Mie plasmons: modes volumes, quality factors and coupling strengths (Purcell factor) to a dipolar emitter

Using either quasi-static approximation or exact Mie expansion, we characterize the localized surface plasmons supported by a metallic spherical nanoparticle. We estimate the quality factor $Q_n$ and define the effective volume $V_n$ of the $n^{th}$ mode in a such a way that coupling strength with a neighbouring dipolar emitter is proportional to the ratio $Q_n/V_n$ (Purcell factor). The role of Joule losses, far-field scattering and mode confinement in the coupling mechanism are introduced and discussed with simple physical understanding, with particular attention paid to energy conservation.


I. INTRODUCTION
Metallic nanoparticles support localized surface plasmon-polaritons (SPP) strongly confined at the metal surface ensuring efficient electromagnetic coupling with neighbouring materials, offering a variety of applications such as surface-enhanced spectroscopies 1,2 , photochemistry 3 or optical nanoantennas 4 . This also opens the way towards control of light emission at the nanoscale [5][6][7][8][9][10] . The ratio Q/V ef f is generally used to quantify the coupling strength between a dipolar emitter and a cavity mode (polariton). Q and V ef f refers to the mode quality factor and effective volume, respectively. High ratio Q/V ef f may lead to strong coupling regime with characteristics Rabi oscillations revealing cycles of energy exchange between the emitter and the cavity. In the weak coupling regime, the emitter energy dissipation into the cavity mode is non reversible and follows the Purcell factor: where γ is the emitter spontaneous decay rate into the cavity compared to its free-space value γ 0 , n B is the optical index inside the cavity and λ is the emission wavelength. Spontaneous emission rate in complex systems follows the more general Fermi's golden rule that expresses γ as a function of the density of electromagnetic mode 11 . However, for describing the coupling to a cavity mode, the Purcell factor is usually preferred since it clearly introduces the cavity resonance quality factor Q and the mode extension V ef f on which coupling remains efficient, bringing therefore a clear physical understanding of the coupling process.
In this context, we propose to determine the quality factors and effective volumes of localized SPPs supported by a metallic nanosphere. Indeed, these quantities are usefull parameters to understand and evaluate the coupling mechanisms between dipolar emitters and plasmonic nanostructures 12,13 . This would help for achieving strong coupling regime 14,15 or for designing plasmonic nanolasers [16][17][18] . We have chosen a spherical particle, a highly symetrical system, since it is fully analytical and simple expressions can be derived with clear physical meaning [19][20][21][22][23] . Moreover, our results could be extended to more complex structures 24,25 .
In section II, we focus on the dipolar mode and present in details the derivation of its quality factor and effective volume. We extend our approach to each mode of the particle in section III. Finally, we discuss the coupling efficiency to one of the particle modes in the last section. For the sake of simplicity, all the analytical expressions are derived assuming a particle in air and dipolar emitter perpendicular to the particle surface. The generalisation to arbitrary emitter orientation and a background medium of optical index n B is provided in the appendix. Exact calculations presented in the main text using Mie expansion correctly include both the background medium and dipolar orientation.

II. DIPOLAR MODE
We first characterize the dipolar mode of a spherical particle. For sphere radius R small compared to the excitation wavelength λ = 2πc/ω, the electric field is considered uniform over the metallic particle. The metallic particle is polarized by the incident electric field E 0 and behaves as a dipole where α 1 is the nanoparticle quasi-static (dipolar) polarisability and ǫ m is the metal dielectric constant. The dipole plasmon resonance appears at ω 1 such that ǫ m (ω 1 ) + 2 = 0. In case of Drude metal, the dipolar resonance is ω 1 = ω p / √ 3 with ω p the bulk metal plasma angular frequency. However, expression (3) does not satisfy the optical theorem (energy conservation). It is well-known that this apparent paradox is easily overcome by taking into account the finite size of the particle and leads to define the effective polarisability 26 : The corrective term (2k 3 α 1 /3) is the so-called radiative reaction correction and microscopically originates from the radiation emitted by the charges oscillations induced inside the nanoparticle by the excitation field 27 .
The dipolar polarisability presents a simple shape near the resonance if the metallic dielectric constant follows Drude model 20 (ω p and Γ abs refers to metal plasma frequency and Ohmic loss rate, respectively) 28 ; where Γ 1 is the decay rate of the particle dipolar mode, and includes both the Joule (Γ abs ) and radiative [Γ rad 1 = 2(k 1 R) 3 ω 1 /3] losses rates.

A. Quality factor
The dipolar response can be described by either the extinction efficiency Q ext , proportional to Im(α 1 ), or scattering efficiency Q scatt , proportional to |α 1 | 2 , and therefore follows a Lorentzian profile centered at ω 1 and with a full width at half maximum (FWHM) ∆ω 1 = Γ 1 : The quality factor of this resonance is therefore: As an example, we consider a R = 25 nm silver sphere in air. The Drude model parameters arehω p = 9.1 eV andhΓ abs = 18 meV that lead to a resonance peak athω 1 = 5.2 eV (ω 1 = 7.98 10 15 Hz) and radiative energyhΓ rad 1 = 1 eV . The quality factor is then Q 1 = 5.
Note that the metal optical properties are better described when including the bound electrons in the Drude model: ǫ m = ǫ ∞ − ω 2 p /(ω 2 + iΓ abs ω), (ǫ ∞ = 3.7 for silver, see also the appendix). In that case, we obtainhω 1 = 3.74 eV (ω 1 = 5.7 10 15 Hz),hΓ rad 1 = 0.13 eV and Although Drude model qualitatively explains the shape of the resonance, a more representative value of the quality factor can only be determined using tabulated data for the Silver particle in air. b) Gold particle in air. c) Gold particle in PMMA (optical index n B = 1.5).
The particle radius is R = 25 nm for each case. Fit refers to a Lorentzian fit using parameters indicated on the figure. dielectric constant of the metal 29 . The extinction efficiencies associated to the dipolar resonance are represented in Fig. 1. For silver particle (Fig. 1a), the extinction efficiency closely follows a Lorentzian shape, as expected, with a quality factor Q 1 = 21, in good agreement with the value obtained using Drude model (with the contribution of the bound electrons included). In case of gold, the resonance profile is not well defined due to interband transitions for ω > 4. 10 15 Hz but a quality factor of 7 can be estimated (Fig. 1b). Figure 1c) represents the extinction efficiency for a gold particle embedded in polymethylmethacrylate (PMMA) into which metallic particles are routinely dispersed. This leads to a small redshift of a resonance, avoiding therefore the resonance disturbance by interband absorption and we recover partly the Lorentzian profile 30 .

B. Effective volume
Coupling rate of a dipolar emitter to a dipolar particle expresses for very short emitterparticle distances d (z 0 = R + d is the distance to the particle center) as [19][20][21] : for a dipole emitter orientation perpendicular to the nanoparticle surface. Using Eq. 6, we obtain, in case of a dipolar emitter emission tuned to the dipolar particle resonance In order to determine the dipolar mode effective volume, we now identify the coupling rate γ / γ 0 to the Purcell factor (Eq. 1, assuming n B = 1), so that we obtain If the excitation field is generated by a dipolar emitter (fluorescent molecules, quantum dots, . . . ), it cannot be considered uniform anymore and the dipolar approximation fails.
One needs therefore to consider the coupling strength to high order modes (Fig. 2). Here again, we discuss the mode quality factors and volume first using quasi-static approximation and then discuss their quantitative behaviour using exact Mie theory.
For a Drude metal, the n th resonance appears at ω n = ω p n/(2n + 1). Therefore, higher order modes accumulate near ω ∞ = ω p / √ 2. Moreover, as discussed for the dipolar case, quasi-static expression of the n th mode polarisability (Eq. 15) does not obey energy conservation. Applying the optical theorem, we recently extend the radiative correction to all the spherical particles modes 22 . This leads to

A. Quality factors
It is now a simple matter to generalize the dipolar mode analysis reported in the previous section to all the particle modes. The n th mode polarisablity can be approximated near resonance. A simple expression for the polarisability is achieved considering a Drude metal: Γ n = Γ abs + Γ rad n , Γ rad n = ω n (n + 1)(k n R) 2n+1 n(2n − 1)!!(2n + 1)!! , (k n = ω n /c) , where Γ n is the total decay rate of the n th mode, that includes both ohmic losses and radiative scattering. As expected, for a given mode n, the radiative scattering rate Γ rad n ∝ R 2n+1 increases with the particle size since it couples more efficiently to the far-field. For instance, we obtainhΓ rad 2 = 39 meV (hΓ rad 2 = 1.8 meV with ǫ ∞ = 3.7) for the quadrupolar mode of a R = 25 nm silver particle in air. As expected, the radiative rate of the quadrupolar is strongly reduced compared to the dipolar mode.
The quality factor associated to the n th mode is therefore Figure 3 details the quality factor the two first modes of a silver sphere in PMMA using tabulated value for ǫ m . The quadrupolar mode presents a quality factor almost 5 times higher than the dipolar mode since it has limited radiative losses. Indeed, quadrupolar mode poorly couples to the far-field. Finally, similar Q values (Q n ≈ 50) are obtained for all the higher modes (n ≥ 3). Here again, assuming a Drude metal and in the quasi-static approximation, we qualitatively explains this result. Actually, Q factors of high order modes are absorption loss limited (Γ rad n ≪ Γ abs , so called dark modes) and tend to Q ∞ = ω ∞ /Γ abs . However this overestimates the mode quality factor (Q ∞ ≈ 207) as compared to the value deduced using tabulated value for ǫ m and exact Mie expansion.

B. Effective volumes
Last, we express the total coupling strength of a dipolar emitter to the spherical metallic particle for very short separation distances (kz 0 ≪ 1) 21,22 : So that the coupling strength to the n th mode is easily deduced as where we have used approximated expression (17) for the n th polarisability. The mode effective volume is then straightforwardly deduced from comparison to the Purcell factor (Eq. 1) We obtain for instance quadrupole effective volume V ⊥ 2 = 4πz 8 0 /(9R 5 ) = 3.2 10 −4 µm 3 = (68 nm) 3 for an emitter 10 nm away from a 25 nm radius metallic particle in air. The calculated effective volumes of the three first modes are shown on figure 4. At long distance, the smaller effective volume (most efficient coupling to an emitter) is associated to the dipole mode since it presents the largest extension. When the emitter-particle distance decreases the coupling strength to the quadrupolar (d ≤ 12 nm), then hexapolar (d ≤ 10 nm) mode becomes stronger as revealed by their lower effective volume. Most efficient coupling between an emitter and one of the particle modes obviously occurs at contact since plasmons mode are confined near the particle surface.
Finally, it is worthwhile to note that mode effective volume is generally defined independently on the particle-emitter distance so that it gives an estimation of the mode extension. This is done by evaluating the maximum effective volume available, and for a random emitter orientation. In case of localized SPP, this is achieved for contact (z 0 = R). Since the decay rate of a randomly oriented emitter expresses γ = (γ ⊥ + 2γ )/3, it comes where V 0 = 4πR 3 /3 is the metallic sphere volume and V n refers to a dipole parallel to the sphere surface (see Eq. 36 in appendix, with ǫ B = 1). Dipolar and quadrupolar mode volumes are V 1 = 3/2 V 0 and V 2 = 3/5 V 0 , respectively. The expression (24), derived for a Drude metal, quantifies an extremely important property of localized SPPs; their effective volume does not depend on the wavelength and is of the order of the particle volume. Metallic nanoparticles therefore support localized modes of strongly subwavelength extension. Highest order modes have negligible extension (V n → 0 for n → ∞, see Fig. 5). As a comparison, photonics cavity modes are generally limited by the diffraction limit so that their effective volume is at best of the order of (λ/n B ) 3 (see ref. 32 ). Nevertheless, the extremelly reduced SPP volume is achieved at the expense of the mode quality factor. It should be noticed that a low quality factor indicates a large cavity resonance FWHM so that emitter-SPP coupling can occur on a large spectrum range.
Note that mode effective volume is generally defined by its energy confinement V ef f = ǫ|E(r)| 2 dr/max(ǫ|E(r)| 2 ) 12 . Sun and coworkers used this definition and obtained 33 V n = 6V 0 /(n + 1) 2 that is in agreement with our expression for the dipolar mode volume (V 1 = 1.5V 0 ) but leads to slightly different values for other modes (e.g V 2 ≈ 0.67V 0 instead of (however, he defined the coupling rate to the whole system rather than considering the coupling into a single mode). Oppositely, we adopt here a phenomenological approach where the mode volume is defined so that Purcell factor remains valid. Nevertheless, both methods lead to very similar results for the effective volumes of localized SPPs supported by a nanosphere. Since mode volume is generally a simple way to qualitatively characterize the mode extension, expressions derived here or by Sun et al could be used.

IV. β -FACTOR
Purcell factor quantifies the coupling strength between a quantum emitter and a (plasmon) mode but lacks information on the coupling efficiency as compared to all the others emitter relaxation channels. For the sake of clarity, it has to be mentionned that coupling strength to one SPP mode corresponds to the total emission decay rate induced by this coupling. It does not permit to distinguish radiative (γ rad ) and non radiative (γ N R ) coupling into a single mode. This could be done by numerically cancelling the imaginary part of the metal dielectric function (see also ref. 35 for similar discussion in case of coupling to delocalized SPPs). However, one can determine the coupling efficiency into a single mode as compared to all others modes. This coupling efficiency, or the so-called β-factor, is easily estimated in case of a spherical metallic particle since all the available channels are taken into account in the Mie expansion. Coupling efficiency into n th mode writes where < γ >= (γ ⊥ + 2γ )/3 is the devay rate of a randomly oriented molecule. β-factor is represented on Fig. 6 for the first three modes. A maximum efficiency of 90% can be achieved in the dipolar mode (d ∼ 10 nm) and 87% into quadrupolar mode (d ∼ 15 nm).
The coupling efficiency into the hexapolar mode is lower (∼ 60% around d ∼ 15 nm) since it has a very low extension, as indicated in Fig. 5. For very short distances, all the coupling efficiencies drop down to zero since all the higher order modes accumulate in this region, opening numerous alternative decay channels. Moreover, it is possible to efficiently couple the emitter to either the dipolar or quadrupolar mode, by matching the emitter and mode wavelengths. This is of strong importance when designing a SPASER (or plasmon laser) 5 so that the active mode can be tuned on the dipolar or quadrupolar mode. This last spasing mode would consist of an extremely localized and ultrafast nanosource 18 . wavelength is assumed to match the considered mode. Therefore β 1 , β 2 and β 3 are calculated at λ 1 = 435 nm (dipolar resonance), λ 2 = 387 nm (quadrupolar resonance), and λ 3 = 375 nm (hexapolar resonance), respectively.

V. CONCLUSION
We explicitly determined the effective volumes for all the SPP modes supported by a metallic nanosphere. Their quality factor is also approximated in the quasi-static case or calculated using exact Mie expansion. Rather low quality factors ranging from 10 to 100 can be achieved, associated to extremely confined effective volume of nanometric dimensions.

VII. APPENDIX
In this appendix, we derive the expression of mode quality factor and effective volume for a spherical particle embedded in homogeneous background of optical index n B = √ ǫ B .
All the expressions obtained in this appendix reduce to the simple analytical case discussed in the main text for ǫ ∞ = 1 (bound electrons contribution neglected) and ǫ B = 1 (background medium is air).