Abstract

The trapping properties of a micron-sized, ring-shaped object are examined with an optical model consisting of ray, wave, electromagnetic, and quantum theoretical components. Numerically calculated values for the trapping force vector and the predicted behavior of the ring are in reasonable agreement with reported experimental observations for this object.

© 1997 Optical Society of America

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References

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    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–289 (1986).
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  5. Y. Liu, G. J. Sonek, M. W. Berns, K. Konig, and B. J. Tromberg, “Two-photon fluorescence excitation in continuous-wave infrared optical tweezers,” Opt. Lett. 20, 2246–2248 (1995).
    [CrossRef] [PubMed]
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    [PubMed]
  7. S. M. Block, “Optical tweezers: a new tool for biophysics,” Noninvasive Techniques Cell Biol. 375–402 (1990).
  8. R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equations,” J. Opt. Soc. Am. B 12, 1680–1686 (1995).
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    [PubMed]
  15. R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micromotor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
    [CrossRef]
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  19. B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 3.
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    [CrossRef] [PubMed]
  21. W. H. Wright, G. J. Soney, Y. Tadir, and M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
    [CrossRef]

1995

1994

1993

1992

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

1990

W. H. Wright, G. J. Soney, Y. Tadir, and M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

1986

1970

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

Almaas, E.

Ashkin, A.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–289 (1986).
[CrossRef]

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

Berns, M. W.

Bjorkholm, J. E.

Block, S. M.

Bloom, A. H.

Bravo, G.

Brevik, I.

Chu, S.

Dziedzic, J. M.

Gauthier, R. C.

R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equations,” J. Opt. Soc. Am. B 12, 1680–1686 (1995).
[CrossRef]

R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micromotor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
[CrossRef]

Ghislain, L. P.

Harada, Y.

Hertz, H. M.

Higurashi, E.

Inaba, H.

Konig, K.

Liu, Y.

Malmqvist, L.

Mervis, J.

Mills, L.

Ohguchi, O.

Prentis, M.

Sato, S.

Sonek, G. J.

Soney, G. J.

W. H. Wright, G. J. Soney, Y. Tadir, and M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Svoboda, K.

Tadir, Y.

W. H. Wright, G. J. Soney, Y. Tadir, and M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Tromberg, B. J.

Ukita, H.

Wallace, S.

Waseda, Y.

Webb, W. W.

Wright, W. H.

W. H. Wright, G. J. Soney, Y. Tadir, and M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

Zarinetchi, F.

Appl. Opt.

Appl. Phys. Lett.

R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micromotor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
[CrossRef]

Biophys. J.

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

IEEE J. Quantum Electron.

W. H. Wright, G. J. Soney, Y. Tadir, and M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
[CrossRef]

J. Opt. Soc. Am. B

Opt. Lett.

Phys. Rev. Lett.

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

Other

K. Svoboda and S. M. Block, “Biological applications of optical forces,” Annu. Rev. Biophys. Biomol. Struct. 23, 247–285 (1994).
[PubMed]

B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, New York, 1991), Chap. 3.

A. Rosse, “Human genome project looks to laser tweezers,” Biophoton. Int. 17(1) 18 (1995).

S. Seeger, S. Monajembashi, K. J. Hutter, G. Futterman, J. Wolfrum, and K. O. Greulich, “Application of laser optical tweezers in immunology and molecular genetics,” Cytometry 12, 497–504 (1991).
[PubMed]

S. M. Block, “Optical tweezers: a new tool for biophysics,” Noninvasive Techniques Cell Biol. 375–402 (1990).

J. M. Colon, P. Sarosi, P. G. McGovern, A. Ashkin, J. M. Dziedzic, J. Skurnick, G. Weiss, and E. M. Bonder, “Controlled micromanipulation of human sperm in three dimensions with an infrared laser optical trap: effect on sperm velocity,” Fertil. Steril. 57, 695–698 (1992).
[PubMed]

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Figures (8)

Fig. 1
Fig. 1

Orientation of the ring-shaped micro-object with respect to the coordinate axis. The ring is 5.7 µm thick with inner and outer cylindrical radii of 2.5 µm and 5 µm, respectively. The top and bottom flat surfaces are contained in the (x, y) plane. The laser beam is directed along the z axis.

Fig. 2
Fig. 2

At the point of incidence [xi(t), yi(t), zi(t)] the incident-photon direction cosines are given by the direction cosines of the radial unit vector , which points from the center of curvature of the wave front to the point of incidence.

Fig. 3
Fig. 3

Geometry used in calculating the intercept point [xi(t), yi(t), zi(t)] when the photons interact with one of the side walls of the micro-object. The propagation parameter t is obtained from Eq. (8) and is used in Eq. (3) to obtain the intercept point.

Fig. 4
Fig. 4

Plot of the radial force F(r) versus the (x, y) coordinates. F(r) is cylindrically symmetric about the coordinate origin and has a positive value in the limits of the plot, indicating that the radial force tends to align the micro-object’s central axis with the axis of the laser beam.

Fig. 5
Fig. 5

Plot of the F(x) force when the y coordinate equals 0 and the position of the minimum waist is offset from the geometrical center of the micro-object. The offset is such that the minimum-waist position approaches the lower surface of the micro-object. When the waist position was greater than 2.85 µm, the photons would pass through the minimum waist before encountering the micro-objects surfaces. Positive forces for negative x values and negative forces for positive x values indicate that the micro-object’s central axis is pushed into the coordinate center and is aligned with the beam’s propagation axis. At the center the force is zero, indicating that in the (x, y) plane this point is a point of stable equilibrium. The centering force is observed to increase with increasing z-axis offset.

Fig. 6
Fig. 6

Plot of the z-axis force as a function of the minimum beam waist W0 and z-axis offset when the radial coordinate r = 0. A minimum in the waist curves for W0 ⩽ 0.94 µm indicates that the micro-object can be trapped in the z direction in the vicinity of the minimum-waist position. For waists greater than 0.94 µm, no minima exist in the curves plotted, indicating that these beam waists should not trap the micro-object as designed. The general features displayed in this plot were also reported experimentally.18

Fig. 7
Fig. 7

This figure shows the average relative magnitude of the photon’s radial wave vector component kr0 and uncertainty dkr as a function of the z-axis distance from the minimum waist of the Gaussian beam (W0=λ= 1 µm). The figure also shows the upper and lower bounds kr0±dkr. On the same scale the waist W(z)/10 is traced, which corresponds to the radial coordinate r used in the generation of the figure.

Fig. 8
Fig. 8

Geometry used to obtain (a) the reflected and (b) the refracted direction cosines given the incident-direction cosines and normal at the point of intercept.

Equations (52)

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

I(x, y, z)=2PπWz2exp-2(x2+y2)Wz2.
Wz=W01+(z/z0)2.
x(t)=xs+lt,
y(t)=ys+mt,
z(t)=zs+nt,
rcylinder=x(t)2+y(t)2,-2.85z(t)+2.85,
z(t)=±2.85,5x(t)2+y(t)25.
R=(R-Δz)2+x(t)2+y(t)2,
R=[z(t)+Δz](1+{z0/z(t)+Δz}2).
li(t)=±[x(t)-cx]/R,
mi(t)=±[y(t)-cy]/R,
ni(t)=[z(t)-cz]/R.
t2+txs2+ys2+(xs2+ys2-rcylinder2)=0.
P=k=2πninλ[li(t)xˆ+mi(t)yˆ+ni(t)zˆ],
F=allinteractingphotonsdPdt.
dPr=hninλ{[l(t)-lr(t)]xˆ+[m(t)-mr(t)]yˆ+[n(t)-nr(t)]zˆ}
dPt=hninλ{[l(t)-nrellt(t)]xˆ+[m(t)-nrelmt(t)]yˆ+[n(t)-nrelnt(t)]zˆ},
F=allpointsofinterceptdFi=allpointsofinterceptNi[RavedPr+(1-RavedPt)].
Ni=I(x, y, z)hcλdA.
τ=allpointsofinterceptdτi=allpointsofinterceptr·(dFi).
F(r)=±F2(x)+F2(y),
dPr=dkr=π/W0.
Pr=kr0±dkr,
θin=cos-1[|ali(t)+bmi(t)+cni(t)|].
θref=θin.
xn=xi(t)+ar cos(θin),
yn=yi(t)+br cos(θin),
zn=zi(t)+cr cos(θin),
r=[xi(t)-xs]2+[yi(t)-ys]2+[zi(t)-zs]2.
av=(xn-xs)/[r sin(θin)],
bv=(yn-ys)/[r sin(θin)],
cv=(zn-zs)/[r sin(θin)],
xr=xs+2rav sin(θin),
yr=ys+2rbv sin(θin),
zr=zs+2rcv sin(θin).
lr(t)=[xr-xi(t)]/r,
mr(t)=[yr-yi(t)]/r,
nr(t)=[zr-zi(t)]/r,
r=[xi(t)-xr]2+[yi(t)-yr]2+[zi(t)-zr]2.
xd=xs-atd,
yd=ys-btd,
zd=zs-ctd,
td=axs+bys+czs+d.
dt=r sin(θin)tan[(π/2-θt],
nin sin(θin)=nout sin(θt).
xt=xd+adt,
yt=yd+bdt,
zt=zd+cdt,
lt(t)=[xi-xt(t)]rt,
mt(t)=[yi-yt(t)]rt,
nt(t)=[zi-zt(t)]rt,
rt=[xi(t)-xt]2+[yi(t)-yt]2+[zi(t)-zt]2.

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