Abstract

Within the Rayleigh approximation, we investigate the behavior of an individual ellipsoidal metal nanorod that is optically confined in three dimensions using a single focused laser beam. We focus on the description of the optical torque and optical force acting upon the nanorod placed into a linearly polarized Gaussian beam (scalar description of the electric field) or a strongly focused beam (vector field description). The study comprises the influence of the trapping laser wavelength, the angular aperture of focusing optics, the orientation of the ellipsoidal nanorod, and the aspect ratio of its principal axes. The results reveal a significantly different behavior of the nanorod if the trapping wavelength is longer or shorter than the wavelength corresponding to the longitudinal plasmon resonance mode. Published experimental observations are compared with our theoretical predictions with satisfactory results.

© 2012 Optical Society of America

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2011

P. V. Ruijgrok, N. R. Verhart, P. Zijlstra, A. L. Tchebotareva, and M. Orrit, “Brownian fluctuations and heating of an optically aligned gold nanorod,” Phys. Rev. Lett. 107, 037401 (2011).
[CrossRef]

2010

F. C. Cheong and D. G. Grier, “Rotational and translational diffusion of copper oxide nanorods measured with holographic video microscopy,” Opt. Express 18, 6555–6562 (2010).
[CrossRef]

A. A. R. Neves, A. Camposeo, S. Pagliara, R. Saija, F. Borghese, P. Denti, M. A. Iatì, R. Cingolani, O. M. Maragò, and D. Pisignano, “Rotational dynamics of optically trapped nanofibers,” Opt. Express 18, 822–830 (2010).
[CrossRef]

L. Tong, V. Miljkovic`, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
[CrossRef]

S. H. Simpson and S. Hanna, “First-order nonconservative motion of optically trapped nonspherical particles,” Phys. Rev. E 82, 031141 (2010).
[CrossRef]

G. Baffou, R. Quidant, and F. J. García de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano 4, 709–716 (2010).
[CrossRef]

M. J. Nasse and J. Woehl, “Realistic modeling of the illumination point spread function in confocal scanning optical microscopy,” J. Opt. Soc. Am. A 27, 295–302 (2010).
[CrossRef]

2009

S. Merabia, P. Keblinski, L. Joly, L. J. Lewis, and J.-L. Barrat, “Critical heat flux around strongly heated nanoparticles,” Phys. Rev. E 79, 021404 (2009).
[CrossRef]

P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragò, “Rotation detection in light-driven nanorotors,” ACS Nano 3, 3077–3084 (2009).
[CrossRef]

2008

A. Jonáš and P. Zemánek, “Light at work: the use of optical forces for particle manipulation, sorting, and analysis,” Electophoresis 29, 4813–4851 (2008).

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

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

F. Borghese, P. Denti, R. Saija, M. A. Iatì, and O. M. Maragò, “Radiation torque and force on optically trapped linear nanostructures,” Phys. Rev. Lett. 100, 163903 (2008).
[CrossRef]

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

O. Ekici, R. K. Harrison, N. J. Durr, D. S. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501 (2008).
[CrossRef]

F. Xu, J. A. Lock, G. Gouesbet, and C. Tropea, “Radiation torque exerted on a spheroid: analytical solution,” Phys. Rev. A 78, 013843 (2008).
[CrossRef]

2007

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]

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

F. Borghese, P. Denti, R. Saija, and M. A. Iatì, “Optical trapping of nonspherical particles in the T-matrix formalism,” Opt. Express 15, 11984–11998 (2007).
[CrossRef]

2006

M. Pelton, M. Liu, H. Kim, G. Smith, P. Guyot-Sionnest, and N. Scherer, “Optical trapping and alignment of single gold nanorods by using plasmon resonances,” Opt. Lett. 31, 2075–2077 (2006).
[CrossRef]

B. Chithrani, A. Ghazani, and W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6, 662–668 (2006).
[CrossRef]

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Y. Seol, A. E. Carpenter, and T. T. Perkins, “Gold nanoparticles: enhanced optical trapping and sensitivity coupled with significant heating,” Opt. Lett. 31, 2429–2431 (2006).
[CrossRef]

2005

J. Pérez-Juste, I. Pastoriza-Santos, L. Liz-Marzán, and P. Mulvaney, “Gold nanorods: synthesis, characterization and applications,” Coord. Chem. Rev. 249, 1870–1901 (2005).
[CrossRef]

R. Fan, R. Karnik, M. Yue, D. Li, A. Majumdar, and P. Yang, “DNA translocation in inorganic nanotubes,” Nano Lett. 5, 1633–1637 (2005).
[CrossRef]

W. Shelton, K. Bonin, and T. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E 71, 036204 (2005).
[CrossRef]

C. Sönnichsen and A. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301–304 (2005).
[CrossRef]

C. Rockstuhl and H. Herzig, “Calculation of the torque on dielectric elliptical cylinders,” J. Opt. Soc. Am. A 22, 109–116 (2005).
[CrossRef]

2004

C. Rockstuhl and H. Herzig, “Rigorous diffraction theory applied to the analysis of the optical force on elliptical nano- and micro-cylinders,” J. Opt. A 6, 921–931 (2004).
[CrossRef]

J. Plewa, E. Tanner, D. Mueth, and D. Grier, “Processing carbon nanotubes with holographic optical tweezers,” Opt. Express 12, 1978–1981 (2004).
[CrossRef]

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

2003

Z. Bryant, M. Stone, J. Gore, S. Smith, N. Cozzarelli, and C. Bustamante, “Structural transitions and elasticity from torque measurements on DNA,” Nature 424, 338–341 (2003).
[CrossRef]

D. Higgins and B. Luther, “Watching molecules reorient in liquid crystal droplets with multiphoton-excited fluorescence microscopy,” J. Chem. Phys. 119, 3935–3942 (2003).
[CrossRef]

J. R. Arias-González and M. Nieto-Vesperinas, “Optical forces on small particles: attractive and repulsive nature and plasmon-resonance conditions,” J. Opt. Soc. Am. A 20, 1201–1209 (2003).
[CrossRef]

2002

2001

R. Yasuda, H. Noji, M. Yoshida, K. Kinosita, and H. Itoh, “Resolution of distinct rotational substeps by submillisecond kinetic analysis of F-1-ATPase,” Nature 410, 898–904 (2001).
[CrossRef]

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468–471 (2001).
[CrossRef]

2000

1996

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

1995

1994

1989

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

1959

E. Wolf, “Electromagnetic diffraction in optical systems. 1. An integral representation of the image field,” Proc. R. Soc. A 253, 349–357 (1959).
[CrossRef]

B. Richards and E. Wolf, “Electromagnetic diffraction in optical systems. 2. Structure of the image field in an aplanatic system,” Proc. R. Soc. A 253, 358–379 (1959).
[CrossRef]

1940

H. A. Kramers, “Brownian motion in the field of force and the diffusion model of chemical reactions,” Physica (Amsterdam) 7, 284–304 (1940).
[CrossRef]

Alexander, D. R.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Alivisatos, A.

C. Sönnichsen and A. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301–304 (2005).
[CrossRef]

Arias-González, J. R.

Asakura, T.

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

Baffou, G.

G. Baffou, R. Quidant, and F. J. García de Abajo, “Nanoscale control of optical heating in complex plasmonic systems,” ACS Nano 4, 709–716 (2010).
[CrossRef]

Barrat, J.-L.

S. Merabia, P. Keblinski, L. Joly, L. J. Lewis, and J.-L. Barrat, “Critical heat flux around strongly heated nanoparticles,” Phys. Rev. E 79, 021404 (2009).
[CrossRef]

Barton, J. P.

J. P. Barton and D. R. Alexander, “Fifth-order corrected electromagnetic field components for a fundamental Gaussian beam,” J. Appl. Phys. 66, 2800–2802 (1989).
[CrossRef]

J. P. Barton, D. R. Alexander, and S. A. Schaub, “Theoretical determination of net radiation force and torque for a spherical particle illuminated by a focused laser beam,” J. Appl. Phys. 66, 4594–4602 (1989).
[CrossRef]

Benito, D. C.

S. H. Simpson, D. C. Benito, and S. Hanna, “Polarization-induced torque in optical traps,” Phys. Rev. A 76, 043408 (2007).
[CrossRef]

Ben-Yakar, A.

O. Ekici, R. K. Harrison, N. J. Durr, D. S. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501 (2008).
[CrossRef]

Billaudeau, C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

Bishop, A.

T. Nieminen, H. Rubinsztein-Dunlop, N. Heckenberg, and A. Bishop, “Numerical modelling of optical trapping,” Comput. Phys. Commun. 142, 468–471 (2001).
[CrossRef]

Block, S. M.

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

K. Svoboda and S. M. Block, “Optical trapping of metallic Rayleigh particles,” Opt. Lett. 19, 930–932 (1994).
[CrossRef]

Bonaccorso, F.

P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragò, “Rotation detection in light-driven nanorotors,” ACS Nano 3, 3077–3084 (2009).
[CrossRef]

Bonin, K.

W. Shelton, K. Bonin, and T. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E 71, 036204 (2005).
[CrossRef]

K. Bonin, B. Kourmanov, and T. Walker, “Light torque nanocontrol, nanomotors and nanorockers,” Opt. Express 10, 984–989 (2002).

Booker, G. R.

Borghese, F.

Bottka, S.

L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
[CrossRef]

Bryant, Z.

Z. Bryant, M. Stone, J. Gore, S. Smith, N. Cozzarelli, and C. Bustamante, “Structural transitions and elasticity from torque measurements on DNA,” Nature 424, 338–341 (2003).
[CrossRef]

Bustamante, C.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Z. Bryant, M. Stone, J. Gore, S. Smith, N. Cozzarelli, and C. Bustamante, “Structural transitions and elasticity from torque measurements on DNA,” Nature 424, 338–341 (2003).
[CrossRef]

Calogero, G.

P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragò, “Rotation detection in light-driven nanorotors,” ACS Nano 3, 3077–3084 (2009).
[CrossRef]

Camposeo, A.

Carpenter, A. E.

Chan, W.

B. Chithrani, A. Ghazani, and W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6, 662–668 (2006).
[CrossRef]

Chaumet, P.

Chaumet, P. C.

P. C. Chaumet and C. Billaudeau, “Coupled dipole method to compute optical torque: application to a micropropeller,” J. Appl. Phys. 101, 023106 (2007).
[CrossRef]

Chemla, Y. R.

J. R. Moffitt, Y. R. Chemla, S. B. Smith, and C. Bustamante, “Recent advances in optical tweezers,” Annu. Rev. Biochem. 77, 205–228 (2008).
[CrossRef]

Cheong, F. C.

Chithrani, B.

B. Chithrani, A. Ghazani, and W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6, 662–668 (2006).
[CrossRef]

Cingolani, R.

Cozzarelli, N.

Z. Bryant, M. Stone, J. Gore, S. Smith, N. Cozzarelli, and C. Bustamante, “Structural transitions and elasticity from torque measurements on DNA,” Nature 424, 338–341 (2003).
[CrossRef]

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]

Durr, N. J.

O. Ekici, R. K. Harrison, N. J. Durr, D. S. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501 (2008).
[CrossRef]

Ekici, O.

O. Ekici, R. K. Harrison, N. J. Durr, D. S. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501 (2008).
[CrossRef]

Eversole, D. S.

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L. Oroszi, P. Galajda, H. Kirei, S. Bottka, and P. Ormos, “Direct measurement of torque in an optical trap and its application to double-strand DNA,” Phys. Rev. Lett. 97, 058301 (2006).
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P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragò, “Rotation detection in light-driven nanorotors,” ACS Nano 3, 3077–3084 (2009).
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ACS Nano

P. H. Jones, F. Palmisano, F. Bonaccorso, P. G. Gucciardi, G. Calogero, A. C. Ferrari, and O. M. Maragò, “Rotation detection in light-driven nanorotors,” ACS Nano 3, 3077–3084 (2009).
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J. Phys. D

O. Ekici, R. K. Harrison, N. J. Durr, D. S. Eversole, M. Lee, and A. Ben-Yakar, “Thermal analysis of gold nanorods heated with femtosecond laser pulses,” J. Phys. D 41, 185501 (2008).
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Nano Lett.

L. Tong, V. Miljkovic`, and M. Käll, “Alignment, rotation, and spinning of single plasmonic nanoparticles and nanowires using polarization dependent optical forces,” Nano Lett. 10, 268–273 (2010).
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B. Chithrani, A. Ghazani, and W. Chan, “Determining the size and shape dependence of gold nanoparticle uptake into mammalian cells,” Nano Lett. 6, 662–668 (2006).
[CrossRef]

R. Fan, R. Karnik, M. Yue, D. Li, A. Majumdar, and P. Yang, “DNA translocation in inorganic nanotubes,” Nano Lett. 5, 1633–1637 (2005).
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C. Sönnichsen and A. Alivisatos, “Gold nanorods as novel nonbleaching plasmon-based orientation sensors for polarized single-particle microscopy,” Nano Lett. 5, 301–304 (2005).
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Phys. Rev. E

W. Shelton, K. Bonin, and T. Walker, “Nonlinear motion of optically torqued nanorods,” Phys. Rev. E 71, 036204 (2005).
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Figures (12)

Fig. 1.
Fig. 1.

Comparison of three different descriptions of the incident laser beam. Normalized profiles of the absolute values of the electric field intensity in the incident beam along x (top left), y (bottom left), and z (top right) axis are shown for the vector description [VD—solid, Eqs. (7)–(9)], the scalar description [SD—dot, Eq. (2)], and the fifth-order description [G5—dash, Eq. (27)]. Beam waist w0 (corresponding to the radial distance along y-axis where |E|=|E0|/e, which is denoted by the dash-dotted line) is identical for all considered approximations, and two different angular apertures of the VD beam, Θ=30° (green) and Θ=70° (red), are studied. The electric field is normalized to its absolute value at the beam focus |E0|. The negative gradient (Dz) of the longitudinal field is plotted along the optical axis z (bottom right). The following quantities and values are used in the calculations: vacuum laser wavelength λ0=500nm, nm=1.33.

Fig. 2.
Fig. 2.

Orientation of the nanorod (NR) in the initial coordinate system xyz and the rotated coordinate system xyz. The NR long axis is parallel to x-axis. The axis x denotes the projection of the axis x to the plane xy.

Fig. 3.
Fig. 3.

Spectral dependence of the polarizabilities αL and αS described by Eq. (17). The bottom row presents the same data as the top row with a different scaling of the vertical axis. Spectral regions where (αLαS) does not change its sign are denoted by the horizontal green arrows. In these regions, the NR is aligned parallel to the beam polarization by its long axis (marked as ) or short axis (marked as ). We consider ellipsoidal Au, Ag, and SiO2 NRs with long axis lL=40nm and short axes lS=10nm immersed in water. The values of the refractive indices are taken from [44).

Fig. 4.
Fig. 4.

Critical parameters under which the NR reorients. The ratio of lengths of the short lS and long lL-axis of the ellipsoidal NR determines the wavelength λ0r at which (αLαS=0) and the NR reorients (black dots). For comparison, the combinations of lS/lL and λ0r for which the imaginary part of the NR polarizability αL and αS reaches maximum are denoted by a red × and a green +, respectively. In the case of metal NRs, they correspond to the plasmon resonances along the long or short NR axis, respectively. We again consider Au, Ag, and SiO2 NRs immersed in water and having long axis lengths equal to lL=40nm.

Fig. 5.
Fig. 5.

Longitudinal equilibrium positions z0 of Au NR as a function of the vacuum trapping wavelength λ0 and the angular aperture Θ. The scalar (SD, left) and vector (VD, right) descriptions of the incident Gaussian beam are compared; contours of identical z0 are in logarithmic scale. The vertical green line divides the range of wavelengths into two regions where the NR is oriented by its long axis parallel () or perpendicular () to the beam polarization (along x-axis). No equilibrium positions z0 exist for parameters falling into the white regions. The intensity of the electric field at the beam focus E0 was equal to E0=19MV/m; the Au NR of the dimensions lL=40nm and lS=10nm was immersed in water.

Fig. 6.
Fig. 6.

The work Uz needed to free the Au NR in longitudinal direction as a function of the vacuum trapping wavelength λ0 and the angular aperture Θ. The same conditions and symbols are used as in Fig. 5.

Fig. 7.
Fig. 7.

The work needed to transfer the NR from the optical axis to infinity in the lateral direction as a function of the vacuum trapping wavelength λ0 and the angular aperture Θ. The optical trap is laterally symmetrical in the scalar description (SD) and, therefore, the work is denoted as Ur and expressed by Eq. (42). However, in the vector description, the work generally differs for the NR transport along x and y-axis, denoted as Ux and Uy, respectively. In the case of Ux, two orientations of the NR must be considered, along y-axis (y) and z-axis (z). The same conditions and symbols are used as in Fig. 5.

Fig. 8.
Fig. 8.

The work UR needed to reorient the NR from the equilibrium position by π/2 as a function of the vacuum trapping wavelength λ0 and the angular aperture Θ. The same conditions and symbols are used as in Fig. 5.

Fig. 9.
Fig. 9.

Dependence of the lateral (Kr, Ky) and longitudinal (Kz) stiffnesses on the deflection ψ of the NR from its equilibrium alignment. The NR is deflected around the optical axis z by the azimuthal angle ψ, while the polar angle is kept fixed at θ=0. Results of the scalar (left column, SD) and vector (right column, VD) descriptions of the incident Gaussian beam are compared for two vacuum wavelengths λ0=1064nm (solid curve) and λ0=700nm (dashed curve) corresponding to the NR alignment parallel () and perpendicular () to the beam polarization (along x-axis), respectively. The stiffnesses are normalized to the unit incident power and calculated for Au NR (lL=40nm, lS=10nm) placed on the optical axis at its original stable longitudinal position z0 (if no deflection is considered). The angular aperture is set to Θ=50°.

Fig. 10.
Fig. 10.

Left: spectral dependence of the imaginary part of the polarizability α1 for a gold nanosphere [R=16nm, dashed curve, Eq. (57)] and a gold ellipsoidal nanorod [lL=40nm, lS=10nm, solid curve, Eq. (58)] of the same volumes. Right: spectral dependence of the temperature increase for the same objects as in the left figure. The nanoparticles are immersed in water, and the electric field intensity at the beam focus is E0=19MV/m.

Fig. 11.
Fig. 11.

Temperature increase of Au NR (lL=40nm, lS=10nm) illuminated at the wavelength λ0=850nm under different angles θ. The NR is immersed in water; the electric field at the beam focus is E0=19MV/m. These results were obtained by finite elements methods using Comsol Multiphysics software.

Fig. 12.
Fig. 12.

Comparison between the experimental trap stiffness measurements [23] and the theoretical predictions based on the vector description of the incident beam if the NRs are trapped on the optical axis at z0. The horizontal axis gives the mean value of the NRs radii R=lS/2; both lS and lL of compared NRs are taken from Table 1 in [23]. The left column demonstrates the influence of the spread of the NR sizes on the stiffnesses. Here, ⋄ denotes the theoretical results corresponding to the average measured values of the NR size and Θ=50°; ▿ and ▵ correspond to the smallest and biggest NRs within the experimental error reported in [23], respectively. The right column shows the dependence of the stiffness on the angular aperture Θ (⋄ versus ⊳) and deflection of the NR from its equilibrium orientation corresponding to the reorientation work UR=2kT (⋄ versus ⊲).

Tables (1)

Tables Icon

Table 1. Physical Conditions for the Stable Alignment of the Nanorod Corresponding to its Parallel and Perpendicular Orientation with Respect to the Electric Field E of the Incident Beam

Equations (58)

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NA=nmsinΘ.
Ex(x,y,z)=E0w0w(z)exp((x2+y2)w2(z))exp(ikz+ikx2+y22R(z)iζ(z)),
Ey(x,y,z)=0,Ez(x,y,z)=0,
|Ex(w0,0,0)|=|Ex(0,w0,0)|=|Ex(0,0,0)|/e.
E(0,y,z)=[ExG5(0,y,z),0,0],
E(x,0,z)=[ExG5(x,0,z),0,EzG5(x,0,z)].
Ex(x,y,z)=i2k(I0+I2x2y2x2+y2),
Ey(x,y,z)=ikI2xyx2+y2,
Ez(x,y,z)=kI1xx2+y2,
I0(r,z,Θ)=0ΘA(α)sinα(1+cosα)J0(krsinα)exp(ikzcosα)dα,
I1(r,z,Θ)=0ΘA(α)sin2αJ1(krsinα)exp(ikzcosα)dα,
I2(r,z,Θ)=0ΘA(α)sinα(1cosα)J2(krsinα)exp(ikzcosα)dα,
A(α)=A0cosα.
Ex(0,0,0)=k15A0(83cos5/2Θ5cos3/2Θ).
Fξ=12R{p*·ξE},
Mξ=12R{p*×E}ξ,
αj=αj01ik3αj06πϵ,αj0=ϵ0nm2VLj+1m21,m=npnm,
L1=0abcds2(s+a2)3/2(s+b2)1/2(s+c2)1/2,
α^=|αL+iαL000αS+iαS000αS+iαS|,
R^=|cosθ0sinθ010sinθ0cosθ||cosψsinψ0sinψcosψ0001|,
A^=R^1α^R^,
p=A^(ψ,θ)E(x,y,z).
Fx(x,y,z,ψ,θ)=E02zR2kx2(zR2+z2)2exp[kzR(x2+y2)zR2+z2][a(ψ,θ)zRa(ψ,θ)z],
Fy(x,y,z,ψ,θ)=E02zR2ky2(zR2+z2)2exp[kzR(x2+y2)zR2+z2][a(ψ,θ)zRa(ψ,θ)z],
Fz(x,y,z,ψ,θ)=E02zR24(zR2+z2)3exp[kzR(x2+y2)zR2+z2]×{2ka(ψ,θ)z4+2a(ψ,θ)z3+(kx2+ky24kzR2+2zR)a(ψ,θ)z2[2kzR(x2+y2)2zR2]a(ψ,θ)z[kzR2(x2+y2)+2kzR42zR3]a(ψ,θ)};
Mx(x,y,z,ψ,θ)=0,
My(x,y,z,ψ,θ)=E02zR24(zR2+z2)exp[kzR(x2+y2)zR2+z2](αLαS)sin2θcosψ,
Mz(x,y,z,ψ,θ)=E02zR24(zR2+z2)exp[kzR(x2+y2)zR2+z2](αLαS)sin2ψcos2θ,
a(ψ,θ)=αS+(αLαS)cos2θcos2ψ,
a(ψ,θ)=αS+(αLαS)cos2θcos2ψ.
Fx(0,0,z,ψ,θ)=0,Fy(0,0,z,ψ,θ)=0,Mx(0,0,z,ψ,θ)=0,
Fz(0,0,z,ψ,θ)=E02zR22(zR2+z2)2[a(ψ,θ)k(zR2+z2)a(ψ,θ)za(ψ,θ)zR];
My(0,0,z,ψ,θ)=E02zR24(zR2+z2)(αLαS)sin2θcosψ,
Mz(0,0,z,ψ,θ)=E02zR24(zR2+z2)(αLαS)sin2ψcos2θ.
Fx(0,0,z,ψ,θ)=0,Fy(0,0,z,ψ,θ)=0,Mx(0,0,z,ψ,θ)=0,
Fz(0,0,z,ψ,θ)=12R{[a(ψ,θ)+ia(ψ,θ)]*Ex(0,0,z)*zEx(0,0,z)};
My(0,0,z,ψ,θ)=14|Ex(0,0,z)|2(αLαS)sin2θcosψ,
Mz(0,0,z,ψ,θ)=14|Ex(0,0,z)|2(αLαS)sin2ψcos2θ.
z0=aD2ka,z0=a+D2ka,D=(a)2+4(a)2kzR(1kzR).
Uz=z0z0Fz(0,0,z)dz>kBT.
Uz=14E02[Da(2kzR1)arctanDa(2kzR1)].
Ur=0Fr(r,z)dr=E02zR(azRaz)4(zR2+z2)=KrzR2+z22kzR,
URψ=ψ0ψ0+δMz(0,0,z0,ψ,0)dψ,URθ=θ0θ0+δMy(0,0,z0,0,θ)dθ.
UR=14|Ex(0,0,z0)|2|αLαS|sin2δ.
Kr=E02kzR2azRaz2(zR2+z2)2.
Kx(z)=xFx(x,y,z)|y=0x=0=132k4R{axH1H3*4azH2H2*},
Ky(z)=yFy(x,y,z)|y=0x=0=132k4R{axH1H4*},
Kz(z)=zFz(x,y,z)|y=0x=0=18k4R{ax(H1H6*H5H5*)},
H1(z)=0ΘA(α)sinα(1+cosα)exp(ikzcosα)dα,
H2(z)=0ΘA(α)sin3αexp(ikzcosα)dα,
H3(z)=0ΘA(α)sin3α(1+3cosα)exp(ikzcosα)dα,
H4(z)=0ΘA(α)sin3α(3+cosα)exp(ikzcosα)dα,
H5(z)=0ΘA(α)sinαcosα(1+cosα)exp(ikzcosα)dα,
H6(z)=0ΘA(α)sinαcos2α(1+cosα)exp(ikzcosα)dα.
T(r)=T0+ΔTRr,
ΔT=I(r0)σabs4πRβκ,
α1=m21m2+2,
α1=m213L(m21)+3,

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