Abstract

We calculate stress distribution on the surface of a spherical cell trapped by two counter-propagating beams in an optical stretcher in the ray optics regime. We explain the apparition of peaks in the stress distribution, which were not revealed in the earlier published results. We consider the divergence of the incident beams from the fibers, and express the stress distribution as a function of fiber-to-cell distance. In an appendix, we show that the local scattering stress is perpendicular to the spherical refractive surface regardless of incident angle, polarization, the reflectance and transmittance at the surface. Our results may serve as a guideline for the optimization of experimental parameters in optical stretchers.

© 2006 Optical Society of America

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  1. A. Constable, Jinha Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, "Demonstration of a fiber-optical light-force trap," Opt. Lett. 18,1867-1869 (1993).
    [CrossRef] [PubMed]
  2. J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
    [CrossRef] [PubMed]
  3. M. Wei, K. Yang, A. Karmenyan, and A. Chiou, "Three-dimensional optical force field on a Chinese hamster ovary cell in a fiber-optical dual-beam trap," Opt. Express 14,3056-3064 (2006).
    [CrossRef] [PubMed]
  4. Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
    [CrossRef] [PubMed]
  5. S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
    [CrossRef] [PubMed]
  6. Y.P. Liu, Chuan Li, A.C.K. Lai, "Experimental study on the deformation of erythrocytes under optically trapping and stretching," Mater. Sci. Eng. A 423,128-133 (2006).
    [CrossRef]
  7. A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
    [CrossRef]
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    [CrossRef]
  9. 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]
  10. 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 (1997).
    [CrossRef]
  11. P. J. Rodrigo, I. R. Perch-Nielsen, and J. Glückstad, "Three-dimensional forces in GPC-based counterpropagating-beam traps," Opt. Express 14,5812-5822 (2006).
    [CrossRef] [PubMed]
  12. K.C. Neuman, "Characterization of Photodamage to Escherichia coli in Optical Traps," Biophys. J. 77,2856-2863 (1999).
    [CrossRef] [PubMed]

2006

M. Wei, K. Yang, A. Karmenyan, and A. Chiou, "Three-dimensional optical force field on a Chinese hamster ovary cell in a fiber-optical dual-beam trap," Opt. Express 14,3056-3064 (2006).
[CrossRef] [PubMed]

Y.P. Liu, Chuan Li, A.C.K. Lai, "Experimental study on the deformation of erythrocytes under optically trapping and stretching," Mater. Sci. Eng. A 423,128-133 (2006).
[CrossRef]

Y.P. Liu, Chuan Li, A.C.K. Lai, "Experimental study on the deformation of erythrocytes under optically trapping and stretching," Mater. Sci. Eng. A 423,128-133 (2006).
[CrossRef]

P. J. Rodrigo, I. R. Perch-Nielsen, and J. Glückstad, "Three-dimensional forces in GPC-based counterpropagating-beam traps," Opt. Express 14,5812-5822 (2006).
[CrossRef] [PubMed]

2001

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

2000

R. M. Hochmuth, "Micropipette aspiration of living cells," J. Biomech. 33,15-22 (2000).
[CrossRef]

1999

Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
[CrossRef] [PubMed]

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
[CrossRef] [PubMed]

K.C. Neuman, "Characterization of Photodamage to Escherichia coli in Optical Traps," Biophys. J. 77,2856-2863 (1999).
[CrossRef] [PubMed]

1997

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 (1997).
[CrossRef]

1993

1992

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]

Ananthakrishnan, R.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[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]

Butt, H.-J.

A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
[CrossRef]

Chiou, A.

Chuan Li, Y.P.

Y.P. Liu, Chuan Li, A.C.K. Lai, "Experimental study on the deformation of erythrocytes under optically trapping and stretching," Mater. Sci. Eng. A 423,128-133 (2006).
[CrossRef]

Constable, A.

Cunningham, C.C.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

Gallet, F.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
[CrossRef] [PubMed]

Glückstad, J.

Gotzost, V.

A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
[CrossRef]

Gratzer, W.

Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
[CrossRef] [PubMed]

Guck, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

Hénon, S.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
[CrossRef] [PubMed]

Hochmuth, R. M.

R. M. Hochmuth, "Micropipette aspiration of living cells," J. Biomech. 33,15-22 (2000).
[CrossRef]

Jinha Kim, A.

Karmenyan, A.

Käs, J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

Kasast, S.

A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
[CrossRef]

Khorsandit, M.

A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
[CrossRef]

Lenormand, G.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
[CrossRef] [PubMed]

Liu, Y.P.

Y.P. Liu, Chuan Li, A.C.K. Lai, "Experimental study on the deformation of erythrocytes under optically trapping and stretching," Mater. Sci. Eng. A 423,128-133 (2006).
[CrossRef]

Mahmood, H.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

Moon, T. J.

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

Neuman, K.C.

K.C. Neuman, "Characterization of Photodamage to Escherichia coli in Optical Traps," Biophys. J. 77,2856-2863 (1999).
[CrossRef] [PubMed]

Perch-Nielsen, I. R.

Richert, A.

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
[CrossRef] [PubMed]

Rodrigo, P. J.

Roosen, G.

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 (1997).
[CrossRef]

Simmons, R.

Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
[CrossRef] [PubMed]

Sleep,

Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
[CrossRef] [PubMed]

Wei, M.

Weisenhornt, A. L

A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
[CrossRef]

Wilson, D.

Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
[CrossRef] [PubMed]

Yang, K.

Biophys. J.

Sleep, J. , D. Wilson, R. Simmons, and W. Gratzer, "Elasticity of the red cell membrane and its relation to hemolytic disorders: an optical tweezers study," Biophys. J. 77,3085-3095 (1999).
[CrossRef] [PubMed]

S. Hénon, G. Lenormand, A. Richert, and F. Gallet, "A New Determination of the Shear Modulus of the Human Erythrocyte Membrane Using Optical Tweezers," Biophys. J. 76,1145-1151, (1999).
[CrossRef] [PubMed]

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]

J. Guck, R. Ananthakrishnan, H. Mahmood, T. J. Moon, C.C. Cunningham, and J. Käs, "The Optical Stretcher: A Novel Laser Tool to Micromanipulate Cells," Biophys. J. 81,767-784 (2001).
[CrossRef] [PubMed]

K.C. Neuman, "Characterization of Photodamage to Escherichia coli in Optical Traps," Biophys. J. 77,2856-2863 (1999).
[CrossRef] [PubMed]

J. Biomech.

R. M. Hochmuth, "Micropipette aspiration of living cells," J. Biomech. 33,15-22 (2000).
[CrossRef]

Mater. Sci. Eng. A

Y.P. Liu, Chuan Li, A.C.K. Lai, "Experimental study on the deformation of erythrocytes under optically trapping and stretching," Mater. Sci. Eng. A 423,128-133 (2006).
[CrossRef]

Nanotechnology

A. L Weisenhornt, M. Khorsandit, S. Kasast, V. Gotzost and H.-J. Butt, "Deformation and height anomaly of soft surfaces studied with an AFM," Nanotechnology 4,106-113 (1993).
[CrossRef]

Opt. Commun.

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 (1997).
[CrossRef]

Opt. Express

Opt. Lett.

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Figures (4)

Fig. 1.
Fig. 1.

Incident, reflected and transmitted rays on a spherical object.

Fig. 2.
Fig. 2.

(a) Position of the output ray on the back surface as a function of incidence angle. Intersection of the horizontal lines and curves are the solutions for two incident rays at a same output position. In the example shown in (b), at incident angles ε=78.2° and 84.3° the rays pass through the cell and hit the back surface at the same point with a polar angle of ϕ2 =70°.

Fig. 3.
Fig. 3.

Stress profile as function of the polar angle. Thick line: at the second surface; Thin line: at the first surface. NA=0.11, n1 =1.335 and n2 =1.37, D=39.9µm (w/ρ=1.1) and ρ=3µm.

Fig. 4
Fig. 4

Stress profile (Nm-2) for different distances D with ��=100mW, ρ=3µm, n1 =1.335, n2 =1.37 and NA=0.11.

Equations (15)

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σ = Δ P A Δ t = P i ( P t + P r ) A Δ t = 1 c E i A Δ t n 1 ( a k ( nT a t + R a r ) ) n 1 c 𝓟 A Q
Q front X = exp 2 ( ρ sin ( ϕ 1 ) w ) 2 [ cos ( δ ) nT ( ε ) cos ( ϕ 1 β ) + R ( ε ) cos ( 2 ε ) ]
Q front Y = exp 2 ( ρ sin ( ϕ 1 ) w ) 2 [ sin ( δ ) nT ( ε ) sin ( ϕ 1 β ) R ( ε ) sin ( 2 ε ) ]
Q back X = exp 2 ( ρ sin ( ϕ 1 ) w ) 2 T ( ε ) [ n cos ( ϕ 1 β ) + nR ( β ) cos ( 3 β ϕ 1 ) T ( β ) cos ( ε + ϕ 1 2 β ) ]
Q back Y = exp 2 ( ρ sin ( ϕ 1 ) w ) 2 T ( ε ) [ n sin ( ϕ 1 β ) + nR ( β ) sin ( 3 β ϕ 1 ) + T ( β ) sin ( ε + ϕ 1 2 β ) ]
arctan Q front Y ( ϕ 1 ) Q front X ( ϕ 1 ) = ϕ 1 and arctan Q back Y ( ϕ 1 ) Q back X ( ϕ 1 ) = 2 β ϕ 1 = ϕ 2
Q front X = exp 2 ( ρ sin ( ϕ 1 ) w ) 2 Q front cos ( ϕ 1 )
Q front Y = exp 2 ( ρ sin ( ϕ 1 ) w ) 2 Q front sin ( ϕ 1 )
Q back X = exp 2 ρ 2 sin 2 ( ϕ 1 ) w 2 Q back cos ( 2 β ϕ 1 )
Q back Y = exp 2 ρ 2 sin 2 ( ϕ 1 ) w 2 Q back sin ( 2 β ϕ 1 )
Q tot = Q front + Q back = ( Q front X ( ϕ 1 ) ) 2 + ( Q front Y ϕ 1 ) 2 + ( Q back X ( ϕ 2 ) ) 2 + ( Q back Y ( ϕ 2 ) ) 2
G = sin ( ϕ ) T [ sin ( ϕ ) cos ( β ) cos ( ϕ ) sin ( β ) + 2 cos ( ϕ ) sin ( β ) ] 2 cos ( ϕ ) sin ( β ) sin ( ϕ ) T sin ( ϕ ) cos ( ϕ ) cos ( β ) + [ T sin 2 ( ϕ ) sin β ) + T sin ( β ) 2 T cos 2 ( ϕ ) sin ( β ) ] + 2 cos 2 ( ϕ ) sin ( β )
G = sin ( ϕ ) cos ( ϕ ) T [ sin ( ϕ ) cos ( β ) + cos ( ϕ ) sin ( β ) ] 2 cos ( ϕ ) sin ( β ) T [ sin ( ϕ ) cos ( β ) + cos ( ϕ ) sin ( β ) ] 2 cos ( ϕ ) sin ( β ) = sin ( ϕ ) cos ( ϕ )
G = n sin ( β ϕ 2 ) + n ( 1 T ) sin ( β + ϕ 2 ) + T sin ( ϕ ϕ 2 ) n cos ( β ϕ 2 ) + n ( 1 T ) cos ( β + ϕ 2 ) T cos ( ϕ ϕ 2 )
G = 2 n cos ( β ) sin ( ϕ 2 ) nT [ sin ( β ) cos ( ϕ 2 ) + cos ( β ) sin ( ϕ 2 ) ] + T sin ( ϕ ) cos ( ϕ 2 ) T cos ( ϕ ) sin ( ϕ 2 ) 2 n cos ( β ) cos ( ϕ 2 ) + nT ( cos ( β ) cos ( ϕ 2 ) sin ( β ) sin ( ϕ 2 ) ) T cos ( ϕ ) cos ( ϕ 2 ) + T sin ( ϕ ) sin ( ϕ 2 )

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