Unusual modes and photonic gaps in a Vicsek waveguide network

Abstract

We propose a simple model of a waveguide network designed following the growth rule of a Vicsek fractal. We show, within the framework of real space renormalization group (RSRG) method, that such a design may lead to the appearance of unusual electromagnetic modes. Such modes exhibit an extended character in RSRG sense. However, they lead to a power law decay in the end-to-end transmission of light across such a network model as the size of the network increases. This, to our mind, may lead to an observation of power law localization of light in a fractal waveguide network. The general occurence of photonic band gaps and their change as a function of the parameters of the system are also discussed.

Engineering wave localization in a fractal waveguide network

“…Photonic gaps, apart from materials with a large di- * E-mail: arunava chakrabarti@yahoo.co.in electric constant, can also be observed in waveguide networks, as proposed by several groups over the past years [30][31][32][33][34][35][36][37][38]. Anderson localized eigenmodes are observed inside the photonic gaps and excellent agreement between theory and experiments has been obtained [30].…”

“…The network models are able to localize a propagating wave by virtue of the geometrical arrangement of the waveguide segments. Of particular interest are a wide variety of models based on waveguide networks designed following a deterministic fractal geometry [34][35][36][37][38], where the gaps result from the typical topology exhibited the hierarchical arrangement of the waveguide segments. The present communication also deals with a hierarchically designed (fractal geometry) waveguide network, but addresses a deeper fundamental question regarding wave localization in such systems, as explained below.…”

“…Nevertheless, there can be a countable infinity of extended eigenfunctions with high (or even, perfect) transmittivity, even though there is no translational order in such systems. Once again, this is true for electrons [43][44][45][46][47][48][49], and classical waves as well [35,36]. A curious point, apparently gone unnoticed or un-appreciated so far is that, while a precise determination of the eigenvalues corresponding to the extended eigenmodes is possible in the above cases of hierarchically grown fractal networks, the task seems to be practically impossible when it comes to an exact evaluation of eigenvalues of the localized modes in such hierarchical systems in their thermodynamic limits.…”