Abstract

The theory of the trapping of nonspherical particles in the focal region of a high-numerical-aperture optical system is formulated in the framework of the transition matrix approach. Both the case of an unaberrated lens and the case of an aberrated one are considered. The theory is applied to single latex spheres of various sizes and, when the results are compared with the available experimental data, a fair agreement is attained. The theory is also applied to binary clusters of spheres of latex with a diameter of 220nm in various orientations. Although, in this case we have no experimental data to which our results can be compared, we get useful indications for the trapping of nonspherical particles. In particular, we find substantial agreement with recent results on the trapping of prolate spheroids in aberrated gaussian fields [S. H. Simpson and S. Hanna, J. Opt. Soc. Am. A 24, 430 (2007)].

© 2007 Optical Society of America

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Errata

Ferdinando Borghese, Paolo Denti, Rosalba Saija, and Maria Antonia Iatì, "Optical trapping of nonspherical particles in the T-matrix formalism: erratum," Opt. Express 15, 14618-14618 (2007)
https://www.osapublishing.org/oe/abstract.cfm?uri=oe-15-22-14618

References

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  4. B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. (London) 253, 358-379 (1959).
    [CrossRef]
  5. A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-853 (2001).
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  10. R. Saija, M. A. Iat`ı, A. Giusto, P. Denti and F. Borghese, "Transverse components of the radiation force in the T-matrix formalism," J. Quant. Spectrosc. Radiat. Transfer 94, 163-167 (2005).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. M. I. Mishchenko, L. D. Travis and A. A. Lacis, Scattering, absorption and emission of light by small particles, (Cambridge University Press, New York, 2002).
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  15. E. Fucile, F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, "General reflection rule for electromagnetic multipole fields on a plane interface," IEEE Trans. Antennas Propag. AP 45, 868 (1997).
    [CrossRef]
  16. R. Saija, M. A. Iat`ı, P. Denti, F. Borghese, A. Giusto and O. I. Sindoni, "Efficient light-scattering calculations for aggregates of large spheres," Appl. Opt. 42, 2785-2793 (2003).
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  17. R. Saija, M. A. Iat`ı, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie theory: the transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
    [CrossRef]
  18. The Erratum to Ref. [14] can be found at http://dfmtfa.unime.it/profs/borghese/ferdinandoborghese.html.
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    [CrossRef]
  20. E. M. Rose, Elementary theory of angular momentum, (Wiley, New York, 1956).
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  23. L. Novotny and B. Hecht, Principles of nano-optics, (Cambridge University Press, Cambridge, 2006).
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    [CrossRef]
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    [CrossRef]

2007

2006

2005

R. Saija, M. A. Iat`ı, A. Giusto, P. Denti and F. Borghese, "Transverse components of the radiation force in the T-matrix formalism," J. Quant. Spectrosc. Radiat. Transfer 94, 163-167 (2005).
[CrossRef]

A. Rohrbach and E. H. K. Stelzer, "Trapping forces, force constants, and potential depths for dielectric spheres in the presence of spherical aberrations," Appl. Opt. 41, 2494-2507 (2005).
[CrossRef]

2004

2003

2001

A. Rohrbach and E. H. K. Stelzer, "Optical trapping of dielectric particles in arbitrary fields," J. Opt. Soc. Am. A 18, 839-853 (2001).
[CrossRef]

R. Saija, M. A. Iat`ı, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie theory: the transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
[CrossRef]

M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transfer 70, 811-816 (2001).
[CrossRef]

1997

E. Fucile, F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, "General reflection rule for electromagnetic multipole fields on a plane interface," IEEE Trans. Antennas Propag. AP 45, 868 (1997).
[CrossRef]

1995

1992

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray-optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

1986

1984

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).

1971

P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).

1959

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. (London) 253, 358-379 (1959).
[CrossRef]

1955

D. S. Saxon, "Tensor scattering matrix of the electromagnetic field," Phys. Rev. 100,1771-1775 (1955).
[CrossRef]

Appl. Opt.

Astrophys. J.

R. Saija, M. A. Iat`ı, F. Borghese, P. Denti, S. Aiello and C. Cecchi-Pestellini, "Beyond Mie theory: the transition matrix approach in interstellar dust modeling," Astrophys. J. 559, 993-1004 (2001).
[CrossRef]

Biophys. J.

A. Ashkin, "Forces of a single-beam gradient laser trap on a dielectric sphere in the ray-optics regime," Biophys. J. 61, 569-582 (1992).
[CrossRef] [PubMed]

IEEE Trans. Antennas Propag. AP

E. Fucile, F. Borghese, P. Denti, R. Saija, and O. I. Sindoni, "General reflection rule for electromagnetic multipole fields on a plane interface," IEEE Trans. Antennas Propag. AP 45, 868 (1997).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

M. I. Mishchenko, "Radiation force caused by scattering, absorption and emission of light by nonspherical particles," J. Quant. Spectrosc. Radiat. Transfer 70, 811-816 (2001).
[CrossRef]

R. Saija, M. A. Iat`ı, A. Giusto, P. Denti and F. Borghese, "Transverse components of the radiation force in the T-matrix formalism," J. Quant. Spectrosc. Radiat. Transfer 94, 163-167 (2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev.

D. S. Saxon, "Tensor scattering matrix of the electromagnetic field," Phys. Rev. 100,1771-1775 (1955).
[CrossRef]

Phys. Rev. A

P. L. Marston and J. H. Crichton, "Radiation torque on a sphere caused by a circularly-polarized electromagnetic wave," Phys. Rev. A 30, 2508-2516 (1984).

Phys. Rev. D

P. C. Waterman, "Symmetry, unitarity and geometry in electromagnetic scattering," Phys. Rev. D 4, 825-839 (1971).

Proc. Roy. Soc. (London)

B. Richards and E. Wolf, "Electromagnetic diffraction in optical systems. II. Structure of the image field in an aplanatic system," Proc. Roy. Soc. (London) 253, 358-379 (1959).
[CrossRef]

Other

A. Rohrbach, "Stiffness of optical traps: quantitative agreement between experiment and electromagnetic theory," Phys. Rev. Lett. 95, 168102-1-168102-4 (2005).
[CrossRef]

M. I. Mishchenko, L. D. Travis and A. A. Lacis, Scattering, absorption and emission of light by small particles, (Cambridge University Press, New York, 2002).

F. Borghese, P. Denti and R. Saija, Scattering from model nonspherical particles, 2nd edition (Springer, Berlin, 2007).

L. Novotny and B. Hecht, Principles of nano-optics, (Cambridge University Press, Cambridge, 2006).

W. C. Chew, Waves and fields in inhomogeneous media, IEEE Press Series on Eletromagnetic Waves (Institute of Electrical and Electronic Engineers, Piscataway, N.J., 1990).

E. M. Rose, Elementary theory of angular momentum, (Wiley, New York, 1956).

J. D. Jackson, Classical electrodynamics, 2nd edition (Wiley, New York, 1975).

The Erratum to Ref. [14] can be found at http://dfmtfa.unime.it/profs/borghese/ferdinandoborghese.html.

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Figures (5)

Fig. 1.
Fig. 1.

Coordinate system we adopt to describe the radiation force on a particle at O′. The focus of the optical system coincides with O and the optical axis coincides with the z axis.

Fig. 2.
Fig. 2.

Contour plot (log scale) of the intensity |E(r)|2 of the unaberrated gaussian (top panels) and aberrated gaussian field (lower panels) in the xz (left panels) and in the yz plane (right panels). The origin of the z axis in the lower panels is shifted by ΔF=-4.0µm.

Fig. 3.
Fig. 3.

Qx (x,0,z 0) (blue solid line), Qy (0,y,z 0) (red dashed line), and Qz (0,0,z) (green dotted line) for single spheres with diameter d=850, 1030 and 1660nm as a function of the position of their centers. Trapping both in unaberrated (left panels) and in aberrated beam (right panels) is considered. Note that the origin of the z axis in the right panels is shifted by ΔF=-4.0µm.

Fig. 4.
Fig. 4.

Comparison of calculated asymmetry factors sl (dashed blue line) and sl a (solid red line) with experimental data of Rohrbach [8] (blue circles) and of Zakharian et al. [7] (red dots).

Fig. 5.
Fig. 5.

Qx (x,0,z 0) (blue solid line), Qy (0,y,z 0) (red dashed line), and Qz (0,0,z) (green dotted line) for binary aggregates of latex with diameter d=220nm, as a function of the position of the contacting point. Trapping both in an unaberrated field (left panels) and in an aberrated field (right panels) is considered. In the right panels the origin of the z axis is shifted by ΔF=-4.0µm.

Tables (2)

Tables Icon

Table 1. Calculated and experimental values of sl =1-κx/κy as well as the trapping position for the spheres we deal with in this paper

Tables Icon

Table 2. Calculated values of sl =1-κx /κy for the unaberrated field and of sla for the aberrated field as well as the trapping position for the binary cluster we deal with in this paper

Equations (28)

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F Rad = r 2 Ω r ̂ · T M d Ω ,
T M = 1 8 π Re [ n 2 E E * + B B * 1 2 ( n 2 E 2 + B 2 ) I ]
E I = E 0 u ̂ e i k · r = E 0 u ̂ e i k · ( r + R O ) = E 0 u ̂ e i k · r = E I
F Rad ζ = r 2 16 π Re Ω ( r ̂ · v ̂ ζ ) [ n 2 ( E S 2 + 2 E I * · E S ) + B S 2 + 2 B I * · B S ] d Ω ,
E I = E 0 p l m J lm ( p ) W I lm ( p ) ,
E S = E 0 p l m H lm ( p ) A lm ( p ) ,
A lm ( p ) = p l m S lml m ( p p ) W I l m ( p )
E I E 0 plm Z lm ( p ) ( r ̂ ) W I lm ( p ) ( ) p 1 kr sin [ kr ( l + 1 p ) π 2 ] ,
E S E 0 plm Z lm ( p ) ( r ̂ ) A lm ( p ) exp ( ikr ) kr i l p
F Rad ζ = E 0 2 16 π k v 2 Re [ plm p l m ( A lm ( p ) * A l m ( p ) + A lm ( p ) * A l m ( p ) ) i l l I ζ lml m ( pp ) ]
2 E 0 2 16 π k v 2 Re [ plm p l m ( W I lm ( p ) * A l m ( p ) + W I lm ( p ) * A l m ( p ) )
× sin [ kr ( l 1 + p ) π 2 ] e ikr ( i ) l + p i l l I ζ lml m ( pp ) ]
= F Rad ζ ( Sca ) + F Rad ζ ( Ext ) ,
I ζ lml m ( pp ) = Ω ( r ̂ · v ̂ ζ ) i p p Z lm ( p ) * ( r ̂ ) · Z l m ( p ) ( r ̂ ) d Ω = 4 π 3 μ Y 1 μ * ( v ̂ ς ) K μ ; lml m ( pp ) .
K μ ; lml m ( pp ) = Ω r ̂ i p p Z lm ( p ) * ( r ̂ ) · Z l m ( p ) ( r ̂ ) d Ω .
K μ ; lml m ( pp ) = 16 π 2 3 4 π C ( 1 , l , l ; μ , m , m ) i l l O ll ( pp ) ,
K ζlml'm' (11) = K ζlml'm' (22) ,   K ζlml'm' (12) = K ζlml'm' (21) ,
F Rad ζ ( Sca ) = 2 E 0 2 16 π k v 2 Re plm p l m A lm ( p ) * A l m ( p ) i l l I ζ lml m ( pp ) ,
F Rad ζ ( Ext ) = 2 E 0 2 16 π k v 2 Re plm p l m W I lm ( p ) * A l m ( p ) i l l I ζ lml m ( pp ) ,
E ( r ) = k x 2 + k y 2 k 2 E PW ( k ̂ ) u ̂ k ̂ e i k · r d k ̂ x d k ̂ y ,
f w = exp [ 1 f 0 2 sin 2 ϑ k sin 2 ϑ Max ] .
u ̂ k ̂ = η ( u ̂ k ̂ · u ̂ η k ) u ̂ η k = η c η u ̂ η k ̂
E ( r ) = k F x 2 + k F y 2 k F 2 E FPW ( k ̂ F ) η c η T η ( ϑ k ) u ̂ η k ̂ F e i k F · r d k ̂ F x d k ̂ F y ,
E ( r ) = k F x 2 + k F y 2 k F 2 E FPW ( k ̂ F ) e i k F · R O η c η T η ( ϑ k ) u ̂ η k ̂ F e i k F · r d k ̂ F x d k ̂ F y ,
E = plm J lm ( p ) ( r , k F ) W lm ( p ) ( R O ) ,
W lm ( p ) ( R O ' ) = η c η k F x 2 + k F y 2 k F 2 E FPW ( k ̂ F ) e i k F · R O T η ( ϑ k ) W lm ( p ) ( u ̂ η k ̂ F , k ̂ F ) d k ̂ F x d k ̂ F y ,
E = plm J lm ( p ) ( r , k ) W lm ( p ) ( R O ) ,
W lm ( p ) ( R O ' ) = k F x 2 + k F y 2 k F 2 E PW ( k ̂ ) e i k · R O W lm ( p ) ( u ̂ k ̂ , k ̂ ) d k ̂ x d k ̂ y ,

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