1. Introduction

When the incident laser intensity exceeds a certain threshold, optical breakdown in underwater occurs and results in the generation of plasma [1–6]. Afterward, the expanding plasma drives shock wave propagation and bubble oscillation [2,7,8]. The mechanism and phenomenon of shock wave induced by illumination with a nanosecond laser pulse have been extensively studied due to its various applications including clinical surgery, drug delivery, and low-invasive medical treatments to name a few [9–13]. The visualization of laser-induced optical breakdown subsequent process underwater provides information about the shock wave emission and cavitation bubble collapse [7,14,15].

As early as the 1960s, Prokhorov et al. first observed the generation and propagation of shock waves underwater through light absorption and thermal expansion [16]. After that, the theoretical and experimental studies on laser-induced underwater shock wave phenomenon are endless, and various review papers came into being [17–21]. The equation of state (EoS) of water is obtained to describe the pressure distribution generated by underwater shock waves by Ridah et al. [22]. The Tait EoS is used to describe the compressibility of water, and the relationship between normal shock waves and oblique shock waves is provided. Ritva et al. established the modified Rayleigh-Plesset equations and described the dynamics of underwater shock waves and cavitation bubbles in detail [23]. The sonoluminescence caused by cavitation bubble expansion and other nonlinear effects is ignored in this study. On the other hand, Docchio et al. reported the movements of the shock wave during plasma formation underwater by using streak photographs [24]. Although underwater shock wave emission has been proven repeatedly, the quantification of shock wave parameters (pressure, energy, etc.) in the process of plasma evolution is still insufficient. These parameters are also important for the propagation and effective control of underwater shock waves.

In 2016, the multiple shock wave dynamics induced by a single focused pulse laser were observed by Tagawa et al. [14], who stated that the peak pressure distribution is not spherically symmetric, the discrepancy derived from multiple breakdowns. Indeed, the phenomenon of multiple breakdowns has been observed decades ago. Zheng et al. [25] have summarized the reasons for multiple breakdowns, including the moving-breakdown effect, impurities, self-focusing effect, etc. In our experiment, the multi-point breakdown effect is removed as much as possible utilizing an aplanat lens. Therefore, only the propagation of a single shock wave is considered in the corresponding optical breakdown region. In addition, compared with a single excitation beam, the multi-excitation beam used in this study accompanying the formation of multiple plasmas, cavitation bubbles, and shock waves emission results in a more interesting and transient dynamic process. However, the use of this underwater multi-excitation model and the subsequent shock wave characterization are rarely mentioned. Partial studies are focused on the influence of external factors on the propagation of shock waves, such as the influence of different incident laser energy on shock wave propagation. To name a few, Vogel et al. [26] found that at high irradiance, the evolution of shock waves is not completely symmetric due to the generation of conical plasmas. When focusing the high power beam onto the submerged metal surface, the shock wave of 230 kbar was determined from the shock wave of 7000 m/s underwater by Bell et al. [19]. Similarly, shadowgraphs of the shock wave propagation under different delay times are also given here.

As to bubble formation, a single bubble and strings of bubbles were originally described as "The black circle is the shadow of the slowly expanding region of the liquid, which is susceptible to thermal diffusion processes" [20]. Carome et al. [18] discovered a single cavitation bubble and its peripheral propagation of shock waves by using shadowgraphs. Afterward, the complete period of phenomena including bubble expansion and collapse after the laser-induced breakdown was first defined by Ellis et al. [21]. However, the study was only a simple image of the bubbles, with no further analysis of the dynamics of the bubbles. Likewise, Lau et al. [27] studied the underwater optical breakdown process and the dynamical evolution of the bubbles by high-speed photography and later high-speed holography. In [28], the excitation measurements of Carome [18] were confirmed and extended by high-speed photography of the laser-generated bubbles as well as the emitted shock waves. In 2021, Bao et al. characterized the interaction process of three underwater cavitation bubbles induced by a pulsed laser by using the Schlieren method [7]. Nonetheless, it is still difficult to find good parameters for the breakdown of the shock wave and bubble simultaneously, mainly because complex internal processes such as heat conduction and mass transfer (evaporation and condensation) in the bubble are not considered in the numerical simulations.

Unlike cavitation bubbles, both single-point optical breakdown and conical plasmas produce shock wave propagation that is almost independent of one another. The corresponding shock wave is simulated as a spherical shock wave. In general, the emission of shock wave arises from a close bubbles cloud, generated by a short intense focused pulse incident underwater, which has been observed via high-speed imaging and acoustically detected with a needle-type hydrophone [29–34]. In Ref. [30], the shock wave signals generated by high-power laser in water are measured experimentally, and the results show that the selection of pulse width has a great influence on the formation and propagation of shock waves (the shape and intensity of the shock wave and so on). Moreover, during the interaction of the femtosecond pulse with water, the deposition energy is low and tends to produce a point source of sound, In contrast, picosecond pulses deposit much more energy and produce an extended source. Localized deposition of laser energy in water depends on the optical breakdown of water to form luminous plasma and is related to high volume energy density [5], temperature [4], and pressure [6]. Therefore, with an isotropic environment, almost perfectly spherical shock waves can be generated in water. Furthermore, Liang et al. [35] have summarized the distribution path of absorbed energy, including shock wave emission [32], bubble oscillation [7], vaporization energy [36], plasma irradiation [6], and energy dissipation [35]. Specifically, the absorbed energy that goes into evaporation is beneficial for laser surgery and soft tissue injuries, whereas the absorbed energy into shock wave emission and bubble oscillation contributes to the optical breakdown process [5,7]. However, the energy carried by shock wave propagation during optical breakdown remains unclear and requires further study. On the other hand, images of individual shock waves and plasma propagation in the air were recorded by Olga et al. [37], who stated that the initial peak pressure loading in the GPa range and subsequently dropped down to MPa [37]. We here concentrate on the underwater scenario and similar results are given. On top of that, Sankin et al. reported that the spatial distribution of multiple breakdowns results in the intensity and propagation of multiple shock wave pressures varying with the direction of extension [38]. Based on this, Potemkin et al. found that a cylindrical shock wave could be created underwater by overlapping multiple shock waves induced by a tightly focused femtosecond laser [39]. However, in the process of multiple independent underwater optical breakdowns, the propagation of shock wave and the information it carries, the relationships between multiple shock waves, and their interaction are still unclear and need further study.

To date, as applications continue to emerge and expand, especially in the medical field, we have seen a large number of studies on a laser-induced optical breakdown in water with bubbles and shock wave formation [2,7,14]. Meanwhile, many articles concerning laser-induced optical breakdown include the understanding and discovery of shock waves and cavitation bubbles [40–42]. In the present study, multiple points of excitation in deionized water were achieved by splitting the nanosecond laser pulse with a triple-spot splitter DOE device. Afterward, we measured and characterized the shock wave propagation distance and the interaction of adjacent shock waves caused by optical breakdown. The effective energy carried by the shock wave is quantified by fitting the experimental results. Further, numerical simulations with an analytic model using the distance between adjacent breakdown locations as input and effective energy as fit parameters, the velocity distributions of shock waves at different time delays, and the peak pressure are obtained. In addition, due to a series of energy losses in the process of shock wave propagation and cavitation bubbles expansion, the temperature in the backward direction during shock wave propagation is also given accordingly. Finally, the influences of mutual interaction among shock waves on the emission and the velocity of shock waves during time delay were discussed.

2. Theory

As mentioned in the introduction, once the laser intensity $I$ exceeds a certain threshold, optical breakdown occurs underwater and a dense thermal plasma is generated, followed by a series of explosive evaporation in a local focusing area [1,4,5,7]. In that case, the high-power laser and water medium interaction process can be associated with a strong explosion in a homogeneous atmosphere, in which the energy generated in the explosion process will dissipate in a short time, further leading to the formation of shock waves. This shock wave resembles a piston movement, expanding at supersonic speed under the counter pressure of the surrounding liquid, producing a shocking phenomenon and compressing the liquid in front of it into a black rim while changing the refractive index of the liquid in front of the shock wave [43]. Therefore, we adopted the classical Sedov-Taylor expansion model (point blast wave theory) to characterize the shock waves generated in water. It reads [43,44]

Given the point explosion effect caused by the high absorption energy during the optical breakdown and the rapid expansion process of the shock wave, we tried to characterize the water temperature during the propagation of the transient shock wave. In hydrodynamics, knowledge of two-phase variables, such as internal energy, specific entropy, pressure, and temperature, is sufficient to describe the whole process of underwater shock wave generation and expansion. The equation of state (EoS) for water connects these variables. However, many of the already existing EoS are too time-consuming and complex to solve. In the context of bubble dynamics, this shortcoming was first addressed by Denner et al. [47], who combined the Gilmore equation and Noble-Abel stiffened-gas (NASG) EoS [39,48] to derive a simple, robust, and accurate formulation as follows [41](The specific derivation process can be found in Refs. [48,49])

3. Experimental setup

The optical system is based on a pump and probe microscopy layout in this study for investigating the dynamics of laser-induced multiple symmetrically aligned shock waves; see Fig. 1. Underwater shock waves are induced by a 1064 nm, 10 ns (full width at half-maximum) laser pulse (Nd: YAG laser; maximum energy: 1J; repetition frequency = 10 Hz) focused through an aplanat to three points inside a water tank. The specific incident laser energy and power density in the experiment are mentioned in Sec. 5.1. The single laser pulse energy is varied between 100 mJ and 500 mJ. The fluctuations of the laser pulse energy are approximately $\pm 5\%$. Specifically, to generate shock waves or subsequent cavitation bubbles, the pump laser beam is divided into two parts using the beam splitter first. One of these is reflected in the energy meter to monitor the energy of the incident laser pulse in real time. The other one is modulated into three beams using a triple-spot splitter DOE (Diffractive Optical Element; Holo/Or TS-245-I-Y-A), which is an optical diffraction element designed based on the fluctuating nature of light and spatial phase gradient [51]. More specifically, a laser beam passing through a triple-spot splitter DOE device emits three non-parallel beams at a specific separation angle with an error of fewer than 0.03 mrad (see Fig. 1(b)). More detailed information about the DOE device is provided in Supplement 1.

 

Fig. 1. Schematic of the experimental setup used to measure the shock waves emission induced by a nanosecond laser. (a) Top view. (b) Triple-spot splitter DOE device separates the beam path. (c) Side view. (d) The image of the laser-induced shock waves underwater. Scale bar 1 mm. The 3D experimental setup is illustrated in Fig. S1 (Supplement 1).

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On the other side, these three non-parallel beams are reflected onto an aplanat ($f = 25 \rm \;{\textrm{mm}}$), and the aplanat lens is utilized to focus on smaller areas to further obtain regular shock waves and bubbles. Note that the three reflection points are aligned at the intersection of the plane of incidence, thereby the breakdown regions caused by the three split laser pulses are aligned on the focal plane. Deionized water is placed in a water tank (fused silica, $50 \rm \; {\textrm{mm}} \ast 50 \rm \;{\textrm{mm}} \ast 50 \rm \; {\textrm{mm}}$) as a breakdown material. Additionally, we can block the split beam to get one or two shock waves. Experimentally, we obtained the distance between the adjacent breakdown regions by setting different angles of the three split pulses and different distances between the mirrors. On the other hand, the combined observation system consists of a short-duration laser generator (Nd: YAG, wavelength = 1064 nm, pulse duration = 2.5 ns) and a single frame CCD (Imi,imc7071g) to visualize the fast process of shock waves. The frequency of the former is doubled (wavelength = 532 nm) using a KTP frequency doubling crystal, and then the new probe beam enters the aperture and the collimating expanding lens system to achieve the energy attenuation and expansion of size, resulting in wide field illumination at the back focal plane of the water tank. Afterwards, the probe beam which carries the information of shock waves propagates in bulk water and enters the CCD camera. We make sure that the pump beam is tightly focused on bulk water while imaging the focused areas using the probe beam, the latter is perpendicular to the pump beam and the focal line. The digital delay generator (Stanford Research Systems Inc., DG535) is used to trigger the lasers and the CCD camera. The accurate time of the CCD image acquisition is further obtained by collecting the time delay between the plasma flash detected by the photodiode and the probe beam. Each measurement was repeated more than five times under the same experimental conditions.

4. Methodology

In this work, the parameters that we focused on include the spread radius, velocity, temperature, and peak pressure of the shock wave front. The radius of the shock wave at a specific time delay is obtained by a series of shadow images. To characterize the propagation and interaction of adjacent shock waves, a dimensionless stand-off distance $\gamma$ is introduced. It reads

 

Fig. 2. (a) The maximum radius of an isolated single bubble, which is closely related to the incident energy, and the value is approximately 1.5 mm in this study. (b) The initial optical breakdown locations at $\gamma = 0.45 (D_0 = 1.35 \rm \;{\textrm{mm}})$.Ignoring the deviation of the beam during propagation, the equal spacing condition is satisfied between the three breakdown locations. (c) Snapshot of the shock waves emission and interaction at $0.5\; {\mu} \textrm{s}$ time delay. (d) Geometrical configuration during multiple points excitation.

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In our experiments, three shock waves were observed, propagating from the area around the focal spot to the distance in the form of spherical waves. Figures 2(c) and 2(d) show a shock wave propagation shadow image, upon multi-pulse irradiation, and its schematic representation. The bubble and shock wave fronts are presented as black and red circumferences. We define the distance between adjacent breakdown locations as $D_0$, the chord length (CL) at the junction of the shock waves as $L(t)$, and the radius vector $\mathbf {R(t)}$ that points at the ends of the flat shock. The angle between $\mathbf {R(t)}$ and $D_0$ is defined as $\theta$. The radial velocity vector at the junction of the shock waves is also given by $\mathbf {v}_r$. By utilizing the multi-point excitation approach, we can determine the motion of the emitted shock waves. To this end, the shock wave velocity is calculated at different radii using the Sedov-Taylor expansion model (Eq. (1)) [45]. Peak pressure and temperature are obtained using the NASG EoS (Eq. (2)) [48] based on the radial position and velocity, as well as the invariant $p_0$ of the hydrostatic pressure. Interestingly, the method adopted in this paper is similar to the Lagrangian tracking approach proposed by Fabian et al. [41] based on the Kirkwood-Bethe hypothesis [52]. Here, the starting time of each shock wave was almost simultaneous, whereas a delay in starting time existed among shock waves during Fabian et al.’s research. Hence, the velocity of the shock wave can be calculated directly from our experimental diagram.

5. Results and discussion

5.1 Velocity evolution of shock wave with time delay

When multiple pump pulses reach the focusing region, the optical breakdown in water occurs after a short time. The formation and expansion of multiple plasmas at high temperatures and pressures caused the formation of multiple initial shock waves. These initial shock waves then began to expand soon. The interaction areas among shock waves are captured in our experiment (Sec. 3). Figure 3 illustrates the evolution of shock waves induced by nanosecond laser pulses in bulk water (see Fig. 1), captured at varying time intervals following the initial passage of the incident pump pulse through the water tank. The shock waves in bulk water were generated using pump pulses with energies and intensities of ${\rm 9}.6\pm 0.3\;{\textrm{mJ}}$ and ${\rm \approx} 4.8 \times 10^{13}\; {\rm W}/{\rm{cm}}^2$, respectively. The mathematical relationship between the laser incident energy and the power density can be found in Ref. [37]. Initially, we observed three dark areas running parallel to each other at the locations of the optical breakdown at $t = 0.0127\; {\mu} \textrm{s}$. As noted before, these regions are closely related to the plasma generated during breakdown. Subsequently, at longer time delays, the dark areas are accompanied by shock waves in the bulk water, which then collide and creates regions of impact. The onset of collision at this specific laser irradiation is observed at $t = 0.393\;{\mu} \textrm{s}$ after the laser excitation. At $t = 0.490\;{\mu} \textrm{s}$, as the fronts of the shock waves radially expand, the resultant elongated CL in the region of impact. This process continues until three sets of shock waves encounter the dark areas and reflect, where rarefaction waves merge (see Visualization 1). A similar phenomenon was observed in Ref. [53]. From the dynamics of the CL, we can pinpoint the location and calculate the propagation velocity of the shock waves.

The shock wave expansion is quantified by the shock wave front radii ${\mathbf{R(t)}}$ and velocity ${\mathbf{v}_{\rm r}}$, as depicted in Fig. 2(d). By measuring the distance between adjacent breakdown locations $D_0$ and the CL ${\rm L}(t)$ in the region of impact, we define the modulus of the radius vector, as follows:

 

Fig. 3. Nanosecond laser-induced optical breakdown, electron plasmas, and shock waves emission during multiple points excitation. The more emission of consecutive frames is shown in Visualization 1. The excitation light propagates from left to right. The distance of adjacent optical breakdown initial locations corresponds to $D_0 = 1.35\rm \; {\textrm{mm}} (\gamma =0.45)$.

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Fig. 4. (a) The crossing lengths of shock waves (black circles) and shock waves radii (deep cyan circles) as a function of the time delay, where the red line represents fitting with the experimental results. (b) The shock wave front radial velocity (light blue curve) is a function of time, in which the brown curve and dark blue dashed curve denote the vertical and horizontal modular components of the velocity vector, respectively.

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Besides, the shock waves are normally modeled as spherical shock, which originates from the corresponding plasma at the onset. Hence, by substituting experimental results from Eq. (5) into Eq. (1), we can obtain the modulus of the shock wave front velocity vector as a function of time by placing the time derivative, as follows:

Reviewing Fig. 2(d), we also indicate the modular components of the radial velocity vector in both the horizontal $(v_r^t)$ and vertical direction $(v_r^v)$. To optically trace the formation, propagation, and interaction of underwater shock waves that are induced by nanosecond laser pulses. We set that both $v_r^t$ and $v_r^v$ must satisfy condition $v_{u+l}^{v(r)}\geq c_0$, where $v_{u+l}^{v(r)}$ represents the sum of the $v_r^t (v_r^v)$ of the shock wave front velocities, derived from the upper $(u)$ and $(l)$ laser focusing position:

5.2 Energy transported by the underwater shock waves

In this section, we first elucidate the distribution path of absorbed energy in the underwater optical breakdown region, then deduce the effective energy carried by the shock wave by fitting the experimental results with the Sedov-Taylor expansion model (Eq. (1)), and finally compare with the existing results.

From an energy point of view, the energy carried by the evaporation of the medium typically contributes to the soft tissue cutting and thermal ablation of the material, while the energy carried by the shock wave generation and the cavitation bubble oscillation contributes to the ionization and optical breakdown of the medium [4,5]. The latter is advantageous in certain circumstances (such as invisible sonic scalpel [9] and targeted therapy [10]), whereas it is also typically a source of undesirable side effects. Estimation of the effective energy carried by the shock wave during the optical breakdown is thus a prerequisite for the choice of an optimal parameter in a particular application. In general, the laser energy absorbed during optical breakdown will be released in various forms, where resulting in the temperature rise of the medium and the change of refractive index. Besides, the absorbed energy partitioning pathways for the laser-induced optical breakdown in the bulk water are separated into five parts [36,50]: shock wave emission, cavitation bubble expansion, vaporization energy, the energy of plasma irradiation, and the rest of the energy dissipates. The laser-induced cavitation bubble energy and the energy assigned to the vaporization are given by Giorgia et al. [54], who stated that most of the energy absorbed by the plasma is dissipated ($80\% \sim 90\%$) as the cavitation bubble collapse. In effect, this conclusion is not entirely convincing. Previous studies have shown that [36,45] the ratio of shock wave energy to cavitation bubble energy is approximately constant (between 1.5:1 and 2:1) during optical breakdown. Excluding the absorbed energy converted to mechanical energy, the absorbed energy going into shock wave propagation is $\approx 25\%$ (pulse duration $= 30\rm \; {\textrm{ps}}$, incident energy $= 1\rm \; {\textrm{mJ}}$), and this fraction increases with increasing laser pulse duration [55]. On the other hand, the energy partitioning during cavitation bubble collapse resembles the process after the optical breakdown, especially since a large share of the energy is radiated out by the shock wave during the rebound process [7,35]. Therefore, to be precise, most of the absorbed energy entering the cavitation bubble oscillation radiates outward and dissipates in the form of sound waves. In addition, the effective energy carried by the shock wave emission in the near-field is typically considered as [14,55]:

On the other side, the shock wave effective energy close to the plasma is almost impossible to measure in the experiment and thus obtained through numerical simulations based on the Sedov-Taylor expansion model (Eq. (1)) [43] in this study. More specifically, the Sedov-Taylor model proposed in Eq. (1) is used to fit the experimental data (i.e., the shock wave propagation radius ${\rm R}(t)$ at each time delay as shown in Fig. 4(a)) to obtain the average effective energy in the process of shock wave emission, see the red line in Fig. 4(a). The fitting provides an estimate of the shock wave energy equal to $3400\pm 208\;{\mu} \textrm{J}$, which accounts for $\approx 35.4\%$ of the incident laser energy (see Table 1). One should note that if experimentally energy fluctuations are considered, then the shock wave energy can also occupy at least $\approx 32\%$ of the incident laser energy in our study(not shown in Table 1). Compared with the far-field measurements made experimentally by Jukna et al. [30] using hydrophones (the distance between the excitation point and the detection point is $= 38\;{\rm cm}$), our results are more inclined to the prediction of shock wave propagation energy in the near-field ($t < 1\;{\mu} \textrm{s}$ and ${\rm R} < 1.5\; {\textrm{mm}}$). Similar results have been reported by Cole et al. [55], where the shock waves are induced by a wavelength = 1064 nm, pulse duration = 6 ns, and energy = 10 mJ laser pulse. Likewise, assuming no extra heat conduction processes, the energy transported by the underwater shock wave accounts for $\approx 41.9 \%$ of the incident laser energy. Given the discrepancies in absorbed energy in the optical breakdown region under different laser incidence conditions (the absorbed energy decreases as the incident laser energy decreases, etc.), a comparison of the predictions in this study with the results of other studies is placed in Table 1.

Given that the absorbed energy accounts for $80\rm \%$ of the incident laser energy, which is the same hypothesis as Vogel et al. [56,57]. Our result is slightly lower than those calculated by Cole et al., who used the Gilmore model to calculate the shock wave energy in the near field (Eq. (8)). The main reason may be that the shock waves measured in Ref. [55] are closer to the excitation point ($R/r = 6, r \sim 100\;{\mu} \textrm{m}$ [58]) and have lower energy dissipation. As noted before, shock wave energy dissipates quickly during propagation (overcoming the viscosity, tension, etc., of the surrounding liquid). Meanwhile, the energy carried at a shock wave radius of 10 mm is given in Table 1, which is provided by Vogel et al. [36]. The comparison reveals that the energy carried by the shock wave is very low ($622\;{\mu} \textrm{J}$) in the far field, where the shock wave may have evolved into acoustic radiation.

Table 1. Effective energy and ratio of shock wave obtained by different methods. The results in brackets are the proportion of the shock energy occupying the incident laser energy. ${\rm R},{\rm r}$ stands for the shock wave radius and the initial plasma radius, respectively.

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In addition, the incident energy is placed in Table 1 is the average energy (${\rm 9}.6\;{\textrm{mJ}}$) achieved in the experiment, to ensure that it is consistent with the incident laser condition of other studies, and reduce the error as much as possible. In the meantime, measurement errors in experiments are taken into account, we present the average value for conversion efficiency in Table 1. This result is relatively satisfactory considering the potential impact of various energy dissipation and distribution path uncertainties [35] on the shock wave energy during optical breakdown.

5.3 Pressure and temperature of shock waves propagation

When the laser pulse penetrates the bulk water, the optical breakdown occurs, then the cavitation bubbles are induced and the shock waves are launched in laser-excited locations. Previous studies [14,59] have shown that the early peak pressure of shock wave decreases approximately proportional to $r^{-2}$ ($r$ is the radii of the shock waves) when the shock wave propagating with sonic wave speed for rapid energy dissipation and the peak pressure decays nearly proportional to $r^{-1}$ [50]. Hence, on a larger time delay, the peak pressure of shock wave decay is slower than that at the initial excitation location. To put it differently, the peak pressure measurement is more accurate as the time delay (after the laser passes through the focusing region) increases. In our experiments, we used a CCD camera to capture the emission of the shock waves at different delay times. We take images of the shock waves, trace their positions (Fig. 3), estimate their velocity (Fig. 4(b)) and derive their pressure (see Fig. 5(a)). Given the conservation of momentum at a shock wave front, the peak pressure and velocity of the shock wavefront can be given by Eq. (3) and Eq. (6). In our numerical simulation, we have calculated the peak pressure for each shock wave image collected at different time delays (see Fig. 3), which are plotted in Fig. 5(a) (blue curve). Within the maximum time delay obtained, the pressure of the shock wave exhibits exponential decay, decreasing gradually from the initial value $\approx 1\;{\textrm{GPa}}$ to $\approx 80\;{\textrm{MPa}}$. It is worth noting that the pressure of the shock wave front exhibits similar time-dependent decay trends even in water/air interface [14,60]. Although our calculation results differed from that obtained by Yoshiyuki et al. [14], who stated that the distribution of peak pressure of the shock wave is not spherically symmetric, the discrepancy is mainly due to the multi-point breakdown effect caused by the pumped laser in the focusing area. In our experiment, we avoided the multi-point breakdown effect in the focusing area by utilizing an aplanat lens (see Fig. 1), whereas at least 5 sets of shock waves were in the same place at the same time during Yoshiyuki et al.’s research. Therefore, numerically, the peak pressure of the shock wave that we calculate is lower.

 

Fig. 5. (a) The pressure (blue curve) and temperature (red curve) of the shock wave front with different time delays. (b) The temperature as a function of the pressure of the shock wave front, obtained with the NASG EoS (grey curve), and the Tait EoS [55] (green curve), compared against the IAPWS R6-95(2018) standard (orange curve). The inset shows the Tait EoS prediction of temperature under high pressure.

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On the other hand, temperature is also an important parameter to characterize the effect of nanosecond laser-induced optical breakdown on the medium. From the perspective of electron-ion heating kinetics, the ground state electron first absorbs the energy of the incident photon through reverse bremsstrahlung, whereby achieving the transition from the ground state to the excited state. Afterward, the excited state electrons transfer energy to low-energy electrons or molecules (or dissociated atoms) through collision and recombination processes, resulting in avalanche ionization in the breakdown region. This energy transfer process typically lasts for a few picosecond [50]. For the ultrashort laser pulse case, this energy transfer process cannot be completed during the laser pulse, which means that the equilibrium temperature develops only after the laser pulse. In contrast, the duration of the nanosecond pulse laser is longer than the energy transfer time, thereby the system has reached temperature equilibrium during the pulse period, which means that the energy density is higher in the breakdown region and the subsequent effect on the medium is stronger. As noted before, the pathways for the distribution of absorbed laser energy are through evaporation, shock wave emission, and cavitation bubble expansion. While measuring the energy of a cavitation bubble is relatively straightforward, determining the energy of a shock wave can be much more challenging. Fu et al. [2] experimentally predicted the energy and conversion efficiency of the cavitation bubble, but did not provide meaningful shock wave energy and temperature. Denner et al. [41] tried to evaluate the underwater shock wave temperature (1000 K for 1 GPa) by utilizing the Tait EoS, which proved to overpredict the temperature compared to the International Association for the Properties of Water and Steam (IAPWS) standard [41,61], see Fig. 5(b) (green curve). Indeed, when the water is heated above $\approx$ 520 K [62], the state of water beyond the liquid-vapor equilibrium is metastable, accompanied by the generation of cavitation bubbles. This is similar to the situation when a liquid is stretched by its weight [63] (or mechanically increases its volume with a bellow to pull a liquid in the action region [64] until the pressure in the region of action is lower than the saturated vapor pressure $P_{sat}$, thereby creating cavitation bubbles [65]). Therefore, the temperature of the water caused by shock wave propagation is typically below the bubble generation threshold ($T_{cav} \approx 520\rm \; \textrm{K}$). Assuming an isentropic flow, we put to use the NASG EoS (Eq. (2) in Sec. 2) to calculate the shock wave front temperature in this study.

Figure 5(a) (red curve) depicts the temperature of the shock wave front as a function of time delay. In the same way as the pressure of the shock wave front, these values represent the average of the temperature, along with their standard deviation obtained from laser energy fluctuations. Each error bar corresponds to the shock wave image at each time delay in Fig. 3. As noted before, the propagation of the shock wave with supersonic speed is accompanied by dissipation of energy, as well as a decrease in pressure. The former also includes heat transfer processes with the surrounding environment. As the shock wave pressure and energy begin to decrease, the temperature of the shock wave front at different time delays also gradually decays until it approaches room temperature (see Fig. 5(a)). The estimated pressure and temperature of the shock wave front at $t = 0.80\;{\mu} \textrm{s}$, both the pressure and temperature of the shock wave front restore the values of ambient water ($\approx 10^5\;{\textrm{Pa}}$ and $\approx 293\;{\rm K}$). In addition, we also gave the temperature as a function of pressure under different numerical models, and then compared it against the IAPWS R6-95(2018) standard, as shown in Fig. 5(b). The results show that the temperature obtained in this study (NASG EoS) is in close agreement with the IAPWS R6-95(2018) standard [61]. Besides, the temperature increases linearly when the pressure exceeds 0.18 GPa, which means that the shock wave may cause more thermal damage to the medium in the near field. In contrast, in the far field, the pressure measurement is more accurate and easier to perform, because the small temperature rise does not cause any interference to the acquisition of shock wave images, as shown in Fig. 6.

 

Fig. 6. The nanosecond laser-induced shock waves emission images at $\gamma =0.8$ (a) and $\gamma =1.0$ (b) with different time delays.

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5.4 Analysis of shock wave propagation and interaction under different $\gamma$ values

In this section, we performed the shock wave velocity and pressure calculations by adjusting the distance between adjacent excitation locations, namely, changing the value of $\gamma$ in Eq. (4). Subsequently, the interaction between shock waves is also analyzed using the obtained results.

Figure 6 shows that the shock waves emission shadow images for different values of $\gamma$ ($D_0 = 2.4\;{\textrm{mm}}$, $\gamma = 0.8$ in Fig. 6(a); $D_0 = 3.0\;{\textrm{mm}}$, $\gamma = 1.0$ in Fig. 6(b)). At $\gamma = 0.8$, the shock waves collide at an earlier time delay ($t = 0.746\;{\mu} \textrm{s}$) than at $\gamma = 1.0$ ($t = 0.942\;{\mu} \textrm{s}$), which arises from a longer average distance between two adjacent breakdown locations, the shock waves take longer to achieve the collision. However, at $\gamma = 1.0$, we can obtain the shock wave propagation scenarios of the breakdown regions at longer time delays, including the velocity and pressure of the front of the shock wave (see Fig. 6(b)). Note that all the images are obtained at the same laser energy. In addition, the velocity and pressure of the shock wave front are also calculated at different $\gamma$, as shown in Fig. 7. In terms of velocity, both $\gamma = 0.8$ and $\gamma = 1.0$ are close to the speed of sound in water (1483 m/s), which due to the propagation of the shock wave is already in the far-field range at the microsecond time delay, where the shock wave propagating as a sonic wave for energy dissipation [2]. Likewise, the pressure of the shock wave front gradually decays with increasing time delay until the shock wave propagates as a sonic wave and the pressure tends to one atmosphere (100 kPa). In contrast to $\gamma = 0.45$, $\gamma = 0.8$ and $\gamma = 1.0$ can obtain shock waves propagation scenarios with larger time delays, which means that we are capable of predicting the information carried by the shock wave in the far field (velocity, pressure, energy, and temperature, etc.). In practice, this provides a wider range of time choices for laser surgery.

 

Fig. 7. The comparison of shock wave front velocity (a) and pressure (b) at $\gamma =0.8$ and $\gamma =1.0$.

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On the other hand, an interesting phenomenon can be observed from a velocity point of view, as shown in Fig. 7(a). More specifically, at the same time delay $t$, the propagation velocity of the shock wave formed by the optical breakdown points with larger spacing is higher than that of the shock wave formed by the optical breakdown points with smaller spacing. Note that our measurements are made a few microseconds after the shock waves overlap and before the shock waves reflect again as sparse waves (see Fig. 3 and Fig. 6). Hence, the maximum time delay varies with different spacing $\gamma$ values, for instance, ${\rm t}_{\textrm{max}} \approx 0.8\; \mu \textrm{s}$ for $\gamma = 0.45$, ${\rm t}_{\textrm{max}} \approx 1.2\; \mu \textrm{s}$ for $\gamma = 0.8$, and ${\rm t}_{\textrm{max}} \approx 1.8\; \mu \textrm{s}$ for $\gamma = 1.0$. In fact, the shock wave front velocity always decreases exponentially with the propagation time and distance, whether a single underwater shock wave propagation process [14] or the interaction of multiple shock waves in this study (see Fig. 7(a)). In other words, with the same time delay, the underwater shock wave generated by the incident laser of the same wavelength and energy should have the same propagation velocity. However, it appears that the situation is different during multi-shock wave interactions. In general, for the same time delay, a larger value of $\gamma$ corresponds to a shorter chord length ($L(t)$ in Fig. 2(d)), which represents the duration of the shock wave interaction. Once the interaction between shock waves occurs, the pressure within the region of shock wave interaction rises for a short time owing to the overlap of tension [53,55]. In addition, different from the shock wave propagation in air [37,66,67], the propagation process of underwater shock waves can be approximated as isentropic flow, and their interaction is not affected by additional ionization procedures. Therefore, the pressure in the interaction region is uniformly distributed [53]. This pressure increase and uniform distribution continue until the shock wave bounces off the cavitation bubble ($t = 0.780\; {\mu} \textrm{s}$ in Fig. 3). At this point, the tension carried by these reflected shock waves produces a stronger oscillation in the area of interaction, reducing the tension change over the shortest distance. Recalling Fig. 7(a), the above overlap of tensions and the uniform distribution of pressure processes provide a good explanation for our results (“Weird behavior” of velocity with the same time delay $t$ and different stand-off distances $\gamma$). With approximately the same delay time $t$, the shock wave interaction time corresponding to $\gamma = 0.8$ is longer than that of $\gamma = 1.0$, which means that the longer the interaction region tensions overlap, the more pronounced the barrier to the shock wave velocity. As can be seen from $\gamma = 0.8$ (Fig. 7(a)), the shock wave velocity is also lower under the same time delay compared with $\gamma = 1.0$. Similar phenomena have been found in the air by Zhang et al. [66], which demonstrated that a freely propagating shock wave in the air has a much higher speed than a hindered shock wave, where this hindrance comes mainly from the regenerating shock wave at the optical breakdown location. For $\gamma = 0.45$, owing to the shock wave being closer to the optical breakdown region, the shock wave velocity is always higher than the shock wave velocity under $\gamma = 0.8$ and $\gamma = 1.0$ during the time delay. Nevertheless, the overlap tension caused by the shock wave interaction still affects the shock wave velocity. It is not difficult to see that in the case of $\gamma = 0.45$, the average velocity of the shock wave is reduced by $\approx 800\; {\rm m}/{\rm s}$ within the $\approx 400\; {\textrm{ns}}$ delay time (see Fig. 4(b)). The same attenuation also appeared in the case of $\gamma = 0.8$ and $\gamma = 1.0$ corresponding to the same time delay ($400\; {\textrm{ns}}$), the shock wave velocity decreased by $\approx 50\; {\rm m}/s$ and $\approx 30\; {\rm m}/{\rm s}$, respectively. Therefore, we believe that the overlap of tensions induced by the interaction between shock waves hinders the expansion of the shock wave. Meanwhile, the closer the distance between shock waves (i.e., the smaller the stand-off distance $\gamma$), the greater the obstacle caused by the interaction, and the faster the shock wave velocity decreases. The specific distribution of this tension and its evolution over time is out of the scope of this paper. Besides, Pedro et al. [53] reported that the temperature increase in the process of shock wave overlap was derived from the adiabatic compression process. However, the adiabatic process should follow the shape of pressure waves [55]. Hence, the shock wave interaction should not significantly affect temperature when the thermodynamic process is viewed as adiabatic expansion.

6. Conclusion

The shock waves emission, the time evolution of the shock waves, and the detailed information carried by the shock waves during optical breakdown were measured simultaneously to investigate the multiple points excitation in bulk water induced by the focused nanosecond laser pulses. According to the results, this unique multi-excitation method allows us to well characterize the characteristics (pressure, temperature, etc.) of underwater shock wave propagation, as well as the effective energy of the shock wave near the optical breakdown point. The latter has been hardly obtained quantitatively in previous experiments with a single excitation beam.

More specifically, according to shock wave images, we quantified the propagation radii of the shock waves at different time delays. And the effective energy carried by the shock wave is quantified by the Sedov-Taylor model fitting with experimental results. We found the average energy ($\approx 3.4\;{\textrm{mJ}}$) carried by the shock waves occupies about $35.4\%$ of the incident laser energy, and occupies about $43.1\%$ of the absorbed energy. The results are in good agreement with those of Cole et al., considering the different locations of the shock waves and the dissipation of energy during shock wave propagation. Afterwards, we further derive the peak pressure and velocity of the shock wave as a function of time. The results show that the shock wave exhibits asymmetry in both its horizontal and vertical velocity and pressure distributions, thereby completing the transition from a supersonic shock wave to an acoustic wave. Considering the energy transfer process, we found approximate linear dependence among the shock wave pressure, time delay, and temperature of the shock wave front. This finding may be useful for laser eye surgery and the subsequent cavitation bubbles emission. Finally, we compared the effect of different values of $\gamma$ on the shock wave emission process. Compared to $\gamma = 0.45$, $\gamma = 0.8$ and $\gamma = 1.0$ can obtain the shock wave emission with a longer time delay. This is useful for tracing the shock wave’s evolution underwater. Moreover, by comparing the shock wave velocities under different $\gamma$ values, it is found that the interaction between shock waves will hinder the emission of shock waves, and the closer the location of the optical breakdown (corresponds to the smaller the $\gamma$ in this study), the stronger the interaction between the shock waves in a short time delay and the faster the velocity decrease. In addition, we show that the use of a multiple points excitation beam opens new horizons for studying the nature of underwater shock waves. We appeal to our method to reveal information about the hydrodynamics of ns laser-induced supersonic and subsonic shock waves underwater. This approach contributes to understanding the physical processes involved in research areas such as biological systems and laser-based surgery.