Abstract

Locating and steering entire ensembles of microscopic objects has become extremely practical with the emergence of holographic optical tweezers. Application of this technology to single molecule experiments requires great accuracy in the spatial positioning of optical traps. This paper calculates the theoretical position resolution of a single holographic beam, predicting that sub-nanometer resolution is easily achieved. Experimental corroboration of the spatial resolution’s inverse dependence on the hologram’s number of pixels and phase levels is presented. To at least a nanometer range position resolution, multiple optical tweezers created by complex superposition holograms also follow the theoretical predictions for a single beam.

© 2005 Optical Society of America

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References

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  24. Due to total internal reflection of rays at the glass coverslip / water interface, the real NA of the objective (1.45) is reduced to 1.33.

Biophys. J. (3)

J. Guck, R. Ananthakrishnan, H.Mahmood, T. J. Moon, C. C. Cunningham, and J. Kas, “The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells,” Biophys. J. 81, 767–784 (2001).
[CrossRef] [PubMed]

M. Lang, C. L. Asbury, J. Shaevitz, and S. M. Block, “An Automated Two-Dimensional Optical Force Clamp for Single Molecule Studies,” Biophys. J. 83, 491501 (2002).
[CrossRef]

G. J. Wuite, R. J. Davenport, A. Rappaport, and C. Bustamante, “An integrated laser trap/flow control video microscope for the study of single biomolecules,” Biophys. J. 79, 1155–1167 (2000).
[CrossRef] [PubMed]

Contemp. Phys. (1)

D. McGloin and K. Dholakia, “Bessel beams: diffraction in a new light,” Contemp. Phys. 46, 15–28 (2005).
[CrossRef]

J. Mod. Opt. (1)

G. Sinclair, P. Jordan, J. Leach, and M. J. Padgett, “Defining the trapping limits of holographical optical tweezers,” J. Mod. Opt. 51, 409–414 (2004).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424, 810–816 (2003).
[CrossRef] [PubMed]

New J. Phys. (1)

G.Whyte and J. Courtial, “Experimental demonstration of holographic three-dimensional light shaping using a Gerchberg-Saxton algorithm,” New J. Phys. 7, 1–12 (2005).
[CrossRef]

Opt. Commun. (2)

J. Liesener, M. Reicherter, T. Haist, and H. J. Tiziani, “Multi-functional optical tweezers using computergenerated holograms,” Opt. Commun. 185, 77 (2000).
[CrossRef]

J. E. Curtis, B. A. Koss, and D. G. Grier, “Dynamic holographic optical tweezers,” Opt. Commun. 207, 169–175 (2002).
[CrossRef]

Opt. Express (5)

Opt. Lett. (3)

Optik (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of the phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Phys. Rev. Lett. (1)

T. Gustavson, A. Chikkatur, A. Leanhardt, A. Gorlitz, S. Gupta, D. Pritchard, and W. Ketterle, “Transport of Bose-Einstein condensates with optical tweezers,” Phys. Rev. Lett. 88, 020,401 (2001).
[CrossRef]

Rev. Sci. Instr. (3)

K. C. Neuman and S. Block, “Optical Trapping,” Rev. Sci. Instr. 75, 2787–2809 (2004).
[CrossRef]

E. R. Dufresne, G. C. Spalding, M. T. Dearing, S. A. Sheets, and D. G. Grier, “Computer-generated holographic optical tweezer arrays,” Rev. Sci. Instr. 72, 1810–1816 (2001).
[CrossRef]

E. R. Dufresne and D. G. Grier, “Optical tweezer arrays and optical substrates created with diffractive optical elements,” Rev. Sci. Instr. 69, 1974–1977 (1998).
[CrossRef]

Other (1)

Due to total internal reflection of rays at the glass coverslip / water interface, the real NA of the objective (1.45) is reduced to 1.33.

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Figures (5)

Fig. 1.
Fig. 1.

Schematic of the HOT apparatus. The setup uses two different objectives for imaging and focusing the trapping laser.

Fig. 2.
Fig. 2.

(Left) Phase structure of 3 phase gratings. The solid lines represent the ideal non-pixelated phase gratings. The horizontal bars show the phase levels for pixelation with N = 6 pixels and g =5 phase levels. Grating (a) shifts the beam by δf , (b) by 2δf and (*) by 1.5δf . The inset shows the phase mask for (a). The HOT position resolution is determined by δf /kmax , where kmax is proportional to the shaded area divided by (2π/g). (Right) Movement of a single OT over a total distance of 2δf using a step size of δf /200 (2 nm). The overlying line shows trap positions smoothed by adjacent averaging (20 frames).

Fig. 3.
Fig. 3.

Comparison of two series of holograms that shift the position of an OT by a total distance δf in 400 steps. The squares have fixed N = 12 and the circles have fixed g = 2. The difference between the experimentally determined 400 trap positions and an ideal continuous shift is given by the rmsd. The dotted line shows the fitted data given by rmsdδf /kmax = δf /(Ng /2). The inset shows the laser focus for (a) N = 512, g = 2 and (b) N = 12, g= 85.

Fig. 4.
Fig. 4.

Superpositions of prism holograms (with N = 512 and g = 130) are used to trap four polystyrene particles with 2 μm diameter, as shown in the inset. The three outer particles are moved radially with variable step sizes, resulting in different speeds and total displacement (traps 1–3 have theoretical steps of 16 nm, 8 nm, 4 nm). The data points show mean positions of 50 frames. The bead trapped in the center is held in the zeroth order beam.

Fig. 5.
Fig. 5.

Detailed motion of optical trap 2 as shown in Fig. 4. The predicted step size using Eq. 2 is δf /50 = 8 nm, in agreement with the experimental measurement of a mean displacement of 8 nm. The line shows trap positions smoothed by adjacent averaging of 20 frames, and the horizontal bars represent the mean position of the trap during 100 frames of each hologram.

Equations (3)

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ϕ ( r ) = 2 π λf ( f 1 f 2 ) ρ · r mod ( 2 π ) ,
δ f / k max ,
k max ( πN ) × ( g / ( 2 π ) ) = Ng / 2 ·

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