Abstract

Transverse and axial trapping forces are calculated in the ray optics regime for a multiple-beam fiber-optic light-force trap for dielectric microspheres located both on and off axis relative to the beam axis. Trap efficiencies are evaluated as functions of the effective index of refraction of the microspheres, normalized sphere radius, and normalized beam waist separation distance. Effects of the linear polarization of the electric field and of beam focusing through microlenses are considered. In the case of a counterpropagating two-beam fiber-optic trap, using microlenses at the distal ends of the fiber to focus the beams may somewhat increase the trapping volume and the axial stability if the fiber spacing is sufficiently large but will greatly reduce the stiffness of the transverse force. Trapping forces produced in a counterpropagating two-beam fiber-optic trap are compared with those generated in multiple-beam fiber-optic gradient-force traps. Multiple-beam fiber-optic traps use strong gradient forces to trap a particle; therefore they stabilize the particles much more firmly than do counterpropagating two-beam traps.

© 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, J. E. Bjorkholm, S. Chu, “Observation of s single-beam gradient force optical trap for dielectric particles,” Opt. Lett. 11, 288–290 (1986).
    [CrossRef]
  3. A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
    [CrossRef]
  4. K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. and Biomol. Struc. 23, 247–285 (1994).
    [CrossRef]
  5. G. Roosen, C. Imbert, “Optical levitation by means of two horizontal laser beams: a theoretical and experimental study,” Phys. Lett. A 59, 6–8 (1976).
    [CrossRef]
  6. G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
    [CrossRef]
  7. G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
    [CrossRef]
  8. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
    [CrossRef] [PubMed]
  9. T. C. B. Schut, G. Hesselink, B. G. Degrooth, J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry 12, 479–485 (1991).
    [CrossRef]
  10. 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]
  11. E. R. Lyons, G. J. Sonek, “Confinement and bistability in a tapered hemispherically lensed optical fiber trap,” Appl. Phys. Lett. 66, 1584–1586 (1995).
    [CrossRef]
  12. The directions of ĝ here and in Ashkin’s paper (Y axis in Fig. 3 of Ref. 8) are opposite. The value of qg in Ref. 8, denoted by Qg, is always less than zero (negative) under the definition given by Eqs. (2), and (3), but Ashkin switches the direction of Fg in his Fig. 2(b) and also plots - Qg in his Fig. 3. We defined our qg as qg = -Qg and changed the direction of ĝ correspondingly. This is also consistent with the definition in Refs. 5 and 6. Under this definition, when a ray strikes a small region of the surface of a microsphere, the component of the force exerted on the microsphere in the direction perpendicular to the ray axis is such that it pulls the microsphere toward the ray axis, provided that n2 < n1.
  13. We used θ, not αi as is done in Ref. 10 for the same purpose, because the transverse distribution of the light intensity I given in Eq. 3 is defined in a plane perpendicular to the beam axis, even though the individual rays of the beam hit the microsphere at certain nonzero angles with respect to the beam axis.

1995 (1)

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

1994 (1)

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. and Biomol. Struc. 23, 247–285 (1994).
[CrossRef]

1993 (1)

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

1992 (2)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

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]

1991 (1)

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

1986 (1)

1979 (1)

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

1977 (1)

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[CrossRef]

1976 (1)

G. Roosen, 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, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

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

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

Bjorkholm, J. E.

Block, S. M.

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. and Biomol. Struc. 23, 247–285 (1994).
[CrossRef]

Brevik, I.

Chu, S.

Constable, A.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

Degrooth, B. G.

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

Dziedzic, J. M.

Greve, J.

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

Imbert, C.

G. Roosen, 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.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

Lindmo, T.

Lyons, E. R.

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

Mervis, J.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

Prentiss, M.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

Roosen, G.

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[CrossRef]

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

Schut, T. C. B.

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

Sonek, G. J.

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

Svoboda, K.

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. and Biomol. Struc. 23, 247–285 (1994).
[CrossRef]

Zarinetchi, F.

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

Ann. Rev. Biophys. and Biomol. Struc. (1)

K. Svoboda, S. M. Block, “Biological applications of optical forces,” Ann. Rev. Biophys. and Biomol. Struc. 23, 247–285 (1994).
[CrossRef]

Appl. Phys. Lett. (1)

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

Biophys. J. (1)

A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. 61, 569–582 (1992).
[CrossRef] [PubMed]

Can. J. Phys. (1)

G. Roosen, “La levitation optique de spheres,” Can. J. Phys. 57, 1260–1279 (1979).
[CrossRef]

Cytometry (1)

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

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

Opt. Commun. (1)

G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. 21, 189–194 (1977).
[CrossRef]

Opt. Lett. (2)

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

A. Constable, J. Kim, J. Mervis, F. Zarinetchi, M. Prentiss, “Demonstration of a fiber-optical light-force trap,” Opt. Lett. 18, 1869–1871 (1993).
[CrossRef]

Phys. Lett. A (1)

G. Roosen, 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 (2)

The directions of ĝ here and in Ashkin’s paper (Y axis in Fig. 3 of Ref. 8) are opposite. The value of qg in Ref. 8, denoted by Qg, is always less than zero (negative) under the definition given by Eqs. (2), and (3), but Ashkin switches the direction of Fg in his Fig. 2(b) and also plots - Qg in his Fig. 3. We defined our qg as qg = -Qg and changed the direction of ĝ correspondingly. This is also consistent with the definition in Refs. 5 and 6. Under this definition, when a ray strikes a small region of the surface of a microsphere, the component of the force exerted on the microsphere in the direction perpendicular to the ray axis is such that it pulls the microsphere toward the ray axis, provided that n2 < n1.

We used θ, not αi as is done in Ref. 10 for the same purpose, because the transverse distribution of the light intensity I given in Eq. 3 is defined in a plane perpendicular to the beam axis, even though the individual rays of the beam hit the microsphere at certain nonzero angles with respect to the beam axis.

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Figures (14)

Fig. 1
Fig. 1

Gradient and scattering forces exerted by a single ray on a dielectric sphere when the sphere is optically more dense than the surrounding medium: ,ĝ, and ŝ, the unit vectors in the direction of the surface normal, the gradient force, and the scattering force, respectively; αi, angle of incidence.

Fig. 2
Fig. 2

Geometry of a dielectric sphere hit at point P by a ray coming from a source at point C. The beam waist is located in thexy plane. The beam axis CA ¯ is away from the zaxis by a distance d along the xaxis. π - θ and φ are the polar and azimuthal angles, respectively.

Fig. 3
Fig. 3

Geometry of the plane of incidence, showing the direction of the scattering force ŝ, and the definition of the distance r from the beam axis to the small region hit by the ray.

Fig. 4
Fig. 4

Geometry of the plane of incidence, showing the direction of the gradient-force ĝ.

Fig. 5
Fig. 5

Geometry of counterpropagating dual Gaussian beams illuminating a dielectric sphere.

Fig. 6
Fig. 6

Variation of the transverse trapping efficiencyQxt as a function of the transverse offset D with (a) the beam waist separation S, (b) the sphere radiusRo, and (c)the effective index of refraction N as a parameter. Parameters not shown in the legends are S = 30,Ro= 1, and N = 1.2.

Fig. 7
Fig. 7

Same as Fig. 6, except that the axial trapping efficiency Qzt is plotted versus the axial offsetZo′ = Zo- S/2 = zo/wo -S/2.

Fig. 8
Fig. 8

Variation of the axial trapping efficiencyQzt as a function of the axial offset Zo′ when N = 1.2.

Fig. 9
Fig. 9

Variation of the axial trapping efficiencyQz1 of beam 1 (Fig. 5) as a function of the axial offsetZo whenN = 1.2.

Fig. 10
Fig. 10

Variation of the axial trapping efficiencyQzt as a function of the axial offset Zo′ when Ro = 1.5 andN = 1.2.

Fig. 11
Fig. 11

Dependence of the transverse trapping efficiencyQxt on the state of incident beam polarization. The solid curve is for a circularly polarized beam, and the long- and the short-dashed curves are for the beams linearly polarized in the x and they directions, respectively. Other parameters areS = 20, Ro = 1, and N = 1.2.

Fig. 12
Fig. 12

Dependence of the transverse trapping efficiencyQxt on the waist of incident beam when (a) S = 20 and (b)S = 60. Other parameters arero= 1 and N = 1.2.

Fig. 13
Fig. 13

Dependence of the axial trapping efficiencyQzt on the waist of incident beam when (a) S = 20 and (b)S = 60. Other parameters arero= 1 and N = 1.2.

Fig. 14
Fig. 14

Dependence of beam field radiusW =w/wo on the beam waist.

Equations (38)

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d F s = ŝ   n 1 c   q s d P ,
d F g = ĝ   n 1 c   q g d P ,
q s = 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 g = - 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   cos   θ   d S = 2 P o π w o 2   exp - 2 r 2 / w 2   cos   θ   d S ,
R c = z p 1 + ρ z p 2 = z p 1 + π n 1 w o 2 λ o z p 2 ,
ŝ = x ˆ r o   sin   θ   cos   φ - d + ŷ r o   sin   θ   sin   φ + z ˆ R z / R c ,
ĝ = R c   tan   γ - 1 x ˆ r 0   sin   θ   cos   φ - d + d R c a   cos   γ + ŷ r 0   sin   θ   sin   φ + z ˆ R z - R c R z + r 0   cos   θ a   cos   γ .
w 2 = w 0 2 1 + z p / ρ 2 ,     r 2 = d 2 + r 0   sin   θ 2 - 2 dr 0   sin   θ   cos   φ ,     d S = r 0 2   sin   θ   d θ   d φ .
F = n 1 P 0 c   4 r 0 2 π   0 π   d φ   0 θ max   d θ   sin   θ × cos   θ   ŝ q s + ĝ q g   exp - 2 r 2 / w 2 w 2 ,
F z = n 1 P 0 c   Q z ,
F x = n 1 P 0 c   Q x ,
Q z = 2 r 0 2 π   0 π   d φ   0 θ max   d θ   sin   2 θ   exp - 2 r 2 / w 2 w 2 R c × q s R z + q g tan   γ R z - R c R z + r 0   cos   θ a   cos   γ ,
Q x = 2 r 0 2 π   0 π   d φ   0 θ max   d θ   sin   2 θ   exp - 2 r 2 / w 2 w 2 R c × q s r 0   sin   θ   cos   φ - d + q g tan   γ × r 0   sin   θ   cos   φ - d 1 - R c a   cos   γ .
Q z = 2 r 0 2   0 θ max   d θ   sin   2 θ   exp - 2 r 2 / w 2 w 2 × q s   cos   γ - q g   sin   γ = 2 r 0 2   0 θ max   d θ   sin   2 θ   exp - 2 r 2 / w 2 w 2 R c × q s R z - q g r o   sin   θ ,
α i = cos - 1 R z   cos   θ - r 0   sin 2   θ R c = γ + θ ,
Q z = 2 r 0 2   0 θ max   d θ   sin   2 θ   exp - 2 r 2 / w 2 w 2   q z ,
q z = cos α i - θ + R   cos α i + θ - T 2   cos α i + θ - 2 α r + R   cos α i + θ 1 + R 2 + 2 R   cos   2 α r .
Q z 2 r o 2 / w o 1 + z o / ρ 2   0 π / 2   d θ q s   sin   2 θ .
F zt = F z 1 - F zt = n 1 c   P 1 + P 2   Q zt = n 1 c   P 1 + P 2   P 1 P 1 + P 2   Q z 1 - P 2 P 1 + P 2   Q z 2 ,
F xt = F x 1 + F x 2 = n 1 c   P 1 + P 2 Q xt = n 1 c   P 1 + P 2   P 1 P 1 + P 2   Q x 1 + P 2 P 1 + P 2   Q x 2 .
Z o = Z o - S / 2 = z o / w o - S / 2 ,
x p = r o   sin   θ   cos   φ ,     y p = r o   sin   θ   sin   φ ,     z p = z o - r o   cos   θ ,  
x c = d ,     y c = 0 ,     z c = - R z - z o - r o   cos   θ ,
R z = ± R c 2 - r 2 1 / 2 .
ŝ = x ˆ x p - x c + ŷ y p - y c + z ˆ z p - z c / R c ,
γ = cos - 1 R c 2 + a 2 - r o 2 2 aR c ,
a = d 2 + R z + r o   cos   θ 2 1 / 2 .
x b = OB ¯   sin   β = d 1 - R c a   cos   γ ,     y b = 0 ,     z b = z o - OB ¯   cos   β = z o - 1 - R c a   cos   γ R z + r o   cos   θ .
ĝ = x ˆ x p - x b + ŷ y p - y b + z ˆ z p - z b / PB ¯ ,
a 2 = r o 2 + R c 2 - 2 r o R c   cos π - α i .
α i = cos - 1 d 2 + R z + r o   cos   θ 2 - r o 2 - R c 2 2 r o R c .
d 2 + R z + r o   cos   θ 2 - r o 2 - R c 2 = 0
Ax + By + Cz + D = 0 ,
A = r o   sin   θ sin   φ R z + r o   cos   θ ,     B = dr o   cos   θ - r o   sin   θ   cos   φ R z + r o   cos   θ ,     C = dr o   sin   θ   sin   φ ,     D = dz o r o   sin   θ   sin   φ .
f s = sin 2 ϕ x = A 2 A 2 + B 2 + C 2 ,
f s = sin 2 ϕ y = B 2 A 2 + B 2 + C 2 ,
f p = 1 - f s .

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