Here we study theoretically the optical responses of hybrid structures composed of dielectric nanostructures and quantum emitters with magnetic dipole transitions. Coherent couplings between magnetic dipole transitions and magnetic modes can occur, leading to giant modifications of the extinction spectra of the constituents in the hybrid structures. For a given hybrid structure, the extinction-cross-section spectra show linear or nonlinear behaviors depending on the strength of the excitation field. For a weak excitation, the extinction of the quantum emitters is greatly enhanced. The hybrid structure shows a dip on its extinction spectrum. For a strong excitation, the resonant extinction of the quantum emitters is weakly enhanced while the extinction spectrum is broadened obviously. The hybrid structure shows a Fano-like line shape on its extinction spectrum, which is different from that with a weak excitation. This difference is highly related to the behaviors of the magnetic polarizabilities of the quantum emitters in the hybrid structure. The optical responses of hybrid structures can be largely tuned by varying the geometric and material parameters.
© 2019 Chinese Laser Press
Dielectric nanostructures with high refractive index have recently drawn lots of interest as they exhibit strong magnetic and electric resonances while their material losses are low [1–3]. With these properties, they can find many nanophotonic applications, such as metamaterials , metasurfaces [5,6], structural colors , magnetic mirrors , and optical nanoantennas [9–11]. The strong optical responses of dielectric nanostructures are usually accompanied by considerable electric and/or magnetic near-field enhancements around the entire structure volume. These near-field enhancements could enable strong near-field couplings between different dielectric nanostructures [9,12–16] as well as couplings between the plasmonic and dielectric nanostructures [17–21]. The couplings of different dielectric nanostructures can induce the phenomena of hybridizations of electromagnetic modes [12,13] and Fano resonances [14–16]. The combinations of plasmonic and dielectric structures can also strongly modify their optical responses, where potential applications have been demonstrated including optical nanoantennas [20–22].
The hybrid photonic structures of dielectric nanoresonators and quantum emitters have begun to attract research interest recently. The strong near-field couplings between the electromagnetic modes of dielectric nanostructures and excitons have been reported [23–27], where the excitons can be excited in molecules or two-dimensional materials. This kind of coupling usually results in peak splitting on the scattering spectrum of a hybrid system. The emission properties of electric dipole emitters coupled to dielectric nanostructures have also been investigated [10,28–33]. Both decay rate and fluorescence enhancements have been realized experimentally [10,31,32]. The modifications of those photon–emitter interactions are based on the electric near-field enhancements in dielectric structures. Many of these studies are carried out by analogy to the plasmon–emitter hybrid systems, where strong plasmon–exciton couplings [34–43] and the modifications of the emission properties of emitters [44–46] have been extensively studied in plasmonic systems. It is well known that magnetic modes with magnetic near-field enhancements can be readily excited in a simple dielectric nanostructure. This feature makes dielectric structures attractive for enhancing the interactions between light and magnetic quantum emitters which exhibit magnetic dipole (MD) transitions. Strong MD transitions can be supported by many rare-earth ions [47–49]. Tailoring the emissions of MD emitters by dielectric nanostructures has begun to be studied [29,30,50–53]. The decay rate enhancement and directionality modification have been demonstrated. Note that plasmonic structures can also be used to modify the emission properties of MD emitters [54–57]. But this usually requires complex geometries because plasmonic nanostructures with simple shapes do not support efficient magnetic-mode resonances. Furthermore, plasmonic structures have higher material losses.
Here we study the optical responses of a hybrid system consisting of a dielectric nanostructure and MD emitters. The MD emitters are rare-earth ions with magnetic dipole resonances, which are taken as two-level emitters. The dielectric nanostructure is a silicon (Si) nanosphere. The modified magnetic dipole moments of the ions coupled with the dielectric sphere can be analytically obtained based on the master equation for the density matrix elements. With the above results, the extinction (or scattering) cross sections of the hybrid structure can be analytically calculated. Efficient magnetic near-field interaction can occur, which leads to giant modifications of the extinction spectra of the hybrid structures. For a given hybrid system, its extinction cross-section spectrum will show linear or nonlinear behaviors depending on the strength of the excitation field. For a weak excitation, the extinction of the ions is greatly enhanced. The hybrid structure (or the Si sphere) shows a pronounced dip on its extinction spectrum. For a strong excitation, the resonant extinction of the ions is only weakly enhanced, but the extinction spectrum becomes broader. The hybrid structure (or the Si sphere) shows a Fano-like line shape on its extinction spectrum. The effects from geometric and material parameters will also be considered.
2. THEORETICAL APPROACH
We consider a hybrid system composed of a dielectric nanosphere and an ion cluster, as is shown in Fig. 1. We take the commonly used Si as the high-refractive-index dielectric material. The radius of the sphere is . The ion cluster is placed at the center of the Si nanosphere. The whole system is excited by a plane wave light. We use the classical Mie scattering theory to describe the optical responses of a Si sphere. For the optical responses of the ion cluster, we use the density matrix equation based on quantum optics. The working frequency is near the magnetic dipole resonance of the Si sphere, and the ion cluster only exhibits MD transitions at this frequency. The response spectrum of the ion cluster is much smaller than that of the Si sphere. We take a single magnetic ion as a simple two-level system. The energies of the ground and excited states are and , respectively. The frequency and matrix element of the MD transition of an ion are and , respectively.58,59]. If the number of the ions is , the magnetic dipole moment of the ion cluster can be written as . and are used to separate the high-frequency part. The magnetic field of incident light as a function of time can be written as . The magnetic field felt by the Si sphere is . The magnetic field felt by ion cluster can be written as
The above matrix elements should satisfy the master equation ,58,60]. From Eq. (3), we have 5) satisfies . We obtain 61]. The magnetic dipole moment of ion cluster can be expressed as . can also be written as , is the magnetic dipole polarizability of an ion cluster. Thus, according to Eq. (4), the polarizability of an ion cluster can be written as 58]. The corresponding FWHM is now .
For a Si nanosphere with the relative refractive index , is approximately satisfied, but is not satisfied. Its extinction and scattering cross section can be written as and , respectively, based on the Mie theory [22,61,62]. Here the electric dipole mode also contributes to the optical responses of the Si sphere . The electric and magnetic dipole moments can be written as and , respectively. is the vacuum dielectric constant, and and are the electric dipole and magnetic dipole polarizabilities of the Si sphere, respectively. According to the Mie theory, and can be written asA for the relevant discussion).
3. RESULTS AND DISCUSSION
For numerical calculations, the radius of the Si sphere is chosen to be , and its refractive index is taken from the Palik’s book . The resonance of its magnetic dipole mode is near . The MD transition of the rare-earth ions is chosen to be at , which is spectrally near the resonant position of Si sphere. The transverse homogeneous lifetime and the longitudinal homogeneous lifetime are both taken to be 0.2 ns. The matrix element for the MD transition is ( is the Bohr magneton). The number of ions is . From Eq. (6), the magnetic polarizability of the ionic cluster varies with and varies with the incident , which can cause the system to show nonlinear optical responses. When the intensity of excitation light is weak, . The magnetic polarizability of the ion cluster is approximately written as . The magnetic ion cluster exhibits approximately linear optical response, namely the extinction spectrum [Eq. (7)] does not vary with the intensity of the light ( can be represented by , , , and ). When the light intensity increases so that cannot be satisfied, the ion exhibits nonlinear optical response. In order to quantify the linear and nonlinear response regimes of the system, we define as the threshold from the linear to nonlinear region. Based on this threshold, the magnetic field of the excitation light for the above system is calculated to be , which corresponds to the light intensity of .
We first consider a case with a relatively low intensity of the excitation light (). Figure 2(a) shows extinction spectra of the individual and coupled ion cluster. The inset of Fig. 2(a) illustrates the response spectrum of . The absorption of the ions is greatly enhanced (more than 200 times) with the Si sphere. This can be understood with the behavior of and the magnetic field enhancement of the Si sphere. With a weak excitation, is very close to 1. This means that an ion is probably at the ground state and it can easily absorb energy. At the same time, the Si sphere provides an efficient magnetic field enhancement for the ion cluster. Thus, the absorption (extinction) cross section of the ion cluster can be significantly enhanced. Quantitatively, the resonant extinction of coupled ion cluster based on Eq. (7) becomes8)] is reduced to 2). The extinction spectra of the coupled Si sphere and the hybrid structure are shown in Fig. 2(b). Pronounced dips appear on the spectra of the hybrid structure and the coupled Si sphere, respectively. The inset shows the extinction spectrum of the whole system for a wider energy regime. We will discuss the dips in detail later. Here, it is interesting to note that the results in the linear regime can be well reproduced by the common finite-difference time-domain (FDTD) simulations, where the ions are taken as a nanosphere which has a classical magnetic Lorentz model for its permeability (see the relevant results in Appendix B).
We now turn to a high light intensity (). Figure 2(c) shows extinction spectra of the individual and coupled ion clusters. The resonant extinction (absorption) cross section shows a weak increment (about 5 times). For , the value for individual ions is much closer to 0 compared to that with weak excitations. And the of coupled ions is close to 0, which means that it is more difficult for the ions to be excited compared to the case of weak excitation. Thus, the ions cannot efficiently absorb more light even when there is enhanced magnetic field around it. For such a high light intensity, we have , and the expression for the resonant extinction of a coupled ion cluster based on Eq. (7) can now be written as8)] is reduced to 11)] as . Note that the high intensity excitation () condition is a special situation of the nonlinear regime (). Figure 2(d) shows extinction spectra of the coupled Si sphere and the whole system. The variation of the extinction value for the coupled system (or coupled Si sphere) is much smaller than the linear case, while the spectrum shows a typical Fano-like line shape.
The variation from a dip to a Fano-like line shape for the extinction spectrum of the coupled Si sphere (or the whole structure) can be understood by considering the magnetic polarizability of the ions and the total magnetic field felt by the ions . Based on Eq. (9), the variation of the extinction of the coupled Si sphere is mainly related to and . And it can be easily checked that is much higher than and . Therefore, the main factor that causes the change of the extinction of the Si sphere is , where is fixed for a given system, and it is dominated by a real value (see Table 1). The magnetic dipole moment of the ions is dependent on the polarizability of the ions and the total magnetic field felt by the ions . Figures 3(a)–3(c) show the spectra of , , and under a weak excitation. The and show a peak and an antisymmetric line shape, respectively. The magnitude of is higher than that of , and is several times larger than . Thus, is mainly dependent on , which has a dip line shape. The negative value of means that and the excitation field are out of phase, and their destructive coupling leads to the appearance of a dip on the spectrum of the coupled Si sphere (and the whole system). For a strong excitation, the and also show a peak and an antisymmetric line shape, respectively, while is much larger than [Fig. 3(d)]. The shows similar behavior [Fig. 3(e)] to that with the weak excitation. Thus, is now mainly dependent on , whose line shape is determined by as is hardly changed with frequency around the . As a result, the extinction line of the coupled Si sphere has a Fano-like line shape which is similar to [Fig. 3(f)]. Here, it can also be seen that the is close to and it is much larger than the excitation field under both the weak and strong excitations.
Figures 4(a) and 4(b) show the extinction spectra of the coupled ion cluster and hybrid structure with different excitation intensities, respectively. The resonant extinction cross section of the coupled ion cluster decreases with the light intensity in the nonlinear regime. This is because the absorption of the ions is becoming saturated with high excitation intensity as discussed before. Quantitatively, the resonant extinction is approaching , which will decrease with . The FWHM of the spectrum is also getting larger as expected in the nonlinear regime [Eq. (13)]. In Fig. 4(b), the extinction spectrum of the hybrid structure varies gradually from a dip to a Fano resonance shape with the light intensity. Moreover, the response spectrum becomes broader and the variation becomes smaller, which are in consistent with the responses of the ion cluster.
The effects from the MD matrix element are investigated. We first consider the weak light intensity . The resonant extinction cross section of the ions first increases and then decreases with the from to [Fig. 5(a)]. This can be understood based on the expression of the resonant extinction of the ions [Eq. (10)], and it is in proportion to . decreases with . When is small (e.g., ), the coupling effect between the ion cluster and Si sphere is relatively small. The variation of is small (see the relevant results in Appendix B), so the resonant extinction increases with . When is larger than , is approaching 0 and the decreasing of is a dominant factor. Thus, the resonant extinction decreases. The dip on the extinction spectrum of the hybrid structure becomes deeper and broader with . The variation of the responses of the coupled Si sphere is mainly related to the , and the depends largely on in the linear region as discussed before. is in proportion to [Eq. (6)]. The variation of is larger than that of even for a large . So the variation of the extinction spectrum of the hybrid structure increases with . () increases (decreases) with (Appendix B), so the FWHM [Eq. (11)] will also increase.
Figures 5(c) and 5(d) show the responses of the coupled ions and the hybrid structure with a high light intensity , respectively. Different from the linear region, the resonant extinction of the ions hardly changes with the , because Eq. (10) for resonant extinction of ions in the high light intensity regime still holds here () and it is independent of . The spectrum of the ions becomes broader with , because its FWHM [Eq. (13)] increases significantly with as is very close to in the limit and it almost does not change with (Fig. 3 and Appendix B). For the hybrid structure, the antisymmetric Fano-like line shape becomes more pronounced with [Fig. 5(d)]. The variation of the extinction of the hybrid system is mainly dependent on . For the typical positions of , the will increase with as almost does not change with .
The MD matrix element of ion cluster is also dependent on the number of ions . Thus, the effects from the number of ions are also investigated [Figs. 5(e)–5(h)]. For a weak light intensity [Figs. 5(e) and 5(f)], the behaviors of the ions and the hybrid structure are similar to the case with varying the [Figs. 5(a) and 5(b)]. The results can also be explained in a similar way to that for Figs. 5(a) and 5(b). For a high light intensity , the resonant extinction of ions increases with the number of ions [Fig. 5(g)], which is different from the case with varying the [Fig. 5(c)]. This is because the resonant extinction of ions [Eq. (12)] increases with while it is independent of . The FWHM of the extinction spectrum [Eq. (13)] is independent of , and is very close to in the limit (Appendix B). The variations of the extinction of the hybrid system [Fig. 5(h)] and the corresponding explanation are similar to that with varying the [Fig. 5(d)]. Here, it should be pointed out that for small or under weak excitation, although the variation of the spectra for the hybrid structure is quite small, the absorption of the ions is still greatly enhanced.
The lifetime of the ions may change with the external environment, for example, temperature [64,65]. Thus the influence from the transverse homogeneous lifetime and the longitudinal homogeneous lifetime is also considered. For simplicity, and are kept to be the same for each case. We also consider the weak and strong excitation cases. The other parameters are the same as that in Fig. 2. Figures 6(a) and 6(b) show the results with light intensity . The resonant extinction of the ion cluster first increases and then decreases with the lifetimes of and varying from 0.1 ns to 1.6 ns. The resonant extinction [Eq. (10)] is proportional to , while decreases with () (Appendix B). For small (), the coupling effect (, ) is relatively small. The variation of () is larger than that of . So the resonant extinction increases with (). When () is larger than 0.8 ns, the coupling is strong and is approaching 0. The variation of is larger than that of (). Thus, the resonant extinction decreases with (). The variation of the extinction of the coupled Si sphere is mainly related to . For the resonant position of , . The variation of () is larger than that of (Appendix B) for each , so the dip of the coupled Si and the corresponding hybrid structure becomes deeper with (). It is interesting to note that the ions show relatively large extinction cross sections which are comparable to that of the coupled Si sphere for large ().
For a strong light intensity , the resonant extinction of the ion cluster decreases with the lifetime () [Fig. 6(c)]. This is because the resonant extinction [Eq. (12)] is inversely proportional to . Its FWHM [Eq. (13)] does not change with the lifetime (). The Fano-like spectrum of the hybrid structure almost does not change with the lifetime [Fig. 6(d)], and the extinction is much larger than that of the ion cluster. The variation of the response of the Si sphere depends largely on with the strong excitation. For the typical positions of , , which are not changed with the lifetime () (the does not change with under the strong excitation, see Appendix B). Therefore, the Fano spectral response of the hybrid structure does not change with the lifetime ().
The magnetic field enhancement of an individual Si sphere varies with the location inside the structure. This will affect the coupling between the ions and the Si sphere. The magnetic field enhancement reaches maximum at the sphere center and decreases with the distance between the location and the center of the sphere. The corresponding coupling coefficients and decrease with (see Table 1). Thus, the coupling strength in the hybrid structure becomes weaker with . We also consider the weak and strong excitation cases. Figures 7(b) and 7(c) show the extinction spectra of the coupled ion cluster and the hybrid structure with different distances . The other parameters are the same as that in Figs. 2(a) and 2(b). The light intensity corresponds to the linear regime. The resonant extinction of the ion cluster [Eq. (10)] decreases with the distance . This is because the total magnetic field felt by the ions , which is related to the and , becomes smaller with the . The FWHM is almost invariable ( is close to ). The dip of the extinction spectrum of the hybrid structure becomes weaker with the distance as the destructive coupling strength between the ions and the Si sphere decreases with .
Figures 7(d) and 7(e) show the results for a high light intensity . The resonant extinction of the ion cluster is almost unchanged with the . The reason is that its value [Eq. (12)] is independent of the and . The FWHM of the ions [Eq. (13)] becomes smaller significantly. This is caused by the decreasing of with . For the Si sphere, the variation of its extinction depends mainly on , where in the nonlinear regime. For the typical positions of , the . It can be easily verified that the variation of is much smaller than that of . Thus, the extinction of Si decreases with the decreasing of .
The above calculations are based on the Si as the high refractive index material, and we also consider the other materials for the dielectric sphere. Figure 8 shows a case with GaP as the material. The GaP sphere has a refractive index of 3.5. The radius is taken to be 71 nm, so that the corresponding magnetic dipole resonance is near 520 nm too. The ion cluster is the same as that in Fig. 2. Figures 8(b)–8(e) show the extinction spectra of the GaP-based hybrid structure with weak and strong excitations. The line shapes of the response spectra are similar to that of the Si-based hybrid structure. This is because the GaP sphere also shows a magnetic dipole resonance similar to the Si sphere, and the magnetic field enhancement inside the GaP sphere is nearly the same as that of the Si sphere.
In conclusion, we have investigated the optical properties of hybrid structures consisting of dielectric nanospheres and quantum emitters with MD transitions. For a given hybrid structure, the extinction-cross-section spectra of the quantum emitters and dielectric nanospheres show linear or nonlinear behaviors depending on the incident light intensity. For a low light intensity, the extinction of the quantum emitters is greatly enhanced, and a dip appears on the extinction spectrum of the hybrid structure. For a high light intensity, the resonant extinction of the quantum emitters does not show obvious enhancement while the extinction spectrum is broadened. A Fano-like line shape appears on the extinction spectrum of the hybrid. The different spectral responses of the hybrid structure are highly related to the behaviors of the magnetic polarizabilities of the quantum emitters. The effects from the geometric and material parameters of the hybrid structure are considered, which include the MD matrix element, the number of ions , the relaxation time of ions, and the materials of the dielectric sphere. The optical responses of the coupled structures can be tuned by these parameters. Our results reveal the efficient couplings between MD transitions and magnetic modes of dielectric structures with considerable magnetic field enhancements. For the experimental realizations, the samples may be prepared based on chemical synthesis or laser ablation [23,27]. If the ions are uniformly distributed in the dielectric spheres, the measured results should be comparable to our predictions according to the results in Fig. 7. The measurement should be done under a low temperature environment and it requires a spectrometer with high spectral resolution. We expect that more efficient magnetic coupling effects can be obtained in other carefully designed dielectric nanostructure–MD emitter hybrids.
APPENDIX A: CALCULATIONS OF AND FOR THE HYBRID STRUCTURE
- 1. For . represents the proportional coefficient between the magnetic field produced by the Si sphere and its magnetic dipole moment from the article. We use the Mie scattering theory to get the magnetic field of the Si sphere . The magnetic fields inside and outside the Si sphere are expanded by vector spherical harmonics.
As shown in Fig. 9, the origin of the coordinate is located at the center of the Si sphere, and the radius of the Si sphere is . The excitation light propagates in the -axis, and its polarizability is in -axis.
The total magnetic field inside the Si sphere is (this is the total field consisting of the magnetic field of the light and the scattered field of the Si sphere). The scattered magnetic field outside Si sphere is (this is only the scattered field of the Si sphere):
For a location on the -axis, it corresponds to , , , and is the distance from the Si center to the location. The total magnetic field inside the Si sphere can be written as
- 2. For . Because the Si sphere is relatively larger than the ion cluster, the magnetic field excited by the ion cluster is inhomogeneous for the Si sphere. Obviously, this contribution is proportional to the magnetic dipole moment of the ion cluster. When a magnetic dipole excites a Si sphere near the wavelength of the magnetic dipole mode of the Si sphere, a magnetic dipole resonance on the Si sphere will be excited. We can equivalently take the excitation of the magnetic dipole moment of the ion cluster as a plane wave light excitation with the magnetic field . The magnetic dipole moment of the excited Si sphere is . can be calculated if the magnetic dipole moment of the excited Si sphere is known. For a system consisting of an excitation magnetic dipole and a Si sphere, the radiation enhancement of the whole system is , where is the radiated power of the whole system in the far field, and is the radiated power of the individual magnetic dipole, .
The magnetic dipole of the whole system is , so the radiated power of the whole system in the far field can be written as . Thus, we can get the radiation enhancement factor of the whole system as66] 10. and are the dielectric constant and permeability of the Si sphere, respectively. and are dielectric constant and permeability in vacuum, respectively. , , ( is the distance between the magnetic dipole and the center of the Si sphere), and A3)–(A5), one can get the .
- 1. Comparison between the FDTD simulation and analytical results (Fig. 11).
In FDTD software (Lumerical FDTD), the ion is taken as a classical sphere, whose permeability is a magnetic Lorentz model. The permeability of the magnetic Lorentz model is .
The radius of the magnetic Lorentz sphere is 5.5 nm. The resonance of the absorption is set to 525 nm (2.3667 eV), and the width of the resonance is taken to be 0.46 nm. The dimensionless oscillator strength is . The sphere is placed at the center of the Si sphere with a radius of 60 nm. For the corresponding analytical calculations, the and for ions are both approximately taken as 0.5 ps, the matrix element for the magnetic dipole transition is 0.3, the number of ions is , and the system is in the linear regime with a weak incident light. The FDTD simulation results agree well with the analytical calculations.
- 2. The total magnetic fields of the ion cluster with different , , and (Fig. 12).
National Natural Science Foundation of China (11704416); Natural Science Foundation of Hunan Province (2017JJ3408).
1. A. I. Kuznetsov, A. E. Miroshnichenko, M. L. Brongersma, Y. S. Kivshar, and B. Luk’yanchuk, “Optically resonant dielectric nanostructures,” Science 354, aag2472 (2016). [CrossRef]
2. M. Decker and I. Staude, “Resonant dielectric nanostructures: a low-loss platform for functional nanophotonics,” J. Opt. 18, 103001 (2016). [CrossRef]
3. Z.-J. Yang, R. Jiang, X. Zhuo, Y.-M. Xie, J. Wang, and H.-Q. Lin, “Dielectric nanoresonators for light manipulation,” Phys. Rep. 701, 1–50 (2017). [CrossRef]
4. S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016). [CrossRef]
5. M. Khorasaninejad and F. Capasso, “Metalenses: versatile multifunctional photonic components,” Science 358, eaam8100 (2017). [CrossRef]
6. E. Arbabi, A. Arbabi, S. M. Kamali, Y. Horie, and A. Faraon, “Controlling the sign of chromatic dispersion in diffractive optics with dielectric metasurfaces,” Optica 4, 625–632 (2017). [CrossRef]
7. X. Zhu, W. Yan, U. Levy, N. A. Mortensen, and A. Kristensen, “Resonant laser printing of structural colors on high-index dielectric metasurfaces,” Sci. Adv. 3, e1602487 (2017). [CrossRef]
8. W. Liu, “Generalized magnetic mirrors,” Phys. Rev. Lett. 119, 123902 (2017). [CrossRef]
9. R. M. Bakker, D. Permyakov, Y. F. Yu, D. Markovich, R. Paniagua-Domínguez, L. Gonzaga, A. Samusev, Y. Kivshar, B. Luk’yanchuk, and A. I. Kuznetsov, “Magnetic and electric hotspots with silicon nanodimers,” Nano Lett. 15, 2137–2142 (2015). [CrossRef]
10. M. Caldarola, P. Albella, E. Cortes, M. Rahmani, T. Roschuk, G. Grinblat, R. F. Oulton, A. V. Bragas, and S. A. Maier, “Non-plasmonic nanoantennas for surface enhanced spectroscopies with ultra-low heat conversion,” Nat. Commun. 6, 7915 (2015). [CrossRef]
11. Y. Yang, V. A. Zenin, and S. I. Bozhevolnyi, “Anapole-assisted strong field enhancement in individual all-dielectric nanostructures,” ACS Photonics 5, 1960–1966 (2018). [CrossRef]
12. U. Zywietz, M. K. Schmidt, A. B. Evlyukhin, C. Reinhardt, J. Aizpurua, and B. N. Chichkov, “Electromagnetic resonances of silicon nanoparticle dimers in the visible,” ACS Photonics 2, 913–920 (2015). [CrossRef]
13. J. van de Groep, T. Coenen, S. A. Mann, and A. Polman, “Direct imaging of hybridized eigenmodes in coupled silicon nanoparticles,” Optica 3, 93–99 (2016). [CrossRef]
14. Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014). [CrossRef]
15. J. Yan, P. Liu, Z. Lin, H. Wang, H. Chen, C. Wang, and G. Yang, “Directional Fano resonance in a silicon nano sphere dimer,” ACS Nano 9, 2968–2980 (2015). [CrossRef]
16. A. E. Miroshnichenko and Y. S. Kivshar, “Fano resonances in all-dielectric oligomers,” Nano Lett. 12, 6459–6463 (2012). [CrossRef]
17. H. Wang, P. Liu, Y. Ke, Y. Su, L. Zhang, N. Xu, S. Deng, and H. Chen, “Janus magneto-electric nanosphere dimers exhibiting unidirectional visible light scattering and strong electromagnetic field enhancement,” ACS Nano 9, 436–448 (2015). [CrossRef]
18. R. Guo, E. Rusak, I. Staude, J. Dominguez, M. Decker, C. Rockstuhl, I. Brener, D. N. Neshev, and Y. S. Kivshar, “Multipolar coupling in hybrid metal dielectric metasurfaces,” ACS Photonics 3, 349–353 (2016). [CrossRef]
19. T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118, 173901 (2017). [CrossRef]
20. Y.-H. Deng, Z.-J. Yang, and J. He, “Plasmonic nanoantenna-dielectric nanocavity hybrids for ultrahigh local electric field enhancement,” Opt. Express 26, 31116–31128 (2018). [CrossRef]
21. Y. Yang, O. D. Miller, T. Christensen, J. D. Joannopoulos, and M. Soljačić, “Low-loss plasmonic dielectric nanoresonators,” Nano Lett. 17, 3238–3245 (2017). [CrossRef]
22. Q. Zhao, Z. J. Yang, and J. He, “Fano resonances in heterogeneous dimers of silicon and gold nanospheres,” Front. Phys. 13, 137801 (2018). [CrossRef]
23. H. Wang, Y. Ke, N. Xu, R. Zhan, Z. Zheng, J. Wen, J. Yan, P. Liu, J. Chen, J. She, Y. Zhang, F. Liu, H. Chen, and S. Deng, “Resonance coupling in silicon nanosphere-J-aggregate heterostructures,” Nano Lett. 16, 6886–6895 (2016). [CrossRef]
24. J. Yan, C. Ma, P. Liu, C. Wang, and G. Yang, “Generating scattering dark states through the Fano interference between excitons and an individual silicon nanogroove,” Light Sci. Appl. 6, e16197 (2017). [CrossRef]
25. S. Lepeshov, M. Wang, A. Krasnok, O. Kotov, T. Zhang, H. Liu, T. Jiang, B. Korgel, M. Terrones, Y. Zheng, and A. Alù, “Tunable resonance coupling in single Si nanoparticle-monolayer WS2 structures,” ACS Appl. Mater. Inter. 10, 16690–16697 (2018). [CrossRef]
26. S.-D. Liu, J.-L. Fan, W.-J. Wang, J.-D. Chen, and Z.-H. Chen, “Resonance coupling between molecular excitons and nonradiating anapole modes in silicon nanodisk-J-aggregate heterostructures,” ACS Photonics 5, 1628–1639 (2018). [CrossRef]
27. Q. Ruan, N. Li, H. Yin, X. Cui, J. Wang, and H.-Q. Lin, “Coupling between the Mie resonances of Cu2O nanospheres and the excitons of dye aggregates,” ACS Photonics 5, 3838–3848 (2018). [CrossRef]
28. A. E. Krasnok, A. E. Miroshnichenko, P. A. Belov, and Y. S. Kivshar, “All-dielectric optical nanoantennas,” Opt. Express 20, 20599–20604 (2012). [CrossRef]
29. M. K. Schmidt, R. Esteban, J. J. Sáenz, I. Suarez-Lacalle, S. Mackowski, and J. Aizpurua, “Dielectric antennas—a suitable platform for controlling magnetic dipolar emission,” Opt. Express 20, 13636–13650 (2012). [CrossRef]
30. P. Albella, M. Ameen Poyli, M. K. Schmidt, S. A. Maier, F. Moreno, J. J. Sáenz, and J. Aizpurua, “Low-loss electric and magnetic field-enhanced spectroscopy with subwavelength silicon dimers,” J. Phys. Chem. C 117, 13573–13584 (2013). [CrossRef]
31. D. Bouchet, M. Mivelle, J. Proust, B. Gallas, I. Ozerov, M. F. García Parajó, A. Gulinatti, I. Rech, Y. De Wilde, N. Bonod, V. Krachmalnicoff, and S. Bidault, “Enhancement and inhibition of spontaneous photon emission by resonant silicon nanoantennas,” Phys. Rev. Appl. 6, 064016 (2016). [CrossRef]
32. R. Regmi, J. Berthelot, P. M. Winkler, M. Mivelle, J. Proust, F. Bedu, I. Ozerov, T. Begou, J. Lumeau, H. Rigneault, M. F. García Parajó, S. Bidault, J. Wenger, and N. Bonod, “All-dielectric silicon nanogap antennas to enhance the fluorescence of single molecules,” Nano Lett. 16, 5143–5151 (2016). [CrossRef]
33. A. F. Cihan, A. G. Curto, S. Raza, P. G. Kik, and M. L. Brongersma, “Silicon Mie resonators for highly directional light emission from monolayer MoS2,” Nat. Photonics 12, 284–290 (2018). [CrossRef]
34. W. Zhang, A. O. Govorov, and G. W. Bryant, “Semiconductor-metal nanoparticle molecules: hybrid excitons and the nonlinear Fano effect,” Phys. Rev. Lett. 97, 146804 (2006). [CrossRef]
35. R. D. Artuso and G. W. Bryantt, “Optical response of strongly coupled quantum dot–Metal nanoparticle systems: double peaked Fano structure and bistability,” Nano Lett. 8, 2106–2111 (2008). [CrossRef]
36. R. D. Artuso and G. W. Bryant, “Strongly coupled quantum dot-metal nanoparticle systems: exciton-induced transparency, discontinuous response, and suppression as driven quantum oscillator effects,” Phys. Rev. B 82, 195419 (2010). [CrossRef]
37. A. Manjavacas, F. J. García de Abajo, and P. Nordlander, “Quantum plexcitonics: strongly interacting plasmons and excitons,” Nano Lett. 11, 2318–2323 (2011). [CrossRef]
38. W. Zhang and A. O. Govorov, “Quantum theory of the nonlinear Fano effect in hybrid metal-semiconductor nanostructures: the case of strong nonlinearity,” Phys. Rev. B 84, 081405 (2011). [CrossRef]
39. P. Torma and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78, 013901 (2015). [CrossRef]
40. G. Zengin, M. Wersäll, S. Nilsson, T. J. Antosiewicz, M. Kall, and T. Shegai, “Realizing strong light-matter interactions between single-nanoparticle plasmons and molecular excitons at ambient conditions,” Phys. Rev. Lett. 114, 157401 (2015). [CrossRef]
41. R. Chikkaraddy, B. de Nijs, F. Benz, S. J. Barrow, O. A. Scherman, E. Rosta, A. Demetriadou, P. Fox, O. Hess, and J. J. Baumberg, “Single-molecule strong coupling at room temperature in plasmonic nanocavities,” Nature 535, 127–130 (2016). [CrossRef]
42. R. Liu, Z.-K. Zhou, Y.-C. Yu, T. Zhang, H. Wang, G. Liu, Y. Wei, H. Chen, and X.-H. Wang, “Strong light-matter interactions in single open plasmonic nanocavities at the quantum optics limit,” Phys. Rev. Lett. 118, 237401 (2017). [CrossRef]
43. D. G. Baranov, M. Wersäll, J. Cuadra, T. J. Antosiewicz, and T. Shegai, “Novel nanostructures and materials for strong light matter interactions,” ACS Photonics 5, 24–42 (2017). [CrossRef]
44. V. Giannini, A. I. Fernández-Domínguez, S. C. Heck, and S. A. Maier, “Plasmonic nanoantennas: fundamentals and their use in controlling the radiative properties of nanoemitters,” Chem. Rev. 111, 3888–3912 (2011). [CrossRef]
45. P. Biagioni, J.-S. Huang, and B. Hecht, “Nanoantennas for visible and infrared radiation,” Rep. Prog. Phys. 75, 024402 (2012). [CrossRef]
46. A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science 329, 930–933 (2010). [CrossRef]
47. C. M. Dodson and R. Zia, “Magnetic dipole and electric quadrupole transitions in the trivalent lanthanide series: calculated emission rates and oscillator strengths,” Phys. Rev. B 86, 125102 (2012). [CrossRef]
48. T. H. Taminiau, S. Karaveli, N. F. van Hulst, and R. Zia, “Quantifying the magnetic nature of light emission,” Nat. Commun. 3, 979 (2012). [CrossRef]
49. M. Kasperczyk, S. Person, D. Ananias, L. D. Carlos, and L. Novotny, “Excitation of magnetic dipole transitions at optical frequencies,” Phys. Rev. Lett. 114, 163903 (2015). [CrossRef]
50. T. Feng, Y. Xu, Z. Liang, and W. Zhang, “All-dielectric hollow nanodisk for tailoring magnetic dipole emission,” Opt. Lett. 41, 5011–5014 (2016). [CrossRef]
51. J. Li, N. Verellen, and P. Van Dorpe, “Enhancing magnetic dipole emission by a nano-doughnut-shaped silicon disk,” ACS Photonics 4, 1893–1898 (2017). [CrossRef]
52. M. Sanz-Paz, C. Ernandes, J. U. Esparza, G. W. Burr, N. F. van Hulst, A. Maître, L. Aigouy, T. Gacoin, N. Bonod, M. F. García Parajó, S. Bidault, and M. Mivelle, “Enhancing magnetic light emission with all-dielectric optical nanoantennas,” Nano Lett. 18, 3481–3487 (2018). [CrossRef]
53. T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic magnetic Purcell effect,” ACS Photonics 5, 678–683 (2017). [CrossRef]
54. S. M. Hein and H. Giessen, “Tailoring magnetic dipole emission with plasmonic split-ring resonators,” Phys. Rev. Lett. 111, 026803 (2013). [CrossRef]
55. M. Mivelle, T. Grosjean, G. W. Burr, U. C. Fischer, and M. F. Garcia-Parajo, “Strong modification of magnetic dipole emission through diabolo nanoantennas,” ACS Photonics 2, 1071–1076 (2015). [CrossRef]
56. K. Yao and Y. Liu, “Controlling electric and magnetic resonances for ultracompact nanoantennas with tunable directionality,” ACS Photonics 3, 953–963 (2016). [CrossRef]
57. D. G. Baranov, R. S. Savelev, S. V. Li, A. E. Krasnok, and A. Alù, “Modifying magnetic dipole spontaneous emission with nanophotonic structures,” Laser Photonics Rev. 11, 1600268 (2017). [CrossRef]
58. A. Yariv, Quantum Electronics (Wiley, 1975).
59. N. R. Brewer, Z. N. Buckholtz, Z. J. Simmons, E. A. Mueller, and D. D. Yavuz, “Coherent magnetic response at optical frequencies using atomic transitions,” Phys. Rev. X 7, 011005 (2017). [CrossRef]
60. L. Allen and J. H. Eberly, Optical Resonance and Two-Level Atoms (Courier Corporation, 1987).
61. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, 2008).
62. A. Garcia-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express 19, 4815–4826 (2011). [CrossRef]
63. E. D. Palik, Handbook of Optical Constants of Solids, E. D. Palik, ed., Academic Press Handbook Series (Academic, 1985).
64. R. M. Macfarlane and R. M. Shelby, “Homogeneous line broadening of optical transitions of ions and molecules in glasses,” J. Lumin. 36, 179–207 (1987). [CrossRef]
65. F. Konz, Y. Sun, C. W. Thiel, R. L. Cone, R. W. Equall, R. L. Hutcheson, and R. M. Macfarlane, “Temperature and concentration dependence of optical dephasing, spectral-hole lifetime, and anisotropic absorption in Eu3+:Y2SiO5,” Phys. Rev. B 68, 085109 (2003). [CrossRef]
66. H. Chew, “Transition rates of atoms near spherical surfaces,” J. Chem. Phys. 87, 1355–1360 (1998). [CrossRef]