Metamaterials constitute a relatively new field which is very promising
because it exhibits properties that may not be readily found in nature. One
of promising lines of research is the study of novel characteristics of the prop-
agation of electromagnetic (EM) waves in gradient refractive index (GRIN)
lenses. In this Thesis, three geometrical optics methods as well as a wave
numerical method have been developed for the investigation of EM waves
propagation through media with certain refractive indices. Furthermore, we
study the propagation of EM waves through specific geometrical configura-
tions as well as through complex random networks of GRIN lenses, such as
Luneburg (LL) and Luneburg Hole (LH) lenses. We show that waveguides,
which are formed by LLs, offer the capability of better controlling the prop-
agation characteristics of EM waves.

In addition, we show that branched
flows and extreme events can arise in such complex photonic systems.
In addition to GRIN lenses networks, we use the discrete nonlinear Schr ̈o-
dinger equation to investigate the propagation of an EM wavepacket through
certain configurations of optical fiber lattices and investigate the effects of
randomness and nonlinearity in the diffusion exponent.

Finally, we study surface plasmon polaritons (SPPs). We investigate how
the presence of active (gain) dielectrics change the dispersion relation and
enhance the propagation length of SPPs. We finally show that the use of an
active dielectric with gain, which compensates for metal absorption losses,
enhances substantially the plasmon propagation.

We present numerical simulations for the propagation of surface plasmon polaritons (SPPs) in a dielectric-metal-dielectric waveguide using COMSOL multiphysics software. We show that the use of an active dielectric with gain that compensates metal absorption losses enhances substantially plasmon propagation. Furthermore, the introduction of the active material induces, for a specific gain value, a root in the imaginary part of the propagation constant leading to infinite propagation of the surface plasmon. The computational approaches analyzed in this work can be used to define and tune the optimal conditions for surface plasmon polariton amplification and propagation.

We use a simple dynamical model and explore coherent dynamics of wavepackets in complex networks of optical fibers. We start from a symmetric lattice and through the application of a Monte–Carlo criterion we introduce structural disorder and deform the lattice into a small-world network regime. We investigate in the latter both structural (correlation length) as well as dynamical (diffusion exponent) properties and find that both exhibit a rapid crossover from the ordered to the fully random regime. For a critical value of the structural disorder parameter ρ≈0.25 transport changes from ballistic to sub-diffusive due to the creation strongly connected local clusters and channels of preferential transport in the small world regime.