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Virtual Journal for Biomedical Optics

Virtual Journal for Biomedical Optics

| EXPLORING THE INTERFACE OF LIGHT AND BIOMEDICINE

  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 4 — Apr. 1, 2009

Optics InfoBase > Virtual Journal for Biomedical Optics > Volume 4 > Issue 4 > Page 2661

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

Timo A. Nieminen, Alexander B. Stilgoe, Vincent L. Loke, Norman R. Heckenberg, and Halina Rubinsztein-Dunlop  »View Author Affiliations


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

We would like to thank the authors for correcting our error in equation (10) in Ref. [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]

]. As they point out, the code itself is correct; Crichton and Marston’s expression [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 Conf. 04, 37–50 (2000).

] is used literally, after converting the vector spherical wavefunction (VSWF) [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]

] amplitudes from our convention to theirs. The reader should beware that there are at least three different normalizations, at least four different sign conventions, and the choice of exp(imϕ) versus cos(mϕ) and sin(mϕ), giving a multitude of possible conventions for the VSWFs; the error noted above resulted from an error in conversion between such conventions.

However, while we do not dispute their comments on the calculation of spherical Hankel functions (the actual values depend on the combination of version and platform), we believe that their conclusions concerning the impact of errors in the calculation of spherical Hankel functions using the standard Matlab (The Mathworks, Natick, MA) function besselh are demonstrably wrong. It should be noted that the convergence criteria cited by the authors are intended for particles of moderate size, while the example case given by the authors is for a size parameter x = ka = 0.118, or a particle of radius a = 0.0188λ. A particle of this size is generally accepted as being sufficiently small so that the Rayleigh approximation—taking only the electric dipole term in the Mie coefficients or T-matrix as non-zero—is reliable [3

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

].

The convergence of the calculations can be easily tested, by changing the n max at which the T-matrix (or Mie coefficients) is truncated [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]

]. For the example case of 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

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

] gives 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

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]

], a relatively large round-off error can result when the force is sufficiently small. This limits the smallest feasible particle radius to about an atomic radius.

Table 1.  Convergence of trap strength Q max and equilibrium position z 0 with increasing n max. The differences between the calculated values and the final converged values (Q max = 2.22×10-5 and z 0 = 2.92×10-4 λ) is shown.
n max ΔQ max ΔQ max/Q max Δz 0(×λ)
12.16 × 10-9 9.73 × 10-5 5.92 × 10-8
22.67 × 10-12 1.20 × 10-7 4.50 × 10-11
32.34 × 10-16 1.06 × 10-11 0
4000
5000

The genesis of the erroneous conclusion that the results presented in [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]

] are in error seems to be an interpretation that the white regions in Fig. 2 in Ref. [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]

] are where the optical forces do not result in trapping. This is incorrect—these regions are where either the optical forces do not result in trapping, or the trap strength is very small (for example, in Fig. 2(c) in Ref. [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]

], the lowest contour is a trap strength of 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.

It is also worth noting that the trap strength is not necessarily the most useful parameter describing how well Rayleigh particles are trapped. The trapping potential—the energy required for the particle to escape from the trap, which can be compared with the thermal energy—is often far more informative [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]

].

Finally, errors in the calculation of the special functions can cause serious problems. Errors for very large 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 9, 5192–S203 (2007). [CrossRef]

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 Conf. 04, 37–50 (2000).

3.

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

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]

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

  1. 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]
  2. 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).
  3. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996). [CrossRef]
  4. 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]
  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]

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