The driving experiment of SiO_{2} microspheres in a water environment was carried out by using tapered fiber microstructures to transmit short pulse lasers. The fiber microstructure can generate plasma and spherical shock waves to drive SiO_{2} microspheres. Through theoretical simulation, the propagation characteristics of shock waves and the dynamic characteristics of microspheres were studied. In the experiment, a high-speed COMS camera was used to capture the images of shock wave diffusion and microsphere motion. A linear relationship between the driving behavior of microspheres and the laser energy distribution is observed. The driving behavior of microspheres is attributed to the resultant force caused by spherical shock wave diffusion. We find that the initial driving velocity approximately follows the inverse quadratic function of the radius ratio of the spherical wave, which is consistent with the experimental results. Compared with the traditional technology, this method has the advantages of directional stability, good security, anti-interference, and so on. It can be used for stable directional driving of micron objects in a water environment.

## I. INTRODUCTION

Pulsed laser-induced plasma shock waves have attracted much attention^{1–3} due to their applications in laser shock hardening,^{4,5} laser propulsion,^{6,7} ocean environmental cleaning,^{8,9} rocket launch,^{10} etc. In these studies, the dynamic expansion of plasma shock waves is an important research direction.^{8} When the energy of the pulsed laser is focused to a critical level where it is capable of penetrating air or water, the target environment is ionized and a high pressure, high temperature plasma^{11} is generated. This plasma continues to absorb energy and expand,^{12} compressing the surrounding gas or liquid into a shock wave, which then rapidly spreads into the target environment.

Chen *et al.*^{13} reported the effect of laser pulses focused on the surface of an aluminum target in an aqueous environment and analyzed the kinetics of bubble generation in the tail of the aluminum target. When the surface of an aluminum target is irradiated by a pulsed laser focus, the high-power laser energy is concentrated in the water and generates plasma, shock waves, and bubbles, which propel the aluminum target. Li *et al.*^{14,15} reported the underwater propulsion of microspheres using a fiber-optic airgun focused pulsed laser, investigated the kinetics of fiber-optic laser-induced bubble generation on a microscale, which emits bubbles to propel polystyrene particles, and observed a linear relationship between particle propulsion and energy distribution. The traditional space laser^{16} focusing and fiber pulse focusing both attribute the main force to the vacuole pulsation force^{17} and the force of microjet generated by vacuole collapse,^{18} and the effect of the plasma shock wave generated by laser focusing to drive the target is less studied.

For the research of microscale laser propulsion, from the use of fiber coupled pulse lasers in air to the use of fiber coupled lasers in water. From the flat end tapered fiber to the tapered fiber that can control the focus, it continues to develop in the microscale laser propulsion.^{19–21}

In this paper, a method of tapered fiber microstructure propulsion based on short pulse laser is proposed, and the interaction between shock wave generated by the laser pulse and microsphere in a water environment is studied. Through simulation and experiment, the underwater laser propulsion process and shock wave propagation characteristics are studied. The experimental results show that with the increase of laser energy, the maximum diameter of the shock wave increases, and more energy is distributed on the microspheres, which improves the propulsion efficiency. At the same time, experiments were carried out with microspheres with different diameters to deduce the relationship between the diameter ratio of the shock wave and the propulsion effect of microspheres and analyze the targeting in the process of propulsion. The laser propulsion system using tapered fiber microstructure has the characteristics of simple operation and accurate positioning. Based on these characteristics, the research of using tapered fiber microstructure to transmit laser to promote SiO_{2} microspheres has potential application prospects in underwater laser cleaning and underwater manipulation of micrometer objects.

## II. EXPERIMENT

Figure 1(a) shows the schematic diagram of the experimental device for underwater tapered fiber conducting short pulse laser driving microsphere. The tapered fiber drive system mainly consists of a laser device (Q-switched Nd: YAG laser), a tapered fiber, a microscope objective, a detection device, and a viewing device. The working mode is to couple the pulsed laser into the tail of the tapered fiber using a 4× optical objective (NA = 0.10). Measure the amount of laser energy emitted by aligning the laser light coming out of the tapered fiber with an energy meter (Field Max II-Top, the resolution is 0.01 *μ*J). The tapered fibers were pulled by manual power generation of multimode fibers by a fiber fusion machine (NT-400H+). The diameter of the tapered tip was reduced from 125 to 10 *μ*m over a length of 400 *μ*m. The experiments were performed using a laser beam with a wavelength of 532 nm, a pulse width of 10 ns, and a repetition frequency of 1 Hz. The SiO_{2} (density: 2.2 g/cm^{3}, diameter: 50–100 *μ*m) microsphere is suspended in water on the surface of the magnesium fluoride (MgF_{2}) substrate. The estimated variation of microsphere diameter is less than 5%. To ensure horizontal alignment between the tapered fiber tip and the microsphere, we use 3D displacement (resolution is 1 *μ*m) to control the microsphere. A high-speed CMOS camera (I-SPEED 7 SERIES) records the dynamic motion of a SiO_{2} microsphere driven by a pulsed laser at 20 000–200 000 frames per second.

Figure 1(b) shows the schematic diagram of the experiment of a plasma shock wave formed by pulsed laser focusing in water through tapered fiber in order to study the relationship between the maximum radius of the shock wave and laser focusing energy. Figure 1(c) shows the experiment of driving SiO_{2} microspheres by the plasma shock wave formed by the focusing of the pulsed laser through the tapered fiber in order to explore the influence of the shock wave size on the directional driving effect of the microspheres.

## III. RESULTS AND DISCUSSION

Figure 2(a) shows the experimental process of a shock wave generated by a 62.3 *μJ* pulsed laser focused in deionized water through tapered fiber. At 0 *μ*s, the pulsed laser converges on the fiber cone. At 10 *μ*s, a bright plasma is generated with a diffuse shock wave. At 20 *μ*s, the spherical shock wave reaches 290 *μm*. Then the shock wave dissipates and causes fluctuations in the water environment, leaving several tiny bubbles. The shape of the spherical shock wave generated by tapered fiber pulsed laser is similar to that of a cavitation bubble generated by traditional space laser focusing, but from the perspective of time scale, the diffusion time of the shock wave is far less than the expansion time of the cavitation bubble, and from the perspective of diameter, the diameter of the shock wave is also far less than the diameter of the cavitation bubble.^{22} In plasma accelerator research, this phenomenon is referred to as a plasma cavity, which represents a highly nonlinear wakefield. The plasma shock wave corresponds to a concentration of plasma at the boundary of this cavity, which is theoretically infinitesimally thin. When subjected to the intense nonlinear effects of a pulsed laser, the laser-driven plasma displaces electrons from their initial positions, forming a spherical cavity devoid of electrons. In this paper, the shock wave corresponds to a collection of plasma on the boundary of an infinitely thin cavity in this theory.^{23} Fig. 2(b) shows the relationship between the maximum diameter of the spherical shock wave and the laser focusing energy. The experimental image shows that with the increase of laser energy, the diameter of the spherical shock wave also increases. It can be observed that the spherical shock wave will be blocked by the tapered fiber cone, resulting in a little depression in the blocking part, indicating that the shock wave will be blocked by the existing object.

In the experiment, we used ultrasound to disperse 50–120 *μ*m microspheres in water and formed centimeter-sized droplets on magnesium difluoride tablets. The microspheres were suspended in the droplets. According to the diameter of the shock wave and the diameter of the microspheres, it can be inferred that it is non-boundary relative to the droplets, so only the diffusion of the spherical shock wave needs to be considered.^{24} The laser emitted through the tapered fiber focuses on the center of the droplet, and there will be no phenomenon similar to the microjet due to the existence of the boundary.

^{3}

*m*/

*s*. With the increase in propagation time, the energy of the shock wave is gradually transferred to the surrounding medium, and the velocity gradually decreases to the sound velocity and finally dissipates in the surrounding medium. In fact, the duration of shock wave generation and outward expansion is very short, which conforms to the “point burst ball symmetry” model. Therefore, the classical Sedov–Taylor theory can be used to calculate the relationship between shock wave expansion distance and time,

^{25}

*ξ*≈ 1 is the adiabatic indices of water vapor as the coefficient of the shock wave formula,

*ρ*

_{0}= 1 kg/m

^{3}is the density of water,

*t*is the propagation time of the shock wave, and

*E*

_{0}is the energy of the shock wave. According to the formula, we calculated the function relationship between the propagation radius of the shock wave and time when the shock wave energy is as shown in Fig. 2(c). It can be seen from the figure that the propagation radius (i.e., propagation distance) gradually increases with the increase of the energy carried by the shock wave. At the same time, combined with the experimental data in Fig. 2(b), it is concluded that the maximum diameter of the shock wave is proportional to the laser energy in Fig. 2(d).

Figure 3(a) shows the experiment of driving a microsphere by a plasma shock wave generated by a pulsed laser through tapered fiber aggregation. The laser energy is 40.6 *μ*J, and the radius of the microsphere is 25 *μ*m. The diffusion of plasma and shock wave can be observed at 10 *μ*s. The spherical shock wave becomes similar to a peanut shape due to the obstruction of the microsphere, and some of the obstructed shock waves generate a driving force on the microsphere. At 20 *μ*s, the microsphere is driven forward by the force of the shock wave, and the obstructed part of the shock wave also diffuses forward. Then the microsphere continues to drive forward and decelerates under the effect of resistance until it is stationary. The shock wave dissipates, and tiny bubbles remain in the water. Figure 3(b) shows the whole process of the experiment of different laser energy driving microspheres with different diameters. For case II, *E* = 40.6 *μJ*, *R*_{P} ≈ 50 *μm*; for case III, *E* = 43.7 *μJ*, *R*_{P} ≈ 25 *μm*; for case IV, *E* = 43.7 *μJ*, *R*_{P} ≈ 50 *μm*. It can be observed that the microspheres are driven by the force of a shock wave. It can be observed from case I and case II that the shock wave radius generated by the same laser energy is similar, and the moving distance of microspheres with larger radii is relatively small. Through case I and case III, it can be observed that when the radius of microspheres is the same, larger laser energy will drive the microspheres farther. It can also be observed from the experimental image in Fig. 3 that different initial positions will lead to different driving directions of microspheres.

In order to study the relationship between laser energy, microsphere size, and driving distance direction, we established a driving model as shown in Fig. 4(a). The radius of the microsphere is *R*_{P}, the distance between the center of the initial position of the microsphere and the tip of the tapered fiber is *L*_{0}, and the angle is *φ*_{0}, the driving distance of the microsphere is *L*_{1}, and the angle of the end position of the microsphere relative to the initial position is *φ*_{1}.

Figure 4(b) shows the experimental image relationship between the movement distance and time of the four groups of microspheres in Fig. 3. It is observed that the motion of the microspheres is relatively high at the initial velocity and then gradually decelerates until stationary. Moreover, we can obtain the moving distances of the microsphere Δ*R* and the time between two frames Δ*t* by processing the images, so the velocity v can be calculated by *v* = Δ*R*/Δ*t*. Therefore, the slopes in Fig. 4(b) can be expressed as the velocity of the microsphere, and we find that the velocity is proportional to the laser energy. This can happen due to the plasma density monotonically increasing with the laser energy, and the microsphere gains more momentum from the plasma according to the law of conservation of momentum. When the laser energy increases, the laser pulse length will increase, which will produce more dense plasma at the same focus point, so as to improve the plasma density. The increase in microsphere displacement in the experiment reflects the enhancement of the shock wave effect.^{26}

*C*

_{m}=

*m*

_{p}

*v*

_{0}/

*E*

_{0}, where

*m*

_{p}represents the particle mass. The energy of the shock wave involves both growth and pulsation processes, which are evaluated via numerical approximation and defined by the energy distribution efficiency

^{27}

*η*

_{s},

The specific impulse and thrust are independent in definition, but they jointly reflect the performance of laser thrusters. The larger the specific impulse is, the greater the impulse can be generated when the propeller consumes the same mass of propellant, so the less fuel it needs to carry in theory. Thrust directly determines the acceleration and velocity changes that can be generated by the propeller. In this experiment, the difference between the mass of the laser propellant is small, so the thrust is used to express the propulsion effect.

Figure 4(d) shows that the driving direction of the microsphere in the experiment is related to the angle of the microsphere, and the initial angle *φ*_{0} and the static angle *φ*_{1} of the microsphere are sinusoidal. The tan *φ*_{0} and tan *φ*_{1} obtained from the experiment form a function diagram, and the dotted line of the function image with tan *φ*_{0} = tan *φ*_{1} is added for comparison. It is found that the orientation of the microsphere driving is relatively good, and the error is less than 5%. Figure 4(e) shows the simulation of the propulsion process in water by using fluid simulation. The simulation image shows that the shock wave is blocked under the action of microspheres to form a waveform similar to the experimental results, and the diffusion speed is consistent with the shock wave diffusion calculation in theory. When the transient spherical shock wave is formed, the pressure at different positions is different. In order to calculate the resultant force of the shock wave on the microsphere, the functional relationship between the ratio of the shock wave force at different radius positions and the maximum force at the center under different energies is obtained according to the formula^{28} $F=k\u2217(E0/RS3)1/3$ in Fig. 4(f). The farther away from the shock wave center, the smaller the shock wave force. According to the force analysis of the microsphere subjected to the shock wave in Fig. 4(f), the force area of the microsphere subjected to the shock wave is the surface area of the microsphere at the contact part, and the force at different positions on this surface area is different. The force on the microsphere can be obtained by splitting it into different concentric circles for force analysis and integral calculation.

Figure 5(a) shows the normalized calculation results of the resultant force for different microsphere shock wave radius ratios. According to different radius ratios of microsphere shock wave, there are different force areas, so the formula for calculating the resultant force is different, and the limit is $RP/RS,max=2/2$. When $RP/RS,max<2/2$, the coverage of the shock wave is greater than half of the surface area of the microspheres. Since the other half of the microspheres cannot be affected by the shock wave, the impact area of the shock wave is half of the surface area of the microspheres. In the first type of hemispherical area promotion, at that time, the part of the shock wave acting on the microsphere was recorded as half of the surface area of the microsphere, which is recorded as $S1=2\pi RP2$. When $RP/RS,max>2/2$, the part of the microsphere subjected to the shock wave force is the surface area of the spherical cap at the interface between the shock wave and the microsphere, and the calculation formula is *S*_{2} = 2*πR*_{P}*h*. Among them, the height of the spherical crown obtained by calculation is *h* = *R*_{S,max}/2*R*_{P}, so the resultant surface area is $S2=\pi RS,max2$. Through calculation, it is found that under the same spherical shock wave radius, the force area of the microspheres is the same, independent of the diameter of the microspheres. However, during the shock process, the shock wave forces at different positions are different, as shown in Fig. 4(f). The resultant force is calculated by integrating the forces at each point.

*F*

_{j1}to $Fj1*$ and set

*R*

_{P}/

*R*

_{S,max}to

*P*for easy calculation,

*δ*is the normalization coefficient and $Fj1*max$ is set to 1. When $RP/RS,max>2/2$, the resultant force calculation formula is expressed as

*F*

_{j2}to $Fj2*$ and set

*R*

_{S,max}to 1 for easy calculation,

*x*

_{0}/Δ

*t*

_{0}, where Δ

*x*

_{0}is the initial 10

*μ*s movement distance of the microsphere and Δ

*t*

_{0}= 10

*μ*s. The calculation formula for initial speed is

*F*

_{j}Δ

*t*

_{0}/

*m*

_{P}, and the mass of microspheres is $mP=4\pi RP3/3$. Two formulas of the relationship between the initial velocity and the ratio of spherical wave diameter are obtained,

Figure 5(b) shows the functional relationship between the initial velocity driven by the microsphere and the radius ratio of the spherical wave and adds the experimental data to the functional relationship for comparison. The results show that the relationship between the initial velocity of microspheres and the ratio of spherical wave radius is an inverse power of two. The larger diameter microspheres are subjected to a larger force, but the initial driving speed is smaller. In the experiment, microspheres with three diameters (50, 60, and 100 *μ*m) are driven, and the results are in line with the relationship curve. However, due to the unstable size of the shock wave and the resistance caused by viscosity in the fluid, the experimental data are scattered and generally low.

## IV. CONCLUSION

In summary, we propose an underwater pulse laser tapered fiber shock wave directional propulsion system and study the interaction between SiO_{2} microspheres and nanosecond laser pulses in the water environment. Compared with the existing propulsion methods, this method has the advantages of low cost, simple structure, good directionality, and easy adjustment. Through quantitative analysis of time and space, we revealed the propagation characteristics of shock waves at different energies and explained the mechanism by which shock waves target and drive microspheres. The combined force of the propulsion of the microspheres mainly by the spherical transient shock wave is analyzed. According to the different radius ratios of the spherical waves, the combined force of the shock wave of the microspheres at different energies and the initial speed of the microspheres are calculated and compared with the initial speed of the microspheres in the experiment. In these cases, the relationship between the initial velocity and the radius ratio of the spherical wave is inversely proportional to the quadratic power obtained by the quantum scale. The findings of this letter contribute to the design of fiber-optic microsystems for targeted target operations, such as blood-targeted drug delivery and marine micro-supply.

## ACKNOWLEDGMENTS

This work was supported by the National Natural Science Foundation of China (Grant No. 52271344) and the Natural Science Foundation of Heilongjiang Province of China (Grant No. LH2021E032).

## AUTHOR DECLARATIONS

### Conflict of Interest

The authors have no conflicts to disclose.

### Author Contributions

**Yang Ge**: Funding acquisition (equal); Resources (equal). **Gaoqian Zhou**: Conceptualization (equal); Methodology (equal); Writing – original draft (equal). **Xulong Yang**: Project administration (equal). **Ying Chen**: Software (equal). **Xianqi Tang**: Formal analysis (equal). **Hangyang Li**: Investigation (equal).

## DATA AVAILABILITY

The data that support the findings of this study are available from the corresponding author upon reasonable request.