SEMI-ANALYTICAL SOLUTIONS OF THE 3-D HOMOGENEOUS HELMHOLTZ EQUATION BY THE METHOD OF CONNECTED LOCAL FIELDS

We advance the theory of the two-dimensional method of connected local fields (CLF) to the three-dimensional cases. CLF is suitable for obtaining semi-analytical solutions of Helmholtz equation. The fundamental building block (cell) of the 3-D CLF is a cube consisting of a central point and twenty six points on the cubeʹs surface. These surface points form three symmetry groups: six on the planar faces, twelve on the edges, and eight on the vertices (corners). The local field within the unit cell is expanded in a truncated spherical Fourier-Bessel series. From this representation we develop a closed-form, 3-D local field expansion (LFE) coefficients that relate the central point to its immediate neighbors. We also compute the CLF-based FD-FD numerical solutions of the 3D Greenʹs function in free space. Compared with the analytic solution, we found that even at a low three points per wavelength spatial sampling, the accumulated phase errors of the CLF 3D Greenʹs function after propagating a distance of ten wavelengths are well under ten percent.