Abstract

Traditional solid geometry ray-tracing method is complex in analyzing the orientation of gradient forces and calculating incident angle of optic rays upon a microsphere. We present a new ray-tracing methodology based on spatial analytic geometry in the ray-optic model. For a single ray upon a microsphere, the directions of transmission and trapping forces are depicted by spatial vectors in a Cartesian coordinate system. At the same time, the polarized direction of a single focused ray can be transformed by a matrix of rotational coordinates. According to the relations of vectors, the trapping forces can be expressed identically. We use this new method to discuss differences of trapping forces in the cases of various states of unpolarized and polarized beams, and also show the reasons for differences in transverse force between measurement and theoretical results. Our simulative results show that this method can be applied identically to calculating both transverse and axial trapping forces, and also for different polarizations of a laser beam.

© 2008 Optical Society of America

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References

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  1. G. Roosen and S. Slanksy, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341-346 (1979).
    [Crossref]
  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]
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    [Crossref]
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    [Crossref]
  6. A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95, 168102 (2005).
    [Crossref] [PubMed]
  7. M. Gu, P. C. Ke, and X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666-3668 (1997).
    [Crossref]
  8. J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
    [Crossref] [PubMed]
  9. X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  13. Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
    [Crossref]
  14. S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
  21. S. H. Xu, Y. M. Li, and L. R. Lou, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. 15, 1391-1397 (2006).
    [Crossref]

2008 (1)

2007 (2)

P. B. Bareil, Y. Sheng, Y. Q. Chen, and A. Chiou, “Calculation of spherical red blood cell deformation in a dual-beam optical stretcher,” Opt. Express 15, 16029-16034(2007).
[Crossref] [PubMed]

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

2006 (3)

2005 (3)

S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Axial optical trapping forces on two particles trapped simultaneously by optical tweezers,” Appl. Opt. 44, 2667-2672 (2005).
[Crossref] [PubMed]

A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95, 168102 (2005).
[Crossref] [PubMed]

2004 (2)

2003 (1)

2002 (1)

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

2000 (2)

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

1997 (1)

M. Gu, P. C. Ke, and X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666-3668 (1997).
[Crossref]

1996 (1)

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, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B 9, 1922-1930 (1992).
[Crossref]

1979 (1)

G. Roosen and S. Slanksy, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341-346 (1979).
[Crossref]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

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]

Axner, O.

Bareil, P. B.

Block, S. M.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Brevik, I.

Chang, Y. R.

Chen, Y. Q.

Cheng, B. Y.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

Chi, S.

Chiou, A.

Chon, J. W. M.

S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
[Crossref]

Cunningham, C. C.

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

Fällman, E.

Gan, X. S.

S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
[Crossref]

M. Gu, P. C. Ke, and X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666-3668 (1997).
[Crossref]

Gong, Z.

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

Gouesbet, G.

Grehan, G.

Gu, M.

S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
[Crossref]

M. Gu, P. C. Ke, and X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666-3668 (1997).
[Crossref]

Guck, J.

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

Guo, H. L.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

Gussgard, R.

Han, X. H.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

Hsu, L.

Im, K. B.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Kas, J.

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

Ke, P. C.

M. Gu, P. C. Ke, and X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666-3668 (1997).
[Crossref]

Kim, H. I.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Kim, P. S.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Kress, H.

Kuriakose, S.

S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
[Crossref]

Lee, D. Y.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Li, Y. M.

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. 15, 1391-1397 (2006).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Axial optical trapping forces on two particles trapped simultaneously by optical tweezers,” Appl. Opt. 44, 2667-2672 (2005).
[Crossref] [PubMed]

Li, Z. L.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

Liao, G. B.

Lindmo, T.

Lou, L. R.

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. 15, 1391-1397 (2006).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Axial optical trapping forces on two particles trapped simultaneously by optical tweezers,” Appl. Opt. 44, 2667-2672 (2005).
[Crossref] [PubMed]

Moon, T. J.

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

Neuman, K. C.

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[Crossref]

Oh, C. H.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Park, B. C.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Ren, K. F.

Rohrbach, A.

Roosen, G.

G. Roosen and S. Slanksy, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341-346 (1979).
[Crossref]

Sheng, Y.

Sheng, Y. L.

Slanksy, S.

G. Roosen and S. Slanksy, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341-346 (1979).
[Crossref]

Song, S. H.

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

Stelzer, E. H. K.

Wang, Z.

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

Xu, S. H.

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. 15, 1391-1397 (2006).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Axial optical trapping forces on two particles trapped simultaneously by optical tweezers,” Appl. Opt. 44, 2667-2672 (2005).
[Crossref] [PubMed]

Yao, X. C.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

Zhang, D. Z.

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

Appl. Opt. (5)

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]

Chin. Phys. (2)

X. C. Yao, Z. L. Li, H. L. Guo, B. Y. Cheng, X. H. Han, and D. Z. Zhang, “Effect of spherical aberration introduced by water solution on trapping force,” Chin. Phys. 9, 824-826(2000).
[Crossref]

S. H. Xu, Y. M. Li, and L. R. Lou, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. 15, 1391-1397 (2006).
[Crossref]

J. Appl. Phys. (1)

S. Kuriakose, X. S. Gan, J. W. M. Chon, and M. Gu, “Optical lifting force under focused evanescent wave illumination: a ray optics model,” J. Appl. Phys. 97, 083103 (2005).
[Crossref]

J. Korean Phys. Soc. (1)

K. B. Im, D. Y. Lee, H. I. Kim, C. H. Oh, S. H. Song, P. S. Kim, and B. C. Park, “Calculation of optical trapping forces on microspheres in the ray optics regime,” J. Korean Phys. Soc. 40, 930-933 (2002).

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

Opt. Commun. (2)

G. Roosen and S. Slanksy, “Influence of the beam divergence on the exerted force on a sphere by a laser beam and required conditions for stable optical levitation,” Opt. Commun. 29, 341-346 (1979).
[Crossref]

Z. Gong, Z. Wang, Y. M. Li, L. R. Lou, and S. H. Xu, “Axial deviation of an optically trapped particle in trapping force calibration using the drag force method,” Opt. Commun. 273, 37-42 (2007).
[Crossref]

Opt. Express (3)

Phys. Rev. Lett. (2)

J. Guck, R. Ananthakrishnan, T. J. Moon, C. C. Cunningham, and J. Kas, “Optical deformability of soft biological dielectrics,” Phys. Rev. Lett. 84, 5451-5454 (2000).
[Crossref] [PubMed]

A. Rohrbach, “Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory,” Phys. Rev. Lett. 95, 168102 (2005).
[Crossref] [PubMed]

Rev. Sci. Instrum. (2)

M. Gu, P. C. Ke, and X. S. Gan, “Trapping force by a high numerical-aperture microscope objective obeying the sine condition,” Rev. Sci. Instrum. 68, 3666-3668 (1997).
[Crossref]

K. C. Neuman and S. M. Block, “Optical trapping,” Rev. Sci. Instrum. 75, 2787-2809 (2004).
[Crossref]

Other (1)

M. Born and E. Wolf, Principles of Optics, 7th (expanded) ed. (Cambridge U. Press, 1999).

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Figures (5)

Fig. 1
Fig. 1

Directions of trapping forces for a single ray striking a bead.

Fig. 2
Fig. 2

Tracing a single ray using spatial vectors. (a) Scheme of a single ray in the coordinate system. Unit vector n 1 indicates the direction of transmission of a single focused ray and points P, O, and O 0 are the bead center, objective focus, and the center of the objective entrancing aperture, respectively. (b) Correlative vectors for calculating trapping forces. Incident point is M and the corresponding outward normal is unit vector n 2 . Unit vectors n 3 , n 4 , and n 5 indicate directions of gradient trapping force, ray polarization, and normal of the incident plane, respectively. The vectors n 1 , n 2 , and n 3 are located in the incident plane.

Fig. 3
Fig. 3

Trapping forces varying with the (a), (b) transverse and (c) axial displacements for different filled states: (a) transverse force, (b) axial force, (c) axial force.

Fig. 4
Fig. 4

Axial trapping force depending on different transverse displacement for a bead moving along the z axis.

Fig. 5
Fig. 5

(a) Transverse and (b) axial forces varying with transverse displacement for circular and linear polarizations.

Equations (19)

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

F z = F scat = n m P c { 1 + R cos ( 2 θ ) T 2 [ cos ( 2 θ 2 γ ) + R cos ( 2 θ ) ] 1 + R 2 + 2 R cos ( 2 γ ) } ,
F y = F grad = n m P c { R sin ( 2 θ ) T 2 [ sin ( 2 θ 2 γ ) + R sin ( 2 θ ) ] 1 + R 2 + 2 R cos ( 2 γ ) } .
n 1 = ( n 1 x , n 1 y , n 1 z ) = ( sin α cos β , sin α sin β , cos α ) .
x n 1 x = y n 1 y = z n 1 z = t ,
( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 = r bead 2 .
t min = n 1 x x 0 + n 1 y y 0 + n 1 z z 0 D 1 / 2 ,
D = ( n 1 x x 0 + n 1 y y 0 + n 1 z z 0 ) 2 x 0 2 y 0 2 z 0 2 + r bead 2 .
n 2 = ( n 2 x , n 2 y , n 2 z ) = ( n 1 x t min x 0 , n 1 y t min y 0 , n 1 z t min z 0 ) / r bead ,
n 1 · n 2 = cos ( π θ ) .
H = [ ( n 1 x t x 0 ) 2 + ( n 1 y t y 0 ) 2 + ( n 1 z t z 0 ) 2 ] 1 / 2 .
n 3 = ( n 3 x , n 3 y , n 3 z ) = ( n 1 x t min x 0 , n 1 y t min y 0 , n 1 z t min z 0 ) / H .
t = n 1 x x 0 + n 1 y y 0 + n 1 z z 0 .
( Q x , Q y , Q z ) = Q scat n 1 + Q grad n 3 .
E = Rot 1 · Lens · Rot · E 0 ,
E 0 = ( cos ϕ sin ϕ 0 ) ,
Rot = ( cos β sin β 0 sin β cos β 0 0 0 1 ) ,
Lens = ( cos α 0 sin α 0 1 0 sin α 0 cos α ) .
Q scat = f s Q scat , s + f p Q scat , p ,
Q grad = f s Q grad , s + f p Q grad , p .

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