Pressure dynamics of an internal shock wave emission inside a water droplet and potential cavitation

Shock reflected by a spherical interface is involved in shock-droplet interaction. Here, we investigate the pressure dynamics of an internal spherical shock wave and the potential cavitation inside a spherical water droplet. We conduct hydrodynamic simulation and employ the linear geometrical acoustics approximation to analyze the negative pressure and the wavefront at different intervals d between the source point of the shock and the droplet surface. Studies indicate that the negative pressure at a very large d is primarily attributed to the focusing effect of the droplet surface, whereas the collision of the reflected waves contributes to negative pressure at a small d. The caustic, which is the locus of the cusps (singular points) on the shock wavefront is determined by the parametric equations. Notably, the caustic also has singular points owing to the off-axis reflection. Finally, we evaluate the cavitation inside the droplet. Cavitation occurs on the opposite side of the droplet and moves away from the surface with d increasing, which agrees with previous experimental results. Additionally, we demonstrate that cavitation can occur with less damage to the droplet compared to a planar shock impact on a water droplet.


Introduction
Pulsed laser-droplet interactions have been investigated extensively due to their potential applications in a variety of fields, such as biomedical research [1,2], fog clearing [3,4], combustion engines [5], extreme ultraviolet (EUV) nanolithography [6], and laser micromachining [7].For enhancing the performance of these applications, it is essential to conduct a comprehensive study of laser-induced droplet dynamic deformation.In EUV nanolithography, for instance, precise and complete control of the size and shape of the prepulse-formed target droplet is crucial for reproducible and stable maximization of conversion efficiency [8].
Recently, there have been significant advancements in droplet shaping detection techniques induced by laser irradiation, such as stroboscopic imaging [9], bright-field microscopy [10], and laser-based molecular spectroscopy [11].These methods have allowed for qualitative and quantitative investigation of spatiotemporal deformation and fragmentation in droplets as well as their correlation with incident laser parameters like energy, pulse width, and beam waist [12,13].However, some phenomena still require a detailed investigation as they have not been studied through existing experiments.These phenomena include cavitation by a focused reflected wave in acoustically levitated water droplets irradiated by a laser pulse [13], cavitation by shock wave focusing in eye-like experimental configurations [14], which is indirectly detected by high-speed camera shadowgraphy, and a small jet on the surface of a dyed water droplet in an experiment involving a laser pulse [9,15], which is attributed to potential cavitation spots caused by shock focusing.Cavitation inside a water droplet is determined by the pressure dynamic of the shock emission.Based on these driving forces, researchers have concentrated on the early stage of the shock emission inside the droplet.The early stage of planar shock emission (induced by an x-ray laser pulse) inside the droplet and the resulting cavitation were investigated through numerical simulations and experiments [16,17].A laser-induced shock inside a water droplet leading to cavitation in another side of the droplet was observed in the experiment [18]; however, the shock dynamics were not investigated in detail.Additionally, cavitation inside the droplet was observed in the interaction between a planar shock and the droplet [19], followed by the study of the pressure dynamic of the transmitted wave and cavitation [20,21].In brief, internal spherical shock emission and potential cavitation have not received sufficient attentions and lack a detailed investigation, despite their presence in many experiments.
In this study, we investigate the pressure dynamic of an internal shock wave and potential cavitation inside a water droplet.We first focus on the mechanism for generating intense negative pressure using the numerical simulation.Subsequently, we use the ray-tracing method to further analyze the wavefront of the waves.Finally, we conclude with an investigation of potential cavitation inside the droplet.Our results provide a useful reference for related experiments and applications and can help further understand droplet deformation in laser-droplet interaction.
The outline of this study is as follows.In section 2, we first provide a brief introduction to the hydrodynamic method.In section 3, we present our numerical results and analyze the pressure evolution.In section 4, we use the ray-tracing method to derive the parametric equations for the wavefront and caustic and conduct a detailed analysis of the wavefront.Section 5 is devoted to the cavitation inside the droplet.Finally, in section 6, we present our conclusion.

Hydrodynamic method
Laser pulses can induce shock inside a droplet; we focus on the pressure dynamics without consideration of the laser-droplet interaction.For a shock dynamic of shock-droplet interaction, the flow is assumed to be inviscid because of the high value of Reynolds number, and the surface tension is ignored owing to the low tension compared to the shock front and the insignificant deformation of the surface at an early stage.The same assumptions are conducted in some studies [17,21,22].The axisymmetric compressible Euler equations governing such immiscible multicomponent flows are expressed as follows: where U is the vector of generalized density, F and H represent flux, S corresponds to a source vector generated by the transformation from Cartesian coordinates to cylindrical coordinates, and the subscripts t, r, and z indicate the time, radial and axial direction, respectively.The vectors U, F, H, and S are expressed as follows: where ρ is the density, u is the radial velocity, v is the axial velocity, P is the pressure, and e denotes the specific total energy of the fluid.To close the equations, we employ the stiffened Equation of state as follows: ρ where γ is the ratio of specific heats, P ∞ is a prescribed pressure-like constant, and c represents the speed of sound.The γ and P ∞ are typically taken as constants determined by the medium.For perfect gases, we have P ∞ = 0 and γ = 1.4; for water, we have γ = 5.5 and P ∞ = 4921.15bars [23].A γ-based model has been developed to achieve interface-capturing [24].In the model, γ and P ∞ obey the advection equations as follows: where Γ = 1/(γ − 1) and Π ∞ = γP ∞/ (γ − 1).Equations ( 1)-( 4) form a quasi-conservative system, which can be used to simulate compressible multicomponent flow problems.Following the method proposed by Johnsen and Colonius [23], the fifth-order weighted essentially non-oscillatory (WENO) scheme is first used to reconstruct the primitive variables.Subsequently, the numerical flux is computed based on the primitive variable using HLLC approximate Riemann solver.Finally, the third-order Runge-Kutta scheme is used for time marching.The same numerical method was used to simulate the bubble dynamic, and the validity of this numerical method has been verified in [25].Additional validation is presented in the appendix to verify our hydrodynamic code and mesh independence further.The geometric model is illustrated in figure 1.A water drop with a radius R d = 1 mm is located at the center of the computational domain, which has a dimension of 4R d × 2R d .There is a high-pressure bubble with a radius of R b = 25 µm at an interval d between the left surface and the bubble center.The initial pressure is set as 1 bar except for the bubble.The density is 996.6 kg m −3 for water and 1.29 kg m −3 for ambient gas.For boundary conditions, a symmetric boundary is used for the z-axis, and outflow boundaries are used for the others.In fact, except for the z-axis, the waves can not reach boundaries within our calculated time owing to low wave speed in ambient gas.Based on the numerical model described above, it is convenient to calculate the dynamic characteristics of droplets for different bubble parameters (such as the interval d and initial pressure).The initial conditions are listed in table 1.

Pressure dynamics inside a water droplet
Dynamic pressure inside the droplet under different intervals d = 0.3 mm and 0.8 mm with an initial pressure of 0.5 GPa are illustrated in figure 2. The black line represents the droplet surface.To provide a better understanding of pressure patterns at d = 0.3 mm, the schlieren results (magnitude of density gradient) are presented at the bottom parts of figures 2(d)-(i), in which a numerical cut-off to the density gradient is applied to enhance the visualization of the pressure wave.As a result, the bubble (black semicircle) has a thick edge.The wave that has been reflected k times is indicated as Γ k , where odd and even values of k correspond to rarefaction and compressible waves, respectively.At d = 0.8 mm, the shock undergoes the reflection on the surface and thus turn into a rarefaction wave (Γ 1 ), which ultimately focuses on the z-axis (shown in figure 2(c)), leading to intense negative pressure.At d = 0.3 mm, as the waves propagate into the right, unlike the smooth wavefront of the reflected wave at d = 0.8 mm, there are two singular points (cusps) on it, as indicated by the arrows in figures 2(e) and (f).These two cusps separate the wave into three parts: the first being convergent, the second being divergent, and the third being less divergent than the first part, as shown by the numbers in figures 2(e) and (f).The first part focuses on the z-axis; one of the segments of the second part collides with the wave from the other side of the z-axis owing to the symmetry, and the remaining has the same trend as the third part.The third part is reflected again and thus transitions to a compressible wave Γ 2 .Since the Γ 2 is from the different parts of Γ 1 , one of the cusps is transferred to Γ 2 after the emergence of Γ 2 , resulting in a γ-shape being induced on Γ 2 shown in figure 2(h).Finally, Γ 2 collides with the wave of Γ 2 from the other side of the symmetry axis, resulting in a complex pressure dynamic because of its proximity to the right surface.In detail, any Γ k that has reached the z-axis results in a new wave propagating to the surroundings, e.g. the wave indicated by a dashed arrow in figure 2(i), part of which will be reflected by the right surface, thereby influencing subsequent pressure evolution, especially for the waves on the z-axis.This phenomenon leads to intense negative pressure,  discussed below.Additionally, there is an intersection between the first and third parts owing to the large time contrast in the occurrence of reflection between the right and left surfaces, as shown in figure 2(f).
In collusion, compared to the pressure dynamic at d = 0.8 mm, there are two significant differences at d = 0.3 mm.The first is the presence of two cusps, which will be discussed below.The second difference is that the main contribution to the intense pressure is the collision rather than the focusing effect.
To further estimate the influence of the focusing and collision on the pressure dynamic, we show the minimum pressure over time for d = 0.3, 0.6, and 0.8 mm in figure 3. The intense negative pressure is attributed to the focusing or collision of the pressure waves.At d = 0.8 mm, the pressure initially decreases rapidly and then remains slightly below 0 MPa until the wave of Γ 1 reaches the z-axis (not shown).At t = 1.21 µs, an extremum of −214 MPa appears owing to the focusing of Γ 1 ; then, the pressure evolution becomes slight.At d = 0.6 mm, the pressure evolution of the early stage is similar to that at d = 0.8 mm, and an extremum of −116 MPa appears at t = 1.27 µs.Unlike d = 0.8 mm, the extremum is generated by the collision of Γ 1 , and the focusing effect is so weak that no intense negative pressure is generated by itself.At d = 0.3 mm, two extrema are from the collisions of Γ 1 and Γ 2 , respectively.The first extremum appears at t = 1.40 µs and is less intense than that at d = 0.6 mm; the second extremum emerges at t = 1.71 µs and is much more intense.Note that the pressure evolution is similar between d = 0.6 and 0.3 mm to some extent if the second extremum is ignored.The prominent difference between them is that although Γ 2 appears at  The shock wave is generated by a small bubble.Therefore, it is necessary to consider the influence of the bubble.The bubble can reflect a part of the waves.In figures 2(e)-(h), the bubble-reflected waves behind the primary waves can be observed.However, the numerical results suggest that the influence is insignificant unless d is very small.
Figure 4 shows the minimum pressure over time at d = 0.2 mm with different initial pressures.We note that the lines exhibit a similar trend, with a downward shift at high initial pressure.Although the initial pressure is different, the wave dynamics are similar, and the reasons for the extrema are both collisions between the pressure waves.The increase in the initial pressure implies a more intense collision and, thus, causes more negative extrema.

Wavefront inside a droplet
We consider a shock wave emission from an approximate point source inside a water droplet.The pressure wave rapidly weakens because of the geometrical factor of spherical expansion and energy dissipation at the shock front.As a result, the reflection on the droplet surface can be described approximately by geometrical acoustics.A detailed wavefront evolution is calculated under the assumption that the waves propagate at a constant sound speed of c 0 .Since we focus on the wavefront before it reaches the z-axis, except for the high shock speed in the very early stage, the shock speed is slightly larger than the sound speed (c/c 0 < 1.1).Furthermore, the rarefaction wave Γ 1 is slower than the shock.In our calculation, the wavefront is obtained using a ray-tracing method, which is the same as geometrical optics but with a different propagation velocity.This method has been used to address similar problems of shock-droplet interaction in [16,20], which provides a direct and accessible interpretation of the wave patterns.
Since our interest is the wavefront evolution inside the droplet, we ignore the transmitted wave and do not estimate the strength of rays. Figure 5 illustrates the geometry of our problem.The complete rays trace the path of wave propagation, including the shock wave and the reflected waves.The red circle corresponds to the bubble in the numerical simulation; the center point S is the source of the rays, which is specified by an interval d between the bubble center and the left surface.A ray is denoted by an angle θ to the z-axis.When a ray experiences the reflection and thereby reaches the z-axis, a distance l to the droplet center and an angle γ to the z-axis denote the intersection M, which are determined by the following: (5)  where β is the incident angle, and n is the number of reflection times required for a ray to reach the z-axis, which depends on a parameter ϑ as follows: where n ′ is a tentative parameter of reflection times.For convenience, we use n-rays to represent the rays that need to be reflected n times to reach the z-axis.For every n ′ that meets the condition ϑ < (1 − d), n is the maximum positive integer of n ′ at a given angle θ, and n ′ -rays are existent at some angle θ. Figure 6 shows the correlation of ϑ and θ at different n ′ .In the case of n ′ = 2, ϑ descends as θ elevates and attains the minimum value of 1/3 at θ = 180 • , which means that 2-rays can be observed when d < 2/3 mm.For larger n ′ , the emergence of n ′ -rays requires a small d, e.g.3-rays for d < 0.2 mm and 4-rays for d < 0.1 mm.Unlike n ′ = 2, the curves are U-shaped, with the bottoms below 1 and the minimum near 90 • .As n ′ increases, the minimum of ϑ increases; thus, n ′ -rays appear in a narrower region of θ, e.g. from 73.3 • to 39.5 • when n ′ increases from 3 to 5.
The complete path of a ray to the z-axis consists of several successive segments separated by the reflection points P k and the points S and M. According to the geometric analysis, the length L k of segment k is provided as follows: Since P k is on the droplet surface, the coordinates can be easily obtained as long as the angles of line P k O to the z-axis are determined.Thus, the coordinates of P k are expressed as The wavefront from the ray-tracing method is given by equation ( 10) and depicted by a curve.
Considering the points S and M belong to the rays, we combine them with the reflection points to build a new point set N as follows: Based on the parameters above, the coordinates of the wavefront at a given time t can be estimated by the following: where c is the velocity of wave propagation, and m specifies the segment in which the ray is propagating and meets the following condition: Combining equations ( 7)-( 11), the analytic wavefront can be obtained.Figure 7 depicts the wavefront evolution of d = 0.2 mm.The top half of each pattern displays numerical schlieren images obtained from our hydrodynamic simulation.The bottom half shows the wave pattern predicted by equation (10).The secondary waves generated by the reflection on the bubble surface can be observed behind the primary waves, which cannot be included in the ray-tracing method.Except for this difference, the comparison indicates a remarkable agreement between the theoretical and numerical results in the morphology and the location of the wavefront.Besides, the bubble-generated waves influence the primary waves insignificantly.Hence, it is reasonable to employ geometrical acoustics (ray-tracing method) to further analyze the characteristics of pressure evolution.
Additionally, it is easy to observe the two cusps in figures 7(c) and (d).As per the results of the ray-tracing method, the wavefront located on one side of a cusp is focusing, but another is divergent.Besides, these regions of the cusps have the highest ray density.This implies that the cusps are the focus.
Figure 8 shows the curves of (γ, l) at different d.Avoiding overly detailed figures, the curves only having the branch of 1-rays are displayed in figure 8(a).To obtain a clear comparison, we also show d = 0.65, 0.6, 0.4, and 0.3 mm in figure 8(b).The focus of 1-rays in a region with a length of 0.04 mm at d = 0.9 mm; in As mentioned above, although Γ 2 appears at d = 0.3 and 0.6 mm, the second extremum emerges at d = 0.3 mm but vanishes at d = 0.6 mm.In figure 8(b), the inclination angle is also close to 90 • for d = 0.3 mm compared to d = 0.6 mm.In general, a frontal collision of pressure waves induces a more intense pressure than an oblique collision.

Cusps inside a droplet
Our issue concerns the shock emission from an off-center point.The curvature of the wavefront upon reaching the droplet surface is determined by the angle θ, while the curvature of the droplet remains constant.There are three scenarios to consider if only one reflection is considered.First, when θ is small, the curvature of the wavefront is small enough to make the wavefront convergent, leading to the focus being inside the droplet.Second, the curvature gradually drops as θ increases, which increases the focal distance, resulting in the focus being outside the droplet.Third, when θ is large, the curvature is not small enough to make the reflected wavefront converge; therefore, there is no focus before the wave is reflected again.In the first scenario, the locus of the focus is a curve known as the caustic.
To obtain the caustic of Γ 1 , we consider another ray, with the parameters θ 1 = θ + dθ and β 1 = arcsin(asin(θ + dθ)), where a = 1 − d.The coordinates of the reflection points P 1-0 and P 1-1 can be obtained from equation (8).Based on the geometrical relation, the slopes of the segments OP 1-0 and OP 1-1 are k = tan(θ + β) and k 1 = tan(θ 1 + β 1 ), respectively.Subsequently, the intersection point between the two lines (segments) is as follows: where b 0 and b 1 are intercepts of the two segments.The focus can be obtained as dθ approaches 0. The first derivative of the numerator and denominator is calculated as follows: Equation ( 13c) is equal to 0 as Substituting equation ( 14) into (13a) yields The formula (b 1 − b) ′ yields a result of 0 when a = 0.33 or 0.46, but it leads to an imaginary value of θ, which means the numerator and denominator cannot be equal to 0 simultaneously.Therefore, the focus is located in infinity under the condition given in equation (14).Additionally, since the square of the sectant sec(θ + 2β) exists in the numerator and the denominator, x F will not be infinite when θ + 2β = π/2, which is also valid in y F .
According to equations ( 12) and ( 13), the coordinates x F and y F of the focus are as follows: ) , Figure 9 shows the locus of the foci (x F , y F ) and the rays predicted using the ray-tracing method.The first and second segments of the rays are displayed.The location of the caustic is in accordance with the region of maximum ray density.The caustic can be separated into two branches, and the intersection point between them is singular.This singular point is at the angle θ = π/2, where the wavefront transitions from leftward convex to rightward convex.If the angle θ is not equal to 0 or π, the focusing of the droplet surface is off-axis.Therefore, the wavefronts with different orientations are focused on the different branches of the caustic, leading to the emergence of two cusps.Note that when a planar shock transmits to a water droplet, the wavefront of the transmitted wave has only one cusp and caustic being smoot [20].

Potential cavitation
When water is subject to a stretch from negative pressure, cavitation occurs if the pressure is sufficiently strong.According to the basic nucleation theory, the thin wall approximation predicts a cavitation pressure of −128 MPa for pure water at 300.0 K [26].Since we do not consider phase changes, we cannot evaluate the influence of the cavitation bubble on the pressure dynamic.Our simulated results are valid before cavitation occurs.However, this does not affect our consideration of the occurrence of cavitation by pressure below some threshold value.The same consideration has been employed in [20].
As mentioned above, two extrema of negative pressure are at small d, and the second is far below the first.The second extremum is invalid if the first extremum is below the cavitation pressure.The first extremum is nevertheless not negative enough to induce cavitation if the initial pressure is not too intense; thus, cavitation first occurs in the region of the second extremum.Note that the first extremum will be more negative as the initial pressure increases and eventually fall below the threshold pressure.As a result, the consequence of the second extremum is unknown.However, the cavitation bubble at normal temperature is weak.Therefore, it is impossible that the second extremum can still induce cavitation even if cavitation has occurred at the first extremum; in other words, it is possible that two cavitation regions may exist simultaneously inside a droplet.
In figure 3(b), we note that the easiest to induce cavitation is near d = 1.0 mm.Although regions have less intense negative pressure, the concern of inducing cavitation is unnecessary because more negative pressure can be achieved by increasing the initial pressure.Instead, the position of cavitation is of interest.Figure 10 depicts the distance z of the extrema to the right surface against d.Since the first and second extremum could cause cavitation, we show the two extrema together.Note that d > 0.5 mm does not have a second extremum.The distance of the first extremum decreases with decreasing d; however, the minimum distance is still greater than 0.6 mm.This indicates that potential cavitation is far away from the surface.In contrast, the distances of the second extrema are completely below 0.4 mm, and the minimum is lower than 0.2 mm.It is clear that potential cavitation from the second extremum is closer to the surface.When a cavitation bubble is close to the surface between air and water, the surface may be deformed [27,28].These phenomena will be insignificant if the cavitation bubble is far from the surface.A jet caused by large acoustic cavitation bubbles at the droplet surface was observed in the experiment of a laser-induced breakdown inside a water droplet; however, it disappears as d increases [18].Additionally, in the experiment [18], the cavitation region presents an inward movement as d increases, which agrees with the overall trend of the extrema qualitatively.However, the quantitative comparison requires more detailed experiments, which are challenging owing to the spherical interface.
Additionally, the maximum radius of the bubble at an initial pressure of 500 MPa is approximately 0.5 mm.Our scenario has a smaller ratio between the radii of the droplet and bubble compared to the experiment in [13], where a water droplet of R = 1.419 mm containing a bubble of a maximum radius with 0.92 mm remains intact surface during the bubble expansion and collapse.Thus, the damage to the droplet is much weaker, and thereby the droplet can keep a complete surface unless the initial bubble is very close to the surface.In the experiment where a planar shock impacts a water column, cavitation occurs as when the incident shock has a Mach number of 2.4, where the breakup of the droplet can be observed.Besides, the experiments show that the breakup of a millimeter-scale droplet occurs at a lower Mach number [29,30].Therefore, internal shock-induced cavitation has less damage to the droplet.
Another noteworthy point is that, as mentioned earlier, if the source of shock is slightly off-center, the focusing of the reflected wave cannot induce an intense negative pressure to cause cavitation.In the experiment [16], an internal planar shock is reflected by the water droplet surface, leading to a focusing rarefaction wave to induce cavitation; the negative pressure is estimated to be below −100 MPa at an initial pressure on the order of 100 GPa.For a planer shock, the main contribution to intense negative pressure is the focusing of the droplet surface.However, in a planer shock impact on a water droplet, the transmitted shock wave is only several MPa and thus causes cavitation [19][20][21].Similarly, our numerical results show that the pressure has dropped to tens of MPa as the shock propagates to the central region of the droplet.Thus, it is suggested that if the shock wavefront does not match the droplet surface, the pressure wave collision will easily induce cavitation.Additionally, the transmitted wave in the shock impact is not a planar wave but an approximately spherical wave with a larger radius, and the virtual source is outside the droplet, which distinguishes the internal shock emission from the planar shock impact.

Conclusion
In this study, we investigated the pressure dynamic of droplet interaction with an internal shock wave and evaluated cavitation inside the droplet, revealing the origin of the negative pressure and the correlation between cavitation position and interval d.Our findings provide insights into the mechanism of the subsequent droplet deformation and the involved applications.
We first evaluate the pressure dynamic inside the water droplet at different intervals d between the droplet surface and the bubble center using hydrodynamic simulation.Subsequently, further analysis was conducted using the ray-tracing method to provide more accessible and clear results compared to the numerical simulations.Finally, we evaluated potential cavitation inside the droplet.Our study revealed several significant conclusions.First, the shock wave undergoes the reflection on the droplet surface, resulting in intense negative pressure on the z-axis near the right surface.The focusing of the reflected waves is the primary contributor to the negative pressure at large d; however, the collision of pressure waves plays an important role at small d.The attenuation of the focusing is attributed to the significant caustic at small d.
Second, we derived the parametric equations of the wavefront using the ray-tracing method.The wavefront from these equations is in agreement with the numerical results.Using this method, we proved that the locus of the cusps on the shock wavefront observed in the two methods is caustic.Furthermore, we gave the parametric equations of the caustic and found that the caustic can be separated into two branches by a singular point because of the off-axis reflection.
Third, we analyzed the potential cavitation inside the droplet.It is not difficult to induce cavitation due to negative pressure extrema, which is closer to the surface with d decreasing.Additionally, cavitation induced by an internal shock does less damage to the droplet than that caused by shock-droplet impact; it is more likely for the pressure collision to induce cavitation compared to the focusing effect if the shock front does not match the droplet surface.
Our findings can be used to better understand the experiments of laser-droplet interaction, which have applications in various fields, such as the biomedical field, fog clearing, combustion engines, and EUV nanolithography.

Figure 1 .
Figure 1.Schematic of the problem geometry.The dash-dot line indicates the computational domain.

Figure 2 .
Figure 2. The pressure evolution at (a)-(c) d = 0.8 mm and (d)-(i) d = 0.3 mm.The tops of (d)-(i) are pressure patterns, and the bottoms are schlieren images.

Figure 3 .
Figure 3. (a) The minimum pressure over time at different d: red line for 0.3 mm, black for 0.6 mm, and blue for 0.8 mm.(b) The minimum pressure against d.

Figure 4 .
Figure 4.The minimum pressure over time of d = 0.2 mm with the different initial pressure.

Figure 3 (
b) shows the minimum pressure over d.As d decreases from 0.8 to 0.5 mm, the minimum pressure increases from −214 MPa to −97 MPa owing to the rapid attenuation of the focusing effect of Γ 1 .When d ranges from 0.5 to 0.2 mm, the minimum decreases from −97 MPa to −159 MPa with d decreasing because of the growing influence of the collision of Γ 1 and Γ 2 .The minimum pressure is directly from the focusing at a large d; for a small d, it is the first extremum generated by the collision of Γ 1, e.g. at d = 0.6 mm, or the second extremum resulting from the collision of Γ 2 , e.g. at d = 0.3 mm.As d < 0.2 mm, the minimum pressure increases as d declines due to the complex pressure evolution resulting from the emergence of Γ k with k > 2. We conclude that the differences between different intervals d are the influence of Γ k and the main contributor to the intense pressure.

Figure 5 .
Figure 5. Ray diagram showing the multiple reflections of a ray from the source point.

Figure 6 .
Figure 6.The parameter ϑ varying with the angle θ and n ′ .The black dashed line denotes the maximum for physical realization.

Figure 7 .
Figure 7.Comparison of the wavefront between the ray-tracing method (bottom half) and hydrodynamic simulation (top half).The wavefront from the ray-tracing method is given by equation (10) and depicted by a curve.

Figure
Figure The distribution of (γ, l) at different d.(a) and (b) Including multiple d and (c) d = 0.15 mm, and (d) d = 0.1 mm.The dotted lines in (c) and (d) are to discriminate the different parts of 2-rays line.

Figure 9 .
Figure 9.The caustic and rays at d = 0.8 mm.Only the first reflection is considered, and the caustic is given by equation (16).The blue and red lines indicate the rays and the caustic, respectively.

Figure 10 .
Figure 10.The distance of the extrema to the right surface at different d.

Figure A1 .
Figure A1.The caustic and rays at d = 0.1 mm.The different colors of the caustics correspond to the different segments Γn.

Figure A2 .
Figure A2.(a) Comparison of numerical and experimental data [31] for RR → MR transition angle.(b) The errors for different meshes of 2 N × N.

Table 1 .
The initial conditions in our numerical simulations.