Abstract

We studied a system of a core-shell nanowire array in the liquid crystal (LC) matrix 5CB supposing that the nanowire shell permittivity is anisotropic with radially power-law dependent components. The nanowire core material is either silver or gold. We calculated the effective permittivity tensor of the core-shell nanowire array in the LC matrix, and show that the system under consideration can possess two hyperbolic metamaterial (HMM) areas: where the real part of the effective permittivity component parallel to the nanowire is negative, and the real part of the effective permittivity component perpendicular to the nanowire is negative. Dependence of the frequency position for these areas on the core-shell nanowire parameters and the LC matrix orientational state is studied.

© 2017 Optical Society of America

1. Introduction

In the last decades, hyperbolic metamaterials (HMMs) gained a central role among other metamaterials [1–5] due to their unique ability to manipulate electromagnetic fields [6–15]. As a rule, for generation of the hyperbolic dispersion relation, two structures are generally used: a stack of sub-wavelength alternating metallic and dielectric layers, or a lattice of metallic nanowires embedded in a dielectric matrix, called a nanowire array. The particular properties of HMMs are based on the excitation of coupled surface plasmons in the metal layers or nanowires inside of HMMs [16–22]. The electric field of surface plasmons decays very fast with distance from the interface between the dielectric and metal, and is defined by distances shorter than the wavelength. When the metallic/dielectric layers or metallic nanowires in a dielectric matrix are periodically arranged with the sub-wavelength separation, the electromagnetic fields of the individual plasmonic interfaces overlap and couple, giving rise to a collective response. Such an optical response of these media can be homogenized via an effective medium theory [23–27] and interpreted as arising from the bulk effective media with a hyperbolic effective permittivity tensor.

There is a growing interest in introducing a possibility of reconfiguration or tunability of HMM structures, allowing a tuning of their properties and functions. In this respect, liquid crystals (LCs) stand out as the preferred means of achieving tunability, as they possess the broadband transparency, large optical birefringence, and easy susceptibilities to applied fields. An analysis of the aligned nematic LC cells containing the nanospheres was made in papers [28,29]. Authors of the papers have shown that these media provided a new type of metamaterial whose index of refraction is tunable from negative, through zero, to positive values. In paper [30] it was shown that LCs containing randomly dispersed metallic spherical nanoparticles exhibit hyperbolic properties in the visible spectrum. It also demonstrated the possibility of tunable wavelength properties by choosing the dielectric-metal core-shell nanoparticles. The nanoparticle shell anisotropy was observed in phospholipid vesicles (∼50 nm radius) [31] and the SiO2@Au core–shell nanoparticles with 50 nm core diameter and 10 nm shell thickness [32]. Tuning of the plasmon resonance in the nanoribbon arrays with the adjacent LC layer was studied in papers [33,34]. The nanosphere dispersed LCs as a hyperbolic metamaterial for wide-angle negative–positive refraction and reflection were proposed in paper [35], where it was shown that the real parts of effective ordinary and extraordinary permittivities of such media can have opposite signs in the near IR spectral region. A voltage-tunable hyperbolic dispersion metamaterial consisting of silver nanowires and nematic LCs was studied in paper [36]. The metamaterial showed an electro-optic effect, in which the dispersion relation switched between elliptic and hyperbolic dispersions in the visible waveband with an external electric field. It was analyzed the optical refraction in a fluid system, which contained silica-coated gold nanorods dispersed in silicone oil under an external electric field [37]. Because of the formation of a chain-like or lattice-like structure of dispersed nanorods along the electric field, the fluid showed hyperbolic properties resulting in optical negative refraction for transverse magnetic waves.

In our paper, we study a system of the core-shell nanowire array in the LC supposing that the nanowire shell permittivity is anisotropic with radially power-law dependent components. The paper is organized as follows. In Section 2 we derive the analytical expressions for the effective permittivity components of the system under consideration. In Section 3 we find the frequency areas of the HMM when the nanowires core is either Ag or Au and the nanowires are embedded in the LC 5CB. Dependence of the frequency position of these areas on the core-shell nanowire parameters are also analyzed. Section 4 contains results showing influence of the LC orientational state on the HMM areas position. Some conclusions of our study are presented in Section 5.

2. Effective permittivity tensor of the LC matrix with the cylindrical core-shell nanowires

Consider a system of cylindrical nanowires embedded in a nematic liquid crystal (LC). The nanowires are periodically arranged in the LC matrix with the sub-wavelength separation.

Frequency ω and wave vector k=(kx,ky,kz) of the extraordinary electromagnetic wave propagating in HMMs must satisfy to the dispersion relation (see, for example [6],)

kx2+kx2εxx+kz2εzz=ω2c2,
where the HMM permittivity components εxx=εyy=ε,εzz=ε and εε<0.

To provide the cylindrical symmetry of the LC matrix containing the nanowires, we suppose that there are strong forces aligning the LC director along the nanowires. It can be an external field orienting the LC director along the nanowires or the planar director anchoring with the nanowire surface.

The nanowire array in the LC matrix creates a heterogeneous medium. Using an effective medium theory we can describe the electric properties of such a system by the effective permittivity. To obtain the effective permittivity components of the LC with the cylindrical nanowires, ε,ε, we can use, for example, the results obtained in paper [38] for the LC filled with ellipsoidal particles. In our case the particle long axis length is set to be infinite, and we denote the nanowire permittivity components perpendicular and parallel to the nanowire axis byεcyl,, and εcyl,, respectively; we arrive at the following expressions for the effective permittivity components of the LC with the cylindrical nanowires:

ε=εLC(1f)εLC+(1+f)εcyl,(1f)εcyl,+(1+f)εLC,
ε=εLC+f(εcyl,εLC),
where εLC, εLC are the LC permittivity components perpendicular and parallel to the LC director, respectively; f is the volume fill fraction of the nanowires, i.e. the ratio of the cross sectional area of the nanowires to the cross sectional area of the LC cell containing the nanowires.

Now we suppose that each nanowire contains a metallic core with a radius b and a permittivity εccovered by the anisotropic dielectric shell with an external radius a(Fig. 1). We consider the case of the anisotropic nanowire shell with the permittivity tensor main components εr,εφ,εz (in the cylindrical coordinate system), supposing the z-axis to be directed along the nanowire. We also assume that the components εr,εφ of the shell permittivity are radially power-law dependent,

εr=εr0(r/a)m,εφ=εφ0(r/a)m,
whereris a distance from the nanowire axis, and m is a positive or negative exponent.

 

Fig. 1 Cross-section of cylindrical nanowire with radially inhomogeneous anisotropic permittivity of the shell, which covers the metallic core with permittivityεc;εr,φare radially power-law dependent components of the shell permittivity;aand bare outer and inner radiuses of the shell.

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Since the core-shell nanowires are optically inhomogeneous we need to perform the so-called “internal homogenization” [39–41]. In the case of the core-shell nanowires with radially anisotropic and power-law dependent components εr,εφof the shell permittivity, the problem is analytically tractable if the external media (the LC matrix) also has the cylindrical symmetry.

In the cylindrical coordinates the electric displacement vector takes the form H1=1,H2=ρ,H3=1gradf=fρeρ+1ρfφeφ+fzez. Setting the external electric field E0 directed perpendicular to the cylindrical axis (the z-axis), the solution to the equation divD=0for the potential functionϕ (E=ϕ) yields:

ϕ(r,φ)=Arcosφ,r<bϕ(r,φ)=(Brt1+Crt2)cosφ,bra,ϕ(r,φ)=(Dr1E0r)cosφr>a
where, taking into account Eq. (4),

t1,2=12[m±m2+4εφ0εr0]

The boundary conditions at r=a and r=b for the nanowire embedded in the LC with the director parallel to the nanowire reduce to the following equations:

Ab=Bbt1+Cbt2Bat1+Cat2=F/a2E0aεcA=εr0bm1(Bt1bt1+Ct2bt2).εr0am1(Bt1at1+Ct2at2)=εLC(2F/a3E0)

The electric potential created by a cylinder of radius a polarized by an external electric field E0 is ϕ=α(a2cosφ/2r)E0, where α is a polarizability of the cylinder in direction perpendicular to the cylinder long axis. This expression can be compared to the expression forϕ in Eq. (5) for r>a, yielding an expression for α=(2F/a2)E0, where α is now regarded as the effective polarizability of the core-shell cylinder.

The two dimensional generalization of the Clausius-Mossotti formula (see [42]), relating the polarizability of a cylinder and its permittivity εcyl,, is α=2(εcyl,εm)/(εcyl,+εm), where εm is a permittivity of the surrounding medium in direction perpendicular to the cylinder and in our case εm=εLC. This permits the derivation of an expression for the effective permittivity εcyl, of the core-shell cylinder in terms of α=(2F/a2)E0. The constant Fis found by solving Eqs. (7). Omitting the details of the calculation, we obtain the following expressions for the effective permittivity component of the cylindrical core-shell nanowire εcyl,:

εcyl,=εr0t2[t1εr0εc(ba)m1](ba)t1t2t1[t2εr0εc(ba)m1][t1εr0εc(ba)m1](ba)t1t2[t2εr0εc(ba)m1].
The longitudinal component of the core-shell nanowire effective permittivity can be described as follows (see [6])
εcyl,=ρεc+(1ρ)εz.
where ρ=b2/a2 is the metallic fill fraction.

We consider the cases when the nanowire core material is Ag or Au. For these materials the frequency dependence of the permittivity εc is very well described by the function, which accounts for the interband transitions [43]

εc(ω)=εD+εCP1+εCP2,
where
εD=εωp2ω2+iωΓp,
εCP1=A1[ωg1ω012(ω+iΓ1)2ln(1(ω+iΓ1ω01)2)+2ωg1(ω+iΓ1)2tanh1(ωg1ω01ωg1)ω+iΓ1ωg1(ω+iΓ1)2tan1(ωg1ω01ω+iΓ1ωg1)ω+iΓ1+ωg1(ω+iΓ1)2tanh1(ωg1ω01ω+iΓ1+ωg1)].
εCP2=A22(ω+iΓ2)2ln(1(ω+iΓ2ω02)2).
Here εDis the Drude term, εCP1,εCP2are the interband contributions to the dielectric function from two critical points (the Van Hove singularities in the electron density of states); ε,ωp,Γp,ωg1,ω01,Γ1,A1,ω02,Γ2,A2are fitting parameters presented in the Table 1.

As the LC matrix, we choose the typical nematic LC 5CB. The permittivity frequency dispersion of the LC 5CB was measured by the authors of paper [44] and fit by the following analytical expressions:

εLC=(Ao+Boλ2+Coλ4)2,
εLC=(Ae+Beλ2+Ceλ4)2,
where λ is the electromagnetic field wavelength in micrometers, and the fitting parameters are presented in the Table 2.

Furthermore, we calculate the real parts of the effective permittivity components (2), (3), which determine the frequency regions of HMM characterized by the sign of the inequalityReε(ω)Reε(ω)<0. For this, we use Eqs. (8), (9) for the core-shell nanowire effective permittivity, Eqs. (11a), (11b) for the LC 5CB permittivity, and Eqs. (10) for the permittivity of the Ag or Au cores.

3. Frequency position of the HMM areas

In Fig. 1 we present real parts of the effective permittivity components,ReεandReε, versus the wavelength for the LC 5CB matrix contained the core-shell nanowires with the Ag core (Figs. 2(a), 2(b)) and the Au core (Figs. 2(c),2(d)). As an example, we show results obtained when the nanowire shell possesses: i) isotropic permittivity (m=0,εr=εφ=εz), ii) anisotropic permittivity with radially power-law dependent components εr,εφ(m0).

 

Fig. 2 Real parts of the effective permittivity components, Reε(solid lines) and Reε(dashed lines), versus the wavelength for the system of the shell-core nanowires with the Ag core (a), (b) and Au core (c), (d) in the LC 5CB matrix for two values of the exponent m;εr0=5,εφ0=5,εz=5,b/a=0.5,f=0.4.

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It is seen from Fig. 2 that the system under consideration can possess two HMM areas: an area where Reε<0(and Reε>0) and an area where Reε<0(andReε>0). The area Reε<0appears at λ>860nmin the case of the Ag core and atλ>870nm in the case of the Au core. As we can see from Eqs. (3), (9) a position of the areaReε<0 depends on the metallic fill fraction ρ=b2/a2, the volume fill fraction of the nanowires f, the core metal permittivity εc, and the εz component of the shell permittivity. It does not depend on the shell permittivity components εr,εφ and the exponent mvalue

The HMM area Reε<0 is observed at smaller wavelengths and is more narrow than the area Reε<0. Its position depends on the ratio b/a, the fill fraction f, the shell permittivity components εr,εφ and, therefore, on the exponentmvalue, but it does not depend on the εz component of the shell permittivity (see Eqs. (2), (8)). It also depends on the type of the core metal as well (see also Fig. 2).

The frequency dependences of the Au and Ag permittivity, εc(ω), significantly differ (see Fig. 3). As a result, the position of the HMM areas in the cases of the core-shell nanowires with the Ag core and the Au core are different. In particular, in the case of the Au core the area Reε<0does not appear when the nanowire shell is isotropic (Fig. 2(c), m=0).

 

Fig. 3 Real (a) and imaginary (b) parts of the nanowire core permittivity versus the wavelength: Ag core - solid lines, Au core - dashed lines. For plotting, we used Eq. (10) and Table 1 obtained in paper [43].

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In Fig. 4 we show the dependence of the width of the HMM area Reε<0 (λmin,λmax) on the exponent mvalue for nanowires with the Ag or Au cores in the LC 5CB matrix. It is seen that the area Reε<0 exists only in the limited interval of the exponent mvalue. The width of this interval depends on the values of the shell permittivity parametersεr0 and εφ0,fand ratio b/a.

 

Fig. 4 The width (λmin,λmax) of the HMM area Reε<0 versus the exponent mvalue for nanowires with the Ag and Au cores.εr0=5,εφ0=5,εz=5,b/a=0.5,f=0.4.

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Dependence of the width of the area Reε<0 (λmin,λmax) and the short-wave border of the area Reε<0on the nanowire fill fractionffor nanowires with the Ag core in the LC 5CB matrix is presented in Fig. 5. The width of both areas increases with an increase off. A decrease of the exponent m leads to the shift of the area Reε<0 to the long-wave side and an increase of its width. The similar dependence takes also place for nanowires with the Au core and m<0.

 

Fig. 5 The width (λmin,λmax) of the HMM areas Reε<0 (a) and Reε<0(b) versus the volume fill fraction fof the nanowires with Ag core. εr0=5,εφ0=5,εz=5,b/a=0.5. For Reε<0: m=0,2,2- solid, dashed, dot-dashed lines, respectively.

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The width of the area Reε<0 and the short-wave border of the area Reε<0increases with an increase of the nanowire shell external to internal radii ratio. As an example, we show this in Fig. 6 for the case of the nanowire with the Ag core. However, as it is seen from Fig. 6(a) the dependence of the position and width of the area Reε<0on the exponentmbecomes small for the nanowire shells with small thickness (b/a1).

 

Fig. 6 The width (λmin,λmax) of the HMM areas Reε<0(a) and Reε<0(b) versus the external to internal radii ratio of the nanowire shell with the Ag core. εr0=5,εφ0=5,εz=5;f=0.4. ForReε<0: m=0,2,2- solid, dashed, dot-dashed lines, respectively.

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Dependence of the width of the area Reε<0 on the shell permittivity parameters εr0, εφ0 for the nanowires with an Ag core in the LC 5CB matrix is shown in Fig. 7 at several values of the exponent m. One can see that the width of the HMM areaReε<0increases with an increase of the parameterεr0 (Fig. 7(a)) and with a decrease of the parameter εφ0 (Fig. 7(b)). The same dependence is obtained for nanowires with the Au core and m<0.

 

Fig. 7 The width (λmin,λmax) of the HMM area Reε<0 versus the parameter εr0 (a) at fixed εφ0=5and the parameter εφ0(b) at fixedεr0=5 in the case of the nanowire shell with Ag core. m=0,2,2- solid, dashed, dot-dashed lines, respectively. b/a=0.5,f=0.4,εz=5.

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In Fig. 8 we present ReεandReεversus the wavelength for the LC 5CB containing the core-shell nanowires with either Ag core or Au core for different values of the shell permittivity components εr,εφ, but with the same value of the shell permittivity tensor trace. Such a situation can take place when the nanowire shell is created by molecules undergoing the trans-cis transitions. In this case, using such transitions we can switch a position of the HMM frequency interval Reε<0(the case of the Ag core in Figs. 8(a), 8(b)) or create this interval (the case of the Au core in Figs. 8(c), 8(d)). However, it is necessary to note that for the nanowires with the Au core the interval Reε<0can appear only if the exponent mis negative (see Fig. 4(b)) with sufficiently large |m|.

 

Fig. 8 Real parts of the effective permittivity components, Reε(solid lines) and Reε(dashed lines), versus the wavelength in the case of the shell permittivity tensor trace to be constant. Nanowires with Ag core - (a), (b); with Au core – (c), (d); εr0=2,εφ0=8- (a), (c); εr0=8,εφ0=2- (b), (d). b/a=0.5,f=0.4,εz=5.

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4. Influence of the LC orientational state on the HMM area position

By heating we can transfer the LC into the isotropic state. The cylindrical symmetry of the system is conserved and we can use the formulae presented above putting here εLC=εLC=εisLC. As εisLCone can use εisLC=(εLC+2εLC)/3. In Fig. 9 we demonstrate an influence of the orientational state of the LC 5CB comparing the positions of the HMM areaReε<0 when the LC director is parallel to the nanowires and when the LC director is disordered (an isotropic state) for nanowires with the Ag core (Fig. 9(a)) or the Au core (Fig. 9(b)). It is seen that in both cases the area Reε<0are sensitive to the LC orientational state: it broadens into the long wavelength side when the LC becomes isotropic. Influence of the LC orientational state on a position of the HMM area Reε<0 is very weak.

 

Fig. 9 Real part of the effective permittivity components Reεversus the wavelength for the system of the core-shell nanowire array in the LC matrix 5CB: Ag core - (a), Au core - (b); the LC director parallel to the nanowires (solid lines), the LC in an isotropic state (dashed lines). εr0=8,εφ0=2,εz=5,b/a=0.5.

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5. Conclusions

We studied a system of the core-shell nanowire arrays in the LC matrix 5CB, supposing that the nanowire shell permittivity is anisotropic with radially power-law dependent permittivity componentsεr,εφ. The nanowire core material is assumed to be Ag or Au. We show that this system can possess two HMM areas: an area where Reε<0 and an area where Reε<0. A position of the area Reε<0depends on the metallic fill fraction ρ=b2/a2, the volume fill fraction of the nanowires f, the εz component of the shell permittivity, and weakly on the type of the core metal, Ag or Au. It does not depend on the shell permittivity components εr,εφ and the exponent mvalue. The HMM area Reε<0 is observed at smaller wavelengths and is more narrow than the area Reε<0. Its position depends on the metallic fill fractionρ, the volume fill fraction of the nanowiresf, the shell permittivity components εr,εφ, and strongly on the exponent mvalue and the type of the core metal, Ag or Au. It does not depend on the εz component of the shell permittivity. In the case of the Au core the HMM area Reε<0does not appear for the exponent m0.

We show that transition of the LC matrix director from the ordered state to the isotropic state influences a position of the HMM area Reε<0 weakly, while the area Reε<0are more sensitive to the LC orientational state: it broadens into the long wavelength side when the LC becomes isotropic.

Thus, choosing the parameters of the core-shell nanowires we can obtain a system with HMM properties in a desired visible or infrared spectral region. By changing the orientational state of the LC matrix, one can tune the position of the HMM area.

Funding

European Office of Aerospace Research and Development (EOARD) (118007).

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37. Z. Su, J. Yin, Y. Guan, and X. Zhao, “Electrically tunable negative refraction in core/shell-structured nanorod fluids,” Soft Matter 10(39), 7696–7704 (2014). [CrossRef]   [PubMed]  

38. S. M. Shelestiuk, V. Yu. Reshetnyak, and T. J. Sluckin, “Frederiks transition in ferroelectric liquid-crystal nanosuspensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4 Pt 1), 041705 (2011). [CrossRef]   [PubMed]  

39. H. Kettunen, H. Wallén, and A. Sihvola, “Cloaking and magnifying using radial anisotropy,” J. Appl. Phys. 114(4), 044110 (2013). [CrossRef]  

40. U. K. Chettiar and N. Engheta, “Internal homogenization: effective permittivity of a coated sphere,” Opt. Express 20(21), 22976–22986 (2012). [CrossRef]   [PubMed]  

41. V. Yu. Reshetnyak, I. P. Pinkevych, T. J. Sluckin, and D. R. Evans, “Cloaking by shells with radially inhomogeneous anisotropic permittivity,” Opt. Express 24(2), A21–A32 (2016). [CrossRef]   [PubMed]  

42. A. Sihvola, “Particular properties of the dielectric response of negative-permittivity scatterers,” PIERS ONLINE 3(3), 246–247 (2007). [CrossRef]  

43. D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014). [CrossRef]  

44. V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006). [CrossRef]  

References

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  7. A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
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  8. T. Tumkur, G. Zhu, P. Black, Yu. 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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  9. 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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  10. C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
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  11. Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, ““Applications of hyperbolic metamaterial substrates,” Adv,” OptoElectro. 2012, 452502 (2012).
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    [Crossref]
  25. V. Yu. Reshetnyak, T. J. Sluckin, and S. J. Cox, “Effective medium theory of polymer dispersed liquid crystal systems: II. Partially ordered bipolar droplets,” J. Phys. D Appl. Phys. 30(23), 3253–3266 (1997).
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  26. V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
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  27. J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
    [Crossref]
  28. I. C. Khoo, D. H. Werner, X. Liang, A. Diaz, and B. Weiner, “Nanosphere dispersed liquid crystals for tunable negative-zero-positive index of refraction in the optical and terahertz regimes,” Opt. Lett. 31(17), 2592–2594 (2006).
    [Crossref] [PubMed]
  29. I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
    [Crossref]
  30. D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
    [Crossref]
  31. A. Erbe and R. Sigel, “Tilt angle of lipid acyl chains in unilamellar vesicles determined by ellipsometric light scattering,” Eur Phys J E Soft Matter 22(4), 303–309 (2007).
    [Crossref] [PubMed]
  32. P. A. A. De Beule, “Surface scattering of core-shell particles with anisotropic shell,” J. Opt. Soc. Am. A 31(1), 162–171 (2014).
    [Crossref] [PubMed]
  33. Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
    [Crossref]
  34. V. Yu. Reshetnyak, T. J. Bunning, and D. R. Evans, “Tuning surface plasmons in graphene ribbons with liquid crystal layer,” Proc. SPIE 9940. Liq. Cryst. XX, 994019 (2016).
  35. G. Pawlik, K. Tarnowski, W. Walasik, A. C. Mitus, and I. C. Khoo, “Liquid crystal hyperbolic metamaterial for wide-angle negative-positive refraction and reflection,” Opt. Lett. 39(7), 1744–1747 (2014).
    [Crossref] [PubMed]
  36. Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
    [Crossref]
  37. Z. Su, J. Yin, Y. Guan, and X. Zhao, “Electrically tunable negative refraction in core/shell-structured nanorod fluids,” Soft Matter 10(39), 7696–7704 (2014).
    [Crossref] [PubMed]
  38. S. M. Shelestiuk, V. Yu. Reshetnyak, and T. J. Sluckin, “Frederiks transition in ferroelectric liquid-crystal nanosuspensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4 Pt 1), 041705 (2011).
    [Crossref] [PubMed]
  39. H. Kettunen, H. Wallén, and A. Sihvola, “Cloaking and magnifying using radial anisotropy,” J. Appl. Phys. 114(4), 044110 (2013).
    [Crossref]
  40. U. K. Chettiar and N. Engheta, “Internal homogenization: effective permittivity of a coated sphere,” Opt. Express 20(21), 22976–22986 (2012).
    [Crossref] [PubMed]
  41. V. Yu. Reshetnyak, I. P. Pinkevych, T. J. Sluckin, and D. R. Evans, “Cloaking by shells with radially inhomogeneous anisotropic permittivity,” Opt. Express 24(2), A21–A32 (2016).
    [Crossref] [PubMed]
  42. A. Sihvola, “Particular properties of the dielectric response of negative-permittivity scatterers,” PIERS ONLINE 3(3), 246–247 (2007).
    [Crossref]
  43. D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
    [Crossref]
  44. V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
    [Crossref]

2016 (6)

V. A. Markel, “Introduction to the Maxwell Garnett approximation: tutorial,” J. Opt. Soc. Am. A 33(7), 1244–1256 (2016).
[Crossref] [PubMed]

J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
[Crossref]

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
[Crossref]

V. Yu. Reshetnyak, T. J. Bunning, and D. R. Evans, “Tuning surface plasmons in graphene ribbons with liquid crystal layer,” Proc. SPIE 9940. Liq. Cryst. XX, 994019 (2016).

Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
[Crossref]

V. Yu. Reshetnyak, I. P. Pinkevych, T. J. Sluckin, and D. R. Evans, “Cloaking by shells with radially inhomogeneous anisotropic permittivity,” Opt. Express 24(2), A21–A32 (2016).
[Crossref] [PubMed]

2015 (1)

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

2014 (6)

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

D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
[Crossref]

Z. Su, J. Yin, Y. Guan, and X. Zhao, “Electrically tunable negative refraction in core/shell-structured nanorod fluids,” Soft Matter 10(39), 7696–7704 (2014).
[Crossref] [PubMed]

G. Pawlik, K. Tarnowski, W. Walasik, A. C. Mitus, and I. C. Khoo, “Liquid crystal hyperbolic metamaterial for wide-angle negative-positive refraction and reflection,” Opt. Lett. 39(7), 1744–1747 (2014).
[Crossref] [PubMed]

D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

P. A. A. De Beule, “Surface scattering of core-shell particles with anisotropic shell,” J. Opt. Soc. Am. A 31(1), 162–171 (2014).
[Crossref] [PubMed]

2013 (5)

2012 (5)

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]

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]

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 metamaterial substrates,” Adv,” OptoElectro. 2012, 452502 (2012).

U. K. Chettiar and N. Engheta, “Internal homogenization: effective permittivity of a coated sphere,” Opt. Express 20(21), 22976–22986 (2012).
[Crossref] [PubMed]

2011 (3)

S. M. Shelestiuk, V. Yu. Reshetnyak, and T. J. Sluckin, “Frederiks transition in ferroelectric liquid-crystal nanosuspensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4 Pt 1), 041705 (2011).
[Crossref] [PubMed]

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

T. Tumkur, G. Zhu, P. Black, Yu. 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]

2010 (1)

I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
[Crossref]

2007 (4)

A. Erbe and R. Sigel, “Tilt angle of lipid acyl chains in unilamellar vesicles determined by ellipsometric light scattering,” Eur Phys J E Soft Matter 22(4), 303–309 (2007).
[Crossref] [PubMed]

A. Sihvola, “Particular properties of the dielectric response of negative-permittivity scatterers,” PIERS ONLINE 3(3), 246–247 (2007).
[Crossref]

A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials (Amst.) 1(1), 2–11 (2007).
[Crossref]

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[Crossref]

2006 (3)

2005 (2)

S. A. Ramakrishna, “Physics of negative refractive index materials,” Rep. Prog. Phys. 68(2), 449–521 (2005).
[Crossref]

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005).
[Crossref]

2003 (1)

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

1997 (1)

V. Yu. Reshetnyak, T. J. Sluckin, and S. J. Cox, “Effective medium theory of polymer dispersed liquid crystal systems: II. Partially ordered bipolar droplets,” J. Phys. D Appl. Phys. 30(23), 3253–3266 (1997).
[Crossref]

1996 (1)

V. Yu. Reshetnyak, T. J. Sluckin, and S. Cox, “Effective medium theory of polymer dispersed liquid crystal droplet systems, I. Spherical droplets,” J. Phys. D Appl. Phys. 29(9), 2459–2465 (1996).
[Crossref]

Abbate, G.

V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
[Crossref]

Alekseyev, L. V.

Atkinson, J

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

Atwater, H. A.

S. A. Maier and H. A. Atwater, “Plasmonics: Localization and guiding of electromagnetic energy in metal/dielectric structures,” J. Appl. Phys. 98(1), 011101 (2005).
[Crossref]

Barnakov, Yu. A.

T. Tumkur, G. Zhu, P. Black, Yu. 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]

Barnes, W. L.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Belov, P.

A. Poddubny, I. Iorsh, P. Belov, and Yu. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–967 (2013).
[Crossref]

Belov, P. A.

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
[Crossref]

Besner, S.

D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
[Crossref]

Black, P.

T. Tumkur, G. Zhu, P. Black, Yu. 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]

Bonner, C. E.

T. Tumkur, G. Zhu, P. Black, Yu. 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]

Bunning, T. J.

V. Yu. Reshetnyak, T. J. Bunning, and D. R. Evans, “Tuning surface plasmons in graphene ribbons with liquid crystal layer,” Proc. SPIE 9940. Liq. Cryst. XX, 994019 (2016).

Cao, Z.

Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
[Crossref]

D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

Chen, G.

J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
[Crossref]

Chen, K.-P.

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
[Crossref]

Chettiar, U. K.

Cheypesh, Ye. I.

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
[Crossref]

Chu, B.

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]

Chulkov, E. V.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[Crossref]

Ciuchi, F.

V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
[Crossref]

Cortes, C. L.

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 metamaterial substrates,” Adv,” OptoElectro. 2012, 452502 (2012).

Cox, S.

V. Yu. Reshetnyak, T. J. Sluckin, and S. Cox, “Effective medium theory of polymer dispersed liquid crystal droplet systems, I. Spherical droplets,” J. Phys. D Appl. Phys. 29(9), 2459–2465 (1996).
[Crossref]

Cox, S. J.

V. Yu. Reshetnyak, T. J. Sluckin, and S. J. Cox, “Effective medium theory of polymer dispersed liquid crystal systems: II. Partially ordered bipolar droplets,” J. Phys. D Appl. Phys. 30(23), 3253–3266 (1997).
[Crossref]

De Beule, P. A. A.

De Stefano, L.

V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
[Crossref]

Dereux, A.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Diaz, A.

I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
[Crossref]

I. C. Khoo, D. H. Werner, X. Liang, A. Diaz, and B. Weiner, “Nanosphere dispersed liquid crystals for tunable negative-zero-positive index of refraction in the optical and terahertz regimes,” Opt. Lett. 31(17), 2592–2594 (2006).
[Crossref] [PubMed]

Drachev, V. P.

Ebbesen, T. W.

W. L. Barnes, A. Dereux, and T. W. Ebbesen, “Surface plasmon subwavelength optics,” Nature 424(6950), 824–830 (2003).
[Crossref] [PubMed]

Echenique, P. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[Crossref]

Engheta, N.

Erbe, A.

A. Erbe and R. Sigel, “Tilt angle of lipid acyl chains in unilamellar vesicles determined by ellipsometric light scattering,” Eur Phys J E Soft Matter 22(4), 303–309 (2007).
[Crossref] [PubMed]

Evans, D. R.

V. Yu. Reshetnyak, T. J. Bunning, and D. R. Evans, “Tuning surface plasmons in graphene ribbons with liquid crystal layer,” Proc. SPIE 9940. Liq. Cryst. XX, 994019 (2016).

V. Yu. Reshetnyak, I. P. Pinkevych, T. J. Sluckin, and D. R. Evans, “Cloaking by shells with radially inhomogeneous anisotropic permittivity,” Opt. Express 24(2), A21–A32 (2016).
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J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
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I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
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P Shekhar, J Atkinson, and Z Jacob, “Hyperbolic metamaterials: fundamentals and applications,” Nano Convergence 1, 14 (2014).

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C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
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Y. Guo, W. Newman, C. L. Cortes, and Z. Jacob, ““Applications of hyperbolic metamaterial substrates,” Adv,” OptoElectro. 2012, 452502 (2012).

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Kidwai, O.

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S. M. Prokes, O. J. Glembocki, J. E. Livenere, T. U. Tumkur, J. K. Kitur, G. Zhu, B. Wells, V. A. Podolskiy, and M. A. Noginov, “Hyperbolic and plasmonic properties of silicon/Ag aligned nanowire arrays,” Opt. Express 21(12), 14962–14974 (2013).
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A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
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Kivshar, Yu.

A. Poddubny, I. Iorsh, P. Belov, and Yu. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–967 (2013).
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Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
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L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
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Liang, X.

Liou, J.

I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
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Liu, Y.

D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
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Liu, Z.

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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Livenere, J. E.

Lu, J. Y.

J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
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Ma, Y.

I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
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Markel, V. A.

Mazur, E.

D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
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Mazzulla, A.

V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
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D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
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Mitus, A. C.

Molesky, S.

C. L. Cortes, W. Newman, S. Molesky, and Z. Jacob, “Quantum nanophotonics using hyperbolic metamaterials,” J. Opt. 14(6), 063001 (2012).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
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Muñoz, P.

D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
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Narimanov, E.

Narimanov, E. E.

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]

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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Newman, W.

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 metamaterial substrates,” Adv,” OptoElectro. 2012, 452502 (2012).

Noginov, M. A.

S. M. Prokes, O. J. Glembocki, J. E. Livenere, T. U. Tumkur, J. K. Kitur, G. Zhu, B. Wells, V. A. Podolskiy, and M. A. Noginov, “Hyperbolic and plasmonic properties of silicon/Ag aligned nanowire arrays,” Opt. Express 21(12), 14962–14974 (2013).
[Crossref] [PubMed]

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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T. Tumkur, G. Zhu, P. Black, Yu. 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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Peng, Z.

Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
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Pitarke, J. M.

J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
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Poddubny, A.

A. Poddubny, I. Iorsh, P. Belov, and Yu. Kivshar, “Hyperbolic metamaterials,” Nat. Photonics 7(12), 948–967 (2013).
[Crossref]

Poddubny, A. N.

A. N. Poddubny, P. A. Belov, and Y. S. Kivshar, “Spontaneous radiation of a finite-size dipole emitter in hyperbolic media,” Phys. Rev. A 84(2), 023807 (2011).
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J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
[Crossref]

Reshetnyak, V. Yu.

V. Yu. Reshetnyak, I. P. Pinkevych, T. J. Sluckin, and D. R. Evans, “Cloaking by shells with radially inhomogeneous anisotropic permittivity,” Opt. Express 24(2), A21–A32 (2016).
[Crossref] [PubMed]

V. Yu. Reshetnyak, T. J. Bunning, and D. R. Evans, “Tuning surface plasmons in graphene ribbons with liquid crystal layer,” Proc. SPIE 9940. Liq. Cryst. XX, 994019 (2016).

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
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S. M. Shelestiuk, V. Yu. Reshetnyak, and T. J. Sluckin, “Frederiks transition in ferroelectric liquid-crystal nanosuspensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4 Pt 1), 041705 (2011).
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V. Yu. Reshetnyak, T. J. Sluckin, and S. J. Cox, “Effective medium theory of polymer dispersed liquid crystal systems: II. Partially ordered bipolar droplets,” J. Phys. D Appl. Phys. 30(23), 3253–3266 (1997).
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V. Yu. Reshetnyak, T. J. Sluckin, and S. Cox, “Effective medium theory of polymer dispersed liquid crystal droplet systems, I. Spherical droplets,” J. Phys. D Appl. Phys. 29(9), 2459–2465 (1996).
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Rioux, D.

D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
[Crossref]

Shekhar, P

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

Shelestiuk, S. M.

S. M. Shelestiuk, V. Yu. Reshetnyak, and T. J. Sluckin, “Frederiks transition in ferroelectric liquid-crystal nanosuspensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4 Pt 1), 041705 (2011).
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H. Kettunen, H. Wallén, and A. Sihvola, “Cloaking and magnifying using radial anisotropy,” J. Appl. Phys. 114(4), 044110 (2013).
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A. Sihvola, “Metamaterials in electromagnetics,” Metamaterials (Amst.) 1(1), 2–11 (2007).
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A. Sihvola, “Particular properties of the dielectric response of negative-permittivity scatterers,” PIERS ONLINE 3(3), 246–247 (2007).
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J. M. Pitarke, V. M. Silkin, E. V. Chulkov, and P. M. Echenique, “Theory of surface plasmons and surface-plasmon polaritons,” Rep. Prog. Phys. 70(1), 1–87 (2007).
[Crossref]

Sipe, J. E.

Sluckin, T. J.

V. Yu. Reshetnyak, I. P. Pinkevych, T. J. Sluckin, and D. R. Evans, “Cloaking by shells with radially inhomogeneous anisotropic permittivity,” Opt. Express 24(2), A21–A32 (2016).
[Crossref] [PubMed]

S. M. Shelestiuk, V. Yu. Reshetnyak, and T. J. Sluckin, “Frederiks transition in ferroelectric liquid-crystal nanosuspensions,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 83(4 Pt 1), 041705 (2011).
[Crossref] [PubMed]

V. Yu. Reshetnyak, T. J. Sluckin, and S. J. Cox, “Effective medium theory of polymer dispersed liquid crystal systems: II. Partially ordered bipolar droplets,” J. Phys. D Appl. Phys. 30(23), 3253–3266 (1997).
[Crossref]

V. Yu. Reshetnyak, T. J. Sluckin, and S. Cox, “Effective medium theory of polymer dispersed liquid crystal droplet systems, I. Spherical droplets,” J. Phys. D Appl. Phys. 29(9), 2459–2465 (1996).
[Crossref]

Smolyaninov, I. I.

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]

Stinger, M. V.

I. C. Khoo, A. Diaz, J. Liou, M. V. Stinger, J. Huang, and Y. Ma, “Liquid crystals tunable optical metamaterials,” IEEE J. Sel. Top. Quantum Electron. 16(2), 410–417 (2010).
[Crossref]

Su, C.-W.

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
[Crossref]

Su, Z.

Z. Su, J. Yin, Y. Guan, and X. Zhao, “Electrically tunable negative refraction in core/shell-structured nanorod fluids,” Soft Matter 10(39), 7696–7704 (2014).
[Crossref] [PubMed]

Tarnowski, K.

Tkachenko, V.

V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
[Crossref]

Tumkur, T.

T. Tumkur, G. Zhu, P. Black, Yu. 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]

Tumkur, T. U.

S. M. Prokes, O. J. Glembocki, J. E. Livenere, T. U. Tumkur, J. K. Kitur, G. Zhu, B. Wells, V. A. Podolskiy, and M. A. Noginov, “Hyperbolic and plasmonic properties of silicon/Ag aligned nanowire arrays,” Opt. Express 21(12), 14962–14974 (2013).
[Crossref] [PubMed]

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]

Vallières, S.

D. Rioux, S. Vallières, S. Besner, P. Muñoz, E. Mazur, and M. Meunier, “An analytic model for the dielectric function of Au, Ag, and their alloys,” Adv. Opt. Materials 2(2), 176–182 (2014).
[Crossref]

Vita, F.

V. Tkachenko, G. Abbate, A. Marino, F. Vita, M. Giocondo, A. Mazzulla, F. Ciuchi, and L. De Stefano, “Nematic liquid crystal optical dispersion in the visible-near infrared range,” Mol. Cryst. Liq. Cryst. 454(1), 263–271 (2006).
[Crossref]

Walasik, W.

Wallén, H.

H. Kettunen, H. Wallén, and A. Sihvola, “Cloaking and magnifying using radial anisotropy,” J. Appl. Phys. 114(4), 044110 (2013).
[Crossref]

Weiner, B.

Wells, B.

Werner, D. H.

Wu, C.

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
[Crossref]

Xiang, X.

Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
[Crossref]

D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

Xuan, L.

Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

Yang, C.

Z. Cao, X. Xiang, C. Yang, Y. Zhang, Z. Peng, and L. Xuan, “Analysis of tunable characteristics of liquid-crystal-based hyperbolic metamaterials,” Liq. Cryst. 43(12), 1–7 (2016).
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D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

Yang, J.-H.

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
[Crossref]

Yang, Z.-H.

Y.-C. Hsiao, C.-W. Su, Z.-H. Yang, Ye. I. Cheypesh, J.-H. Yang, V. Yu. Reshetnyak, K.-P. Chen, and W. Lee, “Electrically active nanoantenna array enabled by varying the molecular orientation of an interfaced liquid crystal,” RSC Advances 6(87), 84500–84504 (2016).
[Crossref]

Yao, L.

D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

Yin, J.

Z. Su, J. Yin, Y. Guan, and X. Zhao, “Electrically tunable negative refraction in core/shell-structured nanorod fluids,” Soft Matter 10(39), 7696–7704 (2014).
[Crossref] [PubMed]

Zhang, H.

D. Jia, C. Yang, X. Li, Z. Peng, Y. Liu, Z. Cao, Q. Mu, L. Hu, D. Li, L. Yao, X. Lu, X. Xiang, H. Zhang, and L. Xuan, “Optical hyperbolic metamaterials based on nanoparticles doped liquid crystals,” Liq. Cryst. 41(2), 207–213 (2014).
[Crossref]

Zhang, T.

J. Y. Lu, A. Raza, N. X. Fang, G. Chen, and T. Zhang, “Effective dielectric constants and spectral density analysis of plasmonic nanocomposites,” J. Appl. Phys. 120(16), 163103 (2016).
[Crossref]

Zhang, X.

L. Ferrari, C. Wu, D. Lepage, X. Zhang, and Z. Liu, “Hyperbolic metamaterials and their applications,” Prog. Quantum Electron. 40, 1–40 (2015).
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Figures (9)

Fig. 1
Fig. 1 Cross-section of cylindrical nanowire with radially inhomogeneous anisotropic permittivity of the shell, which covers the metallic core with permittivity ε c ; ε r,φ are radially power-law dependent components of the shell permittivity;aand bare outer and inner radiuses of the shell.
Fig. 2
Fig. 2 Real parts of the effective permittivity components, Re ε (solid lines) and Re ε (dashed lines), versus the wavelength for the system of the shell-core nanowires with the Ag core (a), (b) and Au core (c), (d) in the LC 5CB matrix for two values of the exponent m; ε r0 =5, ε φ0 =5, ε z =5 , b/a=0.5,f=0.4 .
Fig. 3
Fig. 3 Real (a) and imaginary (b) parts of the nanowire core permittivity versus the wavelength: Ag core - solid lines, Au core - dashed lines. For plotting, we used Eq. (10) and Table 1 obtained in paper [43].
Fig. 4
Fig. 4 The width ( λ min , λ max ) of the HMM area Re ε <0 versus the exponent mvalue for nanowires with the Ag and Au cores. ε r0 =5, ε φ0 =5, ε z =5 , b/a=0.5,f=0.4 .
Fig. 5
Fig. 5 The width ( λ min , λ max ) of the HMM areas Re ε <0 (a) and Re ε <0 (b) versus the volume fill fraction fof the nanowires with Ag core. ε r0 =5, ε φ0 =5, ε z =5 , b/a=0.5 . For Re ε <0 : m=0,2,2 - solid, dashed, dot-dashed lines, respectively.
Fig. 6
Fig. 6 The width ( λ min , λ max ) of the HMM areas Re ε <0 (a) and Re ε <0 (b) versus the external to internal radii ratio of the nanowire shell with the Ag core. ε r0 =5, ε φ0 =5, ε z =5 ; f=0.4 . For Re ε <0 : m=0,2,2 - solid, dashed, dot-dashed lines, respectively.
Fig. 7
Fig. 7 The width ( λ min , λ max ) of the HMM area Re ε <0 versus the parameter ε r0 (a) at fixed ε φ0 =5 and the parameter ε φ0 (b) at fixed ε r0 =5 in the case of the nanowire shell with Ag core. m=0,2,2 - solid, dashed, dot-dashed lines, respectively. b/a=0.5 , f=0.4 , ε z =5 .
Fig. 8
Fig. 8 Real parts of the effective permittivity components, Re ε (solid lines) and Re ε (dashed lines), versus the wavelength in the case of the shell permittivity tensor trace to be constant. Nanowires with Ag core - (a), (b); with Au core – (c), (d); ε r0 =2, ε φ0 =8 - (a), (c); ε r0 =8, ε φ0 =2 - (b), (d). b/a=0.5,f=0.4, ε z =5 .
Fig. 9
Fig. 9 Real part of the effective permittivity components Re ε versus the wavelength for the system of the core-shell nanowire array in the LC matrix 5CB: Ag core - (a), Au core - (b); the LC director parallel to the nanowires (solid lines), the LC in an isotropic state (dashed lines). ε r0 =8, ε φ0 =2, ε z =5,b/a=0.5 .

Tables (2)

Equations (15)

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k x 2 + k x 2 ε xx + k z 2 ε zz = ω 2 c 2 ,
ε = ε LC ( 1f ) ε LC +( 1+f ) ε cyl, ( 1f ) ε cyl, +( 1+f ) ε LC ,
ε = ε LC +f( ε cyl, ε LC ),
ε r = ε r0 ( r/a ) m , ε φ = ε φ0 ( r/a ) m ,
ϕ( r,φ )=Arcosφ,r<b ϕ( r,φ )=( B r t 1 +C r t 2 )cosφ,bra, ϕ( r,φ )=( D r 1 E 0 r )cosφr>a
t 1,2 = 1 2 [ m± m 2 +4 ε φ0 ε r0 ]
Ab=B b t 1 +C b t 2 B a t 1 +C a t 2 =F/ a 2 E 0 a ε c A= ε r0 b m1 ( B t 1 b t 1 +C t 2 b t 2 ). ε r0 a m1 ( B t 1 a t 1 +C t 2 a t 2 )= ε LC ( 2F/ a 3 E 0 )
ε cyl, = ε r0 t 2 [ t 1 ε r0 ε c ( b a ) m 1 ] ( b a ) t 1 t 2 t 1 [ t 2 ε r0 ε c ( b a ) m 1 ] [ t 1 ε r0 ε c ( b a ) m 1 ] ( b a ) t 1 t 2 [ t 2 ε r0 ε c ( b a ) m 1 ] .
ε cyl, =ρ ε c +(1ρ) ε z .
ε c (ω)= ε D + ε CP1 + ε CP2 ,
ε D = ε ω p 2 ω 2 +iω Γ p ,
ε CP1 = A 1 [ ω g1 ω 01 2 (ω+i Γ 1 ) 2 ln( 1 ( ω+i Γ 1 ω 01 ) 2 )+ 2 ω g1 (ω+i Γ 1 ) 2 tan h 1 ( ω g1 ω 01 ω g1 ) ω+i Γ 1 ω g1 (ω+i Γ 1 ) 2 tan 1 ( ω g1 ω 01 ω+i Γ 1 ω g1 ) ω+i Γ 1 + ω g1 (ω+i Γ 1 ) 2 tan h 1 ( ω g1 ω 01 ω+i Γ 1 + ω g1 ) ].
ε CP2 = A 2 2 (ω+i Γ 2 ) 2 ln( 1 ( ω+i Γ 2 ω 02 ) 2 ).
ε LC = ( A o + B o λ 2 + C o λ 4 ) 2 ,
ε LC = ( A e + B e λ 2 + C e λ 4 ) 2 ,

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