Abstract

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

1. INTRODUCTION

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 [13]. With these properties, they can find many nanophotonic applications, such as metamaterials [4], metasurfaces [5,6], structural colors [7], magnetic mirrors [8], and optical nanoantennas [911]. 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 [9,1216] as well as couplings between the plasmonic and dielectric nanostructures [1721]. The couplings of different dielectric nanostructures can induce the phenomena of hybridizations of electromagnetic modes [12,13] and Fano resonances [1416]. The combinations of plasmonic and dielectric structures can also strongly modify their optical responses, where potential applications have been demonstrated including optical nanoantennas [2022].

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 [2327], 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 [10,2833]. 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 [3443] and the modifications of the emission properties of emitters [4446] 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 [4749]. Tailoring the emissions of MD emitters by dielectric nanostructures has begun to be studied [29,30,5053]. 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 [5457]. 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 R. 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 E1 and E2, respectively. The frequency and matrix element of the MD transition of an ion are ω0=(E2E1)/ and μMD, respectively.

 

Fig. 1. (a) Schematic of a hybrid system under study. (b) Quantum transition of a MD emitter.

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The Hamiltonian for an ion in the hybrid system can be written as [58,59]

H^MD=ω0a^a^μMDBMDa^μMDBMD*a^.
a^ and a^ are the creation and annihilation operators of the excited state, respectively. BMD is the total magnetic field felt by the ion. In the coupled system, the total magnetic field BMD consists of the magnetic field B0 of the excitation plane wave and the magnetic field of the Si sphere, namely BMD=B0+XmSi, where XmSi is the contribution of the Si sphere, mSi is the magnetic dipole moment of the Si sphere, and X represents the proportional coefficient between the magnetic field produced by the Si sphere and its magnetic dipole moment mSi. The magnetic dipole moment can be written as mSi=μ01αMBSi, where BSi is the total magnetic field felt by the Si sphere, αM is the magnetic polarizability of the Si sphere, and μ0 is the permeability of vacuum. The total magnetic field felt by the Si sphere consists of the magnetic field B0 of the plane wave and that from the magnetic dipole of the ion. Thus, we have BSi=B0+YmMD, where YmMD is the contribution of the magnetic ion and mMD is the magnetic dipole moment of the ion. Note that the Si sphere is much larger than the ion. Thus, the magnetic field felt by the Si sphere, which is produced by the ion, is not inhomogeneous. Here, for the sake of simplicity, we assume that there is an equivalent uniform magnetic field contributed from the ion, and it is written as YmMD, which is proportional to the magnetic dipole of moment mMD of the ion, where Y is the proportional coefficient. We calculate the magnetic dipole moment of a magnetic ion using the density matrix μMD(ρ12+ρ21). For an ion cluster, one can take it as an equivalent ion with a larger dipole moment [58,59]. If the number of the ions is N, the magnetic dipole moment of the ion cluster can be written as mMD(t)=NμMD(ρ¯12eiωt+ρ¯21eiωt)=(mMD/2)eiωt+(mMD*/2)eiωt. ρ¯12 and ρ¯21 are used to separate the high-frequency part. The magnetic field of incident light as a function of time can be written as B(t)=(B0/2)eiωt+(B0/2)eiωt. The magnetic field felt by the Si sphere is BSi(t)=(B0/2+NYμMDρ¯21)eiωt+(B0/2+NYμMDρ¯12)eiωt. The magnetic field felt by ion cluster can be written as
BMD(t)=μMD[(Ω+Gρ¯21)eiωt+(Ω*+G*ρ¯12)eiωt],=(BMD/2)eiωt+(BMD*/2)eiωt
where Ω=μMD(1+XαMμ0)B02, G=μMD2XYNαMμ0.

The above matrix elements should satisfy the master equation [58],

dρijdt=i[ρ,HMD]ijΓijρij,
where the relaxation elements of diagonal and nondiagonal elements correspond to Γ22=Γ11=1/T1 and Γ12=Γ21=1/T2, respectively, where T1 is relaxation time of ions at equilibrium (the longitudinal homogeneous lifetime) and T2 is the time about loss of phase coherence (the transverse homogeneous lifetime) [58,60]. From Eq. (3), we have
dρ21dt=iω0ρ21+iμMDBMD(t)(ρ11ρ22)ρ21T2,d(ρ11ρ22)dt=2iμMDBMD(t)(ρ21ρ21*)(ρ11ρ22)(ρ11ρ22)0T1,
where (ρ11ρ22)0 is the initial population difference, which is taken to be (ρ11ρ22)0=1. We write Δ=ρ11ρ22, ρ¯12=C+iD, ρ¯21=CiD. With the rotating wave approximation, we find
dCdt=CT2+(ωω0)D(ΩI+GICGRD)Δ,dDdt=DT2(ωω0)C(ΩR+GRCGID)Δ,dΔdt=1ΔT1+4ΩIC+4ΩRD+4GI(C2+D2),
where ΩR and ΩI are the real and imaginary parts of Ω, respectively. GR and GI are the real and imaginary parts of G, respectively. In the steady-state situation, the left-hand side of Eq. (5) satisfies dC/dt=dD/dt=dΔ/dt=0. We obtain
4ΔT2ΩI2+ΩR2(1T2+GIΔ)2+[(ωω0)+GRΔ]2=1ΔT1,C=ΔΩR[(ωω0)+ΔGR]+ΔΩI(1T2+GIΔ)(1T2+GIΔ)2+[(ωω0)+GRΔ]2,D=ΔΩI[(ωω0)+ΔGR]ΔΩR(1T2+GIΔ)(1T2+GIΔ)2+[(ωω0)+GRΔ]2.
The extinction cross sections of the system of the ions and Si sphere can be calculated based on the above results. As the size of the ion cluster is much smaller than the wavelength of light, the extinction and scattering cross section can be expressed as (σext)MD=μ0kIm(mMDBMD*)/B02=kIm(αMD)|BMD2|/B02, (σscat)MD=k4|mMD2|/6πμ02B02=k4|αM2||BMD2|/6πB02, respectively, where k=2π/λ is a wave vector and λ is the wavelength of light [61]. The magnetic dipole moment of ion cluster mMD can be expressed as mMD=NμMD[(C+iD)eiωt+(CiD)eiωt]. mMD can also be written as mMD=μ01BMDαMD, αMD is the magnetic dipole polarizability of an ion cluster. Thus, according to Eq. (4), the polarizability of an ion cluster can be written as
αMD=Nμ0μMD2(ω0ω)T22+T2i1+(ωω0)2T22+μMD22|BMD|2T1T2.
We obtain the analytical formula of the extinction cross section of ion cluster as
(σext)MD=kNμ0μMD2T21+(ωω0)2T22+μMD22|BMD|2T1T2|BMD|2B02.
The full width at half-maximum (FWHM) of the extinction spectrum is
Δω=2T22|BMD1|2|BMD0|21+μMD22|BMD1|2T1T2,
where BMD1 is the magnetic field at the frequency ω=ω0±Δω/2 (the magnitudes of the two magnetic fields at positions of the full width at half-maximum are close to each other, so they are approximately taken as the same value), and BMD0 is the magnetic field at the frequency of the MD transition ω0. For an individual magnetic ion cluster which corresponds to X=Y=0, its extinction cross section becomes
(σext)MD0=kNμ0μMD2T21+(ωω0)2T22+μMD22B02T1T2,
which agrees with the reported results [58]. The corresponding FWHM is now Δω=21+μMD2B02T1T2/2/T2.

For a Si nanosphere with the relative refractive index m=n1+in2, kR1 is approximately satisfied, but y=mkR1 is not satisfied. Its extinction and scattering cross section can be written as (σext)Si=μ0kIm(mSiBSi*)/B02+kIm(PSiESi*)/ε0E02 and (σsca)Si=(k4/6π)(|αM2||BSi2|/B02+|αE2||ESi2|/E02), respectively, based on the Mie theory [22,61,62]. Here the electric dipole mode also contributes to the optical responses of the Si sphere [62]. The electric and magnetic dipole moments can be written as PSi=ε0αEESi and mSi=μ01αMBSi, respectively. ε0 is the vacuum dielectric constant, and αE and αM are the electric dipole and magnetic dipole polarizabilities of the Si sphere, respectively. According to the Mie theory, αE and αM can be written as

αE=αE(0)1ik3π3αE(0),αM=αM(0)1ik3π3αM(0),
where
αE(0)=6πk3tanα1,αM(0)=6πk3tanβ1,tanαn=m2jn(y)[xjn(x)]jn(x)[yjn(y)]m2jn(y)[xyn(x)]yn(x)[yjn(y)],tanβn=jn(y)[xjn(x)]jn(x)[yjn(y)]jn(y)[xyn(x)]yn(x)[yjn(y)],
where jn(x) and yn(x) are the spherical Bessel and Neumann functions, respectively. As BSi=B0+YmMD, the analytical formula of extinction cross section of a coupled Si sphere can be written as
(σext)Si=μ0kIm(mSiBSi*)B02+kIm(αE)=kIm(αM){[B0+Re(YmMD)]2+[Im(YmMD)]2}B02+kIm(αE).
The scattering cross section can also be written as (σscat)Si=(k4/6π)|αM2|{[B0+Re(YmMD)]2+[Im(YmMD)]2}/B02+(k4/6π)|αE2|. Although the electric dipole mode contributes the responses of a Si sphere, this mode does not interact with the ion cluster. In the spherical system we considered, X can be calculated analytically according to the Mie theory. Y can be equivalently obtained by using the theory of magnetic dipole radiation enhancement (see Appendix A for the relevant discussion).

3. RESULTS AND DISCUSSION

For numerical calculations, the radius of the Si sphere is chosen to be R=60nm, and its refractive index is taken from the Palik’s book [63]. The resonance of its magnetic dipole mode is near λ=520nm. The MD transition of the rare-earth ions is chosen to be at λ=525nm, which is spectrally near the resonant position of Si sphere. The transverse homogeneous lifetime T2 and the longitudinal homogeneous lifetime T1 are both taken to be 0.2 ns. The matrix element for the MD transition is 0.3μB (μB is the Bohr magneton). The number of ions is N=300,000. From Eq. (6), the magnetic polarizability of the ionic cluster varies with BMD and BMD varies with the incident B0, which can cause the system to show nonlinear optical responses. When the intensity of excitation light is weak, μMD2|BMD|2T1T2/21. The magnetic polarizability of the ion cluster is approximately written as αMD=(Nμ0μMD2T2)/[(ω0ω)T2i]. The magnetic ion cluster exhibits approximately linear optical response, namely the extinction spectrum [Eq. (7)] does not vary with the intensity of the light (BMD/B0 can be represented by X, Y, αMD, and αM). When the light intensity increases so that μMD2|BMD|2T1T2/21 cannot be satisfied, the ion exhibits nonlinear optical response. In order to quantify the linear and nonlinear response regimes of the system, we define μMD2|BMD|2T1T2/2=0.1 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 B0=3.1×103T, which corresponds to the light intensity of I0=1.1554×105W/cm2.

We first consider a case with a relatively low intensity of the excitation light I0=104W/cm2 (B0=9.1453×104T). Figure 2(a) shows extinction spectra of the individual and coupled ion cluster. The inset of Fig. 2(a) illustrates the response spectrum of Δ=ρ11ρ22. The absorption of the ions is greatly enhanced (more than 200 times) with the Si sphere. This can be understood with the behavior of Δ and the magnetic field enhancement of the Si sphere. With a weak excitation, Δ 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) becomes

(σext)MD1(kNT2μ0μMD2|BMD0|2)/(B02),
under the weak excitation condition μMD2|BMD|2T1T2/21. BMD0 is close to B1 and it is much larger than B0 [B1 is the magnetic field at the center of the individual Si sphere at the frequency ω0, B1=(1+XαM/μ0)B0]. Thus, the resonant extinction of the coupled ions is |BMD0|2/|B0|2 times higher than that of the individual ions (kNμ0μMD2T2/). The FWHM of the extinction spectrum [Eq. (8)] is reduced to
Δω(2/T2)2|BMD1|2/|BMD0|21.
It is slightly larger than the individual case 2/T2 (the value of BMD will be shown later in Fig. 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).

 

Fig. 2. Extinction spectra of a coupled system in the weak and strong light intensity regimes. (a), (c) The extinction spectra of the individual and coupled ion cluster in the weak (a) and strong (c) light intensity regime. The insets are the population difference spectra of the individual ions and the coupled ions. (b), (d) The extinction spectra of the coupled Si sphere and the hybrid structure in the weak (b) and strong (d) light intensity regime. The insets are the extinction spectra for a wider frequency regime.

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We now turn to a high light intensity I0=108W/cm2 (B0=9.1453×102T). 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 ω=ω0, the value Δ for individual ions is much closer to 0 compared to that with weak excitations. And the Δ 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 μMD2|BMD|2T1T2/21, and the expression for the resonant extinction of a coupled ion cluster based on Eq. (7) can now be written as

(σext)MD2kNμ0/(B02T1).
The FWHM of the extinction spectrum [Eq. (8)] is reduced to
Δω2μMD|BMD1|T1T2/(T2).
It is larger than that of the individual ions (2/T2) and the linear case [Eq. (11)] as μMD2|BMD|2T1T2/21. Note that the high intensity excitation (μMD2|BMD|2T1T2/21) condition is a special situation of the nonlinear regime (μMD2|BMD|2T1T2/2>0.1). 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 αMD and the total magnetic field felt by the ions BMD. Based on Eq. (9), the variation of the extinction of the coupled Si sphere is mainly related to Re(YmMD) and Im(YmMD). And it can be easily checked that B0 is much higher than Re(YmMD) and Im(YmMD). Therefore, the main factor that causes the change of the extinction of the Si sphere is Re(YmMD), where Y 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 mMD=μ01BMDαMD. Figures 3(a)3(c) show the spectra of αMD, BMD, and mMD under a weak excitation. The Im(αMD) and Re(αMD) show a peak and an antisymmetric line shape, respectively. The magnitude of Im(αMD) is higher than that of Re(αMD), and Im(BMD) is several times larger than Re(BMD). Thus, Re(mMD) is mainly dependent on Im(αMD)×Im(BMD), which has a dip line shape. The negative value of Re(mMD) means that Re(mMD) and the excitation field B0 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 Im(αMD) and Re(αMD) also show a peak and an antisymmetric line shape, respectively, while Re(αMD) is much larger than Im(αMD) [Fig. 3(d)]. The BMD shows similar behavior [Fig. 3(e)] to that with the weak excitation. Thus, Re(mMD) is now mainly dependent on Re(αMD)×Re(BMD), whose line shape is determined by Re(αMD) as BMD is hardly changed with frequency around the ω0. As a result, the extinction line of the coupled Si sphere has a Fano-like line shape which is similar to Re(αMD) [Fig. 3(f)]. Here, it can also be seen that the |BMD| is close to B1 and it is much larger than the excitation field B0 under both the weak and strong excitations.

 

Fig. 3. Magnetic polarizabilities αMD, magnetic fields BMD, and magnetic dipole moments mMD for the ionic cluster. (a), (d) The magnetic polarizabilities αMD in the weak and strong light intensity regimes. (b), (e) The magnetic fields BMD in the weak and strong light intensity regimes. (c), (f) The magnetic dipole moments mMD in the weak and strong light intensity regimes.

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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 kNμ0/B02T1, which will decrease with B0. 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.

 

Fig. 4. Extinction spectra of (a) the ion cluster and (b) the whole system with different light intensities.

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The effects from the MD matrix element are investigated. We first consider the weak light intensity I0=104W/cm2. The resonant extinction cross section of the ions first increases and then decreases with the μMD from 0.1μB to 1.2μB [Fig. 5(a)]. This can be understood based on the expression of the resonant extinction of the ions (σext)MD1(kNT2μ0μMD2|BMD0|2)/B02 [Eq. (10)], and it is in proportion to μMD2|BMD0|2. |BMD0| decreases with μMD. When μMD is small (e.g., 0.1μB), the coupling effect between the ion cluster and Si sphere is relatively small. The variation of |BMD0| is small (see the relevant results in Appendix B), so the resonant extinction increases with μMD. When μMD is larger than 0.9μB, |BMD0| is approaching 0 and the decreasing of |BMD0| is a dominant factor. Thus, the resonant extinction decreases. The dip on the extinction spectrum of the hybrid structure becomes deeper and broader with μMD. The variation of the responses of the coupled Si sphere is mainly related to the Re(YmMD), and the Re(mMD) depends largely on Im(αMD)×Im(BMD) in the linear region as discussed before. αMD is in proportion to μMD2 [Eq. (6)]. The variation of μMD2 is larger than that of BMD even for a large μMD. So the variation of the extinction spectrum of the hybrid structure increases with μMD. BMD1BMD0 (BMD0) increases (decreases) with μMD (Appendix B), so the FWHM [Eq. (11)] (2/T2)2|1+(BMD1BMD0)/BMD0|21 will also increase.

 

Fig. 5. Extinction spectra of the ion cluster and the whole system with varying (a)–(d) the MD matrix element μMD and (e)–(h) the number of ions N. (a), (c) The extinction spectra of the coupled ion cluster in the weak (a) and strong (c) light intensity regime. (b), (d) The extinction spectra of the hybrid structure in the weak (b) and strong (d) light intensity regime. Panels (e)–(h) show the same contents as that in panels (a)–(d), respectively, with varying the N.

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Figures 5(c) and 5(d) show the responses of the coupled ions and the hybrid structure with a high light intensity I0=108W/cm2, respectively. Different from the linear region, the resonant extinction of the ions hardly changes with the μMD, because Eq. (10) for resonant extinction of ions in the high light intensity regime still holds here (μMD2|BMD|2T1T2/21) and it is independent of μMD. The spectrum of the ions becomes broader with μMD, because its FWHM Δω2μMD|BMD1|T1T2/T2 [Eq. (13)] increases significantly with μMD as BMD1 is very close to B1 in the μMD2|BMD|2T1T2/21 limit and it almost does not change with μMD (Fig. 3 and Appendix B). For the hybrid structure, the antisymmetric Fano-like line shape becomes more pronounced with μMD [Fig. 5(d)]. The variation of the extinction of the hybrid system is mainly dependent on Re(mMD)μ01×Re(αMD)×Re(BMD). For the typical positions of ω0±Δω/2, the Re(mMD)NT2μ0μMDRe(BMD1)/(2|BMD1|T1T2) will increase with μMD as BMD1 almost does not change with μMD.

The MD matrix element of ion cluster mMD is also dependent on the number of ions N. Thus, the effects from the number of ions N are also investigated [Figs. 5(e)5(h)]. For a weak light intensity I0=104W/cm2 [Figs. 5(e) and 5(f)], the behaviors of the ions and the hybrid structure are similar to the case with varying the μMD [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 I0=108W/cm2, the resonant extinction of ions increases with the number of ions N [Fig. 5(g)], which is different from the case with varying the μMD [Fig. 5(c)]. This is because the resonant extinction of ions [Eq. (12)] increases with N while it is independent of μMD. The FWHM of the extinction spectrum [Eq. (13)] is independent of N, and BMD1 is very close to B1 in the μMD2|BMD|2T1T2/21 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 μMD [Fig. 5(d)]. Here, it should be pointed out that for small μMD or N 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 T2 and the longitudinal homogeneous lifetime T1 is also considered. For simplicity, T2 and T1 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 I0=104W/cm2. The resonant extinction of the ion cluster first increases and then decreases with the lifetimes of T2 and T1 varying from 0.1 ns to 1.6 ns. The resonant extinction [Eq. (10)] is proportional to T2|BMD0|2, while |BMD0|2 decreases with T2 (T1) (Appendix B). For small T2 (T1), the coupling effect (X, Y) is relatively small. The variation of T2 (T1) is larger than that of |BMD0|2. So the resonant extinction increases with T2 (T1). When T2 (T1) is larger than 0.8 ns, the coupling is strong and |BMD0|2 is approaching 0. The variation of |BMD0|2 is larger than that of T2 (T1). Thus, the resonant extinction decreases with T2 (T1). The variation of the extinction of the coupled Si sphere is mainly related to Re(mMD). For the resonant position of ω=ω0, Re(mMD)Nμ0μMD2T2Im(BMD0)/. The variation of T2(T1) is larger than that of BMD0 (Appendix B) for each T2, so the dip of the coupled Si and the corresponding hybrid structure becomes deeper with T2 (T1). 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 T2 (T1).

 

Fig. 6. Extinction spectra of the ion cluster and the hybrid structure with different lifetimes T1 and T2(T1=T2). (a), (c) The extinction spectra of the coupled ion cluster in the weak (a) and strong (c) light intensity regime. (b), (d) The extinction spectra of the hybrid structure in the weak (b) and strong (d) light intensity regime.

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For a strong light intensity I0=108W/cm2, the resonant extinction of the ion cluster decreases with the lifetime T2 (T1) [Fig. 6(c)]. This is because the resonant extinction [Eq. (12)] is inversely proportional to T1. Its FWHM [Eq. (13)] does not change with the lifetime T2 (T1). 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 Re(αMD)×Re(BMD) with the strong excitation. For the typical positions of ω=ω0±Δω/2, Re(αMD)NT2μ0μMD/(2|BMD1|T1T2), which are not changed with the lifetime T2 (T1) (the BMD does not change with T2 under the strong excitation, see Appendix B). Therefore, the Fano spectral response of the hybrid structure does not change with the lifetime T2 (T1).

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 d between the location and the center of the sphere. The corresponding coupling coefficients X and Y decrease with d (see Table 1). Thus, the coupling strength in the hybrid structure becomes weaker with d. 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 d. The other parameters are the same as that in Figs. 2(a) and 2(b). The light intensity I0=104W/cm2 corresponds to the linear regime. The resonant extinction of the ion cluster [Eq. (10)] decreases with the distance d. This is because the total magnetic field felt by the ions BMD, which is related to the X and Y, becomes smaller with the d. The FWHM is almost invariable (|BMD0| is close to |BMD1|). The dip of the extinction spectrum of the hybrid structure becomes weaker with the distance d as the destructive coupling strength between the ions and the Si sphere decreases with d.

 

Fig. 7. Extinction spectra of the ion cluster and the hybrid structure with different distance d between the ion cluster and the center of the Si sphere. (a) Schematic of a hybrid structure with a distance d. (b), (d) The extinction spectra of the coupled ion cluster in the weak (b) and strong (d) light intensity regime. (c), (e) The extinction spectra of the hybrid structure in the weak (c) and strong (e) light intensity regime.

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Figures 7(d) and 7(e) show the results for a high light intensity I0=108W/cm2. The resonant extinction of the ion cluster is almost unchanged with the d. The reason is that its value [Eq. (12)] is independent of the X and Y. The FWHM of the ions [Eq. (13)] becomes smaller significantly. This is caused by the decreasing of BMD with d. For the Si sphere, the variation of its extinction depends mainly on YRe(mMD), where Re(mMD)μ01×Re(αMD)×Re(BMD) in the nonlinear regime. For the typical positions of ω=ω0±Δω/2, the Re(mMD)NT2μ0μMDRe(BMD1)/(2|BMD1|T1T2). It can be easily verified that the variation of Re(BMD1)/BMD1 is much smaller than that of Y. Thus, the extinction of Si decreases with the decreasing of Y.

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.

 

Fig. 8. Extinction spectra of the GaP-based hybrid structure. (a) Schematic of the system with a GaP sphere. (b), (d) The extinction spectra of the coupled ion cluster in the weak (b) and strong (d) light intensity regime. (c), (e) The extinction spectra of the hybrid structure in the weak (c) and strong (e) light intensity regime.

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4. CONCLUSIONS

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 N, 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 X AND Y FOR THE HYBRID STRUCTURE

  • 1. For X. X 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 [61]. 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 R. The excitation light propagates in the z-axis, and its polarizability is in x-axis.

    The total magnetic field inside the Si sphere is Hin=k1ωμ0n=1En(dnMe1n(1)+icnNo1n(1)) (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 Hout=kωμ0n=1En(anMe1n(3)+ibnNo1n(3)) (this is only the scattered field of the Si sphere):

    Me1n=1sinθsinϕPn1(cosθ)Zn(ρ)eθcosϕdPn1(cosθ)dθZn(ρ)eϕ,No1n=Zn(ρ)ρsinϕn(n+1)Pn1(cosθ)ersinϕdPn1(cosθ)dθ1ρddρ[ρZn(ρ)]eθ+cosϕPn1(cosθ)sinθ1ρddρ[ρZn(ρ)]eϕ,
    where k1=2πm/λ (m is the complex refractive index of Si), En=E0(2n+1)in/n(n+1), and E0 is the electric field of the light, Me1n and No1n are vector spherical harmonics, and Zn(ρ) represents jn(ρ) and hn1(ρ) in Me1n(1) (No1n(1)) and Me1n(3) (No1n(3)), respectively. ρ=kr; n is the polar eigennumber; and er, eθ, and eϕ are basis unit vectors of the spherical coordinate system. an and bn are the electric and magnetic Mie coefficients, respectively. cn and dn are the magnetic and electric inner coefficients, respectively. an, bn, cn, and dn are
    an=m2jn(mx)[xjn(x)]jn(x)[yjn(y)]m2jn(mx)[xhn1(x)]hn1(x)[yjn(y)],bn=jn(mx)[xjn(x)]jn(x)[yjn(y)]jn(mx)[xhn1(x)]hn1(x)[yjn(y)],cn=jn(x)[xhn1(x)]hn1(x)[xjn(x)]jn(mx)[xhn1(x)]hn1(x)[yjn(y)],dn=mjn(x)mhn1(x)[xjn(x)]m2jn(mx)[xhn1(x)]hn1(x)[yjn(y)],
    where y=mx, x=2πR/λ, hn1(x)=jn(x)+iyn(x), and jn(x) and yn(x) are the spherical Bessel and Neumann functions, respectively. hn1(x) is the spherical Hankel function.

    For a location on the +y-axis, it corresponds to θ=π/2, ϕ=π/2, r=d, and d is the distance from the Si center to the location. The total magnetic field inside the Si sphere can be written as

    Hin=k1ωμ0n=1(2n+1)in+1E0cnjn(k1d)k1dPn1(0).
    The scattered magnetic field outside the Si sphere is
    Hout=kωμ0n=1(2n+1)in+1E0bnhn1(kd)kdPn1(0).
    In our selected frequency range, the scattered magnetic field of the Si sphere is almost all provided by the magnetic dipole mode of the Si sphere. Thus, they can be written as Hin=3E0c1j1(k1d)P11(0)/μ0dω, Hout=3E0b1h11(kd)P11(0)/μ0dω. The scattered magnetic field inside the Si sphere is Hscat(in)=HinH0, where the excitation field is E0=cμ0H0, and we have the magnetic dipole moment of the Si sphere mSi=αMH0=μ01αMB0. So one can obtain X:
    X=Bscat(in)mSi=μ0αM[3c1j1(k1d)P11(0)kd1],(d<R),
    X=Bscat(out)mSi=μ0αM3b1h11(kd)P11(0)kd.(d>R).

  • 2. For Y. 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 μ 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 μY. The magnetic dipole moment of the excited Si sphere is mSi=μ01αSiμY. Y can be calculated if the magnetic dipole moment of the excited Si sphere is known. For a system consisting of an excitation magnetic dipole μ and a Si sphere, the radiation enhancement of the whole system is F=Prad/P0, where Prad is the radiated power of the whole system in the far field, and P0 is the radiated power of the individual magnetic dipole, P0=μ0μ2ω4/12πc3.

    The magnetic dipole of the whole system is μ+μ01αMμY, so the radiated power of the whole system in the far field can be written as Prad=(μ2|μ0+αMY|2ω4)/12πμ0c3. Thus, we can get the radiation enhancement factor of the whole system as

    F=PradP0=|1+αMμ0Y|2.
    For the sphere structure, the analytical formula for the radiation enhancement factor of the magnetic dipole can be written as [66]
    F=PradP0=9|μ1ε0μ2ε12ρ12j12(y1)y12D12|,(d<R),
    F=PradP0=9|j1(kd)b1h11(kd)kd|2.(d>R).
    The formulas are applicable when the direction of the magnetic dipole is parallel to the line between the center of the Si sphere and the magnetic dipole, as shown in Fig. 10. ε1 and μ1 are the dielectric constant and permeability of the Si sphere, respectively. ε0 and μ0 are dielectric constant and permeability in vacuum, respectively. ρ1=2πε1R/λ, y1=2πε1d/λ, ρ0=2πε0R/λ (d is the distance between the magnetic dipole and the center of the Si sphere), and
    D1=μ1j1(ρ1)[ρ0h11(ρ0)]μ0h11(ρ0)[ρ1j1(ρ1)].
    With Eqs. (A3)–(A5), one can get the Y.

Tables Icon

Table 1. X and Y at Different Distance d between the Location and the Center of the Sphere (Radius of the Si Sphere Is 60 nm)

 

Fig. 9. Spherical polar coordinates of the Si sphere.

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Fig. 10. Magnetic dipole μ exciting a Si sphere.

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Fig. 11. FDTD and analysis results of the extinction spectra of the system. (a), (b) The extinction spectra of the magnetic Lorentz sphere and the Si sphere by FDTD simulations. (c), (d) The extinction spectra of the ion cluster and the Si sphere by analytical calculation (I0=104W/cm2).

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Fig. 12. Total magnetic field felt by the ion cluster with different μMD, N, and T2. (a), (b) The magnetic field BMD with different MD matrix element μMD under weak (a) and strong (b) light excitation. (c), (d) The magnetic field BMD with different number of ions N under weak (c) and strong (d) light excitation. (e), (f) The magnetic field BMD with different lifetime T2 (T1) under weak (e) and strong (f) light excitation.

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APPENDIX B

  • 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 μ(ω)=1+χm+Δμωm2ωm22iδmωω2.

    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 Δμ=0.01. The sphere is placed at the center of the Si sphere with a radius of 60 nm. For the corresponding analytical calculations, the T2 and T1 for ions are both approximately taken as 0.5 ps, the matrix element for the magnetic dipole transition is 0.3μB, the number of ions is N=120,000,000, 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 BMD with different μMD, N, and T2 (Fig. 12).

Funding

National Natural Science Foundation of China (11704416); Natural Science Foundation of Hunan Province (2017JJ3408).

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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]  

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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]  

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2018 (8)

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]

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]

Q. Zhao, Z. J. Yang, and J. He, “Fano resonances in heterogeneous dimers of silicon and gold nanospheres,” Front. Phys. 13, 137801 (2018).
[Crossref]

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]

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]

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]

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]

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]

2017 (14)

T. Feng, W. Zhang, Z. Liang, Y. Xu, and A. E. Miroshnichenko, “Isotropic magnetic Purcell effect,” ACS Photonics 5, 678–683 (2017).
[Crossref]

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]

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]

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]

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]

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]

T. Feng, Y. Xu, W. Zhang, and A. E. Miroshnichenko, “Ideal magnetic dipole scattering,” Phys. Rev. Lett. 118, 173901 (2017).
[Crossref]

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]

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]

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]

M. Khorasaninejad and F. Capasso, “Metalenses: versatile multifunctional photonic components,” Science 358, eaam8100 (2017).
[Crossref]

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]

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]

W. Liu, “Generalized magnetic mirrors,” Phys. Rev. Lett. 119, 123902 (2017).
[Crossref]

2016 (11)

S. Jahani and Z. Jacob, “All-dielectric metamaterials,” Nat. Nanotechnol. 11, 23–36 (2016).
[Crossref]

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]

M. Decker and I. Staude, “Resonant dielectric nanostructures: a low-loss platform for functional nanophotonics,” J. Opt. 18, 103001 (2016).
[Crossref]

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]

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]

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]

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]

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]

K. Yao and Y. Liu, “Controlling electric and magnetic resonances for ultracompact nanoantennas with tunable directionality,” ACS Photonics 3, 953–963 (2016).
[Crossref]

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]

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]

2015 (9)

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]

P. Torma and W. L. Barnes, “Strong coupling between surface plasmon polaritons and emitters: a review,” Rep. Prog. Phys. 78, 013901 (2015).
[Crossref]

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]

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]

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]

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]

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]

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).
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2014 (1)

Y. Yang, I. I. Kravchenko, D. P. Briggs, and J. Valentine, “All-dielectric metasurface analogue of electromagnetically induced transparency,” Nat. Commun. 5, 5753 (2014).
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2013 (2)

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).
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S. M. Hein and H. Giessen, “Tailoring magnetic dipole emission with plasmonic split-ring resonators,” Phys. Rev. Lett. 111, 026803 (2013).
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2012 (6)

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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).
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A. E. Miroshnichenko and Y. S. Kivshar, “Fano resonances in all-dielectric oligomers,” Nano Lett. 12, 6459–6463 (2012).
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2011 (4)

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2010 (2)

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).
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2008 (1)

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).
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2006 (1)

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).
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2003 (1)

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A. Yariv, Quantum Electronics (Wiley, 1975).

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Figures (12)

Fig. 1.
Fig. 1. (a) Schematic of a hybrid system under study. (b) Quantum transition of a MD emitter.
Fig. 2.
Fig. 2. Extinction spectra of a coupled system in the weak and strong light intensity regimes. (a), (c) The extinction spectra of the individual and coupled ion cluster in the weak (a) and strong (c) light intensity regime. The insets are the population difference spectra of the individual ions and the coupled ions. (b), (d) The extinction spectra of the coupled Si sphere and the hybrid structure in the weak (b) and strong (d) light intensity regime. The insets are the extinction spectra for a wider frequency regime.
Fig. 3.
Fig. 3. Magnetic polarizabilities αMD, magnetic fields BMD, and magnetic dipole moments mMD for the ionic cluster. (a), (d) The magnetic polarizabilities αMD in the weak and strong light intensity regimes. (b), (e) The magnetic fields BMD in the weak and strong light intensity regimes. (c), (f) The magnetic dipole moments mMD in the weak and strong light intensity regimes.
Fig. 4.
Fig. 4. Extinction spectra of (a) the ion cluster and (b) the whole system with different light intensities.
Fig. 5.
Fig. 5. Extinction spectra of the ion cluster and the whole system with varying (a)–(d) the MD matrix element μMD and (e)–(h) the number of ions N. (a), (c) The extinction spectra of the coupled ion cluster in the weak (a) and strong (c) light intensity regime. (b), (d) The extinction spectra of the hybrid structure in the weak (b) and strong (d) light intensity regime. Panels (e)–(h) show the same contents as that in panels (a)–(d), respectively, with varying the N.
Fig. 6.
Fig. 6. Extinction spectra of the ion cluster and the hybrid structure with different lifetimes T1 and T2(T1=T2). (a), (c) The extinction spectra of the coupled ion cluster in the weak (a) and strong (c) light intensity regime. (b), (d) The extinction spectra of the hybrid structure in the weak (b) and strong (d) light intensity regime.
Fig. 7.
Fig. 7. Extinction spectra of the ion cluster and the hybrid structure with different distance d between the ion cluster and the center of the Si sphere. (a) Schematic of a hybrid structure with a distance d. (b), (d) The extinction spectra of the coupled ion cluster in the weak (b) and strong (d) light intensity regime. (c), (e) The extinction spectra of the hybrid structure in the weak (c) and strong (e) light intensity regime.
Fig. 8.
Fig. 8. Extinction spectra of the GaP-based hybrid structure. (a) Schematic of the system with a GaP sphere. (b), (d) The extinction spectra of the coupled ion cluster in the weak (b) and strong (d) light intensity regime. (c), (e) The extinction spectra of the hybrid structure in the weak (c) and strong (e) light intensity regime.
Fig. 9.
Fig. 9. Spherical polar coordinates of the Si sphere.
Fig. 10.
Fig. 10. Magnetic dipole μ exciting a Si sphere.
Fig. 11.
Fig. 11. FDTD and analysis results of the extinction spectra of the system. (a), (b) The extinction spectra of the magnetic Lorentz sphere and the Si sphere by FDTD simulations. (c), (d) The extinction spectra of the ion cluster and the Si sphere by analytical calculation (I0=104W/cm2).
Fig. 12.
Fig. 12. Total magnetic field felt by the ion cluster with different μMD, N, and T2. (a), (b) The magnetic field BMD with different MD matrix element μMD under weak (a) and strong (b) light excitation. (c), (d) The magnetic field BMD with different number of ions N under weak (c) and strong (d) light excitation. (e), (f) The magnetic field BMD with different lifetime T2 (T1) under weak (e) and strong (f) light excitation.

Tables (1)

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Table 1. X and Y at Different Distance d between the Location and the Center of the Sphere (Radius of the Si Sphere Is 60 nm)

Equations (27)

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H^MD=ω0a^a^μMDBMDa^μMDBMD*a^.
BMD(t)=μMD[(Ω+Gρ¯21)eiωt+(Ω*+G*ρ¯12)eiωt],=(BMD/2)eiωt+(BMD*/2)eiωt
dρijdt=i[ρ,HMD]ijΓijρij,
dρ21dt=iω0ρ21+iμMDBMD(t)(ρ11ρ22)ρ21T2,d(ρ11ρ22)dt=2iμMDBMD(t)(ρ21ρ21*)(ρ11ρ22)(ρ11ρ22)0T1,
dCdt=CT2+(ωω0)D(ΩI+GICGRD)Δ,dDdt=DT2(ωω0)C(ΩR+GRCGID)Δ,dΔdt=1ΔT1+4ΩIC+4ΩRD+4GI(C2+D2),
4ΔT2ΩI2+ΩR2(1T2+GIΔ)2+[(ωω0)+GRΔ]2=1ΔT1,C=ΔΩR[(ωω0)+ΔGR]+ΔΩI(1T2+GIΔ)(1T2+GIΔ)2+[(ωω0)+GRΔ]2,D=ΔΩI[(ωω0)+ΔGR]ΔΩR(1T2+GIΔ)(1T2+GIΔ)2+[(ωω0)+GRΔ]2.
αMD=Nμ0μMD2(ω0ω)T22+T2i1+(ωω0)2T22+μMD22|BMD|2T1T2.
(σext)MD=kNμ0μMD2T21+(ωω0)2T22+μMD22|BMD|2T1T2|BMD|2B02.
Δω=2T22|BMD1|2|BMD0|21+μMD22|BMD1|2T1T2,
(σext)MD0=kNμ0μMD2T21+(ωω0)2T22+μMD22B02T1T2,
αE=αE(0)1ik3π3αE(0),αM=αM(0)1ik3π3αM(0),
αE(0)=6πk3tanα1,αM(0)=6πk3tanβ1,tanαn=m2jn(y)[xjn(x)]jn(x)[yjn(y)]m2jn(y)[xyn(x)]yn(x)[yjn(y)],tanβn=jn(y)[xjn(x)]jn(x)[yjn(y)]jn(y)[xyn(x)]yn(x)[yjn(y)],
(σext)Si=μ0kIm(mSiBSi*)B02+kIm(αE)=kIm(αM){[B0+Re(YmMD)]2+[Im(YmMD)]2}B02+kIm(αE).
(σext)MD1(kNT2μ0μMD2|BMD0|2)/(B02),
Δω(2/T2)2|BMD1|2/|BMD0|21.
(σext)MD2kNμ0/(B02T1).
Δω2μMD|BMD1|T1T2/(T2).
Me1n=1sinθsinϕPn1(cosθ)Zn(ρ)eθcosϕdPn1(cosθ)dθZn(ρ)eϕ,No1n=Zn(ρ)ρsinϕn(n+1)Pn1(cosθ)ersinϕdPn1(cosθ)dθ1ρddρ[ρZn(ρ)]eθ+cosϕPn1(cosθ)sinθ1ρddρ[ρZn(ρ)]eϕ,
an=m2jn(mx)[xjn(x)]jn(x)[yjn(y)]m2jn(mx)[xhn1(x)]hn1(x)[yjn(y)],bn=jn(mx)[xjn(x)]jn(x)[yjn(y)]jn(mx)[xhn1(x)]hn1(x)[yjn(y)],cn=jn(x)[xhn1(x)]hn1(x)[xjn(x)]jn(mx)[xhn1(x)]hn1(x)[yjn(y)],dn=mjn(x)mhn1(x)[xjn(x)]m2jn(mx)[xhn1(x)]hn1(x)[yjn(y)],
Hin=k1ωμ0n=1(2n+1)in+1E0cnjn(k1d)k1dPn1(0).
Hout=kωμ0n=1(2n+1)in+1E0bnhn1(kd)kdPn1(0).
X=Bscat(in)mSi=μ0αM[3c1j1(k1d)P11(0)kd1],(d<R),
X=Bscat(out)mSi=μ0αM3b1h11(kd)P11(0)kd.(d>R).
F=PradP0=|1+αMμ0Y|2.
F=PradP0=9|μ1ε0μ2ε12ρ12j12(y1)y12D12|,(d<R),
F=PradP0=9|j1(kd)b1h11(kd)kd|2.(d>R).
D1=μ1j1(ρ1)[ρ0h11(ρ0)]μ0h11(ρ0)[ρ1j1(ρ1)].

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