1. Introduction

Optical tweezers [1–3] that employ the radiometric forces originating from the transfer of photon momenta to particles are considered to be indispensable tools in biology [4–6], physical chemistry [7,8], and condensed matter physics [9,10]. Although optical tweezers work well for majority of the transparent particles, the absorbing particles experience a considerably high absorption force that can be used to often destroy the stable optical traps. Photophoretic force [11] is used as an alternative mechanism for trapping the absorbing particles. Extensive study related to the absorbing particles that have been suspended in gas media has laid the groundwork for handling absorbing particles, for example, by trapping [12–14], manipulating [15–17], rotating [18,19], and the recent synergistic manipulation of optical and photophoretic forces [20], thereby providing extensive possibilities in optical driving and energy conversion applications. Photophoretic forces are generally neglected while manipulating the objects in liquids, hindering the manipulation and investigation of the accessible absorbing particles in liquid environments. The major difficulty that is associated with the utilization of photophoretic forces in liquids for trapping the strongly absorbing particles is that considerable absorption can be observed on the illuminated side; a positive photophoretic force is usually induced [21], thereby pushing away the absorbing particles from the high-intensity region of the laser beam (laser source). Apart from the trapping of weakly (partly) absorbing particles in liquid media [22,23], which is similar to that observed in the radiation pressure trapping of the so-called low-index transparent particles [24,25], to the best of our knowledge, this is the first report of the optical trapping of the strongly absorbing particles in a liquid medium on the basis of photophoresis.

2. Principle of Δα photophoretic force

We placed a light transparent microparticle in a liquid medium that was illuminated by laser radiation, as depicted in Fig. 1(a). The scattering force that originates from the light radiation pressure effect is sufficient to push the transparent microparticle far away from the laser source. At this point, the temperature gradient of the particle and the surrounding liquid environment is observed to be minimal and can be neglected. When the transparent microparticle is replaced with a light-absorbing (completely opaque) microparticle (see Fig. 1(b)), compared to the radiation pressure introduced by the transparent microparticle, the radiation pressure introduced by the light-absorbing (completely opaque) microparticle is drastically reduced, which because light absorption by particles results in the appearance of a strong liquid–dynamic force that can be referred to as the photophoretic force.

 

Fig. 1 (a) Schematic of an optical radiation pressure force that is exerted on a transparent particle; (b) schematic of a ΔT-type photophoretic force that is exerted on an absorbing particle; (c) schematic of a thermophoretic force that is exerted on an absorbing particle; (d) schematic of a Δα-type photophoretic force that is exerted on an absorbing particle; (e) schematic of the momentum exchange between the microparticle surface and glycerol molecules, where T_i denotes the temperature of the liquid molecules that are incident on the particle surface (before momentum exchange), T_s denotes the temperature of the particle surface, and T_r denotes the temperature emitted by the particle after the momentum exchange is completed.

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Photophoretic force is used as an alternative mechanism for trapping the absorbing particles. When the particle absorbs the laser power, the laser energy is converted into the thermal energy of the particle. Further, the thermal energy is expressed as the temperature gradient distribution in the particle. The photophoretic force is collinear with the heat source distribution inside the particle originating from the absorption of light [21,26–29]. The magnitude of the photophoretic force is in the scale of μN and is observed to be normally 10⁶ times larger than that of the radiation pressure force [17]. In case of the strongly absorbing (black and opaque) particles, considerable absorption can be observed on the illuminated side; therefore, a positive photophoretic force is usually induced, pushing the absorbing particle away from the high-intensity region of the laser beam. Usually, thermal diffusion occurs in a liquid environment, where a heated microparticle is located. This indicates that the thermal energy that is obtained by converting the laser power is thermally diffused by the particles to the surrounding liquid environment, and the temperature gradient distribution is generated in the liquid near the particle (see Fig. 1(c)). Thus, thermophoresis is induced by the thermal gradients. The thermophoretic force is dependent on the temperature gradient of the surrounding liquid molecules. The liquid molecules that originate from the hot side possess a greater velocity when compared to that possessed by those on the cool side. Therefore, the considerable momentum that is received from the hot side pushes the thermophoretic force to the cool side in the direction of decreasing temperature, thereby leading to the overall movement of the absorbing particle to the cool side. Thus, the particle is provided with an unbalanced impulse that is directed from the hot side. Further, the magnitude of the thermophoretic force remains the same as that of the photophoretic force. In general, after a particle absorbs laser power, the particle itself and the surrounding liquid will both produce a thermal gradient, indicating that both thermophoresis and photophoresis play useful roles in the movement of particles. In addition to heat exchange, momentum exchange is also observed to occur between the particle and the surrounding liquid molecules (see Figs. 1(d) and 1(e)). This momentum exchange can be characterized using the thermal accommodation coefficient α [30] that satisfies the condition of T_r-T_i = α(T_s-T_i). T_i denotes the temperature of the liquid molecules that are incident on the particle surface (before momentum exchange), T_s denotes the temperature of the particle surface, and T_r denotes the temperature emitted by the particle after the momentum exchange is completed. The particle motion that is induced by this momentum exchange can be attributed to photophoresis and can be referred to as Δα-type photophoresis. Correspondingly, the photophoresis based on thermal gradient (heat exchange) can be referred to as ΔT-type photophoresis. Further, Δα-type photophoresis is induced by the difference between the momentum exchange of the liquid and the particle surface, and it is induced in the direction from the higher to lower accommodation coefficient. Therefore, it is necessary to construct physical conditions for ensuring that the negative Δα-type photophoresis dominates the particle motion for constructing a negative optical force that attracts an absorbing particle. Further, there are situations in which the surrounding medium not only assists in the creation of the desired motion but also serves as the essential element for achieving negative mobility. Indeed, novel manifestations of light–matter interactions can be observed when the environment is designed to exhibit certain exotic properties. There are different methodologies for modifying the properties of the environment to generate a backflow of particulate matter. Further, we should obtain suitable environment media that can satisfy the conditions that the momentum diffusivity between the liquid molecules and the absorbing particle dominates the temperature variations during heat transfer when compared with thermal diffusivity. The thermodynamic indicators of a liquid should satisfy the conditions that the liquid should have almost no convection or Laminar flow in the vicinity of the heated absorbing particle (Pe << 1 and Re << 1); when compared with thermal diffusivity, the momentum diffusivity between the glycerol molecules and the absorbing particle dominates the temperature variations during heat transfer (Pr >> 1). Further, the thermodynamic indicators of liquid glycerol (see Table 1) satisfy the aforementioned conditions, and we employ pure liquid glycerol for constructing negative Δα-type photophoresis to attract the absorbing particle in the liquid.

Table 1. Basic parameters of liquid glycerol

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3. Trap of absorbing particles

We perform contactless three-dimensional optical trapping on the basis of the thermodynamic properties of pure liquid-phase glycerol and adjust the axial shift of a strongly absorbing particle (see Figs. 2(a), 2(d), and Visualization 1) using the Δα-type photophoresis introduced by a divergent Gaussian beam, without the requirement for specially designed optical bottle structures [16,17,31] or gravity-assisted balance [14,32–34]. Furthermore, we perform three-dimensional trapping and adjust the axial shift of multiple absorbing particles by replacing the Gaussian beam with a Bessel-like beam because of the non-diffraction and self-healing properties of the Bessel-like beam (see Figs. 2(b), 2(e), and Visualization 2). Further, the Bessel-like beam also attracts the absorbing particles over long distances, and the effective tracing range can reach ~1 mm (see Visualization 3). The strongly absorbing particle (AP) that is used in the experiment is a solid silica-doped carbon black sphere, whose mass and volume ratios are 15% and 10%, respectively. The diameter of the microsphere is 6 μm. It is completely opaque in case of the 980-nm laser. The complex refractive index of the AP is m = 1.908 + 0.519i. The thermal conductivity of the AP is k_p = 3.758 [W/(m·K)]. We obtain the complex refractive index and thermal conductivity of AP from the manufacturer.

 

Fig. 2 (a) Schematic of a single absorbing particle (AP) trap based on a normal Gaussian beam; (b) schematic of a multiple AP trap based on a Bessel-like beam; (c) schematic of the single AP trap based on the normal Gaussian beam; (d) experimental image of a single absorbing particle (AP) trap based on a normal Gaussian beam; (e) experimental image of the multiple AP trap based on the Bessel-like beam.

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For the absorbing particle (AP) in the experiment, the force of gravity (F_g) is 2.52 pN along the y-axis (see Fig. 2(c)). The buoyancy (F_b) is 1.40 pN along the y-axis. Because of the large viscous coefficient (η = 1.5 Pa∙s at T = 300 K) of glycerol, the AP is in an equilibrium state along the vertical direction, satisfying F_g + F_b + F_η = 0, implying that the direction of F_η is along the y-axis. Further, when the spherical AP is trapped inside the beam, the radiation pressure and the photophoretic force are directed exactly parallel to the beam axis because of the perfect axial symmetry of the problem. When the incident laser power is 1 mW, the magnitude of the radiation pressure (F_rp) is in the scale of pN, and the direction is along the z-axis. The magnitude of the photophoretic force, which is normally 10⁶ times larger than that of F_rp [17], is in the scale of μN along the z-axis. Two types of photophoretic forces exist: F_∆T, resulting from a temperature gradient, and F_∆α, resulting from different thermal accommodation coefficients [30,35]. When compared with F_g, F_b, F_η, and F_rp, the photophoretic forces (F_∆α and F_∆T) that describe the momentum transfer between the AP and the surrounding liquid dominate the trap and axial shift manipulation and expose the mechanism. In case of strongly absorbing (black and opaque) particles, absorption occurs on the illuminated side; therefore, positive F_∆T is usually induced (along the z-axis direction), thereby pushing the APs away from the high-intensity region of the laser beam and precluding the optical trapping and manipulation with beams that possess a Gaussian intensity profile. In our experiment, the thermal conductivity of the AP is observed to be large (k_p = 3.758 W/(m·K)), and the temperature gradient approaches 0, leading to a nearly disappearing F_∆T. However, F_∆α may dominate even when the particle is evenly heated (F_∆T→0). F_∆α is induced by the difference in heat exchange between the liquid and the particle surface, and it points in the direction of higher to lower accommodation coefficients, which depends on the particle structure and surface properties. Furthermore, the thermodynamic properties of pure glycerol dominate F_∆α. The thermodynamic indicators (Pe, Re, and Pr) of liquid glycerol (see Table 1) denote that pure glycerol has almost no convection or Laminar flow in the vicinity of the heated AP; when compared with thermal diffusivity, the momentum diffusivity between the glycerol molecules and the AP dominates the temperature changes during heat transfer. Subsequently, there is almost no convection or Laminar flow in the vicinity of the heated AP, and the thermophoretic force F_∆Th is nearly vanishing. Therefore, in pure liquid glycerol, F_Δα dominates the motion of the AP based on the momentum accommodation.

F_Δα can be described as [36]

where σ denotes the scattering cross section of the AP and where I is determined by the incident laser irradiation. For a certain liquid (glycerol), the direction of F_Δα is determined by Δα/$\overline{\alpha }$ (Δα/$\overline{\alpha }$~J₁/($\overline{T}-T_i$), see [36]). Even though J₁ < 0 (where F_∆T provides repulsion), F_Δα will still provide an attractive force if $\overline{T}-T_i$ > 0. J₁ represents the weighted integration of the heat source distribution over the particle volume; here, J₁ = −1/2. Thus, F_∆α is determined by the difference between the average temperature of the particle surface ($\overline{T}$) and the initial temperature of the glycerol molecule near the AP (T_i). The two temperatures are related to the heat absorbed by the particle, which is essentially related to the power density of the laser to which the particle is exposed. Therefore, we can investigate the light field distribution of the output beam for assessing the laser power density.

Because of its non-diffractive and self-healing properties, a Bessel-like beam can trap and manipulate multiple APs simultaneously and achieve long-distance attraction as a “tractor beam” [37–40]. We spliced a single mode fiber (SMF) with an arbitrary length (~1 m) and a step index multimode fiber (MMF) with a given length (1138 μm) coaxially for exciting the all-fiber Bessel-like beam (see Fig. 2(b)). Further, the profile of the light field distributions of the output beam shows the approximate non-diffraction characteristics of a Bessel-like beam. Because of its self-healing characteristics, the F_∆α introduced by the Bessel-like beam can simultaneously trap six APs (see Fig. 2(e) and Visualization 2). According to the experimental results, the interval between each AP is observed to be different. This can be attributed to the different reform distances l_sm [4] introduced by different propagating mode groups. In our experiment, according to the structural parameters of SMF and MMF, the output light field of the Bessel-like beam can be regarded as a beam superposed by 24 LP_0m mode groups (m = 1, 2…24). Further, the effective reform distance (l_s) can be obtained by superposing 24 reform distances with the corresponding coefficients (see Fig. 3(f)). Because l_sm is related to the beam propagating parameters, such as the propagating length, before it is blocked, l_s will change along the position in which the AP is trapped. Specifically, when the first AP is trapped by F_Δα at a position z₁, the difference between the temperature field near the illuminated part and the dark part of the AP satisfies ${\sum }_{1\mu }\left[J_1/\left({\overline{T}}_{1\mu s}-T_{1\mu i}\right)\right]+{\sum }_{1\nu }\left[J_1/\left({\overline{T}}_{1\nu s}-T_{1\nu i}\right)\right]=0$, where “1μ” represents the illuminated part and “1ν” represents the dark part of the first AP. Thus, the F_Δα that is exerted on the first AP approaches zero, thereby achieving stable trapping of the first AP. Naturally, the Bessel-like beam is blocked by the first trapped AP at position z₁, and it must propagate a distance, l_1s ($l_{1s}={\Sigma }_{m=0}^{23}\left(c_{1m}\cdot l_{sm}\right)$), to reform the light field distribution so that it becomes similar to that at position z₁. When the light field distribution at position z₂ (z₂ = z₁ + l_1s) again satisfies the condition${\sum }_{2\mu }\left[J_1/\left({\overline{T}}_{2\mu s}-T_{2\mu i}\right)\right]+{\sum }_{2\nu }\left[J_1/\left({\overline{T}}_{2\nu s}-T_{2\nu i}\right)\right]=0$, where “2μ” represents the illuminated part and “2ν” represents the dark part of the second AP; further, F_Δα will trap the second AP at position z₂, thereby achieving the simultaneous trapping of two APs. Consequently, the force analyses for the remaining particles can be performed in a similar manner. Briefly, the Bessel-like beam must propagate a distance z_{n + 1} (z_{n + 1} = z_n + l_ns) to reform the light field distribution for trapping the next particle.

4. F_Δα measurement and analysis

We tested and verified the F_Δα that is exerted on the AP trapped by the Gaussian beam (see Figs. 3(a) and 3(b)) and the first two APs that are trapped by the Bessel-like beam (see Figs. 3(c) and 3(d)) using the fluorescence dye method [41]. The temperature measurement and forces calculated methods are provided in [42]. The calculated results for the axial (z-axis) and transverse (x-axis) directional F_Δα on the APs are consistent with the experimental results. When the incident laser powers of the Gaussian beam are P_G1 = 1.8 mW, P_G2 = 0.9 mW, P_G3 = 0.6 mW, P_G4 = 0.3 mW, and P_G5 = 0.1 mW, the corresponding axial trapping positions of the APs are z₁ = 3.2 μm, z₂ = 5.1 μm, z₃ = 6.9 μm, z₄ = 8.2 μm, and z₅ = 11.4 μm, respectively. The axial trapping position of the AP is inversely proportional to the incident laser power. The corresponding transverse trapping positions of the APs are x₁ = x₂ = x₃ = x₄ = x₅ = 0 μm. The value of the incident laser power does not affect the transverse trapping position, and the transverse F_Δα causes the AP to be in equilibrium along the main axis of the fiber, thereby keeping the radial (transverse) F_Δα = 0.

 

Fig. 3 (a) Simulated and experimental results of the axial Δα photophoretic forces introduced by the Gaussian beam; (b) simulated and experimental results of the transverse Δα photophoretic forces introduced by the Gaussian beam; (c) simulated and experimental results of the axial Δα photophoretic forces introduced by the Bessel-like beam; (d) simulated and experimental results of the transverse Δα photophoretic forces introduced by the Bessel-like beam. In Fig. 2(d), the results on the left side provide the transverse Δα photophoretic forces that are exerted on the first trapped absorbing particle (AP), while the results on the right side provide the transverse Δα photophoretic forces that are exerted on the second trapped AP. (e) Schematic diagram to show the procedure we measure the temperature near the particle; T_inf is the room temperature 298K (25°C). (f) calculated results of the self-healing light field distribution of the Bessel beam.

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For the two particles trapped by the Bessel-like beam, the calculated results for the axial (z-axis) and transverse (x-axis) directional F_Δα on the APs are also consistent with the experimental results. The incident laser powers of the Bessel-like beam are P_B1 = 48.7 mW, P_B2 = 47.6 mW, P_B3 = 44.5 mW, P_B4 = 42.1 mW, and P_B5 = 40.6 mW, while the corresponding axial trapping positions of the first AP are z₁₁ = 3 μm, z₁₂ = 7.8 μm, z₁₃ = 11.9 μm, z₁₄ = 15.4 μm, and z₁₅ = 18.9 μm, respectively, and the corresponding axial trapping positions of the second AP are z₂₁ = 47.3 μm, z₂₂ = 58.8 μm, z₂₃ = 68.5 μm, z₂₄ = 79.8 μm, and z₂₅ = 90.3 μm, respectively. Further, the axial trapping position for each AP is inversely proportional to the incident laser power. The corresponding transverse trapping positions for the two APs are 0 μm. The transverse F_Δα causes the AP to be in the equilibrium state along the main axis of the fiber.

Figures 3(a)–3(d) indicate that the axial positions of the trapped APs can be adjusted by varying the incident laser power. The relations between the incident laser power density exerted on the AP and the position of the trapped AP introduced by the Gaussian beam and the Bessel-like beam are presented in Figs. 4(a) and 4(b), respectively. The particle motion can be introduced using the Gaussian beam as follows. When the incident laser power is 1.8 mW, the optical fiber probe can trap one AP at the 0-μm position. Further, when we slowly decrease the incident laser power from 1.8 to 0.1 mW, the trapping position of the AP correspondingly increases from 0 to 11.4 μm. The experimental results indicate that the controllable and repeatable axial shifting range ranges from 0 to 11.4 μm. Based on the experimental results, the trapping position (z) of different APs is observed to be inversely proportional to the incident laser power density (see Fig. 4(c), the effective incident laser power exerted on the scattering cross section (σ) of the AP).

 

Fig. 4 (a) Experimental results of the relation between the incident power and the trapping position for an absorbing particle (AP) introduced by the Gaussian beam; (b) experimental results of the relation between the incident power and trapping positions for multiple APs introduced by the Bessel-like beam; (c) calculated results for laser power density exerted on the AP introduced by the Gaussian beam; (d) calculated results for laser power density exerted on an AP introduced by the Bessel-like beam. S.R. denotes the simulated results, while E.R. denotes the experimental results; (e) the arrangement of 7 APs in the shape of letter “H”; (f) the arrangement of 8 APs in the shape of letter “E”; (g) the arrangement of 7 APs in the shape of letter “U.”

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The particle motion can be introduced using the Bessel-like beam as follows. When the incident laser power is 30 mW, the optical fiber probe can trap six APs, and the effective trapping range can become as long as 1000 μm. Further, the trap is non-invasive (even the first AP does not contact the fiber tip). Thereafter, the trapped APs move toward the fiber tip when the incident laser power is increased. When the incident laser power becomes 70 mW, the APs will move toward the fiber tip, and the distance between each AP becomes the minimum. Further, the effective trapping range shrinks to ~550 μm. Subsequently, when the incident laser power is slowly decreased from 70 to 0 mW, the trapped APs retreat from the fiber tip. According to the experimental results presented in Fig. 3b, the trajectories of the trapped APs are inversely proportional to the effective incident laser power density, and the relation between the incident laser power density and the trapping positions of the APs can be expressed as $Z_n\left(z\right)=P_{in}{\left[{\left(1-q\right)}^{n}f\left(z_1\right)f\left(z_2-z_1\right)\mathrm{...}{f}_n\left(z_n-z_{n-1}\right)\right]}^{-1}+z_n$, where n = 1, 2, 3…5. In this equation, f(z) denotes the effective power density (see Fig. 4(d)); further, q denotes the energy absorbed by one AP; $z_n$ is determined using the self-healing propagating distance (l_s) of the Bessel-like beam.

F_∆α is determined using the power density that is exerted on the AP. By considering that one AP is stably trapped at position z₁ when the power density is P₁, the power density exerted on the AP will increase when we suddenly increase the incident laser power; further, the AP will move along the direction in which the power density P₁ remains unchanged. According to the results of the power density distribution introduced by the Gaussian beam presented in Fig. 4(c), the power density increases in the range of z = 0 to z = 12 μm. In this range, when the incident laser power suddenly increases, the trapped AP will move toward the fiber tip for maintaining the power density at the same level, indicating that an increase in laser power causes a blue shift in the trapped AP. Similarly, according to the results of the power density distribution introduced by the Bessel-like beam presented in Fig. 4(d), the power density increases in the range of z = 0–1000 μm. In this range, when the incident laser power suddenly increases, the trapped APs will move toward the fiber tip for maintaining the power densities at the same levels, indicating that an increase in laser power causes a blue shift in the trapped APs, with different shift lengths for each AP. The farther the distance of the AP from the fiber tip, the smaller will be the power density exerted on it; therefore, the particle shift will increase. Essentially, the trajectory of each AP follows the contour line of laser power density.

Based on the controllable adjustment functionality of the positions of trapped particles, we integrate three SMF–MMF probes in a linear shape for producing a micro “typewriter,” which may array multiple APs into specific shapes in a two-dimensional space, such as the letters “H,” “E,” and “U.” In addition, we may integrate multiple probes in a square-shape, which may array multiple APs in a three-dimensional space to produce a micro three-dimensional typewriter.

5. Conclusion

The manipulation of the APs in liquid media using laser beams, as reported in this study, provides diverse practical applications. In particular, the simple Gaussian or Bessel-like beam optical pipeline allows this manipulation method to be extensively applied in several research fields. This method is suited for an extensive range of light-absorbing materials; thus, it can be applied in the experimental studies of various light-APs, including large (several microns in diameter), heavy (solid filled), or opaque (J₁ = −1/2), such as metal particles [43–45] and graphite pieces [46,47], and other novel light-absorbing materials. Tailoring the trapping and manipulation of APs in liquid environments will open various avenues for investigating special but unexplored characteristics.