## 6 QD Emitters Near a Flat Interface
### Non-Radiative Losses to a Metal Substrate  
If the substrate is lossy, e.g., a metal, several additional decay channels arise. Mathematically, they are related to the imaginary part of the Fresnel coefficients. Direct evaluation of the integrals in Equations (17) and (18) yields the overall effect of all these channels. However, their relative contributions depend on the distance and the real and imaginary parts of \(\varepsilon_{2}\). At very small distances (\(k_{1}h<<1\)), the dominating contribution is due to Ohmic losses [70; 200],  
\[\frac{\gamma_{\text{OI}}^{(V,\,H)}}{\gamma_{0}}=\frac{3}{(2k_{1}h)^{3}}\text{ Im}\left(\frac{\varepsilon_{2}-\varepsilon_{1}}{\varepsilon_{2}+\varepsilon_{1}} \right)\,. \tag{19}\]  
The coupling to propagating surface plasmons mathematically is described by the pole of \(r^{(p)}\), which yields an additional imaginary part of the integral from \(1\) to \(\infty\) in Equations (17) and (18). Indeed, the equation \((r^{(p)})^{-1}=0\) determines the dispersion relation of \(p\)-polarized SPs. The other Fresnel coefficient, \(r^{(s)}\), has no poles, and accordingly there are no \(s\)-polarized SPs at a metal-dielectric interface. If we neglect the imaginary part of \(\varepsilon_{2}\), the SP wavevector is real. The resonant coupling occurs for  
\[q_{\text{SP}}=\frac{\omega}{c}\sqrt{\frac{\varepsilon_{2}^{\prime}\varepsilon _{1}}{\varepsilon_{2}^{\prime}+\varepsilon_{1}}}\]  
(where \(\varepsilon_{2}^{\prime}\varepsilon_{1}<0\) and \(\varepsilon_{2}^{\prime}+\varepsilon_{1}<0\)). The pole contribution to the integral from \(1\) to \(\infty\) in Equations (17) and (18) is calculated explicitly:  
\[\frac{\gamma_{\text{SP}}^{(V)}}{\gamma_{0}} = 3\pi s_{\text{SP}}^{5}\frac{\sqrt{-u}}{1-u}\exp\left(-\frac{2k_{ 1}h\text{ssp}}{\sqrt{-u}}\right)\,, \tag{20}\] \[\frac{\gamma_{\text{SP}}^{(H)}}{\gamma_{0}} = \frac{3\pi}{2}s_{\text{SP}}^{5}\frac{1}{\sqrt{-u}(1-u)}\exp\left( -\frac{2k_{1}h\text{ssp}}{\sqrt{-u}}\right), \tag{21}\]  
where \(u=\varepsilon_{2}^{\prime}/\varepsilon_{1}\) and \(s_{\text{SP}}=q_{\text{SP}}/k_{1}\).  
Taking \(\varepsilon_{2}(\omega)\) within the simple Drude model for bulk gold, we calculated \(\gamma_{\text{OI}}\), \(\gamma_{\text{SP}}\), and the radiative decay rate (neglecting \(\varepsilon_{2}^{\prime\prime}\)). They are presented in Figure 7. As expected, at small distances the Ohmic losses dominate (see Figure 7, lower panel). However, as the distance to the interface increases, for \(k_{1}h\sim 1\), the coupling to SPs becomes the main mechanism that determines the lifetime (see Figure 7, upper panel). Although this effect decreases with the distance exponentially, this decrease can be rather slow, as \(|\varepsilon_{2}^{\prime}|/\varepsilon_{1}>>1\) and \(s_{\text{SP}}\sim 1\), so it is nearly constant over tens of nanometres. Of course, at still large distances \(\gamma_{\text{tot}}\to\gamma_{0}\).  
Thus, for intermediate distances of the order of a hundred nanometres the radiative decay is mostly due to the coupling to SPs within a broad spectral range (approximately from 2.5 to 4.5 eV). In this range, relevant for colloidal QDs, one can expect also plasmon-enhanced FRET. Even though the enhancement is moderate, the coupling to SPs is not dissipative unless we approach the electrostatic SP resonance frequency, \(\omega_{spr}=\omega_{p}/\sqrt{\varepsilon_{2}^{\infty}+\epsilon_{1}}\) (here, \(\omega_{p}\) is the bulk plasma frquency and \(\varepsilon_{2}^{\infty}\) is the background dielectric constant of the metal). In other words, excitation can be transferred from the QD to SPs and back many times without dissipation. It can lead to strong coupling regime characteristic of plasmonic microcavities [41; 204].