Abstract

We investigate theoretically the experimentally observed nature of the optical trapping force present on elongated, cylindrically shaped micro-objects. The objects chosen have either flat or spherical ends and elongated cylindrical bodies of various length-to-radii ratios. The trapping force, if present, occurs when the objects are placed in the focal region of a highly divergent laser beam. Results indicate that cylindrical objects can be trapped aligned with the laser beam axis.

© 1997 Optical Society of America

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References

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  1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. 24, 156–159 (1970).
    [CrossRef]
  2. A. Ashkin, J. M. Dziedzic, E. J. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  3. E. Higurashi, O. Ohguchi, and H. Utika, “Optical trapping of low-refractive-index microfabricated objects using radiation pressure exerted on their inner walls,” Opt. Lett. 20, 1931–1933 (1995).
    [CrossRef] [PubMed]
  4. R. Lewis, “Special delivery for sperm,” Photon. Spectra (July 1996), p. 44.
  5. J. M. Colon, P. Sarosi, P. G. McGovern, A. Ashkin, J. D. 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,” Fert. Ster. 57, 695–698 (1992).
  6. Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fert. Ster. 52, 870–873 (1989).
  7. J. Newitt, “Human genome project looks to laser tweezers,” Biophoton. Intl. (January/February 1995), pp. 17–18.
  8. S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).
    [CrossRef]
  9. R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
    [CrossRef] [PubMed]
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    [CrossRef]
  11. 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]
  12. R. C. Gauthier, “Ray optics model and numerical computations for the radiation pressure micro-motor,” Appl. Phys. Lett. 67, 2269–2271 (1995).
    [CrossRef]
  13. R. C. Gauthier, “Theoretical model for an improved radiation pressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 (1996).
    [CrossRef]
  14. R. C. Gauthier, “Trapping model for the low-index ring-shaped micro-object in a focused lowest-order Gaussian laser beam profile,” J. Opt. Soc. Am. B 14, 782–789 (1997).
    [CrossRef]
  15. B. Saleh and M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
  16. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
    [CrossRef] [PubMed]
  17. S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bacterial polyhooks measured with optical tweezers,” Cytometry 12, 492–496 (1991).
    [CrossRef] [PubMed]
  18. S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
    [CrossRef]
  19. S. M. Block, “Optical tweezers: a new tool for biophysics,” Noninvasive Techniques Cell Biol., 375–402 (1990).
  20. A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
    [CrossRef]
  21. K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by “multiple” optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
    [CrossRef]

1997 (1)

1996 (1)

R. C. Gauthier, “Theoretical model for an improved radiation pressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 (1996).
[CrossRef]

1995 (3)

1994 (1)

E. Higurashi, H. Utika, H. Tanaka, and O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

1993 (1)

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by “multiple” optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[CrossRef]

1992 (1)

J. M. Colon, P. Sarosi, P. G. McGovern, A. Ashkin, J. D. 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,” Fert. Ster. 57, 695–698 (1992).

1991 (3)

S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).
[CrossRef]

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bacterial polyhooks measured with optical tweezers,” Cytometry 12, 492–496 (1991).
[CrossRef] [PubMed]

1989 (2)

S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fert. Ster. 52, 870–873 (1989).

1987 (2)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

1986 (1)

1970 (1)

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

Appl. Phys. Lett. (3)

E. Higurashi, H. Utika, H. Tanaka, and O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
[CrossRef]

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

R. C. Gauthier, “Theoretical model for an improved radiation pressure micromotor,” Appl. Phys. Lett. 69, 2015–2017 (1996).
[CrossRef]

Cytometry (3)

R. W. Steubing, S. Cheng, W. H. Wright, Y. Numajiri, and M. W. Berns, “Laser induced cell fusion in combination with optical tweezers: the laser cell fusion trap,” Cytometry 12, 505–510 (1991).
[CrossRef] [PubMed]

S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bacterial polyhooks measured with optical tweezers,” Cytometry 12, 492–496 (1991).
[CrossRef] [PubMed]

K. Visscher, G. J. Brakenhoff, and J. J. Krol, “Micromanipulation by “multiple” optical traps created by a single fast scanning trap integrated with the bilateral confocal scanning laser microscope,” Cytometry 14, 105–114 (1993).
[CrossRef]

Electron. Lett. (1)

S. Sato, M. Ishigure, and H. Inaba, “Optical trapping and rotational manipulation of microscopic particles and biological cells using higher-order mode Nd:YAG laser beams,” Electron. Lett. 27, 1831–1832 (1991).
[CrossRef]

Fert. Ster. (2)

J. M. Colon, P. Sarosi, P. G. McGovern, A. Ashkin, J. D. 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,” Fert. Ster. 57, 695–698 (1992).

Y. Tadir, W. H. Wright, O. Vafa, T. Ord, R. H. Asch, and M. W. Berns, “Micromanipulation of sperm by a laser generated optical trap,” Fert. Ster. 52, 870–873 (1989).

J. Opt. Soc. Am. B (2)

Nature (London) (2)

S. M. Block, D. F. Blair, and H. C. Berg, “Compliance of bacterial flagella measured with optical tweezers,” Nature (London) 338, 514–518 (1989).
[CrossRef]

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
[CrossRef]

Opt. Lett. (2)

Phys. Rev. Lett. (1)

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

Science (1)

A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science 235, 1517–1520 (1987).
[CrossRef] [PubMed]

Other (4)

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

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

R. Lewis, “Special delivery for sperm,” Photon. Spectra (July 1996), p. 44.

J. Newitt, “Human genome project looks to laser tweezers,” Biophoton. Intl. (January/February 1995), pp. 17–18.

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Figures (17)

Fig. 1
Fig. 1

Flat-end-face cylindrical micro-object of length L and radius R oriented along the X axis. The incident laser beam is directed along the Z axis and comes to a focus at a distance Zw before passing the plane containing the cylinder axis. In the analysis the cylinder may also contain spherical end faces and may be oriented along the Z axis.

Fig. 2
Fig. 2

Plot of the input beam power needed to maintain the cylinder in the trap and at a position R+0.5 µm from the minimum waist. The lower two traces are for the flat-end cylinders; the upper two traces, with the insert, are for the spherical-end cylinders. The length L=0 µm for the spherical-end cylinders corresponds to a perfect sphere in the laser beam. Squares, X-oriented and flat end faces; diamonds, Z-oriented and flat end faces; triangles, X-oriented and spherical end faces; solid line, Z-oriented and spherical end faces; insert, extension of the Z-oriented spherical end-face cylinder to length L=125 µm.

Fig. 3
Fig. 3

Fy versus (X, Y) coordinates for the spherical-end cylinder oriented along the X axis with L=16 µm and R=5 µm. The figure shows that Fy>0 for Y<0, Fy<0 for Y>0, and Fy=0 at Y=0, indicating that the Y-axis force acts as a trapping-force component.

Fig. 4
Fig. 4

Fy versus (X, Y), first quadrant, when L=0 µm and R=5 µm, a sphere of 5-µm radius. The sphere experiences a Y-axis trapping force (see caption Fig. 3).

Fig. 5
Fig. 5

Fx versus (X, Y) coordinates for the spherical-end-face cylinder oriented along the X axis with L=16 µm and R =5 µm. When the beam is located close to one of the spherical ends, the cylinder’s center is pushed into the central region of the beam. The Fx force component acts as a trapping force for X-direction displacements of the cylinder. The local maxima and minima contained in the dominant profiles are artifacts of the grid size used to generate the figure.

Fig. 6
Fig. 6

Fx versus (X, Y) first quadrant for a spherical-end-face cylinder with L=1 µm and R=5 µm. This force component acts as trapping force, holding the cylinder in the central region of the beam.

Fig. 7
Fig. 7

Radial force Fr for a spherical-end-face cylinder of length L=16 µm and R=5 µm. The radial force acts to hold the cylinder in the beam. The cylinder is free to slide along the central maximum line (stable line) of the figure. The symmetry of the Fr curve reflects the cross-sectional symmetry of the spherical-end-face cylinder located in the laser beam. The local maxima and minima contained in the dominant profiles are artifacts of the grid size used to generate the figure.

Fig. 8
Fig. 8

Y-axis torque component τy for the spherical-end-face cylinder of R=5 µm. Rotation angles are measured between the X axis and the central axis of the cylinder. For L=0 µm, τy=0 for the cylinder equivalent to a sphere. For L>0 µm the torque present increases the angular misalignment between the central axis and the X-coordinate axis.

Fig. 9
Fig. 9

Fy force component for a flat-end cylinder when L =1 µm and R=5 µm. The Fy force component acts as a trapping force, aligning the cylinder with the X axis. This curve is similar to Fig. 4 for the spherical-end-face cylinder of length L =0 µm.

Fig. 10
Fig. 10

Fx force component versus (X, Y) coordinates for the flat-end-face cylinder when L=16 µm and R=5 µm. The force component tends to center the cylinder in the beam. The origin of the two minima and the zero force at X=L/2 is discussed in the text.

Fig. 11
Fig. 11

Fx force component for positions in the first quadrant for the flat-end-face cylinder with L=1 µm and R=5 µm. Fx is greater than zero for the nonaxial positions shown, indicating that the cylinder is pushed out of the beam and is not trapped.

Fig. 12
Fig. 12

Plot of Fx along the X axis for flat-end cylinders with R=5 µm and lengths 0.25L8 µm. Stable 3-D trapping is possible for those cylinders with Fx<0 when X>0. Short cylinders with lengths less than 1.25 µm cannot be trapped for the design parameters used in our model system.

Fig. 13
Fig. 13

Y-axis torque component τy versus rotation angle of the flat-end cylinder in the (X, Z) plane. Stable torque equilibrium exists for cylinders with lengths in the 1.25–2.00-µm range.

Fig. 14
Fig. 14

Plot for Fr versus r for the spherical-end-face cylinder of radius R=5 µm. The radial force is a trapping force because Fr<0 for r>0 and Fr=0 at r=0.

Fig. 15
Fig. 15

Plot of Fr versus r for the flat-end cylinder of radius R=5 µm. The radial force is a trapping force because Fr <0 for r>0 and Fr=0 at r=0.

Fig. 16
Fig. 16

Y-axis torque component present on a spherical-end-face cylinder (R=5 µm) rotated in the (X, Z) plane. A torque equilibrium does not exist because τy>0 for θy>0 and τy <0 for θy<0.

Fig. 17
Fig. 17

Stable torque rotation angle versus cylinder length L. Diamonds represent calculated points with the curve as the best fit. Stable Z-axis-oriented cylinders exist when the angle is 90°, whereas stable X-axis-oriented cylinders exist when the angle is 0°. For all other angles the cylinders find a stable orientation with the central axis inclined to the beam-propagation angle.

Equations (4)

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dPr= 2πλnin[(l0-lr)x+(m0-mr)y+(n0-nr)z],
dPt= 2πλnin[(l0-nrellt)x+(m0-nrelmt)y+(n0-nrelnt)z].
F=allpointsofinterceptdFi=allpointsofinterceptNi[RavedPr+(1-Rave)dPt],
τ=allpointsofinterceptdτi=allpointsofinterceptri×dFi

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