Abstract

Near-field radiative heat transfer between isotropic, dielectric-based metamaterials is analyzed. A potassium bromide host medium comprised of silicon carbide (SiC) spheres with a volume filling fraction of 0.4 is considered for the metamaterial. The relative electric permittivity and relative magnetic permeability of the metamaterial are modeled via the Clausius-Mossotti relations linking the macroscopic response of the medium with the polarizabilities of the spheres. We show for the first time that electric and magnetic surface polariton (SP) mediated near-field radiative heat transfer occurs between dielectric-based structures. Magnetic SPs, existing in TE polarization, are physically due to strong magnetic dipole resonances of the spheres. We find that spherical inclusions with radii of 1 μm (or greater) are needed in order to induce SPs in TE polarization. On the other hand, electric SPs existing in TM polarization are generated by surface modes of the spheres, and are thus almost insensitive to the size of the inclusions. We estimate that the total heat flux around SP resonance for the metamaterial comprised of SiC spheres with radii of 1 μm is about 35% greater than the flux predicted between two bulks of SiC, where only surface phonon-polaritons in TM polarization are excited. The results presented in this work show that the near-field thermal spectrum can be engineered via dielectric-based metamaterials, which is crucial in many emerging technologies, such as in nanoscale-gap thermophotovoltaic power generation.

© 2011 OSA

1. Introduction

Electromagnetic metamaterials are artificial materials, made of sub-wavelength functional inclusions, displaying unusual electric and magnetic properties, such as negative index of refraction [15]. Metamaterials research is motivated by various potential applications such as superlenses [59] and optical cloaking [5,1012]. The possibility of controlling and designing the electric and magnetic responses of materials also paves the way to tailoring media with unique thermal radiative properties. So far, efforts in this area have been restricted to designing selective thermal radiation emitters and absorbers in the far-field [1322], while, to the best of our knowledge, only three papers have investigated near-field radiative heat transfer between metamaterials [2325].

Thermal radiation is in the near-field regime when the bodies are separated by distances of the same order of magnitude as, or less than, the dominant emission wavelength [26,27]. In that case, radiative heat transfer can exceed the blackbody predictions due to tunneling of evanescent waves. Moreover, quasi-monochromatic radiant heat exchange is possible when materials support surface polaritons (SPs) in the infrared, such as surface phonon-polaritons in silicon carbide (SiC) [28] and surface plasmon-polaritons in doped silicon [29]. When dealing with non-magnetic materials, SPs can only be excited in TM polarization [30,31]. Near-field thermal radiation is an emerging area of heat transfer engineering which may find applications in imaging [32], energy conversion [3337] and thermal rectification [38,39] to name only a few. In many cases, it is imperative to have some degree of control over the near-field thermal spectrum emitted and absorbed. For example, poor performance of nanoscale-gap thermophotovoltaic power generation was recently predicted due to the overheating of the cell [37]. It was concluded that this problem could be circumvented by fine tuning the near-field thermal spectrum transferred between the radiator and the cell via thermal excitation of SPs, which can potentially be accomplished using metamaterials with negative permittivity and/or negative permeability at prescribed frequencies.

Joulain et al. [23] studied near-field radiative heat transfer between two hypothetical metamaterials made of an artificial array of thin wires and split ring resonators (SRRs), such that negative permittivity, negative permeability and negative refraction were obtained in some specific spectral bands. Results showed that SP mediated heat transfer in both TM and TE polarizations dominated the radiative flux. Indeed, a medium with negative permeability can support SPs in TE polarization, thus opening an additional channel through which near-field radiant energy can flow. Zheng and Xuan [24,25] provided a comprehensive procedure for solving near-field radiative heat transfer problems in one-dimensional geometry with layers of arbitrary permittivity and permeability. The method was tested for the same metamaterials discussed in reference [23].

Metamaterials based on metallic constitutive elements such as SRRs, thin wires, rods or fishnet structures [5] present several drawbacks. Fabrication of these intricate structures requires sophisticated lithography techniques. Additionally, due to the anisotropy of the metallic inclusions, the macroscopic response of the metamaterial is also anisotropic, unless complicated three-dimensional structures are employed. Finally, high losses are observed in these metal-based metamaterials at optical frequencies [4043].

The general objective of this paper is to investigate near-field radiative heat transfer between realistic, isotropic metamaterials exhibiting negative permittivity and permeability in the infrared. This task is achieved by considering Mie resonance-based dielectric metamaterials [5,4051]. In this work, spherical dielectric inclusions are considered due to a relatively simple fabrication process. Moreover, bulk three-dimensional dielectric sphere-based metamaterials exhibit isotropic macroscopic electric and magnetic responses since spheres are inherently isotropic [40]. The electric and magnetic dipole resonances of a sphere can be conceptualized as the meta-atoms making up the metamaterial. A large collection of these dielectric spheres, within a dielectric host, having strong enough electric and magnetic dipole resonances can thus constitute a material with macroscopic negative permittivity and permeability for specific frequencies. Indeed, even if both the host medium and the inclusions are dielectric, negative permeability may be induced via sub-wavelength particles with very high permittivity supporting strong resonances with a large displacement current [5,42]. In this paper, SiC spheres are chosen as dielectric inclusions since the electric permittivity of this material is very large in the infrared.

The rest of the paper is structured as follows. In the next section, the problem under consideration, namely two bulk metamaterials exchanging thermal radiation and separated by a sub-wavelength vacuum gap, is described; the mathematical details required for calculating the near-field radiative heat flux, based on fluctuational electrodynamics, are provided. Subsequently, the effective permittivity and permeability models used to describe the macroscopic electric and magnetic responses of the metamaterials made of SiC spheres are discussed. In the fourth section, spectral distributions of near-field radiative heat flux are calculated and analyzed thoroughly. Concluding remarks are provided in the last section.

2. Near-field thermal radiation modeling between two bulk metamaterials

The geometry of the problem considered is schematically depicted in Fig. 1 , where two bulk materials modeled as planar half-spaces (media 0 and 2) with perfectly smooth and parallel surfaces are separated by a vacuum gap of thickness d (medium 1). The system is invariant along the x-y plane such that only the z-component of the radiative heat flux is considered.

 

Fig. 1 Schematic representation of the problem under consideration: two planar half-spaces maintained at temperatures T 0 and T 2 and separated by a vacuum gap of thickness d are exchanging thermal radiation.

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It is assumed that the media are in local thermodynamic equilibrium, homogeneous, isotropic and described by frequency-dependent relative electric permittivity ε (= ε+iε) and relative magnetic permeability μ (= μ+iμ) local in space. The problem is solved at steady-state, where media 0 and 2 are maintained at constant and uniform temperatures T 0 and T 2, respectively, via external heating or cooling devices. Such an approximation is physically realistic since the thermal resistance due to near-field radiative heat transfer in the vacuum gap is much higher than the thermal resistance due to conduction in layers 0 and 2. For example, assuming that both layers are 1 mm thick and made of SiC, we estimated that the radiative thermal resistance, in the extreme case where d = 10 nm, is about 180 times the conduction resistance for a temperature gradient of 300 K between media 0 and 2.

Near-field radiative heat transfer is predicted using fluctuational electrodynamics, where the microscopic randomly oscillating charges generating the thermal radiation field are modeled in Maxwell’s equations via macroscopic stochastic currents [52]. When dealing with non-magnetic media, a stochastic current J r , e is added in Ampère’s law in order to model thermal radiation emission due to electric dipole oscillations [52,53]. For magnetic media, as considered in this paper, a stochastic current J r , m should also be included in Faraday’s law to model thermal emission due to fluctuating magnetic dipoles [52]. Assuming a time-dependence of exp(-iωt), the electric and magnetic fields thermally generated can be written as follows [54]:

E(r,ω)=iωμμvVG¯¯Ee(r,r,ω)Jr,e(r,ω)dVVG¯¯Em(r,r,ω)Jr,m(r,ω)dV
H(r,ω)=VG¯¯He(r,r,ω)Jr,e(r,ω)dV+iωεεvVG¯¯Hm(r,r,ω)Jr,m(r,ω)dV
where ε and μ are respectively the relative permittivity and permeability of the emitting medium, while εv and μv are, also respectively, the free space permittivity and permeability. The term G¯¯E(H)e(m) is the dyadic Green’s function (DGF) relating the electric E (magnetic H) field with frequency ω observed at r to an electric e (magnetic m) source located at r '.

The radiative heat flux along the z-direction observed at r due to a source located at r ' within an emitting medium of volume V is derived by computing the z-component of the Poynting vector Sz(r,ω)=2Re{ExHy*EyHx*}, where denotes an ensemble average and * is the complex conjugate operator. The evaluation of the Poynting vector requires computation of terms EiHj*, written as follows using Eqs. (1a) and (1b):

EiHj*=iωμμvVdVVdVGiαEe(r,r,ω)GjβHe*(r,r,ω)Jαr,e(r,ω)Jβr,e*(r,ω)                                     +iωεεvVdVVdVGiαEm(r,r,ω)GjβHm*(r,r,ω)Jαr,m(r,ω)Jβr,m*(r,ω)                                     +kv2εμVdVVdVGiαEe(r,r,ω)GjβHm*(r,r,ω)Jαr,e(r,ω)Jβr,m*(r,ω)                                     VdVVdVGiαEm(r,r,ω)GjβHe*(r,r,ω)Jαr,m(r,ω)Jβr,e*(r,ω)
where the ensemble average is applied to the spatial correlation function of the stochastic currents. In Eq. (2), kv is the magnitude of the wavevector in vacuum, while α and β involve a summation over the three orthogonal components. The ensemble average of the spatial correlation function of currents can be written as a function of the local temperature of the emitting medium via the fluctuation-dissipation theorem (FDT) [52]:

Jαr,e(r,ω)Jβr,e*(r,ω)=ωεvεπΘ(ω,T)δ(rr)δαβ
Jαr,m(r,ω)Jβr,m*(r,ω)=ωμvμπΘ(ω,T)δ(rr)δαβ
Jαr,e(r,ω)Jβr,m*(r,ω)=Jαr,m(r,ω)Jβr,e*(r,ω)=0

In the above, Θ (=ω/[exp(ω/kbT)1]) is the mean energy of a Planck oscillator in thermal equilibrium. Equation (3c) shows that the electric and magnetic source currents are not spatially correlated. Substitution of Eq. (2) into the z-component of the Poynting vector, after application of the FDT and assuming the ergodic hypothesis where an ensemble average is equivalent to time averaging [55], gives the following general expression for the near-field radiative heat flux:

qω(r)=Sz(r,ω)=2kv2Θ(ω,T)πRe{iμεVdV[GxαEe(r,r,ω)GyαHe*(r,r,ω)GyαEe(r,r,ω)GxαHe*(r,r,ω)]+iεμVdV[GxαEm(r,r,ω)GyαHm*(r,r,ω)GyαEm(r,r,ω)GxαHm*(r,r,ω)]}

In the absence of fluctuating magnetic currents, the above equation reduces to the general expression for the near-field radiative heat flux for non-magnetic media given by Eq. (4) in reference [56].

The net near-field radiative heat flux between media 0 and 2 is calculated by computing the difference between the flux absorbed by medium 2 due to the emitting bulk 0 at temperature T 0 and the flux absorbed by medium 0 due to the emitting bulk 2 at T 2. An expression for the near-field radiative heat flux in terms of the Fresnel reflection coefficients can be derived provided that the DGFs for the layered system shown in Fig. 1 are known. These DGFs were reported by Joulain et al. [23] and are not repeated here. After some algebra, the (net) monochromatic near-field radiative heat flux between media 0 and 2 is explicitly given by:

qω,02=Θ(ω,T0)Θ(ω,T2)π2                                                         ×γ=TE,TM[0kvkρdkρ(1|r10γ|2)(1|r12γ|2)4|1r10γr12γe2ikz1d|2+kvkρdkρe2kz1dIm(r10γ)Im(r12γ)|1r10γr12γe2kz1d|2]
where kρ (=|kxx^+kyy^|) is the component of the wavevector parallel to the surfaces of the layers and is consequently a pure real number, while kzj (= kzj+ikzj) is the perpendicular component of the wavevector, such that the magnitude of the wavevector in medium j is kj=kρ2+kzj2. The Fresnel reflection coefficients in TM and TE polarizations are respectively given by rijTM=(εjkziεikzj)/(εjkzi+εikzj) and rijTE=(μjkziμikzj)/(μjkzi+μikzj). It is worth noting that Eq. (5) has the exact same form as the near-field radiative heat flux for non-magnetic materials; the influence of the relative magnetic permeability comes into the picture in the z-component of the wavevector and in the Fresnel reflection coefficients.

Before closing this section, it should be emphasized that extra care must be taken when evaluating the z-component of the wavevector in the metamaterial. For a given kρ, the z-component of the wavevector is calculated as kzj=nj2kv2kρ2, where nj is the refractive index of medium j. The real part of the refractive index of a metamaterial may become negative for some frequencies. Using the complex plane representation, the relative permittivity and permeability can be written respectively as ε=|ε|eiϕε and μ=|μ|eiϕμ, where ϕε and ϕμ take values between 0 and π since ε and μ are always positive. Using n=εμ, it is easy to show that the real part of the refractive index n=|ε||μ|cos[(ϕε+ϕμ)/2] becomes negative when (ϕε + ϕμ) > π. In the frequency bands where n' < 0, Skaar [57] showed that for absorbing materials, the signs of kz and kz for any real value of kρ should be the same as those of n' and n”, respectively. Consequently, the z-component of the wavevector in medium j should be calculated as kzj=-nj2kv2-kρ2 when (ϕε + ϕμ) > π.

The only remaining step for calculating the near-field heat flux is to develop a model describing the relative permittivity and permeability of the metamaterial; this is discussed next.

3. Relative permittivity and permeability of the metamaterial

We assume a dielectric host material containing spherical dielectric inclusions of radius rs. The spheres are arranged in a periodic manner in a simple cubic lattice, where the length of the edge of a unit cubic cell is given by the lattice constant a. The number of inclusions per unit volume is given by N = 1/a 3. Alternatively, the concentration of inclusions can be characterized by the volume filling fraction f = NV, where V is the volume of a sphere. Combining these last two expressions, the volume filling fraction of the spherical inclusions can be written as f=(4π/3)(rs/a)3. The macroscopic electric and magnetic responses of the metamaterial made of a large collection of dielectric spheres can be described via an effective electric permittivity and an effective magnetic permeability [43,44]. This is done by averaging the fields in the long-wavelength limit, roughly defined as λh >> a > 2rs, where λh is the wavelength in the host medium [40,43].

The Clausius-Mossotti model [5,43,58] is employed to describe the macroscopic effective permittivity and permeability of the metamaterial as a function of the polarizabilities of the particles. Note that this model is applicable in the long-wavelength limit and for volume filling fractions substantially smaller than unity [5,40]. The electric and magnetic polarizabilities of a sphere are given respectively by αe=6πia1/kh3 and αm=6πib1/kh3, where a 1 and b 1 are the first order Mie coefficients (i.e., dipole terms) while kh is the magnitude of the wavevector in the host medium. In the general case of a pole of order l, the Mie coefficients are given by [59]:

al=m˜ψl(m˜X)ψl(X)ψl(X)ψl(m˜X)m˜ψl(m˜X)ξl(X)ξl(X)ψl(m˜X)
bl=ψl(m˜X)ψl(X)m˜ψl(X)ψl(m˜X)ψl(m˜X)ξl(X)m˜ξl(X)ψl(m˜X)
where ψl and ξl are Riccati-Bessel functions (the primes indicate differentiation with respect to the argument), m˜ is the ratio of the refractive index of the sphere ns over the refractive index of the host medium nh, while X is the size parameter defined as 2πnhrs/λv, where λv is the wavelength in vacuum.

Substitution of the electric and magnetic polarizabilities of a sphere into the Clausius-Mossotti relations leads to the following effective macroscopic properties [43]:

εeffεh=1+N[kh3i6π(1a11)N3]1
μeffμh=1+N[kh3i6π(1b11)N3]1
where εeff and μeff are respectively the effective relative permittivity and permeability of the metamaterial while εh and μh are the relative permittivity and permeability of the host medium. A scattering correction was applied to αe and αm when deriving Eqs. (7a) and (7b). Indeed, in the long-wavelength regime, infinite periodic lattices should not experience losses due to scattering. These losses, included in the polarizabilities, are corrected by performing the substitution 1/αe,m1/αe,m+ikh3/6π [43]. It is also important to note that despite the periodic arrangement of the spherical inclusions, Wheeler et al. [40,43] have shown that the effective relative permittivity and permeability given by Eqs. (7a) and (7b) can be considered as isotropic in the long-wavelength regime. From now on, for simplicity, the terminology “effective” is dropped when referring to the relative permittivity and permeability of the metamaterial.

Spheres with large magnetic dipole resonances are required to induce a macroscopic magnetic response from the metamaterial. Magnetic dipole resonance is determined by imposing b 1 → ∞. In the long-wavelength limit, the dipole term b 1 can be approximated as follows [43,44]:

b12iX33F(m˜X)1F(m˜X)+2
where

F(m˜X)=2[sin(m˜X)m˜Xcos(m˜X)][(m˜X)21]sin(m˜X)+m˜Xcos(m˜X)

Using Eq. (8), magnetic resonance arises when F(m˜ X) = -2. Substitution of this condition into Eq. (9) leads to sin(m˜ X) = 0, such that the solutions are given by m˜ X = (l = 1, 2, 3, …). The fundamental resonant wavelength (l = 1) in the host medium λh,resm can therefore be written as follows:

λh,resm2rsm˜

Assuming that the long-wavelength limit is about λhm/2rs>~10, strong magnetic dipole resonance occurs when |εs/εh|>~100, where εs is the relative permittivity of the sphere [43,48]. From a thermal radiation point of view, we are interested in metamaterials with magnetic response in the infrared, such that the permittivity of the spherical inclusions should be very large within this spectral band. SiC, which is a polar crystal, fulfills this requirement. The relative electric permittivity of SiC is modeled via a damped harmonic oscillator:

εs=ε(ω2-ωLO2+iΓωω2-ωTO2+iΓω)
where ε is the high frequency limit of the permittivity, ωLO is the longitudinal optical phonon frequency, ωTO is the transverse optical phonon frequency and Γ is the damping factor. For SiC, the following parameters are used: ε = 6.7, ωLO = 1.825×1014 rad/s, ωTO = 1.494×1014 rad/s and Γ = 8.966×1011 s−1 [60]. The real part of the relative permittivity of SiC reaches a value close to 300 near ωTO, such that strong magnetic dipole resonance is achievable using this type of inclusion.

For a given host medium, Eqs. (6) and (7) show that the electric and magnetic responses of the metamaterial depend upon the concentration of inclusions as well as the size and optical properties of the spheres. Engineering the macroscopic electric and magnetic responses of the metamaterial, and therefore the near-field thermal spectrum, is made possible by varying these parameters. In this paper, we restrict our attention to a host medium of potassium bromide (KBr) comprised of SiC spheres with a volume filling fraction fixed at 0.4. Around a wavelength of 10 μm, the refractive index of KBr can be considered frequency-independent with a value of 1.5 [41].

The real part of the refractive index n' and the sum of the phases of the relative permittivity and permeability (ϕε + ϕμ) of the metamaterial are shown in Fig. 2 for rs = 1 μm. Note that due to the fact that the relative permittivity and permeability model is valid in the long-wavelength regime, the spectral analysis is restricted to the band of from 1×1014 rad/s to 2×1014 rad/s, which corresponds to a host medium wavelength λh band of from 6.3 μm to 12.6 μm. For f = 0.4 and rs = 1 μm, the lattice constant a is about 2.2 μm, such that the minimum λh is about three times the size of a.

 

Fig. 2 Real part of the refractive index and sum of the phases of the relative permittivity and permeability of the metamaterial for rs = 1 μm, nh = 1.5 and f = 0.4.

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As discussed in section 2, the real part of the refractive index is positive within the spectral band under consideration since the condition (ϕε + ϕμ) > π is never satisfied. In fact, it can be shown that a metamaterial made of a single type of spherical inclusions cannot have a negative permittivity and a negative permeability simultaneously [43].

4. Spectral distributions of near-field radiative heat flux and resonance analysis

In all simulation results presented in this section, medium 0 is maintained at 300 K while medium 2 is a heat sink at 0 K. The relative permittivity and permeability of media 0 and 2 are given respectively by ε 0 = ε 2 = εeff and μ 0 = μ 2 = μeff (see Eqs. (7a) and (7b)). Spectral distributions of radiative heat flux for gaps d of 10 nm, 100 nm and 1 μm are reported in Fig. 3 for SiC spheres having a radius of 1 μm; the profiles are compared with the flux in the far-field regime and the blackbody predictions.

 

Fig. 3 Monochromatic radiative heat flux between two metamaterials made of SiC spheres (rs = 1 μm, nh = 1.5, f = 0.4) separated by a vacuum gap of thickness d. The near-field profiles are compared with far-field and blackbody predictions.

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It can be seen in Fig. 3 that the radiative flux for small gaps d is concentrated around two resonances arising in the vicinity of 1.42×1014 rad/s and 1.69×1014 rad/s. This nearly monochromatic, marked enhancement of the near-field flux can be analyzed more closely by investigating the conditions for which SPs exist. In the problem treated here, SPs exist at both interfaces delimiting the metamaterial and the vacuum, such that coupling of these resonant surface waves occurs within the gap. For thick media, despite SP coupling in the gap, the resonance of the flux is always located at the resonant frequency of the material-vacuum interface [61]. As such, the resonance analysis is performed hereafter by considering a single material-vacuum interface.

SP dispersion relations in TE and TM polarizations are derived by finding the poles of the Fresnel reflection coefficients r01TE and r01TM [30]. Also, throughout the resonance analysis, losses in the relative permittivity and permeability of the metamaterial are neglected (i.e., ε0=ε0 and μ0=μ0). In TE polarization, r01TE → ∞ when μ1kz0+μ0kz1 = 0. SPs are surface waves characterized by an evanescent electromagnetic field decaying in both media 0 and 1, such that kz0 and kz1 are both pure imaginary numbers. Using kzj=iγj, the SP dispersion relation in TE polarization can thus be written as follows:

γ0μ0+γ1μ1=0
where γj=kρ2εjμjkv2. Equation (12) can be satisfied if and only if μ0μ1<0 since γj>0. By combining the definition of γj with Eq. (12), the SP dispersion relation can also be written in terms of the parallel wavevector kρ:

kρ2=kv2μ0μ1(ε1μ0ε0μ1)μ02μ12

Substitution of the dispersion relation given by Eq. (13) into γj leads to:

γj2=kv2μj2ε1μ1ε0μ0μ02μ12

Specializing for the case where medium 1 is a vacuum, such that μ0 < 0 for SP to exist, Eq. (14) is modified as follows:

γj2=kv2μj21+ε0|μ0|μ021

Note that Eq. (15) is valid only if μ0 < 0, while no restriction is imposed on the sign of ε0. Using Eq. (15), γj2>0 is fulfilled when either of the following conditions is satisfied:

|μ0|>1andε0>1|μ0|
|μ0|<1andε0<1|μ0|

Equations (16a) and (16b) define the domain of existence of SPs in TE polarization [62,63]. In the special case where ε0 > 0, the numerator of Eq. (15) is always positive, such that SPs can be excited in TE polarization when μ0<1 and ε0>0 This last condition is the one prevailing for the SiC sphere-based metamaterial since the relative permittivity and permeability are never simultaneously negative for a given frequency, as discussed in section 3. Similarly, it is easy to show that SPs exist in TM polarization when ε0<1 and μ0>0.

Resonance of the near-field radiative flux arises when kρ is very large within a narrow spectral band. For TE and TM polarizations, kρ → ∞ when μ0 = -1 and ε0 = -1, respectively. When SPs are excited by plasma oscillations or transverse optical phonons, the terminologies surface plasmon-polaritons and surface phonon-polaritons are respectively employed. Here, SPs exist due to the Mie resonances of the dielectric spherical inclusions, such that the generic terminology SP is used. Also, from now on, SPs in TE and TM polarizations will be respectively referred to as magnetic SPs and electric SPs.

The real part of the relative permittivity and permeability of the metamaterial studied in Fig. 3 are plotted in Fig. 4(a) , while the TM and TE evanescent contributions to the near-field heat flux are shown for d = 10 nm in Fig. 4(b). In both plots, the spectral bands where SPs exist are identified.

 

Fig. 4 (a) Real part of the relative electric permittivity and magnetic permeability of the metamaterial for rs = 1 μm, nh = 1.5 and f = 0.4. (b) Comparison between the TM and TE evanescent components of the radiative heat flux for the case shown in Fig. 3 when d = 10 nm.

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It can be seen in Fig. 4(a) that μ0=1 and ε0=1 at 1.42×1014 rad/s and 1.69×1014 rad/s, respectively. Therefore, the lowest frequency resonance of the flux observed in Fig. 3 is due to magnetic SPs, while the second at a higher frequency is due to electric SPs. Moreover, Fig. 4(b) clearly shows that the resonance around 1.42×1014 rad/s emerges only in TE polarization, while the resonance near 1.69×1014 rad/s arises in TM polarization. In the far-field regime, it can be seen in Fig. 3 that the radiative heat flux is quite low within the spectral bands supporting SPs. Indeed, Fig. 2 shows that the real part of the refractive index in these bands is low while the imaginary part (n=|ε||μ|sin[(ϕε+ϕμ)/2]) is high, thus resulting in highly reflecting zones.

The near-field radiative heat flux also exhibits secondary peaks that are not due to SPs (see Figs. 3 and 4(b)). Outside the spectral bands supporting SPs, the near-field radiative heat flux is saturated by evanescent waves generated by total internal reflection at the metamaterial-vacuum interface. These modes, contained within the region kv < kρ < n'kv, correspond to electromagnetic waves propagating within the material but evanescent within the vacuum gap [64]. Therefore, the secondary peaks of the flux arising at some specific frequencies are due to the fact that the real part of the refractive index also exhibits peaks at these frequencies.

The influence of the size of the inclusions is studied in Figs. 5(a) and 5(b). The magnitude of the Mie coefficients a 1 and b 1 are shown in Fig. 5(a) for radii rs of 600 nm, 800 nm and 1 μm. Spectral distributions of near-field radiative heat flux at d = 10 nm are calculated for these rs values and are presented in Fig. 5(b); the profiles are compared with the predictions between two bulks of SiC.

 

Fig. 5 (a) Magnitude of the Mie coefficients a 1 and b 1 for SiC spheres of radii rs of 600 nm, 800 nm and 1 μm in a host medium with nh = 1.5. (b) Monochromatic radiative heat flux between two metamaterials made of SiC spheres (rs = 600 nm, 800 nm and 1 μm, nh = 1.5, f = 0.4) separated by a 10 nm thick vacuum gap. The profiles are compared with the predictions obtained between two SiC bulk materials.

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Although not shown in Fig. 5(a), the quadrupoles a 2 and b 2 have also been calculated. For rs = 1 μm, the quadrupoles are more than an order of magnitude smaller than the dipole terms, such that it is justified to account strictly for a 1 and b 1 when modeling the relative permittivity and permeability of the metamaterial. It can be seen in Fig. 5(a) that the size of the particles affects in a non-negligible manner the strength of the magnetic dipole resonance b 1 as well as its spectral location. This observation is also true when looking at the low-frequency resonance of the flux in Fig. 5(b). The resonant wavelength of b 1 was derived in section 3 (Eq. (10)). By neglecting the losses in the relative permittivity of the spheres, the combination of Eqs. (10) and (11) leads to the following approximation for the resonant frequency of b 1:

ωresmωLO2+Ω2(ωLO2+Ω2)24Ω2ωTO22
where Ω=πcv/εrs. The above approximation, derived for an isolated sphere, is sufficient for predicting the low-frequency resonance of the near-field flux for modest values of the volume filling fraction f [43]. Using Eq. (17), resonance of b 1 for rs of 600 nm, 800 nm and 1 μm are estimated at 1.47×1014 rad/s, 1.45×1014 rad/s and 1.43×1014 rad/s, respectively. These predictions are in good agreement with the results of Figs. 5(a) and 5(b). The resonance given by Eq. (17) is size-dependent and occurs near the transverse optical phonon frequency ωTO where the real part of the permittivity of the sphere is very large. Note that the low-frequency resonance of the flux for rs values of 600 nm and 800 nm are not due to magnetic SPs, but rather due to evanescent waves experiencing total internal reflection at the material-vacuum interface. Indeed, when rs = 600 nm, μ0 is always greater than zero, while for rs = 800 nm, the real part of the magnetic permeability becomes negative within a small spectral band but is never smaller than -1. These results therefore show that a very strong resonance of the magnetic dipole b 1 is necessary in order to induce magnetic SPs. This resonance can be enhanced by maximizing the ratio of the real part of the permittivity of the sphere over the permittivity of the host medium and by increasing the size of the inclusions.

The significant enhancement of the near-field radiative heat flux via electric SPs is physically due to surface electric dipole modes of the SiC particles [43]. The resonant frequency of a surface mode in small spherical particles is given by εs(ωrese)=2εh [31,59]. Using Eq. (11), this resonant frequency is given by:

ωrese2εhωTO2+εωLO2ε+2εh
which is roughly approximated as being size-independent. Using Eq. (18), the electric dipole a 1 resonant frequency is estimated at 1.70×1014 rad/s, which is in reasonable agreement with the results of Fig. 5(a). It is also possible to estimate the near-field radiative heat flux resonance by letting ε0=1. Using a procedure similar to the one described in [43], the resonance of the flux can be approximated as follows:

ωreseεωLO2εhωTO2(f4f+2)εεh(f4f+2)

The resonant frequency of the near-field radiative heat flux is estimated to be 1.72×1014 rad/s, which is again in reasonable agreement with the results of Fig. 5(b). Note that the approximation given by Eq. (19) does not capture the small spectral shift of the resonance toward a higher frequency as the size of the inclusions decreases.

The resonant modes of the flux can also be determined by first calculating ε0 and μ0 via Eqs. (7a) and (7b), and then simply determining the frequencies for which these two values are equal to -1. The goal of the above analysis was to link the resonances of the near-field radiative heat flux with the physical mechanisms responsible for this augmentation.

Finally, it is worth noting that the radiative heat flux integrated within the spectral band from 1×1014 rad/s to 2×1014 rad/s when d = 10 nm is about 35% greater for the metamaterials made of SiC spheres with a radius of 1 μm, where both electric and magnetic SPs are excited, than the flux predicted between two bulk materials of SiC, where only surface phonon-polaritons in TM polarization are excited.

5. Conclusions

Near-field radiative heat transfer between dielectric-based metamaterials separated by a vacuum gap has been analyzed. Metamaterials made of spherical inclusions of silicon carbide (SiC) within a dielectric host medium of potassium bromide (KBr) have been considered. We have shown for the first time that electric and magnetic surface polariton (SP) mediated near-field radiative heat transfer between dielectric-based metamaterials is possible. Spectral distributions of flux for sub-wavelength vacuum gaps show a low-frequency resonance due to excitation of magnetic SPs in TE polarization and a resonance at a slightly higher frequency attributable to electric SPs in TM polarization. The strength and spectral location of the low-frequency flux resonance, due to magnetic dipole resonance of the particles, is quite sensitive to the size of the spherical inclusions. For particles with radii smaller than 1 μm, the magnetic dipole resonance of the spheres is not strong enough to induce SPs in TE polarization. Still, a high near-field radiative flux is observed at low-frequency due to evanescent waves generated by total internal reflection at the metamaterial-vacuum interface. On the other hand, the frequency and strength of the high-frequency resonance of the flux, due to electric surface modes of the spheres, are almost insensitive to the size of the inclusions.

The results presented in this paper show that it is possible to control the near-field thermal spectrum via metamaterials. Mie resonance-based metamaterials may also have significant impacts in tuning far-field emission. For example, excitation of SPs in the far-field via a grating can result in a coherent thermal source. Additionally, the presence of SPs on a flat surface induces highly reflective spectral bands, such that it becomes possible to control precisely the thermal radiative properties of materials. In future studies, it would be interesting to analyze closely the impacts of the shape and the concentration of inclusions, as well as the materials used for the host medium and the particles, on heat exchange.

References and links

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2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000). [CrossRef]   [PubMed]  

3. 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]  

4. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University Press, 2009).

5. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).

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

7. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004). [CrossRef]   [PubMed]  

8. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005). [CrossRef]   [PubMed]  

9. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007). [CrossRef]  

10. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006). [CrossRef]   [PubMed]  

11. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007). [CrossRef]  

12. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009). [CrossRef]   [PubMed]  

13. C. J. Fu, Z. M. Zhang, and D. B. Tanner, “Planar heterogeneous structures for coherent emission of radiation,” Opt. Lett. 30(14), 1873–1875 (2005). [CrossRef]   [PubMed]  

14. M. Maksimović and Z. Jakšić, “Emittance and absorptance tailoring by negative refractive index metamaterial-based Cantor multilayers,” J. Opt. A-Pure Appl. Opt. 8, 355–362 (2006). [CrossRef]  

15. F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007). [CrossRef]  

16. B. J. Lee, L. P. Wang, and Z. M. Zhang, “Coherent thermal emission by excitation of magnetic polaritons between periodic strips and a metallic film,” Opt. Express 16(15), 11328–11336 (2008). [CrossRef]   [PubMed]  

17. C. J. Fu and Z. M. Zhang, “Further investigation of coherent thermal emission from single negative materials,” Nanosc. Microsc. Therm. 12(1), 83–97 (2008). [CrossRef]  

18. Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009). [CrossRef]  

19. C. J. Fu and Z. M. Zhang, “Thermal radiative properties of metamaterials and other nanostructured materials: A review,” Front. Energy Power Eng. China 3(1), 11–26 (2009). [CrossRef]  

20. L.-G. Wang, G.-X. Li, and S.-Y. Zhu, “Thermal emission from layered structures containing a negative-zero-positive index metamaterial,” Phys. Rev. B 81(7), 073105 (2010). [CrossRef]  

21. J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011). [CrossRef]  

22. L. P. Wang and Z. M. Zhang, “Phonon-mediated magnetic polaritons in the infrared region,” Opt. Express 19(S2Suppl 2), A126–A135 (2011). [CrossRef]   [PubMed]  

23. K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81(16), 165119 (2010). [CrossRef]  

24. Z. Zheng and Y. Xuan, “Theory of near-field radiative heat transfer for stratified magnetic media,” Int. J. Heat Mass Tran. 54(5-6), 1101–1110 (2011). [CrossRef]  

25. Z. Zheng and Y. Xuan, “Near-field radiative heat transfer between general materials and metamaterials,” Chin. Sci. Bull. 56(22), 2312–2319 (2011). [CrossRef]  

26. S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009). [CrossRef]  

27. J. R. Howell, R. Siegel, and M. P. Mengüç, Thermal Radiation Heat Transfer (Fifth Edition, CRC Press, 2011), Chap. 16.

28. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002). [CrossRef]  

29. S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer 132(2), 023302 (2010). [CrossRef]  

30. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005). [CrossRef]  

31. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).

32. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006). [CrossRef]   [PubMed]  

33. R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001). [CrossRef]  

34. M. D. Whale and E. G. Cravalho, “Modeling and performance of microscale thermophotovoltaic energy conversion devices,” IEEE Trans. Energ. Convers. 17(1), 130–142 (2002). [CrossRef]  

35. M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006). [CrossRef]  

36. K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008). [CrossRef]  

37. M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energ. Convers. 26(2), 686–698 (2011). [CrossRef]  

38. C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010). [CrossRef]   [PubMed]  

39. S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98(11), 113106 (2011). [CrossRef]  

40. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005). [CrossRef]  

41. M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009). [CrossRef]  

42. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009). [CrossRef]  

43. M. S. Wheeler, A scattering-based approach to the design, analysis, and experimental verification of magnetic metamaterials made from dielectrics, PhD thesis, University of Toronto (2010).

44. L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65–68 (1947).

45. S. O’Brien and J. B. Pendry, “Photonics band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002). [CrossRef]  

46. C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003). [CrossRef]  

47. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005). [CrossRef]   [PubMed]  

48. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73(4), 045105 (2006). [CrossRef]  

49. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007). [CrossRef]   [PubMed]  

50. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010). [CrossRef]  

51. R.-L. Chern and X.-X. Liu, “Effective parameters and quasi-static resonances for periodic arrays of dielectric spheres,” J. Opt. Soc. Am. B 27(3), 488–497 (2010). [CrossRef]  

52. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields (Springer, 1989).

53. M. Francoeur and M. P. Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008). [CrossRef]  

54. L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994). [CrossRef]  

55. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).

56. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s function and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009). [CrossRef]  

57. J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31(22), 3372–3374 (2006). [CrossRef]   [PubMed]  

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59. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2004).

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References

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  1. V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
    [Crossref]
  2. D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
    [Crossref] [PubMed]
  3. 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]
  4. L. Solymar and E. Shamonina, Waves in Metamaterials (Oxford University Press, 2009).
  5. W. Cai and V. Shalaev, Optical Metamaterials: Fundamentals and Applications (Springer, 2010).
  6. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
    [Crossref] [PubMed]
  7. D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
    [Crossref] [PubMed]
  8. N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
    [Crossref] [PubMed]
  9. V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
    [Crossref]
  10. J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
    [Crossref] [PubMed]
  11. W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
    [Crossref]
  12. J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
    [Crossref] [PubMed]
  13. C. J. Fu, Z. M. Zhang, and D. B. Tanner, “Planar heterogeneous structures for coherent emission of radiation,” Opt. Lett. 30(14), 1873–1875 (2005).
    [Crossref] [PubMed]
  14. M. Maksimović and Z. Jakšić, “Emittance and absorptance tailoring by negative refractive index metamaterial-based Cantor multilayers,” J. Opt. A-Pure Appl. Opt. 8, 355–362 (2006).
    [Crossref]
  15. F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007).
    [Crossref]
  16. B. J. Lee, L. P. Wang, and Z. M. Zhang, “Coherent thermal emission by excitation of magnetic polaritons between periodic strips and a metallic film,” Opt. Express 16(15), 11328–11336 (2008).
    [Crossref] [PubMed]
  17. C. J. Fu and Z. M. Zhang, “Further investigation of coherent thermal emission from single negative materials,” Nanosc. Microsc. Therm. 12(1), 83–97 (2008).
    [Crossref]
  18. Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
    [Crossref]
  19. C. J. Fu and Z. M. Zhang, “Thermal radiative properties of metamaterials and other nanostructured materials: A review,” Front. Energy Power Eng. China 3(1), 11–26 (2009).
    [Crossref]
  20. L.-G. Wang, G.-X. Li, and S.-Y. Zhu, “Thermal emission from layered structures containing a negative-zero-positive index metamaterial,” Phys. Rev. B 81(7), 073105 (2010).
    [Crossref]
  21. J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011).
    [Crossref]
  22. L. P. Wang and Z. M. Zhang, “Phonon-mediated magnetic polaritons in the infrared region,” Opt. Express 19(S2Suppl 2), A126–A135 (2011).
    [Crossref] [PubMed]
  23. K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81(16), 165119 (2010).
    [Crossref]
  24. Z. Zheng and Y. Xuan, “Theory of near-field radiative heat transfer for stratified magnetic media,” Int. J. Heat Mass Tran. 54(5-6), 1101–1110 (2011).
    [Crossref]
  25. Z. Zheng and Y. Xuan, “Near-field radiative heat transfer between general materials and metamaterials,” Chin. Sci. Bull. 56(22), 2312–2319 (2011).
    [Crossref]
  26. S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
    [Crossref]
  27. J. R. Howell, R. Siegel, and M. P. Mengüç, Thermal Radiation Heat Transfer (Fifth Edition, CRC Press, 2011), Chap. 16.
  28. J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002).
    [Crossref]
  29. S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer 132(2), 023302 (2010).
    [Crossref]
  30. K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
    [Crossref]
  31. S. A. Maier, Plasmonics: Fundamentals and Applications (Springer, 2007).
  32. Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
    [Crossref] [PubMed]
  33. R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
    [Crossref]
  34. M. D. Whale and E. G. Cravalho, “Modeling and performance of microscale thermophotovoltaic energy conversion devices,” IEEE Trans. Energ. Convers. 17(1), 130–142 (2002).
    [Crossref]
  35. M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
    [Crossref]
  36. K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
    [Crossref]
  37. M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energ. Convers. 26(2), 686–698 (2011).
    [Crossref]
  38. C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
    [Crossref] [PubMed]
  39. S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98(11), 113106 (2011).
    [Crossref]
  40. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
    [Crossref]
  41. M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
    [Crossref]
  42. Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
    [Crossref]
  43. M. S. Wheeler, A scattering-based approach to the design, analysis, and experimental verification of magnetic metamaterials made from dielectrics, PhD thesis, University of Toronto (2010).
  44. L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65–68 (1947).
  45. S. O’Brien and J. B. Pendry, “Photonics band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002).
    [Crossref]
  46. C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
    [Crossref]
  47. V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
    [Crossref] [PubMed]
  48. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73(4), 045105 (2006).
    [Crossref]
  49. J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007).
    [Crossref] [PubMed]
  50. M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
    [Crossref]
  51. R.-L. Chern and X.-X. Liu, “Effective parameters and quasi-static resonances for periodic arrays of dielectric spheres,” J. Opt. Soc. Am. B 27(3), 488–497 (2010).
    [Crossref]
  52. S. M. Rytov, Y. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics 3: Elements of Random Fields (Springer, 1989).
  53. M. Francoeur and M. P. Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
    [Crossref]
  54. L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
    [Crossref]
  55. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, 1995).
  56. M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s function and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
    [Crossref]
  57. J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31(22), 3372–3374 (2006).
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2011 (6)

J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011).
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L. P. Wang and Z. M. Zhang, “Phonon-mediated magnetic polaritons in the infrared region,” Opt. Express 19(S2Suppl 2), A126–A135 (2011).
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Z. Zheng and Y. Xuan, “Theory of near-field radiative heat transfer for stratified magnetic media,” Int. J. Heat Mass Tran. 54(5-6), 1101–1110 (2011).
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Z. Zheng and Y. Xuan, “Near-field radiative heat transfer between general materials and metamaterials,” Chin. Sci. Bull. 56(22), 2312–2319 (2011).
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M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energ. Convers. 26(2), 686–698 (2011).
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S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98(11), 113106 (2011).
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2010 (8)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
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R.-L. Chern and X.-X. Liu, “Effective parameters and quasi-static resonances for periodic arrays of dielectric spheres,” J. Opt. Soc. Am. B 27(3), 488–497 (2010).
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M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two silicon carbide films,” J. Phys. D Appl. Phys. 43(7), 075501 (2010).
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E. Rousseau, M. Laroche, and J.-J. Greffet, “Radiative heat transfer at nanoscale: Closed-form expression for silicon at different doping levels,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1005–1014 (2010).
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C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
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S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer 132(2), 023302 (2010).
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K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81(16), 165119 (2010).
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L.-G. Wang, G.-X. Li, and S.-Y. Zhu, “Thermal emission from layered structures containing a negative-zero-positive index metamaterial,” Phys. Rev. B 81(7), 073105 (2010).
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2009 (7)

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
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J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
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Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
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C. J. Fu and Z. M. Zhang, “Thermal radiative properties of metamaterials and other nanostructured materials: A review,” Front. Energy Power Eng. China 3(1), 11–26 (2009).
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M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s function and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
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M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
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Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
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2008 (4)

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
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M. Francoeur and M. P. Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
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B. J. Lee, L. P. Wang, and Z. M. Zhang, “Coherent thermal emission by excitation of magnetic polaritons between periodic strips and a metallic film,” Opt. Express 16(15), 11328–11336 (2008).
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C. J. Fu and Z. M. Zhang, “Further investigation of coherent thermal emission from single negative materials,” Nanosc. Microsc. Therm. 12(1), 83–97 (2008).
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2007 (4)

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
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V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
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F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007).
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J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007).
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2006 (6)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73(4), 045105 (2006).
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J. Skaar, “On resolving the refractive index and the wave vector,” Opt. Lett. 31(22), 3372–3374 (2006).
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Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
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M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
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J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
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M. Maksimović and Z. Jakšić, “Emittance and absorptance tailoring by negative refractive index metamaterial-based Cantor multilayers,” J. Opt. A-Pure Appl. Opt. 8, 355–362 (2006).
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2005 (5)

C. J. Fu, Z. M. Zhang, and D. B. Tanner, “Planar heterogeneous structures for coherent emission of radiation,” Opt. Lett. 30(14), 1873–1875 (2005).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
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V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
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M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
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2004 (1)

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
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2003 (2)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
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S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225(4-6), 233–240 (2003).
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2002 (3)

S. O’Brien and J. B. Pendry, “Photonics band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002).
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M. D. Whale and E. G. Cravalho, “Modeling and performance of microscale thermophotovoltaic energy conversion devices,” IEEE Trans. Energ. Convers. 17(1), 130–142 (2002).
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J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002).
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2001 (2)

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
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R. A. Shelby, D. R. Smith, and S. Schultz, “Experimental verification of a negative index of refraction,” Science 292(5514), 77–79 (2001).
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2000 (3)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85(18), 3966–3969 (2000).
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D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
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R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277(1), 61–64 (2000).
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1994 (1)

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
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1968 (1)

V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
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1947 (1)

L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65–68 (1947).

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
[Crossref]

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73(4), 045105 (2006).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

Albuquerque, E. L.

F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007).
[Crossref]

Avitzour, Y.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[Crossref]

Baker-Jarvis, J.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
[Crossref]

Bartal, G.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[Crossref] [PubMed]

Basu, S.

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98(11), 113106 (2011).
[Crossref]

S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer 132(2), 023302 (2010).
[Crossref]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
[Crossref]

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[Crossref]

Ben-Abdallah, P.

K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81(16), 165119 (2010).
[Crossref]

Brongersma, M. L.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007).
[Crossref] [PubMed]

Cai, W.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Carminati, R.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002).
[Crossref]

Chen, J. I. L.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
[Crossref]

Chen, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

Chern, R.-L.

Chettiar, U. K.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

Choy, H. K. H.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Cravalho, E. G.

M. D. Whale and E. G. Cravalho, “Modeling and performance of microscale thermophotovoltaic energy conversion devices,” IEEE Trans. Energ. Convers. 17(1), 130–142 (2002).
[Crossref]

Darmanyan, S. A.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225(4-6), 233–240 (2003).
[Crossref]

de Medeiros, F. F.

F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007).
[Crossref]

De Wilde, Y.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

DiMatteo, R. S.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Drevillon, J.

K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81(16), 165119 (2010).
[Crossref]

Fan, S.

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Fang, N.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Finberg, S. L.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Fonstad, C. G.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Formanek, F.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

Francoeur, M.

M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energ. Convers. 26(2), 686–698 (2011).
[Crossref]

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98(11), 113106 (2011).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two silicon carbide films,” J. Phys. D Appl. Phys. 43(7), 075501 (2010).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s function and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[Crossref]

M. Francoeur and M. P. Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
[Crossref]

Fu, C. J.

C. J. Fu and Z. M. Zhang, “Thermal radiative properties of metamaterials and other nanostructured materials: A review,” Front. Energy Power Eng. China 3(1), 11–26 (2009).
[Crossref]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
[Crossref]

C. J. Fu and Z. M. Zhang, “Further investigation of coherent thermal emission from single negative materials,” Nanosc. Microsc. Therm. 12(1), 83–97 (2008).
[Crossref]

C. J. Fu, Z. M. Zhang, and D. B. Tanner, “Planar heterogeneous structures for coherent emission of radiation,” Opt. Lett. 30(14), 1873–1875 (2005).
[Crossref] [PubMed]

Gralak, B.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

Greffet, J.-J.

E. Rousseau, M. Laroche, and J.-J. Greffet, “Radiative heat transfer at nanoscale: Closed-form expression for silicon at different doping levels,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1005–1014 (2010).
[Crossref]

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
[Crossref]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002).
[Crossref]

Greiff, P.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Holloway, C. L.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
[Crossref]

Jakšic, Z.

M. Maksimović and Z. Jakšić, “Emittance and absorptance tailoring by negative refractive index metamaterial-based Cantor multilayers,” J. Opt. A-Pure Appl. Opt. 8, 355–362 (2006).
[Crossref]

Joulain, K.

K. Joulain, J. Drevillon, and P. Ben-Abdallah, “Noncontact heat transfer between two metamaterials,” Phys. Rev. B 81(16), 165119 (2010).
[Crossref]

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002).
[Crossref]

Kabos, P.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
[Crossref]

Kildishev, A. V.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
[Crossref]

King, W. P.

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[Crossref]

Kooi, P.-S.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
[Crossref]

Kuester, E. F.

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
[Crossref]

Laroche, M.

E. Rousseau, M. Laroche, and J.-J. Greffet, “Radiative heat transfer at nanoscale: Closed-form expression for silicon at different doping levels,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1005–1014 (2010).
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M. Laroche, R. Carminati, and J.-J. Greffet, “Near-field thermophotovoltaic energy conversion,” J. Appl. Phys. 100(6), 063704 (2006).
[Crossref]

Lau, W. T.

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Lee, B. J.

S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer 132(2), 023302 (2010).
[Crossref]

B. J. Lee, L. P. Wang, and Z. M. Zhang, “Coherent thermal emission by excitation of magnetic polaritons between periodic strips and a metallic film,” Opt. Express 16(15), 11328–11336 (2008).
[Crossref] [PubMed]

Lee, H.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Lemoine, P.-A.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

Leong, M.-S.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
[Crossref]

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L. Lewin, “The electrical constants of a material loaded with spherical particles,” Proc. Inst. Electr. Eng. 94, 65–68 (1947).

Li, G.-X.

L.-G. Wang, G.-X. Li, and S.-Y. Zhu, “Thermal emission from layered structures containing a negative-zero-positive index metamaterial,” Phys. Rev. B 81(7), 073105 (2010).
[Crossref]

Li, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[Crossref] [PubMed]

Li, L.-W.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
[Crossref]

Lippens, D.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Liu, X.-X.

Maksimovic, M.

M. Maksimović and Z. Jakšić, “Emittance and absorptance tailoring by negative refractive index metamaterial-based Cantor multilayers,” J. Opt. A-Pure Appl. Opt. 8, 355–362 (2006).
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Marquier, F.

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
[Crossref]

Masaki, M. M.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Mason, J. A.

J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011).
[Crossref]

Mauriz, P. W.

F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007).
[Crossref]

Mengüç, M. P.

M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energ. Convers. 26(2), 686–698 (2011).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two silicon carbide films,” J. Phys. D Appl. Phys. 43(7), 075501 (2010).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s function and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[Crossref]

M. Francoeur and M. P. Mengüç, “Role of fluctuational electrodynamics in near-field radiative heat transfer,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 280–293 (2008).
[Crossref]

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
[Crossref]

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73(4), 045105 (2006).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

Moroz, A.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
[Crossref] [PubMed]

Mulet, J.-P.

Y. De Wilde, F. Formanek, R. Carminati, B. Gralak, P.-A. Lemoine, K. Joulain, J.-P. Mulet, Y. Chen, and J.-J. Greffet, “Thermal radiation scanning tunnelling microscopy,” Nature 444(7120), 740–743 (2006).
[Crossref] [PubMed]

K. Joulain, J.-P. Mulet, F. Marquier, R. Carminati, and J.-J. Greffet, “Surface electromagnetic waves thermally excited: Radiative heat transfer, coherence properties and Casimir forces revisited in the near field,” Surf. Sci. Rep. 57(3-4), 59–112 (2005).
[Crossref]

J.-P. Mulet, K. Joulain, R. Carminati, and J.-J. Greffet, “Enhanced radiative heat transfer at nanometric distances,” Microscale Thermophys. Eng. 6(3), 209–222 (2002).
[Crossref]

Nemat-Nasser, S. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
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Nevière, M.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225(4-6), 233–240 (2003).
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S. O’Brien and J. B. Pendry, “Photonics band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002).
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Otey, C. R.

C. R. Otey, W. T. Lau, and S. Fan, “Thermal rectification through vacuum,” Phys. Rev. Lett. 104(15), 154301 (2010).
[Crossref] [PubMed]

Ozin, G. A.

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
[Crossref]

Padilla, W. J.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Park, K.

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[Crossref]

Pendry, J. B.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
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D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

S. O’Brien and J. B. Pendry, “Photonics band-gap effects and magnetic activity in dielectric composites,” J. Phys. Condens. Matter 14(15), 4035–4044 (2002).
[Crossref]

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

Rousseau, E.

E. Rousseau, M. Laroche, and J.-J. Greffet, “Radiative heat transfer at nanoscale: Closed-form expression for silicon at different doping levels,” J. Quant. Spectrosc. Radiat. Transf. 111(7-8), 1005–1014 (2010).
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R. Ruppin, “Surface polaritons of a left-handed medium,” Phys. Lett. A 277(1), 61–64 (2000).
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Schuller, J. A.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007).
[Crossref] [PubMed]

Schultz, S.

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]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Schurig, D.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

Shalaev, V. M.

W. Cai, U. K. Chettiar, A. V. Kildishev, and V. M. Shalaev, “Optical cloaking with metamaterials,” Nat. Photonics 1(4), 224–227 (2007).
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V. M. Shalaev, “Optical negative-index metamaterials,” Nat. Photonics 1(1), 41–48 (2007).
[Crossref]

Shelby, R. A.

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]

Shvets, G.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[Crossref]

Skaar, J.

Smith, D. R.

J. B. Pendry, D. Schurig, and D. R. Smith, “Controlling electromagnetic fields,” Science 312(5781), 1780–1782 (2006).
[Crossref] [PubMed]

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

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]

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Smith, S.

J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011).
[Crossref]

Sun, C.

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Tanner, D. B.

Taubner, T.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007).
[Crossref] [PubMed]

Urzhumov, Y. A.

Y. Avitzour, Y. A. Urzhumov, and G. Shvets, “Wide-angle infrared absorber based on a negative-index plasmonic metamaterial,” Phys. Rev. B 79(4), 045131 (2009).
[Crossref]

Vaillon, R.

M. Francoeur, R. Vaillon, and M. P. Mengüç, “Thermal impacts on the performance of nanoscale-gap thermophotovoltaic power generators,” IEEE Trans. Energ. Convers. 26(2), 686–698 (2011).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Spectral tuning of near-field radiative heat flux between two silicon carbide films,” J. Phys. D Appl. Phys. 43(7), 075501 (2010).
[Crossref]

M. Francoeur, M. P. Mengüç, and R. Vaillon, “Solution of near-field thermal radiation in one-dimensional layered media using dyadic Green’s function and the scattering matrix method,” J. Quant. Spectrosc. Radiat. Transf. 110(18), 2002–2018 (2009).
[Crossref]

Valentine, J.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[Crossref] [PubMed]

Vasconcelos, M. S.

F. F. de Medeiros, E. L. Albuquerque, M. S. Vasconcelos, and P. W. Mauriz, “Thermal radiation in quasiperiodic photonic crystals with negative refractive index,” J. Phys. Condens. Matter 19(49), 496212 (2007).
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V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of ε and μ,” Sov. Phys. Usp. 10(4), 509–514 (1968).
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Vier, D. C.

D. R. Smith, W. J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz, “Composite medium with simultaneously negative permeability and permittivity,” Phys. Rev. Lett. 84(18), 4184–4187 (2000).
[Crossref] [PubMed]

Wang, L. P.

Wang, L.-G.

L.-G. Wang, G.-X. Li, and S.-Y. Zhu, “Thermal emission from layered structures containing a negative-zero-positive index metamaterial,” Phys. Rev. B 81(7), 073105 (2010).
[Crossref]

Wasserman, D.

J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011).
[Crossref]

Whale, M. D.

M. D. Whale and E. G. Cravalho, “Modeling and performance of microscale thermophotovoltaic energy conversion devices,” IEEE Trans. Energ. Convers. 17(1), 130–142 (2002).
[Crossref]

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coupled magnetic dipole resonances in sub-wavelength dielectric particle clusters,” J. Opt. Soc. Am. B 27(5), 1083–1091 (2010).
[Crossref]

M. S. Wheeler, J. S. Aitchison, J. I. L. Chen, G. A. Ozin, and M. Mojahedi, “Infrared magnetic response in a random silicon carbide micropowder,” Phys. Rev. B 79(7), 073103 (2009).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Coated nonmagnetic spheres with a negative index of refraction at infrared frequencies,” Phys. Rev. B 73(4), 045105 (2006).
[Crossref]

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B 72(19), 193103 (2005).
[Crossref]

Wiltshire, M. C. K.

D. R. Smith, J. B. Pendry, and M. C. K. Wiltshire, “Metamaterials and negative refractive index,” Science 305(5685), 788–792 (2004).
[Crossref] [PubMed]

Xuan, Y.

Z. Zheng and Y. Xuan, “Near-field radiative heat transfer between general materials and metamaterials,” Chin. Sci. Bull. 56(22), 2312–2319 (2011).
[Crossref]

Z. Zheng and Y. Xuan, “Theory of near-field radiative heat transfer for stratified magnetic media,” Int. J. Heat Mass Tran. 54(5-6), 1101–1110 (2011).
[Crossref]

Yannopapas, V.

V. Yannopapas and A. Moroz, “Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges,” J. Phys. Condens. Matter 17(25), 3717–3734 (2005).
[Crossref] [PubMed]

Yeo, T.-S.

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
[Crossref]

Young-Waithe, K. A.

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

Zakhidov, A. A.

S. A. Darmanyan, M. Nevière, and A. A. Zakhidov, “Surface modes at the interface of conventional and left-handed media,” Opt. Commun. 225(4-6), 233–240 (2003).
[Crossref]

Zentgraf, T.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[Crossref] [PubMed]

Zhang, F.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Zhang, X.

J. Valentine, J. Li, T. Zentgraf, G. Bartal, and X. Zhang, “An optical cloak made of dielectrics,” Nat. Mater. 8(7), 568–571 (2009).
[Crossref] [PubMed]

N. Fang, H. Lee, C. Sun, and X. Zhang, “Sub-diffraction-limited optical imaging with a silver superlens,” Science 308(5721), 534–537 (2005).
[Crossref] [PubMed]

Zhang, Z. M.

L. P. Wang and Z. M. Zhang, “Phonon-mediated magnetic polaritons in the infrared region,” Opt. Express 19(S2Suppl 2), A126–A135 (2011).
[Crossref] [PubMed]

S. Basu, B. J. Lee, and Z. M. Zhang, “Near-field radiation calculated with an improved dielectric function model for doped silicon,” J. Heat Transfer 132(2), 023302 (2010).
[Crossref]

S. Basu, Z. M. Zhang, and C. J. Fu, “Review of near-field thermal radiation and its application to energy conversion,” Int. J. Energy Res. 33(13), 1203–1232 (2009).
[Crossref]

C. J. Fu and Z. M. Zhang, “Thermal radiative properties of metamaterials and other nanostructured materials: A review,” Front. Energy Power Eng. China 3(1), 11–26 (2009).
[Crossref]

C. J. Fu and Z. M. Zhang, “Further investigation of coherent thermal emission from single negative materials,” Nanosc. Microsc. Therm. 12(1), 83–97 (2008).
[Crossref]

K. Park, S. Basu, W. P. King, and Z. M. Zhang, “Performance analysis of near-field thermophotovoltaic devices considering absorption distribution,” J. Quant. Spectrosc. Radiat. Transf. 109(2), 305–316 (2008).
[Crossref]

B. J. Lee, L. P. Wang, and Z. M. Zhang, “Coherent thermal emission by excitation of magnetic polaritons between periodic strips and a metallic film,” Opt. Express 16(15), 11328–11336 (2008).
[Crossref] [PubMed]

C. J. Fu, Z. M. Zhang, and D. B. Tanner, “Planar heterogeneous structures for coherent emission of radiation,” Opt. Lett. 30(14), 1873–1875 (2005).
[Crossref] [PubMed]

Zhao, Q.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Zheng, Z.

Z. Zheng and Y. Xuan, “Near-field radiative heat transfer between general materials and metamaterials,” Chin. Sci. Bull. 56(22), 2312–2319 (2011).
[Crossref]

Z. Zheng and Y. Xuan, “Theory of near-field radiative heat transfer for stratified magnetic media,” Int. J. Heat Mass Tran. 54(5-6), 1101–1110 (2011).
[Crossref]

Zhou, J.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today 12(12), 60–69 (2009).
[Crossref]

Zhu, S.-Y.

L.-G. Wang, G.-X. Li, and S.-Y. Zhu, “Thermal emission from layered structures containing a negative-zero-positive index metamaterial,” Phys. Rev. B 81(7), 073105 (2010).
[Crossref]

Zia, R.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett. 99(10), 107401 (2007).
[Crossref] [PubMed]

Appl. Phys. Lett. (3)

R. S. DiMatteo, P. Greiff, S. L. Finberg, K. A. Young-Waithe, H. K. H. Choy, M. M. Masaki, and C. G. Fonstad, “Enhanced photogeneration of carriers in a semiconductor via coupling across a nonisothermal nanoscale vacuum gap,” Appl. Phys. Lett. 79(12), 1894–1896 (2001).
[Crossref]

S. Basu and M. Francoeur, “Near-field radiative transfer based thermal rectification using doped silicon,” Appl. Phys. Lett. 98(11), 113106 (2011).
[Crossref]

J. A. Mason, S. Smith, and D. Wasserman, “Strong absorption and selective thermal emission from a midinfrared metamaterial,” Appl. Phys. Lett. 98(24), 241105 (2011).
[Crossref]

Chin. Sci. Bull. (1)

Z. Zheng and Y. Xuan, “Near-field radiative heat transfer between general materials and metamaterials,” Chin. Sci. Bull. 56(22), 2312–2319 (2011).
[Crossref]

Front. Energy Power Eng. China (1)

C. J. Fu and Z. M. Zhang, “Thermal radiative properties of metamaterials and other nanostructured materials: A review,” Front. Energy Power Eng. China 3(1), 11–26 (2009).
[Crossref]

IEEE T. Microw. Theory (1)

L.-W. Li, P.-S. Kooi, M.-S. Leong, and T.-S. Yeo, “Electromagnetic dyadic Green’s function in spherically multilayered media,” IEEE T. Microw. Theory 42(12), 2302–2310 (1994).
[Crossref]

IEEE Trans. Antenn. Propag. (1)

C. L. Holloway, E. F. Kuester, J. Baker-Jarvis, and P. Kabos, “A double negative (DNG) composite medium composed of magnetodielectric spherical particles embedded in a matrix,” IEEE Trans. Antenn. Propag. 51(10), 2596–2603 (2003).
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IEEE Trans. Energ. Convers. (2)

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Figures (5)

Fig. 1
Fig. 1

Schematic representation of the problem under consideration: two planar half-spaces maintained at temperatures T 0 and T 2 and separated by a vacuum gap of thickness d are exchanging thermal radiation.

Fig. 2
Fig. 2

Real part of the refractive index and sum of the phases of the relative permittivity and permeability of the metamaterial for rs = 1 μm, nh = 1.5 and f = 0.4.

Fig. 3
Fig. 3

Monochromatic radiative heat flux between two metamaterials made of SiC spheres (rs = 1 μm, nh = 1.5, f = 0.4) separated by a vacuum gap of thickness d. The near-field profiles are compared with far-field and blackbody predictions.

Fig. 4
Fig. 4

(a) Real part of the relative electric permittivity and magnetic permeability of the metamaterial for rs = 1 μm, nh = 1.5 and f = 0.4. (b) Comparison between the TM and TE evanescent components of the radiative heat flux for the case shown in Fig. 3 when d = 10 nm.

Fig. 5
Fig. 5

(a) Magnitude of the Mie coefficients a 1 and b 1 for SiC spheres of radii rs of 600 nm, 800 nm and 1 μm in a host medium with nh = 1.5. (b) Monochromatic radiative heat flux between two metamaterials made of SiC spheres (rs = 600 nm, 800 nm and 1 μm, nh = 1.5, f = 0.4) separated by a 10 nm thick vacuum gap. The profiles are compared with the predictions obtained between two SiC bulk materials.

Equations (25)

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E ( r , ω ) = i ω μ μ v V G ¯ ¯ E e ( r , r , ω ) J r , e ( r , ω ) d V V G ¯ ¯ E m ( r , r , ω ) J r , m ( r , ω ) d V
H ( r , ω ) = V G ¯ ¯ H e ( r , r , ω ) J r , e ( r , ω ) d V + i ω ε ε v V G ¯ ¯ H m ( r , r , ω ) J r , m ( r , ω ) d V
E i H j * = i ω μ μ v V d V V d V G i α E e ( r , r , ω ) G j β H e * ( r , r , ω ) J α r , e ( r , ω ) J β r , e * ( r , ω )                                       + i ω ε ε v V d V V d V G i α E m ( r , r , ω ) G j β H m * ( r , r , ω ) J α r , m ( r , ω ) J β r , m * ( r , ω )                                       + k v 2 ε μ V d V V d V G i α E e ( r , r , ω ) G j β H m * ( r , r , ω ) J α r , e ( r , ω ) J β r , m * ( r , ω )                                       V d V V d V G i α E m ( r , r , ω ) G j β H e * ( r , r , ω ) J α r , m ( r , ω ) J β r , e * ( r , ω )
J α r , e ( r , ω ) J β r , e * ( r , ω ) = ω ε v ε π Θ ( ω , T ) δ ( r r ) δ α β
J α r , m ( r , ω ) J β r , m * ( r , ω ) = ω μ v μ π Θ ( ω , T ) δ ( r r ) δ α β
J α r , e ( r , ω ) J β r , m * ( r , ω ) = J α r , m ( r , ω ) J β r , e * ( r , ω ) = 0
q ω ( r ) = S z ( r , ω ) = 2 k v 2 Θ ( ω , T ) π Re { i μ ε V d V [ G x α E e ( r , r , ω ) G y α H e * ( r , r , ω ) G y α E e ( r , r , ω ) G x α H e * ( r , r , ω ) ] + i ε μ V d V [ G x α E m ( r , r , ω ) G y α H m * ( r , r , ω ) G y α E m ( r , r , ω ) G x α H m * ( r , r , ω ) ] }
q ω , 02 = Θ ( ω , T 0 ) Θ ( ω , T 2 ) π 2                                                           × γ = T E , T M [ 0 k v k ρ d k ρ ( 1 | r 10 γ | 2 ) ( 1 | r 12 γ | 2 ) 4 | 1 r 10 γ r 12 γ e 2 i k z 1 d | 2 + k v k ρ d k ρ e 2 k z 1 d Im ( r 10 γ ) Im ( r 12 γ ) | 1 r 10 γ r 12 γ e 2 k z 1 d | 2 ]
a l = m ˜ ψ l ( m ˜ X ) ψ l ( X ) ψ l ( X ) ψ l ( m ˜ X ) m ˜ ψ l ( m ˜ X ) ξ l ( X ) ξ l ( X ) ψ l ( m ˜ X )
b l = ψ l ( m ˜ X ) ψ l ( X ) m ˜ ψ l ( X ) ψ l ( m ˜ X ) ψ l ( m ˜ X ) ξ l ( X ) m ˜ ξ l ( X ) ψ l ( m ˜ X )
ε e f f ε h = 1 + N [ k h 3 i 6 π ( 1 a 1 1 ) N 3 ] 1
μ e f f μ h = 1 + N [ k h 3 i 6 π ( 1 b 1 1 ) N 3 ] 1
b 1 2 i X 3 3 F ( m ˜ X ) 1 F ( m ˜ X ) + 2
F ( m ˜ X ) = 2 [ sin ( m ˜ X ) m ˜ X cos ( m ˜ X ) ] [ ( m ˜ X ) 2 1 ] sin ( m ˜ X ) + m ˜ X cos ( m ˜ X )
λ h , r e s m 2 r s m ˜
ε s = ε ( ω 2 - ω L O 2 + i Γ ω ω 2 - ω T O 2 + i Γ ω )
γ 0 μ 0 + γ 1 μ 1 = 0
k ρ 2 = k v 2 μ 0 μ 1 ( ε 1 μ 0 ε 0 μ 1 ) μ 0 2 μ 1 2
γ j 2 = k v 2 μ j 2 ε 1 μ 1 ε 0 μ 0 μ 0 2 μ 1 2
γ j 2 = k v 2 μ j 2 1 + ε 0 | μ 0 | μ 0 2 1
| μ 0 | > 1 a n d ε 0 > 1 | μ 0 |
| μ 0 | < 1 a n d ε 0 < 1 | μ 0 |
ω r e s m ω L O 2 + Ω 2 ( ω L O 2 + Ω 2 ) 2 4 Ω 2 ω T O 2 2
ω r e s e 2 ε h ω T O 2 + ε ω L O 2 ε + 2 ε h
ω r e s e ε ω L O 2 ε h ω T O 2 ( f 4 f + 2 ) ε ε h ( f 4 f + 2 )

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