1. Green’s functions for the Analysis of One Dimensional (1-D) Source Excited EM wave Problems
(a) Introduction to Sturm Lioville form of differential equations for one dimensional (1-D) wave problems.
(b) Formulation of the solution to a Sturm-Lioville problem in terms of the 1-D Green’s function approach via the theory of linear operators in Hilbert space. Use of adjoint operators with Hilbert inner product together with Green’s second identity in 1-D. Adjoint 1-D Green’s function. Formal solution in terms of the adjoint Green’s function.
(c) Relation between the Green’s function and its adjoint.
(d) Conditions for symmetry of Green’s functions and its adjoint.
(e) General procedure for constructing a 1-D Green’s function. Examples.
(f) On the existence and uniqueness of the Green’s functions.
(g) Eigenfunction expansion representation for the Green’s functions. Adjoint eigenfunctions and biorthogonality relationships between eigenfunctions of the Sturm Lioville operator and its adjoint. Proper and improper eigenvalue spectra. Limit point problems and conditions at infinity. Discrete and continuous eigenfunctions and their adjoints.Relations between eigenfunctions (and eigenvalues )with their adjoints. Self adjoint problems. Examples. Continuos eigenfunctions as the limit of discrete eigenfunctions as an end point receeds to infinity. Examples.
(h) Delta function completeness relation in terms of Eigenfunctions of the Sturm Lioville operator. Finding complete set of eigenfunctions from a complex contour integration of the closed form representation of the Green’s function. Development of a variety of Fourier like series and also a variety of integral transforms via the delta function completeness theorem and pertinent Green’s functions.
2. Application of 1-D Green’s function approach for the analysis of single and coupled set of source excited transmission lines
(a) Analytical formulation for a single transmission line made up of two conductors.
(b) Wave solution for the two conductor lines when there are no impressed sources anywhere within the lines. Source/load conditions only at ends of the line.
(c) General Sturm Lioville type 1-D transmission line Green’s function based solution for sources impressed within the lines and source/load conditions at ends of the line. Examples.
(d) Excitation of a two conductor transmission line by an externally incident EM wave within the differential mode current approximation in EMC/EMI applications. Solution via 1-D Sturm Lioville type transmission line Green’s function.
(e) Extension to treat coupled set of transmission lines. Matrix formulation for N coupled transmission lines. Two methods of solution using a matrix of Sturm Lioville problems. In one method the problem is decoupled via a similarity transformation to arrive at N independent Sturm Lioville problems each being solved by the usual 1-D Green’s function approach. The second method solves a coupled Sturm Lioville matrix Green’s function problem using an eigenfunction expansion. Examples.
3. Analysis of Source Excited 2-D and 3-D Scalar Wave Problems via Green’s Functions for Higher Dimensions
(a) Formulation of solution to 2-D and 3-D scalar wave problems via adjoint Green’s functions and Green’s second identity in higher dimensions.
(b) Relationship between the Green’s function and its adjoint in higher dimensions.
(c) Construction of higher dimensional Green’s functions via a convolution approach. Green’s functions in rectangular, cylindrical and spherical coordinates.
(d) Alternative representations of higher dimensional Greens functions emphasizing propagation in preferred coordinate directions. Convergence properties of alternative eigenfunction representations. Examples of source excited rectangular interior and exterior wave problems. Scalar source excited circular cylindrical scatterers and interior guided wave configurations. Scalar source excited wedge and sphere configurations. Alternative representations for the above examples.
4. Analysis of Source Excited 2-D and 3-D EM Wave Problems via Potential Theory and Scalar Green’s Functions in Higher Dimensions
(a) Formulation of solution to 3-D EM wave problems in terms of TE and TM potentials with respect to electric and magnetic point sources aligned along PREFFERED directions in space. EM fields are in terms of TE and TM potentials. Solution to potentials possible by scalarization of partial differential equations governing TE and TM potentials. Scalar 3-D Green’s functions used to solve for TE and TM potentials.
(b) Extension to treat excitation by EM point sources which are ARBITRARILY oriented (rather than along a preferred direction) can be developed directly from the solutions in part (a) above via an application of the reciprocity theorem.
(c) Identification of a dyadic Green’s function for EM source excited problems from part (b) above simply via INSPECTION.
(d) On the completeness of the eigenfunction representation of the EM dyadic Green’s function in the source region.
(e) Use of dyadic Green’s second identity and adjoint EM dyadic Green’s functions to provide solutions to EM wave problems with arbitrary sources of finite extent (instead of point sources). Expressing the EM solution in terms of the EM dyadic Green’s function by developing the relation between the EM dyadic Green’s function and its adjoint.
(f) Examples. EM source excitation of interior rectangular and circular PEC walled waveguides and cavities. Exterior EM source excitation of circular cylindrical, wedge and spherical PEC surfaces. Extension to exterior EM source excited circular cylindrical and spherical surfaces with an impedance boundary condition. EM source excited planar stratified media. Microstrip EM dyadic Green’s function. Identification of pole waves, and branch cut waves, from the Green’s dyadic for above examples.