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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 18, Iss. 24 — Nov. 22, 2010
  • pp: 24632–24647
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Composite THz materials using aligned metallic and semiconductor microwires, experiments and interpretation

Anna Mazhorova, Jian Feng Gu, Alexandre Dupuis, Marco Peccianti, Ozaki Tsuneyuki, Roberto Morandotti, Hiroaki Minamide, Ming Tang, Yuye Wang, Hiromasa Ito, and Maksim Skorobogatiy  »View Author Affiliations


Optics Express, Vol. 18, Issue 24, pp. 24632-24647 (2010)
http://dx.doi.org/10.1364/OE.18.024632


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Abstract

We report fabrication method and THz characterization of composite films containing either aligned metallic (tin alloy) microwires or chalcogenide As2Se3 microwires. The microwire arrays are made by stack-and-draw fiber fabrication technique using multi-step co-drawing of low-melting-temperature metals or semiconductor glasses together with polymers. Fibers are then stacked together and pressed into composite films. Transmission through metamaterial films is studied in the whole THz range (0.1-20 THz) using a combination of FTIR and TDS. Metal containing metamaterials are found to have strong polarizing properties, while semiconductor containing materials are polarization independent and could have a designable high refractive index. Using the transfer matrix theory, we show how to retrieve the complex polarization dependent refractive index of the composite films. Finally, we study challenges in the fabrication of metamaterials with sub-micrometer metallic wires by repeated stack-and-draw process by comparing samples made using 2, 3 and 4 consecutive drawings. When using metallic alloys we observe phase separation effects and nano-grids formation on small metallic wires.

© 2010 OSA

1. Introduction

In this paper we describe a novel method for the fabrication of THz metamaterial films via fiber drawing and subsequent pressing of the metallic or dielectric microwire arrays into composite films. With this technique we demonstrate metal wire based metamaterial films having strong polarization properties, and chalcogenide-based metamaterial films with designable high refractive index. We would like to note that mass production of the vertically aligned extremely long metallic micro/nanowires using fiber drawing technique was recently detailed in [14

14. X. Zhang, Z. Ma, Z.-Y. Yuan, and M. Su, “Mass-productions of vertically aligned extremely long metallic micro/nanowires using fiber drawing nanomanufacturing,” Adv. Mater. 20(7), 1310–1314 (2008). [CrossRef]

]. In application to THz wire-grid polarizers there were two recent reports where the authors suggested the use of several aligned fibers containing wire arrays to build large area polarizers [15

15. A. Tuniz, B. T. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. C. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett. 96(19), 191101 (2010). [CrossRef]

,16

16. A. Mazhorova, J. F. Gu, S. Gorgutsa, M. Peccianti, T. Ozaki, R. Morandotti, M. Tang, H. Minamide, H. Ito, and M. Skorobogatiy, “THz metamaterials using aligned metallic or semiconductor nanowires” We-P.31, Proceedings of IEEE34th International Conference on Infrared, Millimeter, and Terahertz Waves, IRMMW-THz 2010.

]. For applications in the visible spectral range there were several recent reports of micro- and nanowire fabrication by co-drawing copper or gold with glasses [17

17. J. Hou, D. Bird, A. George, S. Maier, B. T. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibres,” Opt. Express 16(9), 5983–5990 (2008). [CrossRef] [PubMed]

,18

18. H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. S. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010). [CrossRef] [PubMed]

]. However, all these works mainly focused on the fabrication and optical properties of the individual fibers containing metallic wires, and no attempt was made of fabricating a continuous planar sample or a metamaterial film. Also, fabrication of the high refractive index composites made of chalcogenide wire arrays was not reported before. Additionally, no study has been reported on the quality of micro-wire structure after a large number of consecutive drawings. Finally, although characterization of the dielectric tensor of microwire arrays was attempted in earlier works [9

9. Y. Minowa, T. Fujii, M. Nagai, T. Ochiai, K. Sakoda, K. Hirao, and K. Tanaka, “Evaluation of effective electric permittivity and magnetic permeability in metamaterial slabs by terahertz time-domain spectroscopy,” Opt. Express 16(7), 4785–4796 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4785. [CrossRef] [PubMed]

], no rigorous effort was made to compare experimentally obtained metamaterial parameters with predictions of a theoretical model.

In this paper, we also report a densification technique where individual metamaterial fibers are first aligned and then pressed at elevated temperature to form a continuous uniform metamaterial film. Transmission measurements through such films are then reported and a transfer matrix-based analytical model of transmission trough a 3-layer metamaterial film is then used to extract the complex frequency dependent effective refractive index and permittivity of a composite medium. Various practical challenges are detailed when trying to infer the metamaterial parameters from the transmission data. Finally, we demonstrate strong polarization effects of the metal containing metamaterials, as well as ability to design high refractive index composite materials using chalcogenide nanowires.

Right from the start we would like to point out two main distinctive features of our metamaterials. First of all, most of the existing experimental studies of the THz wire array materials deal with highly ordered arrays of relatively large diameter wires (hundreds of microns). For such materials comparison with analytical theories (Pendry’s formula, for example) developed for strictly periodic structures is relatively straightforward. In our case, due to specifics of the fabrication technique, we create arrays of parallel, while randomly positioned microwires of variable from wire to wire diameters on the order of 1-10 microns. Therefore, applicability of formulas developed for the periodic wire arrays to our metamaterials is questionable if at all justified. Instead, to describe our metamaterials one has to use effective random media theories such as Maxwell-Garnett or Bruggeman theories. Moreover, in most experimental work on the wire-array media one typically uses commercial wires made of good metals such copper or gold with dielectric constants that can be well fitted with a Drude model. In our case, again due to specifics of the fabrication technique, we can only use low melting temperature alloys which are thermo-mechanically compatible with plastics used during fiber drawing. In particular, we use alloys of semi-metals Sn and Bi with dielectric constants that cannot be generally described by a simple Drude model.

2. Fabrication process

The microwire arrays are made by the stack-and-draw fiber fabrication technique using co-drawing of low melting temperature metals or amorphous semiconductor glasses together with polymers. The fabrication process of both metal and chalcogenide glass (As2Se2) samples is similar and can be divided into several steps. First, we prepare microstructured fibers containing microwire arrays. For that we start by filling a single PC (Polycarbonate) tube with a liquid melt (Tm ~130°C) of Bismuth and Tin alloy (42% Bi, 58% Sn). After the first preform is made it is drawn at 190°C into a polymer-clad wire with diameter of ~260 µm (first drawing). We then stack hundreds of wires together, clad them with another PC tube and draw it again under the same temperature. A scanning-electron-micrograph of the cross-section of a composite fiber after the second drawing is shown in Fig. 1(a)
Fig. 1 SEM pictures of the cross-sections of fabricated wire-array metamaterial fibers. a) metal wire fiber after 2nd drawing, b) metal wire fiber after 3rd drawing, c) semiconductor wire fiber after 4th drawing. Insets show magnification of individual wires. Inset of c) shows cluster of nanowires with the individual fiber diameters as small as 200nm.
. Inset is a magnification of an individual wire of size ~10 µm. Note that after the first drawing, metallic wire is surrounded by a polymer cladding of thickness comparable to the size of a wire. Relatively thick cladding is needed to preventing the individual wires from coalescing with each other during the following consecutive drawings.

Figure 2(c,d) presents an optical micrograph displaying the top view of a metamaterial film, where the aligned metal (c) or chalcogenide (d) wires can be clearly seen. From the SEM images of the pressed films we see that the wire size fluctuates in the range 0.5–4 µm. To create semiconductor nanowires we follow the same fabrication procedure but start with As2Se3 semiconductor chalcogenide glass rods cladded with a PSU (PolySulphone). Fibers were then drawn at 300°C and subsequently arranged and pressed into films at temperatures 280°C and pressures of several thousands of PSI.

A single fiber drawing yields hundreds of meters of a uniform diameter fiber. Cutting, stacking, and pressing such fibers into films enables fabrication of large area composite films. In our experiments we have produced films having an area of several cm2, with the main limitation being the size of a hot press. The main challenge in pressing the composite films is to guarantee the uniformity of a metamaterial layer. Because of the relatively thick plastic cladding of the individual wires, most of the pressed films show stripes of metal wires separated by the stripes of clear plastic. Press parameters have to be chosen carefully to guarantee that during the compactification process the wires are free to intermix in the horizontal plane without the loss of alignment.

3. Transmission measurements

Transmission spectra of the films were then studied in the whole THz range (0.1–20 THz) using a combination of Terahertz Time-Domain Spectroscopy (THz-TDS) and Fourier-Transform Infra-Red spectroscopy (FTIR). In Fig. 3(a)
Fig. 3 FTIR transmission spectra (0.1–20 THz) of a metamaterial film containing ordered (a) metal and (b) semiconductor wires. Strong polarization dependence of transmission spectrum is observed for metallic wire arrays.
we present the FTIR transmission spectra of a metamaterial film containing ordered metal wires (in the range 0.1–20 THz) for two different polarization of the incoming light. Red color corresponds to polarization of the electric field parallel to the wires, whereas blue color corresponds to polarization perpendicular to the wires. As seen from Fig. 3(a) metamaterial films are strongly polarization sensitive for frequencies up to 5 THz, while polarization sensitivity persists up to 10 THz. For comparison, in Fig. 3(b) we present transmission spectra through a metamaterial film containing As2Se3 semiconductor glass wires, where no polarization sensitivity is observed.

Polarization dependence of THz light transmission through metamaterial samples is clearly seen in Fig. 3(a); however, the lack of phase information makes it difficult to retrieve the corresponding effective refractive indices for the two polarizations. As shown in Fig. 3(a), light polarized parallel to the metal wires is blocked, while light polarized perpendicular to the wires is largely transmitted. The wire-grid polarizer consists of an array of slender wires arranged parallel to one another. The metal wires provide high conductivity for the electric fields parallel to the wires. Such fields produce electric currents in the wires, like the microwave dipole receiver antenna. The energy of the fields is converted into energy of currents in the wires. Such currents are then partially converted to heat because of the small but significant electrical resistance of the wires, and partially irradiated back. The physical response of the wire grid is, thus, similar to that of a thin metal sheet. As a result, the incident wave is mostly absorbed or reflected back and only weakly transmitted in the forward direction. Because of the nonconducting spaces between the wires, no current can flow perpendicular to them. So electric fields perpendicular to the wires produce virtually no currents and lose little energy, thus, there is considerable transmission of the incident wave. In this respect, the wire grid behaves as a dielectric rather than a metal sheet.

Alternatively, for the parallel polarization of incoming wave, response of a wire-grid medium can be modeled as that of a metal with an effective plasma frequency defined solely by the geometrical parameters of the medium such as wire size and inter-wire separation. Thus, at lower frequencies below the effective plasma frequency of a wire medium, electromagnetic waves are effectively blocked by such a material. However, at frequencies above plasma frequency polarization sensitivity of the wire grid medium is greatly reduced as both polarizations can go through the medium. At even higher frequencies when the inter-wire spacing and individual wire size (which are comparable in our samples) become comparable to the wavelength of light the metamaterial approximation breaks down. In our samples we observe that polarization sensitive transmission through the wire-medium is lost completely at frequencies higher than 10 THz.

In order to obtain complex permittivity of the samples, complex transmission spectra (amplitude (Fig. 4 a,c
Fig. 4 Transmission spectra (a,c) and phase difference (b,d) of THz light through metamaterial film containing (a,b) ordered metallic wires, (c,d) ordered semiconductor wires.
) and phase (Fig. 4 b,d)) through metamaterial films were measured with a THz-TDS setup. The THz-TDS setup uses a frequency-doubled femtosecond fiber laser (MenloSystems C-fiber laser) as a pump source. Identical GaAs dipole antennae were used as THz source and detector yielding a spectrum ranging from 0.1 to 3 THz. The FWHM spot size of the THz beam at the focal point of the parabolic mirrors is roughly 3 mm. For convenient handling of samples, the metamaterial films were cut into 1cm x 1cm pieces and placed at the focal point of the parabolic mirrors. For each film sample, two transmission spectra were measured corresponding to the parallel and perpendicular polarizations of the THz beam with respect to the wire direction.

To ensure constant input illumination conditions during rotation of samples, we used a rotation mount (RSP1 from Thorlabs). It utilizes two precision bearings for smooth, backlash-free rotation, and a small iris was inserted in the center of the rotation mount. The metamaterial film was then placed over the iris. Normalized transmittance through the metamaterial film was calculated by dividing the transmission data with, and without, the metamaterial film.

4. Interpretation of the experimental data

In order to infer the optical parameters of metamaterial layers, one has to assume a particular geometry of a composite film, as well as optical parameters of the constituent materials. As mentioned previously, the fabricated films can be viewed as a three-layer system (Fig. 2 a, b), where a wire metamaterial layer is sandwiched between the two polymer layers (either PC or PSU for metal or chalcogenide wire films, respectively). Therefore, we first characterize refractive indexes and absorption losses of the pure PC and PSU polymers. We then use this data to fit the parameters of a metamaterial film.

4.1. Procedure for retrieving the refractive index and losses of pure polymers

Characterization of refractive indices of pure plastics is done with THz-TDS setup using thick polymer slabs with parallel interfaces. The samples were prepared by pressing the PC or PSU tubes into thick films at temperatures similar to those used during the fabrication of metamaterial films. Transmission spectra and phase difference of THz light through a plastic slabs is shown at Fig. 6 a
Fig. 6 Transmission spectra (a) and phase difference (b) of THz light through a plastic slabs.
),b).

We retrieve the refractive index and absorption losses of pure plastics by fitting the predictions of a transfer matrix model to the experimental transmission data [19

19. M. Skorobogatiy, and J. Yang, Fundamentals of photonic crystal guiding (Cambridge University Press, 2009)

21

21. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002). [CrossRef] [PubMed]

]. Particularly, using the + and – superscripts to denote forward and backward traveling waves, the field amplitudes before and after a plastic slab are:
(Ein+Ein)=i=12Mi1,1(Eout+0)=[T11mT12mT21mT22m](Eout+0)
(1)
The complex field transmission coefficient through a single layer is then given by
t=1T11m=ta,ptp,aexp(inpωdp/c)1+ra,prp,aexp(2inpωdp/c)
(2)
where ra,p,rp,a and ta,p,tp,a are the complex reflection and transmission Fresnel coefficients of the interface between air (subscript a) and plastic (subscript p), as well as between plastic and air respectively. These coefficients have the same form for both s and p polarizations due to normal angle of radiation incidence. Denoting dp to be the plastic layer thickness, and npto be the refractive index of a plastic layer, we can write:

ta,p=2nana+np
(3)
ra,p=nanpna+np
(4)

For the calculation of refractive index and absorption losses of pure plastics we can assume that slabs are thick enough, so that multiple reflections within a sample (Fabry-Pérot reflections) can be neglected. In this case we can assume that the denominator of Eq. (2) simply equals 1, which allows us to write:
t=T(ω)eiφ=4nreal(ω)(nreal(ω)+1)2exp(iωdpc(nreal1))exp(12αdp)
(5)
The transmittance T(ω) and phase shift φ at each frequency are obtained simultaneously by the THz-TDS measurements. Equation (5) easily separates into two equations respectively for the real and imaginary parts of the refractive index [22

22. W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Material thickness optimization for transmission-mode terahertz time-domain spectroscopy,” Opt. Express 16(10), 7382–7396 (2008). [CrossRef] [PubMed]

]. For nrealand α we get:

nreal(ω)=1+cφωdp
(6)
α=2nimω/c=2dpln((nreal(ω)+1)2T(ω)4nreal(ω))
(7)

It is important to realize that imaginary part of the refractive index of common plastics in THz frequency range can be several percent from the value of the real part of the refractive index. Therefore, to guarantee the third digit accuracy in the value of the real part of the extracted refractive index, Eq. (6,7) may no longer be valid, and the use of the complete formula Eq. (5) may be necessary. Experimentally the thicknesses of the two polymer slabs were 5 mm. In fact, the samples should be thick enough so that multiple Fabry-Pérot reflections are suppressed via absorption of the reflected waves on the length scale of a sample width. On the other hand, the sample should not be too thick, otherwise absorption losses even after one passage would be too large to make a reliable measurement of a transmitted field. The value of the fitted refractive index of PC is almost constant across the whole THz range and it equals to n=1.64, which is in good agreement with prior reports [22

22. W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Material thickness optimization for transmission-mode terahertz time-domain spectroscopy,” Opt. Express 16(10), 7382–7396 (2008). [CrossRef] [PubMed]

]. Absorption loss of plastics are a rapidly increasing function of frequency which can be fitted as PC: αpc[cm1]=3.77f+6.65f2, PSU: αpsu[cm1]=5.03f+20.25f2,where f is frequency in [THz]. In this work, for the first time to our best knowledge, we also report the THz refractive index of PolySulphone n=1.74, which is also effectively constant across the whole THz range. Figure 7
Fig. 7 a) real part of the refractive index and b) absorption losses of pure plastics PC, PSU.
details frequency dependence of the PC and PSU refractive indices (a) and absorption losses (b) obtained by fitting the experimental transmission data.

4.2. Procedure for retrieving the refractive index and permittivity of metamaterial layers

As mentioned earlier, we model the experimental films as a three-layer system (Fig. 5 b
Fig. 5 Schematic of (a) one layer of plastic and (b) a metamaterial film modeled as a three layer system. The subscripts indicate the layer number, while the + and the – signs distinguish incoming and outgoing waves, respectively.
)) where a wire metamaterial layer is sandwiched between two polymer layers (either PC or PSU for metal or chalcogenide-based wire arrays). We start with the metal wire-based metamaterial films. The total thickness of such films is 150 µm. Thickness of a metamaterial layer alone is 35 ± 5 µm. The layer comprises metal wires of 1.3 ± 0.5 µm average diameter.

(Ein+Ein)=i=14Mi1,1(Eout+0)=D01[i=13DiPiDi1]D4(Eout+0)=[T11mT12mT21mT22m](Eout+0)
(8)

Each side of an interface is represented by the corresponding transmission matrix Di, whereas the propagation inside the bulk material of each layer is represented by its propagation matrix Pi
Di=1ti1,i[1ri1,iri1,i1]
(9)
Pi=[exp(iniωdic)00exp(iniωdic)]
(10)
where ri1,iand ti1,i are the complex reflection and transmission Fresnel coefficients:
ti1,i=2ni1ni1+ni
(11)
ri1,i=ni1nini1+ni
(12)
of the interface, diis the thickness of i layer, and niis the complex refractive index. In our case we made only transmission measurements, thus we only have amplitude and phase information about the forward traveling waves Ein+ and Eout+. The complex transmission coefficient of the multilayer is given in terms of the system transfer matrix elements Tij as Ttheor.=1/T11m. The THz-TDS setup allows to obtain information about amplitude Eout+ and phase φ of the signal transmitted through the sample for all THz frequencies in a single measurement. The measured complex transmission coefficient is obtained from the ratio of the sample and reference fields,
Tmeasured=T(ω)eiφ=Eout+(ω)/Eref+(ω)ei(φφref)
(13)
The reference field amplitude Eref+ and phase φrefare obtained by measuring the signal without sample. Assuming that multilayer geometry and refractive index of a host material are known, complex refractive index of the metamaterial layer is found by taking the difference between the measured and theoretical transmission coefficients to be zero, and then finding the roots of a resultant equation:
TmeasuredTtheory(Re(nmeta),Im(nmeta))=0
(14)
Note that Eq. (14) is effectively a system of two purely real equations for the real and imaginary parts of the complex transmission coefficient. This system of equations has two unknowns, which are the real and imaginary parts of the metamaterial refractive index. Therefore, Eq. (14) is well defined.

The theoretical transmission coefficient of the multilayer depends on the knowledge of the exact thicknesses of the layers and their complex refractive indices. While the complex refractive indices of the host polymers, as well as the total thicknesses of a multilayer have been accurately measured, the thickness of a metamaterial layer must be assumed to be an adjustable parameter since its accurate estimation from the micrographs is not possible due to ambiguity in the definition of a boundary between metamaterial and a plastic cover. As it will become clearer in the following section, the exact choice for the value of the thickness of a metamaterial layer can only be made if an additional optimization criterion is established. Without a supplemental restriction on Eq. (14), this equation can be resolved for any given value of the metamaterial layer thickness. To solve numerically Eq. (14) we use a nonlinear Newton method for the real and imaginary parts of the refractive index at each frequency point. Another subtlety in solving Eq. (14) comes from the possibility of multiple solutions of this equation. For the cases with more than one solution, unphysical solutions are eliminated by verifying their compatibility with the Kramers–Kronig relations. Finally, the effective permittivity is calculated from the extracted complex refractive index of the metamaterial in order to facilitate the interpretation of the metamaterial behaviour and the analysis via the effective medium theory.

4.3. Extracted refractive index and permittivity of the films

Alternative interpretation for the operational principle of the wire grid polarisers is in terms of the effective plasma frequency of a metamaterial. This was proposed by Pendry et al. [24

24. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]

] to explain wave propagation through a 2D array of wires. He demonstrated that such an array behaves like a Drude metal with effective plasma frequency which is only a function of the geometrical parameters of a wire array ωp2=2πc2/(a2ln(a/r)), where c is the light speed in vacuum, a is the inter-wire spacing, and r is the wire radius. Several more complicated but precise derivations of the effective plasma frequency for the wire medium have been developed [25

25. S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, “Wire media with negative effective permittivity: a quasi-static model,” Microw. Opt. Technol. Lett. 35(1), 47–51 (2002). [CrossRef]

] and a comparison between the models has been conducted in [26

26. P. Markoš and C. M. Soukoulis, “Absorption losses in periodic arrays of thin metallic wires,” Opt. Lett. 28(10), 846 (2003). [CrossRef] [PubMed]

]. Since the effective plasma frequency in a wire medium can be tuned by adjusting the medium’s geometrical parameters, the spectral region for the desired permittivity values can be engineered to occur at practically any frequency.

At this point it would be interesting to estimate theoretically the plasma frequency of our metamaterial using Pendry’s model. However, in our case we believe that the results of such a prediction would have a limited validity. Particularly, Prendry’s plasma frequency is derived for a periodical network of wires of a fixed diameter. From the SEM pictures of a typical cross-section of the wire-array metamaterial fibers after 3rd drawing (see Fig. 1) we see that the wire diameter fluctuates in the range 0.5–4 µm, with an average wire size being around 1 µm. The distance between wires could be estimated from the optical micrograph of the film containing metal microwire array (Fig. 2), we see that it varies between 5 µm and 50 µm. Thus, when using Pendry’s formula we get estimates to the plasma frequency ranging from 1 THz (d = 0.5 µm, a = 50 µm) to over 25 THz (d = 4 µm, a = 5 µm). Therefore, applicability of the Pendry’s formula to random wire arrays is not obvious.

In passing we mention that there is a growing body of work dedicated to the study of influence of disorder in the metal wire arrays on their transmission properties. Thus, small disorder typically results in the shift of plasma frequency to lower frequencies [27

27. M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, “C. M. Soukoulis “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals”,” Phys. Rev. B 64(19), 195113 (2001). [CrossRef]

]. Disorder also leads to increase in the attenuation of the wire medium, which is, moreover, highly sensitive to the particular realisation of a disorder. In general, in disordered systems plasmonic response becomes less pronounced [28

28. A. Rahachou and I. Zozoulenko, “Light propagation in nanorod arrays,” J. Opt. A, Pure Appl. Opt. 9(3), 265–270 (2007). [CrossRef]

]. In metamaterials consisting of split ring resonators, small disorder yields to narrowing and reduction of the resonance gap [29

29. K. Aydin, K. Guven, N. Katsarakis, C. Soukoulis, and E. Ozbay, “Effect of disorder on magnetic resonance band gap of split-ring resonator structures,” Opt. Express 12(24), 5896–5901 (2004). [CrossRef] [PubMed]

], while large disorder could lead to a complete suppression of resonances [30].

Second reason is the non-Drude behavior of the semimetals and their alloys from which our metamaterial is made. Particularly, it is suggested in several references that spectral behavior of the terahertz conductivity of the poor conductors (such as semi-metals Sn, Bi and their allows), as well as materials near the metal-insulator transition is described by a Drude-Smith model which is a generalization of the Drude model, that can exhibit a substantially different frequency dependence from a classical Drude model [31

31. N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). [CrossRef]

]. In our samples we use Bismuth Tin alloy, bulk electrical conductivity of which is only around 2% of the conductivity of the pure Copper. Moreover, it is known that in micro and nanostructures (such as microwires in our case) Bismuth Tin alloy exhibits phase separation into pure metals, which greatly complicates description of the electrical properties of such alloys [33

33. S. H. Chen, C. C. Chen, and C. G. Chao, “Novel morphology and solidification behavior of eutectic bismuth–tin (Bi–Sn) nanowires,” J. Alloy. Comp. 481(1-2), 270–273 (2009). [CrossRef]

]. In our case phase separation manifests itself in formation of the nano-sized grids as presented in Fig. 10
Fig. 10 Microwires after 2nd drawing featuring nanogrids. Microwires were obtained by dissolving PC polymer matrix. In the inset, shown is a pure Bi, Sn lamellar structure due to phase separation of an alloy.
of the last section. There are several works about refractive index and conductivity of nanostructured Bi in the form of thin (sub-micron) Bismuth layers [34 35

35. W. S. Boyle and A. D. Brailsford, “Far infrared Studies of Bismuth,” Phys. Rev. 120(6), 1943–1949 (1960). [CrossRef]

,]. In these works they conclude that the plasma frequency of the nanosized Bi films is on the order of νpl = 100 THz THz. At the same time, plasma frequency of the monocrystal bismuth was reported to be νpl=4.86 and5.64 THz, depending on the anisotropy axis orientation [36

36. V. A. Podolskiy, L. V. Alekseyev, and E. E. Narimanov, “Strongly anisotropic media: the THz perspectives of left-handed materials,” J. Mod. Opt. 52(16), 2343–2349 (2005). [CrossRef]

], which is more than 20 times different from the plasma frequency of Bi thin films. From this we conclude that microstructure of the semimetal alloys can influence dramatically its electrical properties. To remedy this complexity in our future work we will try to find the material combination of plastics and pure low melting temperature metals such as In, Sn or Bi, to fabricate microwire arrays with electrical properties that would be easier to interpret. The only reason for the use of Tin Bismuth alloy in this work was that it showed outstanding compatibility with the plastic host material during drawing. Finally, additional experiments are needed to study optical properties of the pure low melting temperature metals in the vicinity of 1 THz, which are still not available in the literature.

Finally, for comparison with the metal wire arrays, we present the refractive index and permittivity of the metamaterial layer containing chalcogenide glass wire array. The values calculated from the transmission spectra of Fig. 4(b) are shown in Fig. 9
Fig. 9 Extracted (a) refractive index, and (b) permittivity of a metamaterial layer containing chalcogenide glass wires
. Due to the low conductivity of the chalcogenide wires, the wire-array does not act as an effective metal and there is no difference between the polarizations parallel or perpendicular to the wires. In this case, the chalcogenide glass inclusions merely form a dielectric composite material having a significantly higher refractive index compared to the pure PSU plastic host.

5. Discussion

5.1. Effective medium theory

It is interesting to note that according to the effective medium theories [3

3. W. Cai, and V. Shalaev, Optical metamaterials. fundamentals and applications (Springer, 2010)

,4

4. C. Brosseau, “Modelling and simulation of dielectric heterostructures: a physical survey from an historical perspective,” J. Phys. D Appl. Phys. 39(7), 1277–1294 (2006). [CrossRef]

,37], components of the dielectric tensor of a wire array metamaterial are not independent. They are, in fact, well defined functions of the filling factor and dielectric constants of the metal and host materials. Therefore, one can, in principle, design an algorithm that can check consistency between the experimentally measured parallel and perpendicular components of the metamaterial dielectric tensor. The goal, for example, is to determine validity of various assumptions for the geometrical and material parameters that are made in the data interpretation algorithms, thus improving their reliability.

εeff(theory)(ω)=fεm+(1f)εplastic
(16)

In order to find the optimal value of the filling factor and thickness of a metamaterial layer that one uses in the data interpretation algorithms, one can proceed by simultaneous minimization of the two weighting functions:

Q||(d,f)=ωminωmaxdω|εeff(theory)||(ω)εeff(exp)||(ω)|2
(17)
Q(d,f)=ωminωmaxdω|εeff(theory)(ω)εeff(exp)(ω)|2
(18)

Ideally, if the metamaterial layer thickness and filling fractions are chosen correctly one expects Q(d,f)=0 and Q||(d,f)=0, meaning that consistency between the two components of the dielectric tensor is achieved via their underlying relation to the geometrical and optical parameters of the constituent materials. An obvious disadvantage of this algorithm is the necessity of knowledge of the dielectric constants of metal and host material used in the metamaterial sample. In our particular case, we could not apply the consistency algorithm as the optical data for the Tin Bismuth alloy in THz region is not available.

Finally, one can be tempted to use effective medium theory in the attempt to invert Eqs. (15) and (16) in order to extract, for example, the permittivity of a semimetal alloy from the measured data. For example, from Eq. (16) it follows that:

εm=εeff(theory)(ω)(1f)εplasticf
(19)

When trying to use this relation, especially in the case of low volume fractions where this relation is most applicable, one quickly finds that small measurement errors are greatly amplified by the f1 factor, which renders the inversion algorithm practically unusable.

5.2 Nanostructured inclusions

We note in passing, that metallic wires made of multi-components alloys can exhibit an additional self-organized nanostructure. Particularly upon solidification, the metals of Bi42%Sn58% alloy have been found to separate into different pure metal phases [33

33. S. H. Chen, C. C. Chen, and C. G. Chao, “Novel morphology and solidification behavior of eutectic bismuth–tin (Bi–Sn) nanowires,” J. Alloy. Comp. 481(1-2), 270–273 (2009). [CrossRef]

]. To illustrate this, Fig. 10 presents a scanning-electron-micrograph of the eutectic alloy wires obtained after two subsequent drawings. These wires were extracted from the composite by dissolving the PC cladding. On the wire surfaces one clearly sees formation of nanogratings. Stoichiometric studies using TOF-SIMS measurements (Time-of-Flight Secondary Ion Mass Spectrometry) showed that the lamellar structure on the surface of the individual wires is made of pure Bi (Bismuth) and Sn (Tin) parts.

6. Conclusion

This paper presents an in-depth study of the challenges and practical issues encountered when trying to extract from the THz optical transmission data both the material and geometrical parameters of the deeply subwavelength wire arrays making a metamaterial film. Particularly, we detail a novel fabrication method for the large area THz metamaterial films using a two step fabrication process. First, we draw microstructured fibers containing ordered arrays of micro- or nano-wires made from either metal or dielectric materials. We then arrange such fibers parallel to each other and compactify them into a composite film by using a hot-press technique. Characterization of the THz transmission through metal wire composite films demonstrates strong polarization sensitivity of their optical properties, promising application of such films as polarizers or high pass filters for THz radiation. Furthermore, composite films containing high refractive-index dielectric wires in place of metallic wires showed no polarization sensitivity, while the effective refractive index of such metamaterial films could be adjusted in a broad range. Using the transfer matrix theory, we then demonstrate how to retrieve the polarization dependent complex refractive index and complex effective permittivity of the metamaterial films from the THz optical transmission data. Finally, we detail the selfconsistent algorithm for the validation of such parameters as filling factor and thickness of the metamaterial layers by using an effective medium approximation.

References and links

1.

Y. Ma, A. Khalid, T. D. Drysdale, and D. R. S. Cumming, “Direct fabrication of terahertz optical devices on low-absorption polymer substrates,” Opt. Lett. 34(10), 1555–1557 (2009). [CrossRef] [PubMed]

2.

R. Yahiaoui, H. Nemec, P. Kužel, F. Kadlec, C. Kadlec, and P. Mounaix, “Broadband dielectric terahertz metamaterials with negative permeability,” Opt. Lett. 34(22), 3541 (2009). [CrossRef] [PubMed]

3.

W. Cai, and V. Shalaev, Optical metamaterials. fundamentals and applications (Springer, 2010)

4.

C. Brosseau, “Modelling and simulation of dielectric heterostructures: a physical survey from an historical perspective,” J. Phys. D Appl. Phys. 39(7), 1277–1294 (2006). [CrossRef]

5.

S. Li, H.-W. Zhang, Q.-Y. Wen, Y.-Q. Song, Y.-S. Xie, W.-W. Ling, Y.-X. Li, and J. Zha, “Micro-fabrication and properties of the meta materials for the terahertz regime,” Infrared Phys. Technol. 53(1), 61–64 (2010). [CrossRef]

6.

F. Miyamaru, S. Kuboda, K. Taima, K. Takano, M. Hangyo, and M. W. Takeda, “Three-dimensional bulk metamaterials operating in the terahertz range,” Appl. Phys. Lett. 96(8), 081105 (2010). [CrossRef]

7.

A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials (Amst.) 2(1), 1–17 (2008). [CrossRef]

8.

K. Takano, K. Shibuya, K. Akiyama, T. Nagashima, F. Miyamaru, and M. Hangyo, “A metal-to-insulator transition in cut-wire-grid metamaterials in the terahertz region,” J. Appl. Phys. 107(2), 024907 (2010). [CrossRef]

9.

Y. Minowa, T. Fujii, M. Nagai, T. Ochiai, K. Sakoda, K. Hirao, and K. Tanaka, “Evaluation of effective electric permittivity and magnetic permeability in metamaterial slabs by terahertz time-domain spectroscopy,” Opt. Express 16(7), 4785–4796 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4785. [CrossRef] [PubMed]

10.

K. Takano, T. Kawabata, C.-F. Hsieh, K. Akiyama, F. Miyamaru, Y. Abe, Y. Tokuda, R.-P. Pan, C.-L. Pan, and M. Hangyo, “Fabrication of terahertz planar metamaterials using a super-fine ink-jet printer,” Appl. Phys. Express 3(1), 016701 (2010). [CrossRef]

11.

T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42(Part 2, No. 4A), 373–375 (2003). [CrossRef]

12.

G. L. Hornyak, C. J. Patrissi, and C. R. Martin, “Fabrication, Characterization, and Optical Properties of Gold Nanoparticle/Porous Alumina Composites: The Nonscattering Maxwell-Garnett Limit,” J. Phys. Chem. B 101(9), 1548–1555 (1997). [CrossRef]

13.

A. Huczko, “Template-based synthesis of nanomaterials,” Appl. Phys., A Mater. Sci. Process. 70(4), 365–376 (2000). [CrossRef]

14.

X. Zhang, Z. Ma, Z.-Y. Yuan, and M. Su, “Mass-productions of vertically aligned extremely long metallic micro/nanowires using fiber drawing nanomanufacturing,” Adv. Mater. 20(7), 1310–1314 (2008). [CrossRef]

15.

A. Tuniz, B. T. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. C. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett. 96(19), 191101 (2010). [CrossRef]

16.

A. Mazhorova, J. F. Gu, S. Gorgutsa, M. Peccianti, T. Ozaki, R. Morandotti, M. Tang, H. Minamide, H. Ito, and M. Skorobogatiy, “THz metamaterials using aligned metallic or semiconductor nanowires” We-P.31, Proceedings of IEEE34th International Conference on Infrared, Millimeter, and Terahertz Waves, IRMMW-THz 2010.

17.

J. Hou, D. Bird, A. George, S. Maier, B. T. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibres,” Opt. Express 16(9), 5983–5990 (2008). [CrossRef] [PubMed]

18.

H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. S. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010). [CrossRef] [PubMed]

19.

M. Skorobogatiy, and J. Yang, Fundamentals of photonic crystal guiding (Cambridge University Press, 2009)

20.

M. Born, and E. Wolf, Principles of optics (Cambridge University Press, 7th edition, 1999)

21.

C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002). [CrossRef] [PubMed]

22.

W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Material thickness optimization for transmission-mode terahertz time-domain spectroscopy,” Opt. Express 16(10), 7382–7396 (2008). [CrossRef] [PubMed]

23.

Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513 (2006).

24.

J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs I, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]

25.

S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, “Wire media with negative effective permittivity: a quasi-static model,” Microw. Opt. Technol. Lett. 35(1), 47–51 (2002). [CrossRef]

26.

P. Markoš and C. M. Soukoulis, “Absorption losses in periodic arrays of thin metallic wires,” Opt. Lett. 28(10), 846 (2003). [CrossRef] [PubMed]

27.

M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, “C. M. Soukoulis “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals”,” Phys. Rev. B 64(19), 195113 (2001). [CrossRef]

28.

A. Rahachou and I. Zozoulenko, “Light propagation in nanorod arrays,” J. Opt. A, Pure Appl. Opt. 9(3), 265–270 (2007). [CrossRef]

29.

K. Aydin, K. Guven, N. Katsarakis, C. Soukoulis, and E. Ozbay, “Effect of disorder on magnetic resonance band gap of split-ring resonator structures,” Opt. Express 12(24), 5896–5901 (2004). [CrossRef] [PubMed]

30.

N. Papasimakis, V. A. Fedotov, Y. H. Fu, D. P. Tsai, and N. I. Zheludev, “Coherent and incoherent metamaterials and order-disorder transitions,” Phys. Rev. B 80, 041102 (2009). [CrossRef]

31.

N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). [CrossRef]

32.

J.-H. Peng, J.-J. Yang, M. Huang, J. Sun, and Z.-Y. Wu, “Simulation and analysis of the effective permittivity for two-phase composite medium,” Front. Mater. Sci. China 3(1), 38–43 (2009). [CrossRef]

33.

S. H. Chen, C. C. Chen, and C. G. Chao, “Novel morphology and solidification behavior of eutectic bismuth–tin (Bi–Sn) nanowires,” J. Alloy. Comp. 481(1-2), 270–273 (2009). [CrossRef]

34.

L. Harris and J. Piper, “Optical and Electrical Properties of Bismuth Deposits,” J. Opt. Soc. A 53, 1271 (1963). [CrossRef]

35.

W. S. Boyle and A. D. Brailsford, “Far infrared Studies of Bismuth,” Phys. Rev. 120(6), 1943–1949 (1960). [CrossRef]

36.

V. A. Podolskiy, L. V. Alekseyev, and E. E. Narimanov, “Strongly anisotropic media: the THz perspectives of left-handed materials,” J. Mod. Opt. 52(16), 2343–2349 (2005). [CrossRef]

37.

M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42, 065415 (2009). [CrossRef]

38.

J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006). [CrossRef]

39.

M. G. Silveirinha, ““Nonlocal Homogenization Model for a Periodic Array of ε-Negative Rods,” Phys. Rev. E - Statistical,” Nonlinear Soft Matt. Phys. 73(4), 046612 (2008). [CrossRef]

OCIS Codes
(230.5440) Optical devices : Polarization-selective devices
(300.6495) Spectroscopy : Spectroscopy, teraherz

ToC Category:
Optical Devices

History
Original Manuscript: September 7, 2010
Revised Manuscript: October 29, 2010
Manuscript Accepted: November 2, 2010
Published: November 10, 2010

Citation
Anna Mazhorova, Jian Feng Gu, Alexandre Dupuis, Marco Peccianti, Ozaki Tsuneyuki, Roberto Morandotti, Hiroaki Minamide, Ming Tang, Yuye Wang, Hiromasa Ito, and Maksim Skorobogatiy, "Composite THz materials using aligned metallic and semiconductor microwires, experiments and interpretation," Opt. Express 18, 24632-24647 (2010)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-24-24632


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References

  1. Y. Ma, A. Khalid, T. D. Drysdale, and D. R. S. Cumming, “Direct fabrication of terahertz optical devices on low-absorption polymer substrates,” Opt. Lett. 34(10), 1555–1557 (2009). [CrossRef] [PubMed]
  2. R. Yahiaoui, H. Nemec, P. Kužel, F. Kadlec, C. Kadlec, and P. Mounaix, “Broadband dielectric terahertz metamaterials with negative permeability,” Opt. Lett. 34(22), 3541 (2009). [CrossRef] [PubMed]
  3. W. Cai, and V. Shalaev, Optical metamaterials. fundamentals and applications (Springer, 2010)
  4. C. Brosseau, “Modelling and simulation of dielectric heterostructures: a physical survey from an historical perspective,” J. Phys. D Appl. Phys. 39(7), 1277–1294 (2006). [CrossRef]
  5. S. Li, H.-W. Zhang, Q.-Y. Wen, Y.-Q. Song, Y.-S. Xie, W.-W. Ling, Y.-X. Li, and J. Zha, “Micro-fabrication and properties of the meta materials for the terahertz regime,” Infrared Phys. Technol. 53(1), 61–64 (2010). [CrossRef]
  6. F. Miyamaru, S. Kuboda, K. Taima, K. Takano, M. Hangyo, and M. W. Takeda, “Three-dimensional bulk metamaterials operating in the terahertz range,” Appl. Phys. Lett. 96(8), 081105 (2010). [CrossRef]
  7. A. Boltasseva and V. M. Shalaev, “Fabrication of optical negative-index metamaterials: Recent advances and outlook,” Metamaterials (Amst.) 2(1), 1–17 (2008). [CrossRef]
  8. K. Takano, K. Shibuya, K. Akiyama, T. Nagashima, F. Miyamaru, and M. Hangyo, “A metal-to-insulator transition in cut-wire-grid metamaterials in the terahertz region,” J. Appl. Phys. 107(2), 024907 (2010). [CrossRef]
  9. Y. Minowa, T. Fujii, M. Nagai, T. Ochiai, K. Sakoda, K. Hirao, and K. Tanaka, “Evaluation of effective electric permittivity and magnetic permeability in metamaterial slabs by terahertz time-domain spectroscopy,” Opt. Express 16(7), 4785–4796 (2008), http://www.opticsinfobase.org/abstract.cfm?URI=oe-16-7-4785 . [CrossRef] [PubMed]
  10. K. Takano, T. Kawabata, C.-F. Hsieh, K. Akiyama, F. Miyamaru, Y. Abe, Y. Tokuda, R.-P. Pan, C.-L. Pan, and M. Hangyo, “Fabrication of terahertz planar metamaterials using a super-fine ink-jet printer,” Appl. Phys. Express 3(1), 016701 (2010). [CrossRef]
  11. T. Kondo, T. Nagashima, and M. Hangyo, “Fabrication of wire-grid-type polarizers for THz region using a general-purpose color printer,” Jpn. J. Appl. Phys. 42(Part 2, No. 4A), 373–375 (2003). [CrossRef]
  12. G. L. Hornyak, C. J. Patrissi, and C. R. Martin, “Fabrication, Characterization, and Optical Properties of Gold Nanoparticle/Porous Alumina Composites: The Nonscattering Maxwell-Garnett Limit,” J. Phys. Chem. B 101(9), 1548–1555 (1997). [CrossRef]
  13. A. Huczko, “Template-based synthesis of nanomaterials,” Appl. Phys., A Mater. Sci. Process. 70(4), 365–376 (2000). [CrossRef]
  14. X. Zhang, Z. Ma, Z.-Y. Yuan, and M. Su, “Mass-productions of vertically aligned extremely long metallic micro/nanowires using fiber drawing nanomanufacturing,” Adv. Mater. 20(7), 1310–1314 (2008). [CrossRef]
  15. A. Tuniz, B. T. Kuhlmey, R. Lwin, A. Wang, J. Anthony, R. Leonhardt, and S. C. Fleming, “Drawn metamaterials with plasmonic response at terahertz frequencies,” Appl. Phys. Lett. 96(19), 191101 (2010). [CrossRef]
  16. A. Mazhorova, J. F. Gu, S. Gorgutsa, M. Peccianti, T. Ozaki, R. Morandotti, M. Tang, H. Minamide, H. Ito, and M. Skorobogatiy, “THz metamaterials using aligned metallic or semiconductor nanowires” We-P.31, Proceedings of IEEE34th International Conference on Infrared, Millimeter, and Terahertz Waves, IRMMW-THz 2010.
  17. J. Hou, D. Bird, A. George, S. Maier, B. T. Kuhlmey, and J. C. Knight, “Metallic mode confinement in microstructured fibres,” Opt. Express 16(9), 5983–5990 (2008). [CrossRef] [PubMed]
  18. H. K. Tyagi, H. W. Lee, P. Uebel, M. A. Schmidt, N. Joly, M. Scharrer, and P. S. Russell, “Plasmon resonances on gold nanowires directly drawn in a step-index fiber,” Opt. Lett. 35(15), 2573–2575 (2010). [CrossRef] [PubMed]
  19. M. Skorobogatiy, and J. Yang, Fundamentals of photonic crystal guiding (Cambridge University Press, 2009)
  20. M. Born, and E. Wolf, Principles of optics (Cambridge University Press, 7th edition, 1999)
  21. C. C. Katsidis and D. I. Siapkas, “General transfer-matrix method for optical multilayer systems with coherent, partially coherent, and incoherent interference,” Appl. Opt. 41(19), 3978–3987 (2002). [CrossRef] [PubMed]
  22. W. Withayachumnankul, B. M. Fischer, and D. Abbott, “Material thickness optimization for transmission-mode terahertz time-domain spectroscopy,” Opt. Express 16(10), 7382–7396 (2008). [CrossRef] [PubMed]
  23. Y.-S. Jin, G.-J. Kim, and S.-G. Jeon, “Terahertz Dielectric Properties of Polymers,” J. Korean Phys. Soc. 49, 513 (2006).
  24. J. B. Pendry, A. J. Holden, W. J. Stewart, and I. Youngs, “Extremely low frequency plasmons in metallic mesostructures,” Phys. Rev. Lett. 76(25), 4773–4776 (1996). [CrossRef] [PubMed]
  25. S. I. Maslovski, S. A. Tretyakov, and P. A. Belov, “Wire media with negative effective permittivity: a quasi-static model,” Microw. Opt. Technol. Lett. 35(1), 47–51 (2002). [CrossRef]
  26. P. Markoš and C. M. Soukoulis, “Absorption losses in periodic arrays of thin metallic wires,” Opt. Lett. 28(10), 846 (2003). [CrossRef] [PubMed]
  27. M. Bayindir, E. Cubukcu, I. Bulu, T. Tut, E. Ozbay, and C. Soukoulis, “C. M. Soukoulis “Photonic band gaps, defect characteristics, and waveguiding in two-dimensional disordered dielectric and metallic photonic crystals”,” Phys. Rev. B 64(19), 195113 (2001). [CrossRef]
  28. A. Rahachou and I. Zozoulenko, “Light propagation in nanorod arrays,” J. Opt. A, Pure Appl. Opt. 9(3), 265–270 (2007). [CrossRef]
  29. K. Aydin, K. Guven, N. Katsarakis, C. Soukoulis, and E. Ozbay, “Effect of disorder on magnetic resonance band gap of split-ring resonator structures,” Opt. Express 12(24), 5896–5901 (2004). [CrossRef] [PubMed]
  30. N. Papasimakis, V. A. Fedotov, Y. H. Fu, D. P. Tsai, and N. I. Zheludev, “Coherent and incoherent metamaterials and order-disorder transitions,” Phys. Rev. B 80, 041102 (2009). [CrossRef]
  31. N. V. Smith, “Classical generalization of the Drude formula for the optical conductivity,” Phys. Rev. B 64(15), 155106 (2001). [CrossRef]
  32. J.-H. Peng, J.-J. Yang, M. Huang, J. Sun, and Z.-Y. Wu, “Simulation and analysis of the effective permittivity for two-phase composite medium,” Front. Mater. Sci. China 3(1), 38–43 (2009). [CrossRef]
  33. S. H. Chen, C. C. Chen, and C. G. Chao, “Novel morphology and solidification behavior of eutectic bismuth–tin (Bi–Sn) nanowires,” J. Alloy. Comp. 481(1-2), 270–273 (2009). [CrossRef]
  34. L. Harris and J. Piper, “Optical and Electrical Properties of Bismuth Deposits,” J. Opt. Soc. A 53, 1271 (1963). [CrossRef]
  35. W. S. Boyle and A. D. Brailsford, “Far infrared Studies of Bismuth,” Phys. Rev. 120(6), 1943–1949 (1960). [CrossRef]
  36. V. A. Podolskiy, L. V. Alekseyev, and E. E. Narimanov, “Strongly anisotropic media: the THz perspectives of left-handed materials,” J. Mod. Opt. 52(16), 2343–2349 (2005). [CrossRef]
  37. M. Scheller, S. Wietzke, C. Jansen, and M. Koch, “Modelling heterogeneous dielectric mixtures in the terahertz regime: a quasi-static effective medium theory,” J. Phys. D Appl. Phys. 42, 065415 (2009). [CrossRef]
  38. J. Elser, R. Wangberg, V. A. Podolskiy, and E. E. Narimanov, “Nanowire metamaterials with extreme optical anisotropy,” Appl. Phys. Lett. 89(26), 261102 (2006). [CrossRef]
  39. M. G. Silveirinha, ““Nonlocal Homogenization Model for a Periodic Array of ε-Negative Rods,” Phys. Rev. E - Statistical,” Nonlinear Soft Matt. Phys. 73(4), 046612 (2008). [CrossRef]

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