Geometry of Dirac Operators

Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M \hat{\mathcal{A}}(TM)\wedge \text{ch}(\mathcal{V}) \] where $\mathcal{V} =\text{Hom}_{\mathbb{C}l(TM)}(\slashed{\mathsf{S}},S)$. We prove the above statement with the means of the heat kernel of the heat semigroup $e^{-tD^2}$. The first outstanding result is the McKean-Singer theorem that describes the index in terms of the supertrace of the heat kernel. The trace of heat kernel is obtained from local geometric information. Moreover, if we use the asymptotic expansion of the kernel we will see that in the computation of the index only one term matters. The Berezin formula tells us that the supertrace is nothing but the coefficient of the Clifford top part, and at the end, Getzler calculus enables us to find the integral of these top parts in terms of characteristic classes.

Optical properties of chiral thin films and microparticles

In this work I study the optical properties of helical particles and chiral sculptured thin films, using computational modeling (discrete dipole approximation, Berreman calculus), and experimental techniques (glancing angle deposition, ellipsometry, scatterometry, and non-linear optical measurements). The first part of this work focuses on linear optics, namely light scattering from helical microparticles. I study the influence of structural parameters and orientation on the optical properties of particles: circular dichroism (CD) and optical rotation (OR), and show that as a consequence of random orientation, CD and OR can have the opposite sign, compared to that of the oriented particle, potentially resulting in ambiguity of measurement interpretation. Additionally, particles in random orientation scatter light with circular and elliptical polarization states, which implies that in order to study multiple scattering from randomly oriented chiral particles, the polarization state of light cannot be disregarded. To perform experiments and attempt to produce particles, a newly constructed multi stage thin film coating chamber is calibrated. It enables the simultaneous fabrication of multiple sculptured thin film coatings, each with different structure. With it I successfully produce helical thin film coatings with Ti and TiO_{2}. The second part of this work focuses on non-linear optics, with special emphasis on second-harmonic generation. The scientific literature shows extensive experimental and theoretical work on second harmonic generation from chiral thin films. Such films are expected to always show this non-linear effect, due to their lack of inversion symmetry. However no experimental studies report non-linear response of chiral sculptured thin films. In this work I grow films suitable for a second harmonic generation experiment, and report the first measurements of non-linear response.