A novel three-dimensional time domain method is developed to study interactions between finite-sized electromagnetic sources and infinite periodic structures. The method is based on a periodic finite difference time domain method combined with the spectral expansion of electromagnetic sources. Using this method, only a single periodic cell needs to be modeled in finite difference time domain simulations. The convergence, guidelines on using the algorithm, and the acceleration scheme for the algorithm are discussed. Several periodic structures are simulated by this proposed method. It is shown that this method can significantly reduce the required computer memory and computational time.
 Periodic structures have many electromagnetic applications ranging from frequency selective surfaces to composite meta-materials [Munk, 2000; Vardaxoglou, 1997; Ouchetto et al., 2006]. Rigorous simulations are often used to understand interactions between electromagnetic signals and these structures [Dechant and Elezzabi, 2004; Chae et al., 2004]. To take advantage of the nature of periodic structures, periodic boundary conditions have been developed and implemented in the finite difference time domain (FDTD) method [Taflove, 2000; Kao and Atkins, 1996; Roden et al., 1998; Ren et al., 1994; Marek and MacGillivray, 1993; Ko and Mittra, 1993]. However, all these implementations assume that periodic structures are illuminated by plane wave incidences. That is, these methods are only valid when the incident electromagnetic signal is a plane wave. For some applications where electromagnetic responses from finite-sized sources are required, “brute force” FDTD simulations were performed [Luo et al., 2002a, 2002b]. In the “brute force” simulations, rather than using a single periodic element, many repetitive cells were used to approximate the structure's infinite extension. Oftentimes, at least 30 unit cells are required in directions where infinite repetitions exist. However, such approximation truncates the actual structures, which could lead to significant reflections from the truncated boundary. In addition, this approach requires significantly more computer memory and CPU time compared to the modeling of a single periodic element.
 The purpose of this paper is to present a novel FDTD method to analyze the behaviors of arbitrary finite-sized electromagnetic sources over infinite periodic structures. Only a single periodic element needs to be modeled using this method and the field distribution at any location of the structure can be obtained. This approach not only eliminates the simulation domain truncation errors associated with the “brute-force” approach, it also leads to significant savings in both computer memory and CPU time. This approach is based on a source spectral expansion method combined with a spectral FDTD method [Cangellaris et al., 1993; Aminian and Rahmat-Samii, 2005, 2006; Yang et al., 2006; Qiang et al., 2006, 2007a, 2007b; Kokkinos et al., 2006; Li and Sarris, 2008]. In this approach, the finite-sized electromagnetic source is first expanded into its spectral components. Multiple spectral-FDTD simulations in the complex-domain are carried out using spectral electromagnetic sources. The final results obtained from multiple spectral-FDTD simulations are superposed to restore the electromagnetic signals from the original source. In addition to describing the basic principles of this technique, the guidelines of how to use this new method, which cover the convergence analysis, the selection of spectral sampling rate, and the acceleration schemes of this algorithm are also presented. Numerical examples are presented to validate this new method.
 The remainder of this paper is organized as follows. In section 2, the methodology of this approach is presented in detail, which includes the source expansion and the spectral-FDTD method with complex periodic boundary conditions. In section 3, some properties of this algorithm are discussed. This includes the convergence analysis, selection of appropriate spectral sampling rate, and acceleration scheme for this method. Following several numerical examples in section 4, conclusions are given in section 5.
 Consider, as an example, the structure in Figure 1, where Px and Py are the periodicity along the x and y directions of this periodic structure. An electromagnetic current source with distribution of (x, y, z, t) is located above the periodic structure. In the first step of this approach, the source is expanded into its spectral components in both the x and y directions.
2.1. Source Spectral Expansion
 In the source spectral expansion, the original source is expanded into a series of planar spectral sources with wave numbers of kx and ky in the x and y directions as
Here, (kx, ky, z, t) is the spectral component of the original source. Each spectral component of the source is now a planar electromagnetic source, which will generate planar electromagnetic incidences. It should be noted that after the expansion, the spectral sources are complex numbers. For example, if a z-polarized electric current with a constant magnitude of I0 resides over a rectangular volume with a side length of Δx, Δy and Δz at (x0, y0, z0), a specific planar source with wave numbers of kx and ky in the spectral domain can be obtained by
where U(z − z′) is a step function defined as
Using the spectral sources defined in equation (1), the original source can now be represented by a series of planar electromagnetic sources as
In practical applications, only a finite number of kx and ky can be used in the integral of equation (4). For example, for the unit electric current considered earlier, the right side of equation (4) becomes
where M and N are the number of terms to be used in the spectral expansion and kx,N/2 and ky,M/2 are the maximum values of wave numbers to be used in the spectral expansion. The choice of values for these numbers needs to be determined based on specific applications and will be discussed in the following section. Equations (4) and (5) also indicate that any arbitrary electromagnetic source can be considered as the summation of a series of planar electromagnetic sources with phase delays related to the wave numbers kx and ky in x and y directions, respectively.
2.2. Computation With FDTD Method Using Complex Periodic Boundary Conditions
 Since the expanded electromagnetic source includes phase delays in both x and y directions, this planar electromagnetic source can be implemented into the spectral FDTD method with the complex periodic boundary condition [Aminian and Rahmat-Samii, 2005, 2006]. In the spectral FDTD method, instead of using a split-field scheme which requires modifying the conventional FDTD equations, we can include the complex phase shifts in the field components computation in the time domain. Therefore, the periodic boundary condition applied on the electric and magnetic fields with this method becomes
The procedure of this algorithm can be described as: (1) in the main computational domain, the conventional FDTD method is used; (2) at periodic boundaries, equation (6) is used to update boundary values. Following this procedure, the dispersion relation of the conventional FDTD method is preserved. In addition, no modification in the absorbing boundary condition implementation is required. Furthermore, the time step size of this method is not related to any incident angle compared to the widely used split-field method [Roden et al., 1998; Yang et al., 2006]. Using this method, we can obtain the electromagnetic field distributions of (kx, ky, z, t) due to the plane wave source of (kx, ky, z, t).
 Once (kx, ky, z, t) due to a planar source (kx, ky, z, t) is obtained by a single FDTD simulation using the periodic boundary condition in equation (6), the total electric field located at (x, y, z) within a periodic cell is simply the integration of all spectral responses, given by
For an observation point that is offset m periodic cells in the x direction and n periodic cells in the y direction, the electric field can be obtained by
Therefore, once the field information within one periodic cell is known, the fields in any other periodic unit can be extrapolated directly from equation (8).
3. Numerical Properties of the Algorithm
 Before this algorithm is applied to analyze practical structures, the numerical properties of this algorithm are first discussed. The convergence of the algorithm, the required spectral sampling rate for practical simulations, and an acceleration scheme are discussed in this section. This is done by analyzing the electric field distribution in free space due to the radiation of an electric current as shown in Figure 2. For simplicity, the electric current used here is an infinitely long line current in the y direction. The cross-section of the current is assumed to be infinitely thin (in actual simulations, the current is assumed to be uniformly distributed in the cross-section). Since the current is infinitely long along the y direction, we can perform spectral expansion in the x direction only. In this particular simulation, the operating frequency is 3 GHz sinusoidal signal so that steady state field plot can be obtained to compare the field distribution. The assumed cubic periodic element with a side length of 1 m was used in the simulation. The infinitely long current source passes through the cubic element at the center location.
 As indicated in equation (5), in order to obtain the solution due to a finite electromagnetic source over an infinite periodic structure, one needs to determine the maximum values for wave numbers to be used in each expansion direction. For the example here, we will only discuss the maximum value of the wave number to be used in the x direction and its limit is selected from −100 rad/m to 100 rad/m. Furthermore, we set dkx equal to 1 rad/m and the current strength as 1 A. Figure 3 shows the sampled electric fields obtained by the analytical solution and by the proposed approach along the AA′ and BB′ lines as indicated in Figure 2. If the center of the cubic element (with the current source) is located at (0,0,0) m, the two coordinates for the AA′ line are (0,0,0) m and (0.5, 0,0) m. BB′ line is defined as the duplicate of AA′ line that is shifted −1 m in the z direction. As a result, AA′ corresponds to the electric field along a line perpendicular to the current source while BB′ corresponds to the line that is offset by one unit of the AA′ line along the z direction as shown in Figure 2. As we can see from Figure 2, when these parameters are used in the simulations, a good agreement between the analytical solution and the numerical approach is obtained.
 To understand the effects of the convergence of this algorithm, we show a normalized error as a function of distance from the line source and the maximum value of kx used in the integral of equation (7) or (8). The error is defined as the
Figure 4a shows the error function defined in equation (9). The error bar on the right side clearly indicates that as the observation point gets closer to the source, a larger number of spectral expansion terms should be used in order to obtain accurate solution. In addition, for any distance away from the source, the error generally decreases as the maximum value of kx increases. However, for an observation point more than 0.05 m (which is equivalent to half of a wavelength) away from the source, increasing the maximum value beyond 63 rad/m for kx does not lead to significant error reduction. This maximum value of 63 rad/m is determined by ω. When kx has a value that is larger than the frees pace wave number, the plane wave will have exponential decay in the z direction. Such evanescent behavior will not given further error reduction if the observation point is far away from the source. This indicates that to evaluate propagating waves in far field region, the maximum value of kx in the expansion can be approximately obtained by ω. However, for near field calculation, larger spectral values are needed to represent the near field evanescent wave behavior. To further investigate the this error plot, Figure 4b shows the normalized error in logarithmic scale along three lines that is 0.01 m, 0.21 m, and 0.41 m away from the source. As we can see from Figure 4b, for the two observation points that are relative far away from the source, increasing the maximum value of kx beyond 63 rad/m does not significantly reduce the normalized error, since most evanescent waves cannot propagate that far. However, for the observation point that is closer to the source, even when maximum value of kx reaches 100 rad/m, the normalized error is still around −30 dB. To further reduce this error, more terms maybe necessary dependent on the required numerical accuracy.
3.2. Spectral Sampling Rate
 In the previous analysis, we used a fixed spectral domain sampling rate of 1 rad/m in the convergence analysis. In this section, we briefly discussed the effect of spectral sampling rate on the time domain simulation results. Here, the line source emitted a modulated-Gaussian pulse. For the purpose of comparison, conventional FDTD simulations were also carried out. Figure 5 shows the comparison between the time domain signals generated by the FDTD method and this proposed approach. The observation point here is 0.05 m away from the line source in the x direction. For this simulation, we used the maximum value of 100 rad/m for kx and a sampling rate of 1 rad/m. As we can see from Figure 5, the results of these two approaches agree well with each other up to 20 ns. In Figure 6, we show the time domain wave form when we reduce the spectral sampling rate from 1 rad/m to 3 rad/m. Initially, the result obtained by this approach agrees well with the FDTD approach, but a spurious waveform occurs at the time instance around 10 ns. If we further reduce the sampling rate to 5 rad/m, we observed that the spurious waveform will be even closer to the original waveform. The appearance of the spurious waveform can be explained by the sampling theorem. According to the sampling theorem, the aliasing will occur when the spectral sampling rate is not sufficient. In our algorithm, aliasing appears as image sources located at 2π/Δk away from the original sources (Δk is the spectral sampling rate). Therefore, for large value of Δk, the image sources will be close to the actual electromagnetic source causing errors in FDTD results. Consequently, for accurate simulations, a sufficiently large sampling rate shall be used to guarantee the non-aliasing effect.
3.3. Acceleration Scheme
 In section 2, we indicate that we need to perform spectral sampling for kx and ky from −∞ to ∞. Spectral FDTD simulation should be carried out at every (kx, ky) pair because different kx and ky require different boundary conditions as indicated in equation (6). However, also indicated in equation (6), if we have two different values of kx1 and kx2, and they are related by kx2 = kx1 + 2πn/Px, where n is an arbitrary integer, then both kx1 and kx2 will satisfy equation (6). In other words, if two spectral plane waves whose horizontal wave number kx are offset by the multiples of 2π/Px, both will satisfy the boundary condition described in equation (6).
 Since both of them satisfy the same boundary condition, we can launch two excitations simultaneously in spectral FDTD simulations. In light of this, we can launch multiple plane waves during a single spectral FDTD simulation provided their horizontal wave numbers are offset by multiples of 2π/Px in the x direction and multiples of 2π/Py in the y direction.
 If we assume that the finite-sized electromagnetic source is a source located at (x0, y0) and its variation in the z direction is described by Z(z), we can write this in the form of
Following the Fourier series expansion of impulse functions in two-dimensional spatial domain,
This is equivalent to
Equation (14) indicates that including all kx terms that are offset by multiples of 2π/px in a single simulation can be considered as a different way to perform the source expansion. Rather than performing the planar spectral source expansion, we can also expand the source in terms of the periodic source arrays corresponding to different wave number modes kx and ky. Effectively, approach is similar to the array-scanning-method (ASM) [Munk and Burrell, 1979; Qiang et al., 2007b; Li and Sarris, 2008]. However, it should be pointed out that the convergence and sampling rate criteria of ASM and plane wave expansion approaches can be significantly different due to adding more spectral sources into a single spectral FDTD simulation.
4. Numerical Examples
 To demonstrate the effectiveness of our proposed method, two numerical demonstrations are given below.
 The first example considered here is a single horizontal electric dipole over the center of an infinite microstrip patch array shown in Figure 7. The periodicity in both directions are given by a = 15 mm and b = 15 mm. The length (l) of the strip is 12 mm and the width (w) of the strip is 3 mm. The microstrip patch array resides on a perfect electric conductor (PEC) backed dielectric substrate whose relative dielectric constant is 4. The thickness (h) of the substrate is 5 mm. The dipole source is operating at 9 GHz and polarized in the z direction. It is located 15 mm above the center point of one of the conductor metal strip.
 Both the conventional FDTD simulation and the proposed method were used to obtain the electric near field distribution in the vicinity of the microstrip. For the conventional FDTD simulation, 21 unit elements in both directions are required. However, using the proposed approach, only a single unit element needs to be modeled. Figure 8 shows the electric field (z component) distribution at the top of the microstrip array structure. Simulation results from the FDTD simulation were also given for comparison. As we can clearly see from Figure 8, the results obtained by these two approaches agree well with each other in the center regions of this structure. The discrepancy between the FDTD method and the proposed approach near the edges of the structure is caused by reflection due to the structural truncation in the FDTD simulations. Again, we would like to point out that using the proposed approach, the required computer memory is less than 0.5% of that required by the conventional FDTD method. With such reduction in the memory requirement, it is possible to use this method to investigate many periodic structures where extremely fine mesh is required in the discretization.
 The second example exhibited here is a periodic dipole array with 81 × 81 elements (to approximate infinitely large array) shown in Figure 9. Each element includes a thin wire antenna with a length of 27 mm (λ/2) and its operating frequencyis at 5.55 GHz. The center spacing between dipole antennas are 45mm and 30mm in the x and y directions, respectively. In the simulation, FDTD method used a mesh size of 3 mm in all x, y and z directions.
 Unlike previous example in which there is only one excitation source, this array structure is driven by multiple source, e.g., in a 21 × 21 active region as indicated in Figure 9b. Outside this region, antenna elements are passive although they can also affect the antenna pattern due to mutual coupling. Two different types of excitation sources are used in the example. In the first case, each antenna in the active region is fed by an identical source at 5.55 GHz. In the second case, excitation sources are uniform in the z direction and have a linear roof-top variation in the x direction.
 Using this proposed algorithm, all excitation sources in the active region can be expanded and lumped together into spectral FDTD simulations. Both near field and far field generated by different feedings can be evaluated with simulations performed in a single unit element. Here, only the far field radiation patterns are used as shown in Figure 10. As indicated in Figure 10, the results obtained by this proposed approach agree well with the conventional FDTD method. In addition to using only single unit element in the simulation, we also observed a reduction in CPU time for this particular example. To obtain the results as shown in Figure 10, the proposed method only takes about 0.5 hour on regular CPUs. However the conventional FDTD method requires 10 hours of CPU time on the same PC to analyze the 81 × 81 elements. More importantly, the FDTD simulation already reaches the memory limit on a 32-bits machine and it cannot be used to model larger structures (for example, the 100 × 100 elements array).
 A novel method is proposed for the analysis of the interaction between finite-sized electromagnetic source and infinite periodic structures. The method is based on combining the source spectral expansion with a spectral FDTD method. Some properties of this proposed algorithm are presented in the paper. Several numerical examples are used to demonstrate the effectiveness of the proposed approach. It is observed that this proposed method can reduce the required computer memory by a factor over 100 in most applications.