1. Introduction

Optical tweezers (OTs) have been widely used in biology, colloidal science and physics since they were invented by Ashkin et al. in 1986 [1], for examples, study of mechanical properties of biological cell membrane [2,3], manipulation of the living cell [4], measurement of electrostatic interaction between colloidal particles [5,6] and nano-fabrication [7,8], etc. OTs has been experienced intensive development in the past 30 years. A recent article has reviewed in detail the latest optical trapping configurations and their applications in bioscience, as well as recent advances down to the nanoscale [9]. Conventional optical tweezers (COTs), which bases on the tightly focusing laser beam by a high NA objective, have normally small work distances difficult the trapping of the particle in the depths of the sample. Conversely, optical fiber tweezers (OFTs) have high manipulation flexibility and enable trapping the particle in depth [10], besides easy miniaturization and easy fabrication. Tapered optical fiber is the most importance configuration of the OFTs. The focal length of the tapered fiber is smaller than that of an objective. Besides that, the outputted beam from a cone fiber tip is a non-diffraction Bessel beam with greatly reduced diffraction and smallest optical confinement zones. These features of tapered fiber enable a stronger trapping force and a larger trapping distance from the fiber tip. Therefore, the tapered fiber tweezers can be applied to trapping nanoscale particle [11,12] and multi-particle manipulation such as organization of microscale particles [13], cell patterning [14,15], and so on. Polishing [16,17], thermal pulling [18,19], and chemical etching [10,20,21] are the most importance fabrication method of OFTs. Chemical etching offers a simple and much less expensive technique allowing batch fabrication of fiber tip. Generally, a single-tapered fiber tip has been formed by the chemical etching such as Turner method [22]. Depending on the kind of the organic protective layer, the corresponding taper angle varies from 8 to 41 degrees [23]. Light outputted from a tapered fiber tip is focused near the surface of the tip apex because of the small taper angle, which results in the trapped particles are directly in contact with the fiber tip [18,21] and will inevitably cause mechanical damage to the trapped particles especially to the biological particles. Furthermore, due to the contact capture, the impurities in sample solution are also attracted to the surface of the fiber tip and thereby stain it. In order to achieve contactless trapping as in COTs [1], an annular-distribution multi-core fiber probe [17], and a modified tapered fiber with an abruptly tapered region, and a protruding tip [19] have been applied. Recently, a OFTs based on Bessel-like LP_0n (n is the radial index) modes excited in multimode fiber have been employed for the non-contact trapping of multi-particles [24].

In fact, the tip angle has a significant effect on the focusing of light outputted from the fiber tip. The position of the focus on the beam axis can be adjusted by varying the tip angle and thereby enable contactless capture of particles. Comparing with the single-tapered fiber tip, the tip angle of fiber tip with double tapers can vary over a wider range (e.g., for 90 deg) [25,26]. In this work, a double-tapered optical fiber tweezers (DOFTs) with large tip angle (i.e. the second taper angle, STA, see Fig. 1b) was fabricated by a chemical etching which is called interfacial layer etching by us. This fabrication method is relatively simpler and it is easy to control the STA by the interfacial layer etching time. After that, the contactless trapping of particles by the DOFTs was demonstrated. The distribution of the optical field from the double-tapered tip with different STA and the optical force on the particle at the optical axis were numerically calculated.

 

Fig. 1 Optical microscope images (a) (b), and SEM image (c), of the tapered SMF tip etched by 40% HF acid at temperature 19°C. (a) a single-tapered fiber tip obtained by 63-minutes meniscus etching, (b) a double-tapered fiber tip obtained by 63-minutes meniscus etching and next 37- minutes interfacial layer etching. The second taper can be seen clearly in (b) and (c). θ₁ and θ₂ are the first and second taper angles, respectively.

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The paper is organized as follows. In Sec. 2, the fabrication method was introduced in detail. A 5-μm yeast cell was trapped by the DOFTs with different STA in Sec. 3. The viscous drag coefficient on the particle and effects of STA on the axile trapping force and trapping position were discussed in Sec. 4. Sec.5 gives the conclusions.

2. Fabrication of double-tapered fiber tip

First, the single mode fiber (SMF) with a 125-μm cladding and cut-off wavelength at 920 nm was etched completely by Turner method [22]. The etching solution is 40% HF acid having an organic protective layer of Liquid Paraffin. A single-tapered tip was formed due to the gradually reducing height of the meniscus as the fiber diameter was reduced by the etchant solution. Figure 1(a) shows the resultant tip after 63-minuts corrosion at ambient temperature 19°C. In the case, the height of meniscus decreased to zero, thus the meniscus etching terminated automatically. But the etching is not completely stopped because there are other etching mechanisms except for meniscus etching. As we know, there is a transitional interfacial layer between HF acid and protective overlay. HF concentration gradient was formed in the interfacial layer due to the diffusion of HF molecules. Due to the high volatility of HF, the thickness of the interface layer is about a few microns and so it is suit to etch the apex of the single-tapered tip in this layer. The tip apex locates at the bottom of the interfacial layer when the complete tip was formed. If the tip has not be pulled out the etching solution in this stage, the single-tapered tip will be etched again (i.e. the second etching, the first etching is meniscus etching in Turner method) in the interfacial layer and then a second taper at the tip apex can be formed due to HF concentration non-uniformity which resulted in the difference of the etching rate in the direction perpendicular to the interfacial layer. We call the second etching as the interfacial layer etching because the meniscus disappeared. Here, the formation mechanism of the second taper is different from that in Haber’s work [27]. The formation of the second or multiple taper is attributed to the meniscus distortion caused by the movement of the fiber in their work. Figures 1(b) and 1(c) shown the results obtained by meniscus etching of 63 minutes and next by interfacial layer etching of 37 minutes. The double-tapered profile with the first and second taper angles of 23 deg and 63 deg respectively can be seen clearly. Some defects on the tip can be seen in Fig. 1(c), which probably caused by the vibrations and disturbance of air current during the etching. An imperfect cone shape may cause some effect on the ability of trapping because of the distortion of outputted beam. So the ambient disturbance should be minimized during the etching and the fiber tip selected to trap the particle should be as symmetrical as possible.

Controlling the interfacial layer etching time can vary the STA over a large range. Experiments demonstrated STA increases rapidly first and then slowly with the interfacial layer etching time as shown in Fig. 2. In the figure, the squares were experimental data and the solid line was the fitting line. Four double-tapered fiber tips were fabricated under the same interfacial layer etching time, but in different batches. Therefore the value of STA fluctuated with the external conditions (such as vibration). The largest fluctuation reached ± 5° as shown with the error bars in Fig. 2. During the interfacial layer etching, the tip apex contracted gradually to the upper edge of the interfacial layer where the HF acid concentration and its gradient went to zero. As a result, the change of STA became more and more slow. From Fig. 2, the STA would barely change from 30 min to 70 min. To get a larger STA, the interfacial layer etching time must be extended.

 

Fig. 2 The second taper angle (STA) vs. the interfacial layer etching time. Square markers with error bars indicated the data measured. The error resulted from the STA fluctuation of four fiber tips fabricated under the same interfacial layer etching time but in different batches. The blue solid line was fitting line.

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3. Trapping of yeast cells

In order to investigate the performance of the DOFTs, we carried out trapping experiments. The experimental setup consists of a laser source with SMF fiber coupled output, a double-tapered fiber tip for trapping particles, an inverted microscope for particles observation and a computer connected charge coupled device (CCD) for real-time monitoring and images capture. The laser source is capable of producing the maximum power outputs of 300mW at wavelength 980nm. Through a SMF coupler with coupling ratio of 1:99, the laser power was monitored by the 1 percent end which connected to an optical power-meter. The other end of the coupler was then connected to the fiber tip. The fiber tip was mounted on a mechanical micromanipulator so as to move it in three dimensions and to rotate it for changing the inserted angle. A sample chamber contained aqueous solution of yeast cells was located at the object stage which can move automatically at one dimension by a servo motor with an accuracy of 20nm per pulse (i.e. one step).

In our experiment, the trapping forces and trapping positions of five double-tapered fiber tips having different STA were investigated. The first taper angle (FTA), which depends on the kinds of the organic protective layers [23], of these tips are all 23 deg. Their STAs are 23 deg, 48 deg, 63 deg, 70 deg and 90 deg, respectively. The STA of 23 deg means that the tip is actually single-tapered tip. The experimental results demonstrated the yeast cells trapped by these tips are all contact with the tip except for the one having STA of 90 deg. As an example, Fig. 3(a) shows the contact capture of a 5-μm (typical sizes) yeast cell using the double-tapered fiber tip with STA of 48 deg (See also Visualization 1). The yeast cell trapped only by the tip with STA of 90 deg is non-contact with the tip as shown in Fig. 3(b) and the trapped cell can be freely moved to the forward and backward or left and right directions (See also Visualization 2). When the object stage is stopped to move, the gap between the tip and the trapped cell is about 1 μm (Fig. 3b). From Visualization 2, we can see that the cell is always trapped in a non-contact manner during the translation. This implies that the non-contact capture is stable because the trapped cell withstood the disturbance resulted from the translation. On the other hand, the stability issue caused by the Brownian motion will be discussed in Sec. 4.2. We also find that the trapping force of the fiber tip with STA of 90 deg is obviously weaker than one of the other tips under the condition of the same trapping power. These mean that STA has significant effect on the trapping performances (trapping force and position) of DOFTs. In next section, we will discuss this effect in detail.

 

Fig. 3 Images of a trapped 5-μm yeast cell (see Visualization 1 and Visualization 2, respectively). The STAs of the tips in (a) and (b) are 48 deg and 90 deg, respectively. The trapping powers are 20mW in (a) and 50mW in (b). The insert: optical microscope photograph of the tip.

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4. Discussions

4.1 The viscous drag coefficient

The trapping force per Watt, that is trapping efficiency ζ, is one of the important parameters that measure the trapping performances. In most applications, forces are measured against viscous drag exerted by fluid flow. For a micron-sized object with quite small Reynolds number, the drag force on it can be written as $F=\beta v$, where β is the viscous drag coefficient and v is the fluid relative velocity to the object. For an isolated sphere of radius a, β is given by Stokes’ law, i.e. $\beta =6\pi \eta a$ (so called Stokes drag coefficient), where η is the fluid viscosity. But when a spherical particle was trapped by the tapered fiber tip, the sphere and the tip composed a compounded configuration as shown in Fig. 4. In other words, the fiber tip will affect the flow field around the particle. In the case, Stockes’ law is no longer applicable, except in case of a very large gap D. Accordingly, the drag coefficient β is not equal to the Stokes drag coefficient in the measurement of the trapping force. However, to our knowledge, the force measurements reported in literatures were almost based on the Stokes’s law, e.g. Refs. 10, 18 and 21, which probably resulted in a large error of the measurement. In order to calculate the viscous drag coefficient β on the sphere of the compounded configuration, we carry out a 3D finite element simulation with the help of commercial software COMSOL Multiphysics. Firstly, value β on the sphere was calculated when the spherical particle wascontacted with the fiber tip (D = 0, the contact trapping). The parameters used in the calculation are as follow: the sphere radius a = 2.5μm (corresponding to the radius of the trapped cell), the water viscosity η = 1.0 × 10⁻³ Pa.s at 19°C, θ₁ (FTA) = 23° and θ₂ (STA) = 63°. For convenience, we defined a ratio as f = β/β_∞, where β_∞ is the drag coefficient when the gap D tends to infinity. Obviously, β_∞ is just the Stokes drag coefficient. In fact, the ratio f is the correction factor of the Stokes drag coefficient β_∞ caused by the compounded configuration. This means that the forces measured on the base of Stokes’ law must be corrected by this factor. f is 0.5 calculated by us in the case of contact trapping, which means that this compounded configuration weakened the drag force on the spherical particle. In order to understand the weakening effect, we calculated the distribution of the flow fields near the static compounded configuration and an isolated sphere, respectively. The results were shown in Fig. 5 where the blue line and the red arrow indicated the streamline and the direction of the flow respectively. From the figure, it can be found that the clockwise and counter-clockwise vortex flows were formed respectively in the regions A and B (Fig. 5a) which connected the fiber tip and the spherical particle, but the vortex did not exist in the case of isolated sphere (Fig. 5b). These vortex flows caused a backward shear force on the sphere, which weakened the viscous drag force.

 

Fig. 4 Geometrical model of the DOFTs. D is the gap between the particle and the tip. θ₁ and θ₂ are the first and the second taper angles, respectively.

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Fig. 5 Distributions of the flow fields near (a) the compounded configuration composed of the sphere and the tip, and (b) an isolated sphere. Black areas indicated the fiber tip and the sphere. The blue line and the red arrow indicated the streamline and the direction of the flow, respectively.

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Effects of the gap D and the STA on the drag coefficient β were also calculated. The results were shown in Fig. 6. We can see that the factor f (thus β) increases with the gap D. The value of β is approximately equal to Stokes drag coefficient β_∞ when D is larger than 120μm. But this condition is difficult to satisfy because the realistic gap D is often much less than this value when the particle was trapped by the tapered fiber tip. Effect of the STA on the drag coefficient is relative small. The factor f decreased from 0.59 to 0.48 as the STA increased from 23 deg to 90 deg (Fig. 6). In our experiments that trapped a 5-μm yeast cell, the gap D≈0 or 1 μm (See Fig. 3), so the drag coefficient β ≈0.5β_∞ according to Fig. 6, which means that the forces measured directly by use of Stokes’ law will enlarge two times.

 

Fig. 6 The correction factor f vs. the gap D (θ₂ = 63 deg) and the STA, θ₂ (D = 0), respectively.

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4.2 Effects of STA on the axile trapping force and trapping position

When the yeast cell moved to the trapping range of DOFTs, the transverse optical force on the particle will pull it toward the beam axis. Therefore, only the optical force on the axis shall be discussed here. In the trapping experiments of the yeast cell, the trapping forces of the DOFTs with different STA are obviously different, as motioned in Section 3. The maximal axile optical forces F_max of the five double-tapered fiber tips used in the experiments were measured respectively using the escape-force method [28], i.e. $F_{\max }=\beta v_{\text{esc}}$, where v_esc is the velocity of the objective stage at which the particle escaped and β ≈0.5β_∞ as mentioned above. Figure 7 shows the measured forces of the five double-tapered fibre tips as a function of the input power in fiber probe. The color markers and the solid lines are the experimental and the linear fitted data, respectively. The results demonstrated the linear dependence of theaxile trapping force on the input power. The slope of the fitted line indicates the maximum axile trapping efficiency ζ_max (i.e. axile trapping force per Watt) of the DOFTs. The ζ_max values of the five DOFTs were listed in Table 1. According to the table, ζ_max is largest when the STA is 48 deg and then decreases with STA increasing, which shows that there is an optimal ζ_max when the STA was changed. Unfortunately, DOFTs with the optimal trapping efficiency is not contactless. Among the five DOFTs, the trapping efficiency of the one that enables the contactless capture is lowest. To better understand the effects of STA on the trapping efficiency and trapping position, the distribution of the optical field outputted from the fiber tip with different STA (Fig. 8) and the axile trapping efficiency vs. the position of the particle (Fig. 9) were calculated using COMSOL Multiphysics. In the calculation, a geometrical model of DOFTs was constructed as shown in Fig. 4. The optical force on the particle was obtained by integrating the Maxwell’s stress tensor. The diameters of the yeast cell and the core of the SMF are 5μm and 3.6μm, respectively. The refractive indices of the cell and the cladding of the SMF are 1.4 and 1.4507, respectively. The NA of the SMF is 0.2. The wavelength λ is 980nm. The FTAs of the five double-tapered fiber tips are all 23 deg.

 

Fig. 7 The maximal axile trapping forces of the five double-tapered fiber tips vs. the input power in fiber probe. The color markers and the solid lines are the experimental and the linear fitted data, respectively.

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Table 1. Maximal Axile Trapping Efficiency ζ_max and Trapping Position D₀^a

View Table

 

Fig. 8 (a) – (e) The normalized electric intensity |E|² of the double-tapered fiber tip with different STA. The five figures are normalized individually. (f) The focal length (the distance from the focus to the surface of the tip, normalized to λ) of the double-tapered fiber tip vs. the STA. In which the negative focal length indicates the focus locates inside of the tip.

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Fig. 9 The axile trapping efficiency ζ vs. the gap D. In which the FTAs of the five fiber tips are all 23 deg. (a) Different STA. The inset shows the maximum trapping efficiencies ζ_max as a function of the STA. (b) Different particles’ radius r_p. The STA is 48 deg.

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In Figs. 8(a) – 8(e), the dimensions of the focal spot of the tip with STA 48 deg, 0.74 × 4.59λ² (i.e. the product of the transverse and axial full width at 1/e² maximum, normalized to λ), is the smallest among the five tips and then it is larger and larger especially for the axial dimensions as the STA increasing. Besides, the positions of the focal spot shifted gradually from inside to outside of the fiber tip with the increase of the STA, which caused the trapping positions shifting correspondingly. Figure 8(f) shows the focal length (the distance from the focus to the surface of the tip) as a function of θ₂, in which three data with θ₂ = 30°, 40° and80° were added. The negative focal length indicates the focus locates inside of the tip, which is mainly attributed to the total internal reflection (TIR) at the side of the second taper. The optical confinement has become increasingly weak as shown in Figs. 8(a) – 8(e) as the positive focal length increases. So the large STA weakens the focusing and the trapping abilities of the fiber tip. On the other hand, when the focus locates inside of the tip, the trapping efficiency is generally larger (seen from the insert of Fig. 9a) because the evanescent field caused by TIR is dominant in the trapping. From Fig. 8(c), we can see that the focal spot’s position just right located the surface of the tip when θ₂ (STA) is 63 deg. The focal spot shifted completely out of the tip when θ₂ is 90 deg (Fig. 8e), which suggests that the non-contact trapping should be able to realize.

The dependences of the axile trapping efficiency of DOFTs with different STA and particle’s size on the gap have been simulated and the results were given in Fig. 9. In which, the minus indicates the optical force on the particle is attractive. From Fig. 9(a), we can get the maximum trapping efficiencies ζ_max and the trapping positions D₀ (the gap between the fiber tip and the trapped particle when the axile trapping force reaches the stable equilibrium) of the five fiber tips. These data were listed in Table 1. For comparison, the measured data were also listed. We can see that the experimental results are good agreement with the theoretic ones. The ζ_max of the tip with STA 48 deg is four times of the one of tip with STA 90 deg. But only for the tip with STA 90 deg, the stable equilibrium point of the force is outside of the tip, which results in the non-contact capture (trapping position D₀ = 0.37μm theoretically or ~1 μm experimentally). Now, the issue on the stability of the non-contact trapping caused by the Brownian motion can be discussed. According to the black solid line in Fig. 9(a), the relation between the trapping force F and the gap D can be fitted linearly from D = 0.37μm (i.e. equilibrium point) to D = 1.5μm (but at D = 2.0μm, F = F_max). So we have$F=-\partial U/\partial D$, where U is the potential energy of the particle in the trap. And then has$U/k_{B}T=3.5\times {10}^3$ per Watt after integrated this equation in the interval [0.37μm, 1.5μm], where k_B is the Boltzmann constant and T is temperature of the medium surrounding a particle. A necessary and sufficient condition for stable trapping is that the potential well should be much larger than the kinetic energy of the Brownian particles [1], i.e. Boltzmann factor$\exp (-U/k_{B}T)\ll 1$. As a practical form, this condition is set to be $U/k_{B}T\ge 10$ [1]. Obviously, our result can satisfy the condition. So the non-contact trapping from the DOFTs with STA 90 deg is stable. For STA equals to 23 deg, 48 deg, 63deg and 70deg respectively, the trapping force will very rapidly increase to the maximum value ζ_max with the increase of D when D < 0.3μm. This suggests the large trapping stiffness of DOFTs and thus the good sensitivity in the application of the force sensor. Furthermore, the five tips have almost the same trapping range (about 16 μm), which means the STA has no effect on it. The insert in Fig. 9(a) shows the ζ_max as a function of the STA. In which, other four simulation data with θ₂ = 30°, 40°, 50° and 80° were added. In the insert, an optimal ζ_max can be found at θ₂ = 48°. This result may be explained in terms of total internal reflection (TIR). We assume that the light travels parallel to the axis of the fiber from the main fibre taper to the second taper. When the light hit on the apex side, TIR happens if${\theta }_2<180°-2{\theta }_c$, where ${\theta }_c={\sin }^{-1}{n}_{\text{w}}/n_{\text{co}}$(n_w and n_co are the refraction index of the water and fiber core, respectively) is the critical angle. The θ_c is 65.3°, so TIR happens on the apex side if the condition θ₂ < 49.4° is satisfied. Just because of TIR, the focus of the fiber taper with ${\theta }_2\le 48°$locates inside of the tip as shown in Fig. 8. Under this condition, the smaller the θ₂ is, the larger the incident angle on the apex side is. The light with a larger incident angle is analogous to the lower order modes in optical fiber, so the light power percentage outside of the fiber tip and then the trapping ability are lower for a smaller θ₂. For example, the focal spot of the taper fiber with STA 23 deg is almost enclosed inside of the tip. This also explains the change of ζ_max in the left curve in the insert. When θ₂ > 49.4°, TIR and dominant evanescent field disappear. This results in the maximum trapping force decrease as shown in the right curve in the insert.

Finally, the effect of the particle’s size will be briefly discussed. In Fig. 9(b), four different sizes, i.e. the particle’s radius r_p = 1.75μm, 2.5μm, 3.5μm and 5.0μm, were given. The FTA and STA of the DOFTs are 23 deg and 48 deg, respectively, and the other parameters are same as the Fig. 9(a). The maximum axile trapping efficiencies ζ_max on the four particles are almost the same as well as the positions where ζ reaches its maximum. This means that the size of the particle has little effect on the maximum trapping force for the DOFTs with STA 48 deg. It is probably because the evanescent field resulted from TIR is not sensitive to the size of the particle. But when D > 2μm, the larger the particle is, the stronger the trapping force on it because the evanescent field almost disappears over there.

5. Conclusions

DOFTs was fabricated using the interfacial layer etching. By controlling the etching time, the different STA was achieved. A typical 5-μm yeast cell was trapped respectively by the DOFTs with STA 23 deg, 48 deg, 63 deg, 70 deg and 90 deg. The experimental results demonstrated the non-contact capture of the cell by the fiber tip with STA 90 deg and the contact capture by the other tips. Furthermore, the trapping force per Watt (i.e. trapping efficiency) of the DOFTs reaches a maximal value when the STA is 48 deg and then it decreases as the STA increasing. To explain the experimental results, the electrical intensities of the five double-tapered fiber tips and the axile trapping efficiency vs. the gap between the particle and the fiber tip were calculated theoretically. Good agreement between the experimental and theoretical results was also found. Furthermore, during the measurement of the trapping force, the fiber tip can affect the distribution of the flow field around the particle and thus weaken the viscous drag coefficient. The simulated result shown the viscous drag coefficient on the particle is about half of the Stokes drag coefficient, which result in a more accurate force measured.

Benefitting from the controllable STA, a DOFTs with proper STA which has a shorter focal length (such as shown in Figs. 8(e) and 8(d)) is promising for the trapping of the nanoparticles in addition to the employment of the probe-microlens structure [12]. On the other hand, the fiber probe with a large STA can acquire high transmission when was used in scanning near field optical microscopy [29,30].