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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 23 — Nov. 18, 2013
  • pp: 27602–27610
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Strong coupling between a dipole emitter and localized plasmons: enhancement by sharp silver tips

Stefania D’Agostino, Filippo Alpeggiani, and Lucio Claudio Andreani  »View Author Affiliations


Optics Express, Vol. 21, Issue 23, pp. 27602-27610 (2013)
http://dx.doi.org/10.1364/OE.21.027602


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Abstract

In this work sharp silver nanotips are analyzed and proposed as useful plasmonic tools to reduce the threshold for the onset of strong coupling in the electromagnetic interaction of a point-like emitter with localized surface plasmons. If compared to similarly-sized spherical nanoparticles, conically-shaped nanoparticles turn out to be extremely useful to reduce the oscillator strength requirements for the emitting dipole, a reduction of the threshold by one sixth being obtained in a double cone configuration. Moreover the transition to the strong coupling regime is analyzed for several cone apertures, revealing a nonmonotonic behavior with the appearance of an optimal cone geometry. The emitted-light spectrum is obtained from the computation of the perturbative decay rate and photonic Lamb shift in the classical framework of the Discrete Dipole Approximation. This combined classical-quantum electrodynamics treatment is useful for the theoretical investigation on nonperturbative light-matter interactions involving complex shaped nanoparticles or aggregates.

© 2013 OSA

1. Introduction

In the study of the spontaneous emission of light by an excited dipolar emitter, several different regimes can be recognized [1

1. S. Haroche and J.-M. Raimond, Exploring the Quantum : Atoms, Cavities and Photons (Oxford University, 2006). [CrossRef]

]. In the perturbative or weak-coupling regime, deexcitation of the dipole is described by means of a decay rate calculated with Fermi golden rule. On the other hand, when the strength of interaction overcomes radiative and nonradiative losses, the system could enter the non-perturbative or strong coupling regime, where the excitation is reversibly exchanged between the emitter and the electromagnetic field. A third recently investigated scenario is the ultrastrong coupling regime [2

2. C. Ciuti and I. Carusotto, “Input-output theory of cavities in the ultra-strong coupling regime: the case of time-independent cavity parameters,” Phys. Rev. A 74, 033811 (2006). [CrossRef]

4

4. R. Stassi, A. Ridolfo, O. Di Stefano, M. J. Hartmann, and S. Savasta, “Spontaneous conversion from virtual to real photons in the ultrastrong-coupling regime,” Phys. Rev. Lett. 110, 243601 (2013). [CrossRef]

], characterized by an interaction strength comparable with the dipole transition energy and a relevant role played by the nonresonant terms of the Hamiltonian.

As a general rule, the strong coupling regime can be entered when the interaction of the emitter with radiation is significantly increased due to photonic density localization and enhancement phenomena. For this reason, localized surface plasmons of metal nanosystems represent promising candidates for the investigation of strong coupling effects, thanks to the possibility of concentrating optical energy on the sub-wavelength scale and to enhance the local density of states of radiation, producing an increase of the spontaneous decay rate sometimes comparable with high-Q optical cavities. The reversible exchange of energy between the emitter and electromagnetic modes has interesting potentialities for several applications which span from quantum information to bio-sensors; for this reason, nonperturbative light-matter interaction continues to attract much interest in the nanophotonics community. Strong coupling effects in proximity to metal nanostructures have been the subject of several theoretical [10

10. A. Trügler and U. Hohenester, “Strong coupling between a metallic nanoparticle and a single molecule,” Phys. Rev. B 77, 115403 (2008). [CrossRef]

17

17. M. M. Dvoynenko and J. K. Wang, “Revisiting strong coupling between a single molecule and surface plasmons,” Opt Lett. 38, 760–762 (2013). [CrossRef] [PubMed]

] and experimental works [18

18. Y. Sugawara, T. A. Kelf, J. J. Baumberg, M. E Abdelsalam, and P. N. Bartlett, “Strong coupling between localized plasmons and organic excitons in metal nanovoids,” Phys. Rev. Lett. 97, 266808 (2006). [CrossRef]

23

23. A. E. Schlather, N. Large, A. S. Urban, P. Nordlander, and N. J. Halas, “Near-field–mediated plexcitonic coupling and giant Rabi splitting in individual metallic dimers,” Nano Lett. 13(7), 3281–3286 (2013). [CrossRef]

].

In order to maximize the coupling between the dipole and plasmonic modes, we consider a dipole located very close to an ideally sharp conical tip, since it has been shown that the enhancement of the dipole decay rate (and, in general, the strength of light–matter interaction) is greatly lowered with the increasing of the particle–dipole distance and the radius of curvature of the tip [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

]. Continuous development of fabrication techniques has brought the investigated configuration within the range of state-of-art technology [29

29. F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A Hybrid Plasmonic-Photonic Nanodevice for Label-Free Detection of a Few Molecules,” Nano Lett. 8, 2321–2327 (2008). [CrossRef] [PubMed]

31

31. F. De Angelis, R. Proietti Zaccaria, M. Francardi, C. Liberale, and E. Di Fabrizio, “Multi-scheme approach for efficient surface plasmon polariton generation in metallic conical tips on AFM-based cantilevers,” Opt. Express 19, 22268 (2011). [CrossRef] [PubMed]

]; some discussion of non-ideal configurations is also given. Our computational results reveal the possibility to enter the strong coupling regime with a dipole oscillator strength as low as a few units for the double cone configuration, a value which is compatible with those of organic molecules or quantum dots. Moreover, it represents one of the lowest thresholds in the current literature [10

10. A. Trügler and U. Hohenester, “Strong coupling between a metallic nanoparticle and a single molecule,” Phys. Rev. B 77, 115403 (2008). [CrossRef]

,11

11. E. Waks and D. Sridharan, “Cavity QED treatment of interactions between a metal nanoparticle and a dipole emitter,” Phys. Rev. A 82, 043845 (2010). [CrossRef]

,13

13. C. Van Vlack, P. T. Kristensen, and S. Hughes, “Spontaneous emission spectra and quantum light-matter interactions from a strongly coupled quantum dot metal-nanoparticle system,” Phys. Rev. B 85, 075303 (2012). [CrossRef]

15

15. A. Salomon, R. J. Gordon, Y. Prior, T. Seideman, and M. Sukharev, “Strong coupling between molecular excited states and surface plasmon modes of a slit array in a thin metal film,” Phys. Rev. Lett. 109, 073002 (2012). [CrossRef] [PubMed]

,17

17. M. M. Dvoynenko and J. K. Wang, “Revisiting strong coupling between a single molecule and surface plasmons,” Opt Lett. 38, 760–762 (2013). [CrossRef] [PubMed]

]. The effects of nonradiative losses are discussed in view of real far-field spectroscopy measurements.

We stress that the results are subject only to the approximation of macroscopic electrodynamics, since the experimental dielectric function of silver [32

32. E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

] is used and retardation effects are taken into account. Combining the computational DDA method with a quantum electrodynamical treatment of radiation-matter interaction results in a powerful approach, that can be applied to nanoparticles of any shape and for any given dispersive dielectric function.

2. Method

In the rotating wave approximation (RWA), the power spectrum (averaged over the solid angle) emitted to the far-field by a point-like dipole located at position r0 in proximity to a metal nanostructure is given by the expression [16

16. F. Alpeggiani, S. D’Agostino, and L. C. Andreani, “Surface plasmons and strong light-matter coupling in metallic nanoshells,” Phys. Rev. B 86, 035421 (2012). [CrossRef]

]
S(ω)=h¯ω2πq(ω)Γ(ω)S(ω),
(1)
S(ω)=|1i[ω0ωδω(ω)]+12Γ(ω)|2,
(2)
where ω is the emitted frequency, ω0 the dipole transition frequency, q(ω) the quantum yield of the system, Γ(ω) the perturbative dipole decay rate (calculated from the Fermi golden rule), and δω(ω) the photonic (anomalous) Lamb shift. The quantum yield is defined as the ratio between the radiative decay rate ΓR(ω) and the total decay rate Γ(ω). Nonperturbative dynamical effects are entirely contained in the term S′(ω), here named dipole spectrum, which depends on both radiative and non-radiative plasmonic decay channels. In principle, the dipole spectrum can be directly measured with near field experiments. On the other hand, the prefactor containing the quantum yield depends only on the radiative decay channel and accounts for the propagation of light from the emitter to the far-field detector.

The decay rate and Lamb shift are both functions of the frequency and assume the forms
Γ(ω)=2ω2h¯ε0c2p˜0ImGtot(r0,r0,ω)p˜0;
(3)
δω(ω)=1πh¯ε0c2𝒫0dωω2p˜0ImGsc(r0,r0,ω)p˜0ωωω2h¯ε0c2p˜0ReGsc(r0,r0,ω)p˜0,
(4)
where we have extended the lower integration limit to − and used Kramers–Kronig relations. G⃡tot = G⃡free + G⃡sc is the dyadic Green function containing the free–space term and the scattering term originating from the electromagnetic response of the metal. As it can be inferred from Eqs. (3) and (4), our expression for the far-field spectrum in Eq. (1) is the RWA equivalent of that reported in [33

33. Peijun Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009). [CrossRef]

], the main notation difference being that we explicitly write the perturbative decay rate and the Lamb shift instead of the dyadic Green function. The magnitude of Γ and δω depends on the dipole moment of the emitter 0; in this work, however, we equivalently employ the dipole oscillator strength f=2meω0p˜02/(e2h¯). Notice that Γ and δω, being quadratic with 0, become linear with the oscillator strength f.

In the computational framework of the DDA, the dyadic Green function can be calculated from the field scattered back by the nanoparticle to the dipole location:
Gsc(r0,r0,ω)p˜0=c24ω2E˜sc(r0).
(5)
Thus, we can use Eq. (5) to obtain the decay rate and the Lamb shift and calculate the full emission spectrum with Eq. (1). All the simulations done in this work have been realized with an interdipole distance of d = 1/16 nm for the metallic scatterers, this discretization being enough to ensure a good convergence level [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

].

3. Results

Before moving to the full computational results, we present a qualitative estimation of the magnitude of light-matter interaction. In Fig. 1(a) the decay rates Γ (calculated with the DDA for a f = 1 dipole) are plotted for Ag cones with five different tip apertures as a function of the dipole frequency and compared to the one obtained for an Ag sphere of the same size. For these sharp cones excited in an extremely near region, total decay rates are strongly peaked around a single surface plasmon resonance and result about 10 times larger than those which can be achieved with rounded tips with a curvature radius of 3 nm [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

]. If we suppose, for simplicity, that the peaks are of Lorentzian lineshape, the spectrum in Eq. (1) can be worked out analytically and it is found that the threshold condition for the onset of a vacuum Rabi splitting is [16

16. F. Alpeggiani, S. D’Agostino, and L. C. Andreani, “Surface plasmons and strong light-matter coupling in metallic nanoshells,” Phys. Rev. B 86, 035421 (2012). [CrossRef]

]γ < 4fΓmax, where γ and Γmax are the full width at half maximum and the maximum height of the peak, respectively, this being due to the linear dependence of the total decay rate on f. Thus, for each plasmonic system characterized by γ and Γmax, we can define the threshold oscillator strength fth = γ/(4Γmax), which provides a good starting estimate of the minimal coupling required to enter the nonperturbative regime.

Fig. 1 (a) Perturbative decay rates Γ calculated with the DDA for 20 nm Ag cones with several tip angles θ (sketch in inset), ranging from π/21 to π/3, and for a silver nanosphere with the diameter of 20 nm. The dipole (f = 1) is put 2 nm away from the metal–vacuum interface and it is oriented along the symmetry axis for the cones and radially for the sphere. (b) Threshold values for the oscillator strength required to enter the strong-coupling regime (calculated by fitting curves in (a) with Lorentzian lineshapes, as explained in text) for the different geometries.

The threshold oscillator strengths for the systems under consideration are reported in Fig. 1(b). The θ = π/13 tip shows an important decrease of the oscillator strength with respect to the spherical geometry. However, a non-monotonic behavior of fth as a function of the aperture can be noticed: the oscillator strength decreases with decreasing aperture until an optimal value of π/13 is reached, but further reducing the aperture reverses the trend and increases again the threshold oscillator strength. The trend of Fig. 1(b) is mostly determined by the behavior of the linewidth γ, which has a minimum for the π/13 aperture, as shown in Fig. 1(a). This is in relation with the fact that small-aperture cones, in spite of presenting a stronger localization of the electric field (the well-known superfocusing effect), are characterized also by a wider frequency distribution of the plasmonic modes, which results in a broader lineshape. For large-aperture cones, on the other hand, the linewidth increases towards the limiting value for a flat surface.

The present treatment allows a qualitative estimation of the effect of varying the distance between the dipole and the conical tip. As it can be inferred from [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

], the maximum of the total decay rate depends on the dipole–tip distance d with an approximate d−4 dependence, i.e., a simple doubling of the dipole–tip distance increases the threshold oscillator strength by a factor ∼ 16. As a consequence, very close proximity of the emitter to the metallic nanostructure is crucial for the investigation of nonperturbative effects. From this point of view, the conical geometry has a clear advantage, as it is naturally suitable to a raster scanning configuration [30

30. F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nature Nanotech. 5, 67–72 (2010). [CrossRef]

].

Fig. 2 The dipole spectrum S′ (ω) as a function of the oscillator strength, for (a) 20 nm diameter sphere and (b) cone with a θ = π/13 semi-aperture, for an emitter frequency ω0 fixed respectively to 3.66 eV and 1.60 eV. Insets on the left represent the analyzed systems in which the dipole is put at 2 nm from the nanoparticles.

In Fig. 3 the results obtained for a double cone configuration are reported. More in detail, the dipole is assumed to emit between two identical tips whose aperture is θ = π/13, which corresponds to the optimal one (sketch in Fig. 3). The dipole is 2 nm distant from each cone and oriented along the symmetry axis. Numerical simulations, reported elsewhere [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

], revealed the excitation of plasmonic resonances that are stronger and red-shifted with respect to the single cone one, this being probably due being due to a bonding combination of the resonances reported in Fig. 1. The higher decay rate registered in this case is responsible for a much lower threshold in the oscillator strength (fth ≃ 6) with respect to the single cone configuration (fth ≃ 13) with an anti-crossing behavior of the peak maxima in the emitted spectrum well visible at f = 20 [Fig. 3(b)]. The double-cone effect depends on the distance between the two cones, as discussed in [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

]: when the second cone is more than a few nanometers apart, the single-cone effect is recovered.

Fig. 3 The dipole spectrum S′ (ω) as a function (a) of the emitter frequency ω0 and (b) of the oscillator strength f for a double cone configuration, represented in the inset on the left. The oscillator strength in (a) is f = 20, while the dipole transition energy used in (b) is ω0 = 1.57eV.

Finally, in Fig. 4 we report the far-field emitted-power spectra S(ω) and compare them with the dipole spectra S′(ω) for the three different structures here analyzed (sphere, cone and double cone) and for several oscillator strengths, in the range 10 – 50. This comparison is presented in order to shed light on the effect of the plasmonic quantum yield, in view of real far-field spectroscopy measurements. As in most plasmonic systems, the quantum yield for the systems under consideration is low, due to the prevalence of non-radiative modes at short distance. This effect competes with the strong enhancement of light–matter coupling induced by the localization of the plasmonic field. The resulting radiative decay rate, however, remains significantly higher than the free-space decay rate — up to a few orders of magnitude for the double cone configuration [27

27. S. D’Agostino, F. Della Sala, and L. C. Andreani, “Dipole-excited surface plasmons in metallic nanoparticles: Engineering decay dynamics within the discrete-dipole approximation,” Phys. Rev. B 87, 205413 (2013). [CrossRef]

], — allowing in principle the experimental detection of the emission spectrum with a sufficient signal-to-noise ratio.

Fig. 4 The far-field emitted-power spectrum S(ω) [solid line] compared with the dipole spectrum S′(ω) [red dashed line], for different geometries treated in the work: (a) a 20 nm-diameter silver sphere, (b) a θ = π/13 silver nanocone, (c) the double cone configuration of Fig. 3. Each curve has been calculated for a different oscillator strength (in the range 10 – 50) and normalized independently of the other. Dipole frequencies are the same as in Figs. 2 and 3.

On the other hand, although the onset of the strong coupling regime depends on the total strength of plasmon-dipole interaction in both channels, radiative and non-radiative, the quantum yield induces a modulation of the far-field emission spectrum [see Eq. (1)], which can result in a significant suppression of strong coupling effects. This is evident in the case of the nanosphere (a), where the maximum of the radiative emission is centered onto the l = 1 dipolar surface plasmon at ω ≃ 3.5eV. However, the optimal resonance frequency for the transition to the strong coupling regime is around ω ≃ 3.66eV, as an effect of the excitation of higher-order strongly non-radiative plasmon modes. As a consequence, the vacuum Rabi splitting results strongly quenched in the far-field for the spherical geometry. This is not the case of the cone (b) and double cone (c) systems, for which the maximum of the radiative decay rate coincides with the maximum of the field enhancement, as a result of the excitation of the strongly radiative mode. In these cases, the fingerprint of vacuum Rabi splitting, even if partially modulated by the quantum yield, survives even in the far field spectrum.

The present results in terms of threshold oscillator strength are much closer to experimental realization than those reported in [16

16. F. Alpeggiani, S. D’Agostino, and L. C. Andreani, “Surface plasmons and strong light-matter coupling in metallic nanoshells,” Phys. Rev. B 86, 035421 (2012). [CrossRef]

], on two respects: first, the calculations of this work are performed by taking realistic dielectric functions of Ag, which have a much higher dissipation (almost an order of magnitude) than for the Drude model assumed in [16

16. F. Alpeggiani, S. D’Agostino, and L. C. Andreani, “Surface plasmons and strong light-matter coupling in metallic nanoshells,” Phys. Rev. B 86, 035421 (2012). [CrossRef]

]. Second, the conical shape of the nanoparticles results nowadays well accessible by an experimental point of view [29

29. F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A Hybrid Plasmonic-Photonic Nanodevice for Label-Free Detection of a Few Molecules,” Nano Lett. 8, 2321–2327 (2008). [CrossRef] [PubMed]

] and the configuration with the dipole outside the nanoparticle is also favourable for experimental probing by raster scanning of a nanotip [30

30. F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nature Nanotech. 5, 67–72 (2010). [CrossRef]

]. Thus, the present calculations are believed to represent a significant step in the direction of tailoring the interaction between a dipole and a metal nanoparticle, in view of achieving the strong coupling regime.

4. Conclusion

Employing the numerical DDA approach together with quantum electrodynamics turns out to be a powerful predictive method, that can be applied to metal nanoparticles of any shapes. The systems here analyzed involve emitters and nanoparticles which can be realized with state-of-art techniques, and this could open new avenues towards the real observation of the reversible exchange of energy between a dipole emitter and the electromagnetic field in plasmonic nanosystems.

Acknowledgment

This work was funded by Fondazione Cariplo under project no. 2010-0523. For computational facilities S. D. thanks NNL-CNRNANO, the Italian Institute of Technology (IIT), and the CINECA Consortium (ISCRA Award N. HP10C1ZQTO, 2011).

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F. De Angelis, M. Patrini, G. Das, I. Maksymov, M. Galli, L. Businaro, L. C. Andreani, and E. Di Fabrizio, “A Hybrid Plasmonic-Photonic Nanodevice for Label-Free Detection of a Few Molecules,” Nano Lett. 8, 2321–2327 (2008). [CrossRef] [PubMed]

30.

F. De Angelis, G. Das, P. Candeloro, M. Patrini, M. Galli, A. Bek, M. Lazzarino, I. Maksymov, C. Liberale, L. C. Andreani, and E. Di Fabrizio, “Nanoscale chemical mapping using three-dimensional adiabatic compression of surface plasmon polaritons,” Nature Nanotech. 5, 67–72 (2010). [CrossRef]

31.

F. De Angelis, R. Proietti Zaccaria, M. Francardi, C. Liberale, and E. Di Fabrizio, “Multi-scheme approach for efficient surface plasmon polariton generation in metallic conical tips on AFM-based cantilevers,” Opt. Express 19, 22268 (2011). [CrossRef] [PubMed]

32.

E. D. Palik, Handbook of Optical Constants of Solids (Academic, 1985).

33.

Peijun Yao, C. Van Vlack, A. Reza, M. Patterson, M. M. Dignam, and S. Hughes, “Ultrahigh Purcell factors and Lamb shifts in slow-light metamaterial waveguides,” Phys. Rev. B 80, 195106 (2009). [CrossRef]

OCIS Codes
(240.6680) Optics at surfaces : Surface plasmons
(270.5580) Quantum optics : Quantum electrodynamics

ToC Category:
Plasmonics

History
Original Manuscript: July 29, 2013
Revised Manuscript: October 10, 2013
Manuscript Accepted: October 11, 2013
Published: November 4, 2013

Citation
Stefania D’Agostino, Filippo Alpeggiani, and Lucio Claudio Andreani, "Strong coupling between a dipole emitter and localized plasmons: enhancement by sharp silver tips," Opt. Express 21, 27602-27610 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-23-27602


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