Simultaneously obtaining high photon emission rate and collection efficiency is highly desirable for applications of single photon sources. However, it remains great challenging and is seldom reported before. Here, we demonstrate that highly enhanced radiation of the emitter and efficient collection of the emitted photons can be simultaneously fulfilled in a hybrid photonic-plasmonic cavity which comprises of an Au nanorod dimer and a photonic crystal nanobeam cavity with a collecting waveguide, where the resonance wavelength of nanobeam cavity is red-detuned from that of the Au nanorod dimer. Our calculations show that the spontaneous emission rate of a single emitter can be enhanced by 5060 -folds, correspondingly, the far-field radiation efficiency and collection efficiency into a dielectric waveguide reaches ~97% and ~67%, respectively. The proposed mechanism paves the way towards the practical applications in ultra-bright on-chip single photon sources and plasmon-based nanolasers.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
Manipulation and efficient collection of single photon emission is not only important to fundamental science but also vital to many applications such as quantum cryptography  and quantum information processing [2–4]. Since the spontaneous emission rate can be greatly enhanced by placing quantum emitters in engineered electromagnetic environments , various artificial electromagnetic structures, such as optical microcavities [6–8] and plasmonic nanocavities [9–11], have been extensively explored to enhance spontaneous emission rate of the emitter. However, constrained by the diffraction limit, optical microcavities have relatively large mode volumes compared to the light wavelength [12–14]. In order to enhance the emission rate of the emitter, ultrahigh Q-factors for the optical microcavities are required to compensate for their relatively large mode volumes. Unfortunately, these ultrahigh-Q cavities require good spectral matching between a narrowband emitter and the narrowband cavity resonance, which is still a challenge for nanofabrication and an obstacle to the practical applications. On the other hand, localized surface plasmon resonance originating from the collective oscillation of the conductive electrons in metal can highly confine the electromagnetic fields into a nanoscale space, yielding a ultrasmall mode volume [15–17]. Although large enhancements in the total decay rate have been observed in these plasmonic structures, the nonradiative quenching is dominant owing to the inherent large ohmic losses [18,19]. In addition, in these cavities based techniques, the subsequent coupling of the emitted photons into a single-mode optical fiber or waveguide may reduce the actual collection efficiency. In the view of the ability to directly collect the emitted photons, optical fibers or waveguides would be particularly promising and have been extensively explored [20–23]. However, the maximum enhancement of emission rates in these photonic nanostructures can reach only several ten times.
To overcome these challenges mentioned above, various hybrid structures have been extensively investigated theoretically and experimentally [24–26]. Although the spontaneous emission rate of the emitter placed into these hybrid structures can be enhanced, the ohmic losses of the metal antenna are still large due to the fact that the optical cavity resonances with the metal antenna. For example, Zhang et al.  have demonstrated lasing in a plasmonic-photonic crystal coupled nanolaser, but the laser thresholds of hybrid structures are higher than that of the bare photonic crystal cavity without metal antenna due to the large ohmic losses. Furthermore, Doeleman et al.  have theoretically demonstrated that the spontaneous emission rate of the emitter can be enhanced in antenna-cavity hybrid system and the collection of the emitted photons into a waveguide can be efficient. However, they have not proposed a specific geometry for the collection of emission, which is necessary for on-chip optical circuits. In the metal nanostructure-nanowire hybrid systems [28–30], the nanowires can play a role for the collection of photons. However, the resonant absorption of the metal reduces the radiation and collection efficiency, which is inevitable in these hybrid systems. In addition, the metal nanowires are not suitable for the transmission of photons . Very recently, manipulation of quench has been demonstrated in nanoantenna-emitter hybrid systems via an external detuned cavities , which is significant for improving the collection of emission.
Here, we propose a hybrid photonic-plasmonic cavity consisting of an Au nanorod dimer and a photonic crystal nanobeam cavity with a collecting waveguide. This hybrid cavity simultaneously possesses high Q-factor and ultrasmall , which consequently results in a significantly enhanced spontaneous emission rate of the emitter placed into the gap of the Au nanorod dimer. More importantly, the emitted photons can be efficiently collected and guided by the collecting waveguide. In addition, the emission wavelengths of the emitter in these hybrid cavities are detuned far from the resonant wavelengths of the Au nanorod dimer. Therefore, the nonradiative quenching has been reduced significantly, which leads to improved radiation efficiency and collection efficiency. Our numerical calculations demonstrate that the far-field radiation efficiency and collection efficiency into a dielectric waveguide reach up to ~97% and ~67%, respectively. Simultaneously, the corresponding enhancement of the spontaneous emission rate still remains up to 5060 -folds. These results suggest that the hybrid photonic-plasmonic cavity proposed here is the best candidate for significantly enhancing the spontaneous emission rate of the emitter and simultaneously obtaining large radiation and collection efficiency.
2. Theory and methods
The photonic-plasmonic hybrid cavity proposed here is schematically shown in Fig. 1(a). An Au nanorod dimer is coupled with a Si photonic crystal nanobeam. In addition, a quantum emitter is inserted in the gap formed by the nanorods and the nanobeam, as shown in Fig. 1(b). The nanobeam cavity consists of two mirror sections, a tapering region and a collecting waveguide. Where, the two mirrors have asymmetric structure. Right mirror contains 17 air holes with a fixed radius and a fixed lattice constant . The left mirror contains few air holes with the same radius and lattice constant as the right mirror and the number N of air holes of the left mirror is tunable. The tapering region is made of 12 air holes, such that their radii were linearly decreased from to and the tapering region has the same lattice constant as the mirrors. The Si nanobeam has a width and a thickness , and lays on a silica semi-infinite substrate. The background dielectric is air. In the calculations, the refractive index of silica is set to be 1.45, the frequency-dependent refractive index of gold and silicon are obtained by interpolating the experimental data [32,33]. Note that when the wavelength is in the range of , the refractive index of silicon is set to be 3.45. All results are calculated by COMSOL Multiphysics except Fig. 1(e) by RSoft. In the simulations using the COMSOL Multiphysics, a cylindrical model with radius of and length of () is configured, and a perfectly matched layer with thickness of 800 nm is introduced to minimize boundary reflections.
Figure 1(e) presents the energy band structures of the nanobeam with perfect lattice structure placed on a semi-infinite silica substrate. The black lines represent the energy band structure for the nanobeam with , which includes a big bandgap. When the radii of the air holes increase up to , as the red lines show, the corresponding energy bands shift. Therefore, by tapering the air holes, a resonant condition with which corresponds to the communication wavelength, is created in the photonic crystal bandgap for the case of . The electric field distribution of eigenmode with in the bare nanobeam cavity is shown in Fig. 1(d). Obviously, the electromagnetic energy is mainly confined into the space of the nanobeam cavity. The electric field is polarized in the x-y plane with the component dominating, which can couple well with the waveguide mode.
The Au nanorod dimer is placed on the nanobeam and between the two central air holes as shown in Fig. 1(b). The cross section length of the nanorod is set to a fixed value of , the length of the nanorod and the gap distance of the dimer are tunable and strongly influence the performance of the Au nanorod dimer. In order to understand the features of the Au nanorod dimer, we calculated the extinction spectrum of the Au nanorod dimer on Si substrate with and . As shown in Fig. 1(f), the resonance wavelength of the bare Au nanorod dimer is , and the electric field of the eigenmode is significantly confined into the gap domain. The electric field in the dimer gap is mainly polarized along the dimer axis. Therefore, the Au nanorod dimer is placed on the nanobeam with dimer axis along the x-axis in the hybrid system. When a dipole emitter is placed in the dimer gap, it should polarize along the dimer axis.
When an Au nanorod dimer is placed on the nanobeam cavity, the individual modes of the two subsystems become strongly hybridized, generating new eigenmodes. With the coupling between the two subsystems, interference between different scattering paths can largely affect the total spontaneous emission rate of the emitter. Importantly, when the mode of the bare nanobeam cavity is red-detuned from the mode of the bare dimer, the spontaneous emission rate can be boosted, which can outperforms that in the subsystems . The spontaneous emission rate of a quantum dipole emitter placed at some position can be obtained by :35] , where is the projected local density of states in the hybrid cavity and is density of state in vacuum, , and are the transition dipole moment and its unit direction vector, respectively. Therefore, when the Purcell factor is determined, the spontaneous emission rate of the quantum emitter can be obtained by Eq. (1).
Large Purcell enhancements have recently been demonstrated in similar hybrid systems [36,37]. However, for practical applications, such as ultra-bright on-chip single photons sources, the collection of emission via a single-mode waveguide is necessary and vital beside large spontaneous emission rate. In our hybrid cavity, we propose the nanobeam cavity with a single-mode waveguide, where the mode in nanobeam cavity is compatible with the waveguide mode. Therefore, the photon emission can be coupled well into the waveguide and collected by the waveguide port (denoted by the red arrow in Fig. 1(a)), which results in large collection efficiency. The collection efficiency is defined as 
For the simulation, the efficiency and can be obtained by and , respectively , where is the total emitted power, which can be obtained by calculating the surface integration of the nanosphere enveloping the emitter over the energy flows in the hybrid system; is the far-field radiation energy of the emitter obtained by calculating the surface integration of the closed surface enveloping the whole hybrid cavity over the energy flows; is the energy collected by the waveguide from the port denoted by the red arrow in Fig. 1(a).
3. Results and discussion
Guided by the theories discussed above, we calculate the Purcell factor for the quantum emitter placed in the hybrid photonic-plasmonic cavity and in the two corresponding subsystems, where the parameters are set as: , and . In calculations, the quantum emitter is located in the center of the gap of the Au nanorod dimer in the hybrid cavity or bare Au nanorod dimer, and it’s dipole moment parallels to the Au nanorod as shown in Fig. 1(b). The distances from the emitter to the surface of the nanobeam and Si substrate are the same and set to be . For the bare nanobeam cavity, the emitter is placed between the central air holes and above the surface of the nanobeam. The dipole moment of the emitter parallels to the x-axis. Figure 2(a) shows the Purcell factor for the quantum emitter in the two subsystems. The blue line in Fig. 2(a) shows as a function of the emission wavelength for the quantum emitter in the bare Au nanorod dimer. This Au nanorod dimer has a single mode behavior over a wide range of wavelength and the peak appears at , consistent with the extinction peak, and the maximum reaches up to 652. The magenta and orange lines in the same figure present as a function of emission wavelength for the bare nanobeam. The bare nanobeam has two eigenmodes in the wavelength range of interest, the eigenmode with longer wavelength denoted by symbol a in Fig. 2(a) has a Q-factor of 1936 and a maximum of 249. The eigenmode with shorter wavelength denoted by symbol b in Fig. 2(a) has a Q-factor of 160 and a maximum of 15. In contrast, the hybrid cavity has different behavior from the two subsystems. Figure 2(b) presents of the hybrid eigenmodes. It can be seen that the new hybrid eigenmodes have obvious Fano line shape, manifesting themselves as a broaden resonance and a narrow resonance, respectively . The narrow resonance close to the cavity mode of the nanobeam slightly red-shifts and broadens due to the interaction between the dipole mode of the nanorod dimer and the nanobeam cavity mode [38,39]. As mentioned earlier, when the mode of the bare nanobeam cavity is red-detuned from the mode of the bare dimer, due to constructive interference between different scattering paths, the Fano peak close to the nanobeam cavity resonance can obtain large spontaneous emission enhancement. However, quite the opposite occurs when the mode of the bare nanobeam cavity blue-detuned from the mode of the bare dimer, as the case of the mode B in Fig. 2(b). The calculation results show that the hybrid eigenmode denoted by symbol A in Fig. 2(b) has a large reaching up to 5290, which outperforms that of the Au nanorod dimer at resonance by more than a factor of 8 and that of the nanobeam cavity by more than a factor of 21, respectively. In the remainder of this paper, only the mode A is considered.
The reason why the Fano-resonance mode A demonstrates much larger than those of the two bare subsystems is that it inherits the advantages of the two subsystems. In other words, the hybrid eigenmode A has a high Q-factor (having been discussed above) and an ultrasmall mode volume. To further understand the underlying mechanism to enhance the spontaneous emission rate, we can estimate the mode volume of the eigenmode A according to the definition written as:Eq. (4) is just an approximation, because the Eq. (4) is not strictly valid for the plasmonic system [40,41]. To validate this result, we also calculate of the eigenmode A using the standard Purcell factor calculation [41,42]. The effective mode volume can be written asFig. 2(b), we can obtain . Obviously, the results obtained via the two methods are consistent. Figure 3 shows the electric field distribution of the hybrid eigenmode A. It can be seen that the electric field is mainly confined in the gap of the Au nanorod dimer, which results in an ultrasmall effective mode volume . Importantly, the electromagnetic energy can effectively couple into the collecting waveguide, as can be seen from Fig. 3(c). The photons can be further guided by the collecting waveguide and used for other applications.
In order to obtain the optimal performance of the hybrid cavity, in the following sections, we would investigate the influence of the geometrical parameters on the performance of the hybrid cavity. Here, a dielectric spacer with lower refractive index is placed between the Au nanorod dimer and nanobeam cavity, as shown in Fig. 1(c). This dielectric spacer can improve the coupling between the Au nanorod dimer and the nanobeam cavity, leading to a better performance of the hybrid cavity. The physical origin is that the dielectric spacer can make the electric fields localized in the nanobeam cavity and the nanorod dimer redistribute, which improves an overlap between the two evanescent electric fields. The width of the spacer layer is set to be a fixed value . The parameter d is set to be and is set to be 1, and their influence on the performance of the hybrid cavity will be discussed later. Note that the emitter is placed in the gap of the nanorod dimer and and above the surface of the Si nanobeam for the case with and , respectively. Figure 4(a) shows the influence of the nanorod length and the thickness hs of the spacer layer on . As the black circle dot line shows, when the Au nanorod dimer is directly placed on the nanobeam, i.e., without spacer layer between the Au nanorod dimer and the nanobeam, the Purcell factor increases first and then decreases with the increase of . Similar phenomena can also be observed in the presence of a spacer layer ( and ). According to Eq. (5), the Purcell factor is proportional to . When the nanorod length increases, the resonance wavelength of the bare nanorod dimer red-shifts, which leads to the reduced wavelength detuning between the bare nanobeam cavity and the nanorod dimer. With the reduced wavelength detuning, the nanorod can more strongly confine the electric field, resulting in the reduced mode volume. On the other hand, with the reduced wavelength detuning, the Q-factor reduces due to the enhanced ohmic loss and scattering of the nanorod dimer. When the nanorod length is small, the Q-factor reduces slightly as increases, which results in the enhanced Purcell factor. However, when the nanorod length is large, the Q-factor degrades significantly, which causes the Purcell factor to decrease gradually . With a spacer layer was placed between the Au nanorod dimer and the nanobeam, can reach larger value than that without spacer layer, as can be seen from the magenta hexagon dot line and the cyan rhomboidal dot line.
Figures 4(b) and 4(c) show the influence of both the nanorod length and the thickness hs of the spacer layer on the radiation efficiency and collection efficiency , respectively. In our paper, all efficiencies are calculated at the wavelengths of maximum Purcell enhancement. As shown in Fig. 4(b), decrease with the increase of for the cases with and without spacer layer. The system with has larger than that without spacer layer and with for the same . Interestingly, when the length of the nanorod is short, specially, , the radiation efficiencies are very large and reach ~79%, ~97% and ~98% for the cases without spacer layer, with and , respectively. These large radiation efficiencies, which are much larger than those in many plasmonic nanostructures [18,43], arises from the reduced ohmic losses because that the emission wavelength of the emitter is detuned far from the resonant wavelength of the bare Au nanorod dimer. With the decrease of , the wavelength detuning between the bare nanorod dimer and the emission wavelength of the emitter increases, which results in enhanced radiation efficiency. The enhanced radiation efficiency can increase the photons that can potentially be collected by the collecting waveguide, which consequently results in enhanced collection efficiency. Figure 4(c) presents as functions of and hs. reduces as increases and has largest value for the hybrid cavity with for the same . It is worth to note that can obtain high value when the nanorod length is short, and the value of is approximately equal for the cases of and . In order to simultaneously obtain large collection efficiency and high spontaneous emission rate, selecting geometrical parameters of and is a reasonable consideration.
The gap distance of the Au nanorod dimer can also significantly influence the performance of the hybrid cavity. Here we set the geometrical parameters and , which are optimal parameters as discussed above. The parameter is set to be 1. The influence of on the performance of the hybrid cavity will be discussed later. As can be seen from Fig. 5(a), the Purcell factor significantly reduces with the increase of the gap distance . However, the radiation efficiency and collection efficiency change little as varies, as shown in Fig. 5(b). In range of interest, and can keep above 94% and 64%, respectively. Therefore, the nanorod dimer with small prepared experimentally is vital for enhancing the spontaneous emission rate and simultaneously maintaining large radiation and collection efficiency.
Figure 6 shows that how the air hole number of the left mirror influences the performance of the hybrid cavity. In calculations, the parameters of and are set to be and, respectively, which is optimal parameters as discussed above. The parameter is set to, which is a moderate and experimentally reasonable value. It can be seen from Fig. 6(a) that as increases, increases significantly for the hybrid cavity. The increase of arises from the improved Q-factor of the bare nanobeam cavity due to the increase of and the resultant improved Q-factor of the hybrid cavity. Figure 6(b) presents the radiation efficiency and collection efficiency as a function of. As can be seen, and reduce with the increase of. This is also attributed to the enhanced Q-factor, because that when the Q-factor increases, although the hybrid mode dissipates more rapidly, the energy guiding through the collecting waveguide becomes less efficient, consequently, a substantial portion of emitted photons is reabsorbed by the Au nanorod dimer. Therefore, for the practical applications, it should select an optimal value of so that it can compromise between the spontaneous emission rate and collection efficiency.
In conclusion, by proposing the hybrid photonic-plasmonic cavity composed of an Au nanorod dimer and a Si photonic crystal nanobeam cavity with a collecting waveguide, we have simultaneously obtained high enhancement of the spontaneous emission rate and large collection efficiency. Our numerical calculations show that the collection efficiency of the emitted photons into a dielectric waveguide and the far-field radiation efficiency reach up to ~67% and ~97%, and the corresponding enhancement of the spontaneous emission rate still remains up to 5060 -folds. This phenomenon arises from three facts: The first is that the hybrid photonic-plasmonic cavity proposed here simultaneously possesses the merits of the two subsystems: high Q-factor and ultrasmall mode volume, consequently leading to high . The second is that the collecting waveguide can couple well with the hybrid mode which results in improved . The third is that the nonradiative quenching is significantly reduced because the emission wavelength of the emitter is detuned far from the resonant wavelength of the Au nanorod dimer. The mechanism proposed here is promising for significantly enhancing the spontaneous emission rate of the emitter and simultaneously obtaining large collection efficiency and paves the way towards the applications of the ultra-bright on-chip single photon sources and the plasmon-based nanolasers.
Ministry of Science and Technology of China (Grant No. 2016YFA0301300), the National Natural Science Foundations of China (Grants No. 11334015, No. 11364001, No. 11504050 and No. 61675237).
We thank Dr. J. M. Liu in Guangdong University of Education, Dr. H. X. Jiang in South China Normal University and Dr. Y. C. Yu in Foshan University for fruitful discussions.
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