Abstract

We explore a technique which we term light-assisted templated self-assembly. We calculate the optical forces on colloidal particles over a photonic crystal slab. We show that exciting a guided resonance mode of the slab yields a resonantly-enhanced, attractive optical force. We calculate the lateral optical forces above the slab and predict that stably trapped periodic patterns of particles are dependent on wavelength and polarization. Tuning the wavelength or polarization of the light source may thus allow the formation and reconfiguration of patterns. We expect that this technique may be used to design all-optically reconfigurable photonic devices.

© 2011 OSA

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  7. K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).
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2010 (3)

B. K. Wilson, T. Mentele, S. Bachar, E. Knouf, A. Bendoraite, M. Tewari, S. H. Pun, and L. Y. Lin, “Nanostructure-enhanced laser tweezers for efficient trapping and alignment of particles,” Opt. Express 18(15), 16005–16013 (2010).
[CrossRef] [PubMed]

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82(2), 1767–1791 (2010).
[CrossRef]

2009 (4)

J. H. Lee, Q. Wu, and W. Park, “Metal nanocluster metamaterial fabricated by the colloidal self-assembly,” Opt. Lett. 34(4), 443–445 (2009).
[CrossRef] [PubMed]

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of gold nanoparticles with surface plasmon polaritons: evidence of enhanced optical force from near-field coupling between gold particle and gold film,” Nano Lett. 9(7), 2623–2629 (2009).
[CrossRef] [PubMed]

M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[CrossRef]

2008 (4)

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[CrossRef] [PubMed]

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[CrossRef]

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

2006 (4)

M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl. Phys. Lett. 89(25), 253114 (2006).
[CrossRef]

M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl. Phys. Lett. 89(25), 253114 (2006).
[CrossRef]

A. Rahmani and P. C. Chaumet, “Optical trapping near a photonic crystal,” Opt. Express 14(13), 6353–6358 (2006).
[CrossRef] [PubMed]

P. J. Reece, V. Garces-Chavez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88(22), 221116 (2006).
[CrossRef]

2003 (1)

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

2002 (2)

G. M. Whitesides and B. Grzybowski, “Self-assembly at all scales,” Science 295(5564), 2418–2421 (2002).
[CrossRef] [PubMed]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[CrossRef]

2001 (2)

Y. Xia, B. Gates, and Z.-Y. Li, “Self-assembly approaches to three-dimensional photonic crystals,” Adv. Mater. (Deerfield Beach Fla.) 13(6), 409–413 (2001).
[CrossRef]

Y. Yin, Y. Lu, B. Gates, and Y. Xia, “Template-assisted self-assembly: a practical route to complex aggregates of monodispersed colloids with well-defined sizes, shapes, and structures,” J. Am. Chem. Soc. 123(36), 8718–8729 (2001).
[CrossRef] [PubMed]

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]

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60(4), 2363–2374 (1999).
[CrossRef]

1997 (2)

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Optical bistability in colloidal crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3613–3618 (1997).
[CrossRef]

A. van Blaaderen, R. Ruel, and P. Wiltzius, “Tempate-directed colloidal crystallization,” Nature 385(6614), 321–324 (1997).
[CrossRef]

1987 (1)

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

1986 (1)

Antonoyiannakis, M. I.

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60(4), 2363–2374 (1999).
[CrossRef]

Ashkin, A.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

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(5), 288–292 (1986).
[CrossRef] [PubMed]

Bachar, S.

Barth, M.

M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl. Phys. Lett. 89(25), 253114 (2006).
[CrossRef]

M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl. Phys. Lett. 89(25), 253114 (2006).
[CrossRef]

Bendoraite, A.

Benson, O.

M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl. Phys. Lett. 89(25), 253114 (2006).
[CrossRef]

M. Barth and O. Benson, “Manipulation of dielectric particles using photonic crystal cavities,” Appl. Phys. Lett. 89(25), 253114 (2006).
[CrossRef]

Bermel, P.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Bjorkholm, J. E.

Chaumet, P. C.

Chu, S.

Crozier, K. B.

K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of gold nanoparticles with surface plasmon polaritons: evidence of enhanced optical force from near-field coupling between gold particle and gold film,” Nano Lett. 9(7), 2623–2629 (2009).
[CrossRef] [PubMed]

Dholakia, K.

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82(2), 1767–1791 (2010).
[CrossRef]

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).

P. J. Reece, V. Garces-Chavez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88(22), 221116 (2006).
[CrossRef]

Dickinson, M. R.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, and T. Yamane, “Optical trapping and manipulation of single cells using infrared laser beams,” Nature 330(6150), 769–771 (1987).
[CrossRef] [PubMed]

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(5), 288–292 (1986).
[CrossRef] [PubMed]

Eftekhari, F.

M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[CrossRef]

Erickson, D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Fan, S.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[CrossRef]

Fejer, M. M.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Garces-Chavez, V.

P. J. Reece, V. Garces-Chavez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88(22), 221116 (2006).
[CrossRef]

Gates, B.

Y. Xia, B. Gates, and Z.-Y. Li, “Self-assembly approaches to three-dimensional photonic crystals,” Adv. Mater. (Deerfield Beach Fla.) 13(6), 409–413 (2001).
[CrossRef]

Y. Yin, Y. Lu, B. Gates, and Y. Xia, “Template-assisted self-assembly: a practical route to complex aggregates of monodispersed colloids with well-defined sizes, shapes, and structures,” J. Am. Chem. Soc. 123(36), 8718–8729 (2001).
[CrossRef] [PubMed]

Girard, C.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[CrossRef] [PubMed]

Gordon, R.

M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[CrossRef]

Grier, D. G.

D. G. Grier, “A revolution in optical manipulation,” Nature 424(6950), 810–816 (2003).
[CrossRef] [PubMed]

Grigorenko, A. N.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[CrossRef]

Grzybowski, B.

G. M. Whitesides and B. Grzybowski, “Self-assembly at all scales,” Science 295(5564), 2418–2421 (2002).
[CrossRef] [PubMed]

Harris, J. S.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Huo, Y.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Ibanescu, M.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Joannopoulos, J. D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

S. Fan and J. D. Joannopoulos, “Analysis of guided resonances in photonic crystal slabs,” Phys. Rev. B 65(23), 235112 (2002).
[CrossRef]

Johnson, S. G.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Juan, M. L.

M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[CrossRef]

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]

Klug, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Knouf, E.

Lee, J. H.

Lee, W. M.

K. Dholakia and W. M. Lee, “Optical trapping takes shape: the use of structured light fields,” Adv. At. Mol. Opt. Phys. 56, 261–337 (2008).

Li, Q.

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Optical bistability in colloidal crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3613–3618 (1997).
[CrossRef]

Li, Z.-Y.

Y. Xia, B. Gates, and Z.-Y. Li, “Self-assembly approaches to three-dimensional photonic crystals,” Adv. Mater. (Deerfield Beach Fla.) 13(6), 409–413 (2001).
[CrossRef]

Lidorikis, E.

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Optical bistability in colloidal crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3613–3618 (1997).
[CrossRef]

Lin, L. Y.

Lipson, M.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

Lu, Y.

Y. Yin, Y. Lu, B. Gates, and Y. Xia, “Template-assisted self-assembly: a practical route to complex aggregates of monodispersed colloids with well-defined sizes, shapes, and structures,” J. Am. Chem. Soc. 123(36), 8718–8729 (2001).
[CrossRef] [PubMed]

Mentele, T.

Moore, S. D.

A. H. J. Yang, S. D. Moore, B. S. Schmidt, M. Klug, M. Lipson, and D. Erickson, “Optical manipulation of nanoparticles and biomolecules in sub-wavelength slot waveguides,” Nature 457(7225), 71–75 (2009).
[CrossRef] [PubMed]

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]

Oskooi, A. F.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Pan, J.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Pang, Y.

M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[CrossRef]

Park, W.

Pendry, J. B.

M. I. Antonoyiannakis and J. B. Pendry, “Electromagnetic forces in photonic crystals,” Phys. Rev. B 60(4), 2363–2374 (1999).
[CrossRef]

Petrov, D.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[CrossRef] [PubMed]

Povinelli, M. L.

J. Pan, Y. Huo, K. Yamanaka, S. Sandhu, L. Scaccabarozzi, R. Timp, M. L. Povinelli, S. Fan, M. M. Fejer, and J. S. Harris, “Aligning microcavity resonances in silicon photonic-crystal slabs using laser-pumped thermal tuning,” Appl. Phys. Lett. 92(10), 103114 (2008).
[CrossRef]

Pun, S. H.

Quidant, R.

M. L. Juan, R. Gordon, Y. Pang, F. Eftekhari, and R. Quidant, “Self-induced back-action optical trapping of dielectric nanoparticles,” Nat. Phys. 5(12), 915–919 (2009).
[CrossRef]

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[CrossRef] [PubMed]

Rahmani, A.

Reece, P. J.

P. J. Reece, V. Garces-Chavez, and K. Dholakia, “Near-field optical micromanipulation with cavity enhanced evanescent waves,” Appl. Phys. Lett. 88(22), 221116 (2006).
[CrossRef]

Righini, M.

M. Righini, G. Volpe, C. Girard, D. Petrov, and R. Quidant, “Surface plasmon optical tweezers: tunable optical manipulation in the femtonewton range,” Phys. Rev. Lett. 100(18), 186804 (2008).
[CrossRef] [PubMed]

Roberts, N. W.

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
[CrossRef]

Roundy, D.

A. F. Oskooi, D. Roundy, M. Ibanescu, P. Bermel, J. D. Joannopoulos, and S. G. Johnson, “MEEP: a flexible free-software package for electromagnetic simulations by the FDTD method,” Comput. Phys. Commun. 181(3), 687–702 (2010).
[CrossRef]

Ruel, R.

A. van Blaaderen, R. Ruel, and P. Wiltzius, “Tempate-directed colloidal crystallization,” Nature 385(6614), 321–324 (1997).
[CrossRef]

Sandhu, S.

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[CrossRef] [PubMed]

Nano Lett. (1)

K. Wang, E. Schonbrun, and K. B. Crozier, “Propulsion of gold nanoparticles with surface plasmon polaritons: evidence of enhanced optical force from near-field coupling between gold particle and gold film,” Nano Lett. 9(7), 2623–2629 (2009).
[CrossRef] [PubMed]

Nat. Photonics (1)

A. N. Grigorenko, N. W. Roberts, M. R. Dickinson, and Y. Zhang, “Nanometric optical tweezers based on nanostructured substrates,” Nat. Photonics 2(6), 365–370 (2008).
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[CrossRef]

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Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics (1)

E. Lidorikis, Q. Li, and C. M. Soukoulis, “Optical bistability in colloidal crystals,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 55(3), 3613–3618 (1997).
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Rev. Mod. Phys. (1)

K. Dholakia and P. Zemánek, “Colloquium: gripped by light: optical binding,” Rev. Mod. Phys. 82(2), 1767–1791 (2010).
[CrossRef]

Science (1)

G. M. Whitesides and B. Grzybowski, “Self-assembly at all scales,” Science 295(5564), 2418–2421 (2002).
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Figures (6)

Fig. 1
Fig. 1

Light-assisted templated self-assembly using a photonic crystal slab.

Fig. 2
Fig. 2

(a) Normalized transmission of a silicon slab on a silica substrate for a height of 0.6a and hole radius of 0.2a, (b) Force in the z-direction for a sphere of radius 0.1a placed 0.25a above the slab surface. The sphere is positioned equidistant from the four closest holes.

Fig. 3
Fig. 3

Normalized intensity for normally incident x-polarized light for R1, R2, and R3, respectively at a height of (a)–(c) 0.25a above the surface of the slab and (d)–(f) 0.1a above the surface of the slab. White circles indicate hole positions.

Fig. 4
Fig. 4

(a)–(c) Force in the x-z plane at y = 0, (d)–(f) Force in the x-y plane at z = 0.25a for R1, R2, and R3, respectively. The red dots represent stable trapping points and the black circles indicate hole positions.

Fig. 5
Fig. 5

(a) Pattern for x-polarization, (b) Pattern for y-polarization.

Fig. 6
Fig. 6

(a) Schematic of the system (b) Transmission spectrum of photonic-crystal slab with and without particles above (c) Fx c/ϕ on particle 1 for 3 different positions of particle 2 (d) Fy c/ϕ on particle 1 for 3 different positions of particle 2. The legend indicates the position of particle 2 in (b), (c), and (d).

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