Anthony J. Devaney’s research while affiliated with Northeastern University and other places

1. Radiation and initial value problems for the wave equation; 2. Radiation and boundary value problems in the frequency domain; 3. Eigenfunction expansions of solutions to the Helmholtz equation; 4. Angular spectrum and multipole expansions; 5. The inverse source problem; 6. Scattering theory; 7. Surface scattering and diffraction; 8. Classical inverse scattering and diffraction tomography; 9. Waves in inhomogeneous media; 10. Time reversal imaging for systems of discrete scatterers; 11. The electromagnetic field; Appendices; Index.

The central goal of this project was the investigation of the enhancements in antenna performance and in imaging capability by embedding a source or scatterer in a background medium including metamaterials. This effort was motivated mostly by the possibility of embedding antennas in designer background media so as to obtain radiation performance that would not be possible for comparable antennas in free space. This problem was treated in the present work within a general and non-device-specific framework whose predictions (such as performance bounds) under normalized resources are fundamental. The results were discussed addressing separately the cases of small versus large or resonant antennas, with the overall conclusion that for small antennas one can significantly enhance the radiated power or compress source size via antenna substrates under normalized antenna resources, while for larger antennas the use of substrates can significantly enhance both radiated power and directivity (related to the number of essentially independent field modes that can be radiated effectively) under the given resources.

The theory of the inverse source problem is employed to compute a class of continuously distributed and compactly supported three-dimensional (volume) sources that radiate the scalar wavelets investigated by Kaiser as well as certain electromagnetic generalizations of these scalar fields. These efforts have shown that the scalar wavelet fields can be radiated by a distributional source (generalized function) supported on a circular disk of radius a or an oblate spheroid surrounding that disk. Our main goal here is to replace this distributional source by a more conventional volume source that radiates the same wavelet field outside its support volume. The equivalent volume sources computed in this paper are supported on (three-dimensional) spherical shells whose outer radius a₊ > a and inner radius a_- < a ₊ are arbitrary. These sources are analytic functions of position within their support volumes for any finite, but arbitrarily large temporal frequency omega, and possess minimum L² norm among all possible solutions to the inverse source problem with the given support volume constraint. Electromagnetic versions of the wavelet sources and fields are shown to be easily derived from their scalar wave counterparts.

The filtered back-propagation algorithm (FBP algorithm) is a computationally fast and efficient inversion algorithm for reconstructing the 3D index of refraction distribution of weak scattering samples in free space from scattered field data collected in a set of coherent optical scattering experiments. This algorithm is readily derived using classical Fourier analysis applied to the Born or Rytov weak scattering models appropriate to scatterers embedded in a non-attenuating uniform background. In this paper, the inverse scattering problem for optical diffraction tomography (ODT) is formulated using the so-called distorted wave Born and Rytov approximations and a generalized version of the FBP algorithm is derived that applies to weakly scattering samples that are embedded in realistic, multiple scattering ODT experimental configurations. The new algorithms are based on the generalized linear inverse of the linear transformation relating the scattered field data to the complex index of refraction distribution of the scattering samples and are in the form of a superposition of filtered data, computationally back propagated into the ODT experimental configuration. The paper includes a computer simulation comparing the generalized Born and Rytov based FBP inversion algorithms as well as reconstructions generated using the generalized Born based FBP algorithm of a step index optical fiber from experimental ODT data.

The purpose of this paper is to provide new and simplified statements of local Paley–Wiener theorems on the (n − 1)-dimensional unit sphere realized as a subset of n = 2, 3 Euclidean space. More precisely, given a function , whose restriction to an n − 1 sphere is analytic, we establish necessary and sufficient conditions determining whether f is the Fourier transform of a compactly supported, bounded function . The essence of this investigation is that, because of the local nature of the problem, the mapping f → F is not in general invertible and so the problem cannot be studied via a Fourier integral. Our proofs are new.

There are many methods for identifying the shape and location of scatterers from far field data. We take the view that the connections between algorithms are more illuminating than their differences, particularly with regard to the linear sampling method [D. Colton and A. Kirsch, Inverse Probl. 12, No. 4, 383–393 (1996; Zbl 0859.35133)], the point source method [R. Potthast, Point sources and multipoles in inverse scattering theory. London: Chapman & Hall/CRC (2001; Zbl 0985.78016)], and the MUSIC algorithm [A. J. Devaney, IEEE Trans. Antennas Propag. 53, 1600–1610 (2005)]. Using the first two techniques we show that, for a scatterer with Dirichlet boundary conditions, there is a nontrivial incident field that does not generate a scattered field. This incident field, written as an expansion of eigenfunctions of the far field operator, is used in the MUSIC algorithm to image the shape and location of the obstacle as those points z where the incident field is orthogonal to the far field pattern due to a point source located at z. This has two intriguing applications, one for inverse scattering and the other for signal design. Numerical examples demonstrate these ideas.

The scalar wave inverse source problem (ISP) is investigated for the case where the source is embedded in a non-homogeneous medium with known index of refraction proÞle n(r). It is shown that the solution to the ISP having minimum energy (so-called minimum energy source) can be obtained via a simple method of constrained optimization. This method is applied to the special case when the non-homogeneous background is spherically symmetric (n(r )= n(r)) and yields the minimum energy source in terms of a series of spherical harmonics and radial wave functions that are solutions to a Sturm-Liouville problem. The special case of a source embedded in a spherical region of constant index that differs from the background is treated in detail and results from computer simulations are presented for this case.

A recently developed inverse scattering method based on the distorted-wave Born approximation (DWBA) that applies to objects embedded in known background media [ Inverse Probl. 19, 855 (2003); 20, 1307 (2004) ] is implemented for the special case of circularly symmetric scatterers embedded in circularly symmetric backgrounds. The newly developed scheme is applied in a computer-simulation study of optical diffraction tomography (ODT), and the results are compared and contrasted with reconstructions obtained using the filtered backpropagation algorithm (FBP algorithm). Unlike the DWBA-based inversion algorithm, the FBP algorithm does not take into account multiple scattering within the known background, and it is found that the newly implemented scheme yields reconstructions much superior to those obtained using the FBP algorithm. The research reported applies to a number of important applications that include ultrasound nondestructive evaluation testing of cylinders for defects as well as ODT.

The treatment of time-reversal imaging of multiply scattering point targets developed by the present authors in Gruber et al. ["Time-reversal imaging with multiple signal classification considering multiple scattering between the targets," J. Acoust. Soc. Am., 115, 3042-3047 (2004)] is reformulated and extended to the estimation of the target scattering strengths using the Foldy-Lax multiple scattering model. It is shown that the time-reversal multiple signal classification (MUSIC) pseudospectrum computed using the background Green function as the steering vector yields accurate estimates of the target locations, even in the presence of strong multiple scattering between the targets, and that the target scattering strengths are readily computed from the so-determined target locations using a nonlinear iterative algorithm. The paper includes computer simulations illustrating the theory and algorithms presented in the paper. (c) 2005 Acoustical Society of America.

A recently developed inverse scattering algorithm [ A. J. Devaney and M. Dennison, Inverse Probl., 19, 855 (2003) and M. Dennison and A. J. Devaney, Inverse Probl., 20, 1307 (2004) ] is described and applied in a computer simulation study of optical diffraction tomography (ODT). The new algorithm is superior to standard ODT reconstruction algorithms, such as the filtered backpropagation algorithm, in applications employing a limited number of scattering experiments (the so-called limited-view case) and also in cases where multiple scattering occurs between the object being interrogated and the (known) background in which the object is embedded. The new algorithm is compared and contrasted with the filtered backpropagation algorithm in a computer simulation of ODT of weakly inhomogeneous cylindrical objects being interrogated in a limited number of scattering experiments employing incident plane waves. Our study has potential applications in biomedical imaging and tomographic microscopy.