Abstract

An enhanced ray optics model is applied to the study of the optical levitation and trapping properties of a glass cubic object. It is found that for certain highly symmetric orientations simultaneous force and torque equilibrium can exist in the lowest-order TEM00 laser beam profile. For analytical purposes, the square surfaces of the cube are divided into two identical triangular surfaces, and the interaction of the rays with these triangular surfaces simplifies the computation of the total force and torque on the cube. The technique developed can easily be extended to the study of other regular or complex structures.

© 2000 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, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
    [CrossRef]
  3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  4. R. C. Gauthier, “Trapping model for a 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]
  5. E. Higurashi, H. Ukita, H. Tanaka, O. Ohguchi, “Optically induced rotation of anisotropic micro-objects fabricated by surface micromachining,” Appl. Phys. Lett. 64, 2209–2210 (1994).
    [CrossRef]
  6. R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B 14, 3323–3333 (1997).
    [CrossRef]
  7. R. Lewis, “Special delivery of sperm,” Photon. Spectra44–45 (July1996).
  8. S. C. Kuo, M. P. Sheetz, “Optical tweezers in cell biology,” Trends Cell Biol. 2, 117–118 (1992).
    [CrossRef]
  9. W. H. Wright, G. J. Sonek, Y. Tadir, M. W. Berns, “Laser trapping in cell biology,” IEEE J. Quantum Electron. 26, 2148–2157 (1990).
    [CrossRef]
  10. Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).
  11. E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
    [CrossRef]
  12. R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922–1930 (1992).
    [CrossRef]
  13. T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
    [CrossRef]
  14. R. C. Gauthier, M. Ashman, “Simulated dynamic behavior of single and multiple spheres in the trap region of focused laser beams,” Appl. Opt. 37, 6421–6430 (1998).
    [CrossRef]
  15. E. Higurashi, R. Sawada, T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett. 73, 3034–3036 (1998).
    [CrossRef]
  16. B. E. A. Saleh, M. K. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]

1998 (2)

E. Higurashi, R. Sawada, T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett. 73, 3034–3036 (1998).
[CrossRef]

R. C. Gauthier, M. Ashman, “Simulated dynamic behavior of single and multiple spheres in the trap region of focused laser beams,” Appl. Opt. 37, 6421–6430 (1998).
[CrossRef]

1997 (3)

1996 (1)

R. Lewis, “Special delivery of sperm,” Photon. Spectra44–45 (July1996).

1994 (1)

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

1992 (2)

1991 (1)

T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef]

1990 (2)

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

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

1986 (1)

1971 (1)

A. Ashkin, J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[CrossRef]

1970 (1)

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

Asch, R. H.

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

Ashkin, A.

A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
[CrossRef] [PubMed]

A. Ashkin, J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[CrossRef]

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

Ashman, M.

Bakker Schut, T. C.

T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef]

Berns, M. W.

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

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

Bjorkholm, J. E.

Brevik, I.

Chu, S.

Dziedzic, J. M.

Gauthier, R. C.

Greve, J.

T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef]

Grooth, B. G.

T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef]

Gussgard, R.

Hesselink, G.

T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef]

Higurashi, E.

E. Higurashi, R. Sawada, T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett. 73, 3034–3036 (1998).
[CrossRef]

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

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

Ito, T.

E. Higurashi, R. Sawada, T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett. 73, 3034–3036 (1998).
[CrossRef]

Kuo, S. C.

S. C. Kuo, M. P. Sheetz, “Optical tweezers in cell biology,” Trends Cell Biol. 2, 117–118 (1992).
[CrossRef]

Lewis, R.

R. Lewis, “Special delivery of sperm,” Photon. Spectra44–45 (July1996).

Lindmo, T.

Ohguchi, O.

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

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

Ord, T.

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

Saleh, B. E. A.

B. E. A. Saleh, M. K. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Sawada, R.

E. Higurashi, R. Sawada, T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett. 73, 3034–3036 (1998).
[CrossRef]

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

Sheetz, M. P.

S. C. Kuo, M. P. Sheetz, “Optical tweezers in cell biology,” Trends Cell Biol. 2, 117–118 (1992).
[CrossRef]

Sonek, G. J.

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

Tadir, Y.

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

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

Tamamura, T.

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

Tanaka, H.

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

Teich, M. K.

B. E. A. Saleh, M. K. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

Ukita, H.

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

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

Vela, O.

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

Wright, W. H.

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

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

Appl. Opt. (1)

Appl. Phys. Lett. (3)

E. Higurashi, R. Sawada, T. Ito, “Optically induced angular alignment of birefringent micro-objects by linear polarization,” Appl. Phys. Lett. 73, 3034–3036 (1998).
[CrossRef]

A. Ashkin, J. M. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. 19, 283–285 (1971).
[CrossRef]

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

Cytometry (1)

T. C. Bakker Schut, G. Hesselink, B. G. Grooth, J. Greve, “Experimental and theoretical investigation on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
[CrossRef]

Fertil. Steril. (1)

Y. Tadir, T. Ord, W. H. Wright, R. H. Asch, O. Vela, M. W. Berns, “Force generated by human sperm correlated to velocity and determined using laser generated optical trap,” Fertil. Steril. 53, 945–947 (1990).

IEEE J. Quantum Electron. (1)

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

J. Appl. Phys. (1)

E. Higurashi, O. Ohguchi, T. Tamamura, H. Ukita, R. Sawada, “Optically induced rotation of dissymmetrically shaped fluorinated polyimide micro-objects in optical traps,” J. Appl. Phys. 82, 2773–2779 (1997).
[CrossRef]

J. Opt. Soc. Am. B (3)

Opt. Lett. (1)

Photon. Spectra (1)

R. Lewis, “Special delivery of sperm,” Photon. Spectra44–45 (July1996).

Phys. Rev. Lett. (1)

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

Trends Cell Biol. (1)

S. C. Kuo, M. P. Sheetz, “Optical tweezers in cell biology,” Trends Cell Biol. 2, 117–118 (1992).
[CrossRef]

Other (1)

B. E. A. Saleh, M. K. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Representation of the incident, reflected, and transmitted photons specified in the (X, Y, Z) coordinate system. The five degrees of freedom of the cube are also specified relative to the coordinate systems.

Fig. 2
Fig. 2

Torque versus rotation angle in a plane that brings the cube from face down at 0 deg, corner down at 54.731560 deg, and edge down at 90 deg. The orientation of the cube at the three zero-torque values is shown in a top view. Only the corner-down orientation is stable for this plane of rotation.

Fig. 3
Fig. 3

Dot product difference between the beam propagation axis and the cube body diagonal computed before and after rotation of the cube based on the computed torque vector. Over the -10 to +10 deg range shown, the values are always negative, indicating that the corner-down orientation is preferred. [At 0 deg (corner-down orientation) the dot product difference is 0, implying stability).] Inset, relation of the planes of rotation (-10 to +10 deg) to the cube shown in a top view.

Fig. 4
Fig. 4

Radial force versus (X, Y, 0) offset of the corner-down cube. At (X, Y) = (0, 0) the radial force is zero, and for small offsets in the plane the radial force is negative, implying that the cube would be pushed into axial alignment with the beam propagation axis.

Fig. 5
Fig. 5

Axial force versus axial z displacement of the corner-down cube. The force goes to zero for Z = 0 µm, which suggests the possibility of trapping, and at Z = 27 µm, which suggests levitation.

Fig. 6
Fig. 6

Dynamic modeling of the cube’s trajectory as a function of time when the cube is launched below the minimum waist of the beam. The cube comes into alignment with the beam axis corner down and slows its upward motion into the (0, 0, 0) position, where trapping is possible with a lower-power beam. (Note: The cube is plotted 10 times smaller in size than the trajectory coordinates.)

Fig. 7
Fig. 7

Same as Fig. 3 but for the corner-down cube at (0, 0, 27), the possible levitation location. For a few degrees of rotation about the corner-down orientation, the cube prefers the corner-down orientation, and stability exists. For larger angular deviations the dot product difference may be positive, the corner-down orientation is not restored, and the cube enters an unstable region.

Fig. 8
Fig. 8

Axial force versus axial position of the corner-down cube in the dual-beam system. The F z = 0 µm crossing at Z = -5 µm close to the minimum waist location of the two beams suggests the possibility of dual counterpropagating trapping. Gravity is included in the plot and accounts for the downward shift in the force curve.

Fig. 9
Fig. 9

Same as Fig. 4 but for the dual-beam system. The plot indicates radial trapping of the cube in the corner-down orientation.

Fig. 10
Fig. 10

Same as Figs. 3 and 7 but for the corner-down cube in the dual-beam system. The corner-down orientation is preferred.

Fig. 11
Fig. 11

Dynamic modeling of the cube launched at (3, 3, 50) in the dual-beam system. The cube rapidly centers onto the beams’ axes, orients itself corner down, and proceeds to come to rest at the trap position.

Fig. 12
Fig. 12

Original object and a triangular representation of its surface for cube, an actuator, and a red-blood cell.

Fig. 13
Fig. 13

Single triangular surface, with incident, reflected, and transmitted rays shown. A valid intercept is determined by comparison of corner-angle pairs.

Equations (23)

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

Pi=ki=hλlixˆ+miyˆ+nizˆ
Pr=kr=hλlrxˆ+mryˆ+nrzˆ
Pt=kt=hλltxˆ+mtyˆ+ntzˆ
dPr=hninλ0li-lrxˆ+mi-mryˆ+ni-nrzˆ,
dPt=hninλ0li-nrelltxˆ+mi-nrelmtyˆ+ni-nrelntzˆ,
Rave=RTE2+RTM2/2,
Ni=Ixi, yi, ziEdA.
Ix, y, z=2PπWz2exp-2x2+y2Wz2,
Wz=W01+z/z021/2,  z0=π/λW02.
Fper surface=APIdFi=API NiRavedPr+1-RavedPt,
τper surface=APIdτ=APIri×dFi.
xt+dt=xt+vtxdt, yt+dt=yt+vtydt, zt+dt=zt+vtzdt,
vxt+dt=vtx+Fx-bxvtxmdt, vyt+dt=vty+Fy-byvtymdt, vzt+dt=vtz+Fz-bzvtzmdt,
dβ=|τ|2Idt2.
I=i ri2dmi.
Ax+By+Cz+D=0
xi=x0+lit, yi=y0+mit, zi=z0+nit.
t=-Ax0+By0+Cz0+DAli+Bmi+Cni,
rr=2ri·nˆnˆ+ri.
lr=xi-x0|ri|1+2A cosθin, mr=yi-y0|ri|1+2B cosθin, nr=zi-z0|ri|1+2C cosθin,
rt=ninnout-1ri·nˆnˆ+ninnoutri.
lt=xi-x0|ri|cosθoutcosθinninnout+ninnout-1A cosθin, mt=yi-y0|ri|cosθoutcosθinninnout+ninnout-1B cosθin, nt=zi-z0|ri|cosθoutcosθinninnout+ninnout-1C cosθin,
nin sinθin=nout sinθout.

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