An efficient finite difference time domain (FDTD) algorithm with a simple periodic boundary condition (PBC) is developed to analyze reflection and transmission properties of general periodic structures with arbitrary incident angles. In this new approach, the FDTD simulation is performed by setting a constant horizontal wave number instead of a specific incident angle. The principle of the FDTD/PBC algorithm is discussed and two important implementation issues are addressed, namely, the excitation of plane wave and the suppression of horizontal resonance. The validity of the approach is demonstrated through several numerical examples on dielectric slabs and frequency selective surfaces.
 Periodic electromagnetic structures have received increased attentions in recent years for their applications in the design of frequency selective surfaces (FSS), electromagnetic band gap (EBG) structures, and negative index materials. The finite-difference time-domain (FDTD) method has been used to analyze these structures. To take advantage of the periodic replication of these structures, periodic boundary conditions (PBC) was developed and implemented in various forms so that only a single unit needs to be simulated [Maloney and Kesler, 2000]. Some representative PBCs include the Sin-Cosine method [Harms et al., 1994], the unit cell shifting method [Holter and Steyskal, 1999], the split-field method [Roden et al., 1998], and the spectral FDTD method [Aminian and Rahmat-Samii, 2004].
 In this paper, a simple and efficient FDTD/PBC algorithm is presented to analyze the reflection and transmission properties of periodic structures. Compared to many previous PBC techniques, a distinguish feature of the proposed algorithm is that periodic structures are simulated under a constant horizontal wavenumber instead of at a given incident angle. The idea of FDTD computation with constant wavenumbers was originated in the analysis of guided wave structures and eigenvalue problems [Xiao et al., 1992; Cangellaris et al., 1993], and then expanded to the plane wave scattering problems in [Aminian and Rahmat-Samii, 2004]. The novelty of this paper is the direct computation of E and H fields rather than the indirect calculation using auxiliary fields [Aminian et al., 2005]. Thus, the conventional Yee's scheme is used to update the E and H fields. As a consequence, the proposed method possesses several advantages, such as the simplicity in the algorithm implementation, the same stability condition and numerical errors as those of the conventional FDTD method, and good computational efficiency near the grazing incident angles.
 The implementation procedure of the proposed FDTD/PBC algorithm is discussed and two key issues are addressed. The first one is the plane wave excitation in this algorithm. Since the new algorithm uses a constant horizontal wavenumber instead of an incident angle, the conventional plane wave incident approach is no longer applicable. Alternatively, a one-field plane wave excitation approach is proposed in this paper. For example, only horizontal E incident field is incorporated for the TEz plane wave case and only horizontal H incident field for the TMz case. The second implementation issue is the horizontal resonance associated with the numerical scheme. Due to the implementation of periodic boundary conditions (PBC), the energy exits from one side PBC will re-enter the computational domain from the opposite side PBC. Thus, if a wave propagates horizontally, its energy will not be absorbed by the perfectly matched layers (PML). As a consequence, a horizontal resonance occurs and the fields never decay to zero in the time domain. It is revealed in this paper that the horizontal resonance can be suppressed by properly designing the excitation waveforms.
 The developed algorithm is used to analyze several periodic structures such as a dielectric slab and a frequency selective surface (FSS) consisting of dipole elements. The simulation results are compared with analytical results, results from a split field FDTD method, and results from commercial software such as Ansoft HFSS and Designer. Good agreement between these results is observed, which demonstrates the validity and accuracy of the new algorithm.
2. FDTD/PBC Algorithm
 The novel FDTD/PBC algorithm is developed based on the comprehension of the kx-frequency plane. In this section, we first discuss the reflection coefficient in the kx-frequency plane, and then explain the rationale of the constant kx method. The implementation of the FDTD/PBC algorithm is described afterward.
2.1. Reflection Coefficient in the kx-Frequency Plane
 A dielectric slab is first analyzed as an example to illustrate the reflection coefficient in the kx-frequency plane. Figure 1 shows a TMz plane wave illuminating a dielectric slab of thickness h and dielectric constant ɛr. Assuming the incident angle is θ, the horizontal wave number kx is determined as below:
where k0 = 2πf is the free space wave number. The reflection coefficient of the dielectric slab is calculated analytically and plotted in the kx-frequency plane, as shown in Figure 2. The horizontal axis represents the wave number along the x direction (kx), and vertical axis represents the frequency. The magnitude of the reflection coefficient (Γ) is represented by different dark scales as indicated by the scale bar on the right. The kx-frequency plane is separated into the evanescent wave region (kx2 > k02) and plane wave region (kx2 < k02) by the light line (kx = k0). For our applications, only the reflection coefficient in the plane wave region needs to be considered. The normal incidence and an oblique incidence cases are also denoted by the dashed lines in Figure 2.
 It is observed that the total transmission, represented by dark scales, occurs in two regions. The first region starts at 10 GHz when kx equals to zero (normal incidence). The frequency increases to 12 GHz as the horizontal wave number increases. The second region is around a tilted line in the kx-frequency plane, which corresponds to an incident angle of 58°, the Brewster angle for this case. It is clear that the reflection coefficient plot in the kx-frequency plane provides a complete and clear vision on the scattering property of the dielectric slab for all incident angles.
2.2. Rationale of the Constant kx Method
 Using this kx-frequency plane, we introduce the rationale of the constant kx method and compare it with various PBCs. For a periodic structure with periodicity of a along the x direction, the periodic boundary condition of the electric and magnetic fields in the frequency domain is expressed as below:
Here, kx is the horizontal wave number determined by both the frequency and incident angle through (1). When (2) is directly transformed into the time domain,
where C is the free space wave speed. It is noticed that the field data in the future time step are needed to update electric and magnetic fields in the current time step, which creates a fundamental challenge in formulating periodic boundary conditions (PBC).
 For a normal incident case, we have θ = 0 and kx = 0. Since there is no need for time advancement in this situation, the periodic boundary condition for normal incidence can be implemented in FDTD method without any difficulties. To simulate oblique incidences the Sin-Cosine method was developed in [Harms et al., 1994], but only a single frequency is calculated per simulation as indicated by a plus sign in Figure 2. To improve the computation efficiency and obtain a wideband response per simulation, the split field method [Roden et al., 1998] was developed based on field transformation technique [Veysoglu et al., 1993]. This method is capable to calculate oblique incidence on a tilted line in Figure 2, but the time step size used in such FDTD simulations needs to be decreased as the incident angle increases. In addition, the split field method needs relatively complex formulations for the updating scheme and absorbing boundary conditions.
 An alternative approach to formulate the PBC is to choose a constant horizontal wave number instead of a specified incident angle in the FDTD simulations. The relationship of the horizontal wave number, frequency, and incident angle is described in (1). When a constant kx is chosen, change of frequency leads to corresponding change of the incident angle. The advantage of choosing a constant kx is recognized from the transformation of (2) into the time domain, where a very simple periodic boundary condition (PBC) can be obtained:
Note that exp(jkxa) is a constant number in (4). Therefore, no time delay or advancement is required in this equation. Compared to traditional methods, this technique calculates the reflection coefficient along a vertical dashed line in the kx-frequency plane, as show in Figure 2. It also should be pointed out that both E and H fields have complex values in the FDTD computation because of the PBC in (4).
 The physical meaning of this technique can be understood using the principle of superposition. For a given kx, time domain equation (4) is true for each frequency component. Thus, when a wideband pulse is launched into a linear system, (4) still holds by the superposition of all frequency components.
2.3. Plane Wave Excitation
 The implementation of the FDTD/PBC algorithm is simple and straightforward. The electric and magnetic fields are computed as follows:
 1. In the interior of computational domain, the conventional Yee's scheme is used to update the electromagnetic fields [Taflove and Hagness, 2000].
 2. At the periodic boundaries of the computational domain, (4) is used to update the EM fields. Perfectly matched layers (PML) are used to truncate the computational domain along the z direction, as shown in Figure 3. Note that periodic boundary condition (4) is also applied in the PML regions.
 One important issue related to the proposed algorithm is the excitation of plane waves in the computational domain. The traditional approach is to use the total-field/scattered field technique [Taflove and Hagness, 2000], where both incident electric and magnetic fields need to be incorporated on the interface between the total field and scattered field regions. When this technique is applied in the constant kx method, a difficulty occurs regarding the incident angle. For example, if a TMz wave illuminates upon a periodic structure, the tangential incident fields Hyinc and Exinc are expressed below:
where η is the wave impedance in free space. It is noticed that Ex component depends on the incident angle. Since in the proposed algorithm, kx is fixed and the incident angle θ varies with frequency, it is not easy to transform the Ex in (5) into the time domain. To solve this difficulty, the traditional Total-Field/Scattered-Field (TF/SF) technique is modified: only the Hy component is added on the excitation plane for the TMz case. As a consequence of the one-field excitation technique, the plane wave is excited to propagate not only into z < z0 region but also into z > z0 region. Thus, the entire computational domain becomes the total field region and there is no scattered field region. The scattered field can be calculated by the difference of total field and incident field. Similar strategy applies for TEz case: only the Ey component is added on the excitation plane. For general polarizations, it is required to break it up into TE and TM components, but both components can be excited and computed simultaneously.
 On the excitation plane, the incident field adopts a modulated Gaussian waveform and the phase delay along the x direction should be inserted:
The parameters of the modulated Gaussian waveform are the time delay t0, center frequency f0, and pulse width σt. The time delay t0 is set to a value between 3σt to 5σt for a smooth transition in time domain excitation. For convenience, a frequency bandwidth BW is defined below:
From the definition above, the signal magnitude at f0 ± (BW/2) is 40 dB lower than the signal level at the center frequency f0.
2.4. Horizontal Resonance
 The developed excitation technique is first used to stimulate the propagation of a plane wave in free space in order to demonstrate its validity. The center frequency is set to 10 GHz and the bandwidth is set to 20 GHz. Two horizontal observation planes are set 12.5 mm apart to study the propagation behavior. Figures 4a–4c show the real parts of electric fields at two observation planes for kx = 0, 100.6, and 174.2 radian/m, respectively. When kx = 0, the plane wave is normally incident into the computation domain and a constant time delay is observed between two observation plane. When kx = 100.6 radian/m, the plane wave is obliquely incident into the computational domain and the incident angle varies with frequency. A time delay is also observed, but more importantly an oscillation appears at the late time. Similar phenomenon is observed in Figure 4c, where kx = 174.2 radian/m. The E field levels at both observational planes do not decay to zero even though the excitation signal vanishes at a late time. As a consequence, the calculation of the reflection and transmission coefficients in the frequency domain will be deteriorated by these oscillations.
 To understand this problem, the behavior of the oscillation is analyzed and it is found that the oscillation frequencies in Figures 4b and 4c are 4.8 GHz and 8.3 GHz, respectively. At these frequencies, k0 = kx and θ = 90°. Thus, the plane wave propagates horizontally. A feature of the horizontal wave is that the E fields on two observation planes should be in phase with each other, which is noticed in Figures 4b and 4c. The energy of the horizontal wave continuously re-enters the computational domain through the periodic boundary condition instead of being absorbed by the top and bottom PML. As a consequence, the fields do not decay to zero and an oscillation is observed. The magnitude of the oscillation is determined by the strength of the excitation signal at the resonant frequency. Since 8.3 GHz is closer to the center frequency 10 GHz and a stronger signal is excited, the oscillation magnitude in Figure 4c is larger than that in Figure 4b.
 To suppress the horizontal resonance, various approaches can be implemented such as designing a frequency notch waveform. In this paper, a simple method is used by shifting the center frequency of the modulated Gaussian waveform. Figure 4d presents the FDTD simulated E fields with f0 = 16 GHz, BW = 20 GHz, and kx = 100.6 radian/m. The resonant frequency (4.8 GHz) is now outside the Gaussian bandwidth. Therefore, the horizontal resonance is small enough to be neglected and an accurate transmission coefficient can be obtained in the frequency domain. In practice, the center frequency can be chosen using following equation:
where fr is the resonant frequency of the horizontal wave. In free space, the fr is calculated from the horizontal wave number kx and free space wave speed C. Equation (8) also indicates that the signal strength is concentrated in the frequency range above the light line, which is in the plane wave region as shown in Figure 2.
3. Numerical Examples and Discussions
3.1. Example 1: A Dielectric Slab
 The developed algorithm is first used to analyze the dielectric slab shown in Figure 1 because an infinite dielectric slab can be considered as a periodic structure with arbitrary periodicity. To obtain a good resolution in the kx-frequency plane, reflection coefficients are calculated at 100 distinct kx values ranging from 0 to 419 (radian/m). The FDTD simulation is repeated for 100 times and each simulation outputs the reflection coefficient on a vertical line in the kx-frequency plane. The obtained reflection coefficient as a function of both kx and frequency is plotted in Figure 5a, where the half wavelength transmission and the Brewster angle transmission can be clearly visualized. Figure 5b compares the analytical and FDTD computed reflection coefficients at several kx values. The good agreement demonstrates the validity of the new algorithm.
3.2. Example 2: A Dipole Frequency Selective Surface (FSS)
 The constant kx method is then applied to characterize the reflection coefficient of a frequency selective surface (FSS) that consists of dipole elements [Munk, 2000]. The thin periodic dipole array resides on a dielectric slab with a thickness h and dielectric constant ɛr, as shown in Figure 6. The parameters of the periodic structure are provided in the caption, which are the same as those given in [Gianvittorio et al., 2003]. A TEz plane wave impinges on the periodic structure in the x-z plane. The FDTD simulation is performed 100 times using different values of kx sampled along the axis of horizontal wave number. The calculated reflection coefficients are plotted in Figure 7. Strong reflection is represented by the white color and transmission region is denoted by the dark zone. The first reflected region occurs around 9 GHz and the frequency of total reflection slightly decreases as kx increases, which agrees with the observation in [Pelton and Munk, 1979]. The first transmission region starts at 16 GHz for normal incidence (kx = 0). When kx increases, both the transmission frequency and the magnitude of the transmission coefficient decrease. It is also observed from Figure 7 that the second reflection region immediately follows the first transmission region. Transmission and reflection at multiple Floquet modes occur at frequencies higher than 16 GHz.
 One can also extract curves of the reflection coefficient versus frequency at any given incident angles from the data on kx-frequency plane using an interpolation scheme. For a give incident angle θ and frequency f, the horizontal wavenumber kx can be calculated from (1). Assuming that kx1 and kx2 (kx1 < kx< kx2) are sampled horizontal wavenumbers closest to kx, the reflection coefficient Γ at kx is calculated as follows:
For example, the reflection coefficients for the incident angles of 0°, 30°, 60°, and 85° are extracted and plotted in Figure 8.
 To verify the accuracy of the new algorithm, the same structure is analyzed using three different programs. One is the split field program developed by H. Mosallaei at University of California, Los Angeles [Mosallaei and Rahmat-Samii, 2003]. The other two are Ansoft HFSS program and Ansoft Designer program. The split field program is based on the FDTD method, the HFSS program is based on the Finite Element Method (FEM), and the Designer program is based on the Method of Moment (MoM). All three programs compute the reflection coefficient at given incident angles. Good agreements between different methods are observed, especially at small incident angles. As the incident angle increases, the HFSS results become unstable. When the incident angle is close to 90°, the split field FDTD method cannot give a converged result.
 It is also interesting to compare the memory usage and computational time of different methods. The memory used by our proposed algorithm, HFSS program, and Designer program are 4 Mb, 230 Mb, and 81 Mb, respectively, whereas the memory usage for the split field method is not reported by the program. In term of the computational time, our program spends 75 seconds for each kx line on a computer with an Intel Xeron 3.00 GHz CPU, and total calculation time for the complete kx-frequency plane (100 kx) is around 2 hours. Using the same computer, the HFSS and Designer spend around 750 and 670 seconds for each incident angle (including 29 frequency points). The computational time of the split field method varies with the incident angle. As the angle increases from 0° to 30° and 60°, the time increases from 778 seconds to 950 seconds and 2589 seconds. It should be pointed out that the above comparisons can only be used as a reference because the programs are developed by different agents.
3.3. Advantages of the Proposed FDTD/PBC Algorithm
 The validity of the new FDTD/PBC algorithm has been demonstrated in the preceding examples. It is worthwhile to emphasize several important advantages of this new approach. First, the new algorithm is very easy to implement. In contrast to the field transformation methods that use auxiliary fields [Roden et al., 1998; Aminian et al., 2005], the new approach computes E and H fields directly. Thus, no complicated discretization formulas need to be derived and the traditional Yee's updating scheme is used to calculate field values, as outlined in section 2.3. In addition, the perfectly matched layers remain unchanged, which also helps to simplify the implementation procedure.
 Another advantage of the new algorithm is the efficiency in calculating the scattering at large incident angles. When the field transformation methods such as the split field technique are used to calculate the oblique incidence, the time step size (Δt) needs to be decreased as the incident angle (θ) increases. To satisfy the stability constrains, the time step size and incident angle for square cells and free space should follow the inequality below [Maloney and Kesler, 2000],
Δx is the size of the FDTD grid cell. The time step size approaches zero when the incident angle goes to 90°. For example, when Δx is 1 mm in the FSS analysis, Δt < 1.925 × 10−12 (second) is required for normal incidence. For a large incident angle such as 85°, Δt < 7.323 × 10−15 (second) is necessary. Therefore, the simulation time becomes prohibitively long for the latter one.
 In contrast, the constant kx method presented in this paper uses a standard Yee's scheme to update the E and H fields. The stability condition remains unchanged regardless of the horizontal wave number (kx) or incident angle, as expressed below [Taflove and Hagness, 2000]:
For the FSS example, a constant Δt = 1.667 × 10−12 < 1.925 × 10−12 (second) is used for all kx values. Therefore, the new approach has a good computational efficiency, especially for plane wave scattering at large incident angles. This analysis also explains the computation time differences between the constant kx method and the split field method discussed in section 3.2.
 This paper presents a novel technique to model the periodic boundary condition (PBC) in the FDTD method. By choosing a constant kx value in FDTD simulations, a wide band response of the reflection coefficient can be obtained. The reflection coefficient in the kx-frequency plane is computed through several rounds of FDTD simulations using different kx values. The new approach is easy to implement and efficient in analyzing plane wave scattering from arbitrary incident angles. The validity of the approach has been demonstrated through the good agreements between the FDTD simulation results and the data obtained from analytical method and various numerical programs.
 The author would like to thank the reviewers for the constructive comments that helped to improve the quality of the paper.