Understanding Grating Lobes in the Context of Periodic Boundary Conditions
The purpose of this article is to discuss grating lobes in connection with periodic boundary conditions (PBC) in FEKO. Simple equations are derived to determine the frequency where grating lobes start to form, as well as the grating lobe observation angles in the farfield. These equations are verified by means of a Jerusalemcross frequency selective surface (FSS) example. The effect of grating lobes on the computation of the reflection and transmission coefficients will also be shown.
In order to derive these equations, consider the infinite one dimensional array shown in Figure 1 (similar for 2D). The element spacing is d and the incident plane wave excitation is at an angle θi. We are interested in the farfield at the observation angle θ. The PBC assumes that the current distributions on all array elements are identical (except for the phase increment). To compute the farfield we sum the contribution of all the elements to obtain:
with the wave number k=2π/λ, the phase term above is:
Figure 1: Infinite One Dimensional Array
All elements will radiate in phase when m = 0, 1, 2 for:
Therefore, in the farfield we will have in phase radiation at the observation angles:
For each solution θ there is a corresponding solution 180°  θ, since sin(180°  θ) = sin(θ).
Looking at this last equation, some important properties can be seen:
 The main beams correspond to m=0 at the observation angles θ = θi and θ = 180°  θi.
 The first grating lobe correspond to m=1. It will only exist for real angles θ, when the magnitude of the argument of arcsin is smaller than unity.
 No grating lobes exist if the element spacing is smaller than d ≤ 0.5 λ (for all incident angles θi)
 As the spacing d is increased in terms of wavelength λ, grating lobes will start to form along the end fire directions at θ = ± 90°.
Two more equations can be derived from the above equations at the onset of grating lobes.

The incident angle where grating lobes start from [1, eq. 8]

The frequency where grating lobes start from [2, eq. 14]
where:
 m = 1, 2, etc. (m = 0 gives no grating lobes)
 θi is the incident plane wave excitation angle
 d is the element spacing in the periodic
array
 c is the speed of light
Example: Jerusalem Cross with PBC Unit Cell
Consider the Jerusalemcross FSS [3] with the unit cell shown in Figure 2. This example is used to verify the predicted frequency where grating lobes start to form, as well as the predicted observation angles for the main beams and grating lobes in the farfield. The influence of grating lobes on the computation of the reflection and transmission coefficients will also be shown.
Figure 2: Jerusalem cross FSS with PBC unit cell
Parameters of interest: plane wave excitation in the xzplane with incident angle θi=30°, and square 2D periodic lattice with element spacing d = 15.2 mm. The dimensions of the FSS can be found in [3].
The first predicted frequency where grating lobes start to form is: f = c / d / (1 + sin(θi)) = 13.2 GHz.
Below this frequency there will be no grating lobes, only the two main beams at θ = 30°,150° (they are always present at all frequencies). Above this frequency grating lobes start to form in addition to the two main beams, i.e.
 At f = 13.5 GHz there are two grating lobes at the predicted angles θ = 73.9°, 253.9°.
 At f = 14.0 GHz there are two grating lobes at the predicted angles θ = 65.3°, 245.3°.
To validate the predicted observation angles for the main beams and grating lobes, the farfield radiation pattern of a large finite 51x51 FSS array was analyzed in FEKO using the PBC. This was done at the frequencies f = 13.0 GHz, 13.5 GHz and 14.0 GHz as shown in Figures 35. A cut in the xzplane of the farfield pattern is superimposed on the FSS unit cell in all these figures. The predicted main beam angles are identical to those computed at θ = 30°,150°. Predicted grating lobe angles are within 0.6° at 13.5 GHz and within 0.3° at 14.0 GHz of the computed angles.
Figure 3: Farfield at
13.0 GHz (no grating lobes)
Figure 4: Farfield at 13.5 GHz, computed grating lobes at θ = 73.9°, 253.9°.
Figure 5: Farfield at 14.0 GHz, computed grating lobes at θ = 65.3°, 245.3°.
To determine the reflection and transmission coefficients of the FSS, the near electric fields are computed at a large distance away from the FSS (i.e. at ten wavelengths). The FSS in Figure 2 is located at z=0 and the reflection coefficient is determined from the scattered nearfield computed along the positive zaxis. Similarly, the transmission coefficient is determined from the total nearfield computed along the negative zaxis. In the absence of grating lobes, only the main beam contributes to the nearfields. However, as the frequency is increased, grating lobes will form that will also contribute to the nearfields. For an infinite 2D periodic structure these main beams and grating lobes are propagating plane waves at the observation angles θ (predicted by our simple equation). The effect of grating lobes on the computation of the transmission coefficient can be seen in Figure 6. Here the nearfields below (f = 13.0 GHz) and at the onset (f = 13.2 GHz) of grating lobes is computed along the negative zaxis. At 13.0 GHz only a single propagating plane wave exists (corresponding to the main beam) and the amplitude becomes constant for large z values. At 13.2 GHz there will be two propagating plane waves in the lower half space (corresponding to the main beam and the grating lobe). The interference between these two plane waves causes the oscillation in the amplitude of the nearfield, also for large z distances. So when grating lobes exist, one cannot simply compute the reflection or transmission coefficients by looking at the nearfields at a single point. This procedure is only valid if a single propagating plane wave exists, i.e. for frequencies below 13.2 GHz.
Figure 6: Nearfield along the negative zaxis
Figure 7 shows the computed reflection and transmission coefficients of the FSS versus frequency. The nearfields are computed at two single observation points at z = ±0.1 m. As mentioned previously, these results are only valid for frequencies up to 13.2 GHz (before the onset of grating lobes) where a small spike can be seen. The rapid variation at 9.4 GHz is correct (this is not a grating lobe) and was verified by using the multilevelfastmultipole method to analyze the complete finite array. Also shown in Figure 8 is the reflection coefficient for a normal incident plane wave. Good agreement between FEKO and the published results can be seen. The PBC analyses show that the performance of the Jerusalemcross FSS degrades for θi =30° incidence around 9.4 GHz.
Figure 7: Reflection and transmission coefficients for θi = 30° (only valid below 13.2 GHz)
Figure 8: Reflection coefficient [3] for θi = 0°
References
[1]  V. Lomakin and E. Michielssen, “Beam Transmission Through Periodic Subwavelength Hole Structures”, IEEE Trans. Antennas Propagat., vol. 55, no. 6, pp. 15641581, June 2007. 
[2]  E. E. Kriezis and D. P. Chrissoulidis, “EMWave Scattering by an Inclined Strip Grating”, IEEE Trans. Antennas Propagat., vol. 41, no. 11, pp. 147314, November 1993. 
[3]  I. Stevanovic, P. CrespoValero, K. Blagovic, F. Bongard and J. R. Mosig, “IntegralEquation Analysis of 3D Metallic Objects Arranged in 2D Lattices Using the Ewald Transformation”, IEEE Trans. Microwave Theory and Techniques, vol. 54, no. 10, pp. 36883697, October 2006. 