Single-photon sources with a high emission rate^[1,2] are crucial components of rapidly developing on-chip quantum devices. The spontaneous emission of an emitter can be increased by the optical cavity mode^[3–6], known as the Purcell effect^[7], and it is one of the basic principles to realize single-photon sources^[8]. If coupling with cavity mode, the enhancement of photons emitted from an emitter can be expressed as the Purcell factor^[9], that is, PF=γ/γ0 (γ0 is the spontaneous emission rate in vacuum). Both a large quality factor and a small mode volume^[10,11] are beneficial to enhance the Purcell factor, i.e., PF∝Q/V. To meet the above requirements, researchers have been exploring various micro-nanophotonic structures. Plasmonic nanoparticles can achieve spontaneous emission rates as high as 103γ0–104γ0^[12–14] due to the strong field localization. However, their inherent absorption ratio is large in the emission photons, which limits their application in single-photon sources to a degree. Hence, dielectric nanoparticles^[15–19] without absorption have been proposed to overcome the shortage. Their Mie resonances^[15,16] confine the electric or magnetic field inside and around the nanostructure, resulting in a small mode volume. Nevertheless, these Mie resonances bring great scattering loss^[20,21], i.e., lowering the quality factor, which results in the Purcell factor being ∼102^[22,23]. To address the issue of spectral broadening, high-Q whispering gallery microcavities (WGMs)^[24,25] with an ultra-localized electric field inside the cavity have attracted attention. Unfortunately, the Purcell factors of dielectric WGMs are only tens to hundreds^[26–28] due to the large mode volume.

To overcome these limitations, hybrid micro-nano structures provide possible solutions, which have been used in field enhancement^[29–31], strong coupling^[32], and nonlinearity^[33]. The hybrid structures formed by high-Q WGMs and small mode volume metal nanoparticles can realize higher emission enhancement with an adjustable linewidth than the bare cavity^[34,35]. However, the absorption part of the emission photons still occupies a high proportion^[34], which is not conducive to the practical application of single-photon sources. In addition, the ohmic loss of metal materials^[36] may weaken the performance of plasma-based devices. Based on the above limitations, we consider the construction of all-dielectric hybrid structures. The bare dielectric nanoantennas own a small mode volume, but their photons are scattered around, which are difficult to collect^[21]. In the following, to solve the problem of scattering, a natural photon collection channel is built in nanoantenna-microtoroid structures to guide the scattered photons into the high-Q WGMs. As a result, both suppressed scattering loss and small mode volume of the nanoantenna are obtained, respectively, which allows the achievement of narrow-linewidth Purcell enhancement without absorption.

In this Letter, we propose an all-dielectric hybrid structure, i.e., two microtoroids containing a nanoantenna [Fig. 1(a)]. By combining the advantages of high-Q microtoroids and ultra-small mode volume of nanoantennas, we theoretically achieve large Purcell enhancement with a narrow linewidth. The scattered photons can be guided into the high-Q microtoroids so that the linewidth of the hybrid structure is maintained at the order of hundreds of picometers. Compared to the linewidth of the bare nanoantenna, it can be suppressed by 3 orders in dielectric nanoantenna-microtoroid hybrid structures. Meanwhile, the ultra-small mode volume of the nanoantenna remains almost unchanged. As a result, we obtain a Purcell factor up to 1000–1700 when placing a quantum emitter at the gap of the nanoantenna. The spontaneous emission rate of the hybrid structure is an order of magnitude higher than that of the single dielectric nanoantenna and microtoroid. By suppressing scattering loss and maintaining ultra-small mode volume of the nanoantenna, this kind of Purcell enhancement provides a direction for low-loss systems to enhance the interaction between micro-nanoscale light and matter^[37,38]. The proposed mechanism holds promise for applications in on-chip single-photon sources^[39,40] and low-threshold nanolasers^[41–43].

The Purcell factor is related to the quality factor and the mode volume, that is, PF∝Q/V^[7]. The dielectric nanoantenna supports electric dipole resonance^[15], where the electric field is mainly localized at the gap and the ends of the nanoantenna [Fig. 1(c)]. However, the photons are scattered all around, resulting in a great linewidth broadening with several hundred nanometers. Hence, the Purcell factor of the nanoantenna is at the order of several hundred due to the lower quality factor. Then, we place a dielectric high-Q microtoroid at each end of the dielectric nanoantenna [Fig. 1(a)], which creates a natural photon collection channel to guide photons scattered from the nanoantenna into the microtoroids. This helps suppress scattering loss so that the hybrid structure can still maintain a high-quality factor. Therefore, we utilize the high-Q microtoroids along with the small mode volume nanoantenna to achieve large Purcell enhancement with a narrow linewidth.

The schematic diagram of the proposed system is shown in Fig. 1(a), where the dielectric nanoantenna is coupled to the microtoroids with a nanoscale gap D=20nm. Two WGMs are both microtoroids of r1=1000nm and r2=300nm. The dielectric constant of the microtoroids is ε1=3.13, which can be made by Al2O3 material^[44]. In Fig. 1(b), the whispering gallery TM mode can be described by two mode numbers: radial mode number q=1 and azimuthal mode number m=21. Its electric field distribution is primarily localized in the microtoroid, exhibiting a high-quality factor. A dielectric nanoantenna with radius R=20nm, length a=80nm, and gap d=5nm is inserted into two microtoroids. Its dielectric constant is ε2=12, whose material can be considered as GaP^[45]. An x-polarized quantum emitter is placed at the gap of the nanoantenna. The local field at both ends of the nanoantenna is coupled with the evanescent field of the microtoroids so that the photons originally scattered are bound in the microtoroids [Fig. 2(b)], which suppresses the linewidth of the Purcell factors. It needs to be noted that all the hybrid modes we have obtained are even-symmetric modes due to the placement of the emitter at the geometric center.

Three-dimensional finite element simulations are performed using the COMSOL multiphysics software to calculate the Purcell factors. The model is wrapped in a cylinder with a radius of 5 µm and a height of 3 µm. A perfect matching layer with a thickness of 400 nm is introduced to reduce the boundary reflection to simulate infinite free space. The Purcell factors can be obtained by PF=γ/γ0=W/W0, where W and W0 are the radiation powers of the emitter embedded in the optical cavity and vacuum, respectively. By enveloping the emitter in a small sphere and performing a surface integral of the Poynting vector on the sphere, the radiated power of the emitter is obtained: W=∬S·d∑, where S is the Poynting vector and Σ is the surface of a small sphere surrounding the emitter.

We now show the result of large Purcell enhancement in an all-dielectric nanoantenna-microtoroid structure. As we know, the Purcell factor of the emitter is related to its surrounding electric field^[46]. In our hybrid structure, at the gap of the nanoantenna, it shows a stronger field enhancement than that of other positions [Fig. 2(c)]. Hence, when the emitter is placed at the gap of the nanoantenna, the maximum Purcell factor γ/γ0 can reach 1700 at the azimuthal mode number m=20 [Fig. 2(a)]. The resonance wavelength blueshifts when the m increases. For m ranging from 17 to 22, the Purcell factors of the hybrid structure are all greater than 1000 due to the small mode volume and the larger Q in the hybrid structure. For m ranging from 13 to 16, the whispering gallery mode itself has a weaker confinement ability, which is insufficient to confine local scattered photons in the microtoroids. Hence, the Purcell factors decrease but are still larger than 500. A single dielectric nanoantenna achieves a maximum Purcell factor of only 180 when the emitter is located in the gap. For the bare whispering gallery mode at wavelengths of 500–800 nm [Fig. 2(a)], when the emitter is placed close to the outer surface, the maximum Purcell factor reaches 350. Therefore, we can achieve stronger Purcell enhancement in the hybrid structure than in an individual dielectric nanoantenna or dielectric microtoroids. Also, compared to the Purcell factors of 155 for a single GaAs microdisk^[26], 580 for a microtoroid^[27], and 47 for silicon nanoantenna arrays^[47], in the present work, we demonstrate a significant increase in the spontaneous emission rate. It is noteworthy that we do not calculate the conditions of m greater than 22 due to the limitation of computing resources. More importantly, the shift of the nanoantenna mode and the microtoroid mode will be smaller with the increase of m, which is not beneficial for improving Purcell enhancement. The influence of resonance shift on Purcell enhancement will be further discussed in Sec. 4.

We further demonstrate the origins of the Purcell enhancement in the hybrid structure in detail. Since the Purcell factor is related to quality factor Q and mode volume V^[7], we calculate the specific values of Q, V, and Q/V in the hybrid structure, the bare microtoroid, and the bare nanoantenna (Table 1). The quality factor can be obtained by Q = (center wavelength)/(full width at half-maxima), while the mode volume can be obtained by^[48]V=∫ε|E(r)|2dVmax[ε|E(r)|2],(1)where ε is the dielectric constant and E(r) is the electric field at the position r. The results indicate that the nanoantenna-microtoroid structure combines the advantages of the high Q of the microtoroid mode and small V of the nanoantenna mode. Furthermore, the results of Q/V in Table 1 are consistent with the analysis in the previous paragraph qualitatively. Therefore, we demonstrate large Purcell enhancement with a narrow linewidth in all-dielectric nanoantenna-microtoroid structures compared with the bare microtoroid and bare nanoantenna.

In the dielectric nanoantenna-microtoroid structure, the linewidth of the nanoantenna is reduced to the order of hundreds of picometers [inset in Fig. 2(a)]. The photons are difficult to collect and utilize in a single nanoantenna [Fig. 1(c)], which leads to the broadening of the spontaneous emission spectrum. In the hybrid structure, the microtoroid alters the electromagnetic environment around the nanoantenna, which creates a natural photon collection channel for the scattered photons. As shown in the energy flow line of the inset in Fig. 2(b), the scattered photons emitted can be guided into the microtoroids. Thus, we suppress the linewidth of the nanoantenna from hundreds of nanometers to hundreds of picometers, which is at the same order of the bare microtoroid^[28]. The spontaneous emission spectrum with a narrow linewidth is conducive to the realization of low-threshold nanolasers^[41–43].

When the nanoantenna and microtoroid are on- or off-resonance, more detailed results on the Purcell factors are explored in this section. This resonance can be controlled by adjusting the radius R of the dielectric nanoantenna. As the radius R increases, the maximum value of the Purcell factor decreases [Fig. 3]. The larger size of the nanoantenna leads to greater disruption of the whispering gallery mode, namely, the reduction in the quality factor of the hybrid structure. This will lead to an increase in the coupling efficiency between the nanoantenna mode and microtoroid mode, which is difficult to quantitatively calculate due to the absence of mode splitting and the inaccuracy of the resonance wavelength of the nanoantenna mode. The wavelength of the electric dipole redshifts when the radius R increases, resulting in on-resonance between the nanoantenna and the microtoroids. On resonance, there is a strong energy exchange between the nanoantenna and microtoroids, which subsequently disrupts the light confinement of the microtoroids due to the large scattering of the nanoantenna (insets in Fig. 3). As a result, the Purcell enhancement of the hybrid structure is significantly weakened [Fig. 3(c)]. To achieve significant Purcell enhancement, an appropriate resonance shift is beneficial. This is because the resonance shift in a certain range will result in the reduction of the destruction to the microtoroid mode, although accompanied by the sacrifice of the scattering cross-section.

From an experimental perspective, there will be some deviation in the polarization of the nanoantenna. Next, we discuss the influence of the rotation of the nanoantenna around the z axis by angle θ on the spontaneous emission spectrum of the hybrid structure. As shown in Fig. 4(a), the Purcell factor of the emitter polarized along the x axis decreases with the increase of θ. This is because the polarization overlapping between the quantum emitter and the electric dipole of the nanoantenna is reduced. When the two polarizations are consistent, that is, θ=0° [Fig. 2(a)], the largest Purcell enhancement is obtained. Although the rotation of the nanoantenna weakens the Purcell enhancement, the scattered photons can still be guided into the microtoroids through interaction with the evanescent field of microtoroids [Figs. 4(b)–4(d)]. When the whispering gallery mode of the hybrid structure remains well, the linewidth of the hybrid structure is almost unchanged.

Finally, we consider the possibility of experimental realization of large Purcell enhancement with a narrow linewidth in an all-dielectric nanoantenna-microtoroid structure. At present, both the microtoroid and nanoantenna can be fabricated by the nanotechnology. The whispering gallery mode in the microtoroid in the visible spectrum has already been fabricated by reflowing the microdisk using laser pulse heating^[27]. To make a nanoantenna, one could use techniques such as lithography and electron-beam-induced deposition^[45]. Single quantum emitters can be realized in various systems^[49], such as atoms, molecules, and quantum dots^[50]. Furthermore, the precise positioning technology for quantum dots enables accurate placement of quantum emitters in nanoscale gaps^[51]. By collecting photoluminescence spectra using a fiber taper coupled to the microtoroid, the Purcell factor can be obtained^[52]. Therefore, it is possible to experimentally demonstrate Purcell enhancement with a narrow linewidth in the near future.

In summary, we have demonstrated large Purcell enhancement with a narrow linewidth in all-dielectric nanoantenna-microtoroid structures. By combining the advantages of the high-quality factor of the microtoroid and an ultra-small mode volume of the nanoantenna, the spontaneous emission rate has been enhanced. At the same time, we built a natural photon collection channel in hybrid structures to guide the scattered photons from the nanoantenna into the microtoroids, resulting in the picometer order linewidth of the spontaneous spectrum. Compared to metal-based hybrid structures, our structure has no absorption, which provides a direction for low-loss systems to enhance the interaction between micro-nanoscale light and matter. The mechanism proposed here will provide a practical application for realizing high-emission-rate single-photon sources^[1,2] and low-threshold nanolasers^[37–39].