Chapter 1
Introduction

The purpose of this chapter is to provide an overview of the book. First, the concept of electromagnetic inverse scattering problems (ISPs) is introduced, which is followed by their scientific and real-world applications. Second, we address the forward scattering problem, also known as the direct problem. Third, the fundamental properties of electromagnetic ISPs, including the existence, uniqueness, and stability of the solution, are presented. The inherent nonlinearity of ISPs is emphasized and the classification of ISPs is discussed. Finally, the scope of the book is specified. The topics covered by the remaining chapters are overviewed, which is followed by extension of the methods presented in the book to other areas. Other related topics that are not covered by the book are briefly mentioned.

1.1 Introduction to Electromagnetic Inverse Scattering Problems

The electromagnetic scattering problem deals with determining the scattered field generated by a given scatterer when it is illuminated by incoming electromagnetic waves. This is also called the forward or direct problem. The opposite of the forward problem is called the inverse problem. Electromagnetic inverse scattering is concerned with determining the nature of an unknown scatterer, such as its shape, position, and material, from knowledge about measured scattered fields.

Figure 1.1 shows a schematic diagram of inverse scattering problems. An unknown scatterer is located in the domain , referred to as the domain of interest (DOI), and is illuminated by incoming waves generated by transmitters labelled Tx1, Tx2, . For each illumination, the scattered fields are measured by an array of receivers labelled Rx1, Rx2, . The goal of the inverse scattering problem is to determine the shape, position, and material of the scatterer from the measured scattered fields.

Figure 1.1 Schematic diagram of inverse scattering problems.

Using electromagnetic waves to probe obscured or remote regions, the imaging techniques based on electromagnetic ISPs are suitable for a wide range of applications. For example, in nondestructive evaluation (NDE), the ISP has been applied to detection of possible cracks in civil and industrial structures [1-4]. In geography, this is used in remote detection of subsurface inclusions, such as detecting unexploded ordnance and mines [5, 6]. In the oil industry, it is used for oil and gas exploration [7]. In medicine, it is used for the detection of the early stages of breast cancer [8-12]. In security checks, it is applied to concealed weapon detection [13]. It can also be used for material characterization, such as the determination of constituents and evaluation of porosity [14]. Some real-world applications of inverse scattering in the microwave range can be found in chapter 10 of [15]. In physical science, the interpretation of Rutherford's gold foil experiment, which discovered the atomic nucleus, is also an inverse scattering problem.

From this short and incomplete list, it is apparent that the scope of electromagnetic ISP is extensive and its applications are diverse and important. Nevertheless, compared with its increasing importance, research in inverse scattering technique is still in the nascent stage. The purpose of this book is to introduce several computational methods for solving electromagnetic ISPs. Before discussing the inverse problem, we have to give the rudiments of the corresponding forward problem, which is the topic of the next section.

1.2 Forward Scattering Problems

Electromagnetic scattering theory is based on Maxwell's equations. Maxwell's equations are four partial differential equations that describe the electric and magnetic fields arising from distributions of electric charges and currents. Electromagnetic scattering occurs when scatterers are illuminated by a radiation source. The perturbation field due to the presence of scatterers is referred to as the scattered field; that is, the scattered field is the difference between the fields with and without the scatterers. Since the scattering problem is formulated in an unbounded domain, the boundary condition at infinity is called the radiation boundary condition, which requires the scattered field to be a local plane wave that propagates outward.

Broadly speaking, scatterers can be categorized into two types: Penetrable and impenetrable scatterers. For penetrable scatterers, the wave field is not zero inside the scatterers and satisfies the wave equation that depends on the constitutive parameters of the scatterer. At the interface between a penetrable scatterer and the background medium, continuity of certain components of electric and magnetic fields should be satisfied. For an impenetrable scatterer, the wave field is zero inside the scatterer and the total field satisfies a certain boundary condition, such as the Dirichlet (or sound-soft [16]) boundary condition, Neumann (or sound-hard [16]) boundary condition, or the impedance boundary condition. In this book, scatterers made of nonmagnetic dielectric material are penetrable scatterers, and scatterers made of perfect electric conductors (PEC) are chosen for impenetrable scatterers. In solving ISPs, the values of permittivity of dielectric scatterers have to be reconstructed, whereas the boundary of PEC scatterers has to be determined. In the applied mathematical community, scattering problems involving penetrable and impenetrable scatterers are often referred to as the medium and obstacle problem, respectively [16, 17].

This book deals with time-harmonic waves; that is, monochromatic waves. We do not specify any particular frequency range; for example, radio frequency, microwave, millimeter wave, or optical wave. Instead, we are interested in expressing dimensions and positions in terms of wavelength. The mathematical methods, both theoretical and numerical ones, used to investigate the forward and inverse scattering problems depend heavily on the operating frequency of the wave. For scatterers whose dimensions are much larger than the wavelength, the mathematical methods used to study their scattering phenomena are very different from those used for scatterers whose dimensions are much smaller than, or comparable to, the wavelength. This book is primarily concerned with the forward and inverse scattering problems associated with the scatterers whose dimensions are much smaller than, or comparable to, the wavelength.

The theories, formulations, and computational methods for the (forward) scattering problem are provided in Chapter 2.

1.3 Properties of Inverse Scattering Problems

Following the definition by Hadamard [18], a problem is well posed if its solution exists, is unique, and depends continuously on data. If one of these conditions is not satisfied, the problem is ill- or improperly posed. It is obvious that the first two properties, that is, the existence and uniqueness, should be discussed when the data is noise-free. Otherwise, for example, for a given set of measurement data that are contaminated with noise (such as measurement error and background noise), if there is no candidate acting as an input to the problem that produces an output exactly matching the measured data, then the solution to the problem does not exist. The last property, referred to as continuity (or stability), essentially means that a small perturbation of the data results in a small perturbation of the solution. Mathematical techniques known as regularization methods have been developed to construct a stable approximate solution of an ill-posed problem. More details on ill-posedness and regularization can be found in Appendix A.

For electromagnetic inverse scattering problems, we will address the following questions: the existence, uniqueness, and stability of the solution, the inherent nonlinearity, and classifications.

For electromagnetic inverse scattering problems, the question about existence is trivially confirmative since the measured scattering data for an inverse scattering problem must be generated by a certain scatterer and obviously this scatterer is an automatic solution to the inverse scattering problem. Turning to the question of uniqueness, [19] and Section 7.1 of [16] proved the uniqueness theorem under certain conditions for dielectric and PEC scatterers, respectively. The conclusion for dielectric scatterers is that, under certain conditions, for a fixed wavenumber and all directions of incidence and all polarizations of the electric field, the knowledge of the electric far field pattern for all angles uniquely determines permittivity. The conclusions for PEC scatterers are that (1), for a fixed wavenumber, the electric far field patterns for all incidence direction and all polarizations uniquely determine the PEC scatterer; and (2), for one fixed incidence direction and polarization, the electric far field pattern for all wavenumbers contained in some interval uniquely determines the PEC scatterer.

It is important to note that this book concentrates mainly on computational methods that solve inverse scattering problems with a unique solution. Inverse scattering problems that do not have a unique solution are not considered in this book. In fact, the conditions of non-uniqueness are rather stringent, and thus in practice such inverse scattering problems are not often encountered. For example, for anisotropic scatterers, if the permittivity and permeability are allowed to be zero or infinite, then it is possible to have infinite solutions to the inverse scattering problem. One of the applications of such non-uniqueness is invisibility and cloaking,...