**2. Plasmonic RGWN components**

The operation of RGWNs is based on two basic components: power splitting elements and isolated waveguides. While the waveguides could easily be implemented using dielectric waveguides, the power splitting elements at the intersection of two such waveguides could not be achieved using dielectrics alone. Nevertheless, this splitting operation, which is the key enabler of this technology, is native to the intersection of two plasmonic waveguides. Consequently, a possible implementation of a RGWN is by using plasmonics via a mesh of intersecting sub-wavelength air gaps in a metal matrix.

Surface plasmon polaritons (referred here to as plasmons for brevity) are slow surface waves that propagate at metal-dielectric interfaces. Adding another metal-dielectric interface to this system, results in a metal-insulator-metal (MIM) waveguide, which supports a highly confined plasmon wave (the lowest order transverse magnetic mode - TM0) that does not get structural cut-off as the dielectric gap between the metal layers becomes vanishingly small. The existence of this lowest-order plasmonic mode in MIM waveguides allows for such plasmonic components as power splitters (Feigenbaum 2007-1) and high transmission sharp waveguide bends (for a review of MIM waveguides and their possible applications see Feigenbaum 2007-2). However, the existence of metal in the MIM

designing this network layout reshapes the interference pattern and the optical function of the RGWN, as will be exemplified later in this chapter. The 2x2 RGWN consists of one closed loop resonance, however larger two- or three-dimensional networks can support

Fig. 1. Schematic illustration of (a) a 4-terminal equal power-splitting element and (b) a local

Although the concept of RGWNs is quite general, we will first illustrate the underlying physics of this paradigm using plasmonics since it allows for a simple topological implementation. After introducing this implementation, in the following sections we will demonstrate how the local wave interference can be designed to engineer small (2x2) energy storage RGWN resonators, and also how we can program the optical transmission function of inhomogeneous RGWNs using transfer matrix formalism. We will also address how the same design principles can be utilized to control the optical dispersion properties of infinitely large RGWNs that behave like artificial optical materials. After addressing other possible implementation and practical issues we will conclude with possible future

The operation of RGWNs is based on two basic components: power splitting elements and isolated waveguides. While the waveguides could easily be implemented using dielectric waveguides, the power splitting elements at the intersection of two such waveguides could not be achieved using dielectrics alone. Nevertheless, this splitting operation, which is the key enabler of this technology, is native to the intersection of two plasmonic waveguides. Consequently, a possible implementation of a RGWN is by using plasmonics via a mesh of

Surface plasmon polaritons (referred here to as plasmons for brevity) are slow surface waves that propagate at metal-dielectric interfaces. Adding another metal-dielectric interface to this system, results in a metal-insulator-metal (MIM) waveguide, which supports a highly confined plasmon wave (the lowest order transverse magnetic mode - TM0) that does not get structural cut-off as the dielectric gap between the metal layers becomes vanishingly small. The existence of this lowest-order plasmonic mode in MIM waveguides allows for such plasmonic components as power splitters (Feigenbaum 2007-1) and high transmission sharp waveguide bends (for a review of MIM waveguides and their possible applications see Feigenbaum 2007-2). However, the existence of metal in the MIM

directions and a more detailed comparison to other optical design paradigms.

multiple resonances, which give rise to more design possibilities.

resonance in a 2x2 RGWN.

**2. Plasmonic RGWN components** 

intersecting sub-wavelength air gaps in a metal matrix.

waveguide configuration does add a source of a modal attenuation to the system as a result the usual loss mechanisms present in any real metal-containing system. This results in a trade off between the compression of the modal cross-section and the modal attenuation as the air gap size is decreased. Since the loss in metals is strongly frequency and material dependent, the focus here will be on RGWNs composed of Au-air-Au MIM waveguides operating at telecommunication frequencies where the modal propagation lengths are on the order of tens of microns, which are substantially larger than the propagation lengths at visible frequencies. The optical properties of the materials throughout this chapter are based on tabulated data (Palik 1998).

In this implementation, the intersection of two sub-wavelength MIM waveguides forms an X-junction that functions as the power splitting elements in the network (Feigenbaum 2007- 1) and the MIM segments between the intersections serve as the isolated waveguides connecting the X-junctions. Through this implementation, X-junctions can be tuned to split power equally at infrared wavelengths both for continuous waves and for short pulse waves consisting of only a few optical cycles while conserving the shape of the input signal. The observed equal-power split is a result of the subwavelength modal cross-section of the input plasmonic waveguide that excites the junction with a broad spectrum of plane waves. As such, equal four-way optical power splitting is enabled for transmission lines (e.g., MIM and coaxial configurations) but cannot be easily achieved using purely dielectric waveguides due to their half-wavelength modal cross-section limit. Thus, through a plasmonic implementation, the strong coupling to all four neighboring X-junctions gives the plasmonic RGWN structure an optical response different from a cross-coupled network of purely dielectric waveguides, where most of the power would be transmitted in the forward direction, with only weak coupling to perpendicular waveguides.

Fig. 2. Power splitting properties of the emerging pulses in an X-junction: (a) intensity relative to the exciting pulse, and (b) phase difference at *λ0*=1.5µm (Feigenbaum 2010).

As the MIM waveguide air gap thickness is varied, the power-split between the X-junction terminals can be tuned both in terms of amplitude and phase (Feigenbaum 2010). This, in addition to determining the phase accumulation in the waveguide segments, sets independent controls in designing the interference pattern that governs the operation of a RGWN. The power splitting in the Au-air X-junction was investigated using the 2D finitedifference time-domain (FDTD) method with short pulse excitation and two equal thickness intersecting MIM waveguides. Through this study, it was found that for small (0.25µm) MIM gaps, these plasmonic X-junctions exhibit equal power splitting with the reflected

Resonant Guided Wave Networks 49

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

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).

indicating that the resonator Q-factor is primarily limited by the material loss.

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),

3D network topologies.

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