by both stong positive and negative pemittivity egimes. This can make polaitonic mateials a pefect eplacement of eithe metals o high index dielectics in the THz egime. Stuctuing thus popely such mateials, one can achieve metamateial popeties like negative effective pemeability [13, 1] o negative efactive index [12], and/o manipulate the dispesion of THz waves in unique ways, exploiting the inteplay between mateial and stuctue (geomety) esonances. Among the peculia metamateial popeties that can be obtained with polaitonic mateials, as we show in this wok, is a hypebolic dispesion elation [1 22]. A geat possibility offeed by hypebolic dispesion elation stuctues is the possibility to achieve subwavelength esolution imaging [21 2], and even imaging with magnification [22]. This is based on the fact that hypebolic dispesion elation does not have an uppe limit in the value of the popagating wave-numbes that it can suppot; thus waves that ae evanescent in fee space (such waves cay the finest details of a souce object) can couple to popagating waves in the hypebolic dispesion stuctues and tansfeed without loss to the image plane. Moeove, shaping popely the dispesion elation in such stuctues one can lead to negative efaction fo the popagating modes, achieving thus both evanescent and popagating modes convegence at the image plane. The typical example of hypebolic dispesion elation systems is uniaxial anisotopic systems whee one of the pemittivity (o one of the pemeability) components is negative and the othes positive. Such a uniaxial system (known as indefinite medium) can be ealized, unde cetain conditions, using a two-dimensional peiodic system of metallic nanowies o a layeed (lamella) metal-dielectic system (see, e.g. Ref. [2] and efs. thee in). Indeed, up to now, metamateials with hypebolic dispesion elation [1 22] have been demonstated in the optical egime using metallic nanood aays [2] o popely shaped metal-dielectic layes, leading also to imaging with magnification [21, 22]. Hee we will demonstate such dispesion in the THz egime using systems of polaitonic ods in a host. A lage obstacle in the cuent eseach and applications of metamateials in THz, is the difficulty in the fabication of the equied stuctues, which should be of length scale fom micometes to nanometes. The most common todays fabication appoaches ae lithogaphic appoaches, which ae time consuming, expensive, and ae mainly esticted to plana geometies [2 3]. One pomising way to go beyond the estictions of the lithogaphic appoaches fo the ceation of THz and optical metamateials, is to employ self-oganization appoaches [31 3]. Self-oganization appoaches ae usually simple, inexpensive and can be used fo an easy and lage scale poduction. Such a self-oganization appoach which can be poved extemely suitable fo achieving polaitonic metamateial stuctues is, as shown in this wok, the diectional solidification of eutectic mixtues [32 3]. Using this appoach one can easily obtain self-oganized systems of 1D, 2D o 3D symmety, and of a lage vaiety of geometical pattens of the basic building blocks. Diectionally solidified eutectics ae composites with fine and homogeneous micostuctues fabicated fom melt. The micostuctue and hence some of the mateial popeties can be contolled by the solidification paametes and it is usually of fibilla o lamella mophology [3, 3]. The dimension of the single cystalline phases anges fom hundeds of micometes to tens of nanometes depending on the gowth ate. The volume filing faction is fixed at the eutectic composition so phase size and intephase spacing ae bound magnitudes in eutectics. Alignment along the solidification diection of the constituent cystalline phases induces anisotopic popeties in othewise isotopic composites. Depending of the mateials composing the eutectic mixtue (which can include any type of #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

5 mateial, e.g. metals, magnetic mateials, semiconductos etc.), the eutectics diectional solidification appoach can be used fo ceation of metamateial micostuctues of a ich vaiety of shapes and configuations [3], giving many possibilities in the metamateials fabication eseach. In this pape we demonstate the potential of the eutectics self-oganization appoach to give two-dimensional (2D) peiodic systems of polaitonic ods embedded in a polaitonic host mateial, of vaying system length scale (od diamete fom tens of micons to sub-micon). Moeove, detemining the electomagnetic popeties of the systems obtained, both with simulations and measuements, we show that pat of those systems ae chaacteized by a hypebolic dispesion elation, which makes them suitable fo THz supelensing and sensing applications. The paticula polaitonic systems that we demonstate hee ae alkali-halide systems made of LiF ods in KCl host and LiF ods in NaCl host [33]. The pape is oganized as follows: In Section 2, we discuss the pepaation (fabication) of the eutectic samples along with the main popeties of the component mateials. In Section 3 we pesent the expeimental, computational and theoetical chaacteization appoaches that we use to analyze the electomagnetic wave popagation in those samples. In Section we pesent and discuss the expeimental and computational eflection studies of the samples that have been obtained, and we compae them with esults obtained fom an effective medium desciption of the samples. Finally, in Section we discuss the hypebolic dispesion esponse of the samples and in Section we pesent ou conclusions. 2. Eutectic metamateial samples obtained As was mentioned in the intoduction, two diffeent sets of eutectic metamateial samples wee fabicated and studied: Samples of LiF ods in KCl host, whee the LiF volume filling faction is.%, and samples of LiF ods in NaCl, with LiF filling faction of 2%. All thee polaitonic mateials involved in these samples, i.e. LiF, NaCl and KCl, possess phonon-polaiton esonances (photon induced excitations of tansvese phononic modes within the cystal) in the THz egion of the electomagnetic spectum. In Fig. 1 we have plotted both the eal and imaginay pats of the dielectic esponse function fo these thee mateials at the fequency egime of the phonon-polaiton esonance. The open cicles in Fig. 1 epesent the expeimental data taken fom Palik [3], while the continuous lines epesent a fit (using least squaes method) of the data, using a Loentzian fomula given by ε(ω) = ε (ε ε )ωt 2 ω 2 ωt 2. (1) + iωγ In Eq. (1), ω T is the phonon-polaiton esonance fequency, and ε and ε ae the limiting values of the dielectic function at fequencies much lage than (ε ε ) 1/2 ω T, and at zeo fequency, espectively. The fitting paametes obtained fom the fitting pocedue fo the thee mateials ae shown in Table 1. In Fig. 1 we see that LiF possesses a phonon-polaiton esonance close to THz (bottom gaph), while fo KCl (middle gaph) and NaCl (top gaph) the phonon-polaiton esonances ae at neighboing fequencies:.2 THz fo KCl and. THz fo NaCl Sample pepaation The eutectic samples wee pepaed by the diectional solidification technique, using the Bidgman method [33]:.% pue LiF (Alfa Aesa),.% pue KCl (Mek) and.% pue NaCl (Alfa Aesa) wee used as stating powdes. They wee mixed in thei eutectic composition: 1 wt% of KCl and wt% of LiF fo the LiF ods in KCl (samples named below as LK#), and 1 wt% NaCl and 2 wt% LiF fo the LiF in NaCl (samples named LN#). The numbes #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

6 NaCl Re[ε] & Im[ε].21 THz.2 THz KCl LiF.22 THz 1 2 Fequency f [ THz ] Fig. 1. Real (blue) and imaginay (ed) pats of the dielectic esponse function fo LiF, KCl and NaCl polaitonic mateials. The open cicles epesent expeimental measued data obtained fom Palik [3], and the continuous lines esult fom the fitting of those data using Eq. (1) and the paametes in Table 1. Table 1. Fitting paametes of dielectic function fo LiF, NaCl and KCl, using a Loentzian fomula. Polaitonic mateials Fitting paametes LiF NaCl KCl ε...3 ε f T = ω T /2π [THz] γ = γ/2π [THz].2.1 in the sample aconym indicate the intephase spacing. The gowth was done in cabon-glass cucibles unde an A atmosphee, pulling them out fom the hot egion of the funace though a themal gadient of C/cm, at diffeent pulling ates. Modifying the pulling ate has as a esult the modification of size and inte-spacing of the LiF ods fomed. Lage pulling ates esult to smalle length-scale systems. The volume pecent of fibes (LiF) emains fixed by the eutectic composition (.% fo LiF in KCl and 2% fo LiF in NaCl). Tansvese and longitudinal slices of the pepaed samples wee cut fom the ingots, and polished with abasive gain size of 1 µm o µm, fo micostuctual chaacteization. Unde the naked eye the slices looked as in Fig. 2(a). They appea athe tanspaent to tansmitted light along the solidification diection, as coesponds to well aligned micostuctues. Figues 2(c) (tansmission optical micogaph) and 2(d) (scanning electon micoscopy (SEM) image) show images of typical tansvese coss sections of both eutectics, whee the dak phase is LiF #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

7 and the bight one is NaCl [Fig. 2(c)] and KCl [Fig. 2(d)]. As expected, the micostuctue (see Fig. 2) consists of LiF ods embedded in a KCl o NaCl matix. In Fig. 2(b) we show a SEM image, whee one can appeciate the continuity of the LiF ods coming out of a humidity cooded longitudinal section of KCl-LiF. In fact, LiF fibes as long as seveal millimetes esult [33,3], coesponding to lage eutectic gains. Figues 2(c) and 2(d) indicate a nealy hexagonal aangement of the LiF ods (shot ange ode), foming though a athe polycystalline system. (a) (b) (c) (d) Fig. 2. (a) Slices of the eutectic NaCl-LiF cut tansvese to the ingot and polished. The high tanspaency is the esult of a good alignment of the fibes along the solidification diection; (b) SEM micogaph of longitudinal cut of KCl-LiF sample, patially cooded by ambient wate vapo; (c) tansvese coss-section of sample LN. (NaCl-LiF eutectic, tansmission optical micogaph); (d) tansvese coss-section of sample LK. (KCl-LiF eutectic, SEM micogaph). Thee ae seveal technological issues that may esult in diffeent scale of long-ange odeing, od sepaation and alignment acoss one sample (ingot). The best long ange odeing is obtained when petubations of the gowth font ae minimum, that is, intemediate gowth ates fo a paticula gowth equipment and method, that pevents constitutional undecooling o excessive macoscopic cuvatue of the gowth font (that might aise at lage gowth ate o at the shotest spacing [3]); o tempeatue o mechanical instabilities (that might aise at the slowest gowth ate o lagest intephase spacing, fo example in sample LK23.3, see below). Statistics of intephase spacing (distance between the centes of the LiF ods) and od diametes have been obtained by measuing them on diffeent SEM micogaphs, though a full tansvese coss-section sample (i.e. sample cut pependicula to the od axes). Specifically, the LiF od cente to cente distances wee detemined fom fast Fouie tansfom (FFT) of the gey scale SEM images of tansvese coss section of samples. In Fig. 3(a) we show a chaacteistic diamete distibution of ou samples, showing a single peak in a quite naow distibution, indication of the good sample quality. The only deviation fom this esult is encounteed in samples of intephase spacing lage than 2 µ m (gown at slowe pulling ates). Such a case is shown in Fig. 3(b), concening a LiF-KCl sample of intephase spacing 23.3 µ m, whee the diamete distibution shows clealy two peaks, indicating two diffeent populations of diametes. Samples with diffeent intephase spacing (lattice constant of the hexagonal aangement), #32 - $1. USD (C) 212 OSA Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

8 !"#$%&'(!"##$%& (b) Fequency (counts) Fequency (counts) (a) 3 2!"#$%&'( 3!"%'$'& Diamete [µm]. 3 Diamete [µm] Fig. 3. Diamete distibutions in KCl-LiF samples of intephase spacing (a).2 µ m and (b) 23.3 µ m. anging fom aound 3 to 23 µ m, wee chosen fo optical chaacteization. The geometical featues of the samples chosen ae listed in Table 2 fo the KCl-LiF samples and in Table 3 fo NaCl-LiF ones. The eos indicate standad deviation of the coesponding magnitude. In the case of KCl-LiF systems, the volume filling faction of LiF was.% and the diametes wee within the ange µ m -. µ m, while the sepaation distances between cylindes wee within the ange 2. µ m 23.3 µ m (see Table 2). Fo the NaCl-LiF systems the volume filling faction of LiF was 2%, with diametes of the cylindes between 2. µ m. µ m and sepaation distances 3. µ m 2.3 µ m (see Table 3). Standad deviation of the evaluated micostuctual magnitudes ae lage in LiF-KCl than in LiF-NaCl. This is paticulaly evident fo the intephase spacing as obtained fom FFT images. Also, the almost hexagonal odeing tends to extend a bit futhe in distance in LiF-NaCl than in LiF-KCl. This is consistent with qualitative obsevations in othe fibous diectionality solidified eutectics, suggesting that small volume of the dispesed phase (as in LiF-KCl o MgO-CaF2 []) esults always in pooe long ange odeing of the micostuctue. Table 2. Cylinde diamete and intephase distances (aveage distance between neaest neighbos od centes) in the KCl-LiF samples. Eos ae standad deviations of the image analysis. Sample aconym LK23.3 LK.2 LK.1 LK. LK2. LiF ods embedded in KCl host Intephase spacing [ µ m ] Rod diamete [ µ m ] 23.3 ± 3.. ± ± ±.3.1 ± ±. ± ±.1 2. ± When the samples ae gown at a sufficiently slow pulling ate (leading to lage od diametes [33]), the cylindical LiF fibes actually do not exhibit exactly a cicula coss section, athe a hexagonal-like one. Howeve, fo simplicity, in ou simulations we teat the ods as cicula cylindes, knowing that the influence of the hexagonal geomety in the coss-section of the cylindes will be a slightly blue-shift in the cylinde esonances. #32 - $1. USD (C) 212 OSA Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

9 Table 3. Cylinde diamete and intephase distances (aveage distance between neaest neighbos od centes) in the NaCl-LiF samples. Eos ae standad deviations of the image analysis. LiF ods embedded in NaCl host Sample aconym Intephase spacing [ µm ] Rod diamete [ µm ] LN ±.. LN.. ±.3. LN.1.1 ± 3.3 ± LN3. 3. ±.3 2. ±.1 3. Expeimental setup and both theoetical and numeical tools employed 3.1. Expeimental setup used to measue eflectance The expeimental electomagnetic chaacteization of ou samples was pefomed using specula eflectance measuements at nealy nomal incidence. The specula eflectance was measued in the ange of 3.3 THz to THz, using a commecial Buke IFS v/s FT-IR spectomete. Longitudinal slices (i.e. cuts paallel to the ods) of thickness aound 1 mm wee cut and dy-polished fo optical measuements. Typically the suface of the slices was mm 2. Only the sample LN3. had smalle aea (aound mm 2 ), since the well aligned egion fo this fast-pulled out sample had a smalle size. The eflectance fo both paallel polaization (incident electic field, E inc, paallel to the od axes) and pependicula polaization (E inc pependicula to the od axes) was measued, as indicated in Fig. (a) and (b) espectively, with the wave vecto of the incident adiation, k inc, being always pependicula to the axes of the ods, i.e. in the plane of peiodicity. The eflectance measuements wee pefomed at incidence angle θ inc =13 in espect to the vecto nomal to the inteface (see Fig. ), which is the smallest achievable angle of incidence of the instument. (a) H inc k inc H ef E inc θ k ef inc θ inc E ef (b) H inc E inc k inc E ef H ef kef Fig.. Schematics of (a) paallel and (b) pependicula polaization configuations measued Models used fo numeical calculations of the eflectance The numeical chaacteization of the samples was done mainly though calculations of tansmission and eflection fom finite thickness systems. We modeled each eutectic system as a set of infinitely long paallel cicula cylindes in 2D hexagonal aangement, as shown in Fig.. #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS

10 We consideed popagation in the plane of peiodicity (i.e. in the plane pependicula to the cylinde axes) and we calculated the tansmission and eflection unde nomal incidence, fo both paallel and pependicula polaization [see Fig. (a)]. The calculations wee pefomed using the commecial softwae CST Micowave Studio, which solves numeically Maxwell equations in both time- and fequency-domain, employing the Finite Integation Technique based on the space and time discetization of Maxwell s equations in thei integal fom. (a) (b) E inc H inc kinc E inc k inc H inc k inc Fig.. (a) Sketch of the model fo the eutectic metamateial system used in numeical calculations. The cylindes wee consideed as LiF cicula ods with hexagonal aangement, embedded eithe in KCl o NaCl host. (b) A tansvese cut of the computational cell employed in most of the calculations pesented hee. The cell consists of unit cells along popagation diection, while peiodic bounday conditions along the othe diections have been consideed. a is the unit cell size (lattice constant) and d is the od diamete. The dielectic functions used to chaacteize ods and hosts duing the calculations wee the fitted Loentzian expessions given by Eq. (1), with fitting paametes those of Table 1. Fo each polaization and sample studied, we pefomed simulations modifying the length of the computational cell along popagation diection, to guaantee convegence of the esults assuing that they epesent coectly the behavio of an optically thick system. The esults pesented hee have been obtained fo a system of seven unit cells along popagation diection (ΓK diection of the hexagonal lattice), as shown in Fig. (b), while peiodic bounday conditions along the othe diections have been employed Analytical model: effective medium appoach The idea beneath an effective medium appoach is to emove the highly oscillating electomagnetic fields inside a system by a suitable aveaging pocedue, and thus eplacing the inhomogeneous system by a homogeneous one chaacteized by effective esponse functions. Such an appoach is extemely useful in the electomagnetic chaacteization of metamateials, as it can give a simple way to chaacteize and undestand the behavio and the possibilities of each paticula system. The validity though of such appoach is guaanteed only in the limit of low filling faction of the scatteing units compising the system o in the long wavelength limit compaed to the inte-spacing of these scatteing units, whee stong multiple scatteing and diffaction of the waves is quite esticted. The pecise fequency egime whee these conditions ae fulfilled fo any paticula system is not easy to be identified and it is highly dependent on the system itself (component mateials, filling factions, etc.). #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 12

11 A useful appoach to identify the applicability of effective medium desciptions fo uniaxial anisotopic systems, as the ones discussed hee, has been poposed ecently in Ref. [1], whee a couple of conditions, elated with the insensitivity of the effective mateial paametes fom the system thickness and popagation diection, have been povided. An impotant esult of Ref. [1] fo uniaxial anisotopic systems is that once these conditions ae fullfilled fo polaization puely pependicula and fo polaization puely paallel, then the validity of homogeneous effective medium desciption fo abitay polaization and popagation diection is ensued. In this wok, we test the validity of homogeneous effective medium desciption fo ou samples when the popagation is in the plane of peiodicity, fo both pependicula and paallel polaization. The effective medium desciption that we employ when the electic field (E) is pependicula to the cylindes axes is the well known Maxwell Ganett model, suitable fo dispesed paticles inside a matix [2, 3]. The Maxwell Ganett fomula fo the effective dielectic esponse function in two dimensions is given by ε eff (ω) = ε host(ω) (1+ϕ)ε cyl(ω)+(1 ϕ)ε host (ω) (1 ϕ)ε cyl (ω)+(1+ϕ)ε host (ω), (2) whee ϕ is the volume filling faction of the cylindes, and ε host and ε cyl ae the pemittivities of the host and the cylindes, espectively. When E is paallel to the cylinde axes, the appopiate fomula fo the effective dielectic function is the aveage dielectic function [], given by ε eff (ω) = ϕ ε cyl(ω)+(1 ϕ)ε host (ω). (3) The way to test the validity of the above esponse functions in ou systems is to examine if these functions can epoduce the eflection chaacteistics obtained fom the expeiment and the simulations. Once ensued the applicability of these paticula effective medium models in ou systems, we can futhe use these models to pedict popagation and dispesion chaacteistics which ae of paticula meit fo metamateial applications.. Electomagnetic chaacteization esults and discussion Using the appoaches descibed in the pevious section, we measued and calculated the eflectance fom the two diffeent sets of eutectic metamateials obtained: LiF ods embedded in a KCl matix, with LiF filling faction of.% (five diffeent samples), and LiF ods embedded in a NaCl matix (fou diffeent samples), with LiF filling faction of 2%..1. fom KCl-LiF eutectic metamateial systems In Fig. we pesent the eflection esults obtained fo the LiF ods in KCl samples LK.2, LK.1, LK. and LK2. (see details of each sample in Table 2). We epot the analysis of sample LK23.3 sepaately, since this sample diffes fom the othes in the sense that its diamete distibution show clealy two peaks [see Fig. 3(b)]. In Fig. we have split the esults into two columns. The left column coesponds to paallel polaization in espect to the axes of cylindes, while the ight column to pependicula polaization. On top of each column we show a plot of the effective electical pemittivity as a function of fequency, both eal (blue) and imaginay (ed) pats, coesponding to each polaization case (obtained fom Eq. (2) fo the ight column and Eq. (3) fo the left column). We have indicated with an oange-shaded egion the fequency egion whee the eal pat of the effective dielectic pemittivity is negative fo each polaization, which is the fequency ange whee one expects to have lage eflectivity fom the samples. Fo paallel polaization (left column), we see two sepaate egions whee Re[ε] < : aound THz and aound THz. The lowe-fequency egion coesponds to the eflectance due to #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 13

12 (a) ε eff (ω)=ϕε LiF KCl Cyl + (1 ϕ)ε host (b) ε KCl eff (ω)=ε host (1+ϕ)ε LiF KCl Cyl + (1 ϕ)ε host (1 ϕ)ε LiF KCl Cyl + (1+ϕ)ε host E ods E ods E ods E ods E ods E ods E ods E ods Fig.. Compaison between expeimentally-measued eflectance and both simulation esults and effective medium pedictions fo the KCl-LiF systems. Left column (a) shows esults fo paallel polaization and ight column (b) fo pependicula polaization in espect to the axes of cylindes in the sample. Fist ow shows the eal and imaginay pats of the effective electical pemittivity fo each polaization, and oange-shaded egions highlight the fequency egimes whee the eal pat of the pemittivity is negative. #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

13 the KCl matix, while the highe-fequency egion is due to the contibution of the LiF ods. Fo the pependicula polaization (ight column), we obseve that Maxwell Ganett fomula pedicts Re[ε] < in a single boade fequency egion, fom.2 THz up to.2 THz, with a peak at aound. THz, whee Re[ε] is less negative and the losses ae smalle than those fo smalle fequencies. In Fig., below the effective pemittivity plots, we show both the measued and simulated eflectance fo each sample, as a function of fequency. We have supeimposed to the expeimental eflectance plot (shown in ed and measued at an incidence angle of 13 ), in each case and fo both polaizations, the esults fom the simulations (shown in dash-blue and obtained at nomal incidence). As a geneal tend, we see that fo all samples in paallel-polaization configuation (left column) thee is a vey good ageement between the positions of the measued eflectance peaks with the simulated ones. Moeove, thee is a good ageement in the low-fequency higheflectance egime (centeed aound THz) between the pediction of the effective medium fomula and both the expeimental and simulation esults. Howeve, this simple effective medium model pedicts accuately the position of the second eflectance peak, coming fom the cylindes, only in the case of the smalle lattice constant sample, LK2.. Fo samples with lage lattice constant, the pediction of the effective medium model becomes less and less accuate, as the second eflectance peak appeas at lowe fequencies. This inaccuacy does not signify though failue of the samples to be descibed as homogeneous effective media but athe the failue of the paticula effective medium appoach employed. The eason is that this paticula model (aveage pemittivity and Maxwell Ganett appoach) has been obtained in the limit whee k host R, k cyl R and k eff R ae all much smalle than unity (k denotes the wavenumbe in each mateial and R the cylinde adius). In this limit the main contibution to the cylinde scatteing in the paallel polaization case is the isotopic scatteing tem (of zeo angula momentum) and no cylinde esonance exists in fequencies below the LiF polaitonic-esonance fequency ( f T =.22 THz). Indeed this is the case fo scattees of adius smalle than. µm, as shown in Fig. (a), whee we show the single-cylinde extinction coss-section [] fo a LiF cylinde of adius,, 1. and 3.3 µm, (fo E paallel to cylinde axis). As can be seen fom Fig. (a), as we go to cylindes of lage adii, the lowest-fequency esonance (coming fom the isotopic scatteing), which fo vey small adius is located at f T, moves to fequencies below f T and additional esonances stat to appea in the close-by egime. This is due to the lage values of the LiF pemittivity below f T, which make the quantity k cyl R (= (ω/c) ε cyl R = 2πR/λ cyl ) compaable to unity (leading to esonances in this fequency egime) even fo quite small cylinde adius, binging thus the system beyond the egime of validity of the aveage effective pemittivity and the Maxwell Ganett appoach. In such cases of high index ods in lowe index host, and in the limit whee k host R, k eff R << 1 but k cyl R 1, a coect homogenization appoach should take into account the full singlecylinde scatteing (not only the fist/isotopic tem) and not the limit k cyl R. Such an appoach, known as extended Maxwell Ganett appoach [,], has been applied aleady in the case of spheical polaitonic scattees [1], and has pedicted even magnetic activity (leading to negative pemeability) in those systems in the fequency egimes of high dielectic constant of the sphees. An analogous appoach, based on pope field-aveaging [], has been applied in the case of SiC ods in a host and has pedicted not only magnetic activity but simultaneously negative pemittivity and pemeability, leading to negative index of efaction, a behavio demonstated also expeimentally [12]. Negative pemeability and negative efactive index behavio though equies ods of quite high pemittivity and lage filling faction. In ou systems neithe the LiF pemittivity just below the polaitonic esonance is high enough no the LiF #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

14 (a) C ext E od (b) ε eff E od Fequency [THz] Fequency [THz] Fig.. (a) Single LiF cylinde extinction coss-section (nomalized with the cylinde diamete) in a host with ε = 2 fo paallel polaization. Legends show the adius of the cylinde. The numbes l close to some extinction peaks denote the ode of the cylindical hamonic modes which ae esponsible fo those peaks. (b) Effective pemittivity calculated using the appoach of Ref. [] fo a system of LiF cylindes of adius R in a host with ε = 2, with LiF volume faction.%. filling faction is sufficient to obtain negative pemeability, as is confimed applying the fieldaveaging appoach of Ref. [] in ou systems. What is confimed though [see Fig. (b)] is the shift of the negative effective pemittivity egime oiginated fom the ods to lowe fequencies as the od adius is inceased. Fo ods of adius 1. µm, as in ou LK.2 system, the negative pemittivity egime pedicted by the aveaging appoach lies between and THz, coinciding with the second eflection peak of the sample LK.2 (see Fig. ), and showing that even in that system an homogeneous effective medium appoximation can be applied. The above mentioned appoaches, although can nicely descibe the behavio of ou composites, they ae not associated though simple analytical fomulas allowing an easy paametic investigation and intepetation of the behavio of the systems. This is the eason why we focus hee mostly on Maxwell Ganett and aveage pemittivity fomulas. Fo sample LK2., as was aleady mentioned, the Maxwell Ganett and aveage pemittivity models pedict satisfactoily the position and intensity of the eflection spectum, fo both polaizations. This is demonstated in moe detail in Fig., whee we have supeimposed the expeimental eflection data and the eflection pedicted fom the effective medium models. Retuning to Fig., anothe obsevation is that fo paallel polaization (left column) the elative intensity of the second eflectance band (coming fom the LiF-ods) vesus the intensity of the KCl-matix band is smalle in the expeimental than in the simulation esults. The diffeence inceases as the od diamete (o intephase spacing) inceases. This is in ageement with the fact that the second band is a esult of a esonance, thus it is expected to fade away when the LiF od diamete is no longe unique but distibutes with a modeate standad deviation (see Tables 2 and 3). In Fig., in the eflectance data obtained fom the simulations we also see quick oscillations, clealy seen in the lowe fequency ange fom 3.3 THz to THz fo both polaizations, and also in the highe-fequency ange, fom THz to THz, fo the pependicula polaization (ight column in Fig. ). These oscillations incease in fequency as we incease the length of the computational cell in the popagation diection, and in this sense they ae the esult of intefeence phenomena fom the intenal eflections at both ends of the sample, that is, Faby-Péot #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

15 esonances. This was confimed by modifying slightly the total length of the computational cell and examining the change in the spectal position of those oscillations. It can be also obseved in Fig. that the fist lowe-enegy measued eflectance peaks ae systematically naowe than both the simulated eflectance peaks and what is expected fom the effective medium models; this is moe evident fo the pependicula polaization (ight column in Fig. ). We believe that this effect is due to the fact that we ae using a fit of the expeimental data (Palik [3]) fo the dielectic esponse function of the polaitonic mateials, without consideing the contibution of multiphonon tansitions to the dielectic function (highe ode modes). Although these highe-ode modes ae faint in the dielectic esponse function, they give ise to absoptions on the high-fequency egion of the dispesion and a bit less negative eal dielectic esponse function, both effects diminishing the eflectance intensity. The above explanation is confimed in Fig., whee the eflection obtained fom the homogeneous effective slab has been calculated using both the Loentz fitted data fo the dielectic esponse function of the mateials involved (see Table 1) and diectly the expeimental Palik [3] data. The simulated specta using the data tabulated by Palik epoduce moe accuately the measued eflectance fo the KCl-LiF system, while the ones using the Loenz fitted data lead to boade esonances. (a) E ods Fesnel+MG+Loentz fit Fesnel+MG+Palik data Measued data (b) E ods Fesnel+MG+Loentz fit Fesnel+MG+Palik data Measued data Fequency [THz] Fequency [THz] Fig.. Compaison between measued eflectance data fo the system LK2., and the eflectance pedicted by applying Fesnel fomulas in a homogeneous effective slab of the same thickness, whee the effective paametes have been calculated using both Loentzfitted data and Palik data fo the pemittivities of KCl and LiF. fo (a) paallel and (b) pependicula polaization in espect to the axes of cylindes. We pesent and discuss next, the esults fo sample LK23.3, whee the ageement between expeimental and theoetical esults is poo, indicating the limitation of both ou numeical chaacteization appoach and the effective medium theoies. Sample LK23.3 was gown at the slowest pulling ate, as discussed in Section 2, and pesent diffeent featues than the est of the samples, by showing clealy a two-peaks diamete distibution [see Fig. 3(b)]. Even thought this sample can not be chaacteized by a single aveage diamete of the LiF fibbes, we attempt to pefom numeical simulations of its eflectance by consideing thee diffeent scenaios: We consideed thee diffeent values of a single aveage diamete of the cylindes and aveage sepaation distances between them (while keeping fixed the volume filling faction), assuming peiodicity fo each calculation fo feasibility of the calculations. The simulations wee pefomed fo (i) d=. µm [the aveage diamete of the left peak of the distibution shown in Fig. 3(b)], (ii) d=. µm (the aveage value of the diamete fo the entie distibution) and (iii) d=. µm [the ight peak in Fig. 3(b)]. The esults ae pesented in Fig., sepaated in #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

16 two columns: left one fo paallel polaization and ight one fo pependicula polaization. As in Fig., the fist ow coesponds to the eal (ed) and imaginay (blue) pats of the effective dielectic pemittivity fo each polaization case, using aveage and Maxwell Ganett fomulas. We clealy see in Fig. that the ageement between expeiment and numeical esults is poo, mainly fo the paallel polaization (left column in Fig. ), and that the eflection esults deviate stongly fom the pedictions of the simple homogeneous effective medium models employed. In fact the pesence in the simulations of many distinct, naow and closely aligned eflectance peaks indicates (not confims though) failue of any homogeneous effective medium appoach, thus being clea that this specific sample does not possesses a hypebolic dispesion elation. (a) (b) LiF KCl ε eff (ω ) = ϕ εcyl + (1 ϕ ) ε host Re[ε] & Im[ε] Re[ε] & Im[ε] 2!"#ε"$%& &% '(#ε"$%&% 2!"#ε"$%& '(#ε"$%&% E ods )*+",-("./1%2/% 3,"".%2(-.%:;1.. E ods!"<"c>c"@(#"$"<"de>e"@("" E ods 2!"<"=>?"@(#"$"<"AB>?"@("" )*+",-("./1%2/% 3,"".%2(-.%:;1 E ods. E ods!"<">f"@(#"$"<"d>d"@("" LiF KCl (1+ ϕ ) εcyl + (1 ϕ ) ε host LiF KCl + (1+ ϕ ) ε host (1 ϕ ) εcyl E ods. KCl ε eff (ω ) = ε host 2 )*+",-("./1%2/% 3,"".%2(-.%:;1.. Fequency f [THz] Fequency f [THz] Fig.. Compaison between expeimentally-measued eflectance and both simulation esults and analytical models fo sample LK23.3. Left column (a) shows esults fo paallel polaization and ight column (b) fo pependicula polaization, in espect to the axes of cylindes in the sample. The simulated esults concen the cylinde diametes (d) and lattice constants (a) shown in the legends. Filling faction is always.%. The top ow shows the effective pemittivity vesus fequency and the oange-shaded egions highlight the egimes whee that pemittivity is negative. #32 - $1. USD (C) 212 OSA Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

17 .2. fom NaCl-LiF eutectic metamateial systems The main diffeence between the NaCl-LiF systems and the KCl-LiF systems discussed hee, is in the volume filling faction of the LiF ods; while in the NaCl-LiF case the LiF filling faction is 2%, fo KCl-LiF case it is close to %. This implies that, since the sepaation distances between neighboing cylindes ae simila to the KCl-LiF case, the diametes of the cylindes ae lage in the NaCl-LiF case (see Tables 2 and 3). The study of the NaCl-LiF systems is summaized in Fig., whee we show togethe the expeimental and simulated (using Micowave Studio) eflection data fo fou epesentative samples: LN2.3, LN., LN.1 and LN3.. In the fist ow of Fig. we have also included the effective dielectic esponse function of the system, obtained though Eqs. (3) and (2), indicating with an oange-shaded egion the fequency egimes whee Re[ε] < fo both paallel and pependicula polaizations, in espect to the axes of the cylindes. Fo a diect compaison of the eflection esults with the effective medium pedictions, we have added in the last ow of Fig. the eflection fom a thick effective homogeneous slab (geen line), with the effective paametes obtained though Eqs. (3) and (2), using in this case the LiF and NaCl pemittivities diectly (without fit) fom Palik data [3]. As a geneal tend, we see that both expeimental and simulated eflectance show a moe complicated spectal dependence than in the KCl-LiF system. Fo paallel polaization and fo samples LN., LN.1 and LN3. (thid, fouth and fifth ows in left column in Fig. ), we have easonable ageement between the expeimental eflection peak coming fom the matix mateial (peak at aound.2 THz) and the simulated data. Fo the smalle-scale samples we also see good ageement of the eflection with what is expected fom the effective medium models. Fo the lage-scale samples, the polydispesity in the od diametes and in the oientation of the polycystals in the samples, esults in the elimination (smoothing out) of the naow-band eflection chaacteistics in the expeimental data, obseved in the simulations. As in the KCl-LiF system case, we also see hee the Faby-Péot intefeence phenomenon supeimposed in the simulated esults fo both polaizations. Anothe similaity between the two systems is that systematically the expeimental peaks ae naowe than the simulated ones. We believe that the explanation is what it was mentioned befoe fo the KCl-LiF case, i.e. due to the fact that we ae using a fit of the dielectic esponse function fo both NaCl and LiF polaitonic mateials, and this fit does not epoduce accuately the pemittivity in the egimes whee Re[ε].. Polaitonic systems as indefinite media In the pevious section we have pesented the eflectance specta (expeimental and simulated) fom slabs of the eutectic systems KCl-LiF and NaCl-LiF. These alkali-halide eutectics pesent a hexagonal aangement of aligned LiF ods (of.% volume filling faction in the KCl matix and 2% in the NaCl matix), with vaying lattice paamete. We also showed that the effective medium model epesented by Eqs. (2) and (3) (Maxwell Ganett and aveage pemittivity model) pedicts easonably well, both in band position and elative intensity, the eflectance specta fo the samples with the smalle length-scale (like samples LK2. and LN3.) fo both polaizations in espect to the axes of the ods. Moeove, accoding to the theoetical esults pesented in Ref. [1], the effective paametes obtained though Eqs. (2) and (3) ae independent of the angle of incidence and theefoe they constitute a valid effective medium desciption of those samples. Thus these eutectic samples will be fully descibed by an effective pemittiv- #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

18 (a) (b) LiF KCl ε eff (ω ) = ϕ εcyl + (1 ϕ ) ε host 2!"#$%"&$' ()#$%"&$'$ 2!"#$%&' *+," *+,"-.)"/12$31$.)"$3)1./$$2$.)"!"#$%&'. E odss *+,"-.)"/12$31$.)"$3)1./$$2$!")$%*' E ods ds!")$%*' E odss. *+,"-.)"/12$31$.)"$3)1./$$2$.. E odss *+,"-.)"/12$31$.)"$3)1./$$2$!"(%)' ds E ods!"(%)'!"#$%"&$' ()#%"&$'$ *+,"-.)"/12$31$.)"$3)1./$$2$.. E odss *+,"-.)"/12$31$ -":;"/<$3)1./$$2$ -"=/"2>?@>A12.B$31$!"&%(' 1 2 *+,"-.)"/12$31$.)"$3)1./$$2$ LiF KCl + (1 ϕ ) ε host (1+ ϕ ) εcyl LiF KCl (1 ϕ ) εcyl + (1+ ϕ ) ε host E ods KCl ε eff (ω ) = ε host 2 Re[ε] & Im[ε] Re[ε] & Im[ε] ds E ods!"&%(' *+,"-.)"/12$31$ -":;"/<$3)1./$$2$ -"=/"2>?@>A12.B$31$.. Fequency f [THz] Fequency f [THz] Fig.. Compaison between expeimentally-measued eflectance and both simulation esults and analytical models fo the LiF ods in NaCl host systems. Left column (a) shows esults fo paallel polaization and ight column (b) fo pependicula polaization, in espect to the axes of the ods in the sample. In both cases the popagation is in the plane of peiodicity. Fist ow shows the eal and imaginay pats of the effective dielectic pemittivity fo each polaization. The oange-shaded egions highlight the negative effective pemittivity egimes. #32 - $1. USD (C) 212 OSA Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun June 212 / Vol. 2, No. 13 / OPTICS EXPRESS 1

19 ity tenso of the fom and dispesion elations, ω(k), εeff εeff ε eff () εeff ω 2 c 2 = k2 x + ky 2 + kz, 2 ω 2 c 2 = k2 x + k 2 y ε eff + k2 z ε eff fo the odinay (left equation) and the extaodinay (ight equation) wave, espectively (consideing the cylinde axes along ẑ-diection). In Fig. we eplot the eal pat of the effective pemittivity tenso components εeff and ε eff (obtained fom Eqs. (2) and (3)) as a function of fequency, fo both systems: Fig. (a) fo KCl-LiF and Fig. (b) fo NaCl-LiF. () (a) Re[ε eff ] E Rods E Rods (b) Re[ε eff ] 2 2 E Rods E Rods 12 Fequency f [ THz ] 12 Fequency f [ THz ] Fig.. Effective dielectic esponse function fo (a) KCl-LiF system and (b) NaCl-LiF system, as a function of fequency, fo both polaizations: paallel and pependicula to the axes of the LiF cylindes. Fo the KCl-LiF case [Fig. (a)], the oange-shaded egion at aound THz denotes the ange of fequencies whee the dielectic function is negative fo E paallel to the ods (metallike behavio), while it is positive fo E pependicula to the ods (dielectic behavio). In this egion thus, the sample will behave as an anisotopic uniaxial medium with a negative pemittivity component (indefinite medium), and thus it will be chaacteized by a hypebolic dispesion elation. Hypebolic dispesion elation, as was discussed in the intoduction, gives geat possibilities fo subwavelength imaging applications. Such a dispesion elation has been discussed and ealized so fa only in the case of metallodielectic systems, while it has been discussed only ecently [1] and not ealized at all in the case of polaitonic systems. Polaitonic systems offe a natual and easy way to achieve hypebolic dispesion elation in the THz and fa-infaed pat of the electomagnetic spectum, avoiding the stong spatialdispesion effects [], that one has to face in analogous systems made of metallic components (spatial dispesion effects act detimentally to the hypebolic dispesion elation). Moeove, in polaitonic composites one can achieve, in the same system, fequency egimes of negative (effective) pemittivity, of vey high pemittivity values and/o of egula pemittivity, and both the pemittivity values and the associated fequency egimes can be tuned by changing the od size, giving thus unique possibilities fo dispesion engineeing. Fo example, fo the case of Fig., when the diamete of the cylinde is inceased the negative pemittivity egime which is at. THz in sample LK2. is shifted downwads. The shift is as lage as 1. THz going fom the sample LK2. to the sample LK.2 (diamete fom µm to 3.2 µm see Table 2). #32 - $1. USD Received 12 Jan 212; evised 1 Ap 212; accepted 23 Ap 212; published 1 Jun 212 (C) 212 OSA 1 June 212 / Vol. 2, No. 13 / OPTICS EXPRESS