Study of underwater laser propulsion using different target materials

Hao Qiang; Jun Chen; Bing Han; Zhong-Hua Shen; Jian Lu; Xiao-Wu Ni

doi:10.1364/OE.22.017532

1. Introduction

Laser propulsion was first proposed by Kantrowitz [1] in 1972. Up to now, many studies have been dedicated to laser propulsion in atmospheric [2, 3] and vacuum [4, 5] environment. Recently, Han et al. [6] proposed laser propulsion in water environment. For underwater laser propulsion, the sources of the propelling force including the plasma shock wave, bubble oscillating shock waves and the final bubble collapse impact including liquid jet and splash, and the latter two are the main sources [6]. Besides, the larger the maximum radius of bubble is during its first oscillation, the more fiercely the bubble collapses [7]. So bubble is important to laser propulsion in water, which is different from laser propulsion in atmospheric and vacuum environment. The momentum coupling coefficient C_m and the optimum coupling fluence Φ_opt are two critical parameters for laser propulsion [8]. C_m is the ratio of the momentum I_T obtained by the target to the incident laser energy E. And Φ_opt is the laser fluence when C_m is maximized. For laser propulsion in both atmosphere [9] and water [10], C_m increases with the laser fluence first, and then decreases after the maximum due to the laser plasma shielding.

Different targets are studied for enhancing C_m. To the laser propulsion in atmospheric, many studies have investigated a lot of targets with different geometries [11–14] and materials [5, 15–18]. Among them, T. Yabe [16] found that the water-covered metal target can improve C_m effectively, and the enhanced coupling theory for the metal targets covered with transparent and heavy materials is given [19]. To the laser propulsion in water, Han et al. have studied many targets with different geometries [20, 21] and interfaces [22]. But to the authors’ knowledge, there have not been detailed descriptions about the influence of target materials on the laser propulsion in water.

In this paper, targets of three different materials, including aluminum (Al), titanium (Ti) and copper (Cu), for underwater laser propulsion are studied. The enhanced coupling theory of laser propulsion in atmosphere using metal target covered with transparent and heavy materials is applied to deduce the theoretical model of the target momentum for underwater laser propulsion using metal target, since metal target for underwater laser propulsion can be considered as the case that mental target is covered with a lot of water for laser propulsion in atmosphere. The target momentum is investigated by high-speed photography method and numerical simulation method to verify the theoretical results. Besides, the momentum coupling coefficient C_m and the optimum coupling fluence Φ_opt of the three targets are also measured in this paper.

2. Theory

For laser propulsion in atmosphere, targets covered with transparent overlays have been used as a way of enhancing the momentum coupling coefficient [23]. To give the enhanced coupling theory for this kind of targets, Yabe [19] considered a situation that there was a metal target covered with transparent and heavy materials, and a laser beam penetrated the transparent layer and deposited energy at the interface. To this situation, Yabe thought that two shock waves were generated in the two materials after the energy was transmitted through the layer on the laser side and absorbed at the interface. And Yabe analyzed it using a simple estimate [24]. Suppose the two shock waves are driven in two directions and the momentum flow carried by the two shock waves is [24]:

ρ_{a} U_{a}^{2} = ρ_{b} U_{b}^{2}

where ρ is the density, U is the velocity. Subscripts a and b represent the two shock waves. Thus the flow of kinetic energy W carried by each shock waves is [24]:

\frac{W_{a}}{W_{b}} = \frac{ρ_{a} U_{a}^{3}}{ρ_{b} U_{b}^{3}} = {(\frac{ρ_{b}}{ρ_{a}})}^{1 / 2}

For underwater laser propulsion, the process that a laser beam is focused on the propelling surface of metal target in water can be taken as the situation considered by Yabe, in which there is a metal target covered with transparent and heavy materials, and a laser beam penetrates the transparent layer and deposits energy at the interface. So we think that Eq. (2) is also suitable for underwater laser propulsion with metal target.

To underwater laser propulsion, after bubble collapses, the main parts carrying kinetic energy are metal target, water, and the metal splash. The kinetic energy of the former two is mainly transmitted by bubble since most of the propelling force comes from the bubble impact. And the metal splash is the product of the ablation of target surface. So the kinetic energy of the former two and the metal splash can be characterized by the bubble size and the morphology of ablation respectively. We think that the three can be divided into two parts: the metal target and the mixture of water and metal splash. Their kinetic energy can be considered to be transmitted by two shock waves mentioned in Yabe’s theory [24]. So their kinetic energy E can be described by:

\frac{E_{1}}{E_{0}} = {(\frac{ρ_{0}}{ρ_{1}})}^{1 / 2}

where subscript 1 represents metal target, and subscript 0 represents the mixture of water and metal splash. Both the bubble size and the morphology of ablation are not the same for different metal targets, which will be confirmed in Section 5.1.1 and Section 5.2 respectively. So for another metal target indicated by subscript 2, there is:

\frac{E_{2}}{E_{0}'} = {(\frac{ρ_{0}}{ρ_{2}})}^{1 / 2}

By dividing Eq. (3) by Eq. (4), we can get:

\frac{E_{1}}{E_{2}} = \frac{E_{0}}{E_{0}'} {(\frac{ρ_{2}}{ρ_{1}})}^{1 / 2}

The kinetic energy E and the mass m of the target can be described as:

E = \frac{I^{2}}{2 m}

m = ρ V

where I is the momentum, and V is the volume. Assuming that the two targets are of same shape and size, Eq. (8) can be obtained by combining Eqs. (5)–(7):

\frac{I_{1}}{I_{2}} = {(\frac{E_{0}}{E_{0}'})}^{1 / 2} {(\frac{ρ_{1}}{ρ_{2}})}^{1 / 4}

To simplify the equation, coefficient k is given by:

k = {(\frac{E_{0}}{E_{0}'})}^{1 / 2}

So there is:

\frac{I_{1}}{I_{2}} = k {(\frac{ρ_{1}}{ρ_{2}})}^{1 / 4}

This expression describes the momentum relationship between the two targets with different materials, but same shape and size under the same incident laser energy. If the bubble size and the morphology of ablation are same for the two targets, which means that E₀ = E₀', we can obtain that k = 1. So from Eq. (10), it can be inferred that there are mainly two factors affect the momentum of targets. One is the bubble size and the metal splash. And the other is the density of target material. Besides, the greater influence the former has on the target momentum, the farther the value of k is away from 1. And the greater influence the latter has on the target momentum, the closer the value of k is to 1. So this equation can be used to analyze the momentum of targets with different materials, but same shape and size for underwater laser propulsion. In the meantime, relevant experiment and numerical simulation are applied to verify that the enhanced coupling theory, which was developed originally for laser propulsion in atmosphere, is indeed applicable to underwater laser propulsion with metal targets.

3. Experimental set-up

The high-speed photography method is shown by Fig. 1. A Q-switched Nd: YAG laser (wavelength 1.06μm, pulse duration 7ns) was focused on the surface of the propelled target in a glass filled with distilled water. Due to the optical breakdown in water, the quasi-semispherical bubbles were generated together with plasma, its shock wave, bubble and its collapse, thus the target was propelled along + x direction. The movement of target was captured by high-speed camera, and the images were fed into a computer. In this experiment, the energy meter was used to monitor the energy of Nd: YAG laser pulse. In order to prevent linear breakdown in water, the pulsed laser beam was first expanded by a concave-convex lens group, then focused on the propelled surface. The target was suspended on a solid bar by two threads as a pendulum, which was shown in the dotted circle of Fig. 1. The glass cuvette was large enough and rotated a small angle to neglect the reflection of water waves from the glass walls and prevent the reflected laser light from the glass walls from damaging the optical device. A 532nm laser used as flash light was expanded by a concave-convex lens group to expose the target. Timing of Nd: YAG laser, flash light, and high-speed camera was controlled by digital delay/ pulse generator DG535.

Fig. 1 Experimental set-up based on high-speed photography method: (1) Nd: YAG laser; (2) beam splitter; (3) concave-convex lens group; (4) focusing lens; (5) glass cuvette; (6) target; (7) energy meter; (8) flash light; (9) concave-convex lens group; (10) high-speed camera; (11) computer; (12) DG535 digital delay/pulse generator. Perspective view of suspended target is shown in dotted circle.

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For facilitating the suspension of target and avoiding the rotation of target while moving, the structure of target was designed. The three views of target structure are showed in Fig. 2, including (a) main view, (b) left view, and (c) top view of target structure. And the unit is mm. The grooves shown in Fig. 2(b) were processed for the convenience of drilling holes. Figure 2(d) shows the photo of the targets, including Al, Ti and Cu, and their densities are 2.6702g/cm³, 4.4486g/cm³, 8.7096g/cm³, respectively. Figure 2(e) shows the images of moving Cu target captured by high-speed camera when the incident laser energy is 35.5mJ. In this experiment, time is set to 0 when laser pulse reaches the surface of target. The recording time of three images is 2.85ms, 22.85ms, and 42.85ms, respectively. The curve of the distance traveled by target and time can be obtained by processing the images captured by high-speed camera, shown in Fig. 3(a). Figure 3(b) shows the images of target captured by high-speed camera for the former 6 data points in Fig. 3(a). It can be seen that the displacement of target is unstable at the beginning of the interaction of laser and target due to the oscillation of bubble, but has linear relationship with time after bubble collapses. And the slope represents the average velocity of target in the period of time, which is considered as the initial velocity of target. So the initial velocity of target can be obtained through the image processing, further the momentum coupling coefficient can be calculated. In this paper, every experimental data is the average of 5 times of measurements.

Fig. 2 (a) The main view of target structure. (b) The left view of target structure. (c) The top view of target structure. (d) Photo of three targets. The unit is mm. (e) Images of moving target captured by high-speed camera.

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Fig. 3 (a) Distance traveled by Ti target that varying with time when incident laser energy is 41.8mJ. (b) Images of Ti target captured by high-speed camera for the former 6 data points in Fig. 3(a).

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4. Numerical simulation model

The model used in the numerical simulation is shown by Fig. 4. The unit of the spatial coordinates is mm. A, B, C are the non-reflecting boundaries. Region D is the computational domain, filled with water. And the pressure of water is set to 1.01 × 10⁵ Pa. z is the rotational symmetric axis of the whole model, r is the radial coordinate axis. Point (0, 0) is the laser focus, meaning that it is also the center of the bubble. In the model, the shape of the targets is simplified as cylindrical, and the densities of the targets are modified to ensure that the mass of the targets is consistent with experimental values. To the materials of targets used in the model, including Al, Ti and Cu, the shock equation of state (SEOS) [25] is chosen as the state equation due to the existence of the strong discontinuity, namely the shock wave, and the Steinberg-Guinan strength model [26] is chosen as the constitutive model because that the targets will be deformed under the impact of the shock wave and the bubble final collapse, namely the cavitation erosion. Besides, the Euler algorithm and the Lagrange algorithm are used for region D and target, respectively. The simulations start from the bubble growth, and the initial conditions of the bubbles are taken from experimental results.

Fig. 4 Model used in the numerical simulation.

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The numerical simulation is based on the conservation laws, including the mass conservation Eq. (11), the momentum conservation Eqs. (12) and (13), and the energy conservation Eq. (14) [27]:

\frac{\partial ρ}{\partial t} + \frac{\partial}{\partial z} (ρ u) + \frac{1}{r} \frac{\partial}{\partial r} (r ρ v) = 0

\frac{\partial}{\partial t} (ρ u) + \frac{\partial}{\partial z} (ρ u^{2}) + \frac{1}{r} \frac{\partial}{\partial r} (r ρ u v) + \frac{\partial ρ}{\partial z} = 0

\frac{\partial}{\partial t} (ρ v) + \frac{\partial}{\partial z} (ρ u v) + \frac{1}{r} \frac{\partial}{\partial r} (r ρ v^{2}) + \frac{\partial ρ}{\partial r} = 0

\frac{\partial E}{\partial t} + \frac{\partial}{\partial z} [u (E + p)] + \frac{1}{r} \frac{\partial}{\partial r} [r v (E + p)] = {\begin{matrix} \frac{\partial I}{\partial z} t_{d} < t < t_{p}, (z, r) \in Ω \\ 0 o t h e r w i s e \end{matrix}

where ρ is the local density, t is the time, u and v are velocities along axises z and r respectively, E is the local energy per unit volume, p is the pressure, I is the laser intensity, t_d is the moment when the shock wave appears, t_p is the laser pulse width, and Ω is the area of the laser beam.

5. Results and discussion

5.1 Experiment and numerical simulation analysis of target momentum

Since the bubble impact is the most important resource of propelling force for underwater laser propulsion, the study about the influence of targets with different materials on target momentum can be divided into two main parts. One is to study the influence of different target materials on the bubbles generated on the propelled surface under same laser energy. The other is to investigate the momentum of the targets with different materials when the bubbles are of same size. The analysis of the two parts is as below:

5.1.1 The bubble generated on the target surface

For underwater laser propulsion, the greatest contribution to the propelling force is the bubble impact. So it can be inferred that the propelling force is characterized by the initial energy of bubble. Assuming that the flow field is infinite with a constant pressure, and the condensability and viscosity are neglected, the initial energy E_B and the collapse time T_C of a spherical bubble can be described as Eqs. (15) and (16), respectively [28]:

E_{B} = \frac{4 π}{3} (P_{\infty} - P_{v}) R_{\max}^{3}

T_{C} = 0.915 R_{\max} \sqrt{\frac{ρ}{P_{\infty} - P_{v}}}

where P_∞ is the environmental pressure, P_v is the saturated vapor pressure, R_max is the maximum bubble radius, ρ is the flow density. In this paper, P_∞ is 1.01 × 10⁵ Pa, P_V is 2.33 × 10³ Pa, ρ is 1 × 10³ kg/m³. For the bubble that generated on the propelling surface of target in our experiment, its shape is quasi-semispherical, and its initial energy E_B' can be described as [22]:

E_{B}' = \frac{1}{2} E_{B} = \frac{2 π}{3} (P_{\infty} - P_{v}) R_{\max}^{3}

Equation (17) shows that E_B' is proportional to the third power of R_max. So we can roughly think that the value of the propelling force is characterized by R_max.

In our experiment, time is set to 0 when laser pulse reaches the surface of target. The bubble radius at t = 100μs under different incident laser energy is experimentally measured for the three targets and shown by Fig. 5.It can be seen that the bubble radius increases with the laser energy, but the increasing trend slows gradually due to the laser plasma shielding for all the three targets. And at t = 100μs, the bubble radius of Al target is largest, followed by Ti target, and the smallest is Cu target. We consider that the maximum bubble radius R_max has the same size relationship for the three targets. The explanation is as below:

Fig. 5 The bubble radius R at t = 100μs under different incident laser energy E obtained by the experiment. Squares, triangles, and circles represent the cases of Al target, Ti target and Cu target, respectively. Each point is an average of 5 measurements.

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If the bubble oscillates near a rigid boundary, Eq. (16) is not suitable. Rattrary’s study shows that the collapse time of a bubble near rigid boundary is longer than the case of infinite liquid [29]. Thus, the collapse time T_CR in this paper is longer than the collapse time T_C, which is calculated from Eq. (16). The minimum bubble radius in Fig. 5 is 1.44 mm. Assume that it is the maximum bubble radius R_max during its first oscillation, its T_C is 132μs, calculated by Eq. (16). If the bubble is collapsing at t = 100μs, T_CR should satisfy T_CR < 100μs. But in this case, R_max must be larger than 1.44mm, and T_CR should also satisfy: T_CR > 132μs >100μs. We find that the two relational expressions of T_CR are inconsistent, so the bubble is expanding at t = 100μs, and it can be deduced that all the bubbles in Fig. 5 are expanding at t = 100μs. Besides, the higher the initial energy of bubble is, the faster the bubble expands, and vice versa. Therefore, from Eq. (17), we consider that R_max under different laser energy can be reflected by Fig. 5.

So from Fig. 5, it can be inferred that R_max are different for targets with different materials, but same shape and size. In this paper, R_max of the three targets satisfy: R_max(Al) > R_max(Ti) > R_max(Cu). Besides, we find that at t = 100μs, the bubble radius of Ti target is very close to that of Al target when the laser energy is larger than 45mJ, which means that their R_max are very close in this case. Figure 5 also shows that for the three targets, their R_max all increase with the laser energy, and then begin to level out as the laser energy is larger than 45mJ because of the laser plasma shielding.

The difference of R_max among the three targets may be caused by the different features of their plasmas. The absorption coefficient of the inverse bremsstrahlung absorption in the process of plasma formation is related to the atomic number Z of metal [30], so Cu target absorbs more energy, which means that the liquid in the laser focused point absorbs litter energy, resulting in that the energy of the plasma is lower, and the bubble generated on the surface of Cu target is smaller.

5.1.2 Momentum analysis of targets under same bubbles

This section will investigate the momentum of targets with different materials under the condition that their bubbles are of same size. From Section 5.1.1, we know that the maximum bubble radiuses are different for targets with different materials under same laser energy. It is difficult to make the maximum bubble radiuses same for the three targets, so numerical simulation method is employed in this section.

For comparing with the experimental results and further investigating the influence of the bubble size on target momentum I_T under same laser energy, the bubble size in the numerical model is set to the value of the bubble generated on the propelling surface of Cu target. The reason is that the bubble of Cu target is smallest in experiment, and the numerical simulation results under this setting are easier to be compared with the experimental results. In addition, the numerical simulation is focused on the interaction between collapsing bubble and target since the bubble collapse impact is the main resource of propelling force.

Figure 6 shows the numerical results of I_T with same bubbles when laser energy is 45.4mJ. Table 1 shows the comparison between two numerical simulation results of I_T when laser energy is 45.4mJ. Here, numerical simulation 1 represents the case that the bubbles are same for the three targets, equal to the size of the bubble generated on the surface of Cu target. Numerical simulation 2 represents the case that all parameters, except the density, are same for the three targets, equal to the values of Cu in numerical simulation 1.

Fig. 6 The numerical results of the momentum obtained by the targets in the case of same bubbles when incident laser energy is 45.4mJ. The solid line, dashed line and dotted line represent the cases of Al target, Ti target and Cu target, respectively.

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Table 1. The comparison between the two numerical simulation results^a

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The concrete analysis is as follows:

From Table 1, we can see that there is little difference between the results of numerical simulation 1 and numerical simulation 2, which means that I_T is determined only by the density of target material when the bubbles are same. In addition, Fig. 6 shows that I_T of Cu target is largest, followed by Ti target, and the smallest is Al target, which is consistent with the size relationship of their densities. So we consider that in the case of same bubbles, the larger the density of target material is, the larger I_T is.

5.1.3 Analysis of experimental and numerical simulation results on target momentum

Figure 7 shows the experimental results of target momentum I_T under different laser energy E. From Fig. 7, it can be seen that when E is smaller than 33mJ, I_T of Al target is largest, followed by Ti target, which is the same as the size relationship of their bubble sizes. It means that the bubble size plays a main role when E is smaller than 33mJ. But with E increases, the growth rate of the Cu target momentum is largest, followed by Ti target, and the smallest is Al target. When E is larger than 45mJ, I_T of Cu target is largest, followed by Ti target, and the smallest is Al target. In this case, we find that the size relationship of I_T is consistent with that of target density, which means that I_T is determined primarily by the density of target material when E is larger than 45mJ.

Fig. 7 The momentum I_T of the targets under different incident laser energy E obtained by experiment. Squares, triangles, and circles represent the cases of Al target, Ti target and Cu target, respectively. Each point is an average of 5 measurements.

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Besides, Table 2 shows the comparison between experimental and numerical simulation results of I_T for the three targets when E is 45.4mJ. Here, the numerical simulation is the numerical simulation 1 in Table 1. The relative errors of I_T between experimental and numerical simulation results for Ti target and Al target are primarily caused by the difference of the bubble size between the experimental and the simulation values. From Table 2, we can see that the relative errors are acceptable, meaning that the difference of the bubble size contributes little to I_T when E is 45.4mJ. So we consider that when E is larger than 45mJ, the influence of the bubble size on I_T does exist, but is small.

Table 2. The comparison between experimental and numerical simulation results^b

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Above all, we think that there are two primary parameters impacting on I_T. One is the size of bubble generated on the propelling surface, which is determined by the atomic number Z of the target metal, and the other is the density of target material. The atomic number Z plays a more important role when E is smaller than 33mJ, while the density plays a more important role when E is large larger than 45mJ in this paper.

5.2 Verification of the theory on target momentum

The analysis results of experiment and numerical simulation on target momentum I_T have been given in Section 5.1.3. In this section, the experimental data will be added to the theoretical formula mentioned in Section 2. And the similar conclusion will be obtained to verify that the theory is also applicable to underwater laser propulsion with metal targets.

The concrete analysis is as below:

For Ti target and Al target, Cu target and Al target, Cu target and Ti target, Eq. (10) can be respectively written as:

{\begin{matrix} \frac{I_{T i}}{I_{A l}} = k_{1} {(\frac{ρ_{T i}}{ρ_{A l}})}^{1 / 4} \\ \begin{array}{l} \frac{I_{C u}}{I_{A l}} = k_{2} {(\frac{ρ_{C u}}{ρ_{A l}})}^{1 / 4} \\ \frac{I_{C u}}{I_{T i}} = k_{3} {(\frac{ρ_{C u}}{ρ_{T i}})}^{1 / 4} \end{array} \end{matrix}

Figure 8 shows the coefficient k under different laser energy E for Ti target and Al target, Cu target and Al target, Cu target and Ti target. First of all, from Section 5.1.1, we know that the maximum bubble sizes R_max on the three targets all begin to level out as E is larger than 45mJ due to the laser plasma shielding, which means that their I_T will stop growing and the value of k will no longer change when E is larger than 45mJ. So from Fig. 8, it can be inferred that the value of k increases with E, then tends to a certain value as E continues to increase. When E is larger than 45mJ, it can be calculated that k₁ = 0.985, k₂ = 0.862, k₃ = 0.875. The three values are all close to 1, which means that I_T is primary determined by the density of target material when E is larger than 45mJ. Besides, we can see that the value of k declines sharply and is much less than 1 when E is smaller than 33mJ. It means that the bubble size and the metal splash have a major influence on I_T under this condition.

Fig. 8 The coefficients including (a) k₁, (b) k₂ and (c) k₃ under different incident laser energy E.

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For further investigating the influence of the bubble size and the metal splash on I_T, following analyses are given:

Figure 9 shows the photos of the three targets after experiment. It can be seen that Ti and Cu are similar and both better than Al in the performance of the laser damage resistance. It means that the kinetic energy of the metal splash for Al target is much larger than that for Ti target and Cu target. From Section 5.1.1, we know that the maximum bubble sizes of Ti target and Al target are very close when E is larger than 45mJ, meaning that the kinetic energy of the water for Ti target is very close to that for Al target. But for Ti target and Al target, k₁ = 0.985 when E is larger than 45mJ, which means that the kinetic energy of the mixture of water and metal splash for Ti target is very close to that for Al target. So it can be inferred that the kinetic energy of the metal splash is much smaller than that of the water when E is larger than 45mJ. In addition, we know that the degree of ablation increases with E. Thus, we consider that the kinetic energy of the metal splash is much smaller than the water in the whole laser energy range. And when E is smaller than 33mJ, the primary factor impacting on I_T is the bubble size. For Cu target and Ti target, they are similar in the performance of the laser damage resistance. So we consider that the value of k₃ can reflect the extent of influence of the bubble size on I_T. When E is larger than 45mJ, k₃ = 0.875, it means that the influence of bubble size on I_T does exist, but is small under this condition. Thus, we consider that the influence of the metal splash on I_T is very small, can be neglected. And when E is larger than 45mJ, the influence of the bubble size on I_T is small, but cannot be neglected. This is why k₁ is nearer to 1 than k₂ and k₃, and k₂ and k₃ are very adjacent.

Fig. 9 Photos of targets after experiment.

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In conclusion, we think that there are two primary parameters impacting on I_T. One is the size of the bubble generated on the propelling surface. From Section 5.1.1, we know that it is determined by the atomic number Z of the target metal. The other is the density of target material. The atomic number Z plays a more important role when E is smaller than 33mJ, while the density plays a more important role when E is larger than 45mJ in this paper. This conclusion is consistent with the analysis results of experiment and numerical simulation in Section 5.1.3. It verifies that the enhanced coupling theory mentioned in Section 2, which was developed originally for laser propulsion in atmosphere using metal target covered with transparent and heavy materials, is also applicable to underwater laser propulsion with metal target.

5.3 Momentum coupling coefficient of the target

Figure 10 shows the momentum coupling coefficient C_m under different laser fluence for underwater laser propulsion in the cases of Al, Ti and Cu target. It can be seen that for all three targets, C_m increases with the laser fluence first, and then decreases after the maximum due to the laser plasma shielding. And the optimum coupling fluence Ф_opt exists for all three targets. In this paper, for underwater laser propulsion, Ф_opt of Al target is 155.32kJ/m², which is similar to the result 158.53 kJ/m² reported in [10]. Ф_opt of Ti target is 156.24kJ/m², and Ф_opt of Cu target is 160.26kJ/m². Besides, we find that Ti target achieves highest C_m among the three targets, which is 100.11dyn/W in this paper. Figure 9 shows that Ti and Cu are similar and both better than Al in the performance of the laser damage resistance. So we think that Ti is a suitable target material for laser propulsion in water.

Fig. 10 The momentum coupling coefficient C_m under different laser fluence Ф for underwater laser propulsion. Squares, triangles, and circles represent the cases of Al target, Ti target and Cu target, respectively. The dashed lines represent their fit curves. Ф_opt is the optimum coupling fluence. Each point is an average of 5 measurements.

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6. Conclusion

The enhanced coupling theory of laser propulsion in atmosphere using metal target covered with transparent and heavy materials is proved to be applicable for underwater laser propulsion using metal target. Besides, we find that there are two primary parameters impacting on I_T. One is the maximum radius R_max of the bubble generated on the propelling surface, which is determined by the atomic number Z of the target metal. The increase in Z would result in the decrease in R_max. The other primary parameter is the density of the target material. The atomic number Z plays a more important role when laser energy E is smaller than 33mJ, while the density plays a more important role when E is larger than 45mJ in this paper.

In addition, for underwater laser propulsion, the optimum coupling fluence Φ_opt of Al target, Ti target and Cu target are respectively 155.32kJ/m², 156.24kJ/m² and 160.26kJ/m². Ti target achieves highest C_m, which is 100.11dyn/W in this paper, and plays well in the performance of the laser damage resistance. Thus, it is recommended as the target material for underwater laser propulsion.

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	I_Cu/μN·s⁻¹	I_Ti/μN·s⁻¹	I_Al/μN·s⁻¹
Numerical Simulation 1	45.40	42.18	36.11
Numerical Simulation 2	45.40	42.57	36.40
δ	—	0.92%	0.80%

	I_Cu/μN·s⁻¹	I_Ti/μN·s⁻¹	I_Al/μN·s⁻¹
Experiment	45.76	44.98	37.45
Numerical Simulation	45.40	42.18	36.11
δ_e	0.79%	6.22%	3.58%

	I_Cu/μN·s⁻¹	I_Ti/μN·s⁻¹	I_Al/μN·s⁻¹
Numerical Simulation 1	45.40	42.18	36.11
Numerical Simulation 2	45.40	42.57	36.40
δ	—	0.92%	0.80%

	I_Cu/μN·s⁻¹	I_Ti/μN·s⁻¹	I_Al/μN·s⁻¹
Experiment	45.76	44.98	37.45
Numerical Simulation	45.40	42.18	36.11
δ_e	0.79%	6.22%	3.58%

Study of underwater laser propulsion using different target materials

Abstract

1. Introduction

2. Theory

3. Experimental set-up

4. Numerical simulation model

5. Results and discussion

5.1 Experiment and numerical simulation analysis of target momentum

5.1.1 The bubble generated on the target surface

5.1.2 Momentum analysis of targets under same bubbles

5.1.3 Analysis of experimental and numerical simulation results on target momentum

5.2 Verification of the theory on target momentum

5.3 Momentum coupling coefficient of the target

6. Conclusion

References and links

Cited By

Figures (10)

Tables (2)

Equations (18)

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