As observed in section II.3, the emitter orientation strongly influences the interaction between one dipole and the nanohybrid. Therefore we expect it to also affect the plasmon-mediated dipole-dipole coupling. We study here how the dipoles distribution and orientation affects the eigenvalues.
To this end we perform statistics on the dipoles position and orientation. In the experi-mental study of chapter V, the emitters are randomly distributed around the nanohybrid and their orientation is preferentially tangential to the particle surface. Here we consider N = 50 dipoles placed on a d= 18 nmthick silica shell. We perform statistics on their position and orientation in the plane tangential to the nanohybrid, for 1000 configurations. From each configuration, we estimate 3 superradiant states andN−3 = 47 subradiant states which are represented in figure II.7.
Furthermore in our model the free-space interaction in Eq. II.46 does not account for the Förster resonance energy transfer (FRET) [129] which is a non-radiative energy transfer between the molecules due to van der Waals interactions. For the molecules studied experi-mentally the FRET is efficient for molecules separated by less than 5 nm, see chapter V. We thus set a 5 nm minimum separation between the dipoles.
We first see in Fig. II.7 that both the superradiant states (red dots) and the subradiant states (black pluses) can exhibit either positive or negative energy shifts, depending on the configuration. Note that the subradiant energy shifts can reach large values as compared to the superradiant energy shifts. In accordance with the discussion paragraph II.4.b.i, we identify the states exhibiting the largest shifts as the darkest subradiant states.
Then we observe that the distinction between the superradiant and the subradiant states is not as clear as for the fully symmetrical case. Indeed in Fig. II.7 the dipoles orientation and position strongly alter the system symmetry. In particular we observe that the average superradiant state has a decay rate of about < γ >= 8.8γ0r and varies between 6.9γ0r and 12.2γ0rwith a standard deviation of 8.8%. As expected a larger standard deviation (16%) was observed when the statistics includes non-tangential orientations. In particular each dipole experiences a different Purcell effect and thus a different coupling to the plasmonic modes, depending on its orientation. We deduce that the position and orientation of the dipoles strongly influence the collective optical properties.
Figure II.7: 50dipoles are randomly distributed around a 60 nm diameter gold sphere coated by ad= 18nm thick silica shell. The dipoles have a quantum yield of η0 = 90% and are randomly oriented in a plane tangential to the particle nanohybrid. The minimum separation distance between the dipoles is 5 nm. A statistics over 1000 configurations was performed. The superradiant (the 3000 red dots) and subradiant (black pluses) modes are shown as a function of their normalized energy shift ∆/γ0r and normalized decay rate γ/γ0r. The histograms of the superradiant decay rates and energy shifts are shown in blue b).
II.4.b.iv Influence of the number of emitters for different distances to the metal core
In Ref. [37] V. Pustovit and T. Shahbazyan reported that the superradiant decay rate scales with the number of emitters in the fully symmetrical configuration. This trend is a signature of a collective emission and is thus expected in plasmon-mediated superradiance. In this last section we investigate whether this trend survives when one averages over the emitters position and orientation, as it is performed experimentally.
To this end we derive the average superradiant decay rate for the system that will be investigated experimentally in chapter V. Figure II.8 shows the evolution of the average superradiant decay rate as a function of the number of dipoles N, for different distances to the plasmonic core surface d. The plotted values are the average of the three superradiant decay rates, averaged over the dipoles position and orientation in a plane tangential to the particle surface. A minimum separation distance of 5 nm was set between the dipoles.
First we observe in Fig. II.8 that whatever the distance to the core, the superradiant decay rate scales with the number N of dipoles. This signature of superradiance thus survives in average. Then as the dipoles are closer to the metal core the slope with N increases.
This shows that the interaction between dipoles is enhanced as the dipole-plasmon coupling
Figure II.8: N dipoles are randomly distributed around a 60 nm diameter gold sphere coated by a d thick silica shell. The dipoles have a quantum yield ofη0= 90%and are randomly oriented in a plane tangential to the nanohybrid surface. The minimum separation distance between the dipoles is5nm. For each case (N, d), a statistics over the dipoles position and orientation was performed. We show the evolution of the average superradiant decay rate (dots with error bar) as a function of the number of dipoles N and the distance to the core surface d. Dashed lines are linear fits, as guides for the eyes.
increases.
II.5 Conclusion
To conclude we developed an explicit and efficient classical approach to derive the response of large collections of molecules near a spherical metallo-dielectric nanohybrid.
The optical properties of a metal nanosphere are fully described by the Mie theory and are depicted by plasmon modes. The higher the order of the mode, the more confined is the electric field at the particle surface. Hence the dipole mode is mainly a radiative mode while the contribution of the higher order modes is mainly non-radiative. The position and bandwidth of the plasmon modes are strongly affected by the MNP environment, and notably by the presence of a shell.
The radiation of a single dipole close to a nanohybrid consists in an enhancement of the radiated power as it couples to the dipole plasmon mode while the coupling to the higher order modes mainly leads to Ohmic losses. Below 5 nm from the metal surface all the energy is lost in heat dissipation (quenching effect). The Purcell effect is optimal when the dipole is oriented normally to the particle surface.
The collective radiation ofN emitters distributed near a metal sphere is characterized by 3 superradiant modes, exhibiting the largest decay rate, and N-3 subradiant states. The 3 superradiant states correspond to the 3 dimensions of the sphere. The system decay chan-nels strongly depend on the dipoles position and orientation. The largest decay rates being observed for a fully symmetrical configuration: the dipoles are uniformly distributed and ori-ented normally to the particle surface. We identified that the formation of pure superradiant states is observed when the dipole-dipole interaction is dominated by the dipole LSPP mode.
Then the higher order plasmon modes mainly contribute to promote the interaction between neighbor dipoles, via non-radiative coupling. We deduced that the optimal plasmonic core diameter for the observation of superradiant states is 60 nm. We saw that the superradiant decay rate is proportional to the number of emitters and predicted that this trend survives when averaging over the emitters position and orientation. Furthermore we showed that the associated slope increases if the emitters get closer to the plasmonic core.
The here presented model makes no assumption on the emitters position and orientation.
Furthermore it introduces the complete Mie theory, accounting for the depolarization effect and the presence of a dielectric layer between the dipoles and the metal core. This model is thus an extension of V. Pustovit and T. Shahbazyan model [37]. However the free-space interaction does not describe the FRET between neighbor dipoles. The dipole-dipole sepa-ration distance is thus limited to 5 nm minimum. Besides we assumed that the dipoles are identical. Finally this classical model does not account for pure dephasing processes.
Plasmonic superradiance with a metal nanorod
Contents
III.1 Generalization of the theoretical formalism using Quasi-Normal Modes . . . . 66 III.1.a Quasi-Normal Modes . . . 66 III.1.b Generalization of the theoretical formalism . . . 66 III.1.c Example of a metal nanorod . . . 68 III.2 Dipole doublets near a metal nanorod . . . . 68 III.2.a System eigenfrequencies . . . 69 III.2.b Spectral analysis . . . 70 III.3 100-dipoles radiation near a metal nanorod . . . . 72 III.3.a Spectra and eigenfrequencies . . . 72 III.3.b Induced dipole-moments distributions . . . 74 III.4 Ensemble average . . . . 76 III.5 Conclusion . . . . 78
T
he numerical modeling of coupled systems with large collections of classical oscillators represents a major challenge in computational electrodynamics [36,121]. We showed in chapter II that the repeated calculation of the Green function at different frequencies may be avoided by diagonalizing the problem from Eq. II.44. This diagonalization was feasible thanks to some simplifications: we proposed to expand the free-space contribution while neglecting the frequency dependency of the nanohybrid-mediated interaction. The latter approximation may not be valid for cavities with high quality factor (or similarly oscillators with low quality factor). However an analytical expression of the Green dyadics would allow to identify more suitable simplifications. As a consequence the formalism, that was introduced in chapter II, may be used to efficiently analyze collective effects in ensemble of dipoles that are coupled by any electromagnetic cavity.In this chapter, we aim at providing a more general formalism without making any hypothesis on the electromagnetic cavity, which could be photonic cavities or plasmonic nanoresonators, possibly with an overlapping of several resonances. To this end section III.1 introduces the concept of Quasi-Normal Modes as a way to describe an arbitrary cavity. In particular we show how this approach enriches the theory developed in chapter II. As an example we then consider a metal nanorod. The simple case of dipole doublets near the nanorod is treated in section III.2. This system allows us to clearly distinguish the contribu-tion of the free-space interaccontribu-tion and the plasmon-mediated interaccontribu-tion. We also show how the scattering and absorption spectra are related to the eigenvalues. Section III.3 investigates the cooperativity of the superradiant and subradiant states by studying the induced dipole moments distribution for 100 dipoles. Finally we investigate how the predictions made for the nanosphere in section II.4.b.iii of chapter II are valid in the case of a metal nanorod.
This work was performed in close collaboration with the group of Philippe Lalanne in the LP2N, and in particular with Spyridon Kosionis during his postdoctoral fellowship. I was involved in studying how the non-linear eigenproblem may be diagonalized as well as performing the calculation and analyzing the results.