# Generating Airborne Ultrasonic Amplitude Patterns Using an Open Hardware Phased Array
## 5 Results
### Effect of Phase, Amplitude, and Spatial Resolution  
We carried out multiple simulations using the algorithm that is described in Section 4 with different parameters for emitter size, phase emission resolution and amplitude emission resolution. All of the target amplitude fields were generated 16 cm above the array, since we tested that, at that distance, the best results were obtained. The default simulation parameters are those from the SMD board, i.e., _emitterSize_ = 10 mm, _phaseResolution_ = 32, and no amplitude modulation. One parameter was varied at a time and the mean square error (MSE) of the obtained imaged was obtained. The results can be seen in Figure 9.  
Figure 8: Amplitude slices obtained for different patterns, plotted using the function imagesc of Matlab. The first row is the target, the second one is the simulated slice, and the third row is the experimental measurement.  
Figure 9: Simulated amplitude fields at 16 cm from the array for different target patterns and array parameters. (First column) target amplitude field. (Second column) obtained amplitude field when the emitter array is the surface mounted device (SMD) board presented in the paper, i.e., _emitterSize_ = 10 mm (transducer diameter), _phaseResolution_ = 32 and no amplitude modulation. (Third column) obtained amplitude fields with an array with _emitterSize_ = 2 mm, _phaseResolution_ = 32 and _amplitudeResolution_ = 16. (Fourth column) mean square error (MSE) as the emitter size decreases. (Fifth column) MSE as phase resolution increases, _emitterSize_ = 10 mm. (Sixth column) MSE as the amplitude resolution increases, _emitterSize_ = 10 mm.  
The patterns employed were: the flag of Brazil (non-binary image), the letter A, a Dove, and a smiley face. In general, it can be seen that as the emitter size decreases (i.e., more spatial resolution), the quality of the images improves. It is important to note that significant reductions of MSE are obtained, even when emitters get smaller than half-wavelength (4.3 mm), and that no further improvement is obtained below 2 mm (1/4 of the wavelength); this is different from the generation of regular focal points that do not increase its amplitude once the emitters are reduced below half-wavelength size [13]. The phase resolution significantly improves the pattern quality, but quickly plateaus when the phase resolution reaches eight divisions per period; this is in accordance with the simulations performed for simple focal points [41]. For amplitude resolution, it is clear that having amplitude modulation reduces the MSE by half even when only four different amplitudes can be emitted. In summary, the sweet-spot is obtained with a phase resolution of eight divisions per period and amplitude resolution of four divisions; the MSE improves as the emitters get smaller (i.e., more spatial resolution), but no improvement is found once the emitter size reaches quarter-wavelength.  
These findings could be specific to the patterns that were selected in the study and to our setup characteristics (e.g., wavelength, number of emitters or distance to the target slice); however, the code was made public, so that other researchers could run simulations for their specific setups (e.g., operating in water or with static metamaterials).