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Abstract
We theoretically and experimentally study the optical trapping of two different microparticles by a double-tapered optical fiber probe (DOFP) which is fabricated by the interfacial etching method. A SiO2 microsphere and a yeast, or two SiO2 microspheres with different diameters, are trapped. We calculate and measure the trapping forces on the two microparticles, discuss the impacts of the geometrical size and refractive index on the trapping forces. Both the theoretical calculation and experimental measurements indicate that if the two particles have the same refractive index, the larger the second particle is, the larger the trapping force is. Whereas, if the two particles have the same geometrical size, the smaller the refractive index is, the lager trapping force is. Trapping and manipulation of different multiple microparticles by a DOFP enhance the application of optical tweezers, especially in biomedical engineering and material science.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical trapping of dielectric particles with a single highly focused laser beam was reported by Ashkin in $1986$ [1], which is known as optical tweezers and have been used in many fields such as atomic physics [2,3], biomedical science [4–6], materials science [7,8], neuroscience [9] and so on. The conventional optical tweezers based on high numerical aperture objectives have some limitations, such as short work distance, difficult portability, and high cost. Conversely, the optical fiber tweezers (OFTs) have separative trapping light and observing light which make them easy to manipulate microparticles in complex environments. The use of optical fibers significantly decreases the cost of the trapping system and miniaturizes the system. The OFTs with two optical fibers were presented first in $1993$[10], later a single lensed fiber realized the $3D$ trapping and manipulation of microparticles [11]. Since then, the optical fiber tips with various geometries and structures have been used in single optical fiber tweezers to realize different trapping [12–15]. These single optical fiber tweezers have more flexibility.
On the other hand, optical trapping and manipulating multiple microparticles are required for cell biological investigation and diagnosis of disease [16], cell-cell interaction and communication [17,18] and so on. A tapered single mode fiber (SMF) probe fabricated by chemical etching was used to trap a chain of polystyrene microspheres with the aid of optical binding [19]. The various multicore-fibers with tip micro-processing were applied to simultaneously trap multiple microparticles [20,21]. In order to obtain multiple contactless traps by a single fiber probe, the complex light sources and nanofabricated fiber tips (probes) were applied. For example, one end of a $1310$ nm SMF was offset spliced with a $980$ nm SMF, which was connected to a laser source. The $LP_{11}$ (high) mode was excited in the SMF, together with the $LP_{01}$ (fundamental) mode were used as the trapping light. The other end of the $1310$ nm SMF was first chemically etched to a cone and then ground to a flat end, the resulted fiber tip was used as the optical probe. Based on mode-division multiplexing technology, two yeasts were simultaneously trapped in the axial direction [22]. Offset splicing a few-mode fiber with a step-index multimode fiber (MMF) also can obtain the $LP_{01}$ and $LP_{11}$ modes as trapping light. A high refractive index microsphere was stuck to the flat end of the step-index MMF to form a fiber probe. The OFT consisted of this trapping light and fiber probe can trap and arrange multiple E. coli [23]. A Bessel beam exicted by coaxially splicing a SMF and a step-index MMF was used as trapping light; the MMF tip was first ground and polished to a cone and then was made as a semi-ellipsoid shape by a discharge fusion molding procedure. This trapping light and fiber probe can simultaneously trap three yeasts [24]. In these works, the trapped multiple particles have the same geometry, size and refractive index. However, the simultaneous trapping and manipulation of different microparticles are proposed in material science and biomedical engineering. Recently, a Bessel beam excited by coaxially splicing a step-index MMF and a SMF was used as trapping light, the fiber tip stuck with a high refractive index microsphere was used as the fiber probe, and triethylene glycol (TEG) was used as the background solution. The simultaneous trapping of a polystyrene sphere and a yeast was reported [25]. In this paper, we focus on the simultaneous trapping of different microparticles by a single optical fiber tweezer with a fundamental Gaussian beam, a chemical etching fiber tip and normal sample solution.
We theoretically and experimentally study the trapping of two different microparticles by a double-tapered optical fiber probe (DOFP). The two microparticles are different in geometrical sizes or refractive indexes, such as a SiO$_2$ microsphere and a yeast, or two SiO$_2$ microspheres with different diameters. The light source is a fundamental Gaussian beam, the DOFP is fabricated by the interfacial etching method [26,27] with a $980$ nm SMF and the sample solution is water. We simulate and analyze the output fields of a DOFP by finite element method, calculate the optical forces on the two microparticles and discuss the dynamics of these two microparticles in the output fields. We observe the optical trapping of two different microparticles by the fabricated DOFP and measure the trapping forces on the microparticles. This paper is organized as follows. The outgoing fields of a DOFP and the forces on the trapped microparticles are simulated and analyzed in Sec. 2. The experiments of trapping two different microparticles by the fabricated DOFP are discussed in Sec. 3. The conclusion is given in Sec. 4.
2. Simulation the outgoing field of DOFP
The schematic diagram of a double-tapered optical fiber tip (DOFT) is shown in Fig. 1, $\theta _{1}$ denotes the first tip angle and $\theta _{2}$ the second tip angle. We computed the outgoing field of a DOFT by the finite element method with the software COMSOL and plotted them in Fig. 2(a). In our computation, the refractive index of the fiber core $n_{core}$ is $1.46$; the core diameter is $3.6\;\mathrm{\mu}$m; $\theta _{1} = 22^{\circ }$ and $\theta _{2} = 55^{\circ }$. The light source is a Gaussian beam with a wavelength of $980$ nm and the input power is $1$ W. These data are all taken according to our tapping experiments (See Sec. 3). From Fig. 2(a), we can see that the output focus is right at the fiber tip.
Fig. 1. schematic diagram of a double-tapered optical fiber tip. $\theta _{1}$ denotes the first tip angle and $\theta _{2}$ the second tip angle. The core diameter is $3.6\;\mathrm{\mu}$m, the cladding diameter is $125\;\mathrm{\mu}$m, the refractive index of the fiber core $n_{core}$ is $1.46$.
Fig. 2. (a) Outgoing field of a DOFT. (b) Outgoing field of a DOFT with a trapped SiO$_2$ sphere. $n_{core} = 1.46$; the core diameter is $3.6 \;\mathrm{\mu}$m; the cladding diameter is $125 \;\mathrm{\mu}$m; $\theta _{1} = 22^{\circ }$ and $\theta _{2} = 55^{\circ }$. The light source is a Gaussian beam with a wavelength of $980$nm and the input power is $1$ W. The refractive index of the SiO$_2$ sphere is $n_1 = 1.45$ and the sphere diameter is $3\;\mathrm{\mu}$m.
If place a SiO$_2$ sphere with the refractive index $n_1 = 1.45$ and a diameter of $3\;\mathrm{\mu}$m near this DOFT, the optical radiation force on the sphere can be calculated by the Maxwell stress tensor integrals:
Fig. 3. (a) Axial forces on a $3\;\mathrm{\mu}$m SiO$_2$ sphere. $Z$ denotes the distance between the fiber tip and the sphere center. The origin is taken at the fiber tip. The dotted line denotes the critical position where the axial force is zero and the left region of this dotted line is the trapping region. The inset: a $3\;\mathrm{\mu}$m SiO$_2$ sphere is trapped at a DOFT. (b) Axial forces on the second microsphere. $Z$ denotes the distance between the fiber tip and the center of the second sphere. The first one is a $3\;\mathrm{\mu}$m SiO$_2$ sphere. The second one is a $2\;\mathrm{\mu}$m SiO$_2$ sphere (black solid line), or a $3\;\mathrm{\mu}$m SiO$_2$ sphere (red dashed line), or a $4\;\mathrm{\mu}$m SiO$_2$ sphere (green solid line), or a $4\;\mathrm{\mu}$m yeast (blue dot-dashed line), respectively. The inset: two microspheres are trapped at a DOFT.
The origin is taken at the fiber tip and the sphere center is on the fiber axis. $Z$ denotes the distance between the fiber tip and the sphere center. The negative forces point to the fiber tip; the positive forces are in $z$ direction. It can be seen from Fig. 3(a) that when $Z \ge 7\;\mathrm{\mu}$m, the SiO$_2$ sphere will be pushed away; when $Z < 7\;\mathrm{\mu}$m, the sphere will be attracted to the fiber tip. The black dotted line in Fig. 3(a) denotes the critical position $Z = 7\;\mathrm{\mu}$m where the axial force is zero and the left region of this dotted line ($Z < 7\;\mathrm{\mu}$m) is the trapping region. The maximum axial trapping force on the sphere at the fiber tip is $62.9$ pN.
We computed and plotted the outgoing fields of the DOFP with a $3\;\mathrm{\mu}$m SiO$_2$ sphere trapped at the fiber tip in Fig. 2(b). The reflective and refractive lights on the trapped $3\;\mathrm{\mu}$m SiO$_2$ sphere together with outputting fields from the DOFP result in the interference field distribution. The interference inside the probe is much weaker than that outside the probe, and it does not affect the particle trapping. It can be seen in Fig. 2(b) that there is a strong spot on the optical axis in front of the $3\;\mathrm{\mu}$m SiO$_2$ sphere, which is caused by interference outside of the probe. It plays a key role in trapping the second particle. This trapped SiO$_2$ sphere acts as a lens and refocuses the light beam as shown in Fig. 2(b). Another microsphere can be trapped by this refocused field.
We calculated the axial forces on the second microsphere at different positions on the fiber axis and plotted them in Fig. 3(b). $Z$ denotes the distance between the fiber tip and the center of the second sphere. The first trapped microsphere is a $3\;\mathrm{\mu}$m SiO$_2$ sphere. In order to study the impacts of the size of the second sphere, we calculated the axial forces on the second sphere when it is a $2\;\mathrm{\mu}$m SiO$_2$ sphere (black solid line), or a $3\;\mathrm{\mu}$m SiO$_2$ sphere (red-dashed line), or a $4\;\mathrm{\mu}$m SiO$_2$ sphere (green solid line), respectively. It can be seen that the maximum axial trapping force on the $2\;\mathrm{\mu}$m SiO$_2$ sphere at the fiber tip is $13.6$ pN. And the maximum axial trapping force on the $3\;\mathrm{\mu}$m yeast sphere is $32.1$ pN. Comparing the black solid line, the red dashed line and the green solid line, it can be seen that when the two trapped microspheres have the same refractive index, the larger the second sphere is, the larger the maximum trapping force and trapping region are. In order to study the impacts of the refractive index of the second sphere, we calculated the axial forces on the second sphere when it is a $4\;\mathrm{\mu}$m SiO$_2$ sphere (green solid line) or a $4\;\mathrm{\mu}$m yeast (blue dot-dashed line). Note that a yeast is normally a spheroid, but the difference between its major and minor axes is so small ($\sim 0.2\;\mathrm{\mu}$m) that we take it as a sphere in the theoretical calculation. It can be seen that the maximum axial trapping force on the $4\;\mathrm{\mu}$m SiO$_2$ sphere at the fiber tip is $34.9$ pN and on the $4\;\mathrm{\mu}$m yeast is $38.8$ pN. Comparing the green solid line and the blue dot-dashed line, it can be seen that if the two particles have the same size, the smaller the refractive index of the second sphere is, the larger the maximum trapping force and trapping region are. Note that the maximum axial forces of the second microsphere are much smaller than those of the first one in Fig. 3(a).
We computed and plotted the axial forces on the second particle versus the relative refractive index $n_{21}$ in Fig. 4(a). $n_{21} =n_{2}/n_{1}$, $n_{1}$ is the refractive index of the first particle, $n_{2}$ is the refractive index of the second particle. The first trapped particle is a $3\;\mathrm{\mu}$m (the black solid circles and line) or a $2\;\mathrm{\mu}$m (the red triangles and the red-dashed line) SiO$_2$ sphere, the second trapped particle is $4\;\mathrm{\mu}$m microsphere. The symbols are numerical computation results, the black solid lines and the red-dashed lines are fitted. The yeast, SiO$_2$ and polystyrene (PS) microspheres are mostly used in the optical trapping. Because $n_{SiO_{2}}=1.4507$, $n_{yeast}=1.4$, and $n_{PS}=1.59$, we focused on the region $0.96<n_{21}<1.1$. Consider the case that the first particle is a $3\;\mathrm{\mu}$m SiO$_2$ sphere, see the black solid line in Fig. 4(a). The dotted line denotes the critical position where the axial force is zero and the left region of this dotted line is the trapping region. And when $0.96<n_{21}<1.027$, the larger the $n_{21}$ is, the smaller the trapping force is. When $n_{21}>1.027$, the DOFP can not trap the second particle. For example $n_{PS}/ n_{SiO_{2}} >1.027$, if the first particle is a $3\;\mathrm{\mu}$m SiO$_2$ sphere and the second particle is a $4\;\mathrm{\mu}$m PS sphere, the DOFP can not trap the second one. From the black fitted line, the dependence of the axial forces $F$ on the relative refractive index can be approximated by
Compare the black solid and red-dashed lines in Fig. 4(a), we can see that at the fixed $n_{21}$, the smaller the first spherical size is, the larger the axial force is.
Fig. 4. (a) Axial forces on the second particle versus the relative refractive index. $n_{21}$ denotes the relative refractive index $n_{21} =n_{2}/n_{1}$, $n_{1}$ is the refractive index of the first particle, $n_{2}$ is the refractive index of the second particle. The diameter of the second particle is $4\;\mathrm{\mu}$m. The black solid circles and line: the first trapped particle is a $3\;\mathrm{\mu}$m SiO$_2$ sphere. The dotted line denotes the critical position where the axial force is zero and the left region of this dotted line is the trapping region. The red triangles and the red-dashed line: the first trapped particle is a $2\;\mathrm{\mu}$m SiO$_2$ sphere. The symbols are numerical computation results, the black solid line and the red-dashed line are fitted.(b) Axial forces on the second particle versus the relative radius. $r_{21}$ denotes the relative radius $r_{21} =r_{2}/r_{1}$, $r_{1}$ is the refractive index of the first particle, $r_{2}$ is the refractive index of the second particle. And the refractive index of the second particle is $1.4507$.
We also computed and plotted the axial forces on the second particle versus the relative radius $r_{21}$ in Fig. 4(b). $r_{21} =r_{2}/r_{1}$, $r_{1}$ is the refractive index of the first particle, $r_{2}$ is the refractive index of the second particle. The first trapped particle is a $3\;\mathrm{\mu}$m(the black solid circles and line) or a $2\;\mathrm{\mu}$m (the red triangles and the red-dashed line) SiO$_2$ sphere, the second trapped particle is $4\;\mathrm{\mu}$m microsphere. It can be seen that when $0.6 <r_{21} <3$, the larger the relative radius, the larger the trapping force on the second particle. From the black fitted line, the dependence of the axial forces $F$ on the relative radius can be approximated by
Compare the black solid and red-dashed lines in Fig. 4(b), we can see that at the fixed $r_{21}$ and $0.6<r_{21}<3$ , the smaller the first spherical size is, the larger the axial force is.
3. Optical trapping of multi-microspheres
We used the optical fiber (Nufern, 980-HP) to prepare a DOFP by the interfacial layer etching method. The single mode fiber has an operating wavelength of $980$ nm, a core refractive index of $1.46$ and a core diameter of $3.6\;\mathrm{\mu}$m. The etching solution is $40 \%$ HF acid and the organic protective layer is the paraffin wax. The fiber was immersed vertically into the etching solution. After a complete taper was formed due to Turner etching [28], the interfacial layer formed by molecules diffusing etched the fiber tip again to form the second taper. The angle of the second taper can be controlled by the etching parameters, such as the concentration of the etching solution, the etching time, the temperature and so on. The details about fabrication of a DOFP by the interfacial layer etching method can be found in our formal works [26,27]. Here we used the interfacial layer etching method to prepare a DOFT as shown in Fig. 1 with $\theta _{1} = 22 ^{\circ } , \theta _{2} = 55 ^{\circ }$. The etching procedure took $126$ minutes at $15.2 ^{\circ } C$ . The sample solution is a water solution consisting of SiO$_2$ spheres and yeasts. $\theta _{1} = 22 ^{\circ }$ is determined by the interfacial layer etching time, HF concentration, the overlay and so on, details see our formal work [26]. The optimal $\theta _{2}$ for different microparticles in series are not the same. We chose $\theta _{2} = 55 ^{\circ }$ which is not the optimal angle for any certain microparticle series, but it works well for both two SiO$_2$ microspheres with different radius and a SiO$_2$ microsphere together with a yeast in series.
The optical trapping system based on this DOFP is shown in Fig. 5(a). A laser beam with a wavelength of $980$ nm passes through the beam splitter with a splitting ratio of $1:99$. $1\%$ of the light beam goes to an optical power meter; the remaining $99\%$ are coupled into the double-tapered fiber probe which is fixed to a $4D$ micromanipulator. The laser power can be adjusted from $0$ to $400$ mW. An inverted microscope with a charge-coupled device (CCD) connected to a personal computer (PC) is used to observe and record the experiments. The fiber probe adjusted by a $4D$ micromanipulator can immerse in a water suspension of SiO$_2$ spheres and yeast cells. The translation stage controlled by a motion controller can move in $3D$.
Fig. 5. (a) Schematic of the experimental setup. A laser beam passes through a beam splitter. $1\%$ light beam is coupled to an optical power meter; $99\%$ are coupled into the double-tapered fiber probe which is fixed to a $4D$ micromanipulator. An inverted microscope with a charge-coupled device (CCD) connected to a personal computer (PC) is used to observe and record the experiments. (b) Trapping of two microspheres on the fiber axis at the fiber tip (see Visualization 1). The first trapped particle (marked by ‘Particle A’) is a $3\;\mathrm{\mu}$m SiO$_2$ sphere; the second trapped particle (marked by ‘Particle B’) is a $\sim 4\;\mathrm{\mu}$m yeast cell.
We trapped two microspheres on the fiber axis at the fiber tip, as shown in Fig. 5(b). The first trapped particle (marked by ‘Particle A’) was a $3\;\mathrm{\mu}$m SiO$_2$ sphere; the second trapped particle (marked by ‘Particle B’) was a $\sim 4\;\mathrm{\mu}$m yeast cell. The trapping of a yeast by the DOFPs with different $\theta _{2}$ was discussed in detail in our previous work [26]. The whole trapping process is shown in Visualization 1. Turned on the laser source and moved the probe close to a SiO$_2$ sphere. When the sphere was in the trapping region, it would be attracted to the fiber tip. Then moved the fiber probe with the trapped SiO$_2$ sphere together towards to a yeast cell. When the yeast cell was in its trapping region, it would be attracted to the SiO$_2$ sphere. Two microparticles moved as the fiber probe moved. Turned off the laser, the two particles separated and gradually moved away from the tip. We replaced the second particle (‘Particle B’) by SiO$_2$ spheres with different diameters and repeated the experiments.
After the DOFP traps the two microparticles in series, we move the translation stage and the aqueous solution on it by a motion controller, while the probe does not move. We measure the velocity of a reference particle in the solution, which is the velocity of particle B relative to the solution. We gradually increase the velocity of the translation stage by the motion controller, until particle B escapes. We define the escaping velocity of particle B as the maximum velocity of particle B relative to the solution before escaping. We measured the escape velocity $v_{cse}$ of the particle B and calculated the trapping force on it by using the escape method [29]. The axial trapping force is given by
where $F$ is the axial trapping force, $v_{cse}$ is the velocity of the particle when it escapes from the fiber probe and $\beta$ is the viscous drag coefficient. Since the absorption of water at 980 nm is very weak [30], the effect of heat and force generated by it is negligible in the calculation. For an isolated sphere of radius $r$, $\beta$ can be given by Stokes law: $\beta =6\pi \eta {r}$, where $\eta$ is the fluid viscosity. The viscous resistance of the solution is temperature dependent and the viscosity coefficient of water is $1.005\times 10^{-3}kg/(m \cdot s)$ at temperature $20 ^{\circ }$C. There are other forces in the experiment, such as the gravity, the friction at the bottom of the slide and so on. But they are much smaller than the optical trapping forces and are neglected.Recall that the first trapped particle (particle A) is a $3\;\mathrm{\mu}$m SiO$_2$ sphere. The trapping forces on the particle B versus the laser power are plotted in Fig. 6. The blue squares are the measured forces when the particle B is a yeast cell; the black triangles are the measured forces when the particle B is a $3\;\mathrm{\mu}$m SiO$_2$ sphere; the red circles are the measured forces when the particle B is a $2\;\mathrm{\mu}$m SiO$_2$ sphere. The fitted lines show that the axial trapping forces monotonically increase with the input power. The slopes of the fitted lines indicate the axial trapping efficiency, i.e., the axial trapping force per unit input power. From the fitted lines, we obtain that the maximum axial trapping efficiency is $20.9$ pN/W for a yeast cell, $12.1$ pN/W for a $2\;\mathrm{\mu}$m SiO$_2$ sphere, and $15.3$ pN/W for a $3\;\mathrm{\mu}$m SiO$_2$. The measured trapping forces are smaller than the theoretical values. It might result from that the Stokes equation is not modified for multiple particles closed to slide bottom and that the possible friction between the particle and the slide bottom, the fluid motion , and the solution impurities have been not taken into account. Comparing the black solid line with the red dashed line, we can see that when the two trapped particles have the same refractive index, the larger the second particle is, the larger the trapping efficiency is. Comparing the blue dotted line with the black solid line, it can be seen that the smaller the refractive index of particle B is and the larger the particle B is, the larger the trapping efficiency is. These two results are qualitatively consistent with the theoretical calculations (see Fig. 3 in Sec. 2).
Fig. 6. Axial trapping forces on particle B versus the laser power. Particle A is a $3\;\mathrm{\mu}$m SiO$_2$ sphere. The color markers are experimental data, the blue squares for a yeast cell; the black triangles for a $3\;\mathrm{\mu}$m SiO$_2$ sphere and the red circles for a $2\;\mathrm{\mu}$m SiO$_2$ sphere. The slopes of the fitted lines indicate the axial trapping efficiency, i.e., the axial trapping force per unit input power.
4. Conclusion
A DOFP was fabricated by the interfacial layer etching method with the first taper angle $\theta _{1} = 22 ^{\circ }$ and the second taper angle $\theta _{2} = 55 ^{\circ }$. The outgoing optical fields of this DOFP were simulated by the finite element method. The axial forces on two microparticles at the fiber tip were calculated by the Maxwell stress tensor integrals. The first particle was a $3\;\mathrm{\mu}$m SiO$_2$ , the second one was a $2\;\mathrm{\mu}$m, or $3\;\mathrm{\mu}$m, or $4\;\mathrm{\mu}$m SiO$_2$ , or a yeast. The results predicted that this DOFP can trap two different microparticles. And if the two particles have the same refractive index and $0.6<r_{21}<3$, the larger the second particle is, the larger the trapping force is. Whereas, if the two particles have the same size and $0.96<n_{21}<1.027$, the smaller the refractive index of the second particle is, the larger the trapping force is. The single optical fiber tweezer based on this DOFP was built. The trapping of a SiO$_2$ microsphere and a yeast, or two SiO$_2$ microspheres with different diameters, was observed. The trapping forces on the second microparticle were measured, when the first trapped particle was a $3\;\mathrm{\mu}$m SiO$_2$ microsphere. In the experiments, the phenomenon is qualitatively consistent with the theoretical result. The magnitude of the measured trapping forces is smaller than that of the theoretical calculation, due to the impurities in the solution, the geometry of yeast cells and other influencing factors in the experiment. Our simulation and experiments can be also applied for other microparticle series, such as different polystyrene microspheres in series. Trapping and manipulation of different multiple microparticles by a single optical fiber probe enhance the application of optical tweezers, especially in biomedical engineering (e.g. cells assembly and sorting), soft matter colloidal science (e.g. brownian motion and diffusion, particle interaction potential) and material science (artificial lattice).
Funding
National Natural Science Foundation of China (62061136002).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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