In this thesis a quite realistic complex model for describing
light behavior in paraxial linear and nonlinear regimes in nematic
liquid crystal planar cells is studied. Both one and two
dimensional models for the optical field are shown in these
micrometric structures. Thus the influence of nonlocal
nonlinearity and the effects of anisotropy are presented.
Particularly it has been considered the case for $XZ$ anisotropy
for the one dimensional case (which means anisotropy in the plane
of light propagation) and also the $XY$ anisotropy which yields
vectorial modes in the transverse plane. Besides, new regime due
to non symmetric boundary conditions for the one dimensional
orientation of the molecules is discovered. This new configuration
exhibits a strong focusing linear regime since a deep graded index
guide is induced by the molecular orientation at the boundary.
Moreover the number of graded index guides inside the nematics can
be tuned by the asymmetric boundary condition. Finally the
behavior of light under linear and nonlinear regimes for this
asymmetric setup is studied.

From the numerical point of view,a finite difference method is
employed to solve all the partial differential equations involved.
This numerical method yields systems of nonlinear equations which
are solved with standard iterative techniques such as
Newton-Raphson method. Advanced numerical techniques such as
alternating direction implicit schemes or MultiGrid techniques are
employed to improve the computational performance of our codes.
Besides, we introduce a novel transparent boundary condition which
is used successfully to solve the two dimensional electric field
equation and the twist angle distribution of the nematic over the
cell. This improves a bit more the computational cost needed to
solve the complex optical devices we deal with.

Finally, with respect to applications, we use the codes developed
in this thesis to simulate the energy transfer between the
channels of an optical directional coupler. We also design a new
electro-optic device which profits the versatile properties of
nematic liquid crystal cells to guide light in the linear regime.
The electric voltage applied to the cell is used as a degree of
freedom that allows us to control the trajectory of light.