1. Introduction

Optical trapping and manipulation have become powerful techniques to capture and manipulate, nearly non-invasively and without mechanical contact, microscopic particles of size ranging from tens of nanometer to a few ten microns including living cells, bacteria, and cell organelles. Optical trapping and acceleration of micro-particles in liquid were first demonstrated by Ashkin in 1970 using a pair of mildly focused counter-propagating laser beams [1]. In this dual-beam configuration, the particle is confined on the common optical axis of the two trapping beams by the co-operative transverse gradient forces and stabilized at a point on the optical axis where the two laser scattering forces balance each other. In 1993, Constable et al. [2] used two laser beams exiting from a pair of well-aligned single-mode optical fibers to trap and manipulate polystyrene particles with diameter in the range of 0.1μm ∼ 10 μm, as well as living yeast cell. In 2000, Guck et al. [3] used the fiber-optical dual-beam trap to trap and stretch red blood cells (RBCs) for non-invasive study of the visco-elastic property of a single RBC in buffer solution.

Although optical tweezers first demonstrated by Ashkin et al. in 1986 [4] are much more popular than the fiber-optical dual-beam trap, the latter has certain potential advantages. 1. The potential optical damage to the trapped particle (especially to living biological samples) can be less severe because strong focusing is not required, and hence, the optical intensity on the trapped particles is much weaker. 2. It can be used not only for trapping, but also for stretching of elastic micro-particles including living cells [3]. The main disadvantage of the fiber-optical dual-beam trap, in comparison with optical tweezers, is the necessity to align the two beams with high precision. Besides, both axial and transverse optical forces in a fiber-optical dual-beam trap are often weaker, compared with the corresponding forces associated with optical tweezers, which is essentially a direct consequence of weaker optical intensity. We have used a fiber-optical dual-beam trap to successfully capture and manipulate silica micro-particles of various sizes and Chinese hamster ovary cells in PBS (phosphate buffered saline) solution, and analyzed the three-dimensional optical force field on the trapped particle.

Optical forces on a trapped particle can be measured by either dragging the trapped particle against the optical forces [5] or by tracking and analyzing the Brownian motion of the trapped particle [6]; the latter is particularly suited of the mapping of three-dimensional optical force field in the parabolic potential approximation. In this paper, we report the determination of optical force constants (k_x, k_y, and k_z) of the three-dimensional optical force field on silica particles and on CHO cells in a fiber-optical dual-beam trap by tracking and analyzing the Brownian motion of a trapped particle. A micro-particle trapped in a three-dimensional parabolic potential well [E (x, y, z) = (k_xx² + k_yy² + k_zz²)/2] of a fiber-optical dual-beam trap [6, 7] is unavoidably driven by thermal force to execute Brownian motion. The extent of the thermal fluctuation of a trapped particle depends on a variety of factors including the distance between the two fiber end-faces, trapping laser power, trapped particle size, laser wavelength, the temperature of solution, etc. We can use either the fluorescence emission (in the case of a fluorescent particle) [8] or the trapping laser light scattered by the trapped particle [9] to track the Brownian motion of the particle in a fiber-optical dual-beam trap. In our experiment, we used a pair of orthogonal quadrant photodiodes to track the three-dimensional Brownian motion of the trap particle via the laser light scattered by the trapped particle in a fiber-optical dual-beam trap [10] and analyzed the optical force field on the trapped particles including silica micro-particles of various sizes and CHO cells. The experimental configuration, procedure, and results are discussed in the following sections.

2. Experimental setup and procedure

A schematic diagram of our experimental setup is illustrated in Fig. 1. A laser beam (cw, λ=532nm from a Nd : YVO₄ laser) was expanded and collimated via a 3X beam-expander (3X BE) through a half-wave plate (λ/2) and a polarizing beam splitter (PBS) cube to split into two beams with equal optical power, and each coupled into a single-mode fiber (NA=0.12) via a single-mode fiber coupler (SMFC). The output ends of the two fibers were aligned so that the two laser beams exiting from the fibers formed a pair of counter-propagating beams along a common optical axis inside a sample chamber where micro-particles and CHO cells were trapped in PBS solution. The distance between the two fiber end-faces was kept at 125μm. Portions of the trapping beams scattered by the trapped particle were collected by a pair of orthogonally oriented long-working distant objectives (LOB I and LOB II, Mitutoyo, 50X, NA=0.42) and projected onto a pair of quadrant photodiodes (QPD I and QPD II, On-Trak, PSM2-10Q). LOB I /QPD I and LOB II /QPD II collected the scattered lights and tracked the position of the trapped particle projected on the xz-plane and the yz-plane, respectively; the xyz co-ordinate system is depicted in the upper left corner in Fig. 1. Besides, the LOB I was also used to collect the light from an illuminating lamp for incoherent imaging of the trapped particle on a CCD camera (752 x 582 pixels, WAT-100N).

 

Fig. 1. A schematic diagram of our experimental setup

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Two independent calibration methods were applied to obtain the conversion factor that convert the output voltage of the QPD into the particle displacement. In the first approach, we trapped the micro-particle of interest on the optical axis in the middle of the end-faces of two optical fibers via equal laser power from the two fibers [Fig. 2(a)], and then momentarily turned off one of the laser beams to drive the particle (by the remaining single beam) along the optical axis towards the opposite end-face of the optical fiber; Fig. 2(b) illustrates the particle driven to the right after the beam on the right hand side was turned off. The second beam was subsequently turned on, to drive the particle back along the optical axis towards the original equilibrium position at the middle of the fiber end-faces.

 

Fig. 2. (a) A 2.58 μm silica particle trapped in a fiber-optical dual-beam trap, (b) the silica particle driven to the right by a single beam after the beam on the right hand side was turned off.

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As the micro-particle was driven back along the optical axis under the illumination of both beams, the corresponding output voltage of the quadrant photodiode and the position of the micro-particle on the CCD camera (Fig. 2) were recorded simultaneously. An illustrative example of such a calibration curve (i.e., QPD output voltage vs. particle position recorded on the CCD) is depicted in Fig. 3 which shows a linear dependence with a slope of approximately 0.986μm/V for particle moving from a point about 3.4 μm off-set from the equilibrium position to the final equilibrium position (i.e., the origin of the horizontal axis in Fig. 3). Besides the Brownian fluctuation, the accuracy of this approach is mainly limited by the determination of the particle position from the incoherent image of the illuminated particle on the pixelated digital CCD camera. The calibration method described above was carried out during the return trip of the particle from a point offset from the center to the center equilibrium position when both laser beams were on to ensure that the particle was evenly illuminated from both sides.

 

Fig. 3. Calibration for the conversion of the QPD output voltage to the particle displacement on the optical axis via incoherent imaging on a CCD camera.

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Although the calibration method described above allows us to calibrate directly the QPD output voltage to the axial displacement of trapped particle, it is difficult, if not impossible, for the calibration of the particle displacement perpendicular to the optical axis. To overcome this problem, we followed a second method reported by Ghislain and Webb in 1993 [11] which applied the Fourier transform of the QPD output voltage to obtain its power spectrum and fitted the power spectrum to the following Lorentzian form :

where K_B is the Boltzman constant, T is the absolute temperature, β is the voltage-to-displacement conversion factor, η is the viscosity of the buffer solution (PBS in this case), r is the radius of the trapped particle, fc is the corner frequency (a characteristic frequency of the system), and f is the frequency. In this particular example, the best fitted value of the conversion factor β (fitted in the region of the power spectrum where f≫fc) was β = 0.8 μm/V as is illustrated in Fig. 4. The power spectrum method described above allows us to conveniently obtain the voltage-to-displacement conversion factor for the trapping of each micro-particle in a fiber-optical dual-beam trap under each specific set of experimental conditions. Besides, in contrast to the dragging approach, the results obtained by this method do not rely on the resolution of the incoherent image on the pixilated CCD camera. Hence, the power spectrum method was used to calibrate the QPD output voltage to the particle displacement for all the experimental results presented in the following sections.

 

Fig. 4. Calibration for the conversion of the QPD output voltage to the particle displacement via the power spectrum of the particle thermal fluctuation and the theoretical fitting (the red line) to a Lorentzian form.

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3. Experimental results

We tracked the three-dimensional Brownian motion of a micro-particle trapped in a fiber-optical dual-beam trap, and analyzed its position distribution to obtain the optical force field approximated by a three-dimensional parabolic potential well [6]. Under the parabolic potential approximation and the classical Boltzmann statistics the associated optical force constant along each axis can be calculated from the following two Eqs. [6] :

where ρ(z) is the probability function of the trapped particle position along the z -axis, C is the normalization constant, E(z) is the potential energy function along the z-axis, K_B is the Boltzmann constant, and kz is the optical force constant along the z-axis. Identical set of Eqs. also apply for the x-axis and the y-axis. As a specific example, a set of experimental data representing the parabolic potential E(x) and E(z) of the optical force field on a 2.58μm diameter silica particle in a fiber-optical dual-beam trap (total trapping power = 22mW, distance between the fiber end-faces = 125 μm,) is depicted in Fig. 5. The solid lines represent the theoretical curves based on Eq. (3) given above; the corresponding optical force constants, defined by Eq. (3), were kx = 0.161 pN/μm, and kz = 0.0443 pN/μm. The force constant along the optical axis is weaker than those along the transverse axes, which is consistent with the theoretical results reported earlier [7]. This is also true in the case of optical tweezers [9].

 

Fig. 5. Experimental data representing the parabolic optical force fields E(x) “□” (in black) and E(z) “□” (in blue) on a 2.58μm silica particle. The solid lines represent the theoretical fits.

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We trapped different sizes of silica particles (diameter = 2.58 μm, 5.06 μm, and 9.63 μm) and Chinese hamster ovary (CHO) cell (diameter ∼ 15 μm) (see Fig. 6) in a fiber-optical dual-beam trap, tracked the Brownian motion of the particle (or cell), and calculated the optical force constant along each axis. Our results, which are summarized in Table I below, indicate that, in general, the optical force constant increases with the diameter of trapped particle for particle diameter in the range of approximately 2μm to 15μm. In the last row of Table I, we list the optical force constants for the case of CHO cell treated with 1 μM latrunculin, a toxic chemical known to disrupt the cytoskeleton (actin filament) of the CHO cell [12]. The resulting force constants were reduced approximately by a factor of 2 compared with the corresponding results for the CHO cell without latrunculin treatment. The disruption of the cytoskeleton is expected to render the cell more elastic and more easily subjected to morphological deformation such as stretching [13]. Although we have observed the stretching of a CHO cell in a fiber-optical dual-beam trap-and-stretch in a separate experiment (Fig. 6), whether the reduction of the force constant is related to the change in cellular viscoelasticity and/or the morphological deformation requires further investigation.

Table. 1. A comparison of optical force constants for silica particles of different sizes and for CHO cell with and without latrunculin treatment.

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Fig. 6. CHO cell trapped and stretched in a fiber-optical dual-beam trap. Total trapping (and stretching) optical power: (a) 22mW, (b) 27mW, (c) 38mW, (d) 50mW.

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4. Summary and conclusion

We trapped CHO cell (diameter ∼ 15μm) and silica particles of various sizes (diameter = 2.58μm, 5.06μm, and 9.63μm) in a fiber-optical dual-beam trap, tracked the three-dimensional Brownian motion of the trapped particle via a pair of orthogonally oriented quadrant photodiodes, analyzed the particle position distribution, and calculated the optical force constants along each axis under the parabolic potential approximation. In general, the optical force constant increases with the particle size. Beside, the force constants of the optical force field on CHO cell with disrupted cytoskeleton (caused by latrunculin treatment) were found to be smaller than the corresponding values for the case of an untreated CHO cell by almost a factor of 2.