Super-Planckian thermal emission from a hyperlens
C. Simovski, S. Maslovski, S. Tretyakov, I. Nefedov, S. Kosulnikov, P. Belov
aa r X i v : . [ phy s i c s . op ti c s ] J un Super-Planckian thermal emission from a hyperlens
C. Simovski,
1, 2
S. Maslovski,
3, 2
S. Tretyakov, I. Nefedov, S. Kosulnikov, and P. Belov Department of Radio Science and Engineering, Aalto University, P.O. Box 13000, FI-00076, Aalto, Finland Laboratory of Metamaterials, University for Information Technology,Mechanics and Optics (ITMO), St. Petersburg 197101, Russia Departamento de Engenharia Electrot´ecnica, Instituto de Telecomunicac¸ ˜oes,Universidade de Coimbra, P´olo II, 3030-290 Coimbra, Portugal
We suggest and theoretically explore a possibility to strongly enhance the steady thermal radiation of a smallthermal emitter using an infrared hyperlens. The hyperbolic metamaterial of the hyperlens converts emitter’snear fields into the propagating waves which are efficiently irradiated from the hyperlens surface. Thus, withthe hyperlens, emitter’s spectral radiance goes well beyond the black-body limit for the same emitter in freespace. Although the hyperlens can be kept at a much lower temperature than the emitter, the whole structuremay radiate, in principle, as efficiently as a black body with the same size as that of the hyperlens and the sametemperature as that of the emitter. We believe that this study can lead to a breakthrough in radiative cooling atmicroscale, which is crucial for microlasers and microthermophotovoltaic systems.
PACS numbers: 44.40.+a, 42.25.Bs, 78.20.nd
In the classical theory of thermal radiation the power radiated from a unit surface of an optically large body in freespace per unit interval of frequencies is given by Planck’s formula: P FS ω = πe s ( ω ) B ω = ~ ω e s ( ω )4 π c h e ~ ω/ ( k B T ) − i − , (1)where k B and ~ are Boltzmann’s and Planck’s constants, respectively, T is the temperature of the body surface and e s ( ω ) < is the spectral emissivity of body’s material. For a black body (BB) emitting, in accordance to the Planckiantheory, maximal thermal radiation to free space, e s ( ω ) ≡ . For a material having no optical losses at frequency ω ,i.e. for transparent media, e s ( ω ) = 0 and the emission is absent. It is commonly accepted that the thermal radiation isnon-coherent and its spatial distribution is isotropic. Both these factors result in Lambertian pattern for a radiating half-space. However, recent investigations have shown that thermal radiation can be partially coherent [1], directive [2], andcombining coherence and directionality [3, 4]. These deviations originate from intrinsic properties of metamaterials [5].Especially, the so-called hyperbolic metamaterial (HMM) shows interesting responses to thermal radiation (see e.g. in[6–9]). In this paper we theoretically reveal a possibility for a sample of HMM to strongly enhance the far-field radiationfrom small (several micrometers) emitters exceeding the BB limit defined for the emitters of the same size in free space.To our knowledge, in previous works related with applications of HMMs for radiative heat transfer these materials wereused only to control near-field thermal flows. Here, we use these media to enhance far-field radiation.Since classical works by Kirchhoff and Planck, the BB has been considered as a perfect thermal emitter whose spectralradiance cannot be exceeded in far-field zone (so-called Planckian limit). Despite that the photonic density of states(PDOS) and, consequently, the rate of spontaneous emission responsible for the thermal radiation may be enhancedconsiderably (e.g. in [10] by one order of magnitude) there is a belief that the photons that occupy the extra availablestates cannot be emitted out of a medium with high PDOS [11] due to the total internal reflection (TIR). However, it is notgenerally true.Long ago, in work [12], a possibility to exceed the BB limit for a hot particle having resonant sizes at infrared waspointed out. Recently, the authors of [13] have demonstrated super-Planckian radiation from a macroscopic emitterachieved due to a transparent dielectric dome. For a hemispherical emitter this idea is illustrated in Fig. 1 (left half). As itfollows from Eq. (1), filling free space with an isotropic transparent medium with refractive index n = √ ε h [respectively, c is replaced by c/n in Eq. (1)] increases the radiated power by n , provided that e s ( ω ) stays unchanged. If such transparentmedium forms a lens in a shape of a hemispherical dome as is shown in Fig. 1, the emitted waves impinge on the lenssurface and, after being partially reflected, pass onto free space. When the dome radius R is much larger than the emitterradius r , the power transmittance to the free space for these waves can be found in geometric optics (GO) approximationas for normally incident rays: t GO = 4 n/ ( n + 1) . Thus, the total gain in the power irradiated to the far zone due to thepresence of the dome equals G = n × t GO = 4 n / ( n + 1) . From here it may seem that one can achieve arbitrary highgain when n → ∞ . However, this is not true, because when n & R/r some of the incident rays start experiencing TIR atthe output interface of the dome. This effect also ensures that the apparent diameter of the emitter as is seen from outsideof the dome never exceeds the diameter of the dome, which sets an obvious upper bound for the total gain in this structurewhen R ≫ λ : G < R /r , i.e., the whole structure may not radiate more than a BB with radius equal to the outer radiusof the dome.Because realistic thermal sources have e s < , and because the known transparent materials in the infrared range haverather small refractive indices n . , the thermal lens [13] can hardly offer gain G BB which would exceed . Here, G BB is the ratio of power emitted by a realistic thermal source covered with a transparent dome to the power emittedby an uncovered BB with the same size and temperature as the original source. In [13] the gain G BB ≈ . has beenexperimentally demonstrated for a thermal lens of centimeter size with n ≈ . . The above estimations predict G BB ≈ . for this case, when the emitter is an ideal BB.The study that we are going to present next has been motivated by the following question: Since it is possible to enhancethe thermal radiation of an emitter by 2–5 times by using a hemisphere of a transparent isotropic dielectric, can we gofurther using more advanced materials? Namely, can we approach the GO bound: G max = R /r with these materials?Note that here we are interested in the case when R ≈ λ or greater, because bodies with R . λ can outperform thisbound [12]. We show that a dome made of a hyperbolic metamaterial theoretically allows one to increase the spectralradiance of small emitters by up to two orders of magnitude, as compared to the limit dictated by Planck’s law for BBemitters of the same size in vacuum. Hyperbolic metamaterials (HMM) which we propose for this purpose are uniaxialdielectric composites with the permittivity tensor defined by two components: transverse ε ⊥ and axial ε k , such that Re( ε k )Re( ε ⊥ ) < , Re( ε k , ⊥ ) ≫ Im( ε k , ⊥ ) . The isofrequency surfaces (also called wave-dispersion surfaces) for HMMrepresent hyperboloids. A hot unit volume inside a HMM sample emits much more electromagnetic energy than a unitvolume of a conventional lossy medium at the same temperature. This effect results from high Purcell’s factor of a dipolelocated inside HMM. The concept of Purcell’s factor (the gain in the spontaneous emission rate) historically referred to thecase when the dipole radiation was enhanced by a closely located resonator (see e.g. in [14]). However, in work [15] thenotion of Purcell’s factor was extended to any environment of the dipole source different from free space. Purcell’s factorof HMM dramatically exceeds the Purcell’s factor F P , diel = n of a usual dielectric. In the lossless HMM without internalgranularity the radiation resistance of a point dipole oriented orthogonally to the optical axis tends to infinity [16, 17]because all the power is irradiated in the form of propagating waves. Therefore the thermal radiation of a unit hot volumein such an ideal HMM should be infinite. For realistic (lossy and internally granular) HMM the thermal radiation of a unithot volume is finite but strongly super-Planckian as compared to vacuum [6, 7].The excessive super-Planckian radiative heat in HMM is contained in the modes with high transverse wavenumbers q = 2 π/ Λ tr ≥ ω/c . In flat uniaxial HMM slabs with the optical axis oriented orthogonally to the interface plane thesemodes experience TIR at the interface with free space and, thus, are confined inside the HMM (note that the coupling ofsuch modes with free space can be carried out in asymmetric HMM [18–20], where the optical axis is tilted to the slabinterface). As a result, the thermal radiation from such slabs into free space does not exceed the BB limit. However, it canbe very close to it, and it is known that a half-space of HMM mimics the BB [5–7].On the other hand, in locally uniaxial radially symmetric HMM samples the eigenmodes which are characterized withhigh local wavenumbers q ( r ) ≫ ω/c close to the center of the sample, may attain q ( R ) < ω/c at enough large radialdistance R ≫ r , because in these modes, roughly, q ( r ) ∝ /r . Therefore, in radially symmetric HMM these modes cancouple to the free space propagating waves if the radius R is large enough. This effect is known as hyperlensing [21–23].Hyperlenses (HLs) were previously designed for obtaining magnified images of subwavelength objects.In fact, HL is also a matching device for the radiation propagating from its central part to free space [24]. Here, we sug-gest to use a dome of radially symmetric HMM which operates as an infrared HL to extract the excessive super-Planckianheat otherwise confined within emitter’s near field in the modes with high transverse wavenumbers. An implementation ofsuch HMM in the infrared range is, for example, an optically dense array of aligned metal nanowires called wire medium(WM). WM is a spatially dispersive implementation of HMM [25]. The spatial dispersion of HMM for our purpose is nota harmful factor. On the contrary, in accordance to our estimations the spatial dispersion helps to match the hyperlens tofree space.Performing our emitter as a lossy dielectric body placed inside a transparent dielectric dome both comprising radiallydivergent nanowires, we arrive at the structure sketched in Fig. 1 (right half). For better matching of the HL to free space FIG. 1: Two hemispherical structures which offer super-Planckian thermal radiation from a finite emitter (just a half of each structure isshown). Left: Thermal lens analogous to the one considered in Ref. [13]. Right: Thermal hyperlens (HL) comprising radially divergingnanowires. The lens and HL’s host are made of transparent glass with permittivity ε ≡ ε h . The emitter is formed by a lossy mediumwith complex permittivity ε and is partially filled with nanowires. The emitter is set under high temperature T ≫ T , where T isthe ambient temperature. the ends of the nanowires can be made free-standing as it is shown in the figure. To prevent direct thermal contact, thenanowires in the emitter may be separated from the HL by a sufficiently narrow nanogap. It is critical that the nanowiresare radially oriented in the whole structure and that their density decreases with radial coordinate. The divergence angle φ between adjacent nanowires should be small enough so that the properties of HMM in the emitter volume are preserved,however, large enough so that the best possible matching to free space is achieved.Formulas for the effective permittivity of WM operating at infrared can be found in [25, 26]. We use highly radiallyanisotropic HMM, in which Re( ε ⊥ ) > , Re( ε k ) < , | ε k | ≫ | ε ⊥ | , and the energy propagates roughly in the radialdirection, independently on the value of q . Thus, we notice that in this regard the situation is similar to the case of thesimple dielectric dome considered previously, with a difference that when estimating the power transmittance through theouter interface of the HL we must use the effective complex index of refraction in the WM in the vicinity of the outerinterface: n ⊥ , out = √ ε ⊥ , out . Hence, t HL = 4Re( n ⊥ , out ) / | n ⊥ , out + 1 | .Note that still the modes with q ( R ) ≥ ω/c experience TIR at the dome-air interface and for these modes t HL = 0 . Fromrecent studies of the dipole radiation in WM [27, 28] it is known that the irradiated wave beam is nearly as narrow as theWM period a . Therefore, in order to obtain high transmittance to free space (nearly as high as t HL ) for the dominant partof the spatial spectrum exited within HL, the divergence angle φ should be such that the separation between the nanowiresat the outer surface of the HL is about λ/ or larger. Hence, φ & λ/ (2 R ) .Let us now estimate how large can be the gain G HL in the HL configuration of Fig. 1. First, we note that inclusion ofnanowires into a dielectric host increases the power radiated by an elementary dipole placed inside this medium by F P times, where F P is the Purcell factor for uniaxial WM. This factor was calculated in Ref. [17]. Being averaged over allpossible locations of a transversely oriented electric dipole, F P equals F trP ≈ k k log (cid:20) K m k (cid:21) , (2)where k h = √ ε h ω/c and k p = p π/ log[ a / r ( a − r )] /a are the wavenumber in the host medium and the plasmawavenumber in WM [29], respectively, where a is the WM period and r is the wire radius. In Eq. (2), K m is thespatial spectrum cut-off parameter, which equals √ π/a in unbounded uniaxial WM [17]. Because only the modes with q < ω/c are irradiated from HL’s outer surface, here we must limit this parameter by K m = min[2 √ π/a, ( ω/c )( R/r )] .Strictly speaking, Eq. (2) refers to the case when the wires are perfectly conducting, however in [17] the estimations weredone also for lossy wires and it was shown that (2) was applicable to realistic metal nanowires if they were thick enough(practically, their radius r should be larger than the skin depth). For dipoles parallel to the wires the Purcell factor is muchsmaller than F trP and can be neglected. Since a hot elementary volume of a lossy medium surrounded by nanowires can betreated as a set of three identical mutually orthogonal dipoles emitting thermal radiation, in thermal emission calculationswe must use the average F HLP = 2 F tr P / , where F tr P is given by (2).Under these conditions, the total gain due to the effect of the HL dome can be estimated as follows: G HL ≈ F HLP × Re( n ⊥ , in ) × t HL × e − α ( R − r ) ≈ F trP × Re( n ⊥ , in ) Re( n ⊥ , out )3 | n ⊥ , out + 1 | e − α ( R − r ) , (3)where n ⊥ , in is the effective refractive index of the HL in the vicinity of the emitter (at the inner interface of the HL), and α ≈ ( ω/c )Im( √ ε ⊥ ) is the decay factor due to the loss in the nanowires. Note that the structure suggested in this papercannot radiate more than a BB with the same size as that of the dome and the same temperature as that of the emitter when R ≫ λ . Moreover, because the WM-based HL interacts mostly with P -polarized waves, the actual upper bound for thegain in this case is G maxHL ∼ . R /r .Due to optical losses in the metal the decay of thermal radiation over the path R ≫ λ is not negligible. This factorrestricts the radius R by dozens of λ . However, Eq. (3) does not take into account the thermal radiation of heated wiresinside the hyperlens. This additional emission can significantly increase G HL compared to (3), so that it may approachthe GO bound: G max = R /r . In the same time, (2) slightly overestimates the Purcell factor for realistic nanowires. So,the implementation of our thermal HL with macroscopic dimensions (e.g. with R = 1 cm like in [13]) is disputable. Inthe present study we deal with a microscopic hyperlens with radius R = 10 µ m and an emitter of radius r = 0 . µ m.Estimations of the factor G HL were done using Eq. (3) and the effective-medium model of infrared WM [26]. Thematerial parameters of gold were taken from Ref. [30]. We considered a hemispheric structure with concentric hemisphericemitter located on a perfect mirror as in Fig. 1. The HL is performed of non-tapered gold nanowires with the thickness r = 50 nm located in a matrix with ε = 3 . (chalcogenide glass transparent in the range 50–150 THz). We calculatethe gain G HL in the range 100–140 THz, where r > δ ( δ is the skin-depth of gold), and formula (2) for Purcell’s factoris applicable. Internal ends of nanowires are located at r = 0 . µ m from the geometrical center of the structure. Anemitter comprises the hemisphere r = 0 . µ m and is partially filled with nanowires. The emitter is assumed to be a lossydielectric which is well impedance-matched with the HL. The distance between the centers of nanowires at the surface r = 0 . µ m equals nm and within the emitter the averaged period of the WM equals a = 125 nm. Nanowires divergewith the angle φ ≈ ◦ . This angle is small enough to neglect the divergence of nanowires when calculating the effectivepermittivity of HMM in the domain of the emitter and its Purcell factor F trP . However, it is large enough to offer goodmatching of the HL to free space, because for φ & . ◦ the distance A between the axes of nanowires at the outer surfaceof the HL exceeds λ/ at frequencies 100–140 THz.Following to (2) Purcell’s factor of the medium of parallel nanowires with the period a = 125 nm for a transverseelectric dipole decreases from F trP ≈ to F trP ≈ . over the range 100–140 THz. Then the relative enhancement G HL of the power spectrum radiated by an arbitrary dipole p located in between the wires near the internal surface of the HLin accordance to (3) is within approximately . . . over this frequency range. The range 100–140 THz is around themaximum of emission for the emitter temperatures T of the order 700–800 ◦ C. Higher temperatures are hardly actual forour HL since nanowires can melt. For lower temperatures thermal radiation is concentrated in lower frequencies wherePurcell’s factor is higher. For example, at 50 THz in unbounded WM F tr P ≈ . The same divergence angle at thisfrequency implies larger R needed for matching the HL to free space. The condition A > λ/ holds at 50 THz for R = 20 µ m. Then, taking into account the decay we obtain using (3) the gain G HL ≈ at 50 THz. So, for emitterswith temperatures T < ◦ C the thermal radiation of the emitter within HL may exceed the BB limit for the sameemitter in vacuum by two orders of magnitude.To check our estimations of the gain G HL we performed extensive numerical simulations. We studied a HL excitedby a transverse dipole located in the middle between the ends of adjacent nanowires either at the surface of the centralnanocavity or displaced from this surface – either embedded into the WM up (to 250 nm from the central cavity) orlocated inside it. The parameters of the HL in these simulations are as above besides one replacement – we substitutedgold nanowires by perfectly conducting ones. This replacement dramatically reduced the computation time needed forthe structure comprising many hundreds of metal nanowires and made simulations realistic. Simulations were performedusing the CST Studio Suit software.Although replacing gold by perfect conductor we removed the decay factor exp( − αR ) , this is still a reasonable modelof a HL. The decay factor is not the most relevant parameter and can be easily taken into account analytically. The absenceof absorption makes the relative enhancement of radiation into free space equivalent to Purcell’s factor. This equivalenceallows us to concentrate on the hyperlensing effect, i.e., on emission enhancement and matching of our structure to freespace. Our model source is a very short dipole antenna of perfectly conducting wire with bulbs mimicking the Hertziandipole at the simulation frequency. YZ MAX
FIG. 2: Electric field amplitude distribution in the H -plane (vertical cross section orthogonal to the dipole) produced by a dipole locatedin between the internal ends of perfect nanowires of radius r = 25 nm forming our HL. Its host material (between r = 0 . µ m and R = 10 µ m) is glass.
100 110 120 130 14010 Frequency, THz R e l a ti v e E nh a n ce m e n t TheoryFixed current, radiated power ratioRatio of input resistancesG = 0.5 max (R/r) FIG. 3: Relative enhancement of radiation by a transverse dipole due to the presence of a HL of perfect wires calculated 1) directly viathe radiated power spectrum (green curve) and 2) through the input resistance of a short wire dipole (red curve). The structure is thesame as in Fig. 2. The theoretical blue dashed curve and the green dotted line are explained in the main text.
First, we calculate the field distributions to inspect if the wave beam divergence is sufficient to prevent strong reflectionsfrom the effective surface of the HL. For divergence angles within the range φ = 6 – ◦ the concept of HL turned out tobe fully adequate, and the result weakly depends on φ over this interval of values. The result weakly depends on theexact location of the transverse dipole embedded into the WM up to 250 nm from the internal cavity r . However, if thedipole is moved to the central cavity to the distance more than 250 nm, Purcell’s factor drops to unity. Also, the radiationdecreases if φ < ◦ i.e. when the HL approaches to a block of parallel nanowires. In Fig. 2 a color map illustrates thehyperlensing of the dipole radiation for optimal divergence angle φ = 10 ◦ . The horizontal dipole is located on top of theinternal cavity r in between two central nanowires and radiates at the frequency 120 THz which is between the bandsof Fabry-Perot resonances. In both E - and H -planes we observed a sufficient width of the main radiation beam. Thereflection from the effective surface of the HL in our simulations fits the estimation t ≈ t HL .For a lossless HL G HL can be calculated in two ways: via the input resistance of the antenna and via the far-zoneradiated power. Both these values were calculated and normalized to the corresponding values simulated for the samedipole when the HL is absent. In the first case we keep the same input voltage of the antenna in the absence or presenceof the HL. In the second case we fix the antenna current. The coincidence of two results is expected at low frequencieswhere the short wire antenna is close to the Hertzian dipole. This equivalence is seen in Fig. 3 at 100–130 THz where thered and green curves nearly coincide (besides the small ripples of the red curve which are numeric errors). In this plot weobserve several Fabry-Perot resonances at which the HL gain reaches very high values. These values are, however, hardlyrelevant for the thermal radiation because the emitter mimicking the BB will absorb all incoming waves. Therefore, theFabry-Perot resonances in the HL in a more realistic configuration will be greatly suppressed. The blue curve shows thetheoretical estimation for the HL gain calculated in accordance to Eq. (3) with the factor (2 /
3) exp( − αR ) excludedbecause only single orientation of the dipole in a lossless HL is considered in the simulations. In the range 100–130 THzour theoretical estimation agrees with the simulated gain when averaged between the Fabry-Perot resonances.To conclude, we have suggested a structure that greatly enhances the radiative heat power produced by a small thermalemitter, which may go far beyond the limit enforced by Planck’s law for the same radiator in free space. This is achievedby centering the emitter at the focal point of a hyperlens, which transforms emitter’s near field into propagating waveswhich are matched well to free space and efficiently irradiated. However, the structure suggested in this paper still radiatesless than a BB with the same size as that of the hyperlens and the same temperature as that of the emitter. A theoreticalpossibility to overcome this restriction for bodies of constrained radius is reserved for a future work (see [31]). [1] R. Carminati and J. J. Greffet, Phys. Rev. Lett. , 1660 (1999).[2] J. J. Greffet, R. Carminati, K. Joulain, J. P. Mulet, S. Mainguy, and Y. Chen, Nature , 61 (2002).[3] Y.-B. Chen and Z.M. Zhang,
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