## Local stress distribution on the surface of a spherical cell in an optical stretcher

Optics Express, Vol. 14, Issue 25, pp. 12503-12509 (2006)

http://dx.doi.org/10.1364/OE.14.012503

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### 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

## 1. Introduction

## 2. Theory

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]

12. K.C. Neuman, “Characterization of Photodamage to Escherichia coli in Optical Traps,” Biophys. J. **77**, 2856–2863 (1999). [CrossRef] [PubMed]

*λ*=1.064µm in our calculation. For a RBC cell of radius around

*ρ*=3µm, the ray optics regime criterion (2π

*ρ*/

*λ*≫1) is satisfied. Any near-infrared wavelength can be used in our theory. Only the calculated results are expected to change accordingly.

*P*=

*n*/

_{x}E*c*[9

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]

*n*is the refractive index of the buffer medium,

_{x}*E*is the beam energy and

*c*the speed of light. We denote the momentum of the incident, transmitted and reflected rays by

*P⃗*

_{i},

*P⃗*

_{t}and

*P⃗*

_{r}, and their directional unit vectors by

*a⃗*,

_{k}*a⃗*and

_{t}*a⃗*respectively. Then, according to the law of momentum conservation, Δ

_{r}*P⃗*=

*P⃗*-(

_{i}*P⃗*+

_{t}*P⃗*), the stress

_{r}*σ⃗*applied to the cell’s refractive surface is expressed as:

*A*the area covered by the beam,

*n*=

*n*/

_{2}*n*, with

_{1}*n*and

_{1}*n*being the index of the medium surrounding the cell and inside the cell, respectively,

_{2}*T*and

*R*are the Fresnel transmittance and reflectance, respectively, and Q⃑ is the dimensionless momentum transfer vector defined by Eq. (1). The problem is thus reduced to finding Q⃑. For a stable trap [10

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]

*ρ*, (

*w*/

*ρ*)>1. In Ref. [2

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]

*x*-axis in Fig. 1). We consider a diverging Gaussian beam from the fiber hitting the front surface (left half) of the sphere. For a given distance from the fiber end to the cell center

*D*, there is a unique relationship between the incident point defined by polar angle

*ϕ*and the incident angle

_{1}*ε*. In fact, as shown in Fig. 1

*ϕ*=

_{1}*ε*-

*δ*with tan

^{-1}(

*ρ*sin

*ϕ*/(

_{1}*D*-

*ρ*cos

*ϕ*

_{1})). The polar angle

*ϕ*is the incident angle when the beam is parallel to the x-axis.

_{1}*β*is determined by the Snell’s law

*n*

_{1}sin

*ε*=

*n*

_{2}sin

*β*. After the first refraction, the transmitted ray hits the back surface (right-half) of the sphere from the inside of the cell at a point defined by polar angle

*ϕ*=2

_{2}*β*-

*ϕ*. The angle of the reflected ray to the

_{1}*x*-axis is

*π*-(3

*β*-

*ϕ*

_{1}, and that of the transmitted ray is

*ε*+

*ϕ*-2

_{1}*β*, as shown in Fig. 1. As the reflectance is on the order of 10

^{-3}at normal incidence for the refractive index

*n*=1.33 for the buffer, and

_{1}*n*≈1.38 for the cells, the third and subsequent reflections inside the cell would have relatively weak power and result in weak stress, which can be neglected. Once all angles are found, one can deduce

_{2}*Q⃗*. For the front surface, we have:

*ρ*

^{2}sin

^{2}(

*ϕ*)/

_{1}*w*

^{2}] is a Gaussian beam correction factor. One should note that we made no assumption thus far about the state of polarization, which can affect the reflectance and the transmittance.

*Q⃗*

_{front}and

*Q ⃗*

_{back}are perpendicular to the spherical surface. In the appendix we analytically prove, using Eq. (1) and Eqs. (2–5), that:

*R*and transmittance

*T*and therefore of the incident beam polarization. With the proof of the orthogonality we can write:

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]

**81**, 767–784 (2001). [CrossRef] [PubMed]

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]

*x*-axis and can be analyzed in the

*x-y*plane only.

*ϕ*

_{2}=2 sin

^{-1}[(

*n*

_{1}/

*n*

_{2})sin

*ε*]-

*ε*+

*δ*, where

*n*<

_{1}*n*, as a function of the incidence angle

_{2}*ε*. For small

*ε*,

*ϕ*increases with

_{2}*ε*monotonically. Then, the increase of

*ϕ*is slowed down and finally

_{2}*ϕ*decreases with increasing

_{2}*ε*for

*ε*>

*, as shown in Fig. 2, where*ε ˜

*, corresponding to a maximum output polar angle*ε ˜

*ϕ*(

_{2}*), can be computed by the derivative of*ε ˜

*ϕ*with respect to

_{2}*ε*. Consequently, there is an upper limit of the output polar angle

*ϕ*that depends on the indices

_{2}*n*and

_{1}*n*, the ratio

_{2}*w*/

*ρ*, the fiber

*NA*and the cell radius

*ρ*. As an example, for

*NA*=0.11,

*n*=1.335,

_{1}*n*=1.37,

_{2}*D*=39.9µm (

*w*/

*ρ*=1.1) and

*ρ*=3µm,

*ϕ*=71°. There is no incident ray at the front surface, whose refracted ray can hit the back surface at a position of polar angle higher than the upper limit. Below the upper limit there is a range of

_{2}*ϕ*, where the same output position

_{2}*ϕ*can be reached by two different incident angles, as shown in Fig. 2(a). This range is 65°≤

_{2}*ϕ*≤71° in our example.

_{2}*ϕ*=70° the cell is also hit by a third incident ray from the counter-propagating laser beam. At that position, the contributions of the three rays should be added up. One can therefore separate both the front and back surfaces into 4 regions. In the first region, one incident ray from outside and one ray from inside the cell hit the same point on the surface. This region is limited by the polar angle between 0°<

_{2}*ϕ*<65° in our example. In the second region, three rays, one incident ray from outside and two rays from inside of the cell hit the same point at the surface. This region is limited by the polar angle between 65°<

*ϕ*<71° in our example. In the third region only one external ray hits the cell and exerts a stress on the cell surface, and no other ray can hit this point from inside the cell. The third region is limited by the polar angle 71°<

*ϕ*<87° in our example. The fourth region is limited by the highest position that a ray can hit on the surface for a given fiber

*NA*and fiber-to-cell distance

*D*. Thus, no stress is applied for

*ϕ*>87° in the example. In the first and second regions, to calculate the stress applied to the front surface at a position

*ϕ*, we need to find the incidence angle of the ray (

_{1}*ε*) and

_{2}*ϕ*(

_{2}*ε*) coming in the -

_{2}*x*direction, which will make a polar angle of

*ϕ*=(2

_{1}*β*-

*ϕ*) to the -

_{2}*x*direction when hitting the position

*ϕ*. In other words, we need to solve equation on 2

_{1}*β*-

*ϕ*

_{2}(

*ε*

_{2})=

*ϕ*

_{1}(

*ε*

_{1}) to find

*ϕ*for a given

_{2}*ϕ*. Then we compute Eqs. (2–3) with

_{1}*ϕ*, Eqs. (4–5) with

_{1}*ϕ*and we add the results using Eq. (8):

_{2}_{back}and Q

_{front}is depicted in Fig. 3, where the front surface is denoted as the first surface and the back surface as the second for the incident beam in +

*x*direction, and vice versa for the incident beam propagating in -

*x*direction. Indeed, in the region 0°<

*ϕ*<65° the stress is the addition of contributions of the two rays. In the region 65°<

*ϕ*<71°, the stress is the addition of contributions of the three rays and is thus much more intense. In the region

*ϕ*>71° only the rays hitting the surface from outside of the cell contribute. Finally, in the region

*ϕ*>87° the stress profile at the first surface is cut to zero.

*D*from the fibers end to the cell center. We can see the peaks in the stress distribution located at about 60°, 120°, 240° and 300° positions, which were not revealed in the earlier results [2

**81**, 767–784 (2001). [CrossRef] [PubMed]

*ρ*the stress strength is proportional to the intensity of the input beam, which decreases as 1/

*D*. However the Gaussian beam correction factor exp[-2

^{2}*ρ*

^{2}sin

^{2}(

*ϕ*)/

_{1}*w*

^{2}] increases as the beam size

*w*or the distance

*D*increases. This is why at greater incident angles, close to 90° and 270°, the stress profiles are less sensitive to the distance. We see in Fig. 4 that the surface region near 90°, where no stress is applied, gets smaller as the distance

*D*increases, and that the width of the peak stays almost the same at all distances.

## 3. Conclusion

*NA*. We have shown that the focusing power of the spherical cell concentrates the refracted rays to a smaller area on the second interface, resulting in peaks on the stress distribution around certain angular positions. In addition, we have demonstrated that the optical stress is perpendicular to the spherical surface independently on the incident angle, polarization of the incident beam, reflectance and transmittance at the cell surface.

## Appendix

*ϕ*and having an incident angle

*ε*, we can rotate the

*x*-

*y*axes by

*δ*=

*ε*-

*ϕ*(see Fig. 1) such that the incident ray is parallel to the new

*x*-axis. Thus, the incident angle is equal to the polar angle

*ε*=

*ϕ*with the new axes. We consider G=Q

_{front y}/Q

_{front x}. Using the Snell’s law

*n*=

*n*/

_{2}*n*=sin(

_{1}*ϕ*)/sin(

*β*) and redistributing the parentheses, we obtain

^{2}(ϕ)sin(

*β*). Equation (A1) can then be simplified to obtain A2:

*ϕ*, where the negative sign means the stress is in +

*y*direction and -

*x*direction for 0<

*ϕ*<π/2, i.e. the stress is directed away from the cell, and thus stretching the cell.

_{back y}/Q

_{back x}and we use

*ϕ*=2

_{2}*β*-

*ϕ*to express the angles. We have:

_{2})/cos(ϕ

_{2}). Again, arctan(G)=

*ϕ*=

_{2}*2*-

_{β}*ϕ*. The positive sign meaning the stress is in +

*y*direction and +

*x*direction for 0<

*ϕ*<π/2. Thus, the local scattering force is perpendicular to the spherical refraction surface.

## Acknowledgments

## References and Links

1. | A. Constable, Jinha Kim, J. Mervis, F. Zarinetchi, and M. Prentiss, “Demonstration of a fiber-optical lightforce trap,” Opt. Lett. |

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. |

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 |

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. |

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. |

6. | Y.P. Liu, Chuan Li, and A.C.K. Lai, “Experimental study on the deformation of erythrocytes under optically trapping and stretching,” Mater. Sci. Eng. A |

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 |

8. | R. M. Hochmuth, “Micropipette aspiration of living cells,” J. Biomech. |

9. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the ray optics regime,” Biophys. J. |

10. | G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. |

11. | P. J. Rodrigo, I. R. Perch-Nielsen, and J. Glückstad, “Three-dimensional forces in GPC-based counterpropagating-beam traps,” Opt. Express |

12. | K.C. Neuman, “Characterization of Photodamage to Escherichia coli in Optical Traps,” Biophys. J. |

**OCIS Codes**

(000.0000) General : General

(140.7010) Lasers and laser optics : Laser trapping

**ToC Category:**

Trapping

**History**

Original Manuscript: September 11, 2006

Revised Manuscript: November 20, 2006

Manuscript Accepted: November 24, 2006

Published: December 11, 2006

**Virtual Issues**

Vol. 2, Iss. 1 *Virtual Journal for Biomedical Optics*

**Citation**

Paul B. Bareil, Yunlong Sheng, and Arthur Chiou, "Local scattering stress distribution on surface of a spherical cell in optical stretcher," Opt. Express **14**, 12503-12509 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-25-12503

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### References

- 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]
- 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]
- 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]
- 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]
- 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]
- 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]
- R. M. Hochmuth, "Micropipette aspiration of living cells," J. Biomech. 33,15-22 (2000). [CrossRef]
- 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]
- 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]
- 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]
- K.C. Neuman, "Characterization of Photodamage to Escherichia coli in Optical Traps," Biophys. J. 77,2856-2863 (1999). [CrossRef] [PubMed]

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