1. Introduction

The method of using light to control the position of a miniscule object without physical contact is called light manipulation. Ashkin was the first to investigate the radiation pressure exerted by light [1,2]. Currently, light manipulation has wide applications in fields such as biology (capture and manipulate a miniscule living body [3]), micro science, engineering (rotation of fine particles [4,5]), mechanical engineering (wing rotor [6]), and medicine (elasticity measurement of red blood cells [7]).

Glass can be used as an optical device by transforming its optical characteristics. For instance, the addition of metal to glass has been reported for applications such as metal-doped glass laser [8,9], light amplifier [10], and optical filter consisting of metal in laminated glass [11]. Additionally, femtosecond pulse laser illumination onto the glass interior partially increases the refractivity around the focal point to apply an optical waveguide [12].

This paper proposes a new method of internal glass processing using a CW laser. This method is called CW laser backside irradiation (CW-LBI), and involves manipulating metal spheres implanted into the glass to transform its optical characteristics by exploiting the change in its thermal history around the trajectory and the metal diffusion from the metal sphere to the glass [13]. In CW-LBI, the metal sphere implantation mechanism and movement are as follows. First, a glass, metal foil, and heat insulator are clamped with a jig to form the sample. Second, a laser illuminates the sample in the direction of the glass. Third, the metal foil is heated and melts owing to the high transmittance of the glass to the laser illumination. Forth, the heated metal foil heats the glass and softens it. Finally, a metal sphere from the metal foil is implanted into the glass and moved toward the laser source during the laser illumination. Then, the movement of the metal sphere occurs at a velocity below 0.1 mm/s, which herein is termed “slow movement.” Moreover, in CW-LBI, the metal sphere absorbs the laser beam and the heated surface passes heat to the sphere and increases the temperature of the other surface. The interfacial tension between the metal and glass decreases as the temperature increases, and the resultant force between the surfaces generates a driving force [14]. When the glass matrix is a silica glass, the metal sphere moves, generating periodical stripes in a trajectory [14]. When the glass matrix is CaO-Al₂O₃-SiO₂, the CaO concentration decreases and SiO₂ concentration increases in the center of the metal sphere’s trajectory [15].

However, as for metal sphere implantation, the modification of the glass matrix has been discovered and is under study.

We found that the metal sphere moves fast at a velocity above ~10 mm/s. As compared to a conventional metal sphere (slow movement) at a velocity below 0.1 mm/s, the velocity of the metal sphere is 100 times faster. Herein, the new rapid phenomenon with a velocity above 1 mm/s is termed “fast movement.” The object velocity under light manipulation by means such as optical tweezers is limited to microns per second [2]. Thus, fast movement is a novel and interesting phenomenon, and has a colored trajectory, unlike slow movement.

If the trajectory of fast movement contains metal elements, such as iron, a trajectory with a diameter of approximately 100 µm can be applied to an electrical conduction path or an optical element. Clarifying the conditions needed for the occurrence of fast movement can lead to the development of such applications. Additionally, the amount of metal added to a trajectory can increase the breadth of the application field. Therefore, it is essential to clarify the fast movement mechanism. To the best of our knowledge, the fast movement mechanism, or the conditions for its occurrence, has not been reported so far.

In this study, the conditions for, and mechanism of, fast movement are revealed through observations and a microscopic analysis of fast movement’s trajectory. Therefore, first, the in situ observation of fast movement with a high-speed camera reveals the differences in relation to fast movement and the conditions necessary for its occurrence. Second, a cross-sectional observation of the fast movement trajectory with optical microscopy and scanning electron microscopy (SEM), as well as the elemental analysis of the trajectory, is used to investigate the characteristics of the fast movement trajectory. Third, the slow and fast movement temperatures are measured with a spectroscope to detect the emissions. This paper discusses the fast movement mechanism based on the obtained results.

2. Method

Figure 1 shows a schematic of the experimental apparatus [16]. A borosilicate glass, a metal foil, and a heat insulator were clamped by a jig to form the sample. A metal sphere was implanted by laser illumination. Here, the thickness of the borosilicate glass (Pyrex, Corning 7740, Corning Inc., NY, USA), wherein the metal sphere was implanted, and that of the heat-insulating quartz glass (Grade ES, Tosoh Corp., Tokyo, Japan) were both 5 mm. The thickness of the metal foil (stainless steel, SUS304 foil, #753173, Nilaco Corp. Tokyo, Japan) was 10 µm. The laser oscillator (RFL-C020/A/2/A, Wuhan Raycus Fiber Laser Technologies Co., Ltd., Hubei, China) emitted CW irradiation with a wavelength of 1,064 nm. The beam shape had a Gaussian distribution and diameter of 4 mm. A convex lens (NYTL-30-40PY1, Sigma Koki Co., Ltd., Saitama, Japan) with a focal length of 40 mm converged the laser beam. The laser output when the metal sphere was implanted into the glass was 12 W. The focal point was located 3 mm away from the metal foil, and the laser spot diameter was 300 µm at the metal foil. The metal sphere moved toward the laser source when laser illumination was applied to the metal sphere and after being implanted into the borosilicate glass. Then, the distance between the metal sphere and focal point of the laser beam increased and the laser spot diameter decreased. Thus, the laser power density also decreased as the metal sphere moved toward the laser source. At a certain point, the metal sphere stopped moving, owing to defocusing.

 

Fig. 1 Schematic of the experimental apparatus.

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The following operation was conducted to illuminate the laser beam onto the metal sphere with a diameter of 80 µm. First, metal spheres were placed 2 mm apart from the metal foil by laser illumination. Second, the sample was moved to a location where the beam diameter was 120 µm in motorized positioning stages. Finally, the differences in the movement of the metal sphere depending on the laser power density were observed using high-speed camera (Phantom V7.3, Vision Research Inc., NJ, USA). The shutter was located between the laser oscillator and the sample. Additionally, the timing corresponded to the moment when the laser illumination onto the metal sphere commenced.

Each phenomenon at each laser power density was observed three times to demonstrate that the phenomenon can be reproduced under the same condition. During the observation, a metal halide lamp with an emission peak of 440 nm (LS-M250 Sumita Optical Glass Inc., Saitama, Japan) illuminated the metal sphere and glass from a position opposite to the high-speed camera. A band-pass filter with a center transmission wavelength of 440 nm (10BPF10-440, Newport Corp., Irvine, CA, USA) was located between the sample and the high-speed camera to cutoff the strong light resulting from the movement of the metal sphere, that is, the light scattered by the laser illumination and the heat radiation from the heated metal sphere and glass.

As shown in Fig. 1, the emission from the metal sphere was split between the high-speed camera and a spectroscope (SR-193i, DH334T-18FCH-03, ANDOR, Belfast, Northern Ireland) by a half mirror, to detect the emission. Then, the band-pass filter between the sample and the high-speed camera was detached to ensure that the emission was not cut off.

The measurement of glass temperature near the fast metal sphere movement can clarify the metal sphere movements because the viscosity of the glass depends on temperature. Therefore, the temperature of the metal sphere with slow and fast movements was calculated to detect the emission. Moreover, the high-powered laser illumination onto the glass optical fiber caused a flash that moved toward the illumination source at approximately 1 m/s. This phenomenon is termed as fiber fuse [17].

In our experiment, the fiber fuse was observed at a high-power laser output. The temperature of the area wherein the fiber fuse occurred was calculated using the same method as that used for the movement of the metal sphere. In this experiment, the following starting times were synchronized with a delay generator: emission detection using a spectrograph, laser illumination onto the metal sphere controlled with a shutter, and observation using a high-speed camera.

The temperature of the moving metal spheres was obtained by fitting the emission detected using a spectrograph to Plank’s law, which is expressed as follows:

Here, E_bλ (W/m² μm) is the monochromatic radiation of a black body, T (K) is the temperature of an object surface, c₁ is 3.7413 × 10⁸ (W(μm)⁴/m²), and c₂ is 1.4388 × 10⁴ (μm K).

The trajectory of the fast movement was observed using a confocal microscope (LEXT OLS4000, Olympus, Tokyo, Japan) and a scanning electron microscope (JSM-6510A, JEOL Ltd., Tokyo, Japan), after polishing the sample to expose the trajectory. After the SEM observation, elemental analysis was conducted using energy-dispersive X-ray spectrometry (EDS).

3. Results and discussion

3.1 In situ observation

Figure 2 shows time-lapse images from Visualization 1 and Visualization 2, and outlines the difference in the movements of the metal spheres with different laser power densities: (1) 53 kW/cm² and (2) 62 kW/cm². After the laser illumination, the metal sphere absorbed the illumination and began to accelerate to the maximum velocity (v_Max) at a certain time, before decelerating. The slow and fast movements were defined as v_Max and were below 0.1 mm/s and over 1 mm/s, respectively. The phenomenon by which the velocity was between that of the slow and fast movements, that is, where v_Max was greater than 0.1 mm/s and less than 1 mm/s, was not observed under this condition. Figure 2(1) shows the slow movement at a laser power density I of 53 kW/cm². The metal sphere moved toward the laser source with a transparent trajectory, and v_max was 0.3 mm/s. Conversely, Fig. 2(2) shows the fast movement at a laser power density I of 62 kW/cm². The metal sphere moved toward the laser source with a black trajectory and strong emission, and v_max was 16 mm/s.

 

Fig. 2 Time-lapse images of laser-irradiated metal sphere: (1) slow movement at power density of 53 kW/cm² (Visualization 1); (2) fast movement at power density of 62 kW/cm² (Visualization 2); (1-a), (2-a) metal sphere heated by laser illumination; (1-b)-(1-d) metal sphere moving slowly in the laser source direction with a transparent trajectory and without emission; (2-b),(2-d) metal sphere moving fast toward the laser source with a black trajectory and emission; (2-d) weakened emission.

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3.2 Relation between time t and velocity v

Figure 3 shows the relation between the time t and velocity v of the metal sphere. In Fig. 3(a), the maximum velocity of the slow movement v_{slow movement max} increased to 45, 250, and 600 µm/s as the power density was increased to I = 53, 62, and 71 kW/cm², respectively. By increasing I, the time interval t from the laser illumination to the maximum velocity decreased to t = 2.0, 0.4, and 0.3 s. In Fig. 3(b), the maximum velocity of the fast movement, v_{Fast movement max}, increased to 16, 20, and 26 mm/s as the power density was increased to I = 62, 71, and 80 kW/cm², respectively. By increasing I, the time interval t from the laser illumination to the maximum velocity decreased to t = 140, 85, and 40 ms, and exhibited the same tendency as that exhibited by slow movement. From the abovementioned results, the maximum velocity of fast movement was determined to be approximately 30−600 times that of slow movement. However, with fast movement, the power density and heat input to the metal sphere was only 1.5 times as large as that with slow movement. The energy E, which is a product of t and P, in slow movement was larger than that in fast movement. The maximum ratio of slow movement to fast movement was 60.

 

Fig. 3 Relation between time t and velocity v of metal sphere: (a) slow movement and (b) fast movement.

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Figure 4 shows the relation between the laser power density I and the metal sphere’s maximum velocity v_Max in each experiment. In each trial, the maximum velocity of slow movement tended to increase as the laser power density was increased. In fast movement, the tendency was the same as that in slow movement, with a maximum value of approximately 100 mm/s. Additionally, the metal sphere exhibited the following behavior depending on I: slow movement at 44 kW/cm² ≦ I ≦ 71 kW/cm², fast movement at 53 kW/cm² ≦ I ≦ 150 kW/cm², and dissipated movement at 141 kW/cm² ≦ I ≦ 159 kW/cm². In the experimental apparatus, a high-powered laser illumination with a laser output P of 19 W (laser power density I = 159 kW/cm²) caused the following phenomena: the metal sphere moved faster than that in fast movement, strikingly dissolved in the trajectory, and then disappeared to generate a black trajectory, which is termed as dissipated movement. Dissipated movement is not discussed in this paper because our objective is to clarify the movement mechanism of the metal sphere. Additionally, considerable higher-powered laser illumination with a laser output P of 19 W and a laser spot diameter d of 13.5 μm (laser power density I = 13 MW/cm²) caused a high-intensity flash to move toward the laser source at a high speed of 1.5 m/s (Fig. 5).

 

Fig. 4 Relation between laser power density I and maximum velocity v_Max of the metal sphere in each experiment. A higher laser power density increased v_Max and caused a transition from slow movement to fast movement to full dissipation.

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Fig. 5 Snapshots of fiber fuse at power density of 13 MW/cm² with a velocity of 1.5 m/s.

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Fiber fuse, wherein a high-intensity flash moves at a high speed of 1.8 m/s, was reported by Dianov [18]. In a previous study, we also observed fiber fuse with a high-power laser beam and an experimental apparatus similar to that used in this study [19]. The fiber fuse observed in this study originated from the metal sphere, as shown in Fig. 5.

Additionally, in the same experimental trial, as I increased, this phenomenon manifested as slow movement, fast movement, and dissipated movement. However, compared to the results obtained by different trials, both slow and fast movements were observed at 53 kW/cm² ≦ I ≦ 71 kW/cm². This fluctuation seems to be caused by the following two factors. (1) Uniformity of the metal spheres causing the fluctuation in the heat input to the metal spheres, such that the movement conditions were different. (2) An optical miscorrespondence of the laser spot and the metal sphere center. The gap between the laser spot and the metal sphere center in perpendicular direction from the x-axis to the metal sphere movement could not be adjusted exactly, owing to the observation direction (Fig. 1). Thus, the occurrence of slow and fast movement conditions depended on the laser power density, and both phenomena changed within a certain power range.

Thus, during the movement, the diameter of the laser spot increased. Then, the heat input to the metal sphere decreased as the metal sphere moved under constant laser power. In fast movement, the velocity of the metal sphere decreased with the movement toward the laser source. Then, at a certain point, the bright emission disappeared and the trajectory became transparent, in the same manner as in slow movement.

Here, the slow movement occurring after the fast movement (time t = T ms) is termed as Case 1, and the slow movement at the beginning of the movement with a low laser power density (time t = 0 ms) is termed as Case 2. To investigate whether Case 1 and Case 2 are the same phenomenon, their heat inputs were compared. Figure 6 shows the laser illumination condition for the metal sphere before and after the movement. Here, R is the radius of the metal sphere, which was calculated by the metal sphere’s diameter measured from the observation snapshots, and ω(z) is the laser spot radius. Before the laser illumination, R was 40 μm and ω(z) was 60 μm. The metal sphere moved toward the laser source and absorbed the laser illumination. Moreover, R decreased and ω(z) increased as the distance of the metal sphere increased. At the point of transition from the fast movement to the slow movement, the distance from the starting point to the switching point is defined as z₀. After passing this point, the metal sphere moved slowly. The laser intensity distribution is expressed as follows [20]:

 

Fig. 6 Laser illumination condition for metal sphere before and after the movement: R is the radius of the metal sphere, ω(z) is the laser spot radius, ω(z₀) is the laser spot radius at the point of transition from the fast movement to slow movement.

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Figure 7 shows the heat input (Case 1) to the metal sphere at ω(z₀), the point of transition from the fast movement to slow movement, and a comparison with the heat input in Case 2 calculated using Eq. (3). Here, the horizontal and vertical axes represent, respectively, the laser spot diameter ω(z) (μm) and the heat input to the metal sphere from R = 0 μm to 40 μm. $P_{Case1(n)}$ is the heat input for each trial (n = 1, 2, 3) in Case 1 and $P_{Case2(n)}$ is the heat input for each experimental trial (n = 1, 2, 3) in Case 2. As shown in Fig. 7, when R = 40 μm at ω(z₀), the values are as follows:

 

Fig. 7 Heat input (Case 1) to metal spheres at ω(z₀), point of transition from fast movement to slow movement, and comparison with the heat input in Case 2. The triangle, square, and circle symbols indicate the first, second, and third experimental trials, respectively. The open and closed figure symbols indicate Case 1 and Case 2, respectively. Only the maximum heat input is indicated with open figure symbols.

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3.3 Emission spectrum analysis

During slow movement, the metal sphere had a weak white glow that could be observed in a dark-lighted room. Conversely, in fast movement with a high laser power density, the metal sphere emitted strong light, which could be observed in a lighted room. The temperature of the fast movement was assumed to be higher than that of the slow movement because the thermal radiation intensity increased with temperature. Moreover, the fiber fuse and glass plasma (Fig. 5) were observed at a higher laser power density, compared to the condition for the occurrence of fast movement.

Figure 8 shows the slow movement, fast movement, and fiber fuse spectra. The black line indicates the spectrum detected using a spectroscope. The red line indicates the curve fitting to the spectrum using Plank’s law. The laser power densities for the slow movement, fast movement, and fiber fuse were I = 5.3 × 10⁻², 0.13, and 13 MW/cm², respectively. The emissions from the fast movement and fiber fuse were similar. In previous studies, the fiber fuse temperatures were calculated by comparing the fiber fuse spectra with those of black body emission [18,21]. Thus, the temperatures were calculated by comparing Plank’s law with the normalized spectra curves. Then, the wavelength band in the range of 425 nm ≦ λ ≦ 600 nm was used, owing to the high spectroscope sensitivity and linearity. The temperatures of the slow movement, fast movement, and fiber fuse were T_Slow = 1,900 K, T_Fast = 2,900 K, and T_{Fiber fuse} = 3,600 K, respectively. Compared to the slow movement, the spectra of the detected emission and the fitting curve obtained by Plank’s law had various differences at approximately λ = 500 nm, in the same manner as in the fast movement and fiber fuse. Here, the fiber fuse (Fig. 8 (c)) and fast movement (Fig. 8 (b)) emission spectra fit each other at 520 nm ≦ λ ≦ 600 nm. Moreover, the spectra were similar and exhibited a shoulder shape at λ ≈ 500 nm.

 

Fig. 8 Spectra of (a) slow movement, (b) fast movement, and (c) fiber fuse. The black line indicates the emission observed with the spectroscope; the red line indicates the Plank’s law fitting of the measured emission. The metal sphere temperatures T were as follows: T_Slow = 1,900 K, T_Fast = 2,900 K, T_{Fiber fuse} = 3,600 K. The fast movement and fiber fuse emission spectra had similar shapes.

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3.4 Cross-sectional observation

In the previous study, although the fiber fuse spectrum indicated heat radiation, a specific peak in it was not discussed [18,21]. Thus, the fast movement emission was generated in the fiber fuse, owing to the strong emission (Figs. 2(2) and 5) and the similarity of the emission spectra. This study detected the fiber fuse emission spectrum for borosilicate glass containing silicon dioxide and boron as additive elements (Fig. 8(b)). However, the fiber fuse emission spectrum for quartz glass was detected in the previous study. Note that the additive element may cause spectral differences at λ ≈ 500 nm.

Figure 9 shows the optical and SEM micrographs for the fast movement trajectory cross-section at I = 62 kW/cm² (Fig. 1 (b)). Figure 9(d) shows the location around the metal sphere placed before laser illumination (Fig. 2(2-a)), which caused the metal sphere to move rapidly with the emission and pass through as shown in Fig. 9(c) (Fig. 2(2-b), 2(2-c)). Moreover, the continuous illumination caused the metal sphere to move continuously toward the laser source. The laser power density decreased, owing to defocusing. Near the area shown in Fig. 9(b), the emission weakened, the velocity decreased, and the metal sphere moved toward the laser source with a transparent trajectory, in the same manner as in the slow movement (Fig. 2(2-d)). Figure 9(e) shows an enlarged picture of Fig. 9(d), where the metal sphere moved rapidly with the emission. Additionally, Fig. 9(f) is a partly enlarged picture of Fig. 9(e), where many fine particles with diameters of approximately 1 µm were in the trajectory of the fast movement. Moreover, the trajectory and fine particles contained iron.

 

Fig. 9 (a-d) Optical and (e, f) SEM micrographs of fast movement trajectory; micrographs of black trajectory in fast movement.

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3.5 Mechanism of metal sphere movement

In the slow movement, the emitted light was weak for the following reason. Generally, the silicon dioxide in the glass was converted into silicon monoxide when the laser illumination (wavelength of λ = 1,064 nm) heated the glass above 2,300 K [22]. Then, the temperature of silicon monoxide increased because its absorption was high; the surrounding glass was also heated. When the thermal input surpassed the threshold, the silicon oxide thermally decomposed, became plasma, and generated fiber fuse [22].

In a previous study, the fiber fuse temperature was calculated as T = 10⁴ K [22]. During the slow movement, the temperature was estimated as 1,900 K, which is lower than the temperature needed to generate silicon oxide (T = 2,300 K). Hence, the fiber fuse did not occur and a strong emission from the plasma was not detected.

The fast movement emitted a strong light because the temperature of the glass surrounding the metal sphere in fast movement, T_Fast, was approximately 2,900 K. Here, T_Fast = 2,900 K was higher than T = 2,300 K, which is the temperature at which the fiber fuse occurred. In a previous study, the temperature of the fiber fuse depended on the laser output and ranged from 4,500 K (I = 24 MW/cm²) to 10,500 K (I = 300 MW/cm²) [18]. The laser power density in the fast movement (I_Fast = 0.13 MW/cm²) was 100 times lower compared to that observed in the previous study, namely T = 4,500 K. Moreover, T_Fast = 2,900 K was lower than that reported in the previous study. Our results revealed that, unlike the fiber fuse, the metal sphere migrated. Therefore, the required power density and temperature were different to those of the fiber fuse, and the strong emission of the fiber fuse observed in the fast movement was caused by plasma.

At high power, a metal sphere could not be implanted into the glass. Only a heated spot moved rapidly toward the laser source and increased three- and four-membered rings of the glass by rapid heating and quenching [23]. Conversely, at proper power, a metal sphere was implanted into the glass.

In this study, the CW laser (power density 62 kW/cm², laser radiation time 0.4 s) did not induce ablation processing but fast movement (Fig. 2(2-d)) because laser illumination with an interaction time of ~1 s and laser light intensity of ~10⁵ W/cm² was lower than the acquired energy for ablation. Laser intensity in this power range only melts the materials and is appropriate for welding or cladding [20].

The slow movement trajectory contained few fine metal particles. In the slow movement, the metal sphere was molten by the laser illumination because T_Slow = 1,900 K was higher than the melting point of iron, i.e., 1,800 K [24]. The interfacial tension between the molten metal sphere and the glass retained its spherical shape during the movement. Therefore, the metal did not diffuse into glass and the metal sphere generated a transparent trajectory without metal elements.

In contrast, the fast movement trajectory contained fine metal particles because the metal sphere in the fast movement was molten by laser illumination, owing to T_Fast = 2,900 K being higher than the melting point of SUS304 [24]. The liquid metal sphere was surrounded by glass plasma in the front surface and by softened glass in the rear surface. Some confusion arose in the area of the glass plasma and liquid iron, as will be discussed below.

First, in the front area of the metal sphere, the glass plasma was fused with liquid iron. The compound flew out to the trajectory through the migration of the metal sphere, and cooled when the metal sphere moved away. Thus, the iron particles precipitated in the trajectory. In the rear area of the metal sphere, the laser was not absorbed by the glass because it was blocked by the metal sphere. Hence, the glass was not plasma but had rather melted. The interfacial tension at the rear side was weak because the temperature was higher than that in the slow movement.

Figure 10 shows the mechanical modeling of the movement considering the results of the in situ and cross-sectional observations. For comparison, the mechanical modeling of the slow movement is also shown in the figure, wherein the force against the metal sphere at maximum velocity can be seen. Moreover, the driving force to the rear surface (F_{Rear int}) was equal to the resistance on the front surface (F_Vis, F_{Front int}) because the acceleration was zero. The surface absorbing the laser illumination of the metal sphere and the opposite surface are termed as the front surface and rear surface, respectively. F_{Rear int} and F_{Front int} are the interfacial tensions between the metal sphere and the glass on the rear surface and front surface, respectively. The direction of the metal sphere movement is defined as positive direction. F_{Rear int} and F_{Front int} decreased as the temperatures of the rear surface (T_Rear) and front surface (T_Front) increased, respectively. F_Vis is the viscous resistance applied to the metal sphere by the glass. Additionally, the slow flow around the sphere can be considered as a Stokes flow and expressed as follows:

 

Fig. 10 Mechanism of metal sphere movements: (a) slow movement; (b) fast movement.

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The mechanism of the slow movement in the glass is outlined below. In the slow movement, the metal sphere moved because the interfacial tension decreased as the temperature increased. Specifically, the temperature of the front surface was higher than that of the rear surface because the front area was heated by laser illumination. However, the rear area was heated by the heat conduction from the heated front area. Thus, because the interfacial tension on the front surface was lower than that on the rear surface, the resultant force caused the metal sphere to move toward the laser source [14]. When the velocity of the metal sphere reached a maximum value, the interfacial force on the rear surface became equal to the interfacial force and the viscous resistance on the front surface became F_{Rear int} = F_{Front int} + F_Vis. The maximum velocity was calculated using Eq. (4). Here, the glass viscosity μ_Slow depended on the glass temperature. When T_Slow = 1,900 K, μ_Slow was estimated as 46.6 Pa·s, which is lower than the viscosity of the glass softening point μ_{Softening point} = 10^7.5 dPa·s [26]. Therefore, the metal sphere provided sufficient heating to move the surrounding glass. Additionally, when the density of the surrounding ρ was 2.2 × 10³ kg/m³ [24] and the velocity of the metal sphere v was 45 μm/s, Re was 1.7 × 10⁻⁷ < 1. Therefore, the convection around the metal sphere was considered as a Stokes flow. In the slow movement, the resistance on the metal sphere $F_{vis}$ was 1.6 μN, as determined using Eq. (4).

Figure 10 (b) shows the mechanical modeling of the fast movement. Under the illumination with a high laser power density, the metal sphere moved rapidly with strong emission and generated a black trajectory containing iron. The maximum velocity v_Max was 3 mm/s $\le$ v_Max $\le$ 88 mm/s. The front surface vanished because the glass absorbed the laser illumination and became plasma, and the interfacial tension on the front surface also vanished. Consequently, when the velocity of the metal sphere reached a maximum value, the interfacial force on the rear surface became equal to the viscous resistance on the front surface: F_{Rear int} = F_Vis. The maximum velocity was calculated using Eq. (4).

Compared to the slow movement, in the fast movement, the viscous resistance on the front surface vanished and the viscosity significantly decreased. Thus, the velocity v in the fast movement had to increase when the viscous resistance on the front surface F_Vis was equal to the interfacial tension on the rear surface F_{Rear int}. Hence, the maximum velocity in the fast movement was higher than that in the slow movement. When T_Fast = 2,900 K, the glass viscosity in the fast movement μ_Fast was estimated as 1.37 Pa·s. Therefore, the metal sphere provided sufficient heat to move the surrounding glass. Additionally, when the density of the surrounding ρ was 2.2 × 10³ kg/m³ [24] and the velocity of the metal sphere v was 56 mm/s, Re was 7.3 × 10⁻³ < 1. Therefore, the convection around the metal sphere was considered as Stokes flow. In the fast movement, the resistance on the metal sphere $F_{vis}$ was 58 μN, as determined by Eq. (4).

The resistance in the fast movement was 36 times higher than that in the slow movement for reasons that will be discussed below. Equation (4) assumes that the viscous fluid flow along the sphere generates a certain amount of shear force. However, in the fast movement, the shear force was lower than the value calculated using Eq. (4) because a certain part of the metal sphere was plasma without a shear force. Thus, the resistance on the metal sphere of the actual situation was lower than that assumed.

Finally, the objective of this study was to clarify the slow and fast movement mechanisms. Thus, the phenomena occurring in the critical state between the slow and fast movements with a medium laser power density should be investigated in the future.

4. Conclusion

In this study, the movement mechanism of metal spheres was clarified through the in situ observation of the trajectory generated by a fast-moving metal sphere, as well as the emission spectra analysis conducted for the slow and fast movements of the metal spheres. The following results were obtained (I refers to the laser power density). The phenomenon with regard to the metal spheres depended on I: slow movement at 44 kW/cm² ≦ I ≦ 71 kW/cm², fast movement at 53 kW/cm² ≦ I ≦ 150 kW/cm², and dissipated movement at 141 kW/cm² ≦ I ≦ 159 kW/cm². The metal sphere moved slowly at low I at the starting point and at low I after the fast movement. Fine particles with a diameter of approximately 1 µm were observed in the black trajectory of the fast movement, and mainly contained iron. The temperature of the metal sphere T was as follows: T_Slow = 1,900 K in the slow movement, T_Fast = 2,900 K in the fast movement, and T_{Fiber fuse} = 3,600 K in the fiber fuse. The emission spectra shapes of the fast movement and fiber fuse were similar. At a low temperature with low laser power density, the metal sphere moved slowly without plasma. Conversely, at high temperature with high laser power density, the metal sphere moved fast with plasma.