Abstract

We analyze the characteristics of the radiation force that is generated when a highly focused unpolarized Gaussian beam interacts with a nonabsorbing microsphere whose refractive index exhibits a first-order dependence on the beam intensity. The behavior of the force exerted on the sphere is analyzed as a function of beam power, axial distance, sphere radius, refractive-index difference between the sphere and the surrounding liquid, and wavelength. The force characteristics are compared with those of the radiation force that is generated when the electro-optic Kerr effect is absent. Our results show that a reversal in the net force direction is introduced when the Kerr effect becomes significant, which occurs at sufficiently high beam intensities.

© 1997 Optical Society of America

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  1. T. Bakker Schut, E. Schipper, B. Grooth, J. Greve, “Optical-trapping micromanipulation using 780-nm diode lasers,” Opt. Lett. 18, 447–449 (1993).
    [CrossRef]
  2. S. Kawata, T. Sugiura, “Movement of micrometer-sized particles in the evanescent field of a laser beam,” Opt. Lett. 17, 772–774 (1992).
    [CrossRef] [PubMed]
  3. K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Matsuhara, “Pattern formation and flow control of fine particles by laser scanning micromanipulation,” Opt. Lett. 16, 1463–1465 (1991).
    [CrossRef] [PubMed]
  4. A. Ashkin, J. Dziedzic, T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature (London) 330, 769–771 (1987).
    [CrossRef]
  5. A. Ashkin, J. Dziedzic, J. Bjorkholm, S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef] [PubMed]
  6. J. Barton, D. Alexander, S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  7. R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. A 9, 1922–1930 (1992).
    [CrossRef]
  8. C. Saloma, X. Mei, “The dielectric microsphere in a single plane polarized Gaussian beam: characteristics of the radiation force,” Optik 94, 173–176 (1993).
  9. W. Wright, G. Sonek, M. Berns, “Parametric study of the forces on microspheres held by optical tweezers,” Appl. Opt. 33, 1735–1748 (1994).
    [CrossRef] [PubMed]
  10. C. Saloma, M. Cambaliza, “Single-Gaussian-beam interaction with a dielectric microsphere: radiation forces, multiple internal reflections, and caustic structures,” Appl. Opt. 34, 3522–3528 (1995).
    [CrossRef] [PubMed]
  11. P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
    [CrossRef]
  12. P. Prasad, “Nonlinear optical effects in organic materials,” in Contemporary Nonlinear Optics, G. Agrawal, R. Boyd, eds. (Academic, New York, 1992), Chap. 7, pp. 265–295.
    [CrossRef]
  13. V. Chebotayev, “Nonlinear laser spectroscopy: saturation resonances,” in Contemporary Nonlinear Optics, G. Agrawal, R. Boyd, eds. (Academic, New York, 1992), Chap. 9, pp. 367–410.
    [CrossRef]
  14. S. McCall, E. Harris, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
    [CrossRef]
  15. K. Boller, A. Imamoglu, S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
    [CrossRef] [PubMed]
  16. D. Chowdhury, P. Barber, S. Hill, “Energy density distribution inside large nonabsorbing spheres by using Mie theory and geometrical optics,” Appl. Opt. 31, 3518–3523 (1992).
    [CrossRef] [PubMed]
  17. J. Lock, E. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. B 8, 1541–1552 (1991).
    [CrossRef]
  18. D. Burkhard, D. Shealy, “Formula for the density of tangent rays over a caustic surface,” Appl. Opt. 21, 3299–3306 (1982).
    [CrossRef] [PubMed]
  19. J. Marion, M. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1990), pp. 168–169.
  20. F. Jenkins, H. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 526.
  21. A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.
  22. W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).
    [CrossRef]
  23. W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, U.K., 1986), pp. 254–262.

1995 (1)

1994 (1)

1993 (2)

T. Bakker Schut, E. Schipper, B. Grooth, J. Greve, “Optical-trapping micromanipulation using 780-nm diode lasers,” Opt. Lett. 18, 447–449 (1993).
[CrossRef]

C. Saloma, X. Mei, “The dielectric microsphere in a single plane polarized Gaussian beam: characteristics of the radiation force,” Optik 94, 173–176 (1993).

1992 (3)

1991 (3)

K. Sasaki, M. Koshioka, H. Misawa, N. Kitamura, H. Matsuhara, “Pattern formation and flow control of fine particles by laser scanning micromanipulation,” Opt. Lett. 16, 1463–1465 (1991).
[CrossRef] [PubMed]

K. Boller, A. Imamoglu, S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef] [PubMed]

J. Lock, E. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. B 8, 1541–1552 (1991).
[CrossRef]

1989 (2)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Barton, D. Alexander, S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

1987 (1)

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

1986 (1)

1982 (1)

1969 (1)

S. McCall, E. Harris, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Alexander, D.

J. Barton, D. Alexander, S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Ashkin, A.

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

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

Bakker Schut, T.

Barber, P.

Barton, J.

J. Barton, D. Alexander, S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Berns, M.

Bjorkholm, J.

Boller, K.

K. Boller, A. Imamoglu, S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef] [PubMed]

Brevik, I.

R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. A 9, 1922–1930 (1992).
[CrossRef]

Burkhard, D.

Cambaliza, M.

Chebotayev, V.

V. Chebotayev, “Nonlinear laser spectroscopy: saturation resonances,” in Contemporary Nonlinear Optics, G. Agrawal, R. Boyd, eds. (Academic, New York, 1992), Chap. 9, pp. 367–410.
[CrossRef]

Chowdhury, D.

Chu, S.

Dziedzic, J.

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

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

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, U.K., 1986), pp. 254–262.

Gambogi, W.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).
[CrossRef]

Gerstadt, W.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).
[CrossRef]

Greve, J.

Grooth, B.

Gussgard, R.

R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. A 9, 1922–1930 (1992).
[CrossRef]

Harris, E.

S. McCall, E. Harris, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Harris, S.

K. Boller, A. Imamoglu, S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef] [PubMed]

Heald, M.

J. Marion, M. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1990), pp. 168–169.

Hill, S.

Hovenac, E.

J. Lock, E. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. B 8, 1541–1552 (1991).
[CrossRef]

Imamoglu, A.

K. Boller, A. Imamoglu, S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef] [PubMed]

Jenkins, F.

F. Jenkins, H. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 526.

Kawata, S.

Kitamura, N.

Koshioka, M.

Lindmo, T.

R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. A 9, 1922–1930 (1992).
[CrossRef]

Lock, J.

J. Lock, E. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. B 8, 1541–1552 (1991).
[CrossRef]

Mackara, S.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).
[CrossRef]

Marion, J.

J. Marion, M. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1990), pp. 168–169.

Matsuhara, H.

McCall, S.

S. McCall, E. Harris, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Mei, X.

C. Saloma, X. Mei, “The dielectric microsphere in a single plane polarized Gaussian beam: characteristics of the radiation force,” Optik 94, 173–176 (1993).

Misawa, H.

Prasad, P.

P. Prasad, “Nonlinear optical effects in organic materials,” in Contemporary Nonlinear Optics, G. Agrawal, R. Boyd, eds. (Academic, New York, 1992), Chap. 7, pp. 265–295.
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, U.K., 1986), pp. 254–262.

Saloma, C.

C. Saloma, M. Cambaliza, “Single-Gaussian-beam interaction with a dielectric microsphere: radiation forces, multiple internal reflections, and caustic structures,” Appl. Opt. 34, 3522–3528 (1995).
[CrossRef] [PubMed]

C. Saloma, X. Mei, “The dielectric microsphere in a single plane polarized Gaussian beam: characteristics of the radiation force,” Optik 94, 173–176 (1993).

Sasaki, K.

Schaub, S.

J. Barton, D. Alexander, S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Schipper, E.

Shealy, D.

Siegman, A.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

Sonek, G.

Sugiura, T.

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, U.K., 1986), pp. 254–262.

Vetterling, W.

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, U.K., 1986), pp. 254–262.

Weber, A.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).
[CrossRef]

White, H.

F. Jenkins, H. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 526.

Wright, W.

Yamane, T.

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

Yeh, P.

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

Appl. Opt. (4)

IEEE J. Quantum Electron. (1)

P. Yeh, “Two-wave mixing in nonlinear media,” IEEE J. Quantum Electron. 25, 484–519 (1989).
[CrossRef]

J. Appl. Phys. (1)

J. Barton, D. Alexander, S. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

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

R. Gussgard, T. Lindmo, I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. A 9, 1922–1930 (1992).
[CrossRef]

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

J. Lock, E. Hovenac, “Internal caustic structure of illuminated liquid droplets,” J. Opt. Soc. Am. B 8, 1541–1552 (1991).
[CrossRef]

Nature (London) (1)

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

Opt. Lett. (4)

Optik (1)

C. Saloma, X. Mei, “The dielectric microsphere in a single plane polarized Gaussian beam: characteristics of the radiation force,” Optik 94, 173–176 (1993).

Phys. Rev. (1)

S. McCall, E. Harris, “Self-induced transparency,” Phys. Rev. 183, 457–485 (1969).
[CrossRef]

Phys. Rev. Lett. (1)

K. Boller, A. Imamoglu, S. Harris, “Observation of electromagnetically induced transparency,” Phys. Rev. Lett. 66, 2593–2596 (1991).
[CrossRef] [PubMed]

Other (7)

J. Marion, M. Heald, Classical Electromagnetic Radiation, 2nd ed. (Academic, New York, 1990), pp. 168–169.

F. Jenkins, H. White, Fundamentals of Optics, 4th ed. (McGraw-Hill, New York, 1976), p. 526.

A. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), Chap. 17.

W. Gambogi, W. Gerstadt, S. Mackara, A. Weber, “Holographic transmission elements using improved photopolymer films,” in Computer and Optically Generated Holographic Optics; 4th in a Series, I. Cindrich, S. H. Lee, eds., Proc. SPIE1555, 256–267 (1991).
[CrossRef]

W. Press, B. Flannery, S. Teukolsky, W. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, U.K., 1986), pp. 254–262.

P. Prasad, “Nonlinear optical effects in organic materials,” in Contemporary Nonlinear Optics, G. Agrawal, R. Boyd, eds. (Academic, New York, 1992), Chap. 7, pp. 265–295.
[CrossRef]

V. Chebotayev, “Nonlinear laser spectroscopy: saturation resonances,” in Contemporary Nonlinear Optics, G. Agrawal, R. Boyd, eds. (Academic, New York, 1992), Chap. 9, pp. 367–410.
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Interaction of a focused Gaussian beam and a nonabsorbing microsphere of radius a and refractive index n 2 whose center of mass (point 0) is located a distance z from the beam focus of radius w 0.

Fig. 2
Fig. 2

Angular dependence of reflectance R(θ′) for both s- and p-polarization states: (a) n 1 = 1.33, n 2 (0) = 1.5, n 2 (1) = 0; (b) n 2 (1) = 6.5 × 10-5 m2/W; (c) n 1 = 1.33, n 2 (0) = 1.16, n 2 (1) = 6.5 × 10-5 m 2/W. We used the following parameters: peak power P = 1 × 105 W, a = 17 µm, z = 10 µm, NA of 1.25, and λ = 1.06 µm.

Fig. 3
Fig. 3

Dependence of average force 〈 F T〉 on peak power P. The parameters we used are a = 17 µm, z = 10 µm, NA of 1.25, n 1= 1.33, n 2 (0) = 1.5, and λ = 1.06 µm.

Fig. 4
Fig. 4

Dependence of 〈F T〉 on axial position z of the sphere’s center of mass at point 0. The parameters we used are P = 1 × 105 W, a = 17 µm, NA of 1.25, n 1 = 1.33, n 2 (0) = 1.5, and λ = 1.06 µm.

Fig. 5
Fig. 5

Dependence of 〈FT〉 on Δn = n 2 (0) - n 1. The parameters we used are P = 1 × 105 W, a = 17 µm, NA of 1.25, and λ = 1.06 µm.

Fig. 6
Fig. 6

Dependence of 〈F T〉 on sphere radius a. The parameters we used are P = 1 × 105 W, z = 10 µm, NA of 1.25, n 1 = 1.33, n 2 (0) = 1.5, and λ = 1.06 µm.

Fig. 7
Fig. 7

Dependence of 〈F T〉 on beam wavelength λ. The parameters we used are P = 1 × 105 W, a = 17 µm, z = 10 µm, NA of 1.25, n 1 = 1.33, n 2 (0) = 1.5.

Equations (11)

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Rsα1, α2=sin2α2-α1sin2α2+α1,
Rpα1, α2=tan2α2-α1tan2α2-α1.
NθΔt=2Pλπhcw021+z-1 cos θzR2dS×exp-2w02a2 sin2 θ1+z-a cos θzR2,
n2=n20+n21ITθ,
Iθ=2Pπw021+z-a cos θzR2×exp-2w02a2 sin2 θ1+z-a cos θzR2
n20+Iθn211-sin2α2-α1sin2α2+α1sin α2=n1 sin α1,
a2+c2sin6 α2+2cd-2absin5 α2+2ce+d2+b2-a2sin4 α22cf+2de+2absin3 α2+2df+d2-b2sin2 α2+2ef sin α2+f2=0,
a=2n20+4n21Iθsin α1 cos α1,  b=2n1 sin2 α1 cos α1,  c=n20sin2 α1-cos2 α1,  d=n1sin α1 cos2 α1-sin3 α1,  e=-n20 sin2 α1,  f=n1 sin3 α1.
n20+Iθn211-tan2α2-α1tan2α2+α1sin α2=n1 sin α1.
c2 sin10 α2-2cd sin9 α2+d2-2c2sin8 α2+4cd sin7 α2+2ce+c2-2d2+4a2sin6 α2+2cf-2de-2cd-8absin5 α2+d2-2ce-2df+4b2-4a2sin4 α2+2de-2cf+8absin3 α2+2df+e2-4b2sin2 α2+2ef sin α2+f2=0,
a=n20 sin 2α1/2+n20Iθsin 2α1,  b=n1 sin α1 cos 2α1/2,  c=n20,  d=n1 sin α1,  e=ccos 4α1-1/8, f=-dcos 4α1-1/8.

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