Combined Method for the Computation of the Doubly Periodic Green's Function

Comparative Analysis of Acceleration Techniques for 2-D and 3-D Green's Functions in Periodic Structures Along One and Two Directions

“…The convergence criterion to be met is (14) where is the required relative error. A more restrictive test is proposed in [33], but actually we did not encounter any problem in using (14). The more pronounced the oscillations in the partial sums, the more impressive is the acceleration; nevertheless, a loss of accuracy is possible due to cancellation errors in the denominator of (12), and the method can fail at some isolated points.…”

“…In addition, they cannot be implemented iteratively for further accelerations. An acceleration procedure that overcomes these limits, found at the beginning of 1900, was investigated by Shanks and Winn (see [26], [27]- [29]) and is widely used in electromagnetic problems (see [4], [30]- [33]). It is a nonlinear transformation, which can be applied to oscillating series like (5).…”

“…1(b)]. The basic spatial Green's function is (33) Poisson's formula gives (34) with exponential asymptotic behavior. From (34) the presence of "blind angles" (in this case corresponding to logarithmic-kind branch points) is evident if such that .…”

“…The spectral series in (35) is also accelerated with Shanks or near the axis; an appropriate region was found to be . The series (33) can also be computed with Ewald's method; the relevant calculations are reported in the Appendix II. As a result we obtain the following expression: (38) with (39) and (40) The proper or improper feature of the terms is determined by the choice of the determination of the logarithmic branch cut of the function.…”

Optimization of metallic bi-periodic photonic crystals application to compact directive antennas

“…In order to circumvent this numerical problem, it has been assumed that the grids are periodic (infinite extent along the x and y axis parallel to the ground plane), the number of unknowns is then reduced to the first cell. To speed up the computations, it has been developed a numerical technique for the efficient computation of Green's functions that are periodic in two directions 25 . Then, structures made of a source surrounded by a bi-periodic metallic photonic crystal above a ground plane are investigated: the ground is taken into account with the image theory, and the field diffracted by the source is expanding in a plane wave packet using a bi-dimensional Fast Fourier Transform (FFT) 26 , so a grating problem has to be solved for each plane wave.…”