Imaging of objects with canonical shape by Huygens method

According to this thesis, a study is done about Huygens Principle. In this study, Huygens Principle is used for solving inverse scattering problems of canonical shaped objects. Goal in inverse scattering problems are obtaining the physical properties of the object that is desired to be imaged, such as location, size and electromagnetic features by obtaining and examining the scattered field with various methods by illuminating the space which includes the object with electromagnetic source or plane wave. In order to apply Huygens method, firstly scattered field needs to be calculated. For this purpose, firstly direct scattering problems are considered. According to this purpose, firstly, an example of two-dimensional electromagnetic scattering problem which is scattering from an eccentric perfectly conducting cylinder coated by a dielectric cylinder problem is solved with analytical method. Afterwards, this problem is expanded to two and three layered cylinder problem and solved for dielectric and perfectly electric conducting cases of eccentric cylinder. Secondly, three-dimensional electromagnetic scattering problem is considered. To achieve this purpose, scattering from an eccentric perfectly conducting sphere coated by a dielectric sphere problem is considered and solved with analytical method. After that, the problem is expanded to two and three layered sphere problem and solved for dielectric and perfectly conducting cases of eccentric sphere. In the third phase, Huygens principle is considered. For this purpose, Huygens method is applied to the scattered fields that are obtained in the first and second phases and the image of eccentric objects is tried to be captured for cylinder and sphere cases. These analyzes are done theoratically for sphere cases and are done theoratically for cylinder cases except one analysis which is done with experimental measurement values. Finally, numerical results and comments are considered and pros and cons of Huygens method is mentioned. In the first chapter, primarily, analytic solution of scattering from a dielectric cylinder including a little eccentrically placed cylinder inside of it is done. The little eccentric cylinder can be a perfectly electric conducter or dielectric. Analyzes are done for both cases. Eccentric cylinder occurs a complication for the application of boundary conditions on the surface of the cylinders. In order to eliminate this complication, coordinate transformations are made. This transformation is done by using the translational addition theorems for cylindrical wave functions. After using the addition theorems, the wave functions that are defined for shifted axis, are transformed into the original axis which is the outer cylinder's axis. Then, boundary conditions are applied easily on the surface of the cylinders and unknown coefficients are found. Later, this method is expanded for two and three layered cylinder problem. For two layered geometry, two dielectric cylinders are nested and the eccentric cylinder is located in the innermost cylinder. Two dielectric cylinders' centers are located in the main axis, but the eccentric cylinder's is located in the shifted axis. Boundary conditions are applied straightforward for the two dielectric cylinders. For eccentric cylinder, as mentioned above, translational additions theorems are used, and its coordinate axis is transformed into the main coordinate axis. Then, boundary conditions are applied and unknown coefficients are found. At the final stage of this chapter three layered case is mentioned. The geometry of three layered looks like as two layered geometry. Three dielectric cylinders are nested and the eccentric cylinder is located in the innermost cylinder. Three dielectric cylinders' centers are located in the main axis, but the eccentric cylinder's is located in the shifted axis. Boundary conditions are applied straightforward for the three dielectric cylinders. Again, for the eccentric cylinder, coordinate transformation is made and its coordinate transformed into the main coordinate. Afterwards, boundary conditions are applied and unknown coefficients are found. It should be emphasized that in the three cases, cylinders are illuminated by the line source not plane wave. In the second chapter, firstly analytic solution of the scattering from a dielectric sphere including an eccentrically localized sphere inside of it. Eccentric sphere can be a dielectric and perfectly conducter. Both cases of eccentric sphere is examined. For the sake of ease, eccentric sphere is shifted only in z axis, otherwise the problem gets more complicated. Like the cylinder case, eccentricity occurs a complication about using the boundary conditions. In order to overcome this complication, translational addition theorems for spherical wave functions are applied. Once transformation is done, the wave functions that defined with the shifted axis will be defined with the original axis. Then, boundary conditions are applied on the surface of the spheres and unknown coefficients are found. Afterwards, this one layered problem expanded into a two layered problem. For two layered geometry, two dielectric spheres are nested and the eccentric sphere is located in the innermost sphere. Two dielectric spheres' centers are located in the main axis, but eccentric sphere's is located in the shifted axis. Boundary conditions for the two dielectric spheres are straightforward since their centers are the same. For the eccentric sphere, in order to apply boundary conditions, the wave functions that defined with shifted axis must be transformed and defined with the original axis. For this purpose, translational addition theorems are applied. Once axis transformation is done, boundary conditions are applied easily and unknown coefficients are found. At the last stage of this chapter, three layered case is examined. The geometry of the three layered case is similar to the two layered case. In this geometry, three dielectric spheres are nested and the eccentric sphere is located in the innermost sphere. Three dielectric spheres' centers are located in the main axis, but eccentric sphere's is located in the shifted axis. Boundary conditions are applied straightforward for the three dielectric spheres. Again, for the eccentric sphere, coordinate transformation is done and its coordinate transformed into the main coordinate. Afterwards, boundary conditions are applied and unknown coefficients found like two layered case. As a conclusion of this chapter, it is important to say that in the three cases, spheres are illuminated by the point source, not plane wave. In the third chapter, Huygens Principle is mentioned and its equations are given for cylinder and sphere cases. In this method, firstly, the total field of the cylinders and spheres are calculated at the their surfaces by using the equations that are mentioned in the first and second chapters. Then, Huygens Principle is applied these fields and their source is tried to be found. It is important to say that this method cannot find the real field values whereas it finds a data about the inner field. This data has the information about the mismatch location in the innerfield. Then, this procedure is applied for multi frequencies and multi sources. All the datas that are obtained for each frequency and each source are summed and the image function is created with this sum. This technique is done for cylinders and spheres. For cylinders, Bessel function is used in order to obtain backward field and for the spheres, Dyadic Green function is used for the same purpose. More information about this section can be found in chapter three. In the final chapter, some numerical and experimental examples are given. It is obviously seen that Huygens Principle can capture the image of objects in the specific scatterers. Especially, the success of capturing the brain tumor and breast cancer examples show that this method can be enchanced and might be used in the medical imaging in the future.