# Theory of single- and few-mode lightguides

## Date

1984

## Authors

Black, Richard James

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## Abstract

The thesis is divided into three main parts. After introducing the
thesis with some background material in Chapter 1, Part I (Chs.2-3) sets
the scene by examining how lightguide studies tie in with other physical
theories, and studying some one-dimensional examples. In Part II (Chs.
4-6) we examine certain waveguiding effects for specific fiber profile
shapes, in particular, higher mode "splitting" due to polarization and
noncircularity. In Part III (Chs.7-8) we investigate some techniques
for analyzing fibers of arbitrary profile shape. We conclude with
Chapter 9 which further develops the themes of Parts II and III. A more
detailed outline is as follows.
In Chapter 2 we provide a simple and unified formalism for the
analogy between fiber-optics and mechanics. It is based on the scalar
theory of light for an optical fiber with longitudinally-independent
refractive index, and the mechanics of a particle in a time-independent
potential. Illustrative examples are given. We also mention the
conceptual analogies that arise when we include polarization and
longitudinal- or time-dependent variations. This should prove useful
both in teaching and as a bridge between research areas.
In Chapter 3 we examine planar lightguides which are of interest in
integrated optics and as simple models for fiber effects considered
later in the thesis. In addition to some scalar solutions for specific
profiles, we examine methods for general profiles and for "exact"
numerical solutions recommend use of an adaptation of the Sammut-Pask
shooting-extrapolation method developed originally for fibers. We also
consider a Green function method and the planar lightguide form of the
Gaussian approximation. Then we consider the inclusion of polarization,
and a one-dimensional model of a visual photoreceptor.
In Chapter 4 we examine the theory of few-mode polarization effects
on circularly symmetric fibers. General equations are found which
determine arbitrary order corrections to the scalar wave equation. The
higher-order corrections are of particular interest when there is a
degeneracy at lower orders, and are also required to increase the
accuracy for large numerical aperture fibers. The TEom and TMom modes are the ones for which an excitation dependent polarization splitting
occurs. These modes have corrections to all orders determined by
knowledge of the scalar solutions: we obtain expressions suitable for
computer algebraic evaluation. Explicit polarization splittings are
found for the infinite-parabolic and clad power-law profiles. In
particular we find that in contrast to polarization splittings for the
fundamental modes, those between the TEQm and TMQm modes are highly
profile dependent.
In Chapter 5 we consider the application of few-mode polarization
effect studies to absorption in visual photoreceptors. In particular,
we investigate the direction, wavelength dependence and magnitude of the
polarization dependence of absorption in photoreceptors, assuming as our
model, bound mode theory of uniformly absorbing dielectric waveguides.
This also has application to few-mode optical fibers. First, a physical
understanding is given in terms of simple concepts from plane-wave
theory. Then, in undertaking a modal analysis, we find (in the spirit
of the Gaussian approximation) that the infinite-parabolic profile
provides a simple qualitative understanding of trends. Quantitative
numerical results are given for the step profile.
In Chapter 6 we provide an understanding of the order in which
modes of noncircular lightguides are cutoff, discuss their eigenvalues,
and note some interesting degeneracies.
In Chapter 7 we develop a comparison method which identifies a
lightguide with properties that are well known and very similar to the
one under investigation. It has a very simple mathematical basis which
allows a trivial derivation of the moment method for circular crosssection
fibers including W-fibers, and an extension to non-circular
lightguides. We first use the scalar approximation, and then extend to
the full vector theory which accounts for polarization. Our concern is
with fundamental mode propagation constants and higher mode cutoffs.
In Chapter 8 we firstly extend the moment method to symmetric slab
lightguide, thus providing a simple description of both the fundamental
mode and the second mode cutoff point. Secondly, we further develop the
theory of equivalent step index fibers. The equivalent step method
rests on the concept that properties of the fundamental mode are not very sensitive to refractive index profile details. We show how this
idea may be incorporated into the original variational scheme of Snyder
and Sammut in order to greatly simplify the calculations and to provide
wavelength independent equivalent step parameters. The new approach
uses the effective waveguide parameter and moments of the profile shape
function.
In Chapter 9 we firstly show that highly effective single-mode
single polarization ( SMSP) fibers can be made with comparitively small
values of birefringence, provided that the profile heights Δₓ,Δy differ
and the fiber is bent. The effect is enhanced in anisotropic W-fibers.
Secondly, „e consider a transformation of the scalar wave equation to an
integral equation using the Green function for a step reference profile,
give results for circular profiles as examples, and examine 1
generalization to noncircular lightguides.

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Thesis (PhD)

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