**8. Conclusions and future directions**

RGWNs offer a different approach for designing dispersive photonic materials. Whereas photonic crystals rely on the formation of Bloch wave states by interference of waves diffracted from an array of periodic elements, a truly non-local phenomenon; RGWNs rely on the coherent superposition of power flowing along isolated waveguides and splitting at X-junctions. Furthermore, in photonic crystals, the interference pattern of the diffracted waves depends on the nonlocal periodic spatial arrangement of the diffracting elements; and in RGWNs it is the local network topology that determines the dispersion and resonance features. For example, in a RGWN, the coherent wave propagation through the network is determined only by the total path length along the waveguide and the phase shift added at a power splitting event, having no restriction on whether the waveguides are straight or curved. Metamaterials also feature a design approach based on the attributes of localized resonances, but their dispersive properties do not depend on any length scale between resonant elements – thus differing substantially from RGWNs. Arrays of coupled resonator optical waveguides (CROWs) feature discrete identifiable resonators that act as the energy storage elements, and dispersion occurs as modes of adjacent resonators are evanescently coupled. By contrast, in RGWNs, energy is not stored resonantly in discrete resonators, but rather in the network of waveguides that are designed to exhibit a collective resonant behaviour.

The operation of RGWNs was demonstrated in this chapter using plasmonics, which allowed for a simple layout and broadband range of operation; however, this implementation also brought about substantial attenuation due to the fundamental loss of plasmonic modes. As indicated above, the plasmonic MIM modes used here have typical propagation lengths of about 50 microns due to metal loss. Since the RGWN scope is broader than the field of plasmonics, it calls for an all-dielectric implementation to mitigate the losses brought on by plasmonics. Implementing RGWNs using photonic circuitry would also address the coupling loss associated with the difference in the modal overlaps between the plasmonic modes in the RGWN and the interfacing dielectric optics.

This new design paradigm is based on different underlying physics and thus opens up new directions for the design of artificial optical materials and devices. Since the RGWN design relies on the interference of local waves, we can use these accessible design parameters to program optical functions directly onto the network. Furthermore, the constraints on the propagation and coupling of the local waves in RGWNs allow for the device operation to reduce to a simple mathematical representation using S-matrix formalism. This allows for the network programming to take the form of an optimization procedure over a relatively small parameter space. The RGWN S-matrix representation was demonstrated here where the inputs were given and the S-matrix of device was designed to give a desired output (e.g.,

**Part 2** 

**Photonic Crystals and Applications** 

routing, mode converting). However, this formalism could be extended to different type of functions, such as sensing, in which the inputs are given and the output changes are monitored. In this chapter, dichroic and trichroic RGWN color routing was demonstrated as a proof of concept; however, incorporating more components into the RGWN and therefore increasing the possible degrees of freedom, could allow for more complex devices or alternatively for devices with enhanced performance. Furthermore, we exemplified the RGWN design paradigm using plasmonics, nesting a split element simply by intersecting waveguides, still the concept is broad and implementing the concept using photonic component could open new opportunities in the design of photonic circuitry devices.
