Singularimetry: phase measurement using vortices and caustics

2017-03-01T04:56:13Z (GMT) by Eastwood , Samuel Anthony
This thesis investigates singularimetry in both optical and matter wave fields. The utility of optical singularities for performing phase measurements was demonstrated using a three–beam interferometer. Three–wave interference was used to generate a uniform lattice of optical vortices, which was distorted by the presence of an object inserted into one arm of the interferometer. Using theoretical ideas from singular wave optics, a proportionality between the transverse displacement of the vortices and the phase shift in the object wave was derived and experimentally tested. Tracking the vortices permitted the phase of the object to be reconstructed. We demonstrated the method experimentally using a simple lens and a more complex object, namely the wing of a common house fly. Since the technique is implemented in real space, it is capable of reconstructing the phase locally. By studying the extreme opposite of nodal singularities, an alternative phase retrieval technique was designed and numerically tested, which utilizes the natural intensity singularities of caustics. Using catastrophe theory, we showed that, given the formation of a fold caustic in the wave field, the functional form of the wave’s phase may be expressed as a truncated Taylor series. We then outlined how all expansion coefficients in the series may be determined by quantifying unfolding of the caustic through focus, thus framing this otherwise ill–posed inverse problem into a well–posed one. The method was then successfully implemented on simulated data. Possibilities of extending the technique to higher order catastrophes are also discussed. The relationship between caustics and vortices in matter waves was also investigated. By inducing aberrations in the magnetic lenses of a conventional transmission electron microscope, it is shown that diffraction catastrophes may be created in the electron beam. As a consequence of the duality of caustics and vortices—a central theme to the thesis—electron vortices were created in the electron beam in the vicinity of the caustics. To measure and explore the quantized anomalous Gouy shift of focused astigmatic electron waves, caustics were again generated by inducing aberrations in the lenses and the Gouy phase shift determined by phase retrieval. Multiple theoretical descriptions of the Gouy phase anomaly are presented and discussed. It is shown that these various interpretations may be unified into a single theoretical framework. Lastly, we investigated caustics in the context of order–parameter manifolds. We show that caustic surfaces also appear when a real or complex field is mapped to its order–parameter manifold. We exemplify these structures in the context of spin–1/2 fields, where the order– parameter manifold is the Bloch sphere. These generic structures are a manifestation of catastrophe theory and are stable with respect to perturbations. The corresponding field configurations are also stable and represent a new type of topological defect, which we call order–parameter catastrophe defects. Equations governing the conditions for the existence and unfolding of the defects are derived.