Quantum emitters coupled to circular nanoantennas for high-brightness quantum light sources

Single quantum emitters are at the heart of various quantum applications such as quantum computation, encryption, simulations, communications, metrology among many others. [1, 2] As candidates for such applications, many quantum emitters have been studied [3], including trapped atoms [4], single molecules [5], impurity centres (e.g silicon vacancies (SiV) in diamond [6]), nanocrystal colloidal quantum dots (NQDs) [7] and self-assembled quantum dots [8].

Two limitations face free-standing quantum emitters, however. The first lies in their isotropic angular emission pattern, which limits the ability to efficiently collect their emitted photons. The second is that typical solid-state quantum emitters can have low emission rates that will limit the speed of future technologies. In the past decade, there have been considerable efforts to solve these deficiencies by modifying the photonic environment near the quantum emitter. [9, 10] To achieve this, emitters were embedded in or near to various nanostructures. For simplicity, in this work we will refer to any nanostructure that modifies the emission rate or directionality of a quantum emitter as an antenna.

One approach for improving the directionality and emission rate of quantum emitters is the use of metallic antennas. These include metal nanoparticles (MNPs) [9], plasmonic patch antennas [11–13], metallic nanoslit arrays [14], Yagi-Uda nanoantennas [15, 16] and circular bullseye plasmonic nanoantennas [17, 18]. The advantage of using such structures is that plasmonic modes have low mode volumes accompanied with low-quality factors enabling spontaneous emission lifetime shortening and emission redirection over broad spectral ranges, which is beneficial for room-temperature broadband sources such as colloidal quantum dots and colour centres in diamond [19]. On the other hand, for efficient coupling, the emitter has to be placed in the vicinity of the plasmonic structure which will increase non-radiative recombination rates and if placed too close will cause quenching of the emission all together [9].

Another approach is to use pure dielectric antennas such as microcavities [8] and photonic crystals [20–22] that feature high directionality, high radiative enhancement factors and low-loss [23, 24]. Despite these advantages, however, dielectric antennas usually come with a limiting narrow frequency bandwidth that is usually unsuitable for room-temperature quantum emitters [25], and are much more complex to fabricate.

Therefore, a hybrid metal-dielectric nanoantenna that combines the advantages of metallic and dielectric antennas but with much less of their drawbacks was developed. In such a design, the emitter can be placed at a large distance from the metal and still produce high directionality in a broad spectral range [26, 27]. Our recent experiments demonstrated that the single photons emitted from a single nanocrystal quantum dot (NQD) positioned in such a hybrid circular ('bullseye') antenna can be collected with an efficiency of around $37 \%$ into a moderate numerical aperture (NA) of 0.65. Furthermore, the ability of positioning a single NQD at the hotspot of the antenna was developed which enables fabrication of highly directional room-temperature single-photon sources [27, 28].

While this points in a promising direction, the system should be further optimised to enhance both the collection efficiency and decay rate of a quantum emitter. This optimisation is crucial for applications such as quantum key distribution, single-shot readouts of nitrogen vacancy (NV) centres, sensing applications and random number generation [3, 29–31], some of which will be discussed further below. It is therefore pivotal to develop an antenna that combines both high collection efficiencies and high decay rate enhancement. Another drawback of the previous design was the plasmonic emission due to the metal corrugation near the quantum emitter which reduced the single-photon purity of the source. [27] Consequently, it would be desirable for single-photon sources at least, that the central area lying within the focus of the exciting laser be flat.

In this work, we introduce a new hybrid circular antenna where the emitter is positioned in proximity to a central metal cone at the centre of the bullseye antenna, and use a finite-difference time-domain (FDTD) method [32, 33] to optimise the brightness of the emitter into low NAs, with the goal to have a high-brightness broadband source of single photons directly coupled to a fibre, without the need of any additional collimating optics. This is achieved by simultaneously optimising the emission directionality and the enhancement of the emission rate of a dipole emitter. The particle swarm algorithm [34] was used for the optimisation process.

In section 2 we define relevant quantities to be optimised, such as the collection efficiency and the enhancement factors of the emitters brightness. In section 3 and 4 we will introduce two optimised designs for a bullseye antenna without and with a central enhancing element, respectively.

The decay rate of a dipole emitter Γ in an inhomogeneous environment will depend on the orientation of the dipole moment with respect to the polarisation of the electromagnetic modes. Thus, in general, the response of an antenna will vary for different dipole orientations and therefore we will label all subsequent quantities with an index $i=x,y,z$ that represents the dependence on the dipole orientation with respect to some coordinate system. For an emitter in free space, the local density of states is isotropic and the decay rate reduces to ${{\rm{\Gamma }}}_{0i}$. This intrinsic decay rate in general includes both radiative (${{\rm{\Gamma }}}_{0i}^{r}$) and non-radiative (${{\rm{\Gamma }}}_{0i}^{{nr}}$) decay rates. Intrinsic non-radiative loss mechanisms vary for different emitters and include for example phonon-scattering [35], Auger recombination of multiexcitons in colloidal quantum dots, [36] and losses to lattice defects in colour centres [6].

In a modified environment, the decay rate (${{\rm{\Gamma }}}_{i}$) changes and the Purcell factor is defined as [37]

$$\begin{eqnarray}&&{F}_{i}=\displaystyle \frac{{{\rm{\Gamma }}}_{i}}{{{\rm{\Gamma }}}_{0i}}.\end{eqnarray} \tag{ 1 }$$

where ${F}_{i}=F({\mu }_{i})$ and ${{\rm{\Gamma }}}_{i}={\rm{\Gamma }}({\mu }_{i})$ and so on represents the dependence of these quantities on the dipole orientation as discussed above. This factor determines the reduction (${F}_{i}\gt 1$) or increase (${F}_{i}\lt 1$ ) of the intrinsic lifetime of the emitter due to the modified electromagnetic environment, and in general does not equal the enhancement of the emitter's emission into the far field since part of the emission can be channelled into non-radiative losses in the antenna (such as Joule losses in metals). Thus, we define a radiative enhancement factor ${F}_{i}^{r}$ as the net increase of total photon flux while the non-radiative enhancement factor ${F}_{i}^{{nr}}$ depicts the leakage of emission into lossy modes of the antenna (therefore modifying the non-radiative decay rate). Note that the antenna in most cases does not affect the intrinsic non-radiative decay rate, thus we can write the total decay rate as

$$\begin{eqnarray}&&{{\rm{\Gamma }}}_{i}={F}_{i}{{\rm{\Gamma }}}_{0i}=\mathop{\underbrace{{F}_{i}^{r}{{\rm{\Gamma }}}_{0i}^{r}}}\limits_{{{\rm{\Gamma }}}_{i}^{r}}+\mathop{\underbrace{{F}_{i}^{{nr}}{{\rm{\Gamma }}}_{0i}^{r}+{{\rm{\Gamma }}}_{0i}^{{nr}}}}\limits_{{{\rm{\Gamma }}}_{i}^{{nr}}}\end{eqnarray} \tag{ 2 }$$

In free space, the intrinsic quantum yield ${{QY}}_{0i}={{\rm{\Gamma }}}_{0i}^{r}/{{\rm{\Gamma }}}_{0i}$ defines the probability that the photon is emitted into the far field. The presence of an antenna in general will modify the radiative and non-radiative decay rates and therefore the overall quantum yield of an antenna-emitter system ${{QY}}_{i}={{\rm{\Gamma }}}_{i}^{r}/{{\rm{\Gamma }}}_{i}$ is altered and can be written as

$$\begin{eqnarray}&&{{QY}}_{i}=\displaystyle \frac{{F}_{i}^{r}}{{F}_{i}}{{QY}}_{0i}\end{eqnarray} \tag{ 3 }$$

The effect of the antenna on the quantum yield depends on the intrinsic features of the dipole (${{\rm{\Gamma }}}_{0i}^{r},{{\rm{\Gamma }}}_{0i}^{{nr}}$) and antenna (${F}_{i}^{r},{F}_{i}^{{nr}}$) in a coupled fashion because F_i is defined as in equation (2). In this work, we will be considering a dipole emitter with unity intrinsic quantum yield and therefore F_i reduces to ${F}_{i}^{r}+{F}_{i}^{{nr}}$ and we can define an antenna quantum efficiency ${AE}={F}_{i}^{r}/{F}_{i}$ which is independent of the properties of the dipole emitter.

Another interesting property of the emitter-antenna system is the far-field brightness. An emitter with an intrinsic radiative decay rate (${{\rm{\Gamma }}}_{0i}^{r}$) will have a total far-field brightness denoted by ${{\rm{\Phi }}}_{0i}^{{tot}}$ (tot refers to integration over all angles), which is dependent on the excitation power. We define the brightness of the emitting system as the flux that is emitted into a certain collection cone of given NA, given by

$$\begin{eqnarray}&&{{\rm{\Phi }}}_{i}({\theta }_{{NA}})={{\rm{\Phi }}}_{0i}^{{tot}}{F}_{i}^{r}{\eta }_{i}({\theta }_{{NA}})\end{eqnarray} \tag{ 4 }$$

where ${\eta }_{i}({\theta }_{{NA}})$ is the collection efficiency which quantifies the directionality of the emitting system and can be written in terms of the flux per unit solid angle or intensity ${I}_{i}(\theta ,\phi )$ as

$$\begin{eqnarray}&&{\eta }_{i}({\theta }_{{NA}})=\displaystyle \frac{{\int }_{0}^{2\pi }d\phi {\int }_{0}^{{\theta }_{{NA}}}d\theta \ {I}_{i}(\theta ,\phi )}{{\int }_{0}^{2\pi }d\phi {\int }_{0}^{\pi }d\theta \ {I}_{i}(\theta ,\phi )}\end{eqnarray} \tag{ 5 }$$

It is also useful to define the brightness enhancement, ${\xi }_{i}({\theta }_{{NA}})$, as the ratio of the brightness of a dipole emitter into a certain NA to the brightness of a free dipole emitter into the same NA, i.e.,

$$\begin{eqnarray}&&{\xi }_{i}({\theta }_{{NA}})=\displaystyle \frac{{{\rm{\Phi }}}_{i}({\theta }_{{NA}})}{{{\rm{\Phi }}}_{0i}({\theta }_{{NA}})}={F}_{i}^{r}\displaystyle \frac{{\eta }_{i}({\theta }_{{NA}})}{{\eta }_{0i}({\theta }_{{NA}})}\end{eqnarray} \tag{ 6 }$$

where ${\eta }_{0i}({\theta }_{{NA}})$ refers to the collection efficiency of a free dipole.

The coupling of an emitter to an antenna has a few important consequences as shown in equations (3) and (4). First, the quantum efficiency of the emitter, or more generally, the quantum efficiency of certain emission processes of the quantum emitter, will be modified. For example, it has been shown that coupling colloidal quantum dots with metal nanoparticles will increase the biexciton quantum efficiency [38]. This could be detrimental or beneficial depending on the application (e.g., single-photon sources versus bi-photon sources). The other two consequences are the modification of the emission pattern and decay rate which are the main focus of this paper.

The structure studied in this section is based on our previous work in [27, 28], as is shown schematically in figures 1(a) and (c). It consists of a bullseye circular grating with 15 gratings, period Λ and a central cavity of radius R embedded in a dielectric layer of thickness h (realised by a polymer in [27, 28]). The emitter is considered as an oscillating dipole at the centre of the circular grating and at height d from the metallic slab.

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The large central cavity in the current antenna is introduced since we previously found that the presence of metallic corrugation near the emitter leads to efficient excitation of surface plasmons on the metal-dielectric interface which then results in a noticeable broadband random emission of photons, resulting not from the quantum emitter itself, but rather from these surface plasmons. This excess random emission degraded the performance of a single-photon device [27].

The operation mechanism of this system is as follows. When a quantum emitter (having a spectral band around the central emission wavelength ${\lambda }_{0}$) at the centre of the bullseye structure is excited it emits photons preferentially into the dielectric layer due to the higher index of refraction. The dielectric layer is designed to have modes that support this band of wavelengths (see supplementary information A available online at stacks.iop.org/QST/2/034004/mmedia) with corresponding propagation constants $\beta (\lambda )$. As the photons propagate in the radial direction, they encounter a diffraction grating with period Λ that is designed to match the propagation constant ($\beta \approx 2\pi /{\rm{\Lambda }}$), which diffracts the light. The diffracted photons that hit the air-dielectric surface with angles less than the critical angle couple out while all other photons continue to the next grating. Eventually, nearly all the light is diffracted into free space modes and the interference between the various diffracted beams leads to a low divergence beam in the far field.

In addition to the geometric parameters of the antenna, the dipole orientation has a profound role in the performance of the device. Due to the azimuthal symmetry of the antenna, we only need to consider two independent dipole orientations: the in-plane dipole (P_x), which will be represented by a dipole along the x axis (see figure 1(a)), and the out of plane dipole (P_z ). Since the P_z dipole has a polarisation that is predominantly in the z direction, it will mostly couple to the z component of the ${{TM}}_{0}$ electric field for the metal-dielectric-air (MDA) slab waveguide (see supplementary information A). The electric field of this component is maximum near the metal surface, thus coupling into this mode requires either thicker dielectric layers which introduces higher-order modes, or placing the dipole closer to the metal where diffraction is limited to the first few gratings, thus lowering the redirection effect caused by constructive interference. On the other hand, the P_x dipole couples into the ${{TE}}_{0}$ and the x-component of the ${{TM}}_{0}$ mode, which both have field maxima near the centre of the dielectric layer. This leads to better constructive interference into lower angles, due to the diffraction caused by multiple circular grooves. It is clear from this discussion and from our simulations that the emission from the P_x dipole will have a better coupling with the antenna and thus better directionality. Moreover, due to azimuthal symmetry, an unpolarised source will have a probability of 2/3 to emit from an in-plane dipole (see supplementary information B). For these reasons, we have chosen to optimise the structure considered in this section for the P_x dipole.

The performance of the device will be contingent on the geometrical parameters of the nanoantenna (see figure 1(c)) and the dipole orientation. Hence, an optimisation for these parameters for the P_x dipole was conducted using a particle swarm algorithm [34] in a 3D FDTD Lumerical^® simulation [33]. The figure of merit for this optimisation procedure was the system's brightness, i.e., its total flux, into an ${NA}=0.22$ (${{\rm{\Phi }}}_{x}(0.22)$) in equation (4), corresponding to a typical NA of multi-mode fibre. The total flux was chosen rather than the collection efficiency to incorporate the radiative enhancement factor. The parameters are all normalised to the central wavelength of the source (${\lambda }_{0}$) since the optimisation shows that the structure scales well with wavelength in the spectral range between 600 and 800 nm (see supplementary information C). The dipole emitter was chosen to have a central wavelength of ${\lambda }_{0}$ = 650 nm. The optical parameters of the dielectric layer was chosen to match that of PMMA (Polymethyl methacrylate), which is optically transparent in the desired spectral range and is used for standard fabrication procedures [28].

The optimised structure described in this section has a normalised central radius $R/{\lambda }_{0}=0.97$, a normalised grating period ${\rm{\Lambda }}/{\lambda }_{0}=0.86$, a normalised dielectric layer thickness $h/{\lambda }_{0}=0.41$ and a normalised dipole height $d/{\lambda }_{0}=0.29$. The normalised ring width and height were set to 0.25 and 0.15, respectively.

From the simulations, we can visualise both the near-field and far-field distributions. Figure 2(a) displays the simulated electric field inside the optimised structure in the plane normal to the dipole orientation (yz plane). In this plane, the emission couples into the ${{TE}}_{0}$ mode of the waveguide and we can observe that the standing waveguide modes have effective wavelengths that match the periodicity of the gratings. In the air layer above, the fields propagating out of the structure are visible. These fields have phases resulting in constructive interference in the far field into a narrow cone about $\theta =0$. The far-field spectral response of the antenna for wavelengths around the resonance wavelength ($\lambda =650$ nm) is displayed in figure 2(b). This far-field spectral intensity distribution ($I(\theta ,\lambda )$) is plotted for the yz plane ($\phi =0$). Angular cross sections for the resonance wavelength ($\lambda =650$ nm) and two detuned wavelengths ($\lambda =600$ and 700 nm) are plotted in figure 2(e). From these figures, we can clearly see the dispersion resulting from the circular grating as the first orders (±1) combine around $\lambda =650\,\mathrm{nm}$ to form a narrow directional beam. To better quantify the directionality of the emission, the collection efficiency (defined in equation (5)) for the same three wavelengths is shown in figure 2(d). In figure 2(c), the collection efficiency as a function of wavelength is displayed for three collection NAs (0.12,0.22 and 0.50) corresponding to the NA of a single-mode fibre and two commercially available multi-mode fibres, respectively. From this figure and the overlap of the modes in figure 2(b), it is evident that the antenna's response is broadband ($\gt 20$ nm), which is beneficial for coupling to a room-temperature emitter. It is clear that detuning the emitter's wavelength from the designed wavelength will result in a lower collection efficiency at low NA corresponding to the formation of sidebands as seen in figure 2(e).

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Next, we turn our attention to evaluating the performance of the device in general and in particular to the dependence on dipole orientation. Due to the low-loss of the device we will be interested in the collection efficiency (equation (5)) and brightness enhancement (equation (6)). In this convention, the radiative enhancement factor ${F}_{i}^{r}$ is just ${\xi }_{i}(\pi )/2$. The collection efficiency and the brightness enhancement for a dipole oriented along the x axis (P_x) and for an unpolarised dipole (UnP) are displayed in figures 3(a) and (b), respectively. These are compared with the corresponding values for free dipoles (dashed curves). The brightness for the unpolarised emitter is calculated from the two polarised ones, P_x and P_z (not shown here) as discussed in the supplementary information B. Using this brightness, one can directly calculate the collection efficiency.

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One very promising result is that around $74 \%$ of the emission of both the P_x and unpolarised emitters is collected already into a low, multi-mode fibre compatible numerical aperture of ${NA}=0.22$, and an even higher collection efficiency of $82 \%$ is found for ${NA}=0.5$, corresponding to a larger NA commercially available multi-mode fibre. This high coupling efficiencies into such low NAs should enable direct coupling of the emission into a commercial multi-mode fibre. Moreover, as a result the confinement introduced by the antenna, there is a net radiative enhancement of 2.4 and 1.7 for the P_x and unpolarised emitters, respectively, which together with the improved collection, results in a brightness enhancement (figure 3(b)) of 11 (6) into the corresponding NAs compared with that of a free-standing dipole. This means a large increase in the usable photon rate from the source. It is visible that the brightness enhancement is especially high for low NAs, which is understandable due to the poor collection efficiency from the free-standing dipoles at low collection angles. It is worth noting that the brightness enhancement of the P_x and unpolarised dipoles overlap completely for nearly all the collection angles.

Another important point is the near identical collection efficiency curve of the P_x and unpolarised dipoles. This is due to the enhancement of the emission of P_x dipole and suppression of the P_z dipole so that the unpolarised dipole emission will result primarily from an in-plane dipole, especially for low collection angles. This has the profound effect of transforming an emitter that originally emitted from all dipole orientations to an emitter that primarily emits from a dipole in a plane perpendicular to the nanoantenna. There is, however, a probability of 1/3 to emit from a P_z dipole during each excitation cycle and the longer lifetime of this transition will result in a less time-deterministic source. Therefore, it is advantageous to use a source that is originally polarised and whose dipole can be oriented in the plane of the nanoantenna. Such a source will enable the full use of all the benefits of this design while maintaining the deterministic nature of the source. One interesting candidate is the colloidal quantum rod reported by Sitt et al [39]. In these quantum rods, the symmetry is broken by using elongated shells (dot in rod) or elongated cores and shells (rod in rod) and the preferred transition has a dipole moment along the long axis of the quantum rod yielding degrees of polarisation reaching $83 \%$. [40, 41]. When spin-coated, the rods align in-plane making them ideal for implementation in this new antenna design. Recently, there has also been some significant progress in controlling the orientation of the emission dipole in NV centres in diamond during chemical vapour deposition growth [42], which can also make them suitable for the proposed scheme.

Finally, we address the effect of varying the thickness of the waveguide layer. The angular intensity distribution and the collection efficiency for three thicknesses, namely $h/{\lambda }_{0}=0.3,0.41,$ and 0.5 are shown in figure 3(c) and d, respectively. Slightly changing the waveguide thickness will change the propagation constant of the fundamental ${{TE}}_{0}$ and ${{TM}}_{0}$ modes. Increasing the thickness further will result in the contribution of higher-order modes (see supplementary information A) The former is the cause for the ranges of thicknesses in figures 3(c) and (d). The mismatch between the propagation constant and the grating Bragg wavevector leads to the formation of sidebands. The effect of the other geometrical parameters are discussed in the supplementary information section C.

One particularly interesting implementation (and perhaps test) of such designs is for single-shot readout of spin states in nitrogen vacancy centres in diamond. It was theoretically established that moderate Purcell factors might lead to a significantly enhanced SNR. [30]. As shown above, such moderate Purcell factors can be realised. In fact, it is expected that for diamond waveguides the Purcell factor will be higher than reported above due to the higher index of refraction which leads to tighter confinement. This is confirmed by initial calculations (not shown here) where radiative enhancement factors as high as eight where achieved. Furthermore, the SNR of the readout of the spin state also benefits from efficient collection of the emitted photons, which is an added advantage of this design.

It should be noted that the fabrication and analysis of bullseye structures around single NV centres has recently been reported with purely dielectric bullseye structure [43]. The predicted collection efficiency, however, was only $13 \%$ into an ${NA}=0.7$. Other advances in the fabrication of photonic and plasmonic structures in diamond have been recently reported such as plasmonic apertures [44] and periodic nanocrystal arrays [45]. Silicon vacancies also provide for an interesting application, particularly due to their near lifetime limited linewidths [46, 47]. Therefore, implementing this design will open up an opportunity towards a highly efficient source of indistinguishable single photons necessary for quantum logic gates [31].

As discussed before, in addition to high collection efficiency, it is desirable to enhance the decay rate of a quantum emitter. The design of the previous section had a rather small increase of the intrinsic radiative rate of the emitter, due to the poor confinement of the optical mode around the dipole, and the low LDOS of the essentially dielectric waveguide mode. To overcome this intrinsic limitation, we have considered introducing a plasmonic structure at the centre of the bullseye nanoantenna. An interesting candidate is the silver or gold nanocone. These nanocones have a longitudinal mode along the cone's axis and a transverse mode along the base [48]. The longitudinal mode in particular offers high field enhancement due the lightning rod effect near the rod's tip [49]. Recent theoretical studies on silver nanocones showed that the plasmonic resonance can be tuned across the optical spectrum by changing the apex angle of the cone [50–52]. Furthermore, the study demonstrated that the largest quantum yields occur for larger nanocones of heights around 160 nm and angles of $\pi /6$ [50]. A similar theoretical study on gold nanocones also displayed the shift of the plasmonic resonance with tip angle at fixed cone heights [53]. A detailed study of the electric field enhancement of these nanocones and their dependence on the cone parameters can be found in, e.g., [51, 52, 54, 55].

The difficulty of using nanocones for emission rate enhancement presides in the fact that the emitter must be placed with sub 10 nm precision near the tip of the nanocone. Furthermore, the vertical separation between the emitter and the tip needs to be of the order of a few nanometres due to the high localisation of the field. Recently, both problems were addressed by chemically binding NQDs preferentially to the tip of gold nanocones [56]. The molecular binding and the shell of the NQD provided a vertical separation of around 4 nm [56].

As we are interested in exciting the longitudinal mode of the nanocone, a dipole oriented along the z-axis is considered in this section. First, we consider a nanocone of height h_c and apex angle α on a metal slab embedded in a dielectric (PMMA) layer without the presence of the bullseye rings (see figure 4(a)). We vary the cone apex angle (α) and dipole (${\lambda }_{0}=650$ nm) location in the vertical direction while keeping the PMMA thickness (h = 620 nm) and cone height (${h}_{c}=170\,\mathrm{nm}$ ) fixed as in the optimised structure discussed later (figures 4(b)–(d). It can be seen in figure 4(b) that locating the dipole within a few nanometres from the cone tip is essential in achieving high enhancement factors. The dipole height is fixed at 4 nm due to fabrication limitations in subsequent calculations. Figure 4(c) displays the redshift of the longitudinal plasmonic resonance as the apex angle is reduced as discussed above. An important aspect is that the enhancements are broadband with FWHM ∼80 nm which is essential for broadband room-temperature emitters and fits well with the the broadband emission redirection offered by the bullseye nanoantenna. Furthermore, we find that the antenna quantum efficiency is best for larger nanocones, as has been reported previously (figure 4(d)) [50, 53].

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Next, we place the nanocone at the centre of the bullseye nanoantenna. A close-up view of the structure is shown schematically in figure 5(a) with a zoom-in display of the nanocone in the inset. The different dipole orientation and effective dipole height force a new optimisation for both collection efficiency and brightness enhancement factor. Thus, we use the same figure of merit as in section 3 which is the far-field brightness for an NA of 0.22 (i.e. ${{\rm{\Phi }}}_{z}(0.22)$). The optimisation is carried out for a dipole emitter at central wavelength ${\lambda }_{0}=650\,\mathrm{nm}$ and which is oriented along the cone axis (P_z). The optimisation clearly pushes towards thicker polymer layers and lower groove heights due to the preferential coupling of the nanocone-emitter sub-system into the TM modes of the waveguide layer as discussed in section 3. The parameters for the bullseye nanoantenna are the following: normalised central radius $R/{\lambda }_{0}=0.99;$ normalised grating period ${\rm{\Lambda }}/{\lambda }_{0}=0.86;$ normalised polymer thickness $h/{\lambda }_{0}=0.95;$ and normalised groove heights and widths equal to 0.167 and 0.500, respectively. A nanocone of height ${h}_{c}=170\,\mathrm{nm}$ and apex angle $\alpha =50^\circ$ is placed at the centre of the bullseye structure and the dipole emitter is located 4 nm above the tip of the nanocone. The far-field angular intensity distribution is displayed in figure 5(b), featuring a broad resonance and narrow angular emission pattern. At resonance ($\lambda =650$ nm), the angular pattern has a prominent angular ring at an angle of $\sim 4.5^\circ$. As shown in figure 5(c), this pattern results in a collection efficiency into an ${NA}=0.22$ (0.5) of around $61 \%$ ($78 \%$), respectively, for both the P_z and unpolarised dipole orientations. Due to high enhancement of one dipole orientation over the other, the flux of an unpolarised emitter acquires the dependency of the P_z dipole with an overall flux of about 1/3 that of the P_z dipole. Thus, the collection efficiency of the unpolarised and P_z dipoles are essentially identical.

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A brightness enhancement factor into these NAs are ∼1000 (∼600) for an unpolarised emitter (figure 5(d)). The extremely high-brightness enhancement at low NA for the P_z dipole is due to the extremely poor collection efficiency of the corresponding free dipole (see figure 5(c)). Moreover, the overall radiative enhancement factor is ∼240 with an antenna efficiency of around $31 \%$.

The implementation of this design will vary depending on the particular emitter in question. A particularly interesting application would be for NQDs, as the ability to bond these nanocrystals to the tips of the nanocone has been developed [48, 56]. In particular, the biexciton quantum efficiency will be increased due to the low intrinsic quantum efficiency of the biexciton emission (equation (3)), [38], giving rise to a potential bi-photon source.

For sources where the single-photon purity is retained even when the decay rate is enhanced such as colour centres in diamond, this design will permit the fabrication of a bright and efficient single-photon source. Such a source will improve quantum key distribution systems, which currently use faint laser pulses with an average photon number of $\langle n\rangle \approx 0.5$ per pulse. An ideal single-photon source will have $\langle n\rangle \approx 1$, which requires both a high single-photon purity and a high collection efficiency ($\gt 0.5$) [31]. Furthermore, the emission rate must be in the GHz regime to compete with the current technology which requires high radiative enhancement factors. Because SiV centres already have intrinsic lifetimes on the order of ∼1 ns and display good single-photon purities [47], it is not a long shot to satisfy these requirements using the composite bullseye-nanocone antenna. Small nanodiamonds containing SiV centres may be placed in the proximity of the nanocone by a scanning probe tip using a pick and place technique [57].

Our new design and FDTD calculations suggest an optimised bullseye nanoantenna with over $82 \%$ collection efficiency into an NA of a multi-mode fibre and a corresponding source brightness that is more than 10 fold that of a dipole of free space. We have also investigated a composite structure that combines both high collection efficiency ($\eta ({NA}=0.22)=0.61$) and high radiative enhancement (${{\rm{\Gamma }}}_{z}^{r}=240$) using a silver nanocone at the centre of a bullseye nanoantenna.

These results are especially promising for the production of a high-rate, room-temperature on-chip single-photon sources that can be directly coupled to optical fibres without any additional optics. This structure can be also used for example along with colour centres in diamond (e.g. SiV and NV centres) enabling applications spanning various fields including quantum communication and magnometric sensing [42]. It is important to emphasise that these structures are not limited to any specific quantum emitter and can be applied universally, limited only by challenges in fabrication.

This work is supported in parts by The Einstein Foundation Berlin; The US Department of Energy: Office of Basic Energy Sciences, Division of Materials Sciences and Engineering; The European Cooperation in Science and Technology through COST Action MP1302 Nanospectroscopy; The Ministry of Science and Technology, Israel.