# Cross-Medium Photoacoustic Communications: Challenges, and State of the Art
## 2 Photoacoustic Signal Generation
### Breakdown and Bubble Collapse Shockwaves  
The laser-induced breakdown and bubble collapse produce shockwaves during the nonlinear PA process. First, the breakdown shockwave emission occurs during the plasma formation. Subsequently, the generated cavitation bubbles expand, collapse, and re-expand, creating additional in-water acoustic signals of various frequencies, depending on the bubbles' size. In [63], the breakdown shockwave velocity (\(u_{s}\)) has been measured using time-resolved photography and then determined the shock pressure as,  
\[p_{s}=c_{1}p_{0}u_{s}\bigg{(}10^{\frac{u_{s}-c_{0}}{c_{2}}}-1\bigg{)}+p_{\infty} \tag{7}\]  
Here, \(\rho_{0}\) is the water density before being compressed by the shockwave, \(c_{0}\) is the normal sound velocity in the water, \(c_{1}=5190\) ms\({}^{-1}\), \(c_{2}=25306\) ms\({}^{-1}\), and \(\rho_{\infty}\) is the hydrostatic pressure. However, the determination of shock pressure is accurate only in a small region of less than one mm\({}^{3}\) around the emission center, where the difference between shockwave and sound velocities can be easily detectable. Figure 4 shows experimentally measured shockwave velocities for nanosecond and picosecond pulse durations where we can observe the velocity is higher for nanosecond pulses than the picosecond pulses. Nanosecond pulses produce a higher electron density than picosecond pulses, which results in a higher plasma absorption coefficient, stronger plasma radiation, higher plasma temperature, and stronger mechanical effects such as shockwave emission and cavitation bubble expansion [55]. Figure 3: Plasma sphericity index with laser pulse energy and focusing angle, redrawn from [62]. When the breakdown shockwave separates from the plasma, a volume of high pressure and temperature remains, which is out of equilibrium with the surroundings. Such volume expansion is slower than the breakdown shockwave release. Both work against the ambient static pressure (\(p_{stat}\)), which in underwater steps is a combination of the hydrostatic pressure due to the weight of the water column and the atmospheric pressure on the free surface. The expanding volume is then filled with water vapor, exerting an internal pressure \(p_{v}\) and leading to the formation of a spherical bubble of maximum radius \(R_{max}\). The energy stored in the cavitation bubble is directly proportional to the maximum bubble radius raised to the third power [63],  
\[E_{B}=\frac{4\pi}{3}(p_{stat}-p_{v})R_{max}^{3} \tag{8}\]  
The interior pressure is \(p_{v}\) (2.33 kPa in water at 20 \({}^{\circ}\)C) is much smaller than the outer pressure \(p_{stat}\) (typically about 100 kPa) after reaching the maximum radius of the bubble [64]. Then the bubble is compressed back to a small volume, and a shockwave generates upon the collapse of the bubble. The bubble expands again to a smaller maximum radius because of the energy lost by the previous bubble collapse shockwave and the rebound bubble is no longer spherical [64]. The maximum bubble radius is related to its oscillation period \(T_{B}\) which is twice the collapse time (\(T_{c}\)) by Rayleigh equation [65],  
\[R_{max}=T_{B}/2\times 0.915\left(\frac{\rho_{0}}{\rho_{\infty}-\rho_{v}}\right) ^{0.5} \tag{9}\]  
Figure 5 illustrates the breakdown shockwave and bubble expansion and collapse shockwaves generated by our experiment with 1064 nm Nd:YAG laser with 6 ns pulse duration. We can observe that each successive shockwave has less peak pressure because the previous shockwave containing mechanical energy is already emitted, and heat is exchanged with the surrounding environment. Therefore, the remaining mechanical energy is reduced because no new energy is imparted from the laser source during this time. Thus, the breakdown shockwave has higher peak pressure than bubble expansion and  
Figure 4: Experimentally determined shockwave velocity for 6 ns and 30 ps pulse duration, redrawn from [63]. collapse shockwaves.