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

By H.-W. Chang and S.-Y. Mu

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Abstract:
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 cube's 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 Green's 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 Green's function after propagating a distance of ten wavelengths are well under ten percent.

Citation:
H.-W. Chang and S.-Y. Mu, "Semi-Analytical Solutions of the 3-D Homogeneous Helmholtz Equation by the Method of Connected Local Fields," Progress In Electromagnetics Research, Vol. 142, 159-188, 2013.
doi:10.2528/PIER13060906
http://www.jpier.org/PIER/pier.php?paper=13060906

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