No abstract

When a fan beam of monoenergetic γ rays is emitted from a point source into a plane and the scattered photons are recorded by a point detector, the locus of points over which singly Compton-scattered photons suffer the same energy loss is a circle passing through the source and detector points. Thus, the number of scattered photons recorded at a particular detector and energy can be expressed as a weighted line integral of the electron density over a circular path uniquely determined by the energy and the detector location. This defines a novel tomographic reconstruction problem in which, by recording the number of scattered photons as a function of energy and detector position, an image of electron density can be reconstructed from measurements of its line integrals over many overlapping circular paths. It is shown here that this image reconstruction problem has an analytical solution which bears some resemblance to the filtered-backprojection algorithm used in conventional (transmission) computed tomography (CT). A tomographic imaging scheme based on this idea would have several potential advantages over conventional CT systems in the field of industrial nondestructive evaluation, e.g., the ability to image from one side of the object, and the absence of moving parts.

If a two-dimensional acoustic reflectivity function is defined within the interior of a circle, reflectivity data may be acquired by transmitting acoustic pulses from isotropic elements distributed around the circumference of the circle and recording the resulting backscattered sound as a function of time. If the reflectivity function fulfills the conditions of an ’’idealized’’ weakly reflecting medium, the resulting pulse–echo data may be regarded as the line integrals of this function defined over circular arcs centered at points lying on the circumference of the enclosing circle. In this paper we show that on the basis of these line integrals the unknown reflectivity in the interior of the circle can be expressed explicitly in terms of its line integrals defined over the set of paths consisting of all circular arcs whose centers lie on the circumference of the enclosing circle. We propose that the resulting reconstruction equations could provide the foundation for a new approach to reflectivity tomography. A numerical example is also given.

We examine the problem of reconstructing a 2-D vector field v(x, y ) throughout a bounded region D from the line integrals of v(x, y ) through D. This problem arises in the 2-D mapping of fluid-flow in a region D from acoustic travel-time measurements through D. For an arbitrary vector field, the reconstruction problem is in general underdetermined since v(x, y ) has two independent components, v,(x, y) and v,(x, y). However, under the constraint that v is divergenceless ( V -v = O ) in D , we show that the vector reconstruction problem can be solved uniquely. For incompressible fluid flow, a divergenceless velocity field follows under the assumption of no sources or sinks in D.A vector central-slice theorem is derived, which is a generalization of the well-known 'scalar' central-slice theorem that plays a fundamental role in conventional tomography. The key to the solution to the vector tomography problem is the decomposition of the field v(x, y) into its irrotational and solenoidal components: v = V@ + V x I , where @(x, y) and Y(x, y ) are scalar and vector potentials. We show that the solenoidal component V X Y can be uniquely reconstructed from the line integrals of v through D , whereas the irrotational component V@ cannot. However, when the field is divergenceless in D , the scalar potential @ solves Laplace's equation in D and can be determined by the values of v on the boundary of D. An explicit formula €or @ from the boundary values of v is derived. Consequently, v(x, y ) can be uniquely recovered throughout the region of reconstruction from the following information: line-integral measurements of v through this region and v measured on the boundary of this region.

The spatial and spectral responses of the plasmonic fields induced in the gap of 3-D Nanoshell Dimers of gold and silver are comprehensively investigated and compared via theory and simulation, using the Multipole Expansion (ME) and the Finite Element Method (FEM) in COMSOL, respectively. The E-field in the dimer gap was evaluated and compared as a function of shell thickness, inter-particle distance, and size. The E-field increased with decreasing shell thickness, decreasing interparticle distance, and increasing size, with the error between the two methods ranging from 1 to 10%, depending on the specific combination of these three variables. This error increases several fold with increasing dimer size, as the quasi-static approximation breaks down. A consistent overestimation of the plasmon’s FWHM and red-shifting of the plasmon peak occurs with FEM, relative to ME, and it increases with decreasing shell thickness and inter-particle distance. The size-effect that arises from surface scattering of electrons is addressed and shown to be especially prominent for thin shells, for which significant damping, broadening and shifting of the plasmon band is observed; the size-effect also affects large nanoshell dimers, depending on their relative shell thickness, but to a lesser extent. This study demonstrates that COMSOL is a promising simulation environment to quantitatively investigate nanoscale electromagnetics for the modeling and designing of Surface Enhanced Raman Scattering (SERS) substrates.