Direct and inverse electromagnetic scattering problems in spherically layered media

The direct and inverse scattering problems related to radially inhomogeneous spheres have interesting mathematical properties alongside practical value. In the direct scattering problem, the acoustic or elecromagnetic parameters of the spherical scatterer, which are arbitrary functions of the radial distance only, are assumed to be known; and the aim of the problem is to determine the scattered field in the whole space. On the other hand, for the inverse problem, these parameters constitute the unknowns of the problem, and they are determined through the value of the scattered field supposed to be measured on a measurement surface in the outside region. For the solution of the direct scattering problem, different techniques, with certain advantages and limitations, exist in the literature. The analytical techniques such as dyadic Green's function are only valid for certain types of profile, whereas numerical solution techniques such as method of moments have limited accuracy and they are in general computationally expansive. For the inverse problem, established methods such as Newton's method or the contrast source inversion technique can be applied directly to the three dimensional problem; however, the computational effort necessary for this type of procedure will put a limit to its practical value. Therefore, it can be concluded that the problems involving radially inhomogeneous spherical scatterer is still open to contributions. In this thesis, a method to reduce the original three dimensional acoustic and electromagnetic problems into one dimensional forms has been developed. It has been demonstrated that such a method would be compatible with the available alternatives, and it will require less computational effort than the three dimensional solution techniques. It should be noted that although the orginal problem is a three dimensional one, the homogeneity along the angular direction enables one to replace it with one dimensional object and data equations involving only radial functions. For this dimension reduction procedure, the orthogonality of the spherical harmonics over the unit spherical surface have been used. In the acoustic case, the scalar acoustic field has been expressed as a series expansion in terms of scalar spherical harmonics. Since those are functions of the angular terms, and the geometry is spherically symmetrical, their orthogonality is preserved within the original three dimensional object equation. Therefore, it is possible to eliminate the angular terms via orthogonality relation, and to obtain one dimensional reduced integral equations involving the coefficients of the series expansion for the acoustic field. For the solution of the direct scattering problem these coefficients can be determined by a simple discretization of the one dimensional integrals along the radial direction. On the other hand, a Newton based iterative scheme has been formulated for the solution of the inverse scattering problem. In this formulation, the one dimensional equations are solved using an initial guess for the unknown parameters, and the coefficients of the measured scattered field constitute the data that is used to update the initial guess iteratively. The procedure for the electromagnetic case is similar to the acoustic one. However, since the problem is a vectorial one in this case, the electric field is expanded in terms of vector spherical harmonics. These vectorial functions also satisfy the orthogonality condition over a spherical surface, therefore it is possible to reduce the vectorial electric field integral equation into a system of one dimensional integral equations. The integrals once again will only contain the scalar coefficients of the series expansion for the electric field, which are functions of the radial distance only. The resulting system of equations can be solved simultaneously to obtain the coefficients. The Newton based algorithm that was used in the acoustic case can be adapted to electromagnetic inverse problem in a straightforward manner. Therefore, same technique will be applied for the solution of the electromagnetic profile inversion problem. The results of the numerical tests demonstrate that the method is quite reliable for determining the interior and scattered field in the case of acoustic or electromagnetic direct scattering problem. It is compatible with alternative methods, and it is computationally effective. Unlike analytical techniques which can only be used for piecewise homogeneous spheres, the method can be easily adapted to any kind of profile.Moreover, its accuracy is higher compared to the computationally more expansive numerical techniques. Therefore it is safe to assume that it can be used for practical applications involving complex scatterers such as head models. The acoustic or electromagnetic scattering problems involving radially inhomogeneous spheres are frequently encountered in the research fields such as biomedical engineering or material science. The method developed in this thesis can be reliably used in these problems. For the inverse problem, the method yielded quite satisfactory results for slowly varying continuous profiles, provided that an appropriate initial guess is chosen. Other than the initial guess for the unknown acoustic or electromagnetic parameters, the most important point that effects the outcome of the method is the initial parameters of the iterative process itself. Through various numerical tests, the optimal values of these parameters have been determined and presented in the thesis. However, it was observed that even for these values, the success of the method is considerably lower for more rapidly varying profiles. Especially for layered profiles, the method can only provide a smoothed approximation. Numerical tests also demonstrated that the method produces useful results for relatively low contrast values. It should be noted that these are well-known limitations of Newton based algorithms, and the method performs reasonably well for an iterative profile inversion technique.