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Green's function analysis of periodic structures in computational electromagnetics

  • Author(s): Van Orden, Derek
  • et al.

Periodic structures are used widely in electromagnetic devices, including filters, waveguiding structures, and antennas. Their electromagnetic properties may be analyzed computationally by solving an integral equation, in which an unknown equivalent current distribution in a single unit cell is convolved with a periodic Green's function that accounts for the system's boundary conditions. Fast computation of the periodic Green's function is therefore essential to achieve high accuracy solutions of complicated periodic structures, including analysis of modal wave propagation and scattering from external sources. This dissertation first presents alternative spectral representations of the periodic Green's function of the Helmholtz equation for cases of linear periodic systems in 2D and 3D free space and near planarly layered media. Although there exist multiple representations of the periodic Green's function, most are not efficient in the important case where the fields are observed near the array axis. We present spectral-spatial representations for rapid calculation of the periodic Green's functions for linear periodic arrays of current sources residing in free space as well as near a planarly layered medium. They are based on the integral expansion of the periodic Green's functions in terms of the spectral parameters transverse to the array axis. These schemes are important for the rapid computation of the interaction among unit cells of a periodic array, and, by extension, the complex dispersion relations of guided waves. Extensions of this approach to planar periodic structures are discussed. With these computation tools established, we study the traveling wave properties of linear resonant arrays placed near surfaces, and examine the coupling mechanisms that lead to radiation into guided waves supported by the surface. This behavior is especially important to understand the properties of periodic structures printed on dielectric substrates, such as periodic microstrip lines and leaky wave antennas. We proceed to analyze "twisted" linear periodic arrays, consisting of particles that are sequentially and discretely rotated about the array axis by a constant angle. Such periodic structures support two transverse wave modes with distinct polarization and radiation behavior, which couple to circularly polarized sources with opposite handedness. These structures offer a new dimension of polarization control to linear waveguiding structures

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