## Calculation of optical forces on an ellipsoid using vectorial ray tracing method |

Optics Express, Vol. 20, Issue 14, pp. 14928-14937 (2012)

http://dx.doi.org/10.1364/OE.20.014928

Acrobat PDF (861 KB)

### Abstract

For a triaxial ellipsoid in an optical trap with spherical aberration, the optical forces, torque and stress are analyzed using vectorial ray tracing. The torque will automatically regulate ellipsoid’s long axis parallel to optic axis. For a trapped ellipsoid with principal axes in the ratio 1:2:3, the high stress distribution appears in x-z plane. And the optical force at x-axis is weaker than at y-axis due to the shape size. While the ellipsoid departs laterally from trap center, the measurable maximum transverse forces will be weakened due to axial equilibrium and affected by inclined orientation. For an appropriate ring beam, the maximum optical forces are strong in three dimensions, thus, this optical trap is appropriate to trap cells for avoiding damage from laser.

© 2012 OSA

## 1. Introduction

3. G. Volpe, G. P. Singh, and D. Petrov, “Dynamics of a growing cell in an optical trap,” Appl. Phys. Lett. **88**(23), 231106 (2006). [CrossRef]

4. D. P. Cherney, T. E. Bridges, and J. M. Harris, “Optical trapping of unilamellar phospholipid vesicles: investigation of the effect of optical forces on the lipid membrane shape by confocal-Raman microscopy,” Anal. Chem. **76**(17), 4920–4928 (2004). [CrossRef] [PubMed]

5. J. C. Loudet, A. G. Yodh, and B. Pouligny, “Wetting and contact lines of micrometer-sized ellipsoids,” Phys. Rev. Lett. **97**(1), 018304 (2006). [CrossRef] [PubMed]

6. Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. **124**(5-6), 529–541 (1996). [CrossRef]

7. L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. **79**(4), 645–648 (1997). [CrossRef]

8. K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. **35**(15), 2702–2710 (1996). [CrossRef] [PubMed]

15. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance,” J. Appl. Phys. **65**(8), 2900–2906 (1989). [CrossRef]

16. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the Ray Optics Regime,” Biophys. J. **61**(2), 569–582 (1992). [CrossRef] [PubMed]

21. J. H. Zhou, H. L. Ren, J. Cai, and Y. M. Li, “Ray-tracing methodology: application of spatial analytic geometry in the ray-optic model of optical tweezers,” Appl. Opt. **47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

13. S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A **84**(5), 053808 (2011). [CrossRef]

21. J. H. Zhou, H. L. Ren, J. Cai, and Y. M. Li, “Ray-tracing methodology: application of spatial analytic geometry in the ray-optic model of optical tweezers,” Appl. Opt. **47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

## 2.Vectorial ray tracing

16. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the Ray Optics Regime,” Biophys. J. **61**(2), 569–582 (1992). [CrossRef] [PubMed]

### 2.1. Force on an interface

*n*to other medium with refractive index

_{m}*n*, the force on this interface is express as [22

_{p}22. P. B Bareil, Y. L. Sheng, A. Chiou, P. B Bareil, Y. L. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express **14**(25), 12503–12509 (2006). [CrossRef] [PubMed]

*P*is power,

*θ*and

*γ*are the incident and refractive angles, respectively,

*R*and

*T*are Fresnel reflection and transmission coefficients of energy flow [23], the unit vector

**n**denotes interface surface normal towards

*n*. The force can be described by trapping efficiency

_{m}*Q*, which is a dimensionless factor of

_{Q=Fc/nmP}.

### 2.2. Beam profile

16. A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the Ray Optics Regime,” Biophys. J. **61**(2), 569–582 (1992). [CrossRef] [PubMed]

21. J. H. Zhou, H. L. Ren, J. Cai, and Y. M. Li, “Ray-tracing methodology: application of spatial analytic geometry in the ray-optic model of optical tweezers,” Appl. Opt. **47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

24. E. Aspnes, T. D. Milster, and K. Visscher, “Optical force model based on sequential ray tracing,” Appl. Opt. **48**(9), 1642–1650 (2009). [CrossRef] [PubMed]

25. X. Sheng-Hua, L. Yin-Mei, and L. Li-Ren, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. **15**(6), 1391–1397 (2006). [CrossRef]

26. P. Liu and B. Lu, “Phase singularities of the transverse field component of high numerical aperture dark-hollow Gaussian beams in the focal region,” Opt. Commun. **272**(1), 1–8 (2007). [CrossRef]

_{A=P22n+2(2n)!πω0nmε0c},

*n*is the beam order,

*ω*is the beam waist width. The Gaussian beam and ring beam can be described as

_{0}_{n=0}and

_{n>0}, respectively. So the power of surface element

*Δs*can be written as

### 2.3Vectorial ray tracing for an ellipsoid

**47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

17. R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B **14**(12), 3323–3333 (1997). [CrossRef]

18. K. Shima, R. Omori, and A. Suzuki, “Forces of a single-beam gradient-force optical trap on dielectric spheroidal particles in the geometric-optics regime,” Jpn. J. Appl. Phys. **37**(Part 1, No. 11), 6012–6015 (1998). [CrossRef]

20. D. H. Li, J. X. Pu, and X. Q. Wang, “Radiation forces of a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. **285**(7), 1680–1683 (2012). [CrossRef]

27. 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**(13), 2667–2672 (2005). [CrossRef] [PubMed]

22. P. B Bareil, Y. L. Sheng, A. Chiou, P. B Bareil, Y. L. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express **14**(25), 12503–12509 (2006). [CrossRef] [PubMed]

28. 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**(24), 16029–16034 (2007). [CrossRef] [PubMed]

24. E. Aspnes, T. D. Milster, and K. Visscher, “Optical force model based on sequential ray tracing,” Appl. Opt. **48**(9), 1642–1650 (2009). [CrossRef] [PubMed]

*O*and the objective optic axis as

*z*-axis, as shown in Fig. 1(a) . The center of ellipsoid is located at

*P(x*. The objective entrance aperture is in the

_{0},y_{0},z_{0})*O*plane, where

_{0}*R*is the radius of entrance aperture,

_{obj}*f*is the objective focus length. When a single ray located at

_{obj}*(r,β*in the

_{0})*O*plane is deflected by the objective, the ray meets the relation of

_{0}_{sinα1=r/fobj}since the objective obeys the sine condition [29

29. 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**(10), 3666–3668 (1997). [CrossRef]

*n*) and water(

_{g}*n*), the actual incident ray is further deflected in the interface and goes across a point

_{w}*G(0,0,Δz)*. The direction of the incident ray striking the ellipsoid is described as the unit vectorwhere the angle

*α*meets Snell’s law of

_{2}*n*sin

_{g}*α*sin

_{1}= n_{w}*α*. In those relationships,

_{2}_{fobj=ngRobj/NA}, where

*NA*is the numerical aperture of an objective. The deviation

*Δz*varies with a depth

*z*, which indicates the distance from trap center to coverslip. According to geometric relations,The incident ray is described as a linear equation by means of the position of point

_{cg}*G*and vector

**n**

_{1}. So the position of the first incident point can be solved with the standard equation of an ellipsoid and a linear equation of incident ray.

*O*and each semi-principal axis (

*a,b,c*) is parallel to coordinate axis, as shown in Fig. 1(b), the i-th incident point

_{Mi(xi,yi,zi)}and incident ray

_{ni(nix,niy,niz)}meet the Eq. (6). The vector

**n**indicates the outward normal of the interface at

*M*. By solving the parameter

_{i}*t*, the position

*M*and

_{i }_{n(βMi,γMi)}are determined. According to

**n**

_{i}and

**n**, both the incident and refractive angles can be calculated at

*M*. The force at

_{i}*M*can be calculated using Eq. (1).Since the reflective ray

_{i }**n**

_{i + 1}is in the incident plane, the orientation of vector

_{ni+1(βi+1,γi+1)}can be easily determined by incident angle and

**n**

_{i}after rotating coordinate system with the normal

**n**as z’-axis (see in Fig. 1(c)). Then vector

**n**

_{i + 1}is finally determined by inversing coordinate system. Then the next incident point and corresponding force can be calculated by repeating previous process.

**n**

_{i}and position

*M*can be also calculated by mean of coordinate translation. If the semi-principal axes of ellipsoid are inclined to an arbitrary orientation (

_{i}*β*,

_{axis}*γ*) (in Fig. 1(d)), we can calculate force in a new coordinate system by rotating the semi-principal axes parallel to coordinate axes. The final force is the summation of inversed force at each incident point.

_{axis}### 2.4. Senkrecht and parallel components of a single ray

_{fs=fp=0.5}. For a linear polarized beam, the power ratio can be determined by the intersection angle between the polarized direction at incident point and the normal of incident plane [21

**47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

30. P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive-indexes - an integral-representation,” J. Opt. Soc. Am. A **12**(2), 325–332 (1995). [CrossRef]

**E**indicates the direction and intensity of a single ray after a reflection or refraction from an interface, the polarization vector

_{np=E/|E|}. Since the incident ray

**n**

_{i}and the interface normal

**n**compose the incident plane, the normal of which can be expressed as

_{ninci=n×ni}. The power ratio of s-component

_{fs=cos2Ω}, which Ω meets

_{cosΩ=np•ninci}. Accordingly,

_{fp=1−fs}for the p-component. The power of the single ray decreases with many times of reflection and refraction. Before depletion of the ray,

*f*and

_{s }*f*will vary with incident point.

_{p}### 2.5. Stress and torque

*β*,

*γ,*). If the distance from a surface element to the center is

*r*(Fig. 1(e)), the area of reference sphere with radius

_{i}*r*can be expressed as

_{i}_{ri2dβdγ}. Here the sign

*i*indicates the i-th surface element. Since there is an intersection angle of

*α*between the normal of ellipsoidal surface element and the normal of spherical surface element (the angle between vectors

**r**

_{i}and

**F**

_{i}), the surface element can be expressed as

_{dA=ri2dβdγ/cosα}. So the stress

_{σ=Fi/dA=dQnmP/(cdA)}, where

*F*indicates the total force of all rays.

_{i}_{T=∑iri×Fi}. While the ellipsoid locates in an arbitrary orientation, the torque can be calculated with rotating coordinate system as mentioned above.

## 3. Results

_{ξ=ω0/Robj}. Without detailed explain, the beam is random polarization and Gaussian beam with

_{ξ=1}. Some parameters in below calculation are the same,

_{Robj=3}

*mm*,

_{P=10}

*mW,*the depth

_{zcg=−10}

*μm*, NA is 1.25, refractive index

_{nparticle=1.59},

_{nwater=1.33},

_{nglass=1.51}. Taking a sphere with

_{rbead=3}

*μm*as a reference, the fractional radii can be express as

_{εx=a/rbead},

_{εy=b/rbead}and

_{εz=c/rbead}.

### 3.1. Torque of an Ellipsoid

_{βaxis=0}and

_{γaxis=0}, the torque of the ellipsoid is zero. The torque of the ellipsoid with

*β*= 0 is calculated in Tab.1 . In the case of the ellipsoid with

_{axis}_{εx=1},

_{εy=2}and

_{εz=3}, the torque has only component of

*T*while

_{y}_{0<γaxis<π/2}, and

_{Ty<0}. It indicates that the torque will rotate the ellipsoid to the orientation of

_{γaxis=0}. While

_{γaxis>π/2},

_{Ty>0}, it indicates that the torque will rotate the ellipsoid to the orientation of

_{γaxis=π}. However,

_{γaxis=0}and

_{γaxis=π}are in the same orientation. Though the torque is zero while

_{γaxis=π/2}, the ellipsoid in this orientation is not stable, because the torque induced by very little deviation of

*γ*will rotate ellipsoid to enlarge this deviation. In the case of the ellipsoid with

_{axis}_{εx=3},

_{εy=2}and

_{εz=1}, the torque of the ellipsoid is zero while

_{γaxis=0}and

_{γaxis=π}, but very little deviation of

*γ*will induce the particle to speed up this deviation. Finally the ellipsoid is stable at

_{axis}_{γaxis=π/2}. So the torque will automatically regulate orientation of the ellipsoid to long axis parallel to optic axis (

*z*-axis).

### 3.2 Stress of an ellipsoid

*β*and

*γ*on an ellipsoid surface are divided into

_{64×80}. Figure 2 indicates the stress distribution of a sphere (

_{εx=εy=εz=1}) and an ellipsoid (

_{εx=1,}

_{εy=2,}

_{εz=3}) in an optical trap with spherical aberration. The stress of the sphere at a specific

*γ*has the same value for different

*β*. For a sphere at original point

*O*, the stress distribution is severe at a narrow region (160°-180°) with large peak at

_{γ=165°}as this surface (the front surface, 90°-180°) suffers incident focused rays at first. Since the dielectric sphere is similar to a lens, the refractive rays across on the front surface are further deflected. As a result, the striking region of incident rays on the back surface (0°-90°) is larger than that on the front surface. Furthermore, the rays striking on the back surface are diffused due to the spherical aberration of particle. So the stress distribution on the back surface is low and gentle.

*O*andits longest axis is parallelto optic axis, the stress

*σ*varies with orientation

*β*as an ellipsoid is not symmetric for revolution. While

_{β=0}(in

*x-z*plane) in Fig. 2, stress

*σ*is relatively strong on the back surface with a large peak at

_{γ=~55°}, but it is weak on the front surface. However, the overall distribution of stress is very weak on both front and back surfaces while

_{β=π/2}(in

*y-z*plane). Those differences are originated from the astigmatism when rays go across an aspheric surface of ellipsoid. As a whole, the stress distribution on the surface of small fractional radius (along x-axis,

_{β=0}) will stronger than on the surface of large fractional radius (along y-axis,

_{β=π/2}). For soft particles such as red blood cells or vesicles, the surface of small fractional radius can be expanded more easily due to high stress.

### 3.3 Optical Forces on an Ellipsoid

_{εx=1},

_{εy=2}and

_{εz=3}as an example, we analyze what factors affect the optical forces.

**47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

_{|Qtr|max}) in Fig. 3(a), however, the force at x-axis is much lower than at y-axis whether the beam is polarized or not. So does axial pushing force (

_{Qz>0}) in Fig. 3(b) while the ellipsoid has transverse displacement. Figure 3(c) shows the axial force at z-axis, there is a very small difference near

_{Qz=0}in the two cases of polarization. So the polarization affects the forces very little compared with deformable ratio.

**47**(33), 6307–6314 (2008). [CrossRef] [PubMed]

31. F. Merenda, G. Boer, J. Rohner, G. Delacrétaz, and R. P. Salathé, “Escape trajectories of single-beam optically trapped micro-particles in a transverse fluid flow,” Opt. Express **14**(4), 1685–1699 (2006). [CrossRef] [PubMed]

32. 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**(1), 37–42 (2007). [CrossRef]

*γ*varies, the corresponding torques are shown in Tab. 1, and the optical forces in the two cases are shown in Fig. 5 . For type-A (in Fig. 5(a)),

_{axis}_{|Qx|max}increases within creasing of

*γ*, and its corresponding location is far away from the trap center. Thus, the experimental maximum displacement at x-axis for inclined ellipsoid will be beyond in the case of

_{axis}*γ*. For type-B (in Fig. 5(b)), however,

_{axis}= 0_{|Qy|max}decreases with increasing of

*γ*, and its corresponding location changes very little. If we assay the transverse forces, the maximum force at y-axis is weakened but enhanced at x-axis due to the inclined pose of an ellipsoid. While the ellipsoid moves along axes at appropriate inclined angle, the gap between

_{axis}_{|Qx|max}and

_{|Qy|max}can be small. From Figs. 5(a) to 5(b), the maximum transverse forces on an ellipsoid are both larger than those forces on a sphere. For axial force at

*z*-axis, there is no difference in two cases with same inclined angle. In Fig. 5(c), the axial restoring force decreases with increasing of

*γ*, this will weaken the axial stability, as mentioned in section 3.1.

_{axis}## 4. Summary

_{ξ>0.4}can improve axial trapping stability, but the appropriate beam profile does not weaken the maximum transverse optical forces, such as

_{ξ=0.6}(

_{n=1}). Due to the effect of shape size,

_{|Qy|max}is much larger than

_{|Qx|max}in theoretical calculations for both a Gaussian and ring beams. while an ellipsoid departs laterally from trap center,

_{|Qx|max}will be enhanced and

_{|Qy|max}will be weakened by inclined pose. Thus, the measurement of force depends on the shape size and orientation. To manipulate a living cell, the laser power of an optical trap should be reduced as low as possible for avoiding the damage from laser. However, the optical forces and axial stability will be weakened at the same time. For the optical trap formed by an appropriate ring beam, since the maximum transverse displacement can be enlarged and the axial stability is improved, the maximum transverse forces are enhanced. Thus, this optical trap is more appropriate to trap cells directly.

## Acknowledgments

## References and links

1. | J. G. Wu, Y. M. Li, D. Lu, Z. Liu, Z. D. Cheng, and L. Q. He, “Mesurement of the membrane elasticity of red blood cell with osmotic pressure by optical tweezers,” Cryo Lett. |

2. | S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of biological cells using plane-polarized Gaussian beam optical tweezers,” J. Mod. Opt. |

3. | G. Volpe, G. P. Singh, and D. Petrov, “Dynamics of a growing cell in an optical trap,” Appl. Phys. Lett. |

4. | D. P. Cherney, T. E. Bridges, and J. M. Harris, “Optical trapping of unilamellar phospholipid vesicles: investigation of the effect of optical forces on the lipid membrane shape by confocal-Raman microscopy,” Anal. Chem. |

5. | J. C. Loudet, A. G. Yodh, and B. Pouligny, “Wetting and contact lines of micrometer-sized ellipsoids,” Phys. Rev. Lett. |

6. | Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun. |

7. | L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett. |

8. | K. F. Ren, G. Gréhan, and G. Gouesbet, “Prediction of reverse radiation pressure by generalized Lorenz-Mie theory,” Appl. Opt. |

9. | T. A. Nieminen, H. Rubinsztein-Dunlop, N. R. Heckenberg, and A. I. Bishop, “Numerical modeling of optical trapping,” Comput. Phys. Commun. |

10. | F. Xu, K. F. Ren, G. Gouesbet, X. S. Cai, and G. Gréhan, “Theoretical prediction of radiation pressure force exerted on a spheroid by an arbitrarily shaped beam,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

11. | S. H. Simpson and S. Hanna, “Optical trapping of spheroidal particles in Gaussian beams,” J. Opt. Soc. Am. A |

12. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. Soc. Am. A |

13. | S. H. Simpson and S. Hanna, “Computational study of the optical trapping of ellipsoidal particles,” Phys. Rev. A |

14. | Z. J. Li, Z. S. Wu, and Q. C. Shang, “Calculation of radiation forces exerted on a uniaxial anisotropic sphere by an off-axis incident Gaussian beam,” Opt. Express |

15. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal fields of a spherical particle illuminated by a tightly focused laser beam: Focal point positioning effects at resonance,” J. Appl. Phys. |

16. | A. Ashkin, “Forces of a single-beam gradient laser trap on a dielectric sphere in the Ray Optics Regime,” Biophys. J. |

17. | R. C. Gauthier, “Theoretical investigation of the optical trapping force and torque on cylindrical micro-objects,” J. Opt. Soc. Am. B |

18. | K. Shima, R. Omori, and A. Suzuki, “Forces of a single-beam gradient-force optical trap on dielectric spheroidal particles in the geometric-optics regime,” Jpn. J. Appl. Phys. |

19. | J. S. Kim and S. W. Kim, “Dynamic motion analysis of optically trapped nonspherical particles with off-axis position and arbitrary orientation,” Appl. Opt. |

20. | D. H. Li, J. X. Pu, and X. Q. Wang, “Radiation forces of a dielectric medium plate induced by a Gaussian beam,” Opt. Commun. |

21. | J. H. Zhou, H. L. Ren, J. Cai, and Y. M. Li, “Ray-tracing methodology: application of spatial analytic geometry in the ray-optic model of optical tweezers,” Appl. Opt. |

22. | P. B Bareil, Y. L. Sheng, A. Chiou, P. B Bareil, Y. L. Sheng, and A. Chiou, “Local scattering stress distribution on surface of a spherical cell in optical stretcher,” Opt. Express |

23. | M. Born and E. Wolf, “Reflectivity and transmissivity,” in |

24. | E. Aspnes, T. D. Milster, and K. Visscher, “Optical force model based on sequential ray tracing,” Appl. Opt. |

25. | X. Sheng-Hua, L. Yin-Mei, and L. Li-Ren, “Systematical study of the trapping forces of optical tweezers formed by different types of optical ring beams,” Chin. Phys. |

26. | P. Liu and B. Lu, “Phase singularities of the transverse field component of high numerical aperture dark-hollow Gaussian beams in the focal region,” Opt. Commun. |

27. | S. H. Xu, Y. M. Li, and L. R. Lou, “Axial optical trapping forces on two particles trapped simultaneously by optical tweezers,” Appl. Opt. |

28. | 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 |

29. | 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. |

30. | P. Török, P. Varga, Z. Laczik, and G. R. Booker, “Electromagnetic diffraction of light focused through a planar interface between materials of mismatched refractive-indexes - an integral-representation,” J. Opt. Soc. Am. A |

31. | F. Merenda, G. Boer, J. Rohner, G. Delacrétaz, and R. P. Salathé, “Escape trajectories of single-beam optically trapped micro-particles in a transverse fluid flow,” Opt. Express |

32. | 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. |

**OCIS Codes**

(350.4855) Other areas of optics : Optical tweezers or optical manipulation

(080.5692) Geometric optics : Ray trajectories in inhomogeneous media

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: April 18, 2012

Revised Manuscript: May 23, 2012

Manuscript Accepted: June 5, 2012

Published: June 19, 2012

**Virtual Issues**

Vol. 7, Iss. 9 *Virtual Journal for Biomedical Optics*

**Citation**

Jin-Hua Zhou, Min-Cheng Zhong, Zi-Qiang Wang, and Yin-Mei Li, "Calculation of optical forces on an ellipsoid using vectorial ray tracing method," Opt. Express **20**, 14928-14937 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-14-14928

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### References

- J. G. Wu, Y. M. Li, D. Lu, Z. Liu, Z. D. Cheng, and L. Q. He, “Mesurement of the membrane elasticity of red blood cell with osmotic pressure by optical tweezers,” Cryo Lett.30, 89–95 (2009).
- S. Bayoudh, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of biological cells using plane-polarized Gaussian beam optical tweezers,” J. Mod. Opt.50, 1581–1590 (2003).
- G. Volpe, G. P. Singh, and D. Petrov, “Dynamics of a growing cell in an optical trap,” Appl. Phys. Lett.88(23), 231106 (2006). [CrossRef]
- D. P. Cherney, T. E. Bridges, and J. M. Harris, “Optical trapping of unilamellar phospholipid vesicles: investigation of the effect of optical forces on the lipid membrane shape by confocal-Raman microscopy,” Anal. Chem.76(17), 4920–4928 (2004). [CrossRef] [PubMed]
- J. C. Loudet, A. G. Yodh, and B. Pouligny, “Wetting and contact lines of micrometer-sized ellipsoids,” Phys. Rev. Lett.97(1), 018304 (2006). [CrossRef] [PubMed]
- Y. Harada and T. Asakura, “Radiation forces on a dielectric sphere in the Rayleigh scattering regime,” Opt. Commun.124(5-6), 529–541 (1996). [CrossRef]
- L. Novotny, R. X. Bian, and X. S. Xie, “Theory of nanometric optical tweezers,” Phys. Rev. Lett.79(4), 645–648 (1997). [CrossRef]
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