**1. Introduction**

44 Photonic Crystals – Introduction, Applications and Theory

Magath, T. & Serebryannikov, A.E. (2005). Fast Iterative, Coupled-Integral-Equation

Nojima, S. (2007). Long-Sojourning Light in a Photonic Atoll , *J. Opt. A, Pure Appl. Opt.*, Vol.

Petit, R., Ed. (1980). *Electromagnetic Theory of Gratings*, Springer, Berlin Heidelberg New York Plum, E.; Fedotov, V.A. & Zheludev, N.I. (2009). Planar Metamaterial with Transmission

Rüter, C.E.; Makris, K.G., El-Ganainy, R., et. al. (2010). Observation of Parity-Time

Saado, Y.; Golosovsky, M., Davidov, A. & Frenkel, A. (2005). Near-Field Focusing by a Photonic Crystal Concave Mirror, *J. Appl. Phys.*, Vol. 98, No. 6, 063105 Sakoda, K. (2001). *Optical Properties of Photonic Crystals*, Springer, Berlin Heidelberg New York Scalora, M.; Dowling, J.P., Bowden, C.M. & Bloemer, M.J. (1994). The Photonic Band Edge

Schleuer, J. & Yariv, A. (2004). Circular Photonic Crystal Resonators, *Phys. Rev. E*, Vol. 70,

Schurig, D. & Smith, D.R. (2003). Spatial Filtering Using Media with Indefinite Permittivity and Permeability Tensors, *Appl. Phys. Lett*., Vol. 82, No. 14, pp. 2215-2217 Serebryannikov, A.E.; Magath, T. & Schuenemann, K. (2006). Bragg Transmittance of *s*-

Serebryannikov, A.E. & Ozbay, E. (2009). Unidirectirectional Transmission in Nonsymmetric

Serebryannikov, A.E. (2009). One-Way Diffraction Effects in Photonic Crystal Gratings Made

Serebryannikov, A.E.; Ozbay, E. & Usik, P.V. (2010). Defect-Mode-Like Transmission and

Shadrinov, I.V.; Fedotov, V.A., Powell, D.A., et al. (2011). Electromagnetic Wave Analogue

Singh, R.; Plum, E., Menzel, C. et. al. (2009). Terahertz Metamaterial with Asymmetric

Smigaj, W. (2007). Model of Light Collimation by Photonic Crystal Surface Modes, *Phys. Rev.* 

Vodo, P.; Parimi, P.V., Wu, W.T. & Sridhar, S. (2005). Focusing by Planoconcave Lens Using

Wang, Z.; Chong, Y.D. & Joannopoulos, J.D. & Soljacic, M. (2008). Reflection-Free One-Way Edge Modes in a Gyromagnetic Photonic Crystal, *Phys. Rev. Lett*., Vol. 100, No. 1, 013905 Wu, L.; Mazilu, M., Gallet, J.-F. & Krauss, T.F. (2005). Dual Lattice Photonic-Crystal Beam

Yablonovitch, E. (1987). Inhibited Spontaneous Emission in Solid-State Physics and

Yu, Z.; Wang, Z. & Fan, S. (2007). One-Way Total Reflection with One-Dimensional Magneto-Optical Photonic Crystals, *Appl. Phys. Lett*., Vol. 90, No. 12, 121133 Yu, Z. & Fan, S. (2009). Complete Optical Isolation Created by Indirect Interband Photonic

Polarized Waves Through Finite-Thickness Photonic Crystals With Periodically

Gratings Containing Metallic Layers, *Opt. Express,* Vol. 17, No. 16, pp. 13335-13345

Localization of Light in Photonic Crystals without Defects, *Phys. Rev. B*, Vol. 82, No.

Symmetry in Optics, *Nature Phys.*, Vol. 6, No. 3, pp. 192-195

Optical Diode, *J. Appl. Phys*., Vol. 76, No. 4, pp. 2023-2026

Corrugated Interface, *Phys. Rev. E*, Vol. 74, No. 6, 066607

of Isotropic Materials, *Phys. Rev. B*, Vol. 80, No. 15, 155117

of an Electronic Diode, *New J. Phys*., Vol. 13, No. 3, 033025

Negative Refraction, *Appl. Phys. Lett*., Vol. 86, No. 20, 201108

Electronics, *Phys. Rev. Lett*., Vol. 58, No. 20, pp. 2059-2062

Transmission, *Phys. Rev. B*, Vol. 80, No. 15, 153104

Splitters, *Appl. Phys. Lett*., Vol. 86, No. 21, 211106

Transitions, *Nature Phot*., Vol. 3, No. 2, pp. 91-94

22, No. 11, pp. 2405-2418

9, No. 9, S425

No. 13, 131901

No. 3, 036603

16, 165131

*B*, Vol. 75, No. 20, 205430

Technique for Inhomogeneous Profiled and Periodic Slabs, *J. Opt. Soc. Am. A*, Vol.

and Reflection that Depend on the Direction of Incidence, *Appl. Phys. Lett*., Vol. 94,

In the last two decades, the development of new photonic material design paradigms has opened up new avenues for designing photonic properties based on different underlying physics. For example, photonic crystals, as described elaborately throughout this book, are based on dispersive Bloch wave modes that arise in periodic index structures. Different in operation than photonic crystals, metamaterials (Smith 2004, Shalaev 2007) are based on subwavelength resonant elements (or "meta-atoms") that interact with incident radiation to give rise to complex refractive indices. In this chapter, we introduce a new approach to optical dispersion control based on resonant guided wave networks (RGWNs) in which power-splitting elements are arranged in two- and three-dimensional waveguide networks.

A possible framework for comparing and classifying photonic design paradigms is according to their basic resonating elements with which light interacts to give the desired artificial dispersion. Under this classification scheme, we can think of materials that operate based on the local interaction of waves with sub-wavelength resonating elements (i.e. metamaterials), structures based on the nonlocal interference of Bragg periodic waves (i.e. photonic crystals), and arrays of coupled resonator optical waveguides (CROWs) where adjacent resonators are evanescently coupled (Yariv 1999). Different from these existing concepts, the dispersion that arises in RGWNs is a result of the multiple closed-path loops that localized guided waves form as they propagate through a network of waveguides connected by wave splitting elements. The resulting multiple resonances within the network give rise to wave dispersion that is tunable according to the network layout. These distinctive properties, that will be described here, allow us to formulate a new method for designing photonic components and artificial photonic materials.

A RGWN is comprised of power splitting elements connected by isolated waveguides. The function of the splitting element is to distribute a wave entering any of its terminals between all of its terminals, as illustrated in Fig. 1a. The waves are then propagated in isolated waveguides between the splitting elements, where the local waves from different waveguides are coupled together. For example, four splitting elements arranged in a rectangular network layout form a 2x2 RGWN (see Fig. 1b). When one of the terminals is excited, the multiple splitting occurrences of the incident wave within the network form closed path resonances that reshape the dispersion of the emerging waves according to the network layout and is different from the dispersion of the individual waveguides. Properly

Resonant Guided Wave Networks 47

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

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

on tabulated data (Palik 1998).

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 multiple resonances, which give rise to more design possibilities.

Fig. 1. Schematic illustration of (a) a 4-terminal equal power-splitting element and (b) a local resonance in a 2x2 RGWN.

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