ASM–FDTD: A Technique for Calculating the Field of a Finite Source in the Presence of an Infinite Periodic Artificial Material

A novel technique is proposed to calculate the field due to an arbitrary impressed source of finite extent in proximity with an infinite periodic structure such as an artificial material. The algorithm is based on a spectral domain finite-difference time-domain (FDTD) method combined with the array scanning method (ASM). Using this approach, only a single periodic cell of the periodic structure needs to be numerically modeled. Examples are used to demonstrate the accuracy and efficiency of the proposed approach


I. INTRODUCTION
P ERIODIC structures such as frequency selective surfaces, electromagnetic bandgap materials, metamaterials, and guiding periodic structures are topics that now receive a lot of attention in both the optics and microwave communities. However, the time-domain (TD) modeling of impressed electromagnetic sources of finite extent in proximity with infinite periodic structures has been to date performed by "brute force," including a large number of unit cells in the simulation domain [1], which requires extensive computational resources. This letter presents a novel TD technique to model the excitation of infinite periodic structures with a finite source. For simplicity, the discussion here is restricted to structures that are periodic in one dimension and are excited by a line source, but the extension to structures that are periodic in 2-D and are excited by finite (e.g., dipole) sources can readily be done. This method is based on the combination of the array scanning method (ASM) [2]- [5] and the spectral-domain FDTD method [6]- [11]. Accordingly, the spectral FDTD method is applied here to only one unit cell, with proper complex periodic boundary conditions, accounting for the impressed line source. The FDTD with complex periodic boundary conditions, used in the spectral FDTD, is different from the traditional delay boundary conditions used in other periodic FDTD methods [12], [13]. No field transformation is necessary in this method and the stability condition of this method is no longer limited by the angle of incidence. Manuscript   II. ASM-FDTD METHOD Consider, as an example, the structure in Fig. 1, where is the period along and is the location of the electric line source that produces an electric field , which is henceforth referred to simply as . The observation point in an arbitrary periodic th cell ( Fig. 1) is denoted by 1 , where is defined in the 0 periodic cell, and 1 is the unit vector along . (The source and observation points are vectors, although vector notation is being suppressed here.) In the frequency domain, an time-dependence is assumed and suppressed, and the ASM representation of the total field at 1 , produced by the single source in the periodic environment is [3] (1) In (1), is the electric field at in the infinite periodic structure produced by a periodic set of sources at 1 0 1 , with a phase shift between adjacent sources. This creates a periodic problem in which there is a phase shift between the left and right periodic boundaries in Fig. 1. The electromagnetic fields in the periodic structure therefore satisfy periodic boundary conditions. For example, the electric field satisfies (2) The integration domain in (1) where the hat denotes TD quantities. Here, in the time domain (3), just as in the frequency domain (2), the field quantities are complex and a periodic boundary condition at the edges of the unit cell are assumed, corresponding to the wavenumber . For example, the electric field satisfies the inverse Fourier transform of (2), which is (4) Equation (4) makes it obvious that in this rather unusual TD application, the TD field is a complex function. The same periodicity condition in the TD has been used already in [9]- [11].
To implement the periodic boundary condition (4) using the FDTD method, the phasing parameter has been discretized in an even number of spectral sampling points, uniformly distributed over the fundamental Brillouin zone. Spectral FDTD simulations are carried out at every spectral sampling point using the boundary condition described by (4) [10]. Since complex values described in (4) are used in the FDTD implementation, both electric and magnetic values are complex. For every spectral sampling point 1 2 1 , with 1 , the computed field is obtained. For simplicity, we use the simple midpoint rectangle rule of integration, which is particularly effective for smooth periodic functions [14] (5) though other schemes may be more appropriate if the integrand is singular. It should be pointed out that if a real-value excitation term is used, the final value of (5) is also real-valued. The detailed implementation of the spectral FDTD is as in [9]- [11].

III. ILLUSTRATIVE EXAMPLES
To validate the method and test the accuracy and convergence of this algorithm, we first simulate a case where an electric line source along the direction is located at the origin of theplane in a free-space environment at a frequency of 3 GHz. In Fig. 2, the magnitude of the electric field is evaluated on a line along the direction, with 0, showing a good agreement between the ASM-FDTD method and the reference analytic solution. The ASM-FDTD solution is evaluated by using 30 spectral sampling points in (5). The ASM is implemented assuming a virtual periodicity 0.4 in the direction. The FDTD Yee cell size here and in the next example is  1 mm. To further study the convergence of this method, the numerical error defined as (6) is shown in Fig. 3 versus the number of spectral points in (5). The error is evaluated along the line , with and equal to either and . As clearly seen in the figure, the numerical error decreases as increases. In addition, the convergence rates in the and directions are different-the numerical result converges faster along the direction.
Using the developed software, we now investigate the electromagnetic field behavior in the vicinity of a line source residing above an EBG material made by three rows of conducting wires as shown in Fig. 1. The wires have diameter 7.2 mm and spacing 18 mm in both the and directions. Fig. 4 shows the electric field (magnitude of component) produced by a line source located at 0.5 , i.e., a distance 0.5 above the axis of the first row of wires (Fig. 1) for a frequency of 5 GHz. The field does not penetrate through the material since it has a bandgap for frequencies 0,8 GHz [4]. The  FDTD Yee cell size is now given by 0.5 mm. Fig. 4 shows a comparison between results obtained using both the standard FDTD method [ Fig. 4(a)] and the ASM-FDTD [ Fig. 4(b)] where 24 spectral points have been used in (5). As seen from the figure, these results are in good agreement with each other and the relative error is only 0.98%, where the error is defined similarly as in (6) by an average over a computation space within 3 3 and 0,5.6 . However, to produce the result using the FDTD method, 21 periodic cells along the direction needed to be included in the simulation domain-otherwise truncation effects would be visible. As a consequence, the standard FDTD method requires over 10 more computer memory. The solution time for this particular example is 29 s (FDTD) and 62 s (ASM-FDTD), respectively. To obtain more accurate results in regions further away from the source (not shown in Fig. 4), the FDTD method requires even larger computational domains. If 50 periodic cells are used along the direction, the FDTD method requires 25 more computer memory than that of the ASM-FDTD method. It also requires 73 s in CPU time.
In Fig. 5, the ASM-FDTD is also used to compute a transient signal for the same geometry as in Fig. 4 (5), is in good agreement with the field obtained via the standard FDTD method.

IV. CONCLUSION
A time-domain version of the ASM has been presented for the first time. This algorithm allows for the solution of a finite-size source excitation of an infinite periodic structure in the time domain. Only a single unit cell needs to be discretized. The algorithm has been implemented by using the spectral FDTD method for complex signals, which is particularly suitable for stable FDTD methods involving incidence angles near grazing, showing good agreement with a reference solution. The memory usage of this ASM-FDTD method is drastically reduced compared to a standard FDTD implementation, which has to discretize a large portion of the periodic structure to avoid truncation effects.