## Three-dimensional analysis of optical forces generated by an active tractor beam using radial polarization |

Optics Express, Vol. 22, Issue 3, pp. 3284-3295 (2014)

http://dx.doi.org/10.1364/OE.22.003284

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### Abstract

We theoretically study the three-dimensional behavior of nanoparticles in an active optical conveyor. To do this, we solved the Langevin equation when the forces are generated by a focusing system at the near field. Analytical expressions for the optical forces generated by the optical conveyor were obtained by solving the Richards and Wolf vectorial diffraction integrals in an approximated form when a mask of two annular pupils is illuminated by a radially polarized Hermite-Gauss beam. Trajectories, in both the transverse plane and the longitudinal direction, are analyzed showing that the behavior of the optical conveyor can be optimized by conveniently choosing the configuration of the mask of the two annular pupils (inner and outer radius of the two rings) in order to trap and transport all particles at the focal plane.

© 2014 Optical Society of America

## 1. Introduction

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

2. T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. **86**, 174101 (2005). [CrossRef]

3. T. Cizmar, M. Siler, and P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B **84**, 197–203 (2006). [CrossRef]

3. T. Cizmar, M. Siler, and P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B **84**, 197–203 (2006). [CrossRef]

4. G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express **15**, 13972–13986 (2007). [CrossRef] [PubMed]

5. M. Siler, P. Jakl, O. Brzobohaty, and P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express **20**, 24304–24318 (2012). [CrossRef] [PubMed]

6. N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A **87**, 063812 (2013). [CrossRef]

7. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. **109**, 163903 (2012). [CrossRef] [PubMed]

7. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. **109**, 163903 (2012). [CrossRef] [PubMed]

8. T. A. Nieminen, N. R. Heckenberg, and H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. **33**, 122–124 (2008). [CrossRef] [PubMed]

9. S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, and P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. **38**, 28–30 (2013). [CrossRef] [PubMed]

10. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. **43**, 4322–4327 (2004). [CrossRef] [PubMed]

11. Z. Chen and D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. **37**, 1286–1288 (2012). [CrossRef] [PubMed]

7. D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. **109**, 163903 (2012). [CrossRef] [PubMed]

## 2. Theoretical background

12. B. Richards and E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A **253**, 358–379 (1959). [CrossRef]

13. K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

*l*

_{0}(

*θ*) is the apodization function that we have assumed that is an order one Hermite-Gauss mode: where

*α*is the angular semi-aperture of the focusing system given by

*α*=

*sin*

^{−1}(

*NA/n*). NA is the numerical aperture and

*β*is the ratio of the pupil radius and the beam waist, n is the refractive index between the high numerical optical system and the sample. Following the definitions given in reference [13

13. K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express **7**, 77–87 (2000). [CrossRef] [PubMed]

*T*(

*θ*) given by: where we have assumed that

*g*1 = 1 and

*g*2 =

*Exp*(

*iξt*), so emergent fields from the rings described in transmission Eq. (3) differ in their relative phase. This linear relative phase

*ξt*difference makes the conveyor work [7

**109**, 163903 (2012). [CrossRef] [PubMed]

*θ*

_{2}= 0.6, the red lines to

*θ*

_{2}= 0.7 and finally green lines correspond to

*θ*

_{2}= 0.8. As can be observed, the first ring is the same in all cases.

- The dependence on
*θ*of the amplitude components of integrals 1 can be approximated to their constant value evaluated at the middle point (*θ*;_{l}*l*= 1, 2) of each annular ring described by Eq. (3).

*θ*variable of Eq. (1) can be analytically solved, so Eq. (1) can be written as:

*k*=

_{sl}*kSin*(

*θ*),

_{l}*k*=

_{cl}*kCos*(

*θ*), and

_{l}*ĥ*=

_{ul}*l*

_{0}(

*θ*)

_{l}*h*(

_{u}*θ*) being

_{l}*u*= (

*r*,

*z*). Equation (4) shows that the focusing of a radially polarized beam by using a high-aperture system with mask function

*T*(

*θ*) composed by two annular rings, generates a superposition of coaxial Bessel beams that produces a sharp focal spot [10

10. C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. **43**, 4322–4327 (2004). [CrossRef] [PubMed]

**109**, 163903 (2012). [CrossRef] [PubMed]

*J*

_{0}beams and the axial field is a superposition of Bessel

*J*

_{1}beams radially symmetric. It is important to note, that using this methodology with field integrals described in reference [12

12. B. Richards and E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A **253**, 358–379 (1959). [CrossRef]

*i*= |

_{r}*e*|

_{r}^{2}, and

*i*= |

_{z}*e*|

_{z}^{2}as the radial and axial intensity of the electric field, respectively. In the same way, we have introduced

*φ*, and

_{r}*φ*as the radial and axial phases of the electric field, respectively.

_{z}### 2.1. Optical forces acting on a nanoparticle

*r*<<

_{p}*λ*/20) for which it is accomplished that the scattering is so weak. Therefore, according to [14

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

*e*,

_{r}*e*,

_{ψ}*e*) exists) can be expressed as:

_{z}^{*}is a complex conjugate value and

*α̂*=

*α*+

_{R}*iα*is the complex value of the polarizability particle, which for the dielectric particles considered can be obtained by [15

_{I}15. M. Nieto-Vesperinas, J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express **18**, 11428–11443 (2010). [CrossRef] [PubMed]

*k*

_{0}the wavenumber in vacuum,

*a*the particle radius and

*ε*the particle dielectric permittivity and

_{p}*ε*the dielectric permittivity of the medium where the dipolar particle is embedded. Introducing Eq. (5) into Eq. (11), we obtain that: In last equation, the terms proportional to

*α*are the gradient force and the terms proportional to

_{R}*α*are the scattering force, so Eq. (13) can be written in a compact form as: where it is shown that the gradient force is proportional to the gradient of total intensity of the electric field and the scattering force depends on the gradient of phases of each component of the field. It is interesting to note that, recently, it has been demonstrated that gradientless optical fields can act as tractor beams along a properly chosen interface of two materials with different refractive indices [16

_{I}16. V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics **7**, 787–790 (2013). [CrossRef]

*R⃗*(

*t*) is the position vector of the particle at time

*t*, m is the particle mass,

*ℱ⃗*(

*t*) is a random function force with time.

*ℱ⃗*(

*t*) has a Gaussian probability distribution with correlation function <

*ℱ*(

_{i}*t*),

*ℱ*(

_{j}*t′*)>= 2

*γK*(

_{B}Tδ_{i,j}δ*t*−

*t′*), where

*k*is Boltzmann’s constant and T is the temperature. Coefficient

_{B}*γ*= 6

*πηa*, where

*η*is the viscosity of the media.

## 3. Design of an optimal active tractor beams

*r*→ 0) are given by

*i*, because

_{z}*i*is null at

_{r}*r*= 0. Thus, the axial intensity is described by:

**109**, 163903 (2012). [CrossRef] [PubMed]

*δ*

_{2}=

*δ*

_{2}

*according to: As can be observed, the width of the second ring depends on the width of the first ring, the ring’s positions and the apodization function evaluated at each ring. Introducing*

_{e}*δ*

_{2}

*in Eq. (16), the axial intensity when the beam ratio is 1:1 at*

_{e}*t*= 0 is given by:

**109**, 163903 (2012). [CrossRef] [PubMed]

*of the conveyor described by the axial intensity 18, can be deduced from the arguments of the Sinc functions (the ones that multiplies the cosine function) by using: For optimizing the energetic response of the optical conveyor, we locate the first ring centered at the position where incident field*

_{z}*l*(

*θ*) shows the maximum value: Moreover, we choose: so that the first ring gives us the maximum aperture angle of the system. By taking these parameter values, it is accomplished that

*θ*

_{1}>

*θ*

_{2}and as consequence

*δ*

_{2}

*(*

_{e}Cos*θ*

_{2}) >

*δ*

_{1}

*Cos*(

*θ*

_{1}). Then, according to Eq. (19), the maximum theoretical axial range of the optical conveyor will be given by: Since, in order to optimize the tractor beam, we have two degrees of freedom,

*δ*

_{1}and

*θ*

_{2}.

### 3.1. Numerical results

*δ*

_{1}= 0.06. For this, the NA system was 1.1, and the refractive index between the lens and the sample is

*θ*

_{1}= 0.942.

*δ*

_{2}

*values for each conveyor configuration can be obtained by introducing in Eq. (17) Eq. (2) together with the numerical values of*

_{e}*δ*

_{1}and

*θ*

_{1}. In Table 1, the obtained results for

*θ*

_{2}values between 0.6 to 0.8 are shown. This analysis is limited to this

*θ*

_{2}-range because at lower values than 0.6,

*δ*

_{2}

*increases and condition*

_{e}*δ*

_{2}<< 1 is not fulfilled. On the other hand, higher

*θ*

_{2}-values as 0.8 originate only one ring for the used numerical values.

*θ*

_{2}value raises (Fig. 3). However, at higher

*θ*

_{2}values, the intensity maxima at axial positions (different to

*z*= 0) decreases (Fig. 3). The maximum transport efficiency will be obtained when particles are axially confined. Then, we are going to analyze the best trapping configuration at the focal plane.

## 4. Conclusions

## References and links

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

2. | T. Cizmar, V. Garces-Chavez, K. Dholakia, and P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. |

3. | T. Cizmar, M. Siler, and P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B |

4. | G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, and P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express |

5. | M. Siler, P. Jakl, O. Brzobohaty, and P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express |

6. | N. Wang, J. Chen, S. Liu, and Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A |

7. | D. B. Ruffner and D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. |

8. | T. A. Nieminen, N. R. Heckenberg, and H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. |

9. | S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, and P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. |

10. | C. J. R. Sheppard and A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. |

11. | Z. Chen and D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. |

12. | B. Richards and E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A |

13. | K. S. Youngworth and T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express |

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

15. | M. Nieto-Vesperinas, J. Sáenz, R. Gómez-Medina, and L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express |

16. | V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, and A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics |

17. | F. Reif, |

18. | M. Borromeo and F. Marchesoni, “Brownian surfers,” Phys. Lett. A |

19. | P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. |

20. | M. Siler, T. Cizmar, A. Jonas, and P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(260.2110) Physical optics : Electromagnetic optics

(260.5430) Physical optics : Polarization

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

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: December 5, 2013

Revised Manuscript: January 4, 2014

Manuscript Accepted: January 14, 2014

Published: February 4, 2014

**Virtual Issues**

Vol. 9, Iss. 4 *Virtual Journal for Biomedical Optics*

**Citation**

Luis Carretero, Pablo Acebal, and Salvador Blaya, "Three-dimensional analysis of optical forces generated by an active tractor beam using radial polarization," Opt. Express **22**, 3284-3295 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-3-3284

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### References

- A. Ashkin, “Acceleration and trapping of particles by radiation presure,” Phys. Rev. Lett. 24, 156–159 (1970). [CrossRef]
- T. Cizmar, V. Garces-Chavez, K. Dholakia, P. Zemanek, “Optical conveyor belt for delivery submicron objects,” Appl. Phys. Lett. 86, 174101 (2005). [CrossRef]
- T. Cizmar, M. Siler, P. Zemanek, “An optical nanotrap array movable over a milimetre range,” Appl. Phys. B 84, 197–203 (2006). [CrossRef]
- G. Milne, H. Dholakia, D. McGloin, K. Volke-Sepulveda, P. Zemanek, “Transverse particle dynamics in a Bessel beam,” Opt. Express 15, 13972–13986 (2007). [CrossRef] [PubMed]
- M. Siler, P. Jakl, O. Brzobohaty, P. Zemanek, “Optical forces induced behavior of a particle in a non-diffracting vortex beam,” Opt. Express 20, 24304–24318 (2012). [CrossRef] [PubMed]
- N. Wang, J. Chen, S. Liu, Z. Lin, “Dynamical and phase study on stable optical pulling force in Bessel beams,” Phys. Rev. A 87, 063812 (2013). [CrossRef]
- D. B. Ruffner, D. G. Grier, “Optical conveyors: a class of active tractor beams,” Phys. Rev. Lett. 109, 163903 (2012). [CrossRef] [PubMed]
- T. A. Nieminen, N. R. Heckenberg, H. Rubinstein-Dunlop, “Forces in optical tweezers with radially and azimuthally polarized trapping beams,” Opt. Lett. 33, 122–124 (2008). [CrossRef] [PubMed]
- S. E. Skelton, M. Sergides, R. Saija, M. Iati, O. Marago, P. H. Jones, “Trapping volume control in optical tweezers using cylindrical vector beams,” Opt. Lett. 38, 28–30 (2013). [CrossRef] [PubMed]
- C. J. R. Sheppard, A. Choudhury, “Annular pupils, radial polarization and superresolution,” Appl. Opt. 43, 4322–4327 (2004). [CrossRef] [PubMed]
- Z. Chen, D. Zhao, “4 π focusing of spatially modulated radially polarized vortex beams,” Opt. Lett. 37, 1286–1288 (2012). [CrossRef] [PubMed]
- B. Richards, E. Wolf, “Electomagnetic diffraction of optical system II. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959). [CrossRef]
- K. S. Youngworth, T. Brown, “Focusing of high numerical aperture cylindrical-vector beams,” Opt. Express 7, 77–87 (2000). [CrossRef] [PubMed]
- P. C. Chaumet, M. Nieto-Vesperinas, “Time-averaged total force on a dipolar sphere in an electromagentic fiel,” Opt. Lett. 25, 1065–1067 (2000). [CrossRef]
- M. Nieto-Vesperinas, J. Sáenz, R. Gómez-Medina, L. Chantada, “Optical forces on small magnetodielectric particles,” Opt. Express 18, 11428–11443 (2010). [CrossRef] [PubMed]
- V. Kajorndenukul, W. Ding, S. Sukhov, C.-W. Qiu, A. Dogariu, “Linear momentum increase and negative optical forces at dielectric interface,” Nat. Photonics 7, 787–790 (2013). [CrossRef]
- F. Reif, Fundamentals of Statistical and Thermal Physics (McGraw-Hill, 1965).
- M. Borromeo, F. Marchesoni, “Brownian surfers,” Phys. Lett. A 249, 199–203 (1998). [CrossRef]
- P. Reimann, “Brownian motors: noisy transport far from equilibrium,” Phys. Rep. 361, 57–265 (2002). [CrossRef]
- M. Siler, T. Cizmar, A. Jonas, P. Zemanek, “Surface delivery of a single nanoparticle under moving evanescent standing-wave illumination,” New J. Phys. 10, 113010 (2008). [CrossRef]

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