## 3 Discussion and Conclusions  
By combining FDTD simulations and analytical theory, we have shown how anapole states can be coupled and transferred among closely-packed nanoparticles. This result--which recalls the coupling properties of standard resonant modes--is at first counter-intuitive due to the non-resonant character of the anapole state. However, as can be demonstrated by performing a Fano-Feshbach partitioning of the anapole resonator, the three-dimensional state is characterized by the superposition of several distinct resonances of the system, which collectively produce a scattering-suppression state at the anapole wavelength and which mutually couple among the ensemble of nanoparticles. The Fano-Feshbach analysis of three-dimensional anapole states--including a detailed analysis of the scattered fields--goes beyond the scope of this work, and it will be the subject of a future specialized work on the topic.  
The ability to control and transfer non-radiating states along optical circuitry, however, opens to intriguing possibilities in the development of advanced integrated photonics platforms. As shown in Figure 4, anapole nanoparticles can in fact be arranged in a compact nanochain with remarkable guiding properties. Due to the efficient near-field coupling between adjacent nanoparticles, the nanochain can support guided modes which propagate without radiative losses at distances of several \(\upmu\)m. In these simulations, we selectively excited the first anapole state in the nanochain (not shown in the figure) and we characterized the propagation of energy along the chain. Such excitation can be achieved experimentally, as single anapole nanoparticles can be excited by  
Figure 3: **Fano-Feshbach partitioning of the anapole state**. The characteristic mode-profile of the anapole state is originated by the superposition of a cylindrically symmetric TM\({}_{020}\) and a quadrupolar TM\({}_{210}\).  
means of a near-field scanning optical microscopy (NSOM) setup [3]. As an appealing alternative, anapole-based waveguides could be combined with integrated nanolasers emitting at the anapole frequency [13].  
As the anapole states are tightly confined in the near-field, optical nano-circuitry based on non-radiating modes is extremely robust to bending and splitting, as shown in Figure 5. In standard photonics applications, wave-guide deformation produces significant radiation losses, in particular when considering 90-degree bends and turns [23]. Conversely, in the case of an anapole nanochain, the near-field properties of the non-radiating state allow for efficient transmission of the guided mode across deformations and bends, such as in the case of wave-guide splitting (Figure 5a) or 90-degree bending and re-routing (Figure 5b). These illustrative examples strongly suggest the possibility of integrating anapole nanoparticles with state-of-the-art integrated photonics applications.  
Figure 4: **Anapole nanochain**. Steady-state electromagnetic energy distribution along a chain of anapole nanoparticles. The centre-to-centre distance is \(d=400\) nm. The external excitation is restricted to the first anapole of the chain (not included in the panel).  
Figure 5: **Robust sub-wavelength guiding via near-field transfer of anapole states**. Due to the near-field confinement produced by the anapole state, the anapole nanochain is robust against bending and splitting of the integrated wave-guide. This opens to the realization of (**a**) integrated splitters and (**b**) 90-degree bends without introducing radiation losses.