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Three-dimensional tracking of Brownian motion of a particle trapped in optical tweezers with a pair of orthogonal tracking beams and the determination of the associated optical force constants

Ming-Tzo Wei and Arthur Chiou

Open Access Open Access

Abstract

We report the first experimental results on quantitative mapping of three-dimensional optical force field on a silica micro-particle and on a Chinese hamster ovary cell trapped in optical tweezers by using a pair of orthogonal laser beams in conjunction with two quadrant photo-diodes to track the particle’s (or the cell’s) trajectory, analyze its Brownian motion, and calculate the optical force constants in a three-dimensional parabolic potential model. For optical tweezers with a 60x objective lens (NA = 0.85), a trapping beam wavelength λ = 532nm, and a trapping optical power of 75mW, the optical force constants along the axial and the transverse directions (of the trapping beam) were measured to be approximately 1.1×10-8N/m and 1.3×10-7N/m, respectively, for a silica particle (diameter = 2.58μm), and 3.1×10-8 N/m and 2.3×10-7 N/m, respectively, for a Chinese hamster ovary cell (diameter ~ 10 μm to 15 μm). The set of force constants (Kx, Ky, and Kz) completely defines the optical force field E(x, y, z) = [Kx x2 + Ky y2 + Kz z2]/2 (in the parabolic potential approximation) on the trapped particle. Practical advantages and limitations of using a pair of orthogonal tracking beams are discussed.

©2005 Optical Society of America

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Figures (6)
Fig. 1.
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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Fig. 2.
Fig. 2. (a) A schematic illustration of QPD; (b) ; Calibration for the conversion of QPD output voltage Vx = {[(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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Fig. 3.
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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Fig. 4.
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.
Fig. 5. Optical force constants Kx, Ky, (upper sets of data and lines) and Kz (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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Fig. 6.
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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Equations (3)

Equations on this page are rendered with MathJax. Learn more.

S v ( f ) = k B T β 2 6 π 3 ηr ( f c 2 + f 2 )
ρ ( x ) = C exp ⁡ [ − E ( x ) k B T ]
E ( x ) = − k B T In ρ ( x ) + k B T In C = K x x 2 2
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