Abstract

We theoretically investigate the three-dimensional (3D) trapping force acting on a microsphere held in a pair of counterpropagating beams produced by the generalized phase contrast (GPC) method. In the case of opposing beams of equal power, we identify the range of beam waist separation s that results in a stable 3D optical potential-well by assessing the dependence of the axial and transverse force curves on s. We also examine how the force curves are influenced by other parameters such as size and refractive index of the microsphere. Aside from force curves of beam tandems with equal powers, we also numerically calculate force curves for cases of beam pairs having disparate relative strengths. These calculations enable us to elucidate the large dynamic range for axial position control of microparticles in GPC-based counterpropagting-beam traps.

© 2006 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. Constable, J. Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, "Demonstration of a fiber-optical light-force trap," Opt. Lett. 18, 1867-1869 (1993).
    [CrossRef] [PubMed]
  3. E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995).
    [CrossRef]
  4. G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: a theoretical and experimental study," Phys. Lett. A 59, 6-8 (1976).
    [CrossRef]
  5. E. Sidick, S. D. Collins, and A. Knoesen, "Trapping forces in a multiple-beam fiber-optic trap," Appl. Opt. 36, 6423-6433 (1997).
    [CrossRef]
  6. P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29, 2270-2272 (2004).
    [CrossRef] [PubMed]
  7. P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005).
    [CrossRef]
  8. I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005).
    [CrossRef]
  9. P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005).
    [CrossRef] [PubMed]
  10. J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996).
    [CrossRef]
  11. J. Glückstad and P. C. Mogensen, "Optimal phase contrast in common-path interferometry," Appl. Opt. 40, 268-282 (2001).
    [CrossRef]
  12. J. W. Goodman, Introduction to Fourier Optics, Second Edition (McGraw-Hill, New York, 1996).

2005 (3)

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005).
[CrossRef]

I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005).
[CrossRef]

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005).
[CrossRef] [PubMed]

2004 (1)

2001 (1)

1997 (1)

1996 (1)

J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996).
[CrossRef]

1995 (1)

E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995).
[CrossRef]

1993 (1)

1976 (1)

G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: a theoretical and experimental study," Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

1970 (1)

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

Ashkin, A.

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

Bøggild, P.

Collins, S. D.

Constable, A.

Daria, V. R.

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005).
[CrossRef]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29, 2270-2272 (2004).
[CrossRef] [PubMed]

Gammelgaard, L.

Glückstad, J.

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005).
[CrossRef] [PubMed]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005).
[CrossRef]

I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005).
[CrossRef]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29, 2270-2272 (2004).
[CrossRef] [PubMed]

J. Glückstad and P. C. Mogensen, "Optimal phase contrast in common-path interferometry," Appl. Opt. 40, 268-282 (2001).
[CrossRef]

J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996).
[CrossRef]

Imbert, C.

G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: a theoretical and experimental study," Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

Kim, J.

Knoesen, A.

Lyons, E. R.

E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995).
[CrossRef]

Mervis, J.

Mogensen, P. C.

Perch-Nielsen, I. R.

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005).
[CrossRef] [PubMed]

I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005).
[CrossRef]

Prentiss, M.

Rodrigo, P. J.

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005).
[CrossRef] [PubMed]

I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005).
[CrossRef]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005).
[CrossRef]

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Real-time three-dimensional optical micromanipulation of multiple particles and living cells," Opt. Lett. 29, 2270-2272 (2004).
[CrossRef] [PubMed]

Roosen, G.

G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: a theoretical and experimental study," Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

Sidick, E.

Sonek, G. J.

E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995).
[CrossRef]

Zarinetchi, F.

Appl. Opt. (2)

Appl. Phys. Lett. (2)

P. J. Rodrigo, V. R. Daria, and J. Glückstad, "Four-dimensional optical manipulation of colloidal particles," Appl. Phys. Lett. 86, 074103 (2005).
[CrossRef]

E. R. Lyons and G. J. Sonek, "Confinement and bistability in a tapered hemispherically lensed optical fiber trap," Appl. Phys. Lett. 66, 1584-1586 (1995).
[CrossRef]

Opt. Commun. (1)

J. Glückstad, "Phase contrast image synthesis," Opt. Commun. 130, 225-230 (1996).
[CrossRef]

Opt. Express (2)

I. R. Perch-Nielsen, P. J. Rodrigo, and J. Glückstad, "Real-time interactive 3D manipulation of particles viewed in two orthogonal observation planes," Opt. Express 18,2852-2857 (2005).
[CrossRef]

P. J. Rodrigo, L. Gammelgaard, P. Bøggild, I. R. Perch-Nielsen, and J. Glückstad, "Actuation of microfabricated tools using multiple GPC-based counterpropagating-beam traps," Opt. Express 13,6899-6904 (2005).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Lett. A (1)

G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: a theoretical and experimental study," Phys. Lett. A 59, 6-8 (1976).
[CrossRef]

Phys. Rev. Lett. (1)

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

Other (1)

J. W. Goodman, Introduction to Fourier Optics, Second Edition (McGraw-Hill, New York, 1996).

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Figures (9)

Fig 1.
Fig 1.

(a) Graphical illustration of a microsphere centered at (x o ,y o ,z o ) illuminated by two counterpropagating beams I + and I -. (b) The microsphere’s cross-section in the plane of incidence associated with a ray hitting the surface at an angle α i measured from the normal.

Fig. 2.
Fig. 2.

3D illustration of an incident ray (red arrow) impinging at point P on the surface of a microsphere centered at point O, which is located at (x o ,y o ,z o ) of the unprimed Cartesian coordinate system.

Fig. 3.
Fig. 3.

Pseudo-color plot of the xz-plane intensity distribution appearing at z>0 for a typical GPC-generated beam directed towards the positive z-axis. At z=0, the beam has a tophat field profile extending from -1.5 µm to 1.5 µm of the x-axis. Red color represents maximum intensity while blue represents minimum intensity.

Fig. 4.
Fig. 4.

Typical dependence of the (a) axial and (b) transverse force components on the respective axial and transverse distances of the microsphere from the midpoint between the two beam waists for different waist separation distances.

Fig. 5.
Fig. 5.

Power-independent stiffness of a symmetric GPC-based CB trap as a function of beam waist separation.

Fig. 6.
Fig. 6.

Typical dependence of the (a) axial and (b) transverse force components on the respective axial and transverse distances of the microsphere from the midpoint between the two beam waists (s=40 µm) for different refractive indices n 2. Spheres with radius a=1.5 µm and equal counterpropagating tophat beams of size R=1.5 µm are considered.

Fig. 7.
Fig. 7.

Typical dependence of the (a) axial and (b) transverse force components on the respective axial and transverse distances of the microsphere from the midpoint between the two beam waists for different sphere radii a. Polystyrene spheres with refractive index n 2=1.59 and equal counterpropagating tophat beams of size R=1.5 µm and separation s=40 µm are considered.

Fig. 8.
Fig. 8.

Plot of the (a) axial and (b) transverse force curves for varied differential powers P +-P -. Insets show the change in (a) axial and (b) transverse trap stiffness at three different points of stable equilibrium. The obtained dynamic range for axial position control is ±Δz corresponding to a depth of ~17.0 µm, in good comparison with experimentally obtained ~20.0 µm [6]. Polystyrene spheres with radius a=1.5 µm and index n 2=1.59, and equal counterpropagating tophat beams of size R=1.5 µm and separation s=40 µm are considered.

Fig. 9.
Fig. 9.

Dependence of GPC-based CB trap’s axial stiffness on beam waist separation for different microsphere radii. The plot shows that a larger critical separation s c (i.e. zero-crossing) is obtained for a larger particle.

Equations (12)

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d F + = e ̂ n 1 c q d P ,
d F + = e ̂ n 1 c q d P ,
q = 1 + R cos 2 α i T 2 cos ( 2 α i 2 α r ) + R cos 2 α i 1 + R 2 + 2 R cos 2 α r ,
q = R sin 2 α i + T 2 sin ( 2 α i 2 α r ) + R sin 2 α i 1 + R 2 + 2 R cos 2 α r ,
d P = I + ( x p , y p , z p ) cos α i d S ,
F + = n 1 a 2 c 0 2 π d φ 0 π 2 d θ ( e ̂ q + e ̂ q ) sin θ cos θ I + ( x p , y p , z p ) ,
F x + = n 1 a 2 2 c 0 2 π d φ 0 π 2 d θ cos φ sin 2 θ q ( θ ) I + ( x p , y p , z p ) ,
F y + = n 1 a 2 2 c 0 2 π d φ 0 π 2 d θ sin φ sin 2 θ q ( θ ) I + ( x p , y p , z p ) ,
F z + = n 1 a 2 2 c 0 2 π d φ 0 π 2 d θ sin 2 φ q ( θ ) I + ( x p , y p , z p ) ,
e + ( x , y ) = { ( P + π R 2 ) 1 2 , ( x 2 + y 2 ) 1 2 R 0 , ( x 2 + y 2 ) 1 2 > R ,
Q xt = c n 1 P t ( F x + + F x ) ,
Q zt = c n 1 P t ( F z + F z ) ,

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