Broadband Purcell effect: Radiative decay engineering with metamaterials
BBroadband Purcell effect: Radiative decayengineering with metamaterials
Zubin Jacob*
Department of Electrical and Computer Engineering,University of Alberta, Edmonton, AB T6G 2V4, Canada *[email protected]
Igor I. Smolyaninov
Department of Electrical and Computer Engineering,University of Maryland, College Park, MD 20742, U.S.A.
Evgenii E. Narimanov
Birck Nanotechnology CenterSchool of Electrical and Computer EngineeringPurdue University, West Lafayette, IN 47906, U.S.A.
Abstract
We show that metamaterials with hyperbolic dispersion supporta large number of electromagnetic states that can couple to quantumemitters leading to a broadband purcell effect. The proposed approachof radiative decay engineering, useful for applications such as singlephoton sources, fluorescence imaging, biosensing and single moleculedetection, also opens up the possibility of using hyperbolic metama-terials to probe the quantum electrodynamic properties of atoms and a r X i v : . [ phy s i c s . op ti c s ] O c t rtificial atoms such as quantum dots. Microcavities and photonic crystals are among the most promising sys-tems for enhancing spontaneous emission by the Purcell effect [1] and tosimultaneously collect the emitted photon in a given quantum state [2, 4].They form the test bed for cavity quantum electrodynamics experiments andaid the major advances in single photon sources . However, the high qualityof the resonance required for the cavity Purcell effect immediately puts arestriction on the spectral width of the emitter and hence on the possiblecompatible sources. For example, the reduced linewidth of quantum dotsat low temperatures which is ideally compatible with a microcavity for thedemonstration of the Purcell effect, is too wide at room temperatures —making such systems unviable for Purcell enhancement [3]. Moreover, otherquantum emitters such as molecules and nitrogen vacancy centers in diamondhave broad bandwidth emission not ideally compatible with cavity technol-ogy – and therefore an alternative, non-cavity based approach is needed [4].The resulting recent interest towards systems which show a broadband Pur-cell effect [5, 6, 7, 8] opened up the route to a number of new applications –from broadband single photon sources [9, 10] to strong coupling of emittersto plasmons [11, 12]. In particular, we proposed [13] that the large numberof electromagnetic states of a hyperbolic metamaterial lead to a divergencein the photonic density of states allowing broadband control over light mat-ter interaction at room temperature. As opposed to conventional methodsbased on closed cavity Purcell enhancement of spontaneous emission [4] or2pen cavity systems based on photonic crystal waveguides [14], our approachrelies on a radiative decay engineering approach using metamaterials [15],engineering the dielectric repsonse of the medium surrounding the emitterto provide new electromagnetic states for in-coupling. This effect has beendemonstrated experimentally [16, 17], and substantial enhancement was ob-served in the spontaneous emission rates of organic dyes and quantum dotswhen positioned near the surface of a metamaterial with hyperbolic disper-sion. In this paper, we expand on our original theoretical prediction of thiseffect [13] and present a quantitivate description of the spontaneous emissionrate enhancement.Metamaterials with hyperbolic dispersion, also known as indefinite media[18] lie at the heart of devices such as the hyperlens [19, 20] and non-magneticnegative index waveguides [21]. In an isotropic medium, the dispersion re-lation k (cid:15) = ω c defines a spherical iso-frequency surface in the k -space (seeFig.1(a)), thus placing an upper cut-off for the wavenumber – so that highwavevector modes simply decay away. In contrast to this behavior, a stronglyanisotropic metamaterial where the the components of the dielectric permit-tivity tensor have opposite signs in two ortogonal directions can support bulkpropagating waves with unbounded wavevectors. This can be most clearlyseen in the case of uniaxial anisotropy ( (cid:15) z ≡ (cid:15) (cid:107) , (cid:15) x = (cid:15) y ≡ (cid:15) ⊥ ) where the dis-persion relation for the extraordinary (TM-polarized) waves k (cid:107) (cid:15) ⊥ + k ⊥ (cid:15) ⊥ = ω c for (cid:15) (cid:107) (cid:15) ⊥ < z ( see Fig. 1(b) ) and thus does not limit the magnitude of the wavenumber.3uch high- k propagating modes allow for subwavelength imaging [22] andsubdiffraction mode confinement [23]. As we demonstrate in the present pa-per, these high wavenumber spatial modes in hyperbolic metamaterials alsohave strong effect on quantum-optical phenomena. In particular, they leadto a substantial enhancement of the spontaneous emission, without the needfor coupling to a slow waveguide mode or a counter propagating wave as ina resonator. As we will show below, this strong effect also circumvents theneed for three dimensional confinement of the emitter [5, 6, 7, 8] to achievethe Purcell effect.In the spirit of the Fermi’s golden rule, an increased number of radiativedecay channels due to the high- k states in hyperbolic media, available for anexcited atom ensures enhanced spontaneous emission. This can increase thequantum yield by overcoming emission into competing non-radiative decayroutes such as phonons. A decrease in lifetime, high quantum yield and goodcollection efficiency can lead to extraction of single photons reliably at a highrepetition rate from isolated emitters [4].We consider the classic example of radiative decay engineering using asubstrate which interacts with an emitter placed above it [24, 25]. The spon-taneous emission rate in this geometry can be obtained by a straightforwardgeneralization of the standard semiclassical theory [26]. In Fig. 2(a) weplot the corresponding emission decay relaxation time as a function of thedistance to the sample, for a hyperbolic metamaterial with the dielectric per-mittivity tensor (cid:15) x = 1 . . i, (cid:15) y = 1 . . i, (cid:15) z = − . i . In agreement4ith the qualitative arguments above, in the close vicinity of the substratethe availability of the large number of photonic states causes the photonsto be preferentially emitted into the metamaterial and the lifetime decreasesconsiderably. Even though the emitter is placed in vacuum and is coupledto the quasi continuum of vacuum states, the large number of states in themetamaterial leads to a Purcell effect without the need for confinement.The available radiative channels for the spontaneous photon emissionconsist of the propagating waves in vacuum, the plasmon on the metama-terial substrate and the the continuum of high wavevector waves which areevanescent in vacuum but propagating within the metamaterial. The cor-responding decay rate into the metamaterial modes when the emitter is ata distance a < d (cid:28) λ (where a is the metamaterial patterning scale) isΓ meta ≈ µ hd √ (cid:15) x | (cid:15) z | (1+ (cid:15) x | (cid:15) z | ) In the close vicinity of the hyperbolic metamaterial, the power from thedipole is completely concentrated in the large spatial wavevector channels(Fig 2(a) inset).The same evanescent wave spectrum when incident on a lossymetal or dielectric would be completely absorbed, causing a non-radiative de-crease in the lifetime of an emitter (quenching). On the contrary, the meta-material converts the evanescent waves to propagating and the absorptionthus affects the outcoupling efficiency of the emitted photons due to a finitepropagation length in the metamaterial.Along with the reduction in lifetime and high efficiency of emission intothe metamaterial, another key feature of the hyperbolic media is the direc-5ional nature of light propagation. Fig. 2(b) shows the field along a planeperpendicular to the metamaterial-vacuum interface exhibiting the beamlikeradiation from a point dipole. This is advantageous from the point of view ofcollection efficiency of light since the spontaneous emitted photons lie withina cone [27]. The group velocity vectors in the medium which point in thedirection of the Poynting vector are simply normals to the dispersion curve[Fig. 1]. For vacuum, these normals point in all directions and hence thespontaneous emission is isotropic in nature. In contrast to this behavior, thehyperbolic dispersion medium allows wavevectors only within a narrow regiondefined by the asymptotes of the hyperbola. Hence the group velocity vec-tors lie within the resonance cone giving rise to a directional spontaneouslyemitted photon propagating within the metamaterial. The beamlike natureof the photon in the metamaterial arising solely due to the hyperbolic dis-persion has to be distinguished from that obtained by the mode properties ofa resonant structure such as a micropost microcavity [4] or that of a guidedmode in a photonic crystal waveguide [14, 28].The actual realization of the hyperbolic metamaterial introduces devia-tions from the effective medium description [29]. Here we consider a practicalrealization of a hyperbolic metamaterial consisting of alternate layers of silver( (cid:15) Ag = − . . i ) and alumina ( (cid:15) Al O = 2 .
7) at a wavelength of λ = 365nm. The system consists of 8 layers, each of thickness a = 8 nm which is eas-ily achievable by current fabrication techniques. We compute and comparethe propagating wave spectrum which is routinely used in ellipsometric mea-6urements for extraction of effective medium parameters. Fig 3(a) shows theplane wave reflection and transmission coefficients computed using transfermatrix techniques in the layered realization superimposed on the effectivemedium prediction. Since a (cid:28) λ , effective medium theory holds in a broadbandwidth (Fig 3 (a) inset). The lifetime of the dipole on top of the layeredh-MM at a distance of d = λ/
20 shows a factor of F p = 10 decrease fromthe free space lifetime . It should be noted that non-radiative contributionto reduced lifetime (also known as quenching due to lossy surface waves) isconsiderably decreased when using thin layer of metal as building blocks ascompared to thick metal [26]. To ascertain the role of metamaterial modeswe compare the lifetime with a single period of Ag / Al O . The hyperbolicsystem has a lifetime lower by a factor of 2 as compared to a thin pieceof metal. This decrease in the lifetime is due to the metamaterial stateswith large wavevectors as shown by the spectrum of power emitted by thedipole in the vicinity of the layered structure [Fig.3(b) inset]. At a distanceof d = λ/
10, the efficiency of emission into metamaterial modes (evanescentin air but propagating within the layered structure) is η meta = 77%. Notethat the layered realization shows transmission of large wavevector stateswhen compared to a high index dielectric (Fig. 3(b)), as expected from theeffective medium theory.In conclusion, we have shown that the electromagnetic states of a hyper-bolic metamaterial lead to a non-resonant Broadband Purcell effect withoutthe need for confined emitters. The proposed device based on hyperbolic7etamaterials is compatible with a wide variety of sources and capable ofroom temperature operation due to the broad bandwidth enhancement ofspontaneous emission and directional photon emission. Our work paves theway for using metamaterials for applications in quantum nanophotonics rang-ing from single photon sources to fluorescence based sensing [30].The work was partially supported by ARO MURI. Z.J. wishes to acknowl-edge support from NSERC Discovery and CSEE POP.8 x k z k x k z (a) (b) Figure 1: a ) Dispersion relation for an isotropic medium. The blue arrowdenotes an allowed wavevector, whereas the normal to the dispersion rela-tion gives the direction of the group velocity (red arrow). ( b ) Hyperbolicdispersion relation allowing large number of electromagnetic states with un-bounded values of the wavevector (blue arrow). The group velocity vectors(red arrow) lie within a cone which implies light propagation in such mediais inherently directional. 9 | Ey | x / λ z/ λ -1 -2 (b) distance/ λ τ / τ (a) Spatial frequency: k x /k P o w e r ( a . u . ) ε x > 0 y ε z < 0 ε > 0 Figure 2: a ) Spontaneous emission lifetime of a perpendicular dipole abovea hyperbolic metamaterial substrate (see inset). Note the lifetime goes tozero in the close vicinity of the metamaterial as the photons are emittednearly instantly. Most of the power emitted by the dipole is concentratedin the large spatial modes (evanescent in vacuum) which are converted topropagating waves within the metamaterial. (inset) b ) False color plot ofthe field of the point dipole in a plane perpendicular to the metamaterial-vacuum interface (see inset of (a)) depicting the highly directional nature ofthe spontaneous emission (resonance cone).10 k x /k | R | , | T | −10 Wavelength (nm) Re ε || Re ε ⊥
10 12 14 −2−1.5−1−0.500.511.5 k x /k (evanescent waves) l o g | T | Spatial frequency (kx/k0) P o w e r ( a . u . ) (a) (b) Figure 3: a ) Comparison of the reflection and transmission amplitudes ofplane waves incident on the planar multilayer realization of the hyperbolicmetamaterial and effective medium theory. The metamaterial system con-sists of 8 alternating subwavelength layers of Ag/Al O . Green and redcircles correspond to reflection and transmission computed using transfermatrix methods and the black superimposed line is calculated from effectivemedium theory. (inset) Hyperbolic dispersion is achieved in a broad band-width around λ = 350 nm as the dielectric constants are of opposite signsin perpendicular directions. b ) Transmission of large wavevector waves bythe layered hyperbolci metamaterial (red solid line) as compared to a highindex dielectric (black dashed line). The inset shows the power spectrum ofthe dipole at a distance of d = λ/
20 from the layered structure. Most of thespontaneous emission occurs into high wavevector states which propagatewithin the h-MM
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