Long-Range Dipole-Dipole Interaction and Anomalous Förster Energy Transfer across Hyperbolic Meta Material
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Long-Range Dipole-Dipole Interaction and Anomalous F¨orster Energy Transfer acrossHyperbolic Meta Material
S.-A. Biehs ∗ Institut f¨ur Physik, Carl von Ossietzky Universit¨at, D-26111 Oldenburg, Germany ∗ Corresponding author: [email protected]
Vinod M. Menon
Department of Physics, City College of New York, 160 Convent Ave New York, NY 10031, USA
G. S. Agarwal
Department of Physics, Oklahoma State University, Stillwater, Oklahoma 74078, USA (Dated: February 22, 2016)We study radiative energy transfer between a donor-acceptor pair across a hyperbolic metamate-rial slab. We show that similar to a perfect lens a hyperbolic lens allows for giant energy transferrates. For a realistic realization of a hyperbolic multilayer metamaterial we find an enhancement ofup to three orders of magnitude with respect to the transfer rates across a plasmonic silver film ofthe same size especially for frequencies which coincide with the epsilon-near zero and the epsilon-near pole frequencies. Furthermore, we compare exact results based on the S-matrix method withresults obtained from effective medium theory. Our finding of very large dipole-dipole interactionat distances of the order of a wavelength has important consequences for producing radiative heattransfer, quantum entanglement etc.
PACS numbers: 78.70.g,78.20Ci,42.50.p
I. INTRODUCTION
Dipole-dipole interactions are at the heart of manyfundamental interactions such as van der Waals forcesand vacuum friction [1, 2], F¨orster (radiative) energytransfer (FRET) [3, 4], radiative heat transfer [2, 5],quantum information protocols like the realization ofCNOT gates [6–8], pairwise excitation of atoms [9–11],and Rydberg blockade [12, 13]. Clearly large numberof problems in physics and chemistry require very sig-nificant dipole-dipole interaction at distances which arenot much smaller than a wavelength. The developmentof plasmonic and metamaterial platforms can consider-ably enhance these fundamental dipole-dipole interac-tions. In an early work [14] it was shown that the dipole-dipole interaction [14, 15] can be quite significant evenat distances bigger than microns if one utilizes whis-pering gallery modes of a sphere. More recently suchsystems have been revisited for their remarkable quan-tum features like squeezing [10]. Further it was shownthat the energy transfer across plasmonic metal filmscan be enhanced [16] and that the long range plasmonsallow for long-range plasmon assisted energy transferbetween atoms placed on plasmonic structures such asgraphene [17–19] and metals [20–24].More elaborate structures possessing more featuresthan simple single-layer plasmonic structures are for ex-ample hyperbolic metamaterials (HMM) [25, 26], whichexhibit a broadband enhanced LDOS [27] allowing forbroadband enhanced spontaneous emission [28–39], hy-perbolic lensing [40–42], negative refraction [43, 44] andbroadband enhanced thermal emission [45–50], for in-stance. These HMM can be artificially fabricated by a periodic layout of sub-wavelength metal and dielec-tric components but they also exist in nature [51–55].Here we concentrate on artificially fabricated HMM sincethe choice of fabrication parameters makes it possible tocontrol the epsilon-near zero (ENZ) and the epsilon-nearpole (ENP) frequencies. At these frequencies the HMMcan show extra-ordinary features as enhanced superradi-ance [56] and supercoupling [57, 58]. zz’ 0 d x z
Donor Acceptor
Figure 1: Sketch of the considered geometry. The metamate-rial slab at the moment is an isotropic one with ǫ and µ asthe dielectric and magnetic permeabilities. The aim of our work is to study FRET across a hy-perbolic multilayer metamaterial (see Fig. 1). We willdemonstrate that similar to the perfect lens effect [59],the dipole-dipole interaction across a HMM can be en-hanced by orders of magnitude compared to the pureplasmonic enhancement by a thin plasmonic metal layer.The organization of our paper is as follows: In Sec. IIwe introduce the elementary relations for FRET betweena donor and an acceptor which are separated by an in-termediate slab. We discuss the possibility of a perfectdipole-dipole interaction which is studied in Sec. III an-alytically and numerically for a thin silver film. Then,in Sec. IV, we discuss the anomalous transmission acrossa hyperbolic metamaterial as an alternative for realiz-ing perfect dipole-dipole interaction. Finally, in Sec. Vwe study FRET across a concrete hyperbolic structureusing exact S-matrix calculations. The exact numericalresults are compared with the approximative results ofeffective medium theory.
II. POSSIBILITY OF PERFECTDIPOLE-DIPOLE INTERACTION
The F¨orster energy transfer rate between an donor-acceptor pair as sketched in Fig. 1 is well-known andcan be written in terms of the dyadic Green’s function G as [4] Γ = Z d ω σ abs ( ω ) T ( ω ) σ em ( ω ) (1)where σ abs and σ em are the absorption and emission spec-tra of the acceptor and donor and T ( ω ) = 2 π ~ (cid:18) ω ǫ c (cid:19) | d D | | d A | | e A · G · e D | (2)introducing the dipole-transition matrix elements d D = | d D | e D and d A = | d A | e A of the donor and acceptor.Making a plane wave expansion of the dyadic Green’sfunction, we obtain G ( r , r ′ ) = Z d κ (2 π ) e i κ · X G ( κ , z ) (3)where κ = ( k x , k y ) t , X = ( x − x ′ , y − y ′ ) t , and G ( κ , z ) = ie i γ v ( z − z ′ ) γ v (cid:20) t s a +s ⊗ a +s + t p a +p ⊗ a +p (cid:21) (4)introducing the vacuum wavevector in z-direction γ v = p k v − κ and the vacuum wavenumber k v = ω/c . Notethat the expression for the Green’s function contains thecontribution of the propagating ( κ < k v ) and evanescent( κ > k v ) waves. The evanescent wavas are exponentiallydecaying in z-direction so that during propagation thesecontributions are lost unless the medium somehow canreverse the decay of such waves. Here t s and t p are thetransmission coefficients of the s- and p polarization and a +s , p are the polarization vectors defined by a +s = 1 κ k y − k x and a +p = 1 κk v − k x γ v − k y γ v κ . (5) The transmission coefficients for a metamaterial film canbe expressed as [25, 59] t s = 4 µγγ v e i( γ − γ v ) d ( γ + µγ v ) − ( γ − µγ v ) e γd , (6) t p = 4 ǫγγ v e i( γ − γ v ) d ( γ + ǫγ v ) − ( γ − ǫγ v ) e γd , (7)where the wavevector component inside the medium in z-direction is given by γ = p ǫµω /c − κ . Note that thedipole-dipole interaction in free space (i.e. if we replacethe film by vacuum) can be obtained from the Green’stensor in Eq. (4) by setting the transmission coefficientsto one t s = t p = 1 (which is the result for ǫ = µ = 1), i.e. G (vac) ( κ , z ) = ie i γ v ( z − z ′ ) γ v (cid:20) a +s ⊗ a +s + a +p ⊗ a +p (cid:21) (8)In this case the dipole-dipole interaction would becomeinfinitely large if the donor and the acceptor would beplaced at the same position, that means if z = z ′ . Froma mathematical point of view this is so because the ex-ponential prefactor equals one for z = z ′ so that the κ -integral in Eq. (3) would diverge due to the fact that aninfinite number of evanescent waves with κ > k v wouldcontribute to the energy transfer. On the other hand, itis well known that the dipole-dipole interaction betweenthe donor and acceptor is proportional to 1 / | r − r ′ | whichdiverges for z → z ′ , since we here have x = x ′ = 0 and y = y ′ = 0 as shown in the Fig. 1.For an ideal left-handed material exhibiting negativerefraction the permittivity and permeability are given by ǫ = µ = −
1. In such a material evanescent plane wavesare amplified as shown by Pendry when he introducedthe concept of a perfect lens [59]. For the here definedtransmission coefficients we obtain for ǫ = µ = − t s = t p = exp( − γ v d ) (9)showing clearly the feature of amplification of evanscentwaves inside the ideal left-handed material. Note, thatthis exponential amplification stems from the exponentialin the denominator of the transmission coefficients. Thisis just the same expression derived by Pendry showingthat for the perfect lens both propagating and evanescentwaves contribute to the resolution of the image [59] whichleads to the perfect lensing effect depicted in Fig. 2.Inserting the expression of the transmission coefficientsin the Green’s tensor describing the dipole-dipole inter-action gives G ( κ , z ) = ie i γ v ( z − z ′ − d ) γ v (cid:20) a +s ⊗ a +s + a +p ⊗ a +p (cid:21) . (10)This Green’s tensor is the same as expression (8) of thevacuum Green’s tensor with the important difference thatnow the argument of the exponential prefactor is differ-ent. As in the case of interaction in free space we havean infinite large energy transfer if the exponential equals zz’ 0 d x z Donor Acceptor
Figure 2: Sketch of the perfect lensing effect. one leading to the condition that z − z ′ = 2 d . Henceby placing the donor-acceptor pair such that this con-dition is fullfilled corresponds to a dipole-dipole interac-tion for two dipoles at the same position leading to aninfinitely large energy transfer rate. This is so, becausefor z − z ′ = 2 d all evanescent waves ( κ > k v ) get focused.We have thus shown that a pefect negative material withzero losses can yield perfect dipole-dipole interaction andperfect energy transfer which is limited only by the donorand acceptor line shapes. III. SILVER FILM AS METAMATERIAL
As pointed out by Pendry [59] for observing the perfectlensing effect it suffices to consider a thin silver film. Inthe quasistatic-limit such a thin silver film mimicks aperfect lens for the p-polarized waves at a frequency ω pl where ǫ ′ ( ω pl ) = −
1, which coincides with the surfaceplasmon resonance frequency of a single metal interface.To see this we take the quasi-static limit ( κ → ∞ ) of thetransmission coefficient t p in Eq. (7) with µ = 1. Weobtain t p → ǫ ( ǫ + 1) − ( ǫ − e γd . (11)The exponential in the numerator vanishes because inthe quasi-static limit γ v ≈ γ ≈ i κ . If we now insert ǫ = − ǫ ′′ , (12)then we arrive at t p → ǫ ′′ − ǫ ′′ (e γd −
1) + 4(i ǫ ′′ − γd . (13)It is now easy to see that for vanishing losses ǫ ′′ → γ v ≈ γ ≈ i κ ) t p = exp( − γ v d ) . (14) Proving that a silver film can mimick a perfect lens. Theonly drawback is that in metals the losses are not neg-ligible so that the perfect lens effect does not persist inthis case. Nonetheless even with losses one can expectto find large F¨orster energy transfer due to the fact thatthe donor and acceptor can couple to the surface wavesinside the silver film. | G zz | / | G zz V a c | λ (nm)z - z’ = 50nmz - z’ = 60nmz - z’ = 70nm (a) Ag, F z | G xx | / | G xx V a c | λ (nm)z - z’ = 50nmz - z’ = 60nmz - z’ = 70nm (b) Ag, F x Figure 3: Plot of F z and F x as function of wavelength for asilver film with thickness d = 30 nm. The donor is placed at z ′ = −
10 nm and the position of the accepor is varied suchthat z − z ′ = 50 nm ,
60 nm ,
70 nm.
That the coupling to the surface plasmons can enhancethe energy transfer has been demonstrated theoreticallyfor different systems like nanoparticles [60], plasmonicwaveguides [23] and films [20, 21] as well as graphene [17–19]. This enhanced energy transfer might be exploitedfor solar energy conversion [22]. Experimentally it hasbeen shown by Viger et al. [61] and Zhang et al. [62] thatplasmonic nanoparticles can enhance the energy transferrate; Andrew and Barnes [16] have proven experimentallythat the F¨orster energy transfer across thin silver filmscan be enhanced by the coupling to the surface plasmonpolaritons demonstrating a long-range coupling betweenthe donor-acceptor pair across films with thicknesses upto 120nm. Although we are here particularly interestedin the long-range energy transfer across a film, surfaceplasmons can also be used to mediate the energy transferalong plasmonic structures as shown theoretically [17–21, 23]. Recently, Bouchet et al. [24] have detected thislong-range energy transfer when both donor and acceptorare placed on a plasmonic platform like a thin metal film.To see how the F¨orster energy transfer is affected bythe presence of a metal film, we consider therefore first athin silver layer with permittivity described by the Drudemodel ǫ Ag = ǫ ∞ − ω ω ( ω + i τ − ) (15)with the paramters ǫ ∞ = 3 . ω p = 1 . · rad / s, τ =0 . · − s. In order to quantify the enhancement ofthe energy transfer we introduce the enhancement factor F i ≡ | G ii | | G (vac) ii | , (16)where i = x, z is the orientation of the dipole-transitionmatrix element of the donor/acceptor and G (vac) ii is thevacuum Green’s function. In Fig. 3 we show our resultsfor the enhancement of the dipole-dipole interaction dueto the presence of a silver film of thickness d = 30 nm withrespect to the case where this film is replaced by vacuum.It can be seen that the dipole-dipole interaction andtherefore the F¨orster energy transfer is especially largefor λ ≈
300 nm where ǫ ′ Ag ≈ − z − z ′ = 2 d = 60 nm no partic-ular effect happens. The F¨orster energy transfer becomesjust less important when the distance z − z ′ between thedonor-acceptor pair is increased. In Fig. 4 we show sim-ilar results for a much thicker film with d = 120 nm. Inthis case there can be still seen an enhancement effectfor z orientation but of course the enhancement effectbecomes less important when the thickness of the sil-ver film is increased due the losses inside the metal film.Therefore another kind of material is needed in orderto overcome the harmful effect of the losses. Nonethe-less such an enhancement due to the surface plasmons ofthe silver film have already been measured in the exper-iment by Andrew and Barnes [16] for film thicknesses of d = 30 nm ,
60 nm ,
90 nm and even d = 120 nm. IV. ANOMALOUS TRANSMISSION FORHYPERBOLIC MATERIALS
Now, let us replace the ideal left-handed material slabby a hyperbolic or indefinite material [26]. The beauty ofsuch hyperbolic materials is that waves with large kappa( κ ≫ k v ) emitted by a donor which would be evanes-cent in free space are homogenous within these materials.This leads to significant fields on the other side of the slabwhich makes hyperbolic materials very advantageous forenergy transfer even for slabs with an appreciable thick-ness. Hyperbolic materials are uni-axial materials which | G zz | / | G zz V a c | λ (nm) Ag (a) Ag, F z | G xx | / | G xx V a c | λ (nm) Ag (b) Ag, F x Figure 4: Plot of F z and F x as function of wavelength. Thedonor and accepor are placed in a distance of 10 nm of eachinterface (i.e. z ′ = −
10 nm and z − d = 10 nm) of a silver filmwith thickness d = 120 nm. do exist in nature [51–55]. But they can also be easilyfabricated by combining alternating layers of dielectricand plasmonic materials in a periodic structure, for in-stance. Here, we consider only the non-magnetic case sothat the permeability tensor is given by the unit tensorand the permittivity tensor is given by diag( ǫ ⊥ , ǫ ⊥ , ǫ k )with respect to the principal axis. In our case the z-axisis the optical axis of the uni-axial material. The disper-sion relations of the ordinary and extra-ordinary modesinside the uni-axial medium are [63] γ ǫ ⊥ + κ ǫ ⊥ = k v and γ ǫ ⊥ + κ ǫ k = k v (17)where γ o ( γ e ) is the z-component of the wavevector of theordinary (extraordinary) mode. Neglecting dissipationfor a moment we can easily define normal dielectric uni-axial materials as materials with ǫ k > ǫ ⊥ > ǫ k ǫ ⊥ <
0, i.e.one of both permittivites is positiv and the other onenegative. Due to the property ǫ k > ǫ ⊥ > ǫ k < ǫ ⊥ > ǫ k > ǫ ⊥ < k z - k x plane.Now we will show that similar to the perfect lensingeffect the evanescent waves can be amplified inside a hy-perbolic slab even in the absence of a magnetic response.To this end, we consider the transmission coefficients ofa uniaxial slab where the optical axis is normal to theinterface, i.e. it is along the z-axis. In this case thetransmission coefficients for the ordinary and extraordi-nary modes (which coincide with the s- and p-polarizedmodes) are given by t s = 4 γ o γ v e i( γ o − γ v ) d ( γ o + γ v ) − ( γ o − γ v ) e γ o d (18) t p = 4 ǫ ⊥ γ e γ v e i( γ e − γ v ) d ( γ e + ǫ ⊥ γ v ) − ( γ e − ǫ ⊥ γ v ) e γ e d . (19)The effect of anisotropy is mainly seen in the transmis-sion coefficient t p because only here both permittivities ǫ ⊥ and ǫ k enter. When considering an epsilon-near-zero(ENZ) material with ǫ ⊥ →
0, then γ o , γ e → t s and t p is one andwe find t s → e − i γ v d and t p → e − i γ v d . (20)That means that similar to the perfect lensing effect,the evanescent waves are amplified by uni-axial mate-rial and in particular by a hyperbolic structure. Note,that this time the exponential which enhances the evanes-cent modes stems from the numerator of the transmissioncoefficient which makes this effect more robust againstlosses. Similarly, when considering an epsilon-near-pole(ENP) material with ǫ k → ∞ , then t s remains unaffectedbecause γ o remains unaltered. But for the extra-ordinarywave we have γ e ≈ √ ǫ ⊥ ω/c . It follows that t p → A e − i γ v d (21)with A ≡ ǫ ⊥ √ ǫ ⊥ k v γ v e i k v √ ǫ ⊥ d (cid:0) k v √ ǫ ⊥ + ǫ ⊥ γ v (cid:1) − (cid:0) k v √ ǫ ⊥ − ǫ ⊥ γ v (cid:1) e k v √ ǫ ⊥ d . (22)Therefore t p ∝ exp( − i γ v d ) and the evanescent modes areagain amplified, if the κ -dependent prefactor A is not toosmall.To summarize, we find that for ENZ and ENP fre-quencies evanescent waves in vacuum are amplified andthe dipole-dipole interaction can at least in principle be-come infinitely large if z = z ′ + d , i.e. if the donor andthe acceptor are both exactly deposited on the surfaceof the hyperbolic film which cannot be achieved in a real setup. But even if this condition is not perfectly met, wecan expect to find an amplified energy transfer across ahyperbolic slab at the ENZ and ENP frequencies. Fur-thermore, from the above derivation suggests that theENZ resonance is more advantageous for transmissionthan the ENP resonance, because the prefactor fullfillsfor ǫ > | A | < c ω c ωε ε k x type I γ e c ω c ωε ε k x type II e γ Figure 5: Sketch of the isofrequency lines of γ e for a dielec-tric uniaxial medium (blue ellipse) and a type I/II hyperbolicmaterial (solid black and light blue line). The asymptotes(dashed red lines) are given by k z = ± k x p | ǫ ⊥ | / | ǫ k | . Note, that both regimes of ENZ and ENP were dis-cussed in the context of diffraction suppressed hyperboliclensing [41], using the canalization regime for hyperboliclensing [64, 65] and directed dipole emission [42]. In thesecases, the advantage of using ENZ and ENP resonanceslies in the resulting very flat iso-frequency line of the ex-traordinary modes so that the group velocity is mainlydirected along the optical axis for a very broad band oflateral wavenumbers κ . This becomes clear when lookingat the asymptotes of the isofrequency lines of γ e as shownin Fig. 5. For large wavenumbers κ (evanescent regime)we have γ e ≈ κ p | ǫ ⊥ | / | ǫ k | . With this result we can eval-uate the ratio of the group velocity along the z-axis andthe group velocity perpendicular to the z-axis and obtain (cid:12)(cid:12)(cid:12)(cid:12) d ω d γ e d ω d κ (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) γ e κ ǫ k ǫ ⊥ (cid:12)(cid:12)(cid:12)(cid:12) ≈ s | ǫ k || ǫ ⊥ | . (23)Hence, at the ENZ and ENP resonance the slope of theisofrequency lines become infinitely small and the groupvelocity becomes mainly directed parallel to the opticalaxis so that the energy transferred between the donorand acceptor flows preferentially along the line connect-ing both, if both are placed along the optical axis. V. S-MATRIX CALCULATION OF THEDIPOLE-DIPOLE INTERACTION
Let us turn to a concrete numerical example. We con-sider a hyperbolic multilayer metamaterial made of alter-nating layers of silver and TiO which has for examplebeen used in the experiment in Ref. [37]. In most treat-ments of such multilayer materials, effective medium the-ory (EMT) is used which describes the multilayer struc-ture as a homogenous but uni-axial material, with an op-tical axis perpendicular to the interfaces. Within EMTthe effective permittivities of the multilayer structure canbe easily calculated and are given by ǫ ⊥ = f ǫ Ag + (1 − f ) ǫ TiO , (24) ǫ k = ǫ Ag ǫ TiO f ǫ TiO + (1 − f ) ǫ Ag , (25)where f is the filling fraction of silver and ǫ Ag / ǫ TiO arethe permittivites of the both constitutents of the mul-tilayer structure. For silver we use the Drude model inEq. (15). TiO is transparent in the visible regime. It’spermittivity ǫ TiO is nearly constant in that regime andcan be well described by the formula [68] ǫ TiO = 5 .
913 + 0 . λ − . . (26)where λ is the wavelength in micrometer. The effectivepermittivties ǫ ⊥ and ǫ k are shown in Fig. 6 for a fillingfraction of f = 0 .
35. It can be seen that at the edges ofthe hyperbolic bands we find the ENP and ENZ pointsat λ = 394 . λ = 551 . | ǫ ⊥ | / | ǫ k | . Bychanging the filling fraction the position of the ENZ andENP frequencies can be shifted. When increasing the fill-ing fraction the dielectric band in between both hyper-bolic bands in Fig. 6 will become smaller until f = 0 . or TiO /Ag multilayer mate-rial [66]. Therefore we will use mainly full S-matrix cal-culations [67] to determine the transmission coefficientsof the multilayer hyperbolic film. Here, we choose silveras first layer which is facing the donor. Hence, the lastlayer facing the acceptor is made of TiO . -80-40 0 40 80 300 400 500 600 700 R e ( ε ) λ (nm) ε || ε - εε type I type II | ε pe r p | / | ε pa r | λ (nm) εε type I type II Figure 6: Top: plot of the real part of the effective permit-tivites ǫ ⊥ and ǫ k for a Ag/TiO multilayer hyperbolic mate-rial with a silver filling fraction of f = 0 .
35. The vertical linesmark the edges of the hyperbolic bands of type I and typeII. Bottom: plot of | ǫ ⊥ | / | ǫ k | manifesting the ENP and ENZpoints at λ = 394 . λ = 551 . The transmission coefficient of the extra-ordinarywaves t p is plotted in Fig. 7 for a Ag/TiO multilayerstructure of thickness d = 120 with a filling fraction of f = 0 .
35 of silver. The calculation was made for N = 24layers, which means that the silver layers have a thick-ness of 3.5nm and the TiO layers have a thickness of6.5nm which is at the lower limit of a realizable struc-ture. In Fig. 7 the different coupled surface modes of thehyperbolic multilayer structure can be seen. Obviouslyin the type I hyperbolic band the coupled surface modescan exhibit negative group velocities which will result innegative refraction [44]. Furthermore it can be niceleyseen that all the coupled modes converge for large κd towards the ENZ and ENP frequencies (vertical lines).Therefore at those ENZ and ENP frequencies we havequite a number of surface modes with zero group velocityin direction of the interfaces of the multilayer structure.These are the waves which will contribute dominantlyto the F¨orster energy transfer having a group velocityrather along the optical axis (z-direction) than perpen-dicular to it as shown in Eq. (23). This is so, because forthe evanescent waves with large κ ≫ k v the exponentialsin the Green’s function in Eq. (4) introduce a cutoff [70]at κ ≈ / ( z − z ′ − d ) at the ENP and ENZ frequencies forthe κ -integral, since t p ≈ exp( κd ) in this case. Thereforethe major conributions stem from κ around 1 / ( z − z ′ − d ).That means in the ideal case z − z ′ = d all evanescentwaves contribute, but in the non-ideal realistic case onlyevanescent waves up to a finite value of κ ≈ / ( z − z ′ − d )will contribute to the energy transfer. type IItype I Figure 7: Plot of the transmission coefficient | t p exp( − κd ) | for the hyperbolic Ag/TiO multilayer structure d = 120 nmand N = 24 with filling fraction of f = 0 .
35. We have multi-plied the transmission coefficient with exp( − κd ) to compen-sate the exponential enhancement of the evanescent waveswith large κ ≫ ωc . The horizontal lines and arrows mark theedges of the hyperbolic bands. In Fig. 8 we show now the results for the enhancementfactors F x and F z of the F¨orster energy transfer by aAg/TiO hyperbolic multilayer structure. It can be seenthat the enhancement is especially large close to the ENZand ENP frequencies. Furthermore, the enhancementis more than two orders of magnitude larger than fora single silver film in Fig. 4, so that this enhancementof energy transfer due to the ENP and ENZ resonancesis much larger than the enhancement due to the thinfilm surface plasmons. Since the latter has already beenmeasured by Andrew and Barnes [16] the ENP and ENZenhancement should be easily measurable. Note, thatby increasing the number of layer N in the multilayerfilm or by increasing the distance of the donor-acceptorpair with respect to the film the exact S-matrix resultconverges to the EMT result.The position of the peaks can be explained be the factthat the dominant contributions to the energy transferstem from κd ≈ d/ ( z − z ′ − d ) = 6 in this case. Thevertical line in Fig. 7 is exactly at this value. It canbe seen that the frequencies at which the coupled surfacemodes cross this vertical line coincide with the resonancesof the energy transmission in Fig. 8. If we would increasethe donor-acceptor distance z − z ′ the vertical line wouldmove to smaller κ values in Fig. 7 so that the resonanceswill shift accordingly. This shifting of the resonances isshown in Fig. 9. Obviously, the enhancement around theENZ and ENP frequencies gets smaller and smaller, whenthe distance of the donor and acceptor with respect to thesurface is increased. However as can be seen in Fig. 10, | G zz | / | G zz V a c | λ (nm) N = 12N = 24N = 36N = 48N = 60EMT (a) Ag/TiO , F z | G xx | / | G xx V a c | λ (nm) N = 12N = 24N = 36N = 48N = 60EMT (b) Ag/TiO , F x Figure 8: Plot of F z and F x as function of wavelength. Againdonor and accepor are placed in a distance of 10 nm of eachinterface (i.e. z ′ = −
10 nm and z − d = 10 nm) of the film ofthickness d = 120 nm. Here the film is given by a Ag/TiO multilayer structure with N = 12 , , , ,
60 layers. Thefilling fraction of silver is f = 0 .
35. For comparision the exactand the EMT results are shown. The vertical lines correspondagain to the edges of the hyperbolic bands as in Fig. 6. there can still be an enhancement of 30 for the energytransfer inside the type I hyperbolic band if the distanceof the donor and acceptor with respect to the surface ofthe hyperbolic medium has relative large values as forexample 100 nm so that z − z ′ = 320 nm. VI. CONCLUSION
In conclusion, we have discussed the perfect lens effectand its pendant for hyperbolic metamaterials in the con-text of F¨orster energy transfer between a donor-acceptorpair placed on each side of the perfect or hyperbolic lens.We have demonstrated that in principle in both cases onecan have an infinitely large dipole-dipole interaction forprecisely defined positions of the donor-acceptor pair andwell-defined frequencies which coincide with the surface | G zz | / | G zz V a c | λ (nm) 10nm20nm30nm Figure 9: Plot of F z for a Ag/TiO multilayer structure with N = 24 and different donor-acceptor positions. The positionsare given by z ′ = −
10 nm , −
20 nm , −
30 nm with correspond-ing z − d = 10 nm ,
20 nm ,
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