Abstract

We calculate the optical forces on Au and Ag nanospheres through a procedure based on the Maxwell stress tensor. We compare the theoretical and experimental force constants obtained for gold and silver nanospheres finding good agreement for all particles with r<80nm. The trapping of the larger particles recently demonstrated in experiments is not foreseen by our purely electromagnetic theory based on fixed dielectric properties. Since the laser power produces a heating that may be large for the largest spheres, we propose a model in which the latter particles are surrounded by a steam bubble. This model foresees the trapping of these particles and the results turn out to be in reasonable agreement with the experimental data.

© 2009 Optical Society of America

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2009

2008

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical manipulation of nanoparticles: a review," J. Nanophoton. 2, 021875 (2008).
[CrossRef]

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient optical trapping and visualization of silver nanoparticles," Nano Lett. 8, 1486-1491 (2008).
[CrossRef] [PubMed]

F. Borghese, P. Denti, R. Saija, M. A. Iati, and O. M. Marago, "Radiation force and torque on optically trapped linear nanostructures," Phys. Rev. Lett. 100, 163903 (2008).
[CrossRef] [PubMed]

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

2007

2006

2005

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

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

2004

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

2003

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-induced heating in optical traps," Biophys. J. 84, 1308-1316 (2003).
[CrossRef] [PubMed]

1998

L. M. Liz-Marz’an and P. Mulvaney, "Au@SiO2 colloids: effect of temperature on the surface plasmons absorption," New J. Chem. 22, 1285-1288 (1998).
[CrossRef]

1994

1989

D. Stroud, "Theory of the intensity-dependent optical activity in dilute composites," J. Appl. Phys. 66, 2585-2588 (1989).
[CrossRef]

1987

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).

1977

U. Kreibig, "Anomalous frequency and temperature dependence of the optical absorption of small gold particles," J. Phys. (Paris) C2, 97 (1977).

1972

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

1964

R. H. Doremus, "Optical properties of small gold particles," J. Chem. Phys. 40, 2389-2396 (1964).
[CrossRef]

R. H. Doremus, "Erratum: Optical properties of small gold particles," J. Chem Phys. 41, 3259 (1964).
[CrossRef]

Aabo, T.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient optical trapping and visualization of silver nanoparticles," Nano Lett. 8, 1486-1491 (2008).
[CrossRef] [PubMed]

Ashkin, A.

Bendix, P. M.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient optical trapping and visualization of silver nanoparticles," Nano Lett. 8, 1486-1491 (2008).
[CrossRef] [PubMed]

Bhatia, V. K.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Bjorkholm, J. E.

Block, M.

Block, S. M.

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Borghese, F.

Bosanac, L.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient optical trapping and visualization of silver nanoparticles," Nano Lett. 8, 1486-1491 (2008).
[CrossRef] [PubMed]

Carpenter, A. E.

Christy, R. W.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Chu, S

Crichton, J. H.

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).

Denti, P.

Dholakia, K.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical manipulation of nanoparticles: a review," J. Nanophoton. 2, 021875 (2008).
[CrossRef]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical manipulation of nanoparticles: a review," J. Nanophoton. 2, 021875 (2008).
[CrossRef]

Doremus, R. H.

R. H. Doremus, "Optical properties of small gold particles," J. Chem. Phys. 40, 2389-2396 (1964).
[CrossRef]

R. H. Doremus, "Erratum: Optical properties of small gold particles," J. Chem Phys. 41, 3259 (1964).
[CrossRef]

Dziedzic, J. M.

Gittes, F.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-induced heating in optical traps," Biophys. J. 84, 1308-1316 (2003).
[CrossRef] [PubMed]

Hansen, P. M.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Harrit, N.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Iati, M. A.

Johnson, P. B.

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Kreibig, U.

U. Kreibig, "Anomalous frequency and temperature dependence of the optical absorption of small gold particles," J. Phys. (Paris) C2, 97 (1977).

Lapotko, D.

Liz-Marz’an, L. M.

L. M. Liz-Marz’an and P. Mulvaney, "Au@SiO2 colloids: effect of temperature on the surface plasmons absorption," New J. Chem. 22, 1285-1288 (1998).
[CrossRef]

Marago, O. M.

F. Borghese, P. Denti, R. Saija, M. A. Iati, and O. M. Marago, "Radiation force and torque on optically trapped linear nanostructures," Phys. Rev. Lett. 100, 163903 (2008).
[CrossRef] [PubMed]

Marston, P. L.

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).

Mazilu, M.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical manipulation of nanoparticles: a review," J. Nanophoton. 2, 021875 (2008).
[CrossRef]

Mulvaney, P.

L. M. Liz-Marz’an and P. Mulvaney, "Au@SiO2 colloids: effect of temperature on the surface plasmons absorption," New J. Chem. 22, 1285-1288 (1998).
[CrossRef]

Neuman, K. C.

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Oddershede, L.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

Oddershede, L. B.

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient optical trapping and visualization of silver nanoparticles," Nano Lett. 8, 1486-1491 (2008).
[CrossRef] [PubMed]

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

S. N. S. Reihani, and L. B. Oddershede, "Optimizing immersion media refractive index improves optical trapping by compensating spherical aberrations," Opt. Lett. 32, 1998-2000 (2007).
[CrossRef] [PubMed]

Perkins, T.

Peterman, E. J. G.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-induced heating in optical traps," Biophys. J. 84, 1308-1316 (2003).
[CrossRef] [PubMed]

Reihani, S. N. S.

Rohrbach, A.

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

Saija, R.

Schmidt, C. F.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-induced heating in optical traps," Biophys. J. 84, 1308-1316 (2003).
[CrossRef] [PubMed]

Schubert, O.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

Selhuber-Unkel, C.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

Seol, Y.

Sindoni, O. I.

Sonnichsen, C.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

Stroud, D.

D. Stroud, "Theory of the intensity-dependent optical activity in dilute composites," J. Appl. Phys. 66, 2585-2588 (1989).
[CrossRef]

Svoboda, K.

Toscano, G.

Zins, I.

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

Biophys. J.

E. J. G. Peterman, F. Gittes, and C. F. Schmidt, "Laser-induced heating in optical traps," Biophys. J. 84, 1308-1316 (2003).
[CrossRef] [PubMed]

J. Appl. Phys.

D. Stroud, "Theory of the intensity-dependent optical activity in dilute composites," J. Appl. Phys. 66, 2585-2588 (1989).
[CrossRef]

J. Chem Phys.

R. H. Doremus, "Erratum: Optical properties of small gold particles," J. Chem Phys. 41, 3259 (1964).
[CrossRef]

J. Chem. Phys.

R. H. Doremus, "Optical properties of small gold particles," J. Chem. Phys. 40, 2389-2396 (1964).
[CrossRef]

J. Nanophoton.

M. Dienerowitz, M. Mazilu, and K. Dholakia, "Optical manipulation of nanoparticles: a review," J. Nanophoton. 2, 021875 (2008).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. (Paris)

U. Kreibig, "Anomalous frequency and temperature dependence of the optical absorption of small gold particles," J. Phys. (Paris) C2, 97 (1977).

Nano Lett.

P. M. Hansen, V. K. Bhatia, N. Harrit, and L. Oddershede, "Expanding the optical trapping range of gold nanoparticles," Nano Lett. 5, 1937-1942 (2005).
[CrossRef] [PubMed]

L. Bosanac, T. Aabo, P. M. Bendix, and L. B. Oddershede, "Efficient optical trapping and visualization of silver nanoparticles," Nano Lett. 8, 1486-1491 (2008).
[CrossRef] [PubMed]

C. Selhuber-Unkel, I. Zins, O. Schubert, C. Sonnichsen, and L. B. Oddershede, "Quantitative optical trapping of single gold nanorods," Nano Lett. 8, 2998-3003 (2008).
[CrossRef] [PubMed]

New J. Chem.

L. M. Liz-Marz’an and P. Mulvaney, "Au@SiO2 colloids: effect of temperature on the surface plasmons absorption," New J. Chem. 22, 1285-1288 (1998).
[CrossRef]

Opt. Express

Opt. Lett.

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. B

P. B. Johnson and R. W. Christy, "Optical constants of the noble metals," Phys. Rev. B 6, 4370-4379 (1972).
[CrossRef]

Phys. Rev. Lett.

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

F. Borghese, P. Denti, R. Saija, M. A. Iati, and O. M. Marago, "Radiation force and torque on optically trapped linear nanostructures," Phys. Rev. Lett. 100, 163903 (2008).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

K. C. Neuman and S. M. Block, "Optical trapping," Rev. Sci. Instrum. 75, 2787-2809 (2004).
[CrossRef]

Other

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

J. D. Jackson, Classical Electrodynamics, 2d ed., (Wiley, New York, 1975).

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

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Figures (7)

Fig. 1.
Fig. 1.

Sketch of the trapping geometry. The particle is located at R O with respect to a cartesian frame of reference whose origin O coincides with the nominal focus F of the lens.

Fig. 2.
Fig. 2.

Extinction cross section of Au (left) and Ag (right) spheres. The refractive index is the one tabulated by Johnson and Christy [19].

Fig. 3.
Fig. 3.

Experimental (from [6]) and calculated stiffness κz /P and κx /P in pN/(nm·W) for gold spheres, both for water immersion (red line and points) and for oil immersion lens (blue lines and points). The points marked by an arrow refer to particles embedded into a steam bubble.

Fig. 4.
Fig. 4.

Experimental (from [7]) and calculated stiffness κz /P and κx /P in pN/(nm·W) for silver spheres, both for water immersion (red lines and points) and for oil immersion lens (blue lines and points). The points marked by an arrow refer to particles embedded into a steam bubble.

Fig. 5.
Fig. 5.

Trapping efficiency for Au spheres with r=25nm (left) and r=77nm (right) at λ=1064nm. The nominal focus of the lens is at z=0. d w≠0 denotes the position of the cover slip for the case of oil immersion objective, and the curve for d w=-2µm is truncated at the position of the cover slip.

Fig. 6.
Fig. 6.

F (Ext) Radz (left) and F (Sca) Radz (right) as a function of z for Au spheres of small (top) and large (bottom) radius

Fig. 7.
Fig. 7.

F (Ext) Radz (left) and F (Sca) Radz (right) as a function of z for Au spheres in a steam bubble.

Tables (3)

Tables Icon

Table 1. Trapping positions of Au and Ag nanospheres. r is the radius of the spheres used in [6, 7], z T is the calculated trapping position, and z E the trapping position determined by the extinction contribution alone. r, z T, and z E are all in nm.

Tables Icon

Table 2. Temperature change with respect to room temperature for the Au and Ag nanospheres used in [6] and [7] at their trapping position. r is the radius (in nm) of the spheres used in [6, 7], σabs (in µm2) is the absorption cross section, and ΔT/P is the change of temperature (in K/W) due to heating by the trapping beam. ΔT has been estimated using the formulas in [8] at 5 nm from the surface of the nanospheres.

Tables Icon

Table 3. Trapping positions of Au and Ag nanospheres embedded into a steam bubble. r is the radius of the spheres used in [6, 7], t the thickness of the steam layer, zT is the calculated trapping position, and zE the trapping position determined by the extinction contribution alone. r, t, zT, and zE are all in nm.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

FRad=r2Ωr̂MdΩ,
TM=18π[n2EE*+BB*12(n2EE*+BB*)I].
FRadζ=FRadζ(Sca)+FRadζ(Ext),
FRadζ(Sca)=2E0216πkv2ReplmplmAlm(p)*Alm(p)illIζlmlm(pp),
FRadζ(Ext)=2E0216πkv2ReplmplmWIlm(p)*Alm(p)illIζlmlm(pp).

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