Solution of the inverse scattering problem by T-matrix completion. II. Simulations

This is Part II of the paper series on data-compatible T-matrix completion (DCTMC), which is a method for solving nonlinear inverse problems. Part I of the series contains theory and here we present simulations for inverse scattering of scalar waves. The underlying mathematical model is the scalar wave equation and the object function that is reconstructed is the medium susceptibility. The simulations are relevant to ultrasound tomographic imaging and seismic tomography. It is shown that DCTMC is a viable method for solving strongly nonlinear inverse problems with large data sets. It provides not only the overall shape of the object but the quantitative contrast, which can correspond, for instance, to the variable speed of sound in the imaged medium.Comment: This is Part II of a paper series. Part I contains theory and is available at arXiv:1401.3319 [math-ph]. Accepted in this form to Phys. Rev.

Nonlinear inverse problem by T-matrix completion. I. Theory

We propose a conceptually new method for solving nonlinear inverse scattering problems (ISPs) such as are commonly encountered in tomographic ultrasound imaging, seismology and other applications. The method is inspired by the theory of nonlocality of physical interactions and utilizes the relevant formalism. We formulate the ISP as a problem whose goal is to determine an unknown interaction potential $V$ from external scattering data. Although we seek a local (diagonally-dominated) $V$ as the solution to the posed problem, we allow $V$ to be nonlocal at the intermediate stages of iterations. This allows us to utilize the one-to-one correspondence between $V$ and the T-matrix of the problem, $T$. Here it is important to realize that not every $T$ corresponds to a diagonal $V$ and we, therefore, relax the usual condition of strict diagonality (locality) of $V$. An iterative algorithm is proposed in which we seek $T$ that is (i) compatible with the measured scattering data and (ii) corresponds to an interaction potential $V$ that is as diagonally-dominated as possible. We refer to this algorithm as to the data-compatible T-matrix completion (DCTMC). This paper is Part I in a two-part series and contains theory only. Numerical examples of image reconstruction in a strongly nonlinear regime are given in Part II. The method described in this paper is particularly well suited for very large data sets that become increasingly available with the use of modern measurement techniques and instrumentation.Comment: This is Part I of a paper series containing theory only. Part II contains simulations and is available as arXiv:1505.06777 [math-ph]. Accepted in this form to Phys. Rev.

Coherently tunable third-order nonlinearity in a nanojunction

A possibility of tuning the phase of the third-order Kerr-type nonlinear susceptibility in a system consisting of two interacting metal nanospheres and a nonlinearly polarizable molecule is investigated theoretically and numerically. It is shown that by varying the relative inter-sphere separation, it is possible to tune the phase of the effective nonlinear susceptibility \chi^{(3)}(\omega;\omega,\omega,-\omega)$in the whole range from 0 to$2\pi$.Comment: 10 pages 5 figure

Multiple Projection Optical Diffusion Tomography with Plane Wave Illumination

We describe a new data collection scheme for optical diffusion tomography in which plane wave illumination is combined with multiple projections in the slab imaging geometry. Multiple projection measurements are performed by rotating the slab around the sample. The advantage of the proposed method is that the measured data can be much more easily fitted into the dynamic range of most commonly used detectors. At the same time, multiple projections improve image quality by mutually interchanging the depth and transverse directions, and the scanned (detection) and integrated (illumination) surfaces. Inversion methods are derived for image reconstructions with extremely large data sets. Numerical simulations are performed for fixed and rotated slabs

Inversion of band-limited discrete Fourier transforms of binary images: Uniqueness and algorithms

Inversion of the two-dimensional discrete Fourier transform (DFT) typically requires all DFT coefficients to be known. When only band-limited DFT coefficients of a matrix are known, the original matrix can not be uniquely recovered. Using a priori information that the matrix is binary (all elements are either 0 or 1) can overcome the missing high-frequency DFT coefficients and restore uniqueness. We theoretically investigate the smallest pass band that can be applied while still guaranteeing unique recovery of an arbitrary binary matrix. The results depend on the dimensions of the matrix. Uniqueness results are proven for the dimensions $p\times q$, $p\times p$, and $p^\alpha\times p^\alpha$, where $p\neq q$ are primes numbers and $\alpha>1$ an integer. An inversion algorithm is proposed for practically recovering the unique binary matrix. This algorithm is based on integer linear programming methods and significantly outperforms naive implementations. The algorithm efficiently reconstructs $17\times17$ binary matrices using 81 out of the total 289 DFT coefficients.Comment: 12 page

Bound whispering gallery modes in circular arrays of dielectric spherical particles

Low-dimensional ordered arrays of optical elements can possess bound modes having an extremely high quality factor. Typically, these arrays consist of metal elements which have significantly high light absorption thus restricting performance. In this paper we address the following question: can bound modes be formed in dielectric systems where the absorption of light is negligible? Our investigation of circular arrays of spherical particles shows that (1) high quality modes in an array of 10 or more particles can be attained at least for a refractive index $n_{r}>2$, so optical materials like TiO$_{2}$ or GaAs can be used; (2) the most bound modes have nearly transverse polarization perpendicular to the circular plane; (3) in a particularly interesting case of TiO$_{2}$ particles (rutile phase, $n_{r}=2.7$), the quality factor of the most bound mode increases almost by an order of magnitude with the addition of 10 extra particles, while for particles made of GaAs the quality factor increases by almost two orders of magnitude with the addition of ten extra particles. We hope that this preliminary study will stimulate experimental investigations of bound modes in low-dimensional arrays of dielectric particles.Comment: Submitted to Physical Review

The natural resources of Bolinas Lagoon: their status and future

This publication is an integral part of the Department's high-priority inventory and assessment of coastal marshland and tideflat resources. It is intended as a guide for citizens, planners, administrators, and all others interested in the use and development of coastal lands and waters. Although the resources and problems of Bolinas Lagoon have probably been the subject of more biological and physical investigations than any small estuarine area of the California coast, many of the pertinent reports and information are not readily available to the public. Consequently, it is one purpose of this report to summarize the lagoon's history, ecological attractions, educational values and the problems facing its continued existence. At the same time, it should provide concerned citizens with a knowledge of the sources of additional and more specific information. Publication of this report is consistent with the obligation of the Department of Fish and Game to do everything in its power to protect and maintain the State's fish and wildlife resources. Therefore, its purpose transcends local issues on pollution and development, and the Department is, in fact, submitting a report to the people on the status and future of part of its inheritance and the dowry of coming generations. The report is the third of a scheduled series. It follows similar releases on Upper Newport Bay (Orange County) and Goleta Slough (Santa Barbara county) in March and June of 1970. Documentation of the resources of other critical areas is in progress. There will be future reports of this nature on Elkhorn Slough, Morro Bay, Tomales Bay, Humboldt Bay, and highly threatened marshlands in southern California. (137 pp.

Homogenization of Maxwell's equations in periodic composites

We consider the problem of homogenizing the Maxwell equations for periodic composites. The analysis is based on Bloch-Floquet theory. We calculate explicitly the reflection coefficient for a half-space, and derive and implement a computationally-efficient continued-fraction expansion for the effective permittivity. Our results are illustrated by numerical computations for the case of two-dimensional systems. The homogenization theory of this paper is designed to predict various physically-measurable quantities rather than to simply approximate certain coefficients in a PDE.Comment: Significantly expanded compared to v1. Accepted to Phys.Rev.E. Some color figures in this preprint may be easier to read because here we utilize solid color lines, which are indistinguishable in black-and-white printin

Bubble-Driven Inertial Micropump

The fundamental action of the bubble-driven inertial micropump is investigated. The pump has no moving parts and consists of a thermal resistor placed asymmetrically within a straight channel connecting two reservoirs. Using numerical simulations, the net flow is studied as a function of channel geometry, resistor location, vapor bubble strength, fluid viscosity, and surface tension. Two major regimes of behavior are identified: axial and non-axial. In the axial regime, the drive bubble either remains inside the channel or continues to grow axially when it reaches the reservoir. In the non-axial regime the bubble grows out of the channel and in all three dimensions while inside the reservoir. The net flow in the axial regime is parabolic with respect to the hydraulic diameter of the channel cross-section but in the non-axial regime it is not. From numerical modeling, it is determined that the net flow is maximal when the axial regime crosses over to the non-axial regime. To elucidate the basic physical principles of the pump, a phenomenological one-dimensional model is developed and solved. A linear array of micropumps has been built using silicon-SU8 fabrication technology, and semi-continuous pumping across a 2 mm-wide channel has been demonstrated experimentally. Measured variation of the net flow with fluid viscosity is in excellent agreement with simulation results.Comment: 18 pages, 18 figures, single colum