Overcoming black body radiation limit in free space: metamaterial superemitter

Here, we demonstrate that the power spectral density of thermal radiation at a specific wavelength produced by a body of finite dimensions set up in free space under a fixed temperature could be made theoretically arbitrary high, if one could realize double negative metamaterials with arbitrary small loss and arbitrary high absolute values of permittivity and permeability (at a given frequency). This result refutes the widespread belief that Planck's law itself sets a hard upper limit on the spectral density of power emitted by a finite macroscopic body whose size is much greater that the wavelength. Here we propose a physical realization of a metamaterial emitter whose spectral emissivity can be greater than that of the ideal black body under the same conditions. Due to the reciprocity between the heat emission and absorption processes such cooled down superemitter also acts as an optimal sink for the thermal radiation --- the"thermal black hole"--- which outperforms Kirchhoff-Planck's black body which can absorb only the rays directly incident on its surface. The results may open a possibility to realize narrowband super-Planckian thermal radiators and absorbers for future thermo-photovoltaic systems and other devices.


I. INTRODUCTION
The ability of a hot body to emit thermal electromagnetic radiation is related to its ability to absorb incident electromagnetic waves at the same frequencies. G. Kirchhoff in 1860 introduced the theoretical concept of an ideal black body, which "completely absorbs all incident rays" [1]. This concept was later adopted by M. Planck [2]. It appears that since that time there has been a general belief that no macroscopic body can emit more thermal radiation than the corresponding same-shape and size ideal black body at the same temperature. For example, in a recent paper [3] one reads, "any actual macroscopic thermal body cannot emit more thermal radiation than a blackbody." Equivalent statements can be found in commonly used text books, for example, in the well-known book by Bohren and Huffman [4] it is stated that "... the emissivity of a sufficiently large sphere is not greater than 1. Thus, if the radiating sphere radius is much larger than the wavelength, the radiation above the black body limit is impossible." On the other hand, recently there has been increasing number of publications discussing so-called super-Planckian thermal radiation, when the power emitted by a hot body per unit area per unit wavelength exceeds the one predicted by Planck's black body law. In a great deal of such works, the thermal emission into the electromagnetic near-field is considered, when the bodies that exchange radiative heat are separated by a distance significantly smaller than the wavelength λ (on the order of λ/10 or less). Such emission can easily overcome the black body limit, because oscillators in bodies separated by subwavelength gaps interact through the near (i.e., Coulomb) electric field, and, when close enough to the emitting object, such a field is much stronger than the wave field.
Besides the near field transfer, there are also works -quite surprising for an unprepared reader -which report super-Planckian emission in far-field. In such scenarios, a body emits into free space. The crucial parameter in this case is emitter's size as compared to the radiation wavelength. Actually, the fact that an optically small body can emit more than a black body of a similar size is not surprising, because a small particle may also absorb much more power than one would expect from its size. Indeed, a particle with radius a ≪ λ may have the absorption cross section σ abs much larger than its geometric cross section A g = πa 2 . For a dielectric or plasmonic sphere, this is understood as a consequence of Mie's (or respective plasmonic) resonances, at which the absorption cross section σ abs may outnumber πa 2 by a large factor. For instance, for a single-mode dipole particle the ultimate absorption cross section equals σ abs max = (3/8π)λ 2 (e.g. Ref. [5]), which is much larger than πa 2 if a ≪ λ. Other known superscatterer realizations are as well subwavelength [6].
Therefore, the particularly interesting case -which we study in this paper -is when emitting body has characteristic dimensions much greater than the radiation wavelength. Note that formulating the scattering problem on a finite size body in terms of a multipole expansion allows one to derive certain bounds [6][7][8] on the scattering and absorption cross-sections of such bodies related to contributions of multipoles with different orders. Although already from these results it can be inferred that the scattering cross section can be arbitrary high when an infinite number of multipoles is taken into account, it is commonly believed (especially in the context of radiative heat transfer problems) that the absorption cross section of an optically large body may not exceed its geometric cross section: σ abs ≤ A g .
The far-field thermal radiation from optically large bodies can be enhanced by covering them with transparent refractive media. In particular, it has been experimentally demonstrated [3] that mid-infrared thermal emission from a blackened disk can be significantly enhanced by covering it with a large hemispherical dome made of a transparent material with refractive index n > 1. When compared to the same disk in free space, the combined structure emits to the far zone a significantly greater total power. Because this power also exceeds the one predicted for a black body of the same dimensions as the emitting disk, the obtained emission is claimed to be super-Planckian [3].
However, such radiation enhancement happens just at the emitting surface which is next to a medium with n > 1. Indeed, in the structure of Ref. [3] the diameter of the disk as is seen through the dome is larger (about n times larger when the radius of the dome is much greater than disk's radius), which results in about n 2 -fold increase in disk's effective absorption area. The same factor appears in Planck's emission formula when the speed of light in vacuum c is replaced by c/n. However, note that independently of how high is n, a magnifying dome may not increase the apparent emitter size beyond the size of the dome itself! This means that the emitted power in this system never exceeds the power radiated by a black body with the radius equal to the outer radius of the dome. Thus, the thermal radiation flux produced by this system at the output of the dome is sub-Planckian. Recently proposed thermal emission enhancement with an impedance matched hyperlens [9] has similar limitations. Furthermore, thermal radiation from an unbounded planar interface with a generic photonic crystal has been numerically studied in Ref. [10], and the results show that the power radiated from an infinite planar surface does not go over the black body limit.
It is therefore questionable if super-Planckian emission into free space environment is possible at all. For instance, in Ref. [11] it is argued that such emission would violate the second law of thermodynamics, but is this indeed the case?
In order to avoid confusion we must first agree on the terminology. As is seen from the above discussion, there is currently no consensus in literature on the meaning of the far field super-Planckian radiation in free space. In this paper, we use this terminology exclusively for bodies of finite dimensions. When such an object acts as a source of thermal radiation, its spectral radiance b λ , i.e., the amount of power d 2 P radiated per wavelength interval dλ, projected emitting area A ⊥ , and solid angle dΩ: is sub-Planckian or super-Planckian depending on the choice of the area A ⊥ . If by definition A ⊥ is chosen to coincide with emitter's geometric projected area: A ⊥ = A g , then one has to admit that the spectral radiance of bodies characterized with σ abs > A g is super-Planckian. In this definition, one can obtain super-Planckian radiation from optically small resonant bodies or from shape irregularities with curvature radius a ≪ λ on a surface of a large body [12]. Another example of importance of the choice of A ⊥ is, actually, the system of Ref. [3]. For this system, if A ⊥ equals disk's area, Eq. (1) results in a super-Planckian spectral radiance value, but if A ⊥ is chosen to be the projected area of the dome, b λ is sub-Planckian. For subwavelength objects -which for obvious reasons do not have a well-defined emitting surface in the geometric optics sense, -equating A ⊥ to the geometric area A g has little physical meaning. For the same reason, the spectral radiance is ill defined for such objects. It was suggested to avoid the use of b λ for such objects, and instead use the wavelength-dependent σ abs in radiative heat transfer calculations [13].
However, it appears perfectly meaningful to compare the radiative heat flux at the real, physical surface of an optically large body with the one produced by Kirchhoff-Planck's black body of the same geometric dimensions. That is, for such bodies it is physically reasonable to select A ⊥ = A g in Eq. (1). In this case it is immediately understood that the thermal radiation flux that leaves the body surface is super-Planckian whenever the effective absorption cross section of the body is greater than its geometric cross section. We have seen that this is possible for subwavelength particles, but can this happen for optically large bodies?
Note that earlier studies establishing the widely accepted limitations are based on an assumption that the ideal Kirchhoff-Planck black body is the ultimately effective absorber. However, such a body perfectly absorbs only the rays which are falling on its surface [1]. In this regard we can recall the known result from the electromagnetic theory which states that there is no upper limit on the effective area of an antenna, even when the physical dimensions of the antenna are constrained [14,15]. In particular, this means that a finite-size antenna loaded with a conjugate-matched load Z load = Z * ant , where Z ant is the complex input impedance of the antenna, in principle, can absorb all the power carried by a plane wave (of infinite extent in space) incident from the direction of antenna's main beam. Although this observation indicates a possibility for existence of macroscopic super-Planckian thermal emitters in the above definition, in the present paper we do not stop on this possibility having in mind a more ambitious goal -to show that such omnidirectional emitters are possible and that their existence does not violate fundamental laws of physics.

II. SUBJECT OF OUR PAPER AND ITS CONTENT
In this paper, unlike previous works on related subjects, we consider a theoretical possibility to obtain free-space omnidirectional super-Planckian radiation from a macroscopic body with dimensions a ≫ λ without adding any shells or other elements that increase its physical size. This implies an important question -if there can exist optically large isotropic emitters with effective spectral emissivity greater than unity, when compared to Kirchhoff-Planck's black body of the same size.
In order to answer this question and the related fundamental questions formulated in Introduction, we go over the usual assumption that an emitting and absorbing body is composed of homogeneous materials with positive permittivities and permeabilities at the wavelength of interest. We show that if these restrictions are removed, there are no compelling reasons why a specially crafted metamaterial [16] object cannot provide σ abs > A g at a given wavelength, even when object's diameter is significantly greater than the wavelength.
We show that even for optically large spherically isotropic bodies it is physically allowed to have σ abs ≫ A g at a given wavelength. At this wavelength, the thermal radiation produced by such bodies is characterized with spectral flux density which is above the black body limit within the range of radial distances r such that λ ≪ a ≤ r ≤ r eff , where a is the physical radius of the body and r eff = √ σ abs /π. We show that the super-Planckian part of the flux is transferred in this region by resonant tunneling of photons associated with high-order, highly reactive spatial harmonics (essentially, dark modes) of emitter's fluctuating field (see Sec. V for details). At distances greater than r eff the radiation flux is sub-Planckian in the sense that its spectral density does not exceed the one produced by Kirchhoff-Planck's black body with radius r = r eff . Nevertheless, note that the effective spectral emissivity of such bodies when compared to Kirchhoff-Planck's black body with the same physical radius is greater than unity. This is physically possible because effective absorptivity of such bodies is as well greater than unity, which just means that such finite-size bodies receive per unit area of their surface more spectral power than a Kirchhoff-Planck's black body of the same radius under the same conditions (see Sec. VII).
Moreover, in this paper we prove that, for a body fitting a given sphere σ abs can be made theoretically arbitrary high at any given frequency, independently of the physical size of the body. Such "thermal black holes" absorb much more power than is incident on their surfaces: In the theoretical limit, they absorb the whole infinite power carried by a plane electromagnetic wave. Furthermore, we prove that existence of such exotic objects contradicts neither the second law of thermodynamics, nor Kirchhoff's law of thermal radiation when the latter is properly amended.
The resonant photon tunneling effect mentioned above is rather narrowband due to highly reactive nature of the electromagnetic field associated with high-order spherical harmonics. In this paper we consider this effect from a purely theoretical perspective aiming mostly at demonstrating that free-space thermal radiation flux characterized with super-Planckian spectral density is in principle physically realizable even at spatial scales much greater than the radiation wavelength.
Regarding practical aspects, let us note that narrowband thermal radiation is the key prerequisite for advanced thermo-photovoltaic systems (TPVS). For instance, reducing relative bandwidth to less than 10% practically eliminates the Shottky-Queisser limit related to the dissipation of the excessive photon energy in semiconductors [17]. The nearly monochromatic thermal radiation is the primary target for solar TPVS, where the narrowband thermal emitters already allow the energy conversion efficiency to approach the thermodynamic limit [18]. Note that in all known works on solar TPVS the spectral maximum of this narrow-band thermal radiation is below the conventional Planckian spectral value [19].
The paper is organized as follows. In Sec. III we outline the equivalent circuit model [20] that we use in radiative thermal flux calculations. It has been proven [20] that this approach is fully equivalent in its predictive power to the more common theories operating with distributed thermal-fluctuating currents. Using this model, in Sec. IV we consider general conditions which maximize the radiative heat flux between a hot body and its environment.
In Sec. V we study the thermal radiation produced by finite-size bodies in free space and introduce the concept of the ideal conjugate-matched emitter. Such a truly super-Planckian emitter is able to radiate efficiently to the entire infinite set of free-space photonic states, infinitely outperforming a black body emitter of the same dimensions.
In Sec. VI we consider plane wave scattering on a finite-size body and prove that its scattering, absorption, and extinction cross sections tend to infinity under the perfect conjugate matching condition of Sec. V, independently of the size of the body.
In Sec. VII we show that the second law of thermodynamics is not violated by finite-size emitters with effective spectral emissivity greater than unity. We also propose an amendment to Kirchhoff's law of thermal radiation in order to incorporate such emitters into the existing theory.
In Sec. VIII we search for a physical realization for the conjugate-matched emitter. A possible realizationwhich we call metamaterial thermal black hole -is obtained in the form of a core-shell double-negative (DNG) metamaterial structure.
In Sec. IX we consider a couple of such structures with realistic material parameters and estimate their super-Planckian performance. Finally, in Sec. X we draw some conclusions.

III. ELECTROMAGNETIC THEORY OF THERMAL RADIATION: CIRCUIT MODEL APPROACH
The approach of Ref. [20] allows one to reduce a full-wave thermal emission problem to a set of circuit theory problems operating with effective fluctuating voltages and currents instead of the electromagnetic fields. This approach is based on expanding the emitted field at a given frequency into a suitable series of linearly independent, orthogonal spatial harmonics, and characterizing each of these harmonics with the equivalent circuit model parameters, such as complex wave impedance, voltage and current. The electromagnetic interaction of a hot emitter with the surrounding space can be expressed in this language at each of the mentioned harmonics with an equivalent circuit shown in Fig. 1(a). In this circuit, Z 1 (ν) represents the equivalent complex impedance of emitter's body for a given spatial harmonic of the radiated field, at the frequency ν = c/λ. Respectively, Z 2 (ν) is the equivalent complex impedance of the surrounding space for the same mode, which, in case of the free space, is simply the wave impedance of the corresponding mode: Z 2 ≡ Z w . The effect of thermal fluctuations in this circuit is taken into account by a pair of fluctuating electromotive forces (EMF) e 1 (ν) and e 2 (ν).
FIG. 1: (a) Equivalent circuit of radiative heat transfer between an emitter (represented by the complex impedance Z 1 ) and its environment (represented by the complex impedance Z 2 ). (b) Equivalent circuit for the particular case of an infinitely large hot body occupying the halfspace z < 0 and radiating into the cold free space domain z > 0. In this geometry, the impedance Z 2 = Z w is either purely real (for propagating plane waves) or purely imaginary (for evanescent waves).
For example, in a geometry where a body occupying halfspace z < 0 emits into empty halfspace z > 0 [ Fig. 1(b)] one may conveniently expand the radiated field over the set of free-space plane waves (both propagating and evanescent) with arbitrary transverse wave vectors k t = (k x , k y ). Such modes split into transverse electric (TE) (or s-polarized) waves and transverse magnetic (TM) (or p-polarized) waves. The wave impedances of these modes Z TE,TM where k 0 = 2πν √ ε 0 µ 0 is the free-space wavenumber, and η 0 = µ 0 /ε 0 is the free-space impedance. Respectively, in the equivalent circuit of Fig. 1 The impedance Z 1 in this case coincides with the input impedance of the halfspace z < 0 for a given plane wave incident from the halfspace z > 0. This impedance can be expressed through the corresponding complex reflection coefficients Γ TE,TM as By applying the fluctuation-dissipation theorem [21] (FDT) to the circuit of Fig. 1(a) one finds the mean-square spectral density of the fluctuating EMF as follows (in this article we use rms complex amplitudes x ν for the timeharmonic quantities x(t) defined by where Θ(ν, T j ) = hν[exp(hν/k B T j ) − 1] −1 is Planck's mean oscillator energy (here, j = 1, 2), k B is Boltzmann's constant, and T j is the absolute temperature of the emitter (when j = 1) or the surrounding space (when j = 2). Actually, Eq. (4) is nothing more than Nyquist's formula for the thermal noise in electric circuits [22] where electrical engineers usually approximate Θ(ν, T j ) ≈ k B T j . Let us note that relation (4) implies that the bodies that exchange radiative heat are kept in thermodynamically equilibrium states, which, strictly speaking, is possible only either when |T 1 − T 2 | ≪ T 1,2 or under the assumption that the internal thermal energy stored in the bodies is infinite. The thermal radiation power within a narrow range of frequencies ν ± dν/2 delivered from the side of the emitter, Z 1 , to the side of the environment, Z 2 , is expressed in our formulation (per each spatial harmonic) simply as In Ref. [20] it is proven that such a circuit model approach based on modal decomposition of the thermal fluctuating field is fully equivalent to the more complicated theories operating with distributed fluctuating currents. However, in contrast to these classical methods, our approach allows us to reduce a heat transfer maximization problem to the well-known circuit theory problem of matching a generator with its load.

IV. MAXIMIZATION OF EMITTED POWER: COMPLEX-CONJUGATE MATCHING VERSUS USUAL IMPEDANCE MATCHING
Having at hand an equivalent circuit representation described above, we may now ask ourselves under which conditions the spectral density of power radiated by a hot body is maximized? Due to the orthogonality property of the spatial harmonics used in the field expansion, in order to maximize the total emission we need to maximize the power delivered by each harmonic separately. As is clearly seen from Eq. (5), for the modes with a non-vanishing real part of the wave impedance: Re(Z 2 ) > 0, the delivered power is maximized under the complex-conjugate matching condition: Z * 1 = Z 2 . Under this condition, the maximal possible emitted power per a spatial harmonic per unit of frequency is, from Eq. (5), Note that for the spatial harmonics characterized with complex wave impedance, the conjugate matching condition is, in general, different from the zero reflection condition Γ = (Z 1 − Z 2 )/(Z 1 + Z 2 ) = 0 in the equivalent circuit of Fig. 1(a), which implies the usual impedance matching Z 1 = Z 2 . Thus, by minimizing reflections for the waves crossing the boundary between the emitter and the surrounding space, one does not necessarily maximize the emission! Indeed, the power spectral density under the usual impedance matching condition Z 1 = Z 2 (in what follows we call it simply "impedance matching") attains [Eq. (5)] Recall that Z 2 is related to the wave impedance of the surrounding space. Therefore, as soon as this environment is characterized with complex impedance (for example, when the surrounding space is filled by a dielectric with loss), an impedance-matched, non-reflecting body -that is the black body in its conventional and intuitive definitionwill not anymore be the one that attains the maximal emissivity. Let us also mention one important case when the impedance matching condition is sufficient to maximize the power emitted from a body to free space. It is the case when the emitting body is so large that it can be modeled with the geometry of Fig. 1(b). As was mentioned in Sec. III, in this case the basis of orthogonal spatial harmonics is composed of propagating and evanescent plane waves. The wave impedances of these modes are given by Eq. (2). The evanescent plane waves with q > k 0 have Re Z TE,TM w = 0 and, thus, in accordance with Eq. (7), do not contribute into the far-field emission at all. The propagating modes with q < k 0 have Re Z TE,TM w > 0 and Im Z TE,TM w = 0. Because the wave impedance of these modes is purely real, the maximum emission condition Z * 1 = Z 2 for these waves coincides with the condition Z 1 = Z 2 . Therefore, the optimal emitter in this case is the half-space with zero reflection: Γ = 0, i.e., the black half-space. This essentially forbids any far-field super-Planckian emission in such geometries. However, it does not follow from here that the same conclusion must hold in geometries involving objects of finite size.

V. FREE SPACE FAR-FIELD THERMAL EMISSION FROM BODIES OF FINITE SIZE
Let us now focus on geometries involving optically large spherical emitters in free space, or, more generally, any finite size emitters which completely fit into a sphere of a fixed radius r = a ≫ λ. An analogous treatment can be applied to cylindrical emitters.
It is well-known that the electromagnetic field produced by the sources that are fully contained within a finite volume can be expanded (in the space outside this volume) over the complete set of vectorial spherical waves, defined with respect to a spherical coordinate system (r, θ, ϕ) whose origin lies within this volume. These modes split into TE waves (with E r = 0) and TM waves (with H r = 0), with the field vectors expressed through a pair of l (x) being the spherical Hankel function of the first kind and order l. The function R l (x) is also known as the Riccati-Hankel function of the first kind. The transverse electric field E t = E θθ + E ϕφ and the transverse magnetic is the wave impedance of the spherical wave harmonic with the polar index l (l = 1, 2, . . . ) and the azimuthal index m (m = 0, ±1, ±2, . . . ± l), which can be expressed as Note that the wave impedance of a mode depends on the radial distance r and the polar index l, and it is independent of the azimuthal index m. We exclude the purely longitudinal mode with l = m = 0 because it does not contribute into the radiated power. Wave impedances of spatial harmonics (8) correspond to spherical waves emitted from an object comprising the point r = 0. For incoming spherical waves (see, e.g., Ref. [23]), the wave impedances are expressed through the Riccati-Hankel functions of the second kindR l (x) = xh (2) l (x), x = k 0 r, and are equal to the complex conjugate of the impedances given by Eq. (8). Such waves are also called anti-causal waves, because they cannot be created just by remote external sources: A presence of a scatterer (which is sometimes called "sink", as opposed to "source" [23]) in the vicinity of point r = 0 is necessary for them to appear. Nevertheless, it is convenient to use such waves to describe the heat transfer from the remote environment to the body surface (we will use such waves in Sec. VI and VII).
The striking difference in the properties of the spherical wave harmonics as compared to the plane wave harmonics discussed in the last paragraph of Sec. IV, is that the wave impedance (8) has a non-vanishing real part Re Z TE,TM w,lm > 0 for the harmonics with arbitrary high indices l and m. Therefore, there are no fully evanescent waves among the spherical wave harmonics: Each mode, whatever high index it has, contributes into the far field. Hence, based on the results of Sec. IV, we may conclude that at any given wavelength there is a possibility to satisfy conjugate matching condition for the entire spectrum of spherical waves that are emitted by a body with a finite radius. In this case, a special emitter must be crafted which will provide the input impedance Z 1 = Z TE,TM w,lm * for all the modes with arbitrary indices l and m. We postpone the discussion of this realization until Sec. VIII.
Under such perfect conjugate matching condition, the total power (per unit of frequency) emitted by the body at a given wavelength (with both TE and TM polarizations taken into account) satisfy i.e., it grows infinitely. Thus, at least from a purely theoretical point of view, there is no upper limit on the power spectral density of the far-zone thermal radiation produced by a body of a constrained radius at a given wavelength. Note that in this consideration we did not make any assumptions regarding the radius-to-wavelength ratio or the internal structure of the body. In order to understand this result, let us compare the perfect conjugate-matched case of Eq. (9) with the case when the emitter is simply impedance matched, which is expressed in our equivalent circuit model by the condition w,lm . Under this condition, the power spectral density per a single spherical harmonic can be expressed from Eq. (7), which leads to where the index p = TE, TM labels the polarization. The factor F lm = Re Z TE,TM w,lm 2 Z TE,TM w,lm 2 on the right-hand side of Eq. (10) is close to unity for l N max = 2πa/λ and decreases to zero very rapidly when l > N max . This is illustrated in Fig. 2. The reason for this is that in the spherical waves with l > N max the electromagnetic energy in the vicinity of emitter's surface is mostly concentrated in the near fields (reactive fields) which decay faster than 1/r with distance and, thus, do not contribute into the far field. Respectively, the wave impedance of these waves , which results in the emissivity cut-off at about l ≈ N max . The same cut-off can be explained also by the fact that on the surface r = a, a spherical wave harmonic with an index l ≫ 1 forms a wave pattern with the characteristic spatial period t ≈ 2πa/l. Therefore, when l > 2πa/λ, this period is less than the wavelength so that the mentioned mode behaves at the surface r = a similarly to an evanescent plane wave. Note that such a cut-off is not present under the conjugate matching condition Z 1 = Z * 2 , because in this case the reactive components in Z 1 and Z 2 have opposite signs and compensate one another, i.e., the conjugate matching condition is essentially a resonant condition in the equivalent circuit of Fig. 1(a). Therefore, when dealing with an impedance matched emitter, we may approximate F lm by unity when l ≤ N max , and by zero when l > N max when N max = 2πa/λ ≫ 1. This excludes the modes with l > N max from summation (10), and we obtain the following closed-form expression for the power spectral density: From this formula, by substituting N max = 2πa/λ and taking into account that N max ≫ 1, we get Recognizing 4πa 2 as the area of the spherical surface, we obtain from Eq. (12) which coincides with the amount of power (per unit frequency and unit area) emitted by a black body sphere kept under temperature T = T 1 : Thus, an optically large impedance matched body is equivalent in its emissive properties to Planck's black body. This is expected, because such a body is typically understood as made of a black, non-reflecting material with impedance matched to the free-space impedance at arbitrary angles of incidence. Black bodies of finite size modeled as apertures in walls of large opaque cavities also behave similarly, due to the full match between the domains inside and outside the cavity.
It is instructive to relate the number of independent spherical harmonics into which a hot body can emit with the number of photonic states in free space. Let a spherical body with radius a ≫ λ be situated in vacuum, and consider a free space gap with thickness h (an empty spherical layer) adjacent to it such that λ ≪ h ≪ a. Then, the number of photonic states within this gap which transfer energy away from the body is where D (3) ν = 8πν 2 /c 3 is the photonic density of states in vacuum. The same quantity can be also expressed by counting independent spherical waves within the same gap: where N max ≫ 1 is the maximal spherical harmonic index up to which the body emits efficiently, and D (1) ν = 4/c is the one-dimensional photonic density of states.
By comparing Eqs. (14) and (15) we find that N max ≈ 2πa/λ, i.e., the same limit as for a black body. However, note that, by definition, the photonic density of states in vacuum accounts only for the states which correspond to propagating waves, i.e., to real photons. Evanescent waves -which correspond to virtual, tunneling photonsare not accounted in such description. Respectively, emission above the black body limit is possible only by such tunneling. For instance, radiating in a mode with index l = κN max , where κ > 1, must involve photon tunneling from the body surface at r = a to the distant surface r = κa, where there is an available photonic state for it.
Thus, the conjugate-matched body can emit significantly higher power per unit of area and per unit of frequency as compared to Planck's black body of the same size only because the conjugate matching condition tunes the emitter at resonance with high-order modes, due to which these modes are excited with a very high amplitude. Although these modes are essentially dark modes (because they are very weakly coupled to free space), the resonance greatly increases probability of photon tunneling from one of such states at emitter's surface to a propagating free space state at some large enough radial distance. Obviously, such a resonant photon tunneling effect is not possible with a body made of a simple absorbing material.
Finally, let us note that the conjugate matching condition at an emitting spherical surface mathematically coincides with the zero-reflectance condition for the anti-causal spherical waves incident on the same surface (see Sec. VI for more detail). In such picture, the conjugate-matched emitter stands out as a perfect sink for these waves. The energy transferred by such waves is totally absorbed at emitter's surface without any reflections. Thus, for finite size emitters, one might try to amend the definition of the ideal black body in such a manner that it would refer to the conjugate-matched emitter rather than to the impedance matched one. One, however, would have to accept in this case that such a redefined ideal black body is characterized by infinite effective absorption cross section independently of its real, physical size. This point is discussed with more detail in the next section.

VI. SCATTERING, ABSORPTION, AND EXTINCTION CROSS SECTIONS OF FINITE SIZE BODIES UNDER CONJUGATE MATCHING CONDITION
The scattering, absorption, and extinction cross sections at a given frequency are, by definition, where Π inc = η −1 0 |E inc | 2 is the power flow density in an incident plane wave with the given frequency, and P sc , P abs , and P ext = P sc + P abs are, respectively, the amounts of power scattered by the body, absorbed within it, and extracted by it from the incident field. Without any loss in generality, we may assume that the incident wave is propagating along the z-axis, and is linearly polarized: E inc =xE inc exp(ik 0 z).
As is well known, an incident plane wave can be expanded into vectorial spherical waves. The notations for such waves vary a lot in literature, but when reduced down to Riccati-Bessel functions and derivatives of the Laplace spherical harmonics the expansion can be written as follows (only the electric field component transverse tor = r/|r| is of our interest): were S l (x) = x j l (x) is the Riccati-Bessel function of the first kind with j l (x) being the spherical Bessel function of the first kind and order l, and ∇ t = ∇ −r(∂/∂r). In the inner summation over m the index acquires just two values: m = −1 and m = 1. The first term in the square brackets of Eq. (17) proportional to S l (k 0 r) is due to the TE-polarized spherical waves and the second one proportional to the derivative of the same function is due to the TM-polarized part of the spectrum. These two contributions are mutually orthogonal. Note that the Riccati-Bessel functions in expansion (17) are simple superpositions of the Riccati-Hankel functions R l (x) andR l (x) (of course, the same refers to their derivatives): Therefore, in the plane wave expansion (17) there are two types of spherical waves -leaving waves propagating towards r = ∞ and incoming ones propagating towards r = 0, having exactly the same magnitudes. Thus, the incident wave expansion (17) is essentially an expansion over standing spherical waves. This is not surprising, as the net power flow through any closed surface (with no enclosed scatterers!) vanishes for any plane wave. This explains our earlier remarks regarding the anti-causality of incoming waves. So, these waves may not be excited exclusively by remote sources alone. Even though such waves seem to arrive from r = ∞, they may be separated from their causal pair only due to scattering on an object located in the vicinity of the point r = 0. The scattering destroys the perfect balance between leaving and incoming waves which exists in the incident field. Hence, the expansion (17) can be seen as composed of the counter-propagating TE-and TM-polarized spherical waves with indices l = 1, 2, . . . and m = −1, 1, and with complex amplitudes where we use letter A to denote waves propagating towards r = 0, and letter B for the oppositely propagating ones. In these notations, Due to the presence of a body in the vicinity of the point r = 0, the waves (20) will scatter on it, producing in this way the scattered field: The amplitudes of the scattered waves C TE,TM l,m can be found through the complex reflection coefficient for the corresponding incident spherical harmonic (see Appendix A): where we have introduced body's input admittance for a given spherical harmonic Y TE,TM has absolutely the same meaning as Z 1 in the equivalent circuit of Fig. 1(a). Note a subtle difference from the more usual reflection formula -the presence of the complex conjugate operation, -which arises from the fact that impedances of the counter-propagating spherical waves are different and equal the complex conjugate of each other (see Appendix A).
Using Eq. (22), we may write for TE wave reflections happening at surface r = a: Respectively, for the TM waves reflecting at the same surface, whereΓ TE lm = R l (k 0 a)/R l (k 0 a) Γ TE lm andΓ TM lm = R ′ l (k 0 a)/R ′ l (k 0 a) Γ TM lm are the reflection coefficients with redefined phase such as if the reflection happened at the point r = 0 (note that R l (k 0 a)/R l (k 0 a) = R ′ l (k 0 a)/R ′ l (k 0 a) = 1). The total scattered power is found by integrating the expression for η −1 0 |E sc t | 2 over the closed spherical surface with infinite radius. In doing so, we use the orthogonality of the Laplace spherical harmonics, and the fact that where dΩ = sin θ dθ dϕ. We also take into account that |R l (k 0 r)| = |R l (k 0 r)| → 1 when r → ∞. Thus, we obtain for the total scattered power Respectively, the expression for the normalized scattering cross section reads Note that our derivation remains valid when k 0 a ≫ 1, thus, we may conclude that even for optically large bodies, the scattering cross section is not limited by the geometric cross section and can be arbitrary high. Moreover, because the total power associated with an incident plane wave is infinite, the power scattered by an object can also be arbitrary high.
Our model allows us to conclude also that an optically large body with radius r = a made of an absorbing material with characteristic impedance close to that of free space will behave similarly to the impedance matched body considered in Sec. V. Namely, for such a body,Γ p lm ≈ 0 for modes with l ≤ N max = 2πa/λ, N max ≫ 1, andΓ p lm ≈ 1 for the modes with l > N max . Respectively, the normalized scattering cross section of such a body is i.e., its scattering cross section coincides with the geometric cross section. The absorption cross section σ abs can be found by considering the balance of powers delivered to the body by the incoming waves and taken away by the outgoing waves. The power associated with each incoming spherical wave is proportional to A TE,TM . The difference of these two amounts represents the absorbed power. Therefore, the total power absorbed in the body at a given wavelength can be expressed as [compare with Eq. (27)] From here, the normalized absorption cross section satisfies Note the apparent similarity of the terms under summation (32) with Eq. (33). From Eq. (32) one can see that because a perfectly conjugate-matched body is characterized withΓ TE,TM lm = 0, the absorption cross section of it is infinite, similarly to the scattering cross section we have found earlier. It is also directly seen that the conjugate matching condition maximizes the summation terms in Eq. (32). Analogously to what have been done earlier, one may verify that the absorption cross section of a large impedance matched body is σ abs = σ sc = πa 2 . Finally, the extinction cross section of an arbitrary body can be found from Eq. (28) and (32) as σ ext = σ sc + σ abs .
To conclude with the study of this section let us try to analyze (now from the point of view of scattering theory) what property makes it possible to achieve the values of the normalized absorption cross sections much greater than unity, which, reciprocally, increases in the same proportion the effective emissivity of a body. In order to do this, let us consider the behavior of the incident field expansion (17) within the spherical domain r ≤ a. Although all terms in expansion (17) produce a non-vanishing contribution within this region, the dominant contribution is due to the modes with polar indices l such that l k 0 a = 2πa/λ. Mathematically, this can be seen from the asymptotic behavior of Riccati-Bessel functions S l (x) at small values of argument: S l (x) ∼ (x/2) l+1 [ √ π/Γ(l + 3/2)] (here, Γ(z) is the Gamma-function; this asymptotic is valid up to x 2 l), therefore, the modes with l ≫ k 0 a quickly decay in this region when x = k 0 r approaches zero. Fig. 3 shows the radial behavior of a few spherical standing waves in the vicinity of this region.
This demonstrates that when an object such as, for example, a ball made of some absorbing material is placed in the region r < a it will mostly interact with the modes with the indices l k 0 a. Thus, the power transport from the remote environment to this object will be mediated by these modes dominantly. Reciprocally, when the object is the source of thermal radiation, the fluctuating currents in the object will excite the same set of modes, so that only the modes with l k 0 a will participate in the reversely directed heat transport. However, it is not hard to imagine that a specially crafted body can be forced to interact also with the higher order modes with l ≫ k 0 a, because, besides , which maximizes the terms of summation (32). Thus, the physical reason for the increased interaction is this resonance.

VII. IMPLICATIONS WITH REGARD TO SECOND LAW OF THERMODYNAMICS AND KIRCHHOFF'S LAW OF THERMAL RADIATION
One may think that the surprising result of Sec. V -which essentially states that a body of an optically large but finite size may emit, theoretically, arbitrarily high power per unit of frequency and per unit of area -contradicts the second law of thermodynamics. For instance, earlier claims of super-Planckian thermal radiation from photonic crystals were rebutted in Ref. [11] on this ground (we discuss this reference in more detail later in this section). This is, however, not our case. From the equivalent circuit of Fig. 1(a) it is immediately understood that the conjugate matching condition that maximizes the radiated power, at the same time maximizes the power delivered from the environment back to the emitting body, i.e., the optimal heat emitter is at the same time the optimal heat sink! The same conclusion can be drawn from the results Sec. VI, in which we have demonstrated that the absorption cross section of a body is maximized under the same conjugate matching condition. Therefore, the conjugate matching condition preserves the balance of radiative heat exchange between the body and its environment when T 1 = T 2 . The symmetry of the equivalent circuit (a consequence of the reciprocity principle), actually, simply forbids obtaining from our theory any result that would violate such thermodynamical heat exchange balance. On the other hand, from the same circuit it immediately follows that when T 1 T 2 the net radiative heat flow is always directed from the side with higher temperature to the side with lower temperature.
In order to study implications of our theory with regard to Kirchhoff's law let us inspect how large is the power dP inc associated with an incoming spherical wave incident from the the side of the remote environment (kept at temperature T 2 ), in a general scenario when the input impedance of the body Z 1 Z TE,TM w,lm * . The environmentfree space in our case -is characterized by impedance Z 2 = Z TE,TM w,lm . The expression for the power dP 21 received by the body from the environment is readily obtained from the equivalent circuit model: This power can be split into the incident and reflected power: In this formula, Z * 2 = Z TE,TM w,lm * is the wave impedance for the spherical wave propagating from r = ∞ towards r = 0, is the power reflection coefficient of this wave at the body surface. Thus, by combining Eqs. (33) and (34), we obtain The above result shows that in a free space environment filled with thermal-fluctuating electromagnetic field characterized with temperature T 2 > 0 every spherical harmonic propagating towards the point r = 0 delivers the same amount of heat power: dP inc = Θ(ν, T 2 ) dν. Thus, the amount of power transported by all such harmonics with arbitrary indices l and m to an object comprising the point r = 0 is infinite. Kirchhoff-Planck's black bodies and ordinary absorbers reflect most of this incident power: Only the power delivered by the incident modes with l ≤ N max (see Sec. VI) can be efficiently received by such bodies. Reciprocally, they radiate back only into the same limited number of outgoing modes. On the contrary, an ideal conjugate-matched body is theoretically able to receive the whole infinite power delivered by all such incident waves, as well as to radiate it back.
Comparing Eq. (35) with Eq. (5) when T 1 = T 2 = T we may write where α = 1 − Eq. (36) is a generalization of Kirchhoff's law of thermal radiation. Indeed, the dimensionless parameter α = 1 −ρ has the meaning of absorptivity of the body for a given spatial harmonic; dP 12 is the emitted power at the same spectral component; and the ratio of these two quantities is a universal function of just frequency and temperature. Note that unlike the classical law of the same name, Eq. (36) is written for a single component of the spatial spectrum of the radiated field. Thus, Eq. (36) complements the principle of detailed balance by making it applicable to separate spatial harmonics of the radiated field.
Thus the classical Kirchhoff law of thermal radiation, which states that: "For a body of any arbitrary material, emitting and absorbing thermal electromagnetic radiation at every wavelength in thermodynamic equilibrium, the ratio of its emissive power to its dimensionless coefficient of absorption is equal to a universal function only of radiative wavelength and temperature -the perfect black body emissive power," will hold for any emitter -being optically small or large, including the ones characterized with σ abs much greater than their geometric cross section, -if we disregard the intuitive definition of the perfect black body as an "ultimate absorber" which attains the absolute maximum in absorptivity (as compared to all other bodies), in favour of defining the ideal black body just as an abstract object characterized with the emissive power (36). Indeed, a conjugate-matched body receives from the environment and absorbs a much greater power than the conventional black body of the same size, so that its effective absorptivity relative to the black body is much greater than unity: σ abs /(πa 2 ) ≫ 1 (see Sec. VI).
Finally, let us consider a thought experiment described in Ref. [11], where a presumably super-Planckian thermal radiator exchanges thermal energy with an ideal black body located in far zone. If both objects are infinite in spatial extent (two parallel infinite slabs), super-Planckian far-field radiation is impossible, as we have proven in Sec. IV, and there is no need for thermodynamic considerations to prove that again. On the other hand, when a pair of finite-size bodies are separated by a distance significantly greater than d = max(σ abs,1 , σ abs,2 )/π the heat exchange flux is also sub-Planckian.
Next, by considering the case of two finite-size bodies separated by a distance smaller than d but still much greater than the wavelength, we note that consideration from Ref. [11] would assume in this case that the black body perfectly absorbs all incident power, while its radiation is, naturally, restricted by the Planck law of black body radiation. This assumption, obviously, implies a violation of the second law of thermodynamics, because it violates the reciprocity of the heat exchange. However, as we have shown above, Kirchhoff-Planck's black body perfectly absorbs only the fully propagating part (ray part) of the incident spatial spectrum. The higher-order spherical harmonics incident on the black body surface which are responsible for the super-Planckian part of the radiative heat, will be reflected from its surface, and only the ray part will be ideally absorbed and re-emitted by the receiver. The reflected super-Planckian part of the radiation will be scattered into the surrounding space. Part of this energy will then be re-absorbed by the conjugate-matched emitter, which acts as the ideal sink for all incident waves.
Obviously, when the two bodies have the same temperature, there is no net heat flow between them, and, thus, the second law of thermodynamics is not violated. One can say that the Nature avoids such violation by totally reflecting the photons which cannot be re-emitted by the receiving body. On the other hand, there is no limitation for emission of those photons into free space from a conjugate-matched emitter which interacts resonantly with the entire infinite spectrum of outgoing spherical waves. Such emission is possible because for such ideal emitters all photonic states in the surrounding empty space are available, by the process of resonant photon tunneling discussed in Sec. V.
It is instructive to note here that in the above scenario it is still possible to realize a super-Planckian heat exchange between an ordinary body (i.e., a real body with loss, as opposed to the ideal Kirchhoff-Planck's black body discussed above) and a specially crafted body which is conjugate matched to the modified environment which takes into account the presence of the first body, i.e., in this case the conjugate-matched emitter must be designed so that it maximizes the probability of photon tunneling between the two bodies. Because the tunneling process is reciprocal, the second law of thermodynamics is not violated although the ordinary body will emit above Planck's limit in this scenario.

VIII. CONJUGATE-MATCHED DNG SPHERE: METAMATERIAL "THERMAL BLACK HOLE"
Let us now discuss the practical implications of our theoretical findings. It is clear that realizing conjugate matching condition for waves with polar index l > 2πa/λ in a practical thermal emitter is impossible in emitters formed by homogeneous dielectrics or magnetics with positive constitutive parameters. For such materials, the standard arguments of Ref. [4] apply.
Hence, here we shall investigate if a magneto-dielectric sphere filled by a material with less restricted parameters ε and µ (e.g., a metamaterial) can be used in realization of the conjugate-matched emitter. It is known that the complex permittivity and permeability of a passive material at a given frequency can have either positive or negative real parts, while the sign of the imaginary part is fixed: Im(ε, µ) ≥ 0. Because our goal is to realize an omnidirectional emitter, the material parameters ε and µ may depend on r, but should not depend on the angles ϕ and θ. Thus, we are ought to find such ε(r) and µ(r) that will make the input impedance of a sphere made of this material to become equal to the complex conjugate of impedance (8) for spherical harmonics with arbitrary indices.
In order to do that we first solve an auxiliary problem: With which uniform material should we fill the domain r > a, so that the input impedance of this domain becomes the complex conjugate of (8)? The answer to such a question can be obtained from Eq. (8) generalized for the case when ε, µ ε 0 , µ 0 , which reads Consider the properties of the radial function R l (x) and its derivative: which hold when Im(x) → 0. When ε and µ are such that ε = −ε 0 (1 − i| tan δ|), µ = −µ 0 (1 − i| tan δ|) with loss tangent | tan δ| → 0, the refractive index becomes n = εµ/(ε 0 µ 0 ) → −1, and we obtain from (37)-(39) for the input impedance of the domain r > a filled with such material: which is exactly the complex conjugate of the wave impedance (8). However, we need to obtain such result not for the input impedance of the domain r > a, but for the input impedance of the domain r < a occupied by the emitter. Hence, we need to amend the above consideration somehow so that it applies to the domain r < a. This can be achieved by applying a proper coordinate transformation to the Maxwell equations. Such transformation should map the domain r > a into the domain r < a while preserving the field equations in their usual form. The latter ensures that after such transformation the input impedance of the transformed domain coincides with the one for the original domain: Z r<a = Z r>a .
Therefore, a spherical emitter with radius r = a made of a DNG metamaterial with parameters will have the input impedance Z 1 ≡ Z r<a coincident with Eq. (40) when | tan δ| → 0. Note that this impedance approaches Z TE,TM w,lm * arbitrarily closely when | tan δ| → 0, for modes with arbitrary indices. Thus, an emitter filled by a material with parameters (41) and (42) constitutes a physical realization of the conjugate-matched emitter introduced in Sec. V.
A similar profile of |ε(r)| ∝ 1/r 2 was used in the theoretical [24] and experimental [25,26] papers where all the materials have positive real parts of the permittivity and permeability. This leads to a spherical object which theoretically fully absorbs all rays incident on its surface, that is, behaves as a Kirchhoff black body. In contrast to our proposed body whose absorption cross section is theoretically infinite, the absorption cross section of the "optical black holes" described in Refs. [24][25][26] equals to the geometric cross section of the body. A quasistatic case in which a cylindrical body appears having a larger radius than its physical radius was considered in Ref. [27].
Following the same terminology, we may designate a body characterized with the parameters (41)-(42) as a metamaterial "black hole". With this, we emphasize the property of this object to intercept rays of light which are not incident directly on its surface. Due to this feature, such an object has an effective radius of ray capture (kind of "Schwartzschild radius") which can be much greater than the geometrical radius of the body. The latter, in the case of the ideal conjugate-matched body, can have arbitrarily small dimensions, much like the mass singularity in a black hole in Einstein's gravitation theory. Indeed, as it has been found in Sec. VI an ideal conjugate-matched object has infinite absorption cross section, independently of its geometric size. Hence, in a geometric optics approximation any incident ray will end up hitting such object.
Note however that due to the inevitable dispersion of the DNG medium, our metamaterial black hole is not "black" in the usual optical sense, as its absorption is frequency dependent. Our black hole is also different from its astrophysical counterpart in that sense that when we heat it up it emits light (actually, real black holes emit radiation and particles as well, but let us not go too far with such analogies). Because of this, and also because such an object behaves as an ideal radiative heat sink we may also attach a label "thermal" to its name.
Although, theoretically, | tan δ| can be arbitrary small while still allowing for a non-vanishing loss within the emitter (which is the necessary condition for thermal emission to occur), in practice, | tan δ| is always finite. Moreover, close to the core of the emitter the parameters (41) and (42) are divergent: |ε|, |µ| → ∞, when r → 0. These factors limit the number of spherical harmonics of the radiated field for which the conjugate matching condition can be fulfilled in practice. Therefore, the situation represented by Eq. (9) can never be achieved in an experiment. One, however, can still expect a significant increase in the emitted power for emitters with parameters resembling those of Eqs. (41) and (42), especially, for emitters characterized with moderate values of the ratio a/λ. For such emitters we can have σ abs ≫ πa 2 , while still σ abs < ∞. We may still designate these more realistic objects as "black holes" in our fancy way of giving names, however, the effective Schwarzschild radius r eff = √ σ abs /π of these holes will be finite, like for black holes of any finite mass in astrophysics. Respectively, the astrophysical counterpart of the ideal conjugate-matched emitter is a black hole of an infinite mass.
Let us study the case πa 2 ≪ σ abs < ∞ in more detail and obtain an expression for the effective absorption cross section of a core-shell metamaterial emitter with parameters ε(r) and µ(r) that follow Eqs. (41) and (42) within a spherical shell a 0 < r < a, and with uniform parameters ε c = ε(a 0 ), µ c = µ(a 0 ) within the core r < a 0 (see Fig. 4). The loss tangent of the DNG metamaterial is assumed to be finite, but small: 0 < | tan δ| ≪ 1.
Let us denote the input impedance of emitter's core for a given spherical harmonic by Z TE,TM c,lm . This impedance can FIG. 4: Geometry of the spherical core-shell emitter composed of a core with radius r 1 = a 0 filled with a uniform material with permittivity ε c and permeability µ c and the shell with radius r 2 = a filled with a DNG medium with radially-dependent parameters ε(r) and µ(r) given by Eqs. (41) and (42). The core parameters are matched with the shell parameters at r = a 0 : ε c = ε(a 0 ), µ c = µ(a 0 ).
be calculated as These relations are analogous to the expressions for the impedance of the incoming spherical waves, with the radial dependence functionR l (x) replaced by S l (x). Next, the input impedance of the whole emitter can be expressed as where Z p 11,lm , Z p 12,lm = Z p 21,lm , and Z p 22,lm , p = TE, TM, are the equivalent Z-matrix parameters of the spherical shell a 0 < r < a for the same spherical harmonic. By using the equivalence between the shell region and the domain a < r < a 2 /a 0 under the transformation r → a 2 /r, these parameters can be calculated from known formulas for Z-parameters of uniformly filled spherical shells (see Appendix A).
The absorption cross section can now be calculated with the help of Eq. (32). It can be easily checked that for a body with ε(r)/ε 0 = µ(r)/µ 0 (which is our case),Γ TE lm =Γ TM lm =Γ lm . Also, due to emitter's symmetry these reflection coefficients are independent of the azimuthal index m. Therefore, from Eq. (32) we obtain the following expression for the absorption cross section: in which the impedances need to be calculated just for a single polarization (it does not matter for which). The numerical results obtained with the help of Eq. (46) are presented in the next section.

IX. NUMERICAL RESULTS
We consider the core-shell emitter depicted in Fig. 4. The shell has the inner radius r 1 = a 0 and the outer radius r 2 = a and is formed by a DNG metamaterial with parameters given by Eqs. (41) and (42). In the first numerical example, we set r 2 /r 1 = 10, and, respectively, ε(r) and µ(r) vary within the shell such that ε(r 1 )/ε(r 2 ) = µ(r 1 )/µ(r 2 ) = 10 2 . The core material has the parameters ε c = ε(r 1 ) and µ c = µ(r 1 ). We calculate the normalized absorption cross section (46) for a set of k 0 a values ranging from k 0 a = 10 up to k 0 a = 100 with a step ∆k 0 a = 10. In this range, emitter's circumference varies in the range from 10λ up to 100λ, which indicates that we deal with an optically large body. The values of the loss tangent are optimized at each k 0 a value (using a numerical optimization procedure) in order to maximize the normalized absorption cross section at each point. The result of this calculation is presented in Fig. 5. In the second numerical example we set r 2 /r 1 = 100 (i.e., in this case the core is 10 times smaller) and repeat the same procedure (see Fig. 5). We observe that for k 0 a = 10 in the first example with r 2 /r 1 = 10, the normalized absorption cross section σ abs /(πa 2 ) > 2, which means that at this point the DNG core-shell emitter is performing at least twice better than a black body emitter. Note that such a result is achieved at a loss level | tan δ| ≈ 10 −2 which is significantly less than in typical optical absorbing materials.   6 shows the power flux density (the Poynting vector) inside and outside the DNG core-shell object under plane wave incidence as defined in Sec. VI. Note that the power flux density in regions inside this object is much higher than in the outside region, which confirms that a metamaterial thermal black hole is able to greatly concentrate the incident power flux. The behavior of the Poynting vector at the left and right sides of the body [i.e., close to the points (y, z) = (±a, 0)] is especially peculiar: The flux inside the object is directed oppositely to the incident flux, as if some power were received from the region of geometric shadow behind the object. Similar behavior was observed in DNG metamaterial waveguides and resonators [16]. One can also see from Fig. 6 that the shadow region has a larger diameter as compared to the diameter of the body which results in σ abs > πa 2 . The situation improves even more when the radius of the core is made smaller. In the second example of Fig. 5 with r 2 /r 1 = 100, the normalized absorption cross section attains σ abs /(πa 2 ) ≈ 2.4 when k 0 a = 10. Note, however, that in this case the range of parameter variation is already too large to remain practical: ε(r 1 )/ε(r 2 ) = µ(r 1 )/µ(r 2 ) = 10 4 .
For reference, in Fig. 5 we also provide the result for the case of a uniformly filled sphere, which occurs when r 1 = r 2 . For example, for k 0 a = 100, the uniform DNG sphere with loss tangent | tan δ| ≈ 5 × 10 −2 provides σ abs /(πa 2 ) ≈ 1.08, i.e., even a uniformly filled sphere may outperform the black body of the same size by about 8% in this case.
In all examples, the optimum loss tangent value decreases with the increase of k 0 a. The values of the normalized absorption cross section also decrease with k 0 a. Nevertheless, even when k 0 a = 100, i.e., when emitter's circumference is 100 wavelength long, we obtain more then 20% gain in emitter's performance as compared to Kirchhoff-Planck's black body of the same size (when r 2 /r 1 = 10), and close to 30% gain when r 2 /r 1 = 100. The required loss tangent values in these two cases stay within reasonable limits, for instance, | tan δ| > 10 −3 for the emitter with r 2 /r 1 = 10, which can be realized in an experiment. In our opinion, this is a remarkable result which shows that even for optically large bodies with k 0 a ∼ 10 2 there exist emitters which noticeably outperform Kirchhoff-Planck's black body.
In general, with further decrease in the core radius and | tan δ|, the achievable values of σ abs /(πa 2 ) become larger, however, they grow very slowly. Thus, we may conclude that approaching theoretical result σ abs /(πa 2 ) → ∞ in a practical DNG emitter will meet with unavoidable obstacles such as unrealistically high material parameters in the core region |ε c |, |µ c | ≫ 1 combined with very low levels of loss: | tan δ| ≪ 1.

X. CONCLUSION
In this paper we have proven, from the point of view of fluctuational electrodynamics which deals with bodies kept in thermodynamically equilibrium states and characterized with infinite internal thermal capacity, that there is no upper limit on the spectral power of thermal radiation produced by hot finite-size bodies. This holds at any given wavelength, and even in a scenario when such a body radiates into unbounded free space. Namely, we have demonstrated that a conjugate-matched emitter with radius a ≫ λ can radiate a much larger (theoretically, infinitely larger) power at the wavelength λ than can be predicted for the same-size body by using Planck's black body emission formula.
This result was obtained by two independent methods: Firstly, by identifying the conditions which maximize the power emitted by a body when it performs as a source of thermal radiation, and, secondly, by maximizing the absorption cross section of a body. As expected, both derivations lead to the same conclusion stated in the previous paragraph. Moreover, we have proven that neither the second law of thermodynamics, nor Kirchhoff's law of thermal radiation (when properly amended) are violated by theoretical existence of such strongly super-Planckian emitters, which we have decided to name as "thermal black holes".
It is important to note that we have not demanded the material of the optimal emitter to be highly absorbing. Instead, physical realization of thermal black holes becomes possible with the use of low-loss media with simultaneously negative permittivity and permeability -DNG media. It is known that such media support strongly resonant surface excitations -surface plasmon-polaritons. In flat DNG slabs of infinite extent, these modes are bound to the surface and the energy associated with them cannot be irradiated into free space. In other words, such modes may not participate in the far-zone thermal transfer in these geometries (on the other hand, these modes play the main role in the near-field super-Planckian thermal transfer). However, the same modes on a curved closed surfacelike a spherical surface -always leak some energy to the free space modes. In our metamaterial thermal black hole such leakage is greatly enhanced by tuning the whole structure at resonance which maximizes the probability of free-space photon emission from such dark states. Thus, we show that a properly designed metamaterial structure may dramatically amplify the edge diffraction effects not considered in original Planck's theory, and that these effects at a given frequency can be made even more significant than the standard, classical effects limited by the geometric optics approximation.
It is quite remarkable that, from a theoretical standpoint, a finite-size emitter with the double-negative material parameters derived in Sec. VIII performs as an ideal thermal black hole that has infinite absorption cross section (independently of its physical size), and, respectively infinite effective emissivity when compared to Planck's black body of a similar size. However, realizing such a truly super-Planckian emitter in practice meets with unavoidable obstacles when radius-to-wavelength ratio increases.
Our numerical examples show that a practical spherical double-negative core-shell emitter can outperform Planck's black body (at a given wavelength) by more than 100% for emitters with circumference on the order of 10 wavelengths, and by about 20-30% for emitters with circumference on the order of 100 wavelengths. However, considering widespread beliefs that optically large bodies can never outperform a black body of the same size when radiating into free space, in our opinion, this still makes a remarkable achievement even in practical terms.
Finally, let us note that the integral power emitted at all wavelengths remains sub-Planckian for any body formed by passive and causal components. This limitation can be readily demonstrated with the known sum rules for optical scatterers, although this is out of the scope of this paper.