Efficient Evaluations of Periodic Green’s Functions Through Imaginary Wavenumber Cancellations

We report an interesting and newly invented technique to efficiently evaluate periodic Green's functions that are widely used for analyzing periodic structure scattering using integral equations. The Green's function at an arbitrary wavenumber is evaluated through subtracting out the Green's function evaluated at a specified imaginary wavenumber and then adding back the same component. The extracted terms are effectively evaluated in terms of spatial series which decay exponentially at imaginary wavenumbers. On the other hand, the terms being added back are represented in spectral series, which not only cancel out the singularity of the periodic Green's function, but also significantly improve the spectral convergence rate of the Green's function. Such a technique constitutes a self-consistent methodology to effectively evaluate periodic Green's functions using ordinary representations without involving complicated integral transformations or transcendental functions. The technique is widely applicable to periodic Green's functions that share the same dimensionality between the problem space and periodicity, which occur frequently in metamaterials, photonic crystals, phononic crystals, and atomic structures. It is also applicable to periodic Green's function where the dimensionality of periodicity is less than the problem space, well representative of gratings and metasurfaces. In the former case, the new representation of the Green's function has very simple wavenumber dependences in a rational multiplicative factor, so it is especially effective to evaluate the Green's function over a broadband of wavenumbers. In the latter case, computing the Green's function when both the field point and source point are located in a plane parallel to the lattice vectors is deemed difficult. We demonstrate the effectiveness of the proposed technique using two-dimensional problems with both two-dimensional and one-dimensional periodicities.