Plasmonic Purcell factor and coupling efficiency to surface plasmons. Implications for addressing and controlling optical nanosources
G. Colas des Francs, J. Barthes, A. Bouhelier, J.C. Weeber, A. Dereux
PPlasmonic Purcell factor and coupling efficiency tosurface plasmons. Implications for addressing andcontrolling optical nanosources
G. Colas des Francs, J. Barthes, A. Bouhelier, J.C.Weeber, A. Dereux
Laboratoire Interdisciplinaire Carnot de Bourgogne (ICB) UMR 6303CNRS-Universit´e Bourgogne Franche-Comt´e 9 Av. A. Savary, BP 47 870F-21078 DIJON Cedex FRANCEE-mail: [email protected]
A. Cuche, C. Girard
Centre d’Elaboration de Mat´eriaux et d’Etudes Structurales (CEMES), CNRS,29 rue J. Marvig, Toulouse F-31055, France.
Abstract.
The Purcell factor F p is a key quantity in cavity quantumelectrodynamics (cQED) that quantifies the coupling rate between a dipolaremitter and a cavity mode. Its simple form F p ∝ Q/V unravels the possiblestrategies to enhance and control light-matter interaction. Practically, efficientlight-matter interaction is achieved thanks to either i) high quality factor Q atthe basis of cQED or ii) low modal volume V at the basis of nanophotonicsand plasmonics. In the last decade, strong efforts have been done to derive aplasmonic Purcell factor in order to transpose cQED concepts to the nanocale,in a scale-law approach. In this work, we discuss the plasmonic Purcell factorfor both delocalized (SPP) and localized (LSP) surface-plasmon-polaritons andbriefly summarize the expected applications for nanophotonics. On the basisof the SPP resonance shape (Lorentzian or Fano profile), we derive closed formexpression for the coupling rate to delocalized plasmons. The quality factor factorand modal confinement of both SPP and LSP are quantified, demonstrating theirstrongly subwavelength behaviour. Submitted to:
J. Opt. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a r lasmonic Purcell factor and coupling efficiency to surface plasmons.
1. Introduction
Nanophotonics permits light-matter interaction at thenanoscale, down to the single photon/single atom level.The motivations are notably sensitive sensing with ap-plications such as nano-optical imaging (surface anal-ysis), environnemental health (air pollutants, infec-tious agents detection), security (explosive detection)or healthcare (cancer early diagnosis, theranostics) andthe miniaturization of photonics components for onchip integrated ultrafast devices. The optical cross-section is a simple way to characterize the efficiencyof light-matter interaction. For a molecule, it is typ-ically σ (cid:39) − m , that has to be compared to thefocus area of a diffraction limited beam, of the orderof ( λ/ (cid:39) − m in the visible domain. This un-suitability between the light confinement and the activesize of the molecule points out the difficulty of so-callednanoscopy [1]. In the last decades, several strategieshave been proposed to increase the efficiency of light-matter interaction. i) Increasing the absorption cross-section by working at low temperature; indeed, in thelimit of very low temperature ( T <
10 K), the molecu-lar absorption cross-section increases up to the diffrac-tion limit σ = 3 λ / π revealing that the molecule ab-sorbs almost all the incoming light of a focused beam[2, 3]. ii) Increasing the duration of the interaction byplacing the molecule inside an optical microcavity pre-senting a high quality factor Q [4]. iii) Confining theexcitation beam below the diffraction limit thanks tonear-field optics [5, 6, 7, 8, 9] or plasmonics [10, 11].Quantitatively, the efficiency of light-matter inter-action can be inferred from the so-called cooperativityparameter C . The meaning of this parameter is eas-ily understood from a classical point of view [12, 13].In free-space, the cooperativity can be expressed asthe ratio between the resonant atomic cross-section σ and the effective area A eff = πw of a gaussian beamwith a beam waist w ; C ≈ σ /A eff . Therefore, thecooperativity quantifies the suitability between the fo-cused spot and the molecule active area. In an opticalFabry-Perot cavity, it increases to C = 4 C F /π where F is the finesse of the cavity. The cavity enhances thefree-space cooperativity by the number of wave roundtrips F /π inside the cavity and an additionnal factorof four accounting for the intensity enhancement at amode antinode. In cavity quantum electrodynamics(cQED), the cooperativity writes for a single atom C = g κ cav n Γ (1)where g is the coupling rate between the atom and thecavity mode. κ cav and Γ refers to the cavity lossesrate and the atom decay rate in vacuum, respectively( n Γ is the decay rate in the homogeneous mediumof optical index n ). The strong coupling regime, g (cid:29) κ cav , Γ ( C (cid:29)
1) leads to a reversible energyexchange between the cavity and the atom. In theweak coupling regime, the cavity opens a new channelfor the atom (irreversible) relaxation with a decay rateΓ = (1+2 C )Γ . The Purcell factor quantifies the effectof the cavity on the atom decay rate and writes F p = Γ cav n Γ = 2 C (2)with Γ cav = Γ − n Γ the modification of the decaydue to the optical microcavity. It also expresses [14] F p = Γ cav n Γ = 34 π (cid:18) λ em n (cid:19) QV eff (3)where Q is the quality factor of the cavity and V eff the effective volume of the cavity mode involved in thecoupling. λ em is the emission wavelength of the atom.This expression is equivalent to the classical descriptionwith the finesse for a Fabry-Perot cavity.Remarkably, the Purcell factor expression (3)points out that spontaneous emission can be efficientlycontrolled in an optical cavity presenting a highquality factor and/or a strongly confined mode.However, high Q cavities are obtained at the priceof low modal (diffraction limited) confinement [4].In this context, molecular plasmonics proposes anew strategy for light–matter interaction [15, 16, 17].The strong confinement of surface plasmon polaritonsinsures efficient coupling at a deeply subwavelengthscale whereas cavity-QED increases the duration ofinteraction. At this point, we have to mention thatthe Q factor entering the Purcell factor is the lower ofthe cavity factor and the atomic resonance. Actually,1 /Q = 1 /Q at + 1 /Q cav where Q at and Q cav arethe quality factor of the atom emission spectrumand cavity, respectively [18]. That is why cQEDgenerally works at low temperature where atomicresonance is sufficiently narrow so that cavity modifiesthe spontaneous emission ( Q at (cid:29) Q cav and Q ≈ Q cav ).On the contrary, low Q factor of plasmon resonancepermits to work at room temperature and let envisionhigh-speed optical devices [19, 20]. This paves the wayto ultrafast control at the nanoscale.The Purcell factor describes the emitter-cavitycoupling as a function of the optical cavity properties,independently of the emitter properties. Particularly,the best coupling efficiency is achieved for high Q/V ratio, that occurs either for a narrow resonance, or adeeply confined mode. The Purcell factor is thereforea key parameter to transpose cQED concepts toquantum plasmonics [15, 21, 22, 23, 24, 25, 26, 27, 28].In this article, we first summarize the derivationof the Purcell factor in an optical microcavity withparticular attention to the underlying hypothesis(section 2). In section 3, we briefly discusssome expected applications of efficient emitter-SPP lasmonic Purcell factor and coupling efficiency to surface plasmons.
2. Purcell Factor
The spontaneous emission at the angular frequency ω em = 2 πc/λ em from the excited state | b (cid:105) to theground state | a (cid:105) of an excited atom presents a decayrate that follows Fermi’s golden ruleΓ( r ) = 2 π (cid:126) (cid:88) k n |(cid:104) a, k n | H I | b, (cid:105)| δ ( ω em − ω kn ) (4) H I = − ˆ p · ˆ E ( r ) is the interaction hamitonian describingthe coupling of an atom to an electromagnetic fieldwithin the dipolar approximation, taken at the position r of the atom. The operators ˆ p and ˆ E refer to theatomic transition dipole moment and the electric field,respectively. In the following, we are interested inthe effect of the cavity on the spontaneous emissionrate. Therefore, we separate the decay rate into thefree-space contribution (in an homogeneous medium ofoptical index n ) and the cavity contribution n Γ = 2 π (cid:126) (cid:88) k n |(cid:104) a, k n | H | b, (cid:105)| δ ( ω em − ω kn ) (5)= n p ω em πε (cid:126) c , and (6)Γ cav = 2 π (cid:126) (cid:88) k n |(cid:104) g, k n | H cav | e, (cid:105)| δ ( ω em − ω kn ) . (7)In addition, we assume a single-mode cavity, resonantat ω c , thereforeΓ cav = 2 π (cid:126) |(cid:104) a, | H cav | b, (cid:105)| δ ( ω em − ω c ) (8) indicates a photon into the cavity mode. Thecoupling rate g between the atom and the cavity mode(see Eq. 1) obeys (cid:126) g = |(cid:104) a, | H cav | b, (cid:105)| . The electric-field operator associated to the single-mode cavitywrites [12, 29]ˆ E cav ( r ) = i (cid:114) (cid:126) ω c ε ε V f ( r )ˆ a + h.c. (9)where V is the quantization volume, ˆ a the bosonoperator and f ( r ) describes the spatial variations ofthe mode into the cavity [ | f ( r ) | = 0 at a node and | f ( r ) | = 1 at an antinode]. To achieve this expression,the classical electric field is expressed E cav ( r , t ) = i (cid:114) (cid:126) ω c ε ε V f ( r ) e − iω c t + c.c (10)and is normalized with respect to the energy (cid:126) ω c = 12 (cid:90) [ ε ε ( r ) E ( r , t ) + µ H ( r , t )] d r (11)= (cid:126) ω c (cid:15) V (cid:90) ε ( r ) | f ( r ) | d r , (12)where we assumed a non dispersive medium. Finally,the mode volume obeys V = 1 ε (cid:90) ε ( r ) | f ( r ) | d r . (13) Spectral shape of the resonance
In the case of a lossycavity, the dirac distribution in (8) is replaced by thedensity of modes per unit angular frequency N ( ω )(unit: s.rad − ). Moreover, the profile of the moderesonance is assumed to be Lorentzian δ ( ω − ω c ) → N ( ω ) = 1 π κ cav / ω − ω c ) + κ cav / . (14)Defining the resonance quality factor Q = ω c /κ cav , wecan rewrite N ( ω ) = 2 Qπω c
11 + 4 Q ( ω − ω c ω c ) . (15) Spatial profile
Inserting the electric field operator(Eq. 9) into the interaction hamiltonian, we achieve |(cid:104) g, | H cav | e, (cid:105)| = p (cid:126) ω c ε ε V | u · f ( r ) | (16)where we introduced the dipole moment orientation u ( i.e p = p u ). Purcell factor
Finally, using eq. (15) and (16) thecavity contribution to the decay rate (eq. 8) simplifiestoΓ cav = 2 p (cid:126) ε ε | u · f ( r ) | QV
11 + 4 Q ( ω em − ω c ω c ) , (17)so that the normalized decay rate writes [using eq. (6)and λ em = 2 πc/ω em ]Γ cav n Γ = 34 π (cid:18) λ em n (cid:19) QV | u · f ( r ) | Q ( ω em − ω c ω c ) . (18) lasmonic Purcell factor and coupling efficiency to surface plasmons. The cavity contribution to thedecay rate obeysΓ cav n Γ = F p | u · f ( r ) | Q ( ω em − ω c ω c ) , with (19) F p = 34 π (cid:18) λ em n (cid:19) QV where F p is the so-called Purcell factor, f ( r ) reveals theposition dependency (from cancellation at a node tomaximum effect at an antinode) and the denominatorfactor [1 + 4 Q ( ω em /ω c − ] shows the effect ofthe detuning between the emission frequency and thecavity resonance. Finally, the emitter couples to modespresenting a polarisation along the dipole moment[quantified by the term | u · f ( r ) | ].In the following, we are interested in definingthe Purcell factor for a dipolar emitter coupled to aplasmonic nanostructure. It is therefore useful to recallthe hypothesis done to demonstrate the Purcell factorexpression (3): • the Purcell factor is associated to a given mode ofthe cavity • the cavity resonance follows a Lorentzian shape • the mode volume can be estimated from expres-sion (13), assuming a non dispersive medium. Itequivalently writes V = (cid:82) ε ( r ) | E ( r ) | d r M ax [ ε | E ( r ) | ] (20)where E ( r ) is the electric field associated to thecavity mode [ f ( r ) = E ( r ) /M ax [ | E ( r ) | ] describesthe mode profile]. The full controlof spontaneous emission in 3D optical cavity isa technological challenge and is bandwidth limitedso that simpler configurations have been proposed.In particular, fluorescence emission into photonicnanowires can be enhanced over a large spectrum rangeso that it has been widely studied in the last decade[30, 31]. To derive the Purcell factor near a waveguide,we define an arbitrary quantization length L . Theelectric-field operator and the density of mode write,respectively [32]ˆ E ( r ) = i (cid:115) (cid:126) ω c ε ε A eff L f ( r )ˆ a + h.c. , with (21) A eff = (cid:82) ε ( x, z ) | E ( r xz ) | dxdzM ax [ ε | E ( x, z ) | ] , and (22) N ( ω ) = Lπ v g , with v g = dωdk g (23) A eff defines the mode effective area and v g is thegroup velocity of the guided mode. N D = 1 /πv g is the density of guided modes. We then proceed aspreviously and the Purcell factor for the guided modesimplifies to F p = Γ guided n Γ = 34 π ( λ em /n ) A eff n g n . (24)The Purcell factor near a photonic nanofiber isgoverned by the group index n g = c/v g of the guidedmode and its transverse confinement A eff . HighPurcell factor necessitates low group velocity ( e.g. near the band-edge of the dispersion relation [33, 34])and/or a highly confined mode. Wefinally consider the one-dimensional (1D) planarwaveguide. The density of guided modes obeys N D = ω/ πn eff n g where n eff refers to the mode effectiveindex. The Purcell factor becomes F p = Γ guided n Γ = 34 ( λ em /n ) L eff n eff n g n , with (25) L eff = (cid:82) ε ( z ) | E ( z ) | dzM ax [ ε | E ( z ) | ] (26) L eff is the mode effective length and characterizes itsconfinement. β -factor) So far, wehave introduced the Purcell factor that quantifies thecoupling strength between a dipolar emitter and aphotonic structure. Since the emitter could relax to itsground state thanks to various channels (cavity mode,leakage, non radiative energy transfer), it is also usefulto define the coupling efficiency to the cavity mode. Itis the so-called β -factor β = Γ cav Γ tot = Γ cav Γ cav + Γ other (27)where Γ tot = Γ cav + Γ other is the total decay ratethat includes all the relaxation channels. In a singlemode cavity, the relaxation channels are generally thecavity mode with the decay rate Γ cav , and coupling toleaky modes. Usually, leakage are of the same orderof magnitude than the initial radiative rate (Γ other (cid:39) n Γ ) so that β (cid:39) Γ cav / (Γ cav + n Γ ) (cid:39) F p / (1+ F p ).For instance, a Purcell factor of about F p ∼ β ∼ β factor is required for low threshold lasering [35]. In presence of a lossy and dispersive medium, themodification of the decay rate can be described eitherwithin classical Lorentz model of an oscillating dipole[36, 37, 38] or within the full quantum description [39].In both cases, this leads to the following expression forthe rate modificationΓ u ( r ) n Γ = 6 πn k Im [ G uu ( r , r , ω em )] (28) lasmonic Purcell factor and coupling efficiency to surface plasmons. G is the Green’s tensor associated to theemitter surroundings and G uu refers to its diagonalcomponents along the dipolar direction u . Thisexpression quantifies the modification of the decayrate in a complex surroundings and will be ourstarting point to determine the effect of plasmonicnanostructures on the fluorescence decay rate. Theexpression (28) is a generalization of the Fermi’s goldenrule to dispersive and lossy surroundings. Indeed, theFermi’s golden rule (4) can be recast in the formΓ( r ) = 2 πg ( r , ω em ) N ( ω em ) , (29) (cid:126) g ( r , ω em ) = |(cid:104) a, | H cav | b, (cid:105)| (30)= (cid:114) (cid:126) ω c ε ε V | p · f ( r ) | (31)where we introduced the coupling strength g anddensity of modes N ( ω ), as discussed above. It is well-known from scattering formalism that the partial localdensity of states (P-LDOS) is related to the Green’sdyad (unit: s.m − ) [38, 40] ρ u ( r , ω em ) = k πω Im [ G uu ( r , r , ω em )] (32)so that Eq. (28) and (4) are fully equivalent in anon-absorbing medium. The P-LDOS also writes fromcQED considerations ρ u ( r , ω ) = f u ( r )2 (cid:15) V N ( ω ) . (33)Therefore, the P-LDOS grasps the spatial dependencyof the coupling to the cavity mode by f u ( r ) and thedensity of available modes for light emission ρ ( ω ) = N ( ω ) /V , per unit angular frequency and unit volume.
3. Plasmonic addressing and control of opticalnanosources
Before discussing the plasmonic Purcell factor, itis worthwile to briefly introduce some expectedapplications of coupled emitter-SPP configurations.We indicate here the main applications and refer thereader to the literature for more complete descriptions.
Surface Enhanced Raman Spectroscopy (SERS) isprobably among the first application of molecularmaterial coupled to plasmonics nanostructures [41] andis now available at the single molecule level [42] sothat ultrasensitive chemical or biosensors are expected[43]. In the context of Purcell factor, we would liketo mention that the SERS efficiency follows a ∝ Q /V law with the LSP properties [44].Following SERS, metal enhanced fluorescence(MEF) leads to enhancement factor of the order oftens [45, 46, 47, 48, 49, 50] with possible applications to nanotheranostics [51, 52]. The fluorescence increaseresults from excitation field enhancement and emissionrate modification (Purcell effect). However, due to thenon radiative energy transfer to metal nanostructures,it is crucial to distinguish the radiative from nonradiative rates. At the end, the critical parameteris the quantum yield of the emitter so that thetotal enhancement is generally limited by the intrinsicquantum yield of the fluorescent molecule [53, 54, 55,56]. Remarkably, coupling an emitter to a plasmonicnanostructures opens the way to the control of thephotophysical processes [57], notably blinking effect[58, 59, 60] and photobleaching [61], but also to modifythe ratio between magnetic and electric allowed dipolartransitions [62, 63]. Since a plasmonics nanostructure efficiently interfacesa single molecule to far-field radiation [64, 65, 66, 67]the concept of optical nano-antenna has emerged sincea decade [68, 69, 70]. Optical nano-antennas rely onplasmonics nanostructures to efficiently redirect thefluorescence emission and cQED-like description givesan insight of the coupling mechanism [15, 71]. Re-cently, a Purcell factor up to 1000 keeping a reason-able quantum yield and with a collection efficiency of84% was demonstrated [72]. Moreover, optical nano-antenna could efficiently interface molecular fluores-cent emission and a nanophotonic waveguide [73] withpossible applications to realize a platform for quantumoptics. In addition, coupling a single photon source toan optical nano-antenna permits to control its emissioncadency [74, 75, 76, 77, 78, 79, 80, 81, 82, 83]. Realiza-tion of indistiguishable single photons is also a majorissue [84, 85].
Taking advantage of the analogy between opticalmicrocavities and plasmonic nanostructures, theconcepts of plasmon nanolaser and amplifier wereproposed [86, 87, 88]. It consists of a gain medium incontact to a metal nanostructure so that stimulatedemission of plasmon occurs. The efficiency of thestimulated emission strongly depends on the Purcellfactor [89, 90, 91, 92].
Finally, since an emitter can be efficiently coupledto a surface plasmon, it has been proposed to useplasmon to couple two emitters for applications suchas long range resonant energy transfer (above 10 nm)[93, 94, 95, 96, 97] or qubits entanglement [98, 99]. lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 1.
Decay rate as a function of the distance to a goldor PEC surface. a) Dipole moment parallel to the surface.b) Dipole moment perpendicular to the surface. The emissionwavelength is λ em = 670 nm.The gold permittivity is taken fromtabulated data [102]. In the following, we discuss the plasmonic Purcellfactor for typical configurations. We first consider thewell-known case of extended metal film that is thesimplest case of delocalized SPP enabling to clearlyidentify the coupling mechanisms [100, 101].
4. Quantum emitter decay rate near a metalmirror
Figure 1 presents the dipolar total decay rate as afunction of the distance to a gold surface [103]. Theperfect mirror (perfect electric conductor - PEC) caseis also shown for comparison [104]. Far from thesurface, we observe a typical interference pattern sincethe driving reflected field has to be in phase withthe dipolar oscillation to enhance the emission rate.When the emitter touches the surface, the decay rate
Figure 2. a) Dipolar emission as a function of the wavectorparallel to the surface k (cid:107) . b) Resonant behaviour of the Au/airSPP. The Lorentzian fit is peaked at the SPP effective index n SPP = 1 .
036 and with a FWHM n (cid:48)(cid:48) SPP = 2 . · − ( L SPP = λ em / πn (cid:48)(cid:48) SPP = 20 µ m). The dipole is perpendicular to thesurface. The emission wavelength is λ em = 670 nm. Direct(free-space) dipolar emission is not included in P ( k (cid:107) ). presents a finite value for the perfect mirror. It fullycancels for a dipole parallel to the surface (Fig. 1a)whereas it doubles for a perpendicular dipole (Fig. 1b),in agreement with the image dipole induced into theconductor. In case of real metal, the behavior is ratherdifferent close to the surface due to the apparition ofnew decay channels such as excitation of SPP and nonradiative energy transfer to the metal film[100, 101]. The various contributionsto the total decay rate are easily determined fromthe Sommerfeld expansion of the dipolar emission,represented in Fig. 2 [101, 103]. Indeed, since themetal/air interface is invariant along r (cid:107) = ( x, y ),it is possible to expand the Green’s dyad over thewavenumber k (cid:107) so that the total decay rate (Eq. 28) lasmonic Purcell factor and coupling efficiency to surface plasmons. u ( d ) n Γ = (cid:90) ∞ P ( k (cid:107) ) dk (cid:107) , with (34) P ( k (cid:107) ) = P ( k (cid:107) ) n P (35)where u = (cid:107) , ⊥ represents the orientation of theemitter and d is the distance to the metal surface. P = ω em | p | / π(cid:15) c is the power radiated bythe oscillating dipole in vacuum. P ( k (cid:107) ) is thedipolar emission power spectrum (in the k (cid:107) -space) andrepresents the dipolar emission at a surface wavector k (cid:107) . P ( k (cid:107) ) is P ( k (cid:107) ), normalized with respect to P (unitof P : meter).It is useful to distinguish the radiative waves forwhich k (cid:107) ≤ n k that contributes to the radiative rateΓ rad and the evanescent waves k (cid:107) > n k . Evanescentwaves corresponds to either SPP or lossy waves (LSW)[100, 101]. Particular attention has to be paid toSPP contribution, shown in Fig. 2b. It presents aLorentzian profile that permits to derive a closed formexpression for the Purcell factor as we will discuss indetails later. Finally, the large wavectors are associatedto electron scattering losses. These so called lossywaves are responsible for fluorescence inhibition closeto the metal surface. Let us mention that an additionalnon radiative energy transfer, namely electron-holepair creation in the metal, could also occur at veryshort separation distances ( d < β (cid:107) factor (Fig. 3c). At the opposite, we observethat a perpendicular dipole is efficiently coupled to aSPP (Fig. 3b). About 90% of the dipolar emissioncouples to the Au/air SPP for a separation distanceof d = 200 nm, see Fig. 3c. This high β -factororiginates from a low radiative rate due to a destructiveinterference between the direct and reflected dipolarfields (Fig. 3b). At shorter distances where the SPPrate is higher, we achieve β ⊥ = 70% for d = 40 nm, seeFig. 3c.At this point, a closed form expression ofthe plasmonic Purcell factor is achievable. Sincethe emitted power follows a Lorentzian profile nearthe SPP resonance, the integration over the SPPcontribution leads to [105]Γ SP P n Γ = π P ( k SP P ) L SP P (36)
Figure 3.
Contributions to the total decay rate as a functionof the distance to the gold surface for a dipole parallel (a) orperpendicular (b) to the surface. c) Coupling efficiency to theSPP. The emission wavelength is λ em = 670 nm. In b), thegreen dots represents the SPP contribution calculated assuminga lossless metal. lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 4. a) Contributions to the total decay rate as a functionof the distance to the gold surface. b) Coupling efficiency to theSPP. The emission wavelength is λ em = 525 nm and the dipolaremitter is perpendicular to the surface. This extends the Purcell factor definition to a (lossy)SPP. Care has to be taken when interpretatingplasmonic Purcell factor, in particular the role of losses.We derive in Appendix A.2 the SPP contributionassuming a lossless metal. It perfectly matches theSPP contribution calculated for a real lossy metal (seegreen dots in Fig. 3b). So the plasmonic Purcell factordoes not depend on the propagation length [105, 106],although it explicitly appears in the denominator ofexpression (36). Mathematically, the integral of theLorentzian resonance gives the number of supportedmodes and does not depend on ohmic losses [107].This is finally not surprising since SPP rate defines thecoupling efficiency to the propagating SPP, no matterof how energy is dissipated afterward.As a consequence, the Purcell factor as well asthe coupling efficiency to a SPP can be high, even inpresence of strong losses. We plot in Fig. 4 the differentcontributions to the decay rate at λ em = 525 nm. Dueto strong losses in gold, it is difficult to separate theSPP and LSW contributions to the total decay rate. Figure 5. a) Radiative and scattering rates at the emissionwavelength λ em = 525 nm. b) Absorption rate (Γ abs =Γ rad − Γ scatt ) as a function of the wavelength. The dipole isperpendicular to the surface. This behaviour is discussed in detail in Appendix A.3.For simplicity, we estimate the SPP decay rate fromthe emitter power integrated over 1 < k (cid:107) /k < . λ em = 670 nm case (compare with Fig. 3).Although we expect strong losses in the metal due tointerband transitions, we still observe β ⊥ = 80% for d = 100 nm, see Fig. 4b. The SPP has an effectiveindex n SP P = 1 .
10 (effective wavelength λ SP P =477 nm) but an extremelly short propagation length L SP P = 640 nm. Therefore, the SPP presents onlya single spatial oscillation over its propagation length.This quasi-mode, although efficiently excited would notbe of interest for the control of a dipolar emission.So far, we only discussed the emission process ofthe dipolar emitter. Another quantity of interest is thecollection efficiency. We plot in figure 5 the scatteringrate Γ scatt , defined as the dipolar power dissipated inthe far-field zone. The difference with the radiativerate Γ rad is due to absorption in the metal but doesn’tdepend on the distance to the metal surface [103] (seealso Appendix A.4). We observe in Fig. 5b that lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 6.
Collection efficiency as a function of the detectionnumerical aperture (NA). The emission wavelength is λ em =670 nm and the dipole is perpendicular to the surface. absorption becomes negligible above λ >
650 nm.As last quantity, we plot in Fig. 6 the collectionefficiency η = Γ scatt ( N A ) / Γ rad as a function of thedetection numerical aperture (NA). Γ scatt ( N A ) refersto the power scattered in a given NA. A NA=0.6air objectif collects 10% of the emitted signal so thatstrategies has to be developed to improve this efficiencysuch as surface plasmon coupled emission (SPCE) [46]or grating decoupler [108, 109, 110].
Micro-optical cavities are generally characterized by theirmode confinement and quality factor. For comparisonpurposes, it is therefore convenient to estimate alsoSPP confinement and quality factor.Firstly, the SPP quality factor is estimated fromthe resonance Lorentzian profile (Fig. 3b). At λ em =670 nm it comes Q = k spp / ∆ k spp = n spp / n (cid:48)(cid:48) spp = 192.Secondly, we would like to characterize the modeconfinement. For this purpose, we identify the SPPrate Γ SP P /n Γ (Eq. 36) to the Purcell factor (Eq.25). Remembering the hypothesis done to achieve thePurcell factor expression, we calculate Γ SP P at thegold/air interface where the SPP field amplitude is thehighest and sum the contributions for dipoles along thethree directions (so that | u · f | = ). Indeed, the decayrate obeys thenΓ cav,x n Γ + Γ cav,y n Γ + Γ cav,z n Γ = F p , (37) ≡
34 ( λ em /n ) L eff n eff n g n . (38)We estimate the SPP effective and group indices n spp = 1 .
036 and n g = 1 .
16, respectively, from thedispersion relation. From identification to the Purcellfactor (Eq. 38), we can attribute the effective length
Figure 7. a) Dipolar emission power spectrum P ( k (cid:107) ) abovea 50 nm thin gold film deposited on a glass substrate (opticalindex n sub = 1 . n spp = 1 . n (cid:48)(cid:48) SPP = 6 . · − , q = 50 and b = − . n spp = 1 . n (cid:48)(cid:48) SPP = 0 . q = 4 . b = 16 .
4. The dipole is perpendicular to the surface. Theemission wavelength is λ em = 670 nm. L eff = 203 nm (cid:39) . λ spp /
2) to the gold/air SPP(Purcell factor of F p = 2 . δ/ λ/ k (cid:113) n spp − L eff = (cid:82) | E ( z ) | dzM ax [ | E ( z ) | ] = (cid:90) ∞ e − z/δ dz = δ δ is a good parameter to estimate the SPPconfinement.For λ em = 525 nm, we achieve Q = 8 and L eff = 223 nm (cid:39) ( λ spp /
2) (and δ/ We now turn to thethin metal film case. We consider a 50 nm goldfilm deposited on a glass substrate, and an emissionwavelength λ em = 670 nm. The Au/air and Au/glassSPP modes couple and to form a leaky and a boundSPP. We determine their characteristics using thereflection pole method [111]. The leaky mode isconfined at the gold/air interface and has an effectiveindex n SP P = 1 . lasmonic Purcell factor and coupling efficiency to surface plasmons. L SP P = 8 . µ m ( n (cid:48)(cid:48) SP P = 6 . · − ). The boundSPP is confined at the gold/glass interface ( n SP P =1 . n (cid:48)(cid:48) SP P = 1 . · − corresponding to L SP P =3 . µ m).Figure 7a presents the dipolar emitted power P ( k (cid:107) ). We observe a similar behaviour than above agold mirror with three different contributions; the ra-diative waves ( k (cid:107) ≤ n k ), the two SPP contributionsand the lossy waves for high wavenumbers k (cid:107) . Notethat the leaky plasmon also contributes to the radia-tive waves via leakage into the glass substrate (so-calledsurface plasmon coupled emission -SPCE) [112]. Weagain observe the strong increase of LSW for a dipolaremitter close to the metal film. Finally, the two peaksnear k (cid:107) = 1 . k and k (cid:107) = 1 . k reveal the SPP con-tributions. Their resonance profiles are shown in de-tails on Fig. 7(b,c). These two modes result from thecoupling of the (leaky) Au/air and (bound) Au/glassSPP of a single interface. Therefore their resonant be-haviour does not follow a Lorentzian profile anymorebut rather a Fano profile [113, 114]. Fano profile ex-presses P ( k (cid:107) ) = P ( k spp ) q + 1 ( x + q ) + b x , with (40) x = k (cid:107) /k − n SP P n (cid:48)(cid:48) SP P . (41)This expression is a generalization of the Fano formulato lossy materials according to ref. [114]. q is theratio between the optical response of the mode ofinterest and the second mode and b introduces anoffset due to losses. The Fano fits perfectly matchthe two SPP resonances (Fig. 7b). The leaky modepresents a high q parameter ( q = 50) so that it closelyfollows a Lorentzian profile. However, the boundmode resonance cannot be fitted with a Lorentzianprofile since it is coupled to a leaky mode that canbe treated as a continuum, hence the Fano behaviour[18]. The SPP contribution is therefore estimated fromthe integral of the Fano resonance. If we remove thecontinuum background contribution, we obtain (seeAppendix B)Γ SP P n Γ = p ( k SP P ) L SP P q + b − q + b π , with (42) p ( k SP P ) = P ( k SP P ) n P . We plot in Fig. 8(a,b), the different contributionsto the total decay rate for a dipolar emitter orientedparallel or perpendicular to the metal surface. Theradiative rate refers to the power integrated over 0 ≤ k (cid:107) ≤ n k (radiative waves in medium 1). Note thatleakages of SPP1 into the substrate contribute to thescattering rate but not to the radiative rate. We discussthis point later. The contribution of surface plasmonto the decay rate remains small for a dipole parallel to the surface. Differently, for a perpendicular dipole, weobserve strong excitation of the gold/air SPP1 (Fig.8b) whereas the gold/glass SPP2 contribution remainssmall due to poor overlap with the dipolar emissionsince the emitter is located in air. This is quantifiedby the coupling efficiency β represented on Fig. 8(c,d).Up to β = 93% coupling efficiency is achieved for avertical dipole d = 200 nm above the gold film. Thishigh β -factor originates from the small radiative rate atthis distance due to an interference effect between thedirect emission and the reflected field. Note that theSPP rate is equal to the free-space rate at this distance(Γ SP P /n Γ (cid:39) π str) by an isolated radiator.Since the gold/air SPP is leaky into the substrate( n SP P < n sub ), it is worthwhile to estimate leakageradiation that are of interest for e.g. leakageradiation microscopy [92, 115, 116] or surface enhancedfluorescence [45, 117]. In order to determine theleakage contribution, we first estimate the leakage andOhmic losses of the gold/air SPP. Indeed, the finitepropagation length of the leaky plasmon originatesfrom i) intrinsic (Ohmic) losses with the rate per unitlength α i and ii) radiative losses into the substratewith the rate per unit length α leak [116, 118]. Thepropagation length expresses L SP P = 1 α i + α leak . (43)The leakage rate is estimated by cancelling theohmic losses [ Im ( ε Au ) put to zero] to α leak =6 . · − µ m − . Therefore SPP leakage contributesfor (cid:82) ∞ α leak e − r (cid:107) /L SPP dr (cid:107) = α leak L SP P = 53% to theSPP rate. Figure 9a presents the power emitted in thefar field Γ scatt as well as the radiative rate and SPPleakage (Γ leak = 53%Γ
SP P ). We also estimate theabsorption in the metal Γ abs = Γ rad + Γ leak − Γ scatt .Only a small part of the radiative emission is absorbedin the metal and doesn’t contribute to the far-fieldemission. This rate practically doesn’t depend on thedistance to the metal film as in the mirror case (see Fig.5). Last, the collection efficiency into the substrateusing an oil immersion objective ( N A > n
SP P ) isestimated as β leak = Γ leak / Γ tot and is shown in Fig.9b. It reaches 50% at 200 nm. Weproceed as previously (see § lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 8.
Contributions to the total decay rate as a function of the distance to the 50 nm gold film for a dipole parallel (a)or perpendicular (b) to the surface. c,d) Coupling efficiency to the leaky (c) and bound (d) SPP. The emission wavelength is λ em = 670 nm. Q L eff (nm) δ/ n g F p leaky 85 159 193 0.92 3.02bound 57 27 76 0.57 5.1 Table 1.
Quality factor and effective length of leaky and boundSPPs. The penetration depth (in air or glass) is indicated forcomparison. The group index and Purcell factor [SPP decay ratecalculated at the gold/air (leaky SPP) and gold/glass (boundSPP) interfaces] used to estimate L eff are also given. to the near-field behaviour of the SPP mode. Wewill observe a similar behaviour for localized surfaceplasmons (see § SPP is a surface wave, intrinsically confined near themetal film. It can be furthermore laterally confinedby distributed Bragg reflectors, forming an in-planeplasmonic cavity as schemed in Fig. 10 [119, 120,121]. Such an open in-plane plasmonic cavity wouldfurther increase the plasmonic Purcell factor, withthe possibility to access the fluorescent emitter. This enables an external control or manipulation of theemitter position (optical trapping, AFM manipulation,...) or emission properties (Stark effect using a STMtip, ...).The two-dimensional dipolar emission can benumerically achieved using the 2D-Green’s dyadtechnique. It expresses [105, 122]Γ u ( x, z ) n Γ = 6 n k (cid:90) + ∞ ImG Duu ( r xz , r xz , k y ) dk y (44)where r xz = ( x, z ) is the emitter position in thetransverse plane ( Oxz ) and k y the component ofthe wavector along the invariant y − axis. Sincethe 2D-Green’s dyad G associated to the gratingstructure can be numerically computed, we can de-fine the normalized dipolar emission power P ( k y ) n P = n k ImG Duu ( r xz , r xz , k y ) as a function of the propaga-tion constant k y . This makes a direct analogy betweenthe dipolar emission in 1D and 2D geometries and allthe above discussion near a flat metal film is easily ex-tended to this configuration.Figure 10 compares the behaviour of a (2D) planar lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 9. a) Radiative and leakage contribution to the scattereddecay rate. The absorption contribution is Γ abs = Γ rad +Γ leak − Γ scatt . b) Coupling efficiency β leak into leakage. The dipolaremitter is perpendicular to the metal surface. plasmonic cavity and a (1D) Fabry-Perot cavity. The2D dipolar emission P ( k y ) and 1D dipolar emission P ( k (cid:107) ) are calculated for a dipolar emitter parallel tothe mirrors and located at the center of the cavity.The dipolar emission significantly increases for cavitysize L cav = (2 p + 1) λ eff / p an integer and λ eff the mode effective wavelength ( λ SP P or λ em /n inthe plasmonic and Fabry-Perot cavities, respectively).This corresponds to the emission into the even modesof the cavity that presents an antinode at the cavitycenter. We observe a cut-off for cavity size below λ eff /
2. The normalized decay rate, calculated asa function of the cavity size, presents very similarbehaviour for these two cavities. This demonstrates thestrong analogy between the in-plane plasmonic cavityand the micro-optical cavity. However, some distinctfeatures appear in the planar cavity. First, sinceSPP are polarized perpendicular to the metal film,we do not observe polarization degeneracy (TE/TM)in the planar cavity. We also note the permanentcontribution of the planar SPP mode at k y /k = Figure 10. a) Power dissipated by a dipolar emitter inside theplasmonic cavity, as a function of cavity length L cav and in planewave vector along the cavity axis k y /k . The glass/gold/airslab contribution is subtracted to characterize the cavity effect.Inset: Decay rate as a function of the cavity length obtainedby integrating the dissipated power over all the k y /k spectrumrange (including the glass/gold/air slab contribution). b) Sameas (a) for a 1D gold/air/gold cavity. The dipolar emitter islocated at the cavity center and parallel to the mirror walls.The 1D cavity modes are indicated on the dispersion curve.Reproduced with permission from ref. [121], APS, copyright2012. n SP P = 1 .
04. The gold/glass SPP at k y /k = 1 .
5. Purcell factor near a plasmonic waveguide
As discussed above, SPP Purcell factor into extendedmetal film can be increased by an in-plane cavity thatconfines laterally the delocalized SPP. However, thelateral confinement is still limited to about λ SP P / lasmonic Purcell factor and coupling efficiency to surface plasmons. x, z ) n Γ = 1 + 3 πn k (cid:90) + ∞ ∆ ρ Du ( r xz , k y ) k y dk y (45)where we have introduced the 2D-LDOS [105, 122]∆ ρ D ( r xz , k y ) = 2 ε k y π Im ∆ G Duu ( r xz , r xz , k y ) (46)and ∆ G D = G D − G D is the difference between thetotal Green’s tensor of the 2D-structure and the free-space Green’s tensor G D . It describes the role of thewaveguiding structure only.Figure 11a represents the variation of the 2D-LDOS near a silver nanowire as a function of thepropagation constant k y . It behaves very similarly tothe mirror case (Fig. 2). In particular, we can againdistinguish three contributions to the decay rate. i)Radiative waves for k y < n k , ii) SPP peaked near theplasmon propagation constant (here k SP P = 2 . k )and iii) lossy waves at large k y . The correspondingdecay channels are calculated by numerical integrationsover the different wave domains.We first discuss the SPP rate. The SPPcontribution follows a Lorentzian profile as above athick plasmon film. It is peaked on the plasmonpropagation constant k SP P = 2 . k and with aFWHM inversely proportional to the propagationlength ( L SP P = 1 . µ m; see the inset of figure 11a).This again leads to a closed form expression of theplasmonic Purcell factor [105, 122]Γ SP P n Γ = 3 πλ n k SP P ∆ ρ Du ( r xz , k SP P ) L SP P (47)= π p ( k SP P ) L SP P , with p ( k SP P ) = P ( k SP P ) n P . It is worth to compare this expression to thelossless case for which the coupling rate to the guidedmode expresses [142, 143]Γ
SP P n Γ = = 3 π(cid:15) c | E u ( d ) | n k (cid:82) A ∞ ( E ∧ H (cid:63) ) · ˆ y d A (48)where ( E , H ) is the electromagnetic field associatedwith the guided SPP mode. For a circular waveguide,an analytical expression is readily obtained (see Figure 11. a) Radial 2D-LDOS near a circular silver nanowireas a function of k y for two distances d to the wire surface.b) Comparison of SPP rate into the plasmonic wire for a reallossy waveguide (LDOS), an ideal lossless waveguide and withinthe quasi-static approximation (in presence of losses). c) Modeeffective area of circular waveguide as a function of its radius.The horizontal line indicates the diffraction limit ( λ/ n ) [inset:Purcell factor (Eq. 55). d) Radiative rate near a circularnanowire ( R = 20 nm). e) Non radiative rate Γ NR and SPPcontribution Γ SPP as a function of distance to the nanowiresurface. The wire radius is R = 20 nm (except in c) and theemission wavelength is λ em = 1 µ m. The permittivity of thewire and surroundings permittivity are ε = −
50 + 3 . i and ε = 2, respectively. Adapted with permission from ref. [105],APS, copyright 2011. lasmonic Purcell factor and coupling efficiency to surface plasmons. ω = f ( k spp )governs the guided mode excited by an emitter nearthe emission angular frequency ω em . Therefore, thedirac distribution in the decay rate Eq. (8) becomes δ ( ω − ω c ) = 1 v g δ [ k spp − k spp ( ω em )] (49)introducing the group velocity v g = ∂ω/∂k spp andassuming a weak dispersion around the emissionwavelength. The dirac distribution δ [ k spp − k spp ( ω em )]is again replaced by a Lorenzian profile for a lossywaveguide so that we achieve F p = 34 π (cid:18) λ em n (cid:19) Q spp V ω em v g k spp . (50)Moreover, the quality factor and effective volume ofthe guided mode are expressed, respectively by Q spp = k guide ∆ k guide = k spp L spp (51)and V = (cid:82) ε ( r ) | E ( r ) | dxdydzM ax [ ε | E ( r ) | ] (52)= (cid:82) ε ( x, z ) | E ( x, z ) | dxdzM ax [ ε | E ( x, z ) | ] × (cid:90) ∞−∞ e −| y | /L spp dy = A eff L spp . (53) A eff defines the mode effective area (see Eq. 22) andwe assume again a non dispersive medium. Finally,the Purcell factor simplifies to the nanofiber expression(24) F p = Γ spp n Γ = 34 π ( λ em /n ) A eff n g n (54)and does not depend on the propagation length.Moreover, the guided SPP mode volume obeys V spp =2 A eff L spp that becomes strongly confined for shortpropagation distances L spp . It makes a bridge betweendelocalized and localized SPP, notably in the quasi-static regime where they present very similar behaviour[122]. This discussion also points out that the role oflosses in the plasmonic Purcell factor must be carriedout carefully.Eventually, the Purcell factor is identical consider-ing a real lossy or an ideal non-lossy plasmonics waveg-uide, so that we can identify the Purcell factor expres-sions (24) and (48) to estimate the SPP confinement. Figure 12.
Purcell factor near a circular or pentagonal silvernanowire ( λ em = 1 µ m). d is the distance to the nanowire surfaceor edge. Reproduced with permission from ref. [105], APS,copyright 2011. As previously, we identify at the nanowire surface,Γ x n Γ + Γ y n Γ + Γ z n Γ = F p , (55) ≡ π ( λ em /n ) A eff n g n . (56)The effective surface of the guided SPP is shownon Fig. 11c as a function of the wire radius. Thisreveals that quantum plasmonics relies on stronglysubwavelength mode confinement with huge Purcellfactor (up to 10 see inset of Fig. 11c) [21, 144]. Weachieve similar values for λ em = 670 nm and a goldnanowire (not shown). The small SPP effective areapermits to decrease the threshold for SPP amplificationcompared to a photonic nanowire of similar cross-section [145].Finally, the radiative and non radiative contribu-tions are presented in Fig. 11(d,e). Above d = 20 nm,the only contribution to the non radiative rate is SPP,due to losses along the propagation. At short distances,electron scattering are responsible of the additionnallosses, leading to fluorescence quenching. Last, a cou-pling efficiency into the guided plasmon of β = 83%is achieved at 20 nm from the nanowire (not shown)[105].For a nanowire above a glass substrate, the guidedSPP becomes leaky. The 2D-LDOS again follows aFano profile (or a Lorentzian profile for weak leakage)and the leakage rate is easily estimated as done in thecase of thin metal film. For instance, β leak = 70% ofthe emission is collected into the substrate for a 100nm silver wire 50 nm above the glass substrate [105].Since the 2D-Green’s dyad associated to aplasmonic waveguide can be numerically computed,this formalism can be applied to arbitrary geometries[107, 146, 147, 148]. Specifically, crystalline silver lasmonic Purcell factor and coupling efficiency to surface plasmons.
6. Localized plasmon
This last section is devoted to localized plasmon forwhich we expect full 3D subwavelength confinement.For the sake of clarity, we focus on a sphericalmetal nanoparticle (MNP) that constitutes a canonicalconfiguration for LSP. We first consider the quasi-static approximation for which we derive analyticalexpressions for the mode volume and quality factorof each mode ( § Let us consider a spherical metalnanoparticle (MNP) of radius R small compared to thewavelength. For clarity, we first discuss the dipolarresponse of the particle. It is characterized by theeffective polarizability α eff = (cid:20) − i k α (cid:21) − α , ( k = 2 π/λ ) , (57) α ( ω ) = ε m ( ω ) − ε m ( ω ) + 2 R , (58)The so-called radiative reaction (2 k α /
3) originatesfrom finite size effect and energy conservationconsiderations [150, 151, 152, 153]. α is thenanoparticle quasi-static (dipolar) polarisability and ε m is the metal dielectric constant. The dipole plasmonresonance appears at ω such that ε m ( ω ) + 2 =0. In case of Drude metal, the dipolar resonance is ω = ω p / √ ω p the bulk metal plasma angularfrequency.If the metal dielectric constant obeys a Drudemodel, the effective polarizability follows a Lorentzianprofile near the LSP resonance [154] ε m = 1 − ω p ω + iκ abs ω ; (59) α eff ( ω ) ∼ ω ω ω − ω ) − iκ R , (60) κ = κ abs + 2( k R ) ω , ( k = ω /c ) . (61) κ is the decay rate of the particle dipolar mode, andincludes both the Joule ( κ abs ) and radiative [ κ rad =2( k R ) ω /
3] losses rates. We can therefore definethe quality factor of the dipolar mode Q = ω /κ that typically ranges from 10 to 25 for gold or silvernanoparticles [151]. The decay rate of a dipolar emitter locatedin the very near-field of a spherical metal particleapproximates to [155, 154, 156]Γ ⊥ Γ ∼ k z Im ( α ) , (62) ∼ ω ω R k z κ ∼ π λ R πz Q (63)Γ (cid:107) Γ ∼ k z Im ( α ) ∼ ω π λ R πz Q (64)for a dipole emitter oriented perpendicular or parallelto the nanoparticle surface and an emission tuned tothe dipolar particle resonance ( λ em = λ = 2 πc/ω ).In order to determine the dipolar mode effectivevolume, we now identify the coupling rate Γ / Γ tothe Purcell factor (Eq. 3). It is worthwile to notethat the Purcell factor is obtained assuming a singlemode cavity. Since the dipolar mode is three-folddegenerated, the SPP rate writes Γ / Γ = 3 F p sothat we can identify the Purcell factor to the SPP rateof a randomly oriented dipolar emitter at the particlesurface ( z = R ). The average decay rate writes (cid:104) Γ (cid:105) Γ = Γ ⊥ + 2Γ (cid:107) ∼ π λ πR Q = 34 π λ Q V (65)where we define the dipolar LSP mode volume V =2 πR = 3 / V . V = 4 πR / The Purcell factor associ-ated to the dipolar mode can be generalized to eachLSP mode. Due to the (2n+1) degeneracy, the cou-pling rate to the n th mode (n=1, dipolar LSP, n=2,quadrupolar LSP, ...) obeys ‡ Γ n Γ = (2 n + 1) F p , with (66) F p = 34 π λ Q n V n , (67) Q n = ω n κ n = ω n κ abs + κ radn , (68) V n = 3 n + 1 V , (69) κ radn = ω n ( n + 1)( k n R ) n +1 n (2 n − n + 1)!! , ( k n = ω n /c ) . (70)The quality factor is governed by ohmic losses κ abs inthe MNP, identical for all the modes, and radiative ‡ We use a different definition for the mode volume in Ref. [151],assuming a randomly oriented dipole. The definition used here,that includes the mode degeneracy is more consistent with usualPurcell factor definition. lasmonic Purcell factor and coupling efficiency to surface plasmons. κ radn ∝ ( k n R ) n +1 ]. Quality factor therefore slightlyincreases from a few 10 for a dipolar LSP ( n = 1) toabout 50 to 100 for high order modes. In addition,the high order modes are more confined than thedipolar LSP ( V = 1 . V , V = V , V n → n →∞ β -factor to a given mode. It strongly depends onthe distance to the MNP but also of the emissionwavelength. For an emission wavelength matching thedipolar LSP, the coupling efficiency reaches β = 90%into the n=1 mode at a distance d = 10 nm. It canbe as high as β = 87% into the quadrupolar mode at d = 15 nm if the wavelength emission matches the n=2LSP resonance (not shown, see ref. [151]). In cQED, the modevolume is defined as the energy confinement (see eq.20). For dispersive materials, this extrapolates to V nrjn = (cid:82) U n ( r ) d r max [ ε ε | E n ( r ) | ] , (71) U n ( r ) = ∂ [ ωε ε ( r , ω )] ∂ω | E n ( r ) | + µ | H n ( r ) | where ( E n , H n ) is the electromagnetic field of the n th mode. In the quasi-static regime, the magneticcontribution is negligible and the electric field isconfined near the particle surface. This definition thenleads to V nrjn = 6 / ( n + 1) V [159]. The effectivevolumes of the (2n+1) degenerated n th LSP modes arerepresented on Fig. 13 (see Appendix C for details).We observe that the (2n+1) degenerated modes presentthe same volume (note that the energy confinementof the (l,m) mode is normalized with respect to the(l,m=0) mode maximum intensity).The ratio between the mode volume deducedfrom Purcell factor ( V P urcell ) and the mode volumeestimated from energy mode confinement assuming alossless metal ( V energy ) V P urcell V energy = n + 12 (72)depends on the mode number n only. We attributethis difference to the (Joule and radiative) losses thatare neglected in the energy confinement but takeninto account in the Purcell volume derivation. Unlikedelocalized SPP for which Purcell factor does notdepend on Joule losses, Purcell factor for a localizedLSP is strongly affected by losses in the metal. Inaddition, we observe that both Purcell and energyconfinement derivations leads to identical values for the Figure 13.
Energy confinement mode volume V l,m as a functionof the mode number l and m = 0 , , . . . l . The curve is the Purcellvolume estimated identifying the decay rate to a given mode andPurcell factor V Purcell = 3 V / ( l + 1). dipolar mode volume. This again shows that the modevolume entering in the Purcell factor is governed by themode near-field behaviour, even for a leaky mode (seealso § So far, we have discussed the Purcell factor within thequasi-static regime. In this section, we generalize it tospherical MNP of arbitrary size using Mie expansion.Particular attention is again devoted to the definitionof the mode volume as the energy confinement, inanalogy with cQED definition. Dipolar emissionnear a spherical particle of arbitrary size is exactlysolved using Mie (modal) expansion [160]. We cantherefore again define a mode volume by identifyingthe contribution to the total decay rate of the n th modeto the Purcell expression (3)[161]. In the following,we define this volume as the Purcell effective volume.We compare it to the mode energy confinement (seeeq. 71). However, the application of definition (71) isdifficult in the retarded regime since LSPs leak in thefar-field. As far as light-matter coupling is concerned,the pertinent parameter is the confinement of themode energy stored inside the cavity. Following thework of Koenderink [162], the intrinsic mode volumeis estimated from the energy confinement, excludingradiative leaks [i.e, ordinate at the origin in Fig.14(a,b)].In figure 14, we compare the mode volumeestimated from Purcell factor and energy definition as lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 14.
Mode energy confinement. (a,b) Mode volume(Eq. 71) as a function of the integration sphere cut-off radius.The silver sphere radius is a = 5 nm (a, dipolar resonance at λ = 325 nm) or a = 40 nm (b, dipolar resonance at λ =365 nm, quadrupolar resonance at λ = 320 nm). The energyenclosed in the integration sphere linearly increases with sphereradius with a slope proportional to the field flux Φ. The modevolume energy is defined as the ordinate at origin. (c,d) Dipolarand quadrupolar LSP volumes (the wavelength depends on theparticle size and considered mode). Purcell and energy referto the mode volume definition. The metal dielectric constantfollows Drude model. Reproduced with permission from ref.[163], EPL, copyright 2012. a function of the MNP size. This again reveals subdiffraction mode volume ensuring efficient light-matterinteraction. For small particle, we recover the quasi-static limit. For large particles, the energy definitionis in qualitative agreement with the Purcell factordefinition. However, large MNP supports quasi-modewith lifetime shorter than the collective oscillation sothat the concept of Purcell factor fails to describe thisregime. Recently, Sauvan andcoworkers define a complex mode volume [164] V n = (cid:82) ˜E n · ∂ ( ωε ε ) ∂ω · ˜E n − µ ˜H dr max [ ε ε [ ˜E n ( r )] ] (73)where ( ˜E n , ˜H n ) is the quasi-normal n th mode. Theconvergency of the volume integral is ensured thanksto phase matching layers. This generalises the Purcellfactor to dissipative cavities such that F p = Γ n n Γ = 34 π (cid:18) λn (cid:19) Re (cid:18) Q n V n (cid:19) . (74)This definition is consistent with the Purcell volumedefinition (Fig. 14) and reconciles the Purcell factorwith the energy-like confinement definition of the modevolume [165, 166]. Similar definitions should work forthe effective mode area or length. Figure 15.
Dipolar (n=1) and quadrupolar (n=2) contributionsto the decay rate near a 80 nm gold MNP as a function of theemission wavelength. The emitter touches the MNP surfaceand is randomly oriented. Dots refer to fits using Eq. (75)with the following parameters. Dipolar mode : F p = 41, λ = 2 πc/ω = 579 nm, Q = 6, Im ( V ) /Re ( V ) = − . F p = 34, λ = 2 πc/ω = 529 nm, Q = 8 . Im ( V ) /Re ( V ) = 0 .
16. the optical index of thesurroundings medium is n = 1 . Moreover, detuning between the emission wave-length and the mode resonance ω n leads to a Fano-likebehaviour [164, 167]. Taking into account the (2n+1)LSPs degeneracy, the decay rate for a randomly ori-ented emitter obeys (cid:104) Γ n (cid:105) n Γ = (2 n + 1)3 F p ( ω n /ω em ) Q δ (cid:101) ω (cid:20) Qδ ˜ ω Im ( V n ) Re ( V n ) (cid:21) δ (cid:101) ω = ω em − ω n ω em (75)and generalizes expression (19) to lossy cavities.Figure 15 presents the dipolar and quadrupolar LSPcontributions to the decay rate near a gold MNP.We observe an excellent agreement with the Fano-likebehaviour. From the fitting parameters, we extractthe Purcell factor, quality factor and effective volumeof the dipolar mode: F p = 41, Q = 6 and V =6 . · − µ m = 0 .
01 ( λ /n ) respectively. Similarly,for the quadrupolar mode: F p = 34, Q = 8 . V = 8 . · − µ m = 0 .
02 ( λ /n ) . The Purcell factor quantifiesthe coupling efficiency into a given LSP mode butdoes not distinguish radiative from Joule losses, as fordelocalized SPPs. Figure 16a presents the radiativeand non radiative contributions to the decay rate neara gold MNP. At very short distances, the decay rate isenhanced by several orders of magnitude. For distancesbelow a few nanometers, the non local response of themetal permittivitty (not included in this work) couldlead to different values [168, 169, 170] but without lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure 16.
Decay rate as a function of distance to a 80 nmgold particule. a) Total, radiative and non radiative decay rates.b) Contribution of the dipolar (n=1), quadrupolar (n=2) andhigh order modes (n=10 and 50) to the total decay rate. Thehorizontal dashed line is the free-space contribution. c) Couplingefficiency β to the dipolar LSP and apparent quantum yield as afunction of the distance to the particle. The emission wavelengthis λ em = 670 nm and the optical index of the surroundingsmedium is n = 1 . significative change in the behaviour. In the very near-field, the main coupling mechanism is non-radiativeenergy transfer, responsible for fluorescence quenching[171]. Above d = 20 nm, the total decay rate isstill enhanced by a factor of about ten but is mainlyradiative. In Fig. 16b, we show the contribution ofthe dipolar, quadrupolar and high order LSP modeto the total decay rate. At short distances, highorder modes play a dominant role since they arestrongly confined near the particle surface, as revealedby their extremelly small effective volume (Fig. 14)[55, 151, 156, 172]. For distances of a few tens ofnanometer, the radiative dipolar mode is responsiblefor the decay rate enhancement. This is summarizedin Fig. 16c where the coupling efficiency to the dipolarmode reaches β = 85% at d ≈
20 nm. The apparentquantum yield η = Γ rad / Γ tot is governed by the dipolar mode in the near-field of the MNP as revealed by thetwo superimposed curve in Fig. 16c.
7. Conclusion
In this work, we transposed the cQED Purcell factorto plasmonics. The Purcell factor characterizes thecapability of a structure to modify and control theemission of a nearby emitter. Both optical cavityand plasmon modes present high Purcell factors thatfavor efficient excitation by a dipolar emitter. Howeverit originates from high quality factors in case ofoptical cavities (but diffraction limited confinements)and on strongly subwavelength confinements forplasmonics (but low quality factors). Therefore thePurcell factor reveals the new paradigm opened byquantum plasmonics for achieving efficient light-matterinteraction at the nanoscale. This permits a scalelaw approach profiting from the strong maturity ofcQED concepts and adapt them to nanophotonics.We also discussed the presence of losses that areinevitable at the nanoscale. The Purcell factor includesOhmic losses that are inherent to the excitation ofplasmon in real metal so that it has to be manipulatedwith care. In the particular case of plasmonicwaveguides, the coupling efficiency to a guided modeis not affected by propagation losses. Obviously,only low loss systems such that crystalline nanowiresor nanoplatelets would permit realistics applications.Finally, hybrid plasmonic/nanophotonic configurationswould profit from the concept of Purcell factor througha common description of the coupling mechanism.
The research leading to these results has receivedfunding from the Agence Nationale de la Recherche(grants QDOTICS ANR-12-BS-008 and PLACOREANR-13-BS10-0007), and the Conseil R´egional deBourgogne (PARI ACTION PHOTCOM).
Appendix A. Decay rate above a flat mirror
Appendix A.1. Green’s tensor and total decay rate
The Green’s tensor above a mirror writes G = G + G refl where G is the Green’s tensor of the infinitehomegeneous medium of optical index n and G refl describes the reflection on the mirror. It can beexpressed thanks to Weyl expansion G refl ( r , r ) = i πε k (cid:90) ∞ [ g s + g p ] e ik z ( z + z ) dk (cid:107) (A.1)with k = n k , k z = ( k − k (cid:107) ) / , [ Im ( k z ) ≥ lasmonic Purcell factor and coupling efficiency to surface plasmons. r − r = ( ρ, ϕ, z − z ) in cylindricalcoordinates ; g sxx = k k (cid:107) k z r s (cid:20) sin ϕJ ( k (cid:107) ρ ) + cos(2 ϕ ) J ( k (cid:107) ρ ) k (cid:107) ρ (cid:21) g pxx = − k z k (cid:107) r p (cid:20) cos ϕJ ( k (cid:107) ρ ) − cos(2 ϕ ) J ( k (cid:107) ρ ) k (cid:107) ρ (cid:21) g sxy = − k k (cid:107) k z r s sin ϕ cos ϕ (cid:20) J ( k (cid:107) ρ ) − J ( k (cid:107) ρ ) k (cid:107) ρ (cid:21) g pxy = w k (cid:107) r p sin ϕ cos ϕ (cid:20) J ( k (cid:107) ρ ) − J ( k (cid:107) ρ ) k (cid:107) ρ (cid:21) g sxz = 0 , g pxz = − ik (cid:107) r p cos ϕJ ( k (cid:107) ρ ) g syx = g sxy , g pyx = g pxy g syy = k k (cid:107) k z r s (cid:20) cos ϕJ ( k (cid:107) ρ ) − cos(2 ϕ ) J ( k (cid:107) ρ ) k (cid:107) ρ (cid:21) g pyy = − k z k (cid:107) r p (cid:20) sin ϕJ ( k (cid:107) ρ ) + cos(2 ϕ ) J ( k (cid:107) ρ ) k (cid:107) ρ (cid:21) g syz = 0 , g pyz = − ik (cid:107) r p sin ϕJ ( k (cid:107) ρ ) g szx = 0 , g pzx = ik (cid:107) r p cos ϕJ ( k (cid:107) ρ ) g szy = 0 , g pzy = ik (cid:107) r p sin ϕJ ( k (cid:107) ρ ) g szz = 0 , g pzz = k (cid:107) k z r p J ( k (cid:107) ρ ) r s ( r p ) refer to the fresnel reflexion coefficient onthe slab for TE (TM) polarized light and J are thecylindrical Bessel function.The total decay rate of a dipolar emitter abovefollows Eq. 28 and writes above a slabΓ u ( d ) n Γ = 32 n / k Re (cid:90) ∞ [ g suu + g p ] e ik z d dk (cid:107) (A.2)with g sxx = g syy = k k (cid:107) k z r s g pxx = g pyy = − k z k (cid:107) r p g szz = 0 , g pzz = k (cid:107) k z r p so that the total decay rate writesΓ u ( d ) n Γ = (cid:90) ∞ P uu ( k (cid:107) ) dk (cid:107) , with (A.3) P uu ( k (cid:107) ) = 32 n k Re [( g suu + g puu ) e ik z d ]that defines the dipolar emission power spectrum usedin Eq. 35. Appendix A.2. SPP contribution near a losslessmirror
In this section, we consider a metal( ε )/dielectric( ε )single interface, and assume a lossless metal [ Im ( ε ) =0]. The coefficient of reflexion writes r p = ε k z − ε k z ε k z + ε k z (A.4) and presents a pole for ε k z + ε k z = 0 leading to the(real) SPP wavector k SP P = k (cid:18) ε ε + ε (cid:19) / . (A.5)The dipolar emission power spectrum near the SPPresonance can be assessed from an expansion of r p nearits pole r p ( k (cid:107) ) ∼ k SPP ε ε ε − ε k SP P k (cid:107) − k SP P (A.6)so that the SPP contribution to the decay rate isdetermined from the residu. We achieve [100]Γ (cid:107) ( d ) n Γ = 3 π ε n SP P ( ε − ε ) | ε | / e − ε / | ε | ) / k SPP d (A.7)Γ ⊥ ( d ) n Γ = 3 πn n SP P ε − ε | ε | / e − ε / | ε | ) / k SPP d Appendix A.3. SPP contribution near a strongly lossymirror
We now consider the strongly lossy configuration atthe emission wavelength λ em = 525 nm. Figure A1apresents the dipolar emitter spectrum as a functionof the in-plane wavector. Due to strong losses inthe metal, lossy wave cannot be separated from theSPP resonance (compare to Fig. 2). Indeed, thepower spectrum presents a very broad resonance-likepower spectrum at high k (cid:107) . Therefore the dipolaremitter can couple to either the SPP or LSW, hencea Fano behaviour. In order to determine the SPPcontribution to the total decay rate, we numericallyintegrate the emitted power in the range 1 < k (cid:107) /k < . d ≈
100 nm). The SPP rate estimated fromFano profile or by direct numerical integration are inqualitative agreement but show some discrepencies,illustrating the diffculty to separate the SPP and LSWcontributions.
Appendix A.4. Scattering rate
The scattering rate is achieved from the powerscattering in the far-field zone. It can be estimatedfrom asymptotic behaviour of the Green’s tensor. WeachieveΓ scatt n Γ = (cid:90) π/ sin θσ ( θ ) dθ + n n (cid:90) ππ/ sin θσ ( θ ) dθ lasmonic Purcell factor and coupling efficiency to surface plasmons. Figure A1. a) Dipolar emission as a function of the wavectorparallel to the surface k (cid:107) ( d = 10 nm). b) Resonant behaviour ofthe Au/air SPP. The Fano fit is peaked at the SPP effectiveindex n SPP = 1 .
10 and with a FWHM n (cid:48)(cid:48) SPP = 6 . · − ( L SPP = λ em / πn (cid:48)(cid:48) SPP = 640 nm). c) Estimation of the SPPcontribution to the total decay rate (see text for details). Thedipole is perpendicular to the surface. The emission wavelengthis λ em = 525 nm. where the differential scattering cross-section expresses σ (cid:107) ( θ ) = 34 −
38 sin θ + 38 (cid:0) | r p | cos θ + | r s | (cid:1) (A.8)+ 34 Re [( − r p cos θ + r s ) e ik z d ] σ ⊥ ( θ ) = 34 + 34 | r p | + 32 Re [ r p e ik z d ] (A.9)above the film, for a dipole parallel or perpendicular tothe surface, respectively and, σ (cid:107) ( θ ) = 38 ε ε | e ik z d | (cid:18) | t p | + | k t s k z | (cid:19) cos θ (A.10) σ ⊥ ( θ ) = 34 ε ε | e ik z d | t p | + | k t p k z | cos θ sin θ (A.11)in the substrate. t p and t s refer to the Fresneltransmission coefficients for p and s polarized light,respectively. This generalises the expression achievedfor a thick mirror [103] to a finite metal slab ormultilayer system. The difference betwen the radiativeand scattering rates originates from the part of theradiative waves absorbed before achieving the far-field(see e.g. Fig. 5) and/or from SPCE leakage (see Fig.9).
Appendix B. Fano profile
Assuming a Fano profile of the emitted power P ( k (cid:107) )(see e.g. Fig. 1b), it follows [114] P ( k (cid:107) ) = P ( k spp ) q + b ( qκ spp + k (cid:107) − k spp ) + bκ spp ( k (cid:107) − k spp ) + κ spp (B.1)= P ( k spp ) q + b ( x + q ) + b x (B.2)with x = k (cid:107) − k spp κ spp = k (cid:107) /k − n spp n (cid:48)(cid:48) spp (B.3)where we note κ spp = 1 / L SP P the losses rate of theSPP.The SPP rate is then estimated from the integralof the emitted power. We first estimate the integralof the Fano resonance (the term − I = (cid:90) + ∞−∞ (cid:20) ( x + q ) + b x − (cid:21) dx (B.4)= ( q + b − (cid:90) + ∞−∞
11 + x dx + 2 q (cid:90) + ∞−∞ x x dx = ( q + b −
1) [ atan ( x )] + ∞−∞ + (cid:2) qln (1 + x ) (cid:3) + ∞−∞ (B.5)= ( q + b − π (B.6)and the SPP rate simplifies toΓ SP P = (cid:90) + ∞−∞ P ( k (cid:107) ) dk (cid:107) (B.7)= 12 L SP P (cid:90) + ∞−∞ P ( x ) dx (B.8) lasmonic Purcell factor and coupling efficiency to surface plasmons. P ( k SP P ) L SP P q + b − q + b π q → ∞ ). Appendix C. Energy confinement of LSPs
The electrostatic potential associated to the n th LSPmode of a spherical MNP follows [150]Φ n,m = n (cid:88) m = − n C n,m (cid:16) rR (cid:17) n P mn (cos θ ) e imϕ , r < R (C.1)Φ n,m = n (cid:88) m = − n C n,m (cid:18) Rr (cid:19) n +1 P mn (cos θ ) e imϕ , r ≥ R (C.2)with the normalisation constant C n,m = ( − m (cid:115) n + 14 π ( n − m )!( n + m )! (C.3)and P mn is the associated Legendre polynomial. Theboundary conditions imposes nε m + ( n + 1) ε = 0 (C.4)that fixes the resonances of the n th LSP. If thedielectric constant of the metal ε m follows a Drudebehaviour, the resonance angular frequency obeys ω n = ω p (cid:112) n/n + ( n + 1) ε . The n th LSP presents adegeneracy g n = (2 n + 1) and the electric field of the(n,m) mode writes E n,m = C n,m (cid:16) rR (cid:17) n − − nP mn (cos θ ) e imϕ e r θ (cid:2) ( n + 1) cos θP mn (cos θ ) − ( n − m + 1) P mn +1 (cid:3) e imϕ e θ , ( r < R ) im sin θ P mn (cos θ ) e imϕ e ϕ (C.5) E n,m = C n,m (cid:18) Rr (cid:19) n +2 − ( n + 1) P mn (cos θ ) e imϕ e r θ (cid:2) ( n + 1) cos θP mn (cos θ ) − ( n − m + 1) P mn +1 (cid:3) e imϕ e θ , ( r ≥ R ) − im sin θ P mn (cos θ ) e imϕ e ϕ (C.6)The effective volume is expressed extrapolatingthe cQED description to a dispersive medium V n,m = (cid:82) U n,m ( r ) d r max [ ε ε | E n , m = | ] , (C.7) U n,m ( r ) = ∂ [ ωε ε ( r , ω )] ∂ω | E n ( r ) | Note that,since the mode is degenerated, we normal-ized the field with respect to maximum of the field con-sidering all the degenerated modes (
M ax | E n ( r ) | andnot ( M ax | E n , m ( r ) | in the denominator of Eq. C.8).Assuming a lossless Drude metal, it comes ∂ [ ωε m ( ω )] ∂ω (cid:12)(cid:12)(cid:12)(cid:12) ω = ω n = 2 + n + 1 n ε (C.8)so that the effective volume is for m=0 [159] V n, = 6 V (2 n + 1)( n + 1) (cid:20) n ( n + 1) ε (cid:21) (C.9) For m (cid:54) = 0, the integration over θ is numericallyevaluated (see Fig. 13). We note a discrepancybetween the mode volume deduced from Purcell factor( V P urcell ) and the mode volume estimated from energymode confinement assuming a lossless metal ( V n,m ).Indeed, we have [151] V P urcelln = 3 V (2 n + 1) (cid:20) n ( n + 1) ε (cid:21) so that the ratio V P urcell V energy = n + 12 (C.10)depends on the mode number n only. [1] Hell S W 2007 Science
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