Phd in MATHEMATICAL MODELS FOR ENGINEERING, ELECTROMAGNETICS AND
NANOSCIENCES

Thesis title: Scattering of Electromagnetic Waves (theory and biological applications)

Electromagnetic (EM) wave scattering is a progressive, multidisciplinary area of research with countless applications in the fields of medical imaging, remote sensing and diagnostics. In particular, the subject of EM scattering have great theoretical and practical challenges need to include multiple scattering effects accurately. This thesis is devoted the result of the research activity carried out in the three-year PhD program, which is stated as the topics concerning EM scattering problems in two section: In the first section, we study the theoretical aspect of EM scattering waves. The basic introduction of EM theory (see in chapter 1), we describe the Maxwell’s equations and its approximated solutions of the EM field can be combine into vector and scalar Helmholtz equations of the EM waves scattering by a particle of a simple geometry. Such problems can be figure out exactly by expanding the fields in terms of scalar or vector waves function in separable coordinates system, depending on the shape of the particle. In the second section, we discuss the basic scattering theory and its application in biological sector. We begin the study of spherical vector wave function using translation Addition Theorem (AT) (see in chapter 2). The first step of the research consisted in an in-depth study of the effect of finite terms on the truncation error of translation AT for spherical vector wave function. A finite sum adopted for the translational AT of the spherical vector wave functions is a key-point for addressing a well-known concept regarding the classical electromagnetic scattering of an incoming wave across a generic disposition of spheres. The AT is essential for the translation of the vector spherical wave function expressed with respect to a coordinates system toward a different reference. According to its general formulation, the dependence of the translation coefficients on the relative direction of displacement linking two distinct references is the core study to be addressed. The inherent analytical aspects have been intensively studied during the last decades, but the respective numerical encoding has been limited to few basic configurations, which provided a partial validity of the theory with quite approximate solutions. In the literature, there are many authors reporting different sets of the vector translation coefficients, among which we mention those calculated by Stein, Cruzan and, Mackowski as the most prominent of them. We have selected the Cruzan formulation of the vector translation coefficients for its structure based on the Wigner 3-j function. We have developed the criteria of truncating the inherent infinite series to achieve convergence with a finite version of the same, which leads to an extremely negligible error. During our numerical tests, we have deeply investigated generic truncation errors and outlined a repeatable procedure to get an acceptable convergence. The procedure relies on the different parameters of the vector function, the propagation constant, and so on. Our method allows to calculate a suitable truncation limit which ensures acceptable results in terms of contained AT truncation errors and simultaneously a convergent series able to efficiently reconstruct the generic displaced vector wave function. We study the translation criteria of scattering from PEC sphere along three-dimensional axes using semi-analytical approach (see in chapter 3). The scattering model is based on a generalized Lorenz-Mie theory framework and ensemble with the vector translation AT for the Vector Spherical Harmonics (VSH). Applying extended Mie theory on a sphere leads to a set of an unknown coefficients by the use of translation AT. We have selected the Cruzan formulation of the vector translation coefficients for its structure based on the Wigner 3-j function. As an illustration, we investigated the numerical simulations of total scattered field of multiple PEC sphere using vector translation AT. We used advance computational tools and approaches for mathematical modeling of an observation. During our numerical test, we have deeply investigated generic truncation criteria in scattered electric field using translation AT. However, we have been obtained numerical validation by using computational approach. We study a homogenization model for cloaking applications (see in chapter 4). In the given model, we consider an isotropic inner layer, which is coated with a multilayer anisotropic circular cylinder. We describe the electrostatic response of a Polarly Radially Anisotropic (PRA) multilayer circular cylinder. It consists of different components of the permittivity in radial and tangential directions for each layer. We have shown the mathematical derivation of the polarizability and effective permittivity of a PRA multilayer circular cylinder. Moreover, we demonstrate the cloaking behavior using a PRA multilayer circular cylinder. Meanwhile, we test our formation with advanced computational approaches. During our numerical test, we have investigated the numerical results of the polarizability of a PRA multilayer circular cylinder. We employ the inner cloak surrounded by a multilayer anisotropic circular cylinder. However, for an ideal cloak, the contrast between the permittivity parameters approaches infinity. We have also presents the Transparent Radially Anisotropic (TRA) cylinder with permittivity dyadic, in the electrostatic settings. The quasi-static polarizability of the TRA cylinder with the values and signs of permittivity components offering different interesting features, involving, absorbance, anomalous losses, resonant singularities, invisibility. We have a special focus on the electrostatic response of a circular cylinder with a certain condition of anisotropic parameters. We study the stimulating behaviour of polarizability of a TRA cylinder in quadrant (plane geometry). We have also computed the TRA cylinder behaves as a cloak using suitable choices of the anisotropic parameters. We study a Spherically Radially Anisotropic (SRA) multilayer sphere with an arbitrary number of layers (see in chapter 5). Within each layer permittivity components are different from each other in radial and tangential directions. Under quasi-static approximation, we have developed a more generalized mathematical model that can be used to calculate polarizability of the SRA multilayer sphere with any arbitrary number of layers. Moreover, the working of the SRA multilayer sphere as a cloak has been investigated. It has been shown that by choosing a suitable contrast between components of the permittivity, the SRA multilayer sphere can achieve threshold required for invisibility cloaking. We describe the characterization of microscopical structural anisotropy of biological tissues by polarization imaging (see in chapter 6, 7). Biological structures such as wood fibers or tissue cells often have a characteristic alignment of their molecular architecture. Therefore cells, tissues, organelles can show birefringence and dichroism with the passage of light, which can be detected and measured accurately by performing polarized light imaging. This technique allows to highlight both micro and nano-structures of the materials, characterizing the anisotropy of the optical properties due precisely to the specific molecular order. Hence, absorption, reflection, refraction and dispersion measures of polarized light by structured materials is a powerful tool for analyzing the morphological order of the organization. We have created images of biological structures with micro- and nano-structures by illuminating the samples with linear polarization light. The wavelength has been chosen from time to time to increase the contrast of the recorded images. The samples were illuminated orthogonally but observed at different angles in order to highlight the different behavior of the diffused light according to the viewing angle. The images were mathematically treated to determine the depolarization anisotropy of the light diffused towards the observer. Jones matrix analysis and Stoke's vector representation of light through the Mueller matrix have been implemented to characterize the final polarization state and increase the contrast.