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Exact Image Theory for the Sommerfeld Half-Space Problem with a Vertical Magnetic Dipole

The classical Sommerfeld problem of a vertical magnetic dipole above a planar interface of two dielectric and possibly lossy media is considered in terms of image principles. It is shown that an exact continuous image can be constructed in closed form, which totally lies in complex space. Unlike all previous image theories, being exact, the present theory can be applied for any distance, height of the source and parameters of the half space. It is demonstrated that the well-known dipole image at complex depth and reflection coefficient method both are special cases of the present method. The significance of the exact image theory lies in better convergence of field integrals over the classical Sommerfeld integrals.