Resonance Coupling between Molecular Excitons and Nonradiating

Article Cite This: ACS Photonics XXXX, XXX, XXX−XXX

Resonance Coupling between Molecular Excitons and Nonradiating Anapole Modes in Silicon Nanodisk-J-Aggregate Heterostructures Shao-Ding Liu,* Jin-Li Fan, Wen-Jie Wang, Jing-Dong Chen, and Zhi-Hui Chen Key Lab of Advanced Transducers and Intelligent Control System of Ministry of Education and Department of Physics and Optoelectronics, Taiyuan University of Technology, Taiyuan 030024, People’s Republic of China S Supporting Information *

ABSTRACT: The nonradiating nature of anapole modes owing to the compositions of electric and toroidal dipole moments makes them distinct from conversional radiative resonances, and they have been suggested for the design of nanophotonic devices such as nanolasers based on light−matter interactions tailor by nanodisks. Therefore, the investigation of resonance coupling between molecular excitons and anapole modes is not only of fundamental interest, but is also promising for practical applications. To this end, a heterostructure composed of a silicon nanodisk and a uniform molecular J-aggregate ring is used to achieve the resonance coupling between the exciton transition and the anapole mode. In contrast with that of the conversional resonances, the resonance coupling is evidenced by a scattering peak around the exciton transition frequency, and the anapole mode splits into a pair of eigenmodes characterized as pronounced scattering dips, which are termed as the formation of two hybrid anapole modes caused by the coherent energy exchange in the heterostructure, and it has been verified by the multipole decompositions and the near-field distributions. An anticrossing behavior with a mode splitting of 161 meV is observed on the energy diagram, indicating that the strong coupling regime is achieved. Furthermore, due to the unique near-field distribution associated with the anapole mode, there is a much larger upper limit value for the width of the J-aggregate ring to enhance the resonance coupling, and the molecules located around the apexes of the disk perpendicular to the incident polarization play the dominate role for the resonance coupling. KEYWORDS: anapole modes, resonance coupling, silicon nanodisks, J-aggregates, Rabi splitting

T

in the gap area and small mode volume, strong coupling with Rabi splitting of hundreds of meV are reported by using single metallic nanoparticle dimers.31−33 Nonradiating losses of plasmon resonances can be suppressed using silver, and strong coupling also can be realized with single Ag nanoparticles.34−36 Furthermore, it is shown that the strong coupling regime can even be achieved with heterostructures composed of a single QE and a single plamonic nanostructure, which would be very useful for many applications such as quantum information operations.37−40 In order to avoid the subtle effects for strong coupling with scattering spectra, the Rabi splitting has been ambiguously identified with photoluminescence spectra.27,34,41 In addition to plasmonic nanostructures, high refractive index dielectric nanostructures have gained considerable attention in recent years.42−46 Due to the intrinsic magnetic responses and the lower optical losses, dielectric nanoparticles have advantages in some aspects compared with that of their metallic counterparts. For example, enhanced nonlinear effects are demonstrated with the formation of strong electromagnetic fields inside of dielectric nanoparticles,47,48 and directional light scattering are realized with single dielectric nanoparticles associated with interferences between Mie multipolar reso-

ailoring light−matter interaction with nanoresonators paves the way for modern nanophotonic devices.1−7 Depending on the coupling strength between quantum emitters (QEs) and nanoresonators, two distinct scenarios, that is, the weak and strong coupling regimes, can be realized. In the first case, the spontaneous decay is modified caused by the variation of the local optical density with the presence of nanoresonators.8 In the second case, two hybrid eigenstates separated by a Rabi splitting energy are generated due to the coherent energy exchange between the two constituents, and the Rabi splitting exceeds the rate of electromagnetic mode damping.9 Since strong mode confinement is required, metallic nanostructures are promising platforms to achieve strong coupling.10 It has been shown that despite the large dissipative losses, the excitation of surface plasmon resonances with very small mode volumes results in strongly enhanced near-field around metallic nanostructures, and strong coupling is observed with heterostructures composed of various plasmonic nanostructures and QEs that possess strong dipole moments.11−23 For example, the visibility of Rabi splitting benefits from a large Q-factor of the resonators, and strong coupling has been demonstrated with surface lattice resonances or propagating surface plasmons in arrays of nanostructures,24−30 where a large number of molecules and nanoparticles are involved in the coupling. Owing to the formation of extremely strong near-field © XXXX American Chemical Society

Received: December 25, 2017 Published: February 14, 2018 A

DOI: 10.1021/acsphotonics.7b01598 ACS Photonics XXXX, XXX, XXX−XXX

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Figure 1. Resonance coupling between the molecular excitons and the nonradiating anapole mode in the Si nanodisk-J-aggregate heterostructures. (a) Cartesian multipole decomposition results of the scattering spectrum for the Si nanodisk with diameter D = 300 nm and thickness h = 70 nm, where the upper-right inset shows the electric field enhancement distributions at the center cross section of the xy plane (upper panel), and the magnetic field enhancement distributions at the center cross section of the yz plane (lower panel) for the scattering peak at 752 nm. (b) Scattering spectra of the Si nanodisk-J-aggregate heterostructures with varied oscillator strengths f for the exciton transition, where the width of the J-aggregate ring t = 40 nm. (c) Cartesian multipole expansion results of the scattering spectrum for the heterostructure and (d) the J-aggregate ring when the oscillator strength f = 0.4. The incidence is propagating along the z-axis, the polarization is along the x-axis, the surrounding medium is supposed to be air (n = 1), and the contributions from the multipoles in (a), (c), and (d) are indicated by the solid lines of different colors.

nances.49,50 Although the mode volumes are relatively large, dielectric nanostructures are also promising candidates to tailor light−matter interactions at the nanoscale.51−56 It has been shown that photoluminescence of QEs can be strongly enhanced with different dielectric nanostructures.51−53 Rabi splitting and even strong coupling has been observed in silicon (Si) nanosphere-J-aggregate core−shell heterostructures.57 Besides the radiative resonances, Fano resonances caused by the excitation of subradiant modes are promising to confine the incident fields around plasmonic or dielectric nanoparticles,58−67 and they have been employed to tailor light−matter interactions.68−70 It is theoretically demonstrated that the strong coupling can be achieved using a dolmen Fano structure.71 Despite the appearance of multipole hybrid resonances due to the asymmetry and lifted degeneracy of the molecular state, the resonance coupling with the Fano resonance is evidenced by a quenching dip at the exciton transition frequency, and the hybrid modes manifest as resonance peaks, which are the same as radiative modes.71 Not long ago, a new kind of subradiant resonance, namely, the anapole mode has been proposed to suppress radiative losses,72 and dielectric nanodisks are promising platforms to generate the anapole mode,73−79 where the scattering spectrum represents an antiresonance behavior, radiative losses can be suppressed effectively, and it is very similar as that of the Fano resonances.74 However, the physical mechanisms are totally different, where the anapole modes are caused by destructive interference between electric (ED) and toroidal (TD) dipole moments, both of which are radiative.73 Therefore, the anapole modes can be directly excited without breaking the structural symmetry. The nonradiating nature makes the anapole modes

distinct from conventional radiative resonances, and they have been used to enhance the nonlinear effect,80−82 the formation of extremely high Q-factor,83,84 and the realization of ideal magnetic dipole scattering.85 Very recently, the nonradiating anapole modes are proposed to engineer a coherent light source, and the so-call anapole nanolasers for intense emission and ultrafast pulse generation are demonstrated.86 Nevertheless, the laser emission is still an example of light−matter interaction in the weak coupling regime, it would be interesting to further investigate resonance coupling between the anapole modes and QEs, and to find out whether the strong coupling regime can be achieved. Therefore, in this work, we theoretically demonstrate that by choosing appropriate parameters, the strong coupling between molecular excitons and anapole modes indeed can be realized with silicon nanodisk-J-aggregate heterostructures. However, in contrast with that of conversional resonances, the resonance coupling is evidenced by a scattering peak around the exciton transition frequency, and the anapole mode splits into a pair of eigenmodes characterized as pronounced scattering dips. Besides that, the effects of the thickness of the J-aggregate ring as well as the location of the molecules on the resonance coupling are very different with that of radiative resonances, which are attributed to the unique near-field distribution associated with the anapole mode.

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RESULTS AND DISCUSSION Nonradiating Anapole Modes in Si Nanodisks. The nonradiating anapole mode can be excited with a silicon nanodisk, where the disk diameter D = 300 nm, and the thickness h = 70 nm (the lower inset, Figure 1a). The black line B


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Figure 2. Electric field enhancement distributions at the center cross section of the xy plane (upper panels), and the magnetic field enhancement distributions at the center cross section of the yz plane (lower panels) for the heterostructure and individual nanoparticles. (a) Anapole mode of the Si disk. (b) Exciton transition and (c) cavity mode of the J-aggregate ring. (d) LB and (e) the UB hybrid anapole modes of the heterostructure caused by the resonance coupling. (f) Quenching dip of the heterostructure related to the cavity mode of the J-aggregate ring. The spectral positions of the near-field distributions for the resonances are schematically denoted by the circular points in Figure 1, and the geometry parameters as well as the optical constants are identical with that of Figure 1c.

scattering contribution is crossing with that of the Cartesian ED moment around 682 nm. Instead of a constructive interference, the Cartesian ED and TD interfere destructively with each other, and the total electric dipole (ED + TD) scattering denoted by the red line is approaching to zero around 675 nm (marked with the vertical dashed line), which is the condition resulting in the formation of the anapole mode.73 The slight wavelength difference between the anapole mode and the crossing point of the Cartesian ED and TD contributions is caused by the material losses.85 Compared with that of the other geometries (e.g., spheres), the magnetic dipole scattering for the disk is weak around the anapole mode (MD, the yellow line), and the electric quadrupole (EQ, the cyan line) and octupole (EO, the magenta line) contributions also can be neglected at the same time. Nevertheless, the total scattering is dominated by the magnetic quadrupole contribution (MQ, the dark yellow line), which is a parasitic factor with excitation of the TD moment, as indicated by the magnetic near-field enhancement distributions for the anapole mode (the lower panel, Figure 2a).83 Resonance Coupling in Si Nanodisk-J-Aggregate Heterostructures. Recently, a new kind of nanolaser based on the light−matter interactions at the anapole modes of semiconductor nanodisks has been proposed for mode-locking and ultrafast pulse generation.86 In addition to the laser emission, light−matter interactions often result in the vacuum Rabi splitting. Various studies have shown that due to the coherent energy transfer between the exciton and a conversional radiative resonance (e.g., the ED and MD modes in plasmonic or dielectric nanoparticles), two polaritonic eigenm-

in Figure 1a denotes its scattering spectrum under normal incidence. A Fano-like resonance is observed in the spectrum, and there is a pronounced scattering dip at about 675 nm, which is termed as the excitation of the anapole mode.73 The corresponding electric field enhancement distributions at the center cross section of the xy plane reveal that there are opposite circular displacement currents on the up- and downsides of the Si disk (the upper panel, Figure 2a), and the generated circular magnetic moment is perpendicular to the disk surface (the lower panel, Figure 2a), thereby forming a TD moment oriented parallel to the incident polarization. When the incidence is around the scattering peak at about 752 nm, the near-field enhancement distributions shown in the upper inset of Figure 1a represent a typical ED response for the dielectric nanoparticles. Both electric and toroidal dipoles are excited with the Si nanodisk, and the physical mechanism to generate the anapole mode can be understood by further investigating the Cartesian multipole contributions into the scattering. Although the farfield radiation patterns of the electric and toroidal dipoles are identical with each other, their contributions can be unambiguously separated from the total scattering using the Cartesian multipole decompositions (the detailed derivations are shown in the Methods).87 The blue line in Figure 1a denotes the TD contributions to the particle scattering, where it is very weak when the incidence is on the long-wavelength side of the scattering dip (>700 nm), and the total scattering is dominated by the Cartesian ED moment around this spectral range (the green line, Figure 1a). However, the TD moment increases with the decreasing of the incident wavelength, and its C


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molecular J-aggregate ring are shown by the solid lines in Figure 1d (f = 0.4), and the dash-dotted line denotes the corresponding total absorption spectrum. A pronounced resonance caused by the exciton transition appears in the total scattering spectrum (∼675 nm, the black line). Compared with that of the molecular J-aggregate shell,57 the exciton transition resonance is dominated by the total electric dipole scattering (ED + TD, the red line). Besides that, it is interesting to find that a strong TD moment is generated at the same time (the blue line), and there is a destructive interference with that of the Cartesian ED mode (the green line), where the total electric dipole scattering is suppressed compared with that of the Cartesian ED contribution. Considering that the anapole mode of the Si nanodisk is also caused by the destructive interference between the Cartesian ED and TD moments, the splitting of the anapole mode for the heterostructures can be possibly attributed to the coupling of the Cartesian ED and TD moments between the Si nanodisk and the J-aggregate ring. The above assumption is confirmed by investigating the multipole expansion results of the heterostructure ( f = 0.4, Figure 1c). When the incidence is below 600 nm, the total scatterings as well as the individual multipole contributions of the heterostructure are similar as that of the Si nanodisk (Figure 1a and 1c). Nevertheless, the multipole contributions are strongly modified when the incidence is approaching to the wavelength of the exciton transition (marked with the vertical dashed line). For the Si nanodisk, there is a broad TD resonance around 660 nm (the blue line, Figure 1a), while it splits into two resonances for the heterostructure around 655 and 700 nm (the blue line, Figure 1c). The newly appeared TD resonance with lower energy is almost crossing with that of the Cartesian ED around 725 nm (the blue and green lines, Figure 1c). The summation of the two contributions around this spectral range is weaker than that of the individual scattering (ED + TD, the red line), which can be attributed to the destructive interference between the Cartesian ED and TD moments, thereby giving rise to the scattering dip at 716 nm. The same as the Si disk, this scattering dip for the heterostructure is dominated by the parasitic MQ moment (the dark yellow line), and there is a slightly shift between the scattering dip and the crossing point of the Cartesian ED and TD moments. As for the other newly appeared TD resonance around 655 nm (the blue line, Figure 1c), its scattering contribution is not crossing with that of the Cartesian ED moment (the green line). However, the total electric dipole scattering is weaker than that of the Cartesian ED around this spectral range (ED + TD, the red line), and the scattering dip at 655 nm for the heterostructure also can be attributed to the destructive interference between the Cartesian ED and TD moments (the black line, Figure 1c). Although the TD cannot compensate the Cartesian ED scattering completely due to the unmatched strength for the higher energy scattering dip, the formation of both scattering dips for the heterostructure share the same physical mechanism as that of the anapole mode generated in the Si disk. In addition, it has been shown that the mode splitting for plexcitonic nanostructures is caused by the hybridization between surface plasmon resonances and molecular excitons, which results in the formation of new hybridized energy states.31 The same as the plexcitonic nanostructures, the two scattering dips are caused by the hybridization between the anapole mode of the Si nanodisk and the exciton transition of the J-aggregate ring, and the

odes characterized as scattering or emission peaks can be observed, and the strong coupling regime can be achieved when the Rabi frequency exceeds the rate of mode damping.11−41,57 Compared with that of the conversional radiative resonances, the nonradiating nature of the anapole modes makes them scattering-free and invisible to the propagation of the electromagnetic fields. Therefore, it would be interesting to investigate the effect of the resonance coupling between the excitons and the nonradiating anapole modes, and to find out whether the strong coupling regime can be achieved. To this end, a heterostructure composed of a Si nanodisk and a uniform molecular J-aggregate ring is used to achieve the coupling between the exciton transition and the anapole mode (the inset, Figure 1b). In order to account for the J-aggregate exciton, the dielectric permittivity is described with a classical one-oscillator Lorentzian model, εex (ω) = ε∞ +

fωex2 ωex2

− ω 2 − iγexω

(1)

where ε∞ denotes the high-frequency component of the dielectric function of the ring, ωex is the exciton transition frequency, γex is the exciton line width, and f is the reduced oscillator strength, which is proportional to the molecule concentration. It is worth to mention that fully quantum mechanical or quantum optics approaches are required to study the dynamic responses of the resonance coupling.2,11 Nevertheless, the classical electrodynamics treatment still gives an adequate description of the static optical responses of the heterostructures, and the one-oscillator Lorentzian model has been proven to be successful to describe the resonance coupling between molecular J-aggregates and plasmonic (or dielectric) nanostructures.4,12,13,26,31,32,34,37,39,57 The parameters of the one-oscillator Lorentzian model are determined by the previously reported experimental results of the J-aggregate excitons, which possess strong transitional dipole moments. ε∞ is fixed as 2.5 corresponded to that of the J-aggregates/PVA mixture, ℏωex = 1.837 eV (∼675 nm) is almost degenerate with that of the anapole mode, and γex is supposed to be 52 meV according to the experimental absorption spectra of the J-aggregates.31 When the width of the ring t = 40 nm, the dash-dotted lines in Figure 1b represent the scattering spectra of the heterostructures with different oscillator strength. For the resonance coupling between molecular excitons and conventional resonances, a quenching dip around the exciton transition frequency caused by the coherent energy transfer often arises in the scattering spectra. However, the spectra shown in Figure 1b indicate that instead of a quenching dip, a scattering peak is generated around the exciton transition frequency for the heterostructures proposed in this study (marked with the vertical dashed line), the peak intensity increases with the increasing of the oscillator strength, and the corresponding spectral position is almost unchanged. At the same time, the anapole mode of the Si disk splits into a pair of pronounced scattering dips for the heterostructures. The two scattering dips can be clearly observed even for the heterostructure possesses a relatively small oscillator strength ( f = 0.1, the blue dash-dotted line), and the splitting is enlarged with the increasing of f. The multipole expansion of the scattering spectra for the heterostructure and individual nanoparticles can be used to understand the origin of the splitting of the anapole mode. The Cartesian multipole contributions to the scattering of the D


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Figure 3. Energy diagrams of (a) the Si disk and (b) the disk-ring heterostructure by manipulating the disk diameter, where the orange and red dotted lines denote the variations of the anapole mode and the exciton transition, respectively. The rest geometry parameters as well as the optical constants are identical with that of Figure 1c.

aggregate with a dielectric ring, and there is only a red shift of the anapole mode when the exciton transition is eliminated (Figure S1 in the Supporting Information). In addition to the splitting of the anapole mode, the spectra of the heterostructures shown in Figure 1b reveal that another scattering dip at the higher energy side of the hybrid anapole modes is generated when the oscillator strength f > 0.2, and it red shift and becomes more obvious with the increasing of f. The physical mechanism leading to the scattering dip is the same as that of the nanosphere-J-aggregate core−shell heterostructures.57 One can find that there is an additional resonance peak at about 630 nm for the J-aggregate ring (marked with the vertical dotted line, Figure 1d), and it coincides with the scattering dip of the heterostructure. Nearfield distributions for the additional resonance of the Jaggregate ring shown in Figure 2c reveal that the electric fields are strongly confined within the ring layer, the magnetic responses are very weak, and this resonance can be termed as the so-call cavity mode.13,57 Due to the strong electric field enhancement, one may have expected that there would be a strong electric dipole moment (a strong scattering) for the cavity mode. However, the incident energy at the cavity resonance is confined inside of the ring, and consequently absorbed by the molecular J-aggregates. As a result, the scattering intensity is weak, and its line width is only about 18 nm, thereby resulting in a relatively large Q-factor for the cavity mode (the black line, Figure 1d). Compared with that of the core−shell heterostructures,57 the cavity mode does not exactly match with an eigenmode of the Si disk, and it only leads to a quenching dip at its resonance frequency for the heterostructures (Figure 2f). Therefore, we will focus on the investigation of the resonance coupling between the exciton transition and the anapole mode in the following studies. Strong Coupling Regime for the Heterostructures. Since the anapole mode of the Si nanodisk splits into a pair of eigenmodes due to the coherent energy exchange in the diskring heterostructure, it would be interesting to find out whether the strong coupling regime can be achieved. To that end, the following two conditions must be satisfied. The first one is the anticrossing behavior on the energy diagram, and the second one is that the Rabi splitting exceeds the rate of electromagnetic mode damping. In order to verify the first criteria for strong coupling, the diameter of the Si nanodisk (D) is continuously adjusted from 160 to 500 nm, and the energy of the anapole mode can be tuned across the exciton transition in this way. The scattering spectra shown in Figure 3a reveal that the anapole mode red

appearance of the two scattering dips can be termed as the excitation of two hybrid anapole modes. In analog to the resonance coupling between the exciton transition and a conventional resonance, the resonance coupling between molecular excitons and the nonradiating anapole mode also leads to the mode splitting. However, it is evidenced by a scattering peak at the exciton transition frequency, and two eigenmodes characterized as pronounced scattering dips are generated due to the coherent energy exchange between the exciton transition and the anapole mode. The above observations and assumptions can be further verified by investigating the near-field optical behaviors of the heterostructure and individual nanoparticles. Figure 2b represents the near-field distributions of the exciton transition mode for the J-aggregate ring, which reveals a typical electric dipole response, and it is consistent with the multipole expansion results shown in Figure 1d. Considering the heterostructure, strong TD moments are generated at both scattering dips (Figure 2d and 2e). For example, the overall electric and magnetic field distributions for the scattering dip around 716 nm are very similar as that of the anapole mode of the Si disk (Figure 2d), excepting that the electric field enhancements on the up and down sides of the Si disk are strongly extended into the J-aggregate ring, and the field distributions cannot be separated from individual nanoparticles. In this situation, a strong TD moment is generated, and it almost compensates the Cartesian ED scattering caused by the destructive interference, which suppresses the radiative losses and gives rise to the formation of the hybrid anaple mode. Figure 2e shows the near-field distributions of the other scattering dip around 655 nm, where the formation of a TD moment also can be clearly observed. However, there are relatively strong field enhancements around the left and the right apexes of the heterostructure, indicating the generation of a strong electric dipole moment. In this case, the TD moment cannot compensate the Cartesian ED moment completely. Nevertheless, the destructive interference still leads to the suppressing of the radiative damping, which results in the formation of the hybrid anapole mode around 655 nm. The same as the resonance coupling between molecular excitons and conventional resonances, the two eigenmodes characterized as scattering dips for the heterostructure can be referred to as the lower branch (LB) and the upper branch (UB) hybrid anapole modes, respectively. The resonance coupling (coherent energy exchange) between the excitons and the Si disk plays the key role for the mode splitting, where we have investigated the optical responses of the heterostructure by replacing the JE


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Figure 4. Energy diagrams of the resonance coupling in the disk-ring heterostructure by adjusting the parameters related to the J-aggregate ring. (a) Energy diagram when the oscillator strength f is increased from 0 to 1. (b) Energy diagram by adjusting the width of the J-aggregate ring t in the range of 0−100 nm. (c) Energy diagram when the line width of the exciton transition γex is increased from 10 to 300 meV. (d) Energy diagram by manipulating the background refractive index n in the range of 1−3. Apart from the varied parameters, all of the geometry parameters and the optical constants for the energy diagrams are identical with that of Figure 1c.

shift linearly with the increasing of the diameter (marked with the orange dotted line), and it is almost crossing with the exciton transition when the diameter D = 300 nm. Besides that, the high-order anapole mode manifested as a narrower scattering dip arises around the short wavelength range of the spectra.81,88 The corresponding energy diagram of the heterostructure is represented in Figure 3b, and the scattering spectra indeed show a distinct anticrossing behavior when the anapole mode of the Si disk is varied across the exciton transition. However, instead of two branches of scattering peaks, two prominent branches of scattering dips appear on the energy diagram, which are termed as the UB and LB hybrid anapole modes as we have mentioned before. The LB or UB dominates when there is a large energy deturning between the anapole mode and the exciton transition, and the LB and UB appear as two distinct subradiant split modes when the anapole mode overlaps with the exciton transition. The Cartesian multipole expansion results for the heterostructures with different disk diameters shown in Figure S2 further confirm that the two hybrid anapole modes are caused by the destructive interference between the Cartesian ED and TD moments. The second criteria for strong coupling can be verified by comparing the values of the Rabi splitting (ℏΩ) and the electromagnetic mode damping,4,12,13,26,31,32,34,37,39,57 ℏΩ >

γopt − γex 2

or ℏΩ >

E± =

2

±

4g 2 + (Eopt − Eex )2 2

(3)

where Eopt and Eex are, respectively, the energies of the anapole mode and the exciton transition, g is the coupling rate, and the Rabi splitting energy (ℏΩ = 2g) can be obtained when Eopt = Eex. The fitting results shown in Figure 3b indicate that the mode splitting ℏΩ is as large as 161 meV for the heterostructures. Considering that the line width of the exciton transition is γex = 52 meV, the strong coupling condition can be determined when we get the line width of the anapole mode. Unfortunately, the anapole mode is a nontrivial resonance that manifests as the complete suppressing of radiative damping, and it is hard to determine its line width. Nevertheless, the anapole mode is caused by the destructive interference between the Cartesian ED and TD moments, which are very similar to that of the Fano resonances. Since the visibility of a Fano-like resonance is governed by the corresponding subradiant dark mode, it would be reasonable to assign the line width of the equivalent dark mode of the Fano-like resonance as the line width of the anapole mode. Then one can get the line width value by fitting the spectrum with a Fano interference model.89 We should emphasize that although the formations of the anapole modes and Fano resonances are related to the destructive interferences, the physical mechanisms are different with each other. Previous studies have shown that the excitation of Fano resonances for dielectric nanoparticles is an intrinsic spectral feature, which can be attributed to the interference between the perfectly reflecting background and the Mie mode of the same particle.90 On the other hand, the radiation fields of ED and TD are identical with each other, and the destructive interference between the ED and TD contributions leads to a strong suppression of the far-field scattering, which is the condition resulting in the formation of the anapole mode, and a resonance is not necessarily required in this situation.73 The

γopt + γex 2

Eopt + Eex

(2)

where γopt is the line width of the optical mode and the energy exchange rate between the exciton and the anapole mode exceeds their respective decay rates in this situation. The Rabi splitting value can be extracted with a coupled harmonic oscillator mode,4,12,13,26,31,32,34,37,39,57 where the eigenvalues (E±) for the hybrid modes by ignoring the dissipation can be written as F


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blue shift of the UB hybrid anapole mode (Figure 4b), but the increasing of t also results in a stronger radiative damping, where the TD moment cannot compensate the Cartesian ED moment for the UB hybrid anapole mode, and the UB becomes weaker and narrower with the increasing of the width of the Jaggregate ring (Figure 4b). Since the line width γex characterizes the damping of the exciton transition, the intensity of the exciton transition resonance for the J-aggregate ring would be reduced with the increasing of γex (Figure S4c), and the scattering peak corresponding to the exciton transition that separating the two hybrid anapole modes becomes shallower for the heterostructures (Figure 4c). At the same time, the energy transferred from the Si disk to the J-aggregate ring will be dissipated more rapidly for the heterostructure with a larger γex. One can expect that when γex is larger than an upper limit value, the coupling strength between the Si disk and the J-aggregate ring cannot compete with the dissipation, and the resonance coupling can no longer be observed in the spectra. The variation of the scattering spectra for the heterostructure indeed reveals that the two hybrid anapole modes are broadening with the increasing of γex, and they are almost merged together and can no longer be identified from each other for a very large γex (Figure 4c). Next, the resonance coupling energy diagram for the heterostructure with varied background refractive index n is demonstrated in Figure 4d. Both LB and UB hybrid anapole modes red shift slightly with the increasing of n, and the resonances blurred out when n > 2. These observations can be understood by investigating the optical responses of the Jaggregate ring and the Si nanodisk. The exciton transition resonance of the J-aggregate ring does not change the spectral position with the increasing of the background refractive index n, which is the same as that of the J-aggregate shell (Figure S4d).57 However, the optical modes of the dielectric nanoparticles are governed by the contrast of the refractive index with that of the surrounding medium, and the damping of the optical modes would be enlarged with the increasing of the background refractive index. As a result, the incident energy cannot be confined effectively around the Si disk, and the anapole mode can no longer be excited when n > 2 (Figure S5). Consequently, the resonance coupling between the exciton transition and the anapole mode for the heterostructure is weakened and fade away with the increasing of the background refractive index (Figure 4d). Finally, one remaining important parameter that affects the resonance coupling is the location of the J-aggregate molecules. Although the heterostructure proposed in the above study possesses a high structural symmetry, and the optical responses would be polarization independent under normal incidence, it does not mean that the molecules located at different positions play the same role for the resonance coupling. For the heterostructures reported in previous studies, it is important to place the molecules at the hot spot areas, where stronger fields can be felt by the molecules to enhance the coupling strength.24−41,57 In that case, the molecules located around the apexes of the nanoparticles along the incident polarization play a dominate role for the resonance coupling. However, the situation is reversed in our case. When the molecular Jaggregates are located around the left- and right-side apexes of the Si disk, the splitting between the two hybrid anapole modes becomes much narrower than that of the perfect heterostructure (Figure 5a). On the contrary, there is a minor change

γopt + γex

ℏΩ > 2 , and the resonance coupling between the anapole mode and the exciton transition enters the strong coupling regime. Resonance Coupling Adjustment. In addition to the optical modes, the resonance coupling is governed by the optical responses of the exciton transitions, and the effects of the parameters corresponding to the J-aggregate ring should therefore deserve to be investigated. For example, the mode splitting can be strongly tuned by adjusting the oscillator strength f, and the variations of the scattering spectra for the heterostructure are demonstrated in Figure 4a. The oscillator strength is proportional to the molecule concentration, so a stronger dipole moment would be generated for the molecules with a larger concentration, and the scattering intensity for the excition transition mode of the J-aggregate ring increases with the increasing of f (Figure S4a). Consequently, it is found that the scattering peak around the exciton transition for the heterostructure becomes stronger and wider (Figure 4a). Due to the same reason, a stronger polarized electromagnetic field generated by the molecular J-aggregates can be felt by the Si nanodisk when the oscillator strength is enlarged, which leads to a stronger resonance coupling between the exciton transition and the anapole mode, and the splitting between the resulted LB and UB hybrid anapole modes is broadening with the increasing of the oscillator strength. The influence of the width of the J-aggregate ring (t) is similar as that of the oscillator strength, where both parameters are proportional to the quantity of the molecules involved in the resonance coupling. Therefore, the scattering spectra of the J-aggregate ring reveal that the intensity of the exciton transition resonance is also enlarged with the increasing of t (Figure S4b), which results in stronger resonance couplings between the exciton transition and the anapole mode, and there is a larger energy splitting between the LB and UB hybrid anapole modes for the heterostructure with a thicker Jaggregate ring (Figure 4b). In addition, it is interesting to find that the LB (UB) hybrid anapole mode red (blue) shift continuously with the increasing of t, and the mode splitting is broadening even when the width of the J-aggregate ring is enlarged to 100 nm, indicating that the coupling strength between the exciton transition and the anapole mode is still enhanced in this situation. This observation is quite different with that of the conversional resonances. Due to the finite penetration depths of the near-field around the plasmonic or dielectric nanoparticles for the conversional optical modes, the resonance coupling will be stabilized by further enlarging the thickness when the outermost molecules are no longer felt the near-field enhancements (e.g., the stabilized thickness is only about 5 nm for the core−shell heterostructures).57 In our case, although the anapole mode shown in Figure 2a possesses a finite penetration depth for the near-field around the Si nanodisk, the field enhancements are strongly extended into the J-aggregate ring for the LB hybrid anapole mode (Figure 2d), which can enhance the coupling strength between the inner and outer nanoparticles. Therefore, the LB of the heterostructure red shift continuously with the increasing of t, and there is only a minor increment of the radiative damping (Figure 4b). On the other hand, the enhanced resonance coupling leads to the G


Article

ACS Photonics

Figure 5. Effect of the location of the molecular J-aggregates on the resonance coupling. (a) Scattering spectrum (the solid line) and (c) the corresponding energy diagram by adjusting the diameter of the Si disk when the molecular J-aggregates are positioned at the left- and right-apexes of the Si disk, and (b) and (d) show the corresponding optical responses when the molecular J-aggregates are located around the upper- and lowerapexes of the Si disk. Scattering spectrum of the heterostructure with a perfect J-aggregate ring are represented in (a) and (c) for comparison (the dashed lines), the insets schematically show the geometries of the asymmetric heterostructures, and the rest parameters for the heterostructures are identical with that of Figure 1c.

features compared with that of the conversional radiative resonances. The calculation results demonstrate that instead of a quenching dip, the resonance coupling is evidenced by a scattering peak around the exciton transition frequency, and the anapole mode of the Si disk splits into a pair of scattering dips for the heterostructures. The multipole decompositions and near-field distributions reveal that both scattering dips are caused by the destructive interference between the Cartesian ED and TD moments, which are termed as, respectively, the UB and LB hybrid anapole modes due to the coherent energy exchange between the J-aggregate ring and the Si disk. The strong coupling regime can be achieved where an anticrossing behavior with a mode splitting of 161 meV is observed on the energy diagram. Furthermore, it is interesting to find that due to the unique near-field distribution associated with the anapole mode, there is a much larger upper limit value for the width of the J-aggregate ring to enhance the resonance coupling, and the molecules located around the apexes of the disk perpendicular to the incident polarization play the dominate role for the resonance coupling. The specific optical responses of the resonance coupling between the exciton transitions and the nonradiating anapole modes can help to further our understanding of light-matter interactions at the nanoscale, and the heterostructure could be a promising platform for future nanophotonic applications such as the information processing and sensing.

for the two hybrid anapole modes when the J-aggregate molecules are located around the up- and down-side apexes (Figure 5b). These optical responses can also be understood by investigating the near-field distributions. The field enhancements around the left- and right-apexes of the Si disk are very weak for the anapole mode caused by the formation of the TD moment (Figure 2a). Therefore, the coupling strength between the J-aggregate ring and the Si disk would be very weak when the molecules are located around the left- and right-side apexes of the Si disk, and the energy diagram shown in Figure 5c reveals that the energy splitting for the hybrid anapole modes is only about 57 meV. On the contrary, when the molecules are located around the up- and down-side apexes of the disk, stronger field enhancements can be felt by the molecules (Figure 2a), thereby resulting in an enhanced coupling strength, and the energy diagram represented in Figure 5d reveals that the energy splitting is enlarged to about 135 meV. These observations indicate that for the disk-ring heterostructure, the molecules located around the apexes of the disk perpendicular to the incident polarization play the dominate role for the resonance coupling, which can be attributed to the unique near-field distributions of the anapole mode. Besides that, due to the nonequivalent role of the molecules at different positions for the resonance coupling, polarization dependent responses can be achieved with the asymmetric heterostructure, and the optical responses can be actively tuned by manipulating the incident polarizations (Figure S6).

■

■

METHODS Electromagnetic Simulation. The optical responses of the Si nanodisks, the J-aggregate rings, and the heterostructures are calculated with the finite-difference time-domain (FDTD) method, where perfectly matched layers (PML) around the nanostructures were used to simulate the open space. The complex dielectric constants of silicon were taken from the measured data,91 and the optical constants of the molecular J-

CONCLUSION In conclusion, the resonance coupling between the molecular excitons and the anapole modes in silicon nanodisk-J-aggregate heterostructures are theoretically investigated in this paper. Due to the nonradiating nature of the anapole modes caused by the destructive interference between the electric and toroidal dipole moments, the resonance coupling reveals distinct spectral H


Article

ACS Photonics aggregates are modeled with the classical one-oscillator Lorentzian model.31,57 A normal incident pulse along the zaxis was used as the excitation source, and the polarization is along the x-axis. Multipole Decompositions. The far-field radiation patterns of the electric and toroidal dipoles are indistinguishable, and the two contributions cannot be identified with the spherical multipole expansion. However, the Cartesian multipole expansion can be used to unambiguously separate the electric and toroidal dipole contributions, which provides a tool to help to understand the formation of the nonradiating anapole modes.73 The Cartesian multipole decomposition up to the third order can be calculated with the induced polarization inside the nanoparticle,87 P = ε0(εp − εd)E

Psca =

+

■

∫

■

∫

⎡

the magnetic dipole (MD), iω m=− [r′ × P(r′)]dr′ 2

∫

and the magnetic quadrupole (MQ), ω {[r′ × P(r′)]r′ + r′[r′ × P(r′)]}dr′ M̂ = 3i where the components of the tensor Ô ″ for EO,

∫

O″βγτ = δβγVτ + δβτVγ + δγτVβ

V=

1 5

∫ {2[r′·P(r′)]r′ + r′2 P(r′)}dr′

3780πε02vdμ0

∑ |Mαβ|2 αβ

2

∑ |Oαβγ |

(14)

αβγ

ASSOCIATED CONTENT

AUTHOR INFORMATION

Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS This work was supported by National Natural Science Foundation of China (NSFC; 11574228, 61471254, and 11304219), the Natural Science Foundation of Shanxi Province (201601D021005), and the San Jin Scholars Program of Shanxi Province.

⎤

∫ r′r′r′[∇·P(r′)]dr′ − Ô″

k 08εd2

k 06εd2 160πε0vd

Shao-Ding Liu: 0000-0003-4809-9815

■

(8)

REFERENCES

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the electric octupole (EO), Ô = −

∑ |Q αβ|2 +

1440πε02vdμ0 αβ

ORCID

(7)

∫ ⎢⎣r′P(r′) + P(r′)r′ − 23 [r′·P(r′)]Û ⎥⎦dr′

k 04εd |m|2 12πε0vd

*E-mail: [email protected].

the electric quadrupole (EQ), Q̂ = 3

+

Corresponding Author

(6)

the toroidal dipole (TD), iω T= {2r′2 P(r′) − [r′·P(r′)]r′}dr′ 10

2

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsphotonics.7b01598. Multipole expansion results of heterostructures, Fano interference model, and additional scattering spectra of Jaggregate rings, Si nanodisks, and asymmetric heterostructures (PDF).

(5)

P(r′)dr′

k 06εd

ikd T vd

S Supporting Information *

the Dirac delta function δ(r − r′) can be expanded in a Taylor series with respect to r′ around the origin. Using the definitions of the individual multipole moments, one can get the irreducible representations for the Cartesian electric dipole (ED), p=

p+

where the first term is the total electric dipole contribution (ED + TD), which is determined by the superposition of the Cartesian ED and TD moments, and the destructive interference leads to the formation of the nonradiating anapole modes.

(4)

∫ P(r′)δ(r − r′)dr′

12πε02vdμ0 +

where ε0, εp, and εd denote the vacuum dielectric constant, the relative dielectric permittivity of the nanoparticle, and the relative dielectric permittivity of the surrounding medium, respectively. The total electric field (E) inside the nanoparticles can be evaluated with the numerical calculations. In order to get the Cartesian multipole expansion of the light-induced polarization, P(r) =

k 04

(9)

(10)

(11)

(12) (13)

Û is the unit tensor, β, γ, or τ = x, y, z, and δβγ(τ) is the Kronecker delta. Then, the total scattering power can be calculated by including up to the third-order multipoles,87 I


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