## 2 Results
### Fano-Feshbach Analysis of the Internal Modes  
The counter-intuitive coupling properties of anapole-anapole systems can be explained by analysing the internal resonances of the system. The resonant properties of the single nanostructures can be extracted from the integrated density of states (DOS) \(\rho(\lambda)\), which can be directly computed from FDTD simulations [14]. The integrated DOS is defined as:  
\[\rho(\lambda)=\int\mathrm{d}\mathbf{x}\,\rho(\mathbf{x},\lambda) \tag{1}\]  
where \(\rho(\mathbf{x},\lambda)\) is the local DOS, which is a function of the spatial position inside the resonator. In the FDTD framework, the local DOS corresponds to the spectral response to a single pulse excitation, and it can be defined separately for each component of the electromagnetic fields. For a generic component \(E_{j}\) of the electric field, as an example, the integrated DOS reads:  
Figure 1: **Mutual coupling of non-radiating anapole states**. **(a)** Near-field coupling between two silicon nanodisks (\(n=3.5\)) excited at the anapole wavelength \(\lambda_{\mathrm{an}}=568\) nm. (inset) The dielectric resonators are mutually displaced by a centre-to-centre distance \(d\) and by an angle \(\alpha\); **(b)** Scattering cross-section \(C_{\mathrm{sca}}\) (blue line) and internal electric energy (orange dotted line) as a function of the incident wavelength \(\lambda\). The anapole state (green-dashed line) is characterized by the simultaneous suppression of the scattering cross-section, and by a strong enhancement of the internal field intensity; **(c)** Coupled electric energy as a function of the rotation angle \(\alpha\) (\(d=450\) nm). The mutual coupling is maximum at \(\alpha=0,\pi\) and negligible at \(\alpha=\pi/2,3\pi/2\); **(d)** Coupled electric energy as a function of the mutual distance \(d\). The results correspond to the angular condition of maximum scattering \(\alpha=2\pi\). \[\rho_{E_{j}}(\omega)=\int\mathrm{d}\mathbf{x}\ \left|\mathcal{F}\left\{E_{j}( \mathbf{x},t)\right\}\right|^{2}, \tag{2}\]  
where \(\mathcal{F}[\cdots]\) stands for Fourier transform and \(E_{j}(\mathbf{x},t)\) is the electric field along the \(j\)-th direction (\(j=x,y,z\)). An analogue definition holds for the magnetic field components \(H_{j}(\mathbf{x},t)\). As can be easily verified by solving [PERSON]'s equations in cylindrical coordinates \((r,\phi)\), the response from the anapole resonator can be fully represented in terms of the \(E_{z}\) and \(H_{z}\) field components [15]. In Figure 2, we report the integrated DOS for the (a) \(H_{z}\) and (b) \(E_{z}\) field components. The integrated DOS are computed by considering a single pulse excitation centred at \(\lambda=700\) nm, and the spectral responses are normalized to the source spectrum. Interestingly, the transverse electric (TE) spectrum exhibits a strong peak exactly at the anapole wavelength (red vertical line), while the transverse magnetic (TM) spectrum shows only a slight enhancement at the anapole frequency (Figure 2b). Further insights on the resonant properties of the anapole state can be obtained by decomposing the electromagnetic fields into a set of orthogonal eigenmodes. By definition, however, the silicon resonator represents an open cavity, and the definition of a set of orthogonal resonator modes is a challenging task [16; 17]. As recently shown in [11], a possible way to circumvent such difficulties is the introduction of a Fano-Feshbach partitioning of the system [18]. In a nutshell, the Fano-Feshbach partitioning consists of the mathematical splitting of the total system into two orthogonal eigenspaces, corresponding to the resonator and environment regions. [PERSON]'s equations are rewritten in the partitioned subspaces, providing a rigorous description of light-matter interaction in open resonators. For a cylinder aligned along the \(z\)-axis, the Fano-Feshbach internal resonances for the fields \(H_{z}\) and \(E_{z}\) correspond to the TE and TM modes of a perfect electric conductor (PEC) cavity [19]. They are expressed as:  
\[