## Calculation of radiation force and torque exerted on a uniaxial anisotropic sphere by an incident Gaussian beam with arbitrary propagation and polarization directions |

Optics Express, Vol. 20, Issue 15, pp. 16421-16435 (2012)

http://dx.doi.org/10.1364/OE.20.016421

Acrobat PDF (1237 KB)

### Abstract

On the basis of spherical vector wave functions and coordinate rotation theory, the expansion of the fields of an incident Gaussian beam with arbitrary propagation and polarization directions in terms of spherical vector wave functions is investigated, and beam shape coefficients are derived. Using the results of electromagnetic scattering by a uniaxial anisotropic sphere, the analytical expressions of the radiation force and torque exerted on a homogeneous absorbing uniaxial anisotropic sphere by the arbitrary incident Gaussian beam. We numerically analyze and discuss the following: the effects of an anisotropic absorbing dielectric on the axial and transverse radiation forces exerted by an off-axis Gaussian beam on a uniaxial anisotropic sphere; the variations in the axial, transverse, and resultant radiation forces (with incident angle *β* and polarization angle *α*) imposed by an obliquely Gaussian beam on a uniaxial anisotropic sphere; and the results on the characteristics of the three components of the radiation forces versus the center-to-center distance between the sphere and beam. Selected numerically results on the radiation torque exerted on a stationary uniaxial anisotropic transparent or absorbing sphere by a linearly polarized Gaussian beam are shown, and the results are compared with those exerted an isotropic sphere. The accuracy of the theory and code is confirmed by comparing the axial radiation forces with the results obtained from references.

© 2012 OSA

## 1. Introduction

1. A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. **24**(4), 156–159 (1970). [CrossRef]

3. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. **11**(5), 288–290 (1986). [CrossRef] [PubMed]

4. 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]

5. P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. **25**(15), 1065–1067 (2000). [CrossRef] [PubMed]

6. A. Rohrbach and E. H. K. Stelzer, “Optical trapping of dielectric particle in arbitrary fields,” J. Opt. Soc. Am. A **18**(4), 839–853 (2001). [CrossRef]

*λ*. Conversely, for a particle much larger than the incident wavelength (typically

7. T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry **12**(6), 479–485 (1991). [CrossRef] [PubMed]

12. S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. **37**(27), 6386–6394 (1998). [CrossRef] [PubMed]

*d*/

*λ*range, a rigorous electromagnetic theory has been developed for a spherical particle interacting with a laser beam. In this theory, expanding an incident beam in terms of spherical vector wave functions (SVWFs) is necessary. Kim and Lee [13

13. J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. **73**(3), 303–312 (1983). [CrossRef]

14. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. **64**(4), 1632–1639 (1988). [CrossRef]

15. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**(10), 4594–4962 (1989). [CrossRef]

16. G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A **5**(9), 1427–1443 (1988). [CrossRef]

17. G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorentz-Mie theory,” J. Opt. Soc. Am. A **7**(6), 998–1007 (1990). [CrossRef]

18. K. F. Ren, G. Gréha, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. **108**(4-6), 343–354 (1994). [CrossRef]

19. 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]

20. J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. **43**(12), 2532–2544 (2004). [CrossRef] [PubMed]

21. Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. **38**(2), 131–141 (2002). [CrossRef]

22. G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. **4**(5), 571–585 (1995). [CrossRef]

23. Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. **43**(20), 3999–4006 (2004). [CrossRef] [PubMed]

24. F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. **39**(1), 34–39 (2007). [CrossRef]

16. G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A **5**(9), 1427–1443 (1988). [CrossRef]

*iωt*) is assumed and suppressed, where

*ω*is the circular frequency.

## 2. Theory

*a*centrally located in a spherical coordinate system. The primary optical axis is coincident with the

*z*-axis. As Fig. 1(a) shows, the particle is obliquely illuminated by an

*x'*-polarized Gaussian beam that propagates in the

*z'*-direction in Cartesian coordinate system

*O'x'y'z'*, with the beam center located at

*O'*. The coordinates of beam center

*O'*are (

*x*

_{0},

*y*

_{0},

*z*

_{0}) in

*Oxyz*. Temporary coordinate system

*Ox”y”z”*is established parallel to

*O'x'y'z'*and is known as the beam system. We denote the propagation direction of the beam as two angles in particle coordinate system

*Oxyz*; that is, incident angle

*β*with reference to the

*z*-axis, and polarization angle

*α*between the

*x*-axis and the propagation direction of the beam on the

*xoy*-plane. Then,

*Oxyz*can be obtained by rotating

*Ox”y”z”*with Euler angles

*β*and

*α*(Fig. 1(b)).

### 2.1. Description of the incident and scattered fields

*Ox”y”z”*as follows [39

39. G. Gouesbet, “T-matrix formulation and generalized lozrenz-Mie theories in spherical coordinates,” Opt. Commun. **283**(4), 517–521 (2010). [CrossRef]

*k*

_{0}= 2π/

*λ*and

*λ*is a surrounding medium wavelength,

*μ*

_{0}denotes the permeability of the surrounding medium, and

*E*is the amplitude of the electric field at the beam center. SVWFs

_{0}34. Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A **26**(8), 1778–1788 (2009). [CrossRef]

40. G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. off-axis beams,” J. Opt. Soc. Am. A **11**(9), 2516–2525 (1994). [CrossRef]

41. A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. **36**(13), 2971–2978 (1997). [CrossRef] [PubMed]

*x*

_{0}

*”*,

*y*

_{0}

*”*,

*z*

_{0}

*”*) are the coordinates of beam center

*Ox”y”z”*. According to the coordinate rotation relationship, (

*x*

_{0}

*”*,

*y*

_{0}

*”*,

*z*

_{0}

*”*) can be derived from (

*x*

_{0},

*y*

_{0},

*z*

_{0}) by:

*Oxyz*and

*Ox”y”z”*, with a rotation relationship determined by Euler angles

*β*and

*α*, the SVWFs have the following relationships [35

35. Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A **27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

42. Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express **15**(2), 735–746 (2007). [CrossRef] [PubMed]

43. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systems,” Opt. Commun. **283**(17), 3218–3225 (2010). [CrossRef]

35. Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A **27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

42. Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express **15**(2), 735–746 (2007). [CrossRef] [PubMed]

43. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systems,” Opt. Commun. **283**(17), 3218–3225 (2010). [CrossRef]

35. Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A **27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

42. Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express **15**(2), 735–746 (2007). [CrossRef] [PubMed]

43. G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systems,” Opt. Commun. **283**(17), 3218–3225 (2010). [CrossRef]

*α*and

*β*to denotes that the incident direction of the Gaussian beam is arbitrary; that is, arbitrarily incident and polarization-oriented

*β*and

*α*, respectively. This depiction not only maintains generality, but also enables the simplification of the problem compared with the depiction in [35

**27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

*Oxyz*:

*m*with

*s*in Eq. (7).

30. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(5), 056609 (2004). [CrossRef] [PubMed]

34. Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A **26**(8), 1778–1788 (2009). [CrossRef]

**27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

*r = a*, the scattering coefficients can be obtained thus [30

30. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(5), 056609 (2004). [CrossRef] [PubMed]

34. Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A **26**(8), 1778–1788 (2009). [CrossRef]

**27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

30. Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. **70**(5), 056609 (2004). [CrossRef] [PubMed]

### 2.2. Derivation of radiation force

15. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**(10), 4594–4962 (1989). [CrossRef]

36. 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 **19**(17), 16044–16057 (2011). [CrossRef] [PubMed]

### 2.3. Derivation of radiation torque

15. J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**(10), 4594–4962 (1989). [CrossRef]

37. S. Chang and S. S. Lee, “Optical torque exerted on a homogeneous sphere levitated in the circularly polarized fundamental-mode laser beam,” J. Opt. Soc. Am. B **2**(11), 1853–1860 (1985). [CrossRef]

38. F. Xu, J. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. A **78**(1), 013843 (2008). [CrossRef]

**r**is the distance vector from the origin to the enclosing surface. If the spherical surface of radius

*r*>

*a*is chosen, then,

37. S. Chang and S. S. Lee, “Optical torque exerted on a homogeneous sphere levitated in the circularly polarized fundamental-mode laser beam,” J. Opt. Soc. Am. B **2**(11), 1853–1860 (1985). [CrossRef]

## 3. Numerical calculations

### 3.1. Validation of theory and codes

**27**(6), 1457–1465 (2010). [CrossRef] [PubMed]

36. 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 **19**(17), 16044–16057 (2011). [CrossRef] [PubMed]

36. 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 **19**(17), 16044–16057 (2011). [CrossRef] [PubMed]

7. T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry **12**(6), 479–485 (1991). [CrossRef] [PubMed]

*w*

_{0}= 0.4 μm” indicates that the uniaxial anisotropic spherical particle can be captured by a focused Gaussian beam, similar to an isotropic sphere. The negative RF only indicates that the direction of the RF is negative but does not reflect the magnitude of the RF.

### 3.2. Effects of anisotropic absorbing dielectric

**70**(5), 056609 (2004). [CrossRef] [PubMed]

*x*

_{0}and

*x*

_{0}and

*x*-direction and

*y*-direction are the same.

### 3.3. Effects of incident angle and polarization angle

*F*change with incident angle

_{e}*β*exerted on a uniaxial anisotropic sphere under only electric anisotropy. The resultant RF

*F*is calculated by

_{e}*xoy*-plane. Transverse RF

*α*also equals zero. Axial RF

*β*= 36°, and its minimum value at

*β*= 90°. Note that the minimum is not zero but –0.0742 pN, which results from the effect of

*β*= 90°. The resultant RF has the maximum at

*β*= 50°. Thus, we can conclude that the RF does not remain constant under different incident angles. Figure 5(b) shows the variations of the direction of the resultant RF

*F*with the incident angle.

_{e}*z*-axis and

*x*-axis, respectively. The azimuth angle is zero because transverse RF

*β*= 0°; in Fig. 5(a), therefore, the values of

*θ*and

*β*= 0°.

*F*with incident angle

_{e}*β*exerted on a uniaxial anisotropic sphere with both electric and magnetic anisotropy. The comparison of Figs. 5(a) and 6(a) shows that transverse RF

*F*with the incident angle are shown in Fig. 6(b). A comparison of Figs. 6(a) with 6(b) indicates that the resultant RF

_{e}*F*has the maximum value of 3755.4 pN at

_{e}*β*= 90°, and its orientation angle

*z*-axis. These properties, which are distinct from those of an isotropic sphere, may pertain only to the uniaxial anisotropic sphere and may be applicable to non-intrusive optical detection of optics material through giving the particle approach the surface of optics material a force vertical to the surface, thereby delivering or capturing the particle. Permittivity tensor elements

*α*for different incident angles

*β*exerted on a uniaxial anisotropic sphere are shown in Fig. 7 . Transverse RFs

*β*= 0°, as expected, and they increase as a whole with increasing incident angle

*β*. However, the shape of the curve exhibits little change. As polarization angle

*α*increases, transverse RF

*β*= 0° and

*β*= 90°, respectively, and exhibit minimum values at

*β*= 90° and

*β*= 0°, respectively. That is, when the incident Gaussian beam is

*x*-axis polarized,

*y*-axis polarized,

*x*-direction and

*y*-direction. Figure 7(c) shows that polarization imposes a minimal effect on axial RF

*β*= 90° is not zero but –0.10202 pN. This result is similar to that shown in Figs. 7(a) and 7(c).

_{2}, characterized by

*ε*= 5.913

_{t}*ε*

_{0}and

*ε*= 7.197

_{z}*ε*

_{0}, is a typical uniaxial anisotropic medium. The transverse and axial RF exerted on a TiO

_{2}sphere by a Gaussian beam with different incident angles as a function of the coordinates of beam center –

*x*

_{0}and –

*z*

_{0}are given in Figs. 8(a) and 8(b), respectively. Transverse RF

*β*increases, axial RF

*F*decreases, whereas transverse RF

_{z}*β*= 0°, that is, the Gaussian beam propagates along the z-axis, the TiO

_{2}sphere can be captured in the direction of the

*x*-axis (see the curve denoted by “

*F*(

_{x}*β*= 0°)” in Fig. 8(a)). However, the particle will not be captured in the

*z*-axis direction (refer to the curve denoted by “

*F*(

_{z}*β*= 0°)” in Fig. 8(b)) because permittivity tensor elements

_{2}sphere. The same conclusion is drawn in [36

**19**(17), 16044–16057 (2011). [CrossRef] [PubMed]

*β*= 90°, that is, the Gaussian beam vertically illuminates the TiO

_{2}sphere, the uniaxial anisotropic sphere can also be captured in the direction of the

*z*-axis (see the curve denoted by “

*F*(

_{z}*β*= 90°)” in Fig. 8(b)); it is not captured in the direction of the

*x*-axis (see the curve denoted by “

*F*(

_{x}*β*= 90°)” in Fig. 8(a)). When the TiO

_{2}sphere is illuminated by an absolutely obliquely incident Gaussian beam (e.g.,

*β*= 45°), the TiO

_{2}sphere is very difficult to capture in both the

*x*-axis and

*z*-axis directions.

### 3.4. Calculation and discussion of radiation torque

*T*is calculated when the beam center moves along

_{x}*y*-axis. Here, the

*y*and

*z*components of the radiation torque zero due to the symmetry of the spherical particle, and are not plotted in Fig. 9. When

*ε*=

_{t}*ε*, the anisotropic sphere is reduced to an isotropic sphere. For lossless isotropic sphere, the

_{z}*x*component of the radiation torque is always equal to zero, as it should be; while for lossless anisotropic sphere the

*x*component of the radiation torque is not always equal to zero. This is because the internal field consists of two EM waves with different velocities and polarization directions, then the orbital angular momentum may be transferred to the anisotropic particle when propagation direction of the beam is not coincident with the primary optical axis of the anisotropic sphere. This radiation torque will cause the anisotropic to rotate with

*x*-axis, thus, the primary optical axis will be changed so that the scattering field and radiation force will be changed. In this case, it may be difficult to capture the uniaxial anisotropic spherical particle in transverse direction. This performance on radiation force and torque is very different from that of an isotropic spherical particle.

*ε*<

_{t}*ε*) or negative(

_{z}*ε*>

_{t}*ε*) uniaxial anisotropic sphere increase in whole and are larger than those exerted on an lossless anisotropic sphere, and the secondary extremum is also eliminated. As the beam center moves away from the sphere center along

_{z}*y*-axis, the lever-arm increases so that the torque increases. When the beam center locates at about the edge of the sphere, the radiation torque procures maxima. The position of the beam center when the maximal radiation torque occurs exerted on an anisotropic sphere is farther from the sphere center than that on an isotropic sphere. Less beam ling flux will incident on the sphere and less angular momentum will be transferred with the increasing of |

*y*

_{0}|, thus the radiation torque decreases. Moreover, it can be found that the radiation torques

*T*for both isotropic and anisotropic sphere are zero, as expected. .

_{x}*β*on the radiation torque. Because the spherical particle is illuminated by an on-axis Gaussian beam, the

*x*and

*z*components of the radiation torques are zero and not plotted in Fig. 10. When the anisotropic sphere is reduced to an isotropic sphere, the radiation torque is invariable and equal to zero in despite of the size of the incident angle (see case1). It can be observed that the negative radiation torque will cause the uniaxial anisotropic to rotate anticlockwise when the incident angle 0 <

*β*<90 ; while the positive radiation torque will cause the uniaxial anisotropic to rotate clockwise when the incident angle 90 <

*β*<180 . For the case that the sphere center and the beam center are the same, the distribution radiation torque is symmetrical with

*β*= 90, and it equals zero when the uniaxial anisotropic sphere is vertically illuminated by a Gaussian beam resulting from the uniaxial anisotropic sphere is symmetrical with

*x*-axis. For the case that

*z*

_{0}≠0, namely, the sphere center and beam center are not the same, the radiation torque is not equal to zero(see case6) when the incident angle equals 90 resulting from that the symmetrical axis of the uniaxial anisotropic sphere and the propagation direction of the beam are not the same. It is also can be found that the radiation torque decreases in whole for loss both electrical anisotropic or electrical and magnetic anisotropic sphere compared with those lossless anisotropic sphere. It is different from that performance that the radiation torque changes with the location of the beam center along

*y*-axis shown in Fig. 9.

## 4. Conclusion

## Acknowledgments

## References and links

1. | A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett. |

2. | A. Ashkin, “Applications of laser radiation pressure,” Science |

3. | A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett. |

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

5. | P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett. |

6. | A. Rohrbach and E. H. K. Stelzer, “Optical trapping of dielectric particle in arbitrary fields,” J. Opt. Soc. Am. A |

7. | T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry |

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

9. | R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B |

10. | R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of force equations,” J. Opt. Soc. Am. B |

11. | T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.) |

12. | S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt. |

13. | J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am. |

14. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys. |

15. | J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. |

16. | G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A |

17. | G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorentz-Mie theory,” J. Opt. Soc. Am. A |

18. | K. F. Ren, G. Gréha, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun. |

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

20. | J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt. |

21. | Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron. |

22. | G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt. |

23. | Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt. |

24. | F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol. |

25. | R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic Scattering from Anisotropic Materials, Part I: General Theory,” IEEE Trans. Antenn. Propag. |

26. | S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A |

27. | W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

28. | J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, “FDTD Analysis of Magnetized Ferrites: Application to the Calculation of Dispersion Characteristics of Ferrite-Loaded Waveguides,” IEEE Trans. Antenn. Propag. |

29. | S. C. Mao, Z. S. Wu, and H.-Y. Li, “Three-dimensional scattering by an infinite homogeneous anisotropic elliptic cylinder in terms of Mathieu functions,” J. Opt. Soc. Am. A |

30. | Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. |

31. | B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. Part I. Homogeneous sphere,” J. Opt. Soc. Am. A |

32. | C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag. |

33. | M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A |

34. | Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A |

35. | Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A |

36. | 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 |

37. | S. Chang and S. S. Lee, “Optical torque exerted on a homogeneous sphere levitated in the circularly polarized fundamental-mode laser beam,” J. Opt. Soc. Am. B |

38. | F. Xu, J. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. A |

39. | G. Gouesbet, “T-matrix formulation and generalized lozrenz-Mie theories in spherical coordinates,” Opt. Commun. |

40. | G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. off-axis beams,” J. Opt. Soc. Am. A |

41. | A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt. |

42. | Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express |

43. | G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systems,” Opt. Commun. |

**OCIS Codes**

(140.0140) Lasers and laser optics : Lasers and laser optics

(160.1190) Materials : Anisotropic optical materials

(290.5850) Scattering : Scattering, particles

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: April 18, 2012

Revised Manuscript: June 8, 2012

Manuscript Accepted: June 22, 2012

Published: July 5, 2012

**Citation**

Zheng-Jun Li, Zhen-Sen Wu, Qing-Chao Shang, Lu Bai, and Chun-Hui Cao, "Calculation of radiation force and torque exerted on a uniaxial anisotropic sphere by an incident Gaussian beam with arbitrary propagation and polarization directions," Opt. Express **20**, 16421-16435 (2012)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-15-16421

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### References

- A. Ashkin, “Acceleration and trapping of particles by radiation pressure,” Phys. Rev. Lett.24(4), 156–159 (1970). [CrossRef]
- A. Ashkin, “Applications of laser radiation pressure,” Science210(4474), 1081–1088 (1980). [CrossRef] [PubMed]
- A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, and S. Chu, “Observation of a single-beam gradient force optical trap for dielectric particles,” Opt. Lett.11(5), 288–290 (1986). [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]
- P. C. Chaumet and M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagnetic field,” Opt. Lett.25(15), 1065–1067 (2000). [CrossRef] [PubMed]
- A. Rohrbach and E. H. K. Stelzer, “Optical trapping of dielectric particle in arbitrary fields,” J. Opt. Soc. Am. A18(4), 839–853 (2001). [CrossRef]
- T. C. B. Schut, G. Hesselink, B. G. de Grooth, and J. Greve, “Experimental and theoretical investigations on the validity of the geometrical optics model for calculating the stability of optical traps,” Cytometry12(6), 479–485 (1991). [CrossRef] [PubMed]
- 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]
- R. Gussgard, T. Lindmo, and I. Brevik, “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am. B9(10), 1922–1930 (1992). [CrossRef]
- R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of force equations,” J. Opt. Soc. Am. B12(9), 1680–1687 (1995). [CrossRef]
- T. Wohland, A. Rosin, and E. H. K. Stelzer, “Theoretical determination of the influence of the polarization on forces exerted by optical tweezers,” Optik (Stuttg.)102(4), 181–190 (1996).
- S. Nemoto and H. Togo, “Axial force acting on a dielectric sphere in a focused laser beam,” Appl. Opt.37(27), 6386–6394 (1998). [CrossRef] [PubMed]
- J. S. Kim and S. S. Lee, “Scattering of laser beams and the optical potential well for a homogeneous sphere,” J. Opt. Soc. Am.73(3), 303–312 (1983). [CrossRef]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, “Internal and near-surface electromagnetic fields for a spherical particle irradiated by a focused laser beam,” J. Appl. Phys.64(4), 1632–1639 (1988). [CrossRef]
- J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys.66(10), 4594–4962 (1989). [CrossRef]
- G. Gouesbet, B. Maheu, and G. Grehan, “Light Scattering from a sphere arbitrarily located in a Gaussian beam,using a Bromwich formulation,” J. Opt. Soc. Am. A5(9), 1427–1443 (1988). [CrossRef]
- G. Gouesbet, G. Gréhan, and B. Maheu, “Localized interpretation to compute all the coefficients gmn in the generalized Lorentz-Mie theory,” J. Opt. Soc. Am. A7(6), 998–1007 (1990). [CrossRef]
- K. F. Ren, G. Gréha, and G. Gouesbet, “Radiation pressure forces exerted on a particle arbitrarily located in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects,” Opt. Commun.108(4-6), 343–354 (1994). [CrossRef]
- 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]
- J. A. Lock, “Calculation of the radiation trapping force for laser tweezers by use of generalized Lorenz-Mie theory. I. Localized model description of an on-axis tightly focused laser beam with spherical aberration,” Appl. Opt.43(12), 2532–2544 (2004). [CrossRef] [PubMed]
- Y. K. Nahmias and D. J. Odde, “Analysis of Radiation Forces in Laser Trapping and Laser-Guided Direct Writing Applications,” IEEE J. Quantum Electron.38(2), 131–141 (2002). [CrossRef]
- G. Martinot-Lagarde, B. Pouligny, M. I. Angelova, G. Grehan, and G. Gouesbet, “Trapping and levitation of a dielectric sphere with off-centred Gaussian beams. II. GLMT analysis,” Pure Appl. Opt.4(5), 571–585 (1995). [CrossRef]
- Y. K. Nahmias, B. Z. Gao, and D. J. Odde, “Dimensionless parameters for the design of optical traps and laser guidance systems,” Appl. Opt.43(20), 3999–4006 (2004). [CrossRef] [PubMed]
- F. L. Mao, Q. R. Xing, K. Wang, L. Y. Lang, L. Chai, and Q. Y. Wang, “Calculation of axial optical forces exerted on medium-sized particles by optical trap,” Opt. Laser Technol.39(1), 34–39 (2007). [CrossRef]
- R. D. Graglia and P. L. E. Uslenghi, “Electromagnetic Scattering from Anisotropic Materials, Part I: General Theory,” IEEE Trans. Antenn. Propag.32(8), 867–869 (1984). [CrossRef]
- S. N. Papadakis, N. K. Uzunoglu, and C. N. Capsalis, “Scattering of a plane wave by a general anisotropic dielectric ellipsoid,” J. Opt. Soc. Am. A7(6), 991–997 (1990). [CrossRef]
- W. Ren, “Contributions to the electromagnetic wave theory of bounded homogeneous anisotropic media,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics47(1), 664–673 (1993). [CrossRef] [PubMed]
- J. A. Pereda, L. A. Vielva, A. Vegas, and A. Prieto, “FDTD Analysis of Magnetized Ferrites: Application to the Calculation of Dispersion Characteristics of Ferrite-Loaded Waveguides,” IEEE Trans. Antenn. Propag.43(2), 350–357 (1995).
- S. C. Mao, Z. S. Wu, and H.-Y. Li, “Three-dimensional scattering by an infinite homogeneous anisotropic elliptic cylinder in terms of Mathieu functions,” J. Opt. Soc. Am. A26(11), 2282–2291 (2009). [CrossRef] [PubMed]
- Y. L. Geng, X. B. Wu, L. W. Li, and B. R. Guan, “Mie scattering by a uniaxial anisotropic sphere,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.70(5), 056609 (2004). [CrossRef] [PubMed]
- B. Stout, M. Nevière, and E. Popov, “Mie scattering by an anisotropic object. Part I. Homogeneous sphere,” J. Opt. Soc. Am. A23(5), 1111–1123 (2006). [CrossRef] [PubMed]
- C. W. Qiu, S. Zouhdi, and A. Razek, “Modified Spherical Wave Functions With Anisotropy Ratio: Application to the Analysis of Scattering by Multilayered Anisotropic Shells,” IEEE Trans. Antenn. Propag.55(12), 3515–3523 (2007). [CrossRef]
- M. Sluijter, D. K. G. de Boer, and J. J. M. Braat, “General polarized ray-tracing method for inhomogeneous uniaxially anisotropic media,” J. Opt. Soc. Am. A25(6), 1260–1273 (2008). [CrossRef] [PubMed]
- Z. S. Wu, Q. K. Yuan, Y. Peng, and Z. J. Li, “Internal and external electromagnetic fields for on-axis Gaussian beam scattering from a uniaxial anisotropic sphere,” J. Opt. Soc. Am. A26(8), 1778–1788 (2009). [CrossRef]
- Q. K. Yuan, Z. S. Wu, and Z. J. Li, “Electromagnetic scattering for a uniaxial anisotropic sphere in an off-axis obliquely incident Gaussian beam,” J. Opt. Soc. Am. A27(6), 1457–1465 (2010). [CrossRef] [PubMed]
- 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. Express19(17), 16044–16057 (2011). [CrossRef] [PubMed]
- S. Chang and S. S. Lee, “Optical torque exerted on a homogeneous sphere levitated in the circularly polarized fundamental-mode laser beam,” J. Opt. Soc. Am. B2(11), 1853–1860 (1985). [CrossRef]
- F. Xu, J. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: Analytical solution,” Phys. Rev. A78(1), 013843 (2008). [CrossRef]
- G. Gouesbet, “T-matrix formulation and generalized lozrenz-Mie theories in spherical coordinates,” Opt. Commun.283(4), 517–521 (2010). [CrossRef]
- G. Gouesbet and J. A. Lock, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. II. off-axis beams,” J. Opt. Soc. Am. A11(9), 2516–2525 (1994). [CrossRef]
- A. Doicu and T. Wriedt, “Computation of the beam-shape coefficients in the generalized Lorenz-Mie theory by using the translational addition theorem for spherical vector wave functions,” Appl. Opt.36(13), 2971–2978 (1997). [CrossRef] [PubMed]
- Y. P. Han, H. Y. Zhang, and G. X. Han, “The expansion coefficients of arbitrary shaped beam in oblique illumination,” Opt. Express15(2), 735–746 (2007). [CrossRef] [PubMed]
- G. Gouesbet, J. J. Wang, and Y. P. Han, “Transformations of spherical beam shape coefficients in generalized Lorenz-Mie theories through rotations of coordinate systems,” Opt. Commun.283(17), 3218–3225 (2010). [CrossRef]

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