Disorder, Path Integrals and Localization

Anderson localization is derived directly from the path integral representation of quantum mechanics in the presence of a random potential energy function. The probability distribution of the potential energy is taken to be a Gaussian in function space with a given autocorrelation function. Averaging the path integral itself we find that the localization length, in one-dimension, is given by (E_{{\xi}}/{\sigma})(KE_{cl}/{\sigma}){\xi} where E_{{\xi}} is the "correlation energy", KE_{cl} the average classical kinetic energy, {\sigma} the root-mean-square variation of the potential energy and {\xi} the autocorrelation length. Averaging the square of the path integral shows explicitly that closed loops in the path when traversed forward and backward in time lead to exponential decay, and hence localization. We also show how, using Schwinger proper time, the path integral result can be directly related to the Greens function commonly used to study localization.Comment: 6 pages, no figure

Propagation of Vortex Electron Wave Functions in a Magnetic Field

The physics of coherent beams of photons carrying axial orbital angular momentum (OAM) is well understood and such beams, sometimes known as vortex beams, have found applications in optics and microscopy. Recently electron beams carrying very large values of axial OAM have been generated. In the absence of coupling to an external electromagnetic field the propagation of such vortex electron beams is virtually identical mathematically to that of vortex photon beams propagating in a medium with a homogeneous index of refraction. But when coupled to an external electromagnetic field the propagation of vortex electron beams is distinctly different from photons. Here we use the exact path integral solution to Schrodingers equation to examine the time evolution of an electron wave function carrying axial OAM. Interestingly we find that the nonzero OAM wave function can be obtained from the zero OAM wave function, in the case considered here, simply by multipling it by an appropriate time and position dependent prefactor. Hence adding OAM and propagating can in this case be replaced by first propagating then adding OAM. Also, the results shown provide an explicit illustration of the fact that the gyromagnetic ratio for OAM is unity. We also propose a novel version of the Bohm-Aharonov effect using vortex electron beams.Comment: 14 pages, 2 figures, submitted to Phys Rev

Fourier, Gauss, Fraunhofer, Porod and the Shape from Moments Problem

We show how the Fourier transform of a shape in any number of dimensions can be simplified using Gauss's law and evaluated explicitly for polygons in two dimensions, polyhedra three dimensions, etc. We also show how this combination of Fourier and Gauss can be related to numerous classical problems in physics and mathematics. Examples include Fraunhofer diffraction patterns, Porods law, Hopfs Umlaufsatz, the isoperimetric inequality and Didos problem. We also use this approach to provide an alternative derivation of Davis's extension of the Motzkin-Schoenberg formula to polygons in the complex plane.Comment: 21 pages, no figure

Analytic Computer Generated Hologram Design

For a given asphere the grating equation is used to derive the design for a Computer Generated Hologram (CGH), sometimes referred to as a diffractive null corrector. The CGH converts a spherical wavefront to the shape appropriate for the given asphere as required for interferometric metrology. The derivation is effectively analytic but may require numerical implementation in certain cases depending on the complexity of the shape of the asphere and the accuracy required.Comment: 21 pages, 8 figure

Nanomanufacturing: A Perspective

Nanomanufacturing, the commercially scalable and economically sustainable mass production of nanoscale materials and devices, represents the tangible outcome of the nanotechnology revolution. In contrast to those used in nanofabrication for research purposes, nanomanufacturing processes must satisfy the additional constraints of cost, throughput, and time to market. Taking silicon integrated circuit manufacturing as a baseline, we consider the factors involved in matching processes with products, examining the characteristics and potential of top-down and bottom-up processes, and their combination. We also discuss how a careful assessment of the way in which function can be made to follow form can enable high-volume manufacturing of nanoscale structures with the desired useful, and exciting, properties