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  • Editor: Gregory W. Faris
  • Vol. 4, Iss. 4 — Apr. 1, 2009
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The effect of Mie resonances on trapping in optical tweezers: comment

Bo Sun and David G. Grier  »View Author Affiliations


Optics Express, Vol. 17, Issue 4, pp. 2658-2660 (2009)
http://dx.doi.org/10.1364/OE.17.002658


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Abstract

Recently, Stilgoe, et al., [Opt. Express 16, 15039 (2008)] reported calculations of the force on an optically trapped sphere performed using the “optical tweezers toolbox”. This software suffers from numerical inaccuracies that lead to qualitative errors in the state diagram for stable trapping, particularly for spheres smaller than the wavelength of light.

© 2009 Optical Society of America

An optical tweezer is a single-beam optical trap that can localize objects in three dimensions using forces exerted by a strongly focused beam of light [1

1. 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, 288–290 (1986). [CrossRef] [PubMed]

]. Recently, Stilgoe et al. reported calculations of the axial trapping efficiency of optical tweezers for dielectric spheres, which emphasized the important role of Mie resonances on the stability and strength of optical traps, particularly for high-index particles [2

2. 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]

]. These calculations are based on the Lorenz-Mie theory of light scattering by spheres [3

3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

], and so should provide exact results for all sphere sizes and compositions. This approach also is numerically efficient, and can be generalized to compute the forces and torques due to arbitrary light fields [4

4. B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008). [CrossRef] [PubMed]

] on objects of arbitrary shape [5

5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

, 6

6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

]. The calculations reported in [2

2. 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]

] were performed with the “optical tweezers toolbox” software suite [7

7. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, Y. Hu, G. Knoener, and A. M. Branczyk, “Optical tweezers toolbox 1.1,” University of Queensland (2007), http://www.physics.uq.edu.au/people/nieminen/software.html.

] described in [5

5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

], which is formally correct, but relies on numerically inaccurate Hankel functions provided by Matlab® (versions R2007a through R2008b, The MathWorks, Natick, MA). Consequently, some of the results reported in [2

2. 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 incorrect, particularly for very small spheres.

The incident light field, E 0, the scattered wave, Es, and the field within the sphere, Ei, are all described in Lorenz-Mie theory as expansions in vector spherical harmonics, [3

3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

]

E0(r)=n=1m=nnanmMnm(2)(kr)+bnmNnm(2)(kr),
(1)
Es(r)=n=1m=nnPnmMnm(1)(kr)+qnmNnm(2)(kr),
(2)
Ei(r)=n=1m=nnCnm(Mnm(1)(mpkr)+Mnm(2)(mpkr))+dnm(Nnm(1)(mpkr)+Nnm(2)(mpkr)),
(3)

where k = 2πnm/λ is the wavenumber of light in a medium of refractive index nm, and where mp = np/nm is the particle’s relative refractive index.

The incident field’s coefficients, anm and bnm, were computed in [2

2. 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]

] through the over-determined point-matching method [8

8. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spec. Rad. Transfer 79–80, 1005–1017 (2003). [CrossRef]

] but can be derived equally well with the Debye-Wolf path integral formalism [4

4. B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008). [CrossRef] [PubMed]

], which we adopt here. The scattering coefficients, pnm and qnm, depend on the on the size and composition of the sphere, and on the structure of the light field illuminating it [2

2. 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

4. B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008). [CrossRef] [PubMed]

, 5

5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

]. The interior field’s coefficients, cnm and dnm, similarly are computed to satisfy continuity conditions at the sphere’s surface [6

6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

]:

Cnm=impαmnξn'(x)ψn(mpx)mpξn(x)ψn(mpx)and
(4)
dnm=impbmnmpξn'(x)ψn(mpx)ξn(x)ψn'(mpx),
(5)

where x = ka, and where Ψn(x) and ξn(x) are Ricatti-Bessel functions.

Both E 0(r) and Es(r) contribute to the Maxwell stress tensor on the sphere’s surface, whose integral yields the force. References [2

2. 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]

] and [5

5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

] follow Crichton and Marston [9

9. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Elec. J. Diff. Eq. Conf. 04, 37–50 (2000).

] by performing the integral analytically. The result for the axial trapping efficiency in an optical trap of total power P = ∑ n=1n m=-n |anm|2 + |bnm|2 is then [9

9. J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Elec. J. Diff. Eq. Conf. 04, 37–50 (2000).

]

Q=2Pn=11n+1m=nn(mn{anm*bnmPnm*qnm}+
[n(n+2)(n+1m)(n+1+m)(2n+1)(2n+3)]12×
{PnmPn+1,m*+qnmqn+1,m*anman+1,m*bnmbn+1,m*}).
(6)
Fig. 1. State diagram for dielectric spheres in an optical tweezer of numerical aperture 1.3. Colors in the main diagram correspond to the maximum axial trapping efficiency at each set of conditions, computed on a 144×60 grid. White regions denote conditions for which particles are not stably trapped. Images (a) and (b) show the magnitude of normalized surface fields for the particular conditions indicated on the diagram. (a) corresponds to an unstable point. The sphere at (b) is stably trapped.

Equation (10) of [5

5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

] misstates this result, inconsistently taking the real part of the second term. The distributed software, however, is formally correct [7

7. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, Y. Hu, G. Knoener, and A. M. Branczyk, “Optical tweezers toolbox 1.1,” University of Queensland (2007), http://www.physics.uq.edu.au/people/nieminen/software.html.

].

The central results of [2

2. 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]

] rely on numerical convergence of the sums in Eq. (6). Unfortunately, the scattering coefficients, pnm and qnm, are notoriously difficult to compute accurately [3

3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

, 6

6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

] because the Ricatti-Bessel functions in which they are expressed have numerically unstable recurrence relations. The expansions’ accuracy can be assessed by checking continuity of the field at the surface using Eqs. (3) through (5

5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

). In our implementation, the relative error in the computed fields is smaller than 10-5, with correspondingly small errors in the force.

Figure 1 shows the maximum axial trapping efficiency, Qmax as a function of the sphere’s radius, a and relative refractive index, m, in an optical tweezer of numerical aperture 1.2. These conditions correspond to those of Fig. 2(b) of [2

2. 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 principal qualitative difference is that Fig. 1 predicts stable trapping for spheres smaller than a = 0.2λ, while [2

2. 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]

] does not. The present results are consistent with an analytic computation of the force on a Rayleigh particle in a Gaussian trap [10

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

], and thus would appear to be correct. They also agree quantitatively with results obtained by numerically integrating the Maxwell stress tensor over the surface of the sphere [4

4. B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008). [CrossRef] [PubMed]

], which confirms the correct implementation of Eq. (6). Although these results suggest that even vanishing small spheres can be trapped, the maximum trapping force is limited in practice by saturation of the sphere’s polarization, a nonlinear effect not considered in the linear light-scattering formalism discussed here and in [2

2. 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]

].

Explicitly computing the surface fields, such as the examples presented in Fig. 1(a) and (b), reveals smoothly varying field distributions with only subtle differences differentiating stable and unstable points. The results in Fig. 1 therefore highlight both the need for numerical accuracy in predicting optical traps’ performance, and also the importance of small differences in optical and material properties for trapping high-index particles.

Major qualitative and quantitative discrepancies we observe with the results of [2

2. 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]

] can be attributed to the computational toolbox’s use of the standard Matlab routine besselh(), which implements the Hankel function Hn (2) (x) that is used to compute the spherical Hankel function ******hn(2)(x)=π/(2x)Hn+12(2)(x)=ξn(x)/x. Taking x = 0.118 as a test case, accurate calculation of Qmax requires the sum in Eq. (6) to be computed to at least n = 5 [3

3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

, 6

6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

], with h (2) 5(0.118) = 2.1996469 × 10-9 -i3.5032872 × 108. The corresponding result obtained with Matlab is 2.7184 × 10-8 - i3.5033 × 108, whose real component is in error by a factor of ten. Such numerical errors diminish with increasing x and are negligible for x > 1. Other quantitative differences between Fig. 1 and Fig. 2(b) of [2

2. 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 likely due to the resolution of the sampling grid and slight differences in the description of the incoming wave.

This work was supported in part by the NSF through Grant Number DMR-0606415 and in part by a grant from the Keck Foundation. B.S. acknowledges support from a Kessler Family Foundation Fellowship.

References and links

1.

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, 288–290 (1986). [CrossRef] [PubMed]

2.

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]

3.

C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).

4.

B. Sun, Y. Roichman, and D. G. Grier, “Theory of holographic optical trapping,” Opt. Express 16, 15765–15776 (2008). [CrossRef] [PubMed]

5.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A. M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical tweezers computational toolbox,” J. Opt. A 9, S196–S203 (2007). [CrossRef]

6.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).

7.

T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, Y. Hu, G. Knoener, and A. M. Branczyk, “Optical tweezers toolbox 1.1,” University of Queensland (2007), http://www.physics.uq.edu.au/people/nieminen/software.html.

8.

T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spec. Rad. Transfer 79–80, 1005–1017 (2003). [CrossRef]

9.

J. H. Crichton and P. L. Marston, “The measurable distinction between the spin and orbital angular momenta of electromagnetic radiation,” Elec. J. Diff. Eq. Conf. 04, 37–50 (2000).

10.

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

OCIS Codes
(050.1960) Diffraction and gratings : Diffraction theory
(140.7010) Lasers and laser optics : Laser trapping
(290.4020) Scattering : Mie theory

ToC Category:
Optical Trapping and Manipulation

History
Original Manuscript: October 29, 2008
Revised Manuscript: December 22, 2008
Manuscript Accepted: December 22, 2008
Published: February 10, 2009

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

Citation
Bo Sun and David G. Grier, "The effect of Mie resonances on trapping in optical tweezers: comment," Opt. Express 17, 2658-2660 (2009)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-17-4-2658


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References

  1. 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, 288-290 (1986). [CrossRef] [PubMed]
  2. A. B. Stilgoe, T. A. Nieminen, G. Knoner, 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]
  3. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley Interscience, New York, 1983).
  4. B. Sun, Y. Roichman, and D. G. Grier, "Theory of holographic optical trapping," Opt. Express 16, 765-776 (2008). [CrossRef] [PubMed]
  5. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, G. Knöner, A.M. Branczyk, N. R. Heckenberg, and H. Rubinsztein-Dunlop, "Optical tweezers computational toolbox," J. Opt. A 9, S196-S203 (2007). [CrossRef]
  6. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption and Emission of Light by Small Particles (Cambridge University Press, Cambridge, 2001).
  7. T. A. Nieminen, V. L. Y. Loke, A. B. Stilgoe, Y. Hu, G. Knoener, and A. M. Branczyk, "Optical tweezers toolbox 1.1," University of Queensland (2007), http://www.physics.uq.edu.au/people/nieminen/software.html.
  8. T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, "Multipole expansion of strongly focussed laser beams," J. Quant. Spec. Rad. Transfer 79-80, 1005-1017 (2003). [CrossRef]
  9. 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).
  10. Y. Harada and T. Asakura, "Radiation forces on a dielectric sphere in the Rayleigh scattering regime," Opt. Commun. 124, 529-541 (1996). [CrossRef]

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