3.1 Observations and pressure measurement

Figure 3 (in which $t$ denotes the elapsed time from the start of illumination with the laser pulse) shows the measurement results obtained with the $10\times$ objective lens. Figure 3(a) shows a snapshot of the plasma luminescence in an elongated area, the major axis of which is in the direction of the laser beam. The image sequence of the plasma confirms that all parts of the plasma emit strong lights within $\pm 5$  ns. A laser-induced bubble then emerges where the plasma was formed (figure 3 b) and its shape is also elongated in the direction of the laser beam. At $t=0.4~\unicode[STIX]{x03BC}\text{s}$ , non-single spherical shocks are observed. In contrast, at $t=2.4~\unicode[STIX]{x03BC}\text{s}$ , the shock could be regarded as a single spherical shock (see figure 3 c). However, enlarged images for $\unicode[STIX]{x1D703}=0^{\circ }$ and $\unicode[STIX]{x1D703}=90^{\circ }$ (figure 3 c $\unicode[STIX]{x1D703}=0^{\circ }$ and $\unicode[STIX]{x1D703}=90^{\circ }$ ) display a clear difference. Two shock waves for $\unicode[STIX]{x1D703}=0^{\circ }$ (figure 3 c $\unicode[STIX]{x1D703}=0^{\circ }$ ), which are different from the single shock wave for $\unicode[STIX]{x1D703}=90^{\circ }$ (figure 3 c $\unicode[STIX]{x1D703}=90^{\circ }$ ). Figure 3(d) shows the temporal evolution of the pressure measured with the two hydrophones placed at different positions. There are two peaks for $\unicode[STIX]{x1D703}=0^{\circ }$ , while there is a single large peak for $\unicode[STIX]{x1D703}=90^{\circ }$ , which is approximately 1.3 times higher than that for $\unicode[STIX]{x1D703}=0^{\circ }$ . Note that this dependence of the peak pressure on the angle $\unicode[STIX]{x1D703}$ is the same as that reported by Sankin et al. (Reference Sankin, Zhou and Zhong2008). For the $5\times$ objective lens, this trend is much more pronounced: four plasma groups and four shock waves separated from each other are evident (see figure 4).

Figure 3. Measurement results for a laser-induced underwater shock wave obtained with a $10\times$ objective lens. (a) Plasma luminescence at $t=5$  ns after the laser is fired. The image is captured by an ultra-high-speed video camera at 200 Mfps. (b) Shock waves and bubbles at $t=0.4~\unicode[STIX]{x03BC}\text{s}$ imaged with an ultra-high-speed video camera at 5 Mfps. (c) Shock waves at $t=2.4~\unicode[STIX]{x03BC}\text{s}$ measured with an ultra-high-speed video camera at 5 Mfps. Enlarged images for the areas of $\unicode[STIX]{x1D703}=0^{\circ }$ and $\unicode[STIX]{x1D703}=90^{\circ }$ are also presented. (d) Shock pressure measured by the two hydrophones arranged at $\unicode[STIX]{x1D703}=0^{\circ }$ (red line) and $\unicode[STIX]{x1D703}=90^{\circ }$ (blue line) with respect to the laser direction. Integrations for the pressure with respect to the elapsed time indicate pressure impulses.

Figure 4. Measurement results for a laser-induced underwater shock wave obtained with a $5\times$ objective lens. Captions for (a–d) are the same as those in figure 3.

3.2 Pressure impulse

Here, we compute the pressure impulse for $\unicode[STIX]{x1D703}=0^{\circ }$ , $P_{0}$ and that for $\unicode[STIX]{x1D703}=90^{\circ }$ , $P_{90}$ . We calculate the pressure impulse from $t=2.5$ to $4.5~\unicode[STIX]{x03BC}\text{s}$ , during which period of time the plasma formation ( ${<}10$  ns) is totally covered, i.e. the time window does not affect the results of pressure impulse. Integrations for the pressure with respect to the elapsed time indicate pressure impulses. Both $P_{0}$ and $P_{90}$ for the shock obtained with the $10\times$ objective lens are shown in figure 3(d) in units of Pa s. $P_{0}$ is in reasonable agreement with $P_{90}$ . Furthermore, the pressure impulse and peak pressure for $\unicode[STIX]{x1D703}=0^{\circ }$ and $90^{\circ }$ were examined for all the other experimental conditions. While the peak pressure for $\unicode[STIX]{x1D703}=0^{\circ }$ and $90^{\circ }$ differ significantly, as shown in figure 5(a), $P_{0}$ was in agreement with the corresponding $P_{90}$ within the experimental uncertainty for a wide range of experimental parameters (figure 5 b). Note that the order of the pressure impulse in this study is of the same order as is required for practical use for drug delivery of cytoplasmic molecules (Kodama et al. Reference Kodama, Hamblin and Doukas2000) and the generation of microjets (Tagawa et al. Reference Tagawa, Oudalov, Visser, Peters, van der Meer, Sun, Prosperetti and Lohse2012, Reference Tagawa, Oudalov, Ghalbzouri, Sun and Lohse2013).

Figure 5. (a) Peak pressure of a laser-induced underwater shock wave measured at $\unicode[STIX]{x1D703}=0^{\circ }$ and $\unicode[STIX]{x1D703}=90^{\circ }$ for all the experimental conditions. Each presented value is a mean of three measurements and the error bars show the standard deviations. (b) Pressure impulse of a laser-induced shock wave measured at $\unicode[STIX]{x1D703}=0^{\circ }$ and $\unicode[STIX]{x1D703}=90^{\circ }$ for all the experimental conditions.

Figure 6. Schematic of a multiple structure model for a laser-induced underwater shock wave. Multiple plasmas emit multiple bubbles and spherical shock waves. The shock pressure at a certain point is the sum of the spherical shock pressures that reach the same point.

3.3 Structure of multiple shock waves

Based on the aforementioned results, we here propose a model for the structure of the laser-induced shock wave: the shock has a multiple structure that consists of multiple spherical shock waves, as depicted schematically in figure 6. We assume that each spherical shock wave originates from the corresponding plasma formation. In addition, since the shock wave behaves acoustically at low pressure ( ${\lesssim}100$  MPa (Vogel et al. Reference Vogel, Busch and Parlitz1996a )), we could apply the superposition principle to analyse the pressure impulse in the far field. The net pressure that is produced by two or more shock waves reaching the same point is the sum of the pressure induced by the individual shock waves. A phenomenon that is analogous to this may be the surface wave observed after one or several stones are thrown into a quiescent pond (the so-called Huygens–Fresnel principle). Note that the Huygens–Fresnel principle does not apply for nonlinear shock-wave propagation.

For both the $10\times$ and $5\times$ objective lenses, this model rationalizes the observations of both pressure peaks and pressure impulse: several peaks for $\unicode[STIX]{x1D703}=0^{\circ }$ and the single large peak for $\unicode[STIX]{x1D703}=90^{\circ }$ while pressure impulse for $\unicode[STIX]{x1D703}=0^{\circ }$ matches the pressure impulse for $\unicode[STIX]{x1D703}=90^{\circ }$ . Results for a wide range of experimental parameters (figure 5) may indicate the universality of the multiple structure model for optical breakdown at low or moderate $NA$ . Note that this scenario includes the well-known spherical shock model. It should be emphasized that, as observed with the $10\times$ objective lens (figure 3), even if just a single plasma or a bubble is observed, the origin of the elongated plasma and bubble is expected to be multiple spots of plasma, which lead to the emergence of multiple spherical shocks, resulting in a notable angular variation of shock pressure. This model for the multiple shock structure could possibly rationalize the results reported in previous research. For instance, Sankin et al. (Reference Sankin, Zhou and Zhong2008) reported that a laser-induced shock wave emitted from an elongated plasma has an angular variation of pressure distribution in the far field. Although the shape of the shock wave appears to be spherical, the elongated plasma may cause a multiple structure of the shock, as observed in the present experiments (see figure 3), which would lead to a pressure peak of the shock which is not spherically symmetric. Besides the aforementioned phenomena in the far field, an anisotropy of the shock wave in the near field of cylindrical plasmas has been reported (e.g. Schoeffmann, Schmidt-Kloiber & Reichel Reference Schoeffmann, Schmidt-Kloiber and Reichel1988), which we discuss in detail in § 3.5.

3.4 Laser-induced plasma and bubbles

Here, attention is given to plasma formation, which is the origin of the multiple shock structure. Figure 7(a) shows the plasma luminescence under all the experimental conditions. Figure 7(b) shows the length of the plasma region in the direction of the laser beam as a function of laser energy for the $5\times$ , $10\times$ and $20\times$ objectives. The length increases with the laser energy for all objectives. With lower magnification, the length of the plasma region becomes longer. The relation between an elongated plasma length (region) and focusing angle or laser pulse energy have been intensively investigated experimentally by Vogel et al. (Reference Vogel, Nahen, Theisen and Noack1996b ). They found that there is a strong dependence of the elongated plasma length on the focusing angle and laser energy: plasma is created in the cone angle of the laser beam proximal to the laser and becomes longer with larger laser energy. Our results are consistent with theirs. Thus multiple plasma formation is dependent on the spherical aberrations of the focusing optics, liquid impurities and the input laser energy (Nahen & Vogel Reference Nahen and Vogel1996; Vogel et al. Reference Vogel, Busch and Parlitz1996a ,Reference Vogel, Nahen, Theisen and Noack b ).

Figure 7. (a) Plasma formation under all experimental conditions (Magnification of objective lens: $5\times$ , $10\times$ , $20\times$ . Laser energy: 2.6, 6.9 and 12.3 mJ). The plasma shape is the most elongated with an input energy of 12.3 mJ and the $5\times$ objective lens, whereas it is rather spherical with an input energy of 2.6 mJ and the $20\times$ objective lens. (b) Length of plasma as a function of the laser energy. The circle plot and error bar show respectively the mean and the standard deviation in 5 trials. Red, blue and green colour bar represent the microscope objective of $5\times$ , $10\times$ and $20\times$ , respectively.

There are two mechanisms which can lead to plasma formation: direct ionization of the medium by multiphoton absorption or avalanche ionization via inverse bremsstrahlung absorption (Vogel et al. Reference Vogel, Nahen, Theisen and Noack1996b ). The mechanism leading to optical breakdown (plasma formation) in this study is avalanche ionization by heating of impurities since the position of plasma formation varies within a certain region. Note that impurities may provide centres for both linear and nonlinear absorption. If the multiphoton process in nonlinear absorption has a lower order than that in pure water, impurities can trigger breakdown. Figure 8 shows snapshots of bubbles (and shock waves) under the same conditions (the same magnification, laser energy, etc.). The number/position of the plasmas vary at every laser shot. If the plasma formation mechanism were purely direct ionization of the medium by multiphoton absorption, an identical plasma shape should be observed for every laser shot in the same focusing optics with the same laser energy.

Note that multiple plasma generation will not be avoided by ‘perfect’ focusing without any spherical aberrations since the plasma occurs in the region where the local energy exceeds the breakdown thresholds and thus the plasmas do not always occur at the perfectly focused point (Vogel et al. Reference Vogel, Nahen, Theisen, Birngruber, Thomas and Rockwell1999a , Reference Vogel, Noack, Hüttman and Paltauf2005). Therefore, the focusing angle is a crucial parameter since it strongly affects the local energy density. In our experiment, there exists a strong relation between the length of the plasma region and the focusing angle.

In the case of linear sound propagation, for an extremely elongated cylindrical (or conical) source, anisotropic emission is expected with more total energy and total impulse in the directions perpendicular to the cylinder than along its axis. However, both bubbles and shock waves in the near field (e.g. figure 1) show that the source is not a single plasma but a collection of point-like plasmas in a conical region. Therefore, even in the case of linear sound propagation, it is not obvious to assume anisotropy of the total energy and impulse.

The number of bubbles depends on the laser energy and the objective magnification, as shown in figure 9. The number of bubbles increases with increasing laser energy and a lowering of the objective magnification. This trend is consistent with the length of plasma region as shown in figure 7. We measure the distances between the bubble centres and count the number of times that a certain distance emerges for all experimental conditions. Figure 10(a–c) represents the histogram of the distances of each bubble centre for the microscope objectives at $5\times$ , $10\times$ and $20\times$ , respectively. The number of times is the sum of the three trials. The average distance for each condition depends on the focusing angle (microscope objective) and the laser energy. Bubble centres are not always equidistant.

Figure 8. Snapshots of laser-induced bubbles and shock waves with $5\times$ , $10\times$ , 6.9 mJ, $t=50$  ns. These results are obtained under the same conditions.

Figure 9. Number of bubbles of as a function of the laser energy. Each colour bar represents the result of one pulsed laser. Red, blue and green colour bars represent the microscope objective of $5\times$ , $10\times$ and $20\times$ , respectively.

Figure 10. Histogram of the distances between bubble centres. Vertical axis is number of times that a certain distance emerges for all experiments; (a–c) show the histogram of the microscope objective of $5\times$ , $10\times$ and $20\times$ , respectively. The colour of the bar represents the energy of a pulsed laser. The blue, green and red bars relate to 2.6, 6.9 and 12.3 mJ, respectively.

3.5 Shock wave in the near field

In this section, we estimate the pressure decay in the near field in order to discuss energy dissipation. The estimation of energy dissipation for multiple plasmas and shock waves is, however, a large challenge since it is hard to disentangle the energy dissipation and nonlinear interaction in the near field.

We now use a high-speed camera for following shock-wave emission from the source in the near field. We take snapshots of the shock wave, follow its position (figure 11 a), derive its velocity (figure 11 b) and estimate the pressure (figure 11 c). To estimate the pressure in a shock front $p_{s}$ , we apply following equation (Vogel et al. Reference Vogel, Busch and Parlitz1996a ),

(3.1) $$\begin{eqnarray}p_{s}=c_{1}\unicode[STIX]{x1D70C}_{0}u_{s}(10^{(u_{s}-c_{0})/c_{2}}-1)+p_{\infty },\end{eqnarray}$$

where $\unicode[STIX]{x1D70C}_{0}$ is the density of water before compression by the shock wave, $c_{0}$ is the normal sound velocity in water, $c_{1}=5190~\text{m}~\text{s}^{-1}$ , $c_{2}=25\,305~\text{m}~\text{s}^{-1}$ and $p_{\infty }$ is the hydrostatic pressure. The results show that, in the near field, the pressure of the shock in the direction perpendicular to the laser beam is higher than that in the direction of the laser beam, as shown in figure 11(c). Note that Vogel et al. (Reference Vogel, Busch and Parlitz1996a ), Noack & Vogel (Reference Noack and Vogel1998) found similar pressure values close to the plasma rim as presented in the present study. The energy dissipation at the fronts of high-pressure shock waves is thus expected to be faster than at the low-pressure fronts since Vogel et al. (Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999b ) showed that the dissipation rate of acoustic energy is proportional to the pressure jump at the shock front. However, the pressure decay related to energy dissipation is slightly slower at the front of a high-pressure shock wave than that at a low-pressure front. Similar results had been reported by Schoeffmann et al. (Reference Schoeffmann, Schmidt-Kloiber and Reichel1988), Vogel et al. (Reference Vogel, Busch and Parlitz1996a ) that the pressure decay is not faster at the fronts of high-pressure shock waves than those of low-pressure fronts. Vogel et al. (Reference Vogel, Busch and Parlitz1996a ) attributed it to the formation process of the shock wave: the pressure maximum is located behind the leading edge of the pressure transient. Our interpretation of this is that, in the near field along the direction perpendicular to the laser beam, a shock wave from one of the plasmas is followed by the other shock waves originating from the other plasmas that eventually overlap and add to the pressure of the shock at larger distances. This might lead to slower pressure decay compared to the case of a single shock wave.

It is noteworthy that high pressures in the near field combined with nonlinear propagation and a strong anisotropy translate into a linear isotropic pressure impulse in the far field. We discuss mechanisms of this interesting finding in this paragraph. For a cylindrical source Schoeffmann et al. (Reference Schoeffmann, Schmidt-Kloiber and Reichel1988) showed an anisotropic shock-wave emission with most energy in the $90^{\circ }$ direction due to a geometrical effect. The anisotropic pressure jump at the initial shock front suggests that the total amount of energy dissipation is higher in the $90^{\circ }$ direction than in the $0^{\circ }$ direction (cf. Vogel et al. Reference Vogel, Noack, Nahen, Theisen, Busch, Parlitz, Hammer, Noojin, Rockwell and Birngruber1999b ). Therefore one likely explanation for the transition is that the initial anisotropy is eroded with increasing propagation distance during nonlinear sound propagation in the transition from the near field to the far field, and an isotropic impulse distribution in the far field can evolve. Another possible explanation is that nonlinear propagation in the near field has limited effects on the isotropic impulse distribution in the far field under the present experimental conditions. In order to address the aforementioned discussion, further experimental evidence of ultra-high-speed recordings would be needed.

Figure 11. (a) The distance between plasma and shock front versus the elapsed time. (b) The velocity of the shock wave versus the elapsed time. (c) The pressure of the shock wave versus the distance between the plasma and shock front. The inserted numbers indicate the local slope of the corresponding curve.