## 6 QD Emitters Near a Flat Interface
### Radiative Lifetime Near Interface  
We shall consider the QD emitter as a point dipole. (See [201] for discussion of possible effects beyond this approximation.) The radiative decay rate in the presence of other bodies is determined by the total field that acts on the emitter (created by itself and scattered by the bodies) and given by [78]  
\[\gamma=\text{Im}[\mathbf{d}^{\star}\cdot(\mathbf{E}_{0}(\mathbf{r})+\mathbf{E}_{ \text{s}}(\mathbf{r}))]/2\hbar\, \tag{15}\]  
where \(\mathbf{E}_{0}\) is the dipole field in infinite space. Here, \(\mathbf{d}^{\star}\) is the complex-conjugate of the classical dipole moment, the necessary adaptation for the quantum mechanical transition dipole matrix element consists in multiplying it by 2 in the final result, \(\mathbf{d}\to 2\mathbf{d}\)[78]. The electric component of the scattered field created by an emitting dipole located in the origin of the reference frame can be written in terms of the Green's dyadic [78], which is specific of system's geometry:  
\[\mathbf{E}_{\text{s}}(\mathbf{r}^{\prime})=4\pi k_{1}^{2}\mathbf{G}(\mathbf{r }^{\prime},\mathbf{0})\cdot\mathbf{d}\,. \tag{16}\]  
The \(3\times 3\) dyadic matrix \(\mathbf{G}(\mathbf{r}^{\prime},\mathbf{r})\) is determined by the Fresnel coefficients, \(r^{(p)}\) and \(r^{(s)}\), and its explicit expression can be found in [197]. From Equation (15), one obtains for a dipole perpendicular to the interface (i.e., \"vertical\", see inset in Figure 6):  
\[\frac{\gamma^{(V)}}{\gamma_{0}}=1+\frac{3}{2}\text{Re}\int_{0}^{\infty}\frac {s^{3}}{\sqrt{1-s^{2}}}r^{(p)}\exp{(2 ik_{1}h\sqrt{1-s^{2}})}ds\, \tag{17}\]  
and for a dipole parallel to the interface (i.e., \"horizontal\"):  
\[\frac{\gamma^{(H)}}{\gamma_{0}}=1+\frac{3}{4}\text{Re}\int_{0}^{ \infty}\frac{s}{\sqrt{1-s^{2}}}\Big{[}r^{(s)}-(1-s^{2})r^{(p)}\Big{]}\] \[\times\exp{(2 ik_{1}h\sqrt{1-s^{2}})}ds\,. \tag{18}\]  
Here, \(\gamma_{0}\) is the decay rate in an infinite medium with dielectric constant \(\varepsilon_{1}\), given by Equation (1). The integrals in Equations (17) and (18) with respect to the normalized in-plane wavevector, \(s=q/k_{1}\), can be divided into two parts, one from 0 to 1 corresponding to propagating waves in the upper half-space and the other from 1 to \(\infty\) representing evanescent waves with imaginary wavevector component along \(z\) axis perpendicular to the interface, \(k_{1z}\). The latter type of waves exist in two cases: (i) if \(\varepsilon_{1}<\varepsilon_{2}\) and both dielectric constants are positive and (ii) if \(\varepsilon_{2}^{\prime}\equiv\mathrm{Re}(\varepsilon_{2})<0\), i.e., when the second medium is a metal. In the case of two non-dispersive dielectrics, \(\gamma\) given by Equations (17) and (18) represents only radiative losses of dipole's energy, renormalized by the back action of the scattered waves. Yet, for \(\varepsilon_{1}<\varepsilon_{2}\) there are EM waves (excited by the dipole) propagating in the lower half-space (with real \(z\)-component of the wavevector inside medium 2, \(k_{2z}\)) and experiencing total internal reflection at the interface with the upper medium, possessing imaginary \(k_{1z}\). The Fresnel coefficient \(r^{(p)}\) has an imaginary part associated with these waves. If the interaction of the dipole with these evanescent waves is neglected (dotted lines in Figure 6), i.e., we set the integral from 1 to \(\infty\) in Equations (17) and (18) equal to zero, then \(\gamma\) becomes independent of dipole's distance from the interface. As can be seen from this figure, the emission decay rate is strongly enhanced, \(\gamma>>\gamma_{0}\), at small distances (\(k_{1}h<1\)) for both dipole orientations because of the strong interaction with the evanescent waves. It has been pointed out for plane interfaces by [PERSON] and [PERSON] [198], and it is similar to the coupling to the whispering gallery modes (a kind of surface EM waves) observed for emitters placed inside a micrometer-size dielectric sphere [202,203]. As we will show in the next section, this coupling can be used for FRET enhancement.