Theory of single- and few-mode lightguides

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.