## Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam |

Optics Express, Vol. 21, Issue 7, pp. 8677-8688 (2013)

http://dx.doi.org/10.1364/OE.21.008677

Acrobat PDF (1113 KB)

### Abstract

Under the framework of generalized Lorenz-Mie theory, we calculate the radiation force and torque exerted on a chiral sphere by a Gaussian beam. The theory and codes for axial radiation force are verified when the chiral sphere degenerates into an isotropic sphere. We discuss the influence of a chirality parameter on the radiation force and torque. Linearly and circularly polarized incident Gaussian beams are considered, and the corresponding radiation forces and torques are compared and analyzed. The polarization of the incident beam considerably influences radiation force of a chiral sphere. In trapping a chiral sphere, therefore, the polarization of incident beams should be chosen in accordance with the chirality. Unlike polarization, variation of chirality slightly affects radiation torque, except when the imaginary part of the chirality parameter is considered.

© 2013 OSA

## 1. Introduction

1. A. Ashkin and J. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. **19**(8), 283–285 (1971). [CrossRef]

2. 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]

3. A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science **235**(4795), 1517–1520 (1987). [CrossRef] [PubMed]

4. G. Roosen, “A theoretical and experimental study of the stable equilibrium positions of spheres levitated by two horizontal laser beams,” Opt. Commun. **21**(1), 189–194 (1977). [CrossRef]

5. 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]

10. R. C. Gauthier and S. Wallace, “Optical levitation of spheres: analytical development and numerical computations of the force equations,” J. Opt. Soc. Am. B **12**(9), 1680–1686 (1995). [CrossRef]

11. 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]

12. G. Gouesbet, B. Maheu, and G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an arbitrary location,” Particle & Particle Systems Characterization **5**(1), 1–8 (1988). [CrossRef]

13. G. Gouesbet, B. Maheu, and G. Gréhan, “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]

14. 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]

15. 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]

16. J. A. Lock, “Calculation of the Radiation Trapping Force for Laser Tweezers by Use of Generalized Lorenz-Mie theory. II. On-Axis Trapping Force,” Appl. Opt. **43**(12), 2545–2554 (2004). [CrossRef] [PubMed]

17. 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]

18. J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a gaussian laser beam,” Opt. Acta (Lond.) **29**(6), 801–806 (1982). [CrossRef]

19. 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]

20. 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), 4594–4602 (1989). [CrossRef]

21. T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. I. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. **142**(1-3), 468–471 (2001). [CrossRef]

22. T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online **3**(3), 338–342 (2007). [CrossRef]

23. R. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express **13**(10), 3707–3718 (2005). [CrossRef] [PubMed]

24. A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A **18**(8), 1944–1953 (2001). [CrossRef] [PubMed]

25. 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]

26. Z. J. Li, Z. S. Wu, Q. C. Shang, L. Bai, and C. H. 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**(15), 16421–16435 (2012). [CrossRef]

27. L. A. Ambrosio and H. E. Hernández-Figueroa, “Radiation pressure cross sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt. **50**(22), 4489–4498 (2011). [CrossRef] [PubMed]

28. L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express **1**(5), 1284–1301 (2010). [CrossRef] [PubMed]

29. D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. **41**(6), 526–533 (2011). [CrossRef]

30. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. **29**(3), 458–462 (1974). [CrossRef]

31. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. **51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

19. 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]

20. 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), 4594–4602 (1989). [CrossRef]

*iωt*) is assumed throughout this paper.

## 2. Theoretical formulations

### 2.1. Scattering of a Gaussian beam by a chiral sphere

*z*-axis. The media of the chiral sphere can be described by the following constitutive relations:where

*ω*in chiral media is always decomposed into two modes: the right-handed circularly polarized (RCP) wave with wave number

30. F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. **29**(3), 458–462 (1974). [CrossRef]

31. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. **51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

32. D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **56**(1), 1102–1112 (1997). [CrossRef]

*x*-polarized (linearly polarized in the

*x*-direction),

*y*-polarized (linearly polarized in the

*y*-direction), RCP and LCP wave incidences when

31. Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. **51**(27), 6661–6668 (2012). [CrossRef] [PubMed]

33. A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. **22**(10), 1242–1246 (1951). [CrossRef]

35. Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. **26**(6), 1393–1401 (1991). [CrossRef]

36. 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]

37. L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A **19**(3), 1177–1179 (1979). [CrossRef]

36. 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]

38. G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf. **112**(1), 1–27 (2011). [CrossRef]

*x*-polarized Gaussian beam (

*x*-polarized at the waist) propagating in the positive

*z*direction with a beam waist radius

36. 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]

39. 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]

40. G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. **27**(23), 4874–4883 (1988). [CrossRef] [PubMed]

42. J. A. Lock and G. Gouesbet, “Rigorous justification of the localized approximation to the beam-shape coefficients in generalized Lorenz-Mie theory. I. On-axis beams,” J. Opt. Soc. Am. A **11**(9), 2503–2515 (1994). [CrossRef]

*y*-polarized Gaussian beam can be readily obtained from those of an

*x*-polarized Gaussian beam:

### 2.2. Expressions of radiation force and torque

13. G. Gouesbet, B. Maheu, and G. Gréhan, “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]

20. 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), 4594–4602 (1989). [CrossRef]

25. 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]

26. Z. J. Li, Z. S. Wu, Q. C. Shang, L. Bai, and C. H. 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**(15), 16421–16435 (2012). [CrossRef]

*S*is an arbitrary surface that encloses the sphere; and

*S*; and

*S*.

*S*is a large spherical surface with the center at the origin. Thus,

25. 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]

26. Z. J. Li, Z. S. Wu, Q. C. Shang, L. Bai, and C. H. 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**(15), 16421–16435 (2012). [CrossRef]

*c*is the light velocity in vacuum;

## 3. Numerical results and discussions

### 3.1. Verification of radiation force

6. 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]

6. 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]

*a*= 3.75 µm; the power of the

*x*-polarized incident Gaussian beam is

*P*= 0.1 W; the wavelength of the beam in vacuum is

*λ*= 0.488 µm; and the beam waist radius is

*w*

_{0}= 1.8 µm. The experimental results and our numerical results are in good agreement, confirming the validity of the theory and codes in this paper.

### 3.2. Analysis of radiation force

*z*-axis, it always experiences a zero transverse radiation force. Figure 2 shows axial radiation force

*Fz*versus sphere location

*z*

_{0}for different chirality parameters, assuming a sphere with radius

*a*= 1.5 μm and refractive index

*n*= 1.59 in a surrounding medium with refractive index

*n*

_{0}= 1.33. The beam center of the

*x*-polarized incident Gaussian beam with wavelength

*λ*= 0.488 μm in vacuum, beam waist radius

*w*

_{0}= 0.5 μm, and power

*P*= 0.1 W is located at the origin. The chirality parameter of the sphere is −0.3, 0.0, 0.3, 0.5, and 0.7, respectively. As shown in Fig. 2, the axial radiation force curve of a chiral sphere is similar to that of an isotropic one. As the sphere moves from the negative

*z*-axis to the positive

*z*-axis, the axial radiation force initially decreases and then increases, during which a stable equilibrium point that can trap the sphere stably appears if negative force occurs. However, all the axial radiation forces in Fig. 2 are positive; thus, the sphere cannot be trapped by this single beam. It seems that a large chirality parameter increases the axial radiation force. Therefore, realizing a negative axial radiation force for a chiral sphere may be more difficult. We find that the curves with chirality parameters −0.3 and 0.3 are coincident. The succeeding numerical results will continue to focus on this and we will try to give some explanations then.

*x*

_{0}is depicted in Fig. 3 for the same conditions as Fig. 2, except that the sphere now moves along the

*x*-axis. The

*x*-component

*Fx*of transverse radiation force is shown in Fig. 3(a) and the

*y*-component

*Fy*in Fig. 3(b). Figure 3(a) shows that as the chirality parameter increases,

*Fx*varies from positive to negative when the sphere is located on the negative

*x*-axis, and varies from negative to positive when the sphere is on positive

*x*-axis. These findings suggest that transverse radiation force

*Fx*toward the beam axis (

*z*-axis) becomes a repellant force that pushes the sphere away from the

*z*-axis when the chirality parameter is large enough. Similar to the case of axial radiation force in Fig. 2, the curves of transverse radiation force

*Fx*versus

*x*

_{0}with chirality parameters −0.3 and 0.3 are coincident. As we known, the

*y*-component of the radiation force

*Fy*of an isotropic sphere at

*x*-axis is always zero. However, as shown in Fig. 3(b), the absolute value of

*Fy*for a chiral sphere does not vanish but increases as the chirality parameter increases. This result indicates that a chiral sphere at the

*x*-axis experiences a force that pushes it away from the

*x-*axis. Besides, in contrast to the

*Fx*and

*Fz*curves, the

*Fy*curves with chirality parameters −0.3 and 0.3 are not equal but opposite. Thus, the corresponding spheres will be pushed toward different

*y*-directions. We conclude from Fig. 3 that trapping a chiral sphere transversely may be more difficult than trapping an isotropic one.

*κ*|, are shown in Figs. 4 and 5 for differently polarized beam incidences. The

*x*-polarized, LCP, and RCP incident Gaussian beams are denoted by “XP,” “LP,” and “RP” in the figures, respectively. Figures 4(a) and 4(b) present axial radiation force

*Fz*versus the chirality parameter at sphere locations of (0 μm, 0 μm, 0 μm) and (0 μm, 0 μm, 10 μm), respectively. Figure 5(a) shows transverse radiation force

*Fx*for a sphere located at (1.0 μm, 0 μm, 0 μm), and Fig. 5(b) shows transverse radiation force

*Fy*for a sphere located at (0.5 μm, 0 μm, 0 μm). Chirality parameter

*κ*varies from −0.9 to 0.9. The other parameters are the same as those above. It can be observed that the axial radiation forces on a chiral sphere are no longer the same for different circularly polarized incident beams. For the chiral sphere located at the beam center, as shown in Fig. 4(a),

*Fz*basically increases with the chirality parameter in all cases. However, Fig. 4(b) shows that for the RCP beam incidence with a negative chirality parameter and the LCP beam incidence with a positive chirality parameter,

*Fz*decreases to its minimum and then increases as chirality |

*κ*| varies from 0.0 to 0.9. For a chiral sphere with an appropriate chirality parameter, therefore, it may be easier to realize a negative axial radiation force by using a circularly polarized beam. Surprisingly, the other

*Fz*curves in Fig. 4(b) show obvious oscillations, which may be due to the special scattering properties of a chiral sphere for circularly polarized incident waves. Similar curves of scattering cross-sections versus the size parameter of a chiral sphere are found in [29

29. D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. **41**(6), 526–533 (2011). [CrossRef]

*x*-polarized beam incidence, RCP beam incidence with negative chirality, and LCP beam incidence with positive chirality, the transverse force

*Fx*on a chiral sphere at

*x*

_{0}= 1.0 μm varies from negative to positive as |

*κ*| increases. In these cases, the chiral sphere with a sufficiently large chirality parameter is pushed away from the

*x*-axis, as the situation shown in Fig. 3(a). However, for the RCP beam incidence with positive chirality, and the LCP beam incidence with negative chirality,

*Fx*is always negative, implying that the force always pulls the chiral sphere toward the beam axis. Thus, to transversely trap a chiral sphere effectively, we should use a beam with appropriate polarization in accordance with the chirality. As shown in Fig. 5(b), a chiral sphere at the

*x*-axis experiences a force in the

*y*-direction for all three types of polarized beam incidences (XP, RP, and LP). As analyzed in Fig. 3(b), this force disrupts the transverse trapping of the sphere, especially when the chirality parameter is large.

*κ*and a LCP incident beam with -

*κ*, the

*Fx*s and

*Fz*s are equal, respectively, but the

*Fy*s are opposite. For an

*x*-polarized beam incidence with opposite chirality parameters, the phenomenon observed is consistent with that depicted in Figs. 2 and 3. It seems that symmetry exists between the radiation forces on chiral spheres with opposite chirality parameters. In fact, as introduced in Section 2, the relationship between the wave numbers of the RCP and LCP waves in chiral media is

*κ*is symmetric physically to a LCP wave scattering from a chiral sphere with chirality parameter -

*κ*. Therefore, the scattering fields in the two cases are the same in magnitude but contrary in RCP components and LCP components. This attribute may account for the symmetry of radiation forces discussed above.

### 3.3. Analysis of radiation torque

*z*

_{0}for differently polarized Gaussian beam incidences. The parameters are as follows: the refractive index of the sphere is

*n*= 1.59 + 0.0003

*i*; the refractive index of the surrounding medium is

*n*= 1.33; the radius of the sphere is

*a*= 1.5 µm; the power of the incident Gaussian beam is

*P*= 0.1 W; the wavelength of the beam in vacuum is

*λ*= 0.488 µm; and the beam waist radius is

*w*

_{0}= 1.0 μm. It can be observed that in contrast to radiation force, the axial radiation torque increases to its maximum at

*z*

_{0}= 0 and then decreases as the sphere moves from the negative to the positive

*z*-axis. The radiation torques on a chiral sphere with

*κ*= 0.2 and an isotropic one (

*κ*= 0) are almost the same, indicating that the effect of chirality parameter on axial radiation torque is minimal. Instead, polarization of the incident beam has a great influence on the torque. RCP beam incidence and LCP beam incidence rotates the sphere toward different direction, respectively. However, if the chirality parameter is a complex quantity (

*κ*= 0.2 + 0.0002

*i*, for example), as shown in the figure, the curve will be quite different from that for

*κ*= 0.2. Loss of a sphere considerably influences on radiation force and torque. According to the expressions of the wave numbers in chiral media

*x*

_{0}for differently polarized beam incidences. All the parameters are the same as previously stated, except for the refractive index of the sphere, which is

*n*= 1.59 + 0.0001

*i*. It can be seen that axial torque

*Nz*increases to its maximum at

*x*

_{0}= 0 and then decreases as the sphere moves from the negative to the positive

*x*-axis. However, values of

*Nz*are tiny, compared with transverse torque

*Nx*or

*Ny*. Again it is found that a real-valued chirality parameter minimally influences both axial and transverse radiation torques. For both a general isotropic sphere and a chiral sphere located at the

*x*-axis in an RCP or a LCP beam, the sphere experiences a transverse torque in the

*x*-direction. The sign of torque

*Nx*depends on beam polarization and sphere location. The curves of transverse torque

*Ny*in all cases are close to one another and have large values compared with

*Nx*and

*Nz*as

*Ny*s are attributed to the asymmetric intensity of the beam with respect to the sphere.

## 4. Conclusion

## Acknowledgment

## References and links

1. | A. Ashkin and J. Dziedzic, “Optical levitation by radiation pressure,” Appl. Phys. Lett. |

2. | 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. |

3. | A. Ashkin and J. M. Dziedzic, “Optical trapping and manipulation of viruses and bacteria,” Science |

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

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

6. | 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 |

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

8. | W. H. Wright, G. J. Sonek, and M. W. Berns, “Radiation trapping forces on microspheres with optical tweezers,” Appl. Phys. Lett. |

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

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

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

12. | G. Gouesbet, B. Maheu, and G. Gréhan, “Scattering of a Gaussian Beam by a Sphere Using a Bromwich Formulation: Case of an arbitrary location,” Particle & Particle Systems Characterization |

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

14. | 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. |

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

16. | J. A. Lock, “Calculation of the Radiation Trapping Force for Laser Tweezers by Use of Generalized Lorenz-Mie theory. II. On-Axis Trapping Force,” Appl. Opt. |

17. | 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. |

18. | J. S. Kim and S. S. Lee, “Radiation pressure on a dielectric sphere in a gaussian laser beam,” Opt. Acta (Lond.) |

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

20. | 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. |

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

22. | T. A. Nieminen, V. L. Y. Loke, G. Knöner, and A. Branczyk, “Toolbox for calculation of optical forces and torques,” Piers Online |

23. | R. Gauthier, “Computation of the optical trapping force using an FDTD based technique,” Opt. Express |

24. | A. G. Hoekstra, M. Frijlink, L. B. F. M. Waters, and P. M. A. Sloot, “Radiation forces in the discrete-dipole approximation,” J. Opt. Soc. Am. A |

25. | 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 |

26. | Z. J. Li, Z. S. Wu, Q. C. Shang, L. Bai, and C. H. 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 |

27. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Radiation pressure cross sections and optical forces over negative refractive index spherical particles by ordinary Bessel beams,” Appl. Opt. |

28. | L. A. Ambrosio and H. E. Hernández-Figueroa, “Fundamentals of negative refractive index optical trapping: forces and radiation pressures exerted by focused Gaussian beams using the generalized Lorenz-Mie theory,” Biomed. Opt. Express |

29. | D. Guzatov and V. V. Klimov, “Chiral particles in a circularly polarised light field: new effects and applications,” Quantum Electron. |

30. | F. Bohren, “Light scattering by an optically active sphere,” Chem. Phys. Lett. |

31. | Z.-S. Wu, Q.-C. Shang, and Z.-J. Li, “Calculation of electromagnetic scattering by a large chiral sphere,” Appl. Opt. |

32. | D. Sarkar and N. J. Halas, “General vector basis function solution of Maxwell's equations,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

33. | A. L. Aden and M. Kerker, “Scattering of electromagnetic wave from concentric sphere,” J. Appl. Phys. |

34. | C. F. Bohren and D. R. Huffman, |

35. | Z. S. Wu and Y. P. Wang, “Electromagnetic scattering for multilayered sphere: Recursive algorithms,” Radio Sci. |

36. | 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. |

37. | L. W. Davis, “Theory of electromagnetic beams,” Phys. Rev. A |

38. | G. Gouesbet, J. A. Lock, and G. Gréhan, “Generalized Lorenz–Mie theories and description of electromagnetic arbitrary shaped beams: Localized approximations and localized beam models, a review,” J. Quant. Spectrosc. Radiat. Transf. |

39. | 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 |

40. | G. Gouesbet, G. Grehan, and B. Maheu, “Computations of the gn coefficients in the generalized Lorenz-Mie theory using three different methods,” Appl. Opt. |

41. | K. F. Ren, G. Gouesbet, and G. Gréhan, “Integral Localized Approximation in Generalized Lorenz-Mie Theory,” Appl. Opt. |

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

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(290.5850) Scattering : Scattering, particles

(160.1585) Materials : Chiral media

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: January 15, 2013

Revised Manuscript: March 10, 2013

Manuscript Accepted: March 15, 2013

Published: April 2, 2013

**Virtual Issues**

Vol. 8, Iss. 5 *Virtual Journal for Biomedical Optics*

**Citation**

Qing-Chao Shang, Zhen-Sen Wu, Tan Qu, Zheng-Jun Li, Lu Bai, and Lei Gong, "Analysis of the radiation force and torque exerted on a chiral sphere by a Gaussian beam," Opt. Express **21**, 8677-8688 (2013)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-21-7-8677

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