Angular and position stability of a nanorod trapped in an optical tweezers

Paul B. Bareil; Yunlong Sheng

doi:10.1364/OE.18.026388

Angular and position stability of a nanorod trapped in an optical tweezers

Paul B. Bareil and Yunlong Sheng

Optics Express
Vol. 18,
Issue 25,
pp. 26388-26398
(2010)
•https://doi.org/10.1364/OE.18.026388

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Paul B. Bareil and Yunlong Sheng, "Angular and position stability of a nanorod trapped in an optical tweezers," Opt. Express 18, 26388-26398 (2010)

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Related Topics
Table of Contents Category
- Optical Trapping and Manipulation
Optics & Photonics Topics
?

The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- General (000.0000)
- Laser trapping (140.7010)
- Nanomaterials (160.4236)
- Nanophotonics and photonic crystals (350.4238)
- Optical tweezers or optical manipulation (350.4855)
- Electromagnetic optics (260.2110)

About this Article
History
- Original Manuscript: October 15, 2010
- Revised Manuscript: November 16, 2010
- Manuscript Accepted: November 17, 2010
- Published: December 1, 2010

Accessible

Open Access

Abstract

We analyze the trap stiffness and trapping force potential for a nano-cylinder trapped in the optical tweezers against its axial and lateral shift and tilt associated to the natural Brownian motion. We explain the physical properties of the optical trapping by computing and integrating the radiation stress distribution on the nano-cylinder surfaces using the T-matrix approach. Our computation shows that the force stiffness to the lateral shift is several times higher than that to the axial shift of the nano-cylinder, and lateral torque due to the stress on the side-face is 1-2 orders of magnitude higher than that on the end-faces of a nano-cylinder with the aspect ratio of 2 – 20. The torque due to the stress on the nano-cylinder surface is 2-3 orders of magnitude higher than the spin torque. We explain why a nano-cylinder of low aspect ratio is trapped and aligned normal to the trapping beam axis.

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References

View by:
Article Order
|
Year
|
Author
|
Publication

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[Crossref]
Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]
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(16), 163903 (2008).
[Crossref] [PubMed]
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. Soc. Am. A 9, S196–S203 (2007).
W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of optically trapped nonspherical birefringent particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 021911 (2006).
[Crossref] [PubMed]
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(10), 4594–4602 (1989).
[Crossref]
T. A. Nieminen, H. Rubinsztein-Dunlop, and N. R. Heckenberg, “Multipole expansion of strongly focussed laser beams,” J. Quant. Spectrosc. Radiat. Transf. 79–80, 1005–1017 (2003).
[Crossref]
G. Videen, Light Scattering from a Sphere Near a Plane Interface (Springer, 2000).
C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. 111(19), 8825 (1999).
[Crossref]
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 (2000).
K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]
M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002), p. 307–319.
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(2), 822–830 (2010).
[Crossref] [PubMed]
M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]
M. Rodriguez-Otazo, A. Augier-Calderin, J.-P. Galaup, J.-F. Lamère, and S. Fery-Forgues, “High rotation speed of single molecular microcrystals in an optical trap with elliptically polarized light,” Appl. Opt. 48(14), 2720–2730 (2009).
[Crossref] [PubMed]
F.-W. Sheu, T.-K. Lan, Y.-C. Lin, S. Chen, and C. Ay, “Stable trapping and manually controlled rotation of an asymmetric or birefringent microparticle using dual-mode split-beam optical tweezers,” Opt. Express 18(14), 14724–14729 (2010).
[Crossref] [PubMed]

2010 (2)

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(2), 822–830 (2010).
[Crossref] [PubMed]

F.-W. Sheu, T.-K. Lan, Y.-C. Lin, S. Chen, and C. Ay, “Stable trapping and manually controlled rotation of an asymmetric or birefringent microparticle using dual-mode split-beam optical tweezers,” Opt. Express 18(14), 14724–14729 (2010).
[Crossref] [PubMed]

2009 (1)

M. Rodriguez-Otazo, A. Augier-Calderin, J.-P. Galaup, J.-F. Lamère, and S. Fery-Forgues, “High rotation speed of single molecular microcrystals in an optical trap with elliptically polarized light,” Appl. Opt. 48(14), 2720–2730 (2009).
[Crossref] [PubMed]

2008 (2)

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(16), 163903 (2008).
[Crossref] [PubMed]

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[Crossref]

2007 (2)

Y. Nakayama, P. J. Pauzauskie, A. Radenovic, R. M. Onorato, R. J. Saykally, J. Liphardt, and P. Yang, “Tunable nanowire nonlinear optical probe,” Nature 447(7148), 1098–1101 (2007).
[Crossref] [PubMed]

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. Soc. Am. A 9, S196–S203 (2007).

2006 (1)

W. Singer, T. A. Nieminen, U. J. Gibson, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Orientation of optically trapped nonspherical birefringent particles,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 73(2), 021911 (2006).
[Crossref] [PubMed]

2003 (1)

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

2000 (1)

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 (2000).

1999 (2)

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

C. H. Choi, J. Ivanic, M. S. Gordon, and K. Ruedenberg, “Rapid and stable determination of rotation matrices between spherical harmonics by direct recursion,” J. Chem. Phys. 111(19), 8825 (1999).
[Crossref]

1998 (1)

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]

1989 (1)

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(10), 4594–4602 (1989).
[Crossref]

Alexander, D. R.

Augier-Calderin, A.

Ay, C.

Barton, J. P.

Borghese, F.

Branczyk, A. M.

Camposeo, A.

Chen, S.

Choi, C. H.

Cingolani, R.

Crichton, J. H.

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 (2000).

Denti, P.

Dholakia, G.

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[Crossref]

Dienerowitz, M.

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[Crossref]

Fery-Forgues, S.

Friese, M. E.

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]

Galaup, J.-P.

Gibson, U. J.

Gordon, M. S.

Heckenberg, N. R.

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

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]

Iatì, M. A.

Ivanic, J.

Kawata, S.

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Knöner, G.

Lamère, J.-F.

Lan, T.-K.

Lin, Y.-C.

Liphardt, J.

Loke, V. L. Y.

Maragò, O. M.

Marston, P. L.

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 (2000).

Mazilu, M.

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[Crossref]

Nakayama, Y.

Neves, A. A. R.

Nieminen, T. A.

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

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]

Okamoto, K.

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Onorato, R. M.

Pagliara, S.

Pauzauskie, P. J.

Pisignano, D.

Radenovic, A.

Rodriguez-Otazo, M.

Rubinsztein-Dunlop, H.

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

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]

Ruedenberg, K.

Saija, R.

Saykally, R. J.

Schaub, S. A.

Sheu, F.-W.

Singer, W.

Stilgoe, A. B.

Yang, P.

Appl. Opt. (1)

Electron. J. Differ. Equations Conf. (1)

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 (2000).

J. Appl. Phys. (1)

J. Chem. Phys. (1)

J. Nanophoton. (1)

M. Dienerowitz, M. Mazilu, and G. Dholakia, “Optical manipulation of nanoparticles: a review,” J. Nanophoton. 2(1), 021875 (2008).
[Crossref]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

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

Nature (1)

Opt. Express (2)

Opt. Lett. (1)

M. E. Friese, T. A. Nieminen, N. R. Heckenberg, and H. Rubinsztein-Dunlop, “Optical torque controlled by elliptical polarization,” Opt. Lett. 23(1), 1–3 (1998).
[Crossref]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Phys. Rev. Lett. (2)

K. Okamoto and S. Kawata, “Radiation Force Exerted on Subwavelength Particles near a Nanoaperture,” Phys. Rev. Lett. 83(22), 4534–4537 (1999).
[Crossref]

Other (2)

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles (Cambridge University Press, 2002), p. 307–319.

G. Videen, Light Scattering from a Sphere Near a Plane Interface (Springer, 2000).

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Fig. 1 Nano-cylinder fixed in the spherical coordinate system for calculating forces and torques.

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Fig. 2 Stress distribution on a nano-cylinder of radius R = 50 nm at equilibrium position with (a) Beam along the z-axis; (b) Beam shifted to x = 50nm; (c) Axial force and axial stiffness as a function of position of the focus in the z-axis for radius R = 300 nm; (d) Lateral foce and lateral stiffness as a function of shift distance from the x-axis for a length H = 1 μm.

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Fig. 3 Axial stiffness as a function of length for (a) radius R = 50 – 100 nm; (b) R = 300nm; as a function of (c) radius for length H = 200–1000 nm; and (d) NA for R = 100nm and H = 1μm.

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Fig. 4 Lateral stiffness as a function of length for (a) radius R = 50-100nm; and (b) R = 300nm; as a function of (c) radius for length H = 200-1000 nm; and (d) NA for R = 100nm and H = 1μm.

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Fig. 5 (a) Spin torque τ_z, (b) Lateral torque τ_y. (c) Stress distribution in the situation of (b) with nano-cylinder R = 100 nm and H = 1 μm; (d) Stress distribution on a nano-cylinder aligned along the x-axis with the beam tilted by β = 40° with respect to the nano-cylinder axis, R = 25 nm and H = 100 nm.

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Fig. 6 Lateral force F_x and grandient dF_x /dx for a nano-cylinder tilted by ± 40° for H = 2.8μm and (a) R = 300 nm, (b) R = 70 nm.

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Fig. 7 Lateral torque as a function of tilt angle for a nano-cylinder of radius R = 50 nm.

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Fig. 8 Lateral torque τ_y and its gradient ∂τ_y /∂β as a function of tilt angle for a nano-cylinder of: (a) R = 50 nm and H = 900 nm; (b) R = 25 nm and H = 100 nm.

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Fig. 9 (a) Spin torque and (b) lateral torque as a function of tilt angle for R = 50nm and H = 500 nm to1.1 μm; (c) lateral and spin torques for R = 25nm and H = 100nm.

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Equations (9)

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\begin{array}{l} E_{i n c} (r) = \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} a_{n m} R g M_{n m}^{} (k r) + b_{n m} R g N_{n m}^{} (k r) \\ E_{int} (r) = \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} c_{n m} R g M_{n m}^{} (k r) + d_{n m} R g N_{n m}^{} (k r) \\ E_{s c a t} (r) = \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} p_{n m} M_{n m}^{(1)} (k r) + q_{n m} N_{n m}^{(1)} (k r) \end{array}

\begin{array}{l} [\begin{array}{l} p \\ q \end{array}] = [\begin{matrix} T^{11} & T^{12} \\ T^{21} & T^{22} \end{matrix}] [\begin{matrix} a \\ b \end{matrix}] \\ [\begin{matrix} c \\ d \end{matrix}] = [\begin{matrix} T I^{11} & T I^{12} \\ T I^{21} & T I^{22} \end{matrix}] [\begin{matrix} a \\ b \end{matrix}] \end{array}

\begin{array}{l} E_{i n c / /} = - E_{s c a t / /} + E_{int / /} \\ H_{i n c / /} = - H_{s c a t / /} + H_{int / /} \end{array}

\begin{array}{l} 〈 \vec{F} 〉 = \frac{1}{2} Re ∯_{S} d S \vec{n} • 〈 \vec{T} 〉 \\ 〈 \vec{τ} 〉 = \frac{1}{2} Re ∯_{S} d S \vec{n} • 〈 \vec{T} 〉 \times \vec{r} \end{array}

〈 \vec{F_{z}} 〉 = \frac{ε_{0}}{2 k_{0}} (\begin{array}{l} \frac{2}{(n + 1)} \sqrt{\frac{n (n + 2) (n - m + 1) (n + m + 1)}{(2 n + 1) (2 n + 3)}} Im (\begin{array}{l} {\tilde{p}}_{n m} {\tilde{p}}_{n + 1 m}^{*} + {\tilde{q}}_{n m} {\tilde{q}}_{n + 1 m}^{*} \\ - {\tilde{a}}_{n m} {\tilde{a}}_{n + 1 m}^{*} - {\tilde{b}}_{n m} {\tilde{b}}_{n + 1 m}^{*} \end{array}) \\ - \frac{2 m}{n (n + 1)} Im (i {\tilde{q}}_{n m} {\tilde{p}}_{n m}^{*} - i {\tilde{b}}_{n m} {\tilde{a}}_{n m}^{*}) \end{array})

〈 \vec{F_{x}} 〉 = \frac{ε_{0}}{2 k_{0}} (\begin{array}{l} \frac{1}{(n + 1)} \sqrt{\frac{n (n + 2) (n + m + 1) (n + m + 2)}{(2 n + 1) (2 n + 3)}} Im (\begin{array}{l} {\tilde{p}}_{n m} {\tilde{p}}_{n + 1 m + 1}^{*} + {\tilde{q}}_{n m} {\tilde{q}}_{n + 1 m + 1}^{*} \\ - {\tilde{a}}_{n m} {\tilde{a}}_{n + 1 m + 1}^{*} - {\tilde{b}}_{n m} {\tilde{b}}_{n + 1 m + 1}^{*} \end{array}) \\ + \frac{1}{(n + 1)} \sqrt{\frac{n (n + 2) (n - m) (n - m + 1)}{(2 n + 1) (2 n + 3)}} Im (\begin{array}{l} {\tilde{p}}_{n + 1 m} {\tilde{p}}_{n m + 1}^{*} + {\tilde{q}}_{n + 1 m} {\tilde{q}}_{n m + 1}^{*} \\ - {\tilde{a}}_{n + 1 m} {\tilde{a}}_{n m + 1}^{*} - {\tilde{b}}_{n + 1 m} {\tilde{b}}_{n m + 1}^{*} \end{array}) \\ - \frac{\sqrt{(n - m) (n + m + 1)}}{n (n + 1)} Im (i {\tilde{b}}_{n m} {\tilde{a}}_{n m + 1}^{*} - i {\tilde{p}}_{n m} {\tilde{q}}_{n m + 1}^{*} + i {\tilde{q}}_{n m} {\tilde{p}}_{n m + 1}^{*} - i {\tilde{a}}_{n m} {\tilde{b}}_{n m + 1}^{*}) \end{array})

τ_{Z} = \frac{ε_{0}}{2 k_{0}} \sum_{n = 1}^{N \max} \sum_{m = - n}^{n} m ({| {\tilde{a}}_{n m} |}^{2} + {| {\tilde{b}}_{n m} |}^{2} - {| {\tilde{p}}_{n m} |}^{2} - {| {\tilde{q}}_{n m} |}^{2})

\vec{τ} = \iint_{S} d S σ (ϕ, z) \vec{r} \times \vec{n}

\vec{σ} = \frac{1}{2} (n_{2}^{2} - n_{1}^{2}) (\frac{n_{1}^{2}}{n_{2}^{2}} E_{1 n}^{2} + E_{1 t}^{2}) \overset{⇀}{n}

Abstract

References

2010 (2)

2009 (1)

2008 (2)

2007 (2)

2006 (1)

2003 (1)

2000 (1)

1999 (2)

1998 (1)

1989 (1)

Alexander, D. R.

Augier-Calderin, A.

Ay, C.

Barton, J. P.

Borghese, F.

Branczyk, A. M.

Camposeo, A.

Chen, S.

Choi, C. H.

Cingolani, R.

Crichton, J. H.

Denti, P.

Dholakia, G.

Dienerowitz, M.

Fery-Forgues, S.

Friese, M. E.

Galaup, J.-P.

Gibson, U. J.

Gordon, M. S.

Heckenberg, N. R.

Iatì, M. A.

Ivanic, J.

Kawata, S.

Knöner, G.

Lamère, J.-F.

Lan, T.-K.

Lin, Y.-C.

Liphardt, J.

Loke, V. L. Y.

Maragò, O. M.

Marston, P. L.

Mazilu, M.

Nakayama, Y.

Neves, A. A. R.

Nieminen, T. A.

Okamoto, K.

Onorato, R. M.

Pagliara, S.

Pauzauskie, P. J.

Pisignano, D.

Radenovic, A.

Rodriguez-Otazo, M.

Rubinsztein-Dunlop, H.

Ruedenberg, K.

Saija, R.

Saykally, R. J.

Schaub, S. A.

Sheu, F.-W.

Singer, W.

Stilgoe, A. B.

Yang, P.

Appl. Opt. (1)

Electron. J. Differ. Equations Conf. (1)

J. Appl. Phys. (1)

J. Chem. Phys. (1)

J. Nanophoton. (1)

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transf. (1)

Nature (1)

Opt. Express (2)

Opt. Lett. (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (1)

Phys. Rev. Lett. (2)

Other (2)

Cited By

Figures (9)

Equations (9)

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