# Methods for Grating Lobe Suppression in Ultrasound Plane Wave Imaging
## 3 Results and Discussion  
Both of the GL suppression methods were verified through computer experiments by obtaining CW beam patterns and PSF images using a 128-element linear array transducer with a center frequency of 5.208 MHz and a pitch of 0.298 \(\upmu\)m (length of the array = 38.1 mm). The CW beam pattern was calculated using MATLAB R2015a (MathWorks, Natick, MA, USA); acoustic field responses of plane waves with different angles, each with a frequency of 5.208 MHz generated from the linear array transducer, were calculated and then compounded with proper delays for a Tx focal point at (\(x\) = 0, \(z\) = 30 mm) [1]. The amplitude of each PW was set to 1.0, and the final synthesized beam pattern was normalized by its maximum value and displayed in logarithmic scale. In the PSF experiments, a single pulse plane wave with a center frequency of 5.208 MHz was transmitted along different directions from the same array transducer. First, RF echo data sets of different plane wave angles reflected from a single point target at (\(x\) = 0, \(z\) = 30 mm) were generated using the Field II simulator [15]. Then, MATLAB was used to obtain the PSF images by reconstructing a B-mode image of the point target from the RF data sets using coherent plane wave beamforming [1], demodulation, and logarithmic compression. In the beamforming process, traditional dynamic Rx focusing was performed with an F-number of 1.0, and all the plane waves were coherently synthesized with proper delays for each pixel to obtain two-way (Tx and Rx) dynamic focusing. In the logarithmic compression, the PSF image was normalized by its maximum value and compressed with a dynamic range of 60 dB. To validate the methods for GL suppression with a uniform \(d_{\alpha}\), CW Tx beam patterns were obtained, varying the interval, \(d_{\alpha}\), that is controlled by the number of PW angles within the range of \([-10^{\circ},10^{\circ}]\). The focus of the beam pattern was at a depth of 30 mm (\(z_{\rm f}\) = 30 mm) on the center scanline (\(x_{\rm f}\) = 0). Figure 4a presents the magnitude of the first GL relative to that of the main lobe as a function of normalized \(d_{\alpha}\), \(\hat{d}_{\alpha}\) (= \(d_{\alpha}/(2\lambda/D)\)); normalization was performed so that \(\hat{d}_{\alpha}\) equals 1 when Equation (5) is satisfied. Figure 4b illustrates the PW propagating regions; region A is a region where no PWs pass through; region B is a region where more than one PW propagate, and region C is a region where all of the collimated beam areas of the PWs overlap. As mentioned above, the larger \(\hat{d}_{\alpha}\) yields GLs closer to the main lobe. Sections A, B, and C shown in Figure 4a indicate that the first GL is located in regions A, B, and C in Figure 4b, respectively. Figure 4c shows the beam patterns of three values of \(\hat{d}_{\alpha}\) from section A, B, and C (from left to right panels). The gray dash-dot line of each panel indicates  
Figure 3: PW angles for synthetic focusing (left panels) and theoretical synthetic beam patterns (right panels) using three angle sets; (**a**) uniform angle set 1: \(N_{1}\) = 6, \(d_{\alpha,1}\) = 0.069; (**b**) uniform angle set 2: \(N_{2}\) = 6, \(d_{\alpha,2}\) = 0.046; (**c**) non-uniform angle set (combination of two uniform sets in (**a,b**)): \(N\) = \(N_{1}\) + \(N_{2}\) = 12.  
the theoretical first GL position (\(x_{\text{GL},1}=\lambda/d_{\alpha}\)). From the left to right panels in Figure 4c, one can observe that the magnitude of GL increases as \(\hat{d}_{\alpha}\) increases. Since none of the PWs propagate with the LWF in region A of Figure 4b (i.e., the GL elimination requirement in Equation (4) is satisfied), the GL level is sufficiently suppressed in section A of Figure 4a. In section B of Figure 4a, as \(\hat{d}_{\alpha}\) increases, the first GL moves closer to the focus. As it is closer to the center scanline (\(x=0\)) across region B of Figure 4b, the number of overlapped collimated beam areas (i.e., the number of PWs preserving the LWF) increases.