## 2 Results  
### Ab-Initio Analysis of Multiple Anapole Systems  
We performed finite-differences time-domain (FDTD) analysis by considering an ensemble of three-dimensional anapole nanoparticles, each composed of a silicon nanodisk of radius \(R=150\) nm and height \(h=50\) nm. Each nanodisk is aligned with the \(z\)-axis, and it is illuminated by an \(E_{x}\) polarized plane wave propagating along the cylinder axis. In order to characterize the mutual coupling of closely-packed anapole nanoparticles, we consider a system of two slightly displaced nanodisks (see Figure 1a). The relative displacement of the identical nanoparticles is described by a centre-to-centre distance \(d\) and by a rotation angle \(\alpha\) measured from the \(x\)-axis (Figure 1a-inset).  
The anapole wavelength \(\lambda_{\mathrm{an}}\) is identified by analysing the scattering cross-section \(C_{\mathrm{sca}}\) of the isolated structures, as reported in Figure 1b (blue line). In the proposed configuration, the anapole state corresponds to \(\lambda_{\mathrm{an}}=568\) nm (green dashed line). Despite the strong reduction of the scattering cross-section, the anapole state is associated with a strong enhancement of the electric field inside the nanodisk. The field enhancement is measured by integrating the electric intensity inside the resonator (Figure 2b, dashed orange line), which exhibits a strong peak at the anapole wavelength \(\lambda_{\mathrm{an}}\). The strong field enhancement associated to the anapole state is a counter-intuitive feature of the non-radiating state, and it has recently been exploited to amplify light-matter processes in semiconductor nanostructures [12; 13]. To characterize the mutual coupling between anapole states, we performed a set of simulations for different rotation angles \(\alpha\) and distances \(d\), whose results are reported in Figure 1c,d. In our numerical experiments, we selectively excited one of the two nanoparticles at the anapole wavelength \(\lambda_{\mathrm{an}}\), and we measured the steady-state electric field intensity \(|\mathbf{E}|\) in the second resonator. In terms of angular displacement \(\alpha\), the anapole coupling is characterized by a symmetric dipolar profile (Figure 1c). The near-field coupling is maximum in the direction of the three-lobe profile of the anapole state (\(\alpha=0,2\pi\)), while it is negligible in the orthogonal direction (\(\alpha=\pi/2,3\pi/2\)). Conversely, in terms of the mutual distance \(d\), the anapole coupling exhibits a more complex profile, with a sharp 20% reduction in less than 150 nm total displacement (Figure 1d). Interestingly, the spatial decay of the anapole coupling does not follow a power law decay, as would be expected from near-field dipolar states. To verify this, we compared the anapole coupling distribution from Figure 1c,d to the scattered field of an isolated nanoparticle (see Supplementary Figure S1). Even in the case of an isolated nanoparticle, the scattered field is mostly dipolar, as multipole-components are negligible at the anapole wavelength [3]. The dipolar angular distribution--associated with a complex decay profile--can be considered as a first signature of a complex modal interaction between closely-packed anapole nanoparticles.