## Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam

Optics Express, Vol. 14, Issue 13, pp. 6316-6321 (2006)

http://dx.doi.org/10.1364/OE.14.006316

Acrobat PDF (158 KB)

### Abstract

Radiation forces exerted upon a dielectric, circular-shaped cylinder of infinite length illuminated by a non-paraxial cylindrical Gaussian beam are considered. Vectorial projections of the radiation pressure force on a dielectric, arbitrary- and circular-shaped cylinder are expressed analytically. In particular, the radiation force is expressed through coefficients of the decomposition of the non-paraxial Gaussian beam into the cylindrical functions. Using numerical examples, a possibility to optically trap a circular-shaped cylinder in a non-paraxial cylindrical Gaussian beam is demonstrated.

© 2006 Optical Society of America

## 1. Introduction

1. 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**, 1427–1443 (1988). [CrossRef]

7. H. Polaert, G. Grehan, and G. Gouesbet. “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun **155**, 169–179 (1998). [CrossRef]

8. G. Gouesbet. “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am A **16**, 1641–1650 (1999). [CrossRef]

9. J. Barton, D. Alexander, and S. Schaub. “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**, 4594–4602 (1989). [CrossRef]

10. R. Gussgard, T. Lindmo, and I. Brevik. “Calculation of the trapping force in a strongly focused laser beam,” J. Opt. Soc. Am B **9**, 1922–1930 (1992). [CrossRef]

1. 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**, 1427–1443 (1988). [CrossRef]

7. H. Polaert, G. Grehan, and G. Gouesbet. “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun **155**, 169–179 (1998). [CrossRef]

9. J. Barton, D. Alexander, and S. Schaub. “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**, 4594–4602 (1989). [CrossRef]

2. G. Gouesbet and J.A. Lock. “A rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz-Mie theory II. Off-axis beams,” J. Opt. Soc. Am A **2**, 2516–2525 (1994). [CrossRef]

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

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

6. H. Polaert, G. Grehan, and G. Gouesbet. “Improved standard beams with applications to reverse radiation pressure,” Appl. Opt. **37**, 2435–2440 (1998). [CrossRef]

9. J. Barton, D. Alexander, and S. Schaub. “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. **66**, 4594–4602 (1989). [CrossRef]

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

12. A. Rohrbach and E.H.K. Stelzer. “Trapping forces, force constant, and potential depths for dielectric spheres in the presence of spherical aberration,” Appl. Opt. **41**, 2494–2507 (2002). [CrossRef] [PubMed]

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

12. A. Rohrbach and E.H.K. Stelzer. “Trapping forces, force constant, and potential depths for dielectric spheres in the presence of spherical aberration,” Appl. Opt. **41**, 2494–2507 (2002). [CrossRef] [PubMed]

13. 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**, 2532–2544 (2004). [CrossRef] [PubMed]

14. 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**, 2545–2554 (2004). [CrossRef] [PubMed]

15. D. Ganic, X. Gan, and M. Gu. “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express **12**, 2670–2675 (2004). [CrossRef] [PubMed]

16. T.A. Nieminen, N.R. Heckenberg, and H. Rubinstein-Dunlop. “Computational modeling of optical tweezers,” Proceedings of SPIE , **5514**, 514–523 (2004). [CrossRef]

17. A. Mazolli, P.A. Maia Neto, and H.M. Nussenzveig. “Theory of trapping forces in optical tweezers,” Proc. R. Soc. Lond. , **459**, 3021–3041 (2003). [CrossRef]

19. R. Pobre and C. Saloma. “Radiation forces on nonlinear microsphere by a tightly focused Gaussian beam,” Appl. Opt. , **41-36**, 7694–7701 (2002). [CrossRef]

20. P.L. Marston and J.H. Crichton. “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A. , **30-5**, 2508–2516 (1984). [CrossRef]

7. H. Polaert, G. Grehan, and G. Gouesbet. “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun **155**, 169–179 (1998). [CrossRef]

21. E. Zimmerman, R. Dandliner, and N. Souli. “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A , **12**, 398–403 (1995). [CrossRef]

22. Z. Wu and L. Guo. “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm,” Progress in electromagnetics research, PIER , **18**, 317–333, (1998). [CrossRef]

23. L. Mees, K.F. Ren, G. Grehan, and G. Gouesbet. “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results,” Appl. Opt. , **38**, 1867–1876 (1999). [CrossRef]

## 2. Radiation pressure force on a microobject

*V*limited by the surface

*S*is given by

*σ*

_{ik}is Maxwell’s stress tensor (

*σ*

_{ik}=

*σ*

_{ki});

**E**,

**H**are the vectors of the electromagnetic field strength in free space, and ε

_{1}is the permittivity of medium.

*ω*is the cyclic frequency) yields, in place of Eq. (1) (

*F*

_{x}=0)

*S*

_{1}is a contour that envelops the object cross-section in the YOZ-plane,

*µ*is the permeability of medium (

*µ*=1), and

*µ*

_{0}is the permeability of vacuum. In the 2D case for TE-polarization (

*H*

_{x}=

*E*

_{y}=

*E*

_{z}=0), the electric field is parallel to the X-axis:

*E*

_{x}≠0,

*z*is the optical axis, with the 2D object presented by an arbitrary-shaped cylinder of infinite length along the X-axis. The YOZ-plane is where light propagates (see Fig.1).

## 3. Diffraction of a cylindrical non-paraxial Gaussian beam by a circular homogeneous cylinder

*E*

_{x}≠0) and TM (

*H*

_{x}≠0) polarization. In homogeneous space, the fields

*E*

_{x}and

*H*

_{x}satisfy the scalar 2D Helmholtz equation. Any function that satisfies the 2D Helmholtz equation can be represented by an integral decomposition into plane waves. Therefore, for TE-polarization, when (

*ρ,φ*) are the coordinates in the plane (Y,Z),

*γ*=arcsin

*q*,

*p*

^{2}+

*q*

^{2}=1,

*p*=cos

*γ*, and

*q*=sin

*γ*,

*ω*

_{0}is the waist’s radius,

*E*

_{0}=const,

*c*is the light speed in free space, and

*λ*is the wavelength. The series expansion of the electric field strength (5) in terms of cylindrical harmonics is

*J*

_{n}

*(x)*is the Bessel function of the

*n*-th order. From Maxwell’s equation we go on to calculate the magnetic field strength,

*H*

_{r}and

*H*

_{φ}. Then, the magnetic field projection of the incident wave takes the form:

*E*⃗

^{S},

*H*⃗

^{s}) functions are:

*=a*

_{n}

*C*

_{n},

*R*is the radius of the cylinder’s circular cross-section,

*ε*is the permittivity of the cylinder. Substituting Eqs. (5)–(11) into Eqs. (3)–(4), integrating over a circumference of radius

*R*′>

*R*and tending the radius

*R*′ to infinity, we can obtain an analytical expression for projections of the radiation pressure force on the circular homogeneous dielectric cylinder (

*µ*=1):

## 4. Numerical simulation

*λ*=1

*µm*,

*ε*=1.2, the particle diameter is 2

*µm*, the Gaussian beam waist’s diameter is 1µm, and the power is 100 mW/m. It should be noted that in our case the confinement factor

*S=1/(kω*

_{0}

*)*=1/π (Ref. [5

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

**66**, 4594–4602 (1989). [CrossRef]

*L*, acting along (

*F*

_{z}) and transversely to (

*F*

_{y}) the light propagation axis. The displacement

*L*is the spacing between the beam’s waist and the cylinder’s center. At L>0, the cylinder’s center is located in the diverging beam and at L<0 the cylinder is in the converging beam (note that in Refs. [5

**35**, 2702–2710 (1996). [CrossRef] [PubMed]

6. H. Polaert, G. Grehan, and G. Gouesbet. “Improved standard beams with applications to reverse radiation pressure,” Appl. Opt. **37**, 2435–2440 (1998). [CrossRef]

*F*

_{z}is less than 5%. In calculation, all the series are truncated to the first 15 terms. Figure 2 demonstrates the optical trapping of a dielectric circular cylinder in a single Gaussian beam.

**35**, 2702–2710 (1996). [CrossRef] [PubMed]

6. H. Polaert, G. Grehan, and G. Gouesbet. “Improved standard beams with applications to reverse radiation pressure,” Appl. Opt. **37**, 2435–2440 (1998). [CrossRef]

*R*=1 µm. This conclusion agrees with the results reported by other researchers (see Fig. 8 in [5

**35**, 2702–2710 (1996). [CrossRef] [PubMed]

**37**, 2435–2440 (1998). [CrossRef]

14. 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**, 2545–2554 (2004). [CrossRef] [PubMed]

^{-10}N/m, which is equivalent to the 3D situation when for the beam power of 0.1W and sphere radius of

*R*=1 µm the order of the force is 10

^{-10}N. This is in compliance with Fig. 2 in Ref. [19

19. R. Pobre and C. Saloma. “Radiation forces on nonlinear microsphere by a tightly focused Gaussian beam,” Appl. Opt. , **41-36**, 7694–7701 (2002). [CrossRef]

## 5. Conclusions

## Acknowledgments

## References and links

1. | 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 |

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

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

4. | G. Martinet-Lagarde, B. Pouligny, M.A. Angelova, G. Grehan, and G. Gouesbet. “Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis,” Pure Appl. Opt. |

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

6. | H. Polaert, G. Grehan, and G. Gouesbet. “Improved standard beams with applications to reverse radiation pressure,” Appl. Opt. |

7. | H. Polaert, G. Grehan, and G. Gouesbet. “Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam,” Opt. Commun |

8. | G. Gouesbet. “Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres,” J. Opt. Soc. Am A |

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

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

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

12. | A. Rohrbach and E.H.K. Stelzer. “Trapping forces, force constant, and potential depths for dielectric spheres in the presence of spherical aberration,” Appl. Opt. |

13. | 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. |

14. | 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. |

15. | D. Ganic, X. Gan, and M. Gu. “Exact radiation trapping force calculation based on vectorial diffraction theory,” Opt. Express |

16. | T.A. Nieminen, N.R. Heckenberg, and H. Rubinstein-Dunlop. “Computational modeling of optical tweezers,” Proceedings of SPIE , |

17. | A. Mazolli, P.A. Maia Neto, and H.M. Nussenzveig. “Theory of trapping forces in optical tweezers,” Proc. R. Soc. Lond. , |

18. | Y.K. Nahmias and D.J. Oddl. “Analysis of radiation forces in laser trapping and laser-guided direct writing application,” IEEE J. daunt. Electr. , |

19. | R. Pobre and C. Saloma. “Radiation forces on nonlinear microsphere by a tightly focused Gaussian beam,” Appl. Opt. , |

20. | P.L. Marston and J.H. Crichton. “Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave,” Phys. Rev. A. , |

21. | E. Zimmerman, R. Dandliner, and N. Souli. “Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach,” J. Opt. Soc. Am. A , |

22. | Z. Wu and L. Guo. “Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm,” Progress in electromagnetics research, PIER , |

23. | L. Mees, K.F. Ren, G. Grehan, and G. Gouesbet. “Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results,” Appl. Opt. , |

24. | L. D. Landau and E. M. Lifshitz, “Brief course in theoretical physics. Mechanics. Electrodynamics,” Moscow, Nauka Publishers, Book 1, (1969). |

**OCIS Codes**

(260.1960) Physical optics : Diffraction theory

(260.2160) Physical optics : Energy transfer

**ToC Category:**

Physical Optics

**History**

Original Manuscript: March 13, 2006

Revised Manuscript: May 19, 2006

Manuscript Accepted: May 31, 2006

Published: June 26, 2006

**Virtual Issues**

Vol. 1, Iss. 7 *Virtual Journal for Biomedical Optics*

**Citation**

V. V. Kotlyar and A. G. Nalimov, "Analytical expression for radiation forces on a dielectric cylinder illuminated by a cylindrical Gaussian beam," Opt. Express **14**, 6316-6321 (2006)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-14-13-6316

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### References

- 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, 1427-1443 (1988). [CrossRef]
- G. Gouesbet and J. A. Lock, "A rigorous justification of the localized approximation to the beam-shape coefficients in the generalized Lorenz-Mie theory II. Off-axis beams," J. Opt. Soc. Am A 2, 2516-2525 (1994). [CrossRef]
- F. Ren, G. Grehan, and G. Gouesbet, "Radiation pressure forces exerted on a particle located arbitrarily in a Gaussian beam by using the generalized Lorenz-Mie theory and associated resonance effects," Opt. Commun. 108, 343-354 (1994). [CrossRef]
- G. Martinet-Lagarde, B. Pouligny, M. A. Angelova, G. Grehan, and G. Gouesbet, "Trapping and levitation of a dielectric sphere with off-centered Gaussian beams, II-GLMT analysis," Pure Appl. Opt. 4, 571-585 (1995). [CrossRef]
- K. F. Ren, G. Grehan, and G. Gouesbet, "Prediction of reverse radiation pressure by generalized Lorenz-Mie theory," Appl. Opt. 35, 2702-2710 (1996). [CrossRef] [PubMed]
- H. Polaert, G. Grehan, and G. Gouesbet, "Improved standard beams with applications to reverse radiation pressure," Appl. Opt. 37, 2435-2440 (1998). [CrossRef]
- H. Polaert, G. Grehan, and G. Gouesbet, "Forces and torques exerted on a multilayered spherical particle by a focused Gaussian beam," Opt. Commun 155, 169-179 (1998). [CrossRef]
- G. Gouesbet, "Validity of the localized approximation for arbitrary shaped beams in the generalized Lorenz-Mie theory for spheres," J. Opt. Soc. Am A 16, 1641-1650 (1999). [CrossRef]
- J. Barton, D. Alexander, and S. Schaub, "Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam," J. Appl. Phys. 66, 4594-4602 (1989). [CrossRef]
- R. Gussgard, T. Lindmo, and I. Brevik, "Calculation of the trapping force in a strongly focused laser beam," J. Opt. Soc. Am B 9, 1922-1930 (1992). [CrossRef]
- A. Rohrbach and E. H. K. Stelzer, "Optical trapping of a dielectric particle in arbitrary fields," J. Opt. Soc. Am A 18, 839-853 (2001). [CrossRef]
- A. Rohrbach, and E. H. K. Stelzer, "Trapping forces, force constant, and potential depths for dielectric spheres in the presence of spherical aberration," Appl. Opt. 41, 2494-2507 (2002). [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, 2532-2544 (2004). [CrossRef] [PubMed]
- 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, 2545-2554 (2004). [CrossRef] [PubMed]
- D. Ganic, X. Gan, and M. Gu, "Exact radiation trapping force calculation based on vectorial diffraction theory," Opt. Express 12, 2670-2675 (2004). [CrossRef] [PubMed]
- T. A. Nieminen, N. R. Heckenberg, and H. Rubinstein-Dunlop, "Computational modeling of optical tweezers," in Optical Trapping and Optical Micromanipulation; K. Dholakia, and G. C. Spalding, eds., Proc. SPIE 5514, 514-523 (2004). [CrossRef]
- A. Mazolli, P. A. Maia Neto, and H. M. Nussenzveig. "Theory of trapping forces in optical tweezers," Proc. R. Soc. London 459, 3021-3041 (2003). [CrossRef]
- Y. K. Nahmias, and D. J. Oddl. "Analysis of radiation forces in laser trapping and laser-guided direct writing application," IEEE J. Qauntum. Electron., 38-2, 1-10 (2002).
- R. Pobre, and C. Saloma. "Radiation forces on nonlinear microsphere by a tightly focused Gaussian beam," Appl. Opt., 41-36, 7694-7701 (2002). [CrossRef]
- P. L. Marston, and J. H. Crichton. "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A., 30-5, 2508-2516 (1984). [CrossRef]
- E. Zimmerman, R. Dandliner, and N. Souli. "Scattering of an off-axis Gaussian beam by a dielectric cylinder compared with a rigorous electromagnetic approach," J. Opt. Soc. Am. A, 12, 398-403 (1995). [CrossRef]
- Z. Wu, and L. Guo. "Electromagnetic scattering from a multilayered cylinder arbitrarily located in a Gaussian beam, a new recursive algorithm," Prog. Electromagn. Res. 18, 317-333, (1998). [CrossRef]
- L. Mees, K. F. Ren, G. Grehan, and G. Gouesbet. "Scattering of a Gaussian beam by an infinite cylinder with arbitrary location and arbitrary orientation, numerical results," Appl. Opt., 38, 1867-1876 (1999). [CrossRef]
- L. D. Landau, and E. M. Lifshitz, "Brief course in theoretical physics. Mechanics. Electrodynamics," (Moscow, Nauka Publishers, Book 1, 1969).

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