The Creative Research Of Sommerfeld Integral In Electromagnetic Theory

Since raised 100 years ago, Sommerfeld integral has been a scientific problem of great practical interests and wide applications in literature.Due to its mathematical complexity, multiple solutions were derived and formulated to characterize the same physical problems in reality and there exists no reference solution which is theoretically complete and numerically unique. By making a coordinate transform between the spherical and planar stratified structures, this proposal aims to obtain an analytic solution of the Sommerfeld integral, which is expected to be very accurate when the radius of the inner sphere in the coordinate transform is electrically large enough.This thesis complete a review of earlier research on the study of Sommerfeld integral,and the existing research methods is analysed in detail, they can mainly be divided into completely numerical method, the asymptotic method and semi-analytical method. Different from the traditional calculation of the Sommerfeld integral or similar analytical method, in this thesis, a large number of research is investigated and innovatively found Sommerfeld integral existed in planar and cylindrical stratified structure, and there is no complex Sommerfeld integral in spherical stratified structure. According to the findings, by establishing the transformation relations between planar and spherical, as well as coordinate transformation relates the spherical coordinates and the Cartesian one, to get a generalized expression of electromagnetic field in planar stratified media, so as to bypass the Sommerfeld integral calculation.Then the electromagnetic field excited by a vertical Hertz dipole in the presence of N-layer spherical stratified structure and corresponding scattering coefficient of the dyadic Green’s function is solved, when solving the scattering coefficient of dyadic Green’s function, method of linear equations is used to avoid the repeated derivation of boundary conditions and iteration solving of scattering coefficient.And the derived formula is applied to two layers and three layers spherical stratified structure, which is considered to be either perfectly electric conductor or Isotropic dielectric, the Gibbs phenomenon existing in the free space dyadic Green’s function is mainly solved in the thesis (which affect discontinuity of the electric field in the sphere of the source point), and then by analyzing the formula, the Gibbs phenomenon is cleverly eliminated using math skills, and the continuous vector expression of green’s function of scattering field excited by electric dipole is derived. Due to expression of electric field consists of infinite series summation, it is necessary to solve the problem of truncation term. Then Mie series formula was referred, and convergence figure is drew to get a new formula of series convergence truncation number. Base on this, then the thesis completes comparison of the planar layered and spherical stratified structure, by analyzing figures,the thesis found that with the radius of inner sphere increased, the calculated scattering field excited by electric dipole in the presence of two-layer planar stratified media and the spherical stratified media is consistent. Sequentially we prove that the new method solving Sommerfeld integral in spherical stratified structure is effective.When applied to three-layer spherical stratified geometry, the actual example is used in this thesis, by comparing scattering field on spherical stratified media with planar stratified media, and there are two main method to compute planar stratified media, one is R.W. P. King and S. S. Sandler, the other is hong-qi zhang and Wei-Yan Pan, to verify that when the layer is consist of isotropic dielectric, the method of spherical stratified structure to calculate scattering field excited by electric dipole is still available and the results is consistent with the results of planar layered structure model. However, it should be pointed out that the method of spherical layered media does not exist constraints, while the calculation method of planar layered structure has a great calculation error if the constraints does not meet.