# Shape Optimization of an Open Photoacoustic Resonator
## 3 Results and Discussion
### Parameter Sweep  
#### 3.2.1 Resonance Cylinder  
A first sweep is performed by changing the resonance cylinder length (\(L_{r}\)) and position along the cavity cylinder (\(D_{r}\)). The length is changed from 0 mm (no resonance cylinder) to 17 mm with 1 mm step size. Since the microphone is located at the end of the resonance cylinder, \(L_{r}\) also indicates the position where the PA signal is detected. The position of the resonance cylinder along the cavity cylinder is described relative to the open resonator end. The position (\(D_{r}\)) is changed from the open end of the cavity cylinder to the end connected to the absorption cylinder by a step size of 1 mm that is between 2 mm and 14 mm. During the sweep, all other resonator dimensions are kept constant and the PA signal for all possible combinations of \(L_{r}\) and \(D_{r}\) is calculated between 48 kHz to 51 kHz. To evaluate the quality of the solution from each parameter combination in comparison to the reference cell, a ratio of the relative PA signal (\(\varsigma\)) is defined  
\[\varsigma\ =\ \frac{S}{S_{ref}} \tag{1}\]  
\(S\) and \(S_{ref}\) are the maximum PA signal of each parameter combination and the reference resonator, respectively. Figure 3: Frequency response of the reference resonator from the viscothermal model with perfectly matched layers (VT-PML) simulation and measurement. The measurement and simulation results are scaled for easier viewing and comparison. The results of the resonance cylinder sweep are presented in Figure 4. Six peaks with \(\varsigma\) of more than 1.5 are identified and the discussion is limited to these six peaks. The peaks are observed when \(L_{r}\) is 1 mm, 8 mm and 15 mm while \(D_{r}\) is at either 4 mm or 5 mm and at 12 mm. The resonance cylinder supports a longitudinal acoustic mode as can be seen in Figure 5. The resonance cylinder can be independently viewed as an open-closed cylindrical resonator whose resonance is analytically calculated using [25]. \[L=\frac{(2m-1)c}{4f} \tag{2}\]  
where \(m~{}=~{}1,2,3\dots\). The terms \(c\) and \(f\) are the speed of sound and the resonance frequency respectively. \(L\) is the length of the resonator together with the so-called end correction. Assuming that the resonance frequency is 50 kHz and the speed of sound is 343 m/s, the resonance cylinder length for \(m~{}=~{}1\) to \(m~{}=~{}5\) is analytically calculated as 1.72 mm, 5.15 mm, 8.58 mm, 12.01 mm and 15.44 mm. Therefore, it can be concluded that resonance amplification is responsible for the peaks at \(L_{r}\) 1 mm, 8 mm and 15 mm. For a given \(D_{r}\), \(\varsigma\) is highest when \(L_{r}\) is 8 mm. However, the difference to the \(\varsigma\) values of \(L_{r}\) at 1 mm and 15 mm is small. When \(D_{r}\) is at either 4 mm or 5 mm and at 12 mm, \(\varsigma\) has a value of more than 1.5. These positions represent the location of an acoustic antinode of the cavity cylinder. By  
Figure 4: Relative photoacoustic (PA) signal of the coarse resonance cylinder sweep (**a**) along with a contour plot (**b**). The color scale gives the values of \(\varsigma\). The peak highlighted in the left plot is the region with the highest relative PA signal. Figure 5: The absolute pressure distribution plots of the six peaks. The plots are normalized by the highest value of the individual plot. Top: resonance cylinder position (\(D_{r}\)) 4 mm (**a**) and 5 mm (**b**,\(\epsilon\)). Bottom: resonance cylinder position (\(D_{r}\) ) 12 mm. The length of the resonance cylinder (\(L_{r}\) ) is 1 mm, 8 mm and 15 mm respectively. placing the resonance cylinder at these positions, the antinode is coupled into the resonance cylinder and thus a strong PA signal is detected. For the same \(L_{r}\), \(\zeta\) is higher when \(D_{r}\) is 12 mm than when at either 4 mm or 5 mm. Therefore, it can be concluded that \(D_{r}\) affects the strength of the signal more than \(L_{r}\). The highest \(\zeta\) value of 2.2 is observed when \(D_{r}\) is 12 mm and \(L_{r}\) is 8 mm (highlighted in Figure 4).