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We investigate the transport properties of a classical wave propagating through a quasi-periodic Fibonacci array of waveguide segments in the form of loops. The formulation is general, and applicable for electromagnetic or acoustic waves through such structures. We examine the conditions for resonant transmission in a Fibonacci waveguide structure. The local positional correlation between the loops are found to be responsible for the resonance. We also show that, depending on the number of segments attached to a particular loop, the intensity at the nodes displays a perfectly periodic or a self-similar pattern. The former pattern corresponds to a perfectly extended mode of propagation, which is to be contrasted to the electron or phonon characteristics of a pure one dimensional Fibonacci quasi-crystal.

We have studied the effects of dimerization on the energy levels of a one-dimensional molecular chain attached between two electrodes. Analytic expressions for the change in energies in the presence of a small perturbing external potential have been obtained for the three limiting cases: (a) uniform, (b) partially dimerized and (c) completely dimerized chains. We find that the presence of dimerization enhances the mixing between low-lying energies in the system resulting in a situation conducive to showing negative differential resistance (NDR) in the current–voltage characteristics. The effect of spin-polarized molecule–electrode couplings on a dimerized chain has also been studied, where both spin-parallel and spin-antiparallel current show NDR behaviour. Strong dimerization however is found to destroy the spin-valve effects that are most essential for spintronic devices.

We investigate theoretically, the character of electronic eigenstates and transmission properties of a one dimensional array of stubs with Cantor geometry. Within the framework of real space re-normalization group (RSRG) and transfer matrix methods we analyze the resonant transmission and extended wave-functions in a Cantor array of stubs, which lack translational order. Apart from resonant states with high transmittance we unravel a whole family of wave-functions supported by such an array clamped between two-infinite ordered leads, which have an extended character in the RSRG scheme, but, for such states the transmission coefficient across the lead-sample-lead structure decays following a power-law as the system grows in size. This feature is explained from renormalization group ideas and may lead to the possibility of trapping of electronic, optical or acoustic waves in such hierarchical geometries

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.

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