**3. Resonators**

After characterizing the properties of the RGWN building blocks, we illustrate the working principles of RGWNs by investigating the dynamics of a compact 2x2 RGWN resonator. In order to form a resonance, the network is designed such that when an X-junction is excited from the internal ports, the exciting waves are out-of-phase, resulting in constructive interference inside the network, as illustrated in Fig. 3. For such out-of-phase excitation the fields in the external terminals interfere destructively, and the power is coupled back into the resonator, enhancing the energy storage quality factor (Q-factor).

Fig. 3. Resonance build-up in a 2×2 RGWN. (a) Two in/out-of phase input pulses result in destructive/constructive interference inside the network. (b) Steady-state of waves resonating in a 2×2 network where each pair of pulses excites the X-junctions out of phase (Feigenbaum 2010).

When the 2×2 RGWN is excited from the lower-left arm (see Fig. 4), after a transient that includes the first five splitting events, the resonant state is reached as pairs of pulses resonate between junctions 1 and 3 (exemplified by snapshot *t6*) and junctions 2 and 4 (exemplified by snapshot *t7*). However, before this steady state is reached, it is instructive to follow the dynamics that lead up to this resonance. Starting with the third power split, this event occurs as junctions 1 and 3 are both simultaneously excited by two waveguides. The incoming pulses arrive at both junctions in-phase, which would result in destructive interference inside the network if the R and S split components of each pulse were exactly π-phase shifted. However, the interference is not completely destructive due to the finite size of the waveguides, which causes the phase difference to deviate from a perfect π-phase shift (in accordance with Fig. 2). This power splitting event determines how much power couples into the network. For all future power splitting events after the third one, the two pulses arriving simultaneously at each junction are out-of-phase and therefore interfere constructively inside the resonator. The trade-off between coupling power into the resonator

pulse being out-of-phase (i.e., approximately π-phase shifted) with respect to the sideways and forward transmitted pulses. As illustrated in Fig. 2a, as the MIM gap size is increased, the optical power flow deviates from equal power splitting between the terminals towards dominant power transmission directly across the X-junction, which resembles the wavelength-scale photonic mode limit. Furthermore, in these calculations, the phase shift between the sideways (S) and the forward (F) transmitted pulses is consistent with the

After characterizing the properties of the RGWN building blocks, we illustrate the working principles of RGWNs by investigating the dynamics of a compact 2x2 RGWN resonator. In order to form a resonance, the network is designed such that when an X-junction is excited from the internal ports, the exciting waves are out-of-phase, resulting in constructive interference inside the network, as illustrated in Fig. 3. For such out-of-phase excitation the fields in the external terminals interfere destructively, and the power is coupled back into

Fig. 3. Resonance build-up in a 2×2 RGWN. (a) Two in/out-of phase input pulses result in destructive/constructive interference inside the network. (b) Steady-state of waves resonating in a 2×2 network where each pair of pulses excites the X-junctions out of phase

When the 2×2 RGWN is excited from the lower-left arm (see Fig. 4), after a transient that includes the first five splitting events, the resonant state is reached as pairs of pulses resonate between junctions 1 and 3 (exemplified by snapshot *t6*) and junctions 2 and 4 (exemplified by snapshot *t7*). However, before this steady state is reached, it is instructive to follow the dynamics that lead up to this resonance. Starting with the third power split, this event occurs as junctions 1 and 3 are both simultaneously excited by two waveguides. The incoming pulses arrive at both junctions in-phase, which would result in destructive interference inside the network if the R and S split components of each pulse were exactly π-phase shifted. However, the interference is not completely destructive due to the finite size of the waveguides, which causes the phase difference to deviate from a perfect π-phase shift (in accordance with Fig. 2). This power splitting event determines how much power couples into the network. For all future power splitting events after the third one, the two pulses arriving simultaneously at each junction are out-of-phase and therefore interfere constructively inside the resonator. The trade-off between coupling power into the resonator

geometrical difference in their pulse propagation trajectories (see Fig. 2b).

the resonator, enhancing the energy storage quality factor (Q-factor).

**3. Resonators** 

(Feigenbaum 2010).

and maintaining it inside suggests that MIM gap sizes that are subwavelength, but not arbitrarily small, will maximize the network resonance. To interpret the FDTD observations and arrive to the conclusion described above, a simplified analytical description of pulse propagation in the network is derived in which only a few parameters are tracked: phase, amplitude, position and direction. The pulses are assumed to travel in the waveguides and split into four new pulses upon arrival at an X-junction. This model also illustrates the compactness of the possible mathematical representation of RGWNs, and the importance of this advantage becomes more substantial when considering the dynamics of larger 2D and 3D network topologies.

Fig. 4. Time snapshots of *Hz* (normalized to the instantaneous maximum value) in a 2x2 plasmonic RGWN recorded at the third to the seventh power splitting events for a 2D-FDTD simulation. The MIM waveguides are 0.25µm thick and 6µm long (Feigenbaum 2010).

Calculating the Q-factor of such 2x2 RGWN resonators (Fig. 5) illustrates the role of interference in generating a strong network resonance, which causes the network Q-factor to be an order of magnitude larger than what would be expected if optical power splitting in the X-junctions operated incoherently, i.e. we lost half the power in each splitting event. Increasing the MIM gap size causes the phase of the interfering waves to deviate from being π-phase shifted, resulting in a degradation of the constructive interference inside the resonator and a decrease in the overall network Q-factor. On the other hand, as the gap size is decreased, the plasmonic mode attenuation increases due to metallic losses in the waveguides. Between these two competing effects, the maximal Q-factor value is obtained for a gap size of 250nm. These RGWN Q-factor values are considerable for plasmonic resonators and even comparable to typical values of wavelength-size dielectric resonators that are dominated by radiation loss (e.g., a cylindrical dielectric cavity of radius 1.3 with a purely real refractive index of n=2.5 surrounded with air has a Q~100). If we were to artificially decrease the Au loss at 1.5µm (or alternatively go to longer wavelengths), the Qfactor of the resonator would increase appreciably (e.g., Q ~ 750 for a 200 nm gap width), indicating that the resonator Q-factor is primarily limited by the material loss.

Resonant Guided Wave Networks 51

loop that has dominant contribution to the RGWN dispersion. Further possibilities for achieving band dispersion control are illustrated in Fig. 6b, showing a flat bands over a wide range of wavevectors at 130 and 170 THz, as well as the formation of a photonic bandgap

The infinitely large RGWN is illustrated in Fig. 6c along with a few schematic resonance orders that represent the resonances that could arise within the network. The operating mechanism of the RGWN is very different from that of photonic crystals composed of metaldielectric alternating materials. Although the schematic layout might look similar, the difference between the two classes of artificially designed optical materials becomes clear when considering the difference in the length scales of their composite elements. Whereas photonic crystals operate based on non-local interaction of Bloch waves with the entire array, RGWNs rely on the interference of local waves. Therefore RGWNs are not sensitive to the actual topology of waveguides between junctions but only to its trajectory length, whereas the properties of photonic crystals would greatly depend on the shape of the periodic metallic islands. Additionally, RGWNs do not necessarily have to be periodic to operate as resonant guided wave networks, and for the same reason, planting a defect in a

Because the underlying physics of RGWNs is based on the interference of local waves, it allows for layouts that are inhomogeneous and non-periodic across the network. Unlike photonic crystals, which are restricted to Bragg wave effects in periodic structures, the flexibility of RGWNs open up design possibilities where the wave properties are varied across the structure. With respect to metamaterials, which could inherently be nonhomogeneous due to the local nature of the interaction between light and the meta-atoms, RGWNs have the advantage of having interference effects within the network, which allows

An additional unique feature of RGWNs relates to the constraints on wave propagation within the structure. Unlike other photonic designs, RGWNs have a limited number of modes that are allowed to propagate within the structure (e.g. only the TM0 mode for the case of the plasmonic implementation described previously). Furthermore, the waves can propagate only inside the waveguides connecting the splitting elements. The different waveguides are coupled only by X-junctions, which each have only a limited number of terminals. This level of control is beneficial for several reasons. First, the interference pattern in the network can be controlled more directly. Second, it allows for a comprehensive mathematical representation of the RGWN by scattering matrix (S-matrix) formalism that greatly reduces the computational complexity of programming the network. Third, since the waveguides are isolated from each other, their only contribution to the network is to serve as phase retardation elements between the splitting elements. As a result, the waveguide length is the only effective parameter in its contour, as long as the bending is not too severe. This waveguide feature allows for the network to maintain its engineered function even when distorted. Additionally, the ability to utilize curved or bent waveguides to accommodate long contours is useful when designing the interference

between 140-160 THz, for appropriately chosen network parameters.

RGWN would not have the same effect as it would in a photonic crystal.

**5. Programming the optical properties of a network** 

for frequency spectrum reshaping designs through these effects.

pattern of RGWNs.

Fig. 5. Q-factor of 2×2 RGWN resonator from simulation results compared with those resulting from incoherent power splitting (Feigenbaum 2010).
