Thesis (Ph. D.)--University of Rochester. The Institute of Optics, 2019.
The term “structured light” is used to refer to light having nontrivial and interesting
intensity, phase and/or polarization distributions. This concept has been at the
center of research efforts ranging from quantum optics to biophysics. It is therefore
important to propose theoretical models that highlight the main properties of such
fields, e. g. their angular spread, polarization structure and orbital angular momentum
content, to facilitate their understanding as well as their design. Throughout this
thesis, several geometrical models are used to study structured light, from the paraxial
scalar to the nonparaxial electromagnetic regimes. These models provide us with
abstract spaces in which optical fields can be represented in an intuitive way.
We start by studying self-similar structured-Gaussian (SG) beams through operator
and ray-based formalisms. These two complementary approaches lead to distinct
geometrical representations on the modal Poincaré sphere (MPS): the operator formalism
leads to the Majorana constellation, which represents an SG beam by a collection
of points on the surface of the MPS; while ray theory uses the Poincaré path,
a curve on the surface of the MPS, to determine the rays that conform the same SG
beam. Both representations provide information about the angular momentum content
and invariances to specific astigmatic transformations through their rotational
symmetries, which can lead to continuous or discrete geometric phases. Nonparaxial fields are then described in terms of their plane-wave amplitude representation providing us with an intuitive space, the sphere of directions, in which the
polarization and directional spread of the field can be easily visualized. This space is
ideal for introducing the concept of complex-focus fields and several complete bases
that can be constructed from them. These exact solutions to the wave equation
are well suited to describe highly focused fields, due to their controllable degree of
focusing and similarity with fields used in experimental settings. Moreover, their
Lorenz-Mie scattering by a spherical particle can be calculated analytically. This, in
turn, provides us with a simple framework to compute the forces and torques exerted
on the scattering particle. This work contributes towards deepening our understanding of structured light
and its many interesting properties. What is more, the geometrical representations
presented here allow us highlight analogies of these optical fields with other physical
systems, clearly showing the reach of the results presented.