Abstract

The force that is produced from the momentum change of a stream of photons incident upon micrometer-sized spheres is developed from a ray-optic model. The resulting force component expressions, axial and radial with respect to the photon stream center and incident direction, are in a form that makes them suitable for computer modeling of the levitation phenomena. Simulated results presented are in excellent agreement with published experimental observations.

© 1995 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. 24, 156–159 (1970).
  2. For a review of the literature on applications see R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," Theor. Phys. Semin. Trondheim (Norway) 7, 1–41 (1991).
  3. 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] [PubMed]
  4. J. S. Kim and S. S. Lee, "Scattering of laser beams and the optical potential well for a homogeneous sphere," J. Opt. Soc. Am. 73, 303–312 (1983).
    [CrossRef]
  5. S. Chang and S. S. Lee, "Optical torque on a homogeneous sphere levitated in circularly polarized fundamental-mode laser beam," J. Opt. Soc. Am. B 2, 1853–1860 (1985).
    [CrossRef]
  6. J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594–4602 (1989).
    [CrossRef]
  7. G. Roosen and C. Imbert, "Optical levitation by means of two horizontal laser beams: A theoretical and experimental study," Phys. Lett. 59A, 6–8 (1976).
  8. G. Roosen and C. Imbert, "The TEM01 mode laser beam—a powerful tool for optical levitation of various types of spheres," Opt. Commun. 26, 432–436 (1978).
    [CrossRef]
  9. G. Roosen, "La lévitation optique de sphère," Can. J. Phys. 57, 1260–1279 (1979).
    [CrossRef]
  10. J. Taylor and C. Zafiratos, Modern Physics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1991), p. 150.
  11. B. E. Saleh and M. Teich, Fundamentals of Photonics (Wiley, New York, 1991).
    [CrossRef]
  12. J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
    [CrossRef]
  13. A. Ashkin and J. Dziedzic, "Optical levitation by radiation pressure," Appl. Phys. Lett. 19, 283–285 (1971).
    [CrossRef]

1989 (1)

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

1986 (1)

1985 (1)

1984 (1)

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

1983 (1)

1979 (1)

G. Roosen, "La lévitation optique de sphère," Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

1978 (1)

G. Roosen and C. Imbert, "The TEM01 mode laser beam—a powerful tool for optical levitation of various types of spheres," Opt. Commun. 26, 432–436 (1978).
[CrossRef]

1976 (1)

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

1971 (1)

A. Ashkin and J. 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. 24, 156–159 (1970).

Alexander, D. R.

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the 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. 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] [PubMed]

A. Ashkin and J. 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. 24, 156–159 (1970).

Barton, J. P.

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

Bjorkholm, J. E.

Brevik, I.

For a review of the literature on applications see R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," Theor. Phys. Semin. Trondheim (Norway) 7, 1–41 (1991).

Chang, S.

Chu, S.

Combemale, Y.

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

Dziedzic, J.

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

Dziedzic, J. M.

Gussgard, R.

For a review of the literature on applications see R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," Theor. Phys. Semin. Trondheim (Norway) 7, 1–41 (1991).

Imbert, C.

G. Roosen and C. Imbert, "The TEM01 mode laser beam—a powerful tool for optical levitation of various types of spheres," Opt. Commun. 26, 432–436 (1978).
[CrossRef]

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

Kim, J. S.

Lee, S. S.

Lindmo, T.

For a review of the literature on applications see R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," Theor. Phys. Semin. Trondheim (Norway) 7, 1–41 (1991).

Papuchon, M.

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

Plantegenest, M.

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

Pochelle, J.

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

Raffy, J.

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

Roosen, G.

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

G. Roosen, "La lévitation optique de sphère," Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

G. Roosen and C. Imbert, "The TEM01 mode laser beam—a powerful tool for optical levitation of various types of spheres," Opt. Commun. 26, 432–436 (1978).
[CrossRef]

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

Saleh, B. E.

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

Schaub, S. A.

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

Taylor, J.

J. Taylor and C. Zafiratos, Modern Physics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1991), p. 150.

Teich, M.

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

Zafiratos, C.

J. Taylor and C. Zafiratos, Modern Physics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1991), p. 150.

Appl. Phys. Lett. (2)

J. Pochelle, J. Raffy, Y. Combemale, M. Papuchon, G. Roosen, and M. Plantegenest, "Optical levitation using single mode fibers and its application to self-centering of microlenses," Appl. Phys. Lett. 45, 350–352 (1984).
[CrossRef]

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

Can. J. Phys. (1)

G. Roosen, "La lévitation optique de sphère," Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

J. Appl. Phys. (1)

J. P. Barton, D. R. Alexander, and S. A. Schaub, "Theoretical determination of the 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. (1)

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

Opt. Commun. (1)

G. Roosen and C. Imbert, "The TEM01 mode laser beam—a powerful tool for optical levitation of various types of spheres," Opt. Commun. 26, 432–436 (1978).
[CrossRef]

Opt. Lett. (1)

Phys. Lett. (1)

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

Phys. Rev. (1)

A. Ashkin, "Acceleration and trapping of particles by radiation pressure," Phys. Rev. 24, 156–159 (1970).

Other (3)

For a review of the literature on applications see R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," Theor. Phys. Semin. Trondheim (Norway) 7, 1–41 (1991).

J. Taylor and C. Zafiratos, Modern Physics for Scientists and Engineers (Prentice-Hall, Englewood Cliffs, N.J., 1991), p. 150.

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

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Figures (12)

Fig. 1
Fig. 1

Stream of photons represented by a ray of light incident upon the lower side of a sphere. The sphere has index of refraction ns, and the surround has index of refraction n0. The ray is shown refracted and reflected at the lower and the upper surfaces.

Fig. 2
Fig. 2

Axial force versus separation between the location of the minimum waist W0 and the lowest point on the sphere. At the position ds = 830 μm the force is 0 pN. The maximum in the force curve occurs at d = 0 μm. Inset: Enlargement of the 700–1000-μm region.

Fig. 3
Fig. 3

Axial force versus separation between the location of the minimum waist W0 and the lowest point on the sphere. At the position ds = 723 μm the force is 0 pN. The maximum in the force curve occurs at d > 0 μm. Inset: Enlargement of the 600–900-μm region.

Fig. 4
Fig. 4

Axial force versus separation between the location of the minimum waist W0 and the lowest point on the sphere. The position d = 297 μm is the stable equilibrium levitation distance, whereas the distance d = 55 μm is the unstable equilibrium position.

Fig. 5
Fig. 5

Power in watts versus sphere radius required for maintenance of a sphere at a distance of 297 μm from the minimum waist of the beam. The curve is not linear because A′ in Eq. (40) is a function of the sphere radius.

Fig. 6
Fig. 6

Radial force versus radial offset. The legend indicates the axial separation, in micrometers, between the beam’s minimum waist position and the lowest point on the sphere. The radial force is always greater than or equal to zero for all axial offsets. The sphere located in the laser beam experiences a centering force pulling it into alignment with the beam’s central axis.

Fig. 7
Fig. 7

Radial force versus radial offset. The legend indicates the axial separation, in micrometers, between the beam’s minimum waist position and the lowest point on the sphere. The radial force is always greater than or equal to zero for all axial offsets. The sphere located in the laser beam experiences a centering force pulling it into alignment with the beam’s central axis.

Fig. 8
Fig. 8

Radial force versus radial offset. The legend indicates the axial separation, in micrometers, between the beam’s minimum waist position and the lowest point on the sphere. The radial force is always greater than or equal to zero for all axial offsets. The sphere located in the laser beam experiences a centering force pulling it into alignment with the beam’s central axis.

Fig. 9
Fig. 9

Axial force versus distance when the relative index ns/n0 is 0.966. The figure indicates that the force may have zero crossing positions giving stable or unstable equilibriums. However, Fig. 10, for the radial force, confirms that both equilibrium distances are unstable because the radial force is always less than or equal to zero for all axial distances.

Fig. 10
Fig. 10

Radial force versus radial offset for a sphere when ns/n0 = 0.966. The legend in the figure indicates the axial separation, in micrometers, between the beam’s minimum waist location and the lowest point on the sphere. The figure shows that the radial force is never greater than zero. This indicates that the sphere is pushed outside the beam and is not levitated.

Fig. 11
Fig. 11

Axial force versus distance for a perfectly reflecting sphere. The zero axis crossing of the force curve tends to indicate that perfectly reflecting spheres can be levitated. However, Fig. 12, for the radial force, confirms that the equilibrium position is unstable because the radial force is always less than zero for all axial positions.

Fig. 12
Fig. 12

Radial force versus radial offset for a perfectly reflecting sphere. The legend in the figure indicates the axial separation, in micrometers, between the beam’s minimum waist location and the lowest point on the sphere. The radial force is never greater than zero, indicating that these spheres are pushed outside the beam and are not levitated.

Equations (41)

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F = d p d t ,
p = h λ 0 n 0 ,
- h λ 0 n 0 [ 1 + cos ( 2 θ 1 ) ] ,
Δ p 1 r z = h λ 0 n 0 [ 1 + cos ( 2 θ 1 ) ] .
Δ p 1 t z = h λ 0 [ n 0 - n s cos ( θ 1 - θ 2 ) ] ,
Δ p 2 r z = h λ 0 n s [ cos ( θ 1 - θ 2 ) + cos ( 3 θ 2 - θ 1 ) ] ,
Δ p 2 t z = h λ 0 { n s cos ( θ 1 - θ 2 ) - n 0 cos [ 2 ( θ 1 - θ 2 ) ] } ,
r 1 2 = | r TE + r TM 2 | 2 ,
r 1 2 = ( n 0 n s ) 2 [ cos 2 ( θ 1 ) - cos 2 ( θ 2 ) ] 2 { n 0 n s [ cos 2 ( θ 1 ) + cos 2 ( θ 2 ) ] + ( n 0 2 + n s 2 ) cos ( θ 1 ) cos ( θ 2 ) } 2 ,
t 1 2 = 1 - r 1 2 ,
r 2 2 = r 1 2 ,
t 2 2 = t 1 2 .
I ( ρ , z ) = 2 P π W ( z ) 2 exp [ - 2 ρ 2 W ( z ) 2 ] ,
W ( z ) = W 0 [ 1 + ( z / z 0 ) 2 ] 1 / 2 ,
W 0 = ( λ 0 z 0 / π ) 1 / 2 ,
ϕ av = N d t .
P = ϕ av E ph .
d F A z = N d p z d t ,
d p z = r 1 2 Δ p 1 r z + t 1 2 Δ p 1 t z + r 1 2 t 1 2 Δ p 2 r z + t 1 2 t 1 2 Δ p 2 t z .
d F A z = 2 P E ph 1 π W ( z ) 2 exp [ - 2 ρ 2 W ( z ) 2 ] d p z d A .
F z = F 1 r z + F 1 t z + F 2 r z + F 2 t z .
F 1 r z = 0 π / 2 π c n 0 [ 1 + cos ( 2 θ 1 ) ] I ( ρ , z ) r 1 2 R 2 sin ( 2 θ 1 ) d θ 1 ,
F 1 t z = 0 π / 2 π c [ n 0 - n s cos ( θ 1 - θ 2 ) ] I ( ρ , z ) t 1 2 R 2 × sin ( 2 θ 1 ) d θ 1 ,
F 2 r z = 0 π / 2 π c n s [ cos ( θ 1 - θ 2 ) + cos ( 3 θ 2 - θ 1 ) ] I ( ρ , z ) × t 1 2 r 1 2 R 2 sin ( 2 θ 1 ) d θ 1 ,
F 2 t z = 0 π / 2 π c { n s cos ( θ 1 - θ 2 ) - n 0 cos [ 2 ( θ 1 - θ 2 ) ] } × I ( ρ , z ) t 1 2 t 1 2 R 2 sin ( 2 θ 1 ) d θ 1 ,
F z net = F z - 4 3 π R 3 ( σ s - σ 0 ) g ,
Δ p 1 r r = - h λ 0 n 0 sin ( 2 θ 1 ) cos ( ϕ ) ,
Δ p 1 t r = h λ 0 n s sin ( θ 1 - θ 2 ) cos ( ϕ ) ,
Δ p 2 r r = h λ 0 n s [ sin ( 3 θ 2 - θ 1 ) - sin ( θ 1 - θ 2 ) ] cos ( ϕ ) ,
Δ p 2 t r = h λ 0 { n 0 sin [ 2 ( θ 1 - θ 2 ) - n s sin ( θ 1 - θ 2 ) ] } cos ( ϕ ) .
d p r = r 1 2 Δ p 1 r r + t 1 2 Δ p 1 t r + r 1 2 t 1 2 Δ p 2 r r + t 1 2 t 1 2 Δ p 2 t r ,
d F A r = 2 P E ph 1 π W ( z ) 2 exp [ - 2 ρ 2 W ( z ) 2 ] d p r d A .
F r net = F 1 r r + F 1 t r + F 2 t r + F 2 t r ,
F 1 r r = - 0 π / 2 0 2 π I ( ρ , z ) n 0 2 c sin ( 2 θ 1 ) r 1 2 R 2 cos ( ϕ ) × sin ( 2 θ 1 ) d ϕ d θ 1 ,
F 1 t r = 0 π / 2 0 2 π I ( ρ , z ) n s 2 c sin ( θ 1 - θ 2 ) t 1 2 R 2 cos ( ϕ ) × sin ( 2 θ 1 ) d ϕ d θ 1 ,
F 2 r r = 0 π / 2 0 2 π I ( ρ , z ) n c 2 c [ sin ( 3 θ 2 - θ 1 ) - sin ( θ 1 - θ 2 ) ] × t 1 2 r 1 2 R 2 cos ( ϕ ) sin ( 2 θ 1 ) d ϕ d θ 1 ,
F 2 t r = 0 π / 2 0 2 π I ( ρ , z ) 2 c { n 0 sin [ 3 ( θ 1 - θ 2 ) ] - n s sin ( θ 1 - θ 2 ) } t 1 2 t 1 2 R 2 cos ( ϕ ) sin ( 2 θ 1 ) d ϕ d θ 1 .
ρ ( θ 1 , ϕ ) = [ a 2 + R 2 sin 2 ( θ 1 ) + 2 a R sin ( θ 1 ) cos ( ϕ ) ] 1 / 2 ,
σ s = 1.1 g / mL , σ 0 = 1.0 g / mL , n s = 1.5468 , n 0 = 1.333 , λ 0 = 0.514 μ m , P = 20 mW .
F ( R ) = R 2 ( A - B R ) ,
P = B A ( R ) R .

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