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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References

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Appl. Phys. A (1)

E.-L. Florin, A. Pralle, E. H. K. Stelzer, and J. K. H. Horber, "Photonic forcemicroscope calibration by thermal noise analysis," Appl. Phys. A 66, 75-78 (1998).
[CrossRef]

Biophysical J. (3)

S. Kulin, R. Kishore, J. B. Hubbard, and K. Helmerson, "Real-time measurement of spontaneous antigen-antibody dissociation," Biophysical J. 83, 1965-1973 (2002).
[CrossRef]

M. J. Lang, C. L. Asbury, J. W. Shaevitz, and S. M. Block, "An automated two-dimensional optical force clamp for single molecule studies," Biophysical J. 83, 491-501 (2002).
[CrossRef]

I. M. Peters, B. G. d. Grooth, J. M. Schins, C. G. Figdor, and J. Greve, "Three dimensional single �?? particle tracking with nanometer resolution," Biophysical J. 69, 2762-2766 (1998).

IEEE J. Sel. Top. Quantum Electron. (1)

A. Ashkin, "History of optical trapping and manipulation of small-neutral particle, atoms, and molecules," IEEE J. Sel. Top. Quantum Electron. 6, 841-856 (2000).
[CrossRef]

Microscopy Research and Technique (1)

A. Pralle, M. Prummer, E.-L. Florin, E. H. K. Stelzer, and J. K. H. Horber, "Three-dimensional high-resolution particle tracking for optical tweezers by forward scattered light," Microscopy Research and Technique 44, 378-386 (1999).
[CrossRef] [PubMed]

Nature (1)

A. Ashkin and J. M. Dziedzic, "Optical trapping and manipulation of single cell using infrared laser beams," Nature 330, 769-771 (1987).
[CrossRef] [PubMed]

Opt. Commun. (1)

N. Malagnino, G. Pesce, A. Sasso, and E. Arimondo, "Measurements of trapping efficiency and stiffness in optical tweezers," Opt. Commun. 214, 15-24 (2002).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett (1)

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. Lett 24, 156-159 (1970).
[CrossRef]

Review of Scientific Instruments (1)

K. C. Neuman and S. M. Block, "Optical trapping," Review of Scientific Instruments 78, 2787-2809 (2004).
[CrossRef]

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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.

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].

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.

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..

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.

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).

Equations (3)

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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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