1. Introduction

Optical tweezers (or single-beam gradient-force optical trap) were first reported by Ashkin et al. in 1986 [1] almost 16 years after the first report on optical acceleration and trapping of micro-particle in a counter-propagating dual-beam configuration by Ashkin et al. in 1970 [2]. Optical forces (on the order of a few pico-Newtons to tens of pico-Newtons) in optical tweezers with near infrared (NIR) laser beam (e.g., at wavelength = 1.06μm) were soon demonstrated for non-invasive trapping and manipulation of a single living cell [3]. With proper force calibration (to be discussed in the next paragraph), optical tweezers can be used as a convenient force transducer for the measurement of biological molecular interactions [4, 5]. Optical trapping and manipulation has since proven to be a useful tool in many research disciplines [6, 7].

Optical forces on a micro-particle trapped in optical tweezers are often calibrated by either one of the following two approaches. In the first approach, a trapped particle is dragged along a direction perpendicular to the optical axis with a gradually increasing viscous force until it escapes from the optical trap [8]. This technique is impractical (or inconvenient) for the measurement of optical force along the axial direction. Besides, it is intrinsically subjected to a significant amount of random experimental errors. The second (and a better) approach is by tracking the three-dimensional Brownian motion of a trapped particle [9] and analyzing its position distribution as was first proposed and demonstrated by Florin et al. [10]. In this paper, we use the second approach to determine the set of force constants (Kx, Ky, and K_z) which completely defines the optical force field E(x, y, z) = [K_x x² + K_y y² + K_z z²]/2 (in the parabolic potential approximation) on the trapped particle. Once the force field is determined for a given experimental condition, optical tweezers can be used as a force transducer to measure the interaction between micro-particles (e.g., antibody/antigen- coated polystyrene beads) [4] or between a micro-particle and a substrate (e.g., kinesin-coated polystyrene bead and microtuble-coated substrate) [5] for potential biological applications.

A micro-particle trapped in optical tweezers is unavoidably driven by Brownian force to undertake a three-dimensional random walk around its equilibrium position within a confined volume. The extent of fluctuation in particle’s position depends on many factors including laser power, numerical aperture of the focusing beam, the temperature & the viscosity of the surrounding medium, and the size (and the weight) of the micro-particle. Under the experimental conditions reported in this paper the fluctuation was found to be in the sub-micron to micron range (as will be explained later in the section on “Experimental Results”). The particle trajectory can be conveniently tracked by an optical position sensor such as a quadrant-photodiode (QPD). In our experiment, we used a pair of orthogonal tracking beams in conjunction with a pair of QPDs to track the thermal fluctuation of an optically trapped particle at a sampling rate on the order of 2×10⁴ samples per second per channel. For each particle trapped in optical tweezers with a specific set of trapping parameters (specified by optical wavelength, optical power, numerical aperture NA, etc.), optical force constant along each of the three orthogonal directions was deduced via a simple algorithm based on the analysis of the position distribution of the particle [10]. The experimental setup and procedure are described in the following sections.

2. Experimental setup

The main components of our experimental setup are illustrated schematically in Fig. 1. A laser beam (λ=532nm) was expanded and collimated via a spatial filter/beam expander (SF/BE) through a two-lens beam scanning system (L1 and L2, with equal focal length f = 150mm) prior to being focused by an objective lens OB I (60X, NA=0.85) for the trapping of a micro-particle (in de-ionized water) or a biological cell (in a proper buffer solution) inside a sample holder (i.e., a thin glass tube with 0.2mm × 0.2mm square cross-section). The scanning lens L1 was mounted on a motorized two-dimensional translation stage for beam scanning as is described in the next section. The transmitted and the forward scattering components of the trapping beam was collected by a second objective lens OB II (20X, NA=0.4) and projected onto the first quadrant photodiode (QPD I) to track the Brownian motion of the trapped particle in the transverse xy plane. A second laser beam (He-Ne laser, λ=632.8nm, optical power ~1mW) orthogonal to the trapping beam was focused on the trapped particle by a long working distance objective lens LOB I (Mitutoyo, 100X, NA=0.55), and its forward scattering light collected by another long working distance objective lens LOB II (Mitutoyo, 80X, NA=0.5) and projected onto the second quadrant photodiode (QPD II) to track the Brownian motion of the trapped particle in the xz plane. The optical power of the tracking He-Ne laser beam was kept as low as possible (~1mW) and a low numerical aperture (NA = 0.55) objective lens was used for weak focusing (compared with that of the trapping beam) so that the perturbation of the tracking beam on the trapped particle was minimized. Note that the displacement of the particle along the x-axis was simultaneously monitored by both QPDs independently. This redundant set of data for the particle position along the x-axis was used for checking the alignment of the second tracking beam with respect to the primary trapping (and tracking) beam as is discussed later in the section on “Experimental Results”.

 

Fig. 1. The main components of our experimental setup; a lab-coordinate system referred to in numerous places through out the text is given in the upper right corner.

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3. Calibration

The calibration required for the conversion from the QPD output voltage into the particle displacement was accomplished by two approaches. In the first approach (a dragging approach), we scanned the trapping laser beam across the focal plane to drag a trapped silica particle (diameter = 2.58μm) along the x-axis (and transversely across the tracking He-Ne laser beam) and recorded the voltage signal V_x from the QPD ∥ {where V_x = [(V1 + V2) -(V3 + V4)]/ [V1 + V2 + V3 + V4], and V_i is the voltage output from the i^th quadrant of the QPD, as is illustrated in Fig. 2(a)} while simultaneously recording the displacement of an image (with predetermined magnification) of the particle on a CCD camera. The trapping beam scanning (and the associated dragging of the particle across the tracking He-Ne laser beam) was accomplished by moving (along the x-axis) the lens L1 mounted on a motor driven translation stage as is illustrated schematically in the lower part of Fig. 1. Although the particle is unavoidably subjected to Brownian motion as it was dragged across the tracking beam through out the calibration scan, the noise due to the Brownian motion seemed to be fairly tolerable. An illustrative example of such a calibration curve is depicted in Fig. 2(b) which shows a linear dependence with a slope of approximately 0.92μm/V for particle moving in the range of about -1.3μm to + 1.3μm. By axial symmetry, we assumed that the same value of β was also valid for the conversion the output voltage V_Z (of QPD II) to the particle displacement along the z-axis. The axial conversion factor that relates the voltage (V_sum = V1 + V2 + V3 + V4) of QPD II to particle displacement along the y-axis (i.e., the axial direction of QPD II) is expected to be different from the value of β given above. We will elaborate more on this point later in the section on “Experimental Results”. We did not measure this axial conversion factor in our experiment because it is not required in our approach using a pair of orthogonal tracking beams. Note also that the value of β (0.92μm/V in this particular example) is expected to vary from experiment to experiment and is meaningful only for each specific experimental condition. The purpose of the dragging approach described above is mainly to check for the linearity range of the QPD for particle displacement along the transverse plane. In the rest of this paper, we use only the value of β obtained from the second approach (the power spectrum approach) described in the next paragraphs since in the power spectrum approach the value of [β was deduced for each experiment directly from the same set of experimental data (of output voltages V_x, V_y of QPD | and V_x, V_z of QPD ∥) recorded for that specific experiment.

In the second approach (the power spectrum approach), we followed the method prescribed by Ghislain and Webb [11] and carried out the Fourier transform of the QPD output voltage to obtain its power spectrum and to fit the power spectrum with the following Lorentzian form :

(via two fitting parameters f_c and β) as is illustrated in Fig. 2(c). In the equation given above, k_B is the Boltzman constant, T is the absolute temperature, β is the voltage-to-displacement conversion factor, r is the radius of the particle, η is the viscosity of the surrounding fluid (water in this case), f_c, the corner frequency, is a characteristic frequency of the system, and f is the frequency. The conversion factor β is deduced from the best fit of the slope of the power spectrum in the region where f≫f_c (the corner frequency) [11]. The best fit value of β= 1.2μm/V agrees with the value of β = 0.92μm/V obtained by the dragging method (described above) to within 30%. We speculate that the discrepancy is partially due to the digitizing errors in estimating the position of the image of the particle on the CCD camera based on the CCD’s pixels. Additional factors that might contribute to the discrepancy are not clearly understood and require further investigation.

 

Fig. 2. (a) A schematic illustration of QPD; (b) ; Calibration for the conversion of QPD output voltage V_x = {[(V1 + V2) - (V3 + V4)]/ Vsum} to the particle x-position by dragging a trapped particle transversely across the tracking beam; (c) a different calibration method (for the conversion of the QPD output voltage Vx to the particle x-position) via the power spectrum of Vx and the associated theoretical fit (the red straight line) to a Lorentzian form [please refer to the text for explanation].

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In what follows we briefly discuss the practical advantages as well as the drawbacks of using a pair of orthogonal tracking beams. In our experimental configuration with a pair of orthogonal tracking beams, each QPD is used to sense the position of the trapped particle only along the transverse plane (of the QPDs) and not along the axial direction. Furthermore, two sets of data (one representing the projection of the particle three-dimensional trajectory over the xy plane, and the other representing the corresponding projection on the xz plane) provide an extra set of redundant data (in particle displacement along the x-axis) which can be used to check for possible misalignment of the second tracking beam relative to the primary trapping (and tracking) beam. Specifically, if the data corresponding to the x-position of the particle obtained from one channel is plotted against the corresponding data obtained from the other channel the ideal result should be a straight line (through the origin) with slope = 1. An illustrative example is given in Fig. 3 (a). Comparatively, Fig. 3(b) is an example of the same plot when the system is either misaligned or unstable or both. This kind of plots [exemplified by Fig. 3(a) and Fig. 3(b)] provide a metric to fine tune the system to minimize the systematic error or noise. In the traditional approach (reported in the literature) [12, 13] using one QPD for three-dimensional tracking of particle trajectory, the motion of the particle along the axial direction of the QPD is obtained from V_sum = V1 + V2 + V3 + V4 (i.e., the sum of the output voltage from the four quadrants of the QPD). It has been well documented [12, 13] that the conversion of the QPD output voltage V_sum = V1 + V2 + V3 + V4 to particle displacement along the axis of the QPD relies on the constructive and the destructive interference of the un-scatted component and the forward scattering component of the tracking beam as the particle moves axially across the focal plane from the upstream to the downstream of the tracking beam. It is not as straight forward as the conversion for displacement along the transverse plane. Specifically, the linear range in which V_sum bears a linear relationship to the particle displacement (along the axial direction of the QPD) depends critically on the numerical aperture of the beam project on the QPD; experimental procedures to improve the linear range along the optical axis have been reported in the literature [12, 13]. In the case of particle tracking using a pair of orthogonal tracking beams, the particle Brownian motion along any one axis can always be obtained from the QPD whose input face is parallel to that axis. When we plot the particle z-position deduced from V_Z = V_sum of QPD| against V_Z = [(V1 + V2) - (V3 + V4)]/ [V1 + V2 + V3 + V4] deduced from QPD∥, the result is shown in Fig 3(c) which indicates that V_sum of QPD| can not be converted to the z-position of the particle via the value of β deduced from the same (power spectrum) algorithm. In addition, Fig. 3 (c) also indicates that the conversion of V_Z to z-position via V_sum = V1 + V2 + V3 + V4 is intrinsically noisier than the conversion via [(V1 + V2) - (V3 + V4)]/ [V1 + V2 + V3 + V4] because the former does not have the degree of freedom to leverage on the normalization by V_sum = V1 + V2 + V3 + V4 (as the later does).

 

Fig. 3. Particle x-positions deduced from one QPD vs. those deduced from the second QPD, (a) when the system is well-aligned; (b) when the system is mis-aligned; (c) Particle z-positions deduced from Vsum of QPD I vs. those deduced from {[(V1 + V2) - (V3 + V4)]/ Vsum} of QPD II.

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Although our approach with a pair of orthogonal tracking beams (in conjunction with a pair of QPDs) exhibits the advantages alluded above, there is one major drawback; when a high NA (e.g., NA > 1) objective lens (such as a water-immersed or an oil-immersed objective lens) is used for trapping, it is extremely difficult (if not impossible) to inject the orthogonal tracking beam through the sample without being blocked by the surrounding obstacles due to the physical constraint imposed by the high NA objective lens.

4. Experimental results

Using the power spectrum approach described in the previous section, we tracked the three-dimensional Brownian motion of a trapped particle, and analyzed its position distribution. Note that all the experimental results reported in this section were obtained from the tracking of the particle Brownian motion with all the tracking beams fixed and without any tracking beam scanning. An example of the distribution of the particle position projected on the xy plane is depicted in Fig. 4(a). We assume that the optical force field can be represented by a three-dimensional parabolic potential and use Boltzmann statistics to calculate the force constant of the optical force field along each axis via the following two equations [10] :

where ρ(x) is the probability function of the particle position along the x-axis, C is the normalization constant, E(x) is the potential energy function along the x-axis, k_B is the Boltzmann constant, and K_x is the spring constant along the x-axis.

Experimental data representing the parabolic potential E(x), E(y), and E(z) of the optical force field along the x-, y-, and z- directions together with the corresponding theoretical fits are depicted in Fig. 4(b) for the case of a 2.58μm diameter silica micro-sphere when the optical trapping power was 75mW (Note: through out this paper, the trapping power refers to the optical power measured at the trapping site), and the numerical aperture (NA) of the focusing objective lens was 0.85 (60x). Under such experimental condition, the corresponding optical force constants along the x, y, and z directions were determined to be K_x = 1.32×10^-7N/m, K_y = 1.39×10^-7N/m, and K_z = 1.1×10^-8N/m. Note that K_x and K_y agree to within 4.5% as is expected from the axial symmetry of a Gausian beam, whereas K_z is weaker than K_x and K_y by almost an order of magnitude, which is a direct consequence of the relatively weak focusing (NA = 0.85) used in this experiment. If an oiled-immersed objective lens with a much higher numerical aperture (e.g., NA = 1.25) had been used, the difference between K_z and K_x (or K_y) would have been much less.

When we increased the optical power of the trapping beam from 7.5mW to 75mW, the optical (parabolic) potential well became steeper and steeper (i.e., the force constant became larger and larger) as is depicted in Fig. 4(c) for the case of E(x). The force constants K_x, K_y, and K_z as a function of optical power (in the range of 7.5mW to 75mW) deduced from the experimental data [of Fig. 4(c)] are plotted in Fig. 5. The linear dependence agrees with the prediction of the ray-optics model (F_x = nPQ/C = K_xX) [6]. Deviation of experimental data from the ideal parabolic potential as the particle is further away from the bottom of the potential well can be clearly observed in both Fig. 4(b) and Fig. 4 (c).

 

Fig. 4. (a) The distribution of the particle position projected on the xy plane, (b) Experimental data, (“25AB;”, “×”, and “◦”), representing the optical parabolic potentials E(x), E(y), and E(z), respectively, of the optical force field along the x-, y- and z- directions along with the corresponding theoretical fits, (c) Parabolic potential E(x) along the x- direction at different optical power..

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Fig. 5. Optical force constants K_x, K_y, (upper sets of data and lines) and K_z (lower set of data and line) as a function of optical power. R² (=Regression Sum Squares / Total Sum Squares) represents a figure of merit of curve fitting; R² = 1 means a perfect fit.

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In addition to optical trapping of a silica micro-particle and the measurement of the optical force field described above, we also trapped a Chinese hamster ovary (CHO) cell [see Fig. 6(a)] and analyzed the three-dimensional optical force field by the same method. Under the same experimental condition (i.e., wavelength = 532nm, optical power =75mW, NA = 0.85), optical parabolic potentials E(x), E(y), and E(z) confining the CHO cell along the x-, y-, and z- directions are depicted in Fig. 6(b); the corresponding force constants were determined to be Kx = 2.49×10^-7N/m, K_y = 2.09×10^-7N/m, and K_z = 3.09×10^-8N/m. CHO cell was chosen because it serves as a convenient cell model for the study of cellular interaction with surrounding bio-molecules. Although we did not observe any physical changes of the CHO Cell from visual inspection via the video monitor, a trapping beam (λ = 532nm) with a numerical aperture of 0.85 and an optical power of 75mW at the trapping site was likely to cause permanent damage to the CHO cell. To the best of our knowledge, our report represents the first experimental result on the direct measurement (via a pair of orthogonal tracking beams) of optical force constants confining a biological cell. Despite the non-uniformity in refractive index of the cytoskeleton and the cytosol (n = 1.46) and anticipated partial absorption of trapping light (λ = 532nm in our case) inside the CHO cell, optical force constants confining the CHO cell is slightly larger than those confining the 2.58μm diameter silica micro-sphere. We speculate that the reduction in optical force due to the refractive index non-uniformity (and partial absorption of light) is more than compensated for by the relatively larger size of the CHO cell, whose diameter is typically in the range of 10 μm to 15μm.

 

Fig. 6. (a) Micrograph of a CHO cell trapped in optical tweezers; (b) experimental data representing optical force fields E(x): “▫◦, E(y): “×”, and E(z): “◦” on a CHO cell (when optical power = 75mW).

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5. Summary and conclusions

We report the mapping of three-dimensional optical force field on a silica micro-particle as well as that on a Chinese hamster ovary cell trapped in optical tweezers. The mapping of the force field is accomplished by tracking and analyzing the three-dimensional Brownian motion of the trapped particle via a pair of orthogonal tracking beams in conjunction with a pair of quadrant photo-diodes (QPDs). We have shown that an extra set of redundant data obtained from this method can be used to check for possible misalignment of the second tracking beam relative to the primary trapping (and tracking) beam.

At a specific trapping condition (i.e., optical wavelength = 532nm, optical power = 75mW, NA = 0.85), the force constants K_x, , K_y and K_z associated with optical parabolic potentials E(x), E(y), and E(z) confining a CHO cell along the x-, y-, and z- directions were determined to be K_x = 2.49×10^-7N/m, K_y = 2.09×10^-7N/m, and K_z = 3.09×10^-8N/m. To the best of our knowledge, our report represents the first experimental results on the direct measurement (via a pair of orthogonal tracking beams) of optical force constants confining a biological cell. Although the force constants may vary significantly from one cell to another even under the same experimental condition, the method reported in this paper is useful for quantitative measurement of cell-cell interactions once the force field on each specific sample was calibrated.