Coherent Energy Transfers between Orthogonal ... - ACS Publications

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Coherent Energy Transfers between Orthogonal Radiant and Weakly Radiant Plasmonic Nanorod Resonators Zhong-Jian Yang* Department of Applied Physics, Chalmers University of Technology, 412 96 Göteborg, Sweden S Supporting Information *

ABSTRACT: The coherent couplings between plasmon modes have attracted significant research interest recently as they can dramatically modify the light−matter interactions and have many applications such as sensor and metamaterial. In this work, the energy transfer properties in a radiant and weakly radiant plasmon mode coupled system are investigated in the time domain. The phase difference between the two modes is found to vary from π to 0 with wavelength, and it is about π/2 near the resonance. The plasmon energy transfers back and forth between the two modes, and the transfer cycles increase with the coupling strength. Therefore, the system can undergo from enhanced absorption to Fano resonance phenomena on the optical spectra. Furthermore, the total energy transfer efficiency from the radiant plasmon mode to the weakly radiant one is also studied, and it shows a Förster resonance energy transfer (FRET)-like behavior.

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INTRODUCTION Understanding the coherent couplings between plasmonic resonators is important for the fundamental study of plasmonics1,2 and also has practical significance for applications such as optical antennas,3,4 biosensors,5,6 plasmon rulers,7 and metamaterials.8,9 Recently, the couplings between bright and dark modes in plasmonic structures have attracted significant research interest. A bright plasmon mode can be excited directly by a plane wave excitation source, while a dark mode, which does not scatter light, usually cannot be excited directly by the same source. A bright mode is usually broader than a dark mode in the spectra. When these two modes are coupled to each other through near-field interactions, some interesting phenomena can be found such as Fano resonance,10−17 electromagnetic induced transparency (EIT),18,19 and enhanced absorption.20−22 These phenomena have been shown in plasmonics by mimicking quantum interference effects. Plasmonic Fano resonance or EIT usually refers to a dip appearing on the extinction (scattering) spectra due to the destructive interference between broad and narrow modes. Plasmonic enhanced absorption corresponds to the phenomenon that the absorption of the whole coupled structure is higher than that of the uncoupled system. Coherent interactions between different plasmonic modes lead to energy transfers between them. Studying the energy transfer properties and related phenomena is therefore important for understanding the physical processes during these interactions as well as for developing applications. However, energy transfer phenomena have seldom been discussed in the context of plasmonics. For example, in some plasmonic structures with Fano resonances or EIT,13−15,18 the two energy transfer pathways I → B and I → B → D → B (I, B, and D are source, bright mode, and dark mode, respectively) © 2015 American Chemical Society

were proposed to explain the Fano interferences, but they were just mentioned as an analogue to the atomic physics concept. Recent developments in ultrafast spectroscopic methods have enabled a means to investigate and manipulate optical processes in plasmonic nanostructures on femtosecond time scales.23−26 Real-time observations of energy transfer processes have inspired and motivated us to carry out a detailed investigation of coherent plasmonic coupling in the time domain. Investigation on how much of the energy transferred from one mode to another one (energy transfer efficiency, ETE) in their interactions is also important for understanding the coupling and has potential applications like nanorulers. In molecules or quantum dots (point dipoles), the transfer efficiency from donor to acceptor shows Förster resonance energy transfer (FRET) behavior.27,28 This property has been used for applications in biology rulers and sensors.29,30 Here in our plasmonic systems, if we can construct two dipolar resonance modes which work similarly to the donor and acceptor, respectively, this could enable purely plasmonic analogues of FRET, where there are much stronger dipole responses than in molecules. In this work, we use a radiant and weakly radiant plasmon modes coupled nanorod system to investigate the coherent energy transfer properties in plasmonic Fano resonance (or EIT) and enhanced absorption phenomena. The structure consists of two orthogonal Au nanorods. One of them is a big rod hosting a radiant broad dipolar plasmon mode, and the other one is a small rod with weakly radiant narrow dipole plasmon mode. Here, the nanorod system is chosen due to the Received: August 20, 2015 Revised: October 31, 2015 Published: October 31, 2015 26079

DOI: 10.1021/acs.jpcc.5b08122 J. Phys. Chem. C 2015, 119, 26079−26085

Article

The Journal of Physical Chemistry C

the software returns the amount of power transmitted through a power monitor or a profile monitor, normalized to the source power. Negative values mean the power is flowing in the negative direction. The transmission is calculated with the formula T(f) = (1/2) ∫ real(P⃗( f)Monitor)·dS⃗/source power where T( f) is the normalized transmission as a function of frequency; P(f) is the Poynting vector; and dS⃗ is the surface normal. By adding the six transmissions and multiplying by the source area one can get the absorption cross section. The scattering cross section can be obtained in a similar way, where the only difference is that the monitors are placed in only the scattered light region (for absorption the monitors are in the total field region). The extinction cross section equals the sum of absorption and scattering cross sections.

facts that the system is common and simple, the plasmon modes are highly tunable with geometry,31,32 and the near-field responses in time domain for different modes are easily obtained, respectively. The last fact can help one to easily get the energy transfer properties between plasmon modes on different nanorods. In this system, the plasmon resonance of the big rod, which can be excited directly, is a bright mode. However, the weakly radiant resonance of the small rod cannot be excited directly due to the excitation configuration. So this behavior is quite similar to a dark-like mode in Fano resonances. Thus, a bright and “dark”-like mode coupled configuration is constructed formally. It is found that the phase difference between the two modes shows a variation from π to 0 with wavelength, and it is about π/2 near resonance peaks. The coherent energy transfer cycles between the two modes varies with their coupling strength. The combination of the two factors about the phase difference and energy transfer cycles can lead to the phenomena from enhanced absorption to EIT (or Fano resonance) just by increasing the coupling strength in this system. On the basis of the above results the ETE from the radiant to the weakly radiant mode was also studied, and it shows a Fö rster resonance energy transfer (FRET)-like behavior.

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RESULTS AND DISCUSSION Figure 1(a) illustrates the geometry of the radiant and weakly radiant mode coupled plasmonic nanostructure. The two Au nanorods are perpendicular to each other. The dielectric of Au is taken from ref 33. The radiant bright mode (B) on the big nanorod (NR1) and the weakly radiant mode (denoted by D as it works similar to a dark-like mode) on the small nanorod (NR2) are both dipolar plasmon resonances with peak positions close to each other. The mode on the NR1 has broad spectra, and the scattering is much larger than absorption; however, the spectra of the NR2 are narrow, and the absorption is higher than scattering (Figures 1(b) and 1(c)). The mode on the NR2 cannot be excited directly by the plane wave excitation source with the given x-axis polarization (Figure 1(a)). The coupling strength between the two modes can be tuned by varying the distance between the two rods. It is noted that some optical properties of similar structures have been investigated such as spectra responses34 and polarization conversion.35 Weak Coupling and Enhanced Absorption. With distance d = 140 nm, the coupling between the B and D mode is weak. The enhanced absorption phenomenon is obtained as shown in Figure 2(a). The absorption of the whole structure is larger than the individual bright rod (NR1). However, the absorption of NR1 becomes smaller (Figure 2(a)), and the extinction peak of the whole system decreases with the coupling (Figure 2(b)). To understand this here we study the energy transfer properties, where an important property is the energy transfer pathways. To get to know this, one can use a pulse plane wave source to excite our system and monitor the electric field responses with time near the two rods (points PB and PD in Figures 1(a) and 2(c), each is 10 nm from its rod end surface) as shown in Figure 2(d). The frequency range of the pulse plane wave is the same as that in Figure 1, which covers only one mode (the dipolar one) on each rod, and the pulse excitation ends at about 20 fs. Here only the Ex of NR1 (Ey of NR2) is shown because in this specific configuration the other field components at the point are too small and can be ignored. It is clearly seen that at first the field on NR1 is excited, and then it drives the field on NR2 (the red line follows the black line with some retardations within one period). Alternatively, one could also get the energy transfer pathway by the field wave packets. As it is seen the wave packet of the NR2 follows that of NR1, and this means the energy wave packet transfers from NR1 to NR2. After a certain time (about 46 fs), the field on NR2 starts to drive the field on NR1, although this signal is very weak. This means there is a weak energy transfer pathway D → B. So the pathways I → B and I → B → D → B with weak D → B are found in this structure now. The whole

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THEORETICAL METHODS The simulations were carried out by using commercial finitedifference time-domain (FDTD) software (Lumerical FDTD). The mesh size near all the simulated structures is 1 × 1 × 1 nm3. The excitation source is the total-field scattered-field (TFSF) plane wave. For all coupled structures, the excitation plane wave polarization is along the x-axis, and the wave vector is along the z-axis (Figure 1). Perfectly matched layer (PML) boundary conditions were used in the simulations. The surrounding index for simulations is n = 1. The absorption cross section for a structure is obtained by normalizing the power flowing into the structure by the plane wave intensity. In calculations, we set a power monitor box surrounding the structure (6 2D monitors in total) and then get the transmission of each monitor. The transmission function in

Figure 1. (a) Configuration of the coupled two nanorod structure. The length, width, and height of the big rod (NR1) are 260, 60, and 30 nm, respectively. For the small rod (NR2), the length, width, and height are 200, 20, and 30 nm, respectively. The distance between NR1 and NR2 is d. The far-field spectra of the individual NR1 and NR2 are shown in (b) and (c), respectively. Each individual rod is excited by the plane wave with polarization along its long axis. 26080


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Figure 3. Phase differences between the two modes on NR1 and NR2. Electric fields of each rod as a function of time under continuous excitation of a single wavelength source at λ = 930, 1035, and 1160 nm are shown in (a), (b), and (c), respectively. The Ex of NR1 (at point PB) and Ey of NR2 (at point PD) are shown in black and red lines, respectively. (d) The phase difference between the fields of NR1 and NR2 as a function of wavelength.

Figure 2. Weak coupling and enhanced absorption. The distance between two rods is d = 140 nm. (a) Absorption spectra for NR1 (green line), NR2 (blue line), and the whole structure (red line). The black line shows the results for the individual NR1. (b) The scattering spectra for coupled structure (blue solid line) and individual NR1 (blue dotted line), and the extinction spectra for coupled structure (red solid line) and individual NR1 (red dotted line). (c) Near-field distribution at resonance λ = 1035 nm. Each one of PB and PD is 10 nm from the rod end. (d) Electric field as a function of time on each rod under short pulse excitation. The Ex of NR1 (at point PB) and Ey of NR2 (at point PD) are shown in black and red lines, respectively. Part of (c) is zoomed in and plotted by the inset to show the field driving property in this region more clearly.

scattered and interact destructively with the original energy (I → B). This energy redistribution between absorption and scattering explains why there could still be enhanced absorption under destructive interference. One can also see why the enhanced absorption requires higher scattering on the bright plasmonic mode, which has also been discussed in ref 21. Our results also show that there is requirement for the D mode to induce enhanced absorption. A more pronounced enhanced absorption can be obtained by decreasing the NR2 size (increasing the ratio of absorption to scattering; see Figure S1a in the Supporting Information). Likewise, if the absorption of NR2 is smaller than its scattering, there is no enhanced absorption phenomenon anymore (Figure S1b). Moderate and Strong Couplings. The coupling strength is increased by decreasing the distance between the two rods. For d = 60 nm, the Fano resonance (or EIT) is obtained as a dip appears on the absorption spectrum (scattering and extinction spectra, see Figure S2) of NR1 or the whole structure (Figure 4(a)). This Fano resonance in other similar structures has also been reported before.36,37 The phase difference as a function of wavelength between the B and D modes is nearly the same as that with the weak coupling situation (Figure 3(d)). The energy transfer pathways can also be obtained from the field responses with time on the two rods under short pulse plane wave excitation as shown in Figure 4(b). Before 30 fs, the energy transfer from B to D is seen. Then there is a clear transfer from D back to B. So there are still two clear pathways I → B and I → B → D → B; however, here the D → B is stronger than that in the weak coupling situation. This means the destructive interference becomes stronger, and as a result the Fano dip appears near the peak of the spectrum. Here there is still a very weak transfer B → D after D → B, but it can be ignored since it almost does not affect the spectrum response. As the coupling strength is further increased to d = 20 nm, besides the undergoing Fano resonance a dip appears on the NR2 absorption spectrum (Figure 4(c)), which is a new feature due to the strong coupling in this structure. It is easy to check that the phase difference between the two modes still keeps

process is visible in Figure 2d despite the fields being strongly damped over time. Another important factor in the energy transfer properties is the phase difference between the optical responses on the two modes as the energy transfer is coherent. In order to get it, we use single wavelength plane waves to excite the structure continuously. After enough time, one could get stable responses. Figures 3(a)−3(c) show the electric field responses with time (at points PB and PD in Figures 1(a) and 2(c)) at wavelengths λ = 930, 1035, and 1160 nm, respectively. Only a short period time is displayed as to clearly show the phase differences. Here we also only show the Ex of NR1 and Ey of NR2 because they dominate at their respective locations. The phase difference between NR1 and NR2 is about π/2 near resonance (λ = 1035 nm). This means that the energy transferred from D back to B will have a π phase difference from the one with I → B excitation. This will cause the destructive interferences near peak. The phase difference as a function of wavelength is shown in Figure 3(d). It goes from π to 0 with wavelength. Comparing the changes of absorption and scattering (Figure 2) due to the coupling, one can see that the absorption is enhanced at the expense of more reduced scattering near resonance. This is due to the fact that the scattering on NR1 is larger than absorption, while on NR2 the situation is reversed (Figure 1). So more of the near-field energy transferred from NR1 to NR2 (B → D) will be absorbed on NR2 (in fact the ratio of absorption to scattering for NR2 here is the same as that of an individual NR2 with plane wave excitation, see the relevant discussion later). When the near-field energy is transferred back from NR2 to NR1 (D → B), more will be 26081


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(Figure 4(a)). The strong coupling corresponding to a dip appears on each spectrum of NR1 and NR2 (Figure 4(c)). The hybridization in this system can also be explained by the energy transfer behavior. To explain this we need to focus on the responses at wavelengths that are away from resonances. In the above discussion for spectra responses, we focused on the resonance wavelength where the phase difference is Δφ ≈ π/2. In fact the hybridization effect here is also seen as the whole system spectrum is broader than the individual NR1 (Figures 2(b), 4(a), and 4(c)). Apart from the resonance center wavelength, Δφ shows a tendency to 0 or π. For Δφ either near 0 or π, the two energy pathways I → B and I → B → D → B show a constructive interference on NR1 as the energy on NR1 from these two pathways shows a 0 or 2π phase difference (similar for NR2 with strong coupling, the situation can also be explained this way). Compared to the uncoupled system this interference causes the enhanced optical responses of the whole structure near Δφ = 0 and even near antibonding mode (Δφ = π). ETE and FRET-Like Behavior. Now let us turn to the ETE from NR1 (donor) to NR2 (acceptor). For FRET in molecules, the ETE can be defined as Et = kET/(kET + kDi), where kET is the energy transfer rate from donor to acceptor and kDi is the instinct energy decay rate of the donor in the absence of acceptor (kET + kDi is the total de-excitation rate of donor). In fact this formula can be used to define the general energy transfer rate for any transfer channel which is not only limited for FRET. To get kDi in our system, we can fit the energy decay of the individual NR1 after a pulse excitation (Figure S3, Supporting Information). The value for kDi is about 1/3.7 fs−1. For the coupled system, the configuration is the same as before (Figure 1). The energy transfer rate kET here can be obtained by measuring the transfer time between different energy wave packets as shown in Figure 4(d). The time for one complete energy transfer cycle Δt can be calculated by the separation between the beginning of one transfer way (it should be after the end of the excitation pulse which is about 20 fs in our system) like D → B and the next adjacent same one. The transfer rate between donor (or B) and acceptor (or D) is obtained by kET = 2/Δt. In this way one can get the ETE for different coupling strengths Et as shown by the blue dots in Figure 5(c). It is seen that Et increases with the coupling strength. The results are only shown to d = 60 nm because there is not a complete transfer cycle on the field responses for weaker couplings. As the time domain and frequency domain

Figure 4. Moderate and strong couplings. (a) and (b) correspond to the situation with d = 60 nm. (a) The absorption spectra for coupled NR1 (green line), NR2 (blue line), and the whole structure (red line). The black line shows the results for the individual NR1. (b) The electric field as a function of time with short pulse excitation. The red (at point PB) and black (at point PD) lines correspond to fields near NR1 and NR2, respectively. (c) and (d) show the same kind of contents as (a) and (b), respectively, with d = 20 nm.

similar behavior to that with weaker couplings. Here the main reason for the new feature should be the energy transfer pathway. The field responses with time near NR1 and NR2 are shown by Figure 4(d). It shows that the energy that goes from D to B now can go back to D again (D → B → D) due to such a strong coupling. Before this, there was also a pathway B → D. So near the resonance wavelength, these two pathways show destructive interference on D, and a Fano dip appears on the spectrum of NR2. It is seen that the energy of the wavepacket can go through more than one cycle between D and B with strong couplings. This is similar to the vacuum Rabi oscillations in atomic and quantum dot systems.38,39 A time evolution of the near-field response for the strong coupling case is illustrated in more detail in the movie in the Supporting Information. Here it is needed to point out that we define the weak, moderate, and strong couplings by the absorption spectra lineshapes of radiant and weakly radiant modes. We take the weak coupling as the absorption spectra for both NR1 and NR2 peaks (Figure 2(a)). For the moderate coupling, the spectrum of NR2 is a peak, but there is a dip on the spectrum of NR1

Figure 5. ETE and FRET-like behavior. (a) Schematic of a general configuration and the geometry parameters for FRET-like property study. The starting point of dx and dy (the green dot) is located at the point which is the half width of NR2 (10 nm) left from the right top corner of NR1. (b) Absorption cross-section spectra of NR2 with different d (dx = 0, dy = d; with dx = 0, the configuration in (a) is the same as that in Figure 1(a)). (c) ETE from NR1 to NR2 obtained by integrated extinction spectra (black dots), field response in time domain (Et, blue dots), and FRET fitting (red line). The distance between the two rods is d. 26082


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the left part of the NR1 region should show the same behavior. The discussion about the situation when dy ≲ 0 should also be quite similar to the above one. Scattered Far-Field Properties. Due to the orthogonal configuration feature in our system, the scattering cross section of each rod in our coupled structure can be obtained from the far-field responses. Let us discuss the case for an individual NR1 first. If we now consider only the individual NR1 in Figure 5(a), there is a dipole plasmon response (along x-axis) on it. Then it will show some basic far-field properties of an electric dipole. On the y−z plane which is perpendicular to the dipole moment, there is only the Ex far field (Ey−z x ), and its amplitude is the same on the plane (with the same distance to the dipole). The scattering cross section of the dipole mode on NR1 can be 2 2 written as40 σscat = Cs × |Ey−z x | /E0, where E0 is the amplitude of the excitation plane wave electric field and Cs a constant, Cs = 8πr2/3 = 8π/3 (r is the far-field monitor sphere radius which is 1 m in all our FDTD calculations; see Figure S5 and relevant discussion in the Supporting Information). Let us now turn to the coupled system. Figure 6(a) shows the calculated far fields on the x−z plane with dy = 35 nm for

are closely related, the results in frequency domain could also be considered to investigate the ETE. The absorption spectra of NR2 with different couplings are shown in Figure 5(b). The area under a spectrum increases with the coupling strength (by decreasing the distance d). Since extinction is more general to describe the energy behavior, we also need to get the scattering of NR2 in the coupled system, but we cannot calculate this part directly in our simulations. So we here use again the fact that in this system the near-field energy transfer from NR1 to NR2 has the same ratio of scattering to absorption as the individual NR2 with plane wave excitation. The value for the ratio is about 0.92 (Figure 1(c)). Now one can get the integrated extinctions of NR2 and then normalize them to the integrated extinction of individual NR1 (donor). The results are shown by the black dots in Figure 5(c). They agree well with the efficiency Et we got before. This means that the spectra integrations can also be used to get ETE (even for weak couplings). To verify if the ETE is FRET, one needs to check the relation between the efficiency and donor-to-acceptor separation distance R. The relation is 1

E= 1+

6

( ) R R0

(1)

where E is FRET efficiency; R0 is the FRET distance; and R is the distance between two point dipoles. In our system, the nanorods are of finite sizes. So here we assume that there is a point on each rod which can represent the location of its dipole resonance (see Figure 5(a)). For the FRET distance R0, we have R60 ∝ κ2, where κ2 is orientation dependence.27 κ2 can be expressed as κ2 = (2 cos θDR̂ ·P̂A + sin θDθ̂D·P̂A)2, where R̂ is the vector from donor and acceptor and θD is the angle from the donor dipole axis to R̂ . In our specific configuration, κ2 is κ2 = (3 cos θD sin θD)2. So we have R60 ∝ (((RxRy)/(R2)))2 = C(((RxRy)/(R2)))2, where C is a constant, Rx = Rx0 + dx, and Ry = Ry0 + dy (the point for dx = dy = 0 is shown by the green dot). Now eq 1 can be written as E=

1 1 + CR10/(R xR y)2

(2)

To get Rx0 and Ry0, in principle one can use any three results with different acceptor locations (for example with dx = 0 and three different dy; this is the case that we investigated above, and dy is dy = d now). The values for them are obtained to be Rx0 ≈ 160 nm and Ry0 ≈ 200 nm. Then the fitting results with eq 2 show perfect agreement with simulations as shown in Figure 5(c). The fitting constant C here is C = 1.6 × 10−15 nm−6. Here it should be noted that the region with −120 nm ≤ dx ≲ 0 (the region between two gray dashed lines in Figure 5(a)) is special. In fact, there is no such corresponding region for point dipoles. The ETE is relatively smaller (even the distance is shorter). Here it can be understood as that the two parts on the donor rod which are left and right of the acceptor will cancel each other when they are coupling with the acceptor. The extreme case is for dx = −120 nm which is the NR1 center. The corresponding efficiency is around 0 as there is almost no coupling between the two rods. Numerically, the ETE can still be fitted by eq 2 with the same Rx0 and Ry0, but the fitting constant C will increase with dx approaching the NR1 center (Figure S4, Supporting Information). For symmetry reasons,

Figure 6. Scattered far-field properties. (a) Electric far-field intensity distribution of three different components on the x−z plane (1 m radius from the structure). λ = 1035 nm. (b) The corresponding farfield Ey intensity response spectra with different d (d = dy).

the two rods. The wavelength is λ = 760 nm. The far field Ey (or Ex−y y ) is clearly seen, and its amplitude is nearly the same on the plane. One can easily extract its intensity and plot it as a function of wavelength as shown by the black line in Figure 6(b). Since the dipole on NR1 does not contribute to it at all, this Ey is only from the dipole resonance on NR2. This directly shows that there is energy transfer from NR1 to NR2. Here as the two dipoles on NR1 and NR2 are perpendicular to each other, the one on NR2 only induces Ey far field in the x−z plane. Similar to the case of only one dipole, the scattering cross section from NR2 in the coupled system can be expressed 2 2 as σ2scat = ((8π)/3) × |Ey−z x | /E0. With this equation, one can easily check that the integration of σ2scat is about 0.92 times the 26083



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ACKNOWLEDGMENTS This work was founded by the Knut and Alice Wallenberg Foundation. The author thanks T. Shegai, M. Käll, and T. Antosiewicz for helpful discussions.

integrated absorption of NR2 for all distances. This ratio was used as an assumption above, and here it is confirmed numerically. The dipole on NR2 does not contribute to the farfield Ex on the y−z plane either. So we have σ1scat = ((8π)/3) × | 2 2 Ey−z x | /E0 for the scattering from only NR1 in the system.

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CONCLUSIONS In conclusion, the energy transfer properties in the radiant and weakly radiant dipole mode coupled plasmonic structures were investigated by the FDTD method. The radiant and weakly radiant dipole plasmon modes are formed by two perpendicular Au nanorods. The phase difference between the two modes varies from π to 0 as a function of wavelength, and it keeps nearly the same with different coupling strengths. Near resonance peaks of the modes, the phase difference is about π/2. The coherent energy transfer cycles between B and D vary with coupling strength, which combined with the phase difference can lead to different features on spectra responses from enhanced absorption to EIT. The hybridization effects in this system can also be explained by this phase-related energy transfer. The ETE from B to D can be obtained by extinction spectra or results in time domain, and it shows FRET-like behavior. In our system, this energy transfer will induce energy redistribution for the far fields with different polarizations. This study of energy transfer behavior provides a powerful way to investigate the coupled radiant and weakly radiant plasmonic cavities from the microscopic point of view. It helps us to understand the spectra phenomena, especially the differences between absorption and scattering (extinction) responses in general. For example, if we now replace NR2 with a quadrupole resonant mode, quite similar spectra behaviors from enhanced absorption to Fano resonance can be reproduced by varying the coupling strength (Figure S6). It could also be used to understand the absorption (or scattering) properties of plasmon and exciton (for example a molecular or quantum dot) coupled systems.41−43 This microscopic way may also be instructive for the study of two bright mode coupled systems and other more complex plasmonic structures. Moreover, the FRET-like energy transfer efficiency could find applications in chemical and biological sensing.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b08122. Absorptions with different geometry parameters of NR2; instinct energy decay rate of the individual donor (NR1); fitting constant C as a function of dx in the FRET equation; relation between scattering cross section and scattering far field of an electric dipole; coupled system with a quadrupole resonant mode on NR2 (PDF) Time evolution of the near-field response for the strong coupling case (MPG)

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

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