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Rutherford backscattering spectrometry (RBS) is shown to be an elegant, powerful tool for the chemical characterization of optical coatings. RBS studies of several thin film materials are presented to illustrate the technique's unique abilities, and to show how RBS is best exploited in investigations of thin film stoichiometry and diffusion. The text begins with an introduction to optical coatings and the practical problems encountered in their implementation. The basic principles of RBS are discussed, and the technique is compared to other popular surface analysis tools. The introductory material concludes with a chapter devoted to specific techniques for RBS data and error analysis, including the derivation of a simple formula for determining the optimum thickness of multi-element samples. The accurate stoichiometric measurements provided by RBS give new insights into the chemical structure of ion-bombarded MgF₂ coatings. The analysis shows that lightly-bombarded coatings contain a small oxygen fraction (< 6%), and the absence of this oxygen in opaque, heavily-bombarded samples implies the oxygen compensates for fluorine deficiencies and is therefore an essential ingredient for transparent films. This beneficial oxygen appears to diffuse into the coatings along columnar voids, and the implied compromise between packing density and transparency is discussed. The final chapter takes advantage of the nondestructive depth-profiles provided by RBS. We present the first direct experimental verification of the interfacial oxide layer responsible for the superior adhesion of aluminum to glass, and show that contrary to popular belief, the layer is not an artifact of oxygen adsorbed during the aluminum's evaporation. We then discuss the diffusion of copper through silver films, and show that the migration is enhanced by exposure to the RBS probe beam. Finally, we consider the diffusion of carbon, from graphite substrates, into the voids of porous coatings during the RBS measurements. This effect, like the enhanced copper diffusion, is consistent with a low temperature, measurement-induced anneal; however, we show that the migrant carbon does not alter the chemical structure of the coatings, but instead serves as a convenient, non-intrusive indicator of film porosity.

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Monopoles are solutions of an SU(2) gauge theory in R3 satisfying a lower bound for energy and certain asymptotic conditions, which translate as topological properties encoded in their charge. Using methods from integrable systems, monopoles can be described in algebraic-geometric terms via their spectral curve, i.e. an algebraic curve, given as a polynomial P in two complex variables, satisfying certain constraints. In this thesis we focus on the Ercolani-Sinha formulation, where the coefficients of P have to satisfy the Ercolani-Sinha constraints, given as relations amongst periods. In this thesis a particular class of such monopoles is studied, namely charge 3 monopoles with a symmetry by C3, the cyclic group of order 3. This class of cyclic 3-monopoles is described by the genus 4 spectral curve X , subject to the Ercolani-Sinha constraints: the aim of the present work is to establish the existence of such monopoles, which translates into solving the Ercolani-Sinha constraints for X . Exploiting the symmetry of the system,we manage to recast the problem entirely in terms of a genus 2 hyperelliptic curve X, the (unbranched) quotient of X by C3 . A crucial step to this aim involves finding a basis forH1( X; Z), with particular symmetry properties according to a theorem of Fay. This gives a simple formfor the period matrix of X ; moreover, results by Fay and Accola are used to reduce the Ercolani-Sinha constraints to hyperelliptic ones on X. We solve these constraints onX numerically, by iteration using the tetrahedral monopole solution as starting point in the moduli space. We use the Arithmetic-GeometricMean method to find the periods onX: this method iswell understood for a genus 2 curve with real branchpoints; in this work we propose an extension to the situation where the branchpoints appear in complex conjugate pairs, which is the case for X. We are hence able to establish the existence of a curve of solutions corresponding to cyclic 3-monopoles.