Introduction

Light matter interaction at the single atom/photon level is extremely weak and strategies have to be developed to enhance this coupling. In a simple picture, the coupling strength depends on the quality factor Q and the effective volume V of the involved mode by the ratio Q/V. High Q/V ratio, hence efficient light–matter interaction, is achieved in cavity-based systems, where large modal volumes are easily compensated by extremely narrow resonances1,2. It has been also proposed to work with diffraction limited volumes by coupling a single-atom to an elongated fiber3. Another downscaling step is made with plasmonics4. Surface plasmon polariton (SPP) results from coupling electromagnetic wave to a surface density of charges. They are therefore naturally confined near a metal surface and are not diffraction limited, offering thus a new tool for interfacing light and matter at the nanometer scale5,6,7. Since the seminal work of Chang et al.8, coupling between a dipolar emitter and a metal nanowire received strong interest. Metal nanowires define 1D plasmonic waveguides with a great potential for integrated optical routing9. They can be chemically synthesized with high crystallinity hence supporting SPP with reduced losses10,11,12. Controlled positionning of quantum emitter(s) near 1D plasmonic guide is now achievable13,14,15, opening the way to realize original nano-optical devices. For instance, SPP propagation can be controlled by a single emitter, leading to the concept of single photon transistor16. Reciprocally, two distant emitters can be interfaced via surface plasmons, with applications such as SPP mediated resonant energy transfer17, remote qubits entanglement18,19 or nano-optical logical gates20.

This report is devoted to a careful analysis of the coupling mechanisms between one or two individual emitters and a metal nanowire. We quantify all the coupling channels (radiative and non radiative coupling, surface plasmon coupled emission, electron scattering). To date, radiative and non radiative rates were estimated within the quasistatic approximation8. Moreover, the expression of the decay rate into plasmon modes were derived in the lossless case only8,21. In this context, we are specifically interested in the role of losses in the emitter-SPP coupling mechanism. Therefore, we investigate two distinct regimes. A first regime taking into account retardation explaining SPP propagation and routing and a second regime focused on quasi-static SPP responsible for an efficient interfacing with dipolar emitters but at the price of strong losses. To this aim, we consider a gold nanowire since both retardation regime and quasi-static pictures are present in the optical domain. Figure 1 represents the studied configuration. A dipolar emitter excites a surface plasmon supported by a gold nanowire. The report is organized as follows. We first carefully describe the modes properties of a 1D plasmonic nanowire in presence of losses. This allows for a rigorous description of the coupled emitter-nanowire system in the retarded regime. We identify all the relaxation channels of the emitter close to 1D plasmonic nanowire and compare our data to previous works. We notably achieve a closed form expression for surface plasmon coupled emission, in presence of losses with a clear link to the Purcell factor. This closed form expression permits to unambiguously define the mode volume of lossy 1D SPP. Then, we discuss SPP mediated dipole-dipole coupling between two emitters. In the last section, we fix the operating at λ = 525 nm, a wavelength corresponding to the quasi-static regime at the band edge of the dispersion relation.

Figure 1
figure 1

Surface plasmon coupled emission of a dipolar quantum emitter close to a metal nanowire of radius R.

We represent the intensity at a given instant of the guided SPP. p refers to the emitter dipole moment and d is the distance to the nanowire surface. and refer to the dielectric constant of the surroundings and the metal nanowire, respectively.

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Results

In order to discuss the relaxation channels available for coupling the emitter into the plasmonic waveguide, we first introduce the concept of local density of modes in link with the density of modes supported by the metal nanowire.

Metal nanowire: modes density, retarded and quasi-static regimes

Figure 2 represents the dispersion relation of the TM01 mode supported by a gold nanowire. Strictly speaking, the stationary mode is defined in absence of dissipation (infinite lifetime), as shown in Fig. 2a. This surface plasmon polariton results from coupling a photon and a surface plasmon so that its dispersion relation closely follows the light line at low frequencies (propagating SPP, retarded regime) and bends towards the quasi-static SPP resonance at larger frequencies. Indeed, in the quasi-static approximation, the SPP resonance condition obeys , satisfied for ω = 3.5 1015 rad.s−1 (λ = 525 nm), in the present case. The SPP presents a zero group velocity in this regime (in absence of loss). The density of guided modes (TM01), proportional to dkz/dω, is represented in Fig. 2b. It reveals a strong accumulation of modes near ω = 3.5 1015 rad.s−1 that defines a common domain with the high-order localized quasi-static modes sustained by a metal nanoparticle. In the following, the quasi-static regime refers to the SPP behaviour near λ = 525 nm.

Figure 2
figure 2

(a) Dispersion relation ω = f(kz) of a gold nanowire of radius 100 nm in absence of losses. kz is the propagation constant of the guided mode at an angular frequency ω. The horizontal asymptote at the angular frequency ω = 3.7 × 1015 rad.s−1 corresponds to (quasi-static regime). (b) DOS of the TM01 mode deduced from dkz/dω (in absence of loss). (c) Dispersion relation taking into account losses. (d) Comparison of the TM01 mode propagation length (LSPP) and spatial period (λSPP). The mode propagation length is shorter than its oscillation period for ω > 3.1015 rad.s−1 (grey area). The gold dielectric constant is taken from Ref. 22. The dielectric constant of the surrounding medium is .

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In realistic conditions, the SPP mode must be defined in presence of absorption as shown in Fig. 2c. This defines a quasi mode with a finite lifetime. We observe a similar dispersion curve except near the quasi-static asymptote where a typical back-bending occurs23. This area corresponds to a mode lifetime shorter than its oscillation period, or equivalently, a propagation length shorter than the spatial oscillation (see Fig. 2d). The concept of modes is then unavailable. However, this part of the dispersion curve is essential to describe non radiative coupling of a dipolar emitter to a metal nanostructure. We therefore abusively associate the term SPP mode to the whole dispersion curve. The retarded and quasi-static regimes are of great interest to guide (propagating SPP) or confine (quasi-static SPP) light at the nanoscale. The mode propagation length is finally represented in Fig. 2d). As expected, the quasi-static SPP is strongly confined on both the transverse [high kz, hence short penetration depth in air ] and longitudinal (low propagation length LSPP) directions. An extremely small modal volume can thus be defined so that efficient coupling to a nearby quantum emitter is expected8,24.

Local density of modes, Purcell factor and relaxation channels near a plasmonic wire

The local density of modes (2D-LDOS) near a plasmonic or optical waveguide reads25

where ΔG2D refers to the Green's dyad part associated with the waveguide structure only (i-e excluding its surroundings), is the dielectric constant at the location and kz is the wavevector along the nanowire axis. This definition allows to evaluate the density of guided modes (that cancels for kz = 0) but also includes all the scattering and non radiative channels that are excited by a nearby dipolar emitter. This is therefore extremely useful to quantify and describe the dipolar emitter relaxation. Figure 3a shows the calculated 2D-LDOS near a gold nanowire as a function of its radius at the wavelength λ = 800 nm. We observe two resonances corresponding to the TM01 and HE11 modes, respectively. A strong increase of the (TM01) 2D-LDOS appears for small radii that corresponds to a quasi-static-like behavior (different from the quasi-static regime that occurs at a specific wavelength and investigated later). The mode characteristics (wave vector and propagation length) are easily extracted from this formalism25. Indeed, for a given radius, the 2D-LDOS presents a Lorentzian profile near SPP resonance. It is peaked at the mode propagation constant and with a full-width-at-half-maximum (FWHM) inversely proportional to the propagation length as shown in Fig. 3b. As expected, the propagation length strongly decreases by increasing the metal absorption and the 2D-LDOS profile flattens. However, the mode effective index nSPP does not depend on the losses. The number of supported modes is given by the area below the 2D-LDOS curve. For a Lorentzian shape, it is proportional to the ratio Δρ2D/LSPP. This ratio is therefore independent on absorption over a very wide range as observed in Fig. 3c. For large losses, the 2D-LDOS does not follows a Lorentzian profile anymore and the concept of mode is then unavailable. In the non-absorbing limit, the 2D-LDOS profile associated to the SPP resonance is a Dirac distribution.

Figure 3
figure 3

(a) 2D-LDOS calculated at distance R/2 from the surface of a gold nanowire as a function of its radius R. The superimposed white lines represents the dispersion relations of the waveguided modes (taking into account the losses. (b) 2D-LDOS as a function of the normalized propagation constant kz/k0 for different metal absorption . The nanowire radius is fixed at R = 10 nm. The 2D-LDOS follows a Lorentzian profile peaked at the mode effective index nSPP. The FWHM ΔnSPP is inversely proportional to mode propagation length LSPP. (c) Evolution of the ratio Δρ2D/LSPP as a function of .

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Following Fermi's golden rule, we can now quantify the modification of the fluorescence rate for a point-like dipolar emitter coupled to the nanowire. Indeed, its spontaneous emission is proportional to the 3D-LDOS at the emitter position. The 3D-LDOS is simply related to the kz-Fourier transform of the 2D-LDOS so that the total decay rate is achieved by integrating the 2D-LDOS on the whole wave-vector spectrum (see methods). Physically, the region |kz| < n1k0 corresponds to propagative waves into the medium 1, so-called radiative waves whereas |kz| > n1k0 corresponds to surface waves propagating along the nanowire (including SPP and so-called lossy surfaces waves)27. The 2D-LDOS beyond |kz| > n1k0 contributes to the non-radiative rate only but a small amount of radiative waves (|kz| < n1k0) could also be absorbed into the metal nanowire. The radiated power is given by the Poynting vector flux. Its computation thanks to the Green's dyad technic involves an integration over the full nanowire volume (or at least on several propagation lengths)28, that is a challenging computing task for the infinite wire considered here. Therefore, the radiative rate is evaluated by integrating the 2D-LDOS on the radiative spectrum (|kz| < n1k0) and cancelling the metal losses and is represented in Fig. 4a. The non-radiative rate is the difference between total and radiative rates and is represented in Fig. 4b. We also represent the results from a quasi-static approximation for comparison29.

Figure 4
figure 4

Normalized radiative (a) and non-radiative (b) rates of a dipolar emitter placed at a distance d to a gold nanowire for three emitter orientations.

The nanowire radius is R = 10 nm and the emission wavelength is λ = 800 nm. The surface plasmon coupled emission contribution to the non radiative rate is also indicated. For orthoradially oriented emitter ΓSPP = 0 [Fig. b,ii)], since guided mode does not possess orthoradial electric component. QS refers to the quasi-static approximation.

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Let us first discuss the radiative rate behavior (Fig. 4a). Γrad is only weakly modified in presence of the metal nanowire. The quasi-static approximation reproduces very well the distance dependence, except for a dipole parallel to the wire axis (Fig. 4a,iii) but the modification remains low. Indeed, quasi-static approximation considers the dipole induced in the nanowire that cancels along the longitudinal axis29. The dependence of radiative decay rate with distance closely resembles to the radiative decay rate near a spherical metal particle30 for both radial (Fig. 4a,i) and orthoradial (Fig. 4a,ii) dipoles since the decay rate is governed by the dipole induced in the metal nanoparticle for both geometries.

The non radiative rate is presented in Fig. 4b. Two main channels contribute to the non radiative coupling26,27: excitation of the guided SPP with a rate ΓSPP and lossy surface waves at large kz (see also Fig. 3a at small radii where high a 2D-LDOS appears for kz = kSPP and above). Although included in the non radiative contribution, the surface plasmon coupled emission exists even for lossless metal and is related to the excitation of a guided mode that can be detected at the nanowire output. This point is discussed in details later. In the case of a realistic absorbing metal, the SPP part contributes to the non radiative channel. SPP absorption originates from the finite conductivity of the metal. It dissipates the energy of the current density associated to the SPP propagation. Therefore, it is referred as Joule (or ohmic) losses. The second dissipation process appearing at large momenta is electron scattering at the lattice impurities. Finally, electron–hole excitation in the metal occurs at very short distances27. Note that lossy surface waves describe the dissipated power from both electron scattering and electron–hole excitation. The modelisation of electron–hole excitation would necessitate to include non-local effects in the permittivity. This process is negligible for separation distances above considered here27.

At very short distances, electron scattering loss dominates (ΓSPP ≪ ΓN Rad). Above 10 nm, the non radiative channel is dominated by coupling into the propagating SPP (Fig. 4b). We also observe a small contribution of the HE11 mode to ΓNrad above 80 nm in case of radially oriented dipole [see the small difference between ΓNrad and ΓSPP in Fig. 4b(i) and Fig. S4 in supplementary information]. For radially (Fig. 4b,i) or longitudinallly (Fig. 4b,iii) oriented emitter, the quasi-static approximation strongly underestimates the non radiative rate since it does not include SPP contribution. In the case of an orthoradial emitter, the SPP contribution is null and the quasi-static approximation is slightly better (Fig. 4b,ii).

The surface plasmon coupled emission finally expresses as26

where both the forward and backward propagation modes are included. This expression extends the definition of the Purcell factor to a guided mode picture making a useful link with cavity quantum electrodynamics. As expected, the coupling rate depends on the overlap with the mode profile (represented by the density of guided mode Δρ2D) so that it decreases with the emitter-nanowire distance (penetration depth in air at λ = 800 nm: dSPP ~ 26 nm). The 1/LSPP dependence is more surprising at first glance. Close inspection of the expression clarifies this point. This Purcell factor (Eq. 2) is constant on a large absorption range, since the ratio Δρ2D/LSPP determines the number of modes and does not depend on the losses (see Fig. 3b,c). This implies that an emitter couples to the waveguide independently on the following dissipations: Joule losses or propagation up to the waveguide output26. Obviously, lower absorption is preferred to achieve a sufficient signal level at the waveguide output. This original result can also be understood from the expression of the Purcell factor

where Q and Veff refer to the mode quality factor and effective volume, respectively. Here, the mode quality factor is Q = kSPP/ΔkSPP = nSPPk0LSPP (typically Q = 12 for R = 10 nm and Q = 62 for R = 100 nm, see supplementary information). The mode volume is estimated as follows. The mode confinement in the transverse plane is characterized by an effective surface Aeff that we explicit later on. Due to the finite propagation length LSPP in the longitudinal direction, the mode volume varies as Veff ~ Aeff × LSPP. Finally, the Purcell factor Fp ∝ Q/Veff ∝ Aeff does not depend on LSPP. Identifying expressions (2) and (3), we can deduce the mode volume of the guided SPP. For this, we assume a randomly oriented emitter located at the nanowire surface (see also Ref. 24). Since the Purcell factor does not depend on Joule losses, the mode area is computed using the lossless coupling rate description (see supplementary information). We represent in Fig. 5a the guided SPP mode volume for nanowire radius varying from 10 nm to 100 nm. Its is significantly subwavelength demonstrating the ability of surface plasmon to confine light at the nanoscale. It is however worthwhile to note that the mode confinement along the longitudinal direction is associated to the finite propagation length LSPP. Since it is compensated by the mode quality factor Q in the Purcell factor, the only parameter of interest is the mode effective area represented in the inset of Fig. 5a. As a partial conclusion, we stress that the plasmonic Purcell factor has to be manipulated with care and cannot be used as a figure of merit of the coupled system without an additionnal analysis of losses. Figure 5b presents the mode confinement at λ = 525 nm for comparison and will be discussed in the last paragraph.

Figure 5
figure 5

Normalized effective volume and area (insets) as a function of nanowire radius at (a) λ = 800 nm and (b) λ = 525 nm.

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Dipole-SPP coupling efficiency and SPP mediated dipole-dipole coupling

Thus far we investigated in details the various relaxations channels for a dipolar emitter coupled to a metal nanowire. Finally, a key parameter is the β-factor that quantifies the coupling efficiency into the guided SPP as compared to all available channels. β = ΓSPP/(ΓNrad + Γrad) is represented in Fig. 6a. At very short distances electron scattering loss dominates. At long distances the plasmonic decay rate becomes negligible compared to the radiative rate. But for emitter-nanowire distances in the 10–30 nm range, a high coupling efficiency above 85% is achieved for both longitudinal and radial orientations. As discussed previously, surface plasmon coupled emission ΓSPP does not depend on Joule losses. Nevertheless, low loss systems such as crystalline nanowire are evidently preferred. We recently demonstrated that the plasmon coupling rate to a pentatwinned crystalline silver nanowire is optimized near the nanowire edges12,26. This means that the plasmon coupling rate ΓSPP is similar, or even better, to the case studied here. Additionally, crystalline nanowire presents lower losses rate10 ΓNR leading to higher β-factor and signal level at the plasmonic waveguide output.

Figure 6
figure 6

(a) β-factor near a R = 10 nm gold nanowire at λ = 800 nm as a function of the emitter-nanowire distance d. (b–d) Collective decay rate Γ12 for two radially oriented dipolar emitters as a function of their separation distance Δz for three distances d to the nanowire and three emitters orientations. The dotted red and dashed-dotted green curves refer to the SPP and radiative contributions, respectively. Distance is normalized with respect to the SPP wavelength.

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Since excellent surface plasmon coupled emission is achieved, we now consider two coupled emitters placed near the nanowire at distance Δz from each other. The coupling rate Γ12 between two radial emitters is represented in Fig. 6b as a function of the emitter's separation distance. If a donor emitter in its excited state is weakly coupled to an acceptor emitter in its ground state, Γ12 refers to the coupling rate between the two emitters. This weak coupling leads to an irreversible resonant energy transfer from the donor to the acceptor31,32,33,34. Differently, if the two emitters are strongly coupled, the system cannot be described from isolated dipolar emitters in their ground (g) or excited (e) state. One rather have to introduce the coupled basis and for identical emitters. The two entangled states |+ > and |− > are of particular interest since they do not possess classical counterparts. Γ12 is the so-called cooperative decay rate of the coupled system and Γ11 ± Γ12 refer to the decay rates of these coupled states. Positive and negative cooperative rates Γ12 correspond to superradiance and subradiance of the two coupled emitters, respectively (note that the physical decay rates Γ11 ± Γ12 are always positives). We represent in Fig. 6b the cooperative decay rate. We also evaluated the radiative and SPP contributions to the collective emission. The highest collective decay rate (Fig. 6b) corresponds to the collective excitation of guided SPP but also lossy waves. The radiative contribution to the collective relaxation is negligible. Therefore the useful part of the collective decay rate is the collective surface plasmon coupled emission, achieved for the highest β–factor, that is for d = 22 nm in the present case (Fig. 6a,c). For emitters close to the nanowire the coupling rate Γ12 presents an oscillatory behavior with a spatial period corresponding to the SPP wavelength (Fig. 6b,c). For larger separation distances, the cooperative decay rate presents a more complex behaviour due to a beating between free-space direct emission and the surface plasmon coupled emission (see Fig. 6d and S8).

Quasi-static regime

In this last section, we investigate the surface plasmon coupled emission at the band edge of the dispersion relation, corresponding to the quasi-static behavior of the SPP (see Fig. 2). Indeed, the mode group velocity strongly slows down in this spectral range and we therefore expect important light-matter interaction enhancement in this regime19,35,36,37.

Radiative and non radiative rates are calculated in Fig. 7 for an emission wavelength λ = 525 nm. The radiative rate is significantly enhanced near the surface plasmon resonance (Fig. 7a). As expected, the quasi-static approximation better describes the non radiative coupling as compared to the retarded regime (compare Figs. 7b and 4b). The non radiative coupling is dominated by electron scattering. For comparison purpose, we also represent the SPP contribution to the non radiative rate, even if the definition of the plasmonic mode becomes questionable in this regime (cf. Fig. 2). Due to strong losses, the 2D-LDOS no more follows a lorentzian profile as in Fig. 3b). Therefore, we estimate the SPP contribution using a lossless guide description since we demonstrated earlier that it correctly describes the coupling rate to the guided mode26. This is however an extrapolation from the propagating regime situation. It is nonetheless helpful for a qualitative discussion.

Figure 7
figure 7

Same as Fig. 4 for the emission wavelength λ = 525 nm.

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The coupling rate into the SPP decreases very fast as a function of the distance (Fig. 7b). This abrupt decay reveals the strong mode confinement at the metal surface. We deduce from the slope the penetration of the mode in air to dSPP = 9 nm. This is confirmed by the calculated mode effective volume or area represented in Fig. 5b. In this quasi-static regime, the SPP mode volume reaches extremely low value Veff ~ 10−4(λ/n1)3 similar to the localized SPP configuration24,38. Nevertheless, strong losses limits the maximum achievable β-factor to β ≈ 30% at d = 15 nm (not shown).

Finally, Fig. 8 presents the cooperative decay rate for two quantum emitters near the metallic nanowire at the emission wavelength λ = 525 nm. Due to strong losses, SPP coupling does not exist anymore. We rather observe a low collective decay rate, governed by radiative contribution since only direct free-space coupling occurs. Therefore the pseudo-periodic behavior of the cooperative decay rate follows the free-space wavelength. It is moreover strongly damped due to high losses in this regime.

Figure 8
figure 8

Collective decay rate Γ12 for two dipolar emitters as a function of their separation distance Δz for three emitters orientations.

The dotted red curves refer to the radiative contribution. The emitter is 11 nm away from the nanowire surface. Distance is normalized with respect to the free-space wavelength.

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Discussion

A 1D plasmonic waveguide efficiently promotes light-matter interaction at the nanoscale. Although the SPP presents a modest quality factor (of the order of 10 to 100), they possess extremely low modal area (or volume) that are not diffraction limited and are thus very promising for enhancing single atom/single photon process. In the retarded regime, we demonstrated modal volume typically ten times below the diffraction limit and β factor up to 90% but that could be still increased if crystalline nanostructures are involved. In this work, we also clarified the effect of Joule losses and proposed a simple physical understanding of the coupling mechanism in surface plasmon coupled emission. We expect that this would be also helpful in defining strong coupling regime conditions39.

Finally, we discussed the cooperative behavior of two coupled quantum emitters and demonstrated an efficient collective excitation of guided SPPs. It has been recently proposed to entangle two remotes qubits18,19 or to realize nano-optical logical gates20 by controlling sub/superradiance. This regime is also of strong interest in the context of superradiant lasers (also called bad cavity lasers) presenting high spectral purity40,41. Due to the poor quality factor of SPP resonances, we envision a fruitful analogy for designing plasmon nanolasers42,43,44. Indeed, the collective relaxation of several identical emitters in a poor cavity can lead to a coherent emission with extremely narrow spectral width40,41. Practically, the limit linewidth of a single mode laser writes45

where κ is the cold-cavity loss rate, Pout the laser output power and Nsp = Ne/(Ne − Ng) the degree of population inversion. In a good cavity regime (), the laser spectral width is governed by cavity loss rate. It follows the Schawlow-Townes limit . However, as far as SPP amplication is concerned, one as to considered the bad cavity regime (): . This defines the superradiant laser for which the electromagnetic coherence originates from the collective behaviour of the gain medium instead of the cavity feedback. Here, the cavity mode is only necessary to collectively couple the atoms forming the gain medium. The laser linewidth is then limited by the gain medium and could be largely below the Schawlow-Townes limit40,41. Note also that a partial population inversion Nsp degrades the linewidth lower limit. This has to be taken into account when describing SPP nanolasers since non radiative energy transfer to the metal limits the population inversion46. Since the SPP guided mode presents a rather low quality factor but mediates efficiently cooperative relaxation, we expect a fruitful analogy with superradiant lasers to achieve a coherent SPP nanosource.

Last, we investigated the quasi-static regime and demonstrated SPP confinement down to 10−4(λ/n1)3 but very low surface plasmon coupled emission due to critically important losses. Although limited to plasmonics nanowires in this work, the rich physics underlying the coupling between single photon source(s) and lossy materials is very general and remains valid for arbitrary geometries as well as metamaterials. Additionally, the extreme mode confinement observed at the band-edge of the dispersion relation is closely related to localized surface plasmon and create a bridge with cavityless quantum electrodynamics.

Methods

Dispersion curve

The dispersion curve of the metal nanowire is achieved by solving the dispersion relation of the TM01 and HE11 modes with Davidenko's algorithm in the complex plane47. This leads to complex propagation constant solutions . The mode propagation length obeys . Davidenko's algorithm is an efficient method to find complex root, including leaky modes (mode effective index below the surrounding optical index). In the present case, we observe that the HE11 mode is not cut-off but its effective index approaches that of the surrounding optical index for small nanowire in agreement with the 2D-LDOS calculation (see Fig. 3) and also with the asymptotic expansion derived in ref. 8 (see also supplementary information).

Green's dyad

The 2D-Green's dyad is numerically computed discretizing the nanowire cross-section with rectangular meshes as detailed in ref. 48. Meshes sizes varies from 0.5 nm to 10 nm for larger radii. Importantly, the Green's dyad variation inside the mesh is exactly included, ensuring accurate evaluation. Arbitrary waveguide profile eventually supported on a wafer, are easily treated with this method26. Symmetrical properties of the Green's tensor are used to limit the numerical evaluation of G2D on the kz ≥ 0 range. Indeed, G2D(r1//, r2//, −kz) = [G2D(r2//, r1//, kz)]T (see supplementary information). The behaviour of G2D near SPP resonance is also discussed in the supplementary information (Fig. S6).

Decay rates

The dipolar total decay rate expresses where r0// = (x, y) refers to the emitter position in the transverse plane and p = pu is the dipolar transition moment. The collective decay rate of two coupled dipoles (p1 = p1u1, p2 = p2u2) writes with r1// and r2// the dipole positions in the transverse plane and Δz = z1 − z2 their separation distance. Radiative rates Γrad and are obtained limiting the integration on the radiative waves (|kz| < n1k0) and cancelling the metal losses []. SPP contributions ΓSPP and are achieved keeping the SPP resonance only into the integration. Thanks to the Lorentzian shape of the resonance, analytical expressions are available (see Eq. 2 and supplementary information).

Relaxation channels in lossless or quasi-static limits

The Purcell factor near a non-absorbing waveguide is given by21,49

where (E, H) is the modal electromagnetic field, * refers to the conjugated form and u is the emitter orientation. For a circular cross-section, it is analytical (see supplementary information).

In the quasi-static limit, radiative and non radiative rates expressions are given in ref. 29.