Abstract

Artificially structured hyperbolic metamaterials (HMMs) - uniaxial materials with opposite signs of permittivity for ordinary and extraordinary waves - are one of the most attractive classes of metamaterials. Their existing in nature counterpart natural (homogeneous) hyperbolic materials (NHMs) has several advantages but has not yet been analyzed extensively. Here, based on literature-available data on permittivity as a function of wavelength, we review materials with naturally occurring anisotropy of permittivity in specific wavelength ranges. We suggest the best choice of materials that may act as NHMs depending on the wavelength, strength of the dielectric anisotropy (SDA), and losses.

© 2015 Optical Society of America

1. Introduction

Hyperbolic metamaterials (HMMs) are one of the most attractive classes of metamaterials [1,2] due to their unique hyperbolic dispersion [3,4] enabling unusual electromagnetic response, such as broadband negative refraction [5,6] without artificial magnetism, which is not very sensitive to structural disorder. HMMs [7,8] are materials with strong dielectric anisotropy. They are characterized by opposite values of the permittivity for the ordinary and extraordinary waves, where the electric-field vector E is perpendicular (transverse electric polarized light - TE) and parallel (transverse magnetic polarized light - TM) to the c-axis, respectively. For an electromagnetic wave propagating in one direction HMMs have metallic properties, Re(εi)<0; while for waves travelling in the other direction it exhibits dielectric properties, Re(εj)>0. As a consequence the electromagnetic wave propagates in such materials with a hyperboloid-like shape of the isofrequency surface [2] instead of ellipsoidal as for standard anisotropic materials or spherical for isotropic materials. The combination of metallic and dielectric behavior and the unusual electromagnetic high k-mode propagation enabled by the open isofrequency surface make HMMs very attractive from the point of view of applications: nano-imaging [9], nano-sensing [10], nano-resonators [11], thermal emission engineering [12], control of emission [13–15], efficient absorbers [16] and sub-diffraction imaging [17–19]. While the non-resonant character of the origin of the effect enables smaller losses, broader spectral range and smaller sensitivity to disorder [6].

Optical properties of a material can be described by the dielectric function 𝟄, which consists of real and imaginary parts of the dielectric permittivity, 𝜀. In general, the dielectric permittivity is a tensor which depends on the crystal symmetry. Through control of the components of the dielectric permittivity tensor of a particular material we can influence its optical properties. Propagation of the electromagnetic wave (k-wavevector) through a material is determined by the dispersion relation, Eq. (1), which describes the dependence of electromagnetic wave velocity (~refractive index) on the frequency.

kx2ε1+ky2ε2+kz2ε3=ω2c2

In anisotropic materials, different permittivities in the x, y and z direction have to be taken into account. If the permittivities in all directions are positive, the solution of this equation in three dimensional space is an ellipsoid. However, if the value of the permittivity is positive in some directions and negative in others then it results in a hyperboloid isofrequency surface extending to infinity. Materials with hyperboloidal character of the dispersion relation introduced due to its composite subwavelength structure are HMMs, while one-phase naturally occuring anisotropic materials can be abbreviated as NHMs. Due to the same character in the x and y direction, the dielectric tensor describing the HMM's permittivity can be simplified to 3 variables, Eq. (2):

ε=[εxx,εyy,εzz] (2),
where εxx = εyy = ε2 and εzz = ε1; ε1>0, ε2 < 0. As a result, the dispersion relation for hyperbolic materials can be described by Eq. (3) [9]:

(kx2+ky2)ε1+kz2ε2=(ωc)2

In HMMs, TM-polarized light is negatively refracted (with the positive phase velocity [5, 20]) while TE-polarized light is positively refracted. The isofrequency surface of the extraordinary wave (TM polarized) propagating in the z-direction in such a medium has the form described by Eq. (3).

Depending on the direction in which the negative value of the dielectric permittivity occurs, HMMs can be classified into type I HMMs or type II HMMs [9] (Fig. 1). The first one has one negative component of the dielectric tensor negative (ε1<0; ε2 = (εxx, εyy)>0), while type II HMMs have two components negative (ε1>0; ε2 = (εxx, εyy)<0). Type II HMMs are more reflective and possess higher losses and higher impedance mismatch with vacuum than type I HMMs due to their larger metallic contribution [21]. Two types of HMMs result in different types of imaging [17, 19, 22].

 

Fig. 1 Scheme presenting types of hyperbolic materials.

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Usually hyperbolic materials are obtained as artificial nanostructures, being the combination of two component materials – metallic and dielectric. The first approach is the multilayer structure consisting of alternating layers of metallic and dielectric character in the same wavelength range [5, 23]. The second technique is related to metallic nanowires embedded in a dielectric matrix [6, 24]. Negative refraction in HMMs has been demonstrated: at ultraviolet wavelengths utilizing Ag/TiO2 [25]; at visible wavelengths in Ag/Al2O3 [6], and at far-infrared wavelengths in InGaAs-AlInAs [5]. Other composite materials have been manufactured and analyzed as hyperbolic materials e.g. in the visible Ag/MgF2 [26]; and in the near-infrared wavelength range TiN/AlN [27], ITO (In2O3:SnO2)/TiN [28].

However, manufacturing hyperbolic metamaterials - artificial anisotropic - layered/fibrous - structures out of two component materials, enabling anisotropy of permittivity [5, 25], is still a challenge. Homogeneous/natural hyperbolic materials - uniaxial materials with opposite signs of permittivity for ordinary and extraordinary waves - are a good alternative [29], however their potential has not yet been analyzed extensively [30,31].

Recently, it has been observed that natural materials possessing a strong anisotropy of dielectric permittivity, Fig. 1, behave as hyperbolic materials [31]. Homogeneous hyperbolic materials [30] (natural single-phase materials) are regarded as a potential alternative to technologically challenging structures of artificial hyperbolic metamaterials (structured composite materials). Only few materials have been considered as natural hyperbolic materials up to now: graphite (C@254 nm) [32], cuprates (i.e. La1.92Sr0.08CuO4@290 nm, YBa2Cu3O7−x@880nm) [33], magnesium diboride (MgB2@420 nm) [33], tetradymites (Bi2Se3@709 nm–1.18 𝜇m and Bi2Te3@310 nm–1.38 𝜇m) [34], ruthenates (Sr2RuO4@1.49 𝜇m and Sr3Ru2O7@1.49 𝜇m) [33], hexagonal boron nitride (hBN @6.2–7.3 𝜇m and 12.1-13.2 𝜇m) [35], sapphire (Al2O3@20 𝜇m) [5, 31], bismuth (Bi@53.7-63.2 𝜇m) [31, 36], triglycine sulfate (TGS@255 𝜇m) [30] and calcite (CaCO3@6.75 𝜇m and 11.33 𝜇m) [31].

Here, using the available data on optical properties of materials [37] we review various natural materials and propose the best ones which fulfill the criteria of natural hyperbolic materials in various wavelength ranges.

2. Methodology

We present a comparison of the materials that may be regarded as NHMs. The literature values of the refractive index, nj, and the extinction coefficient, κj, were used as input parameters [37]. Both parameters were obtained from measurements of uniaxial or biaxial single crystals at room temperature using polarized radiation in two or three characteristic j- directions of the optical constants. In this paper, for uniaxial materials j = 1 will denote the ordinary wave where E ⊥ c [37] (E-vector is normal to the principal plane, which contains both the wave vector and the optic axis c) while j = 2 will denote the extraordinary wave where E II c (E-vector is parallel to the principal plane). This compares with notation used in other papers ε1 = ε and ε2 = ε. The values of the real, Re(𝜀j), and imaginary, Im(𝜀j), parts of the dielectric permittivity (Eq. (4) and Eq. (5) [37]) were determined. Additionally, for each material we provide the quality factor (Qj, Eq. (6)) [5, 31] that enables a comparison between various hyperbolic materials by taking into account the losses in the direction in which the material exhibits metallic character.

Re(εj)=nj2kj2
Im(εj)=2njkj
Qj=Re(εj)Im(εj)

In Ref [37]. the optical constants for 32 uniaxial and 3 biaxial materials are available. In the first part of the supplementary material of this paper, calculated parameters Re(𝜀j), Im(𝜀j) and Qj-factor for 22 chosen uniaxial materials were collected which fulfill the condition of Re(εi)<0, Re(εj)>0, i,j = 1,2. For three biaxial crystals based on optical constants in 3 crystallographic orientations a, b, c we present permittivities: Re(εa), where E II a, Re(εb), where, E II b, Re(εc), where E II c.

Additionally graphs showing the wavelength dependence of Re(𝜀j), Im(𝜀j) and Q-factor for two or three directions of the optical constants are shown. Finally, tables with natural materials of the highest quality factor, Qmax, and materials with the highest strength of the dielectric anisotropy, ∆εmax, are shown. The summary Table 1S, as we show at Data set 1 (Ref [38].), shows the range of opposite values of Re(𝜀j) in 2 (uniaxial materials) or 3 (biaxial materials) optical directions for each of the 25 of analyzed materials. For each range we calculated the maximal value of the quality factor, Qjmax, occurring at a specific wavelength, λQj. Each material in specific wavelength ranges has been assigned as type I or II NHM, which corresponds to type I or II HMM. Table 2S in te Data set 1 (Ref [38].) shows the maximal strength of the anisotropy, ∆εmax at a specific wavelength, λΔε for each material. The Im(εj) and the Im(εi) values are also displayed to provide the information on losses in these materials.

3. Results and discussion

Three parameters were chosen for selection of the natural hyperbolic materials, i.e. crystalline materials with opposite values of permittivity for the ordinary and extraordinary waves: (i) the wavelength range, Δλ, where material exhibits hyperbolic dispersion (Re(εi)<0, Re(εj)>0, i,j = 1,2), (ii) the maximal quality factor, Qmax, which should indicate mainly materials with the smallest electromagnetic losses for Re(εi)<0, i = 1,2, (metallic losses) (iii) Im(ε2) – losses of positive part of permittivity (dielectric losses); and (iv) the strength of the dielectric anisotropy (SDA), Eq. (7):

Δε=Re(εj)Re(εi)
where Re(εi)<0, Re(εj)>0, i,j = 1,2. The detailed data for all the investigated materials is shown in the Data set 1 (Ref [38].) (Fig. S1-S55, Tables. S1-S27).

4.1 High Q natural hyperbolic materials

Figure 2 shows representative potential natural hyperbolic materials at specific wavelength ranges where opposite signs of dielectric permittivity exist for such materials. These materials were selected on the basis of their maximal available Q factors, where materials with Q>5 are denoted in green and those with 3>Q>5 are denoted in blue. For materials with Q>5, the maximal Q values, Qmax, at specific wavelengths are shown in Fig. 3(a), and for materials with 3>Q>5 are shown in Fig. 3(b), other materials were neglected.

 

Fig. 2 Occurrence of natural hyperbolic materials at particular wavelength ranges. Chosen crystalline uniaxial and biaxal (orange stars) materials are shown which possess at least at some wavelength maximal quality factor Q>5 (green colour) or 3<Q<5 (blue colour); Materials with Q<3 have been neglected. The total wavelength range for which these materials exhibits Re(εi)<0, Re(εj)>0 with Q>5 or 3<Q<5 are shown.

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Fig. 3 Natural hyperbolic materials with highest available quality factor, Qmax, demonstrating materials with relatively small losses in the comparison with the magnitude of the negative real part of permittivity. Q = -Re(εi)/Im(εi), where Re(εi)<0, Re(εj)>0. The histograms are presented for chosen crystalline uniaxial and biaxal (orange stars) materials for which their maximal quality factor is: (a) Qmax>5; (b) 3<Qmax<5. Materials with Qmax<3 have been neglected. Δε = Re(εi)-Re(εj) and Im(εj) at this wavelength are also presented.

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Parameter Q provides the information on the metallic losses, while dielectric losses are provided with the Im(ε) - denoted in violet and green in Fig. 3, respectively. For each of these materials the SDA is also reported, the magnitudes of both permittivities shown separately. Table 1 presents NHMs with high quality factors (Q>5 or 3>Q>5) within the range of wavelengths where these materials are characterized with respective Q values. Materials for specific wavelength ranges have been assigned as NHM type I or II, and in Table 2 chosen data for these materials are shown.

Tables Icon

Table 1. Natural hyperbolic materials characterized by quality factor Q>5 or 3<Q<5 at specific wavelength range.

Tables Icon

Table 2. Chosen data, as quality factor, Q, dielectric losses Im(ε), strength of anisotropy, Δε, for chosen materials.

In the case of materials with Q>5 one of the best potential NHMs in the far-infrared wavelength range is zirconium silicate, ZrSiO4, Fig. 4(a), which shows the highest Qmax of 50@10.53μm with Δε = 22.93 and also relatively small losses for the positive part of the dielectric permittivity (Im(ε2) = 3.26). For this material Q decreases at longer wavelength, however it is still high, while the losses in the positive permittivity direction decrease (Q1 = 10.98, Im(ε2) = 0.77 @10.87 µm, and Q1 = 5.2, Im(ε2) = 0.57 @10.99 µm), Table 2. ZrSiO4 is a high refractive index material with very high melting point (2550°C), high thermal conductivity, excellent chemical stability, and it is used for long-term storage of nuclear waste [39], and as a thermal barrier coating in engines [40]. Due to its properties it can be used at high temperatures and in harsh chemical conditions.

 

Fig. 4 Natural uniaxial materials with high quality factor, Q: a) ZrSiO4, b) BaTiO3. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and quality factor (dot line) for ordinary and extraordinary waves.

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The well-known lithium tantalate, LiTaO3, widely used in surface acoustic wave devices due to its piezoelectric properties [41] (e.g. surface acoustic wave biosensors), [42] is the material with the second highest Qmax as well as very high SDA (11.65@32.26μm and Δε = 50.17), but very high dielectric losses, Im(ε2) = 34.38. LiTaO3 has smaller dielectric losses at slightly longer wavelengths (Q2 = 9.99, Im(ε1) = 5.75@35.71μm and Δε = 24.25). LiTaO3 exhibits many other interesting properties as strong ferroelectricity up to about 600°C [43], pyroelectric properties and large electro-optic and nonlinear optical coefficients [41], which led to its use in passive infrared sensors such as motion detectors [44], and terahertz generation and detection [45]. The advantage of using LiTaO3 is also that it is quite easily obtainable via standard growth techniques - single crystals from the melt [46] or as a thin films (e.g. by pulsed laser deposition technique).

Like LiTaO3, barium titanate BaTiO3, belongs to the family of ferroelectric materials (Tc = 130°C) [47] with piezoelectric effects [48]. Due to the large electro-optic coefficients it is extensively used as a photorefractive material [49]. It is also widely used in ceramic capacitors, piezoelectric transducers, pyroelectric detectors, ferroelectric memories [50] and electro-optic devices (e.g. electro-optic modulators) [45, 49]. It is much more difficult to obtain as a single crystal but growth by the Czochralski method was reported [51] as well as growth of thin films by epitaxial techniques [52]. BaTiO3 is also one of the most interesting natural hyperbolic materials, due to relatively high Qmax = 6.73@13.7µm (Fig. 4(b)) together with large Δε = 71.28 and very small dielectric losses (Im(ε2) = 0.29).

Aluminum oxide, Al2O3, is a material with the highest value of dielectric anisotropy, Δε = 107.2@23.26 µm, among materials with Qmax>5 (Qmax = 5.61, Im(ε2) = 13.1), Fig. 5(b). To decrease the dielectric losses one may choose slightly longer wavelengths, 23.81 µm or 24.39 µm, where Im(ε1) = 3.27 and Im(ε1) = 1.63 respectively; however, the quality factor decreases as well (Q2 = 4.09 and Q2 = 2.26, respectively). Also at 20.41 µm Al2O3 (Fig. 5(a)) exhibits high Qmax = 5.67, relatively low Δε = 8, and small dielectric losses (Im(ε2) = 0.6). Al2O3 has already been proposed for use at 20 µm. [5,31,53] Other materials with Qmax>5 and relativity small dielectric losses are SiO2 @20.90 µm (Im(ε2) = 1.39, Δε = 16.31) – Fig. 6(a), and magnesium fluoride, MgF2, @37.04 µm (Im(ε2) = 0.3, Δε = 14.73), Fig. 6(b).

 

Fig. 5 Aluminum oxide, Al2O3, as homogeneous hyperbolic material: a) exhibiting a relatively high Q factor and small losses, b) exhibiting very high dielectric anisotropy, Δε. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and the quality factor (dot line) for ordinary and extraordinary waves.

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Fig. 6 Natural hyperbolic materials with the maximal value of quality factor Qmax>5 and relativity small losses for the positive part of the dielectric permittivity: a) SiO2, b) MgF2. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and quality factor (dot line) for ordinary and extraordinary waves.

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Natural hyperbolic materials with 3<Q<5 include: SiO2, LiTaO3, MgF2, BaTiO3, ZrSiO4, TiO2, KNbO3, NaNO3 and CuGaS2, with detailed data provided in Table 2.

4.2 Broad frequency range homogeneous hyperbolic materials

Natural materials with the widest wavelength range, where Re(εi)<0 and Re(εj)>0 (or vice versa) in the same wavelength range, are NaNO3 (Δλ = 114.3-190.5 µm), Fig. 7(a), BaTiO3, (Δλ = 37.04-52.63 µm, and Δλ = 60-250 µm) Fig. 7(b), LaTiO3 (Δλ = 52.63-66.67 µm), MgF2 (Δλ = 33.33-40 µm), Fig. 2 and Fig. 6(b), Bi (Δλ = 53.7-63.2 µm) [31], and graphite (Δλ = 0.19-61.99 µm). As can be seen the wavelength range where natural materials have opposite signs of permittivity in two different directions can be even tens or hundreds of micrometers. Such a wide operation range has not been reported so far in artificial hyperbolic metamaterials; for example, it is a few micrometers for InGaAs/AlInAs layered structure [5] in the infrared. The intrinsic properties of NHMs can be changed in various ways enabling tuning of the permittivity function and thus also the hyperbolic properties wavelength range. The wavelength ranges where Q>5 or 3<Q<5 are much shorter than total Δλ, where hyperbolic dispersion exists. Typically it is around a few micrometers with the exception of NaNO3 for which it is tens of micrometers, and over a dozen micrometers for BaTiO3, Table 1.

 

Fig. 7 Natural uniaxial materials with opposite signs of permittivity for ordinary and extraordinary waves in the broad range of wavelengths: a) NaNO3, b) BaTiO3. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and the quality factor (dot line) for ordinary and extraordinary waves.

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4.3 High strength of dielectric anisotropy (SDA) in natural hyperbolic materials

Natural materials with the highest ∆ε, ∆εmax>100, are shown in Fig. 8(a), and the wavelength ranges in which these materials fulfill the condition of Re(εi)<0 and Re(εj)>0 are shown in Fig. 8(b) and are summarized in Table 3. The materials here were not selected from the point of view of the smallest losses. Graphite shows huge ∆εmax = 625@61.99μm (Fig. 9), and materials with the next highest ∆εmax are LiTaO3max = 219@66.67μm and εmax = 202 @47.62μm) and KNbO3ma x = 204@100μm).

 

Fig. 8 Natural hyperbolic materials with highest known anisotropy of permittivity, ∆εmax. Δε = Re(εj)-Re(εi), where Re(εi)<0, Re(εj)>0. a) Histograms show materials with Δε>100 at the wavelength of its maximal value. Im(εi) and Im(εj) at the ∆εmax wavelength are also presented. b) Occurrence of natural hyperbolic materials for which Δεmax>100 at particular wavelength ranges. Complete wavelength ranges at which these materials exhibit Δεmax>100 are shown.

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Tables Icon

Table 3. Natural hyperbolic materials characterized by high strength of dielectric anisotropy, Δε, Δεmax>100 at specific wavelength ranges.

 

Fig. 9 Graphite - natural uniaxial material with the highest anisotropy of permittivity, ∆ε. Graph shows real parts of dielectric permittivity for ordinary and extraordinary waves.

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Barium titanate, BaTiO3 is the material with the widest wavelength range (56.18-250 µm) where ∆εmax>100, but it is characterized by high losses, Im(ε1) = 343.44, Im(ε2) = 2.96 at 166.7 µm. The smallest losses for this material, where ∆ε>100, are Im(ε1) = 47.28, Im(ε2) = 18.25 at 60.61 µm.

However when taking into account various applications different parameters of the hyperboloid isofrequency contours can be important. For example for HMM type I the big difference in magnitudes for |Re(εi)<0| >> |Re(εj)>0| results in a flat-band iso-frequency countour, which potentially provides a small phase delay, while |Re(εi)<0| << |Re(εj)>0| results in a narrow-band isofrequency contour, potentially good for very high resolution focusing. Thus not only the magnitudes of the permittivities should be considered but also their differences or ratios, which can be usefully represented as ∆εs = |Re(εj)| - |Re(εi)| or ∆εr = |Re(εj)| / |Re(εi)|. That is why in the Fig. 3 and Fig. 8(a) not only the SDA has been shown but also the magnitudes of |Re(εi)| and |Re(εj)|.

4.4 UV-Vis-NIR natural hyperbolic materials

Up to now only a few materials were reported as exhibiting hyperbolic dispersion in the short wavelength range. In the ultraviolet range, all negative angle refraction was demonstrated in graphite [32], at 254 nm for relatively low Q, around 2.5, and relatively low SDA, around 4. In the case of artificial structures such as Ag nanowires embedded in an Al2O3 matrix, the enhancement of luminescence has been demonstrated at ca. 850 nm also for relatively small SDA, around 5 [14].Thus many natural materials reported by us easily fulfill the material parameters for which the phenomena has already been demonstrated experimentally. Graphite could also exhibit hyperbolic dispersion at 225.4 nm though it could be too limited by the very small Q at this wavelength range (Q1 = 0.4, Im(ε2) = 0.1). In the visible and near-infrared ranges Bi2Se3 and Bi2Te3 [34], were reported as NHMs at relatively broad wavelength ranges. They exhibit relatively high SDA around Δε = 15 (@~590 nm) and Δε = 25 (@~826 nm) respectively, however the Q values are rather small Q = ~0.5 in both cases, which may exclude these materials from applications.

Gallium (II) telluride, GaTe, is a III-V semiconductor which can be obtained as single crystal by conventional crystal growth methods,[54] and as thin film by chemical vapor deposition techniques [55]. It is characterized as having a direct band gap (1.69 eV) [56], that is why p-type semiconducting GaTe has been proposed for gamma-ray detectors [57], switching memories [58] and Schottky devices [59]. It is the only material we found in the short wavelength range with Q>3; its most optimal Q begins in the UV range (279 nm) and extends up to the VIS range (497 nm), Fig. 10. However, unlike all the (uniaxial) materials discussed so far, GaTe is biaxial so three parameters have to be taken into account in this case, and consequently the response of this material may be not trivial. At λ = 497 nm GaTe exhibits high anisotropy with εa, εc <0, εb>0, where Qc max = 3.34 (Δε = 20.37), Qa max = 0.12 (Δε = 18.61), and losses Im(εb) = 3.09. The high anisotropy of this material results from its layered crystal structure where single layers are bonded by weak Van der Waals forces [60].

 

Fig. 10 Gallium telluride, GaTe, biaxial natural material with high dielectric anisotropy in the visible wavelength range. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and quality factor (dot line) for three crystallography axes.

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In the NIR range only cobalt (Co@2,26 µm) fulfills the criteria for NHMs. However, its high losses make it impractical for applications Qmax = 0.35 (Im(ε2) = 58.69, Δε = 22.93).

4.5 Biaxial crystals

In all analyzed materials there are three biaxial crystals with opposite signs of permittivities corresponding to different directions: GaTe with two negative and one positive permittivities (εa, εc <0, εb>0, type II NHM) at the wavelength of interest λ = 497 nm, and two crystals, potassium niobate, KNbO3, and arsenic trisulphide, As2S3, with two positive and one negative permittivities (εa, εc >>0, εb<0, type I NHM) at the wavelengths of interest λ = 100 µm and 71,5 µm, respectively. GaTe has been described above as a material with hyperbolic properties at visible wavelengths. On the other hand, KNbO3 and As2S3 are materials with very high anisotropy, ∆ε>100. At 71.5 µm, where ∆εmax = 103 As2S3 exhibits very low dielectric losses in one direction Im(εa) = 0.15, Im(εc) = 42.24 yet relatively low Q, Qb max = 2.07. KNbO3 has similar characteristics, ∆εmax = 204@100 µm, however the losses in this case are higher Im(εa) = 3.00, Im(εc) = 138.55 and Qb max = 1.20.

4.6 Tunable natural hyperbolic materials

The properties of the natural hyperbolic materials could be at least partially adjusted to needs of particular applications by altering their permittivity functions, for example by thermal treatment or doping. However the tunability of the hyperbolic dispersion including wavelength range, SDA and losses could be much more easily achieved in materials with special properties. For example utilizing piezoelectric properties the dielectric function of LiTaO3 could be tuned by applying mechanical force or an electric field [61], resulting in the change of its NHM properties. The ferroelectric properties of this material also provide broad range tunability via thermal treatment [62] and application of electric fields [63] especially near the Curie temperature. All the special properties of BaTiO3 promise potential tunability of its NHM properties, which could be controlled electrically and mechanically (piezoelectricity). In non-centrosymetric materials such as BaTiO3, application of a constant or varying electric field changes the refractive index of the material exhibiting electro-optic effect by inducing birefringence proportional to the applied voltage [64]. Therefore this effect can be also employed for altering the NHM properties of materials. On the other hand, photorefractive materials such as BaTiO3 are materials exhibiting both electro-optic effect and photoconductivity. By absorption of light at specific wavelengths, photoconductivity enables generation of free carriers, which can move by diffusion of the applied electric field. Thus light provides another way to tune such materials [65].

The majority of the analyzed natural uniaxial and biaxial materials exhibit strong dielectric anisotropy in the infrared range, with both I- and II-types of NHMs available. We recall that, in general, type-I hyperbolic materials, which have the lowest reflectivity, are better for imaging applications, due to taking advantage of both propagating and evanescent waves required for high resolution. However, type-II hyperbolic materials, which reflect propagating waves and transmit evanescent ones, may also be used in imaging in order to reduce the image distortion of subwavelength features caused by propagating waves [66]. On the other hand, optical devices built of type-II hyperbolic materials may offer a better platform for high resolution nano-photolithography and light response probing, especially those working at UV wavelengths (GaTe). Both type-I and type-II natural hyperbolic materials can also be used in fluorescence imaging by manipulating the photonic density of states and radiative rate (emission lifetime) engineering [23]. In the infrared range, natural hyperbolic materials (especially those of broadband response) may find applications in non-invasive sub-surface sensing and imaging, which are especially important for IR fingerprinting and in quality control of integrated circuits to find cracks or defects a few μm in size (not achievable by conventional techniques due to diffraction limits) and Super-Planckian thermal emitters [67].

4. Conclusions

Hyperbolic metamaterials have the highest figure of merit (most effective with the lowest losses) among any demonstrated metamaterials and therefore they are the metamaterials with the highest potential for practical applications. However the originally investigated artificial HMMs, made of interspaced phases, suffer from fabrication limitations resulting in impedance mismatch, surface and interface scattering as well as unit cell dimension limit. On the other hand natural/homogeneous hyperbolic materials (NHMs) have no internal wave scattering and the need for complicated design and fine and often sophisticated fabrication techniques are eliminated. The elimination of the unit cell in homogeneous NHMs enables even higher k modes than in the cell-limited artificial structures, though the losses will not allow for indefinitely high modes.

Simple estimation based on well-known optical parameters of different materials has been carried out to identify natural materials that should exhibit hyperbolic dispersion. Consequently, they can potentially be used as natural hyperbolic materials, if other properties of the examined media, like mechanical properties, possibility of cheap manufacturing, purity and accessibility allow their implementation or integration in various devices. Figure 11 summarizes the already demonstrated composite (artificial) and homogeneous (natural) hyperbolic materials which were proposed previously or in this paper (in bold).

 

Fig. 11 Composite and natural hyperbolic materials at different spectral ranges. Materials shown in bold are proposed in this paper.

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In artificial HMMs the operational wavelength range can be tuned by applying different component materials, structural geometries and sizes. In the case of homogeneous NHMs they operate at the naturally fixed wavelength ranges, which can be very broad; but they may also be influenced by doping, thermal treatment, pressure or voltage (for example in the case of piezoelectric materials). Potential devices based on natural hyperbolic materials may take advantage of the dielectric anisotropy, small losses and all the unusual features related to their natural hyperbolic dispersion. The validity of these predictions should now be verified experimentally. This paper therefore gives hints to experimentalists and theoreticians regarding what materials should be considered for practical realization in a wide range of applications.

Acknowledgments

The authors thank prof. Viktor A. Podolskiy from the University of Massachusetts Lowell for very helpful discussions. The authors thank the MAESTRO Project (2011/02/A/ST5/00471) and HARMONIA Project (2013/10/M/ST5/00650) operated by the National Science Centre for support of this work.

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2. R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292(5514), 77–79 (2001). [CrossRef]   [PubMed]  

3. I. I. Smolyaninov, E. Hwang, and E. E. Narimanov, “Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions,” Phys. Rev. B 85(23), 235122 (2012). [CrossRef]  

4. Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37(14), 2907–2909 (2012). [CrossRef]   [PubMed]  

5. A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007). [CrossRef]   [PubMed]  

6. J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008). [CrossRef]   [PubMed]  

7. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005). [CrossRef]  

8. P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Convergence 1(1), 14 (2014). [CrossRef]  

9. K. G. Balmain, A. A. E. Lüttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1(1), 146–149 (2002). [CrossRef]  

10. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006). [CrossRef]   [PubMed]  

11. J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. 108(28), 11327–11331 (2011). [CrossRef]   [PubMed]  

12. Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012). [CrossRef]  

13. Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband purcell effect: radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012). [CrossRef]  

14. M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010). [CrossRef]   [PubMed]  

15. T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011). [CrossRef]  

16. U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012). [CrossRef]  

17. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007). [CrossRef]   [PubMed]  

18. Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006). [CrossRef]   [PubMed]  

19. A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006). [CrossRef]  

20. D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. 90(7), 077405 (2003). [CrossRef]   [PubMed]  

21. D. Korobkin, B. Neuner 3rd, C. Fietz, N. Jegenyes, G. Ferro, and G. Shvets, “Measurements of the negative refractive index of sub-diffraction waves propagating in an indefinite permittivity medium,” Opt. Express 18(22), 22734–22746 (2010). [CrossRef]   [PubMed]  

22. S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. 7(2), 265–271 (2013). [CrossRef]  

23. Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007). [CrossRef]   [PubMed]  

24. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009). [CrossRef]  

25. T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497(7450), 470–474 (2013). [CrossRef]   [PubMed]  

26. T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101(9), 091105 (2012). [CrossRef]  

27. G. V. Naik, J. L. Schroeder, X. Ni, A. V. Kildishev, T. D. Sands, and A. Boltasseva, “Titanium nitride as a plasmonic material for visible and near-infrared wavelengths,” Opt. Mater. Express 2(4), 478–489 (2012). [CrossRef]  

28. J. Liu, G. V. Naik, S. Ishii, C. Devault, A. Boltasseva, V. M. Shalaev, and E. Narimanov, “Optical absorption of hyperbolic metamaterial with stochastic surfaces,” Opt. Express 22(8), 8893–8901 (2014). [CrossRef]   [PubMed]  

29. R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010). [CrossRef]  

30. L. V. Alekseyev, V. A. Podolskiy, and E. E. Narimanov, “Homogeneous hyperbolic systems for terahertz and far-infrared frequencies,” Adv. Optoelectron. 2012, 267564 (2012). [CrossRef]  

31. V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: New physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013). [CrossRef]   [PubMed]  

32. J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: Graphite,” Appl. Phys. Lett. 98(10), 101901 (2011). [CrossRef]  

33. J. Sun, N. M. Litchinitser, and J. Zhou, “Indefinite by nature: from ultraviolet to terahertz,” ACS Photonics 1(4), 293–303 (2014). [CrossRef]  

34. M. Esslinger, R. Vogelgesang, N. Talebi, W. Khunsin, P. Gehring, S. De Zuani, B. Gompf, and K. Kern, “Tetradymites as Natural Hyperbolic Materials for the Near-Infrared to Visible,” ACS Photonics 1(12), 1285–1289 (2014). [CrossRef]  

35. J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014). [CrossRef]   [PubMed]  

36. E. Gerlach, P. Grosse, M. Rautenberg, and W. Senske, “Dynamical conductivity and plasmon excitation in Bi,” Phys. Status Solidi 75(2), 553–558 (1976). [CrossRef]  

37. E. D. Palik and G. Ghosh, The electronic handbook of optical constants of solids (Academic Press, 1999).

38. K. Korzeb, M. Gajc, and D. A. Pawlak, “Dataset 1,” figshare (2015) [retrieved 2 July 2015], http://figshare.com/articles/Compendium_of_natural_hyperbolic_metamaterials/1471761.

39. H. Friis, A. A. Finch, C. T. Williams, and J. M. Hanchar, “Photoluminescence of zircon (ZrSiO4) doped with REE 3+ (REE = Pr, Sm, Eu, Gd, Dy, Ho, Er),” Phys. Chem. Miner. 37(6), 333–342 (2010). [CrossRef]  

40. R. Terki, G. Bertrand, and H. Aourag, “Full potential investigations of structural and electronic properties of ZrSiO4,” Microelectron. Eng. 81(2-4), 514–523 (2005). [CrossRef]  

41. K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, “Crystal Growth and Low Coercive Field 180° Domain Switching Characteristics of Stochiometric LiTaO3,” Appl. Phys. Lett. 73(21), 3073–3075 (1998). [CrossRef]  

42. K. Länge, B. E. Rapp, and M. Rapp, “Surface acoustic wave biosensors: a review,” Anal. Bioanal. Chem. 391(5), 1509–1519 (2008). [CrossRef]   [PubMed]  

43. A. M. Glass, “Dielectric, thermal, and pyroelectric properties of ferroelectric LiTaO3,” Phys. Rev. 172(2), 564–571 (1968). [CrossRef]  

44. C. B. Roundy and R. L. Byer, “Sensitive LiTaO3 Pyroelectric Detector,” J. Appl. Phys. 44(2), 929–931 (1973). [CrossRef]  

45. T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Sci. 37(1), 317–350 (2007). [CrossRef]  

46. Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, “Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system,” J. Cryst. Growth 197(4), 889–895 (1999). [CrossRef]  

47. Z. Zhao, V. Buscaglia, M. Viviani, M. T. Buscaglia, L. Mitoseriu, A. Testino, M. Nygren, M. Johnsson, and P. Nanni, “Grain-size effects on the ferroelectric behavior of dense nanocrystalline BaTiO3 ceramics,” Phys. Rev. B 70(2), 024107 (2004). [CrossRef]  

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52. H. Guo, L. Liu, Z. Chen, S. Ding, H. Lu, K.-J. Jin, Y. Zhou, and B. Cheng, “Structural and optical properties of BaTiO3 ultrathin films,” Europhys. Lett. 73(1), 110–115 (2006). [CrossRef]  

53. R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010). [CrossRef]  

54. H. S. Güder, B. Abay, H. Efeoglu, and Y. K. Yogurtcu, “Photoluminescence characterization of GaTe single crystals,” J. Lumin. 93(3), 243–248 (2001). [CrossRef]  

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References

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  4. Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37(14), 2907–2909 (2012).
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  7. V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
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  10. T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
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  17. Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
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  24. M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
    [Crossref]
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    [Crossref] [PubMed]
  26. T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101(9), 091105 (2012).
    [Crossref]
  27. G. V. Naik, J. L. Schroeder, X. Ni, A. V. Kildishev, T. D. Sands, and A. Boltasseva, “Titanium nitride as a plasmonic material for visible and near-infrared wavelengths,” Opt. Mater. Express 2(4), 478–489 (2012).
    [Crossref]
  28. J. Liu, G. V. Naik, S. Ishii, C. Devault, A. Boltasseva, V. M. Shalaev, and E. Narimanov, “Optical absorption of hyperbolic metamaterial with stochastic surfaces,” Opt. Express 22(8), 8893–8901 (2014).
    [Crossref] [PubMed]
  29. R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010).
    [Crossref]
  30. L. V. Alekseyev, V. A. Podolskiy, and E. E. Narimanov, “Homogeneous hyperbolic systems for terahertz and far-infrared frequencies,” Adv. Optoelectron. 2012, 267564 (2012).
    [Crossref]
  31. V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: New physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  34. M. Esslinger, R. Vogelgesang, N. Talebi, W. Khunsin, P. Gehring, S. De Zuani, B. Gompf, and K. Kern, “Tetradymites as Natural Hyperbolic Materials for the Near-Infrared to Visible,” ACS Photonics 1(12), 1285–1289 (2014).
    [Crossref]
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    [Crossref]
  37. E. D. Palik and G. Ghosh, The electronic handbook of optical constants of solids (Academic Press, 1999).
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    [Crossref]
  40. R. Terki, G. Bertrand, and H. Aourag, “Full potential investigations of structural and electronic properties of ZrSiO4,” Microelectron. Eng. 81(2-4), 514–523 (2005).
    [Crossref]
  41. K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, “Crystal Growth and Low Coercive Field 180° Domain Switching Characteristics of Stochiometric LiTaO3,” Appl. Phys. Lett. 73(21), 3073–3075 (1998).
    [Crossref]
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    [Crossref]
  44. C. B. Roundy and R. L. Byer, “Sensitive LiTaO3 Pyroelectric Detector,” J. Appl. Phys. 44(2), 929–931 (1973).
    [Crossref]
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    [Crossref]
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2014 (5)

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Convergence 1(1), 14 (2014).
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J. Liu, G. V. Naik, S. Ishii, C. Devault, A. Boltasseva, V. M. Shalaev, and E. Narimanov, “Optical absorption of hyperbolic metamaterial with stochastic surfaces,” Opt. Express 22(8), 8893–8901 (2014).
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J. Sun, N. M. Litchinitser, and J. Zhou, “Indefinite by nature: from ultraviolet to terahertz,” ACS Photonics 1(4), 293–303 (2014).
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M. Esslinger, R. Vogelgesang, N. Talebi, W. Khunsin, P. Gehring, S. De Zuani, B. Gompf, and K. Kern, “Tetradymites as Natural Hyperbolic Materials for the Near-Infrared to Visible,” ACS Photonics 1(12), 1285–1289 (2014).
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J. D. Caldwell, A. V. Kretinin, Y. Chen, V. Giannini, M. M. Fogler, Y. Francescato, C. T. Ellis, J. G. Tischler, C. R. Woods, A. J. Giles, M. Hong, K. Watanabe, T. Taniguchi, S. A. Maier, and K. S. Novoselov, “Sub-diffractional volume-confined polaritons in the natural hyperbolic material hexagonal boron nitride,” Nat. Commun. 5, 5221 (2014).
[Crossref] [PubMed]

2013 (5)

V. P. Drachev, V. A. Podolskiy, and A. V. Kildishev, “Hyperbolic metamaterials: New physics behind a classical problem,” Opt. Express 21(12), 15048–15064 (2013).
[Crossref] [PubMed]

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497(7450), 470–474 (2013).
[Crossref] [PubMed]

S. Ishii, A. V. Kildishev, E. Narimanov, V. M. Shalaev, and V. P. Drachev, “Sub-wavelength interference pattern from volume plasmon polaritons in a hyperbolic medium,” Laser Photonics Rev. 7(2), 265–271 (2013).
[Crossref]

N. Novak, R. Pirc, and Z. Kutnjak, “Impact of critical point on piezoelectric and electrocaloric response in barium titanate,” Phys. Rev. B 87(10), 104102 (2013).
[Crossref]

P. M. Reshmi, A. G. Kunjomana, and K. A. Chandrasekharan, “Electrical and mechanical properties of vapour grown gallium monotelluride crystals,” Int. J. Min. Met. Mater. 20(10), 967–971 (2013).
[Crossref]

2012 (10)

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
[Crossref]

Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, “Applications of hyperbolic metamaterials substrates,” Adv. Optoelectron. 2012, 452502 (2012).
[Crossref]

T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101(9), 091105 (2012).
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G. V. Naik, J. L. Schroeder, X. Ni, A. V. Kildishev, T. D. Sands, and A. Boltasseva, “Titanium nitride as a plasmonic material for visible and near-infrared wavelengths,” Opt. Mater. Express 2(4), 478–489 (2012).
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L. V. Alekseyev, V. A. Podolskiy, and E. E. Narimanov, “Homogeneous hyperbolic systems for terahertz and far-infrared frequencies,” Adv. Optoelectron. 2012, 267564 (2012).
[Crossref]

I. I. Smolyaninov, E. Hwang, and E. E. Narimanov, “Hyperbolic metamaterial interfaces: Hawking radiation from Rindler horizons and spacetime signature transitions,” Phys. Rev. B 85(23), 235122 (2012).
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Y. He, S. He, and X. Yang, “Optical field enhancement in nanoscale slot waveguides of hyperbolic metamaterials,” Opt. Lett. 37(14), 2907–2909 (2012).
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Y. Guo, C. L. Cortes, S. Molesky, and Z. Jacob, “Broadband super-Planckian thermal emission from hyperbolic metamaterials,” Appl. Phys. Lett. 101(13), 131106 (2012).
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Z. Jacob, I. I. Smolyaninov, and E. E. Narimanov, “Broadband purcell effect: radiative decay engineering with metamaterials,” Appl. Phys. Lett. 100(18), 181105 (2012).
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U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
[Crossref]

2011 (6)

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. 108(28), 11327–11331 (2011).
[Crossref] [PubMed]

J. Sun, J. Zhou, B. Li, and F. Kang, “Indefinite permittivity and negative refraction in natural material: Graphite,” Appl. Phys. Lett. 98(10), 101901 (2011).
[Crossref]

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011).
[Crossref]

L. Wu, F. Ling, J. Liu, and J. Yao, “Optical modulation of permittivity for different doped near stoichiometric lithium niobate in the terahertz range,” Proc. SPIE 8330, 83300N (2011).

V. Skoromets, S. Glinšek, V. Bovtun, M. Kempa, J. Petzelt, S. Kamba, B. Malič, M. Kosec, and P. Kužel, “Ferroelectric phase transition in polycrystalline KTaO3 thin film revealed by terahertz spectroscopy,” Appl. Phys. Lett. 99(5), 052908 (2011).
[Crossref]

Z. Yan, Y. Guo, G. Zhang, and J.-M. Liu, “High-performance programmable memory devices based on co-doped BaTiO3.,” Adv. Mater. 23(11), 1351–1355 (2011).
[Crossref] [PubMed]

2010 (5)

R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010).
[Crossref]

H. Friis, A. A. Finch, C. T. Williams, and J. M. Hanchar, “Photoluminescence of zircon (ZrSiO4) doped with REE 3+ (REE = Pr, Sm, Eu, Gd, Dy, Ho, Er),” Phys. Chem. Miner. 37(6), 333–342 (2010).
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D. Korobkin, B. Neuner, C. Fietz, N. Jegenyes, G. Ferro, and G. Shvets, “Measurements of the negative refractive index of sub-diffraction waves propagating in an indefinite permittivity medium,” Opt. Express 18(22), 22734–22746 (2010).
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R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010).
[Crossref]

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

2009 (1)

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

2008 (2)

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

K. Länge, B. E. Rapp, and M. Rapp, “Surface acoustic wave biosensors: a review,” Anal. Bioanal. Chem. 391(5), 1509–1519 (2008).
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2007 (4)

T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Sci. 37(1), 317–350 (2007).
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A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
[Crossref] [PubMed]

Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
[Crossref] [PubMed]

2006 (5)

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

A. Salandrino and N. Engheta, “Far-Field Subdiffraction Optical Microscopy Using Metamaterial Crystals: Theory and Simulations,” Phys. Rev. B 74(7), 075103 (2006).
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T. Taubner, D. Korobkin, Y. Urzhumov, G. Shvets, and R. Hillenbrand, “Near-field microscopy through a SiC superlens,” Science 313(5793), 1595 (2006).
[Crossref] [PubMed]

P. Kŭzel, F. Kadlec, H. Nemec, R. Ott, E. Hollmann, and N. Klein, “Dielectric tenability of SrTiO3 thin films in the terahertz range,” Appl. Phys. Lett. 88(10), 102901 (2006).
[Crossref]

H. Guo, L. Liu, Z. Chen, S. Ding, H. Lu, K.-J. Jin, Y. Zhou, and B. Cheng, “Structural and optical properties of BaTiO3 ultrathin films,” Europhys. Lett. 73(1), 110–115 (2006).
[Crossref]

2005 (2)

R. Terki, G. Bertrand, and H. Aourag, “Full potential investigations of structural and electronic properties of ZrSiO4,” Microelectron. Eng. 81(2-4), 514–523 (2005).
[Crossref]

V. A. Podolskiy and E. E. Narimanov, “Strongly anisotropic waveguide as a nonmagnetic left-handed system,” Phys. Rev. B 71(20), 201101 (2005).
[Crossref]

2004 (1)

Z. Zhao, V. Buscaglia, M. Viviani, M. T. Buscaglia, L. Mitoseriu, A. Testino, M. Nygren, M. Johnsson, and P. Nanni, “Grain-size effects on the ferroelectric behavior of dense nanocrystalline BaTiO3 ceramics,” Phys. Rev. B 70(2), 024107 (2004).
[Crossref]

2003 (1)

D. R. Smith and D. Schurig, “Electromagnetic Wave Propagation in Media with Indefinite Permittivity and Permeability Tensors,” Phys. Rev. Lett. 90(7), 077405 (2003).
[Crossref] [PubMed]

2002 (1)

K. G. Balmain, A. A. E. Lüttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1(1), 146–149 (2002).
[Crossref]

2001 (2)

R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental Verification of a Negative Index of Refraction,” Science 292(5514), 77–79 (2001).
[Crossref] [PubMed]

H. S. Güder, B. Abay, H. Efeoglu, and Y. K. Yogurtcu, “Photoluminescence characterization of GaTe single crystals,” J. Lumin. 93(3), 243–248 (2001).
[Crossref]

2000 (1)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
[Crossref] [PubMed]

1999 (1)

Y. Furukawa, K. Kitamura, E. Suzuki, and K. Niwa, “Stoichiometric LiTaO3 single crystal growth by double crucible Czochralski method using automatic powder supply system,” J. Cryst. Growth 197(4), 889–895 (1999).
[Crossref]

1998 (1)

K. Kitamura, Y. Furukawa, K. Niwa, V. Gopalan, and T. E. Mitchell, “Crystal Growth and Low Coercive Field 180° Domain Switching Characteristics of Stochiometric LiTaO3,” Appl. Phys. Lett. 73(21), 3073–3075 (1998).
[Crossref]

1997 (1)

E. G. Gillan and A. R. Barron, “Chemical vapour deposition of hexagonal gallium selenide and telluride films from cubane precursors: Understanding the envelope ofmolecular control,” Chem. Mater. 9, 3037–3048 (1997).
[Crossref]

1994 (1)

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B Condens. Matter 50(9), 5941–5949 (1994).
[Crossref] [PubMed]

1992 (1)

J. Z. Wan, J. L. Brebner, R. Leonelli, and J. T. Graham, “Optical properties of excitons in GaTe,” Phys. Rev. B Condens. Matter 46(3), 1468–1471 (1992).
[Crossref] [PubMed]

1977 (1)

G. Godefroy, P. Lompre, C. Dumas, and H. Arend, “Pure and doped barium titanate. Crystal growth and chemical composition,” Mater. Res. Bull. 12(2), 165–169 (1977).
[Crossref]

1976 (1)

E. Gerlach, P. Grosse, M. Rautenberg, and W. Senske, “Dynamical conductivity and plasmon excitation in Bi,” Phys. Status Solidi 75(2), 553–558 (1976).
[Crossref]

1974 (1)

W. I. Milne and J. C. Anderson, “Memory switching in gallium telluride single crystals,” J. Phys. D Appl. Phys. 7(11), 1540–1548 (1974).
[Crossref]

1973 (2)

W. I. Milne and J. C. Anderson, “Threshold switching in gallium telluride single crystals,” J. Phys. D Appl. Phys. 6(17), 2115–2123 (1973).
[Crossref]

C. B. Roundy and R. L. Byer, “Sensitive LiTaO3 Pyroelectric Detector,” J. Appl. Phys. 44(2), 929–931 (1973).
[Crossref]

1968 (1)

A. M. Glass, “Dielectric, thermal, and pyroelectric properties of ferroelectric LiTaO3,” Phys. Rev. 172(2), 564–571 (1968).
[Crossref]

Abashin, M.

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497(7450), 470–474 (2013).
[Crossref] [PubMed]

Abay, B.

H. S. Güder, B. Abay, H. Efeoglu, and Y. K. Yogurtcu, “Photoluminescence characterization of GaTe single crystals,” J. Lumin. 93(3), 243–248 (2001).
[Crossref]

Agrawal, A.

T. Xu, A. Agrawal, M. Abashin, K. J. Chau, and H. J. Lezec, “All-angle negative refraction and active flat lensing of ultraviolet light,” Nature 497(7450), 470–474 (2013).
[Crossref] [PubMed]

Alekseyev, L.

A. J. Hoffman, L. Alekseyev, S. S. Howard, K. J. Franz, D. Wasserman, V. A. Podolskiy, E. E. Narimanov, D. L. Sivco, and C. Gmachl, “Negative refraction in semiconductor metamaterials,” Nat. Mater. 6(12), 946–950 (2007).
[Crossref] [PubMed]

Alekseyev, L. V.

L. V. Alekseyev, V. A. Podolskiy, and E. E. Narimanov, “Homogeneous hyperbolic systems for terahertz and far-infrared frequencies,” Adv. Optoelectron. 2012, 267564 (2012).
[Crossref]

Z. Jacob, L. V. Alekseyev, and E. Narimanov, “Optical Hyperlens: Far-field imaging beyond the diffraction limit,” Opt. Express 14(18), 8247–8256 (2006).
[Crossref] [PubMed]

Anderson, J. C.

W. I. Milne and J. C. Anderson, “Memory switching in gallium telluride single crystals,” J. Phys. D Appl. Phys. 7(11), 1540–1548 (1974).
[Crossref]

W. I. Milne and J. C. Anderson, “Threshold switching in gallium telluride single crystals,” J. Phys. D Appl. Phys. 6(17), 2115–2123 (1973).
[Crossref]

Aourag, H.

R. Terki, G. Bertrand, and H. Aourag, “Full potential investigations of structural and electronic properties of ZrSiO4,” Microelectron. Eng. 81(2-4), 514–523 (2005).
[Crossref]

Arend, H.

G. Godefroy, P. Lompre, C. Dumas, and H. Arend, “Pure and doped barium titanate. Crystal growth and chemical composition,” Mater. Res. Bull. 12(2), 165–169 (1977).
[Crossref]

Atkinson, J.

P. Shekhar, J. Atkinson, and Z. Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Convergence 1(1), 14 (2014).
[Crossref]

Balmain, K. G.

K. G. Balmain, A. A. E. Lüttgen, and P. C. Kremer, “Resonance cone formation, reflection, refraction, and focusing in a planar anisotropic metamaterial,” IEEE Antennas Wirel. Propag. Lett. 1(1), 146–149 (2002).
[Crossref]

Barnakov, Y. A.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011).
[Crossref]

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
[Crossref]

Barron, A. R.

E. G. Gillan and A. R. Barron, “Chemical vapour deposition of hexagonal gallium selenide and telluride films from cubane precursors: Understanding the envelope ofmolecular control,” Chem. Mater. 9, 3037–3048 (1997).
[Crossref]

Bartal, G.

J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. 108(28), 11327–11331 (2011).
[Crossref] [PubMed]

J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
[Crossref] [PubMed]

Bernasconi, P.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B Condens. Matter 50(9), 5941–5949 (1994).
[Crossref] [PubMed]

Bertrand, G.

R. Terki, G. Bertrand, and H. Aourag, “Full potential investigations of structural and electronic properties of ZrSiO4,” Microelectron. Eng. 81(2-4), 514–523 (2005).
[Crossref]

Black, P.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011).
[Crossref]

Boltasseva, A.

Bonner, C. E.

T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011).
[Crossref]

M. A. Noginov, H. Li, Y. A. Barnakov, D. Dryden, G. Nataraj, G. Zhu, C. E. Bonner, M. Mayy, Z. Jacob, and E. E. Narimanov, “Controlling spontaneous emission with metamaterials,” Opt. Lett. 35(11), 1863–1865 (2010).
[Crossref] [PubMed]

Bovtun, V.

V. Skoromets, S. Glinšek, V. Bovtun, M. Kempa, J. Petzelt, S. Kamba, B. Malič, M. Kosec, and P. Kužel, “Ferroelectric phase transition in polycrystalline KTaO3 thin film revealed by terahertz spectroscopy,” Appl. Phys. Lett. 99(5), 052908 (2011).
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T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011).
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M. A. Noginov, Y. A. Barnakov, G. Zhu, T. Tumkur, H. Li, and E. E. Narimanov, “Bulk photonic metamaterial with hyperbolic dispersion,” Appl. Phys. Lett. 94(15), 151105 (2009).
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T. U. Tumkur, J. K. Kitur, B. Chu, L. Gu, V. A. Podolskiy, E. E. Narimanov, and M. A. Noginov, “Control of reflectance and transmittance in scattering and curvilinear hyperbolic metamaterials,” Appl. Phys. Lett. 101(9), 091105 (2012).
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U. Tumkur, L. Gu, J. K. Kitur, E. E. Narimanov, and M. A. Noginov, “Control of absorption with hyperbolic metamaterials,” Appl. Phys. Lett. 100(16), 161103 (2012).
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J. Yao, Z. Liu, Y. Liu, Y. Wang, C. Sun, G. Bartal, A. M. Stacy, and X. Zhang, “Optical negative refraction in bulk metamaterials of nanowires,” Science 321(5891), 930 (2008).
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T. Feurer, N. S. Stoyanov, D. W. Ward, J. C. Vaughan, E. R. Statz, and K. A. Nelson, “Terahertz polaritonics,” Annu. Rev. Mater. Sci. 37(1), 317–350 (2007).
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J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. 108(28), 11327–11331 (2011).
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L. Wu, F. Ling, J. Liu, and J. Yao, “Optical modulation of permittivity for different doped near stoichiometric lithium niobate in the terahertz range,” Proc. SPIE 8330, 83300N (2011).

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J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. 108(28), 11327–11331 (2011).
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M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B Condens. Matter 50(9), 5941–5949 (1994).
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Z. Yan, Y. Guo, G. Zhang, and J.-M. Liu, “High-performance programmable memory devices based on co-doped BaTiO3.,” Adv. Mater. 23(11), 1351–1355 (2011).
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J. Yao, X. Yang, X. Yin, G. Bartal, and X. Zhang, “Three-dimensional nanometer-scale optical cavities of indefinite medium,” Proc. Natl. Acad. Sci. U.S.A. 108(28), 11327–11331 (2011).
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Z. Liu, H. Lee, Y. Xiong, C. Sun, and X. Zhang, “Far-field optical hyperlens magnifying sub-diffraction-limited objects,” Science 315(5819), 1686 (2007).
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Y. Xiong, Z. Liu, C. Sun, and X. Zhang, “Two-dimensional imaging by far-field superlens at visible wavelengths,” Nano Lett. 7(11), 3360–3365 (2007).
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Z. Zhao, V. Buscaglia, M. Viviani, M. T. Buscaglia, L. Mitoseriu, A. Testino, M. Nygren, M. Johnsson, and P. Nanni, “Grain-size effects on the ferroelectric behavior of dense nanocrystalline BaTiO3 ceramics,” Phys. Rev. B 70(2), 024107 (2004).
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J. Sun, N. M. Litchinitser, and J. Zhou, “Indefinite by nature: from ultraviolet to terahertz,” ACS Photonics 1(4), 293–303 (2014).
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R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010).
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R. Wang, J. Sun, and J. Zhou, “Indefinite permittivity in uniaxial single crystal at infrared frequency,” Appl. Phys. Lett. 97(3), 031912 (2010).
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Zhou, Y.

H. Guo, L. Liu, Z. Chen, S. Ding, H. Lu, K.-J. Jin, Y. Zhou, and B. Cheng, “Structural and optical properties of BaTiO3 ultrathin films,” Europhys. Lett. 73(1), 110–115 (2006).
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T. Tumkur, G. Zhu, P. Black, Y. A. Barnakov, C. E. Bonner, and M. A. Noginov, “Control of spontaneous emission in a volume of functionalized hyperbolic metamaterial,” Appl. Phys. Lett. 99(15), 151115 (2011).
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Zhu, Y.

M. Zgonik, P. Bernasconi, M. Duelli, R. Schlesser, P. Günter, M. H. Garrett, D. Rytz, Y. Zhu, and X. Wu, “Dielectric, elastic, piezoelectric, electro-optic, and elasto-optic tensors of BaTiO3 crystals,” Phys. Rev. B Condens. Matter 50(9), 5941–5949 (1994).
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ACS Photonics (2)

J. Sun, N. M. Litchinitser, and J. Zhou, “Indefinite by nature: from ultraviolet to terahertz,” ACS Photonics 1(4), 293–303 (2014).
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M. Esslinger, R. Vogelgesang, N. Talebi, W. Khunsin, P. Gehring, S. De Zuani, B. Gompf, and K. Kern, “Tetradymites as Natural Hyperbolic Materials for the Near-Infrared to Visible,” ACS Photonics 1(12), 1285–1289 (2014).
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Z. Yan, Y. Guo, G. Zhang, and J.-M. Liu, “High-performance programmable memory devices based on co-doped BaTiO3.,” Adv. Mater. 23(11), 1351–1355 (2011).
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Proc. SPIE (1)

L. Wu, F. Ling, J. Liu, and J. Yao, “Optical modulation of permittivity for different doped near stoichiometric lithium niobate in the terahertz range,” Proc. SPIE 8330, 83300N (2011).

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[Crossref] [PubMed]

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[Crossref] [PubMed]

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Other (5)

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Figures (11)

Fig. 1
Fig. 1 Scheme presenting types of hyperbolic materials.
Fig. 2
Fig. 2 Occurrence of natural hyperbolic materials at particular wavelength ranges. Chosen crystalline uniaxial and biaxal (orange stars) materials are shown which possess at least at some wavelength maximal quality factor Q>5 (green colour) or 3<Q<5 (blue colour); Materials with Q<3 have been neglected. The total wavelength range for which these materials exhibits Re(εi)<0, Re(εj)>0 with Q>5 or 3<Q<5 are shown.
Fig. 3
Fig. 3 Natural hyperbolic materials with highest available quality factor, Qmax, demonstrating materials with relatively small losses in the comparison with the magnitude of the negative real part of permittivity. Q = -Re(εi)/Im(εi), where Re(εi)<0, Re(εj)>0. The histograms are presented for chosen crystalline uniaxial and biaxal (orange stars) materials for which their maximal quality factor is: (a) Qmax>5; (b) 3<Qmax<5. Materials with Qmax<3 have been neglected. Δε = Re(εi)-Re(εj) and Im(εj) at this wavelength are also presented.
Fig. 4
Fig. 4 Natural uniaxial materials with high quality factor, Q: a) ZrSiO4, b) BaTiO3. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and quality factor (dot line) for ordinary and extraordinary waves.
Fig. 5
Fig. 5 Aluminum oxide, Al2O3, as homogeneous hyperbolic material: a) exhibiting a relatively high Q factor and small losses, b) exhibiting very high dielectric anisotropy, Δε. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and the quality factor (dot line) for ordinary and extraordinary waves.
Fig. 6
Fig. 6 Natural hyperbolic materials with the maximal value of quality factor Qmax>5 and relativity small losses for the positive part of the dielectric permittivity: a) SiO2, b) MgF2. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and quality factor (dot line) for ordinary and extraordinary waves.
Fig. 7
Fig. 7 Natural uniaxial materials with opposite signs of permittivity for ordinary and extraordinary waves in the broad range of wavelengths: a) NaNO3, b) BaTiO3. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and the quality factor (dot line) for ordinary and extraordinary waves.
Fig. 8
Fig. 8 Natural hyperbolic materials with highest known anisotropy of permittivity, ∆εmax. Δε = Re(εj)-Re(εi), where Re(εi)<0, Re(εj)>0. a) Histograms show materials with Δε>100 at the wavelength of its maximal value. Im(εi) and Im(εj) at the ∆εmax wavelength are also presented. b) Occurrence of natural hyperbolic materials for which Δεmax>100 at particular wavelength ranges. Complete wavelength ranges at which these materials exhibit Δεmax>100 are shown.
Fig. 9
Fig. 9 Graphite - natural uniaxial material with the highest anisotropy of permittivity, ∆ε. Graph shows real parts of dielectric permittivity for ordinary and extraordinary waves.
Fig. 10
Fig. 10 Gallium telluride, GaTe, biaxial natural material with high dielectric anisotropy in the visible wavelength range. Graphs show real parts (solid lines) and imaginary parts (dash lines) of dielectric permittivity and quality factor (dot line) for three crystallography axes.
Fig. 11
Fig. 11 Composite and natural hyperbolic materials at different spectral ranges. Materials shown in bold are proposed in this paper.

Tables (3)

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Table 1 Natural hyperbolic materials characterized by quality factor Q>5 or 3<Q<5 at specific wavelength range.

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Table 2 Chosen data, as quality factor, Q, dielectric losses Im(ε), strength of anisotropy, Δε, for chosen materials.

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Table 3 Natural hyperbolic materials characterized by high strength of dielectric anisotropy, Δε, Δεmax>100 at specific wavelength ranges.

Equations (7)

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k x 2 ε 1 + k y 2 ε 2 + k z 2 ε 3 = ω 2 c 2
ε =[ ε xx , ε yy , ε zz ]
( k x 2 + k y 2 ) ε 1 + k z 2 ε 2 = ( ω c ) 2
Re( ε j )= n j 2 k j 2
Im( ε j )=2 n j k j
Q j = Re( ε j ) Im( ε j )
Δε=Re( ε j )Re( ε i )

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