# Corner Reflector Plasmonic Nanoantennas for Enhanced Single-Photon Emission
## 2 Materials and Methods
The emission yield is characterized by the quantum efficiency \(\eta_{\text{Q}}\) of the process: the ratio between the rate of radiative transitions and the total transition rate, including radiative \(\gamma_{r}\) and non-radiative \(\gamma_{nr}\) transitions, and the rate of photons lost in the metals \(\gamma_{l}\)[21],  
\[\eta_{Q}=\frac{\gamma_{r}}{\gamma_{r}+\gamma_{nr}+\gamma_{l}}. \tag{3}\]  
If we assume a highly efficient atom-like emitter, with negligible non-radiative transitions \(\gamma_{nr}<<\gamma_{r}\), the quantum efficiency can be approximated as  
\[\eta_{Q}\simeq\frac{\gamma_{r}}{\gamma_{r}+\gamma_{l}}=\frac{f_{R}}{f_{R}+f_{ L}}, \tag{4}\]  
where we have divided both numerator and denominator of the above expression by \(\gamma_{r,0}\). If we substitute (1) and (2) into (4), the resulting expression coincides with the classical definition of the radiation efficiency of an antenna [15; 16]:  
\[\eta=\frac{p_{rad}}{P_{rad}+P_{loss}}. \tag{5}\]  
The evaluation of the emission properties for the NAs in this study is based on the calculation of \(f_{R}\) and \(f_{L}\) from the FDTD results to gauge the photon emission rate. Both gold and silver plasmonic NA were considered, with their respective optical constants obtained from [26; 27]. The external photon collection capabilities of the hybrid system were evaluated using on-axis directivity, which is determined from the far-field radiation pattern as  
\[D=4\pi\frac{K_{0}}{P_{rad}}, \tag{6}\]  
where \(K_{0}=K(\theta=\pi/2,\phi=\pi/2)\) is the radiation intensity emitted along the \(Y\) axis, i.e., the power-per-unit solid angle in that direction. In Equation (6), \(D\) measures the ratio of the power radiated per unit solid angle along the \(Y\) axis to that of a hypothetical isotropic radiator [15; 16]. The antenna gain, defined as  
\[G=\eta D, \tag{7}\]  
combines the emission efficiency \(\eta\) and directivity \(D\), serving as the key metric for antenna performance in an RF link [15; 16]. This figure of merit is applicable to optical NAs, where it indicates the potential for photon collection from single-photon emitters. We now analyze the values of \(D\), \(f_{R}\), and \(f_{L}\), which are the three parameters that determine \(G\) for silver and gold NAs. The Kildal limit [28] sets a theoretical upper bound on the maximum directivity that can be achieved with an antenna  
\[D_{max}=(kR)^{2}+3, \tag{8}\]  
where \(k\) is the wavenumber, and \(R\) is the radius of the sphere circumscribing the antenna. Antennas having directivities beyond the limits imposed by their physical dimensions are often named superdirective antennas [29]. It was early realized that the implementation of such antennas is subject to physical restrictions [30; 31] such us very high values of \(Q\), i.e., very low bandwidths, that are also associated with very low efficiencies due to very large ohmic losses, as well as very sensitive implementation tolerances [29]. Antenna designs based on dielectric resonators coupled to a dipole emitter have proven particularly advantageous in the pursuit of electrically small, practical superdirective antennas with high efficiency. This design has been theoretically analyzed at the nanoscale [32] and experimentally demonstrated in the UHF band [33]. It is also well established that superdirectivity is not viable for small plasmonic nanoparticles [32]; however, in principle, plasmonic nanostructures can outperform their dielectric counterpartsdespite the intrinsic metal losses they exhibit [34]. The approach in this work aligns with classical antenna design, where using a reflector provides a straightforward solution to spatially shaping the emitted field into a highly directive beam by enlarging the antenna dimension \(R\).