# Methods for Grating Lobe Suppression in Ultrasound Plane Wave Imaging
## Materials and Methods
### GL Suppression Method with Non-Uniformly Distributed PW Angles  
The GL levels can also be reduced using non-uniformly distributed PW angles (i.e., by violating the uniform \(d_{\alpha}\) condition). In this paper, to generate a non-uniform angle set, an approach using two uniform angle sets with different \(d_{\alpha}\) values is presented. We let the uniform angle set 1 and set 2 have \(N_{1}\) PWs with an interval of \(d_{\alpha,1}\) and \(N_{2}\) PWs with an interval of \(d_{\alpha,2}\), respectively. The GLs of set 1 would appear at integer multiples of \(\lambda/d_{\alpha,1}\), while the GLs of set 2 would arise at integer multiples of \(\lambda/d_{\alpha,2}\), according to Equation (2). To suppress the GL level, a non-uniform PW angle set can be obtained by combining the two uniform angle sets. The beam pattern of the non-uniform angle set is given by the sum of the beam patterns of the two uniform angle sets. Note that the main lobes of sets 1 and 2 are both centered at \(x=x_{\rm f}\). Therefore, after the field responses for two uniform sets are summed, the peak value of the resulting main lobe will always be larger than the main lobe peak of each angle set. When \(d_{\alpha,1}\) and \(d_{\alpha,2}\) are chosen to locate the GLs of two uniform sets in different locations (i.e., \(m_{1}\lambda/d_{\alpha,1}\
eq m_{2}\lambda/d_{\alpha,2}\) where \(m_{1}\) and \(m_{2}\) are integer numbers), the GLs of the non-uniform angle set will have smaller magnitudes than the main lobe. In addition, even when the GLs of two uniform sets overlap (i.e., \(m_{1}\lambda/d_{\alpha,1}=m_{2}\lambda/d_{\alpha,2}\)), they can also be reduced after they are combined if the two coincident GLs have opposite phases. For example, when \(N_{1}\) and \(N_{2}\) are both even, the GL at the same location in the beam pattern of the non-uniform set will have a magnitude of:  
\[|\psi|=\big{|}(-1)^{m_{1}}N_{1}+(-1)^{m_{2}}N_{2}\big{|}, \tag{6}\]  
which can be derived by Equation (3). In this case, the magnitude will be decreased to \(|N_{1}-N_{2}|\) if \(m_{1}\) is even and \(m_{2}\) is odd, or vice versa. Figure 3 shows an example of a non-uniform angle set that is obtained by combining two uniform angle sets. The angle distributions (left panels) and synthetic Tx beam patterns (right panels) of the uniform angle set 1 and set 2 are presented in Figure 3a,b, respectively. The number of PWs and angular intervals of set 1 and set 2 are (\(N_{1}=6\), \(d_{\alpha,1}=0.069\)) and (\(N_{2}=6\), \(d_{\alpha,2}=0.046\)), respectively. The combined non-uniform angle set and its synthetic beam pattern are shown in Figure 3c. The synthetic beam pattern was obtained by Equation (1), assuming a center frequency of 5.208 MHz and a sound speed of 1540 m/s (i.e., \(\lambda=0.296\) mm). In Figure 3a,b, the main lobe is located at 0 and the GLs are repeated at a certain interval. The intervals between the GLs of set 1 and set 2 are 4.29 mm and 6.43 mm, respectively, according to Equation (2). The GLs of uniform sets 1 and 2 at different locations are halved in the beam pattern of the non-uniform angle set (see the right panel of Figure 3c). For the chosen parameters (\(d_{\alpha,1}=0.069\), \(d_{\alpha,2}=0.046\)), \(m_{1}\lambda/d_{\alpha,1}\) is equal to \(m_{2}\lambda/d_{\alpha,2}\) when \(m_{1}=3\) and \(m_{2}=2\), which means that the third GL of set 1 coincides with the second GL of set 2, as in Figure 3a,b. As the two GL have the same magnitude with the opposite sign according to the Equation (3), they are canceled out in the combined transmit beam pattern (Figure 3c), as is expected from Equation (6). On the other hand, the next coincident GLs at \(x=25.6\) mm, corresponding to \(m_{1}=6\) and \(m_{2}=4\), have the same sign and thus have the same magnitude as the main lobe peak in the beam pattern of the non-uniform set. However, this theoretical beam pattern is calculated assuming an infinite aperture.