## The effect of Mie resonances on trapping in optical tweezers: reply

Optics Express, Vol. 17, Issue 4, pp. 2661-2662 (2009)

http://dx.doi.org/10.1364/OE.17.002661

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### Abstract

We show that errors in the calculation of spherical Hankel functions for very small size parameters does not affect the calculation of optical trapping forces; predicted forces agree with the Rayleigh formula.

© 2009 Optical Society of America

## 1. Introduction

*n*

_{max}at which the T-matrix (or Mie coefficients) is truncated [1

1. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A **9**, 5192–S203 (2007). [CrossRef]

*x*= 0.118, a relative refractive index of

*m*= 1.59/1.33, and a circularly polarized beam focussed by an objective lens of numerical aperture NA = 1.3, the convergence with increasing

*n*

_{max}is shown in table 1. The error in the calculation of

*h*

^{(1)}

_{5}(

*x*) is entirely irrelevant, since round-off error ensures that it does not contribute to the final result at all. The results obtained with

*n*

_{max}= 1 are sufficient for almost all practical purposes, especially when the likely uncertainties in particle size and refractive index, the effect of aberrations in the optical system, and so on, are considered. The Rayleigh formula [3

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

*Q*

_{max}= 2.22 × 10

^{-5}, the same as the tweezers toolbox, and

*z*

_{0}= 4.42 × 10

^{-4}

*λ*, which is about 1 atomic diameter larger. Since the toolbox finds the optical force by calculating the difference between the incoming and outgoing momentum fluxes [1

1. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A **9**, 5192–S203 (2007). [CrossRef]

4. A. B. Stilgoe, T. A. Nieminen, G. Knöner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express **16**, 15039–15051 (2008). [CrossRef] [PubMed]

4. A. B. Stilgoe, T. A. Nieminen, G. Knöner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express **16**, 15039–15051 (2008). [CrossRef] [PubMed]

4. A. B. Stilgoe, T. A. Nieminen, G. Knöner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express **16**, 15039–15051 (2008). [CrossRef] [PubMed]

*Q*

_{max}= 1.27×10

^{-3}). Although we may have contributed to this by not explicitly stating this in the text, it is indicated in the color scale bars in the figures. We only briefly discussed the limits of trapping of Rayleigh particles due to Brownian motion in the text since this was not relevant to the main topic of the paper, the effect of Mie resonances in optical trapping. In the white region along the left and lower borders of the figures, the toolbox correctly reproduces the forces given by the Rayleigh approximation, and predicts trapping as expected.

5. W. Singer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Collecting single molecules with conventional optical tweezers,” Phys. Rev. E **75**, 011916 (2007). [CrossRef]

*n*and

*m*are not of great concern here, since calculations of optical forces using the toolbox which require such values are, by and large, computationally infeasible, but should be kept in mind when dealing with very large particles. More potentially troublesome is the failure of calculation of spherical Bessel or Hankel functions for strongly absorbing particles of large size. While such particles cannot be trapped in conventional optical tweezers, the forces may well be of interest. The calculated forces on weakly or moderately absorbing particles of moderate size appear to be correct, or at least physically reasonable.

## References and links

1. | T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Brańczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A |

2. | J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Electronic Journal of Differential Equations |

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

4. | A. B. Stilgoe, T. A. Nieminen, G. Knöner, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “The effect of Mie resonances on trapping in optical tweezers,” Opt. Express |

5. | W. Singer, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Collecting single molecules with conventional optical tweezers,” Phys. Rev. E |

**OCIS Codes**

(140.7010) Lasers and laser optics : Laser trapping

(290.4020) Scattering : Mie theory

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

**ToC Category:**

Optical Trapping and Manipulation

**History**

Original Manuscript: January 12, 2009

Revised Manuscript: February 4, 2009

Manuscript Accepted: February 5, 2009

Published: February 10, 2009

**Virtual Issues**

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

**Citation**

Timo A. Nieminen, Alexander B. Stilgoe, Vincent L. Loke, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop, "The effect of Mie resonances on trapping
in optical tweezers: reply," Opt. Express **17**, 2661-2662 (2009)

http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-4-2661

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### References

- T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knoner, A. M. Branczyk, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Optical tweezers computational toolbox," J. Opt. A 9, 5192-S203 (2007). [CrossRef]
- J. H. Crichton and P. L. Marston, "The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation," Electron. J. Differ. Equations Conf. 04, 37-50 (2000).
- Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996). [CrossRef]
- A. B. Stilgoe, T. A. Nieminen, G. Kn¨oner, N. R. Heckenberg and H. Rubinsztein-Dunlop, "The effect of Mie resonances on trapping in optical tweezers," Opt. Express 16, 15039-15051 (2008). [CrossRef] [PubMed]
- W. Singer, T. A. Nieminen, N. R. Heckenberg and H. Rubinsztein-Dunlop, "Collecting single molecules with conventional optical tweezers," Phys. Rev. E 75, 011916 (2007). [CrossRef]

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