Green's function expansion for multiple coupled optical resonators with finite retardation using quasinormal modes

Abstract

The electromagnetic Green's function is a crucial ingredient for the theoretical study of modern photonic quantum devices, but is often difficult or even impossible to calculate directly. We present a numerically efficient framework for calculating the scattered electromagnetic Green's function of a multi-cavity system with spatially separated open cavities (with arbitrary shape, dispersion and loss) and finite retardation times. The framework is based on a Dyson scattering equation that enables the construction of the Green's function from the quasinormal modes of the individual resonators within a few-mode approximation and a finite number of iteration steps without requiring nested integrals. The approach shows excellent agreement with the full numerical Green's function for the example of two coupled dipoles located in the gaps of two metal dimers serving as quasinormal mode cavities, and is easily extended to arbitrarily large separations and multiple cavities.

Figures (4)

• Figure 1: Sketch of the framework for obtaining the $N$-cavity Green's function via a set of $N$ scattering equations. Starting from the single-cavity Green's function $\mathbf{G}^{(1)}$, we iteratively add more cavities with $\mathbf{G}^{(n-1)}$ from the previous step serving as an input for the scattering equation for the Green's function $\mathbf{G}^{(n)}$ [cf. Eq. \ref{['eq:dyson']}]. In each step, the recursive Dyson equation is terminated using the condition from Eq. \ref{['eq:termcond']}.
• Figure 2: Sketch of two metal dimers serving as QNM cavities. The dimers are described by a Drude permittivity [cf. Eq. \ref{['eq:Drude']}], and separated by he center-to-center distance $R_{12}$. Two dipole emitters $\mathbf{r}_a\in \mathcal{V}_1$ and $\mathbf{r}_b\in \mathcal{V}_2$ are placed in the dimer gaps.
• Figure 3: Normalized coupling $g_{ba}(\omega)$ [cf. Eq. \ref{['eq:gba']}] between the two dipoles in Fig. \ref{['fig:sketch']} for $R_{12} = 2020\,{\rm nm}$. The QNM expansion from Eq. \ref{['eq:QNMexp']} together with Eq. \ref{['eq:Bapprox']} ($G^{\rm QNM}$) yields excellent agreement with the full numerical Green's function calculation ($G^{\rm full}$). Using Eq. \ref{['eq:Bpole']} instead of Eq. \ref{['eq:Bapprox']} ($G^{\rm QNM}_{\rm pole}$) does not fully include retardation effects. The single-cavity QNM expansion ($G_1$) [Eq. \ref{['eq:singlecavexp']}] does not match the full numerical results, confirming the coupling of the QNMs. The coupling is normalized to $g^{\rm back}_{ba}(\omega_1)$ (i.e., without the dimers).
• Figure 4: Same as in Fig. \ref{['fig:Gcomp_d2000']}, but with $R_{12} = 760\,{\rm nm}$. Equation \ref{['eq:QNMexp']} together with Eq. \ref{['eq:Bapprox']} ($G^{\rm QNM}$) agrees excellently with the full numerical solution ($G^{\rm full}$). Equation \ref{['eq:Bpole']} ($G^{\rm QNM}_{\rm pole}$) yields better agreement than for larger separations, while the single-cavity QNM expansion ($G_1$) from Eq. \ref{['eq:singlecavexp']} fails to capture the shape and magnitude of the coupling.